hate QJolbgc of Agriculture At (JfnrneU 3Ilni»erHttg atljata, N. ^. Cornell University Library HB3711.M6 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924002610966 ECONOMIC CYCLES: THEIR LAW AND CAUSE THE MACMILLAN COMPANY NEW YORK • BOSTON • CHICAGO • DALLAS ATLANTA • SAN FRANCISCO MACMILLAN & CO., Limited LONDON • BOMBAY • CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, Ltd. TORONTO ECONOMIC CYCLES: THEIR LAW AlVD CAUSE BY HENRY LUDWELL MOORE PUpFESSOB OF POLITICAL ECONOMY IN COLUMBIA UNIVERSITY AUTHOR OF " LAWS OF WAGES " " Noiis croyons en effet, pour notre part, que pour avancer vraiment dans la connaissance 6conomique, il faut s'attaquer directement et d'abord, a des variations, c'est-a-dire 3, la forme dynamique des phfinomSnes, par la voie ex- p&imentale." Francois Simiand. THE MACMILLAN COMPANY 1914 All rights reserved COPTRIQHT, 1914 bt the macmillan company Published December, 1914. TO JANE MOORE A CRITIC WHO NEVER DISHEARTENS A CO-WORKER WHO KEEPS THE FAITH CONTENTS CHAPTER I Introduction CHAPTER II CYCLES OP BAINFALL The Use of Fourier's Theorem 6 Periodogram of Rainfall . . 14 The Equation to the Rainfall Curve 21 Rainfall in the Com Belt 26 CHAPTER III RAINFALL AND THE CROPS The Secular Trend in the Yield of the Crops 35 Critical Periods of Growth 41 Cycles in the Yield of the Representative Crops and the Corresponding Cycles of Rainfall 44 Cycles in the Index of Crop Fluctuations and in the Corre- sponding Index of Mean Effective Rainfall 49 CHAPTER IV THE LAW OF DEMAND The Theory of Demand . 62 Statistical Laws of Demand . 66 The Prediction of Prices . . ... 77 Elasticity of Demand .... ... 82 viii Contents CHAPTER V THE MECHANISM OF CYCLES * The Prices of Agricultural Commodities Correlated with the Yield of the Several Crops 94 Rising and Falling Prices as Related to Yield-Price Curves . 100 The ^'olume of the Crops and the Activity of Industry . . 103 A New Type of Demand Curves 110 The Fundamental, Persistent Cause of Economic Cycles . . 116 CHAPTER VI SUMMAEY AND CONCLUSIONS 135 ECONOMIC CYCLES: THEIR LAW AND CAUSE CHAPTER I INTRODUCTION There is a considerable unanimity of opinion among experts that, from the purely economic point of view, the most general and characteristic phenomenon of a changing society is the ebb and flow of economic Ufe, the alternation of energetic, buoyant activity with a spiritless, depressed and uncertain drifting. During the creative period of the rhythmic change each factor in production receives an augmenting income, and the mutual adjustment of interests in the productive process is brought about in a natural way, primarily through the operation of competitive law. The period of dechne in the cycle presents a sharply contrasted aspect of industry. With the organization of capital and labor at first unchanged, the amount of the product falls; each of the interested factors seeks at least to retain its absolute share of the product; friction and strife ensue with a threatening of the disruption of industry. What is the cause of this alternation of periods of activity and depression? What is its law? These are the fundamental problems of economic dynamics the solution of which is offered in this Essay. PoHtical Economy began to make progress in a rational way when the Physiocrats put forth their doctrine of the dependence of all forms of economic life 1 2 Economic Cycles: Their Law and Cause upon agriculture. Another momentous step was taken in the direction of theoretical development when the EngHsh economists formulated the law of diminishing returns in agriculture and traced its all-pervasive influence in the production and distribution of the product of industry. The desideratum of economic dynamics at the present time is the discovery of a law that shall be to a changing society what the law of dimiinishing returns in agriculture is to a society in a comparatively static condition. The full truth in the old Physiocratic doctrine has not been exploited. The Department of Agriculture of the United States reaffirms the central idea of the doctrine in its motto: "Agriculture is the Foundation of Manufacture and Commerce," and in the spirit of this motto it publishes invaluable statistical data. It is proverbial that the farmer is at the mercy of the weather. If it be true that the explanation of economic cycles is to be found in the law of supply of agricultural products, it is surely wise in a study of rhythmic eco- nomic changes to inquire whether the law of the chang- ing supply of raw material is not associated with a law of changing weather. Is there a well-defined law of chang- ing weather? Supposing that it is possible to discover that the weather passes through cycles of definite periods and definite amplitudes, it will then be necessary to show how the crops are affected by the weather and how the cycles of the weather are reproduced in cycles of the yield of the principal crops. Introduction 3 When the changes in the physical yield of the crops are shown to be dependent upon changes in the weather, the next stage in the investigation is to connect the yield with its value, and this brings one face to face with another unsolved problem in theoretical economics. The most recent phase of economic theory opens with a description of the "law of demand," which from the time of Cournot, Dupuit, and Gossen has been assumed in all theoretical discussions, but there has been no method for finding the statistical equation to the law. It will be necessary to overcome the difficulties of this problem before a solution can be offered of the more fundamental inquiry as to the law and cause of cycles in economic phenomena. When the physical yield of the