A General Equilibrium Study of the Monetary Mechanism A General Equilibrium Study of the Monetary Mechanism David L. Schulze A General Equilibrium Study of the Monetary Mechanism David L. Schulze David L. Schulze A UNIVERSITY OF FLORIDA BOOK THE UNIVERSITY PRESSES OF FLORIDA GAINESVILLE / 1974 A UNIVERSITY OF FLORIDA BOOK THE UNIVERSITY PRESSES OF FLORIDA GAINESVILLE / 1974 A UNIVERSITY OF FLORIDA BOOK THE UNIVERSITY PRESSES OF FLORIDA GAINESVILLE / 1974  EDITORIAL COMMITTEE SocialSciencesMonogrphs WILLIA E. CARTER, Chairmano BENJAMIN L. GORMAN Director, Professor of Sociology Center for Latin American Studies 1RVING0J. ODDOO0 VoYcE A. HIoES Professor ofEconomics Prfessor of Education MANoING J. DAUER HARRY' W. PAUL' Professor of Political Science Aociat Professor of fHisoryo Library of Congross Cataloginog in Publictions Data Schulze, David. 1939- A goeneral equilibrium study of the motarty mechonism. (Universiyof FloidocialSciences m oophno. 51) "A Unoivesity of Floida book." Bibliograophy: p. 1. Money supply. 2. Equilibrsiumo (Ecoomics). 3. Money supply - Mathematlicl msodelo. I. Tile. II. Series Florida. University, Gainsvsille. Univerityof Floridamonorahs. Social sciences, no. 51. HG0221.S358 332.4 74-13495 ISBN 0-8130-0407-1 no. C . 1- COPYRIGHT @l974 oYoTHE BAR OGNS EDITORIAL COMMITTEE SoiSincMoogrphs WILLsAM E. CooTER, Chiromaon DENsMINs L. OoMAN Directors, Professor of Sociology Censter fort Loltin American Studie IRVIN J.lOpsoos VYNCE A. HISoo Prsofessoor of Ecoomso Professors of Educatliono Moososo 2. DAUERo HARR W. PAoL Prfcsor of Political Science Assoioate Poso of Hisory Librarsy of Congress Coataloging in Poblicatioo Dosa Scholzes, David. 1939- A genlqilibr~imstdy of heonetarymechis. (Unieity of Flrdasocl siencs o nographo.51) "A Uiver~lsy of Florida bosk." Bibliograophy: p. 1. Money supply. 2. Eqoilibriuom (Econos). 3. Moneoy supply - Mathematlicl modeso. L Tile. 11. Ser)is Florsida. Unoiversity, OGainsvilles. Universoity of Floida monographs. Social sciencDs, tno. 51. HO221.S33D 332.4 74-13495 ISBN 0-8130-0407-1 3oS. I- C.2- CnOYIGT ct974 BY ToE BOR O EToS 34-5 TH SAT 0O FORD EDITORIAL COMMITTEE SocialSienceso Monsographo WILLIAM E. CARTER, Chirmaoo BENJsAMIN L. ORMAN Directo, Profosor of Sociology Centor foo Latio AmericanD Soudios IRVssINL OFsoMA VyNCsA. HINSs Profesor of Economois Pofessor of Educaiono MANNIsG L. DAUER HARRYo . PAUL Prsofessoooof Poliicl Science Asoioto Professoooofistory Liboooy of Congress Catoging in Publicotioo 0a1a Schotoe, Dosid. 1939- A goeneral eqoilibioum study of tho moneotary moechaoism. (Universiyof Floida oil siencesD onograph no. 51) "A Unisorsity of Floida book.' Bibliography: ps. 1. Money supply. 2. Eqoiibriumo (Eoooos). 3. Moneyooupply - Mathematsical models. 1. Title. 11. Soois:o Ploido. Uoivorsity, Ginesille. Univsitoy of Forda onosograops. Social sciences, no. 51. HO221.S58 332.4 74-13495 ISBN 0-8130-0407-1 COPYRIGHT c01974 BY0THE BOARD 00 REGENTS OFTESTAoE OF FLODA PRINTD By BOYDo BDOTHDRS, INEORP'DOTDPODDDBosDODDIEDO OD DODBYODDooosEooo'ooo PANAMAoo CITY, FLORIDA oiosCooPOIOPnso os ~~D PRINTED By BOYD BROTHERS, INCORPORATED PANAMA CITY, FLORIDA PRINTED By BOYD BROTHERS, INCORPORATED PANAMA C=, FLORIDA  U Acknowledgments SHOULD like to express my appreciation to the National Science Foundation for providing funds to support this work and to Iowa State University for actually allocating the funds under the NSF grant. I am also deeply indebted to Professor Dudley G. Luckett for his guidance throughout the course of the project. Special thanks go to my wife for her cheerful suffering of the effects of my frustrations. Thanks must go also to the Graduate School of the University of Florida for making possible the publication of this monograph. Acknowledgments SHOULD like to express my appreciation to the National Science Foundation for providing funds to support this work and to Iowa State University for actually allocating the funds under the NSF grant. I am also deeply indebted to Professor Dudley G. Luckett for his guidance throughout the course of the project. Special thanks go to my wife for her cheerful suffering of the effects of my frustrations. Thanks must go also to the Graduate School of the University of Florida for making possible the publication of this monograph. Acknowledgments SHOULD like to express my appreciation to the National Science Foundation for providing funds to support this work and to Iowa State University for actually allocating the funds under the NSF grant. I am also deeply indebted to Professor Dudley G. Luckett for his guidance throughout the course of the project. Special thanks go to my wife for her cheerful suffering of the effects of my frustrations. Thanks must go also to the Graduate School of the University of Florida for making possible the publication of this monograph. 4/ tt~ -4 flt Cl   Contents Contents Contents 1. Review of the Literature 2. The Model 3. Solution with a Passive Government 4. Solution with an Active Government 5. Results of the Study Appendix: Definitions Literature Cited 12 68 96 140 145 151 1. Review of the Literature 2. The Model 3. Solution with a Passive Government 4. Solution with an Active Government 5. Results of the Study Appendix: Definitions Literature Cited 68 96 140 145 151 1. Review of the Literature 2. The Model 3. Solution with a Passive Government 4. Solution with an Active Government 5. Results of the Study Appendix: Definitions Literature Cited 68 96 140 145 151   1. Review of the Literature T HE CLASSICAL dichotomy between the cent and monetaey ati- obtes in the economy it, in one form at another, on extremety hardy beatt. One of its milde cr eincarnationt it the ideo thot on examination of the determinants of the ttoch of money it, ot bett, onty an intellectuot game, since the chain of caotatity runs from income cod pricet to the money stock. The demand formoney isvisualized aspimarilytafnc- tionoofe lee ofoaionaltinome, and any coreleation betmeen income and pricet and money it due solely to the "pull" of income on the money stock.' No impoetant feedback from the money ttoch to income and pricec it believed to exst. With the gteat deal of mock done in the 1950t end early 1960s providing a convincing theoetica baiforthe existence of achain of cauaty tunnieg from the moeey stoch to the teat variablet in the economy, ot en mention Keynet' moth (3t, 32), economits began, in the eacty t1960t, to inveetigate more thoroughty the deteemination of the money stach. The forcet affecting the money stach mete important, since the money steach in tote affected the tenet of prices and income. The primary purpote of thitmwock is to examine the proceses through mhich the money stock it determined. to addition, fuethet theoretical tupport will he provided foe the potition that changet in the stach of money affect the tenet of economic activity, and the effects and effec- tinenect of the variousteools of monetary pobicy mdtl be exatmined. The framewoth in mhich chit mitt be carried oat it a generat equilibrium model of the economy compoted of fine tectort-the public, manufac- turing fitrms,banks, onnofinancialtfirms,and the goenment. t. See, toe nempte, Goldsmich (26) end Klein and Gotdbeeteer (331. Fll infoemation on liereatue cited benc on p. t51. Lee definitins at tymbota need in the model, tee Appendix, p. t45. 2. Patnini'eMoney, dnerest, andd'licee (47) teeved acedeh ietone and a stimul or nfurth~ereorkintis are. 1. Review of the Literature T HE CLASSICAL dichotomy between the teat and monetacy vati- abtes in the economy it, in one form at another, an extremety hardy beat. One of its mildee reincatnationt it the idea that an examination of the deteeminants of the stockc of money is, at beste,only an intllectual game, snce the chain of cautaltiy toot from income and pricet to the money stock.The demand for money isiualized astpimarily a func- tion of the tenel of notional income, and any correltion between income and prices and money is doe solely to the "gall" of income on the money stnch.t No impoetant feedbach from the money tach to income and pricet it behieved to exist. With the great deal of moth done in the t950c and early t960s providing a convincing theoceticat basis fat the existence of a chain of cautality tanning from the money stock to the teat variabtet in the economy,2 not to mention Keynes' moth (31, 32), economists began, in the natty 1960si, to innettigate mote thoeroughly the deteemination of the money eaoch. The faeces affecting the money stnch mete important, tince the money stock in taco affected the level of pricet and income. The primary putpote of this moth it to examine the procestet through which the money stock it determined. In addition, furthee theoreeical support wigl be provided foe the potitin that changes in the ttoch of money affect the tenet of economic activity, and the effectt and effec- tivenets of the various tools of monetary policy witl be examined. The frameworh in which this mill be carred ant it a general equilibrium model of the economy composed of five sectots-the public, menuac- toeing firms, banks, nonbank financial firms, and the government. t.nSet, forexcample, Gotdsmith (26) and Klein end Goldberger 133). Fall infoematin on literatuee deted helloon p. 1531. tn, definitinse oi symbols aced in the model, see Appendix, p. 145. 2. Patiolin's Money,Inteet, andPriesc(47) served asboth a milestonead a stimuluorfurther workinctisnaeao. 1. Review of the Literature T HE CLASSICAL dichotomy between the teat and monetaty vaci- ablet in the economy it, in one form attothet, an extremely handy beat. One of its millet reincarnations it the idea that an exemination of the determinants of the tach of money it, at bestonly an intelectual game, tince the chain of causabity cone from income and pricet to the money tach. The demand foe money is isualized at primarily a func- tin of the level of national income, and any correlationobetween income and prices and money is doe tolely to the "pull" of icome on the moneyttoc.' No important feedbacnfrom thecmoneeystockto income and pricet is believed to exist. Withtecgreatedealnofworkdonecin the 1950ts andearlyl1960e providing a convincing theoretical baths foe the existence of a chain of causalty conning from the money tach to the teal variables in the economy, ot en mention Keynes' mock (31, 32), ecoomiests began, in the early 196s, to invetigate mace thaooghly the determination of the moneytstoch. Thecfoces affectingthe moneyteckcwereimporant, since the money tach in taco affected the level of prices and income. The primary purpose of thiswork is to examine the peocesset through which the money tach it determined. to addition, fucther theoretical support will be peovided foe the petition that changes in the tach of money affect the tenet of economic activity, and the effects and effec- tiveness of the various tools of moncety policy will be examined. The frameworkc in which this will be carried one is a general equidibrium model of the economy composed of five sectors-the public, manufac- toeing firms, banks, onbookfinancial firms, and the government. t. See, forexcample, Goldsmith (20) and tlen and Goldbercer (33), Full informatin on literature ciced Online n p. t5t. toe defiiios at avmols aced in the modet, see Appendix, p. 105. 2. Paoelci's Money,dinteestc,and Price(47 second as boamilestneand a  2 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM This wash is primnarily an entension of what is commonly refessed to as "money supply theoey." The hasic idea of she appcoaches so be discussed isto generate expessions fothestock of money instermsof she variahles of whatever economic model is postulated, and so derive statements aboos she effects of changes in chase variables on she stoch of money. These expressions foe she money stock arecalled money sopply equations.t The stody of the sopply of money hogan wish the early mock of C. A. Phillips (48) and oshers (1, 35, 39, 50) in she 1920s and 1930s. Theic mock culminated in the standard seushook money multipliers, wish which we ace so familiar, like AM =(1/c) shoes she oeiginal change in she money stock (where T is she anecage reserve requirement). No coal advance in this area occorred utl she 1960s and she appeacance of she mocks of Milson Fciedman and Anna Schwaetz (24), Phdhlp Cagan (I0), and Karl Brunner and Alan Melter (4, 5, 6, 8, 41). The Eriedman- Schwactz-Cagan and Bounnec-Melter approaches so she money supply ace she hess known today. The Eredmass-Schmacsa-Cagan 4 approch is hosed on Iwo simple defi- nisions. The money stock, M, is equaltso sotl cursency holdings, C, and tocal demand deposits: 2 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM This mock is primarily an extension of what is comsmonly referred so as "money supply theory." The hasic idea of she approchestso he discutsed is to generate expressions foe she clock of money in teems of she variables of whatever economic model is postulated, and to desive statements about the effects of changes in these variables on the stock of money. Theme expresmions for the money stock arecalled money supply eqoations.3 The study of the sopply of money began wish she early mock of C. A. Phillips (48) and others (1, 35, 39, 50) in she 192gs and 1930s. Their mock culminased in she standard senshook money mulsipliers, wish which we ace so familiar, like AM = (1/c) timses she original change in she money stock (wheesr s she average reserve reqoirement). No coal advance in skis 0000 occured motl she 196gs and the appearance of she mocks of Mdcton Friedman and Anna Schwartz (24), Phdhlp Cagan (1g), and Karl Brunnec and Alan Meltzer (4, 5, 6, g, 41). The Eriedman- Schwartz-Cagan and Brunner-Melser opproaches so she money sopply ace the hess known soday. The Ecicdmm-Schmarte-Cagan4 approach is hosed on Iwo simple defi- nitions. The money stock, M, is equal to total cucrency holdings, C, and tosal demand deposits: 2 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM This mock is primaridy an emtension of whas is commonly ceferred to as "money supply sheory." The hasic idea of the appcoaches to he discussed is to generate expressions foe she stock of money in tecms of she variahles of whatever economic model is postulased, and to derive statements abont she effects of changes in these variables on she stock of money. These exprssmions foe the money stock are called money sopply oqoassons.3 The study of she supply of money hegan wish the early mock of C. A. Phillips (48) and others (1, 35, 39, 50) in the 1920s and 19303s. Their mock culminated in she standard textbook money mulsiphiersmish which me ace so familiac, like AM = (1/c) times the original change in the money stock (where r is the anerage resenve requicement). No real advance in skis areao occurred untid she 1960s and she appearance of the mocks of Mdlton Friedman and Anna Schwartz (24), Phillip Cagan (18), and Kaod Brunner and Alan Melterc (4, 5, 6, 8, 41). The Priedmano- Schwartz-Cagan and Beuninec-Melte appeoaches so she money supply ace she host known today. The Psiedman-Schwase-Cagot 4oaprsoach is hosed on swo shmple dog.- nitions. The money stunk, M, is eqoal to tosal coscency holdings, C, and sotl demand deposits: M = Cn+. (1.1) High-powered money, H, defined as she local of all sypes of money shos con he used as corrency or reserves, is simpty Ht=C+R (1.2) whewe R is simply resenves. The basic Esiedmon-Sckwacsn-Cagancresuls is obtained hy simply dinid- 3. They ace not supply eqatons in she nomal sence cc eke seen, since tkey all purpotcoasiveeactualosockhf oneyowhenshevalns ofonheiryarameers ace knows. Ifoktheynwerentrue spynequations,stenctualosocofmoneywoldhbe given, coo by she "supply" eqation slone, hoe ky simultaoneous solution of ohe aggegate demand Ice mcoy eqation and a "true" sanely equakion. En, this resnw choososeaoh of the monetory mechanism inching a smlaeu doeeminakion cote mosey stock, rather skin she suyply of money alone. 4. Cagan's oauoology fon, one money ctonch is slightly different hrow shoecon- sentea is Aypendix 0 to A Monetary History of she U.S., 1867z196n ny Friedman and Sckwartz (24). Cagan's forwulatinn ks eased on eke tautology derioed ey Fsiedman cod Scnwartz aescried in no txt M = C + D. (1.1) M = C +13. (1.1) High-powered money, H, detined as Ike sosal of all types of money shot canbeoused as currency orrsevens, issimply High-powered money, H, dehined as she sotal of alt types of money shas canhbe used asocrency orcreserves, issimply H= C +R (1.2) H=C+R (1.2) whereoRis simplyceserves. The hasic Eciedman-Schmosse-Cogonsresult is obtained hy simply dinid- 3. Tey ass noe supply equaaikns is the normal sense of the coon, since one all curooo so givs cue accual stockhof moneymwheoonohovles ofteipaameters ace known. Ifokhey moon none supply eqaios, see actual ntook of money would e given, not by she "sopply" eqoation alone, bo, hb' simultaneous solton of se oagregat demand fo, money equation and a "true" supply eqnation. Ear skis resnme chooseohea of the monetary meceanism implying asimaltaneous decerminatios of Inc money atock, sashes ohne soucpply of money close. 4. Conan's tausology foe she money scoch is sligetly differnt from seas pre- sented 00 Accendix B to A Mooetury History of see U.S., d867-d960 ey Fredman anduSchwartz(24. Cagn'sformlakionisceased onotheotautlogyderivedkby Friedman and Schwartz descrdbed in ohcoonst. whose BR is simply resernes. The hosic Poicdmam-Sckwacle-Cagancresot is obtained hy simply dinid- 3. They are 000 supply equations in eke nosmal en of te teem, sinonthyl curorton givesteactalosockof moeywenehevaues ofeipaamocosar known. If they were sresucpply equations, tho actual stooh of money woald e aloen, nst by see "sopply" equaon atone, ens ey simulanous solution of se aggregate demand foe money equation and a "true" supply equation. En, this reason owe eoosy seah of see moeneay mechanism implying asmlaeu determinatinnof the mosey sock, rather thas sue supply oflmoney' alouo. 4. Cagac's taucology Inc te money cinch ic slightly differntooehm seas con- mooted in Acpendix B so A Moonetaoy Hisory of nhn U.S., 186-y960 by Friedman and Schwartz(24). Cagan'slformation ihasd ooothetauoogy derioedlby Friedmnancd Schwartz descied in teexnn.  REVIEW OF THE LITERATURE 3 REVIEW OF THE LITERATURE 3 ing Equation 1.1 by Equation 1.2 which yields, after a few simple algebraic manipulations,' ing Equation 1.1 by Equation 1.2 which yields, after a few simple algebraic manipulations,5 M=H D/R (I + D/C) D/R + D/C (1.3) M=H D/R (1 + D/C) D/R + D/C (1.3) Equation 1.3 is a tautology, being derived from the definitions of M and H. In this approach the money stock is determined by the decisions of three sectors: the government, which determines H; the public, by determining its deposit to currency ratio, D/C; and the banks, by determining the deposit to reserve ratio, D/R. Friedman and Schwartz (24) call H, D/R, and D/C the "proximate determinants" of the money stock (p. 791). The factors underlying these proximate determinants are spelled out only vaguely. D/C is said to depend upon the "relative usefulness" of deposits and currency, the costs of holding these assets, and "perhaps income" (p. 787). D/R is a function of legal reserve requirements and precautionary reserves (p. 785). The determinants of H are not spelled out specifically, even though a large portion of the book is devoted to describing and analyzing various actions by the monetary authorities. Brunner and Meltzer actually present two hypotheses-linear and non- linear. Their linear hypothesis is based on the reaction of the banking system to the presence of surplus reserves, defined as the difference between actual and desired reserves, the portfolio adjustments caused by these surplus reserves (4, 8), and the process by which surplus reserves are generated or absorbed. The total portfolio response of the banking system to the presence of surplus reserves is given by Equation 1.3 is a tautology, being derived from the definitions of M and H. In this approach the money stock is determined by the decisions of three sectors: the government, which determines H; the public, by determining its deposit to currency ratio, D/C; and the banks, by determining the deposit to reserve ratio, D/R. Friedman and Schwartz (24) call H, D/R, and D/C the "proximate determinants" of the money stock (p. 791). The factors underlying these proximate determinants are spelled out only vaguely. D/C is said to depend upon the "relative usefulness" of deposits and currency, the costs of holding these assets, and "perhaps income" (p. 787). D/R is a function of legal reserve requirements and precautionary reserves (p. 785). The determinants of H are not spelled out specifically, even though a large portion of the book is devoted to describing and analyzing various actions by the monetary authorities. Brunner and Meltzer actually present two hypotheses-linear and non- linear. Their linear hypothesis is based on the reaction of the banking system to the presence of surplus reserves, defined as the difference between actual and desired reserves, the portfolio adjustments caused by these surplus reserves (4, 8), and the process by which surplus reserves are generated or absorbed. The total portfolio response of the banking system to the presence of surplus reserves is given by REVIEW OF THE LITERATURE 3 ing Equation 1.1 by Equation 1.2 which yields, after a few simple algebraic manipulations,' D=H 0/R (1 + D/C) (1.3) D/R + D/C Equation 1.3 is a tautology, being derived from the definitions of M and H. In this approach the money stock is determined by the decisions of three sectors: the government, which determines H; the public, by determining its deposit to currency ratio, D/C; and the banks, by determining the deposit to reserve ratio, D/R. Friedman and Schwartz (24) call H, D/R, and D/C the "proximate determinants" of the money stock (p. 791). The factors underlying these proximate determinants are spelled out only vaguely. D/C is said to depend upon the "relative usefulness" of deposits and currency, the costs of holding these assets, and "perhaps income" (p. 787). D/R is a function of legal reserve requirements and precautionary reserves (p. 785). The determinants of H are not spelled out specifically, even though a large portion of the book is devoted to describing and analyzing various actions by the monetary authorities. Brunner and Meltzer actually present two hypotheses-linear and non- linear. Their linear hypothesis is based on the reaction of the banking system to the presence of surplus reserves, defined as the difference between actual and desired reserves, the portfolio adjustments caused by these surplus reserves (4, 8), and the process by which surplus reserves are generated or absorbed. The total portfolio response of the banking system to the presence of surplus reserves is given by dE= S (1.4) 5. The derivation of Equation 1.3 is: (1) M/H = C + D/C + R. Multiplying numerator and denominator by D yields (2) M/H = DC + D2/DC + RD. Then the right-hand side of (2) is multiplied by RC/RC, yielding DC+D2 DC D2 D D2 D ( M _ RC RC RC R +RC _R C H DC+RD DC RD D D D D RC-- R nC +RC- R C R C Multiplying both sides by H gives the desired result dE= S (1.4) dE= S (1.4) 5. The derivation of Equation 1.3 is: (1) M/H = C + O/f + R. Multiplying numerator and denominator by D yields (2) M/H = DC + D2/DC + RD. Then the right-hand side of (2) is multiplied by RC/RC, yielding DC+D2 DC D2 2 D o M _ RC RC RC _ RC _ R C ) H DC+RD DC RD 1) D D D RC RC + RC R C C Multiplying both sides by H gives the desired result. 5. The derivation of Equation 1.3 is: (1) M/H = C + D/C + R. Multiplying numerator and denominator by D yields (2) M/H = DC + D2/DC + RD, Then the right-hand side of (2) is multiplied by RC/RC, yielding DC+D2 DC D2 D D D 0 M _R yRC R RC _ RRC H DC+RD DC RD D D D D RC RC +TC R + C R C Multiplying both sides by H gives the desired result.  4 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM where E is the value of the banks' portfolio, S is the amount of surplus reserves, and h is the average loss coefficient (i.e., X measures the amount of surplus reserves lost per dollar of portfolio adjustment). A is less than 1 since the banking system will generate added deposits (and thus reserves) as it attempts to eliminate surplus reserves by buying interest-bearing assets. p is equal to (1 - n)p, where p is the average spillover of deposits from the expanding bank (the one trying to elimi- nate surplus reserves) to other banks and n is a linear combination of average spillover into currency and time deposits. Thus y reduces the average loss coefficient and the term (X - y) ais Brunner and Meltzer's money multiplier for responses to surplus reserves. Surplus reserves are given by the relation S = A dB + dL -A dC + A2 dt. + Au dE - dV; (1.5) B is the monetary base (the amount of money issued by the govern- ment); L is the total of changes in required reserves resulting from changes in the average reserve requirement and from the redistribution of deposits among various classes of banks; E is a parameter measuring the structure of interbank deposits; dC represents changes in the public's demand for currency occurring independently of changes in the public's monetary wealth; dt, represents changes in the public's demand for time deposits occurring independently of the public's wealth; and dV, represents changes in the banks' demand for cash assets in excess of required reserves occurring independently of changes in the level of banks' deposits. The A5 are positive constants. Then the change in Ma (defined as currency plus demand deposits plus time deposits) is given by 4 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM where E is the value of the banks' portfolio, S is the amount of surplus reserves, and h is the average loss coefficient (i.e., X measures the amount of surplus reserves lost per dollar of portfolio adjustment). X is less than 1 since the banking system will generate added deposits (and thus reserves) as it attempts to eliminate surplus reserves by buying interest-bearing assets. p is equal to (1 - n)p, where p is the average spillover of deposits from the expanding bank (the one trying to elimi- nate surplus reserves) to other banks and n is a linear combination of average spillover into currency and time deposits. Thus p reduces the average loss coefficient and the term (N - p)- is Brunner and Meltzer's money multiplier for responses to surplus reserves. Surplus reserves are given by the relation S = A. dB + dL - A, dC + A2 dtn + A3 dE - dV ; (1.5) B is the monetary base (the amount of money issued by the govern- ment); L is the total of changes in required reserves resulting from changes in the average reserve requirement and from the redistribution of deposits among various classes of banks; E is a parameter measuring the structure of interbank deposits; dCo represents changes in the public's demand for currency occurring independently of changes in the public's monetary wealth; dtn represents changes in the public's demand for time deposits occurring independently of the public's wealth; and dVr represents changes in the banks' demand for cash assets in excess of required reserves occurring independently of changes in the level of banks' deposits. The A; are positive constants. Then the change in Mu (defined as currency plus demand deposits plus time deposits) is given by 4 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM where E is the value of the banks' portfolio, S is the amount of surplus reserves, and h is the average loss coefficient (i.e., X measures the amount of surplus reserves lost per dollar of portfolio adjustment). X is less than 1 since the banking system will generate added deposits (and thus reserves) as it attempts to eliminate surplus reserves by buying interest-bearing assets. p is equal to (I - n)p, where p is the average spillover of deposits from the expanding bank (the one trying to elimi- nate surplus reserves) to other banks and n is a linear combination of average spillover into currency and time deposits. Thus a reduces the average loss coefficient and the term (N - )' is Brunner and Meltzer's money multiplier for responses to surplus reserves. Surplus reserves are given by the relation S = A dB + dL -A dC + A2 dtr + A3 dE - dV ; (1.5) B is the monetary base (the amount of money issued by the govern- ment); L is the total of changes in required reserves resulting from changes in the average reserve requirement and from the redistribution of deposits among various classes of banks; E is a parameter measuring the structure of interbank deposits; dC represents changes in the public's demand for currency occurring independently of changes in the public's monetary wealth; dt, represents changes in the public's demand for time deposits occurring independently of the public's wealth; and dV. represents changes in the banks' demand for cash assets in excess of required reserves occurring independently of changes in the level of banks' deposits. The A are positive constants. Then the change in M2 (defined as currency plus demand deposits plus time deposits) is given by dM2 = m2s + q dB (1.6) where m2 is the surplus reserve (or money) multiplier and q is the proportion of a change in the money base that affects bank reserves and deposits simultaneously. The change in M' (M2 - T) is dM' = m's + q dB - dt (1.7) where m' is the money multiplier for the definition of M excluding time deposits. dM2 = m2s + q dB (1.6) dM2 = m2s + q dB (1.6) where m2 is the surplus reserve (or money) multiplier and q is the proportion of a change in the money base that affects bank reserves and deposits simultaneously. The change in M' (M2 - T) is where m2 is the surplus reserve (or money) multiplier and q is the proportion of a change in the money base that affects bank reserves and deposits simultaneously. The change in M' (M2 - T) is dMt = m's + q dB - dto (1.7) dM' = m' s + q dB - dt,, (1.7) where m' is the money multiplier for the definition of M excluding time deposits,. where m' is the money multiplier for the definition of M excluding time deposits.  REVIEW OF THE LITERATURE 5 REVIEW OF THE LITERATURE 5 Replacing s in Equations 1.6 and 1.7 with Equation 1.5 and integrat- ing yields the linear hypothesis' expressions for M' and Ma: M2 = m. + m2(B + L) - m2A, Co + m2Aatn - m2V, (i) (1.8) M' = n + mI(B + L) - mA1C0 - [I - mIA2] to - Replacing s in Equations 1.6 and 1.7 with Equation 1.5 and integrat- ing yields the linear hypothesis' expressions for M' and Ma: M2 = mo + m2(B + L) -m2AC0 + m2 A2t0 - m2V (i) (1.8) Mt = no + m(B + L)- mtA C, - [1 - m A2] to - m1V (i) (1.9) nV (i) (1.9) where B + L is the "extended monetary base," mo and no are positive constants, and the notation V (i) is used to express the dependence of the money stock on interest rates through the impact of interest rates on the banks' asset portfolio. mt and n are the money multipliers. Behind the terms C., to, and V0 lie the public's demands for currency and time deposits, which depend upon the public's money wealth, nonmoney wealth, and all interest rates, as well as the banks' demand for "available cash assets," which depends on the relevant interest rates and the level of deposit liabilities. Again the money stock depends upon the decision of three sectors: the government in determining B + L, the public in determining C, and to, and the banking system in determining Vd (i). Implicit in this hy- pothesis is the assumption, as Fand has pointed out (19), that the marginal propensities to hold time and demand deposits (with respect to changes in M) are constant. Brnner and Meltzer's nonlinear hypothesis centers on the credit market. The money stock and interest rates emerge from the interaction of the public's supply of assets to the banks and the banks' resulting portfolio readjustment. The banks' desired rate of portfolio readjustment, E', is given by where B + L is the "extended monetary base," me and no are positive constants, and the notation V (i) is used to express the dependence of the money stock on interest rates through the impact of interest rates on the banks' asset portfolio. m' and ma are the money multipliers. Behind the terms C., t, and V0 lie the public's demands for currency and time deposits, which depend upon the public's money wealth, nonmoney wealth, and all interest rates, as well as the banks' demand for "available cash assets," which depends on the relevant interest rates and the level of deposit liabilities. Again the money stock depends upon the decision of three sectors: the government in determining B + L, the public in determining C. and to, and the banking system in determining Vd (i). Implicit in this hy- pothesis is the assumption, as Fand has pointed out (19), that the marginal propensities to hold time and demand deposits (with respect to changes in M) are constant. Brunner and Meltzer's nonlinear hypothesis centers on the credit market. The money stock and interest rates emerge from the interaction of the public's supply of assets to the banks and the banks' resulting portfolio readjustment. The banks' desired rate of portfolio readjustment, Ea, is given by REVIEW OF THE LITERATURE 5 Replacing s in Equations 1.6 and 1.7 with Equation 1.5 and integrat- ing yields the linear hypothesis' expressions for Mt and M2: M2 = m0 + m2(B + L) - m'A1C0 + m2At - m2V (i) (1.8) Mt = no + mI(B + L) - mt A, C. - [1 - Int A2} to m Vf(i) (1,9) where B + L is the "extended monetary base," n0 and no are positive constants, and the notation V (i) is used to express the dependence of the money stock on interest rates through the impact of interest rates on the banks' asset portfolio. m and m2 are the money multipliers. Behind the terms C., to, and Vo lie the public's demands for currency and time deposits, which depend upon the public's money wealth, nonmoney wealth, and all interest rates, as well as the banks' demand for "available cash assets," which depends on the relevant interest rates and the level of deposit liabilities. Again the money stock depends upon the decision of three sectors: the government in determining B + L, the public in determining C0 and to, and the banking system in determining V (i). Implicit in this hy- pothesis is the assumption, as Fand has pointed out (19), that the marginal propensities to hold time and demand deposits (with respect to changes in M) are constant. Brunner and Meltzer's nonlinear hypothesis centers on the credit market. The money stock and interest rates emerge from the interaction of the public's supply of assets to the banks and the banks' resulting portfolio readjustment. The banks' desired rate of portfolio readjustment, ts, is given by $'=h(R- Rd) (1.10) where R is actual reserves and Rd is desired reserves. R' = R' (D, T, i, p) (1.11) where i is a vector of all interest rates and p is the discount rate. Excess reserves, Re, are given by R" = R (i, p, D + T). (1.12) t' = h (R - R') where R is actual reserves and Rd is desired reserves. (1.10) t' = h (R - R d) where R is actual reserves and Rd is desired reserves. R' = R (D, T, i, p) (1.10) (1.11) Rd = R' (D, T, i, p) (1.11) where i is a vector of all interest rates and p is the discount rate. Excess reserves, Rt, are given by R* = R*(i, p, D + T). (1.12) where i is a vector of all interest rates and p is the discount rate. Excess reserves, Re, are given by R = R' (i, p, D + T). (1.12)  6 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM R' is assumed to be homogeneous of degree one in D + T so that we can write Re = e (i, p) (D + T). (1.13) The public's rate of supply of assets to the bank, Fd, is given by Fd = f (i, W, E) (1.14) where W is the public's wealth and Ed the public's desired portfolio of liabilities to the banks. The public's desired rates of change in currency holdings and time deposits are given by 00 = q' (kD - C0) (1.15) T = q2 (tD - T) (1.16) where k is the desired currency to demand deposit ratio, t is the desired time deposit to demand deposit ratio, and D is the level of demand deposits. The banks' desired rate of change of indebtedness to the Federal Reserve System is A = a [b (D + T) - A] (1.17) where b is the desired indebtedness ratio. Based on the above, Brunner and Meltzer write: B =A + B (1.18) B =R + C (1.19) R (r + e) (D + T) (1.20) Ce =kD (1.21) 6 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM R' is assumed to be homogeneous of degree one in D + T so that we can write R* = e (i, p) () + T). (1.13) The public's rate of supply of assets to the bank, td, is given by $d = f (i, W, Ed) (1.14) where W is the public's wealth and Ed the public's desired portfolio of liabilities to the banks. The public's desired rates of change in currency holdings and time deposits are given by 0 = qt (kD - C0) (1.15) T = q2 (tD - T) (1.16) where k is the desired currency to demand deposit ratio, t is the desired time deposit to demand deposit ratio, and D is the level of demand deposits. The banks' desired rate of change of indebtedness to the Federal Reserve System is A a [b (D + T) - A] (1.17) where b is the desired indebtedness ratio. Based on the above, Brunner and Meltzer write: B =A+B (1.18) B =R + C (1.19) R (r + e) (D + T) (1.20) C0 =kD (1.21) T =tD (1.22) A =b (D +T) (1.23) E = E (i, W) (1.24) 6 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM R* is assumed to be homogeneous of degree one in D + T so that we can write R* = e (i, p) (D + T). (1.13) The public's rate of supply of assets to the bank, Ed, is given by Fd = f (i, W, Ed) (1.14) where W is the public's wealth and Ed the public's desired portfolio of liabilities to the banks. The public's desired rates of change in currency holdings and time deposits are given by EP = q1 (kD - C0) (1.15) T = q (0t - T) (1.16) where k is the desired currency to demand deposit ratio, t is the desired time deposit to demand deposit ratio, and D is the level of demand deposits. The banks' desired rate of change of indebtedness to the Federal Reserve System is A = a [b (D + T) - A] (1.17) where b is the desired indebtedness ratio. Based on the above, Brunner and Meltzer write: B =A + B' (1.18) B =R + C (1.19) R (r + e) (D + T) (1.20) T =tD A =b(D+T) E =E(i,W) (1.22) (1.23) (1.24) C0 = kD T =sD A =b(D+T) E =E(W) (1.22) (1.23) (1.24)  REVIEW OF THE LITERATURE 7 where B in the adjusted or "relatively exogenous" base and all other symbols have been previously defined. This system of seven equations is then reduced to two through substitution and with the help of the assumption that Ba and W are given exogenously. These equations are: REVIEW OF THE LITERATURE 7 where BA in the adjusted or "relatively exogenous" base and all other symbols have been previously defined. This system of seven equations is then reduced to two through substitution and with the help of the assumption that B' and W are given exogenously. These equations are: M2 = m2 Be (1.25) REVIEW OF THE LITERATURE 7 where BA in the adjusted or "relatively exogenous" base and all other symbols have been previously defined. This system of seven equations is then reduced to two through substitution and with the help of the assumption that B' and W are given exogenously. These equations are: M2 =m2 Ba (m2 - 1) Ba = E (i, W) where int, the money multiplier, is given by m2 = I+k+t (r + e - b) (1 + t) + h (1.25) (1.26) (1.27) (m2 - 1) Ba = E (i, W) where int, the money multiplier, is given by M2 = 1t + kl I (r + e - b) (1 + t) + k (1.26) (1.27) (m - 1) Ba = E (i, W) where in', the money multiplier, is given by l+k+t (r + e - b) (1 + t) + h (1.26) From Equation 1.26, one of the rates of interest, say i' (using their notation), can be determined in terms of the rest of the interest rates, p, W, Ba, r, and k. Then this solution for it can be substituted into 1.25,6 giving M' as a function of interest rates is (s # 1), p, Ba, r, and W. In other words, the solution to their two equations yields M2 and one rate of interest, it. Equations 1.25 and 1.26 are the expression of the nonlinear hypothesis. Of the many other works that might be mentioned briefly," we shall concentrate on the models of Ronald L. Teigen (51) and Frank de Leeuw (16). The Teigen model is based on the proposition that the total level of reserves in the Federal Reserve System, various rules (such as the reserve requirements), and regular behavioral relations (between currency levels and the total money stock, etc.) "determine a maximum attainable money stock at any given time, and that this quantity (M**) can be considered to be the sum of two parts: one part which is considered to be exogenous and is based on reserves supplied by the Federal Reserve System (R'),r and the other based on reserves created by member bank borrowing (B),t and therefore considered endogenous" (p. 478). Teigen's goal is to explain the ratio of the observed money stock (M) to the exogenous segment of the total money supply (M*). He asserts that 6. Since all the terms in m2 are functions of i and/or p. 7. See, for example, Grambley and Chase (27), Meigs (40), Modigliani (44), and Goldfed (25). These and several other studies cited in the bibliography will not be discussed because of their highly specialized nature. 8. This is M*. 9. This is B*. From Equation 1.26, one of the rates of interest, say i' (using their notation), can be determined in terms of the rest of the interest rates, p, W, Ba, r, and k. Then this solution for ie can be substituted into 1.25,6 giving M2 as a function of interest rates is (s# 1), p, Ba, r, and W. In other words, the solution to their two equations yields M2 and one rate of interest, i1. Equations 1.25 and 1.26 are the expression of the nonlinear hypothesis. Of the many other works that might be mentioned briefly,7 we shall concentrate on the models of Ronald L. Teigen (51) and Frank de Leeuw (16). The Teigen model is based on the proposition that the total level of reserves in the Federal Reserve System, various rules (such as the reserve requirements), and regular behavioral relations (between currency levels and the total money stock, etc.) "determine a maximum attainable money stock at any given time, and that this quantity (M**) can be considered to be the sum of two parts: one part which is considered to be exogenous and is based on reserves supplied by the Federal Reserve System (R'),' and the other based on reserves created by member bank borrowing (B),t and therefore considered endogenous" (p. 478). Teigen's goal is to explain the ratio of the observed money stock (M) to the exogenous segment of the total money supply (M*). He asserts that 6. Since all the terms in m2 are functions of i and/or p. 7. See, for example, Grambey and Chase (27), Meigs (40), Modigliani (44), and Goldfeld (25). These and several other studies cited in the bibliography will not be discussed because of their highly specialized nature. 8. This is M'. 9. This is B*. From Equation 1.26, one of the rates of interest, say it (using their notation), can be determined in terms of the rest of the interest rates, p, W, B', r, and k. Then this solution for i can be substituted into 1.25, giving M2 as a function of interest rates is (s # 1), p, B', r, and W. In other words, the solution to their two equations yields M' and one rate of interest, i. Equations 1.25 and 1.26 are the expression of the nonlinear hypothesis. Of the many other works that might be mentioned briefly,' we shall concentrate on the models of Ronald L. Teigen (51) and Frank de Leeuw (16). The Teigen model is based on the proposition that the total level of reserves in the Federal Reserve System, various rules (such as the reserve requirements), and regular behavioral relations (between currency levels and the total money stock, etc.) "determine a maximum attainable money stock at any given time, and that this quantity (M**) can be considered to be the sum of two parts: one part which is considered to be exogenous and is based on reserves supplied by the Federal Reserve System (R'),' and the other based on reserves created by member bank borrowing (B),r and therefore considered endogenous" (p. 478). Teigen's goal is to explain the ratio of the observed money stock (M) to the exogenous segment of the total money supply (M*). He asserts that 6. Since a the terms in m2 are functions of i and/or p. 7. See, for example, Grambley and Chase (27), Meigs (40), Modigliani (44), and Goldfeld (25). These and several other studies cited in the bibliography wilt not be discussed because of their highly specialized nature. 8. This is M*. 9. This is B *.  8 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM this ratio is a function of the profitability of bank lending. The impor- tant conclusions of the Teigen model are derived from his definitions of the money stock and the public's demand for currency and demand deposits (which he assumes are a constant proportion of the actual money stock). k k t M = (R - R) + b (B-- D) (1.28) -c -h 1-c h k where k is the reciprocal of the weighted average reserve ratio, a is the fraction of M held as currency by the public, h is the fraction of M held by the public as demand deposits in nonmember banks, Re is excess reserves, and Dg is U.S. government deposits in member banks. 8 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM this ratio is a function of the profitability of bank lending. The impor- tant conclusions of the Teigen model are derived from his definitions of the money stock and the public's demand for currency and demand deposits (which he assumes are a constant proportion of the actual money stock). k k 1t M = (Rs - Re) + (B- D) (1.28) 1-c-b 1- c- k - cL where k is the reciprocal of the weighted average reserve ratio, a is the fraction of M held as currency by the public, h is the fraction of M held by the public as demand deposits in nonmember banks, Re is excess reserves, and D is U.S. government deposits in member banks. M* = k (Rs) = k*Rs 1- c- h (1.29) (1.30) M* k (R) = k*R 1-c-h and M M* X (r,) (1.29) (1.30) M M* - X (re, r)t 8 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM this ratio is a function of the profitability of bank lending. The impor- tant conclusions of the Teigen model are derived from his definitions of the money stock and the public's demand for currency and demand deposits (which he assumes are a constant proportion of the actual money stock). k k t M = (R' - R") + (B - - D') (1.28) 1 - c-h 1 - c - h k where k is the reciprocal of the weighted average reserve ratio, c is the fraction of M held as currency by the public, h is the fraction of M held by the public as demand deposits in nonmember banks, R' is excess reserves, and D' is U.S. government deposits in member banks. M*= k (Rs) = k*Rs (1.29) 1- c-h and M =~ -X(r ,r) (1.30) where r is a measure of the return on bank loans and ra is a measure of the cost of bank loans. x/3r > 0 indicating that as the return on loans increases, the en- dogenous portion of M increases relative to M*. Sx/Dre < 0 indicating that as the cost of loans increases, M* becomes a larger proportion of M. Thus, Teigen breaks the money stock down into endogenous and exogenous portions and attempts to explain the relation between the actual money stock and the exogenous portion in terms of the returns and costs of bank loans. Changes in these factors presumably change the quantity of loans banks are willing to supply and thus result in portfolio readjustment by the banking system, fueling changes in the actual stock of money. The de Leeuw model is part of the Brookings-SSRC model. His portion of the overall model deals with the financial sector. There are seven markets: bank reserves, currency, demand deposits, time deposits, U.S. securities, "savings and insurance," and private securities. The sectors included are banks, nonbank financial, the Federal Reserve, the Treasury, and the public. This submodel (of the SSRC model) assumes that the value of real variables is known and does not consider the where r is a measure of the return on bank loans and rc is a measure of the cost of bank loans. x/r > 0 indicating that as the return on loans increases, the en- dogenous portion of M increases relative to M*. x/arc < 0 indicating that as the cost of loans increases, M* becomes a larger proportion of M. Thus, Teigen breaks the money stock down into endogenous and exogenous portions and attempts to explain the relation between the actual money stock and the exogenous portion in terms of the returns and costs of bank loans. Changes in these factors presumably change the quantity of loans banks are willing to supply and thus result in portfolio readjustment by the banking system, fueling changes in the actual stock of money. The de Leeuw model is part of the Brookings-SSRC model. His portion of the overall model deals with the financial sector. There are seven markets: bank reserves, currency, demand deposits, time deposits, U.S. securities, "savings and insurance," and private securities. The sectors included are banks, nonbank financial, the Federal Reserve, the Treasury, and the public. This submodel (of the SSRC model) assumes that the value of real variables is known and does not consider the where r is a measure of the return on bank loans and r, is a measure of the cost of bank loans. x/ar > 0 indicating that as the return on loans increases, the en- dogenous portion of M increases relative to M*. x/arc < 0 indicating that as the cost of loans increases, M* becomes a larger proportion of M. Thus, Teigen breaks the money stock down into endogenous and exogenous portions and attempts to explain the relation between the actual money stock and the exogenous portion in terms of the returns and costs of bank loans. Changes in these factors presumably change the quantity of loans banks are willing to supply and thus result in portfolio readjustment by the banking system, fueling changes in the actual stock of money. The de Leeuw model is part of the Brookings-SSRC model. His portion of the overall model deals with the financial sector. There are seven markets: bank reserves, currency, demand deposits, time deposits, U.S. securities, "savings and insurance," and private securities. The sectors included are banks, nonbank financial, the Federal Reserve, the Treasury, and the public. This submodel (of the SSRC model) assumes that the value of real variables is known and does not consider the  REVIEW OF THE LITERATURE 9 effects of changes in the various rates of interest, or the money stock on the real variables in the model. (Their effects are measured elsewhere in the Brookings model.) The model itself is composed of nineteen simultaneous equations, four of which are identities (the reserve identity, etc.) and the rest of which express the desired changes in assets in terms of lagged asset holdings, rates of return, and various short-run constraints on asset holding. Solving this system simultaneously, de Leeuw derives the following expression for the money stock (p. 518): S = RESNBC I_ RDD + 0.84 [RRRDD] [RDD + RDDGF] + 0.82 [RRR00] [RET] + [0.011 RMFR B - REVIEW OF THE LITERATURE 9 effects of changes in the various rates of interest, or the money stock on the real variables in the model. (Their effects are measured elsewhere in the Brookings model.) The model itself is composed of nineteen simultaneous equations, four of which are identities (the reserve identity, etc.) and the rest of which express the desired changes in assets in terms of lagged asset holdings, rates of return, and various short-run constraints on asset holding. Solving this system simultaneously, de Leeuw derives the following expression for the money stock (p. 518): RESNBC M _RDD + 0.84 [RRRDD] [RDD + RDDGF] + 0.82 [RRRDT] [RDT] + [0.011 RMFR - 0.010 RMGBS3 - 0.007] [RDD + RDT] (1.31) 0.010 RMGBS3 - 0.007] [RDD + RoTh] (1.31) where SM is the money supply (private demand deposits [DD] and currency); RDD = DD/SM; RDT = DT/SM; RDDGF = DDGFSM; DT is private time deposits; RESNBC is unborrowed reserves plus currency held by member banks; RRRDD is a weighted average of required reserve ratios against demand deposits; RRRDT is a weighted average of required reserve ratios against time deposits; DDGF is government de- mand deposits; RMFRB is the discount rate; and RMGBS3 is the average market yield on three-month Treasury bills. Substituting the definitions for RDD and RDT into 1.31 and using the ai to replace constants, we have - RESNBC 1 S + a, RRR (DD M ) + a2 RRR1- DT DD + DT (t.32) ( + [a3 RMRR - a4MGSB3 -- 5 which clearly shows the dependence of the right-hand side of Equation 1.31 on S,, supposedly given by Equation 1.31. Solving Equation 1.32 for S, yields where SM is the money supply (private demand deposits [DD] and currency); RDD = DD/SM; RDr T= DT/SM; RDDGF = DD ,/S,; DT is private time deposits; RESNBC is unborrowed reserves plus currency held by member banks; RRRDD is a weighted average of required reserve ratios against demand deposits; RRRET is a weighted average of required reserve ratiOos against time deposits; DDGF is government de- mand deposits; RM B is the discount rate; and RMGBS3 is the average market yield on three-month Treasury bills. Substituting the definitions for RDD and R DT into 1.31 and using the ai to replace constants, we have RESNBC SM I-DD +,R ,D(DD + DDGF )+a R T SM RRRDD M )+a2RRRET DT - j[DD + DT J (-2 T) + [a3 RMR - a4 GSB3 - a] - + which clearly shows the dependence of the right-hand side of Equation 1.31 on S,, supposedly given by Equation 1.31. Solving Equation 1.32 for Sm yields REVIEW OF THE LITERATURE 9 effects of changes in the various rates of interest, or the money stock on the real variables in the model. (Their effects are measured elsewhere in the Brookings model.) The model itself is composed of nineteen simultaneous equations, four of which are identities (the reserve identity, etc.) and the rest of which express the desired changes in assets in terms of lagged asset holdings, rates of return, and various short-run constraints on asset holding. Solving this system simultaneously, de Leeuw derives the following expression for the money stock (p. 518): =M RESNBC M 1-Ro0 + 0.84 [RRRDD] [RDD + RDDG + 0.82 [RRRDT] [RDT1] + [0.011 RMFR B 0.010 RMGBS3 - 0.007] [RDD + RDnI (1.31) where SM is the money supply (private demand deposits [DD] and currency); RDD = DD/SM; ROT = DT/SM; ROOGF DD / ; DT is private time deposits; RESNBC is unborrowed reserves plus currency held by member banks; RRRo0 is a weighted average of required reserve ratios against demand deposits; RRRDT is a weighted average of required reserve ratios against time deposits; DDGF is government de- mand deposits; RMFRB is the discount rate; and RMGBS3 is the average market yield on three-month Treasury bills. Substituting the definitions for RDD and RDT into 1.31 and using the ai to replace constants, we have - RESNBC j SM + a, RRR0, ( D D ) + a2 RRRDT DT DD + DT )+ [a3 RMFRB - O4 RMGSB13 - a5 - [ k 3 M M which clearly shows the dependence of the right-hand side of Equation 1.31 on SM, supposedly given by Equation 1.31. Solving Equation 1.32 for SM yields  10 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM SM = RESNBC + DD - 0.84 [RRR5D] [DD + DDG F - 0.82 RRR51 DT - [0.011 RMFRB - 0.010 RMFBS3 10 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM M = RESNC + DD - 0.84 [RRRDD] [DD + DDGF] - 0.82 RRRDT DT - [0.011 RMFRB - 0.010 RMFBS3 0.007] [DD + DT]. (1.33) 0.007] [DD + DT]. (1.33) 10 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM M = RESNBC + DD - 0.84 [RRRDD] [DD + DDG0 - 0.82 RRRDT DT - [0.011 RMFRB - 0.010 RMFBS3 - 0.007] [DD + DT]. (1.33) Equation 1.33, derived from de Leeuw's expression for the money stock, says that the money supply, SM, is smaller in size than unbor- rowed reserves plus currency held by the banks plus private demand deposits. This is a nonsense result and throws suspicion on the entire de Leeuw model. For the month of April 1969, the appropriate figures (taken from the July 1969 Federal Reserve Bulletin) are (in billions of dollars): Equation 1.33, derived from de Leeuw's expression for the money stock, says that the money supply, S,, is smaller in size than unbor- rowed reserves plus currency held by the banks plus private demand deposits. This is a nonsense result and throws suspicion on the entire de Leeuw model. For the month of April 1969, the appropriate figures (taken from the July 1969 Federal Reserve Bulletin) are (in billions of dollars): Equation 1.33, derived from de Leeuw's expression for the money stock, says that the money supply, SM, is smaller in size than unbor- rowed reserves plus currency held by the banks plus private demand deposits. This is a nonsense result and throws suspicion on the entire de Leeuw model. For the month of April 1969, the appropriate figures (taken from the July 1969 Federal Reserve Bulletin) are (in billions of dollars): Total reserves Borrowings Unborrowed reserves Total demand deposits Government demand deposits Private demand deposits Discount rate Yield on three month bills Time deposits 27.079 .996 26.083 152.8 5.1 147.7 5.5 per cent 6.11 per cent 201.6 Total reserves Borrowings Unborrowed reserves Total demand deposits Government demand deposits Private demand deposits Discount rate Yield on three month bills Time deposits 27.079 .996 26.083 152.8 5.1 147.7 5.5 per cent 6.11 per cent 201.6 Total reserves Borrowings Unborrowed reserves Total demand deposits Government demand deposits Private demand deposits Discount rate Yield on three month bills Time deposits 27.079 .996 26.083 152.8 5.1 147.7 5.5 per cent 6.11 per cent 201.6 Plugging these figures into Equation 1.33 and performing the arithmetic, we find that de Leeuw's equation gives a money supply of $150.64 billion. The actual money supply for April 1969 was $196.7 billion. The difference between de Leeuw's prediction and the actual money stock is primarily the $43.9 billion of currency in circulation in April. As can be seen from Equation 1.33, this component of the money supply has been lost in de Leeuw's formulation. Of the models reviewed here, the de Leeuw model is most similar to the approach taken in this study. The other models tend to be deficient in two respects. First, they are too aggregative in the sense that the economy is broken down into only three sectors-the government, banks, and the public. No distinctions among households, manufacturing firms, and nonbank financial firms are drawn. Second, they all hide the general equilibrium nature of the monetary mechanism. In the first two models reviewed, the behavior functions of the various sectors are not explicitly specified. The Teigen model, while specifying the public's Plugging these figures into Equation 1.33 and performing the arithmetic, we find that de Leeuw's equation gives a money supply of $150.64 billion. The actual money supply for April 1969 was $196.7 billion. The difference between de Leeuw's prediction and the actual money stock is primarily the $43.9 billion of currency in circulation in April. As can be seen from Equation 1.33, this component of the money supply has been lost in de Leeuw's formulation. Of the models reviewed here, the de Leeuw model is most similar to the approach taken in this study. The other models tend to be deficient in two respects. First, they are too aggregative in the sense that the economy is broken down into only three sectors-the government, banks, and the public. No distinctions among households, manufacturing firms, and nonbank financial firms are drawn. Second, they all hide the general equilibrium nature of the monetary mechanism. In the first two models reviewed, the behavior functions of the various sectors are not explicitly specified. The Teigen model, while specifying the public's Plugging these figures into Equation 1.33 and performing the arithmetic, we find that de Leeuw's equation gives a money supply of $150.64 billion. The actual money supply for April 1969 was $196.7 billion. The difference between de Leeuw's prediction and the actual money stock is primarily the $43.9 billion of currency in circulation in April. As can be seen from Equation 1.33, this component of the money supply has been lost in de Leeuw's formulation. Of the models reviewed here, the de Leeuw model is most similar to the approach taken in this study. The other models tend to be deficient in two respects. First, they are too aggregative in the sense that the economy is broken down into only three sectors-the government, banks, and the public. No distinctions among households, manufacturing firms, and nonbank financial firms are drawn. Second, they all hide the general equilibrium nature of the monetary mechanism. In the first two models reviewed, the behavior functions of the various sectors are not explicitly specified. The Teigen model, while specifying the public's  REVIEW OF THE LITERATURE 11 demands for demand deposits and time deposits, does so in terms of the total money stock, and takes the total stock of money as the independ- ent variable in these functions. While such a formulation will probably yield significant empirical results, from a theoretical point of view it seems awkward to visualize the public changing their holdings of de- mand and time deposits in response to a change in M rather than because of changes in income, prices, and interest rates. Hiding the general equilibrium nature of the problem also precludes description of the effects of changes in the money stock on the real variables of the economy. (This can be done in the SSRC model, but not by the de Leeuw submodel itself.) The purpose of this work is, however, not to repudiate any of the existing work in this area, but rather to extend and amplify the analysis begun by these more narrow and specialized studies. REVIEW OF THE LITERATURE 11 demands for demand deposits and time deposits, does so in terms of the total money stock, and takes the total stock of money as the independ- ent variable in these functions. While such a formulation will probably yield significant empirical results, from a theoretical point of view it seems awkward to visualize the public changing their holdings of de- mand and time deposits in response to a change in M rather than because of changes in income, prices, and interest rates. Hiding the general equilibrium nature of the problem also precludes description of the effects of changes in the money stock on the real variables of the economy. (This can be done in the SSRC model, but not by the de Leeuw submodel itself.) The purpose of this work is, however, not to repudiate any of the existing work in this area, but rather to extend and amplify the analysis begun by these more narrow and specialized studies. REVIEW OF THE LITERATURE II demands for demand deposits and time deposits, does so in terms of the total money stock, and takes the total stock of money as the independ- ent variable in these functions. While such a formulation will probably yield significant empirical results, from a theoretical point of view it seems awkward to visualize the public changing their holdings of de- mand and time deposits in response to a change in M rather than because of changes in income, prices, and interest rates. Hiding the general equilibrium nature of the problem also precludes description of the effects of changes in the money stock on the real variables of the economy. (This can be done in the SSRC model, but not by the de Leeuw submodel itself.) The purpose of this work is, however, not to repudiate any of the existing work in this area, but rather to extend and amplify the analysis begun by these more narrow and specialized studies.  2. The Model T HE MODEL is made up of five sectors: public, manufacturing (the firms), banking (the banks), nonbank financial (the intermediaries), and government. This chapter contains details of each sector and the relationships among sectors. The solution to the model will be con- sidered in chapters 3 and 4. The behavioral relations for each sector are given in both implicit and explicit form. For simplicity two assumptions are made: most of the explicit forms are linear and reflect either utility or profit maximizing behavior, and the individual units in each sector are homogeneous so that, in most cases, aggregate levels can be obtained by summing the representative functions. PRODUCTION, INVESTMENT, AND GROWTH Technology is assumed to be characterized by increasing opportunity costs and is constant over time. For simplicity these assumptions are made: 1. There are only two inputs-capital and labor. A unit of labor is indistinguishable from any other unit of labor. Capital is also perfectly homogeneous. 2. There are only two outputs-capital and the consumption good. The consumption good is perfectly homogeneous. 3. Firms fall into two categories-those that produce only the capital good and those that produce only the consumer good. Each firm within each category is identical to every other firm in the group. There is a large enough number of firms in each category so that, coupled with freedom of entry and exit, each firm is a perfect competitor in the output market. 4. Individuals in the economy have identical endowments of capital and labor. No organization controls the supply of either capital or labor. 12 2. The Model THE MODEL is made up of five sectors: public, manufacturing (the firms), banking (the banks), nonbank financial (the intermediaries), and government. This chapter contains details of each sector and the relationships among sectors. The solution to the model will be con- sidered in chapters 3 and 4. The behavioral relations for each sector are given in both implicit and explicit form. For simplicity two assumptions are made: most of the explicit forms are linear and reflect either utility or profit maximizing behavior, and the individual units in each sector are homogeneous so that, in most cases, aggregate levels can be obtained by summing the representative functions. PRODUCTION, INVESTMENT, AND GROWTH Technology is assumed to be characterized by increasing opportunity costs and is constant over time. For simplicity these assumptions are made: 1. There are only two inputs-capital and labor. A unit of labor is indistinguishable from any other unit of labor. Capital is also perfectly homogeneous. 2. There are only two outputs-capital and the consumption good. The consumption good is perfectly homogeneous. 3. Firms fall into two categories-those that produce only the capital good and those that produce only the consumer good. Each firm within each category is identical to every other firm in the group. There is a large enough number of firms in each category so that, coupled with freedom of entry and exit, each firm is a perfect competitor in the output market. 4. Individuals in the economy have identical endowments of capital and labor. No organization controls the supply of either capital or labor. 12 2. The Model THE MODEL is made up of five sectors: public, manufacturing (the firms), banking (the banks), nonbank financial (the intermediaries), and government. This chapter contains details of each sector and the relationships among sectors. The solution to the model will be con- sidered in chapters 3 and 4. The behavioral relations for each sector are given in both implicit and explicit form. For simplicity two assumptions are made: most of the explicit forms are linear and reflect either utility or profit maximizing behavior, and the individual units in each sector are homogeneous so that, in most cases, aggregate levels can be obtained by summing the representative functions. PRODUCTION, INVESTMENT, AND GROWTH Technology is assumed to be characterized by increasing opportunity costs and is constant over time. For simplicity these assumptions are made: 1. There are only two inputs-capital and labor. A unit of labor is indistinguishable from any other unit of labor. Capital is also perfectly homogeneous. 2. There are only two outputs-capital and the consumption good. The consumption good is perfectly homogeneous. 3. Firms fall into two categories-those that produce only the capital good and those that produce only the consumer good. Each firm within each category is identical to every other firm in the group. There is a large enough number of firms in each category so that, coupled with freedom of entry and exit, each firm is a perfect competitor in the output market. 4. Individuals in the economy have identical endowments of capital and labor. No organization controls the supply of either capital or labor. 12  THE MODEL 13 THE MODEL 13 THE MODEL 13 Thus, the capital goad firms see also perfect competitors in the input Thos, the copitat good firms ore also perfect compositors in the input That, the copitl good firms are alto perfect competitors in the inut market aud the taoo market is perfectly competitive. morket sod the laoo maehet is perfectly competitive. markot sod the lahor market is perfectly competitive. Poduction Production Production The aggregate production fonction for the capitol good is giver hy The aggregate prodoction function foe the capital good is given hy The aggregate production founction foe the capitol good is tiara hy Xk= Xk, (Lk, Xcc) (2.1) x: = Xc (Lc, Xkc) (2.1) x: = Xc (Lc, Xcc) (2.1) wheeeLis theamount of laorsed in the podctionf capietlad where L istheoamoountoflaboreused inthe podction of capitaland where Lkis theaounat ofaborsed intheproductionof cpitalsand Xk is thetamoutsf capitalued inthep rdction of cpitl, Xkk istheoamout of capital used in the prdction of cpitl, Xcc is the amount ofcapitalsed intheprdctionf cpita, Xk> 0, k4 >0, 304 <0, Va4 <0, 04 > 0, 04k >o 354 ____ 0 e < 0, k> 0, axk > k3 <0, kt:< sod and tand kx >o 0.xk > 0 a5Xk >0. aLc OX50 < 31,k aX55 < 3L5 aXcc If there ore n firms producing capital, the production function for the If there are sn fiems prodacing capital, the productiorn function foe the If there ore n firms produciug copital, the production function foe the its dividual firm is i~h individual firm is i thindividual firm is X57 =. =Xc (2.2) Xci = -Xk n_'' (2.2) Xci = k (2.2) x:i (L Xs1) GnXk . Xi= . The aggregate production fuction far the conesumer good firmt is given The aggregote production function for the contumer goad firms is given The aggeegate production fusctios foe the consumr good firms is given bly bly hy X = Xc(L,, xk.) (2.3) x.: = Xc(Lc, X5,) (2.3) x:' = Xc(Lc, Xk,) (2.3) where L, is the amouse of laboe used in the production of the car- where Lc is the amount of labor ased in the production of the coo- wnhere L. is the amount of luarosed in the production of the con- sumptinvgoodtand Xk,is the amountf cpitlausd ithe poductiont sampiou good and Xc,is theamountf capitaltusedin the prdction sumptin goaduXk5is theamountof capitlsed inthe prdction of the consumption good. of the consumption good, of the consumption good. >0 >x >0, ax: 0 ax: > 0 1at: < 0 ax: 0, 1x: >01 3042 <0' c 2x: 0 aed and and  14 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM ax0" ay < If shere see mn films prodocieg she consumer good, she psoductios fusctions fos she j'5 fisss is Xd= -X( 0 ,Xk) X, M_ i -In(2.4) The transformation curve is shows is Eigore 1. The sransformasion functios is gives by Xc = T(Xk) (2.5) wher I. T(0) = ac 2. T(ski) = 0 11iis she maximuem production of she consumption good isn period while ak sepresents she maximum psoductios of copisal fee she some peeiod. Defising she transformation fonctioo implicitly we boos T'(X,, X,) = 0. (2.6) Thos X=(5QX, is o foll-employment output vectos if T'(Xk, XV = 0. (2.7) X'= (X',, Vc) hs less thao foll employmens if T'(X'k I X'c) < 0. (2.8) If T' < 0 she unused psoductive potestial of she economy is measured by she negative solos of T. The economy dis full-employment eqoilib- 14 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM s1x2 0. If shere are mn firms producisg she cosuwee good, lbs production function fore sj~h fiem is X =Xc ( I , c Xs mn n -m- (2.4) The transformation cueve is shows is Figure 1. The traosformotion fusction is gives by Xc = T(Xk) (2.5) where 2. T(o0) = 0 3. a,< 0. ais she maximum productionof thecosumpioogood ispesiod i while ski represents she masximom predoction of copisol fee the some pesiod. Defining she transfosmation fuscsion implicisly we hove T'(X, Ic V 0. (2.6) Thus X = (Xk, Xc) is a follfemploymens output vector if T'(0, Xc) = 0. (2.7) K= (X'k, X'c) is less shoe foil employmest if T'(X'0,I X'c) < 0. (2.8) If T' < 0 the unussd productive posential of the ecosomy is measured by she negotice volue of T. The ecosomy is is full-employmentseqoilib- 14 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM a2X ' 0. If there see m firms psoducing she cosomer good, she productios fusction foe she jhfirm is ,= ; = X f ff .)) (2.4) The tsassformatios cone is shows is Figure 1. The tsansformasios fusction is gives by Xc = T(X0) (2.5) wher I. T(0)=~ 2. T(s)O= 3. a,0 . OXc 11iis she maximum peodciosofthe consssmpiosgood inperiod i while a. repeesents she mosximom productios of capital foe she some peeled. Defising she transformation fuscties implicitly we hove T'(Xk, Xc) 0. (2-6) Thos X = (Xs,, X,) is a full-empleymess output vector if T'(X0, X,) = 0. (2.7) = (X',, X'c) is Sews shoe fulR employmess if T'(X,, IK,) < 0. (2.8) If T' < 0 she usnosed productive poentsiol of she econsomy is measured by she segative volue of T. The economy is is full-employmess eqoilib-  THE MODEL 15 niuns if 2.6 is satisfied and if 0c/'k = dXc/dXc. Ens purpases af the madel it is attained that the explicit fasm af 2.5 is X kX( kai (2.9) and aci = k foe all i. That the explicit transformatin case attained is a quarterecircle in the psitive qadrant. Writing 2.9 ina form eqina- THE MODEL 15 siam if 2.6 is satisfied and if PcIPk = dXc/dXc. Fat parposes af the model it is attained that the explicit farm af 2.5 is Fli+Xk = a2k (2.9) and aci=ai foall i.Thstheexpicit tansfomain cuevassumed is a qarter circle in the pasitive qaadrant- Writing 2.9 inca fasm eqeint- Xx THE MODEL 15 siam if 2.6 is satisfied and if Pc/Pk = dXc/dXc. Fee paspases af the madel it is attained that the explicit fasm af 2.5 is + k~ = (2.9) and ac ai far alli. Thsthe eplicttansforatin cureassumed is a qarter circle in the pasitive qaadrant. Writing 2.9 in a fasm eqaina- aK5 aK, aK1 I X, eC, Fig. 1. Thestsansfnsnaincsve lent en 2.2 nwe hate that X\ (X1i, XkD) is a full-employment autput vectas if X'i= 17f X5,I XL 0- (2.10) The marginalsrateof transformation is dX, X, MRT =- - c211 dXc (a2 - X2)1/2 2.1 Whten 2.10 is satisfied, autput is at fall emplayment and where, in additian, 2.11 is equal In the price satin, autpat is also an equilihriuin output. The explicit fasm of 2.7 is simply Xi+ X21 - oaL = 0. (2.12) I I X eeC5 Fig 1. The transfnormannon case lent ta 2.2 we havs that X = (Xc5, XLi ) is a full-emplaymen oatpat vectas if XL=i \f X XL 0- (2.10) The mnarginal sale of transformation is NIT dXc Xk(.1 MRT= c X2)1/2 dXk (n k k)5 (.1 Whten 2.10 is satisfied, output is at fall emplayment and where, in additian, 2.11 is eqal In the grie satin, oatput is nlso an equilihrium output. The explicit farm af 2.2 is tin-ply XLi + XLi - .2 = 0. (2.12) etC5 Fig. t. Ths transorationcuv lent ta 2.2 we have that X'= (X,5, Xki) is a full-emplcoyment output vectcor if x0ki=4U xt 0 (2.10) The marginal eats of transfarmation it MRT= dXc = ,(2.11) Whsen 2.10 is satisfied, oatpnt is as fall employment and whese, in additian, 2.11 is equal to the psice eatia, autput it alsa an equilihrium autput. The explicit foem af 2.7 it simply XL + X k a (2.12)  16 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Growth and Investment The labor force is assumed to grow at the same rate as the population. This rate is assumed to be a function of the rate of change of the real output of the consumption good over time. 16 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Growth and Investment The labor force is assumed to grow at the same rate as the population. This rate is assumed to be a function of the rate of change of the real output of the consumption good over time. dL dX, iL = = h (g ) (2.13) dL dXe rt /L = X = A ( ) (2.13) where X is the rate of growth of the labor force, X(dX,/dt) is functional notation, and where 1 dA > 0. dX d( C) dt The rate of growth of the capital stock, k, is not tied to the rate of growth of the labor force. Gross and net investment are determined by the interaction of the supply and demand for capital. In general, how- ever, k is assumed to be a function of the price of capital, the price of the firms' output, the firms' profit expectations, the various rates of interest, the rate of depreciation, and the existing stock of capital. The demand for capital is composed of both a stock demand for capital, D, and a flow demand, dK.t The stock demand is given by where X is the rate of growth of the labor force, A(dX/dt) is functional notation, and where 1 d >0. dX d( c) dt The rate of growth of the capital stock, k, is not tied to the rate of growth of the labor force. Gross and net investment are determined by the interaction of the supply and demand for capital. In general, how- ever, k is assumed to be a function of the price of capital, the price of the firms' output, the firms' profit expectations, the various rates of interest, the rate of depreciation, and the existing stock of capital. The demand for capital is composed of both a stock demand for capital, D, and a flow demand, dK.t The stock demand is given by 16 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Growth and Investment The labor force is assumed to grow at the same rate as the population. This rate is assumed to be a function of the rate of change of the real output of the consumption good over time. dL dX ((2.13) where A is the rate of growth of the labor force, h(dXc/dt) is functional notation, and where > dO >0. dX d( - ) dt The rate of growth of the capital stock, k, is not tied to the rate of growth of the labor force. Gross and net investment are determined by the interaction of the supply and demand for capital. In general, how- ever, k is assumed to be a function of the price of capital, the price of the firms' output, the firms' profit expectations, the various rates of interest, the rate of depreciation, and the existing stock of capital. The demand for capital is composed of both a stock demand for capital, D, and a flow demand, dK.t The stock demand is given by Dk = D(P, r, 0) (2.14) where r is a vector of interest rates (r = (r, rg, r, etc.)) and 0 is a profit expectation function; also OPk rk <0, and >0. The flow demand for capital, dK, is given by dK = nK (2.15) where n is the rate of depreciation, 0 < n < 1, and K is the existing capital stock. 1. See P. Davidson 113). Dk = D0(Pk, r, 0) (2.14) D = D0(Pk, r, 0) (2.14) where r is a vector of interest rates (r = (r, rt, etc.)) and 0 is a profit expectation function; also ak k <0, and Sn > 0. The flow demand for capital, dK, is given by where r is a vector of interest rates (r = (r, rg, r t, etc.)) and 0 is a profit expectation function; also k -k 0, and >0. OP r 00 The flow demand fee capital, 4K, it given by dK = nK (2.15) dK = nK (2.15) where n is the rate of depreciation, 0 < n < 1, and K is the existing capital stock. 1. See P. Davidson (13). where n is the rate of depreciation, 0 < n < 1, and K is the existing capital stock. 1. See P. Davidson (13).  THE MODEL 17 The supply of capital is also composed of a stock supply and a flow supply. The stock supply, Sk, is simply equal to the existing capital stock, while the flow supply, st, is assumed to be dependent on the price of capital. THE MODEL 17 The supply of capital is also composed of a stock supply and a flow supply. The stock supply, Sc, is simply equal to the existing capital stock, while the flow supply, sk, is assumed to be dependent on the price of capital. Sk Sk 'k (2.16) k k k) (2.16) THE MODEL 17 The supply of capital is also composed of a stock supply and a flow supply. The stock supply, Sk, is simply equal to the existing capital stock, while the flow supply, sc, is assumed to be dependent on the price of capital. k sc't) (2.16) where dsk >0. dPc In Figure 2, D + dK is the market (stock + flow) demand for capital and Sk + sk is the market (stock + flow) supply of capital. Pk is the PcI K st /sK+SK dsk >0. dPc In Figure 2, D + dK is the market (stock + flow) demand for capital and S + s is the market (stock + flow) supply of capital. P is the P S s. S +ss D +dK K K K, K, Fig. 2. Supply and demand for capital equilibrium price of capital. At this price gross investment is equal to K2 - K and net investment equal to K, - K. Note that it is the rates of interest relevant for financing and deter- mining relevant discount rates that, along with profit expectations, determine the exact locations of Dk. As rates fall, Dk shifts outward, ceteris paribus. Although it would be more elegant to consider gross and net invest- dsk > 0. dP, In Figure 2, Dk + dK is the market (stock + flow) demand for capital and Sc + sc is the market (stock + flow) supply of capital. Pc is the Pu SK sK Sn +sg K K K, K-, Fig. 2. Supply and demand for capital equilibrium price of capital. At this price gross investment is equal to K2 - K and net investment equal to K, - K. Note that it is the rates of interest relevant for financing and deter- mining relevant discount rates that, along with profit expectations, determine the exact locations of Dk. As rates fall, Dk shifts outward, ceteris paribus. Although it would be more elegant to consider gross and net invest- T. K K K, K. Fig. 2. Supply and demand for capital equilibrium price of capital. At this price gross investment is equal to K2 - K and net investment equal to K, - K. Note that it is the rates of interest relevant for financing and deter- mining relevant discount rates that, along with profit expectations, determine the exact locations of Dk. As rates fall, D shifts outward, ceteris paribus. Although it would be more elegant to consider gross and net invest-  18 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM meat for each geoup of firms separately, we shall assame that 2.14-16 are defied in sock a wanner that their solution ar shown in Figare 2 tepresents the aggregate levelr of grostaed net inestment far horh geoups of fhrms comhieed. The rate ef growth of the capital stock, k, it, in teems of Figure 2, 18 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM meet fer each group of firms separately, we shall astume that 2.14-16 ate defined ia such a maner that their solatioe at showa in Figare 2 represenrs rhe aggregate levels of gross and et investmet forboth groupstoffirms comined. The rte of growth of the capiral stock, h, is, ia teems of Figure 2, K+K - K K, K K (2.17) kK +Kr K, K K (2.17) We mast now coesider she effects of changes in the capiral stack and the lahor force on she rransformation carve, given that technology' is constant. The questioneishbasically this: if LtK(tgiveoak = ac what will ar aad at equal if Lgrowsthyhtper cenr aed Kgrowshy kt per ceat, Xt >/< k,, e.g., what relation will the transformation cate in a + 1 heat to she cate ia t? Will relatioe 2.9 hold ever lime? If eat, what other asumptions ahout the eature of production must we make to insure that it does? Writing the total differetials of the production functions, we have We mart snow consider the effects of chaeges inthe capital stock and the labor force oa the teaesformation carve, givea that techeology is constant The question isbasically this: if L,Kgive ak act, what wilt acres anda "k, I equal if Lgrowshbyhtperceet andK gowsahy kt per cet, he >/< k5, e.g., what relation will the transfotmation carve inet+ 1 bearto the curve int? Willrelationa2.9 hold ovee timse?Ifaot, what ashes assumptions about the nature of production mart we make so ansare that it doer? Writing she tosal differentials of she production functions, we have 1t EQUILIBRIU M STUDY OF THE MONETARY MECHANISM minI foe each group of firms separately, we shalt assame rhas 2.14-16 ore defiaed in such a maner that their solution as shown ia Figuee 2 represents the aggregate levels of gross and net investment for hash groaps of firms comhined. The sate of growth of rhe capital stock, k, hs, in trtin of Pigare 2, K +K- K K, k = _____ - . (2.17) K K We meal now consider the effects of changes in the capital stock and the lahor force on the transformation curve, given rhas technology is conslant. The question is basically tis: if L,, K, give aks =Oact, what will 0010+5 and ak I I equal if LgrowshbyhXper centand Kgrowshby kt pet cent, h5 >/< k5, e.g., what relation wilt the transformation carve in t + 1 hear ro the carve in t? Wilt relation 2.9 hold over time? If aot, what other assumptions ahout the narure of peoduction mast we make to insane that it does? Writing the total diffesentias of the prodaction fanctions, we have dXc = -c dK+ac dL, (2.18) tKc IL1 sI0 Rk dK k~ dLU. (2.19) From the defiaitions of k and X., k =- -- /K (2.20l) C dL dX= tXc dK+ SdL, dXk = 3k 4Kk + afiX, dLk Pram the definitions of k and Xs, K dK L ds we have dK = kK da and (2.18? (2.19) (2.20) (2.21) (2.22) ax-K ax dXk = -X dKk + aL0 dL0. Prem the definitions of k and X, kK dK / we have dK k - =k (2.18) (2.19) (2.20) (2.21) (2.22) L ds L we have dK k ds and (2.21) (2.22)  THE MODEL 19 (2.23) THE MODEL 19 (2.23) dL XL dt frost which it follows that dK = kKdt dL =XL, from which it follows that dK = kKds aod dL = ltLdt. dL h Lt. Sobstitutiog 2.24 ond 2.25 ioto 2.18 sod 2.19 too hoot dX0 = hx Kdt +1a- XLdt 8K KL (2.24) (2.25) (2.26) (2.27) (2.28) (2.29) a_ ax0 dXk = hKdt + -tkXLdt 8K 8L from which it follows that dX0 ax c ax X dt ax 8L Suhstitutintg 2.24 and 2.25 isto 2.18 cod 2.19 wo hoot d = cx hKdt + a XLdt 8K 8L, and dk= 8 k hKdt ax0k XLdt 8K al, from which it follows that dX = h K + a XL aod at 5K L- (2.24) (2.25) (2.26) (2.27) (2.28) (2.29) THE MODEL dL -XL, ds from which it follows that dK = hKdt and il = XLdI. Suhstitutiog 2.24 aod 2.25 ino 2.18 and 2.19 wt haot dX0 = a__ hKdt + a XLdt 3K UL (2.24) (2.25) (2.26) dk= ax0 hKdt a x0 ~d 3K 8 from which it follows that dxc = aX, hK + aXL dft a x 8L (2.28) (2.29) ax0 ax0 kK k XL. 81t 3K L ax0 = ax0 kK4X XL. 3t 8K L Equatiots 2.28 and 2.29 tell os how tht mooimum possihle outputs of thocoosuntor good and the capitol food chaoge over tint if tho eotirt increase it the stockos of lahor and capitol is used in onesgood at Itoe other. tnoarder far the traosfotmation carat to shift io a pasollel way as a result of the growth of capitol and labor, itisncssary and sufficieot that 2.28 oqual 2.29. Sioct ma are statting from a postiion whort 00 = 0,aoly theraotesof changeneedhbeoqual toisuoretheicreaseoi.nac it equal to the increase an "0. Thus we hoot Equations 2.28 asnd 2.29 toll as how the maximum possihlt outputs of the consumes good aod tht capdtal good change over timt if the entire increase in le stocks of lahor and capital is used intone good or the other. It ordet fat the tsansfosrmation carat to shift it a patadllel way as a totals of the growth of capital and laboit is necessary aod sufficient that 2.28 equal 2.29. Sitar wt art startsing fsom a position ohete a, ",only the rtes of change need he equal to insure the increase is a, is equattotheoincrase inoaThus wehave Equations 2.28 and 2.29 toll as how the maximum possihte outputs of tho coosumot goad and the copitl good change oats limo if the eotirt incoreast an the stochs of lahot tand capital isaused intoone good or the other. tn otder fat tho transfotmation carat to shift inoapaallot way as a result of the growth of capital aod labor, it is ncssary tand sufficiett that 2.28 oqual 2.29. Sioot we are starting ftom a position ohoso ac = "k, onlytheoratesof changeneedhbetqalto isuretheicreseinai oquol to tho inctease an "k. That wt hoot  20 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 20 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM a kK+ a XL= k kK+ x XL. SKa fLe K, EKk (2.30) 3X' kK + a-x' XL S Xk kK+ Xk L K+ aL- akK+ AeL O, Oec aEk SEk (2.30) Equation 2.30 can be rewritten in two ways, both of which express the condition necessary for a parallel shift of the transformation curve. (1) kK( X - aik)+XL( a - k)=0 aKe OK5 OL, Lk Regardless of the relative sizes of k and X and of their signs, if the marginal product of capital in production of the consumer good is equal to its marginal product in producing capital, and if the same is true of the marginal products of labor in its two uses, (1) will be satisfied. If these marginal products are not equal, (2) expresses the condition that must be satisfied for parallel shifts. Number (2) is also derived directly from 2.30. AL ax Xe (2) aK, OK aXe - 5X aLe 0Lk Since (2) places no unrealistic constraints on the production processes, we assume that it is satisfied for all X and k between ± 1. The Labor Market The aggregate supply of labor is a function of the wage rate, P, and the size of the population. If it is assumed that the labor force is a constant percentage of the population, we may write Equation 2.30 can be rewritten in two ways, both of which express the condition necessary for a parallel shift of the transformation curve. (1) kK( fXc - a )+XL( a - ___ 0K, 014k O- OLk Regardless of the relative sizes of k and X and of their signs, if the marginal product of capital in production of the consumer good is equal to its marginal product in producing capital, and if the same is true of the marginal products of labor in its two uses, (1) will be satisfied. If these marginal products are not equal, (2) expresses the condition that must be satisfied for parallel shifts. Number (2) is also derived directly from 2.30. AL ax5 axe (2) = II ax, ax5 Since (2) places no unrealistic constraints on the production processes, we assume that it is satisfied for all h and k between ±1. The Labor Market The aggregate supply of labor is a function of the wage rate, PV, and the size of the population. If it is assumed that the labor force is a constant percentage of the population, we may write 20 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM axe kK + ax, XL= ax kK+ ax5 XL. (2.30) fie OL, OR5 0L5 Equation 2.30 can be rewritten in two ways, both of which express the condition necessary for a parallel shift of the transformation curve. (1) kK( a - 5)+XL(a - a =0 aKa OK fie Lk Regardless of the relative sizes of k and X and of their signs, if the marginal product of capital in production of the consumer good is equal to its marginal product in producing capital, and if the same is true of the marginal products of labor in its two uses, (1) will be satisfied. If these marginal products are not equal, (2) expresses the condition that must be satisfied for parallel shifts. Number (2) is also derived directly from 2.30. XL axe axe (2) = I K hs axc ax5 ae aL, Since (2) places no unrealistic constraints on the production processes, we assume that it is satisfied for all h and k between ± 1. The Labor Market The aggregate supply of labor is a function of the wage rate, P0, and the size of the population. If it is assumed that the labor force is a constant percentage of the population, we may write S = S0(PV, L) where s0 aS2 > Oand > 0. Op0 L (2.31) S = S (PV, L) where ask as0 > 0and >0. OP0 L (2.31) SQ = Sv (PQ, L) where > Oand >0. aPk aL (2.31) At any point in time, the supply of labor may be taken to be a function of only the price of labor. At any point in time, the supply of labor may be taken to be a function of only the price of labor. At any point in time, the supply of labor may be taken to be a function of only the price of labor.  THE MODEL 21 The aggregate demand for labor is composed of the demand of the capital good firms and the consumer good firms, as well as the demands of the banks, government, and intermediaries. No attempt will be made to specify explicitly the demand functions for these sectors. (This is in keeping with the practice of not specifying these sectors' demand for the capital good explicitly.) This demand is included by adding a constant, E, to the sum of 2.32 and 2.33. These demands are THE MODEL 21 The aggregate demand for labor is composed of the demand of the capital good firms and the consumer good firms, as well as the demands of the banks, government, and intermediaries. No attempt will be made to specify explicitly the demand functions for these sectors. (This is in keeping with the practice of not specifying these sectors' demand for the capital good explicitly.) This demand is included by adding a constant, E, to the sum of 2.32 and 2.33. These demands are THE MODEL 21 The aggregate demand for labor is composed of the demand of the capital good firms and the consumer good firms, as well as the demands of the banks, government, and intermediaries. No attempt will be made to specify explicitly the demand functions for these sectors. (This is in keeping with the practice of not specifying these sectors' demand for the capital good explicitly.) This demand is included by adding a constant, E, to the sum of 2.32 and 2.33. These demands are Dk = MPg0P, D = MPg1Pc since all firms are perfect competitors. The aggregate demand is simply DQ = MPgcPc + MP10PXe + E. (2.32) (2.33) (2.34) DR = MP2 P DQ= MPg P since all firms are perfect competitors. The aggregate demand is simply D = MPk Pk + MPQCPXc + E. (2.32) (2.33) (2.34) D MPgk P Dll =MP1P, since all firms are perfect competitors. The aggregate demand is simply DQ = MPg0P + MP5,PX, + E. (2.32) (2.34) Under our assumptions on production, the aggregate demand curve will be downward sloping. Since labor is homogeneous, it must be paid the same wage in each use. Thus we have Figure 3. Pg is the equilibrium Under our assumptions on production, the aggregate demand curve will be downward sloping. Since labor is homogeneous,_it must be paid the same wage in each use. Thus we have Figure 3. Pg is the equilibrium Under our assumptions on production, the aggregate demand curve will be downward sloping. Since labor is homogeneous,_it must be paid the same wage in each use. Thus we have Figure 3. PQ is the equilibrium SD+DL+E I I D +DL DL SCK L L F LK L Fig. 3. Supply and demand foe Ianer Dc+DL+E DL +DL IC K DL C L Le La L Fig 3. Supply and demand Ice lbors 'DL +DL+E DL +DL L L F La L Fig 3. Supply and demand for bos  22 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM peice of lahor. The amouants employed hy each group of fiems can he read feom she diogeom: L is she sotal amoont emnployed, L. she amount ueedhbythe consoumergood firms, and L- L. she amount employed hy she capieai good firms. No eesteictions oee placed oen dPn/dI, e.g., no assumption of mage inflexihiiy is mode. Thos, in one se010, lahor will he folly employed so long os she exiseing PQ is an eqoilihriom peice. If P,0 PQ, involunary ounemploymeot exises. If Pg PV, labor is folly employed, evn though there is a poesiie excess demaod foe laoo. Productios Equilihriom and Foll Employmene These coodieions mose he eoeisfied foe ehe peodoceive seceoe of ehe economy to be in equiliherium: 1. PQ isequal inhothoses. 2. P0 is equal inhboth noes. 3. PV = MpkkPk MPg0P,. 4. Pk = MPkkpk Mpk0Pc. Cooddtion 4 cleaely implies shoe ehe maeginal peodoct of copitol in she prodoction of capisol must he I in eqoiloim. Thie is notesartling, since ehe capitol good fisms moold ohvioosly increase their omn ose of capial if MPkk>l ondereduce iif MPkk P2, involontary onemploymenos exists. If Pg < Pg, laor is folly employed, even shoogh there is o positive excess demand foe lahor. Prodoceion Eqoilihrium aod Foll Employment These conditions moest he sasisfied foe she prodoctive sector of she economytohbe in equiirim: 1. Pg is equal inhbothases. 2. Pk iseqnualinbhses. 3. Pg = MP50P0 = MP50P,. 4. P0 = MPkkPk = Mpk0P,. Coodition 4 cleorly imphies shoe she marginal prndoct of capitel in she prodoction of capital moss he 1 in eqoiiorim. Thisis notsarlng, since she capital good fiems moold ohvoly increase their own one of capial if MP0>lI andsreduceditif MPkk Pg, innoluntary unemployment edists. If Pg < PQ, lahor is fully emaployed, even shough there is a positive excess demand foe lahor. Production Equiliheium and FoPl Employment These coaditions must he satsied fan she produceine sector of she economy so be in equdihlim: I. Pghisequal inhbothses. 2. Pk is equalinhbothuses. 3. Pg = MPli0P0 = MPg0P0. 4. P0 = MP00P0 = Mp0~P,. Condition 4 clearly implies that she marginal product of cupital in she production of capital moss he 1 in equiirim. This is not startling, since she capital good firmsmwould obviously increase theireown ass of capital if Mp0 l 1and reduce it if MP00 k <1 5. The marginal ease of sechnical suhstitution of laoo foe capital equals she input price ratio foe hash groups of herms. a. MRTP~k = P00 P0 . MRTP~,1 = P0 k h. RTk1 = MRTPck/g = 0 When conditions I-h are satisfied, the productive sector is an equilih- rium in she sense thut, given she total amounts of she inpots heing used, it is impoasihle so incease output of she earlier commodity hy redistrih- uting sloe capital and lahor heing used hesmeen the tmo groups of herms  THE MODEL 23 without reducing the output of the other commodity. However, 1-6 are not sufficient to insure that the equilibrium output vector is also a full-employment output vector. This is simply because nothing in these conditions implies that the total stocks of capital and labor are being used. That is, 1-6 may be satisfied under conditions of unemployed labor and/or capital. In this case, even though redistribution of the inputs actually being used cannot increase the output of one commodity without reducing the output of the other, it is entirely possible that increasing the total use of capital and/or labor can lead to an increase in the production of both goods. Thus, another condition must be added to insure that the equilibrium is also a full-employment equilibrium. This condition is simply that the outputs of capital and the consumer goods that satisfy 1-6 also satisfy 7.Xk c , where R, and X. are the outputs resulting from satisfying 1-6. Note that the aggregate level of consumption demand, not yet considered, has an impact on these conditions through its influence on P. and X. and, of course, may prevent condition 7 from being satisfied. THE MANUFACTURING SECTOR (FIRMS) The firms are divided into two groups: one produces only the capital good while the other produces only the consumer good. Each group is assumed to be perfectly competitive. The only interfirm purchases are those of capital goods. Each group will be treated in the aggregate rather than on an individual firm basis. Production and sales for each firm in a group are identical (see the section on production in this chapter). Each firm has a desired level of retained earnings such that the aggregate desired level is given by THE MODEL 23 without reducing the output of the other commodity. However, 1-6 are not sufficient to insure that the equilibrium output vector is also a full-employment output vector. This is simply because nothing in these conditions implies that the total stocks of capital and labor are being used. That is, 1-6 may be satisfied under conditions of unemployed labor and/or capital. In this case, even though redistribution of the inputs actually being used cannot increase the output of one commodity without reducing the output of the other, it is entirely possible that increasing the total use of capital and/or labor can lead to an increase in the production of both goods. Thus, another condition must be added to insure that the equilibrium is also a full-employment equilibrium. This condition is simply that the outputs of capital and the consumer goods that satisfy 1-6 also satisfy where Xk and Xe are the outputs resulting from satisfying 1-6. Note that the aggregate level of consumption demand, not yet considered, has an impact on these conditions through its influence on Pc and X, and, of course, may prevent condition 7 from being satisfied. THE MANUFACTURING SECTOR (FIRMS) The firms are divided into two groups: one produces only the capital good while the other produces only the consumer good. Each group is assumed to be perfectly competitive. The only interfirm purchases are those of capital goods. Each group will be treated in the aggregate rather than on an individual firm basis. Production and sales for each firm in a group are identical (see the section on production in this chapter). Each firm has a desired level of retained earnings such that the aggregate desired level is given by THE MODEL Z3 without reducing the output of the other commodity. However, 1-6 are not sufficient to insure that the equilibrium output vector is also a full-employment output vector. This is simply because nothing in these conditions implies that the total stocks of capital and labor are being used. That is, 1-6 may be satisfied under conditions of unemployed labor and/or capital. In this case, even though redistribution of the inputs actually being used cannot increase the output of one commodity without reducing the output of the other, it is entirely possible that increasing the total use of capital and/or labor can lead to an increase in the production of both goods. Thus, another condition must be added to insure that the equilibrium is also a fall-employment equilibrium. This condition is simply that the outputs of capital and the consumer goods that satisfy 1-6 also satisfy 7. X, = where Xk and Xe are the outputs resulting from satisfying 1-. Note that the aggregate level of consumption demand, not yet considered, has an impact on these conditions through its influence on P, and X, and, of course, may prevent condition 7 from being satisfied. THE MANUFACTURING SECTOR (FIRMS) The firms are divided into two groups: one produces only the capital good while the other produces only the consumer good. Each group is assumed to be perfectly competitive. The only interfirm purchases are those of capital goods. Each group will be treated in the aggregate rather than on an individual firm basis. Production and sales for each firm in a group are identical (see the section on production in this chapter). Each firm has a desired level of retained earnings such that the aggregate desired level is given by E' = uKt + 01I, - I + s[Pet Xet + P5 Xkt]. (2.35) In 2.1, aK, represents depreciation; Pet Xct + Pkt Xkt is, of course, aggregate sales in t; and s is a constant, 0 < s < 1. This term is included to reflect the demand for retained earnings arising from the desire of the firm to insure itself from the unexpected. Such risks are simplistically assumed to grow in proportion to total sales. tI0t _ , is a factor E' = OKt + 01n, + s[Pet X, + Pkt Xkt] (2.35) Er = aKt + l _I + s[P, Xet + P Xkt]. (2.35) In 2.1, aKt represents depreciation; P, Xct + Pkt Xk at, of course, aggregate sales in t; and s is a constant, 0 < s < 1. This term is included to reflect the demand for retained earnings arising from the desire of the firm to insure itself from the unexpected. Such risks are simplistically assumed to grow in proportion to total sales. #1,.. - is a factor In 2.1, aK represents depreciation; Pe Xet + Pr Xr is, of course, aggregate sales in t; and s is a constant, 0 < s < 1. This term is included to reflect the demand for retained earnings arising from the desire of the firm to insure itself from the unexpected. Such risks are simplistically assumed to grow in proportion to total sales. MIt - I is a factor  24 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM reflecting the influence of pace nlee investment on Ike level of retamned earnings, a poetion of which, it ic occamed, aeepe to meet tke demand for financing nt investment in fotoee peeiodc. Foe simpliciey only one previous net investment fignee ic aced in 2.35, although greer reolism could be obeained by perhaps acing averages of severl past periods. It is assumed tha) 0 i greer than zero and less than one. Equation 2.35 olso gives the deeieed level of financial accetc-coch, demond deposiec, lime deposits, goveenment securities, ond depositscininermediaries-in ehe oggregate foe peeiod t, since it ie in ehece foemc chotleetained earnings are held. Al lhe end of each period the actnloand deckred stocks of retained earnings are equalieed by adjusemene of the peofit paymentc to tho owneec of tke fiemc. Only when profic payments are 0010 would it be possible foe the 001001 stock of reetained earnings lo be Iess thoo the deeieed level. In no case will the actool clock exceed the desired level. Before disccecing ehe deeieed dieeeibneion of the clock of reetained earnings, another factar influencing the actual stock most ho dicussed. This is eke relationship hetween desired financing and actual financing. The eeplacement demand foe capktal, Ie - In_ ks assumed tohbepaid for completely out of relamned enings. Only a poetion of net investment, equal to O(111- I , is paid forfrometained earnings. Theeremainderln - 01I,,t - 1, 0100111 tke co-called demand foe financing. Thic domcnd is the hacks foe Ike firms' demand foe bank loans and loanscfeominee- mediodiec and for eheie deckre to iscae moe debe (ehe cupply of films' nonowneechip scunities). We have 24 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM reflecting the influence of pace nee investment on the level of reeained lenings, a portion of which, kt ks accumed, are kepe to weetl ehe demand foe financing net investment in future periods. Foe simplicity only one previouc nel invesemene figare is aced in 2.35, akthough greaee eealism could he obtained by perhaps nsing averages of several past periods. It is accomed ehae I is greatee ehan zeeo and lesc thee one. Equation 2.35 alco givec ehe decired level of financial acceec-cach, demand depositc, lime depocits, government securities, and depoitc in inteemediaries-in ehe aggregate fol peeiod I, since kt ks in tese foems that retained earnings are held. At the end of each period the actual and desired seocks of reetained earnings ace equalized by adjustment of the profit paymenta 10 eke ownees of eke fiems. Only when peofie paymenec ae z110 woald it he possible foe the actual ctock of retained earnings to be lessehan eke desired level. Inno case willheactal sock exceedcthe desired level. Before discscing eke deckred disceibution of eke clock of reetained earnings, anoehee factor influencieg ehe accual ctock wool be discacced. This is the relationship beeween deckred financing and actual financing. The replacement demand for capital, Is- I., ks assumed to bepaid for completely one of retained earnings. Only a portion of net investment, equal t1,t-0 )1 1, is paid foe feom retained earnings. Thebremainder, I. - 401t -i, crees eke ca-called demand for financing. This demand is the basis fob eke fiemc' demand forbhank loanscand loansofrom inter- mediaries and foe thekr desire to kssue more debt (eke supply of firms' nonowneechipsecurities). Wekhave 24 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM refleceing the influence of past net investment on eke level of retained lenings, a poetion of which, it ic allowed, aee kept 10 meet che demand foe financing netlinvestmenteinfutureeperiods. Foreimphicity only one peevious net innestment figuee ic aced in 2.35, alehough geeatee realism could he obtained by perhaps acing averages of several past periods. Ie is assamed that 0 i greater than ero and less than one. Equation 2.35 alco givec the decired level of financial assets-cash, demand deposile, lime deposits, goveenment securities, and deposita in inteemediaries-in Ike aggeegate fob period t, since it ksin chece folms that retained earnings are held. At the end of each period eke actual and desieed stocks of retained earnngs are equalized by adjustment of the peofit paymento 10 eke owneec of the firmc. Only when peofit payments ale zero would it he possible foe the actual stock of retained earnings In be lecs chan Ike decieed level. In eo case wil eke aceual clock exceed te dekeredlenel. Befoee discussing the dekered distribution of eke clock of reetained earnings, anochee factoe influencing eke actual clock mace be dicuceed. Thiskslheelaonhiphetweendesired financngnd actalfinancing. The replacement demand foe capital, 1, - In, ic essumed to be paid foe complecely olt of retained earnings. Only a poetion of net investmene, equal to (101.- 1 ,ki paid foe from retained earnings. The remainder, In - 1n1 - I ,ceeates the co-called demand foc financing. Thic demand ic the bakes foe tke firms' demand for hank loans and loans feom intee- mediaeies and foe theib desiee eo issue more debe (Ike cupply of birine nonowneechipsecurities). We hee F1= I~t - 01~~. (2.36) wheee P5D isthe desired level offinancing in periodt. ft (e t~ (2.37) explicitly, wheee P1D ic eke defired level of financing in period I. explicitly, Ot= I1PF + A,3I (2.36) (2.37) (2.38) whele F13 iseke deckred level of financing in period t. itf0(e, FD) expliciely, Lft= pelt + AF (2.36) (2.37) (2.38) ,,F' + A3 ft t -fl (2.38) whele Oft is fiews' 101al demand for loans in t, A13 iscavectoe of c isane ianectorof allineestraesandP e hetota muto financing dekered. where L D ic firms' total demand foe loans in t, A3 ic a veceae of consancs jfiscavectoreofallineesteraes,andP F hetotal amounnlof financing decieed. wheeeL isfirmseotal demandforeloancint, A3 ksa vector of consanes ifjiscavectoe ofallieeecrates, andPF thetoeal amounteof financing deckred.  THE MODEL 25 (2.39) THE MODEL 25 (2.39) FS = f3(-f,, F ) explicitly, F' = b, F' + A4f, F = f3(f, F' ) explicitly, FS = b, F' + A4h, (2.40) (2.40) THE MODEL 25 F t= (If, FF) (2.39) explicitly, F = b, F' + A4h, (2.40) where F is the supply of firms' securities in t and all other symbols are as defined above. The firms' demand for loans, Lt,, is broken down into a demand for bank loans,Leb, and a d r o mediaries, t, f a demand fat leas from inter- eb = b b ns Le (,ftr' t, Let (2.41) bs =ftr where FS is the supply of firms' securities in t and all other symbols are as defined above. The firms' demand for loans, L, is broken down into a demand for bank loans, L D, and a demand for loans from inter- mediaries, Ln , where FS is the supply of firms' securities in t and all other symbols are as defined above. The firms' demand for loans, Lat, is broken down into a demand for bank loans, L Db ' dewn lte mediaries, L demand f lans frm inter- mei iis ta = Lf (r't, tesDs Lftb _ b rt t ) L = L (rfft> rnfst, L D). In explicit form, Lfb bt nt 1Lt Lt)" = a2(r - rs) + b2 Lr (2.41) (2.42) (2.43) (2.44) LDb _ b (rbt tt' Lt5) Lr = Ln " (r5f t , nft, La) In explicit form, LD b = a, (,bt - rnt) + b L t L" = a2(rft - ret) + b2 L L b~ (5 5 +b L (2.41) (2.42) (2.43) (2.44) L" = L' " (e f,, rnfes, Le). In explicit form, Lrb = ay(rt - rnt) + b La Lft = a2 (rf , - rf.,) + b2 LE, (2.42) (2.43) (2.44) where a1 = -a2, b, + b2 = 1, a < 0, a2 > 0. These restrictions on the constants in 2.43 and 2.44 insure that the sum of 2.43 and 2.44 equals 2.37. The coefficients of LE, are assumed to be constant (and not necessarily equal) to allow for the possibility that the firms may want to borrow different amounts from the two lending sectors even though rfb =rn. This mix of desired borrowing is assumed to be constant over time. If Lft is the net increase in borrowing and F, the net increase in securities outstanding, then where a, = - a, bI + b2 = 1, a < 0, a2 > 0. These restrictions on the constants in 2.43 and 2.44 insure that the sum of 2.43 and 2.44 equals 2.37. The coefficients of Lt are assumed to be constant (and not necessarily equal) to allow for the possibility that the firms may want to borrow different amounts from the two lending sectors even though rfb rn. This mix of desired borrowing is assumed to be constantver time. If Lft is the net increase in borrowing and Ft the net increase in securities outstanding, then where a, = -a2, b, + 2 = 1, a, < 0, a2 > 0. These restrictions on the constants in 2.43 and 2.44 insure that the sum of 2.43 and 2.44 equals 2.37. The coefficients of L, are assumed to be constant (and not necessarily equal) to allow for the possibility that the firms may want to borrow different amounts from the two lending sectors even though r =fr. This mix of desired borrowing is assumed to be constant over time. If L, is the net increase in borrowing and Ft the net increase in securities outstanding, then It - 0I. - 1 - (Lft + Bft) > 0. (2.45) Lt and Ft are determined by the interaction of the demand for loans (the supply of securities) and the supply of loans to firms (the total demand for firms' securities). If 2.41 equals zero, then the financing demand is satisfied completely by increasing the firms' debt. If, how- ever, 2.41 is positive, the difference is made up by a temporary reduc- tion in the stock of retained earnings below their desired level. (Note that the capital good firms do not themselves extend credit to their Int - 1n - 1 - (Lt + B5,) > 0. (2.45) Int - 0Irt - I - (Lft + B,,) > 0. (2.45) Lft and Ft are determined by the interaction of the demand for loans (the supply of securities) and the supply of loans to firms (the total demand for firms' securities). If 2.41 equals zero, then the financing demand is satisfied completely by increasing the firms' debt. If, how- ever, 2.41 is positive, the difference is made up by a temporary reduc- tion in the stock of retained earnings below their desired level. (Note that the capital good firms do not themselves extend credit to their Lft and Ft are determined by the interaction of the demand for loans (the supply of securities) and the supply of loans to firms (the total demand for firms' securities). If 2.41 equals zero, then the financing demand is satisfied completely by increasing the firms' debt. If, how- ever, 2.41 is positive, the difference is made up by a temporary reduc- tion in the stock of retained earnings below their desired level. (Note that the capital good firms do not themselves extend credit to their  26 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM purchaers.) This discrepancy between desired and actual retained earn- ings is made op by a reduction in profit paymente at discueeed befaee. Lee 26 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM purchaser.) Thsdiscepancy between deired andactualetained ean- ings it made ap by a redaction in profit paymente at diecassed befoee. Lee 26 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM purchaer.) This discrepancy between deired andactal etaiedearn- ings it made ap by a redactian in profit paymente at ditcussed before. Let lea - lt - I - (Lt -+ Bit) = AhI(Aft) > 0 (2.46) It- llI~t - 1- (Lft + Br1) = AI(Aft) > 0 (2.46) Icr .t I -~ (Lt + B,,)= Ae(A,) > 0 (2.46) wheee ihI(Af,) represents the unintended change in the fieme' financial aset position caueed by ineafficient financing tn meet investent de- mend. AI(Af,) repeesents a redistribution of income away from the ownert of the coneomer good firnms en the owners of capital good firma. As we shall see, aggregate public income it not reduced. In seminary, whete AI(Aft) reptesents the unintended change in the firms' yinancial asset position caueed by ineufficient financing en meet investmene de- wand. AI(Aft) represents a redisteibution of income away from the owesof the conneameregood firms en the owners of capital good firms. As we shall tee, aggregate pabhic income it net redaced. In summary, EnD .Ea' =n E -+AED (2.47) E mEn, p a -+ AEn (2.47) where E7+ it the actualttockofetied earing(r,eqialently, the actual steach of financial ateeta, An, tteedo eidt a it the actual steach of retained earnings at the beginning of period I, which it equal In the actual and deedred stock of retained earningseat theaed of periodeI- 1; and AE~ Diste detsired change in the stach of retained earnings daring I. t~ can be enpressed at follows: AEa = alet I+ tPerXet + PtXt-et- IXet - I where Ea, is the actual stock of retained earnings (or,equoivalently, the acual stoc of financial asets, A,)attherendaofperiodt;Ea- isthe actual stack of retained ernings at the beginning of periad I, which it equal to the actual and desired stach of retained earningatrtherend of period t - 1; and AE~ Di the desired change in the aloof of retained earnings during t. tk cat he exprested as followt: AE, neD ~ [~rD E X where Sl(Aft) repretents the unintended change in the firms' financial asset petition causedhby insufficient financing to meet investmentede- wand. SI(Aft) repreente a rediseribution af income away from the owners of the container good firms tn the owers of capital good firms. At we thall ser, aggregate public income is notereduced. en sammary, E wEt, -E' -+SAE' (2.47) where E, is the actal stock of retained earnings (or, equivalendly, the actual stockaf financial assetsa,~)at the end of peridCt;E,' dste atalt ock of retained earnings at the beginning of period t, which it equal to the actual end desired stocke of retained earnings atherendof peeiodet- 1; and AE Di trhedeired change in the stockhof retained earningseduring I. SE can be enpressed at follows: SF1 =aInrt I+t[Pet Xet±+PktXkt -Peat 1Xcrt1 IXkt - k I + 0(Iet. - 1 9.t- ) (2.48) Eutermore, Pnt Xet +~a Pk k can be written as afnctin of Kr: Pkt - i Xk1 - 1 1 + 00nt - I - l.t - 2)' (2.48) Pkt - I Xkt - I I + 00nt - I - lnl - J (2.48) Furthermore, Pt Xe+ iPk Xk cantbewritten aea fnctionof Kt Furthermore, Pt Xca + k Xct can be written as afunction of Kt: PetaXet + Per Xk = f3(Kt). (2.49) Pet Xe1 + Per Xct = f3(Kt)- (2.49) Pet X,1 + Pkr Xk e f3(Kt). (2.49) The function f3 it really a reduced form of the production function. The tenet of outpat is determinate, gien the amount of capital need under the assumption that, for a particuar inpat price ratio, the least- coer combination of labor end capital it used. Thee, 2.44 becomet t~ = aler.t +tSff() - f3(Kr -I)] + 0(Ier - It ln - a) (2.50) The function f3 is really a reduced form of the productionnfunction. The evel of oatput it deteerminate, gien the amoane of capirt nsed andr the astamption that, foe a particuar leper price ratio, the leat- cot combination of labor and capit it need. Thee, 2.44 beoomes AE D = 0jn I ~~(r- f0(Kt.9 The function f3 it really a reduced form of the production fanction. The tenet of output it determinae, glora the amount of capital used under the aesumption that, for a pareicuar input price ratio, the Irat- cost combination of labor and capdtal it ated. Thee, 2.44 becomes SE = nt I +[yf(Kt- fs(Kr.-.1)] +  THE MODEL 27 THE MODEL 27THE MODEL 27THMOE THE MODEL =nt112 2+ sf(K2. . 1 -'22 ) f2(K2.-.2 + '0t 2) A K+ ',22- I - '222- 2) 22 - 2)1+ 0)(l'-2I I~ 2- 2) + 0G-2 - I - ln-2)]2 (2.51) (2.52) (2.S3) =o12. 1+ sf3(Kt -. . t -'2 ) -f2(K - 2 + =oalt -.+ns~f2(K2.-. +t.t) - f (K, - 2+ '22 2)]+ 00.('t- 1 - '22 2) +n00m Is-f(1. t's '2 2)52) (2.51) (2.52) (2.53) =-1r - I+ s~f(K.-I.+2 n -U51) - 3K + = 2II s4f3('t '+t - 1) - f (Kt - 2 + '02-2)] + 00.t2- I '22 -2) +=ut -.. Is-2(K.. -14+ 2 2)52) (2.52) (2.53) Q2t2 -IS~l -0 +s~212. 00n )(o- 1 -n 2) -0 (2.54) Letting (a+9)(Ot2.-.1)+ sf3 t - .) equaltf (It1) )we have AED fu('se 00M )s- 2) (2.55) which expeesses the dependency of desired changes in retained earnings on net investment. Summing oncer 11in2.55 (or integrating when lime is assumed to be continuous) leacts to the dependency of the stock of reetuined eaenings on the capital stock us etxpeessed in 2.35. Reweiting 2.35 in an analogons munnet teads to ED s uKt + tf2(Kt) + O(2(2 - Kt - 1) (2.56) or, tetting (a + 0) K2 + tf2 (Kt) = f3 (Kt), ED= f2(Kt) _ )O(Kt _ , (2.57) =-at52.-2 +tSlf3(lt. -) 0f+ Otn2.t~ ts- I 4 ) (2.54) Letting (a + 0) (t - I + sf2 (t2-.1) equal fu (t2- ) we hone AE' = f2(t5 - 2) - (lnl - 2) (2.55) which expees the dependency of desieed changes in eetained earnings on net investment. Summing over t in 2.55 (or integrating when time is assumed to he continnous) leads to the dependency of the stock of retained earnings on the capital stock as expressed in 2.35. Rewiting 2.35 in an analogous mannee leads to t = uKt + sf3 (K)+)(K2 - K2 - ) (2.56) ot, letting (n + is) Kt + sf2 (Kt) = 3(K) =Otn..t t+S~ft. -) 01 + 00.tI - 'n 02 (2.54) Letting (au+ ))(It - I+nsf2 (jt2- 1) equal f4(tn,-2) we hae chEF f4(lt2 - 1) - ('t - 2)(25 which enpeesses the dependency of desieed ohaonges in resumned earnings an net investment. Summing over t in 2.55 (or integrating when time it assumed to he continuous) leads to the dependency of the Htock of eetaioed eatnings on the capital stock as expeessed in 2.35. Rewedting 2.35 in an ansalogous manner leads to tE D a=ut + sf3(Kt) +0g(K - Kt -) (2.56) =(a+ ) K, sf2(KU) - P('S- 022 tetting (a + 0) Kt + tf2 (Kt) =-~ ) E D= f, (K) _ -(K _ 0 (2.57)  28 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM The firms' decision-making process in regard to the size and distribution of retained earnings is visualized in this manner. First a decision is made regarding the desired stock of retained earnings and the necessary adjust- ments of profit payments made to realize this goal. Second, after desired size has been achieved, the firm decides on the desired distribution of retained earnings (financial assets) as described below. Letting DEB represent the desired distribution of retained earnings in t, we have: 28 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM The firms' decision-making process in regard to the size and distribution of retained earnings is visualized in this manner. First a decision is made regarding the desired stock of retained earnings and the necessary adjust- ments of profit payments made to realize this goal. Second, after desired size has been achieved, the firm decides on the desired distribution of retained earnings (financial assets) as described below. Letting DEB represent the desired distribution of retained earnings in t, we have: 28 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM The firms' decision-making process in regard to the size and distribution of retained earnings is visualized in this manner. First a decision is made regarding the desired stock of retained earnings and the necessary adjust- ments of profit payments made to realize this goal. Second, after desired size has been achieved, the firm decides on the desired distribution of retained earnings (financial assets) as described below. Letting DED represent the desired distribution of retained earnings in I, we have: DE m DA D = f4(r, PX) (2.58) where DAB is the desired distribution of financial assets, r is a vector of all interest rates, and PX = Pe X, + Pk Xk. DAD is itelf a vector, the elements of which are cash balances, demand deposits, time deposits, government securities, and deposits in intermediaries. Firms are assumed to hold no debt instruments issued by other firms. Thus, DE D DA' = f4(r, PX) (2.58) DEF a DAB = f4(r, PX) (2.58) where DA D is the desired distribution of financial assets, r is a vector of all interest rates, and PX = Pc X, + Pk XE. DAB Diselfavecrorthe elements of which are cash balances, demand deposits, time deposits, government securities, and deposits in intermediaries. Firms are assumed to hold no debt instruments issued by other firms. Thus, where DAB is the desired distribution of financial assets, r is a vector of all interest rates, and PX = P Xa + Pk X. DAB is itself a vector, the elements of which are cash balances, demand deposits, time deposits, government securities, and deposits in intermediaries. Firms are assumed to hold no debt instruments issued by other firms. Thus, DAB = (CB, DB, TB, G', NB). Obviously, S (DAf) Afs. i= 1 (2.59) (2.60) DAf = (CB, DB, TB, G , NB). Obviously, Z (DAft) = Af. i= 1 (2.59) (2.60) DAB = (CD, D T1, GD, Nat). Obviously, (2.59) Equation 2.53 can be broken down into an interdependent system of equations each giving the desired level of one asset. The desired level of cash balances, CD, is assumed to depend only on the level of sales. Thus, Equation 2.53 can be broken down into an interdependent system of equations each giving the desired level of one asset. The desired level of cash balances, CD, is assumed to depend only on the level of sales. Thus, CB = as [ctXt + PktXkt] (2.61) CB = as PetXet + PktXkt] (2.61) Z (DABt) a AB. (2.60) i= I Equation 2.53 can be broken down into an interdependent system of equations each giving the desired level of one asset. The desired level of cash balances, CD, is assumed to depend only on the level of sales. Thus, CB = as [PetXt + PItXkr] (2.61) where a, is a constant, 0 < a. < 1. Changes in rates of interest are assumed not to affect desired currency balances, although they are assumed to influence the desired level of demand deposits. Equation 2.61 is designed to reflect the assumption that holding currency balances is a nuisance to the firms and such balances are held to an absolute minimum. The desired level of demand deposits is assumed to be a function of the level of sales, the rate of interest on time deposits, and the rate of interest on government securities. Thus, DB = de(PX) + A6. (2.62) where a. is a constant, 0 < a, < 1. Changes in rates of interest are assumed not to affect desired currency balances, although they are assumed to influence the desired level of demand deposits. Equation 2.61 is designed to reflect the assumption that holding currency balances is a nuisance to the firms and such balances are held to an absolute minimum. The desired level of demand deposits is assumed to be a function of the level of sales, the rate of interest on time deposits, and the rate of interest on government securities. Thus, DD = df(PX) + Ash. (2.62) where as is a constant, 0 < a. < 1. Changes in rates of interest are assumed not to affect desired currency balances, although they are assumed to influence the desired level of demand deposits. Equation 2.61 is designed to reflect the assumption that holding currency balances is a nuisance to the firms and such balances are held to an absolute minimum. The desired level of demand deposits is assumed to be a function of the level of sales, the rate of interest on time deposits, and the rate of interest on government securities. Thus, DB = df(PX) + AnTe. (2.62)  THE MODEL 29 THE MODEL 29 The desired levels of time deposits rod govornment secrities ore also fooctions of the same variables. Thur, The desired tevets of time deposits rod govesoment seorities ore also functions of rho tome variables. Thos, TD= t,(PX) + ~r G'= gfUPX) + rt ND= f(PX) + A _i. (2.63) (2.64) (2.65) TD= tf(PX) + A'r GD= gf{PX) +±sr ND= fr(PX) + Ajf'. (2.63) (2.64) (2.65) THE MODEL 29 The desired levels of time deposits rod government sororities ore also frnctions of the tome variabtes. That, T= tf(PX) + A7t;263 GD= gJ(PX) + Aji§; (2.64) ND= fe(PX) + A,-h. (2.65) We row lern to or examination of the sores rod ores of incomo for the firms ir rho consumer good group. Thre are for sorooes of irrome foe these firms: sles, iterest or time deposits, irterest or government sororities, rod interest or deoesits inoitermediaies. Lot Rft be she rrorsipts (income) of those firma in period t. Thor Weenowtutotaneomintioneofthesouces andeuses of income for the fims inthe cosumer good goup. Thereerefour sources of ircome foe these firmsotaes, interest or time deposits, intrest or goveromeot sororities, red intorest or deposits in intermediaries. Lot Rf, he the receipts (income) of there firma ir poriod 5L Thee We stow tore toren examination of the aores red uses of income for the firms in rho corsumet goad group. Those ore for aores of income for these firms: sates, interest or time deposits, interest or goverment sororities, red interest or deposits in intermediaries. Let Rft be the receipts (income) of these firms ir period L. Thor Rear = PXi +Or,G, rtsTr+rnNfcs (2.66) Re.r = P211 +Osifltcs + rt ,5 +rt Nfc. (2.66) Re = PXt + rsGt,, + rrtTfct + inNfcs (2.66) The osos of income include those sir: paymerts to taboe, profit pay- meets ("divideds"), charges in fieancial asset hotdings, tear repay- monts, debt (sorority) retiremet, rod irvestment expenditure. The first two are treated stoictly as residuats rod ore reprerted by Yf, (income received hy the public from corsumer good firms). Charges in financial asset hotdings are eqoal to AEgD Lore repayments equal t (1+oei) ti (I + r e(i))Lfnki) 1l (1+rfi)L 1)+ X;[ k n i=t-r N i t- ns N abbreviated ELf. Dobs retiement equats t (I + ifc(i))P,,i) irt-e N abbreviated XFr. Gross investment equalstI.Lot Uepresetthe sum of oe through five. Thee Tbe eases of income irclte those sirs: paymerts to laoe, profit pry- meets ("dividends"), charges in finaeia asset holdings, lore repay- meets, debt (tearrty) tetirement, rod inestment expendite. The first two mre treated strictly as rsiduals red re represented by Yec (income received by rho public from consumer good firms). Changes infinanciat asset hotdings ore eqalt to AEg LoeepyDt qa l (t + rt bLi t+ (I+ r(i))Lf (i) i=t -on N i~t-vn N abbreviated ILr. Debt retirement equals t (I + ifr(i))Pr(i)1 abbreviated IE0. Gross investment equats t0.. Let Uf. represent the sum of ono throogh five. Thee Tbortses of income irctode these tie: paymerts to labor, profit pay- meets ("dividends"), charges in financial asses holdings, toare opay- meets, debt (sraerity) retiremert, red investment expenditore. The first two ore treated strictly as oesiduals red ore repeented by Yf (income received by she public from consumer good herms). Changes in financial asset holdiogs are equal to AE% LoeD pyes qa 1 b1 + 1()) ti (I + r05(i))Lfn(i) i= -rn N i=t-en N abbreviated ELe. Debt setiremont eqoats X [(t + rtc(i))Pc(i) abrevited Ecr flors investmetequalsI..Lt Uf represn the sum of rone throogh five. Thee U, = Yf~ +AAf +XYL, +XIP(.7 However, R t < U0 since poet of ret investment most hr fionacd. Lot S0 ho total spending power of she consumer good firms in L. Thee Uacr = Yer+ AAt, + IL +ZF. (2.67) U = e + Af~g +OE;t +XPc. (2.67) However, Rf ct < Uf " since part of net investment must be financed. Let Sf, be total spending power of the consumer good firms in t. Then HoevrRosUes ieepot frtivstoematofeaco.However, Rf 5t< Uf sinceprtsofrotinestmentsmusthefinaced. Los 0 h toal peeiegpowr o sh cosemr god irm hrr. heeLt So. be total speoding power of the consumer good firms in t. Thee  30 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 30 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Sfet = Ret + Let + Fet and St = U,. (2.68) (2.69) St = Ret + Let + Fet and S1et =Ufe. (2.68) (2.69) 30 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM St = R, + L, + Fe, and S1. = U". (2.69) Inclusion of the capital good firms allows the elimination of the c subscript in 2.65 to 2.68. It is assumed a typical capital good firm, even though it satisfies its demand for capital goods from its own output, has a financing demand identical to the consumer good firm, and behaves otherwise in the manner described above. Thus, aggregation over all firms yields the detailed statement of 2.69: PXt + ret Gf0 + r, Tt + rqt N, + Lft + Bft= A(Aft) + ELet + EBes + PktXkft + Yft Yft = PXt + r5 G, + rtt Tft + r Nft + Lft + Bft - A(Aft) - ELft - IF, - PktXkt. (2.70) (2.71) Inclusion of the capital good firms allows the elimination of the c subscript in 2.65 to 2.68. It is assumed a typical capital good firm, even though it satisfies its demand for capital goods from its own output, has a financing demand identical to the consumer good firm, and behaves otherwise in the manner described above. Thus, aggregation over all firms yields the detailed statement of 2.69: PX, + rt G, + r Tit + 1n Nft + Lft + Bft= A(Aft) + ELt + EBg, + PskXo, + Yft (2.70) Yft = PXs + rst Gft + rs T, + r. Nft + Lt + Bft - A(Aft) - ELt - IFt - PktXkft. (2.71) THE GOVERNMENT SECTOR All levels of government are treated together-that is, as if there were only one government. No attempt is made to reflect the actual institu- tional constraints under which "government" operates. There are two functions our government performs. 1. Reallocation-All physical production is assumed to take place in the manufacturing sector. The government buys capital and the con- sumption good from the firms. A portion of these goods is consumed by the government and a portion is distributed to the public free of charge. 2. Economic regulation-Fiscal policy is not consciously used to regu- late the level of economic activity. Government spending is limited to the acquisition of the amount of goods necessary for the operation of the government and for making the (exogenously determined) transfer payments. Tax receipts are assumed to be equal to these expenditures plus the interest payments on government securities. Conscious eco- nomic regulation is attempted through monetary policy exclusively. The Inclusion of the capital good firms allows the elimination of the e subscript in 2.65 to 2.68. It is assumed a typical capital good firm, even though it satisfies its demand for capital goods from its own output, has a financing demand identical to the consumer good firm, and behaves otherwise in the manner described above. Thus, aggregation over all firms yields the detailed statement of 2.69: PXt + rgt Gt + rtt Tft + rt Nft + Lft + B~ft A(Aft) + ELt + Eft + PatXkft + Yft (2.70) Yft = PXtft e + r Tft + rs N0t + Lft + Bft - A(A,5) - ELft - EFft - PktXkft- (2.71) THE GOVERNMENT SECTOR All levels of government are treated together-that is, as if there were only one government. No attempt is made to reflect the actual institu- tional constraints under which "government" operates. There are two functions our government performs. 1. Reallocation-All physical production is assumed to take place in the manufacturing sector. The government buys capital and the con- sumption good from the firms. A portion of these goods is consumed by the government and a portion is distributed to the public free of charge. 2. Economic regulation-Fiscal policy is not consciously used to regu- late the level of economic activity. Government spending is limited to the acquisition of the amount of goods necessary for the operation of the government and for making the (exogenously determined) transfer payments. Tax receipts are assumed to be equal to these expenditures plus the interest payments on government securities. Conscious eco- nomic regulation is attempted through monetary policy exclusively. The THE GOVERNMENT SECTOR All levels of government are treated together-that is, as if there were only one government. No attempt is made to reflect the actual institu- tional constraints under which "government" operates. There are two functions our government performs. 1. Reallocation-All physical production is assumed to take place in the manufacturing sector. The government buys capital and the con- sumption good from the firms. A portion of these goods is consumed by the government and a portion is distributed to the public free of charge. 