crops has, on the one hand, been related to the cycles of the weather and, on the other, to the prices of the respective crops, it will then be possible to take the final step and to show how the cycles in the physical yield of the crops produce the cycles in the activity of industry and the cycles of general prices, and how, finally, the law of the cycles of the crops is the law of Economic Cycles. CHAPTER II CYCLES OF RAINFALL "The first thing that in my opinion ought to be done towards making the observations useful for scientific' purposes is to perform that kind of more perfect averaging which is afforded by the har- monic analysis. There is a certain amount of averaging done, but that is chiefly daily averages, with monthly averages, and yearly averages; but the more perfect averaging of the harmonic analysis would give the level of the variation of the phenomenon." — Lord Kelvin, in his testimony before the Meteorological Com- mittee of the Royal Society, 1876. From the point of view of the relation of changing weather to the varying fruitfulness of agriculture, the most important factors that are usually included in the term, weather, are temperature and rainfall. We begin our investigation with this common behef and inquire, in this chapter, whether the varying amount of annual rainfall is subject to any simple law. In order to carry forward the inquiry as to the exist- ence of a law of annual rainfall an analysis must be made of a long record of precipitation. Our choice of a record is Umited by two conditions: First, our object in investigating the periodicity of rainfall is the hope of throwing light upon the periodicity in the yield of the crops, and this expectation obviously makes it desirable that the record of rainfall shall be as repre- sentative as possible of the conditions of precipitation Cycles of Rainfall 5 in our leading crop area; secondly, as the existing meteorological records are of unequal lengths and of varying reliability, it is necessary to take the best long records that can be found within the limits of the crop area. The principal region of grain production in the United States is in the Mississippi Valley, but the meteoro- logical records of the Middle West do not extend through a long period of time. In order to achieve the two ends of having a long record of precipitation and of having the record typical of the conditions in the grain area, the device has been adopted of investigating rainfall in the Ohio Valley — ^which affords the longest record ob- tainable in the neighborhood of the central Mississippi region — and of showing that the rainfall of our lead- ing grain state, Illinois, follows the same law as the rainfall of the Ohio Valley. The stations in the Ohio Valley with long rainfall records are Marietta, Portsmouth, and Cincinnati. Their mean annual rainfall since 1839 is given in Table I ^ of the Appendix to this chapter. The graph of the course of rainfall in the Ohio Valley since 1839 is traced with other graphs on Figures 4, 5, and 6. The problem that must now be faced is the question as to whether the sequence of annual rainfall in the Ohio Valley follows a simple law, and if so, to give a quanti- tative formulation of the law. 1 The data were taken from Bulletin W of the Weather Bureau of the United States and from the Annual Reports of the Chief of the Weather Bureau. 6 Economic Cycles: Their Law and Cause The Use of Fourier's Theorem A preliminary examination of the rainfall data of the Ohio Valley leads to the conclusion that there is prob- ably no secular trend to the data, that is to say, there is probably no tendency of the rainfall to increase con- tinuously or to decrease continuously with the flow of time. It is true that when the amount of rainfall is correlated with time, the coefficient of correlation is r = —.227 ±.075, where the coefficient is three times its probable error and is therefore suggestive of a decrease in the amount of rainfall with the flow of time. More- over, if a straight line is fitted to the data, the indicated annual decrease in the rainfall is seven hundredths of an inch. But these facts are no justification for hold- ing to a secular decrease in the amount of annual rainfall. For, in the first place, if there are cycles in the amount of the rainfall, the low degree of the ob- served correlation might be due to the data of rainfall including incomplete cycles; in the second place, the record is drawn from only three stations and because of the limited number of stations might give an acciden- tal, low degree of correlation between amount of rain- fall and time; and in the third place, improvements in the method of taking the observations might have introduced changes that would account for the ob- served small annual decrease in the amount of rain- fall. In view of these considerations, it is probably best to proceed with our problem on the assumption that there is no secular trend in the amount of annual Cycles of Rainfall 7 rainfall. If this assumption is true, it follows that, in all probability, the course of rainfall in the Ohio Valley, is cyclical, or a combination of cycles. In an inductive treatment of any form of rhythmic or cyclical change it is necessary that the method adopted shall satisfy two conditions: (1) It shall be consistent with recognized mathematical processes; (2) It shall afford means of testing the degree of proba- bility that the results are not chance phenomena. Unless the method rests clearly upon an approved mathematical process, it is scarcely possible to say whether the attained results may not be entirely formal; and unless the findings are tested for the degree of their probabihty, there is no