2. Economic regulation-Fiscal policy is not consciously used to regu- late the level of economic activity. Governmnen spending is limited to the acquisition of the amount of goods necessary for the operation of the government and for making the (exogenously determined) transfer payments. Tax receipts are assumed to be equal to these expenditures plus the interest payments on government securities. Conscious eco- nomic regulation is attempted through monetary policy exclusively. The  THE MODEL 31 standard monetary tools are available opec-market operatiocs, changes ic the discount eate, and changes in reserve requiremens, each of whichs will be considered in detail in chopter 4. Taxer ad Government Spending All laces are assomed to be paid by the individuals in the economy. No taxes are explicitly levied ye Ike backs, firms, or ictermediaries. Alt profits over and above the requitements fat retained earninga to meet fetare ivvestmevt are paid to the individual owners of these enterprises. This iccome is raced at Ike some role as income received from other esores (wage and interest payments). We have thee THE MODEL 31 standard monetary tools are available opec-mashes operations, changes ir Ike discoant rate, and changes in reserve requirements, each of which will be considred in detail in chapter 4. Traces and Govrernment Speding All taxes are assumed to be paid by Ike individuals in lbs economy. Na laxes are ecplicitly levied on the baffle, firms, or intermediaries. All profile aver and above she reqairements far rerained earnings to meet fotre investmenr are paid so the individual owners of these enterprises. This icome is laced at the same rare as icome received from other seors (wage and interest payments). We have then THE MODEL 31 etandard macsleep loots are available opec-masker operatives, changes in the diecount sale, and changes in reserve requirements, each of which will be considered in dredai in chapter 4. Taes red Governcment Spending All laces are assumed to be paid by she individuals ic the economy. Na laces aee explicitly levied on the backs, firmsl, as intermediaries. All profits aver and above the requirements for retained earnings to meet frtars icvestment are paid to the individual owners of thee ecterprises. This icome is raced at she lame sale as icome received from other sources (wage and interest payments). We have then T = rY (2.72) T = eY (2.72) T = eY (2.72) where T is toraltlax receipts, t the lax rare, and Y aggregate public inooms before racer. It is assumed that I is constant to reflect an erther assumptioc of ye conscious fiscal policy. Eurthermore, government spending is given by where T is total lax receipts, t lbs lax role, and Y aggregate public icome before laces. It is assamed that tha coastent to reflect an erther assumption of no conscious fiscal policy. Furthermore, governmenr spending is glaren by where T is blatl teax receipts, I the tax sale, and Y aggregate public icome before laces. Is is assumed that I is constant to reflect ac earlier assumption of ye conscioas fiscal policy. Eurthermore, governmenr spending is glaren by T =fTfl + P5X05 + 1N15s (2.73) T =TF126+ pXk + px (2.73) Tk=O + Poxk5 + C9 (2.73) where rg and G epresent the couponateadaggregaterfacervaluerf goverrervsecusriie outstanding, respectively (see she next section). The awarnts of capitalandltheconsumrgoodpuchaed aredeer- mied residally: where rg and G repsreent hrcoponvrae adaggregatesfacervaluef government secvrities outstandivg, respectively (se Ike next section), The amaunts of capital red lbs consumer good purchased are deter- mined residually: where r. and G repressent she coupon rats and aggregate face vale of governmetlsecuritiesaousadig, respectively (see she next seasive). The amounts of capitl and the consamer goad purchased are drter- wived residually: PXg= g (T - Trb2) PX,= (I - l) (T - Tgd) (2.74) (2.75) PCXkI = 0 (T - 7gG) PeXg = (-0) (T -hG ) (2.74) (2.75) PkXkg = 0(T - TSG) P'~ I- 0) (T~7 (2.74) where 0 is a positive constant less than 1. These relatives insurs that Ike budget is balanced. overnente Secsurities red Mocetary Affars The goernement isaues onlyaonesypef scrity withoe-year marity and fixed facsvaluesof $1.Thr couponraterisfixed atT, Theracrual eels in acy peviod, r., way, of coarse, differ from she coupon rare depending on whether or eel the hood is sold atlits face valor. If P, is where 0 is a positive constant leer thee 1. These relations inscre that the bedget is balanced. overnmect Securitier ad Macetary Affairs The government isores only eve type of security withaoe-yearsmatuity red fixed face valueof $1.The couponvrateeis fixedat '5. Thryactval sae in any peviod, r., may, of care, differ from the coupon sale depending ye whether as vet she hoed is sold at dts face vale. If P. is where 0is a positive constant less thee 1. These relations lasue that the hedges is balavced. Govcernme Securities and Monetary Affairs The government issers only ace type of security milk ave-year materity and fixed facervaueof $1.Thercouponrateis fixed atbl. Theractal rate in any period, r e' may, of coarse, differ from the coupon rate depending ye whether vs eel the hoed is sold at firs face vale. If Pe is  32 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM the price of one security, then 32 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM the price of one security, then 32 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM the price of one security, then FT = Pr. (2.76) (2.77) 7. = Pgrg (2.76) (2.77) r. = Pgrg rg=, (2.76) (2.77) At the end of each year the holder of a bond receives $(1 + lg) payment of interest and principal. The government is not required to buy back unmatured bonds, although they may be freely traded among individuals and corporate entities. The amounts and timing of govern- ment sales and purchases of government securities may be determined either by purely passive reaction to the net demand of the nongovern- ment sectors or may be determined by conscious monetary policy goals. This area will be examined in detail in chapters 3 and 4. In any event, the effect of net changes in the amount of government securities outstanding will be to change the stock of money in the hands of the private sectors. Suppose PgtG, dollars in bonds were issued at the beginning of period t. At the end of the period (1 + g)Gt dollars are paid out in interest and principal. In t + 1, Pgt + Gt , 1 dollars worth of bonds are issued and interest and principal payments are (1 + Fg)G+ , at the end of the period. The initial impact on the money stock in period t + t is given by (1 + g)Gt - Pgt + 1Gt + 1. If this is positive, the refunding increases the money stock by (1 + Fr)Gt - Pgt + G + , times the appropriate multiplier; if negative, the money stock is decreased. The effect of refunding in I + 2 will be given by (t + i 5)Gt + a - Pgt + 2Gt + 2 times the appropriate multiplier, etc. The government's ability to control G, the number of bonds issued, and P (or rg) is the key element in open-market operations. The size of the multiplier and the strength of the relations between the stock of money and real variables determine the effectiveness of this sort of monetary policy (see chapter 4). When solving the model in the absence of discretionary monetary policy, we shall assume that the government's supply of bonds is infinitely elastic at the current rate of interest (price of bonds). In chapter 4, this assumption will be removed. Thus, Gs o GDa (2.78) At the end of each year the holder of a bond receives $(1 + Tg) payment of interest and principal. The government is not required to buy back unmatured bonds, although they may be freely traded among individuals and corporate entities. The amounts and timing of govern- ment sales and purchases of governent securities may be determined either by purely passive reaction to the net demand of the nongovern- ment sectors or may be determined by conscious monetary policy goals. This area will be examined in detail in chapters 3 and 4. In any event, the effect of net changes in the amount of government securities outstanding will be to change the stock of money in the hands of the private sectors. Suppose PgG, dollars in bonds were issued at the beginning of period t. At the end of the period (I + Fg)G, dollars are paid out in interest and principal. In t + 1, Pg e Gt + , dollars worth of bonds are issued and interest and principal payments are (1 + 01g)G , I at the end of the period. The initial impact on the money stock in period t + t is given by (1 + fg)Gt - Pgt + Gt + 1. If this is positive, the refunding increases the money stock by (1 + 5g)Gt - Pgt + AG , I times the appropriate multiplier; if negative, the money stock is decreased. The effect of refunding in I + 2 will be given by (1 + F9)Gt + I - Pgt + 2Gt + 2 times the appropriate multiplier, etc. The government's ability to control G, the number of bonds issued, and P. (or rg) is the key element in open-market operations. The size of the multiplier and the strength of the relations between the stock of money and real variables determine the effectiveness of this sort of monetary policy (see chapter 4). When solving the model in the absence of discretionary monetary policy, we shall assume that the government's supply of bonds is infinitely elastic at the current rate of interest (price of bonds). In chapter 4, this assumption will be removed. Thus, Gs - GOa (2.78) At the end of each year the holder of a bond receives $(1 + i7) payment of interest and principal. The government is not required to buy back unmatured bonds, although they may be freely traded among individuals and corporate entities. The amounts and timing of govern- ment sales and purchases of government securities may be determined either by purely passive reaction to the net demand of the nongovern- ment sectors or may be determined by conscious monetary policy goals. This area will be examined in detail in chapters 3 and 4. In any event, the effect of net changes in the amount of government securities outstanding will be to change the stock of money in the hands of the private sectors. Suppose PgGt dollars in bonds were issued at the beginning of period I. At the end of the period (1 + 7g)Gt dollars are paid out in interest and principal. In t + 1, Pgt + IGt + 1 dollars worth of bonds are issued and interest and principal payments are (I + fg)Gt + I at the end of the period. The initial impact on the money stock in period t + I is given by (1 + fg)G, - P, , 1Gt 1. If this is positive, the refunding increases the money stock by (1 + g)G - Pgt + Gt + times the appropriate multiplier; if negative, the money stock is decreased. The effect of refunding in t + 2 will be given by (1 + i7)Gt + I - Pt + 2 Gt + 2 times the appropriate multiplier, etc. The government's ability to control G, the number of bonds issued, and Pg (or rg) is the key element in open-market operations. The size of the multiplier and the strength of the relations between the stock of money and real variables determine the effectiveness of this sort of monetary policy (see chapter 4). When solving the model in the absence of discretionary monetary policy, we shall assume that the government's supply of bonds is infinitely elastic at the current rate of interest (price of bonds). In chapter 4, this assumption will be removed. Thus, G G (2.71)  THE MODEL 33 where Gs is the supply of bonds and Gso the aggregate demand for new bonds. This implies that the price of bonds (actual rate of interest on government securities) is constant over time. The rediscount mechanism is assumed to operate in the following manner. All loans made by the banks are discounts and are assumed to be eligible paper. The rediscount rate, r, is a percentage of the face value of the notes held by the bank. The bank receives (1 - r)X when it rediscounts a note whose face is X dollars. If the total value of the bank's loan portfolio in any period is Y dollars, the maximum amount of rediscounting is (I - r)Y. The government is assumed to rediscount as much paper as the banks offer at the current rediscount rate. The rate itself is set by the monetary authority (see chapter 4). Thus, for any rediscount rate, THE MODEL 33 where Gs is the supply of bonds and Goa the aggregate demand for new bonds. This implies that the price of bonds (actual rate of interest on government securities) is constant over time. The rediscount mechanism is assumed to operate in the following manner. All loans made by the banks are discounts and are assumed to be eligible paper. The rediscount rate, rd, is a percentage of the face value of the notes held by the bank. The bank receives (1 - r)X when it rediscounts a note whose face is X dollars. If the total value of the bank's loan portfolio in any period is Y dollars, the maximum amount of rediscounting is (1 - r)Y. The government is assumed to rediscount as much paper as the banks offer at the current rediscount rate. The rate itself is set by the monetary authority (see chapter 4). Thus, for any rediscount rate, THE MODEL 33 where Gs is the supply of bonds and GDa the aggregate demand for new bonds. This implies that the price of bonds (actual rate of interest on government securities) is constant over time. The rediscount mechanism is assumed to operate in the following manner. All loans made by the banks are discounts and are assumed to be eligible paper. The rediscount rate, rd, is a percentage of the face value of the notes held by the bank. The bank receives (I - ra)X when it rediscounts a note whose face is X dollars. If the total value of the bank's loan portfolio in any period is Y dollars, the maximum amount of rediscounting is (1 - r)Y. The government is assumed to rediscount as much paper as the banks offer at the current rediscount rate. The rate itself is set by the monetary authority (see chapter 4). Thus, for any rediscount rate, d - d'(r,) (2.79) d = dd(e) (2.79) d m d'(ra) (2.79) where d is the actual amount of rediscounting and dd the quantity of rediscounting demanded at rate re. The effect of rediscounting is, as will be shown, to increase the quantity of bank loans supplied. The government (monetary authority) also establishes, and is free to charge, the reserve requirement, r. It is assumed that both time and demand deposits are subject to the same reserve requirements. Thus, the total amount of required reserves, R, is given by where d is the actual amount of rediscounting and dd the quantity of rediscounting demanded at rate re. The effect of rediscounting is, as will be shown, to increase the quantity of bank loans supplied. The government (monetary authority) also establishes, and is free to charge, the reserve requirement, r. It is assumed that both time and demand deposits are subject to the same reserve requirements. Thus, the total amount of required reserves, R, is given by R = r (D + T) (2.80) R = r (D + T) (2.80) where D is the aggregate level of demand deposits and T the aggregate level of time deposits. All banks in the economy are assumed to be subject to the regulation of the monetary authority. Effects of change in the reserve requirement on the stock of money are discussed in chapter 4. The government is strictly a passive supplier-absorber of currency. Thus, at any point in time, the stock of currency, Ct, is identical to the aggregate demand for currency, CD a. Thus, where D is the aggregate level of demand deposits and T the aggregate level of time deposits. All banks in the economy are assumed to be subject to the regulation of the monetary authority. Effects of change in the reserve requirement on the stock of money are discussed in chapter 4. The government is strictly a passive supplier-absorber of currency. Thus, at any point in time, the stock of currency, C5, is identical to the aggregate demand for currency, Cfa. Thus, where d is the actual amount of rediscounting and dd the quantity of rediscounting demanded at rate r. The effect of rediscounting is, as will be shown, to increase the quantity of bank loans supplied. The government (monetary authority) also establishes, and is free to charge, the reserve requirement, r. It is assumed that both time and demand deposits are subject to the same reserve requirements. Thus, the total amount of required reserves, R, is given by R = r (D + T) (2.80) where D is the aggregate level of demand deposits and T the aggregate level of time deposits. All banks in the economy are assumed to be subject to the regulation of the monetary authority. Effects of change in the reserve requirement on the stock of money are discussed in chapter 4. The government is strictly a passive supplier-absorber of currency. Thus, at any point in time, the stock of currency, C, is identical to the aggregate demand for currency, CEa Thus, Ct mC0a. (2.81) Until monetary policy is considered explicitly, the government is essentially passive in the model. Ct =C a. (2.81) Ct C a. (2.81) Until monetary policy is considered explicitly, the government is essentially passive in the model. Until monetary policy is considered explicitly, the government is essentially passive in the model.  34 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM THE BANKING SECTOR Banks perform two major functions: they accept time and demand deposits from the public, the firms, and the intermediaries, and they make loans to the public and the firms. As adjuncts to these services they also hold currency and government securities as secondary reserves, engage in rediscounting, and hold primary reserves. Time and Demand Deposits Time deposits earn a yearly rate of interest, rt. This rate is paid on all time deposits regardless of their source (public, firm, or intermediary). The banks view all time deposits as homogeneous, regardless of their source. Even though some time deposits (or demand deposits) may be held as compensatory balances, no attempt is made to distinguish this portion of deposits from "ordinary" time or demand deposits. The banks' demand for time deposits is perfectly elastic at the current rate of interest on time deposits. Banks "buy" the total of time deposits willing to be "sold" by the other sectors at the prevailing rate on time deposits. Thus, letting TD represent the banks' demand for time de- posits, we have 34 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM THE BANKING SECTOR Banks perform two major functions: they accept time and demand deposits from the public, the firms, and the intermediaries, and they make loans to the public and the firms. As adjuncts to these services they also hold currency and government securities as secondary reserves, engage in rediscounting, and hold primary reserves. Time and Demand Deposits Time deposits earn a yearly rate of interest, rt. This rate is paid on all time deposits regardless of their source (public, firm, or intermediary). The banks view all time deposits as homogeneous, regardless of their source. Even though some time deposits (or demand deposits) may be held as compensatory balances, no attempt is made to distinguish this portion of deposits from "ordinary" time or demand deposits. The banks' demand for time deposits is perfectly elastic at the current rate of interest on time deposits. Banks "buy" the total of time deposits willing to be "sold" by the other sectors at the prevailing rate on time deposits. Thus, letting TD represent the banks' demand for time de- posits, we have 34 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM THE BANKING SECTOR Banks perform two major functions: they accept time and demand deposits from the public, the firms, and the intermediaries, and they make loans to the public and the firms. As adjuncts to these services they also hold currency and government securities as secondary reserves, engage in rediscounting, and hold primary reserves. Time and Demand Deposits Time deposits earn a yearly rate of interest, r. This rate is paid on all time deposits regardless of their source (public, firm, or intermediary). The banks view all time deposits as homogeneous, regardless of their source. Even though some time deposits (or demand deposits) may be held as compensatory balances, no attempt is made to distinguish this portion of deposits from "ordinary" time or demand deposits. The banks' demand for time deposits is perfectly elastic at the current rate of interest on time deposits. Banks "buy" the total of time deposits willing to be "sold" by the other sectors at the prevailing rate on time deposits. Thus, letting TD represent the banks' demand for time de- posits, we have TD - S + TS + Tn (2.82) TD e T + T, + T (2.82) TD T + T, + T (2.82) where TS, Te, and T0 represent the quantity of time deposits by the three sectors. Tt is the total level of time deposits in period t; thus, where Ta' TS, and Tn represent the quantity of time deposits by the three sectors. Tr is the total level of time deposits in period t; thus, T, = Te - T + Tt + T, (2.83) Tt = T D T5t + TS + T5s, I P r I r (2.83) The rate banks pay on time deposits is assumed to depend on the profitability of loans and the cost of obtaining reserves from alternate sources. The rates of loans to the public (rv) and to the firms (rv) minus the rate on time deposits are surrogates for profitability. The only other source of reserves under the control of the banks is the rediscount mechanism (see below). The rediscount rate (re) measures the cost of reserves obtained in this manner. Thus, The rate banks pay on time deposits is assumed to depend on the profitability of loans and the cost of obtaining reserves from alternate sources. The rates of loans to the public (rv) and to the firms (rbf) minus the rate on time deposits are surrogates for profitability. The only other source of reserves under the control of the banks is the rediscount mechanism (see below). The rediscount rate (rd) measures the cost of reserves obtained in this manner. Thus, whewe TSTan5 s there T, Te, and T0 represent the quantity of time deposits by the three sectors. Tt is the total level of time deposits in period t; thus, Tt = T D T + T5 + Tnt (2.83) The rate banks pay on time deposits is assumed to depend on the profitability of loans and the cost of obtaining reserves from alternate sources. The rates of loans to the public (rbp) and to the firms (rbt) minus the rate on time deposits are surrogates for profitability. The only other source of reserves under the control of the banks is the rediscount mechanism (see below). The rediscount rate (rd) measures the cost of reserves obtained in this manner. Thus, SI =pri(rn, Sm- t, ibf- Simplifying, (2.84) it It - rt, '(1b P bf - rt, rd) Simplifying, (2.84) rt r(rvp rtrbt rtrd). Simplifying, (2.84)  THE MODEL 35 (2.85) THE MODEL 35 (2.85) rt = rt(rp, rbf, e). The explicit form of 2.85 is 't= rt(rhe, ebf, ed). The explicit form of 2.85 is rt = t t - I + a(E - LQe - Le) (2.86) , t t + a(b L _ ) (2.86) where 1 > a > 0. Increases in rbf or rbp make loans more profitable and thus induce the banks to attempt to attract more time deposits by raising r and vice versa. Increases in the rediscount rate tend to reduc rediscounting and thus induce the bank to look elsewhere for reserves to make up for the drop in rediscounting. Demand deposits do not earn a monetary return. Service charges are ignored. The banks accept all demand deposits offered them. Thus, the banks' demand for demand deposits is perfectly elastic. Letting DD represent the banks' demand for demand deposits, we have where t > a > 0. Increases in re or rp make loans more profitable and thus induce the banks to attempt to attract more time deposits by raising rt and vice versa. Increases in the rediscount rate tend to reduce rediscounting and thus induce the bank to look elsewhere for reserves to make up for the drop in rediscounting. Demand deposits do not earn a monetary return. Service charges are ignored. The banks accept all demand deposits offered them. Thus, the banks' demand for demand deposits is perfectly elastic. Letting DD represent the banks' demand for demand deposits, we have THE MODEL 35 rt= rt(rbp, r0,, rd). (2.85) The explicit form of 2.85 is = rr _ + a(Lb - L D ~ L f) (2.86) where 1 > a > 0. Increases in rbf or rbp make loans more profitable and thus induce the banks to attempt to attract more time deposits by raising r, and vice versa. Increases in the rediscount rate tend to reduce rediscounting and thus induce the bank to look elsewhere for reserves to make up for the drop in rediscounting. Demand deposits do not earn a monetary return. Service charges are ignored. The banks accept all demand deposits offered them. Thus, the banks' demand for demand deposits is perfectly elastic. Letting DD represent the banks' demand for demand deposits, we have DD = D t + D' + D. (2.87) The total amount of demand deposits in t, Dt, is given by DO = D + D + Dn. The total amount of demand deposits in t, Dt, is given by D DD =D + D' + Dt. (2.87) (2.88) DO = D + DS + Dn. The total amount of demand deposits in t, Dt, is given by D, = D = D, + Dt + D t . (2.87) (2.88) D -m D = Dt + Dt + Dn. (2.88) Demand and time deposits are the only liabilities of the bank that will be given explicit treatment. The only explicit recognition of capital account items is the assumption that all profits are paid out to the banks' owners. Loans, Reserves, and Rediscounting The legal reserve requirement, r, applies to both demand and time deposits. The total level of required reserves in period t, Rt, is given by Demand and time deposits are the only liabilities of the bank that will be given explicit treatment.? The only explicit recognition of capital account items is the assumption that all profits are paid out to the banks' owners. Loans, Reserves, and Rediscounting The legal reserve requirement, r, applies to both demand and time deposits. The total level of required reserves in period I, Rt, is given by Demand and time deposits are the only liabilities of the bank that will be given explicit treatment?. The only explicit recognition of capital account items is the assumption that all profits are paid out to the banks' owners. Loans, Reserves, and Rediscounting The legal reserve requirement, r, applies to both demand and time deposits. The total level of required reserves in period t, Rt, is given by Rt = rt(Dt + Tt). (2.89) Required services are all held in the form of noninterest-bearing deposits at the monetary authority. For simplicity, vault cash, or cash held by the banks, is not assumed to be part of required, or primary, reserves. Secondary reserves are held in three forms-cash, securities issued by 2. Banks are assumed not to hold demand deposits in other banks. This in effect eliminates the correspondent banking system from consideration in the model. Ri = rt(Dt + T). (2.89) R, = rt(Dt + Tt). (2.89) Required services are all held in the form of noninterest-bearing deposits at the monetary authority. For simplicity, vault cash, or cash held by the banks, is not assumed to be part of required, or primary, reserves. Secondary reserves are held in three forms.-cash, securities issued by 2. Banks are assumed not to hold demand deposits in other banks. This in effect eliminates the correspondent banking system from consideration in the model. Required services are all held in the form of noninterest-bearing deposits at the monetary authority. For simplicity, vault cash, or cash held by the banks, is not assumed to be part of required, or primary, reserves. Secondary reserves are held in three forms-cash, securities issued by 2. Banks are assumed not to hold demand deposits in other banks. This in effect eliminates the correspondent banking system from consideration in the model.  36 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM firms, and governmsent securaties. Desired cash balances, Ca, are given by 36 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM brns, and goveranment securities. Desised cash balances, Ca, sre given by Cb= y(D + T) (2.90) C, = y(D + T) (2.90) where 1 > y > S. lIntesest raaes ace amiated fram 2.90, reflecting the assamption tata banks bald vaalt cash strictly an maca day-an-day withdrawal requirements, and than any cash aver the minimum needed far tbeserequiremns will beusedatoby gvenmentsecitiesas lng as a1 is greases lbhan zeera. This is equivalent In assuming ahat she banha do nat have a speculative demand farcmnny, in Ibis case, cash. The desirad level of governm ent securitirs, G,' is given by whbere 1 > y > 0. Interest cares are omitted from 2.90, reflrcring she assamptin tbat banks bald vault casb strictly an meal day-an-day wiahdrawal reqairements, and that any cash ovee tbe minimam needed far abase requirements will be used an bay government secarities as long as a1 is grater than seen. This is equivalent In assaming tbat the banks do not bane a specalaive demand far money, in ahis case, cash. Tbe dashred level of governmena securities, G,' is given by 36 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM herms, and governmantsecuriaies. Dedired casb balances, Ca, are given by Ca = y(D + T) (2.90) where 1 > y > S. Insterest rates ace amitted fram 2.90, reflecaing she assumptian tat banks bald vaula cash strictly ta meal day-la-day witbdawal requirements, and that any cash aver the minimum needed far these reqnirements mill be used In bay gavernment securitirs as lang as a1 is greater than maca. This is eqaivalena In assnming that the banks da nat have a speculative demand far money, in Ibis case, cash. The dashred lanaI of gavernment securities, Ob I is given by Gn =g(D +T) + A, %, (2.91) G,= p(D +T) +Aoi, (2.91) Gb= pG2 + T) + Aari,, (2.91) where p is a pasitive canstant less than 1. las magnitude is determined by Ike "instituainnal" requirements far secandaay reserves, such as seasonal fluctataionsin deposits, etc. aniapsiivcnsantcrflecing two assumptions: that as she yield an government secuitaies increases, abeam securities became a mare ataracaive farm in whbich to bald secondary aasaavs-e.g., an increase inar causes a ager poraion af secnndary rserves to be held in govrnmentsecueries,ceerispaibs; and chat an increase in a1 also canses abe total amont of desired secondary reserves tn increase, ceteris paribus. b5 a is a negative cansant refleting thefactathatasecondaryareserves will beswichedfrom gven- ment's In firms' securities as tbe yield an lbhace securities rises, ceteris paribus. fsn' d15, and el5 ace also negative constants reflecting two assamptians: tat as a1 rises, prafia margins ae sqauezed, indacing abe banks ta shifa fands fram low-yielding secondary ceserves In higher- yielding loans; and that as the eases banks charge for Inoans incr ease (rae and er), Ike incrased profitabily of lnans alsn lends an redace the levelof desired secondaryceserves. The cnverses nfthe above assnmp- tions see also assumad In bald. The desired level of firms' securities, FbaI is given by where p is a positive constant less than 1. Ira magnitude is determined by abe "insaitutional" requirements for secondary reserves, sack as seasnal flucuatinsin deposits, etc. aanis apsiivcnsanteflcing twn assumptions: that as Ike yield an government securities increases, abase securities became a maca attracirve farm in which an bald secnndary reseeves-eg., an incarease in rg causes a larger pnrtinn of secondary reserves In be held in gvernmentlscriies,ceteis paibus; and that an increase in rgalso causes she total amonn of danred secondary reserves In increase, ceaeris paribus. b,, is a negative constant reflecinglthefactatsecndary reserves willbe switchedfrom gvern- meal's In firms' secarities as she yield an abase securities rises, ceteris paribas. ear' dan, and a1, are also negative constanas reflecting awn assumptions: tat as at rises, aprofia margins are sqneeed, inducing Ike banks an shift funds from law-yielding secnndary ceservesato higher- yielding nayns; and tata as the rases banks charge far loans increase (r and cr), the inceased profitabily of loans alsn lands In reduce Ike level of desired secondary reserves. Tbha cnnveress of the above assump- tians are alan assamed an bald. The desired lavel af firms' sacucries, EnI is given by where p is a pnsitive constant lass than 1. las magnitude is deternmined by Ike "instiautional" reqairemenas far secondary reserves, such as seasonal flctationscin deposits,aetc. anisa psiivecnsanteflecing awn assumpains: ahat as the yield an government securities increases, abase securiaies became a mere atracaive farm in which an bald secondary reserves-ag., an increase in an causes a larger portion of secondary aeserves an ha held in governm ent securities,ceaeris paribs; and that an increrase in r1 also causes thetotal amountf dsied secondary reseaves tonincrease, ceterisparibus. banis anegaivcnsant seflecting the fadt that secondary resrves will be swiached from govern- ment's an firms' secuiies as the yield an these secuiies rises, ceaeris paribas. flo d~o and al arm also negatiar caneaanascaeflectingatwo assnmptions: that as at cises, profit margins are squeezed, inducing the banks an shifa funds from law-yielding secondary reservrs In highec- yielding loas; and that as the rataes banks charge far loans increase (erl and aye), the inceed profitability of loans alan rends an reduce the levelnofdsired secondaryrserves. Thrcnveses ofstheabovessump- tions ace alsn assumed an bald. The desired level of firms' securitis, EnI is given by Fb= (l) + T) + A1 , Fa (2.92) F = p(D+ T) +A, hi,. (2.92) En = p(D + T) + A1 a tmn (2.92) p is a positive constant less tans 1. a, Iis a negaive constant while bi1a is positive. f1,d1,andael arecalso negative constants. Theargnments here sre the sama as those given far the signs of the constana terms in 2.91 save. g is a positive constant less ahan 1. a1 I1is a negative constana while hasI is positive. f,1I, d1, and a11i ace alsn negative constants. The argumenas bee are the same as abase given far the signs of the constant terms in 2.91 save. p1 is a positive constant less than 1. a,11 is a negaive constant while b,1a is positive. Iaa' d1,and a51, are alan negative consanas. The aguments beam ace the same as abase amven for the signs nf the constant atemsin 2.91 save.  THE MODEL 37 THE ~ ~ ~~ H MOEL3O3DTEMOEL3 37 THE MODEL 37 The total level of detieed secontdary reterves is givee by the turn of 2.90, 2.91, and 2.92. Al no time wilt the actual level of tecondary reservet he teat than the detired tenet. tf, however, the bunko ate unabte to wake the total amount of tant they with, actual secondary reeevet msay be greater than the desired level. Letting R'be actual secondary retervet inI The total level of desited secondaey teterves it given by the turn of 2.90, 2.91, and 2.92. At no time will the actual level of secondaey etervet be lesstthan the deiedlevel.If, hwve,the bnksaenale to wake the total amtount of lutne they with, actual tecondary reaserves mtay be greater than the desired level. Letting R St be actual seconary restevetein L R Cne +Obt + Fnl+(Ls -Lo (2.93) = Cv, +CabtO + t+(Lts - L,) (2.93) whee L it the total quantity of loane banks with to woke in I and Lt the awount actually lent by the banks in L. The teem in the parenthetes in 2.93 will be eefeeted to at eueplus reseeves. It it attained that all surpluseervesarte held in the formwof cashso that the banks' actual cath holdings in t are given by whee L it the total quantity of loans banks with to make in t andL the amount actually lent by the banks in t. The teem in the parenutheses in 2.93 wilt be refeeted to at turplut reteeves. It it aumed that all tueplat eevet ate held in the foew of oath to that the banks' actual ash holdings in I ate given by The total level of detired secondaty reeevet it given by the sum of 2.90, 2.91, and 2.92. At no time wilt the actual level of secondaty reservet be lews than the detited level. If, howevee, the banks ate unable to make the total aunt of tant they with, actual secondary reterves way be greater than the desied level. Letting Rs' be actual tecondary eserves in t R a= Cat + Gas + Ft + (LS - Lt) (2.93) whee LtS it the total quantity of loans banha with to make in I and Lt the amount actually tent by the banks in I. The teem in the parenthetet in 2.93 will be refereed to at tuerpint reseeves. It it ataumed that all turplus eerves one held in the form of cath an that the bunks' actual cash holdings in t are given by Ct= Ci+ L4 - Lt (2.94) when Lu - Lt it positive. Cht =Cn + LS- Lt when LS - L, it potitive. C, = Cat (2.94) (2.95) =i Ca, + 14 Lt when Le - L, is potitive. (2.94) when Ln = Le. Surplut reseres ate assumed to be held in cauth eathee thantsecuritiettoerefleatlheir transiory natue. That it, banksfeel that tuck a situation it only temporary and do not witha t witch in and out of secueitiet on a shoet-eun, unpeedictable basis.' Bank toant to hoth the public and firms ae wade foe a period ofn years. Loant madein periodtIcarryaarate of iteetoferb~t ande'ft respectively. The proceedt to the basic of a loan of X dollaet ate (I + ta)X, repaid in a inttallments of (1 a ev)X /n dollaes. Loans to both the public and the fints ate attuwed to be aitktets. (Alternatively, one could think of tate and bt as atepetenting the net return pee dollar lent aftee default and added collection enpenses.) The aggregate amount banks with to loan feom unbuteowed teseeves in Iit given by C ab h (2.95) when L? = Lt. Surplus retervet ate attuwed to be held in cash rathee oh an securities to reflect their teansitory noate. That is, bunks feel that such a situation it only temnporary and do nut wish to switch in and out of seautitiet an a shoet-in, unpredictable batis.t Bank tant to both the public and firms ate made foe a petiod of n yenrs. Loans made inpeidt carryaerateof interest oftrbpt andeneft, tespectively. The peoceeds to the bunk of a loan of X dollars ate (1 + ta)X, repaid in a installments of (I + e0)X /n doltaes. Loans to both the public and the firms ae attained to be rishleat. (Alteenatively, one could think of tate and t a rtepesenting the net return pee dollat lent aftee default and added collection enpentes.) The aggeegate amount bunks wish In loan feom unboetowed reserves in lt isgiven by Ca= Cat (2.95) whnL'= Lt. Surplas reserves ate assumed to be held in cash rather than securities to reflect theirtrIansitory noate. That isbanks feel that such a situation is only temporary and do not wish to switch in and out of securtiies on a thort-tun, unpeedictable basis.3 Bunk tant to bulb the public and fiaws ate made foe a petiod of n years. Lonsmade ipeid tcarryuatrte ofinteesttofar~ andarft respectively. The peoceeds to the bank of a loan of X dollbts ate (I + ta)X, tepaid in n installments of (I + ea)X /n dollaes. Loans to both the public and the fiews see asamed to be tishlees. (Alternatively, one could think of tate and tat, us repesenting the net retuen toer dollae lent after default and added cotlection expenses.) The aggeegate awaunt banks wish to loan feom unboteawed reserves in l ie given by Lte (D + T)+ AI 2. (2.90) 3. Inuan souepe to kanep the models simple asepossible, weohave omittedthe Fedeal Fnds areketaonbughius econizedthatothis s heorot ofstaticon ohaso createdt thes particular financialmarkt. Ls =(1),+ T,)+A50? (2.96) =t e r (Dt + Tt) + A0 1 2'b (2.96) 3. toI as attempt to keep the model at bunkl as possil,uwe aveomitted the Fede.1aFndsbmarete,ataouhaitiserecognizedtataethistheot o f ituation thate aede this patticularfinancintaltrkt. 3. to an attemnt to keep the moetas simpe at posse, wenhaveomitted tet Fedeal undsumaeket,although it istrcogniztatatsiteort ofsituation eae createbthis aticularefiancialwmaret.  38 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM iti ta potive contant less than 1. a,, and b,2 are negatine constanttasice inceeated yieldr on recurities will, cleti paribur, tend to eeduce loars and inreate secondary reeevea. dr, it a poritive conatant since increased cota create preeturefor shifting fromtecondary resrvet to loans. a, and e12a are positive, reflecting the fact that the increared prafitability of loant at interest eater rite will resuit in increared willing- neat to lend.Sarpaeervetin pedI-(CatCrlal increeare the hanha' willingest to lend. The amonar hanht with to loan to the public and to firmt, LSI anE n h ifeec betee end anoe n cL dependonEtedfrnc betwen bpt nd bftandintitutionat factnrs: 38 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM or it a positive consttant lett then 1. a, and h12 are negative constantrince increated yields on seoneities will, cleti paribus, lend In reducetloansrand incereaecondary reeeve. d12isapoitivecnttant since increated cota create prestnre for shifting freon secondary reterves to loant. fl and e12 are potitive, reflecting the fact that the increated profitability of loots at interest ratet rtse wilt retalt in indrented willing- neasto olend. Surplstreerve inpeiod - I(Cjbt - 1C1.-.) cleary increate the banha' willingness In lend. The atnonts banht with to loan In the public and tofto,L and _S, on ,tedfrnc Pt Lt depend Lnas h dfeec between rot and ttdr, and institutional factort: 38 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM fit a potitive conttant lett than 1. a12 and b,, are negative conttantrtsince increated yieldtonecueities wilt, ceterisrparibus,ltend to reduce loors and increase secondary retervet. d 12 it a positive contant tince increated cotst create preetture foe thifting from aecondary reerves to loans. fl and eta. are positive, reflecting the fact that the incrented profitability of toana at interest rater rite will resuit in inceased willint- nes tolend. SrplsreervierinoettodtI(Ct I- CoI-t) cearly increase the banhs' willingnett to tend. The amounts bardcs with ro loan to he ublc nd o frmser and t5, depend on Er, the difference between ar pr and rbft, and institutional facrors: Err = t0ES + a, rop - ever) E~rb = qbtS-n~ro or (2.97) (2.98) ErI = tbELS+ a, 1(rol- e00t) I0 = - o (rorr - tort) (2.97) (2.98) Eb= +bt a, a(entt - it, I) (2.97) where kb0+- t=t1and a1ia positive contant.Instittionl facor, mobh at the deale to meet the demands of enisting customeers, unwilling- neat to loan more or lett than tome percentaget of the total loan portfohio to any one type of boerower, etc., deteemine the relative tizet oftQbad kawell athe abolteieof a3..Themalerna13is, the more end, and Ili mart diffee to induce a wide difference between t and qt> and 0 reprtetent the initial quantity of loans the banks ate willing to supply the public and the firmst from anborrowed resernet. There figures do not necessarily reprerent the actual amount of loanr made. Let L4 and Lft repeent the quantity of banh boats demanded by the pabboc and the firmstin period a. The actual amount ofhbnh loanamade to eachasectorinoperiod t, Ln b L db ra , n fdpends on the qantitiet demanded and the qnantities supplied feom anborrowed re- tervet at well at the banhs' willingneas to engage in, end abdlity to obtain, redinconting. Whe It + t + t 4,no rediscounting occurs. In thir care the fioat amounts lent ate given by where tb + 4= ad 13 i a oitive costant. Instittional factor, suob at the deale to meet the demands of existing cusaomert, unwilling- near to loan more or lets than tome percentages of the total loan portfolio to any onr type of boreower, etc., determrine the relative sizes aft and at well alte abroluteasizeeof a The smallerea,3 it, the more r01 and toe mart differ to induce a wide difference between tir and 40. Lrt andLe represenr the initial quantity of lons the banhs ate willing to supply the public and the firma from onbereowed reserves. There figureedo nat necesariy repreentlthe actual amount of loans mae.L t ~ n It represent the qantriry of hanh loors demanded by the public and the firma in period t. The actual amont of bank loanamade toeachrsectorein priodtI, L, b nLdpnrnh qantities demanded and the quantiriet tupplied fain norrowed re- serves at well at the bonker' willingness to engage in, and ability to obtain, aedhs~onting. Whe L1 t 0 t It L~ no rediscounting occurs. In this care the final amounotslent ate given by =t _1,-a a(rorr -b en) (2.98) whr I tb~ anda3isp aoitivecnstantInttional faorsa suoh at the desire to meet the demanda of exdsting cnstomeers, nwilling- near to loan more or tear than tome percentages of the total loan portfolio to any one type of horrowee, etc., determaine the relative titer oftkb and tb asrwell asthe aboteieofoa . Themaller a13 i,the mIferndomut differ toindce awide diffeence between Lr and 40b ErP ad L7b reprent the initial quantity of boors the banha are willing to supply the public and the firma from anbioreowed reserves. There fignes do nor necessarily represent the actnal amontr of lana made. Let ELl and LD ' represent the quantity of baoh loana demanded hy the public and the firms in period t. The actnal amount of honk loanamade to eachtsector inperiod t, L1 an Ledpb ao h quantities demanded and the qantities supplied from anhorrowed re- sevsat welt at the banhs' willingneas to engage in, and ability to obtain, redisconting. Whe L t + [ t t 40+ , no redisconting aoneurs. In this care the final amounts lent ae given by Le = Lea This situation it illustrated in Table 1. (2.99) (2.100) This situation it illustrated in Table 1. (2.99) (2.100) Li b L Db This situation it illustrated in Table I. (2.99) (2.100)  THE MODEL 39 THE MODEL 39 In this case, even though Let > LI , the entire public loan demand was satisfied by shifting a portion of the initial (and nlent) allocation for loans to firms over to public loans. Firms were also able to borrow the amount they wished. TABLE 1. fL b tSan>fnb eb In this case, even though Ltt > I , the entire public loan demand was satisfied by shifting a portion of the initial (and undent) allocation for loans to firms over to public loans. Firms were also able to borrow the amount they wished. TABLE 1. E fS tb + b LS $100 Tptb = $125 tb = $200 L5 $160 t= $125 Let $160 =t $100 b = $200 t = $125 Lpt = $125 S cb = $160 L t= THE MODEL 39 In this case, even though I b > I , the entire public loan demand was satisfied by shifting a portion of the initial (and unlent) allocation for loans to firms over to public loans. Firms were also able to borrow the amount they wished. TABLE 1. Ep1 + Eon Eptb + fb Ic aloe Lptb = $125 Lpt = $125 =fStb $200 lb = lace Let = $160 0 0 tot L~ S $v105 L =v $108 Lpb + Lftb = $300 t~b + bfl = $8 Lbt +b =$8 When LI, + I < Lb + LI5 the final amounts lent depend on the amount of rediscounting The banks' total demand for rediscounting is given by ot -e - -n + b) d I Le L t ( ft) d bpt dt Le+ t = $300 et + Erb = t85 Lpt + Lt t o85 When f + ELb < Lth + Lb the final amounts lent depend on the amount of rediscounting. The banks' total demand for rediscounting is given by d0 =L + -an + ) d r -( r ) Lt -t (Lt ft bpt dt Let 0tb = 100 Le + b = $285 Lbp + Lt = $285 When LI + I~=S < L0b + fe the final amounts lent depend on the amount of rediscounting. The banks' total demand for rediscounting is given by et +Lb - (Lp5 +Lt)- d(r - ) bpt dt d, ( ). tbft - dt (2.101) d ( d ). tbft -d C0 (2.101) d, ( ). bt - dt (2.101) It is convenient to decompose 2.101 into the banks' demand for rediscounting to make additional loans to the public, dd, and to make additional loans to firms, d . It is convenient to decompose 2.101 into the banks' demand for rediscounting to make additional loans to the public, d , and to make additional loans to firms, d . It is convenient to decompose 2.101 into the banks' demand for rediscounting to make additional loans to the public, d , and to make additional loans to firms, d - 6= [L an an _nD o usd e~b Et=(Le b) d, bpt dt d t o b -(E I tbb _ d, It ft t P, bft - dt where ddot + d t = dtl It - a o on____ (2-12) (2.103) (2.104) t~d tb _ SI Stb _ Db) _ o a t e t -I 'bp_ v dt dt d b _ Stb t, Db) _ , it L5 Lt p(Let Lt bft - dt where d0 + dd d. (.102) (2.103) (2.104) tn = L b tb -D L Let Le - bpt dt It pt pt bft - dt where d + d = d' (2.102) (2.104) The terms -( - t in 2.102 i - It) in.13eter these equations only when they are negative, that is, only when the The terms -(4sb _ Db) in 2.102 and -( _ Db) in 2.103 enter these equations only when they are negative, that is, only when the The terms -(It -nb) in 2.102 and -(n Db) in 2.103 enter these equations only when they are negative, that is, only when the  40 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM quantity of loans demanded by one secors is smnaller than she quantity the books see willing to lend to tbus sectos. This situasion ollows sho banksto "shift"nmoreefsnds inamousntssequalsto-(4, Lit) or -l"- E") to lbs othee sodasr, thus reducing the books' demand foe rediscounting. The pseameters de and d, ae~ Eositive and assumed to be greer eseas 1. Theis sizes desermine how responsive lbs demsnd foe discounting is so differences beeween the Isle(s) of interest an loans sod she redis- count rate. Theclosererdcomsto50(5),ehegeereis d./05 - lae(de/lnes - rdm) snd she smalles is ehe swoons of discounisg she banks see willing en engsge in. Nose ebse eke consenuctions of 2.102 and 2.103 imply thst eke smount of rediscouneing will nst be sufficiene so meee she entire sanest demund foe Issue. The addition of a conseant teem eo these equations would make skis possible foe ceetsin combins- tions ofkhe loanrsses andtheerediscountsrate. These termshave been omited t eflectmoeseongly the bnks'ssumedelucanceeto engage in eediscounting. Based on she previous sssumption, the aceuul amout of rediscounsing is slways equal Is Ike amoune of discouneing demanded: 40 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM qantity of lous demanded by one seceoe is smallee eban eke quantity she banks use willing so lend to thas sector. This situation allows she banksto "shift" mosefunds in amounts equslto -(% S )o -L'- Ee ) so she ashes sectoe, sbus seducing eke bsnks' demund foe rediscounting. The psrameters d. and d, see positive and assumed so be greauter shun 1. Theirsizoes deteremine how responsive she demand foe discounting is to differences besween she else(s) of interess on loans and she sedis- countsrate. The closererdcomes toe50p(s55)lbte geeris d./50 - rtd/bt- 'ae) sod she sinules is she smouns of discounsing she banks see willing so engage in. Nose that she constructions of 2.102 sod 2.103 imply sbas she amount of sediscounsing will not he sufficiens to meet she sulkse excems demand foe losses. The additione of a constant seem so these equations would mske skis Eossible foe ceesain combina- tions of she loan rases and she rediscount ease. Themseemsbhave been omisted so reflect moss strongly she banks' assumed reluctance so engage is rediscosnting. Based on she previous assumption, she acsual amount of sediscounting is always equal so tbs amount of discounting demanded: 40 EQUILIBRIUM STUDY OFTH4E MONETARY MECHANISM quantity of loans demssded by one secor is sinules thsn tke quantity the banks see willing so lend tn thas sector. This sitsstion allows Ike bsnksto "shift"moreefunds in amountssequalto -( 5 -L5)aor -(~ L5)stosthe othersector,sthseducingte banks' demand for rediscounting. The pseameters d. and d, see positive and assumed tobegeatserthan 1. Their dozes deteemine how respossive she demand foe discounting is so differences between she ease(s) of interess on loses and she redis- nouns rte. Tbhs closersed comes so e55(e55), she geatee is d./e005 - eae(ds/esee - emt) and she smallee is she amoons of discosunting the hunks see willing In engage in. Nate that she constructions of 2.102 and 2.10l3 imply that Ike swoons of rediscounting will not be sufficient so mess she entire excess demand foe loans. The addition of a constant seem Is shese equations wosld make skis possible foe cestiun combina- sionsofstheloanoraes andtheeediscounterate. Thesesterms havebeen omissed en reflecs moss strongly the bsnks' assumed reluctance tn engage in rediscosnsing. Based on she previous asmption,sthe actual amount of redisconting is always equal Is she umont of discounting demsnded: d -Id = d', od d (2.1 05) It, -_ d d = rl' , + d' I it" (2.1 05) dtd d = dd5 + d (2.105) We skull use she teems d0 soad d5 s o sed foe bosh the quantities of rediosconing demand us well us she amosnts of rediiscounting made because of she excess demand foe loans from either the public (d d) or eke firms (d d) in period a. Thus, in she came whewe L55 + L55 0 THE MODEL 41 lions 2.106 and 2.107 can also be nsed to express she final amounts lent whnl +I. Lp + 1 , sinceneithertype ofborowereconbe induced to borrow more thon the quantity of loans he initially demonds (L or f~ ). Equations 2.99 ond 2.100 express this in much simpler foem than the more general relationships 2.106 and 2.107. TAB LE 2. -f _ ff <0, iS _ ED >0 TH4E MODEL 41 tions 2.106 nod2.107 cnalso be used to express the finol amounts lent wheoL0 +L L +I14 , sinceoneithrype ofhorrowerecanxbe induced so harrow more than the qantrity of loans be initially demands (L.b Be Er ). Equations 2.99and 2.100 express tbisin much simpler form than tho more general relationships 2.106 and 2.107. LABL 2. Ef < 0, LS - ED > 0 Ers = t0O Ts= too Of 150 ErS-LOf = -5t - 0 P - Ep = 10 Ers =00 -0= 50 TS _ED -5o f Ef Ers 100 OPi = 100 Er 15 ErO fl ED - 50 90 Ep , 10 TfSr L, 200 < E' + TD = 040 =f ~ 200 < ED ,ED 240 dd=90 - 100 (10t0 - 050) _ dxo =-to-o- <0=0 dd0 go- too-r1iot- 1501 - do__ 0-1 d-o < 0 0 rbP - rdt dr d 1SO- - ( 100 - 901- d 5- 10- d =(by ampionon d, b,erd 3 Lp win 19n. toof a (too - tool L"' = win 1 150 - loot 00 90) +O 0 35 wino90, 100 0 0 =EPD 100+0+ 35 =0145 --lo 0 d <0 dd=150 l oo _ (tOO - 9t) - di__ -,S+ -S 0 Lr o+i10 = 240 cid -90 - 10O(100 -150) - d rbP - ad ext - t dd 10 00 100 -90) d 5O0 10 d (by asuwption onot, r, si)c 35 Opt = win 190, 100 01(100 - s50t L~ w in 1o50 t oot 0 100 9 01 + 0 f 0+35 w is o oo ,160 = o =ED = too + 10+035 =145 b 50S - 10 - (by___ lx assawption end11, a1, rdl = 3S Lb wio . ro0, 10+(000 - 050) Lbt m in fl50 - 1001 01(100 - 90) wina90, 100 + 0 =90 =E0 100 +10 +35 s145ED The rates of interest on bank loans in t are given by = t e- Pt-. I) + t teeee- roll = 5rft- 1 + af(E -b E ~ brbt - rnfts1) (2.108) (2.109) The rates of interest on bank loans in I are giver by +01e0.. +Op(Eps I - l bp(rxeet r - 'npt-1) rft = exfte I + at (Er t _ S t) + b(bt1- eere- 1) (2.108) (2.109) The rates of interest an hank loans in t are given by rbt='p-I+ a(LTL e005 ~ Pt I~.) + trtce e - ep b~ 'bft = tbfr.. I + af(l.0r. et 11~ br(ere...rI - 't- ) (2.108) (2.109) where a.and af see positive cosnts. These equations embody this concepturalization: At the beginning of period t, banks adjust their rates where a. and a0 are positive constants. These equations embody this conceptualization: Al the beginning of period t, bankr adjart their eater where ap and a1 are positive constants. These equations embody Ibis conceptualization: Ar the beginning of period t, banks adjust their rates  42 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM either up or down, deponding on whethertShee was an excess demand L.1.b 1 S >0)moran nucessapply of loans (Eb -L 0) in the ptriousaperiod. The amount of the adjustment depends not only on thesizeeof the peious excesspplyaordemand,aut alsaaon she sites of te and a1. Othertetsof interest ate notsexplictly included in 2.108 and 2.109, astIheir imnpact an rr and e51 is contatned in the termaLE~ andLE b Conclnding this section we hone the simple statement that the bnnks' ansets and hiahilities moat he equal. Tt +ODs=CHt+Onlt+Et+ Rt + X U -. 42 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM tithes up or down, depending on whether thete was an eucems demand tU , - L t_> 0)atran excess supplyof loans (LED b -Lb~ < 0) in the previous period. The amoant of the adjustment depends not only on the tie of the preeviout eucess supply at demond, has also on the tizts of ap, and a1.Othersratesof inteest tee not explicitly included in 2.108 and 2.109, as thehr impact on tr and etchi contained in the teems ED b and C~t. Concluding this section we hare the simple statement that the hanhs' amsets and hiahilities mant he equal. Ts +Os= CH +OGb +E5bt + Rt+ %~ 42 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM either np at dawn, depending on whethet thete was an eucess demand - _,5. > 0) at an eucess supply of lans(L - L < 0) in the preevious period. The amount of the adjument depends not only on the size ofheprevious eucesapply ordemand,hbut alsoaon the slzet of ar and a1. Othes rtes of interest aee not euphichtly incladed in 2.108and 2.109, astheir impactanon o ndtrbfis cntained inthn termsaL~ andLf Concluding this sectian we hare the simple statement that the hanht' assnts and liahilities mast he equal. Tt + Ot = Ct+Oust+Et+ Rt + iI C5 - Z di. (2.110) 2 4 (2.110) Z di. (2.1 10) The banks' income statement in petiod t is more complxuas itmust tahe into account the effects of rediscaunting an loon paofitahility and the possihdlityof capital gainsatrlsses incurtedangoenmentrand firms' secneities. Let noel sepsesent the total stneam of profits from loans made in peniod 1 at matnaity. Then, wihaat rediscounting, unden she assumptian of no defalt, The hans' income statement in period a is mate complex adtmst tahe into accont the effects of rediscounting an loan profitahility and thepossihdlityof capital gainsaorlsses incurredaon govrnmentsand firms' secutitins. Let 7rgP1 eepresent the total stream of paofits from loans made in petiod I at maturity. Then, wihout rediscounting, under she assumption of no defahl, Thehbanhsincometstatement in periddtismore complexuasit mst tahe into account the effects of rediscoanting an loan profitahility and the posibility of capital gains at lanses incurred on govrnment and firms' securities. Let ngQPI reptesent the total stream of pnofits from loans made in petiod t at maturity.Then, without tediscounting, undr the assumption of no default, noeli = (1 + rbpn1) Let - Let = trtpLeL (2.111) a011 = (I + snet) Let - Ln1 = ttLe. (2.111) ?fT~p = (1 + runt) Let - LP, = nbr1L01. (2.111) The profit from loans made in period t in any one pediod, again without discounting, is simply Thnprofifrom loans made inperiddtinany onepeid, again without discounting, is simply The profit from loans made inpeiod I in anyone perod,ngain without discounting, is simply (I + tttLet _let.- tnse (2.112) (I + tut)Let - _LI. =iilg (2.1 12) (I + 'b A21 - LPI jbEI E! n n n (2.112) The total stream of profits at matutity from loans made inperiod 1, gluen shot a passion of them ate rediscounted in petiods aftet petiod 1, is giren hy The total stream of profits at maturity feom loans made in petiod 1, giren that a pontion of them ate rediscounted in petiods afar petiod 1, is given hy The total stream of profits at matutity from loans made in period 1, giuen that a portion of them ate rediscounted in peniods after peeiod 1, is giren hy nfl = - (Il+rb)L+d l+[( m [5-(n - m)(l - rd. )(I + ttt) V k = (I+ttt)LI+dnn+1I+[( n m )(I I+e51)] [L1- Odut+it L (n - m)(l - rd. a )(l a b 1. (l0+tul)LI + d.+t+[( n m )(1 + 01) ndnun (2.113) (2.113) (2.113) wheren (I + t51)LI gives the repaymena of interest and ptincipal teceived hy the hanh prion to rediscoanting a poetion of L,; dn a + t is the proceeds of rediscounting a portion of L, in period m + 1; where W (1 + e5)LI gires the repayment of inteeest and psincipal received by the bank peior to rediscounting apoton of LI; d. , c1 is she procneds of nediscoanting a partion of L, in peiod m+l1; whete - (1 + tt1)L, giant the repaymnt of intret and principal teceiedhbythe banh priortoaedscounting apotionuofLI; dr at1, 1i the proceeds of rediscounting a poetion of L, in pediod m +01;  THE MODEL 43 THE MODEL 43 + tnt [b LI - nd. . ,/(n - m)(t - rdnrt )(l + en)] - Lt is the repaynment of interest and principal of she portion of L, not rediscounted in period mn+ 1; and L, is simply the face valor of the loans made in period 1. Equation 2.113 redoces to (nC )( +r'b) [LI - ttdm , ,/(n - m)(1 -erd. , )(1 + rb 1)] - L is the repayment of interest and principal of the portion of 1,not redisconted iperiodm+1; andLiis simply the fcevlue of te totes mtade in period t. Equation 2.1t3 redaces to THE MODEL 435 ( _m)(t + rbi) [LI ndmrir,/(n -m)(t - rdm+,)(tIet) is the repayment of interest and principal of the portion ofbLnot redisconted in periodmn+ 1; and1,isimplyte facenalue ofthe loans mode in period t. Equation 2.t13 reduces to ngtr55jLj+dm [t, - 1-ed (2.114) n~~~~~t~~ =r~5 0n5 1 - rd. (2.1 14) n21 tb5l,+ d. +j, [I1- 1- I I-rd a (2.1 14) The latterm in2.114mwillbe negativeice (Lf X did + m sI nn +d j(I+(n - m)(1 + rrj)(l rdm n+ 1 1 (nr-m)(l -rdmrt ) 1l+ d.nt+ (2.116) I 1 (n -m)(l - Td.nt) l+trdw+I~ (2.1 16) When no rediscoanting occars (din +1, 1 = 0), 2.116 ishobionsly equiva- lent to 2.111. Eqation 2.116 serves as the basis for expressing the profirfrommallloansin any one period. Letting djjreprement the amepsid principal and interest of a jtt period loan rediscounted in period i, we have mr ma 1 +I r i >i(Lj-.X 6 = +m n - + d t1tj(I+ I- (a - m)(1 + rb)(l - rdm + 1) When no rediscounting occurs (din + 1, 1 = 0), 2.116 is ohvously equiva- leer to 2.111. Equation 2.116 serves as the basis for eapressing she profilfromall loans in any one period. Letting djireprement the amepaid principal and interest of a jt period loan rediscounted in period i, we have = ma + ~ i a n dn + dmn+t, j(I1+ (a - m)(l + rbj)(1tri + 1da)  44 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 44 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 44 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM (n- m)(1- rd.+ ) 1- rdm+. (2.117) (n- m)(1- rdm.+) I- rd (2.117) (n- rn)(1- rdm+a) 1- rdm (2.117) Profits on loans to the public, +, is given by using rbpj, deji, and dpm + r . i in 2.117, while n, is obtained by using the corresponding rate and rediscounting measures for loans to the firms. Thus, Profits on loans to the public, na 5, is given by using rbpj, d,55, and dpm + 1 i in 2.117, while nr is obtained by using the corresponding rate and rediscounting measures for loans to the firms. Thus, Profits on loans to the public, nmP+ 1, is given by using rbpj, d,55, and dpm , 1 in 2.117, while n5 is obtained by using the corresponding rate and rediscounting measures for loans to the firms. Thus, S * = 7m+ + n,* . 7r l nor n (2.118) Tt g + I + + OI. (2.118) 7 n p+ I + Tr'*I. (2.118) Profit in period m +I from government securities, r + 1, is simply Profit in period m +1 from government securities, 7. , is simply Profit in period m +I from government securities, n + 1, is simply i~m + rgm + I GbM + 1 (2.119) 7g + rgm + 1 Gbm + 1 (2.119) m + is=gm + 1 Gbm + 1. (2.119) No capital gains or losses are made on government securities as a result of the assumptions of one-year maturity and no intraperiod trading of the securities by the banks. Profit in us + I from firms' securities, Of ' 1, includes both interest and possible capital gains (losses). Thus, im+ 1 -k k i-m - k No capital gains or losses are made on government securities as a result of the assumptions of one-year maturity and no intraperiod trading of the securities by the banks. Profit in m + I from firms' securities, 1 +e7 includes both interest and possible capital gains (losses). Thus, . + + rB + I um riabfi + m +m+1- k k i m k No capital gains or losses are made on government securities as a result of the assumptions of one-year maturity and no intraperiod trading of the securities by the banks. Profit in us +1 from firms' securities, 7 + 1, includes both interest and possible capital gains (losses). Thus, m+_ + 1 r5Bbri + m+ = m -k k im -k (Pbfm + 1 - Pbri) (Fbfm + I - Fbf5) (2.120) (Pbfm + I - Pbri) (Fbfm + I - Fbi) (2.120) (Pbfm + 1 - Fbf) (Fbfm + - Fbfi) (2.120) where k is the maturity of the firms' bonds and Fbf + 1 - Fbi does not enter the equation unless it is negative, e.g., unless the banks actually sell period i bonds in m + 1 to actually realize accrued capital gains or losses. The banks' overall gross profit (before payments to owners, purchase of factors, and payment of interest on time deposits) in us + 1, 7m + ,is given by where k is the maturity of the firms' bonds and Fbfm + I - Fbi does not enter the equation unless it is negative, e.g., unless the banks actually sell period i bonds in m + 1 to actually realize accrued capital gains or losses. The banks' overall gross profit (before payments to owners, purchase of factors, and payment of interest on time deposits) in m + 1, n + , is given by where k is the maturity of the firms' bonds and Fbfn + 1 - Fbi does not enter the equation unless it is negative, e.g., unless the banks actually sell period i bonds in m + 1 to actually realize accrued capital gains or losses. The banks' overall gross profit (before payments to owners, purchase of factors, and payment of interest on time deposits) in m +1, s + i given by S.nmn+ + + +nn 9m + I + n. (2.121) no + nm + I + nSnm or no r (2.121) 7eb =og +7 tr I + 7rm + I + r. (2.121) The only portion of vm + 1 that is not received directly by the public in the form of income in o+I is the portion used to purchase capital, Pcm + Xkbm +1. Thus, the banks' contribution to the public's income in m + 1, Ym + 1, is Y nT+ - Pkus m Xnm. (2.122) The only portion of m + 1 that is not received directly by the public in the form of income in m + l is the portion used to purchase capital, Pkm + 1 Xkbm +1- Thus, the banks' contribution to the public's income in m + 1, Ym , is Y c =cnb Pk. + 1 Xkn .sr (2.122) The only portion of nm + 1 that is not received directly by the public in the form of income in m+I is the portion used to purchase capital, Pkn + 1 Xknm +I. Thus, the banks' contribution to the public's income in m + 1, Y' is - + km + 1 Xkb + 1. (2.122)  THE MODEL 45 THE MODEL 45 THE NONBANK FINANCIAL SECTOR (INTERMEDIARIES) The nonbank financial sector is assumed to have two major functions. It accepts deposits from the public and the firms and lends to each of these sectors. The insurance function of this sector will not be explicitly recognized. Rather, deposits will be taken to include not only the typical savings deposit at, say, a savings and loan institution but also insurance premiums. Payments on insurance claims will be included in any withdrawals of principal plus interest from the "savings" accounts. For simplicity, it is further assumed that the rates of interest paid on these deposits are the same for the public and the firms. The deposits in the nonbank financial sector are not assumed to be part of the stock of money. With the submersion of the insurance function, the major impact of the nonhank financial sector will be on the banking sector with which it competes both for deposits and for loans. The following relations describe the aggregate activities of the non- bank financial sector. THE NONBANK FINANCIAL SECTOR (INTERMEDIARIES) The nonbank financial sector is assumed to have two major functions. It accepts deposits from the public and the firms and lends to each of these sectors. The insurance function of this sector will not be explicitly recognized. Rather, deposits will be taken to include not only the typical savings deposit at, say, a savings and loan institution but also insurance premiums. Payments on insurance claims will be included in any withdrawals of principal plus interest from the "savings" accounts. For simplicity, it is further assumed that the rates of interest paid on these deposits are the same for the public and the firms. The deposits in the nonbank financial sector are not assumed to be part of the stock of money. With the submersion of the insurance function, the major impact of the nonbank financial sector will be on the banking sector with which it competes both for deposits and for loans. The following relations describe the aggregate activities of the non- bank financial sector. THE MODEL 45 THE NONBANK FINANCIAL SECTOR (INTERMEDIARIES) The nonbank financial sector is assumed to have two major functions. It accepts deposits from the public and the firms and lends to each of these sectors. The insurance function of this sector will not be explicitly recognized. Rather, deposits will be taken to include not only the typical savings deposit at, say, a savings and loan institution but also insurance premiums. Payments on insurance claims will be included in any withdrawals of principal plus interest from the "savings" accounts. For simplicity, it is further assumed that the rates of interest paid on these deposits are the same for the public and the firms. The deposits in the nonbank financial sector are not assumed to be part of the stock of money. With the submersion of the insurance function, the major impact of the nonbank financial sector will be on the banking sector with which it competes both for deposits and for loans. The following relations describe the aggregate activities of the non- bank financial sector. N = N' - N. (2.123) N = N' - No. (2.123) N = N' a N' (2.123) The actual level of deposits, N, is identically equal to the demand by the firms and the public for them, i.e., the supply of deposits is perfectly elastic. The actual level of deposits, N, is identically equal to the demand by the firms and the public for them, i.e., the supply of deposits is perfectly elastic. The actual level of deposits, N, is identically equal to the demand by the firms and the public for them, i.e., the supply of deposits is perfectly elastic. N = q4 + L" + G. + D + Bn + Cu + Tu. (2.124) N = I4 + L" + G, + D. + B. + C + T-. (2.124) N = 4 + L + G, + D, + B0 + C5 + Tn. (2.124) This is simply the balance sheet equation for the nonbank financial sector. Note that 2.124 reflects the assumption that the nonbank sector has no direct connection with the banks except to hold demand deposits and time deposits. This is simply the balance sheet equation for the nonbank financial sector. Note that 2.124 reflects the assumption that the nonbank sector has no direct connection with the banks except to hold demand deposits and time deposits. This is simply the balance sheet equation for the nonbank financial sector. Note that 2.124 reflects the assumption that the nonbank sector has no direct connection with the banks except to hold demand deposits and time deposits. In =lnN + A,f Gn gn N + A, sint Dn= dnN + A , F =. bN + A T. (2.125) (2.126) (2.127) (2.128) In =N + A74Y Gn= gnN + A, Jia D = dN + A1 6n. Fp= bn N + Al 7 (2.125) (2.126) (2.127) (2.128) L =litN + A] J. G. = gN + A , D = dnN + A sn Fn= bN +A y, (2.125) (2.126) (2.127) (2.128) C' = CnN. (2.129) Cn' = c N. (2.129) Cn' = enN. (2.129)  46 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Equations 2.125 through 2.129 are the basic decision functions of the nonbank sector. Equation 2.125 gives the maximum aggregate amount of loans the sector is willing to make while 2.126 to 2.129 represent the demand for nonloan assets, given that the nonbank sector is able to loan all it desires. Insufficient demand for loans will result in additional holdings of government securities, firms' securities, cash, and demand deposits as specified below. ., g., d., br, and c are positive constants, while rnp is the rate of interest on loans to the public, rrf the rate on loans to the firms, and rf the rate paid on deposits. All other symbols are as defined previously. The aggregate amount willing to be lent, Lr, is broken down between the firms and public in a manner analogous to that of the banks. 46 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Equations 2.125 through 2.129 are the basic decision functions of the nonbank sector. Equation 2.125 gives the maximum aggregate amount of loans the sector is willing to make while 2.126 to 2.129 represent the demand for nonloan assets, given that the nonbank sector is able to loan all it desires. Insufficient demand for loans will result in additional holdings of government securities, firms' securities, cash, and demand deposits as specified below. Q., g., dr, bn, and c, are positive constants, while rn, is the rate of interest on loans to the public, roe the rate on loans to the firms, and re the rate paid on deposits. All other symbols are as defined previously. The aggregate amount willing to be lent, Er, is broken down between the firms and public in a manner analogous to that of the banks. 46 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Equations 2.125 through 2.129 are the basic decision functions of the nonbank sector. Equation 2.125 gives the maximum aggregate amount of loans the sector is willing to make while 2.126 to 2.129 represent the demand for nonloan assets, given that the nonbank sector is able to loan all it desires. Insufficient demand for loans will result in additional holdings of government securities, firms' securities, cash, and demand deposits as specified below. 2r, g, dn, bn, and c are positive constants, while rue is the rate of interest on loans to the public, rue the rate on loans to the firms, and rf the rate paid on deposits. All other symbols are as defined previously. The aggregate amount willing to be lent, L , is broken down between the firms and public in a manner analogous to that of the banks. 142'=t'E2'+ as8(rnespet j2'r= qf~n .,a (r e) rf (2.130) (2.131) I " = 2L + a, a(rnpt nft) Itn = tE "- a, (rnpt -rft) I" + L =I " (2.130) (2.131) (2.132) [It = t2I + a, s (ret rnft) Is = qn"I - a, 8 (rnpt r,,t) [SI + It" = L". (2.130) (2.131) (2.132) Iln + f " = Le". (2.132) Equations 2.130 and 2.131 are analogous to 2.97 and 2.98 for the banking sector. 2 and q4 represent institutional factors influencing the desired distribution of loans between the public and firms. 2" + k" = 1. a, is a positive constant whose size determines the importance of interest rate differences in loan distribution. Adjustments in the initial breakdown occur if either the public's or the firms' demand for loans from the nonbank sector is less than the initial amount the nonbank sector is willing to lend while the demand from the other sector is greater than the initial amount. Thus, if Lt < s' and L, > Ls, then Lr, the actual amount lent to the public, is equal to L ' and Lf = min (tf + lE - Lp, L- L2D). Similarly, if Le >I, and L < Ir theneLn= m in (i , + 14O- L4s, I - LD). In either case the amounts actually lent in each sector are given by Equations 2.130 and 2.131 are analogous to 2.97 and 2.98 for the banking sector. 2 and q" represent institutional factors influencing the desired distribution of loans between the public and firms. 2" + 02' = 1. a1, is a positive constant whose size determines the importance of interest rate differences in loan distribution. Adjustments in the initial breakdown occur if either the public's or the firms' demand for loans from the nonbank sector is less than the initial amount the nonbank sector is willing to lend while the demand from the other sector is greater than the initial amount. Thus, if Lr L, then Lr, the actual amount lent to the public, is equal to Iw and L r, = min (l ,r+1 - L. p, L- ). Similarly, if LD, > I , and LQ, < I r then Ln, = min (l +L el - L , L - 2). In either case the amounts actually lent in each sector are given by Equations 2.130 and 2.131 are analogous to 2.97 and 2.98 for the banking sector. 2 and R, represent institutional factors influencing the desired distribution of loans between the public and firms. 