assurance that the adduced cycle may not be a chance occurrence. The literature in which rhythmic phenomena are treated in a statis- tical way teems with fallacies and uncertainties that illustrate the need of observing the above conditions; for the method frequently adopted of smoothing the data is so arbitrary that one is at a loss to know whether, after all, the alleged periodicity may not, in fact, be due to the process of smoothing; and, in addition, one is left in doubt as to whether an indefinite number of cycles other than the particular one adduced might not, with equal or greater probabihty, be obtained from the same data. The method that was employed to reach the results of this chapter rests upon the analysis invented by Joseph Fourier,^ which is called, in English treatises, ' The most philosophic exposition of Fourier's theorem is in 8 Economic Cycles: Their Law and Cause harmonic analysis. The perfection of the method whereby the findings may be subjected to the test of probabiUty is the work of Professor Arthur Schuster ^ of Manchester. We may begin the presentation of the method with a definition of a series of terms that constantly recur in the treatment of periodic phe- nomena. Figure 1 will facih- tate the exposition by affording a graphic description of the terms dealt with. Suppose that the point Q moves uniformly in the circle of Figure 1 , that is to say, sup- FiQURE 1. p^gg ^-^^^ ^^^ p^-j^^ Q (describes equal arcs in equal times and, therefore, proportional arcs in different times. Then, if the measurements of the arcs of the circle are made from the point A and the reckoning of time is begun when Q is at E, the angle A E is called the angle at epoch, or simply Fourier's own work: Theorie analytique de la chaleur. In Freeman's English translation the treatment is found on pp. 137-212. ^ The fundamental memoirs of Professor Schuster are "On the Investigation of Hidden Periodicities with Application to a Supposed 26 Day Period of Meteorological Phenomena." Terrestrial Magnetism for March, 1898. "The Periodogram of Magnetic Declination as obtained from the records of the Greenwich Observatory during the years 1871-1895." Cambridge Philosophical Society Transactions, Vol. 18, 1899. "On the Periodicity of Sunspots." Philosophical Transactions of the Royal Society of London, A, Vol. 206, 1906. "The Periodogram and its Optical Analogy." Proceedings of the Royal Society of London, A, Vol. 77, 1906. Cycles of Rainfall 9 the epoch of the uniform cu-cular motion. The radius of the circle is the amplitude of the motion; the time of going once around the circle is the period of the motion; the ratio of A Q to the circumference of the circle is the phase of the mo- tion. If from each position of Q a perpendicular is dropped upon the diameter of the circle, G H, the foot of the perpendicular wiU describe a simple harmonic motion. The ampUtude of the simple harmonic motion is one- half of the range of the motion, that is, one-haK of G H, or the radius of the circle. The period of the simple harmonic motion is the interval between the passing of the point P twice through the same position in the same direction. The distance of the point P from the middle of its range, 0, is a simple harmonic function of the time, O P =y = a sin. (nt+e), where a is the radius of the circle — or the amphtude of the simple harmonic motion — e is the angle of epoch, and n is the angle de- scribed by the moving point Q in the unit of time. The period of the simple harmonic motion is, in the above 27r _^ , . nt + e case, — . Its phase is —^ — . Figure 2 presents a graph of simple harmonic mo- tion. As in Figure 1, the point Q moves uniformly in the circle ; the point P performs simple harmonic motion according to the formula y=a sin {nt-\-e), where a is the amphtude of the motion, or radius of the circle, e is the angle of the epoch, namely, A E, and n is the arc described by Q in the unit of time. If time is meas- 10 Economic Cycles: Their Law and Cause ured upon the line B C, the sinuous curve of Figure 2 is the graph of the function, y = a sin (nt+e). X p '--\ / \ r / o — -'"■'^ A B 1 2 \ J 4 S le / \ / \ J ^""----A- ^ \ y Figure 2. The importance of simple harmonic functions in the study of periodic phenomena grows out of the fact that any periodic curve however complex ^ can be ex- pressed mathematically by a series of simple harmonic functions. By the help of Fourier's analysis a periodic fimction may be put in the form (1) y =Aa + tti cos kt + Oj cos 2 A;i + Oj cos 3 H + . . . + 6i sin kt + 62 sin 2 /:< + 63 sin 3 A;< + . . . If in (1), we put, a^ = 4i sin e^; a^ = A^ sin e^'i (^3 = ^3 sin e^; &c., bi = Ai cos Ci] 62 = A2 cos 62; 63 = A3 cos 63; &c.. We get, (2) y = Ao +^1 sin {kt + gj) + Aj sin (2 A;« + gj) + A3 sin (3fc< + 63) + . . . where y is expressed as a series of sines. In a similar manner, equation (1) may be expressed as a series of cosines, ' The few exceptions to the general rule are discussed in the mathematical texts that develop Fourier's theorem. Cycles of Rainfall 11 (3) y = Ao + Bi cos (kt - € j) + S2 cos (2 fc< - tj) + B3 cos (3 fct-Es) + . . . In the use of Fourier's theorem for the purpose of analyzing periodic phenomena, we follow a process analogous to the use of Taylor's theorem in the simpler demonstrations of mathematical economics. By far the greater part of Cournot's pioneer treatise and of subsequent work of his school is based upon the as- sumption that, if the economic function under investi- gation is y=f{x), then f(x+h) may be expanded by Taylor's theorem, and the first terms of the series may be used as an approximation to the form of /(x). Simi- larly, in our use of Fourier's series, the attention will be focussed upon a few harmonics as a first approximation to the solution of the problem in hand of expressing in mathematical form the periodicity of annual rainfall. Assmning that any periodic function may be ex- pressed as a Fourier series, the problem is presented of determining the values of the coefficients. The series, as we know, is of the form y =f(t) = Ao + ai cos kt + 02 cos 2 kt + . . . + 61 sin kt + b2 sin 2 kt+ . . . What are the values of the first term and of the co- efficients of the sines and cosines? In order to deduce the necessary values, we shall have need of the follow- ing lemma: If m and n are two unequal integers and k is put equal to -y , then 12 Economic Cycles: Their Law and Cause j cos mkt cos nkt dt = 0, o /T sia mkt sin nkt dt = 0, o /T sin mkt cos nfct dt = 0. o The lemma may be proved to be true by evaluating the three integrals according to the usual methods. The first integral, for example, becomes /T /»T COS mkt cos nkt dt=\ / { cos (m—n) kt+cos (m+n) kt \ dt o o rsin (m-n) kt sin (m +n) kt y ^ L 2 (m-n) k "*" 2 {m + n) k L But k = -^, and, consequently, j cos mfc< cos wA:< dt = 0. With the aid of this lemma we may proceed to evalu- ate the coefficients in Fourier's series. If we integrate the series between the Umits o and T, we get, f{t)dt=Ao I dt + ai j cosktdt + bi j siD.ktdt+ . . . O o But all of the terms except the first on the right-hand side of the equation will vanish, and consequently /T fit) jyit)dt=A,j 'dt = A,T, or A,= ° ^ dt /T f{t)dt is the area of the original curve for one o whole period T, the constant term ia Fourier's series is equal to the value of the mean ordinate of the original curve. Cycles of Rainfall 13 To determine the value of Oi, multiply throughout by cos ht and integrate between hmits o and T. /T pi px f (0 COS kt dt = Ao I cos kt dt + Ui j cos^ kt dt o o o /T sin kt cos kt dt + . . . o /T /*T /^T / {t) COS fcJ dt = Oi I cos" A;< d<, since I cos A< di and o o o I sin fc« cos kt dt are both equal to zero and all the other o terms on the right-hand side of the equation, according to our lemma, disappear. But n , . J. r'^ + cos 2 H ,, , r, , sin 2 My T J '^o-'ktdt^j 2 '^*=n* + "2^J„=2 o o and as a result, we have ^t / / it) cos kt dt fli "o = / / (') ^^^ ^* ^*' or Oj = 2 — : yi Therefore Ci is equal to twice the mean value of the [ product /(Ocos A;i. — -^ In a similar manner the value of any other coefficient may be determined. Take, for example, 6„. Multiply throughout by sin nkt and integrate between o and T, /'' . , , , C ■ 7.J. r C'^ I- COS 2 nkt j^ f («) sin nkt dt = h„ I sm^ nkt dt = b„ j ^ dt = O O ^ f,r s in 2 nkt y] , T ^"[i'~^i^lr^''2 /T _ / (0 sin n < < Therefore &„ 14 Economic Cycles: Their Law and Cause is equal to twice the mean value of the product f{t) sin nkt. Having found the algebraic values of the coefficients in Fourier's series, we may now proceed to determine their statistical equivalents in the case of annual rainfall. The Periodogram of Rainfall If the length of a cycle of rainfall were known before- hand, the preceding exposition of Fourier's theorem would suffice to determine, from the data of precipita- tion, the amphtudes and phases of the harmonic con- stituents of the Fourier series descriptive of the rainfall cycle. But in the problem before us of analyzing the rainfall data of the Ohio Valley, we do not know whether there are many cycles or only one cycle or, indeed, whether there are any cycles at all. And there is no short method of solving the problem. Suppose, for example, it were assumed from a priori considerations that the amount of rainfall is affected by sunspots, and, as sunspots are known to occur in periods of about eleven years, suppose it should be in- ferred that the annual rainfall will likewise show a period of eleven years. If the rainfall data of the Ohio Valley are examined for an eleven years period, it will be found that the data yield a definite amplitude and a definite phase for a cycle of eleven years, but this fact is no warrant for holding that there is a true rainfall period of eleven years. Every other grouping of the seventy-two years record will likewise show a definite amphtude Cycles of Rainfall 15 and a definite phase. The questions that one is in- terested to have answered are: (1) What is the law of the distribution of Fourier coefficients when the data are analyzed for all possible periods; and (2) how may the true cycles be separated from the accidental, spurious cycles that are obtained when the data are exhaustively analyzed? In Figure 3 the results of a detailed, laborious ex- amination of the data of annual rainfall in the Ohio Valley are presented in graphic form. On the axis of abscissas are measured, within assigned limits, the possible lengths of cycles in the 72 years of rainfall. By extending the calculations to 36 years, we obtain for the assumed periods a record of possible recur- rences varying from 2, in case of the period of 36 years, to 24, in case of the period of 3 years. On the axis of ordinates are measured the squares of the co- efficients of the first harmonic in the Fourier series corresponding to the lengths of periods recorded on the axis of abscissas. The numerical values of these squares are given in the fourth and eighth columns of Table II in the Appendix to this chapter. The method of deriving the values may be illustrated by taking the cycle of 8 years. Suppose, as a first approximation, that the equation to Fourier's series is put in the alge- braic form y = F{t) = Ao + ai cos kt + 6i sin U = Ao-\- A^ sin {kt + e). Then the corresponding arithmetical values derived from the Ohio rainfall data are 16 Economic Cycles: Their Law and Cause II^J'^!