2 + 2 = 1. a1 is a positive constant whose size determines the importance of interest rate differences in loan distribution. Adjustments in the initial breakdown occur if either the public's or the firms' demand for loans from the nonbank sector is less than the initial amount the nonbank sector is willing to lend while the demand from the other sector is greater than the initial amount. Thus, if L < L on -'> S ren or and E >, then Lp the actual amount lent to the public, is equal to Lq, and Lnt = min( 1 4 - e' no) Simnilary, if L > I e and L < f ,hen L = m i (1 h + 1 f - L2, L4 - L ). In either case the amounts actually lent in each sector are given by Ln = min I4e + ('_" - LDn, L'e Ls = min L~ + (Isr + LDn),L . (2.133) (2.134) L. = min L e + (Ij" - LDn) Le and Lr = min L ,r+ (E'" + LD ),L (2.133) (2.134) LE = min ILn + (I * - LD n), L and Lan r min L0n, + (40 + L D),Lne. (2.133) (2.134) The total amount lent, L., is simply the sum of 2.133 and 2.134. In no case can L be larger than Ln. It may, however, be smaller. In this case The total amount lent, L, is simply the sum of 2.133 and 2.134. In no case can L be larger than L. It may, however, be smaller. In this case The total amount lent, L., is simply the sum of 2.133 and 2.134. In no case can L, be larger than L . It may, however, be smaller. In this case  THE MODEL 47 the holdings of interest esening assets see ioceaoed shoot the tenets goren hy 2.75 throngh 2.t27 as given betow. THE MODEL 47 the holdings of inteest earoing assets see incressed shoot the tenets given by 2275 therough 2.127 as given betow. THE MODEL 47 the hotdings of interest eaening ssets see ineeod shore the tenets given hy 2.75 through 2.127 Os fivn helow. t.0- L5= AG. iferg >st, rf L - L= AB Di if t> t, r. 0. -Ln =ATf' if,, >e,, t,. (2.135) (2.1 36) (2.137) L - L -AG0 ifte >trt,tt t~L0 = ABP if rf > r,,i 4 - L = r~fDift >tng,tf. (2.1 35) (2.1 36) (2.1 37) L0 - L = ABfif rf > r, OnLv = ATr if rt, rf. (2.1 35) (2.1 36) (2.1 37) If two of the totes (r., it, en) see equalsand the tird smaller, the inserter in demand for those assets wilt be equsl vs one-half of L~ - L_. If alg three roves see equalterincresse in demsndforrschoaset will equal one-third of L5, - L.. Since the supplies of government securities snd tite deposits see assumed to be perfectly elassic, no complications srise if either 2.135 or 2.137 hold. If 2.136 holds, it 6s possible that, since the supply of firms' securities is not perfectly elstic, the eovbsnlsector mynotbeshle tosacquireall thefirms' securities it desiresI tis cost, eithr government securities or time deposits will he incressed in on amount equsl to the unsatisfied demand for firms' securities, depending on the relative sires of tg snd it. To describe the determinstion of the votions totes of interest associ- ated with the noehonk sector, we again introduce the subscript t to represent time. Note that this subscript hss heen omitted from she first fifteen equations merely foe convenience. We hove ,tr+tI tots + a, 0(L~r - Lf ') + b, 9(r~ft - thee) (2.138) where a1 > 0 sed it0 < 0. rupt+ e rnt+rsso(LS.rr-LDrt) + b50(rnpt - thee) (2.139) where a~ > 0 snd b2, < 0' If two of the rstes (en, it, rf) see equsl sod the tird smasller, the incres in dmndfotthosessets willbe equal toone-half of L - L_. If sll theree rates see equal, the incease in demand foe each ases wilt equal one-third of Ln - L.Since the spplies ofgoernmetn securities snd time deposits ore asmumed to be perfectly elstic, no comphicstions srise if ethbet 2.135 or 2.137 hold. If 2.136 holds, is it possible thst, since the supply of firms' serities is not perfectly elaseic, the voehsek sector may not be ahle to scquite siltThe firms' securities it desires. In this case, either government securities or time deposits will be incresed in on amountlequl so the nstisfied demand for firms' securities, depending on the reltive sizes of en red rt. To descdibethe dterminstion of thenrioststes of intrestssoci-. ated with the nonhonk sector, we sgsin introduce she suhscript I to represent rinse. Note thatt tis suhscript hoe been omitted from the first fifoeeo equstions merely for coneniec. We hove erot I rtI a, s5(L. - Lt') + b, t(trte - eves) (2.138) where a19 > 0 ted b, < 0. errs=~rr + tt(Ltr -Lpt) + o(rvpe - ever) (2.139) where a20 >0 and b2 < 0. if two of she roles (t5, t, r) ore equst ted she lied smasler, the incresse in demasnd for those aselts will be eqoat to one-haif of Len - Lv. If slt three ratesarerequal,teincese indemand forechsst will equal one-third of Ln - L. Since the supplies of government securities sod time depoits ore ssumed to he perfectly elastic, no complications arhse if either 2.135 or 2.137 hold. If 2.136 holds, 0t it possible thst, since the supply of firms' securites is not perfectly elaseic, the eonhsntcsectoemynothbesabletosacquiresllthe firms'secriiesit desires. In sis csse, either government securties or time deposits wilt hr inctessed in on amount equsl to the usasisfied demand foe firms' securtiies, depending on the relative sizes o g ted r,. To describe the determinstion of she various roles of interest smoci- sled wish the nonbonk sector, we again introduce the subscript t to represent time. Note thtl this subscrips hss heren omitted from the first fifteen equstions merely for convenience. We hsve tsrt I tes + a, 9(L~f - Lf n) + 5(rnft - rrf) (2.138) where tsr >0sndb,5 <0. en ets =tee + so(Lee -Lp0) + bsr(trrr - this) (2.139) where a2r > 0 and h50 < 0. rst I t+ a2 1(L -L D _4?r) (2.140) tetsI rt + ads(I - L LD D (2.140) S D D rnt + I rt + a2 1(L. - Lnp - Lnd (2,140) where s.,, b55 < 0 Poe the noobseit sector to be in equiirim these conditions most he satisfied: rt+i rt+i- 1 (2.141) where a2,, b., < 0. For the nionhonk sector to he in equilihrium these conditions meet he satisfied: eros+ =~05 5n (2.141) where a~l b2, < 0. Por the nonhonk sector tobe inequilibrium these conditionstmustbe satisfied; rnt i r.,+ i- 1(2.141)  48 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 48 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 48 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM rnft + i rnft + i I rupt + i = npt + i -I1 Lr = L'I F, = F' (2.142) (2.143) (2.144) (2.145) raft + i nft + i- I rnpt + i=rnpt + i- I Lnt = L., Ft = F . (2.142) (2.143) (2.144) (2.145) rnft + i = rnit + i -I1 rapt + i = rnpt + i -I L I = L't F, = FD (2.142) (2.143) (2.144) (2.145) Note that desired holdings of time and demand deposits, government securities, and cash will be satisfied under all conditions. The intermediaries' "balance sheet" was given in Equation 2.124. Their "income statement" provides Ye, the contribution of the intermediaries to the income of the public in period L For simplicity, it is assumed that all loans made by the intermediaries have a maturity of n years. Since we do not provide for any governmental sources of reserves for the intermediaries (no rediscounting of their loans), the profit on loans in period I, rt,, is simply n = t 1 r L + t- i1 i=n t - n - 1 n i= t - -1 Note that desired holdings of time and demand deposits, government securities, and cash will be satisfied under all conditions. The intermediaries' "balance sheet" was given in Equation 2.124. Their "income statement" provides Yn, the contribution of the intermediaries to the income of the public in period t. For simplicity, it is assumed that all loans made by the intermediaries have a maturity of n years. Since we do not provide for any governmental sources of reserves for the intermediaries (no rediscounting of their loans), the profit on loans in period I, stn, is simply t- t - 1 ni= t n i= t-n- 1 Note that desired holdings of time and demand deposits, government securities, and cash will be satisfied under all conditions. The intermediaries' "balance sheet" was given in Equation 2.124. Their "income statement" provides Y., the contribution of the intermediaries to the income of the public in period t. For simplicity, it is assumed that all loans made by the intermediaries have a maturity of n years. Since we do not provide for any governmental sources of reserves for the intermediaries (no rediscounting of their loans), the profit on loans in period t, 4tn, is simply t t 1 r -L n. t 1 Tt 2n Z (I noliot T i= t-- n i= t-n-1I rntii ). (2.146) n Profit on government securities in t, 7r'., is given by n Profit on government securities in I, ign, is given by gn = rgtGnt. Profit on firms' securities is (2.146) (2.147) n Profit on government securities in t, at, is given by it. = rgtGnt. Profit on firms' securities is t 1 rfiB,, m + 1 nfn + 2; im+1-k k im k (Psem + 1 - Pbri)(Bnfm + I - Beni) (2.146) (2.147) Profit on firms' securities is t + I r m + 7r, = z ___ + I i Im1- k k i=tm - k (Pbf m + 1 ~ Pbfi) (Bnf m + I - Bnf;) (2.147) t m I r+BI , M + I i-m +1- k k i= k (bfm + 1 - Pbfi)(Bnfm + I - Bnei) (2.148) which is equivalent to 2.120. The intermediaries' gross profit in t, 1t , is Otn no + ot,_ + or. (2.149) (2.148) (2.148) which is equivalent to 2.120. The intermediaries' gross profit in t, on, is =n +0 += t. (2.149) which is equivalent to 2.120. The intermediaries' gross profit in t, ot., is On = ltkn + ±rg + 4ifn. (2.149)  THE MODEL 49 THE MODEL 49 The only portion of nir not received directly by the public in the form of interest payments, labor payments, or dividends is the portion used to purchase the capital good, PeoXe.. Thus, The only portion of el, not received directly by the public in the form of interest payments, labor payments, or dividends is the portion used to purchase the capital good, PktXknt. Thus, THE MODEL 49 The only portion of nr, not received directly by the public in the form of interest payments, labor payments, or dividends is the portion used to purchase the capital good, PktXkat- Thus, Y = a - PktXkat. (2.150) Ytn = "I - PktXknt. (2.150) Yn wt - PktXkat (2.150) THE PUBLIC SECTOR (HOUSEHOLDS) The public sector is composed of all individuals in the economy acting in their roles as consumers and suppliers of factors. Only the aggregate behavior of this sector is considered; no attempt is made to distinguish among different individuals or groups of individuals. The basic relationships for the public sector are: THE PUBLIC SECTOR (HOUSEHOLDS) The public sector is composed of all individuals in the economy acting in their roles as consumers and suppliers of factors. Only the aggregate behavior of this sector is considered; no attempt is made to distinguish among different individuals or groups of individuals. The basic relationships for the public sector are: THE PUBLIC SECTOR (HOUSEHOLDS) The public sector is composed of all individuals in the economy acting in their roles as consumers and suppliers of factors. Only the aggregate behavior of this sector is considered; no attempt is made to distinguish among different individuals or groups of individuals. The basic relationships for the public sector are: Drr= kIYr + A22or (2.151) Dpt = kjYt + A22p cot = k2Yt + A23-fp Tpt = k3 t + A24Tp Cat = cY + A2s f Gpt gYt + A26p L45 flY + AaiT~ + (Lrbt 0 L oY + A27p + t-i - Lrt-) Lt of t + A28rp + t-I - LDt Fpt pt + A297p Nrr neYt + Ase, (2.151) (2.152) (2.153) (2.154) (2.155) (2.156) (2.157) (2.158) (2.159) Drt = kYt + A22FP Cpt = k2 t + A23rp Tpt = k3Yt + A24 p Ce = cY + A2sip Gpt = t + A26p Let r + A27?p + (L2t-7 - Let-) o = y + A2T8r + (Lpt-1 - L1t"-1) Fpt opf t + A29ip Not neY, + A30fp (2.151) (2.152) (2.153) (2.154) (2.155) (2.156) (2.157) (2.158) (2.159) Crt = k2Yt + A23rp Tpt = k3 + A24p C., = cYt + A2 syp Gpt g t + A26Fp Lth - t + Ap +(LE_ I Let) Lt = Q"Yt + A2stp + (Lt - 1) Fpt fpYt + A29-p Ner nYt + A30p (2.152) (2.153) (2.154) (2.155) (2.156) (2.157) (2.158) (2.159) Equations 2.151 through 2.159 express the public's demand for demand deposits, currency, time deposits, the consumption good, government securities, loans, and firms' securities in dollar terms. P5 is not only the price of the consumption good but, since there is by assumption only one composite consumer good, it also serves as the consumers' price index. Equations 2.151 through 2.153 require no further comment. Equation 2.154 expresses the dollar value of the public's purchases of the con- Equations 2.151 through 2.159 express the public's demand for demand deposits, currency, time deposits, the consumption good, government securities, loans, and firms' securities in dollar terms. Pc is not only the price of the consumption good but, since there is by assumption only one composite consumer good, it also serves as the consumers' price index. Equations 2.151 through 2.153 require no further comment. Equation 2.154 expresses the dollar value of the public's purchases of the con- Equations 2.151 through 2.159 express the public's demand for demand deposits, currency, time deposits, the consumption good, government securities, loans, and firms' securities in dollar terms. P. is not only the price of the consumption good but, since there is by assumption only one composite consumer good, it also serves as the consumers' price index. Equations 2.151 through 2.153 require no further comment. Equation 2.154 expresses the dollar value of the public's purchases of the con-  50 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM sumer good, e.g., Cut = PrtXept. It does not include the value of the consumer good transferred to the public sector by the government. Total consumption by the public sector of Xc in period t is given by Cot + PctXcgt (see the section on the government sector). Equations 2.156 and 2.157 express the public's demand for loans; their sum is the aggregate demand. It is assumed, for simplicity, that an unsatisfied demand for loans from one sector will not increase the quantity of loans the public demands from the other sector in the same time period. Unsatisfied loan demand in t causes the quantity of loans demanded in t + I to increase by an amount equal to the excess demand. Equations 2.155 and 2.158 are self-explanatory. Other relations for the public sector include: (1) the definition of the public's gross income, Y,: Yt gt + Ybt + Yn1 + Yft + rgGY s + rtT, + 50 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM sumer good, e.g., Ct = etXcept. It does not include the value of the consumer good transferred to the public sector by the government. Total consumption by the public sector of X. in period t is given by Cat + PctXrgt (see the section on the government sector). Equations 2.156 and 2.157 express the public's demand for loans; their sum is the aggregate demand. It is assumed, for simplicity, that an unsatisfied demand for loans from one sector will not increase the quantity of loans the public demands from the other sector in the same time period. Unsatisfied loan demand in t causes the quantity of loans demanded in t + 1 to increase by an amount equal to the excess demand. Equations 2.155 and 2.158 are self-explanatory. Other relations for the public sector include: (1) the definition of the public's gross income, Yt: t = gt + Ybt + Y, + Yt + rtG t + rttTt + 50 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM sumer good, e.g., C, = PctXcpt. It does not include the value of the consumer good transferred to the public sector by the government. Total consumption by the public sector of Xc in period l is given by Cat + PctXcgt (see the section on the government sector). Equations 2.156 and 2.157 express the public's demand for loans; their sum is the aggregate demand. It is assumed, for simplicity, that an unsatisfied demand for loans from one sector will not increase the quantity of loans the public demands from the other sector in the same time period. Unsatisfied loan demand in t causes the quantity of loans demanded in t + 1 to increase by an amount equal to the excess demand. Equations 2.155 and 2.158 are self-explanatory. Other relations for the public sector include: (1) the definition of the public's gross income, Yt: Yt Yt + Ybt + Ynt + Yft + rgtGpt + rttTpt + rntNpt + reFpt; (2) the definition of disposable income, Yt: Yt = (1 - t)Yt; (see the section on the government sector). (2.160) (2.161) rntNpt + rftFt; (2) the definition of disposable income, Yt: Yt = (1 - t)t; (2.160) (2.161) rttNpt + rftFpt; (2) the definition of disposable income, Yt: Yt = (1 - t)Yt; (see the section on the government sector). (2.160) THE MARKETS In this section I attempt to connect the previous sections by examining the various markets in the model in greater detail and in isolation. Currency Market The price of currency is the opportunity cost of holding it, the income sacrificed by not holding a return-earning asset. For simplicity it is assumed that this cost can be represented by the largest of r, rg, ro, and rt.4 Thus, the price of currency for the public is equal to I + max 4. Perhaps a more theoretically aesthetic way of viewing Pc is this: The price of currency is the opportunity cost of holding currency rather than a return-earning asset, e.g., time deposits, deposits in intermediaries, government securities, and firm scurities. Let Oc be this cost. Then Oc = Oc(rf ig, r , rt) (deposit in intermedi- aries) where the function describes the return that could be earned on an extra THE MARKETS In this section I attempt to connect the previous sections by examining the various markets in the model in greater detail and in isolation. Currency Market The price of currency is the opportunity cost of holding it, the income sacrificed by not holding a return-earning asset. For simplicity it is assumed that this cost can be represented by the largest of rf, rg, in and rt.4 Thus, the price of currency for the public is equal to 1 + max 4. Perhaps a more theoretically aesthetic way of viewing Pc is this: The price of currency is the opportunity cost of holding currency rather than a returnarning asset, e.g., time deposits, deposits in intermediaries, government securities, and firm securities. Let Oc be this cost. Then Oc = Ochr, rg, en, rt) (deposit in intermedi- aries) where the function describes the return that could be earned on an extra THE MARKETS In this section I attempt to connect the previous sections by examining the various markets in the model in greater detail and in isolation. Currency Market The price of currency is the opportunity cost of holding it, the income sacrificed by not holding a return-earning asset. For simplicity it is assumed that this cost can be represented by the largest of rf, rg, r., and rt.4 Thus, the price of currency for the public is equal to I + max 4. Perhaps a more theoretically aesthetic way of viewing Pc is this: The price of currency is the opportunity cost of holding currency rather than a return-earning asset, e.g., time deposits, deposits in intermediaries, government securities, and firm securities. Let 0c be this cost. Then Oc = Oc(rf, rg, ra, rt) (deposit in intermedi- aries) where the function describes the return that could be earned on an extra  THE MODEL 51 rf, rg, ra,rIt while for the fimsitiislI+ max rI, ru,rIf Isince firms do not, by ossomption, hold debt instromeors of other firms. P, for rho inormediories is 1 + max If, r., rtf Iwhie for the books, P. is I+rmaxnIr, rsie b kholdiher tiedpositsnor depositssin inrermedisries. If P, is the pricrof corrency, websavrthetypicl downword-sloping demond core. liqootioos 2.46, 2.90, 2.129, sod 2.154 glee thr demood for currency explicitly. All ennoabler io these eqostiont except moo 0, , t r, r, moss be considered fixed when included in Pigore 4. Increases in sres other shoe she maximum rose shift she demond corves downward, done nsrrowing she difference he- teee interest roses mobkes she security wish the higher rote relotively lesssattractie.tIncrasscihY,D+ T, or PXc + P0X shiftrthecore outward. Noterthat thedemand cregivesthe demand for changes in currency holdings. At prices lower thn P00 the polic (book, firm, or intrmedisry) moors ro increase its curreocy holdings, while or pricer shore Pec, it wants so reduce them. The stock demand for currency can be reoddly found by combining last period's stock, C5..., with te desired chonge in rhis period. Nore rhot C, , estoblishes the lower limit for chonges io corrency holdhngs dosing period t, since she stock of currency connot benegttive. doilear distribted in st sae prenoage as tho prrsent disribotion hetween rime deposits, governent sectrities, otd firms' securitirs. Poe etch mseor e aw 1. 0~ et It Tp + GrO +O~ er + ct +OBfec+Nr TrO+GrO+ rrO+Nc 2. 000 =e It 05 + 0ref f 3- 15c = rt T + 0 t + ttg t5 Fsr simplicity or haoo chosn nor to ue this tetinition sf00c. THE MODEL 51 Ifrrrsr,r m t hilforthe firm tiistI+ maxjI,rI, itrI since firms do net, by asmumption, hold debt instruments of other firms. P. for the intermedisdies is Io+maxInfjr,,mwile fothebnks, Pis 1 + max Ier since banks hold neither rime deposits nor deposits in intermediaries. tf P. is the price of currency, me hsee the sypicol downward-sloping demand core. Eqostions 2.66, 2.90, 2.129, sod 2.154 glee the demond for currency explicitly. All varisbles is these equations except won rf, sg, ru, if } moot be considered fixed when included in Figore 4. tncresses in totes other than the maximum rote shift the demand cores downwsrd, since narroming the difference be- tween interest roles makes the security with the htigher rote retatively hes atrctive. tncreases in Y, D+ T, or PX. + PkXk shift the core outward. Note that the demand core glees the demand for changes in corrency holdings. At prices loots shoe P00 she public (honk, firm, or intermediory) ints to increase its currency holdings, while or prices obove Pe., it moors to reduce them. The stock demand for currency con be reodily found by combining lost period's stock, C, I , wish the desired change in this period. Note thor C5..., establishes the lower limit for changes in currency holdings dosing period I, since the stock of currency cannotbeoegive. dllar didstrte in slit stint prertage or slit prestnt distlri tto rete time deposits, govermmrnt securites, set eirms' seccorities. Per echsoct05or haw: T. " It e5 TrGc+ r + 0tc + 0p+ f Np Trt+Gp +Otrtt+lc 2. 0ci If T rtr Org Trtctt T+OIf OOf Tr Gf rgNf 3. 0ce Itot T+ rOe e Tf +0 oG r Tn + 0 vG r Tf + 0 0f + t Pot simplicity or hase chossn nor to on this definition of 0,,. THE MODEL 51 Irf, to, in, rf odhie for the firms it is 1 + max Irte n, t I since firms do not, by assumption, hold debt instruments of other firos. P. for the intermediaries istI+ mx IIf, rt rj hileforthe bnks,P, is I + max IrI, Is dicc books hold neither rime deposits nor deposits in intermediories. tf Pc is the price of corrency, me hove she typicol downword-sloping demand curve. Equations 2.66, 2.90, 2.129, sod 2.154 give the demand for currency explicitly. All voriables in these equations except max Ir1, e'rl, n, most., be considered fixed when inctoded in Figure 4. Increases in roses other shoe the maximum rose shift the demand cores downword, since norroming the difference be- tween interest roses mokes the security wish the higher rote relosively lesmottractiv.tIncresesin Y, D+T, or P,X, + PkXk shift the come outword. Note shot the demand curve gives the demand for changes in currency holdings. As prices lower thon P_, the public (hook, firm, or intermediary) wools so increase its currency holdings, while as prices shove P, itmwants to reduce them. The stock demand for currency con he readily foond by combining lost period's stock, C, I, with the desired change in this period. Note shot Ct- estoblishes the lower limit for changes in currency holdints dosing period t, since the stock of corrency connot be negative. tonear tistriliuted is the sce percrntrgr rs the present distribtoion Ortoren timr depsitsgovernmentscriies,eandtfrms' securiies. or eachsectorweihore: Tp+Gp +orcr+Np Tp t cO+ rcO+Np 2. 000 =tIt Trb t to Tc OG + ore Bf0r0O 3. 0ct~f =~ G It ~ O Tor~ S0v Tf +1% 0 Of T +0, tr T,, G +0, 0t Pot simpicity or bae chosen ot to so rhthisdfnticono 000  52 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM The supply of currency in this model is not independent of the demand for it, since by assumption we have the government passively issuing or absorbing currency in the aggregate amount demanded by the public, firms, and banks. Thus, we have 52 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM The supply of currency in this model is not independent of the demand for it, since by assumption we have the government passively issuing or absorbing currency in the aggregate amount demanded by the public, firms, and banks. Thus, we have 52 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM The supply of currency in this model is not independent of the demand for it, since by assumption we have the government passively issuing or absorbing currency in the aggregate amount demanded by the public, firms, and banks. Thus, we have D(dC) m S(dC) (2.162) D(dC) m S(dC) (2.162) D(dC) m S(dC) (2.162) for all P, This aggregate demand for currency is obtained by summing the four sectors' demands for currency. Equation 2.162 holds both for each sector and for the aggregate. Thus, the currency market is in perpetual equilibrium. for all P, This aggregate demand for currency is obtained by summing the four sectors' demands for currency. Equation 2.162 holds both for each sector and for the aggregate. Thus, the currency market is in perpetual equilibrium. Pc Pc, P,-- D(dC) for all P. This aggregate demand for currency is obtained by summing the four sectors' demands for currency. Equation 2.162 holds both for each sector and for the aggregate. Thus, the currency market is in perpetual equilibrium. D(dC) $(dC) -C dlC. C Fig. 4. Demand for currency dC -C[ dC Fig. 4. Demand for currency (dC) The Market for Demand Deposits The price of demand deposits, PD, is defined in the same way as the price of currency, i.e., Po= + max r, rg, re, rt for the public, equal to t + max Ir, re, r, for the firms, and equal to 1 + max Ir, rg, rt for the intermediaries. The demand for demand deposits can be viewed in the same way as the demands for currency, with the obvious exception that in this case there is, again by assumption, no bank demand for demand deposits. Figure 5 is analogous to Figure 4. The banks are willing to accept any amount of new demand deposits and cannot prevent their withdrawal. Thus, again the supply of demand deposits is not independent of their demand, and we have The Market for Demand Deposits The price of demand deposits, PD, is defined in the same way as the price of currency, i.e., PD = 1 + max r, r, re, rt for the public, equal to 1 + max Ir, et, for the firms, and equal to I + max Ir, r, rt for the intermediaries. The demand for demand deposits can be viewed in the same way as the demands for currency, with the obvious exception that in this case there is, again by assumption, no bank demand for demand deposits. Figure 5 is analogous to Figure 4. The banks are willing to accept any amount of new demand deposits and cannot prevent their withdrawal. Thus, again the supply of demand deposits is not independent of their demand, and we have Peo PeF,, -- D(dC) ICT I dC,, $(dC) CC2 Fig. 4. Demand for currency The Market for Demand Deposits The price of demand deposits, PD, is defined in the same way as the price of currency, i.e., PD Ig n,, , it for the public, equal to 1 + max , en, t for the firms, and equal to 1 + max Ir, r, rt for the intermediaries. The demand for demand deposits can be viewed in the same way as the demands for currency, with the obvious exception that in this case there is, again by assumption, no bank demand for demand deposits. Figure 5 is analogous to Figure 4. The banks are willing to accept any amount of new demand deposits and cannot prevent their withdrawal. Thus, again the supply of demand deposits is not independent of their demand, and we have  THE MODEL 53 (2.163) THE MODEL 53 (2.163) THE MODEL D(dD) - S(dD) D(dD) - S(dD) D(dD) m S(dD) (2.163) for all PD. The aggregate demand is the sum of the public's, firms', and intermediaries' demands and 2.163 holds for the aggregate market so that the market for demand deposits is also in perpetual equilibrium. The Market for Time Deposits The price of time deposits, P, is also an opportunity cost. Since time deposits earn a rate of return, the price of one dollar in time deposits, Pt, is given by for all P0. The aggregate demand is the sum of the public's, firms', and intermediaries' demands and 2.163 holds for the aggregate market so that the market for demand deposits is also in perpetual equilibrium. The Market for Time Deposits The price of time deposits, Pt, is also an opportunity cost. Since time deposits earn a rate of return, the price of one dollar in time deposits, P, is given by for all PD . The aggregate demand is the sum of the public's, firms', and intermediaries' demands and 2.163 holds for the aggregate market so that the market for demand deposits is also in perpetual equilibrium. The Market for Time Deposits The price of time deposits, Pt, is also an opportunity cost. Since time deposits earn a rate of return, the price of one dollar in time deposits, Pt, is given by Pt = 1 + max Ir, r - rt for the public, Pt = 1 + r - rt for the firms, and Pt = 1 + max Ir, r. - rt for the intermediaries. (2.164) (2.165) (2.166) Pt = 1 + max r, r. - rt for the public, Pt = 1 + rg - rt for the firms, and Pt = 1 + max r, - rt for the intermediaries. (2.164) (2.165) (2.166) Pt = 1 + max r, r - rt for the public, Pt = 1 + r, - rt for the firms, and Pt = 1 + max Ir, rI - rt for the intermediaries. (2.164) (2.165) (2.166) -D_ dD, $(dD) D, Fig. 5. Demand for demand deposits The demand for time deposits is also analogous to the demand for currency and is shown in Figure 6. Summing over the public's, firms', and intermediaries' demands again yields the aggregate demand for time deposits. The banks again are assumed to be willing to supply an amount of time deposits and cannot prevent their withdrawal long enough to affect the analysis. Once again, then, - _dD, D, Fig. 5. Demand for demand deposits D)D) $(dD) D(dD) The demand for time deposits is also analogous to the demand for currency and is shown in Figure 6. Summing over the public's, firms', and intermediaries' demands again yields the aggregate demand for time deposits. The banks again are assumed to be willing to supply an amount of time deposits and cannot prevent their withdrawal long enough to affect the analysis. Once again, then, Fig. 5. Demand for demand deposits The demand for time deposits is also analogous to the demand for currency and is shown in Figure 6. Summing over the public's, firms', and intermediaries' demands again yields the aggregate demand for time deposits. The banks again are assumed to be willing to supply an amount of time deposits and cannot prevent their withdrawal long enough to affect the analysis. Once again, then,  54 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM O(dT) -_ S(dT) (2.167) for P, and She aggaegate araaet fartlime deposits is always in eqai- libriam. The Market for oveernamene Secueities This matkes is in aSl reespects siiar ta the ants already described. We have 54 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM O(dT) =_S(dT) (2.167) fee P, and she aggregate matket fat tiaae depasita is always in equi- lihrium. The Market far Gaeernament Securities Thia market ia in alt respects simidot ta the ants already deacesibed. We have 54 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM O(dT) a_ S(dT) (2.167) fat Pt and the aggegate market fartiOwe deposits is alwayinequi- hirim. The Market far overnment Securities This matkea ia in all rspecta simidar to the ants alteady dlaesried. We have P1 P1 I + max Itetalr - r, forthepublic; I+etf- t1 for thehbanks; It axantneal - r. forthe firms; I + max let, rt - r. forthe inermediaries. (2.168) (2.169) (2.170) (2.171) Pg = I +man {es,er,1 - sg forthe public; P1 =1I +sr - e foreahehbanks; P1 =1 + maxlr.,t r, r. forthefirms; P1 = I +waleessl - a forsthe intermediaries. (2.168) (2.169) (2.170) (2.171) P1 =1 O+man f et I - r1 forethepublhn; P5 =1 Iv+et- eg forthehbanks; P1 1+wmaxltn,adt rfothe fims; P1 = IO+naxs je, sat re forthe intermediaies. (2.168) (2.169) (2.170) (2.171) L T,i Fig 6. Dewandlfoatimaedeposdts Then the demands can he shown in Fignte 7. Aggregate demand is the samn of the fans seclts' demands. The government is a passive stuppliter- absorber of government securities so that in each marhet and ina 16e aggregef O(dO) -_ S(dO) (2.172) foe all Ps. The warket fat governmtnt securities is always in equilibtiam. Fig .Deandforetimetdeposits Then she demands can he shawn in Figure 7. Aggregate demand 6s the sum of the fans sectors' demands. The government is a passive spplier- ahsorher af government securities so that in eachwarket and in the aggregate O(dO) =_ S(dO) (2.172) for all Pg. The masrket fat govenent securities is always in equaiirim. F, F, O(dT) T, T $(dT) Fiw 6. Dewmad foe tine deposits Thea she demands can he shown in Figuee 7. Aggregate demand is the saw of she fans sectors' demands. The goveenment is a passive supplier- ahsoree of goveernment securities so that in each marete and in thse aggregate D(dG) a_ S(dG) (2.172) fat all P.. The maret foe government secueities is always in eqailhibrim.  THE MODEL 55 THE MODEL 55 THE MODEL 55 The Market for Bank Loans This is the first market to boast a supply function that is independent of demand. The price of bank loans, P2, is I + rbp for the public and l + rbf for the firms. The demand for loans is again expressed in terms of desired changes in indebtedness to the banks. Thus, we have in general the situation shown in Figure 8. L- is the total indebtedness to the bank (unpaid principal plus interest on loans) at the beginning of period t, and dL represents the change in indebtedness during t. Note that if Pg = Pgl, the desired change in indebtedness is zero, but that this does not mean that desired new loans in t are also zero. When Pk = PQo, desired The Market for Bank Loans This is the first market to boast a supply function that is independent of demand. The price of bank loans, Pk, is 1 + rbp for the public and I + rtf for the firms. The demand for loans is again expressed in terms of desired changes in indebtedness to the banks. Thus, we have in general the situation shown in Figure 8. I,, is the total indebtedness to the bank (unpaid principal plus interest on loans) at the beginning of period t, and dL represents the change in indebtedness during t. Note that if Pk = Pg0, the desired change in indebtedness is zero, but that this does not mean that desired new loans in t are also zero. When P = Pg, desired The Market for Bank Loans This is the first market to boast a supply function that is independent of demand. The price of bank loans, Pg, is 1 + rbp for the public and 1 + rb, for the firms. The demand for loans is again expressed in terms of desired changes in indebtedness to the banks. Thus, we have in general the situation shown in Figure 8. L 1 is the total indebtedness to the bank (unpaid principal plus interest on loans) at the beginning of period t, and dL represents the change in indebtedness during t. Note that if Pk = Pk., the desired change in indebtedness is zero, but that this does not mean that desired new loans in t are also zero. When PQ 2., desired G1, P P., Gd) P5 C~d) _d $(dG IG, Fig. 7. Demand for government secrities loans in t are equal to ft [1 + rb(r)] L(r)dr/n, the amount of loan repayments in t. Thus, desired new loans are zero when Pg = P22 and the demand for loans is given by D(dk) in Figure 8. A graph like Figure 8 exists both for firms and for the public. At any point in time the aggregate quantity of loans the banks are willing to supply is given by the solution to Equation 2.96. The supply of loans to each sector is based on the aggregate figure. The discussion in the section on the banking sector can be shown graphically as in Figure 9. Here Q; and Qfe represent the initial amounts desired to be loaned to the public and the firms as given by Equations 2.97 and 2.98. In the situation drawn in Figure 9, excess supply exists in both loan markets and the actual amount of loans made will be QDe + QDf. In the next period both rates G7 d, m G I Fig. 7. Demand for government securities $(dC) G v Fig. 7. Demand for government securities $(dC) loans in t are equal to f6 {1 + r(r)] L(r)dr/n, the amount of loan repayments in t. Thus, desired new loans are zero when Pg= P2 and the demand for loans is given by D(d) in Figure 8. A graph like Figure 8 exists both for firms and for the public. At any point in time the aggregate quantity of loans the banks are willing to supply is given by the solution to Equation 2.96. The supply of loans to each sector is based on the aggregate figure. The discussion in the section on the banking sector can be shown graphically as in Figure 9. Here Q,Pj and Qri represent the initial amounts desired to be loaned to the public and the firms as given by Equations 2.97 and 2.98. In the situation drawn in Figure 9, excess supply exists in both loan markets and the actual amount of loans made will be QD, + QDf. In the next period both rates loans in t are equal to f6 [1 + rb(r)] L(r)dr/n, the amount of loan repayments in t. Thus, desired new loans are zero when P8 = Pq2 and the demand for loans is given by D(dk) in Figure 8. A graph lke Figure 8 exists both for firms and for the public. At any point in time the aggregate quantity of loans the banks are willing to supply is given by the solution to Equation 2.96. The supply of loans to each sector is based on the aggregate figure. The discussion in the section on the banking sector can be shown graphically as in Figure 9. Here QPw and Qfi represent the initial amounts desired to be loaned to the public and the firms as given by Equations 2.97 and 2.98. In the situation drawn in Figure 9, excess supply exists in both loan markets and the actual amount of loans made will be QD, + QDr. In the next period both rates  ++ L ++ L / I/I / J I, I I' I I' F L I, / II / I, I, I' -1  THE MODEL 57 will be reduced. An equilibrium situation is shown in Figure 10. Not only do rates fall, but the aggregate amount of desired loans by the banks is also reduced. Figure 11 is a graphical presentation of the adjustment occurring when there is excess demand in one market and excess supply in the other. Here QSff represents the final quantity of loans made to the firms and the two black arrows, ES, the final total excess supply of loans. Equilibrium in the loan market clearly requires QS, = QDr, QS0 =QD, and QD, + QDS QS,. Equilibrium is achieved through adjustments of the rates of interest with its impact both on quantity demanded and quantity supplied. QD, QS I I Id, I I I I I I D(d/)c, m D(d/) Fig. 10. Equilibrium in bank-loan market THE MODEL 57 will be reduced. An equilibrium situation is shown in Figure 10. Not only do rates fall, but the aggregate amount of desired loans by the banks is also reduced. Figure 11 is a graphical presentation of the adjustment occurring when there is excess demand in one market and excess supply in the other. Here QSff represents the final quantity of loans made to the firms and the two black arrows, ESe, the final total excess supply of loans. Equilibrium in the loan market clearly requires QS, =QDe, Sf = QDr, and QDn + QDs = Q'sr Equilibrium is achieved through adjustments of the rates of interest with its impact both on quantity demanded and quantity supplied. QD,=Qs, I | = 1 I THE MODEL 57 will be reduced. An equilibrium situation is shown in Figure 10. Not only do rates fall, but the aggregate amount of desired loans by the banks is also reduced. Figure 11 is a graphical presentation of the adjustment occurring when there is excess demand in one market and excess supply in the other. Here QSff represents the final quantity of loans made to the firms and the two black arrows, ESf, the final total excess supply of loans. Equilibrium in the loan market clearly requires QS0 = QD,, QSf = QDf, and QD, + QDs = QS,. Equilibrium is achieved through adjustments of the rates of interest with its impact both on quantity demanded and quantity supplied. | Q,=QS. I Iqo-qt QS' Id/ I I | IF I Fin O E i in nD(d/)F Fig. 10 Equilibrium in bank-loan market Fig. 10. Equilibrium in bank-loan market Qs . E , ES, QD,,, 9s,. Qs, p 4 9s1, 2s ES, ES, Os, QS' 4 es,, Es, ES, -QD, Qs5. 1 + ,, -- - D(d/), --- D(d/),. __ D(d/), -- D(d/),. QDF P I. 00,, ,, ye, Fig1. Excess demand and excess supply in bank-loan market Fig. 11. Excess demand and excess supply in bank-loan market Fig. 11. Excess demand and excess supply in bank-loan market  58 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM The Market for Intermediary Loans The analysis of the market for intermediary loans is identical to that given above for bank loans and will not be repeated. The Market for the Capital Good See the section on production, investment, and growth, and in particular Figure 2, for a discussion of the market for the capital good. The Market for Firms' Securities Debt instruments issued by the firms are held by the public, by inter- mediaries, and by banks. The price is again defined in an opportunity cost sense. Thus, P,, the price of firms' securities, is taken to be: 58 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM The Market for Intermediary Loans The analysis of the market for intermediary loans is identical to that given above for bank loans and will not be repeated. The Market for the Capital Good See the section on production, investment, and growth, and in particular Figure 2, for a discussion of the market for the capital good. The Market for Firms' Securities Debt instruments issued by the firms are held by the public, by inter- mediaries, and by banks. The price is again defined in an opportunity cost sense. Thus, Pf, the price of firms' securities, is taken to be: 58 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM The Market for Intermediary Loans The analysis of the market for intermediary loans is identical to that given above for bank loans and will not be repeated. The Market for the Capital Good See the section on production, investment, and growth, and in particular Figure 2, for a discussion of the market for the capital good. The Market for Firms' Securities Debt instruments issued by the firms are held by the public, by inter- mediaries, and by banks. The price is again defined in an opportunity cost sense. Thus, PC, the price of firms' securities, is taken to be: P = 1 + max Ir , r, - r for the public; PC = 1 + rg - r for the bank; PC = 1 + max r , r - r, for the intermediaries. (2.173) (2.174) (2.175) P, = 1 + max Ir, r, rI - rf for the public; PC = 1 + rg - r for the bank; P, = 1 + max Ir, rtf - r for the intermediaries. (2.173) (2.174) (2.175) PC = 1 + max Ir, ra, rI - r for the public; PC = 1 + rs - rf for the bank; PC = 1 + max I , r - r for the intermediaries. (2.173) (2.174) (2.175) The demands can be shown by Figure 12, which is completely analogous to Figure 7 (the demand for changes in holdings of government securi- ties). In each period, the supply of securities by the firms is given by the solution of Equation 2.40. The desired change in securities outstand- The demands can be shown by Figure 12, which is completely analogous to Figure 7 (the demand for changes in holdings of government securi- ties). In each period, the supply of securities by the firms is given by the solution of Equation 2.40. The desired change in securities outstand- The demands can be shown by Figure 12, which is completely analogous to Figure 7 (the demand for changes in holdings of government securi- ties). In each period, the supply of securities by the firms is given by the solution of Equation 2.40. The desired change in securities outstand- D(dF) ODrlF) OylE) B 1 e Fig. 12. Demand for firms' securities B, Fig. 12. Demand for firms' securities 11, _ __ a $ B' Fig. 12. Onward foe firme' securitirs  THE MODEL 59 ing can be represented by a vertical line. Thus, combining these flow demands and supplies we have, in the aggregate, Figure 13. A Note on Aggregate Demands Prices in the financial markets described have all been framed in terms of opportunity costs. This results in different prices for the same item in different sectors. For example, the price of government securities for the public is assumed to be 1 + max r, r - r for the public, but 1 + max I r, r1 - rg for the firms. If re > rt, different prices result. When aggregating sector demands, the price is assumed to be 1/1 + r, where r represents the rate on the item in question. This function does not contradict the sector prices since there is a one-to-one relationship between, for example, 1/1 + r and 1 + max r, r - r and 1 + max r rj - r,. THE MODEL 59 ing can be represented by a vertical line. Thus, combining these flow demands and supplies we have, in the aggregate, Figure 13. A Note on Aggregate Demands Prices in the financial markets described have all been framed in terms of opportunity costs. This results in different prices for the same item in different sectors. For example, the price of government securities for the public is assumed to be + max Ir, rI - rg for the public, but 1 + max Ire r - r, for the firms. If rf > r, different prices result. When aggregating sector demands, the price is assumed to be 1/1 + r, where r represents the rate on the item in question. This function does not contradict the sector prices since there is a one-to-one relationship between, for example, 1/1 + r, and 1 + max rI, rj -rs and 1 + max ren r - rn. THE MODEL 59 ing can be represented by a vertical line. Thus, combining these flow demands and supplies we have, in the aggregate, Figure 13. A Note on Aggregate Demands Prices in the financial markets described have all been framed in terms of opportunity costs. This results in different prices for the same item in different sectors. For example, the price of government securities for the public is assumed to be 1 + max Ir, rI - r for the public, but 1 + max It, ra - rg for the firms. If r, > r different prices result. When aggregating sector demands, the price is assumed to be 1/1 + r, where r represents the rate on the item in question. This function does not contradict the sector prices since there is a one-to-one relationship between, for example, 1/1 + rg and I + max r, r - rg and 1 + max irrI -r. S(dF) L-"-F S(dF) users, Fig. 13. Aggregate demand for firms' securities RELATIONS OF THE MODEL We reproduce simply the key relationships for each sector here for convenience.5 Production, Investment, and Growth 1. Aggregate production functions: Fig. 13. Aggregate demand for firms' securities RELATIONS OF THE MODEL We reproduce simply the key relationships for each sector here for convenience.' Production, Investment, and Growth 1. Aggregate production functions: Fig. 13. Aggregate demand for firms' securities RELATIONS OF THE MODEL We reproduce simply the key relationships for each sector heere for convenience.' Production, Investment, and Growth 1. Aggregate production functions: K = Xk(L,,Xk) (2.1) 5. See Literature Cited, items 3, 20, 22, 25, 29, 37, 42, 43, 47, 53, 55, and 57 for a more complete discussion of some of the demand and supply functions presented here. Xa= Xk(LkXkk) (2.1) Xfk = Xk(-.k Xk) (2.1) 5. See Literature Cited, items 3, 20, 22, 25, 29, 37, 42, 43, 47, 53, 55, and 57 for a more complete discussion of some of the demand and supply functions presented here. 5. See Literature Cited, items 3, 20, 22, 25, 29, 37, 42, 43, 47, 53, 55, and 57 for a more complete discussion of some of the demand and supply functions presented here.  60 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM =a XJLr, Xkd. 2. Transfomation functions: Xc= T(Xk) ki ta ki' 3. Full employment: X' (X',, X'0i) it a foI-rerployment outputt vector if x4,i= f<-4jIXk o (!0 4. Rate of groewth of the labor force: dXc ir)dt 5. Stock demand foe capital: D= ekpr, 0)' 6. Flow demand foe copital: dK =oR.( 7. Stock tupply of capitol: St= K,. S. Flow supply of capital: k= r)~Pk). 9. Balaoced growth: S-c kK +SXc XL = OX0 kK + 6- L U. 3K, Slc 3E, 5a (2.3) (2.5) (2.9) 2.10) 2.13) 2.14) 2.15) 60 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM X4 = X~JLc, Xoc). 2. Transformation fonctiont: X, T(Xk) X .i + X5i=. ki =ki" 3. Foll employment: X'= (Xc), Xk) it a-foll-employment ortput rector if i=F - X't, X'k >0. 4. Rate of growth of the labor force: x (dX dt 5. Stock demand for capital: 00 = Oo(pk, r, 0).( 6. Flow demand for capital: dK =nE. 7. Stock sapply of capital: Sct = K,. 0. Flow tupply of capital: k= tk(Pk). (2.3) (2.5) (2.9) 2.10) 2.13) 2.14) 2.15) 60 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Xa = X(L~, Xkd. 2. Transformation fncions: Xc = T(Xk) 3. Fall employment: X'= (Xc), Xki) it a fll-employment oatpur rector if X)=f7 a X- X2,X) 0.( 4. Rate of growth of the labor force: dX, dt 5. Stock demand foe capital: D= 00(Pk, r, 0).( 6. Flow demand foe capital: dE =nK. 7. Stock tapply of capital: tIkt = K,. 8. Flow tupply of capital: k= Sk(Pk). 9. Balanced growth: - kcE+ Xc A L= a kE+ axkXL. SWc SLc SE0 3Lc (2.5) (2.9) 2.15) 2.16) 2.30) 2.16) 9. Balanced growth: SXc k + SX XL = ak W+t XL. SKc 3Lc aK0 3Lk (2.30) 2.30)  THE MODEL 61 TEMDL6 THE MODEL 61 10. Supply of labor: - S Q(PQ, L). 11. Domood for labor: D9- MPkkP, + MP20PX0 + E. The Firms 1. Desired level of rerainedrearniogs: E,'=nK, Ol'rt + S[PctYt+ PkrXkt + OIra-s E= f0(K5) _ g(K,..1) 2. Desired leoelof finacing: 3. Demand for loans: L0b = a ('t- Qlm)+ br I4 2, DrE + A,-h. 4. Supply of securities: ES =bFr + Aarrf. 5. Desired change in retained earnings: AEF D f~rra 6. Desired disrributioo of retained earnings: DAt= (C D D 0,N (Ce, Drr. Trr, 0ra' Nr). (2.31) (2.34) (2.35) (2.52) (2.36) (2.43) (2.44) (2.38) (2.40) (2.55) (2.59) 10. Suppiy of labor: -R Sk(E0, OL. ItI. Demand for labor: D= ME05E5 + NIPcE Xc + E. The Firma 1. Desired levelof retained earnings: E tD= aKt + 9llrr-r + S[E5X0 +EPkaXkr] + P'nr-s Er = f3(I) - ll(K-.9. 2. Desired level of financing: 3. Denmand for loans: 1.0 r =aa ,- rY, +b, t41 45r a(rf,-'t r?+r0b F~ DrE + A 3q. 4. Sapply of secarities: F'a beF D + Auref. 5. Desired change in retained earings: AEr D a~a) l(n~ 6. Desired distribntion of retained earnings: DA? = (CeD, Des,, Ter Dee, Nr) (2.31) (2.34) (2.35) (2.52) (2.36) (2.43) (2.44) (2.38) (2.40) (2.33) (2.59) THE MODEL 10. Snpply of labor: - S (E2, L). 11. Demand for labor: DR= Mpicpk + MpirEXa + E. The Firma 1. Dedired level of retnined earnings: Ees = n K5 + lllnt + S[EX5t + PkXk] + OlIt ErDt= f3(145 - ll(r-t- 1) 2. Desired lesel of financing: EF5D= Ire g'nt-s 3. Demand for loans: 'an = a,(,st - ' t) + b, 141 It' a2(rf~t- rfnt)+ ba4 1.8 = fE{ + A~fe. 4. Snpply of secrities: Ers = beE? + A,,Fe. 3. Desiredchsange in retained earning: 6. Desired distribntion of retained earnings: DA' = (CrD DaT, ,Ne) (2.34) (2.33) (2.52) (2.36) (2.43) (2.44) (2.38) (2.40)  62 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 7. Desired level ofoash (corency) balanes: Cf8 = as[PreXo + PkXkt] = a, PX. 8. Desired level of demand deposits: D8 = d4(PX) + AFh. 9. Desired level of time deposits: ft = r/(PX) + Aj4,.( 10. Desired level of governmentseoeiies: Gf8t =(PX) +AsTf.( It. Desired level of deposits in intermediaries: ND =nf(PX) + AT.( 12. Contribution of firms to publics inome: Y55 = P\ + rgt f +e tTft +e .Nft + 4t + B, - A(Aft) - Zl-et - E,- Pkt~kot The Government 1. Tao reeoeipts: T=tIV.( 2. Governmentrspendintg: T =75G + PkXk, + PrXos.( 3. Supply and demand for governmentsecnurites: G Sm- eDs 2.61) 2.62) 2-63) 2.64) 2.65) 2.71) 2.72) 2.73) 2.78) 62 EQUILIBRtUM STUDY OF THE MONETARY MECHANISM 7. Desired level ofoash (crrency) baanes: C8= a.[PotXrt + PktXkt]( =05, PX. 8. Desired level of demand deposits: 08' = df(PX) + A611. 9. Desired level of time depositE: fT8 tf(PX) + A7Ff. 2.61) 2.62) 2.63) 62 EQUILIBRIUM STUDY OP THP MONETARY MECHANISM 7. Desied level of oath (currency) bolances: 10. Desied levelof goernmeoterities: o8' = gf(PX) + AF4. 11. Desired level of deposits in intermediaries: N8~ = e5(PX) + gf 12. Contribution of firms to poblics incms: Y55 =I'Xt r , +10 tteTft yetN0 +±l- + Bee - d(Aes) - 111t - IF,~ - PksXkft. The Government 1. Tan reoeipts: T = rY. 2. oveenment spending: T =r50 + PX,, + PoXas. 3. Supply anddemand fogoernmentseuities: 0 nO GD. (2.64) (2.65) (2.71) (2.72) (2.73) (2.78) CD5 = a.[P5X,, + PktXkt ft as PX. 8. Desired level of demand deposits: D' = dr(PX) + A~rh. 9. Desired level of rime depositE ft8 te(PX) + Aver. 10. Desired level of governm ent seonrities: ft = t(PX) + 5 r 11. Desired level of deposits in intermediaries: ft = .,/PX) + Aserr- 12. Conseibotion of fiems to public's inmome: Bft - d(Aes) - FI-ft - Z~f - kthrXoes The Government 1. Tan recoeipts: T = tY . 2. Government spending: T = hO + PoXke + PCX 9. 3. Supply and demandforegovenment secriie: 0 G Da (2.62) (2.64) (2.72) (2.73) (2.78)  THE MODEL 63 TEMDL6 THE MODEL 63 4. Reditonting: dm4-d (rd)' 5. Stock of currency: t 6. Rate on government sonorities: g get - 1 - g(CD I - G D 2). The Bonking Sector 1. Demond for time deposits: T= Tt mT,+ TS+T S. 2. The rate of interest on time deposits: rC = it, , + a(40 - T,-0 ~ ) 3. Demand fot demand deposits: D,= D D-D,+ S+Ds 4. tenet of tegai teserves: Rt= rt(Dt + Tt). 5. Desited cunerncy batances: Cb = y(D + T). 6. Desired level of govetrnment secutities: G,= p(D + T) + A, On. 7. Desired level of firms' secnrities: Fb = pA(D + T) + A, 1-b (2.79) (2.81) (2.83) (2.86) (2.88) (2.89) (2.90) (2.91) (2.92) 4. Rediscoonting: d - d (rd). 5. tockof cnerency: Ct iI 6. Rote on government securities: The Banking Sector 1. Demand for time deposits: T= T' nT't+ Ts +Tn 2. The tote of interest on time deposits: 3. Demand for demnnd deposits: 4. Level of legat reserves: Rt = rJDt + Tt. 5. Desired cnrrencypbannces: Cv = y(D + T). 6. Desiredtlevelofgovrnment securdties: Cv = (D +T) + AOF 7. Desired tenel of firms' securities: Fb = ja(D + T) + A, 17b. (2.79) (2.81) (2.83) (2.86) (2.88) (2.89) (2.90) (2.91) (2.92) 4. Redisconting: dn- d (ed). 5. Stock of currency: 6. Rate on government securities: rr=t-I - g(G D I - G D9 The Banking Sector 1. Demand for time deposits: Tt = T>--T8 + T +Tsn 2. The rate of interest on time deposits: t = tI +a(Ev- L P- L~). 3. Demand for demand deposits: D, =o no - S,+ts + Dnt. 4. Levet of legalrervs Rt=r(,+ Tt). S. Desired cnerency balances: Cb = y(D + T). 6. Desired level of government securities: G,=p(D + T) +A, 035 7. Desired tevet of fiems' secneities: Fb = 1,(D + T) + A, Js.- (2.81) (2.86) (2.88) (2.98) (2.91) (2.92)  64 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 8. Luau supply: =ss,(D~ +T5)+A it, (2.96) ES, EL +a 3(r5~ bp s f) (2.97) L I, t a] 3(ebpt - 'bfs). (2.98) 9. Demand for rediscousnting: I Pt '1bpt - d 64 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 8. Luau supply: = (D5 + T ) + Allrb (2.96) ES,=QEs+aI (bI PS 'brI (2.97) ft fft - a, 3(rbpt - 'set). (2.98) 9. Demand for sediscounting: t Pt r~bpt d 64 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 8. Loan sapply: E s (05 + Tt) + A I r5 E= OlEIS +a~~ - re) d-b-b -S + ) d d d= LDI, +Lrsb -(L5 res 1 e) - d, (2. 5bft - rdt d~d , LP~t ES (E~b b)Pt I rbpt - d (2 it5 - I5 - t 'Lft - - (2. 2.96) 2.97) 2.98) 101) 102) 103) errs - 5us I bpt - 'dt It Pt P, s'es - 'at 10. Actaal amounts lent each secto: = mL ~+ tb _ E~b) +dd E~min ILb, qb+ (-' L5+~ 11. Rates of interest on bank loans: 'bps bee-sI + ar(Lrs...Is - -t ) s+ r t 'b It - + af(iT,')) I E15t I) + bf(rsrs-s - rers...). 12. Contribution to public's incmme: +Y =Tm5'tPcl css (2.101) (2.102) (2.103) (2.106) (2.107) (2.108) (2.109) (2.122) esp5 - rdt dd Epp I, I, D5tbEr d. d d pe d t it(q Pt 'bft - 'dt 10. Actual amounts lent each sector: PtL + (Lfb L t) + deo 4tmIin -us-Ss-Ss+-usl d 11. Rates of interest an bank loans: e p(pt-l -a (Lpt-1) I - +) bfrbrs- I - 5ssC5 5)- 12. Contribation so public's income: (2.101) (2.1 02) (2.103) (2.106) (2.107) (2.108) (2.109) (2.1 22) 10. Actual amounts lees each sector: 4t minEp~ 1 ,b + (Ef~t -f d 4t ~mina 1Lqb Ub+.(Lb L)d4 11. Rates of inteest on bank loans: rbf5 = 'bt + a(rU... - )+ b Iess s-5e- s + 12. Contribution to public's income: Y " " ki' b~ (2.106) (2.107) (2.108) (2.109) (2.122)  THE MODEL 65 H ODL6 THE MODEL 65 The Intermsediuries 1. The supply of deposits: N=N5 -N0D 2. Supply of loans: L.S =1 N +A] 4T. PL,,s q1Ls +usrs(eses i4"=o i4J'- ass(,.ses- ses)- 3. Desired levelof goernmentseuiies: G = g.N + A5 . 4. Desred level ofdemaneddeposits: Du =d5lN + A,,0. 5. Desired leveloffirms' securities: Fn5 = b.N+A, 17 n 6. Desired currencybulussces: Cn = en N. 7. Actuluamountslent: [L5O155 1L.e+(LOO -n L.LfeI 8. Roses of interest: s555,=5+uf L t - b s)) (2.123) (2.125) (2.130) (2.131) (2.126) (2.127) (2.128) (2.129) (2.133) (2.134) (2.138) The Jsteremediaries 1. The supply of deposiss: N = N' _N0 2. Supply of louns: QS =QN + Al 4T. Et = 2L.S +uss (s.et -nt 4n= Q12L~n - a,=e -(nt rf) 3. Desiredlesvelof goernmentssecuiies: 4. Desired level ofdemanoddeposis: 5. Desired level of firms' secoeities: P5D = b, N+A 17 .. 6. Desired currencey baluncue 7. Actual amoounts lens: Lsemi I e+LfOn L~),E~ 8. Rules of interest: 'nt s I= ~ t +oa, (Lnt L4I") + (2.123) (2.125) (2.130) (2.131) (2.126) (2.127) (2.128) (2.129) (2.133) (2.134) (2.138) The Inermeediuies 1. The supply of deposiss: 2. Supply of loans: Eii ='N + Al 4C 4n= QlL~ - u, 8(rnp - rf) 3. Desied Inee of governmsentlsecurities: Qn'= g.N +A, -F.- 4. Desiredtlevelofdemand deposits; OnD =d0lN + A1,.- 5. Desired level of firms' secorities: P0D = bnN+ A,7Tn . 6. Desiredcuency blansces: C0 =e nN 7. Actualtaonsooslent: L, e in 10 , e+(L?0 Ce L,f =min JLf + (L" Lp"),LOn 8. Roses of inerest: rnf + = nf + ua(,7 -~t J4~) + bl(.t- rbft) (2.123) (2.125) (2.13 0) (2.131) (2.126) (21127) (2.128) (2.129) (2.13 3) (2.134) (2.138)  66 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM b2o(rne - enp) (2 rnt+ I e r + a2 (1 - L0~p- Ln). (2 9. Intermediaeies' contibution 10 public1 income: t st".- E0,Xk0. (2 The Public Sector 1. Desired lecel of demand deposits: .139) .140) .150) DPt= k1Yt + A2 J0. 2. Desired carrency balances: C05t= k2 Y + A2 f0, 3. Desired Intel of time deposits: T0, = k3Y1 + A240?0. 4. Demand foe the cotnsumption good: Cnt = cY, + A0 5r? 5. Desiked tlee of government sncurtiies: 001 = gY1 + A206?0. 6. Demand for bank leant: LO' =10ft A0?0 7. Demnadfoe intermsediary lons: op,= foYt + A0,?0r. 9- Demand foe firms' securities: (2.151) (2.152) (2.153) (2.154) (2.15 5) (2.156) (2.157) 66 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM nnel + a20(Unp - Lnp) + boo(euo - rbp) (2 f~ n 2I(n- L.Dp - L~1). (2 9. Iotermediaries' conteiboaion to pnblic's inome: Ya=o" - EceXknt- (2. The Public Sector 1. Desired level of demaod deposits: DPt= k1 Y + A20?0. (2. 2. Desiked currency balances: CP t= k2Yt +A2 3fp -(2. 3. Desired level of lime deposit: Tpt =k3Yt+ A2-fp-(2. 4. Demand forthe consamptionogood: Cnt = cYt + A0J0. (2. 5. Desiredlevel ofgoenmetsecuritis: Gp1= gY0 + A26'r (2. 6. Demand foe bank loans: =, tlY1 + A07?0.2 (2. 7. Demand for intermediary loams: Lpt P,+ A00?0r. (2. 8. Demand foe fitms' secaeities: .139) .140) 150) 151) 152) 153) 154) 155) 156) 157) 66 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 0,0,0+ t= 'np + aoo(L5 - LP at (2 9. Intermediaries' conteibation to public's income: 4 = ' - Pk~n'(2 The Public Sector 1. Desired leoot of demand deposits: Dpt= k1'?0 + A00?0. (2 2. Desieed curency'balancs: C00 = k0Y + A03?0. (2 3. Desired level of lime deposits: .139) .140) .150) .15 1) .15 2) T~ ~t+ A00?0. 4. Demand for Ihe consumption good: 5. Desired levelof goernment secuities: 000 gYt I A06?0.- 6. Demaod foe book boos: 7. Demaod foe inteemediary boos: 70 = t + 8 A p?0 8. Demand foe firms' secorities: (2.153) (2.154) (2.155) (2.156) (2.157)  THE MODEL F~t= ' fYt + A29rp. 9. Demand for deposits in intermediaries: Nt = neYt + A30 nrp 10. Gross income: yt grY + Ybt + Y~ + Yrt + rrtcet + entNpt + 4tp + t~t 11. Disposoble income: Y,= (I - t) Y5. 67 (2.15 8) (2.15 9) (2.160) (2.16 1) THE MODEL Pt = fpYt + AgP 9. Demand for deposits in intrmediaries: Net = neYt + A30Jr. 10. Gross incomr: ?J =Y t + Yr±+YiYft + rsopt+nrtNee+ rttTpt + rr5Pt. 11. Disposable icome: Yt= (1 - c) Y , 67 (2.158) (2.15 9) (2.160) (2.16 1) Err =fpYt + A29FP. 9. Demand for deposits in intermediaries: N~ = neYr + Aoo?. 10. Gross income: Y,= Y~t + Yrt sen +Yt + rg0GP,+ rntp + rttTet + rops. 11. Disposable income: yt= (1 - 1) ilt 67 (2.158) (2.159) (2.160) (2.16 1)  3. Solution with a Passive Government T HE EFFECTS of changes in the variables of the model on the stock of money, and vice versa, will be considered here, under the assumption that the government is essentially passive, that is, the govern- ment does not engage in active monetary or fiscal policy. The reserve requirement, r, is fixed at r*; the discount rate is fixed at r; and the government is a passive supplier-absorber of government securities. The solution to the model concentrates on two areas: the effects of changes in the variables in the model on the stock of money, and the effects of changes in the money stock on the variables of the model. In the first case the solution is designed to yield the following expressions: SM/Sr for all r, SM/SY, SM/SE for all P, SM/SX for all X, a total of thirteen expressions (M the dependent variable). In the second case, we consider SX/SM for all X, SY/SM, SE/SM for all P, and r/SM for all r, thirteen more expressions (M the independent variable). Due to the complexity of the model to be solved it is not in general true that, for example, SM/SEX = l/5Pc/SM. Thus, different methods of solution will be used in each case in order to avoid making such (possibly) erroneous assumptions. AN EXPRESSION FOR THE MONEY STOCK Time deposits and deposits in the intermediaries are not considered part of the stock of money. (These could be easily included in the analysis by simply adding T and N to Equation 3.1.) The stock of money in existence in period t, Mt, is thus simply the sum of all currency holdings and all demand deposits: 3. Solution with a Passive Government THE EFFECTS of changes in the variables of the model on the stock of money, and vice versa, will be considered here, under the assumption that the government is essentially passive, that is, the govern- ment does not engage in active monetary or fiscal policy. The reserve requirement, r, is fixed at r*; the discount rate is fixed at r; and the government is a passive supplier-absorber of government securities. The solution to the model concentrates on two areas: the effects of changes in the variables in the model on the stock of money, and the effects of changes in the money stock on the variables of the model. In the first case the solution is designed to yield the following expressions: SM/Sr for all r, SM/SY, SM/SP for all P, SM/SX for all X, a total of thirteen expressions (M the dependent variable). In the second case, we consider SX/SM for all X, SY/SM, SP/SM for all P, and Sr/SM for all r, thirteen more expressions (M the independent variable). Due to the complexity of the model to be solved it is not in general true that, for example, SM/SPX, = 1/SEc/M. Thus, different methods of solution will be used in each case in order to avoid making such (possibly) erroneous assumptions. AN EXPRESSION FOR THE MONEY STOCK Time deposits and deposits in the intermediaries are not considered part of the stock of money. (These could be easily included in the analysis by simply adding T and N to Equation 3.1.) The stock of money in existence in period t, Mt, is thus simply the sum of all currency holdings and all demand deposits: 3. Solution with a Passive Government T HE EFFECTS of changes in the variables of the model on the stock of money, and vice versa, will be considered here, under the assumption that the government is essentially passive, that is, the govern- ment does not engage in active monetary or fiscal policy. The reserve requirement, r, is fixed at r*; the discount rate is fixed at r*; and the government is a passive supplier-absorber of government securities. The solution to the model concentrates on two areas: the effects of changes in the variables in the model on the stock of money, and the effects of changes in the money stock on the variables of the model. In the first case the solution is designed to yield the following expressions: SM/Sr for all r, SM/SY, SM/SE for all P, SM/SX for all X, a total of thirteen expressions (M the dependent variable). In the second case, we consider SX/SM for all X, SY/SM, SP/SM for all P, and Sr/SM for all r, thirteen more expressions (M the independent variable). Due to the complexity of the model to be solved it is not in general true that, for example, SM/SEX, = 1/Sc/SM. Thus, different methods of solution will be used in each case in order to avoid making such (possibly) erroneous assumptions. AN EXPRESSION FOR THE MONEY STOCK Time deposits and deposits in the intermediaries are not considered part of the stock of money. (These could be easily included in the analysis by simply adding T and N to Equation 3.1.) The stock of money in existence in period t, Mt, is thus simply the sum of all currency holdings and all demand deposits: Me = Cpt + Cbt + Cat + Cft t + Dn + Dt. (3.1) Substituting the appropriate expressions from chapter 2 for each of the 68 Mt = Cpt + Cbt + Cnt + Cft + D + D + Dft. (3.1) Mt = Cpt + Cb, + Cn, + Crt + Dp + Dnt + Df. (3.1) Substituting the appropriate expressions from chapter 2 for each of the Substituting the appropriate expressions from chapter 2 for each of the  SOLUTION WITH A PASSIVE GOVERNMENT epesosin 3.1 and simplifying, wI obtain 69 SOLUTION WITH A PASSIVE GOVERNMENT exspressions in 3.1 and simplifying, me obtain 69 M = (a, + dfPX) + (dn + cn)N + (k, + k2)Y + ,y(O + T) + Tr(A2, + A23) + Tf6+ i5A50.' (3.2) (Thet subisceipt has been dropped in 3.2.) Sobstitosing slit enpressions foreNandOD+ Tfrom chapters2yieldsoan expression forMinstetms of PX (tlie saint of goods produced), Y (disposable income), slit various rtes of itres, and the parameters of the model. M = (a5 + df)(PX) + (do + cn)N + (k, + k0)Y+ 'Y(O+ T) +T-r(Aso + A23) +o4A6 TfAl 6' (3.2) (The I subsctipt has been dropped in 3.2.) Substituting the expeessons fortNandD+ T fom chper2yields an expression foreMinsterms of PX (the nalue of goods ptoduced), Y (disposable income), the notions totes of interet, and the patametters of slit model. M =PX(n5 + df+oefd. + ntc, + yd + ytf +ydonf) + Y(dp+ con,+k, + k2+ YdnI+,yk, +yks)+ r (n+ on + yjdn)A30 + (I + 'Y)A22 + A23 + 'YAaa] + Ff(n+ en + yda)At + (I + y)As + 'Y,7] + toJKt + Y)A5061. Using CI,. . ., C. foe slit pasametic tems in 3.4, me bone M = PX C5 + Y C0 + ipC3 + NfC0 + TnC5 (3.3) (3.4) M =PX(5 + d5+nold, + rc, + yd + Yt+ 3dn;) + Yod~ +enn, + to, + k2 + yd,,np +yk, +,yk3)+ Yfe[(dn + cn + ydn)A30 +(1 + Y)A'00 + A33 +'yA3]+ 'r (n+ co + ydn)Ag + (I + 'Y)A6 + -A1+ To[(t + y)A 16]. Using C, . . ., C, foe the pasometric terms in 3.4, me bone M =PX C1 +Y C2 + ? C3 +NFC4+ F.CS. (3.3) (3.4) SOLUTION WITH A PASSIVE GOVERNMENT 69 epesosin 3.1 and simplifying, me obtain M (a. + df)(PX) +(d. + c,)N +(k, +hs)Y + ,y(O +oT) + TP(A22 + A23) + NFA0 + FaAan6- (3.2) (The I subsctipt has been deopped in 3.2.) Sobstituting slt exptessions forsNandOD+ Tfom chaper2 yieldsnan expression forinserssof PX (st value of goods peodoced), Y (disposable income), st narions eases of intesess, and slit paameters of the model. M =PX(a. + df+ nfd + fcn + ydf+ ytf +y-dnnf) + Y(d5 + Co,+hIs +hk2+Tydnnp+ yk, +yk3) + A', [n+ c, + yd,)A30 + (I +y)A22 + A23 +yAlia + eTf[(d, + c,, + ydn)Ag + (1 + y)A6 + 'YAj + in[I+ 7)Ae]. (3.3) Using Ci,...,- C5 fot slit parammtic teems in 34A, sot bane M =PX C5 +Y C2 +T -C3 + NC0C +?5fC5.- (3.4) Tis expeession fot slit money stoch ploys a hey tote in slit solusion of she model. SOLUTION WITH M AS THE DEPENDENT VARIAnLE The solution in Ibis ease begins swith diffetentiation of Eqation 3.4. Tis yields she following eqations: Sm = P CX [P 5p +aP.+akX X0 k 1. Equastions 3.6-3.11 bate tot beet tepaodoced becausa Itoe fotlowia sequoence from Equastion 3.3. Tis expeession foe slit money stock plays a hey sole in slit solotion of slit model. SOLUTION WITH M AS THE DEPENDENT VARIABLE The solution in Ibis case begins mitli diffetentinsion of liqatsion 3.4. Tis yields slit following eqoations: - C, tX, + OP, + 0Xk+ 0PJ1+ C0 by + N Ca 2 C _ + C, bin0.) 5. Equations 3.6-3.11 hsve not toten teptoduced becase toe folow in sequence from Equation 3.5. This enpeession fat alit money teach plays a hey sale in slit soltsion of slit model. SOLUTION WITH M AS THE DEPENDENT VARIAnLE The solotion in Ibis cast begins wli diffetensiation of Equation 3.4. Tis yields slit following equasions: C0 +C5 +C0 -31 a0 (3.5)' Ste Ste Itt are. 1. Equaticons 3.6-3.11 haot ot tees reproduced because Itylo w n iseuece haom Equation 3.5.  z2 + + + It 1I C + C, I~ + As 6 CC + U 2 I Y~ & ~ 5: 55- 5< CC + U 'UK- 5< U 5< 12]5< C 5<  SOLUTION WITH A PASSIVE GOVERNMENT 71 amk (3.17) This system of thirteen equations contains the followinig unknowns: 1. .t y i,j (=I when ij) (56 unknowns); as. 2.- ay r ( 8unknowns); r SE 3.-T V P,r ( 4 unknowns); ar ax 4.- y X, r ( 4tuknowns); 5.- a VP (2 nknownts); Y "ax 7. a v r(28tnknownss); ay' as- S. VXE (~ 2unknowns); api ap. 11. r V , P(1 2unsknowns); 12. ax (~ 2 unknrowns); ax. SOLUTION WITH A PASSIVE GOVERNMENT 71 Sm ax0 .... (3.17) This systrem of thirtreen equations contains thr following unknowns: 1. .2 a vi. j(=I whernti j) (56 unknrownts); Sr. 2.-a Vtr 8 urnknownts); Sr 3.-a VP, r (4 unknowns); T ax 4.-T x, r (4 unnowns); Sr .-a V P (2 unknowns); aT' ax 6. - VX2 unkhnownts); ax' as 7-y-Vr 8Sunknowns); 6. ax VX P ( 4 unknowns); 9. -n- i 4j ( 2rsunknotwns); ax' 10. -VP ( 2 unknownts); as 11. - vr, P (16 usnknowns); 12. ai 5.j (2 nknorwns); ax. SOLUTION WITH A PASSIVE GOVERNMENT 71 SM .... (3.17) axk This system of thirteen rquations corrains thr following unknowns: 1 . a vi~j (=I whrn i'j) (56 unknowns); Sr i ax' 2.-T Vsr (8unknowns); Sr 3.- avP, r ( 4 unknrowns); Sr ax 4.-T Vx,r (4 nkowns); Sr 5. pv P ( 2 nkowns); ax' ax 6. - V x (2nkhnowns); Sr 7. V f' ( 8unknowns); ax- 9. a j ij ( 2 unknoswns); ap i ax' 10. a-VP ( 2 unknownss); SE 11.-T v r, (16 unknowns); 12. ai ij (2unnowns); ax  72 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 72 EQULIBRIU STUDYOF TH MONETRY MECANISM72 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 7 QIIRU TD FTEMNTR EHNS 72 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 13. .- Y F, X ax So 15. -Y , X a6x (4 unknowns); 2onknown); (16 unknowns). 13. aYVP, X ax aY 14. 6- V X ax 15-YX VI, X 4nknowns); (2 unknowns); (16 oarhoow-Is 13. Yi'V , X 14. aY X ax ax ( 4 unknowns); ( 2 unknowns); (16 nnknnonso Expressing the unknowns in (1) above in motixn (nt-n we hove 0000 I.n I 'on rbln 1op, 0,1,0 0rr, ,It rgo 0000 1 toos rbp0 rnf 0opI p101 rgb1 1n01 'too 1 rpbt 1,150 1npbl 0on0 p 1gbnbp r001 1000 p 1 0nlbp 1npbp where r** = r/bi For exmlrfp= "fan.The series of I's down rho prinipal diogonol 0010 the rij. Below 010 the relation, desorihing how Ike 0011001 roles of interest are asoumed 10 chonge over rime: Expressing she unknown, in (1) ahovre in matrix form we hove L y rn In, Ief 05,pf 00Ir 0rpf 1n 00Or 01 'tn - 0nf 'bn 0rf nonn 1100 0100 000,0 't~ 00100 0opnl 1 0001,1 1100 'gno rnnp stnp 00110 1bn 1,10 mono, where oj = Sr/Sri For enomple, 00101 arbr/arno. The series of 101 down rhe prinoipal diagonol are the 1i. Below are rho rolatioso descrihing how rho v00i001 roles of inleetI ore assumed to change over rime: Expressing thr unknowns in (1) above in marin form we hove [1 1n1 101 100 0001 0pf 00010 0,pf 'n 1g I It Ott, 001n 0nfn 0001 0101 110 0001 000 1 000 00010 00100 I nb n 1001'gn 0001 0000 001,0 0000 lop I whome 06 arSo/Srj. For example, 00100 = 5o00/So00. The ser-ies of I's down rho prinoipol diogonal are trho . Below aro the relooions desoribing how the narions 00001 of interest 010 assumedtoochaneovoerime: In, 00 .f-+ arnq(LS.f- o - 405) + j (r'ns.. 1 - 0010...) ,apt = Pt-O 1 1a20 ,( LD j +b2 (0,pt-. - '0p0-1) 000 e01t- + a( - L - D 000d (2.138) (2.139) (2.140) (2.86) ,.ft0,.0 I0 a,10(L00.t 1 - L80') + b 00000010000 I1 av20(Lnpt- I -LDt-1+ o (o'oo.. - rnpt- o) 0n =00t +afL5Q _-P Y- L% itoIttoI + 1' a( ,-Lp se m0=m010(L0 00-Lf) (2-138) (2.139) (2.140) (2.86) sf1 =00f-1, + 01 0(L,10 - I 4000) + hb,9 (in0It 1 - r t -Io) 000000 =o005.. + a20(onpt..- Lf D0t- 1) +0 (ooos... - 'bOo-fl) ,et o1,t +oa(S- L D-4L' It = It -I+ a(!1nm-L P -Lne) (2.139) (2.140) (2.86)  SOLUTION WITH A PASSIVE GOVERNMENT 73 l'bpt = 'bt + ptE0-. I - LUPI.. I) +b ('bpt-~I - 01100-1) (2.108) toll = Gbt + at (140 I - E01..1 )+Ib SOLUTION WITH A PASSIVE GOVERNMENT 73 001= 011+nE1 t- 10-) +b (too.. 1 - rno- 1) 0o =1ft -I + f(Bfst - - BDJ r= r.,..I - g(G5s 1C5GD ) (2.109) (3.18) 000t = bt- I + a f(E01 - I) + b (roI. -I - I., I) 01 = t55..1 - g(G1-1I - G0D.1). (2.108) (2.109) (3.18) (3.19) SOLUTION WITH A PASSIVE GOVERNMENT 73 rbt= rp-I+ a. PtS SO b (loots -,1011.1) (2.108) rb01 = 0001..I + )t ~ -E _ + b, (0001...1 - Gof- I) r =t tf + f(80 1.. - Bo D_) g= g -I-g(G1-I-G ) I GD (2.109) (3.18) (3.19) (3.19) Diffooontiatin of these reltions withi respect to the r's 0000011 nhat tho teems in ohe "interest-inlteraction" matrix depend on the effects of changes in the e's otn tho quantity demanded and quantity nupplied of 10001 ood of films' securities; the qoantity demaondod of government scrte;on institutional linkages 8010000ene rous ra001 (snob no ho- IweRn he ratesoharged by different 1000001 00 loons 1o the public ond theoratoshaogod by tho bonks on loans to tho 0011000 1000001); and on theosensitivity of 00001 00 deposits to either 00 exc011 supply nor demand foe 10001 10 rthe previous period (1100 sizes of 005, 000, 001,0 a, , etc.). Difforontiation of tis system would yield a system of fifty-six equntions in the fifty-sin interest-intenaction looms and 8Y/Oo V r, OX/to V r, X and aP/at V r, P. Differentintion with respect In Y wvill yield expees- sions foe the &r/WY Multiplying this remult by 8Y/OP V P will yield exprersions foe Oc/OP V P, while multiplying the oeiginal result by tV/OX V X gives 81/OX V r, X. This wilt be discusmed hot. The expremsions foe OV/Oe, OV/O, and OV/OX V r con be obtained by differentiating the expresions foe Y, Y =PX +7r b+ 7n +oG + e1T0 + r0nN0 + rIfte with respect In each of the e's, P's, and X's. The expressions foe OP/Oe, OX/Or, OX/OV, OP/OV, nnd 0Pj/0P con be obtained from the imsplicis supply and demand functions foe X, and Xk. These ate Differentiation of those relntons with respect In the o's reveals that the looms in the "interest-interaction" morx depend on the effects of changes in she e's on the qoantity demanded and quantity supplied of loans and of Germs' secunrities; the quantity demsnded of government scrte;on institutional linkages between varions totes (snob as be- tween ohe rates chare d by different sectoos on buans to the pablic and the totes chaeged by the banks 00 10001 to the various sectors); and on rhe sensitivity of roles on deposits In either on excess supply or demand for loans in the previous period (the sines ofn ,00a0,001l a, 00, etc.). Differentiation of tis system wonldl yield a system of fifty-six equntions in the ffy-six interest-interacdion rerms and OV/Or V r, OX/Or V r, X, andO3P/Or eP.Oifferentiatioewithrespecs to Ywdllyieldxprs- sions for the Or/OY. Mulhiplying this result by OY/OP V P will yield expresions for Or/OP V P, while multiplying the original 00108t by OV/OX V X gives Or/OX V r, X. This will be discussed later. The eupesins for OY/Or, OY/OP?, and OV/OX V r can be obtained by differentiating rho eupressions foe Y, Y =PX +eb+e7n+ rG+ rT +1 rN +rfBf- with respect to each of the t'sP's, and X's. The expressions for OP/Oe, OX/Or, OX/OY, OP/OY, and OPi/OP. con be obtainedfromrthe implicitsupply and demandfuctions forXc and Xk Those ore Differentiation 0f these rehlions with respect ro the r's reveals that the teems in the "interest-interaction" matrix depend on the effects of changes in the e's on the qunoty demanded and quantity snpplied of loans and of firms' securitios; the quantity domanded of govornwent scrte;on institotional linhages between various rates (such as be- tween the rares charged by different sectors on loons tn the pnblic and ohe erates char god by the bankos n loans to the various sectors); and on the sensitivity of rates on deposits In either an excess snpply or demand foe loans in the previous period (rho sines of al9, a20, aos' 0,00, oe.). Differentintion of tis system would yield a systom of fify-six equations in she ffy-six interest-intraction teems and OY/Or V r, OX/Or V r, X, and OP/Or V oP. Differentiation withrespectoto Ywill yield expres- dions for the Oe/OV. Multiplying this result by OV/OP V P will yield expressions for Or/OP V P, while multiplying rho original result by OV/OX V X gives Or/OX V r, X. This will be discusmed inter. The oxpressions foe OY/Or, OV/OP, and OY/OX Vtr can be obtained by differentiating the expresions for V, = PX +n0b+e0T+ 1G + rT0 +trNP +vrfB~ - withorespect to each cof she e's, P's, and X's. The expressions foe OP/Or, OX/Or, OX/OY, OP/OY, and OPi/OP can ho obtained from the implicit supply and demand functions foe X. and X., These se  74 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 74 EQULIBRIU STUDYOF TH MONETRY MECANISM74 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 7 QIIRU TD FTEMNTR EHNS 74 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Sk = I(Pk I P.,F) Ot-dk(PE, P.,F) S1 =IJPtl,TBY) Dc =dc( Pe , Y). (3.20) (3.21) (3.22) (3.23) = d.