^'' JO S'SLfOui ui 3pn4i/Ju/s sl/^jo sjenbc Cycles of Rainfall 17 y = F{t) = 41.19-3.13 cos ^ < + 2.69 sin ^ t = 41.19+4.13 sin (y t + 310° 41'). The values of the terms a\, b\, Al are respectively (3.1339)2, (2.6938)2, (4.1325)2, and these values are given in the proper columns of Table II in the Ap- pendix. In Figure 3, the values oi A^ for the several periods are measured on the axis of ordinates. An examination of Figure 3 will illustrate the truth of a statement advanced a moment ago. It is clear from the course of the periodograph ^ that if one were to take any period at random between the limits of 3 years and 36 years, he would in every case obtain a finite value for the amphtude of the selected cycle ; and if, by chance, selection should fall upon, say, 18, or 21, or 29, or 36 years, an argument might be made with some degree of plausibihty that a real cycle had been dis- covered. But, in truth, the real significance of no one cycle taken at random can be judged apart from its place in the distribution of all the cycles that can be derived from the data. This last point is of fundamental importance. The only object of investigating cycles of rainfall or cycles of economic phenomena is that the knowledge of the 1 The terms periodograph and periodogram were coined by Pro- fessor Schuster. The periodograph is the curve tracing the values of A'; the periodogram is the surface between the periodograph and the base line giving the lengths of the periods. Schuster: "The Period- ogram of Magnetic Declination," p. 108. 18 Economic Cycles: Their Law and Cause constant recurrence of the cycles may place one in a position to foresee and utilize the dependent phenomena. But the control of phenomena dependent upon a cycle presupposes that the cycle is itself a real phenomenon with a natural cause, and that consequently it persists with an increase in the number of observations. If, however, an apparent cycle of any length taken at random is obtained from the given data, one would surely misspend his time if he were to set about the search for its cause, and were to derive conclusions based upon the hypothesis of the persistence of the cause. The cycles due to formal, accidental causes must be discriminated from the cycles with natural causes. The separation of true cycles from spurious or accidental cycles is faciUtated by the periodogram ^ of observations. If, following Professor Schuster, we call the square of the amphtude of any given period the "intensity" of the period, then it may be said that the probability of the reaUty of a period is dependent upon the ratio of its intensity to the mean intensity of the periodogram. Or, again following Professor Schuster, if we call the mean intensity of the periodogram the "expectancy," then the reaUty of a period is dependent upon the ratio of its intensity to the expectancy of the periodogram. For instance, if in case of a given period the ratio of intensity to expectancy is, say, 3 to 1, then in about one case in twenty we should expect to obtain by chance a greater amphtude than the amplitude of the particular period in question. If, on the other hand, ' See the preceding note. Cycles of Rainfall 19 the ratio were say, 7 to 1, a greater ratio would not occur by chance once in a thousand times. ^ With these facts in mind, let us again examine Fig- ure 3. It is clear that the principal periods needing attention are those respectively of 8, 29, 33, 36 years. In case of the 8 year cycle there can be very httle doubt as to the existence of a true periodicity approx- imating 8 years in length. The ratio of the square of its amphtude to the mean square amplitude of the periodogram is 6.71 to 1. We may accordingly accept with considerable confidence the existence of a natural period of rainfall in the Ohio Valley approximating 8 years in length. The cycle of 33 years, inasmuch as the ratio of the square of its amplitude to the mean square amplitude of the periodogram is 3.27 to 1 is in all probability a true cycle. The doubt that exists is due to the smallness of the ratio and the few recurrences — only two ^ — 1 Schuster: "The Periodogram of Magnetic Declination," pp. 124- 125. 2 Those who deprecate the use of such meager data should con- sider well the testimony of Lord Kelvin before the Meteorological Committee of the Royal Society, 1876. Question 1710. "The sum which parUament will give for this purpose being a limited sum, do you think that it would be well to reduce the number of observations in order to have more money to spend upon the reduction of observations? / think at all events until one eleven years period, the sun spot period, is completed, it would be wrong to reduce the number of observations." Question 1735. "Supposing that you had one of these analyses calculated for a period of 11 years, would each year's observations and stiU more each period of 11 years observations, require to be introduced into this analysis so that you would have an analysis of 22 years, and an analysis of 33 years, and so on from time to time, 20 Economic Cycles: Their Law and Cause that our data afford. A greater confidence in the exist- ence of a real period of 33 years is given by the fact that Bruckner ^ claims to have found a true period of about 35 years in an examination of a vast mass of rainfall material all over the world. Accordingly, the existence in the Ohio Valley of a real 33 years period of rainfall we shall assume to be very probable. The other two periods of 29 years and 36 years are not easily disposed of. But in the first place, the ratios of the squares of the respective amphtudes to the mean square amphtude of the periodogram are not such as to justify the acceptance, with any degree of confidence, of the existence of true cycles of 29 years and 36 years. In the second place, they are both so close to the period of 33 years as to cause a doubt as to whether they may not be spurious periods that are hkely to appear in the neighborhood of a real period.