(P, P0?' ) 0 =d,(Pk IP,BTY)' (3.20) (3.21) (3.22) (3.23) Sk =kP I P I T) Dn dk(Pk, Pa, I) Sc =Jk 0Pt,' 0, Y) 00=J~k O30P,TY)' (3.22) (3.23) The general technique is In differentciate bosh the supply and demand equationss foe one good wish respet t0 the ets (01 Y) and then impose the equiibrtium conditiou shot S. = D.. Fr exttmple, difforentiathng 3.20 and 3.21 with eespect to qf yieldt aDk ask P, , ad~, 01 +daf -arl ~ ~ k f P rf T ' Ac equilibelum 0S- de1 = aD E de1, so chat ac, lPf aPk 8E, 0aE Te)de The geuneral technique is so differentiase bosh she supply aod demaod equsations for one good with r11p1c1 to the r's (0r Y0) and then impose the equilibrium condition shat S. = D.. Foe extaeple, differeotiotiog 3.20 and 3.21 Rich respect to if yi1141 3k ad t k + d 0 p d - At equilibeium -.- dee = Dkdrf, so shut (850ap OP Isk lPc Ilak 8?f ) de apt orf OP0 lf -61 l The general techniqae is so diffeentsiate both the supply and demand equations for one good with respect to she r's (or Y) and then hmpose she equilbeium condition thus Sn = Du. Forexrample, diffeentiasing 3.20 aud 3.21 wish respnct In ef Fields lot = as, Opk + ask bpL.+±. T2 lot O -a-i Id P 14f 0 As eqailihrium 11kdr = Dkdcc, so that (150 - ap 0 L0 apt art OP0 8cf Of lirf adk 3P, + 84k OF, + adk 3? da Canceling de1 feom both sides, and simplifying, Ia Odka k 3r ak lat 1r1 P - P r It1f ask a dk apt ap0 (3.24) (3,25) (14t Opt + adk IP, + adt -)df Cuncehing 4cf from bash sides, aud simplifying, let ad a k - r ad, ap, 3 p0 (3.24) 3d p dk 1P0 + ad, 0? (1T -2t- + . -' ) de1. Canceling de1 feom botb sides, and simplifying, 1rk Is - ld Opt aPk (3.24) ask atk a y Ilk) a=) -. (3.25) (3.25) prcdsprocedure, Differoiatng .22and .23wit erpec tor~an folowng he ameDifferentiasing 3.22 and 3.23 with eespect so It and following the same Differeentiating 3.22 tnd 3.23 wish respect to It cod following the same procedure,  SOLUTION WITH A PASSIVE GOVERNMENT 75SOUINWTAPASVGOENET 7SLTON IHAPAIEGVRMNT 5 SOLUTION WITH A PASSIVE GOVERNMENT 75 SOLUTION WITH A PASSIVE GOVERNMENT 75 ____ ( ad, se~ aDr ad a6e1 aae ad, ape ape a___k at) apk (ae als) ar (ad a SkE SE- --6' aPk aPT)+' al, ar -a'= (3.2 6) (3.26) a (Sdea ala) +ar ( -d _ as ape = a3t, aP- abk al, r a air as, _ ada -aPC aE0- (3.26) aef ala d saF a pa Equatins 3.25 and 3.26 foam a system af Iwo eqations that can be salved tiamaltaneaatly far the anknawat apk/at1 aad aE,/aet. Repealing thit procedure will yield aP/ac V P, a. Eqaations 3.25 and 3.26 farm a system af twa eqaations that can be solved simaltaneoasly fae the anknawns Sa/Set and aP,/SeL. Repeatiag dti peaceduee will yield aP/ar V P, I. Eqations 3.25 aed 3.26 farm a system af two eqations that can be talved simnaltaneously foe the anknownt apk/arf and aPa/aet. Repealing this peocedure will yield aP/ac V P, e. Fi. 14, ChaneinP andX Fig. 14. Chaanesin Pand X The same system (3.20 to 3.23) it used In solve for ax/ar. We 11111 with an eqailibeiumn situation where s, and d, (the K and C subscripts) bane been omitted sinae the technique it the lame foe bath. See Eigate 14. We then imagine a change in one of the elements oferthat resltstin a shift in both the demand and supply carvet to D. and S..This results in ahanges in bnth X and P. The exptetssion fne SE/ax was develnped in the lat paragraph. The eqailibrium ahange in X it obtained in thit x, = ln(Et ,a) = Sk(EsI ,, r) St St St The tame system (3.2a to 3.23) is used to tolve foe ax/St. We stat with an eqailibtium situation where s, and d, (the K and C subscripts) have been omitted tinge the technique it the same fot both. See Figue 14. We then imagine a change in one of the elements of r that results in a shift in both the demand and supply curves to D2 and S2. Thistresults in changes in both X and E. The exptession fot SE/ax wat developed in the latt paragraph. The equilibrium cange in X is obtained in thit X, = d,(P, + - dt,EaC + a dr, fL....dt) St St 3r Fig. 14. fChanges inPand X The same system (3.20 to 3.23) it nted to tolve foe ax/Se. We stat with an eqnilibrium situationnwheretsand d, (the K and C subsceipts) have been omitted since the techniqae it the tame fat both. See Figue 14. We then imaginetashage inaone of the elements ofthatesltsin a shift in both the demand and sapply cnrves to 00 and S2. This results in changes in both X and E. The expestion foe SE/ax was developed in the latt paragraph. The equilibriam change in X it obtained in thit XI= dk(E1,I P' f0 = Sk(EI , r ) SE Se asr X2 = d(P,+.. dt,EP,+ tde F +- de) St 3r St  76 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM as' as' ar s (P + dr P + dr, if + -dr). The change in X, given the change in r, is simply X2 - X1 or as' as' ar d( s P oc f asi dr as 'r r + -dr) -- dcPs,EPc,S) AX SP SP ar aP SP Sf _= dk ( -dr, 4 dr, -dr) = s(- dr, k dr, -dr). Ar Se Se Sr r ar as (3.27) In the limit as ar - 0, AX/Ar - aX/ar which is still given by either expression in 3.27. Repetition of this process yields aX/ar for all X and r. The expressions for SP/SY and 3X/aY are obtained in an analogous manner which will not be repeated here. The same technique is also used to obtain expressions for SP/5P1. The expressions for aXc/aXc and SXc/SXc can be obtained directly from the transformation function. These relations obtained by differenti- ating the transformation function hold only in a situation of full employment. At less than full employment, these rates of change may approach + -o if the economy begins to utilize previously unused capital and/or labor. The expressions for SX5/SP are obtained in a manner analogous to the above by differentiation of the supply and demand functions and the imposition of the equilibrium condition that quantity supplied equals quantity demanded. Before commenting further on the solution with M as the dependent variable, we will consider the solution with M as the independent variable because of the close similarity of the technique in this case and that used above. SOLUTION WITH M AS THE INDEPENDENT VARIABLE The rates of change we wish to develop here measure the effects of changes in the stock of money on the key variables in the model. Thus, we are interested in obtaining expressions for SXk/SM, aXc/SM, SY/aM, SaE/SM, SP/SM, and Sr/SM V r. As indicated earlier, it is not sufficient to assume that these rates of change are simply the inverses of those obtained earlier due to the complexity of the model. They may be, but in general it cannot be expected that they will be. 76 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM sk C, + Tdr, PE + c dr, y + --dr). The change in X, given the change in r, is simply X2 - X1 or SE SP Sf d,(E5 + dr, P, + dr, + dr) - d i(PiPf) AX SP SE Si SP SE Sf = dk ( --dr, e dr, -dr) = sk(- dr, dr, -dr). Ar orI or or or or or (3.27) In the limit as Ar - 0, AX/Ar -s aX/Sr which is still given by either expression in 3.27. Repetition of this process yields SX/Sr for all X and r. The expressions for SP/Y and 3X/SY are obtained in an analogous manner which will not be repeated here. The same technique is also used to obtain expressions for SP/SP' . The expressions for SXc/SXa and SXc/SXc can be obtained directly from the transformation function. These relations obtained by differenti- ating the transformation function hold only in a situation of full employment. At less than full employment, these rates of change may approach + as if the economy begins to utilize previously unused capital and/or labor. The expressions for 5X5/5P are obtained in a manner analogous to the above by differentiation of the supply and demand functions and the imposition of the equilibrium condition that quantity supplied equals quantity demanded. Before commenting further on the solution with M as the dependent variable, we will consider the solution with M as the independent variable because of the close similarity of the technique in this case and that used above. SOLUTION WITH M AS THE INDEPENDENT VARIABLE The rates of change we wish to develop here measure the effects of changes in the stock of money on the key variables in the model. Thus, we are interested in obtaining expressions for 5X/SM, SXa/SM, SY/aM, SPe/SM, 5P/SM, and Sr/SM V r. As indicated earlier, it is not sufficient to assume that these rates of change are simply the inverses of those obtained earlier due to the complexity of the model. They may be, but in general it cannot be expected that they will be. 76 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM sk(, + dr, P. + a dr, r + -dr). r o6r or The change in X, given the change in r, is simply X2 - X, or dE(P + -dr, Pc + adr, + dr) - d(EI scIT) AX SP SP Sf SP SE ST = dk ( -dr, c dr, -dr) = sk(- dr, dr, -dr). Ar Se Se Se Sr Sr Se (3.27) In the limit as Ar + 0, AX/Ar - SX/Sr which is still given by either expression in 3.27. Repetition of this process yields SX/Sr for all X and r. The expressions for SP/Y and SX/SY are obtained in an analogous manner which will not be repeated here. The same technique is also used to obtain expressions for SPi/SPf The expressions for SXc/SX and SXc/SXc can be obtained directly from the transformation function. These relations obtained by differenti- ating the transformation function hold only in a situation of full employment. At less than full employment, these rates of change may approach + - if the economy begins to utilize previously unused capital and/or labor. The expressions for 5X/P are obtained in a manner analogous to the above by differentiation of the supply and demand functions and the imposition of the equilibrium condition that quantity supplied equals quantity demanded. Before commenting further on the solution with M as the dependent variable, we will consider the solution with M as the independent variable because of the close similarity of the technique in this case and that used above. SOLUTION WITH M AS THE INDEPENDENT VARIABLE The rates of change we wish to develop here measure the effects of changes in the stock of money on the key variables in the model. Thus, we are interested in obtaining expressions for SXk/SM, SXc/1M, SY/SM, 5E5/SM, BPk/SM, and Sr/SM V r. As indicated earlier, it is not sufficient to assume that these rates of change are simply the inverses of those obtained earlier due to the complexity of the model. They may be, but in general it cannot be expected that they will be.  SOLUTION WITH A PASSIVE GOVERNMENT 77 To obtain expressions for aP/3M and aX/OM, the implicit supply and demand functions 3.20 to 3.23 are again used. Differentiation of 3.20 and 3.21 with respect to M yields SOLUTION WITH A PASSIVE GOVERNMENT 77 To obtain expressions for P/bM and OX/tM, the implicit supply and demand functions 3.20 to 3.23 are again used. Differentiation of 3.20 and 3.21 with respect to M yields SOLUTION WITH A PASSIVE GOVERNMENT 77 To obtain expressions for OP/tM and OX/OM, the implicit supply and demand functions 3.20 to 3.23 are again used. Differentiation of 3.20 and 3.21 with respect to M yields S ask + a5k ,OP + ask of liM II ~P, +M tE OM Of OM -aD ady, aP ad, amP adv of k - Odk aP + ad c OP0 + ,dk rr OM P, OM OP, OM a OM (3.28) (3.29) +MO0O OP0 OM ty OM OM ak p + OM Oa + a0 ON -dM kP SM Pc r T7M (3.28) (3.29) OS ask spk t a OM Of k +' P Pk M M rf M (3.28) - + c + . (3.29) OM OPk OM OPc OM 6= aM Again, for equilibrium, aSk/M dM = 0D/OM dM so that, by equating 3.28 and 3.29 and simplifying, we obtain Again, for equilibrium, 0S/OM dM = 3D/BM dM so that, by equating 3.28 and 3.29 and simplifying, we obtain Again, for equilibrium, 0S /3M dM = OD/OM dM so that, by equating 3.28 and 3.29 and simplifying, we obtain a aP adk ask af ad, ask OP- O-d - - + 0 -c -M O( -d - aPc ta Nc lic OM 00 0 atk + 0dk OP0 0Pk (3.30) Oe ad, oP0 - OM IOe0 OM - V 05) + 6--(Odfc - 05=, as +f ad ap OM O 0 Os d0 (3.30) 3e ad as Of (Odk P0 _ OM OP0 OPa M 0 0 M as d as0 0F0 (3.30) Proceeding in the same manner we obtain the expression for 0Pc/OM. This system can then be solved for OPa/aM and OPk/aM in terms of the parameters of the supply and demand functions and 00/aM. The expressions for OX/OM we obtained in the same manner in which those for OX/ar were obtained in the section on solution with M as the dependent variable. Thus, Proceeding in the same manner we obtain the expression for tPc/OM. This system can then be solved for 0Pe/OM and 0Pk/OM in terms of the parameters of the supply and demand functions and 00/OM. The expressions for OX/ON we obtained in the same manner in which those for OX/tr were obtained in the section on solution with M as the dependent variable. Thus, OX0 dk (Pc dM, tcdM, aM dM) Sk tdcdM Fr dN) am 'ON 'ON (3.31) 00 d ( k dM, tcdM, aM dM) 0Pk OP Of Sk (dM, dM' 3MdM) (3.31) Proceeding in the same manner we obtain the expression for 0P0/OM. This system can then be solved for 0Pc/OM and 0Pk/OM in terms of the parameters of the supply and demand functions and a/ON. The expressions for OX/OM we obtained in the same manner in which those for aX/r were obtained in the section on solution with M as the dependent variable. Thus, OX P OP 00 = d ( dM, cdM, dM) OP0 P 00(3.31) = sk ( ddM, dM, dM) and similarly for 0Xc/OM. The expressions for er/OM are obtained from the relations in the section on solution with M as the dependent variable, describing the determination of various rates of interest. Each of these relations is differentiated with respect to M. In general, the expressions for ar/OM depend upon the effects of changes in the money stock on the demand and supply of loans, firms' securities, and government securities. These effects are, in turn, primarily dependent and similarly for 0X0/aM. The expressions for ar/aM are obtained from the relations in the section on solution with M as the dependent variable, describing the determination of various rates of interest. Each of these relations is differentiated with respect to M. In general, the expressions for r/OM depend upon the effects of changes in the money stock on the demand and supply of loans, firms' securities, and government securities. These effects are, in turn, primarily dependent and similarly for X,/OM. The expressions for ar/OM are obtained from the relations in the section on solution with M as the dependent variable, describing the determination of various rates of interest. Each of these relations is differentiated with respect to M. In general, the expressions for r/OM depend upon the effects of changes in the money stock on the demand and supply of loans, firms' securities, and government securities. These effects are, in turn, primarily dependent  78 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM upon thte influences of changes in the money stock on the valiout rates of interest. This procedue yields a system of eight relatione io Se/SM which coo Ihen be solved simnltaneonsly, yielding solutions in 111r01 of SY/SM, ax/SM, otd aP/aM. The enpeession for SY/SM is obtained by differentiating tbe enpression for Y, Y =PX + ib + 7n,+1n0+ e1T1 +1 rN +eerlef - rbfL51 'nfLnfe with resect to M, yielding by axc ap ax, SF, Sef +F Pk+ + x o+ + am M am S IMF SM I m- a- -m- V .+ Se1Bf + fPe Se5 - 5h+ e b am am am1W1a 78 EQUILIBRIUM STUDY HE THE MONETARY MECHANISM upon the influencee of chongee in the money stock on the varioue rotes of interest. This procedure yieldt a system of eight relations in Se/SM which can then be solved simultaoneouely, yielding eoltions in teems of SY/aM, ax/SM, and SF/SM. The enpreseion foe SY/SM is ohtained by diffeeeneiating ehe expreesion foe Y, Y =FX +ib +e7r.+ e1 +e1,T1 + e1N1 + eree, - wilE respect to M, yielding by = ax, ap, ax5 ap Snb ia ...m-3p + 1mxc + am 0P + -a-x0 + +m a'n r -GR+ rgGp+ar 1T + Lr1t+ 1--N, + a NP..r. + - "Bp- +a-P r e SM ( rb b +ab b Sm e m SM m SMa 78 EQUILIBRIUM STUDY HE THE MONETARY MECHANISM upon the inluences of changes in the money seech on the various eates of interese. This proceduee yields a system of eight relations in Se/SM which con then be eolvnd eimnultaneously, yielding solntins in 111101 of SY/aM, ax/SM, and SF/SM. The enpesion foe SY/SM ie obtained by differentiating the enpeession fmr Y, Y =FPx+vb + n +e1OG+e1Tp +erNp + eeB1p - rb~f- en1Une, with respect to M, yielding by axc SF ax0 ap0 Se~b SaM- Sm 5M S xI SM am- S,+aM x5 S -M- "'- + -G- + Se Gp5e' +31 Spr anN SbN Sef 5811 Se~f 5L1 S1ne SI.5+ f (3.32) Snr = by al1. ay Sin by 3L Sy wheree - f( am ) I W=6fl(- - -) S 3(M - M) Smp ' (aM, a 5m-) Sm 'Ym a) SL ap a L Si LSa SM~ f= f1, SMT nmI Une+ ahi'ene) whee am f(S Sm ,-n), Si 5n (3.32) SM- Lne + _ n Inn wher f(S aM - 5 am -S amM 'S am am'SM (3.32) by SC Si Ij Yw-ama 2iif.i VSF by4 D ),and amn TM an) 5L5m Smia SM f8( SM' SM SM. SOP S ay ST by Si SN1 by Si am1 f6(Sby ScM), ~f f,(SMx aLm ), and SC11 aPx SC5' Si Sm f8( Sm- SM'1 SaM  SOLUTION WITH A PASSIVE GOVERNMENT 79 THE "SOLUTION" The preceding two sections together yield a system of simultaneous equations which could be solved yielding expressions for 3 - /aM and 3M/a- (where - represents the variables of interest) in terms of the parameters (the elements of AS, etc.) alone. No attempt has been made to push the solution to this level. The size and complexity of the resulting expressions would obscure rather than illuminate their eco- nomic meaning and significance. Consequently, we will continue to express these relations in terms of partial derivatives, indicating when necessary what variables they depend on. This procedure increases the ease with which the results can be interpreted. The key results from the previous sections are reproduced here. aM = PS aX 3P, 3X0 C [ X + - P5+ Xk+ Pk] + at ar r c r 3r ar, SOLUTION WITH A PASSIVE GOVERNMENT 79 THE "SOLUTION" The preceding two sections together yield a system of simultaneous equations which could be solved yielding expressions for a - /3M and 3M/a- (where - represents the variables of interest) in terms of the parameters (the elements of A,, etc.) alone. No attempt has been made to push the solution to this level. The size and complexity of the resulting expressions would obscure rather than illuminate their eco- nomic meaning and significance. Consequently, we will continue to express these relations in terms of partial derivatives, indicating when necessary what variables they depend on. This procedure increases the ease with which the results can be interpreted. The key results from the previous sections are reproduced here. aM 3P aX 34, aXk -C[ C XC+ 'X Pe+ X0+ P ]+ ate I ar at ace ar SOLUTION WITH A PASSIVE GOVERNMENT 79 THE "SOLUTION" The preceding two sections together yield a system of simultaneous equations which could be solved yielding expressions for a - /3M and 3M/a- (where - represents the variables of interest) in terms of the parameters (the elements of A, etc.) alone. No attempt has been made to push the solution to this level. The size and complexity of the resulting expressions would obscure rather than illuminate their eco- nomic meaning and significance. Consequently, we will continue to express these relations in terms of partial derivatives, indicating when necessary what variables they depend on. This procedure increases the ease with which the results can be interpreted. The key results from the previous sections are reproduced here. aM 3P aX 3Pg 3X0 -C[ ' X,+ C P + X0+ P0]+ are ar ar ar 3rt aY 3F y a? +C arC -P-.+C n (3.5) 23tc ace 3te 3te 3M _ P tX 3k ax -Cs[ c X e c P+ X + P og Yo k g ar ar Ca2~p' +C C +Ca- +C " (3.6) r 4 . r 5rg aY 34 3? 3r C2 +C- -C+ C a r 3ae ar 4 r 3M 3P aX apk 3Xk - C3 X + cp k X + 3 P + ar3 arg ar ± rg a rg C2 + C3 + + C45 + + C, n r P rg a rg a rg 3m 3P ax r k axk - C [ S X + t P+ Xk+ E P + ar r, at.r art. art , Y aF af n C2 - + C3 3 +C4 --P-+Cs C C art rt art art (3.5) (3.6) (3.7) aY f 3? a - +Ce i +C~ ..+Ct.+ 2 e r 3, a C 4 +, Crg 3M = P aX 3P aX -C5[ ' X+ P+ k Xk++-x P 0]+ r ar. or, ar. orn k C2 + C3 3r 5 C4 a! + C 3 r aY . 34 3 3?n 3M 3P aX a~g aX -C [ ' X+ " Pe+ 0X+ P ]+ r rC, , r P r , + r + aY a af tn + L + 2 Cr, 3 C 4 or, or 3M P aX r P a X -C5[ X + c P+ X0+ Pk+ 3Y 34 3? o3? C2 + C -+C4 -P +C5 3e (3.5) (3.6) (3.7) 3M 3P aX aP0 aX0 yC[y x + a Pe 3 X+ P ]+ r, r, 3 4 3? C TrY atr nr 3M P aX aP0 aX0 C { C-ta Xc+ 3c PC + k X+ 35 P + T at 3 t 4 ort alt k C Y + oC , f+a C n C2 +C3 +C -P + C, "& rt art art art (3.8) (3.8)  SO EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Sm SE X axC SE, S b k --C1[ 1 . 'Xk +-, E,+ SPbf ap5, ap51 5rbf a'EP 80 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Sm - E ax[_P S,_X P+akk +aX, k C1b [arj ' b 3 b SapP r k+~ R I + 80 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 3Mb apf a'b ax f r +-5Ey SEaf, ,a, + SX bI't, Sf 5r'bf S'b f 1 ' E1b - C2 by+C +C2 -s-.+C1 5rK I 515f3'f Srbp (3.9) amEp C, [ 3xC X,E, + fkxk + 1',,]k + C2 y SC2 + C2-air + C1 ' 5'SE Sp p SPbp a5rb5 Sm SE ax Sk ax1 SI,f =Cl[ ~Xc X PP + --X k S, y SF SF- SF-,- ,ar SM -. SE - a Sf a x1 Cby + C -.+---r + xc + E,]+ 3M ai ax SE1 ax, ~~X +C[ n.y P+ + Ek+ k] la+ 3 y b~y Cby SM- axC SE, ax, S, y a SF SFf+c ai ,C C2 -5 -, a C2pC -5- (3.10) (3.11) by S Ff SF SF+Cr C2 - C, +C4--+C Sm SE aX Spk Sxk C, [-a- x, + PC+ k+ k Sr, 51, - E4, + C xE + E] C2b 5 -i-sC p rp ab am=C1 [.. S. x, + j~ PP + 'kx, + ~kIl+ Sy a. an arn arn C2-5 +C2,I C,-5C1 S M a S ax SP Sxk C, C2 xes + EPC+-n X1+k + k C2 by + C1 1r+C-aj-+ C, 2n SM nE ax SE., ax ay lay ay SM-p axC as', ax SE, by [x,+-f. E+ SE, + , E1] axC ai-C SF. SF (3.9) (3.10) (3.11) am= C1 [ a__ __+_c C+ak k+b~ k Cby + Ca- ,aIL+ x1+ r k] Trbp rbp 'p arpp by a;f SF SF C2 - C, -C21 +C1 SM - SE a x~ SEf a x1 rpC [ +C + s - C ~+C Xk P. SM . SE ax~ SEp ax, SI SI, r SI C2 + r- C, - + C4 1P + C SI Sp 5'~ SI SM- by SE ax ay ax, C2C x-E +P4,]s C ay bay bCy~ Sm axC SE, axk SC,+ SE E,+ SE _L_ SE + -X ay- a;, aC a C2 +C---C4 SE! + C, E (3.10) (3.12) (3.12) (3.12) (3.13) (3.13) (3.13) (3.14) (3.14) (3.14)  SOLUTION WITH A PASSIVE GOVERNMENT 81 SMk apk ax, ax4 SOLUTION WITH A PASSIVE GOVERNMENT 81 aK ark axk X4 SOLUTION WITH A PASSIVE GOVERNMENT 81 Sm =C arcX _cP ax XE XkP S5pk C1apl ap+ a(PX4 ki aXy -al, a-,. S Tr4 ar ap4 C ar2 (3.15) Sm Si' SPE SXk C .+ X4 +- rkl+ ax, CIaxX ax, ax4 ax4 3ax,, 4, +Cs' , M ar ax arkax axE =Ci[ apx x+ ax c+ axk xk +EPk' + -5k ax4 aX ax a ar, (Sdk - , 5 ) , + , S (S4- ark SM k ar ark M SF, aSk arpk ark SM Ma, p S, -, r am s, SE4 ax4 dk usp dM, l.r dM, ar dM) = 1( ap dM, s.- dM, F-rLdM) Sm 'Sm 'S (3.16) (3.17) (3.30) (3.30a) C2 ar4+C -al4 a-,. SM k SE ap4 ax4ap =C,[ x + x4 + r4] + ax, ax, ax4 ax, C2ax, C, axF, , -a, + a-, SM K a ax ax4 a6x4 =C[~kxs ax4 ra~Xks4] (- -_,) +-( - S4 SM SE 5S, SM SF S --I -a55 SE4 Sd as SF Sd 5, SE, SM k Si, SE 3M SF SF Sm 5,, - d, ax4 dk( ak dM, a dM, SFdM = 1( P- dM, s"'c dM, .2K dM) Sm Sm Sm- (3.16) (3.15) C, asC, +C4 -aI C, r SMc Sx,+ ar4 ax4 a, ax, -K ax ax ax S aC4 aiE~k ax4 x4 ax ax 4r4 C3aP~ ax4 ask a ax4as S, (5p4 - 514) +(d - . SE4 am SMSE S, SM SF S Sak( )d 51 ( S,-Ta) ap apk am 4) (3.16) (3.15) (3.17) (3.30) (3.17) (3.30) (3.30a) Sm 5,c ad, Ep- SEk ax4k d(S4 dM, p dM, 21dM) = SE4 SE SF -p M r M (3.31) (3.31) (3.31)  82 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM SM d( , M , 37, S ) 82 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM SX d(SE SPF af SY) M d( , a_, 5M, Sm 82 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM SX= d(Sak ap , a M - a , -M , am) = s k , _c- - , - SM aM aM aM aY aX SF SXk +k m P- a+ SMX+ P + Xk + b + + -r + G + -t T + SM SM aM s SM SM Sa Sr SN Sr It+ -N +-MP r + BfP + aB art a'Lb + , a f SM S - (M SM rbf+3M L.S + Iwr ,) SM (3.3 1a) SFE SF Sf SY scM- , -, SM am am aM M (3.3 1a) = S(M , - S, -, SM am am am am) (3.3 1a) SY SX SPrSXc 5Pk SM SM X+ c M + Xk+ Snb + + r% + -G + Srt T + SM SM SM ' SM P SM SF' Sr SN Sr,, M t+ N irf+ Bp+ 3Bfp f arbf nb ,f SM r-( aM L-ne+ SM tbf+ oM Lrr+ "- rar) aM (3.32) (3.32) It has been indicated previously how the terms on the right-hand side of these relations can be obtained. We now attempt to examine these relations in more detail and to breathe some economic meaning into them. The expressions SM/Sr depend upon the effects of changes in interest rates on prices, real output, income, and all other interest rates (as well as institutional factors which are represented by the values of the constants C,). Consider an increase in one rate of interest, r*. The following statements consider the effects on P from the supply side only. In general, SP/ar* > 0 when r* is a cost to the firm (rbr, rf, and re fall into this category). On the other hand, when r* represents a return to the firm (rg, rt, r.), the effects of an increase in r* on prices will be less. Conceivably, in some cases, P/Sr* for some P and r* could even be negative. When r* represents a rate not directly related to the firms (rbp or r,,), SP/Sr* will be very near zero. From the demand side, an increase in r* represents a potential increase in Y when r* is rs, ir, rt, or rf. In these cases, increased Y will also increase the demand for goods and thus tend to increase P. When r* is either rbp or r,, the net direct effect on Y of an increase in r* will be zero since increased It has been indicated previously how the terms on the right-hand side of these relations can be obtained. We now attempt to examine these relations in more detail and to breathe some economic meaning into them. The expressions SM/ar depend upon the effects of changes in interest rates on prices, real output, income, and all other interest rates (as well as institutional factors which are represented by the values of the constants Ci). Consider an increase in one rate of interest, r*. The following statements consider the effects on P from the supply side only. In general, P/r* > 0 when r* is a cost to the firm (rbf, r, and rr fall into this category). On the other hand, when r* represents a return to the firm (rg, rt, ra), the effects of an increase in r* on prices will be less. Conceivably, in some cases, SP/ar* for some P and r* could even be negative. When r* represents a rate not directly related to the firms (rbp or rn). SP/ar* will be very near zero. From the demand side, an increase in r* represents a potential increase in Y when r* is rs, r., r, or r . In these cases, increased Y will also increase the demand for goods and thus tend to increase P. When r* is either rbp or re, the net direct effect on Y of an increase in r* will be zero since increased SY IXa 8P 5X5 ay= P + a X + axM Pk p Xk + SM SM SM SM F SM ITr Sr SN Sre a1Prt+ N +MP r. + B, + Br Srbf + r rarnf SM re ( bM Ebx SM rbf+SM Lse+ SM r.) (3.32) It has been indicated previously how the terms on the right-hand side of these relations can be obtained. We now attempt to examine these relations in more detail and to breathe some economic meaning into them. The expressions SM/Sr depend upon the effects of changes in interest rates on prices, real output, income, and all other interest rates (as well as institutional factors which are represented by the values of the constants Ci). Consider an increase in one rate of interest, r*. The following statements consider the effects on P from the supply side only. In general, SP/Sr* > 0 when r* is a cost to the firm (r, r, and rf fall into this category). On the other hand, when r* represents a return to the firm (rg, rt, r), the effects of an increase in r* on prices will be less. Conceivably, in some cases, SP/Sr* for some P and r* could even be negative. When r* represents a rate not directly related to the firms (rbp or r,,), SP/r* will be very near zero. From the demand side, an increase in r* represents a potential increase in Y when r* is rs, r., rt, or rf. In these cases, increased Y will also increase the demand for goods and thus tend to increase P. When r* is either rh or r,, the net direct effect on Y of an increase in r* will be zero since increased  SOLUTION WITH A PASSIVE GOVERNMENT 83 interest payments will result in increased profit distributions to the owners of the financial sectors. (Indirect effects on Y may be positive or negative depending on the responsiveness of actual amounts lent to changes in r* and the multiplier effects of changes in loans on income.) In general, therefore, we would expect OP/Or > 0. We would expect OXk/Or* < 0 when r* was a cost of investment (r, rbf, or ruf), in keeping with standard investment theory. Likewise OXI/ar* < 0 would be expected in this case since r* represents a cost to the firm. See Figure 15. SOLUTION WITH A PASSIVE GOVERNMENT 83 interest payments will result in increased profit distributions to the owners of the financial sectors. (Indirect effects on Y may be positive or negative depending on the responsiveness of actual amounts lent to changes in r* and the multiplier effects of changes in loans on income.) In general, therefore, we would expect OP/Or > 0. We would expect OXk/Or* < 0 when r* was a cost of investment (re, ruf, or rn), in keeping with standard investment theory. Likewise OXr/Or* < 0 would be expected in this case since r* represents a cost to the firm. See Figure 15. SOLUTION WITH A PASSIVE GOVERNMENT 83 interest payments will result in increased profit distributions to the owners of the financial sectors. (Indirect effects on Y may be positive or negative depending on the responsiveness of actual amounts lent to changes in r* and the multiplier effects of changes in loans on income.) In general, therefore, we would expect OP/Or > 0. We would expect OXr/Or* < 0 when r* was a cost of investment (rf, rbf, or r.), in keeping with standard investment theory. Likewise 0Xc/Or* < 0 would be expected in this case since r* represents a cost to the firm. See Figure 15. P=MR X X' Xn Fig. 15. X/Or < 0 Here AC' (> AC') is the average cost curve after an increase in r*. Notice that when the demand side is considered as well, it is necessary to point out that increased r* causes an increase in Y which would tend to increase demand and thus X. This effect in general would not be large enough to offset the reduction in X noted earlier, since that reduction itself causes Y to fall, ceteris paribus. When r* is not directly related to the firm (rbp or re), we assume OX/Or* = 0. When r* represents a source of income to the firm (rg, r, rn), we expect that OX/Or* > 0 (although these effects are probably small). One straightforward way to think about these effects is to note that increases in these rates may reduce the firms' dependence on X' X* Fig. 15. OX/r < 0 Here AC' (> AC') is the average cost curve after an increase in r*. Notice that when the demand side is considered as well, it is necessary to point out that increased r* causes an increase in Y which would tend to increase demand and thus X. This effect in general would not be large enough to offset the reduction in X noted earlier, since that reduction itself causes Y to fall, ceteris paribus. When r* is not directly related to the firm (rbp or rnp), we assume OX/Or* = 0. When r* represents a source of income to the firm (r, r r), we expect that OX/Or* > 0 (although these effects are probably small). One straightforward way to think about these effects is to note that increases in these rates may reduce the firms' dependence on X' X* Fig. 15. X/r <0 Here AC' (> AC') is the average cost curve after an increase in r*. Notice that when the demand side is considered as well, it is necessary to point out that increased r* causes an increase in Y which would tend to increase demand and thus X. This effect in general would not be large enough to offset the reduction in X noted earlier, since that reduction itself causes Y to fall, ceteris paribus. When r* is not directly related to the firm (rbp or r,), we assume OX/Or* = 0. When r* represents a source of income to the firm (rg, r, ri), we expect that OX/Or* > 0 (although these effects are probably small). One straightforward way to think about these effects is to note that increases in these rates may reduce the firms' dependence on  84 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM financing provided by banks and intermediaries and thus permit greater self-financed expansion. We have already referred to the effects of changes in r* on Y as a secondary effect in discussing aP/r* and OX/Or*. It also enters the expressions for 3M/ar* directly. The previous discussion wil not be repeated here. The interest-interaction terms 0r/0r enter the expressions via the last three terms on the right-hand side of 3.5 through 3.13. We hypothesize that Or/Or 0 for all i, j. This is equivalent to saying that all interest rates tend to move in the same direction. Clearly, the size of the expression will vary, depending on the closeness of the relation between the two rates. For some pairs we would expect this relation to be quite strong (such as OrbP/bf), while for others it may be quite weak (such as ort/araf). We now turn to a discussion of the constant terms C, through C. C5 = (as + d + nd. + neC, + ydf + yt, + ydnnf). Table 3 contains the definitions of these terms and their signs. Clearly, C, > 0. The value of C, tells us by how much the money stock in- creases, given a one-dollar increase in PX, as a result of the firms' increase in demand for money (a5 and df) and their deposits in banks and intermediaries, which in turn cause these sectors to increase their demands for money (nd.-the increase in the intermediaries' demands for demand deposits as a result of firms' increasing their deposits in the intermediaries, etc.). The last term, ydnnf, is a "third generation" effect-the increase in banks' demand for currency, caused by an in- crease in intermediaries' demand for deposits, which was in turn a result of an increase in the firms' demand for deposits in the intermediaries. C2 = (dane + can, + k, + k2 + yden + yk, + yk3). Table 4 gives the definitions and signs of the terms in C2 not in C1. Thus, C2 is clearly greater than zero. Its interpretation is analogous to that of C1, except that it measures the effects of an increase in Y on the public's demand for money and the effects of changes in the public's demand on the banks' and intermediaries' demands for money. 84 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM financing provided by banks and intermediaries and thus permit greater self-financed expansion. We have already referred to the effects of changes in r* on Y as a secondary effect in discussing OP/Or* and OX/Or*. It also enters the expressions for 3M/Or* directly. The previous discussion will not be repeated here. The interest-interaction terms ar/ar enter the expressions via the last three terms on the right-hand side of 3.5 through 3.13. We hypothesize that Or/Ora 0 for all i,j. This is equivalent to saying that all interest rates tend to move in the same direction. Clearly, the size of the expression will vary, depending on the closeness of the relation between the two rates. For some pairs we would expect this relation to be quite strong (such as arbp/rbe), while for others it may be quite weak (such as Ors/Ore). We now turn to a discussion of the constant terms C, through C. C, = (a. + df + nfd, + nCv + ydf + ytf + ydenf). Table 3 contains the definitions of these terms and their signs. Clearly, C5 > 0. The value of C, tells us by how much the money stock in- creases, given a one-dollar increase in PX, as a result of the firms' increase in demand for money (a. and d1) and their deposits in banks and intermediaries, which in turn cause these sectors to increase their demands for money (ndv-the increase in the intermediaries' demands for demand deposits as a result of firms' increasing their deposits in the intermediaries, etc.). The last term, ydnf, is a "third generation" effect-the increase in banks' demand for currency, caused by an in- crease in intermediaries' demand for deposits, which was in turn a result of an increase in the firms' demand for deposits in the intermediaries. C2 = (dn, + cne + k, + k2 + ydnnp + yk, + yk3). Table 4 gives the definitions and signs of the terms in C2 not in C1. Thus, C2 is clearly greater than zero. Its interpretation is analogous to that of C1, except that it measures the effects of an increase in Y on the public's demand for money and the effects of changes in the public's demand on the banks' and intermediaries' demands for money. C3 = [(d. + ce + yde)A30 + (1 + y)A22 + A23 + yA24] - 84 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM financing provided by banks and intermediaries and thus permit greater self-financed expansion. We have already referred to the effects of changes in r* on Y as a secondary effect in discussing OP/Or* and OX/r*. It also enters the expressions for OM/Or* directly. The previous discussion will not be repeated here. The interest-interaction terms Or/Or enter the expressions via the last three terms on the right-hand side of 3.5 through 3.13. We hypothesize that ri/Or .>0 for all i, j. This is equivalent to saying that all interest rates tend to move in the same direction. Clearly, the size of the expression will vary, depending on the closeness of the relation between the two rates. For some pairs we would expect this relation to be quite strong (such as Orbp/rb), while for others it may be quite weak (such as art/arf). We now turn to a discussion of the constant terms C, through Cs. C, = (a. + dt + ndn + nrCn + yd, + y-t + ydnnf). Table 3 contains the definitions of these terms and their signs. Clearly, C, > 0. The value of C, tells us by how much the money stock in- creases, given a one-dollar increase in PX, as a result of the firms' increase in demand for money (a. and df) and their deposits in banks and intermediaries, which in turn cause these sectors to increase their demands for money (ndc-the increase in the intermediaries' demands for demand deposits as a result of firms' increasing their deposits in the intermediaries, etc.). The last term, ydn, is a "third generation" effect-the increase in banks' demand for currency, caused by an in- crease in intermediaries' demand for deposits, which was in turn a result of an increase in the firms' demand for deposits in the intermediaries. C2 = (dn + ccn, + k, + k2 + yde + yk + yk3)- Table 4 gives the definitions and signs of the terms in C2 not in C1. Thus, C2 is clearly greater than zero. Its interpretation is analogous to that of C1, except that it measures the effects of an increase in Y on the public's demand for money and the effects of changes in the public's demand on the banks' and intermediaries' demands for money. C3 = [(d. + cn + ydn)A3 0 + (1 + -Y)A2 2 + A2 3 + -tA2 41 - C3 = [(dn + cn + -ydn)A30 + (1 + 'y)A22 + A23 + ItA24] -  SOLUTION WITH A PASSIVE GOVERNMENT 85 TABLE 3. Terns of C5 Term Sign Defitnitioc as >0o CefiientoftXiirs'demadforcurrncy f >0o CfficitoftPXinfim' ded fd em ad deposits r5 > 0 CoefficietoftPX iunrs' denssd fordeposits in f > 0 CofficieteofPX im' demad fortime deposits 7 > 0 CofenbtofDc+T inbanks'dad forcurrecy en > 0 Coefficient ofNinitermdiaies' demand for 0,, >0o CoefficietoftNinuintermediaries'demandtcor demanod deposits Table 0 gives she definitions and signs of she new seems in C3. Thus, wish the etxception of c3 0 and d204, all teems he A3 0, A2 2, A203, and A20 use negative since, wish she enception of these two teems, they repeesent she coefficinss of eates of interest on competing assess foe the public. The interpretation of she actual seems he C3 is straightfoewaed. Ens enample, (du + On + yfd.)A30 gives she impact of changes in the public's demand foe deposits in isntermediaries (resulting feom a change in some elementsofiTP) onhe inermediaies' (dctcnc)AcoI andethe banks' (dnA3u) demands foe money. TABCE 4. Tetms ot C2 Teto Sign Definition up > 0 Ccetficient of V in publtic's demaod for deposits lI ~ > 0 Coeficient oflYinplic'semad forcuecy k2 >0 Coeficient of Yin pulic's demand fordemnd dsepcsits k3 > 0 Coefficient ofTY is publtic's demsand for tint deposits SOLUTION WITH A PASSIVE GOVERNMENT 85 TABLE 3. Tecms of C5 Termt Sign Definition s > 0 Coefticittof PX infirms'demndforcurecy f > 0 Coefficient ot Ptiofirms' demcndfortdemad deposits of >0 Ccetficiensot Ptinstirms' demand fordepositsin intermediaries f> 0 CoetficiensotPX in firms' demand fortme deposits 'y > 0 Coeficiet of Dc+T inans' demadfoc recy n > 0 Coefficiensof N intinermediaries' demand for do >0 Coeticent oftNinointermdiarets'dmand for dtend deposits Tulle 5 gives she definitions and signs of she new teems in C3. Thus, with she exception of C30 and d.4, all seems he A3on' A22, A23, and A2a ass negative since, witk she enception of these Iwo seems, they repsesent the coefficients of eases of inserest on competheg assess foe she puhlic. The interpetasion of she actcal seems he C3 is saightsforwurd. Foe example, (de + cc + y'dn)A30 gives she impact of changes in the puhlic's demand foe deposits in inseemediaries (resulsing feom a change in some element ofre on she hneemediaries' [(d. + fc)Aottl and she bunks' (d,A35) demands foe money. TABLE 4. Torme of C2 Tesm Stan Definition op >0 CoefficiensotlYin pbic's demandtfordeposts k, >0o Coefficient oflYin pulic's dewandfocurrtcy k2 >0 Coefficiet oflYinspubic's demadfordeand deposits k3 ~ >0o Coetficient oftYin publfic's demand to, tioe deposits SOLUTION WITH A PASSIVE GOVERNMENT 85 TABiLE 3 Terms ofC, Teen Shen Definition , > 0 Cotfficint of PXinfirns'deandfocurrtcy f > 0 Coetficietsof PX infirms'demandtfotdemnsd deposits of ~ > 0 Coetficient of PX in firms' demand tot depositssin if>0 Coeficient of PX infirms'demandfortime deposits 'y > 0 Coefficiet offDl+T inbans' demadfocurcecy en >0o Coefficient oftiin iermediaries' demand for do >0o Coefficietoffl initermediaries' deand for demand deposits Table 5 gives lbs definitions and digns of she new teems in C3. Thus, wish she exception of C30 and d.4 all seems he A30, A22, A23, and A. see negative since, with she enception of these Iwo seems, they repeesent she coefficients of eases of interest on competing assess foe the public. The interpretation of she actal seems he C3 is steaightforward. Poe example, (dc + c. + yd.)A3. sines she impact of changes in she public's demand foe deposits in intermediaries (rselting feom a change in some elemens of i5) on she hnermediaries' [(do + c.)A30] and she hanha' (d.A3s) demands for money. TABLEO4. Twoms oftC2 Term tigc Definition ep >0o CoefficintoflYinpubticsdemadfordepsits k, >0 CoefficiensotlYinpulficsdemadfcurrcency k2 > 0 CoefficintofYtinpulic'sdemnd fordemad deposits 00 > 0 Coefficiet otlYinpulic's demand fortime deposits,  86 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM TABLE 5. Terms of C3 86 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM TABLE 5. Terms of C3 86 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM TABLE 5. Terms of C3 Term Sign Definition A30 a30 b3B c30 130 f30 h30 A22 a22 b22 c22 d22 f22 h22 A23 a23 b23 c23 d23 f23 h23 < 0 < 0 > 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 <0 <0 < 0 < 0 < 0 Coefficient of F, in public's demand for deposits in intermediaries Coefficient of if in public's demand for deposits in intermediaries Coefficient of rg in public's demand for deposits in intermediaries Coefficient of rn in public's demand for deposits in intermediaries Coefficient of rt in public's demand for deposits in intermediaries Coefficient of rbp in public's demand for deposits in intermediaries Coefficient of rup in public's demand for deposits in intermediaries Coefficient of ip in public's demand for demand deposits Coefficient of rf in public's demand for demand deposits Coefficient of rg in public's demand for demand deposits - Coefficient of in in public's demand for demand deposits Coefficient of rt in public's demand for demand deposits Coefficient Of rbp in public's demand for demand deposits Coefficient of rap in public's demand for demand deposits Coefficient of rp in public's demand for currency Coefficient of rf in public's demand for currency Coefficient of rg in public's demand for currency Coefficient Of rn in public's demand for currency Coefficient of rt in public's demand for currency Coefficient of rbp in public's demand for currency Coefficient of rup in public's demand for currency A30 232 130 r30 d130 f30 h30 A22 a22 b22 c22 d22 f22 h22 A23 a23 b23 C23 d23 f23 h23 < 0 < 0 > 0 < 0 < 0 < 0 <0 < 0 < 0 <0 <0 < 0 <2 < 0 < 0 < 0 < 0 < 0 Coefficient of r in public's demand for deposits in intermediaries Coefficient of rf in public's demand for deposits in intermediaries Coefficient of rg in public's demand for deposits in intermediaries Coefficient of rn in public's demand for deposits in intermediaries Coefficient of t in public's demand for deposits in intermediaries Coefficient of rbp in public's demand for deposits in intermediaries Coefficient of rap in public's demand for deposits in intermediaries Coefficient of p in public's demand for demand deposits Coefficient of 1 in public's demand for demand deposits Coefficient of rg in public's demand for demand deposits Coefficient of rn in public's demand for demand deposits Coefficient of rt in public's demand for demand deposits Coefficient Of Tbp in public's demand for demand deposits Coefficient Of rnp in public's demand for demand deposits Coefficient of rp in public's demand for currency Coefficient of rf in public's demand for currency Coefficient of rg in public's demand for currency Coefficient of rn in public's demand for currency Coefficient of rt in public's demand for currency Coefficient of rbp in public's demand for currency Coefficient of rnp in public's demand for currency A30 a30 b30 c30 d30 f30 h30 A22 a22 b22 c22 d22 f22 h22 A23 a23 b23 C23 d23 r23 h23 < 0 < 0 > 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 Coefficient of rp in public's demand for deposits in intermediariesp Coefficient of rf in public's demand for deposits in intermediaries Coefficient of rg in public's demand for deposits in intermediaries Coefficient Of in in public's demand for deposits in intermediaries Coefficient of rt in public's demand for deposits in intermediaries Coefficient Of rbp in public's demand for deposits in intermediaries Coefficient of rnp in public's demand for deposits in intermediaries Coefficient of rp in public's demand for demand deposits Coefficient of if in public's demand for demand deposits Coefficient of rg in public's demand for demand deposits Coefficient of rn in public's demand for demand deposits Coefficient of rt in public's demand for demand deposits Coefficient Of rbp in public's demand for demand deposits Coefficient of rn, in public's demand for demand deposits Coefficient of rp in public's demand for currency Coefficient of if in public's demand for currency Coefficient of rg in public's demand for currency Coefficient of rn in public's demand for currency Coefficient of rt in public's demand for currency Coefficient of rbp in public's demand for currency Coefficient of rnp in public's demand for currency  SOLUTION WITH A PASSIVE GOVERNMENT 87 TABLE 5-Continued A24 Coefficient of frp in public's demand for time deposits a24 < 0 Coefficient of rf in public's demand for time deposits 024 < 0 Coefficient of ig in public's demand for time deposits c24 < 0 Coefficient of r, in public's demand for time deposits d24 > 0 Coefficient of rt in public's demand for time deposits f24 ( 0 Coefficient Of rbp in public's demand for time deposits h24 < 0 Coefficient of rp in public's demand for time deposits C4 = [(dn + c, + yd.)A, + (I + y)A6 + yA7]. The new terms in C4 are given in Table 6. Once again, all terms but c, and d7 are negative since they represent rates on competing assets for the firms. The interpretation of the actual terms in C4 is analogous to those of the previous C's. [(d. + cn + yd.)A,], for example, is the effect of the firms' changed demand for deposits in intermediaries on the intermediaries' and banks' demands for money. C, = (1 + y)A16' A16 is the coefficient of T. in the intermediaries' demand for demand deposits. Table 7 gives the signs and definitions of the elements of A16. Thus, Cs indicates the effects of changes in an element of T. on the intermediaries' demand for demand deposits as well as the secondary effect on the banks' demand for currency. The preceding discussion and descriptions provide the necessary mate- rial to interpret any of the expressions for tM/Or* for any r*. The expression for 3M/dY contains the same five constants, C, through C5, described above. Both the derivatives of prices and physical outputs with respect to income will be positive for obvious reasons. The signs of 0r*/OY are given in Table 8. The only sign in Table 8 that can be specified exactly without making further assumptions is that of or/OY, which will be negative as in- SOLUTION WITH A PASSIVE GOVERNMENT 87 TABLE 5-Continued A24 Coefficient of fp in public's demand for time deposits a24 ( 0 Coefficient of r in public's demand for time deposits b,24 ( 0 Coefficient of rg in public's demand for time deposits c24 < 0 Coefficient of rn in public's demand for time deposits d24 > 0 Coefficient of rt in public's demand for time deposits f24 ( 0 Coefficient of rbp in public's demand for time deposits h24 < 0 Coefficient of rp in public's demand for time deposits C4 [(dn + cn + yd.)A, + (1 + y)A6 + yA]. The new terms in C4 are given in Table 6. Once again, all terms but c, and d. are negative since they represent rates on competing assets for the firms. The interpretation of the actual terms in C4 is analogous to those of the previous C's. [(d. + c + ydn)An], for example, is the effect of the firms' changed demand for deposits in intermediaries on the intermediaries' and banks' demands for money. C, = (1 + y)A16. A1 is the coefficient of in in the intermediaries' demand for demand deposits. Table 7 gives the signs and definitions of the elements of A16. Thus, C, indicates the effects of changes in an element of T, on the intermediaries' demand for demand deposits as well as the secondary effect on the banks' demand for currency. The preceding discussion and descriptions provide the necessary mate- rial to interpret any of the expressions for tM/Or* for any r*. The expression for aM/DY contains the same five constants, C, through C., described above. Both the derivatives of prices and physical outputs with respect to income will be positive for obvious reasons. The signs of r*/aY are given in Table 8. The only sign in Table 8 that can be specified exactly without making further assumptions is that of ore/DY, which will be negative as in- SOLUTION WITH A PASSIVE GOVERNMENT TABLE 5-Continued Coefficient of rp in public's demand for time deposits A24 a24 < Coefficient of rf in public's demand for time deposits b24 ( 0 Coefficient of r. in public's demand for time deposits c24 < 0 Coefficient of rn in public's demand for time deposits d24 > 0 Coefficient of rt in public's demand for time deposits f24 < 0 Coefficient of rbp in public's demand for time deposits h24 < 0 Coefficient of rnp in public's demand for time deposits C4n [(d. + ce + yd.)A, + (1 + y)A6 + yA7]. The new terms in C4 are given in Table 6. Once again, all terms but c, and d7 are negative since they represent rates on competing assets for the firms. The interpretation of the actual terms in C4 is analogous to those of the previous C's. [(dn + co + yd,)A,] , for example, is the effect of the firms' changed demand for deposits in intermediaries on the intermediaries' and banks' demands for money. C5 = (1 + y)A16- A16 is the coefficient of Tn in the intermediaries' demand for demand deposits. Table 7 gives the signs and definitions of the elements of A16- Thus, C. indicates the effects of changes in an element of Fn on the intermediaries' demand for demand deposits as well as the secondary effect on the banks' demand for currency. The preceding discussion and descriptions provide the necessary mate- rial to interpret any of the expressions for 3M/ar* for any r*. The expression for 3M/DY contains the same five constants, C, through C, described above. Both the derivatives of prices and physical outputs with respect to income will be positive for obvious reasons. The signs of dr*/dY are given in Table 8. The only sign in Table 8 that can be specified exactly without making further assumptions is that of r,/DY, which will be negative as in-  88 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM TABLE 6. Terms of C4 88 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM TABLE 6. Terms of C4 88 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM TABLE 6. Terms, of C4 Tem Sign Def6iio A6 99 A69 b6 A6 6 96 < 0 < 0 > 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 > 0 Coefficent o F0q in Sims,' demand40or deposits in Coeffiientof6 i091 firms' demadfor de6positsin Coefficien of ibt in firm' demand4 for depsits in Coefficien of r4 1in firm' demand9 for deposts i Coeffcien of rat in firm1, ' deman fo deposits6in intermdiarie Coeffiinto06infirs'demand for demn Coefficien of in,666m' demand4for4deman Coefficient of rbf in fim' 466man9 for deman deposits Coefficient6 of 766 in6 firms' dean for time16 de6posit Coeficiet6of969 i firms9Oj'4demand for time 4deposits Cofiin of~609 1660 in firs' deand frtim deposits6 Term6 Sign 69 9, < 0 09 < 0 69 > 0 49 < 0 669 < 0 69 < 0 6 a6 < 0 b6 < 0 c6 <0 d6 < 0 e6 < 0 96 <0I Coefficient o0f 19 in9996m' 46666699 for deposits in Coefficient9 of1 696666irm' demand96for9deposits1,in Coefficient0 of I. 19 firms' 466mand for deposits in intermediar906ie 966d66690666966 Coefiiet f n n irs' emndfo dpoit i Cintermed9iaries ' 66666 69 699166 Coefficient of, 19 696fir' 4669966 for deposits6in Coefficient6 of rb in firm' d66man6d for deposits6i Coefficient of rnf in firm' 4demand9 for deposits94 Coefficient6 of Tf in firms9' 4deman4 for demand9 Coefficient 66f r, in 06irms' demand6 fordean Coefficient0 of Is in0firms' 46em664 for demand~ Coefficient O0 In6 in firms,' demand9 for 46em994 Coefficient of,, it6 in firm' demand69 for demand6 Coefficient O0 rf in firm' demand6 for 6demand 1660ficient of6 66 in6fir6' 4666666 0or time6 Coefficient of 6if in firms' demand 0966 time Coefficient of rg in firms' demand,6 for time depoi,96 6,9 69 69 A6 a6 6 6 < 0 < 0 > 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 Coefficien 60f 1f in firm' demand6 for deposits in6 C6606616666 60 6616 in6f,66' d66,69nd 096 deposits 166 Coefficient 906 66 06,66rm' demand6 09946,9969619i 1960066666t 6of 66 firms' 46em6664096646,96666166 Coefficient6 of41 r 6inirm' demand9 for deposts i Coefficient6 60 6f in6 firm' demand4 for depositsi Coefficien 66 Ef 1in firms' demand4 for demand Coefficient of0if06666ir6' 46,666406646,66664 Coeffic1ent of6 66 in6firm' 469,964 for4demand Coefficient Of in4 i66 firms' demand6 09646966664 Coefficient 6o6f6 it 6 infirm' demand6 for 4696664 Coefficient O0f 6f infirm' demand9 096 demand Coefficient6 of 6, 16 firms' demand4 f96 time Coefficient of96 r in6firm' 4696666 066 61966 1660661669t of, 16 infirm' 469,69409661im6 4depos1ts 67 d7 < 0 < 0 < 0 > 0 A79 9, < 0 b7 < 0 r7 <0 d,9 > 0  SOLUTION WITH A PASSIVE GOVERNMENT TABLE 6-Continued 89 SOLUTION WITH A PASSIVE GOVERNMENT TABLE 6-Continued 89 SOLUTION WITH A PASSIVE GOVERNMENT TABLE 6 -Continued 89 e7 < 0 Coefficient Of rbf in firms' demand for time deposits g7 < 0 Coefficient of r,,f in firms' demand for time deposits TABLE 7. Elements of A16(C5) Term Sign Definition a16 < 0 Coefficient of rf in intermediaries' demand for demand deposits b,16 < 0 Coefficient of rg in intermediaries' demand for demand deposits c16 < 0 Coefficient of rn in intermediaries' demand for demand deposits d16 < 0 Coefficient of rt in intermediaries' demand for demand deposits g16 < 0 Coefficient of rno in intermediaries' demand for demand deposits h16 < 0 Coefficient of rn, in intermediaries' demand for demand deposits ar* TABLE 8. - N_ Term Sign Term Sign ar arbf aV < ~ ST _ ar0 0 > SY Sy <0 3rn > 0 ren > 0 ST < ~ ST _ S>0 0 art 0 0 > Ely >aY < e7 < 0 Coefficient of rbf in frms' demand for time deposits g7 < 0 Coefficient of rnf in firms' demand for time deposits TABLE 7. Elements of A16(C5) Term Sign Definition a16 < 0 Coefficient Of rf in intermediaries' demand for demand deposits b16 < 0 Coefficient of rg in intermediaries' demand for demand deposits c16 < 0 Coefficient of rn in intermediaries' demand for demand deposits d.16 < 0 Coefficient of rt in intermediaries' demand for demand deposits g16 < 0 Coefficient Of Snf in intermediaries' demand for demand deposits h16 < 0 Coefficient of rnp in intermediaries' demand for demand deposits ar* TABLE 8. ar Term Sign Term Sign ar >0 rbf > ST <0 T a, <0 arbp >0 NY aY < arn > Carnf > aY < n,,, ST >0 STy e7 < 0 Coefficient of rbf in firms' demand for time deposits g7 < 0 Coefficient of rnf in firms' demand for time deposits TABLE 7. Elements of A16(C5) Term Sign Definition a16 ( 0 Coefficient of rf in intermediaries' demand for demand deposits b16 < 0 Coefficient of rg in intermediaries' demand for demand deposits C16 < 0 Coefficient of rn in intermediaries' demand for demand deposits d1 6 ( 0 Coefficient of rt in intermediaries' demand for demand deposits g16 < 0 Coefficient of rnf in intermediaries' demand for demand deposits h16 < 0 Coefficient of trnp in intermediaries' demand for demand deposits TABLE 8. N Term Sign Term Sign arg > arbf >0 <0 <0 <0 >0 Srt >0 STp  90 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM creased demand for government securities will bid their price up and the rate down. The signs of the other terms depend on the relation between the impact of changes in income on the demand for loans and firms' securities and the supply of loans and securities. In a strictly partial equilibrium sense, we can say that if the impact on these demands is greater than on the corresponding supplies, the correspond- ing partial derivative will be positive. If the impact on the supplies is larger than on the demand, the derivative will be negative. The question of the relative sizes of these effects is not one that can be answered without specifying the actual values of the parameters in the appropriate demand and supply functions. Thus, the answer must be provided either 90 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM creased demand for government securities will bid their price up and the rate down. The signs of the other terms depend on the relation between the impact of changes in income on the demand for loans and firms' securities and the supply of loans and securities. In a strictly partial equilibrium sense, we can say that if the impact on these demands is greater than on the corresponding supplies, the correspond- ing partial derivative will be positive. If the impact on the supplies is larger than on the demand, the derivative will be negative. The question of the relative sizes of these effects is not one that can be answered without specifying the actual values of the parameters in the appropriate demand and supply functions. Thus, the answer must be provided either MC AC -P,=MR, 90 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM creased demand for government securities will bid their price up and the rate down. The signs of the other terms depend on the relation between the impact of changes in income on the demand for loans and firms' securities and the supply of loans and securities. In a strictly partial equilibrium sense, we can say that if the impact on these demands is greater than on the corresponding supplies, the correspond- ing partial derivative will be positive. If the impact on the supplies is larger than on the demand, the derivative will be negative. The question of the relative sizes of these effects is not one that can be answered without specifying the actual values of the parameters in the appropriate demand and supply functions. Thus, the answer must be provided either MC AC -P,=MR, P =MR0 XtX, Fig. 16. SXc/Pc and SXk/SPk by assumption on the parameters (which would be only a tentative answer subject to empirical verification) which we have, and will avoid, or by empirical estimation. It could (and will be) argued, however, that, since it is to be expected that increases in income will tend to increase the money stock, if some of the r/SY are in fact negative, they cannot be so negative as to cause the entire expression for SM/SY to be negative. The expressions for SM/SP also contain C, through C. as well as the partial derivatives of Xc, Xk, P, Pk, Y, and the r's with respect to the P's. The terms SXc/SPc and 5Xk/5pk are positive under the assumptions of perfect competition. See Figure 16. SY/SPk and SY/SPc are both positive because of the direct relation between PX and Y, and since the terms SXe/aPc and SXk/Spk are (as argued earlier) positive. The terms XoX, Fig. 16. SXc/Pc and SXk/apk X X XoX Fig. 16. SXe/SPe and SXc/SPk by assumption on the parameters (which would be only a tentative answer subject to empirical verification) which we have, and will avoid, or by empirical estimation. It could (and will be) argued, however, that, since it is to be expected that increases in income will tend to increase the money stock, if some of the Sr/SY are in fact negative, they cannot be so negative as to cause the entire expression for aM/SY to be negative. The expressions for SM/SP also contain Cr through C, as well as the partial derivatives of X, Xk, P, Pic, Y, and the r's with respect to the P's. The terms SXc/SPc and SXk/Spk are positive under the assumptions of perfect competition. See Figure 16. SY/SPk and SY/SPc are both positive because of the direct relation between PX and Y, and since the terms SXc/SP and 5Xk/Spk are (as argued earlier) positive. The terms by assumption on the parameters (which would be only a tentative answer subject to empirical verification) which we have, and will avoid, or by empirical estimation. It could (and will be) argued, however, that, since it is to be expected that increases in income will tend to increase the money stock, if some of the Sr/SY are in fact negative, they cannot be so negative as to cause the entire expression for 3M/SY to be negative. The expressions for SM/aP also contain C5 through C. as well as the partial derivatives of X, X, P, X Pk, IY, and the r's with respect to the P's. The terms SXc/SPr and SXk/Spc are positive under the assumptions of perfect competition. See Figure 16. SY/SPk and SY/SP are both positive because of the direct relation between PX and Y, and since the terms SXr/SP and 5Xk/p are (as argued earlier) positive. The terms  SOLUTION WITH A PASSIVE GOVERNMENT 91 SOLUTION WITH A PASSIVE GOVERNMENT 91 aP./aP. and apk/aP5 are assumed to be positive in deference to the widely observed phenomenon that prices tend to move together. Table 9 lists the remaining terms in the aM/aP and their signs. The six terms whose signs are greater than zero simply reflect the fact that as prices increase, so does output, thus increasing the firms' de- mands for both internal and external financing and thus, ceteris paribus, aM TABLE 9. Terms of Term Sign Term Sign 3rf art aPc >0 ak > 0t 0 bar art 3. >0 3n- 0 Ste Ott art art 0 rbf >0 arbf > 0 aPc aPk 0rb 0 arn - 3,, > 0 ___ >0 arnp 0 3p0 ape aPk the rates of interest paid on the various types of financing. The notation "~ 0" is used for the other terms to indicate that they are "nearly" zero, but must be positive since, by our assumption on the interest- interaction terms, all interest rates move together. The causality may, however, not run directly from a change in P, or Pk to a change in the particular interest rate being considered. Thus, the terms aM/OPc and OM/aPk are positive once the assumption apt/Spa and Spk/3P, are assumed to be positive in deference to the widely observed phenomenon that prices tend to move together. Table 9 lists the remaining terms in the SM/OP and their signs. The six terms whose signs are greater than zero simply reflect the fact that as prices increase, so does output, thus increasing the firms' de- mands for both internal and external financing and thus, ceteris paribus, SM TABLE 9. Terms of Term Sign Term Sign Saf >0 ar >0 0r 0 a,0 an- 0 art 0o art 0 ae ak arbf 0 t0 Satn >0 karn >0 3r~p0 3r .0 Sa c k the rates of interest paid on the various types of financing. The notation "- 0" is used for the other terms to indicate that they are "nearly" zero, but must be positive since, by our assumption on the interest- interaction terms, all interest rates move together. The causality may, however, not run directly from a change in P, or Pk to a change in the particular interest rate being considered. Thus, the terms M/aP and M/aP are positive once the assumption SOLUTION WITH A PASSIVE GOVERNMENT 91 aP,/aPk and aPk/ae are assumed to be positive in deference to the widely observed phenomenon that prices tend to move together. Table 9 lists the remaining terms in the SM/SP and their signs. The six terms whose signs are greater than zero simply reflect the fact that as prices increase, so does output, thus increasing the firms' de- mands for both internal and external financing and thus, ceteris paribus, SM TABLE 9. Terms ofr Term Sign Term Sign St e Ore > org 0 arg , c k S00 D.0 Ser o ark 0 0 artc art arbf > arbf >0 >0 0_ Sate Stk arn > 0 3rf >0 arP 0 p0 the rates of interest paid on the various types of financing. The notation "~ 0" is used for the other terms to indicate that they are "nearly" zero, but must be positive since, by our assumption on the interest- interaction terms, all interest rates move together. The causality may, however, not run directly from a change in P, or Pk to a change in the particular interest rate being considered. Thus, the terms aM/OPc and SM/OP0 are positive once the assumption  92 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM (not a very startling one) is granted that aXk /aP > 1 aX/a |Pk and aXeP/aP, > 1 Se/Pk P Pck aM/aXe and aM/aXk are the last of the key relations in which the five constants C, through C, enter. With the exception of the terms aXe/aXe and aXe/aXe all terms in these two expressions will also be positive for reasons analogous to those given in the previous argument. At less than full employment these two terms can also be positive (as noted), even though when operating on the transformation curve they must both be negative. Once again, there is no ambiguity about the signs of SM/SXc and aM/OXe as both will be positive, even with negative aoX/axi. We turn now to an examination of the expressions in which M appears as the independent variable. The first two of these, aPk/aM and aP,/aM, we have been assured by many, many economists, must be positive. Examination of 3.30 and 3.30a should, we hope, reaffirm the quantity theory. Clearly, the denominators of both of these expressions are positive, since both 5d5/SPk and ada/aPe are negative if the demand curves for Xk and Xe are downward sloping. (These derivatives are simply the change in the quantities demanded given a change in price.) What about the numerators? The two terms ask/af and ase/ad are negative, since increases in the elements of T represent an increase in costs and shift the firms' supply curve (MC) to the left. adk/3r and d,/SF will both be positive as quantity demanded will increase, given the increase in income caused by increases in the elements of F (This effect will be somewhat dampened if increases in r reduce loans signifi- cantly and thus, indirectly, reduce the amounts of Xk and Xc de- manded.) ade/P and Sde/SPk can both be expected to be positive, since increases in the P's will increase income and thus the quantities demanded. So far, all the elements of 3.30 and 3.30a have the proper sign. The only potential source of trouble is in the signs of the elements of S/aM. In general, it is expected that these terms will be negative increases in the money stock, should it tend to reduce rates of interest. Thus, 3.30 and 3.30a will have the "proper" sign (positive) only so long as ape/SM (ade/aPk - 3Se/SPe) > I So/SM (ade/af - ase/ad) I and likewise in the corresponding expression for aPk/SM. There seems to be little reason to think this inequality will not be satisfied, since price effects should be more important than interest rate effects on quantities supplied and demanded. The derivations of the expressions for aXk/aM and OXe/SM require no further comment since all they amount to is plugging in the equilibrium 92 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM (not a very startling one) is granted that X ap > 1 ak/ac 1 k and aXe /aPe > a aXe/aPe I P.E a a aM/dX. and aM/aX0 are the last of the key relations in which the five constants C, through C, enter. With the exception of the terms aXe/aXe and aXe/aXe all terms in these two expressions will also be positive for reasons analogous to those given in the previous argument. At less than full employment these two terms can also be positive (as noted), even though when operating on the transformation curve they must both be negative. Once again, there is no ambiguity about the signs of aM/aXe and SM/aX as both will be positive, even with negative axi/ax . We turn now to an examination of the expressions in which M appears as the independent variable. The first two of these, aPk/SM and aPe/aM, we have been assured by many, many economists, must be positive. Examination of 3.30 and 3.30a should, we hope, reaffirm the quantity theory. Clearly, the denominators of both of these expressions are positive, since both adk/5Pk and ada/aPe are negative if the demand curves for Xk and Xe are downward sloping. (These derivatives are simply the change in the quantities demanded given a change in price.) What about the numerators? The two terms ase/F and dsc/d are negative, since increases in the elements of r represent an increase in costs and shift the firms' supply curve (MC) to the left. d/ar and dde/F will both be positive as quantity demanded will increase, given the increase in income caused by increases in the elements of / (This effect will be somewhat dampened if increases in r reduce loans signifi- cantly and thus, indirectly, reduce the amounts of X, and Xe de- manded.) ad/dE and ode/dPe can both be expected to be positive, since increases in the P's will increase income and thus the quantities demanded. So far, all the elements of 3.30 and 3.30a have the proper sign. The only potential source of trouble is in the signs of the elements of ad/SM. In general, it is expected that these terms will be negative increases in the money stock, should it tend to reduce rates of interest. Thus, 3.30 and 3.30a will have the "proper" sign (positive) only so long as aPe/SM (adeaPk - dSe/dPe) > I Sf/SM (5de/Si - dsc/df) I and likewise in the corresponding expression for 5Pk/SM. There seems to be little reason to think this inequality will not be satisfied, since price effects should be more important than interest rate effects on quantities supplied and demanded. The derivations of the expressions for dXe/dM and aXc/M require no further comment since all they amount to is plugging in the equilibrium 92 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM (not a very startling one) is granted that ax5 /P > 1 aXe/aP ek and aXp/dP > I aXc/Spk I Pe SM/aXe and SM/aX are the last of the key relations in which the five constants C, through C. enter. With the exception of the terms aXe/aX and aXe/axe all terms in these two expressions will also be positive for reasons analogous to those given in the previous argument. At less than full employment these two terms can also be positive (as noted), even though when operating on the transformation Curve they must both be negative. Once again, there is no ambiguity about the signs of SM/aXe and SM/aX as both will be positive, even with negative dx1/axf. We turn now to an examination of the expressions in which M appears as the independent variable. The first two of these, 5Pk/SM and aPe/aM, we have been assured by many, many economists, must be positive. Examination of 3.30 and 3.30a should, we hope, reaffirm the quantity theory. Clearly, the denominators of both of these expressions are positive, since both Sdc/SPe and dde/SP are negative if the demand curves for Xk and X, are downward sloping. (These derivatives are simply the change in the quantities demanded given a change in price.) What about the numerators? The two terms dst/Y and asc/T are negative, since increases in the elements of i represent an increase in costs and shift the firms' supply curve (MC) to the left. 5dk/Sr and ade/d will both be positive as quantity demanded will increase, given the increase in income caused by increases in the elements of F (This effect will be somewhat dampened if increases in r reduce loans signifi- cantly and thus, indirectly, reduce the amounts of Xk and X, de- manded.) d/aP. and 8de/dPe can both be expected to be positive, since increases in the P's will increase income and thus the quantities demanded. So far, all the elements of 3.30 and 3.30a have the proper sign. The only potential source of trouble is in the signs of the elements of ad/SM. In general, it is expected that these terms will be negative increases in the money stock, should it tend to reduce rates of interest. Thus, 3.30 and 3.30a will have the "proper" sign (positive) only so long as SPk/SM (de/dek - aseaPk) > I ST/aM (dde/ST - dse/SF) I and likewise in the corresponding expression for aPk/SM. There seems to be little reason to think this inequality will not be satisfied, since price effects should be more important than interest rate effects on quantities supplied and demanded. The derivations of the expressions for dXk/dM and aXe/aM require no further comment since all they amount to is plugging in the equilibrium  SOLUTION WITH A PASSIVE GOVERNMENT 93 price change and subtracting from that expression the expression for the original quantity demanded or supplied. The signs of these terms depend on the direction of the effect of changes in M on the demand and supply curves, as well as the location and shape of the initial and final curves. I hypothesize that increases in M cause both demand curves to shift to the right, because of the impact of M on Y. In the event that P MC,,,,=5SUPL,,,, x, x, Fig. 17. Effect of dM on MC the supply curve shifts downward, both SXc/SM and 3X,/SM will be positive. An upward shift in the supply curve is a necessary but not sufficient condition for SXc/OM Or aX,,/M to be negative. For nega- tivity the reduction in supply must be appropriately large. Whether the supply curves shift upward or downward depends on whether, on bal- ance, an increase in M increases or reduces average cost (and thus marginal cost). Interest expenses will tend to fall while the costs of labor and capital tend to increase. On the whole it must be concluded that increases in M tend to increase AC and thus to shift the supply curves to the left for at least a portion of the curve. If changes in M shift not only the position of the AC curve but also affect its shape significantly, the new supply curve (MC curve) may lie above the old curve for other ranges. The argument may be made that increases in M cause such significant increases in demand that firms are induced to build larger plants (perhaps through 0, the profits expectations variable, as well as a result of increases in prices) once a situation like Figure 17 consequently results. In the case illustrated in Figure 17, SX/SM is clearly positive. In general, we will assume that aXa/aM and aXk/aM will be positive, although it is clearly not true that in an n-commodity world SX/SM V X need be positive. Indeed, at full employment in even a two- commodity world, increases in M cannot result in changes in the SOLUTION WITH A PASSIVE GOVERNMENT 93 price change and subtracting from that expression the expression for the original quantity demanded or supplied. The signs of these terms depend on the direction of the effect of changes in M on the demand and supply curves, as well as the location and shape of the initial and final curves. I hypothesize that increases in M cause both demand curves to shift to the right, because of the impact of M on Y. In the event that P MC ,=SPPL PY., C. .,, x x Fig. 17. Effect of dM on MC the supply curve shifts downward, both 3X/SM and SXc/SM will be positive. An upward shift in the supply curve is a necessary but not sufficient condition for aXk/SM or SXc/SM to be negative. For nega- tivity the reduction in supply must be appropriately large. Whether the supply curves shift upward or downward depends on whether, on bal- ance, an increase in M increases or reduces average cost (and thus marginal cost). Interest expenses will tend to fall while the costs of labor and capital tend to increase. On the whole it must be concluded that increases in M tend to increase AC and thus to shift the supply curves to the left for at least a portion of the curve. If changes in M shift not only the position of the AC curve but also affect its shape significantly, the new supply curve (MC curve) may lie above the old curve for other ranges. The argument may be made that increases in M cause such significant increases in demand that firms are induced to build larger plants (perhaps through 0, the profits expectations variable, as well as a result of increases in prices) once a situation like Figure 17 consequently results. In the case illustrated in Figure 17, SX/M is clearly positive. In general, we will assume that aXa/3M and aX/SM will be positive, although it is clearly not true that in an n-commodity world SX/SM V X need be positive. Indeed, at full employment in even a two- commodity world, increases in M cannot result in changes in the SOLUTION WITH A PASSIVE GOVERNMENT 93 price change and subtracting from that expression the expression for the original quantity demanded or supplied. The signs of these terms depend on the direction of the effect of changes in M on the demand and supply curves, as well as the location and shape of the initial and final curves. I hypothesize that increases in M cause both demand curves to shift to the right, because of the impact of M on Y. In the event that Fig. 17. Effect of dM on MC the supply curve shifts downward, both SXc/M and SXc/SM will be positive. An upward shift in the supply curve is a necessary but not sufficient condition for aXk/DM or SX/M to be negative. For nega- tivity the reduction in supply must be appropriately large. Whether the supply curves shift upward or downward depends on whether, on bal- ance, an increase in M increases or reduces average cost (and thus marginal cost). Interest expenses will tend to fall while the costs of labor and capital tend to increase. On the whole it must be concluded that increases in M tend to increase AC and thus to shift the supply curves to the left for at least a portion of the curve. If changes in M shift not only the position of the AC curve but also affect its shape significantly, the new supply curve (MC curve) may lie above the old curve for other ranges. The argument may be made that increases in M cause such significant increases in demand that firms are induced to build larger plants (perhaps through 0, the profits expectations variable, as well as a result of increases in prices) once a situation like Figure 17 consequently results. In the case illustrated in Figure 17, aX/3M is clearly positive. In general, we will assume that SXc/SM and aXk/SM will be positive, although it is clearly not true that in an n-commodity world SX/SM V X need be positive. Indeed, at full employment in even a two- commodity world, increases in M cannot result in changes in the  94 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM amounts of both commodities produced. This will be dealt with in more detail. The terms in SY/SM are all positive. Clearly SY/IM is itself positive. Since Y is defined as money rather than real income, this conclusion should be obvious. The relations described and discussed above contain the essential information provided by the model on the determination of the stock of money and of the effects of changes in the stock of money on the variables (prices, physical output, income, and interest rates) of the model. These expressions are, of course, rates of change and do not, by themselves, provide us with the actual amount of change in any particu- lar circumstance. This information is derived in the following manner. We know that the money stock is, by definition, the sum of currency outstanding (C) and total demand deposits (D). Thus 94 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM amounts of both commodities produced. This will be dealt with in more detail. The terms in dY/SM are all positive. Clearly SY/IM is itself positive. Since Y is defined as money rather than real income, this conclusion should be obvious. The relations described and discussed above contain the essential information provided by the model on the determination of the stock of money and of the effects of changes in the stock of money on the variables (prices, physical output, income, and interest rates) of the model. These expressions are, of course, rates of change and do not, by themselves, provide us with the actual amount of change in any particu- lar circumstance. This information is derived in the following manner. We know that the money stock is, by definition, the sum of currency outstanding (C) and total demand deposits (D). Thus 94 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM amounts of both commodities produced. This will be dealt with in more detail. The terms in SY/SM are all positive. Clearly SY/SM is itself positive. Since Y is defined as money rather than real income, this conclusion should be obvious. The relations described and discussed above contain the essential information provided by the model on the determination of the stock of money and of the effects of changes in the stock of money on the variables (prices, physical output, income, and interest rates) of the model. These expressions are, of course, rates of change and do not, by themselves, provide us with the actual amount of change in any particu- lar circumstance. This information is derived in the following manner. We know that the money stock is, by definition, the sum of currency outstanding (C) and total demand deposits (D). Thus M = C + D. Taking the total differential of this expression yields (3.33) M = C + D. Taking the total differential of this expression yields (3.33) M = C + D. Taking the total differential of this expression yields (3.33) SC - SC SC SC SC dM= - dr+ --- dY + - dP, + d -dXc + Sr SY SPi ape Xk SC SD OD SD SD - dX + - de+ dY+ dP +- dP, + aXc a? SY a ii e SC SC SC SC SC dM = d- i+ dY + -dP + dPc + dX, + Sr Y Pi k X, 4 SC OD aD aD bD dXc + _r d-dY+- dP+- dPc + aX, o~r aY aP, ae C xi dX + b dX . Equation 3.34 can be simplified (since D + C = M) to SM S M SM SM dM- dr+ dY+ dXa dXc + Ef -5Y- _ Xk aXe 4 SM SM - dP, + - dP . TP k p, e (3.34) aD aD a dX + dX,. SX Xec Equation 3.34 can be simplified (since D + C = M) to SM S M SM SM dM= dr+ dY+---dXk+-- dX, + of T Y aX dx e dc SM SM d idPi + -- dP. (3.34) aC SC SC SC SC dM = -- di+ - dY + -c dP + -- dPc +-adX + rF Y pi, SP +X + aC 3D - SD OD SD dX + dr+ dY - d _ dP + aXe or o5 Y dY5 Pi,+5 ' bD aD dXk + dXc. (3.34) aXk aXc Equation 3.34 can be simplified (since D + C = M) to SM _ SM SM SM dM= dr+-- dY+- dX +-- dX + a SY Xx + aX d (3.35) (3.35) SM SM dPi + dP?, SFk SF Pc (3.35) The partial derivatives on the right-hand side of 3.35 are the terms developed previously. Equation 3.35 provides us with the vehicle to calculate the change in the money stock resulting from a change in one or any combination of the variables T, Y, X, X, P, or Pk once the size of the change(s) is (are) known. The partial derivatives on the right-hand side of 3.35 are the terms developed previously. Equation 3.35 provides us with the vehicle to calculate the change in the money stock resulting from a change in one or any combination of the variables T, Y, X, X, P, Or Pi once the size of the change(s) is (are) known. The partial derivatives on the right-hand side of 3.35 are the terms developed previously. Equation 3.35 provides us with the vehicle to calculate the change in the money stock resulting from a change in one or any combination of the variables T, Y, X, X, P, or Pk once the size of the change(s) is (are) known.  SOLUTION WITH A PASSIVE GOVERNMENT 95 SOLUTION WITH A PASSIVE GOVERNMENT 95 Equation 3.35 is also a partial differential equation which could be used, given proper boundary conditions, to derive an expression for the money stock. Equation 3.35 is also a partial differential equation which could be used, given proper boundary conditions, to derive an expression for the money stock. SOLUTION WITH A PASSIVE GOVERNMENT 95 Equation 3.35 is also a partial differential equation which could be used, given proper boundary conditions, to derive an expression for the money stock. sfSM -fSM [SM M= Y J mdr+ dY+ dX + Wi of Y Xy [SM _ SM [SM M dr + - dY+j dX5+ f a M dXc + dP + dPec + K, J Xe S I J e 1 8 aM_ o M M Ef dr+ dY+ f dX+ r1 o f a JY oXk J M [SM [SM dXc + J dPk + j dl + K, avc Pap- e -a (3.36) (3.36) (M (SM [SM J dXc + dPc + J dP + K, . e iix- f -ae- J -c (3.36) where K is a composite constant of integration. Since we already have a perfectly good expression for M (see the first part of this chapter), this integration and specification of initial and boundary conditions will not be carried out. A WORD ON FULL EMPLOYMENT AND EQUILIBRIUM The questions of the stability, existence, and uniqueness of equilibrium as well as of full employment are beyond the scope of this work. It has been shown that the model can be solved for the value of prices, outputs, income, interest rates, and the money stock, and the inter- relations between these factors have been developed. What we have not shown (or attempted to show) is whether or not this vector of solutions corresponds to a full-employment vector of output. We shall assume that it is possible for our economy to reach full employment without specifying whether or not this occurs (1) without active monetary policy or (2) in conjunction with an "unacceptable" (however defined) rate of increase in prices. In fact, chapter 4 is concerned with the effects and effectiveness of the various tools of monetary policy under two different assumptions about the state of the economy: that the economy is in an under-full-employment equilibrium (how and how effectively do the various types of monetary policy affect real output or employment?) and that the economy has reached full employment with an unaccept- able rate of price inflation (how and how effectively do the various tools of monetary policy combat inflation?). where K is a composite constant of integration. Since we already have a perfectly good expression for M (see the first part of this chapter), this integration and specification of initial and boundary conditions will not be carried out. A WORD ON FULL EMPLOYMENT AND EQUILIBRIUM The questions of the stability, existence, and uniqueness of equilibrium as well as of full employment are beyond the scope of this work. It has been shown that the model can be solved for the value of prices, outputs, income, interest rates, and the money stock, and the inter- relations between these factors have been developed. What we have not shown (or attempted to show) is whether or not this vector of solutions corresponds to a full-employment vector of output. We shall assume that it is possible for our economy to reach full employment without specifying whether or not this occurs (1) without active monetary policy or (2) in conjunction with an "unacceptable" (however defined) rate of increase in prices. In fact, chapter 4 is concerned with the effects and effectiveness of the various tools of monetary policy under two different assumptions about the state of the economy: that the economy is in an under-full-employment equilibrium (how and how effectively do the various types of monetary policy affect real output or employment?) and that the economy has reached full employment with an unaccept- able rate of price inflation (how and how effectively do the various tools of monetary policy combat inflation?). where K is a composite constant of integration. Since we already have a perfectly good expression for M (see the first part of this chapter), this integration and specification of initial and boundary conditions will not be carried out. A WORD ON FULL EMPLOYMENT AND EQUILIBRIUM The questions of the stability, existence, and uniqueness of equilibrium as well as of full employment are beyond the scope of this work. It has been shown that the model can be solved for the value of prices, outputs, income, interest rates, and the money stock, and the inter- relations between these factors have been developed. What we have not shown (or attempted to show) is whether or not this vector of solutions corresponds to a full-employment vector of output. We shall assume that it is possible for our economy to reach full employment without specifying whether or not this occurs (1) without active monetary policy or (2) in conjunction with an "unacceptable" (however defined) rate of increase in prices. In fact, chapter 4 is concerned with the effects and effectiveness of the various tools of monetary policy under two different assumptions about the state of the economy: that the economy is in an under-full-employment equilibrium (how and how effectively do the various types of monetary policy affect real output or employment?) and that the economy has reached full employment with an unaccept- able rate of price inflation (how and how effectively do the various tools of monetary policy combat inflation?).  4. Solution with an Active Government IN THIS chapter two major questions will be examined. First, how do the major tools of monetary policy work? That is, how do these tools affect the important variables of the model (in particular, the money stock and the various rates of interest)? Second, given the effects of these monetary tools, how do they affect problems of unemployment and inflation? What are the qualitative and quantitative differences among the monetary tools? CHANGES IN RESERVE REQUIREMENTS The reserve requirement, r, enters the model in two places. The level of required reserves and the quantity of bank loans supplied are both affected by changes in r. We have 4. Solution with an Active Government IN THIS chapter two major questions will be examined. First, how do the major tools of monetary policy work? That is, how do these tools affect the important variables of the model (in particular, the money stock and the various rates of interest)? Second, given the effects of these monetary tools, how do they affect problems of unemployment and inflation? What are the qualitative and quantitative differences among the monetary tools? CHANGES IN RESERVE REQUIREMENTS The reserve requirement, r, enters the model in two places. The level of required reserves and the quantity of bank loans supplied are both affected by changes in r. We have 4. Solution with an Active Government IN THIS chapter two major questions will be examined. First, how do the major tools of monetary policy work? That is, how do these tools affect the important variables of the model (in particular, the money stock and the various rates of interest)? Second, given the effects of these monetary tools, how do they affect problems of unemployment and inflation? What are the qualitative and quantitative differences among the monetary tools? CHANGES IN RESERVE REQUIREMENTS The reserve requirement, r, enters the model in two places. The level of required reserves and the quantity of bank loans supplied are both affected by changes in r. We have R = r(D + T) (4.1) Lb = f1(r)(D + T) + A12b (4.2) where we have used the function notation f, (r) to replace the term r in Equation 2.92, since this notation clearly shows the dependence of LS on r. R is, of course, simply the level of required reserves. Expressions 4.1 and 4.2 indicate the effects of changes in r. Consider an increase in r, dr > 0. From 4.1 this has the obvious effect of increasing required reserves: R = r(D + T) L = f,(r)(D + T) + AtoFbi (4.1) (4.2) R = r(D + T) = - f,(r)(D + T) + A,2b? (4.1) (4.2) where we have used the function notation f, (r) to replace the term n in Equation 2.92, since this notation clearly shows the dependence of Lb on r. R is, of course, simply the level of required reserves. Expressions 4.1 and 4.2 indicate the effects of changes in r. Consider an increase in r, dr > 0. From 4.1 this has the obvious effect of increasing required reserves: where we have used the function notation fl(r) to replace the term a in Equation 2.92, since this notation clearly shows the dependence of LS on r. R is, of course, simply the level of required reserves. Expressions 4.1 and 4.2 indicate the effects of changes in r. Consider an increase in I, dr > 0. From 4.1 this has the obvious effect of increasing required reserves: dR = dr(D + T). (4.3) dR = dr(D + T). (4.3) dR = dr(D + T). (4.3) The immediate impact of this increase in R is a reduction in the banks' holdings of currency, government securities, and firms' securities in an amount equal to dR: The immediate impact of this increase in R is a reduction in the banks' holdings of currency, government securities, and firms' securities in an amount equal to dR: 96 The immediate impact of this increase in R is a reduction in the banks' holdings of currency, government securities, and firms' securities in an amount equal to dR: %6  SOLUTION WITH AN ACTIVE GOVERNMENT 97 SOLUTION WITH AN ACTIVE GOVERNMENT 97 SOLUTION WITH AN ACTIVE GOVERNMENT dR = dr(D + T) = d(Gb + Fb + DO) dR = dr(D + T) = d(G, + F, + DO). (4.4) dR = dr(D + T) = d(G + FE + D). (4.4) (4.4) We assume that the banks reduce the levels of these assets in proportion to the coefficients of D + T in the banks' demand for them. That is, We assume that the banks reduce the levels of these assets in proportion to the coefficients of D + T in the banks' demand for them. That is, dGb = P dR p + '+ p dF, = P dR p + '+ yA dCb = ' dR. p+ p+7 (4.5) (4.6) (4.7) dGb = P dR p + '+ y dFb = 1 dR P + -+ yA dCb = dR. p+ y+ 9Y (4.5) (4.6) (4.7) Consider the banks' attempt to reduce their holdings of government securities. These securities may be purchased by either the public, the firms, the intermediaries, or the government (if it wishes to take some action to help offset the "crunch" of a change in the reserve require- ment). We can then write dG = dGe + dGe + dGn + dGe. (4.8) We assume that dG is determined by the government strictly on the basis of economic policy and may range from zero to dGb. The amount of dG not absorbed by the government is divided among the private sectors in this manner: dGn - dG, = dG, + dG, + dG Consider the banks' attempt to reduce their holdings of government securities. These securities may be purchased by either the public, the firms, the intermediaries, or the government (if it wishes to take some action to help offset the "crunch" of a change in the reserve require- ment). We can then write We assume that the banks reduce the levels of these assets in proportion to the coefficients of D + T in the banks' demand for them. That is, dG = P dR (4.5) p+ Y+ y dFb = dR (4.6) p+ + yI dCb = Y dR. (4.7) p+ P + Consider the banks' attempt to reduce their holdings of government securities. These securities may be purchased by either the public, the firms, the intermediaries, or the government (if it wishes to take some action to help offset the "crunch" of a change in the reserve require- ment). We can then write dG = dGe + dG, + dGr + dGs. (4.8) We assume that dG is determined by the government strictly on the basis of economic policy and may range from zero to dG. The amount of dGb not absorbed by the government is divided among the private sectors in this manner: dG - dG = dGe + dG + dG, dGn = dG, + dG + dGr + dGg. (4.8) We assume that dG is determined by the government strictly on the basis of economic policy and may range from zero to dG. The amount of dGb not absorbed by the government is divided among the private sectors in this manner: dG - dG = dGe + dG, + dGn dG, = (dGb - dG ) dGe = f (dGb - dGg) g~9rg,, d1 dGn = "a (dG - dGg), g+ ge+ g d) (4.9) (4.10) (4.11) dGe = g (dGe - dG1) dG = 9 (dG - dG1) dGn = "e (dG, - dG1), g+ g + g (4.9) (4.10) (4.11) dG, = g (dG, - dGg) g + g +g (4.9) dG = 9r (dGb - dGg) (4.10) dG, = g (dG - dG), (4.11) g+ g, + gn where the g's are the major parameters in the sectors' demands for government securities. These three sectors pay for their purchases of government securities by reducing the level of all other (financial) assets they hold (except for where the g's are the major parameters in the sectors' demands for government securities. These three sectors pay for their purchases of government securities by reducing the level of all other (financial) assets they hold (except for where the g's are the major parameters in the sectors' demands for government securities. These three sectors pay for their purchases of government securities by reducing the level of all other (financial) assets they hold (except for  98 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM firms' securities). The reductions in other assets for each sectre are given by dGe d~es + dCes + dN01 + dT (4.12) whee h1p k,0+c3a e ~ (4-13) d~es k2Ic d~e (4.14) k,1 + Ik2 + k3~ + u dNet n,5 dcv (4.15) kc1 + k2+ k3+ ne dTpg k3 dOp (4.16) dGf dO11 + dO15 + dNee + dTf, (4.17) where 98 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM firms' securities). The reductions in other assets for each sector are given by d =dDet + dCes + dNc + dTee (4.12) where 98 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM firms' securities). The reductions in other assets foreach sector are given by dGp = d~er + dCg + dNg + dT (4.12) where d0v1 dCpg dN,,g dTeeg dO1 where d0ee dN,, dTf, k,5 + k,1 + k,3 + n k 2 dOp k, + k,+ k3 +u -kI3 dGp; =dO~g + dC,, + dNf, + dT,, df + a, + rcf + t - at dOf df+ a. +Sef+ttf df + a, + of + t (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) (4.19) (4.20) (4.21) d~ve dCet dN01 dTes dG, ee dNe dTfg k -dGe k+ k2 + k3 + u k+ kc2 + k3+u S np dOp k I3 dOp k,5 + kc2 + k3~ + u =dOf, + dCte + dNee + dTrr - dt dGf df+ a. + nr + t -at dO, dt+ a, + nt+ If df+ a, + elf + t If te f d+as+ nr + tf (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) (4.19) (4.20) (4.21) d~fe dots dNf, dT,, - f 4e df+ a. +nof+ If as dOf df + a +enf+ tf df+ as + nf + It - tt dof df +a.~rs + f I (4.18) (4.19) (4.20) (4.21) and dO, = d~ve + dO55 (4.22) and dO0 = dDoe + dO0 (4.22) dGn = do51 + dn, (4.22)  SOLUTION WITH AN ACTIVE GOVERNMENT 99 SOLUTION WIT AN ACTIVE GVERNMENT 99SOLUTION WITH AN ACTIVE GOVERNMENT 99SOUINWTANCIVGVEMNT 9 SOLUTION WITH AN ACTIVE GOVERNMENT 99 where dO55 = d,, dO d +0C dWnn = Cn dOn dn + C (4.23) (4.24) where dnO = dn dOn d + c dC,1- -5 dO, d~g=dn Sc (4.23) (4.24) whers dKnn d5C dOn 4n +0c (4.23) (4.24) Again, the paramtrs min preceding equations ate fromn the snctors' demands for the nations assets. Ignoring foe a motment the impact on interest sates, the results of this initial sale of G by the honks can be written 01 dDnt dO0, + dO,, + do,,1 +dp +(dO) ~f +d + ' (4.25) whese 40n, means the first change in demand deposits resnlting ferom the honks' sale of government securities. By combhining 4.25 and she system 4.9 through 4.11, we can write dO11 = k, )( 9 ) k,______ +g,+k n p 1+g ( )( on )] (dO5-0 dO,, k. kg + dfgf dng0) k, +k2+hk3 + n d + a. +nf + tf d + c5 (dGn - dGll (4.26) g + gt + g Proceeding in a sumidae masses we can inrite the oxpression foe the initial impact on lime deposits resuleing from the hooks' sale of govern- ment securities as Again, she parnmeters in she prenceding equations ate feom the sectors' demands fat the narions assets. Ignoring foe a moment the impact on interest rtes, the results of this initial sale of 0 hy the hnhs can he written as dO1, dO51 + d~fg + dO51 -k, + k2 +k3+no, +d~n d.+ nf + tf(df + d,, (dOn) dn +0C (4.25) whets d~s, means the fiest change in demand deposits resulting from the honks' sale of government securties. By comhining 4.25 and the system 4.9 thsough 4.11, we con write dO1, k, )q ( h, + k2 + h03 + n0 g + gf + g _______ ) 15f df +oa. + n,+ tf g+ gf+ g n I( n )] (dO5Ob d_ __ n+ n9+g + )n h1hth3n dtg+05+ntf t+ d+c0 k,+k4+k01) ,+a,+n I C Again, she paeameters in she preceding equations ass from she sectors' demands foe the vatious assets. Ignoring foe a moment the impact on interest sales, the results of this initial sale of 0 hy she honks can be written as d~o = dDen + dDi+ d~nI k,+ k 2 + k3 + up ~ df + as + of + tf + n (dn) (4.25) whets d, means the fallst change in demand deposits resuolting from the honks' sale of government securities. By comhining 4.25 and the system 4.9 through 4.11, we can wedse d~ns I k, I ( dn ) ge9 de+ a, + nf+ tf g+ gf+ g. d. + c. g +gf+ g. -~ h kg + dfgf +~ tdn (d~n - d~gn) (4.26) Ig + 1,, g Proceeding in a similas mannes we con write the enpression foe the initial impact on time deposits totalting Item the honks' sole of gassesn- menttsecuities an (d~n - g g + ltf + P5 (4.26) Psoceeding in a similar manner we coo welts the enpression for the initial impact on time deposits resualting from the honks' sale of gonvern- ment securities as  100 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM dTg k~g + 1111 k, 1c+k, + k3+ n, de+ a,+ ne+ If dO0 - dG,, (4.27) g + Be + g. The expeesson fot the initial impact on deposits in intermediories is dN,, n.9________ + nfgf (d~n 4%(4.28) B + Be + g The initiol impact on cureency balances is given by dO1 - k~g + as ge + k, + k,+ k,+n P d + a, +nof+If 1180 (4.29)d We now ennmine the initial impact of the honks' sl of fiems' dP5 A dR =dP + dE (4.30) Peoceeding in an anlogons monnee, dFn = f dE1 (4.31) dE, = b- dEb (4.32) These puechases of fiems' securities by the public and the intennediaries canses a fuethee sedaction in Itie othee assets, given by dE0 = dO0, + dC0f + dN0, + dT,, (4.33) whee 100 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM dT11 ( k~g + tff - ks+lk2a+ k3n+ n df + a. + nf+i d~t, - dO1 ) (4.27) g + Be + g The epessionn for the initial impact on depnsits in inermediaries is dt4g I tt0B + nfgf (d~n - d~e) (4.28) g + Be + te The initial impact on cuseency balances is given by dO11 - 2 ( - + asgt + d.cgXd~ - dOI). (4.29) We vow examine she initial impact of the banks' sole of beats' secarities: dE5 - R dRF + d~n. (4.30) P + /I+7' Proceeding in an onaiogoos manner, dP = f dP5 (4.31) dF n dE5. (4.32) f + b These porchases of firms' securities by the public and she intermediories causes a forther reduction in their other assets, given by dP = d~ne + dC55 + dNPg + dT5, (4.33) whete 100 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM dTg, -( hg + sno dO I - dO1 ) (4.27) B + Be + Bni Tbe epession foe tbe initial impact on deposits in intermediaries is dN11 -I npg + attn (d~b - dOI 4.0 (i+ Bt B The initiai impaci on currency balances it given by dCg~ I ( hg - + attn + k, +1k2 +1k, +nup de +ne +ss OntsX ~ -d (4.29) Bn Ber 9+g + gn We now enamine the initial impact of the banks' sole of firms' securities: dE5 - R 11 dRdE + dE (4.30) Proceeding in n analogoos manner, dP = f dE1 (4.31) d f + b. fo = b d, (4.32) These parchases of beats' snesrities by tbe public and she intermediaries causes a farther redaction in tbeir otber assess, given hy dF1 = dO55 + dC1, + dNee + dT55 (4.33) whee  SOLUTION WITH AN ACTIVE GOVERNMENT =~f k, dEP dC01 -= k dFI dTee = kI3 dE-; dF, dO,, + dC1 wher 101 (4.34) (4.35) (4.36) (4.37) (4.38) SOLUTION WITH AN ACTIVE GOVERNMENT -Df= k dFP dO1 k, + k2 + k3, + a0 dCf= k dFP kC, +a 4k2 +4k3 +nu, dN,,e = n 12..~... dFP k, +k, +k3 +a, -Tf k3 dF; dT,, k+k, +k3 + n, dF,0 = dD,, + W.I. wher 101 (4.34) (4.35) (4.36) (4.37) (4.38) SOLUTION WITH AN ACTIVE GOVERNMENT -~ k dFO kia + k2 + k3 + n dC,1 k, dO,, k,+ ka2 + Ia3 + u =N, np dOP -Tf k dFP dO,,1 dO,,, + dC,,1 where 101 (434) (4.35) (4.36) (4.37) (4.38) don = d,, do 4.9 C dn + a (39 dCce = an dO,(.0 dl + e4n0 The initisl impacta reaultiag feocm the bans' sale ef fiems' scurities ace gives by dO11 = ( k, f + d,,b1 dO5 (4.41) k,+ k, +k, + n dc +c nIF+b,, dTf, = ( k~f )dO5 (4.42) k, + k,+k3 +a1 f+ bc dC11 = ( k2f + ascba F (4.43) k, +k2 + k3 +, dp,+o, f+ bc dN1, = ( a~f ) 00 (4.44) k, +Ik,2+Ik,+sn, f + b The least isitial imspacts on, D, T, C, and N ace obtained by aimply addisg the sappropriate paira of equatiosc and simplifyiog. dO1 4.41 + 4.26 k, (g +D) d,, (g,, + bc)] k, + k2, + k3 + up , ,+ dn d~ce = da do,,439 dn + (439 dC,,1 =- dOn 4.0 dn +0 a(40 The initial impsacts ceculting from lhe bansla' sale of firma' ascitiec ace gives by df, = ( 1~ + ac)n 5 ~ (4.41) dT11 = ( k~f dO5 (4.42) dCf 1 = ( k~f + Cabs F (4.43) k, +Ok2 + k3 +5 dPf+ , fb b The local initial impsacts os 0, T, C, sod N see obtained by simsply adding lhe appropriste psica of eqations and aimplifying. dO1 4.41 + 4.26 = k, (g +Qf) d,, (g +b,,)] k,1+ k,+k3 +n, o,,+ d dO,,1 - d, dOn(.9 r + a 4.9 dCnf = Cn dFn-(.0 dn +, (440 The isicial imcs reslting from tbe bans' sale of firma' seuitiesae gives by dO,1I =( kif +dnb dO0 (4I k,1+ k2 +k3 +aII, d.,, 0Cc IF+b5 dT11 = k~f dO5 (4A2) k+k2k3 I + ob= __ dCfa = ( ~ +____ Os,, (4.43) k,1+k, +k3 +s,, d, +a,, fo+b, dN11 = k,~ - +02+k u F+b (4A4) The local iniial impacts on 0, T, C, sand N are obtsised by simply sddiag lhe sppropciste pairs of equstios and aimplifying. dO, =4.41 + 4.26 k,______ (g +f + d (gc oh,,)] k, +k3 +a n, o+d  102 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM dGb - dG + dFb]+( dg g+ g+g, f+b, d+ a, +n1+ t (dGb- dG8 g + gf+ 0,, dT, = 4.42 + 4.27 k3 (f + g) dGb - dG + dFb k,+k2+k3+ g g+ g5+ g f+bn tfgf dGb - dG) df + a. + nf + tf g + g + g, dC, = 4.29 + 4.43 k2 (f + g) + e(b. + gn) k, + k2 + k, +n, cn + dn F51 dGb - dG ag5 f+h b g+ g+gn d +n15+ tf dG - d~g) 5 + gf + gn dN, = 4.28 + 4.44 n (f + g) Fb + dG - dG k,+k2+ k3+B, f+bn g+ g+g + nfgf dGb - dG, + ( ) ( . d d4-+ as + n5 +1 t Ig+g )+g 102 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM dG -- dG + dFb dfgf g+ g+ g f+b d+ a, +n+ tf 102 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM (4.45) (dGb - dG) 5 + 9f + 0, (4.45) (4.46) dT1 = 4.42 + 4.27 k3 (f + g) dGb - dG + dFb k 2k+k2 +g g+ gf+ g f+bn ( tg1 (dGb - dG). df + as + n5 + t1 g + g, + g dC, = 4.29 + 4.43 _ k2 (f + g) + n(bn + gn) k, + k2 + k3 + n, c + d ( F + dG - dG)+( + asgr f+bn g+ gf+g df+a, + n+ tf dG - dG g + gf+ g. dN, = 4.28 + 4.44 nP (f + g) Fb + dG, - dG k + k2 +k3 + n f + b g+ g,+ gn + ( ) ( + . ) dk s + f + g +g,, 'f + 1g (4.46) dGb - dG[ + dFb ] + ( drg g+ g5,+ g f+ b df+ a. + nt,+ t (dGb - dGg 5 + gf + g dT, = 4.42 + 4.27 k, (f + g) (dGb - dG + dF, kI+k2+k3+ g g+ g8 +g f+ b, ( tfgf dG, - dGg df + as + nf + tf g + gf+ gn dC, = 4.29 + 4.43 k2 (f + g) ea (bn + ) k, + k2 + k3 + n, en + d, Fb + dGb - dGg) + asg, f+ b g+ g+g d+a. +n5+ 11 dG - dG, B + St + 65 dN1 = 4.28 + 4.44 n, (f + g) F, + dG, - dG k, +k2 +k3 + f+b, g+g,+g ng, dG, - dG df + a5 + nf + tf g + g, + gn (4.45) (4.47) (4.47) (4.47) (4.48) (4.48) (4.48) The sum of 4.45 and 4.46 gives the initial reduction in bank deposits. This, of course, results in another reserve deficiency for the banks and thus initiates another round of asset adjustment throughout the econ- omy. The new level of required reserves is given by R new = (r + dr) [) + T - (dD + dT)], The sum of 4.45 and 4.46 gives the initial reduction in bank deposits. This, of Course, results in another reserve deficiency for the banks and thus initiates another round of asset adjustment throughout the econ- omy. The new level of required reserves is given by R new = (r + dr) [D + T - (dD + dT)], The sum of 4.45 and 4.46 gives the initial reduction in bank deposits. This, of course, results in another reserve deficiency for the banks and thus initiates another round of asset adjustment throughout the econ- omy. The new level of required reserves is given by R new = (r + dr) [D + T - (dD + dT)],  SOLUTION WITH A PASSIVE GOVERNMENT 103 while actual reseves are sow R actuaol = (r + dr) (0 + T) - (do + dT), so thatlthe esere deficincy is the diffeene betwen these Iwo R def = (dO + dT) [I - (r + d)]. (4.49) Is terms of Equations 4.45 and 4.46 R def I (hk ( of +3 (g + f))] dG - dO1+ dE0 9 +gf n T+ fb] gf (d + tf) dSG, - dS0 I (e-I + d)t . (4.50) None than Equation 4.50 can be written as R def = [( k, k ) (d~ +dFe)O+ k, Ok2 +k3+nup df+ If dn (dO5 + dF~n)) [I - (r + dn)] ,(4.51) sshene esch teem on lbs eighs-hand side is the sie of ose of the nosbank private sectot's renductiin sf its hslding sf demasd and lime depssits. Funthermoe, we know slims dO, = ) K( ) (de(O + T)) -dO1] (4.52) dG, = gf ( 3 ( (dn(O + T)) -dGn] (4.53) g + ge+ gs P + Y+ P SOLUTION WITH A PASSIVE GOVERNMENT 103 while actualeserves one now R actual = (r + de) (D + T) - (dO + dT), so that the reserve deficiency is the diffeence benween these Iwo expressions, on R def = (do + dT) [1 - (r + dr)] -(4.49) Is teems of Equatioss 4.45 and 4.46 Rdf (kc + k)(g + f) + 4n (n+b R de = f +~643n + I ~ += + ) gf (df + en) dG - d015 11I- (no+ dn)1 . (4.50) Nose than Equation 4.50 can he weitten as R def =[( k,+k ) d +FP+ hI oh oh +k,+ )p ed~) dn + i doto+( dn (dO,, + dE,)] [I - (n + de)), (4.51) where each teem on the night-hood side is she sine of sone of the soohank peivate secton's renduction of its holding of demand sod time deposs. Furthenmoe, we know that dGe = K P [( ) (dn(O + T)) -d~g] (4.52) d~n 3f K( 3 (de(O + T)) -dO1) (4.53) g ~+ n g+g. p +yf p SOLUTION WITH A PASSIVE GOVERNMENT 103 while actunl nesenves ae now R actual = (no+ de) (D + T) - (dO + dT), so that the resenve deficiency is the diffeence between these two R def = (dO o dT) [n - (no+ de)). (4.49) In terms of Eqationn 4.45 and 4.46 R df (k, +okh)(go+f) o+oh O cd (gnr hob)] dO0 - dO1 + __I 9 +gf n f +h 1 K gf (df + tf) dO5 - dO131 1- (no+ 40)1 (4.50) Note shot Eqoation 4.55 can he wedtten as R def = h( k h, + s)(dOP odE 30 k+ k2 k3 + Onp dtfo+ nt, d~n+ ( d, (dO= + dE0)] [I - (no+ dn)], (4.51) where each tenrm son the righn-hand side is thn sine of one of the nonhank prisate secton's nedoction of ill holding of demand and time deposits. Fnrthermone, we hnow than dO1 9 ) K [( (de(O + T)) - dOGI (4.52) = + If n 0 Y+1  104 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM dG,,= (g )K P ) (ds(Oo +T)) - dO1] (4.54) g0R0, +g+g. p +5' + 104 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM gl( g1=0(gg ) [( ) (de(O) + T)) - dO1] (4.54) 104 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM dE = ( f) ( A) de(O)+T) dF =( bn P ) dr( + T). fo+b,P OyO+ 1 (4.55) (4.5 6) 4P =( )( 1 ) dr(Oo+T) P f +hb,, p~+p +/ 46,, = , b do(O)+ T). IF+ b goy + (4.55) (4.56) 6,,g g + ' dF0"=( f) ( ) de(O)+T) 46,,= ( b. )( P ) do(Oo+T). fo+bn 6 +5+6P d~gll (4.54) (4-56) R dof 1010111 0n another reduction of she banks' holdings of gonern- ment secusities, firms' securisies, and currency which, in loon, cue anotherlearrangement of she asses porsfolios of she varOious 1001011. This complen p100011 continues, fooming an infinite series of changes in assess. The solnsion renolves aenand shore Iwo problems: findiog she gonoral enpression foe she series of reoserne deficiencies, and finding whether or not shis Series connerges and, if so, to whas. We introduce she following nosation: R def 1 = do(O + T) kdf ,~h + k2 + k3 + u d+ f d~f +( dn droornea s +n+if c,,+ d, (dOn + 460)] [1 - (en+ do)] . By sabstituting 4.52 throungh 4.56 int 4.51, we ohtain R def 2 = i +1 ohk 5 g I - (r +do) h1 + It3+n g U + grn+g K )RdefI l dO (-] p 1 ) def rde + n.f R def resultssin anosheo reduction of she banks' holdings of gosern- mentlsecuoities, firms' secaritios, and caeoency which, in tnon, caoos anotheroearrangement of she asses porsfolios of she salinas 1001011. This cowplox process continues, fosming an infinise series of changes in assess. The solution oenolves asand obese Iwo psoblems: finding she general enpression (nor she series of reerve deficiencies, and bonding whether or nat shis seies conveoges and, if so, In whas. We insrodace she fallowing notation: R dof 1 = dr(O + T) R def 2' = [( oh+k0 (G dP k,+ k2 + k3 +0up df +0a5 + n10 0,, od (dGe 0 46,,) [I - (r + do)] . By substisusing 4.52 shrough 4.16 int 4.51, we obtain R def 2 k, +h jo g s 1_- @,dr) k, oh02 +hk, +On ( + g +g. [K )+ )R def I1- do1] ( - P ) Rdef + d00s0i p + y + as+d, n+or R dof resultsin another redaction of she banks' holdings of gonern- mentlsecrities, firms' secuoities, and carrency which, in loon, canses anothersearrangement of she nase portfolios of she sariouas sectors. This complex process continuos, fosming an infinite series of changes in assess. The solation resolves aroand these Iwo problems: finding she general esprersion foo she series of rservne deficiencies, and bonding whether or not Ibis series converges and, if so, to whas. We introduco she following nosation: R def 1 = do(O) + T) "Rdef2 =[( k,+k d +FP+ k, + k2 + k3 On up dn F0 d, + If__ dCf + ( )n (dO,, + d~F,, [I - (r + do)].- By sunbstituting 4.1 2 throuagh 4.16 into 4.1 1, we obtain R def 2 k,~ + oh3 9___ I1- (rcods) k, oh k h2 on k3+p g +g +g. 6 f [( ) Rdef- + ' )f+I P + )Rd+ f I a( + )f+or+I  SOLUTION WITH AN ACTIVE GOVERNMENT 105 5 f K [ )(R def 1) 0+1+n +755 +g +g P1~ ) (Rdef) - do1] b SOLUTION WITH AN ACTIVE GOVERNMENT 105 ( f g) [(5 ) (R def 1) -dGg]( +e,+d., ( K____(R def) - d1 + b A ) Rdef) 11 +7Y+P1 SOLUTION WITH AN ACTIVE GOVERNMENT 105 5 f [( )(R defl1) 050e1(. )1+7+1 K5 )(R def) - do1] + f b P +/A +'YI + ( )1R def 5 11+7'Y+1P1 (4.57) (4.57) 5 ) Rdef5 11+'Y+1/I (4.57) In oedee to aimplify She analysis, we asawe that after itl first purchkase of gnvernwmens securties, She ganernment purchases nomr froms She banking systemn while the readjustment process is working itself ent. This assumption petmits as en write the general enpeesion for any reserve deficiency as R def j =[ k, + k3 d5 Fj kr + k, +lk, + n, d I, (dGej)+ nd In ordet to simplify the analysis, we assnwe that aft er its first purchase of gavernment securities, the goveenmena parohases no wale flaw the hacking systew while She readjustent ptocess is woeking itself out. This assumptionnpermitsus to witehe geneal expressian fneany tesetve deficiency as R def j=[( k1 +k0 3 (dO~ + dP)+ d, +nei t (dGfj)s + d In oedel to simsplify the analydis, we assume that aftel its first putohase of goveram eat securities, the govetrnment pastchasesnomr feom the hacking sysoem while Ike seadjustment process is wotking itself out. This assumption permits as In write the general expression foe any teserve deficiesscyas R def j=[( ks+k d~ ~j k, + k+ k3 + eeu e de + a, + nf + If c ,+ d wherehecause of oat assaumptions about She govenenta, dOP 9= ( )R def j -1 d05=( )( )R defj- 1 d~s1= 55 def j- 1 f + 11P Y (4.581) whete, because of ens assumptions about the gaveenment, d01( )( )R def j I dO5i = f 5 5 R def j I gO+0ef+0 11 +7'Y+11- 0e~ g S 11 R defj 1 g + g1+ g1 1+7Y+11I (4.58) (df41+ d5)] [I1- (rt+ d)] , wheee, hecause af eel assamptions abeut the ganeenmene, d05( )( )R def j I 0 + Os + On P1+7'Y+1P 0d 010 11 P71 )R defj- I dO .-( 9 ' ____ R defj- I N 0+Otf+0n 1+7r+1/J (4.58) f 1 dF1 = ( 55 )Rdef j- I f + b p+ y + 1 d~ f 11 df IF +( 55 P )Rdefj+ S  106 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM ff~ = b ) / ) Rdef j- 1. Sabstituting these expressioes isle 4.58 ted dividing eteogh by R defj - 1, we bae R def j - k k KE g ) R2ej + +, + n,5 g + gf + g P + f P A+ ( )+ , ( d.1g P ) + [( b)) 1 p +y+1 b p y+p [I - (t + d)] . (4.59) Tbis testult helds fee all R def j, j > 4. Cell this tatie of rseret deflcejecet Q,. Thee it fellews tbat R def 4 R def 3 = Q, => R def 4 = QRdef 3 R def 4 106 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM efi=(b ) ( A ) Rdef j- 1. Sabstitating thete exeptesstitns iete 4.5 8 aed dividing shroagh by R def j - 1, we bate R def j kt +h k2 R defj - I ki + k2 + k3 +t up Be1 + p f ~ d, ±(t )( )]+ d5a +1t d. R gn 9 + 4y+1 .+cl f+g ( +~ ( )( 11 p + ay+1 b [t - (t + 4sf]. (4.59) This result heleds fee all R def jj > 4. Call this ratie ef resete deficeeces Q,. Thee is followt that R def 4 R def = Q, => Rdefd4= Q,R def 3 R 4f 3 R defd 4 Q, => Rdef 5 -=QRe 106 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM b P dF n) ( - ) Rdef j- 1. Substitutsing shese expeessiess isle 4.58 aed ditidieg throsgh by R 4sfj - 1,we hate R def j k, h+h k o R def j- - lk, +hk2+hk3 +s g + gf+ g 9+7 + fb p- A + ( deft + 8f t P ) e + 8. ( ) + ( -)( ) P+t+ 1 + b p8 [- (t + de)] . (4.59) This sesult helds fet all R def j, j > 4. Call this eatio of tesere deficieseet Qr. Thee it followt thee R 4sf 4 R 4sf 5 R def 4 Qr =~> R 4sf 5 =Q~Rdef 3 R def I = Q => Rdef i= Q-3Rdf3 R def i- t Qrde Relfi- I =Q, => Rdehf i= Q'trd Retefi~~ R~ =deiQ def 3. R def i t  SOLUTION WITH AN ACTIVE GOVERNMENT 107 The expression fee she tetal teseeee deficiency, R def T, can be weitten ns R def T =Rdef1+ R def 2+ Rdef3+ R Rdef i (4.60) =Rdeft +Rdef2+Rdef3+lim QR def3. Thi infinite seieswillcnege t R def 3/ -Qsif I QI <.Since each teem individually in Q, is less shan 1, Q, will he less than I dane is multiplicative stuctute. (If Q, wete greater than 1, any change in the eesetve teqairement moald bane an infinitely large impact en the ecenemy since the tetal excess at deficit in neserves weuld he infinitely large [small]) We can wreite she toal eeseeve deficiency as R defT =R deflI + Rdef 2+ Rdef 3(1 + ) (4.61) twhere Rdcef3 =Q ( , 3 5 k, + f2+k3+n + f+g ( )de(O±+T) +( ) ( ) dn(O+ T)] 0 +'Y+6t f+ bn +7y+6p ( )de(O+ T) +( )J[ )K 0 + y+1 cn + dn 0 +'Y+6I + gf + g hn +5n P'o + o' SOLUTION WITH AN ACTIVE GOVERNMENT 107 The expression fan the tosal reserve deficiency, R def T, can he written as R def T=R defI+ R def 2+Rdef 3+ R Rdef i (4.60) =RdeftI + Rdef 2 +Rdef 3+ im (Y Q def 3. This infinite seres will cenerge te R def 3/1 - rif I Qr I< 1. Since eachteermindiidaly in Q isesthan1, Q,n'dihle less than Iduesto its muldtiplicative structane. (If Qe were gteater than 1, any change in she reserve requirement neald have an infinitely latge imnpact en she econemy since the tosal excess or deficitsintesenves weald be infinitely large [small] .) We can mrite the natal teserve deficiency as R def T=R defI+ R def 2+ Rdef 3 (1+ ) (4.61) whete R def3 =Q k, h kh ___ k, h k +l k+h up ne gf + ge ( )dt(O+ T) +( ) ( P ) dn(O+ T)] as + df~s + r+I g + On + g ( ) dn(OC+T) + dn AR 0 ( gn +( bn f61 ( 1 + gf + gn 5ea+cn P+ Y+o [I - (r + de)]. (4.62) There seems litsle neason to wnite 4.61 eat in fall except te anden- scene the complexity added when a model of she money mechanism is made jest a his mene realistic. SOLUTION WITH AN ACTIVE GOVERNMENT 1037 The expesmion fee the natal nesetve deficiency, R def T, can he written an R def T=R defl1 + Rdef2 +Rdef3+ R Rdef i (4.60) =RdefI+ R def 2 +Rdef3 +hm IQjR def 3. This infinite seres will converge to Rdef 3/1 Q, if I QI 0). Likewise, the quantity of loans demanded by the public and the firms wili be reduced when r is increased, since interest rates will move higher while income, prices, and real output tend to fall as the money stock The quantity supplied of bank loans is reduced by an increase in r, ceteris paribus, for three reasons: an increase in r reduces D + T ((D + T)/r < 0) (this impact on D + T has been discussed); an increase in r reduces the amount that may be lent per dollar of deposits (Sf1/r < 0); and an increase in r increases the rates on securities held as second- ary reserves which, without an increase in loan rates, reduces the amount banks are willing to lend per dollar of deposits (a2 and b, 2 < 0 combined with Sr/Sr and org/ar > 0). Likewise, the quantity of loans demanded by the public and the firms will be reduced when r is increased, since interest rates will move higher while income, prices, and real output tend to fall as the money stock The quantity supplied of bank loans is reduced by an increase in r, ceteris paribus, for three reasons: an increase in r reduces D + T ((D + T)/Sr < 0) (this impact on D + T has been discussed); an increase in r reduces the amount that may be lent per dollar of deposits (Sf5/Sr < 0); and an increase in r increases the rates on securities held as second- ary reserves which, without an increase in loan rates, reduces the amount banks are willing to lend per dollar of deposits (a, 2 and b1 2 < 0 combined with Sr/or and org/r > 0). Likewise, the quantity of loans demanded by the public and the firms will be reduced when r is increased, since interest rates will move higher while income, prices, and real output tend to fall as the money stock  114 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM shrinks. Consequently, the actual amount of loans will fall, which will, in turn, lower the demand for both the consumer and capital good, thus further depressing the economy's level of economic activity. These effects on loans and demands could be calculated exactly in terms of the model, but the chain of causality seems so clear that this will not be done. It is sufficient to say that the expressions developed earlier, when only the asset readjustment effects of a change in r were considered, understate the various impacts of a change in r on the economy since they do not take into account the impact of dr on either the quantity of loans demanded or the quantity supplied. In quick summary, it has been shown that changes in the reserve requirement affect the economy through two major channels: by causing a readjustment in the asset portfolios of the various sectors of the economy, and by influencing the amount of loans made and thus the demand for goods. Working through both these avenues, changes in the reserve requirement are a powerful and diffuse technique for influencing the level of activity in our economy. CHANGES IN THE REDISCOUNT RATE The banks' demand for rediscounting is given by 114 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM shrinks. Consequently, the actual amount of loans will fall, which will, in turn, lower the demand for both the consumer and capital good, thus further depressing the economy's level of economic activity. These effects on loans and demands could be calculated exactly in terms of the model, but the chain of causality seems so clear that this will not be done. It is sufficient to say that the expressions developed earlier, when only the asset readjustment effects of a change in r were considered, understate the various impacts of a change in r on the economy since they do not take into account the impact of dr on either the quantity of loans demanded or the quantity supplied. In quick summary, it has been shown that changes in the reserve requirement affect the economy through two major channels: by causing a readjustment in the asset portfolios of the various sectors of the economy, and by influencing the amount of loans made and thus the demand for goods. Working through both these avenues, changes in the reserve requirement are a powerful and diffuse technique for influencing the level of activity in our economy. CHANGES IN THE REDISCOUNT RATE The banks' demand for rediscounting is given by 114 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM shrinks. Consequently, the actual amount of loans will fall, which will, in turn, lower the demand for both the consumer and capital good, thus further depruing the economy's level of economic activity. These effects on loans and demands could be calculated exactly in terms of the model, but the chain of causality seems so clear that this will not be done. It is sufficient to say that the expressions developed earlier, when only the asset readjustment effects of a change in r were considered, understate the various impacts of a change in r on the economy since they do not take into account the impact of dr on either the quantity of loans demanded or the quantity supplied. In quick summary, it has been shown that changes in the reserve requirement affect the economy through two major channels: by causing a readjustment in the asset portfolios of the various sectors of the economy, and by influencing the amount of loans made and thus the demand for goods. Working through both these avenues, changes in the reserve requirement are a powerful and diffuse technique for influencing the level of activity in our economy. CHANGES IN THE REDISCOUNT RATE The banks' demand for rediscounting is given by da - = Db + j~b - S do. d, rbp -rd bf edt where d 0 n -os -S d d, a>0fwhen (b + D s+ + rr rnp d rd bf d and d d = 0 otherwise. (2.101) d ob + nb _ d d d1 ibp rd 'bf t- dt where -o -o - d + d > .when L +4Db S d rl + d, addrbp -rd bf- rd and d d = 0 otherwise. (2.101)t d -o -Db _ -S do _ di rbp rd rbf - dt where -s-o- d d d>0lwhenLeb Db4 >14+ +n~t s s addarbp rd rbf d and d d = 0 otherwise. (2.101) No matter how great the difference between the rates on loans and the rediscount rate, no rediscounting occurs unless there is an excess de- mand for loans. There are two cases to consider when examining the effects of changes in the rediscount rate. First, there may be an excess supply of loans. In this case, neither increases nor reductions in ra will have any effect on the economy, since the change in re will not cause a change in the actual amount of loans, nor will banks have, solely because of the No matter how great the difference between the rates on loans and the rediscount rate, no rediscounting occurs unless there is an excess de- mand for loans. There are two cases to consider when examining the effects of changes in the rediscount rate. First, there may be an excess supply of loans. In this case, neither increases nor reductions in rd will have any effect on the economy, since the change in rd will not cause a change in the actual amount of loans, nor will banks have, solely because of the No matter how great the difference between the rates on loans and the rediscount rate, no rediscounting occurs unless there is an excess de- mand for loans. There are two cases to consider when examining the effects of changes in the rediscount rate. First, there may be an excess supply of loans. In this case, neither increases nor reductions in r will have any effect on the economy, since the change in rd will not cause a change in the actual amount of loans, nor will banks have, solely because of the  SOLUTION WITH AN ACTIVE GOVERNMENT 115 change inerd, anynreason to changetheerateson loans(een toghtee will lend In fall n01 a10 a1res11 of a change in rd, boe becase of the encesoospply of loana). Second, tbee may be eiehee as encess demand foe lona 01 a sitation of equilibeism is tie book loan maekee. I wane In show tat is this case changes in rd will have an impact on the economy. Notethat thelerms excesa demand and exceslsupply of loans aee sed 00 eefee an abe difference between the aggregate qoantity of poadmneL + 11 and the quantity of inansosupplied ort of onboreowed resernes, La. Wben Lp, + 1 > Lal tbe poasibility exiato tbat the bank will engage in aome rediacounaing; the actual amoont will nor, in general, be equal 10 tbia diffeeoce. Examination of 2.101 sbowaabhat, since dn and d, areposiive consaonts, inceases in rd lowerteamounaofeediscontingbanksae willing 10 engage in wbile reductiona in e1 inceose d d cetresaparibs. Differenaiating 2.101 wdbh reeapectard yielda lila Old a t (rbp -rd)1 dl( hrf -1 lie1 (497) SOLUTION WITH AN ACTIVE GOVERNMENT 115 choange in rd, any reaaon to cbonge the erates on loana (1000 ahoogb tbese willatendatofallnotasa areaultof acbange inrI, butbecauae ofate excemaospply of loans). Second, tbere may be eitber an excems demond foe loaor 01a situation of equilibrium in the bank loon market. Iwant to shnow tat in thia case changes in rd will bane an impact on the economy. Note taa the teems exceaa demand and excesa aopply of loans arsed to refer In ahe diffeee beaween the aggeegate quantity of loana demanded, L0 Db +Lab and tbe qoantity of loana aupplied oua of unbrroed eseve, LS. henL LaD > , tbe poibiiy eista tbat the bank will engage in sowe eediacounaing; the acasal amount will not, is general, be equal ro tbia difference. Eoxamination of 2.101 abows taa, aince d0 and dl are positive conatanta, increases in rd lowee tbe amount of cediacounaing bankca are milling to engage in wbile reducriona in rd bncrease dd , ceaeria paibus. Diffeeentiating 2.101 witk respect to rd yields ad - liLe + li~b 34i~ do lied + Or1 li3li r (enp - r )2 SOLUTION WITH AN ACTIVE GOVERNMENT 115 cbange in rd, any reaaon 10 cbange abe ratea on loana (even tboogb tbeae will rend to fall nor as a result of a cbange is rd, but becamse of the excesupply of loans).lSecond,treemay bceitberean excams demand foe loana 01 a dsation of equiibrium in the bank loan maeket. I moor 10 abow tat in tbis case cbanges is rd will bane on impact on the economy. Nore tat Ike rerma excesm demand and excess spply of loana are sed to aefee to tbe diffeeence beaween Ike aggegate quanaity of lon deadd L 40L , and tbe quanrity of loana supplied out of onboeaowed reaerese,I . Whaen L1 onD -n p +4. >,thpoibiliyeisas rbhat tbe bank mill engage in some rediscounting; tbe acaual amount will not, ingeneral, be eqalothisdiffeenc. Euamination of 2.101 abows tat, Oince do and d, ace positive constanta, incresea in rd lowee Ike amount of rcdiaconting banksar willing 10 engage in wbile raductions in rd incrcaae d1 a eteriapparibus. Differentiating 2.101 witb reapedt to rd yielda (r, - rd) 1) lied (roe - rd)' (4.97) (4.97) (rnc - rd30 adid/0e1 is negatine. 1. lL/Ord and Oh/011 are negative becase increoae in rd rend to increaae 1le and a01, tbus lowering the quanaity of loana demanded; 2. lii/s1l poslinie since increaaea in a00 and rb will increse abe quantity of Inoana aupplied; t3. .1boab lime/Old and 0101/011 are positine boa lesa than or equal The effecaa of boab increases and decreasel in rd under abe pomibdli- aiea of (1) proe equilibrism in the loan market and (2) prioa 100111 demand in the loan maeket will now be considered. a. Ifar liehfli and irhfliad were creater thoe a (this as, or course, or empirical qston)l,wehavetheperere"caseawhrean icrase ird aasnsch alsrong impact or rhe raebanoks charge tat this inducee them an increaser their smount of rediscontng.(Remembwhe aediscussinga satiowhereathareramaybean exesdemanoorloans.aIn sch aestuon0,rthepervere"resltisperhapsless unlikely.) ad/0ir1 is negative. 1. 0140/011 and Oh_ /011 are negatine becauae incrasaes is rd aend Ia increase 1he and af abs lowering tbe quanaity of loana demanded; 2. a4/ie1 ia positine sinceaincraesin rn and rbfwill inceaaeabe quantity of laanas pplied; t3. 1botb lirbp/lea and li1bf/li1 are godtie boa leaa tan or equal The effects of barb increases and decreases in rd undee the possibii liar of (1) prior equilibrium in abe loan market and (2) paine excesa demand in the loan market will now be conidered. a.I ar rn/lirn and abilie1 were greatr tan a (thia as, of coarse, an empirical qustnion), we havo rho "prerre" case where an inreasnr hs such anong impact on rho rares hanks chareo thar this induces them toices00 eraon oflrediscontsing.l(Remomeruwerdiscusia sationhr hererma y boyhan exesdemandfoaroans. rnsch adsation,atheoprerse"rsultis papsress nlikely.) li d /011 is negativea. p/lirdandla4 /lirdareoegative becaseincreasesain rd tend an increase 1np and ace, thbs lowering Ike quantity of loana dcmanded; 2. 04/r1 is posiie aince increases in ane and 1r will inceare the quanaity of loans spplied; 3. boah Olhe/011 and lime/Or1 aae positine boa leaaabhan or equal In I., Tbe effecta of boab increases and decraes in rd undec abe poaaibili- tiet of (1) paine equidibeiom in the loan maeket and (2) prior excems demand inabhe loan marketawill now be consdered. I0. If lirb/Orb nd lima/Ora were ratr than I (this is,of cors,anepical ueston),we havertheopeenrse"'csehere an incraeeinard hasesuch acsrong impacclonrhersrae hanks choarge thoe this induces rham en increase their 000u01 of redisconting. (Remrmher no ace discussing a sation whaeethre mayhbe an exesdemandforeloans. Insch a staonh e "rprrs"aresultaisperaps s ulikely.)  116 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM By equilibrium in the bank loan market it is meant that the actual amount lent to each sector is equal to its quantity of loans de- manded from the bank. This equality may or may not have been achieved through rediscounting by the banks. Consider first an increase in rd coupled with equilibrium in the loan market. If equilibrium were previously achieved without rediscounting, an increase in rd will have no impact on the economy since, ceteris paribus, the actual amounts lent and rates on bank loans will be unchanged. If, however, equilibrium were reached through rediscounting, the increase in re will reduce the amount of rediscounting the banks are willing to engage in, thus causing a reduction in the amounts actually lent to the other sectors. This raises the rates on bank loans and increases the firms' and public's demands for loans from the intermediaries. If the intermediaries are unable to accom- modate this increase in the quantity of loans demanded, total loans in the economy will fall, reducing income and the output of goods. Interest rates tend to rise. In the situation under discussion, a reduction in re will have no effect on the economy since the reduction in re will not cause either the public or the firms to increase the quantity of loans they demand. Consider a situation of excess demand for loans. Here, when re is increased, the actual amount of rediscounting will fall, increasing the excess demand for loans, thereby driving rbp and rbf higher. These increases in turn result in increases in all other rates of interest. As a result, income and the output of goods will fall. On the other hand, a reduction ii r in this situation will increase the amount of rediscount- ing and thus the amount of loans actually made. This has the effect of increasing the demand for both capital and consumer goods, raising prices and output. The reduction in r also reduces the upward pressure on the loan rates which, in turn, tends to dampen down the increases in all other rates of interest. The effect is a general stimulation of eco- nomic activity. In many circumstances, changes in rd have no effect on the economy. Even in the event that it does, the resulting changes in the level of economic activity are likely to be minor unless the change in rd is very large. The effects are also transitory in the sense that any change resulting from a change in rd will be swamped by the effects of other changes occurring in the economy. As shown in Table 10, changes in rd have an impact on the economy only in disequilibrium situations (and only in particular sorts of disequilibria). In cases 6 and 8 in Table 10, rates of interest are being bid up by the excess demand for loans. Unless 116 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM By equilibrium in the bank loan market it is meant that the actual amount lent to each sector is equal to its quantity of loans de- manded from the bank. This equality may or may not have been achieved through rediscounting by the banks. Consider first an increase in re coupled with equilibrium in the loan market. If equilibrium were previously achieved without rediscounting, an increase in rd will have no impact on the economy since, ceteris paribus, the actual amounts lent and rates on bank loans will be unchanged. If, however, equilibrium were reached through rediscounting, the increase in rd will reduce the amount of rediscounting the banks are willing to engage in, thus causing a reduction in the amounts actually lent to the other sectors. This raises the rates on bank loans and increases the firms' and public's demands for loans from the intermediaries. If the intermediaries are unable to accom- modate this increase in the quantity of loans demanded, total loans in the economy will fall, reducing income and the output of goods. Inteerest rates tend to rise. In the situation under discussion, a reduction in re will have no effect on the economy since the reduction in r ill not cause either the public or the firms to increase the quantity of loans they demand. Consider a situation of excess demand for loans. Here, when re is increased, the actual amount of rediscounting will fall, increasing the excess demand for loans, thereby driving rbP and rbf higher. These increases in turn result in increases in all other rates of interest. As a result, income and the output of goods will fall. On the other hand, a reduction in r in this situation will increase the amount of rediscount- ing and thus the amount of loans actually made. This has the effect of increasing the demand for both capital and consumer goods, raising prices and output. The reduction in re also reduces the upward pressure on the loan rates which, in turn, tends to dampen down the increases in all other rates of interest. The effect is a general stimulation of eco- nomic activity. In many circumstances, changes in rd have no effect on the economy. Even in the event that it does, the resulting changes in the level of economic activity are likely to be minor unless the change in rd is very large. The effects are also transitory in the sense that any change resulting from a change in rd will be swamped by the effects of other changes occurring in the economy. As shown in Table 10, changes in rd have an impact on the economy only in disequilibrium situations (and only in particular sorts of disequilibria). In cases 6 and 8 in Table 10, rates of interest are being bid up by the excess demand for loans. Unless 116 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM By equilibrium in the bank loan market it is meant that the actual amount lent to each sector is equal to its quantity of loans de- manded from the bank. This equality may or may not have been achieved through rediscounting by the banks. Consider first an increase in re coupled with equilibrium in the loan market. If equilibrium were previously achieved without rediscounting, an increase in rd will have no impact on the economy since, ceteris paribus, the actual amounts lent and rates on bank loans will be unchanged. If, however, equilibrium were reached through rediscounting, the increase in re will reduce the amount of rediscounting the banks are willing to engage in, thus causing a reduction in the amounts actually lent to the other sectors. This raises the rates on bank loans and increases the firms' and public's demands for loans from the intermediaries. If the intermediaries are unable to accom- modate this increase in the quantity of loans demanded, total loans in the economy will fall, reducing income and the output of goods. Interest rates tend to rise. In the situation under discussion, a reduction in r will have no effect on the economy since the reduction in r will not cause either the public or the firms to increase the quantity of loans they demand. Consider a situation of excess demand for loans. Here, when rd is increased, the actual amount of rediscounting will fall, increasing the excess demand for loans, thereby driving r, and rbf higher. These increases in turn result in increases in all other rates of interest. As a result, income and the output of goods will fall. On the other hand, a reduction in rd in this situation will increase the amount of rediscount- ing and thus the amount of loans actually made. This has the effect of increasing the demand for both capital and consumer goods, raising prices and output. The reduction in re also reduces the upward pressure on the loan rates which, in turn, tends to dampen down the increases in all other rates of interest. The effect is a general stimulation of eco- nomic activity. In many circumstances, changes in rd have no effect on the economy. Even in the event that it does, the resulting changes in the level of economic activity are likely to be minor unless the change in re is very large. The effects are also transitory in the sense that any change resulting from a change in rd will be swamped by the effects of other changes occurring in the economy. As shown in Table 10, changes in rd have an impact on the economy only in disequilibrium situations (and only in particular sorts of disequilibria). In cases 6 and 8 in Table 10, rates of interest are being bid up by the excess demand for loans. Unless  SOLUTION WITH AN ACTIVE GOVERNMENT 117 ris reduced enough to eliminate completely thia excess demand, the total result in each case will be anincrease in theaates of interest and either an unchanged or decreased amont lent by the bank. Only in came 7 in the table does the change ins2d reinforce the major curensa of change in the ecoaomy. Mete wilt be meen of this in comparing the effectiveneas of the various toola of monetary policy. Theme effects are sammarized in Table 10. TA13LE 10. Effects of Aed Sitnation As2 Effect 1.00D ILD 01-1h Ot = 0 4. L OLt > Lb, None 5. Lpb + LtI > E t on 6. 00b vL00> E5 Actual amunts lent f v>iccome,uturpt f; o td =0f intrestesrates j d 0 Oh L Dh+ ">S 0 Actul aosunt len l =>incne,uoutput I 2 > 0 interestes a.Aslongas athere isanoecess denandforsloans, intrestrates tendtoinreas. When rd tails, these increases are less than they would have born witheut the reduction in 53. OPEN-MARKET OPERATIONS To this point the government has been a passive supplier-absorher of government securities. Is has vol consciously attempted to influence the level of economic activity hy huying or selling government securities.' 