^ Considering the short range of our data it would not be properly cautious to press the point of the existence of any definite real cycle. But this much is certain: If there are true cycles in the data of the 72 years of rainfall in the Ohio Valley, there is far greater prob- abiUty that two cycles are those of 8 years and 33 years than of any other round numbers between 3 and or, being done, would it be done once for all? / cannot say whether anything with reference to Terrestrial Meteorology is done once for all. I think probably the work will never be done." ' Edward Bruckner: Klimaschwankungen seit 1700. Bruckner's period fluctuates greatly in length and has an average value of 35 years. 2 Schuster: "The Periodogram of Magnetic Declination," p. 130. Cycles of Rainfall 21 36 years. Moreover, the periods of 8 years and 33 years afford the most probable basis derivable from the data upon which to reason both as to the future course of rainfall in the Ohio Valley and as to the course of the phenomena dependent upon rainfall. Assuming, then, that for the purpose in hand, the 33 years and 8 years periods are the most probable and valuable, we tiun to the consideration of the equation to the graph giving the course of rainfall in the Ohio VaUey. The Equation to the Rainfall Curve It will be helpful to approach the algebraic descrip- tion of the cychcal movement of rainfall in the Ohio Valley, by observing how we obtain an increasingly accurate account of the actual rainfall by superposing the constituent cycles. We shall use, as an index of the relative fit of the several curves, the root-mean-square deviation of the observations from each curve. If, as a preliminary step, the raw data of the course of annual rainfall are examined, it is found that the mean annual rainfall in the Ohio Valley is 41.19 inches, and the root-mean-square deviation about the mean is 5 = 6.70 inches. If the long 33 years cycle is considered by itself, it appears that the root-mean-square deviation about the 33 years curve is S = 6.39 inches. The graph of the 33 years cycle is given in Figure 4. Its equation is y = 41.19 + 2.88 sin (^ t + 328° 7'V 22 Economic Cycles: Their Law. and Cause tH (M "o + a> 0, 1— 1 >, Tt< u II Tff Si S P o i^ Q X o o. a ca , ^■"■^^ (; 13 lO 00 H 00 b: IN R + ;^ O: ^H ^ II Si ■S9t^oui ui ji^ui&j jonuuY Cycles of Rainfall 25 csifoui ui //e^u/pj jenuuy 26 Economic Cycles: Their Law and Cause lingers at the minima and the short period during which it flows in the neighborhood of the maxima.^ Rainfall in the Corn Belt Thus far we have dealt with the law of rainfall only in the Ohio Valley. The object in taking the Ohio data, rather than the data of a state more representa- tive of the leading cereal area, was to make an investiga- tion of a longer meteorological record than is afforded by the data of the central Mississippi Valley. But our purpose in deaUng with meteorological records at aU is to show the dependence of crops upon the cycUcal movement of the elements of the weather. We must, therefore, prove that the cycles of rainfall which we have • I should like to make clear the method I have followed in the derivation of the equations to the curves. My object was to obtain a summary description of the general course of rainfall in order that I might discover, later on, whether the characteristic general fea- tures of the movement of rainfall are reproduced in the changing yield per acre of the crops. As a first step I tried to detect the real cycles in rainfall and I believe I have shown that, if the 72 years record is sufficiently long to reveal the true cycles, then the most probable lengths of the cycles are, in round numbers, 33 years and 8 years respectively. With so short a range of data I regarded it as useless to attempt to calculate the lengths of the periods to a greater degree of precision. I next had to derive the equations to the curves showing the characteristic general course of rainfall, and it seemed to me that, for this purpose, the method described in the text for evaluating the coefficients in a Fourier series might properly be used. K the 33 years cycle were taken as the fundamental cycle, then the 8 years cycle would be approximately the fourth harmonic in the series, and the 4 years cycle would be the eighth harmonic. The arithmetical process for computing the coefficients is indi- cated by Professor Schuster in Hidden Periodicities, pp. 13, 14 and is briefly described by Professor Perry in an article on "Harmonic Analysis" in The Electrician, for February 5, 1892. Cycles of Rainfall 27 discovered for the Ohio Valley are likewise the cycles that exist in the heart of the grain producing area. Among the states of the Middle West, Illinois is probably the most highly representative of American cereal production. It f)roduces the largest crop of corn,^ which is the leading American cereal, and it ranks second in the production of oats. Most of the other cereals that are produced in the upper Mississippi Valley are Ukewise cultivated with success in IlUnois. Another fact that makes IlUnois a desirable state for our purpose is that its meteorological records are fairly long and are obtainable from so many stations as to be representative of the weather conditions in the entire state. This last fact is all-important if the statistics for crop production of the whole state are to be con- sidered in relation to the weather cycles of the state. In Table III of the Appendix to this chapter the record of the annual rainfall in IlUnois is given for a period of 41 years. ^ The ideal direct method with 1 This statement was accurate when it was first written. But in 1912 Iowa gained by a narrow margin the first place among the com producing states. 