2. 1 have dllowed fur gocernment puschasres us sales ofagornmsecurtesi the fleat section in conjunction with oh anger in the reserve requirement. SOLUTION WITH AN ACTIVE GOVERNMENT 117 ris seduced enough to eliminate completely this eucess demand, the total result in each case will be an increasrintherates of intersoand either an unc hanged Ot decreased amount lent by the banks. Only in case 7 in the table does the change in rd reinforce the major current of change in lbhe economy. More will be seen of this in comparing the effectiveness of the various tools of monetary policy. Theme effects sue summarized in Table 10. TABLE 10. Effects of Acrd Situation Ar2 Ettect 1. L~bO L~ DI SoS tone 00 D ,O0D < S None 3. Led eLf >Lb NoSue d = 0 4. I- L >f Lb Noue 2d = 0 0. L b O~ L Actual amounts lent f =>inaumt, outpu t d= 0 interestrstes Va 7.Lf00 oL~0 D Actual amounts lent oumruturptil d2d> 0 interest ratesf 8. L D0 vL~0f Db E ualamounts lret =>inoaumtutput 2 > 0 interes rate, t a. Asrlong as there isanrucss dendfolas, intrestates tndo icease. When rd falls, these increases are less thae they would have hero withut the reduction inrd. OPEN-MARKET OPERATIONS To this point the government has been a passive supplier-absosber of government securities. It has not consciously attempted to influence the level of economic uctivity by buying oo selling government srcursrses.2 2. 1 have allowed for government purchases or soles of covernment securities in the first section in conjunction with changes in the reserve requiremnt. SOLUTION WITH AN ACTIVE GOVERNMENT 117 ris reduced enough to eliminate completely this excess demand, the tutalsresult in each ase wigl br an increaseninthe rates of interestnd either annchanged or decreasd amount lentby thebanks.Only in came 7 in the tahle dues the change in rd reinforce the major currunt of change in the ecunomy. Mote will hr seen uf this in comparing the effectiveness of the various tools of munetary policy. These effects are summurized in Table 10. TABLE 10. Effects of As2d Situatioc Ar2 Effect 1. L oL bL Pb So one p e eLt>Lb Su 4. L D cL Db>I - None 22 f0 22it= 0 ueevs 3 . L F fL F 0 > f l A c t u a l a u n tno s l n t f = in c o m , o u tp u t f ; 8. L D oLt >L 0~b Actualuamounts len t >icome, output t 20 a. Aslong s tereis an excrss demandfur loans, interestrates tndoicease. When rd falls, these increases are less thsu they ould hue heo without rho reductiun in 2. OPEN-MARKET OPERATIONS To Ibis point the govetnment has heren a passive supplier-absorber uf government secureities. It has nor cunsciously attempted to influence she level uf economic activity by buying on selling government secuestsieS. 2. 1 hace sllooed for governmnt purchases or sales of government secueries in rhe firstO section i0 counction wish changes inthenresrvsrquirmentr.  118 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM In this section the economic impact of such activity by the government will be considered. Government purchases or sales of government securities affect the economy through two routes: their impact on the rate of interest on government securities, and their impact on the level of bank reserves. Since these are also the primary routes through which changes in the reserve requirement affect the economy, this analysis will be similar to that in the first section of this chapter. Consider first a sale by the government of G* dollars worth of government securities. For simplicity, it is assumed that this amount is purchased initially by each sector in proportion to the major parameter in its demand function for government securities. Thus, 118 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM In this section the economic impact of such activity by the government will be considered. Government purchases or sales of government securities affect the economy through two routes: their impact on the rate of interest on government securities, and their impact on the level of bank reserves. Since these are also the primary routes through which changes in the reserve requirement affect the economy, this analysis will be similar to that in the first section of this chapter. Consider first a sale by the government of G* dollars worth of government securities. For simplicity, it is assumed that this amount is purchased initially by each sector in proportion to the major parameter in its demand function for government securities. Thus, 118 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM In this section the economic impact of such activity by the government will be considered. Government purchases or sales of government securities affect the economy through two routes: their impact on the rate of interest on government securities, and their impact on the level of bank reserves. Since these are also the primary routes through which changes in the reserve requirement affect the economy, this analysis will be similar to that in the first section of this chapter. Consider first a sale by the government of G* dollars worth of government securities. For simplicity, it is assumed that this amount is purchased initially by each sector in proportion to the major parameter in its demand function for government securities. Thus, G* = dGe + dG, + dGn + dG where dGv= 9 G* P + g, + gn + P dG= t G* g + g, + g, + P dGn = " G* g + gf + g, + P dO5 on. S+g91+ gn+ P (4.98) (4.99) (4.100) (4.101) (4.102) G* = dG, + dGe + dGe + dGb where dG= g G* g + g8 + g + p dG, = f G* g + g, + gn + p5 dG= g" G* "g+g .+g,+ p g + gr + g, + p (4.98) (4.99) (4.100) (4.101) (4.102) G*= dG + dG + dGn + dGb where dG= g G* g + g, + gn +p dG= G* g + 8 + gr + p dGv= 0 G* " + g+g+ P dGb 8 G*. g + gf + gn + p (4.98) (4.99) (4.100) (4.101) (4.102) The increased holdings of G require each sector to reduce the level of other assets it holds. Thus, The increased holdings of G require each sector to reduce the level of other assets it holds. Thus, The increased holdings of G require each sector to reduce the level of other assets it holds. Thus, dGe =dC + dD, + dFP + dT + dN, dG = dC + dDe + dTf + dNe dGe =dC= + dDn + dF, (4.103) (4.104) (4.105) dG =dC + dD + dFP + dT, + dN, dG dC, + dDe + dTf + dNr dG, = dC + dDn + dFP (4.103) (4.104) (4.105) dG =dC, + dD, + dFP + dT + dN, dG, = dC, + dD + dT + dNe dGe = dC. + dD + dF (4.103) (4.104) (4.105) dGb = dC,, + dFb- (4.106) dG = dC, + dFb. (4.106) dG, = dCb + dF,. (4.106)  Il IZ I4 4,- o + + +4 + 41 +4 - + + . s~ s~+ 0 0 0 aZ  120 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 120 EQILIBRIM STUD OF TH MONETRY MECANISM120 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM12EQIBRUSTYOFHEMNAYMCAIM 120 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM dFb dO5 'y+ I' (4.120) dF5b dO5. (4,120) dFb = 11 dO0 'y +11 (4.120) The immediate result of this fiest round eeadjustment in assets is that deposits in the books bane been seduced by d(Oo+T) = d0 + d05 + dO + dT5 + dTe - k, k 3 (dGe) + k,+k k2 +k, + +f0 Tbe immediase result of obis first round readjustsnent in assess is shut deposits in obe banks bane been reduced by d(0) + T) = dOP + dOf + dO5 + dT, + dTf The ismediate rensult of Ibis fiest round readjustment in assets is thus deposits in she banks havn been seduced by d(O + T) = dO5 + dO5 + df)n + dT0 dTf k,+ k, + 1:3 + nP + Pd5 (dO5) + (dO ), a,5+ ds+tsf+ ne c + d + b which eesults in asresnerve deficiency equaltoI R defl=(Il- r) d(Os+T). (4.121) (4.1 22) k, + k2 + k3+ np+f (de df+I dof) + an (dO,), which eesubts in asreserve deficiency equal to R deft=(1- r) d( + T). (4.12 1) (4.122) df I (5 f + d,, (dO,), as+ de+ te+ ns 0 +drl + b which results in aereserve deficienay equal so R defl I ( t - r) d( + T). (4.121) (4.122) The banks use now foeced to seduce teie asses holdings in an amount equal In the eeseren deficiency. Thus, The banks ass now fosced to seduce their asset holdings in an amont equal so she reserve deficiency. Thus, The banks use now fosced so seduce their asset holdings in an awount equal so she reserve deficiency. Thus, R def 1 = dCb5 + dO55 + dFb1. (4.123) R def 1 dCbI + dO05 + dFnF- (4.123) R def 1 = dCb5 + dO55 + dFnF- (4.1 23) Once again, she nonbank-psinate sectors mass be induced to eupand theis holdings of G and F. (Ht is asumed that, in this cuss, the govnent will nut buy any of its secasitiss from the banks.) Offsetting reductions in these sectos' boldings of CO D and T mustresult. Wekhane, then, Once again, the nanbank-peinase sectors wmu be induaed to expand tbeir holdings of 0 and F. (It is assumed that, in Ibis oase, she government wdll aot buy any of its securities from she banks.) Offsetting seductions in shese sectos holdings of C, 0, and T mustsresult. We buns, then, Once again, the nanbank-psinae secors must be induced tn expand theis holdings of 0 and F. (Is is assumed shus, in sis case, she goernmens widl nut buy any of its securities from she banks.) Offsetting reduations in these secos' holdings of C, 0, and T must resuls. We bane, tben, dO51 = dO55 + dO55 + dO,1 dFn = dFes +±dF=, wher 9 + &f + go (4.124) (4.125) (4.126) (4.127) dO05 = dGns + dOf, + dO55 dF50= dF51 + dFnl wher dO1,5 = 9 dO55 O + 0s ± gs dO55 = Os dO55 g + Os 01 Oo (4.124) (4.125) (4.126) (4.127) dO55 = dO55 + d~es + d~nt dF, = dF55 odE, whets dGns = 9 dO01 O 01 Os + 0,, d~f, = g5 dO05 O 01 gs0 +0, (4.124) (4.125) (4.126) (4.127)  SOLUTION WITH AN ACTIVE GOVERNMENT 121 dGnc = gn dot, (4.128) d~c,1 = f dF01 (4.129) fT+ b. dEn = 8, dE1,,. (4.130) The cesultant decreases in ansets of she nonbank-peivale sctnes ace given by dCc,1 = 63 (dOv ~j 411 k, + k, +3 kn, i up 1)(.1 dD,1 = k k (do01 + d~c,1) (4.132) dTc, = k3 (dOp ~I 413 k, 4k, +6k3 +nc up +~c (13 dNc, = 0p (d0c,~ + d~c,,) (4.134) k,1k2 +3- + nup n,-l= s +df+ tf+ n, , (4.135) dO11 = de dO,(.16 dNf5= a nc, dGn (4.137) a, + dc+eef+ Rc, dT11 = tf dO, (4.138) as + df+ If+eor dKni = n~d (do11 + Od) (4.139) dOni = £ n (dGni + dE01). (4.140) SOLUTION WITH AN ACTIVE GOVERNMENT 121 d~ni = Rn dGi-, (4.128) d~c, = Tb.dFi-1 (4.129) dEN = b. dFi-1. (4.130) The resultant deceases in assets nf the nonbank-peinace sectorc are ginen by d~C, = k2 (dGcI + d~c,1) (4.131) 61 + k2 + k3 + n d~,= 1, (do01 + d~c,) (4.132) kI + k2+ k63 + Rc, dT,1 = k3 (4001 + d~c,1) (4.133) k,+ k, + k3 + uc, dN, = n. (4G,+dp 414 k, +6k2 + k3 +~c up 1dc)(14 dC, = a1 dG,(n65 a, + dc,+ tc+ n0 41> dn= df G (4.136) a,+ df +10f + nf O dN, = oe dO,(-17 a1 + df+12f+ nf,~(17 dT1 = If d.......A (4.138)... d a, + df+I, +-n1 418 dCc1 = C +nd (dGnc + d~ni) (4.139) dDn1 = dn (dO, + dE ). (4.140) en + d c SOLUTION WITH AN ACTIVE GOVERNMENT dc, = dn dG, f+ dF,=bn dF,-. f + b The resultant decreases in assets of the nnnbank-prinate sectnrs by dCc,1= k, (dI+ pj k1+6216= +nc (d~c1+ dF,5) d~c,1= k, (dI+ pj d k,1 + k62 6 3 + d ,1dn,1 dNc, = 6 6 - (dGc,1+ d~c1) d~fn = a. dO, a. + df+ If+ n, dO5I = dc, dO01 dNc, = 4001G a, + df+1If+ nf dTn = If..... dO5 a, + d, +- If + nf dO05 = +d (d~ni + dFci) dO01 = 0n (d~nI + dE=1). 121 (4.128) (4.129) (4.130) ace given (4.131) (4.132) (4-133) (4.134) (4. 133 ) (4.136) (4.137) (4.138) (4.139) (4.140)  - ri]  fbI * f() + 1 NAll .lodl d" +~b orf + Rd~ p17+ ref r,- f It 7er Rdffjf (I -R Iderfi-I kb orf +o . . . . . f~ + If 1 kdIj 1 " Ao)( +' N .)l + If + I~3 R. def , - 11- If ) +4 (4A 1 446)1 DiAjdigbhd,.gh, Ry Idfi-I Itdof, k, +kf, beQdl fI r .Q ID 6lk, Ik,+ R --,+g A A4 1 +h1 , SO)LUTIO)Nf ITEI AN ACFET fy C .OVffNNIFNT 123 dIf ). . AIdol I (Tr b d43 . > I ef- I dF,, ( , " -) 1 - 4f hf 3, + l f I SIhflolfII Ill 1 ,111 -deIIOI ofA A 145 -dt~ o ,--o fl 41 1 f, yr. or R A, , - I: Rdf~I- )i + % (d- +I + 1 1 R 6 Af l-f + p (1 3 -) 41 R del I Q, 0 r,+ k,1 + If 1 1(ef , 1(3. +__ 4 1 6) * 71 SOLU IIe IN WI H AN 4(4115h IACIVI3NNEN1 12, dOp = , t __o_)_R1der dO 4 1 1 1f) 1 S~~tT~j~ tes ATsj A- t, 145) It orI3 o Rder ( f(I TI 33 Ft deri- I I A)jfff f , dof) (I +- Rdf, t +-(I6 k1+k -(doi 4 I 3 ±)R d r f 3 d+ .IId. +~, der - I4- Q- 0pk,1+ 1 fe + k r A 13-I[ 4 4  124 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM p +'y+p as +df+ nf+ tf K f p A+ d. g.___ p ) +( b. g f + 8.f, p +7y+ /I f+ b. + A )1 (4.147) Equation 4.147 holds for oll] !3. Consequently, R def 3 R def 2 = G => Rdef 3 =QG. Rdef 2 R dof 4 R def 3 = G =>R def 4 =QG R drf2 124 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM p + 'y + A5 a. + de + nr + tf K gn p )( p A (4.147) Equation 4.147 holds for alt] 3. Corsequtsntly, R def 3 R def 2 = QG => R drf3 = QG R def 2 R def 4 R def 3 = QG'=> Rdef 4 QG R dof2 124 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 9f p + c5d, Kgn+g p p +bI- A )]_ (4.147) Eqsuatiosn 4.147 holds for alj 3. Consequenstly, R dof 3 R def 2 = QG- = R drf3 = Q0 R dof2 R dtf 4 R dtf3 = Q0 => R drf4 = QG R def 2 R de = Q => R drfi = QGQ R def 2. (4.148) R defi- 1 Tht total reserve deficiency is sherefore giveo by R drfT =R defl1+ R drf2+ R Rdf i = RdeflI+ Rdef 2+ QX Q~Rdef 2. As seea earlier, this infinite series conrvtrges to R def 2/1 - Q0. since O < QG. < 1. lhe total reserve deficiency is therefore R def T = R deftI + R def 2 + R deG R def i-t = QG0 => Rdef i= Q i- R def 2. (4.148) Tht total reserve deficitacy is thoerfore given by = deflI+ Rdef 2+ i Q2 R def 2. As seen erther, this infinite seri~et converges to R del 2/1 - Q0, since 0 < QG' < 1. The total reserve deficiency is therefore R def T=R def I+ R drf2 + R-e R defi - =4.148 R defi- I =G Q= >Rdefi G'Q R def2. (418 The total reserve deficiency is therefore given by R defT =R deflI+ R drf2+ R Rdefi =R deflI + Rdef 2+ QI Q~R drf2. As seen erther, this ifinite series converges to R def 2/1 - Q0.- since 0 < QG.<. The total reserve deficiencyis threfore R def T = R deftI + R def 2 + R-e  SOLUTION WITH AN ACTIVE GOVERNMENT 125 SOLUTION WITH AN ACTIVE GOVERNMENT 125 R deflI+ Rdef 2 [1 + 1 - 1.' (4.149) =Rdefl +Rdef2 [I+I IQGI SOLUTION WITH AN ACTIVE GOVERNMENT '=Rdeft +Rdef2 [I+ 1-IQ'I (4.149) (4.149) Once ogots, this expresion will sot be written out he full. Hosing obtined the expeestion foe R def T, we ore once ogoin ina petition to weite out the toent impacte 00 ostet holdins of the goseru- mnt'st cute of 01 dollort woeth of goseenment securitiet. Foe the hotnks, we hose: Once ogois, thit expreess will not he weitten out is foil. Honing obtained the expreetsinforeRdef T,weoaeosceagoinsinto petition to weite out the totol impacts on ottet holdings of the gosern- ment's tote of on dollaes woeth of goveenment tecurities. Foe the honkt, we have: Once ogoin, thit eupess will sot he writtes out is foil. Hosing ohtained the expreetsion foe R def T, we aee once ogain is potition to weite one the tot impucto s octaset holdins of the goseen- ment tole of 01 dolloos woeth of govetrnment scuritiet. Foe the busnks, we hose: Kb Y R def T dose = 0 R defT + 01G dFse = R def T. P0+5Y+ A (4.150) (4.151) (4.152) dN = ' def T dose = 0 R def T+ 01G dO01 = 01 R def T. 0 + '+ A1 (4.150) (4.151) (4.152) N~,= ' def T 0 +5Y+0g dt= P R def T+ 0 01 dFbt = N1Rdef T. +5f+0P (4.150) (4.151) (4.152) The nonhonk-peisote sectoes hose iscreoted theit holdingt of goseesment ond fiems' secueities hy =G~ 01 ___ R def T+ g-g+ g 0 +5Y+01 The nonbonk-peisate sectort hose inceeoed theit hsldins of goseenment ond firmst' tecuritiet by d~p = 9 P~y RdefT + The sonbosh-peisote tectett hose incesed theie holdings of goveennment ond firtnt' teeueitiet hy d0et = 9 P R50 RdefT + 0P 011+ g G d In g ___ df 0 f+g +5Y+0/1 0_G1 + 01 G 0+1001+01n p+g9+ gf+ g (4.153) (4.154) 0 _01P0e- T, 0*1+ 01 o P + g+ 9fg P+ 9 + gf + On 40nt = g0en P R def T+ 0 01~j+ gn on 0 +g+11 +1 0+01 +11 +01 (4.153) (4.154) 0 01 + - 9 G P_____ + 01 f n + g 01g + dO5, = 01 R NdefT + 01+01e+01n P +5- +01 0 0*1+ 01, on P+g9+ g1+ gn 0+0+01t+0 (4.153) (4.154) d0f= [f I RdefT + g + ge+gn 0 +5'Y+01 0 G* + gf G* (4.155) 0 +0g1001e+1n P0+g9+ ge+gn dof= [f I R def T+ 0 011* + 9e G,, (4.155) P0+01+ 9f+ gn P0+01+11 +11 ______ NfI P RdefT + g + ge+gn 0 +5'Y+1 0 G* + 01e on (4.155) 0 +g1+11 +g1 0+g1+1e+1s  126 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 126 EQILIBRIM STUD OF TH MONETRY MECANISM126 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM12EQIBRUSTDOFHEMNAYMCAIM 126 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM dEHt = I__ /I___ R def T] dF - [ ', R Rdef Ti. Summing ovee each 11c101, I(G + FA 9 R defT + p~g+, 01]+ 6 QI P f 6 P+9 f+g + [ R def T] d(0 + F) g- [ R def T - + fq. ~gI+ Rdef T]. (4.156) (4.157) fF~ = Rdef T] f+h,, p+-Y +P dFE = b-, [ 6 R def T]. Summing over each sector, d(01 +Fe),= g 6 R def T + 9 6~ 9 G P+9 f + 6n P+g f+g + [ R def T] d(0 +E -,) [n R defT + 01~j+ g1 G + 6, [ R def T]. f9+ b 9+7'+6A (4.156) (4.157) dF11 = R Rdef T] -~t b. 6 P Rdef T]. Summing aver each 11c151, d(01 + F1), = g1g R defT + 9 f+g +7T+61 p Pg-g1 g,, 0+ g 01 P f 6 n P+9 f+ + [ R def T] =+ I,, P d(Gn Fn~t gn P R defT + P +g9+ gf+ g 01+ 9 + f +e + + Is, 6 R def T1. f +b, 9+7'y+ (4.156) (4.157) (4.158) (4.158) (4.158) (4.159) (4.159) (4.159) dO1, = 4.155. The resulting decreases is nombank-privale sectos' asset holdings are gives by the lame relations as 4.73 throuagh 4.82 and mill sot be repeated here, even though the expessionsfored(0 +15),tandd(0 + F),,1 ace not the tnme as is the first section of Ibis chaptee. The total change in the money stock, dM,5,, is given hy =M - k a , 4(0 + F)1, + 4(0 + F)n, k, + k2 + k3,+5u d0ft = 4.155. The recolting decreases in noohank-peivate 1101011' asset holdings ace given by the tame relations as 4.73 through 4.82 and mill not he repeated here, even though the expressions foe 4(0 + F)1, and 4(0 + F),, are not 1he same as is she first tection of this chapes. The local change in the money stock, dM,,, 6s given by 6, +h1= k, , npd( + F)1, + d(0 +F)nr d~t= 4.155. The resalting decleases in oonhooh-ytivate nectors' asset holdins ale given by the same relations as 4.73 through 4.82 and mill not he repeated here, even though lbhe expressions fol 4(0 + F)11 and 4(0 + F),,, are not the same as in the first section of Ibis chaptes. The total change is the mosey stoch, dM, G Iis gives by -G =, kh2 +1+ k,, d( + F)P, + d( + F).,  SOLUTION WITH AN ACTIVE GOVERNMENT 127 as + d + a, + d, + n, + t p+ +,Y + I SOLUTION WITH AN ACTIVE GOVERNMENT 127 + a.+df dGr+ a.+ d I+nf+ t, p+7y+ y SOLUTION WITH AN ACTIVE GOVERNMENT 127 + as d, dG+ a,+ df+n + tf p+y+p [R def T] + G*. 'Y+p / (4.160) {R def T] + ' G*. 7+p (4.160) [R def T] + ' Ga. Y+p (4.160) All the terms here are the same as those in 4.87 with the exception of (y/y + p) G*, which represents the portion of the banks' initial pur- chases of government securities paid for by drawing down the banks' currency balances. In the case under discussion dMtG* is, ofcourse, negative. The same arguments hold for the situation in which the govern- ment buys securities. Only the signs need be changed to protect the innocent; otherwise, the relations are identical to those above. We turn now to consider the effects of open-market operations on the prices of securities (on the rates of interest). In the case under discus- sion, prices must be lowered (rates increased) on government and firms' securities to induce the nonbank-private sectors to expand their holdings of these securities. The aggregate demands are Ga = gY + A26p + gnN + A,5g5n + g(PX) + Airf (4.161) All the terms here are the same as those in 4.87 with the exception of (y/y + p) G*, which represents the portion of the banks' initial pur- chases of government securities paid for by drawing down the banks' currency balances. In the case under discussion dMtG* is, of course, negative. The same arguments hold for the situation in which the govern- ment buys securities. Only the signs need be changed to protect the innocent; otherwise, the relations are identical to those above. We turn now to consider the effects of open-market operations on the prices of securities (on the rates of interest). In the case under discus- sion, prices must be lowered (rates increased) on government and firms' securities to induce the nonbank-private sectors to expand their holdings of these securities. The aggregate demands are Ga = gY + A26Jp + gnN + Ai + ge(PX) + Aod (4.161) All the terms here are the same as those in 4.87 with the exception of (y/y + p) G*, which represents the portion of the banks' initial pur- chases of government securities paid for by drawing down the banks' currency balances. In the case under discussion dMtG* is, ofcourse, negative. The same arguments hold for the situation in which the govern- ment buys securities. Only the signs need be changed to protect the innocent; otherwise, the relations are identical to those above. We turn now to consider the effects of open-market operations on the prices of securities (on the rates of interest). In the case under discus- sion, prices must be lowered (rates increased) on government and firms' securities to induce the nonbank-private sectors to expand their holdings of these securities. The aggregate demands are G. = gY + A26?P + gnN + A, srn + gf(PX) + A7f Ft = bsN + A, 7rn + nPY + A30ip'- (4.161) (4.162) F = bsN + AssTs + nY + A3TP. (4.162) F = bN + A i + nPY + Aol'. ars n a oe (4.162) We have omitted the banks' demand for government and firms' securi- ties, since the new price must induce the nonbank-private sectors to absorb the appropriate amount of securities, e.g., dGt + dGut + dGft, and dFt + dF,,. Revisiting 4.161 and 4.162 in terms of only the variables directly affected by open-market operations: GaD = A26Jip + gn N + AisJn + A77fr Fa = bN + Aign + A3P.'A Then it must be true that dG? = dGPt + dG, + dGft = A26 p + g dN + A, dir + Ai s dFn = dFP, + dFnt = bndN + A,7d1i + Atadie. We have omitted the banks' demand for government and firms' securi- ties, since the new price must induce the nonbank-private sectors to absorb the appropriate amount of securities, e.g., dGpt + dGnt + dGt, and dFe + dF1. Revisiting 4.161 and 4.162 in terms of only the variables directly affected by open-market operations: G ="A26p + gnN + A sT. + Ai F = bnN + Ai1?, + A3fp. Then it must be true that dG = dGpt + dGnt + dGft = A26dre + g.dN + A7dri + Asinf dFa = dFpt + dnt = b.dN + Atdia, + A3od11. We have omitted the banks' demand for government and firms' securi- ties, since the new price must induce the nonbank-private sectors to absorb the appropriate amount of securities, e.g., dGt + dGat + dGrt, and dFPt + dFat. Revisiting 4.161 and 4.162 in terms of only the variables directly affected by open-market operations: Gr = A26-Tp + g.N + A, 5sin + AFf Ff=bnNs + At 7fn + A30ip Then it must be true that dGf = dGPt + dGnt + dGft = A26dri, + gadN + A di i + Arn dFa = dFet + dFut = bdN + Ai di, + A3 diP.  128 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM WI know that dC, + dC~ + dQf p R defT + Ot +,p + W pg + g1+ g P0+ 9 + f +n +7y+0m 128 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM We know that dO0, + dt + d~ftt R def T+ p G + f + 0.G R def T + G P +g9+ 9f+ g p+-y +pA 128 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM We know that dGpt + d001 +46 =~t R def T + p +7y+ 1 0+g9+g9f+ g + 9 + Cf . *= p R defT + G p +g + g0+g. p +7y+p/ and that dE0t + dFno R def T, p +7y+ p and that dO01 + dF,,1= R dnfT, +7'y+ A and that dO01 + dnt = R def T, 0 + y+0p so that p R def T + Gr= A0d + 8ndN + A,5"n + A, d T and PR doE T - bndN + A,17dTr + A30dr0 p R defT + G*=(a7 +015. +o006)dt + 0 +7y+0/ (b7 + b, + b0dr, and P N def T = bndN + (01 7 + a3o)dtt + O +7y+01 (b17 + b3,)dy. (4.163) (4.164) to that 0 R doe T + G- = A05dE1l + OndN + Al 5% + A, d T and 0 RdtfT =bndN +At7drn + A3odT1 0++ R def T +QG (a7 + a,, +a2 6)do + (b7 + b l + b26)dt1 and R dnf T = bn dN + (a1 7 + a3 0)dtt + (b 17 + b 3n)dr. (4.163) (4.164) to that + A, 7h and R RdefT = bndN + Ald-n + A3ndT p±± R dofT +C*= (a, +a,+ an)drf + (b 7 + b15 + btn)dtg and R RdnfT bndN +(a17 +oa70 )dt + (b,17 + b 3 )d1. (4.163) (4.1 64) (4.165) (4.165) (4.165) (4.166) (4.166) (4.166)  SOLUTION WITH AN ACTIVE GOVERNMENT 129 Solving 4.165 and 4.166 simultaneously yields the changes in the rates of interest (1/dP) necessary to induce the increased holdings of G and F in terms of R def T, dN, and G*. The resulting expressions for dr and dra are quite similar to those derived in the first section, except for the addition of the term G*: dr G.=R defT [ P - a7+a, + a26 p+y+g a1o +a3o SOLUTION WITH AN ACTIVE GOVERNMENT 129 Solving 4.165 and 4.166 simultaneously yields the changes in the rates of interest (1/dP) necessary to induce the increased holdings of G and F in terms of R def T, dN, and G*. The resulting expressions for dr and dr are quite similar to those derived in the first section, except for the addition of the term G*: drG= RdefT [ - a,+a, +a26 s +Y+y a1,+a3o SOLUTION WITH AN ACTIVE GOVERNMENT 129 Solving 4.165 and 4.166 simultaneously yields the changes in the rates of interest (1/dP) necessary to induce the increased holdings of G and F in terms of R def T, dN, and G*. The resulting expressions for drf and drg are quite similar to those derived in the first section, except for the addition of the term G*: drgG*=Rdef T [ - p + Y +y a, 7+ a30 y a + a-ta-. ( )]+dN [( 7 i , 26)b - g.] p + y + y a, + a30 a0 +a1 +a05 b7 + b,5 + b26 7 as 2 6 ) (bi 7 + b3.) a7 + a30 R def T y de* a10 + a30 p +'y + y P (a + aIS + a26) p +Y + y (al7 + ag) P +'Y + y b7 + b s + b26 -(b1 + b30) 6+1 b. + (b7 + b30) a(h as +hnG a (a, + a I5 + a26+ - a1 + a30 . b7 + b, 5 + b2 6 -(b,7 + b30) (b17 + b30)G* (a7 + a3o) [b7 + b,5 + b26 - (b1 + b3)] (4.167) ( )]+ dN[( a7 0 26)b - g] p + Y + + a17 + ao a10a +aa b7 + bls + b26 a 7 is a1 26 ) (bi, + b30) a,, + a3o R def T y deeo.a= 7 + a3n p0 + h + ) [ ] 0++ (a10 j5 a2 )0++ b7 + bis + b26 - (b + b3) dN - b. + (b + b30) a10 +Oaso b (a7 + a,+a6) - On a,7 + a3(o ] + b7 + b, h + b26 (bo + b3) (b i + b3o)G* (a, + a30) [b7 + b15 + b26 - (b17 + b30)] (4.167) p++ a10 +a n ( )] +dN [( a.+1s 2.6) b. - g.] p + 'Y +A "1a7 + a30 - a0+a15+a0 b7 + b]5 + b26 a7 is aI+a26 )(b 17 + b3o) 17 030 R def T y drt0. a1, + a0 p + (Y + t P (a0 + a1, + a26) 0 p + y + y (a, 7+ a30) p++ b 7 + bls + b26 - (b1 + b30) dN - bi, + (b, + b3o) a17 + a30 b (a 0 a 1 0 + a 2 6 ) _ a + a3o ] b7 +1b5 + b2 6 - (b, + bsr) (b + b3)G* (a, + a3) [b7 + its + b26 - (b7 + bso (4.167) (4.168) (4.168) (4.168) As in the first section of this chapter, the expressions for dMtG, drn., and drfo* enable us to calculate the other effects of open- market operations on the economy when combined with the various rats of change calculated in chapter 3. These other effects will now be examined for the three tools of monetary policy. As in the first section of this chapter, the expressions for dMtG*, drgG and drfG enable us to calculate the other effects of open- market operations on the economy when combined with the various ratos of change calculated in chapter 3. These other effects will now be examined for the three tools of monetary policy. As in the first section of this chapter, the expressions for dMtG*, drgG*, and difG, enable us to calculate the other effects of open- market operations on the economy when combined with the various rates of change calculated in chapter 3. These other effects will now be examined for the three tools of monetary policy.  130 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM THE EFFECTS AND EFEECTIVENESS OF THE TOOLS OF MONETARY POLICY In the first three sections of this chapter snere developed, with some rigor, expressions for the effectt of the various tools of monetary policy on the money stock, the sate on governmensseouities, and the rateon firms' sororities. In this section the primary goot is so compore she effects of these toots and their effectiveness in comhatting inflation and unemployment. First, as far as the effects of these toolt are concerned, it it cleat that changes in the rediscount role affect the economy throngh different channels than those throngh which changes in hoth the reserve require- ment and open-market operations work. Changes in the rediscount rate woek primaeily (when they have any impact as all) through their impact on the amount of loans uctually wade hy the honking system. Changes in she amount of leans in turn affect the rates charged hy she hunks as well as the aggregate demand foe hush consumer and capital goods. Chages in the money stock are notoanimportant chnnel through which the rediscount rate affects the economy. On the other hand, changes in both the reserve requirement and open-market operations have as their prime avenues of influence changes insthe money stroth and inethe rates on government and firms' securities. Second, white their effects are seep similar, changes in the reserve requirement have one direct effect not shared hy open-marker opera- buons. Changes in r affect the amount the hank is willng It end per dollar of deposits as welt as she total amount of loans supplied through changes in the amount of deposits. Open-market operations do nor affect the quantity of loans supplied pes dollar of deporits encept insofar as changes in rules of insterest resuftheg from open-market operations may change this figure. (This indirect effect is also shared hy changes in reserve requirements.) Thus, given a particular change in the reserve requimement and an open-market operation that hush canse the same change in hunks' reserves, the change in rresults in a greater change in eke quantity of hank loans supplied than does the open-market opera- tion. Third, an open-market operation results in hoth huying and selling of government securities hy the hanking system, while the honks either only huy or only sell government securities given a change in r. This difference is of little importance. A direct comparison of open-market operations and oh anges in reserve 130 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM THE EFFECTS AND EFFECTIVENESS OF THE TOOLS OF' MONETARY POLICY In she first shree sections of this chapter were developed, wish some rigor, enpressions far the effects of she various snobs of monetary policy on she money stock, the rate on governmensrecuriies, andsthe rateon firms' securities. In this section the primary gout is to cowpare the effects of these loots and their effectiveness in comhatting inflation and unemployment. First, as far as the effects of these tools are concerned, it is clear that changes in the rediscount rate affect she economy through different channels than those through which changes in hark the reserve require- went and open-market oprations work. Changes in the rediscount rule work primarily (when they have any impact at all) through their impact on the amount of loans actually made by the hanking system. Changes in the amount of leans in turn affect the rates charged by she banks as well as she aggregate demand for hash consumes and capital goods. Changes in the money ntook are not an important channel through which the rediscount rate affeces the economy. On the ether hand, changes in hash the reserve requirement and open-masker operations have asetheir prime avenues of influence changes insthemoney stock and in the ratsson government and firms' secarities. Second, while ther effects are very simidar, changes in she reserve requirement have one direct effect nst shared hy open-market opera- lions. Changes in r affect the amount the hank is willing to lend per dollmr of deporits as well as the total amount of loans supplied through changes in Ike amount of deposits. Open-market operations do not affect she quantity of leans supplied per dollar of deposits euceps insofar as changes in rates of insrest resulting from open-market operations may change this figure. (This indirect effect is also shared hy changes in reserve requirements.) Thus, given a particular change in the resmrve requirement and an open-market operation that hash cause the same change in hunks' reserves, the change in teselts in a greater change in the quansisy of hank loans supplied than does the open-marker opera- tion. Third, an open-mashes operation results in bhbhuying end selling of government securities by the honking system, while the hanks eithes only hay or only refl government securities given a change in r. This difference is of little importance. A direct comparison of open-market opesatins and changeis in reserve 130 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM THE EFFECTS AND EFFECTIVENESS OF THE TOOLS OF MONETARY POLICY In she frst three sections of this chapter were developed, hash some rigor, enpremsions for the effects of the naries tools of monetary policy on she money stack, the rate on government securities,sandsthe rate on firms' securities. In this section the priwary goal is to compare the effects of these tools and their effectiveness in combatting inflation and unemployment. First, as far as the effects of these tools are concerned, it is clear that changes in the rediscount rate affect the economy through diffesent channels than those through which changes in hoth the reserve require- ment and open-market operations work. Changes in the redicount rate work primnrily (when they have any impact at all) through their impact on the awount of leans actually made hy the hanking system. Changes in she amount of loans in turn afferot the rates chargedhby thehbanks as well as the aggregate demand for both consumer and capital goods. Changs in the money stock are not animporantchnnelshrough which she rediscount rate affects the ecnnomy. On the othes hand, changes in both the reserve requirement and open-market operations have as their prime avenues of influence changes in themoneynsock and in the rates en governwent and firms' securities. Second, while shedr effects are very similar, changes in she reserve requirement have one direct effect not shared hy open-market opera- tions. Changes in in affect the amount she hank is willing to lend per dollar of deposits as well as the total amount of Inoens snpplied trough changes in the amount of deposits. Open-market operations do net affect the quantity of loans supplied per dollar of deposits encept insofur as changes in eases of interest resulting from open-market operations may change this figure. (This indirect effect is alto shared by changes in ermerve requirements.) Thus, given a particalar change in the reerve requirement and an open-mnrket operation that both cause the sawe change inhbanks' reserves, the change invrresltsin a greater changrin the quantity of hank leans supplied shun does the oprn-market opera- tion. Third, an open-market operation results in both buying and selling of government securities by the banking system, while the banks either only buy or only sell government securities given a change in r. This difference is of little importance. A direct comparison of open-market operations andchangestin reserve  + + + + + + + + !2 0+ + + +0 + + ++ H !R" ZL  132 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM o5,+++t d00 + 'y [Rdef T] + drG = Rdef T[ P a7 + a,, + 12 6 a + a7y+ A 01,+a.) - y+ 1 + 3 b7 + 615 + hi a7+a,( 2 )(b17 + b30) 117 + 130 R def T P b1 b0 ds50. -a17+ a3 p + 'y + 1 -(1 +b 0++1 (+ t, +60 ) 4 1 617 ' +1 +a17 +-3) P+' b- +dN b6-(,7+b 11 +13 b. + (b17 + b30) bn(a7+ a,, +11.6) -g [ 117 a,+1a30 I + bb15 + b26 (b 7 + b30) + (657 + boG (117 +130) [b7 + b15 + b26 - (b17 +6b30)] 132 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM a+ f dGf + ' [R def T] + a, + df+ nf+ If p+,y+ P 132 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM a+ fd01 + 7 [R def T + a+ df+ nf+t If 1+3y+611 (4.160) 3+ A (4.160) 'y G* (4.160) (4.167) de10=R def T[ 11_ 7 +,30a. +_ _ +dN[( a7+111+a26) P + - +/I 117 + 130 bn - gn + 0'] 17 + 01 5 + a26 bo b0 b7 + i 5 +b26 117 + 130 R def T P1 - (b17 +b30) p (a (1+0at.+a..) 1 61+73+11 (11 7 +3a30) P +'y+ol 67 + his + b26 - (617 + b30) dN b.+b1+ba a17 +1a30 60(17 + a,, + a2 6)- 117 +0a30 67 +61it5 + b65 (617 +6b30) + (6 17 +6b30)0' (a,7 + a30) [b7 + 615 + b26 - (b,7 + 630)] do10,; R def T[ (4.167) 17+115,+1a26 117 + 130 S]+ dN [( a7+11, +1a.6 P + T+ 11 117 +1a50 b. g +G*] 07 + b 15 + 03617+ , do R - def T A1 - (617 +630 117 + 030 30)+6 P (7+a.+ 13 6) 61P b 7+bt5s+b60- (6 17 +6 30) - N b.n+(017 +6b30) a17 +0a30 60(17 + 11 5 + 136-) g 1 17 + 1 30+ 67 + b, + 626 - (b, +6b30) + ( 7 + b630)00 (117 + 030) [b67 + hi + 6 26 -(617 + 30)] (4.167) (4.168) (4.1681) (4.1681) Comparison of dMtG* and dMr shows the two expressions to be (superficially) identicol with lhe exception of the term (,y/, + 61) G* in dMtG*. This term is the honks' initial changes in currency holdings as a result of first purchase or sole of government securities. Ifit is surethat eqoal changes in reserves imply equal changes in the nonbank-peinote sectrs' holdings of gonernment and firms' secureities, then it would teem that, even though C' and dr resulted in an equal change in reserves, G Comparison of dMtG. ond dMt, shows the to eupressions to be (superficially) identical with the exception of the teem ('Y(, + 61) 0' in dM1,. This teem is the books' ioitial changes in currency holdings as a rsult of first purchase or sole of governmenr srcurities. If it issourthat equal changes ineresernes imply equal chaoges in the nonbok-pinate sectors' holdings of government and firms' securitieo, then it would seem thot, even though G* and de resulted in an equal change in reservet, Gn Comparison of dM10, and dMer shows the two expressions to be (superficially) identical with the exception of the term (-y/,y + 1) fin in dM10.. This trm is the banks' initial changes in currency holdings asa resulr of first purchase or sole of governmena securitis. If itis surethat equal changes inereserves imply equal changes in the nonbonk-prisate sectors' holdiogs of government and firms' securliis, then it would seem that, mven though G' and de resulted in an equal thong emi reserves, G  SOLUTION WITH AN ACTIVE GOVERNMENT 133 lead a greatersimpact onthe monsey slosh. Tu be surse of this concolusion, she relations betweene the Seems d(0 + F)P d(G + F),~ used dO5 in the ewe equaiones must be examined. These teems see giveen by d(G__ + F____9 1 R defT + f +gf + n P8+7'+8A G___ + 9 Gg 8 + g8+ gn p+g9+ gf+ gn SOLUTION WITH AN ACTIVE GOVERNMENT 133 had agreatereimpaontesbmoney stock. Tobe sue ofthisoconlusion, she relaeions betweene ehe tesms d(O + F)1t, d(O + F) used dO5 in she twe equatiosmust be exeamined. These terms are giveen by -( + [t' R defT + g f+g 8+7Y+8 p G1+ 9 + G P +g9+ gf+ gn p+g9+g1+g SOLUTION WITH AN ACTIVE GOVERNMENT 133 bud a greasee impact use she money stoch. To be ese of ehis conclusione, she relasioes betweene ehe seems d(G + 8)es' d(G + F)05, used dGf ine she Iwo equations mess be examined. These terms use given by d(G +E)psu1 = [ R def T+ _______ +_ - Gfl+ P f + 8n P+9 f+ + [ R def T] (4.158) -f+ bn P+ Y + / + [ R def T] fo+b0 p + y+pj (4.1 58) +- I) R def T] f +h5 p+ Y+ /I (4.158) -( +[~t* g R defT + + [ I -1 R def T) =- 9f I P R def T - 8~sfGfe+fg sr] + 'Y +8 + 8 d( + F)Pt R def T- dG10) g + 8f+g P+' + f- ( -9 R def T) fT b0 P +7Y+8P d( +F.t (g R def T- dG50) + b5 8 R defT) Tf+ bn P + y +P (4.159) (4.155) (4.71) (4.72) -( +[)t* 9 R defT + P + 8 +gf + n p +g + g1+ g + ha' [ R def T] dOf0 = Se R__ NdefT + p G~j + 0' G d(G + FPtr 9 P R def T- 4010) f + 8f+g P+' + R Rdef T) -( +(), = g R def T- dO10) + b0 P 1 R defT) fo+b0 p+-y+ i (4.1 59) (4.155) d(G +8)nsus = n I [ R deflT+ 9 + gf +sn P +7Y+8 _____ 0*1+ - n on + b0 1 1 R def T] dO00. = f [- RdefT + _______ 01 + gnf G 5dE(G + IF,,7=-g 8 R def T- do00) 9 + 8f+g Y+1 + f R def T) f + b 8 +7'Y+81 dG+Fn,= ( R def T- do10) g + g5+ g- +7Y+8g + b0 8 R def T) f + b P + Y +p1 (4.159) (4.I55) (4.71) (4.72)  134 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 134 EQILIBRIM STUD OF TH MONETRY MECANISM134 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM13EQIBRUSTDOFHEMNAYMCAIM 134 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM dG,, 9 ( P R def T- do1,,). g + gc+ g p+y +SA (4.68) d~fi,= f ( P R def T- dO1.). gT + g,, g + y + P (4.68) dO05, = S9Bft (P~y R dsfT - dGo~). (4.68) Compsarison of thete ters s haws unamtbigtsaly that d(O + FpG > d(O +- F)5tc' d(G + P),tG. > d(G a E)at, and dO00,- > d5ae This is not5 sarprisint, as ace of she chief charactesiastics of opec-mashes operatioca is shot they change she cotal assount of govestnmenttsecurities held hy she private sectors, while chacgicg a doss not hose shius effecs (except so the extent she government decides as a masses of policy either so buy orstell governmoent securities its conjusnction with she change in r). Consequenstly, is cac he concloded unamhiguously that, when Ge and ds rsals io she same change ha banks' reserves, she impact of G* on she money stach will he lasges sham thas of do. Earthermore, compaisoc of she expreeinsfoa dsfand dsalso showssthatsde105> drg and dew,- > drr That, it sac he concluded that, evec though hash de and f* rest in she tame chante in resservet of she hanking system, G* (an open-market operasion) mill haoc a greere impact on the level of economic activity since G* ressults in greater changes in M, si., ande1g than does ds.3 The aext question is what combination of de and G* results in an eqal change in banks' resseeves? The expressions foe she total chants in hanks' reserves are R def Tc= R dsfl+ Rdef2,c+R def 3,(l+ Q) (4.158) " def T0, R def lG, + R def 20. P5 + - ] (4.149) QGc where Q, = QO. ds, 3. It isonot meat so stessct heae differences betweecchanaeseinstheeseve reqohremens and open-macket operationsat tohe enpense of istituaional differ- encec which the modes does sot take into account. Same of these differences te the factsthat open-mareteopeaions wrkcsowyelahivestocangesineserve requiremens, she possibitity shat Sank ecs view rseese created Op changes i reererequireents dfscrentlc sham rserves genetated Sy opcs-masket opertiont, Compasison of thesterm sm howt unambiguoasly that d(G + FPG > d(G + F)ecc' d(G + bntcc > d(G + F)nt,, and dO50,- > dGf1,- This is cat surpriting, as one of the chisf chaactesistics of open-market operationt it that thsy change she total amoont of goernmenttsecoritis held hy she peivate tectort, while changing r does moo have this effect (except so she extent the goveenment decidet as a mattes of policy eithee so huy os shl tovernment securities in conjuction wish the change in r). Contequently, is can he concladed unambiguously shoe, when G' and ds cetult in the tome chants in hankcs' reseeves, the impact of C' on the moey mtock will he laegertsham shot of dv. Furthermore, comparison of the expressions foe do5 and de, also thaws shot deans date and de50, > de5,. That, is can he concloded that, even though hoth dsand G*esultsinhesamechangein seves ofsthehanking system, G'* (an open-mashes opeation) mill have a greater impact on she leant of economic activity since G* results in geater changes in M, rf, andsr than doesdr.' The cent qustion it what comhination of ds and fGt sesults in an equal change in honks' aeseroes? The expcessions fartshe total change in hacks' aeserves ass Rdsfc=Reflc Rde~co~efs Qt+ ) (4.158) R def TG, R dsf t, *o+R def 255* [I + - - 1 (4.149) where Q, = Q05 de, requirements and opet-market operadios as the expense ocistitutional differ encs which she model dos oot sake inso accost. Some of shece diffeences ace the fact that open-mashes operations wosk sainty, reltieto changea inttaeev requirements, the poasibidit that bankes dien rseres created by changes is reee equiremestdifferenttan serso generaatedhbyopen-marketoeratios, Comparson of these termt thaws onambigousdy that d(G + F),cao > d(G + E\0t, d(G + F)at0' > d(G + F)t' and dO50,- > dGft,. Thit is not sospeiting, as one of she chief characteistics of open-mashes operations it that they change she total amoant of goovenmentsecorihies held by the private sectors, white changing a dos coo hose this effect (except so she extent she government decides as a masses of policy tithes so hay ar sell government securities in conjunction wish she change in e). Consequently, is can he concluded unambigously shot, when C' and de cetols in she some change in honks' reseaves, she impact of C' an ohs mooey stock will he larger than shot of do. Euetheemore, comparscon of the exptessions foe de5 and dre also thaws that drgco* > dt55 and dtcos > dt55. Thos, is can be conctuded shot, even thouth both da and c rescult in the tame change in reserves of she honking system, Ge (an open-maerket opertion) will have a greater impact an the level of economic activity since Ge rests in greater changes inM, r~, ands sg han does dr.t The next qustion it what combination of do and 0' results in an equal change in hanks' reserves? The expressions foe she total change in banks' reserves ass R defTc=R def1,+ R def2r + Rdef 3e(l+ -, ~Q, R def TGa R cldef 1G, + R def 2G* [1 + whee Q, = QtG de, (4.158) (4-149) 3. It ic moo mentat to ssress these differences between changes is shererv requiremntosand open-marketsopertios atsthe expense ofistitutionalsdiffer- encec whick she model dos sot take 0000 account. Some oc thece dicfercences ass she facs shot opea-market opecationc nook slowly, reltive so changes in rserve aequirementc, she possibility that baskets view reservesacrested kv changes in resetvecreqiemens diveretsly ohan reservec generated Op open-inarkes operatiocs,  SOLUTION WITH AN ACTIVE GOVERNMENT 135 an that when de <50, + >(+ _ ) -Q, l-(3 ' and nine versa when dr > 0. Rather than hate this analytis an a nomplete investigation af R def T, and R def Tr,., we assume that an accurate appeonimation of she answer can he foand in a tomparison of the first changes in the banks' R def te, de(D + T) R dtf l05(1 -e)[ h1 +h k3 9 )G*_ h, +hk,+ k3 + np g+g+ gf+ g df +I te )G* + a, + df+ton+ If P+g +ICf+6r dn ) ( gn )G*___ ______ +n dn+bn P+g s , That, for R def 1, In equaliR def 1tG *Iit mast he Onue thtat de h + g dr (1-r) [( + a3 ______ h, + k2 + ka + np g+g + gf+g, df +t if ) ef a+ de+ner+Ie p+g9+ gf+ g SOLUTION WITH AN ACTIVE GOVERNMENT 135 so that when dt <50, Qv ) Za G+ and vine versa when de 0. Rashes than hase this analysis an a complete investigation af R 4sf T wed R def l%,, we assume that an accuate approximation af she answer can be fond in a comparison of she fist changes insthe banks' R def l1, ds(O + T) "Rdef 1 G =( - r)[ Ic k, 3 9__ ht+ka+hk3+nep p+g + gf+g,, G Thas, far R def 1, en equalR def IG*, it muast he tate that dr k, +hk3 g (l-)[ +)() f a,+ f o +hif5 +a +e + g g+ SOLUTtON WITH AN ACTIVE GOVERNMENT 135 an that when de < 0, t I ) > (1 and vine versa when dr 0. Rashes than Ease this analysis an a templete invstigation af R def T, wed R deS T,*, we assaune that an accurate approximation aS the anwrcan he foand in a comparison af the fist changes in the banhs' R def 1, = de(O + T) R defl0G* (1 - r) [ , 3 )( k, + k2 +hk3+a up + +It g0n a , s~ra+ e plf g Ife+I d.)(g )O*I. c+ da + bn P +9g+ gn, Thus, fat R def 1, tn equal R def 1G*, it mast he true that (l-r [+ )() f a.+ df+r f P +l±+St±+Sn C+ P. + ha P±+S9+ ft+ g (4.1 69) nd )hn p gng5+g I (4.169) d, ) gn A t,,+ dr+Ea 6+l9+If+Ign (4.169) The denominatoe aS 4.169 is cleadly lets than 1. Call it dem. Then, to prnduce the same first change in reserves as a 1 percentage paint thange in r, npen-market opeeations in an amoant equal tn -=(DO+T) no G =.01(DO+T) dem (4.170) .01 dew de The denominator aS 4.169 is cleadly lest than 1. Call is dew. Then, toaprodutethe samefirssthangeineseves as a Iperentnage point change in r, open-marhet operatins in an amount eqal to on t as = D+T) o = .01(DO+T) - (4.170) .01 dew dewt The denominator aS 4.169 is clearly less than 1. Call it dew. Then, toprodacesthe same first change in reseves asalI percentage point change in U, open-markes operations in an amoant equal so on 11 - =(Oa+T) a- r G*= .01(D+ T) d (4.170) .01 dedw  136 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM most be undertaken. The onace size of this 00 depeeds, of course, on the sizes of the paranmetees in the denominotos of 4.169 sod on the size of D + T. Even though it is recognized thee only looking at the berst rosette change probabty pronides an inaccurate answer to the question, 4.170 cestainty undescoret the strength of ohaonges in the rseeve so- quireent at a monetary toot wohen one contiders the oize of Qt necettary to peoduce thre some first chooge insresernes as atIpercntage point change in r We cooclude this compasison of changes io r and opec-mashes opera- tions by noting that an open-market operation smsller than the fi0 in 4.170 will produce the same change in the money stock and in the sates an secusities as does a I percentage point change in r. This is obvious since we hone alreeady shown that a fi0 which cantos the same total change in reseres as a particuar de will hate a greaer impact on M, a0, and rg titan the change in the rsesve sequisement. This obsernation also andeescoses she strenghof d,sinceuafl*oftubstantialiewould have so be undeetaken to prodace equivalent changes in M, rr, and g We sumn now to a consideeation of the effectiveness of these snobs of monetaey policy in combatting the problems of infustion and unemploy- ment. With regard to changes in rho rediscount rate, Table 10 shows that the only circumstances in which is bus any impact on the economy are when these is an ecess demand foe hank loans. Typically, excess demand situations would corsespond to periods of inflation, while one of the characteristics of unemployment would be an excess supply of bank loans. Thus, we can conclude that changes in the rediscount rate will be paeticularly ineffective in combatting unemployment. The effec- tiveness of this tool is limited to combatting inflation. Enen hose, the impact of any changes io rd is likely to be small. This asymmetry of Ike effects of changes in the sediscount rate is not shared by changes in the rsere requirement and open-mashes opera- tions. As indicated, the argumenss preented, though foamed in terms of reductions in total resesnes, hold equally well for changes insrand open-mashes operations aimed at inceasing the seserves of the bunking system. Thor, these is little so choose between changes i sesere requirements and open-market operations when one is faced by eithsr unemployment or inflation.' The equationt below measure the impact of changes in r and open-maeket operations on she nariables of the 4.nxceptitituional actors which, it hsbenrgaed, prevntrfractional and frequnt changes inth rhoeserene requirement. Thse argaments hane no ecotomic validity, hat are probabln valit trm other goits of view. 136 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM must be undertaken. The enac dize of this fi5 depends, of couse, on ths sires of she parameters in she denominator of 4.169 and on she tie of D + T. Enen though it is secognized that only looking at the first reserne change probably pronides an inaccasate answer to the question, 4.170 certainly underscores the strength of changes in she resesne re- quirement asa mntary tool when oneeconsidrsthedsieeoffin necessary to produce the sums fisst change ineernes ass 1 percentage pains change in r We conclude this compasison of changes insr and open-market opera- tions by noting that an open-market operation small er than the fin in 4.170 will produce the same change in she money stock and in Ike sates an secueities us does a 1 percentage point change in r. This is obtious snewe hate already shown shut s fin which causes the same sotal change in resernes as apaticular dr will haveuagreater impactsonM,srf, ands ran the change in theresoerverequisoment. This observtation also underscoses she strength of de, since a Gn of substantislisize would hate to be undertaken to produce equivalent changes inM, sf, and e0. We town now to na cnideration of the effectiveness of thete toolt of monetary policy in combstting the problems of inflation and unemploy- went. Wish regard so changes in she rediscount sate, Table l1t ehows shut she only circumstances in which it bus any impact on the economy see when shore is an excess demand for hank loans. Typically, eowes demand situations would correspond to periods of inflation, whdle one of the charactrstics of unemployment would be an ecess supply of hack losns. Thus, we can conclode that changes in the rediscount rate will be particulsrly ineffective in combatting unemployment. The effec- tiveness of this soul is limited to combatting inflation. Even here, the impact of any changes in to is likely to be small. Thit asymmetty of she effects of changes in the rediscount rate is not shared by changes in the reserve requirement ond open-market opera- tions. As indicated, the argaments presented, though framed in terms of seductions in total resernes, hold eqoally well for changes in r and open-musket operations aimed at increasing the reserves of the banking system. Thus, there is little to choose between changes in rosette requirements and open-market operations when one is faced by either unemployment or inflation.0 The equations below measure the impact of changes incrand open-market operations anl she variables of the 4. Exceptinsttionalfactorsnwhich,irt hsbenargued, grooens tractionaland trequent changes in the reseve requirement. These srgumentt have cc economic validity, hat see prohahly vanid froon other triers ot view. 136 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM most be undertaken. The oeatsidee of this flu depends, of course, on she siet of the parameters in the denominasor of 4.169 and on rho size of 0 + T. Enen though it is recognized that only looking at she fist reserve change probably provides an inaccurate answer to the question, 4.170 certainly underseasres the strsongth of changes in rho reserve re- quirement us a monetsry soot when one eonsiders she doze of fia necessary to produce the same first change in reserves as a 1 percentage point change in r. We conclude this comparison of changes incr and open-market opera- tions by noting that an open-musket operation smaller than ltre fin in 4.170 wdl produce she sums chango in the money stock and in the sates on seensities as does a 1 pecentage point change in r. This is obtious since we hate already shown that a Gn which cases she same total change in rserves as a particular dr will hate a greater impsct on M, so, and rg stuan she change in she rserne requirement. This obtestation also and erscores the nteongh ofdr,dsnceeafl*of subtantialdsie wold have to beundertakento produeoequivalentechangessinhM,rf,oands0. We turn now so a consideration of the effectineness of these tools of monetary pokecy in combatting she problems of inflation and unemploy- ment. Wish regard to changes in she rediscount rate, Table 10 shows that she only circumstances in which it has any impact on the economy are when these is an excsers demand for bank loans. Typically, ecess demand situations would correspond to periods of inflation, while one of she characterisics of unemployment would be ancexceaso spply of bunk loans. Thus, we can conclude shut changes in the reddscaunt sate will he particularly ineffectine hi combatting onemploywent. The effec- sineness of skis tool is limited to combatting inflation. Enen here, the impact of any changes in rd is irkely so be small. This asymmetry of she effects of changes in she rediscount sate is not shared by changes in the rosette requirement and open-market opera- Sions. As indicated, the argoments presented, though framed in teems of seductions in total rsesrves, hold equally well foe changes in r ad open-musket operations aimedt inesing therseres of the banking system. That, there is little to choose between changes in reserve reqoirements and open-market operations when one ifacedbyrither unemployment or inflation .0 The equations below measume the impact of changes in e and open-market operstions on the variables of she 4.Exept institutional factorsowhich,Oithashbeen argued, preentfractionaloand Srequent chsnes in she rectr requirement. These argumns hav or en oc naliity, but ae grohably valid5 tram other goints ot view.  SOLUTION WITH AN ACTIVE GOVERNMENT 137 economy, and illustrate the channels through which these tools operate in achieving their results. The immediate impact on Y as a result of either dr or G* is the sum of three effects: the impact on Y of the change in the money stock resulting from dr or G*; the impact on Y of the change in r, resulting from either dr or G*; and the impact on Y of the change in r. resulting from either dr or G*. Thus, SOLUTION WITH AN ACTIVE GOVERNMENT 137 economy, and illustrate the channels through which these tools operate in achieving their results. The immediate impact on Y as a result of either dr or G* is the sum of three effects: the impact on Y of the change in the money stock resulting from dr or G*; the impact on Y of the change in r resulting from either dr or G*; and the impact on Y of the change in rg resulting from either dr or G*. Thus, SOLUTION WITH AN ACTIVE GOVERNMENT 137 economy, and illustrate the channels through which these tools operate in achieving their results. The immediate impact on Y as a result of either dr or G* is the sum of three effects: the impact on Y of the change in the money stock resulting from dr or G*; the impact on Y of the change in r resulting from either dr or G*; and the impact on Y of the change in r. resulting from either dr or G*. Thus, dyr =bydMr + dr, + br drgr aG dG * + dfG* + Sr drgG*. (4.171) (4.172) dY, = dMr + r dr + r- drgr a M ax' ax' by, S dMG, + bydrfG* + aty dt10.. dyG i-M -r-f -Ste (4.171) (4.172) dYrx= dMr + r dr + drg Sm arf der a1 s (4.171) Similar expressions can be written out for the effects on Xc, Xk, , and P: dXc(k)r(G*) ctk) dMr(G*) + ct dr,(G*) + Sm Srf Similar expressions can be written out for the effects on Xc, Xg, Pc, and Pk: dX(k)r(G*) dMr(G*) + ack) dfr(G*) + Sm St tG' dxg * dMG. + by dfG * + Sr drgG* (4.172) Similar expressions can be written out for the effects on Xc, Xk, Pe, and Pk: dXc(k)r(G*) (kdM)(G*) + ck) fr(G*)+ Sm ar') Se rt' arc(k) Idg(G*) ar d]?ctkt~lr t S-) k dMr( 4 ) + ilhl defG* + Sm r1 art tt' (4.173) (4.174) S-aiai de d c(k ) g(G*) * d~(* dPctat(* (k) .l r(G*)+ deg) dr + .11P5 d Ste~ drr(G*)> (4.173) (4.174) SXat dPc(k)r(G*) SM 'r(G*)+ ck) df(G*) + aP 45 gcLk) reG*) gr (4.173) (4.174) where the terms in the parentheses are used to indicate the other variables for which separate equations could be written. Equations 4.173 and 4.174 actually represent eight different expressions. Rather than reproduce the explicit forms of all the relations 4.171 through 4.174, we will present only one, that for 4.171, as an illustra- tion. dYr c + P X + P + X + + Sm am SM am am SM SM M +r + GP + "r T + rT + -r" N + -3-N laM SM SM T SM SM e SM where the terms in the parentheses are used to indicate the other variables for which separate equations could be written. Equations 4.173 and 4.174 actually represent eight different expressions. Rather than reproduce the explicit forms of all the relations 4.171 through 4.174, we will present only one, that for 4.171, as an illustra- tion. SX SP SXa p P Sn5 Sn dYr [ aP+ aX+ P + X _ + +r + G,-Ic- + rtT + - r + "N + - aSM SM C SM SM SM a SM where the terms in the parentheses are used to indicate the other variables for which separate equations could be written. Equations 4.173 and 4.174 actually represent eight different expressions. Rather than reproduce the explicit forms of all the relations 4.171 through 4.174, we will present only one, that for 4.171, as an illustra- tion. dY,= P+ S k + Sp X, + + +r + G + --T +-- r + N + aSM SM a SM a SM SM P SM  + +> + + 4+ + + +< + + ~ + + + + , + ++ se I se + +++ + +C + + + + + + +~ +- +  SOLUTION WITH AN ACTIVE GOVERNMENT 139 R def T, - (a7 + a, + a26) dfr + Iy a, + a30 ( ) + dN b ( a7 + a, + a526 p + Y + a17 + a3, (7+55~ a, a 1711+a26) b 7 + b+ + b6 - ,6 (b17 + b30) a 1 + a,, SOLUTION WITH AN ACTIVE GOVERNMENT 139 R def T - (a,+ aI + a2) p +Y + 1 a7 + ao ___ )I +dN b. ( a7 +aI + a6) p ++Y + y a,, + a3o SOLUTION WITH AN ACTIVE GOVERNMENT 139 R defTr - ___(a, +a., +6) p +,Y + yI a,, + a3, ( 9 ) + dN b, ( a. +a, + a2) p+7Y + y a , + a3, -. (4.175) b + + b26 - (a7 a,, + a) (b37 + b30) a17 + a3, ] (4.175) b7 + b15 + b26 - (17+115+36) (b17 + b30) a,7 + a3 ]. (4.175) The complexity of this equation speaks for itself. It clearly illustrates, however, the pervasive influence of the changes in M, re, and rg associ- ated with (in this case) a change in the reserve requirement. It should be noted at this point that the relations 4.171 through 4.174 actually understate the effects of dr and G* on the pertinent variables in the model. The basic reason for this is that they ignore such things as, for example, the impact on prices and outputs of changes in the other rates of interest (other than rg and rf) induced by changes in M, r1, and rg. Furthermore, the change in Y itself will induce another round of changes in M and the other variables in the model. These second, third, and higher generation changes are not contained in 4.171 through 4.174. This does not mean that their impacts cannot be mea- sured (although it is a messy job). For example, consider the effects of the original changes in Y, dYr; rg, drgr; and rf, and drr on the money stock. The "second-generation" change in M is given by SM SM SM SMaf dM, = dY,+ -dr +---dr +----dM + r y ar, dr asg sM r The complexity of this equation speaks for itself. It clearly illustrates, however, the pervasive influence of the changes in M, r, and rg associ- ated with (in this case) a change in the reserve requirement. It should be noted at this point that the relations 4.171 through 4.174 actually understate the effects of dr and G* on the pertinent variables in the model. The basic reason for this is that they ignore such things as, for example, the impact on prices and outputs of changes in the other rates of interest (other than rg and rt) induced by changes in M, rf, and r.. Furthermore, the change in Y itself will induce another round of changes in M and the other variables in the model. These second, third, and higher generation changes are not contained in 4.171 through 4.174. This does not mean that their impacts cannot be mea- sured (although it is a messy job). For example, consider the effects of the original changes in Y, dYr; rg, drg,; and r, and dr, on the money stock. The "second-generation" change in M is given by SM SM SM aM f dM, = dY + drf + dr + -dMr+ S Y Ire s Srg g o SM The complexity of this equation speaks for itself. It clearly illustrates, however, the pervasive influence of the changes in M, r, and r, associ- ated with (in this case) a change in the reserve requirement. It should be noted at this point that the relations 4.171 through 4.174 actually understate the effects of dr and G* on the pertinent variables in the model. The basic reason for this is that they ignore such things as, for example, the impact on prices and outputs of changes in the other rates of interest (other than rg and r) induced by changes in M, re, and rg. Furthermore, the change in Y itself will induce another round of changes in M and the other variables in the model. These second, third, and higher generation changes are not contained in 4.171 through 4.174. This does not mean that their impacts cannot be mea- sured (although it is a messy job). For example, consider the effects of the original changes in Y, dYr; rg, drgr; and re, and drf on the money stock. The "second-generation" change in M is given by SM SM SM SM of dM = mdY + -mdr + - dr + - -dMe+ S2 y rr, d r Sr n r oi SM aM of aM SS of a, dfr + rg (4.176) SM af 5M5af -dfr + drgr Stear aS Sr1 (4.176) SM of SMSof - r df, + - drgr Fr Stf SSKr5 k55 (4.176) where the last two terms reflect the impact of changes in other rates on M induced by changes in r. and r, while the third from last term captures the effects of changes in M on M. To answer the question of the total size of the second and higher generation effects, one would have to resort again to finding the general terms for several infinite series and then testing these series for convergence. This has not been done, primarily because the model has, I hope, been formulated in such a way that the majority of the impacts on the economy are captured by the relations 4.171 through 4.174. where the last two terms reflect the impact of changes in other rates on M induced by changes in rg and rf, while the third from last term captures the effects of changes in M on . To answer the question of the total size of the second and higher generation effects, one would have to resort again to finding the general terms for several infinite series and then testing these series for convergence. This has not been done, primarily because the model has, I hope, been formulated in such a way that the majority of the impacts on the economy are captured by the relations 4.171 through 4.174. where the last two terms reflect the impact of changes in other rates on M induced by changes in r. and re, while the third from last term captures the effects of changes in M on M. To answer the question of the total size of the second and higher generation effects, one would have to resort again to finding the general terms for several infinite series and then testing these series for convergence. This has not been done, primarily because the model has, I hope, been formulated in such a way that the majority of the impacts on the economy are captured by the relations 4.171 through 4.174.  5. Results of the Study 5. Results of the Study 5. Results of the Study BY THE TERM "monetary mechanism" I mean the complex web of causal relations through which interest rates, prices, and real phenom- ena react to determine the money stock, as well as the chain of causality running in the opposite direction-the impact of the stock of money on the real and financial variables of the economy. The model constructed in chapter 2 enables examination of both of these facets of the monetary mechanism. Based on the identity for the money stock (see Equation 3.3) expres- sions were derived for the rates of change in M with respect to the key variables of the model. These expressions were of the form SM a(PX) SY an, oau - = C +e -C2 +eC Z 6Z +2 Z 8Z + ZC4 Z +CS ni (5.1) Sz where Z is any interest rate, price, real output, or income. From 5.1, I conclude that a change in any of these variables affects the stock of money through its impact on prices (P), real output (X), income (Y), and the various rates of interest. The terms C, through C, are in a sense "money multipliers." They show by how much the money stock changes, given a change in the variables with which they are associated. (See Tables 3-7 and the associated discussion for a complete description and interpretation of these terms.) I have thus developed a scientific money multiplier for each of the key variables in the model. The Brunner-Meltzer and Teigen models develop only a single money multi- plier, while the C's are only some of several "multipliers" I have developed. (Other multipliers include the Q's developed in the analysis of the effects of changes in reserve requirements and open-market operations.) 140 BY THE TERM "monetary mechanism" I mean the complex web of causal relations through which interest rates, prices, and real phenom- ena react to determine the money stock, as well as the chain of causality running in the opposite direction-the impact of the stock of money on the real and financial variables of the economy. The model constructed in chapter 2 enables examination of both of these facets of the monetary mechanism. Based on the identity for the money stock (see Equation 3.3) expres- sions were derived for the rates of change in M with respect to the key variables of the model. These expressions were of the form SM a(PX) aY C ai e + + z sZ + C aSZ +C5 Sifn Sz (5.1) BY THE TERM "monetary mechanism" I mean the complex web of causal relations through which interest rates, prices, and real phenom- ena react to determine the money stock, as well as the chain of causality running in the opposite direction-the impact of the stock of money on the real and financial variables of the economy. The model constructed in chapter 2 enables examination of both of these facets of the monetary mechanism. Based on the identity for the money stock (see Equation 3.3) expres- sions were derived for the rates of change in M with respect to the key variables of the model. These expressions were of the form SM _ a(PX) F, + a + =C Z +C2 +C a Z a4Z +Cs nf (5.1) Sz where Z is any interest rate, price, real output, or income. From 5.1, I conclude that a change in any of these variables affects the stock of money through its impact on prices (P), real output (X), income (Y), and the various rates of interest. The terms C, through Cs are in a sense "money multipliers." They show by how much the money stock changes, given a change in the variables with which they are associated. (See Tables 3-7 and the associated discussion for a complete description and interpretation of these terms.) I have thus developed a scientific money multiplier for each of the key variables in the model. The Brunner-Meltzer and Teigen models develop only a single money multi- plier, while the C's are only some of several "multipliers" I have developed. (Other multipliers include the Q's developed in the analysis of the effects of changes in reserve requirements and open-market operations.) 140 where Z is any interest rate, price, real output, or income. From 5.1, I conclude that a change in any of these variables affects the stock of money through its impact on prices (P), real output (X), income (Y), and the various rates of interest. The terms C, through C. are in a sense "money multipliers." They show by how much the money stock changes, given a change in the variables with which they are associated. (See Tables 3-7 and the associated discussion for a complete description and interpretation of these terms.) I have thus developed a scientific money multiplier for each of the key variables in the model. The Brunner-Meltzer and Teigen models develop only a single money multi- plier, while the C's are only some of several "multipliers" I have developed. (Other multipliers include the Q's developed in the analysis of the effects of changes in reserve requirements and open-maket operations.) 140  RESULTS OF THE STUDY 141 The neet ssep in she analysis of this facet of she monesary mechanism was an exploration of she specific seems an the tight-hand side of 5.1. 1 showed how each of these teems coold he deemved and span what variables they depend. The thirteen equations of she foem 5.1, coopted with she desceiption of the teems on she sighs-hand sides of these eqoations provided in chaper 3, constitote a complte description of the faces of the monetary mechanism where she chain of causalisy tons from the nariables of she economy to the steach of money. The genteral equilihrium esate of she approach and she addition of mote sectors and varialetprovidea freatter wealh of detadilnaoe compeehensive description of how she money stach sendts so changes in the economy than any of she ashes extant studies of the money supply. IShone syelled out the specific behavioral functions and hudlt from these, whereas the ashe tstsudies only indicate somewhat fozzily span what variahles she teems in sheir money supply functions depend.t The majos advance provided hy this parthos of she stady is the explicit demonstrasion of the links among the variahles in the model and the money stock. I hone cloely displayed these links rathee than hiding she guts of this portion of she monetary mechanism hehind highly simplified identities foe she money stack of she sass developed hy Friedman and Schwaets, Beannee and Meltee, and Teigen, at hehind a logically inconsistent equation as developed hy de Leesm. The addition of an examination of the links munning from she money stock to the vatiahles of the model repeesents an impoetant advance ones extant studies. Exytessions foe SZ/SM have heen descrihed and analysed, where Z again is any nariable (poice, inserest rte, income, output) in Ike model.' This analysis seeves she exsremely useful puepose of persuading as that a study of she monetaey mechanism is important, since changes in M will hone an impact on the coal and financial variahles of she economy. We have shown shat she impact of M on real outputs operates theough she effects of changos in M an income, prices, and instetest sates, that affecting hash she demand and sopply of goods. Even if we assume that all demand fanctions see homogeneous of degree zeo in all theie arguments (which was not done), a change in she money s. Do Letoo's modol is exemyt team most of ths criticism, dine it isa simultaneous model baile as specific behadiorad eqations. 2. One serendlipitous tesate of eke sanlyis as a aiffeents proof afthIe quantitytheortyandoasemostrationethat interestae effecttaonenand topply may waklen else irect reltion between money andeprices (seeochapters3) and theimpotanceof ownaril-sloping demnad curves fortettuaontity theoryto hold. RESULTS OF THE STUDY 141 The next stoy in the analysis of this facet of the monetasy mechanism was an exploration of the specific teems an she eight-hand side of 5.1. 1 showed how each of these teems could ho deemved and upon what variahles they dopend. The thirteen equstions of she fosm 5.1, coopted wish she description of the seems on the eight-hand sides of those equations provided in chapsee 3, constitute a complete desceiption of she facet of she monetary mechanism where she chain of causality sans feom she variables of she economy so the stock of money. The genesal equilihrium atute of the approach and the addition of mooe sectos and variables peovide a greter wealth of detaid sod a mote comprehensive descripsion of how sho money stock otacts so changes in the economy than any of she asher extant stodies of the money supply. Ihave spolled out she specific behavioal funcsions and hails from these, wheseas she oshee studies only indicate somewhat fuzzidy upon what variahles she teems in their money supply functions depend.' The majos advance provided hy this passion of the study is she expicit demonstrasion of she links among she variables in the model and the money stock. I have cloely displayed these links sashes than hiding the guts of this passion of she monesary mechanism behind highly simplified identities foe she money stack of she suet developed hy Priedman and Schwartz, Brunner and Melter, and Teigen, as hehinda logically inconsistent equation as developed hy do Loomw. The addition of an examination of she links running from the money slack so the variahles of the model represents an important advance ones extant studies. Expressions foe SZ/SM have keen described sod analyzed, whose Z again is any variable (price, insterest rte, income, output) in she model.' Thin analydis sesves she exsremely useful purpose of pertsding us that a study of she monetary mechanism is impostant, dsine changes in M will have an impact on she stat and financial variables of the economy. We have shown that the impact of hi on seat outpos operuses shrough she effects of changes in M on income, prices, and intstot eates, shot affecting hash the demand and supply of goods. Even if we assume that all demand fanctions see homogeneous of degree sees in alt their arguments (which was not done), a change in the money 1. Os Leena's model 0s exemps from woot of thit criticism, Oince 0t is simolsuneasus model boils on tpecific behsvioral eqations. 2. One serendipitous tesate of eke andlysis as a difterens prcof of tbs qusntiytheoryeadademnsatonthattintestrae effects ondemand and supply may weakeon the direct solstice besten money a prices (setchauptert3) and thetimportance ofdomnae-stcpiog demnd curvet forethe quantity teoryto RESULTS OP THE STUDY 141 The neat step in she analysis of this fact of she monetary mechanism was an exploation of the specific teems on she tight-hand side of 5.1.1 showed how each of themseterms could he deeived and span what variables they depend. The thirteen equations of the form 5.1, coupled wish the desciption of she teems an the tight-hand Oides of these equations provided in chapter 3, consitute a complete description of she faces of she mnonetaey mechanism whose the chain of causality tans from she variahles of the economy so she stack of money. The general equilihrium noate of the aptroach and she addition of mote secter sod variahles provide a greer wealth of detail and a muse comprehensive description of ham the money stack teases so changes in she economy than any of she ashes extant studies of she money supply. Ihave speledaoutthespecificehavialtfunctions andultfromthes, whereas she ashes studies only indicate somewhat fussily upon what variahles the seems in their money supply functions depend.t The major advance provided hy this portion of the study is she explicit demonstration of the links among she variables in she model and she money stack. I have cleasly displayed these links sashes than hiding she guts of this paotin of the monetary mechanism behind highly simplified identities fat the money stock of the sass developed hy Friedman and Schmarts, Brunnee and Moltet, and Teigen, as hehind a logically inconsistent equation us developed hy de Loamw. The addition of an examination of else links sunning from the money tack to the variahles of the modleepesents an impotant advance ones extant studies. Expemions foe SZ/SM have keen described and analysed, whose Z again is any variable (price, interest sate, income, output) in the model.' This analysis serves she extremely useful purpose of persuading us that a study of the monetary mechanism is important, since changes in M wilt have an impact an the seat and financial variables of the economy. We have shown that the impact of M on seal outputs operates theough the effects of changes in M an income, prices, and inteest eases, shut affecting hash she demand and supply of goads. Even ifine assume that all demand fanctoans mre homogeneous of degee zero in all theit arguments (which mat not done), a change in the money 1. Dt Lenne modal is exempt from moot of this vOiticoo, Oince iti0 simultaneous model built an specific behuviorsl esqutions. 2. One serendlipitus tofeatsr of eke analysis as a different psoof of she quantity teory soil a dlemonstation that interst sats effaes on demandl sa supply may meshes else direct relation between money and priots (seetchapted3 andte importace of donar-sloingemand notates forthe qantitystheoryeto  142 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM stock will result in a shift in demands, since the change in M will not, in general, produce equiproportional changes in the P's, Y's, and r's. The same argument can be made with regard to shifts in supply. Thus, the only circumstances in which the SX/SM's will be zero is if the shifts in supply and demand exactly balance each other out. There is no reason to suppose that, in general, this will be the case. THE TOOLS OF MONETARY POLICY The model was used to analyze the effects and the effectiveness of the major tools of monetary policy-open-market operations, changes in the reserve requirements, and changes in the rediscount rate-in combatting both inflation and unemployment. Changes in the rediscount rate were found to be almost completely ineffective in combatting problems of unemployment (characterized by an excess supply of bank loans from unborrowed reserves). They were only somewhat effective in helping solve problems of inflation (when there is likely to be an excess demand for bank loans). In these situations changes in the rediscount rate affect the amount of redis- counting and thus the actual amount of loans made by the bank. Changes in the amounts of bank, loans made potentially change the demand for the consumer and capital good, thus affecting the level of economic activity. The implication of these conclusions is to cast more doubt on the usefulness of discretionary changes in the rediscount rate as a monetary tool, especially in combatting problems of unemploy- ment. The effects of open-market operations and changes in the reserve requirement on the variables of the model were a result of their primary effects on the money stock, the rate on government securities, and the rate on firms' securities. Their effectiveness in combatting inflation depends on the size of SPa SF SFP SP ( + c)dM+( + c)dr + SM Sm rg org 142 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM stock will result in a shift in demands, since the change in M will not, in general, produce equiproportional changes in the P's, Y's, and r's. The same argument can be made with regard to shifts in supply. Thus, the only circumstances in which the SX/SM's will be zero is if the shifts in supply and demand exactly balance each other out. There is no reason to suppose that, in general, this will be the case. THE TOOLS OF MONETARY POLICY The model was used to analyze the effects and the effectiveness of the major tools of monetary policy-open-market operations, changes in the reserve requirements, and changes in the rediscount rate-in combatting both inflation and unemployment. Changes in the rediscount rate were found to be almost completely ineffective in combatting problems of unemployment (characterized by an excess supply of bank loans from unborrowed reserves). They were only somewhat effective in helping solve problems of inflation (when there is likely to be an excess demand for bank loans). In these situations changes in the rediscount rate affect the amount of redis- counting and thus the actual amount of loans made by the bank. Changes in the amounts of bank loans made potentially change the demand for the consumer and capital good, thus affecting the level of economic activity. The implication of these conclusions is to cast more doubt on the usefulness of discretionary changes in the rediscount rate as a monetary tool, especially in combatting problems of unemploy- ment. The effects of open-market operations and changes in the reserve requirement on the variables of the model were a result of their primary effects on the money stock, the rate on government securities, and the rate on firms' securities. Their effectiveness in combatting inflation depends on the size of SF SF SFP SPF ( k + - -)dM+(_ - + v)dr + SM SM Srg rgt 142 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM stock will result in a shift in demands, since the change in M will not, in general, produce equiproportional changes in the P's, Y's, and r's. The same argument can be made with regard to shifts in supply. Thus, the only circumstances in which the SX/SM's will be zero is if the shifts in supply and demand exactly balance each other out. There is no reason to suppose that, in general, this will be the case. THE TOOLS OF MONETARY POLICY The model was used to analyze the effects and the effectiveness of the major tools of monetary policy-open-market operations, changes in the reserve requirements, and changes in the rediscount rate-in combatting both inflation and unemployment. Changes in the rediscount rate were found to be almost completely ineffective in combatting problems of unemployment (characterized by an excess supply of bank loans from unborrowed reserves). They were only somewhat effective in helping solve problems of inflation (when there is likely to be an excess demand for bank loans). In these situations changes in the rediscount rate affect the amount of redis- counting and thus the actual amount of loans made by the bank, Changes in the amounts of bank, loans made potentially change the demand for the consumer and capital good, thus affecting the level of economic activity. The implication of these conclusions is to cast more doubt on the usefulness of discretionary changes in the rediscount rate as a monetary tool, especially in combatting problems of unemploy- ment. The effects of open-market operations and changes in the reserve requirement on the variables of the model were a result of their primary effects on the money stock, the rate on government securities, and the rate on firms' securities. Their effectiveness in combatting inflation depends on the size of SP SM Sr5 S SMk + cdM+( + - )dr + m S F r9 rg ( ±+ ) dr. (5.2) T tf Sth Thrir effecsiveness in combatting unemployment depends on the rice of ( k + 4 ) drf. SrF rF (5.2) k + c)dr. Sr S rte (5.2) Their effectiveness in combatting unemployment depends on the size of Their effectiveness in combatting unemployment depends on the size of  RESULTS OF THE STUDY 143 ( k+ - - )dM+( k+ c)dr+ SM SM org arg a ax0 ax0 are + as, ) dry, (5.3) where the differentials refer to the changes in M, rg, and r caused by either of the two monetary tools. The sizes of the partial derivatives are independent of the particular monetary tool used; consequently one tool can attain the same results as the other. There were two major differences between the two tools. Open-market operations cause an immediate change in the total size of the private sectors' asset portfolio by changing the total amount of government securities held by the private sectors; changes in the reserve requirement, while indirectly changing the size of the private asset portfolio (through its impact on C, D, and T), do not have this direct effect. Second, changes in the reserve requirement change the quantity of bank loans supplied per dollar of deposits, while open-market operations lack this effect. As a result of these major differences I found that, given an open-market operation and a change in the reserve requirement that produce the same total change in the reserves of the banking system, the open-market operation has a greater impact on both the money stock and on the rate on government securities. I also developed a way of finding combinations of open-market operations and changes in the reserve requirement that produce the same change in total reserves. This formulation added to the belief that changes in reserve requirements are an extremely powerful tool of monetary policy because of the very large open-market operation that would have to be undertaken to produce the same effects as the change in the reserve requirement. SUGGESTED EMPIRICAL TESTING PROCEDURE Empirical testing of the model and its conclusions will be centered about the estimation of the elements of the vectors At, i = 1, 2, . . . , 30 and the other parameters of the model. Once estimates for these parama- 3. This conclusion on interest rate effects seems to contradict the findings of Ascheim (2). He found that given an equal reduction in demand deposits, a higher reserve requirement would produce a greater increase in interest rates than a restrictive open-market operation. It is not clear that his analysis, framed in terms of equal changes in demand deposits, and ours in terms of equal changes in reserves are directly comparable. RESULTS OF THE STUDY 143 ax, axe i ax0 + axe r (am + c)dM+( + )dr + ax, ,axe 3as, +a -1) dre, (5.3) where the differentials refer to the changes in M, rg, and r caused by either of the two monetary tools. The sizes of the partial derivatives are independent of the particular monetary tool used; consequently one tool can attain the same results as the other. There were two major differences between the two tools. Open-market operations cause an immediate change in the total size of the private sectors' asset portfolio by changing the total amount of government securities held by the private sectors; changes in the reserve requirement, while indirectly changing the size of the private asset portfolio (through its impact on C, D, and T), do not have this direct effect. Second, changes in the reserve requirement change the quantity of bank loans supplied per dollar of deposits, while open-market operations lack this effect. As a result of these major differences I found that, given an open-market operation and a change in the reserve requirement that produce the same total change in the reserves of the banking system, the open-market operation has a greater impact on both the money stock and on the rate on government securities.3 I also developed a way of finding combinations of open-market operations and changes in the reserve requirement that produce the same change in total reserves. This formulation added to the belief that changes in reserve requirements are an extremely powerful tool of monetary policy because of the very large open-market operation that woald have to be undertaken to produce the same effects as the change in the reserve requirement. SUGGESTED EMPIRICAL TESTING PROCEDURE Empirical testing of the model and its conclusions will be centered about the estimation of the elements of the vectors A,, i = 1, 2, . . . , 30 and the other parameters of the model. Once estimates for these param- 3. This conclusion on interest rate effects seems to contradict the findings of Ascheim (2). He found that given an equal reduction in demand deposits, a higher reserve requirement would produce a greater increase in interest rates than a restrictive open-market operation. It is not clear that his analysis, framed in terms of equal changes in demand deposits, and ours in terms of equal changes in reserves are directly comparable. RESULTS OF THE STUDY 143 aM + c)dM+( + c)dr + m M arg as1 (ax + axc ) dry, (5.3) ar, ar where the differentials refer to the changes in M, rg, and r caused by either of the two monetary tools. The sizes of the partial derivatives are independent of the particular monetary tool used; consequently one tool can attain the same results as the other. There were two major differences between the two tools. Open-market operations cause an immediate change in the total size of the private sectors' asset portfolio by changing the total amount of government securities held by the private sectors; changes in the reserve requirement, while indirectly changing the size of the private asset portfolio (through its impact on C, D, and T), do not have this direct effect. Second, changes in the reserve requirement change the quantity of bank loans supplied per dollar of deposits, while open-market operations lack this effect. As a result of these major differences I found that, given an open-market operation and a change in the reserve requirement that produce the same total change in the reserves of the banking system, the open-market operation has a greater impact on both the money stock and on the rate on government securities? I also developed a way of finding combinations of open-market operations and changes in the reserve requirement that produce the same change in total reserves. This formulation added to the belief that changes in reserve requirements are an extremely powerful tool of monetary policy because of the very large open-market operation that would have to be undertaken to produce the same effects as the change in the reserve requirement. SUGGESTED EMPIRICAL TESTING PROCEDURE Empirical testing of the model and its conclusions will be centered about the estimation of the elements of the vectors Ai, i = 1, 2, . . . , 30 and the other parameters of the model. Once estimates for these param- 3. This conclusion on interest rate effects seems to contradict the findings of Ascheim (2). He found that given an equal reduction in demand deposits, a higher reserve requirement would produce a greater increase in interest rates than a restrictive open-market operation. It is not clear that his analysis, framed in terms of equal changes in demand deposits, and ours in terms of equal changes in reserves are directly comparable.  144 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM eters are obtained, numerical estimates of the various multipliers can be obtained. The second level of testing will center about estimation of the ac- curacy of the rates of change and total changes predicted by the model. How do the predictions "fit" with changes in M and other variables actually observed? It is felt that regression analysis and one- and two-stage least squares should comprise the bulk of the statistical techniques to be used. The data to which the model will be fit will be those used by the Brookings- SSRC model. Completion of this portion of the work will provide an empirical judgment of the conclusions of the work which, to this point, are based strictly on theory and a few assumptions about the signs, but not the magnitudes, of the parameters of the model. 144 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM eters are obtained, numerical estimates of the various multipliers can be obtained. The second level of testing will center about estimation of the ac- curacy of the rates of change and total changes predicted by the model. How do the predictions "fit" with changes in M and other variables actually observed? It is felt that regression analysis and one- and two-stage least squares should comprise the bulk of the statistical techniques to be used. The data to which the model will be fit will be those used by the Brookings- SSRC model. Completion of this portion of the work will provide an empirical judgment of the conclusions of the work which, to this point, are based strictly on theory and a few assumptions about the signs, but not the magnitudes, of the parameters of the model. 144 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM eters are obtained, numerical estimates of the various multipliers can be obtained. The second level of testing will center about estimation of the ac- curacy of the rates of change and total changes predicted by the model. How do the predictions "fit" with changes in M and other variables actually observed? It is felt that regression analysis and one- and two-stage least squares should comprise the bulk of the statistical techniques to be used. The data to which the model will be fit will be those used by the Brookings- SSRC model. Completion of this portion of the work will provide an empirical judgment of the conclusions of the work which, to this point, are based strictly on theory and a few assumptions about the signs, but not the magnitudes, of the parameters of the model.  Appendix: Definitions T HIS LIST contains the major symbols used in tke model rod rheie definitioos. Aoy Greek lesser or lower rose English lessee ot lisred shore (with or mishoot subsnripts) represeots a parameter insa decisioo foontion foesa porsinulor sentor. r =sloe reserne requiremeot 00 demaod aod lime deposits sf = he role of ioteress oo firms' senurities r =the role of interest on goveenmentssenurisies b =te role of intrest on deposits ioiotermediaries i =the rose of ontsess on rime deposits 'b he rateofinterest onbank loansto frss sb he rose of interest on hook loses so eke poblic In hr rare of inrerest on inrermediary loons so firms ti =the ease of interest on intermediary loans so she public r =the redisnount rose F =the noupoo role oo goveromentlsecurities -fn (rf rii,0,,enf, 0,0) Tl5 = (r, 11, r., r, 0, rb,o, roe) de = a policy desermined nhonge in resereeeeqiremens Ai = (i =1, 2,... 30l)=columnector ofnonstns =(as , bi, ci, di, 00, f5, gi, hi) A = eor of firms' asses Appendix: Definitions T HIS LIST cootaios she mojor symbols used in she model sod their definitioos. Aoy Greek leseer or lower rose English leeter nor lissed aooe (milk or mishout sobscrips) repressors a porameter insa decision fontion foesa poetinulor secsor. r =sthe resereerequirementeon demandoandime deposits r, = eke role of interest on firms' securities sg = he rose of interess onogovernmentssenuriies sn = he rose of ioterest on deposirs ierineemediories it = she ease of intrest on lime deposits rb =ehe rate of intrestoonb oansto firs rbe = tke ease of interest on book boons so she public r = eke rose of interest 00 intermediary boons so films rp = she role of ioterest on intermediary boons so Ike publin sd = he rediscont rote Ik te coupon rose on governmentssecurities ir = (rf, I0, r_ it, rfbI', ~'r5' 0) hrb = (r, rg, 0 fitre r' 0 % (r,eg,er,,,,lrr, roe) oP (rI, rg, in, it, Orb0,10, rnp) dr = a policy determined ckange ineserve reqiremens Ai = (i=1, 2,... 30) =columnectorofnonstants = (a0, h0, n0, di, eI, fi, gi, hi) A1 = vector offirms' ssets Appendix: Definitions T HIS LIST contains she major symbols sed in she model and their definisions. Any Greek lesser oe liner nose English leteer nor bisted shone (mith or wihout subscripts) represents a paeametr in a denision function for a paetinulae sentor. r =sthe reserve requirement on demand sod rime deposis 11 = she rose of inteeresan fiems'securities sk te rate of ineestongoernmentsecuiies ek te rose of itrest on deposis in inteemediaries ek te ease of intereet on rime deposits 101 = eke ease of interest on hook Icoans en firms rp = eke role of inteest on honk loans so rhe pubhin 'rr = theerateof interest on intermediaryloanslto firms rp = she rose of interest on intermediary loans so she poblin rd = the rediscountbrate s = he noopon eate on governmentseseuities hf (rf, 0o, in, it, rf 0 ,eo0,) fb = (r, , 0 t, 1e'1n5'p n01 0) or (rr, rg,in,eet,fl,flr p1 , rn) dr = o policy deteemined change in reserve reqiements A0 = (i=1, 2,. ,30) =colmnecorof onstans =(i,hb, ci,di, e, f0, p0, hi) A5 = vector of firms'ossets  146 APPENDIX: DEFINITIONS C = cureency D = demand deposita D, = stack demand foe capital dK = flow demand foe copitl D Q = demand forelabor DA0D = desired distributioe of films' financial assets DEP = desired distsibutian af fisms' setained easnings d = actual amountlofsediscounting dd = amont of rediscounting demanded dod =amoans of rediscoanting demanded an meet pabtic's demand foe loans dd =amount oferediscounting demendedto meet fitms' demand fot dZ, = change in asset Z resalting feom a change in she honks' holdings of firms' securities dZe = change in asset Z resulting from a change in she honks' holdings of goveenmentssecurities dZnaI = change in secaos k's holdings of asset Z resahting feom a change in sectes k's holdings of assetI dZT = totaltchange in asset Z dZ05 = eoal change in asset Z resalting from a change inrerv requirements dZTG. = sotal change in asset Z resulaing feom an open-maeket opeeation dMT, = tatal change in she money stock resulting feom a change in reserverequiements dT = tatal change in the maney slack resalting feom an open-market operation dr., change in interesterateer.ereslting froma changeineseee requisemenas disc~ = change in interesteateereesling feomanpen-marhet opeation E' actual slack of fitms' resained earnings 146 APPENDIX: DEFINITIONS 146 APPENMIX DEFINITIONS C D dK Mt DA' ME' dZ dZ dZ05 dZg dZ05 dMT1 = currency = demand deposits = stock demandforacapital = flow demand foe capital = demand forlahor = desised disteihusion of firms' financial assets = desired diesihusion of fiems' stamned earnings = actal aemoans of eediscounting = amont of rediscounting demanded = amtount of rediscanting demanded to meet pubhhc's demand fat loans = amont of rediscounting demanded toameetsfirms' demand foe loans = change in asses Z sesulting from a change in the flanks' holdings of firms' secarisies = change in asset Z resulsing feom a change in she hanhe' haldings of government secueities = change in sector k's holdings of asset Z resalting from a change in sector kc's holdings of assetlI = total change in asset Z - tolat change in asset Z resulting feom a change in testerv requirements = tatal change in asset Z resulting feom an open-mashes operation - total change in the money stock resulting from a change in resereequirements C D dK DAD dZ ddk dZ0 dZe dM,, = caseency = demand deposits - stock demand fat capital = flow demand fos capital = demand foe tahoe = desired distribution of firms' fnancial assets = desied distribution of fiems' retained earnings = actual amountofsediscouning = amont of redisconting demanded = amonn of redisconting demanded In meet paublic's demand foe loans = amountsoferediscounting demanded tomeetfirms' demand for loans = change inassetaZeslting faom achange in the flanks' holdings af fiems' securtiies = change in asses Z rselting feam a change in she flanks' haldings of goveenmentlsecurities = change in sectork's holdings of asset Zsresulting from achange in sectorlc's holdings of asset 1 = total change in asset Z = toedl change in asses Z resdlting from a change in reseree reqairemntsl - total change inasset Zseslting froman ope-maket operation basoal change in she money stack eesnlting feom a change in resereeseqaisements dT = totl change in the money soch eesulting fram an open-mashes operation drne = change in intersest eat e rresalting from a change in reseree requirements de5, = change in insteerateer.tesultng feoman open-market operation EPa = actualsock of firms'etsained earnings dMG = tonad change in the money stack eesdlting from an apen-maskes operation dres = change in inteeest rte r. eesulting feom a change in eeserve requirements d. = change in inteest ease I. resdlting from an open-market aperation P5a = actual stockoffirms'eained earnings  APPENDIX: DEFINITIONS 147 Ed = desired stock of firms' retained earnings F = firms' debt instruments (securities) Fd = firms' demand for financing FS = firms' supply of new securities G = government securities G* = an open-market operation G = face value of government securities held by the private sectors GS = supply of government securities G = demand for government securities Ig = gross investment 1 = net investment K = capital stock k = rate of growth of the capital stock L = amount of labor in the economy Lc = amount of labor used to produce the consumer good Lk = amount of labor used to produce the capital good = total bank loans Lu = total intermediary loans L = total loans to the public L = total loans to the firms L = bank loans to the public L = bank loans to the firms S = intermediary loans to the public " = intermediary loans to the firms Lg = firms' aggregate demand for loans Db = firms' demand for bank loans Lr" = firms' demand for intermediary loans L = public's aggregate demand for loans L b = public's demand for bank loans APPENDIX: DEFINITIONS 147 Ed = desired stock of firms' retained earnings F = firms' debt instruments (securities) F = firms' demand for financing FS = firms' supply of new securities G = government securities G* = an open-market operation G = face value of government securities held by the private sectors G = supply of government securities GD = demand for government securities la = gross investment In = net investment K = capital stock k = rate of growth of the capital stock L = amount of labor in the economy L = amount of labor used to produce the consumer good Lk = amount of labor used to produce the capital good Lt = total bank loans L" = total intermediary loans Le = total loans to the public L, = total loans to the firms L = bank loans to the public L = bank loans to the firms L" = intermediary loans to the public L = intermediary loans to the firms LF = firms' aggregate demand for loans LFb: = firms' demand for bank loans L" = firms' demand for intermediary loans L = public's aggregate demand for loans LDb = public's demand for bank loans APPENDIX: DEFINITIONS 14, Ed = desired stock of firms' retained earnings F = firms' debt instruments (securities) Fd = firms' demand for financing FS = firms' supply of new securities G = government securities G* = an open-market operation O = face value of government securities held by the private sectors G = supply of government securities GD = demand for government securities Is = gross investment In = net investment K = capital stock k = rate of growth of the capital stock L = amount of labor in the economy L = amount of labor used to produce the consumer good L = amount of labor used to produce the capital good L = total bank loans L = total intermediary loans LP = total loans to the public L, = total loans to the firms Lb = bank loans to the public L = bank loans to the firms = intermediary loans to the public L = intermediary loans to the firms L = firms' aggregate demand for loans Leb = firms' demand for bank loans L" = firms' demand for intermediary loans LD = public's aggregate demand for loans LDb = public's demand for bank loans  148 APPENDIX: DEFINITIONS L = public's demand for intermediary loans sb = banks'loan supply from unborrowed reserves LSb amount banks are willing to loan the public from unborrowed reserves i amount banks are willing to loan the firms from unborrowed reserves aggregate amount intermediaries are willing to lend I" = aggregate amount intermediaries are willing to lend the public S" aggregate amount intermediaries are willing to lend the firms M =the money stock m = the number of consumer good firms N = deposits in intermediaries n = the number of capital good firms P = the price of the consumer good P = the price of the capital PX = (PcXc+PcXc) QG. = the ratio of successive reserve changes resulting from an open-market operation Qr = the ratio of successive reserve changes resulting from a change in reserve requirements R = required reserves RS = secondary reserves R def = reserve deficiency R defT = total reserve deficiency S = stock supply of capital S = supply of labor sk = flow supply of capital T = time deposits T, = tax receipts t = marginal rate of taxation Xc = the consumer good 148 APPENDIX: DEFINITIONS Le = public's demand for intermediary loans LSb = banks' loan supply from unborrowed reserves Esb = amount banks are willing to loan the public from unborrowed reserves I-sb = amount banks are willing to loan the firms from unborrowed reserves toy = aggregate amount intermediaries are willing to lend In* = aggregate amount intermediaries are willing to lend the public 4n = aggregate amount intermediaries are willing to lend the firms M = the money stock m = the number of consumer good firms N = deposits in intermediaries n = the number of capital good firms Pc = the price of the consumer good Pk = the price of the capital PX = (P Xa + PkXk) QG. = the ratio of successive reserve changes resulting from an open-market operation Qr = the ratio of successive reserve changes resulting from a change in reserve requirements R = required reserves R = secondary reserves R def = reserve deficiency R defT = total reserve deficiency S = stock supply of capital S = supply of labor s = flow supply of capital T = time deposits T, = tax receipts t = marginal rate of taxation Xc = the consumer good 148 APPENDIX: DEFINITIONS L D = public's demand for intermediary loans L = banks'loan supply from unborrowed reserves Esb amount banks are willing to loan the public from unborrowed reserves Ls =amount banks are willing to loan the firms from unborrowed reserves I-sn = aggregate amount intermediaries are willing to lend 4 = aggregate amount intermediaries are willing to lend the public i = aggregate amount intermediaries are willing to lend the firms M = the money stock m = the number of consumer good firms N = deposits in intermediaries n = the number of capital good firms Pc = the price of the consumer good P, = the price of the capital X = (PXc +P0X0) QG* = the ratio of successive reserve changes resulting from an open-market operation Q = the ratio of successive reserve changes resulting from a change in reserve requirements R = required reserves Rs = secondary reserves R def = reserve deficiency R deft = total reserve deficiency S = stock supply of capital S = supply of labor sk = flow supply of capital T = time deposits T, = tax receipts I = marginal rate of taxation X, = the consumer good  APPENDIX: DEFINITIONS 149 APPENDIX: DEFINITIONS 149 X, = the capital good Xk0 = amount of the capital good used in the production of the consumer good Xkk = amount of the capital good used in the production of capital Y = public's disposable money income Y = gross money income Yb = banks' contribution to income Yf = firms' contribution to income Y, = intermediaries' contribution to income Yg = government's contribution to income o = rate of depreciation a,, ak = Xk and Xc intercepts of the transformation curve X = rate of growth of the labor force 0 = profit expectation function for the firms 7nb = total banks' profits kb = banks' profit from loans ob = banks' profit from government securities nfb = banks' profit from firms' securities Tn, = total intermediaries' profit 'Tkn = intermediaries' profit from loans gny = intermediaries' profit from government securities "tr = intermediaries' profit from firms' securities X, = the capital good Xac = amount of the capital good used in the production of the consumer good Xkc = amount of the capital good used in the production of capital Y = public's disposable money income Y = gross money income Y = banks' contribution to income Y, = firms' contribution to income Y = intermediaries' contribution to income Yg = government's contribution to income a = rate of depreciation act k = Xk and X. intercepts of the transformation curve X = rate of growth of the labor force 0 = profit expectation function for the firms 7b = total banks' profits ngb = banks' profit from loans ngb = banks' profit from government securities fb = banks' profit from firms' securities n7 = total intermediaries' profit n0n = intermediaries' profit from loans arg = intermediaries' profit from government securities , = intermediaries' profit from firms' securities APPENDIX: DEFINITIONS 149 Xk = the capital good Xta = amount of the capital good used in the production of the consumer good Xkk = amount of the capital good used in the production of capital Y = public's disposable money income YV = gross money income Yb = banks' contribution to income Y = firms' contribution to income Yn = intermediaries' contribution to income Y = government's contribution to income a = rate of depreciation ac, a - Xa and Xc intercepts of the transformation curve X = rate of growth of the labor force 0 = profit expectation function for the firms rb = total banks' profits Ttb = banks' profit from loans "gb = banks' profit from government securities ef = banks' profit from firms' securities =, = total intermediaries' profit Igk = intermediaries' profit from loans n. = intermediaries' profit from government securities fn = intermediaries' profit from firms' securities   Literature Cited Literature Cited Literature Cited Angell, J. The behavior of money; exploratory studies. New York: McGraw- Hill Book Co., Inc., 1936. Ascheim, J. Restrictive open market operations versus reserve requirement inceases: a reformulation. Economic Journal 73 (1963): 254-66. Baumol, W. The transactions demand for cash: an inventory theoretic ap- proach. Quarterly Journal of Economics 66 (1952): 545-56. Brunner, K. A schema for the supply theory of money. international Eco- nomic Review 2 (1961): 77-88. . Some major problems in monetary theory. American Economic Review 51 (1961): 381-97. Brunner, K., and Meltzer, A. An alternative approach to the monetary mechanism. Washington: U.S. Government Printing Office, 1964. Brunner, K., and Meltzer, A. Predicting velocity: implications for theory and policy. Journal of Finance 18 (1963): 319-54. Brunner,, K., and Meltzer, A. Some further investigation of the demand and supply functions of money. Journal of Finance 19 (1964): 321-59. Cagan, P. The demand for currency relative to the total money supply. Journal of Plitical Economy 66 (1958): 303-28. - . Determinants and effects of changes in the stock of money, 1875-1960. Washington: National Bureau of Economic Research, 1965. Chase, S., and Gramley, L. Time deposits in monetary analysis. Federal Reserve Bulletin 51 (1965): 1380-1406. Curry, L. The supply and control of money in the United States. Cambridge: Harvard University Press, 1934. Davidson, Paul. Money, portfolio balance, capital accumulation, and eco- nomic growth. Econometrica 36 (1968): 291-321. Davidson, Paul, and Smolensky, E. Modigla i on the interaction of real and monetary phenomena. Review of Economics and Statistics 46 (1964): 429-31. Davis, R.G. Open market operations, interest rates, and deposit growth. Quarterly Journal of Economics 79 (1965): 431-54. de Leeuw, F. A model of financial behavior. In The Brookings Quarterly Economic Model of the United States, ed. J. Duesenberry, G. Fromm, L. Klein, and E. Kuh, pp. 465-528. Chicago: Rand-McNally, 1965. Dewald, W. Free reserves, total reserves and monetary control. Journal of Political Economy 71 (1963): 141-53. - . Money supply vs. interest rates as approximate objectives of monetary policy. Washington: National Bureau of Economic Research, 1966. Fand, D. Some implications of money supply analysis. American Economic Review 57 (1967): 380-400. 151 1. 2. 3. 4. 5. 6. 7. 8. 9. 12. 13. 14. 15. 16. 17. 18. 19. Angell, J. The behavior of money; exploratory studies. New York: McGraw- Hill Book Co., Inc., 1936. Ascheim, I. Restrictive open market operations versus reserve requirement increases: a reformulation. Economic Journal 73 (1963): 254-66. Baumol, W. The transactions demand for cash: an inventory theoretic ap- proach. Quarterly Journal of Economics 66 (1952): 545-56. Brunner, K. A schema for the supply theory of money. International Eco- nomic Review 2 (1961): 77-88. --- . Some major problems in monetary theory. American Economic Review 51 (1961): 381-97. Brunner, K., and Meltzer, A. An alternative approach t tthe monetary mechanism. Washington: U.S. Government Printing Office, 1964. Brunner, K., and Meltzer, A. Predicting velocity: implications for theory and policy. Journal of Finance 18 (1963): 319-54. Brunner,:K., and Meltzer, A. Some further investigation of the demand and supply functions of money. Journal of Finance 19 (1964): 321-59. Cagan, P. The demand for currency relative to the total money supply. Journal of.Political Economy 66 (1958): 303-28. --- . Determinants and effects of changes in the stock of money, 1875-1960. Washington: National Bureau of Economic Research, 1965. Chase, S., and Gramley, L. Time deposits in monetary analysis. Federal Reserve Bulletin 51 (1965): 1380-1406. Curry, L. The supply and control of money in the United Stat-. Cambridge:; Harvard University Press, 1934. Davidson, Paul. Money, portfolio balance, capital accumulation, and ec- nomic growth. Eonometrica 36 (1968): 291-321. Davidson, Paul, and Smolensky, E. Modigliani on the interaction of real and monetary phenomena. Review of Economics and Statistics 46 (1964): 429-31. Davis, R. G. Open market operations, interest rates, and deposit growth. Quarterly Journal of Economics 79 (1965): 431-54. de Leeuw, F. A model of financial behavior. In The Brookings Quarterly Economic Model of the United States, ed. J. Duesenberry, G. Fromm, L. Klein, and E. Kuh, pp. 465-528. Chicago: Rand-McNally, 1965. Dewald, W. Free reserves, total reserves and monetary control. Journal of Political Economy 71 (1963): 141-53. --- . Money supply vs. interest rates as approximate objectives of monetary policy. Washington: National Bureau of Economic Research, 1966. Fand, D. Some implications of money supply analysis. American Economic Review 57 (1967): 380-400. 151 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. Angell, J. The behavior of money; exploratory studies. New York: McGraw- Hill Book Co., Inc., 1936. Ascheim, J. Restrictive open market operations versus reserve requirement increases: a reformulation. Economic Journal 73 (1963): 254-66. Baumol, W. The transactions demand for cash: an inventory theoretic ap- proach. Quarterly Journal of Economics 66 (1952): 545-56. Brunner, K. A schema for the supply theory of money. International E- nomic Review 2 (1961): 77-88. --- . Some major problems in monetary theory. American Economic Review 51 (1961): 381-97. Brunner, K., and Meltzer, A. An alternative approach to the monetary mechanism. Washington: U.S. Government Printing Office, 1964. Brunner, K., and Meltzer, A. Predicting velocity: implications for theory and policy. Journal of Finance 18 (1963): 319-54. Brunner, K., and Meltzer, A. Some further investigation of the demand and supply functions of money. Journal of Finance 19 (1964): 321-59. Cagan, P. The demand for currency relative to the total money supply. Journal of Political Economy 66 (1958): 303-28. --- . Determinants and effects of changes in the stock of money, 1875-1960. Washington: National Bureau of Economic Research, 1965. Chase, S., and Gramley, L. Time deposits in monetary analysis. Federal Reserve Bulletin 51 (1965): 1380-1406. Curry, L. The supply and control of money in the United States. Cambridge: Harvard University Press, 1934. Davidson, Paul. Money, portfolio balance, capital accumulation, and eco- nomic growth. Econometrica 36 (1968): 291-321. Davidson, Paul, and Smolensky, E. Modigliani on the interaction of real and monetary phenomena. Review of Economics and Statistics 46 (1964): 429-31. Davis, R.G. Open market operations, interest rates, and deposit growth. Quarterly Journal of Economics 79 (1965): 431-54. de Leeuw, F. A model of financial behavior. In The Brookings Quarterly Economic Model of the United States, ed. J. Duesenberry, G. Fromm, L. Klein, and E. Kuh, pp. 465-528. Chicago: Rand-McNally, 1965. Deald, W. Free reserves, total reserves and monetary control. Journal of Political Economy 71 (1963): 141-53. --- . Money supply vs. interest rates as approximate objectives of monetary policy, Washington: National Bureau of Economic Research, 1966. Fand, D. Some implications of money supply analysis. American Economic Review 57 (1967): 380-400. 151  LITERATURE CITED Feige, E. The demand for liquid assets: a temporal cross section analysis. Englewood Cliffs, N.J.: Prentice-Hall, 1964. Fisher, I. The theory of interest. New York: Augustus M. Kelley, 1965. Friedman, M. The demand for money, some theoretical and empirical results. Journal of Political Economy 67 (1959): 327-5 1. - The lag effect of monetary policy. Journal of Political Economy 69 (1961): 447-66. Friedman, M., and Schwartz, A. A monetary history of the United States, 1867-1960. Princeton: Princeton University Press, 1963. Goldfeld, S. Commercial bank behavior and economic activity; a structural study of monetary policy in the postwar United States. Amsterdam: North- Holland Publishing Co., 1966. Goldsmith, A. Financial intermediaries in the American economy since 1900. Princeton: Princeton University Press, 1964. Grambley, L., and Chase, S. Time deposits in monetary analysis. Federal Reserve Bulletin 51 (1965): 1380-1406. Gurley, J., and Shaw, E. Money in a theory of finance. Washington: The Brookings Institution, 1960. Horwich, G. Elements of timing and response in the balance sheet of banking. Journal of Finance 12 (1957): 310-44. Johansen, L. The role of the banking system in a macroeconomic model. International Economic Papers 8 (1958): 91-110. Keynes, J. A tract on monetary reformt. London: Macmillan and Co., 1923. --- . A treatise on money. New York: Harcourt, Brace and Co., 1930. Klein, L., and Goldberger, A. An econometric model of the United States, 1929-1952. Amsterdam: North-Holland Publishing Co., 1955. Kuh, E. The validity of cross-sectionally estimated behavior equations in time series applications. Econometrica 27 (1959): 197-214. Lavington, F. The English capital market. 2d ed. London: Methuen and Co., Ltd., 1929. Lindbeck, A. A study of monetary analysis. Stockholm: Aloquist and Wikesell, 1963. Lydall, H. Income, assets and the demand for money. Review ofEconomics and Statistics 40 (1958): 1-14. Maddala, G., and Vogel, R. The demand for money, a cross-section study of business firms: comment. Quarterly Journal of Economics 79 (1965): 153-59. Meade, J. The amount of money and the banking system. Economic Journal 64 (1934): 77-83. Meigs, A. Free reserves and the money supply. Chicago: University of Chicago Press, 1962. Meltzer, A. The behavior of the French money supply. Journal of Political Economy 67 (1959): 275-96. --- . The demand for money: a cross-section study of business firms. Quarterly Journal ofEcontomis 77(1963): 405-22. Miller, M., and Orr, D. A model of the demand for mone y b firms. Quarterly Journal ofEconomics 80 (1966): 413-35. Modigliani, F. The monetary mechanism and its interaction with real phe- nomena. Washington: National Bureau of Economic Research, 1963. Morrison, G. Liquidity preferences of commercial banks. Chicago: University of Chicago Press, 1966. Orr, D., and Mellon, W. Stochastic reserve losses and expansion of bank credit. American Economic Review 51 (1961): 614-23. 152 LITERATURE CITED 20. Feige, E. The demand for liquid assets: a temporal cross section analysis. Englewood Cliffs, N.J.: Prentice-Hall, 1964. 21. Fisher, 1. The theory of interest. New York: Augustus M. Kelley, 1965. 22. Friedman, M. The demand for money, some theoretical and empirical results. Journal ofPltical Economy 67 (1959): 327-51. 23. --- . The lag effect of monetary policy. Journal of Political Economy 69 (1961): 447-66. 24. Friedman, M., and Schwartz, A. A monetary history of the United States, 1867-1960. Princeton: Princeton University Press, 1963. 25. Goldfeld, S. Commercial bank behavior and economic activity; a structural study of monetary policy in the postwar United States. Amsterdam: North- Holland Publishing Co., 1966. 26. Goldsmith, A. Financial intermediaries in the American economy since 1900. Princeton: Princeton University Press, 1964. 27. Grambley, L., and Chase, S. Time deposits in monetary analysis. Federal Reserve Bulletin 51 (1965): 1380-1406. 28. Gurley, J., and Shaw, E. Money in a theory of finance. Washington: The Brookings Institution, 1960. 29. Horwich, G. Elements of timing and response in the balance sheet of banking. Journal of Finance 12 (1957): 310-44. 30. Johansen, L. The role of the banking system in a macroeconomic model. International Economic Papers 8 (1958): 91-110. 31. Keynes, J. A tract on monetary reform. London: Macmillan and Co., 1923 32. --- . A treatise on money. New York: Harcourt, Brace and Co., 1930. 33. Klein, L., and Goldberger, A. An econometric model of the United States, 1929-1952. Amsterdam: North-Holland Publishing Co., 1955. 34. Kuh, E. The validity of cross-sectionally estimated behavior equations in time series applications. Econometrica 27 (1959): 197-214. 35. Lavington, F. The English capital market. 2d ed. London: Methuen and Co., Ltd., 1929. 36. Lindbeck, A. A study of monetary analysis. Stockholm: Almquist and Wiksell, 1963. 37. Lydall, H. Income, assets and the demand for money. Review ofEconomics and Statistics 40 (1958): 1-14. 38. Maddala, G., and Vogel, R. The demand for money, a cross-section study of business firms: comment. Quarterly Journal of Economics 79 (1965): 153-59. 39. Meade, J. The amount of money and the banking system. Economic Journal 64 (1934): 77-83. 40. Meigs, A. Free reserves and the money supply. Chicago: University of Chicago Press, 1962. 41. Meltzer, A. The behavior of the French money supply. Journal of Political Economy 67 (1959): 275-96. 42. --- . The demand for money: a cross-section study of business firms. Quarterly Journal of Economics 77(1963): 405-22. 43. Miller, M., and Orr, D. A model of the demand for money by firms. Quarterly Journal of Economics 80 (1966): 413-33. 44. Modigliani, F. The monetary mechanism and its interaction with real phe- nomena. Washington: National Bureau of Economic Research, 1963. 45. Morrison, G. Liquidity preferences of commercial banks. Chicago: University of Chicago Press, 1966. 46. Orr, D., and Mellon, W. Stochastic reserve losses and expansion of bank credit. American Economic Review 51 (1961): 614-23. 152 LITERATURE CITED 20. Feige, E. The demand for liquid assets: a temporal cross section analysis. Englewood Cliffs, N.J.: Prentice-Hall, 1964. 21. Fisher, 1. The theory of interest. New York: Augustus M. Kelley, 1965. 22. Friedman, M. The demand for money, some theoretical and empirical results. JournalofPolitical Economy 67 (1959): 327-51. 23. --- . The lag effect of monetary policy. Journal of Political Economy 69 (1961): 447-66. 24, Friedman, M., and Schwartz, A. A monetary history of the United States, 1867-1960. Princeton: Princeton University Press, 1963. 25. Goldfeld, S. Commercial bank behavior and economic activity; a structural study of monetary policy in the postwar United States. Amsterdam: North- Holland Publishing Co., 1966. 26. Goldsmith, A. Financial intermediaries in the American economy since 1900. Princeton: Princeton University Press, 1964. 27. Grambley, L., and Chase, S. Time deposits in monetary analysis. Federal Reserve Bulletin 51 (1965): 1380-1406. 28. Gurley, J., and Shaw, E. Money in a theory of finance. Washington: The Brookings Institution, 1960. 29. Horwich, G. Elements of timing and response in the balance sheet of banking. Journal of Finance 12 (1957): 310-44. 30. Johansen, L. The role of the banking system in a macroeconomic model. International Economic Papers 8 (1958); 91-110. 31. Keynes, J. A tract on monetary reform. London: Macmillan and Co., 1923. 32. --- . A treatise on money. New York: Harcourt, Brace and Co., 1930. 33. Klein, L., and Goldberger, A. An econometric model of the United States, 1929-1952. Amsterdam: North-Holland Publishing Co., 1955. 34. Kuh, E. The validity of cross-sectionaly estimated behavior equations in time series applications. Econometrica 27 (1959): 197-214. 35. Lavington, F. The English capital market. 2d ed. London: Methuen and Co., Ltd., 1929. 36. Lindbeck, A. A study of monetary analysis. Stockholm: Almquist and Wikesell, 1963. 37. Lydall, H. Income, assets and the demand for money. Review of Economics and Statistics 40 (1958): 1-14. 38. Maddala, G., and Vogel, R. The demand for money, a cross-section study of business firms: comment. Quarterly Journal of Economics 79 (1965): 153-59. 39. Meade, J. The amount of money and the banking system. Economic Journal 64 (1934): 77-83. 40. Meigs, A. Free reserves and the money supply. Chicago: University of Chicago Press, 1962. 41. Meltzer, A. The behavior of the French money supply. Journal of Political Economy 67 (1959): 275-96. 42. --- . The demand for money: a cross-section study of business firms. Quarterly Journal ofEooomics 77(1963): 405-22. 43. Miller, M., and Orr, D. A model of the demand for money by firms. Quarterly Journal of Economics 80 (1966): 413-35. 44. Modigliani, F. The monetary mechanism and its interaction with real phe- nomena. Washington: National Bureau of Economic Research, 1963. 45. Morrison, G. Liquidity preferences of commercial banks. Chicago: University of Chicago Press, 1966. 46. Orr, D., and Mellon, W. Stochastic reserve losses and expansion of bank credit. American Economic Review 51 (1961): 614-23.  47. 48. 49. 50. 51. 52. 53. 54. 55. 56.- 57. LITERATURE CITED 153 Patinkin, D. Money, interest, and prices. 2d ed. New York: Harper and Row, 1965. Phillips, C. A. Bank credit. New York: Macmillan and Co., 1921. Polak, J., and White, W. The effect of income expansion on the quantity of money. International Monetary Fund Sta4fPapers 4 (1954-55): 327-52. Riefler, W. Money rates and money markets in the United States. New York: Harper and Bros., 1930. Teigen, R. Demand and supply functions of money in the United States: some structural estimates. Econometrica 32 (1964): 476-509. Tobin, J. Commercial banks as creators of money. Banking and Monetary Studies. Homewood, IL: Richard D. Irwin and Co., Inc., 1963. --- . Money and economic growth. Econometrica 33 (1965): 671-84. Tobin, J., and Brainard, W. Financial intermediaries and the effectiveness of monetary controls. American Economic Review 53 (1963): 383-400. Vogel, R., and Maddala, G. Cross-section estimates of liquid asset demand by manufacturing corporations. Journal of Finance 22 (1967): 557-76. Warburton, C. Monetary velocity and monetary policy. Review of Economics and Statistics 30 (1948): 304-13. Whalen, E. A cross-section study of business demand for cash. Journal of Finance 20 (1965): 423-43. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. LITERATURE CITED 153 Patinkin, D. Money, interest, and prices. 2d ed. New York: Harper and Row, 1965. Phillips, C. A. Bank credit. New York: Macmillan and Co., 1921. Polak, J., and White, W. The effect of income expansion on the quantity of money. International Monetary Fund Staff Papers 4 (1954-55): 327-52. Riefler, W. Money rates and money markets in the United States. New York: Harper and Bros., 1930. Teigen, R. Demand and supply functions of money in the United States: some structural estimates. Econometrica 32 (1964): 476-509. Tobin, J. Commercial banks as creators of money. Banking and Monetary Studies. Homewood, Ill.: Richard D. Irwin and Co., Inc., 1963. --- . Money and economic growth. Econometrica 33 (1965): 671-84. Tobin, J., and Brainard, W. Financial intermediaries and the effectiveness of monetary controls. American Economic Review 53 (1963): 383-400. Vogel, R., and Maddala, G. Cross-section estimates of liquid asset demand by manufacturing corporations. Journal offFinance 22 (1967): 557-76. Warburton, C. Monetary velocity and monetary policy. Review of Economics and Statistics 30 (1948): 304-13. Whalen, E. A cross-section study of business demand for cash. Journal of Finance 20 (1965): 423-43. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. LITERATURE CITED 153 Patinkin, D. Money, interest, and prices. 2d ed. New York: Harper and Row, 1965. Phillips, C. A. Bank credit. New York: Macmillan and Co., 1921. Polak, J., and White, W. The effect of income expansion on the quantity of money. International Monetary Fund Staff Papers 4 (1954-55): 327-52. Riefler, W. Money rates and money markets in the United States. New York: Harper and Bros., 1930. Teigen, R. Demand and supply functions of money in the United States: some structural estimates. Econometrica 32 (1964): 476-509. Tobin, J. Commercial banks as creators of money. Banking and Monetary Studies. Homewood, Ill.: Richard D. Irwin and Co., Inc., 1963. --- . Money and economic growth. Econometrica 33 (1965): 671-84. Tobin, J., and Brainard, W. Financial intermediaries and the effectiveness of monetary controls. American Economic Review 53 (1963): 383-400. Vogel, R., and Maddala, G. Cross-section estimates of liquid asset demand by manufacturing corporations. Journal offFinance 22 (1967): 557-76. Warburton, C. Monetary velocity and monetary policy. Review ofEconomics and Statistics 30 (1948): 304-13. Whalen, E. A cross-section study of business demand for cash. Journal of Finance 20 (1965): 423-43.   UNIVERSITY OF FLORIDA MONOGRAPHS UNIVERSITY OF FLORIDA MONOGRAPHS UNIVERSITY OF FLORIDA MONOGRAPHS 1. The Whigs of Florida, 1845-1854, by Herbert J. Doherty, Jr. 2. Austrian Catholics and the Social Question, 1918-1933, by Alfred Dia- mant 3. The Siege of St. Augustine in 1702, by Charles W. Arnade 4. New Light on Early and Medieval Japanese Historiography, by John A. Harrison 5. The Swiss Press and Foreign Af- fairs in World War 1, by Frederick H. Hartmann 6. The American Militia: Decade of Decision, 1789-1800, by Job K. Mahon 7. The Foundation of Jacques Mari- tain's Political Philosophy, by Hwa Yol Jung 8. Latin American Population Stud_- ies, by T. Lynn Smith 9. Jacksonian Democracy on the Florida Frontier, by Arthur W. Thompson 10. Holman Versus Hughes: Exten- sion of Australian Commonwealth Powers, by Conrad Joyner 11. Welfare Economics and Subsidy Programs, by Milton Z. Kafoglis 12. Tribune of the Slavophiles: Kon- stantin Aksakov, by Edward Chmie- lewski 13. City Managers in Politics: An Analysis of Manager Tenure and Te- mination, by Gladys M. Kammerer, Charles D. Farris, John M. DeGrove, and Alfred B. Clubok 14. Recent Southern Economic De- velopment as Revealed by the Chang- ing Structure of Employment, by Ed- gar S. Dunn, Jr. 15. Sea Power and Chilean Inde- pendence, by Donald E. Worcester 16. The Sherman Antitrus t Attand Foreign Trade, by Andre Simmons 17. The Origins of Hamilton's Fiscal Policies, by Donald F. Swanson 18. Criminal Asylum in Anglo-Saxon Law, by Charles H. Riggs, Jr. 19. Colonia Bar~n Hirsch, A Jewish Agricultural Colony in Argentina, by Morton D. Winsberg 20. Time Deposits in Present-Day Commercial Banking, by Lawrence L. Crum 21. The Eastern Greenland Case in Historical Perspective, by Oscar Svar- lien 22. Jacksonian Democracy and the Historians, by Alfred A. Cave 23. The Rise of the American Chem- istry Profession, 1850-1900, by Ed- ward H. Beardsley 1. The Whigs of Florida, 1845-1854, by Herbert J. Doherty, Jr. 2. Austrian Catholics and the Social Question, 1918-1933, by Alfred Dia- mant 3. The Siege of St. Augustine in 1702, by Charles W. Arnade 4. New Light on Early and Medieval Japanese Historiography, by John A. Harrison 5. The Swiss Press and Foreign Af- fairs in World War tI, by Frederick H. Hartmann 6. The American Militia: Decade of Decision, 1789-1800, by John K. Mahon 7. The Foundation of Jacques Mari- tain's Political Philosophy, by Hwa Yol Jung 8. Latin American Population Stud- ies, by T. Lynn Smith 9. Jacksonian Democracy on the Florida Frontier, by Arthur W. Thompson 10. Holman Versus Hughes: Exten- sion of Australian Commonwealth Powers, by Conrad Joyner 11. Welfare Economics and Subsidy Programs, by Milton Z. Kafoglis 12. Tribune of the Slavophiles: Kon- stantin Aksakov, by Edward Chmie- lewski 13. City Managers in Politics: An Analysis of Manager Tenure and Ter- mination, by Gladys M. Kammerer, Charles D. Farris, John M. DeGrove, and Alfred B. Clubok 14. Recent Southern Economic De- velopment as Revealed by the Chang- ing Structure of Employment, by Ed- gar S. Dunn, Jr. 15. Sea Power and Chilean Inde- pendence, by Donald E. Worcester 16. The Sherman Antitrust Act and Foreign Trade, by Andre Simmons 17. The Origins of Hamilton's Fiscal Policies, by Donald F. Swanson 18. Criminal Asylum in Anglo-Saxon Law, by Charles H. Riggs, Jr. 19. Colonia Baran Hirsch, A Jewish Agricultural Colony in Argentina, by Morton D. Winsberg 20. Time Deposits in Present-Day Commercial Banking, by Lawrence L. Crum 21. The Eastern Greenland Case in Historical Perspective, by Oscar Svar- lien 22. Jacksonian Democracy and the Historians, by Alfred A. Cave 23. The Rise of the American Chem- istry Profession, 1850-1900, by Ed- ward H. Beardsley 1. The Whigs of Florida, 1845-1854, by Herbert J. Doherty, Jr. 2. Austrian Catholics and the Social Question, 1918-1933, by Alfred Dia- mant 3. The Siege of St. Augustine in 1702, by Charles W. Arnade 4. New Light on Early and Medieval Japanese Historiography, by John A. Harrison 5. The Swiss Press and Foreign Af- fairs in World War 1I, by Frederick H. Hartmann 6. The American Militia: Decade of Decision, 1789-1800, by John K. Mahon 7. The Foundation of Jacques Mari- tain's Political Philosophy, by Hwa Yol Jung 8. Latin American Population Stud- ies, by T. Lynn Smith 9. Jacksonian Democracy on the Florida Frontier, by Arthur W. Thompson 10. Holman Versus Hughes: Exten- sion of Australian Commonwealth Powers, by Conrad Joyner 11. Welfare Economics and Subsidy Programs, by Milton Z. Kafoglis 12. Tribune of the Slavophiles: Kon- stantin Aksakov, by Edward Chmie- lewski 13. City Managers in Politics: An Analysis of Manager Tenure and Ter- mination, by Gladys M. Kammerer, Charles D. Farris, John M. DeGrove, and Alfred B. Clubok 14, Recent Southern Economic De- velopment as Revealed by the Chang- ing Structure of Employment, by Ed- gar S. Dunn, Jr. 15. Sea Power and Chilean Inde- pendence, by Donald E. Worcester 16. The Sherman Antitrust Act and Foreign Trade, by Andre Simmons 17. The Origins of Hamilton's Fiscal Policies, by Donald F. Swanson 18. Criminal Asylum in Anglo-Saxon Law, by Charles H. Riggs, Jr. 19. Colonia Barn Hirsch, A Jewish Agricultural Colony in Argentina, by Morton D. Winsberg 20. Time Deposits in Present-Day Commercial Banking, by Lawrence L. Crum 21. The Eastern Greenland Case in Historical Perspective, by Oscar Svar- lien 22. Jacksonian Democracy and the Historians, by Alfred A. Cave 23. The Rise of the American Chem- istry Profession, 1850-1900, by Ed- ward H. Beardsley  24. Aymara Communities and the Bolivian Agrarian Reform, by Wil- liam E. Carter 25. Conservatives in the Progressive Era: The Taft Republicans of 1912, by Norman M. Wilensky 26. The Anglo-Norwegian Fisheries Case of 1951 and the Changing Law. of the Territorial Sea, by Teruo Ko- bayashi 27. The Liquidity Structure of Firms and Monetary Economics, by Wil. liam J. Frazer, Jr. 28. Russo-Prsian Commercial Rela- tions, 1828-1914, by Marvin L. Bnt- ner 29. The Imperial Policy of Sir Robert Borden, by Harold A. Wilson 30. The Association of Income and Educational Achievement, by Roy L. Lassiter, Jr. 31. Relation of the People to the Land in Southern Iraq, by Fuad Baali 32. The Price Theory of Value in Public Finance, by Donald R. Es, carraz 33. The Process of Rural Develop- ment in Latin America, by T. Lynn Smith 34. To Be or Not to Be ... Existen- t-il-Psychological Perspectives on the Self, edited by Sidney M. Jourard 35. Politics in a Mexican Commu-. nity, by Lawrence S. Graham 36. A Two-Sector Model of Eco- nomic Growth with Technological Progress, by Frederick Owen Goddard 37. Florida Studies in the Helping Professions, by Arthur W. Combs 38. The Ancient Synagogues of the Iberian Peninsula, by Don A. Hal- perin 39. An Estimate of Personal Wealth in Oklahoma in 1960, by Richard Edward French 40. Congressional Oversight of Exec-- utive Agencies, by Thomas A. Hen.- derson 41. Historians and Meiji Statesmen, by Richard T. Chang 42. Welfare Economics and Peak Load Pricing: A Theoretical Appli- cation to Municipal Water Utility Practices, by Robert Lee Greene 43. Factor Analysis in International Relations: Interpretation, Problem Areas, and an Application, by Jack E. Vincent 44. The Sorcerer's Apprentice: The French Scientist's Image of German Science, 1840-1919, by Harry W. Paul 45. Community Power Structure: Propositional Inventory, Tests, and Theory, by Claire W. Gilbert 46. Human Capital, Technology, and the Role of the United States in International Trade, by John F. Mor- rall III 47. The Segregation Factor in the Florida Democratic Gubernatorial Primary of 1956, by Helen L. Jacob- stein 48. The Navy Department in the War of 1812, by Edward K. Eckert 49. Social Change and the Electoral Process, by William L. Shade 50. East from the Andes: Pioneer Settlements in the South American Heartland, by Raymond E. Crist and Charles M. Nissly 51. A General Equilibrium Study of the Monetary Mechanism, by David L. Schulze 24. Aymara Communities and the Bolivian Agrarian Reform, by Wil- liam E. Carter 25. Conservatives in the Progressive Era: The Taft Republicans of 1912, by Norman M. Wilensky 26. The Anglo-Norwegian Fisheries Case of 1951 and the Changing Law of the Territorial Sea, by Terno Ko- bayashi 27. The Liquidity Structure of Firms and Monetary Economics, by Wil- liam J. Frazer, Jr. 28. Russo-Persian Commercial Rela- tions, 1828-1914, by Marvin L. Ent- ner 29. The Imperial Policy of Sir Robert Borden, by Harold A. Wilson 30. The Association of Income and Educational Achievement, by Roy L. Lassiter, Jr. 31. Relation of the People to the Land in Southern Iraq, by Fuad Baali 32. The Price Theory of Value in Public Finance, by Donald R. Es- carraz 33. The Process of Rural Develop- ment in Latin America, by T. Lynn Smith 34. To Be or Not to Be ...Existen- iial-Psychological Perspectives on the Self, edited by Sidney M. Jourard 35. Politics in a Mexican Commu- nity, by Lawrence S. Graham 36. A Two-Sector Model of E- nomic Growth with Technological Progress, by Frederick Owen Goddard 37. Florida Studies in the Helping Professions, by Arthur W. Combs 38. The Ancient Synagogues of the Iberian Peninsula, by Don A. Hal- ped.n 39. An Estimate of Personal Wealth in Oklahoma in 1960, by Richard Edward French 40. Congressional Oversight of Exec- utive Agencies, by Thomas A. Hen- derson 41. Historians and Meiji Statesmen, by Richard T. Chang 42. Welfare Economics and Peak Load Pricing: A Theoretical Appli- cation to Municipal Water Utility Practices, by Robert Lee Greene 43. Factor Analysis in International Relations: Interpretation, Problern Areas, and an Application, by Jack E. Vincent 44. The Sorcerer's Apprentice: The French Scientist's Image of German Science, 1840-1919, by Harry W. Paul 45. Community Power Structure: Propositional Inventory, Tests, and Theory, by Claire W. Gilbert 46. Human Capital, Technology, and the Role of the United States in International Trade, by John F. Mor- rall III 47. The Segregation Factor in the Florida Democratic Gubernatorial Primary of 1956, by Helen L. Jacob- stein 48. The Navy Department in the War of 1812, by Edward K. Eckert 49. Social Change and the Electoral Process, by William L. Shade 50. East from the Andes: Pioneer Settlements in the South American Heartland, by Raymond E. Crist and Charles M. Nissly 51. A General Equilibrium Study of the Monetary Mechanism, by David L. Schulze 24. Aymara Communities and the Bolivian Agrarian Reform, by Wil- liam E. Carter 25. Conservatives in the Progressive Era: The Taft Republicans of 1912, by Norman M. Wilensky 26. The Anglo-Norwegian Fisheries Case of 1951 and the Changing Law of the Territorial Sea, by Teruo Ko- bayashi 27. The Liquidity Structure of Firms and Monetary Economics, by Wil- liam J. Frazer, Jr. 28. Russo-Persian Commercial Rela- tions, 1828-1914, by Marvin L. Ent- ner 29. The Imperial Policy of Sir Robert Borden, by Harold A. Wilson 30. The Association of Income and Educational Achievement, by Roy L. Lassiter, Jr. 31. Relation of the People to the Land in Southern Iraq, by Fuad Baai 32. The Price Theory of Value in Public Finance, by Donald R. Es- carraz 33. The Process of Rural Develop- ment in Latin America, by T. Lynn Smith 34. To Be or Not to Be ...Existen- tial-Psychological Perspectives on the Self, edited by Sidney M. Jourard 35. Politics in a Mexican Commu- nity, by Lawrence S. Graham 36. A Two-Sector Model of Eco- nomic Growth with Technological Progress, by Frederick Owen Goddard 37. Florida Studies in the Helping Professions, by Arthur W. Combs 38. The Ancient Synagogues of the Iberian Peninsula, by Don A. Hal- perin 39. An Estimate of Personal Wealth in Oklahoma in 1960, by Richard Edward French 40. Congressional Oversight of Exec. utive Agencies, by Thomas A. Hen- derson 41. Historians and Meiji Statesmen, by Richard T. Chang 42. Welfare Economics and Peak Load Pricing: A Theoretical Appli- cation to Municipal Water Utility Practices, by Robert Lee Greene 43.- Factor Analysis in International Relations: Interpretation, Problem Areas, and an Application, by Jack E. Vincent 44. The Sorcerer's Apprentice: The French Scientist's Image of German Science, 1840-1919, by Harry W. Paul 45. Community Power Structure: Propositional Inventory, Tests, and Theory, by Claire W. Gilbert 46. Human Capital, Technology, and the Role of the United States in International Trade, by John F. Mor- rall III 47. The Segregation Factor in the Florida Democratic Gubernatorial Primary of 1956, by Helen L. Jacob- stein 48. The Navy Department in the War of 1812, by Edward K. Eckert 49. Social Change and the Electoral Process, by William L. Shade 50. East from the Andes: Pioneer Settlements in the South Amneri can Heartland, by Raymond E. Crist and Charles M. Nissly 51. A General Equilibrium Study of the Monetary Mechanism, by David L. Schulze