2 The raw data were taken from Bulletin W of the Weather Bu- reau of the United States and from the Annual Reports of the Chief of the Weather Bureau. The stations used in computing the mean annual rainfall were: — In Northern lUinois: Aurora, Cambridge, Chicago, TiskUwa, Galva, Kishwaukee, Ottawa, Winnebago, and Henry. In Central Illinois: Charleston, CarlinviUe, Coatsburg, Decatur, Griggsville, KnoxviUe, Havana, LaHarpe, Pana, Peoria, and Springfield. In Southern Illinois: Cairo, Cobden, Carlyle, Golconda, Flora, GreenviUe, McLeansboro, Mascoutah, Mt. Carmel, and Palestine. All of these stations do not present full records for the 41 years. 28 Economic Cycles: Their Law and Cause reference to these data would be to compute the periodogram in the same manner in which it was com- puted in the case of the Ohio Valley data, and then com- pare the periodograms. But this method has not been followed. A less direct, and far less laborious, process has been adopted. We know from the Ohio data that there are two cycles of rainfall, a 33 years cycle and an 8 years cycle, and we know, furthermore, that when the curve for rainfall in the Ohio Valley is computed for the 33 years and 8 years periods and their senuharmonics, a good fit to the data is obtained. The questions that are asked with reference to the Ilhnois data are these: If we assume the existence of a 33 years period and an 8 years period in the Illinois rainfall data, will the rainfall curve fit the Ilhnois data as well as the Ohio curve fits the Ohio data? Will the Ilhnois curve re- produce the characteristic features of the Ohio curve? A presumption in favor of an affirmative answer to these questions is suggested by the fact that the correla- tion between the annual rainfall in the Ohio Valley and the annual rainfall in the state of Ilhnois is r=6.00. The graph of the curve of rainfall in Ilhnois is given in Figure 7. Its equation is j/=38.53+3.03 sin (^ <+325° 35') + 1.87 sin(|^ t+lM" 55') +3.05sin(^H241°52')+1.12sin/^H232''26'), the origm being at 1870. The root-mean-square devia- but in no year were fewer than seven records obtainable while for a large proportion of the years the thui;y records were complete. Cycles of Rainfall 29 ■s3Lf3ui ui iie^iej jBnuuy 30 Economic Cycles: Their Law and Cause tion of the observations from this curve is S =4.20. In case of the Ohio curve the root-mean-square deviation was 5 = 5.29. But this is a better relative fit for the Ilhnois curve than we have a right to claim, because in Ohio the mean annual rainfall is 41.19, while in Ilhnois the mean is 38.53. If we express the relative scatter of the observations about the curve as the ratio of the root-mean-square deviation of the observations to the mean rainfall, we get for Ohio and Illinois, respectively, SI = .128; 51 = .109. In Figure 8, the Ohio curve for 1870-1910 is placed upon the same chart as the Illinois curve for the same flow of time, and the degree of correspondence of the two curves is seen to be so close that, with due allowance for the difference in their mean annual rainfall, they seem to be almost congruent. We may say, therefore, that the two curves fit their respective data equally well. Our problem has now received its solution. Annual rainfall in the chief grain-producing area of the United States has no secular trend, but its mean course is the resultant of causes producing two cycles of 33 years and 8 years respectively. The manner in which these cycles of rainfall produce a rhythmical expansion and contraction in the yield of the crops we shall examine in the next chapter. Cycles of Rainfall 31 'ssi^oui ui ffej-utpj fenuu^ 32 Economic Cycles: Their Law and Cause APPENDIX TABLE I. — Annual Rainfall in the Ohio Valley Stations: Cincinnati, Poktsmouth, Maeietta Yf-ab Rainfall in Inches Year Rainfall in Inches Year Rainfall in Inches 1839 29.92 1863 37.95 1887 38.00 1840 42.84 1864 36.68 1888 46.19 1841 43.94 1865 48.93 1889 37.06 1842 41.89 1866 47.37 1890 55.43 1843 48.20 1867 40.72 1891 40.68 1844 37.95 1868 46.87 1892 36.96 1845 40.11 1869 41.29 1893 40.80 1846 48.39 1870 37.46 1894 31.07 1847 55.26 1871 29.91 1895 29.03 1848 44.97 1872 32.90 1896 39.22 1849 46.37 1873 45.18 1897 44.80 1850 54.77 1874 38.48 1898 45.04 1851 32.54 1875 44.78 1899 40.46 1852 46.73 1876 47.34 1900 33.60 1853 35.67 1877 34.69 1901 31.78 1854 40.30 1878 36.35 1902 39.53 1855 47.89 1879 39.22 1903 37.98 1856 28.98 1880 49.94 1904 28.24 1857 37.95 1881 41.60 1905 42.81 1858 55.48 1882 56.10 1906 41.95 1859 46.68 1883 49.25 1907 46.68 1860 36.00 1884 40.05 1908 33.29 1861 43.81 1885 37.63 1909 41.40 1862 40.26 1886 39.61 1910 36.20 Cycles of Rainfall 33 TABLE II. — ^The Pekiodogram of Rainfall in the Ohio Valley y = F(t) = Ao +ai cos kt + bi sin kt = Ao + Ai sin (kt -\- e) Length OF Fbbiod IN Years 0= 62 a.^+b^=A^ Length OF Pe- BIOD IN Yeaes a' (.2 a2+6''=A2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1.2628 .0003 .0897 .2220 2.1838 9.8215 .0327 .5978 1.0756 .4371 .0044 .1078 .1874 .7691 .9795 2.9332 1.4777 .0294 2.4821 4.5689 .4520 .1403 3.7869 7.2563 .3120 .0190 .6791 .1143 .0007 .1670 .0863 .0424 .0626 .9270 1.9422 1.5961 3.7449 4.5692 .5417 .3623 5.9707 17.0778 .3447 .6168 1.7547 .5514 .0051 .2748 .2737 .8115 1.0421 3.8602 3.4199 1.6255 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 .0046 .2454 .8471 .3551 .2755 .0566 .9692 .6227 4.2657 .6464 .6112 .5776 2.3199 .2017 .0456 .0036 4.4260 2.4237 .8714 .0678 .1327 .0002 .0019 .0300 1.1153 .4767 .5923 1.1168 5.9974 1.7652 1.7914 6.8567 4.4306 2.6691 1.7185 .4229 .4082 .0568 .9711 .6527 5.3810 1.1231 1,2035 1.6944 8,3173 1,9669 1,8370 6,8603 Mean value of A' = 2.5459 34 Economic Cycles: Their Law and Cause TABLE III. — ^Annual Rainfall in Illinois Yf,ar Rainfall in Inches Yeab Rainfall in Inches 1870 29.65 1891 34.11 1871 36.53 1892 44.17 1872 33.98 1893 35.89 1873 41.62 1894 28.99 1874 32.91 1895 32.92 1875 40.34 1896 38.27 1876 45.50 1897 37.44 1877 42.76 1898 49.09 1878 37.61 1899 34.95 1879 36.10 1900 36.19 1880 42.31 1901 27.17 1881 42.32 1902 42.65 1882 49.04 1903 35.97 1883 47.81 1904 39.33 1884 45.83 1905 37.33 1885 40.80 1906 38.10 1886 36.16 1907 40.61 1887 33.40 1908 36.76 1888 39.41 1909 44.74 1889 36.27 1910 34.34 1890 40.34 Mean 38.53 CHAPTER III RAINFALL AND THE CROPS "It is mere weather . . . doing and undoing without end." — William James. In the preceding chapter the course of annual rainfall in the great cereal-producing area of the United States has been shown to move in cycles: There is a ground- swell of thirty-three years in length upon which cycles of eight years in duration are superposed. Our object in studying the rhythmic changes in the volume of rain- fall was to bring these changes into relation with the variations in the yield per acre of the crops, and in the present chapter we shall be able to realize our purpose. The actual course of the varying yield per acre of the crops will be shown to have both a secular and a cyclical movement; these two movements will be separated for representative crops; and the cyclical movements will be shown to be dependent upon the cyclical movements in the weather represented by the cycles of rainfall. The Secular Trend in the Yield of the Crops The state of IlUnois was chosen in the preceding chapter to illustrate the general conditions of rainfall in the Corn Belt of the Middle West, and we shall now examine the statistics of the yield of its most important crops. 35 36 Economic Cycles: Their Law and Cause According to the Yearbook of the Department of Agriculture for 1912, we find the acreage and value of the leading Ilhnois crops as they are given in the subjoined Table: Acreage and Value of Crops in Illinois, 1912 Crop Acreage Value of Crop (1) Corn 10,658,000 $174,791,000 (2) Oats 4,220,000 54,818,000 (3) Hay 2,512,000 41,152,000 (4) Wheat 1,183,000 8,641,000 (5) Potatoes 137,000 8,302,000 (6) Barley 57,000 952,000 (7) Rye 48,000 538,000 (8) Buckwheat 4,000 70,000 (9) Tobacco 900 62,000 It is clear, from this Table, that five crops — corn, oats, hay, wheat, and potatoes — make up the bulk of the crops of Ilhnois, and one could not go far wrong if he based his generaUzations as to the conditions of agricul- ture in the state upon these five crops. But for the purposes we have in view, in this and other chapters, it is not possible to utihze the statistics of wheat produc- tion because both spring and winter wheat are grown iij the state, and the statistics of their relative yield and price are not given in the pubHshed material for the long record covered in our investigation. Accordingly, the crops that have been actually used in our inquiry are corn, oats, hay, and potatoes. These crops total 93.13 per cent, of the crop acreage and 96.45 per cent, of the crop value as these quantities are given in the above Table. Rainfall and the Crops 37 As the yield per acre of the various crops may show a secular as well as a complex cychcal change, it will be necessary, before their cyclical elements can be brought into relation with the corresponding cyclical changes of rainfall, to eliminate from the recorded course of the yield per acre of the several crops the element of change that is secular in character. The method that has been adopted here to effect the elimination of the secular change is simple, but to secure a first approximation, it is adequate. For a period of time covered by the statistics, a change is regarded as a secular change if, for the period of time taken as a whole, the yield per acre of the crop shows a tendency either to increase or to decrease. In order to determine whether there is a secular change in the yield per acre, for a certain period of time, the yield data are correlated with time, and the existence or non-existence of a secular change is inferred from the relative magnitudes of the coefficient of correlation and its probable error. If there be a secular change, the calculation of the coefficient of correlation of the yield with time is then a first-step toward the elimination of the secular element by means of a regression equation in which the co- efficient of correlation is a factor. The method may be illustrated by taking the history of the yield per acre of corn. In Figure 9 the actual yield per acre in IlUnois is plotted for the period 1870- 1910. The straight hne showing the secular trend of the yield is the graph of the regression equation between the yield per acre and time. The correlation of the 38 Economic Cycles: Their Law and Cause 1 r -] r o .sfe O 1-1 •S-S 9 g "a !^o o O CO 0) o; 5 5 IS s + a, 3-^ [^ it ^0 c! C «*-( CQ o •rt o IN + (N o CO -t-3 t^ l4-l 1— I CO + 1-H + a 1=. lor a 03 Jfjoo r, - ^ n bf) o .y-n '''•^— — ^ u ■R ?i -c c <_) L*-^ ^ CQ CO + + ■a ' ■rl ^ ^ n 3 S 1 (N o 00 S Sf o 1> a CO ::< + 0) + O T3 n-1 •*i 0) S "^ G N Iro K o3 b 1 CO ^ i CO 03 _>> .9 S "^ ^ ~ ^ "3 + }i + 2 CO CD CO IM 1-H ■^ CN o CD c6 + + b Ico (M 1 CO b Ico _?tf -/fi -S *•* */■* +£4- ■*-34- * Percenfade chande in the production of oats. Figure 19. The law of demand for oats. y = 8.22 — 1.1904a; — .00663x2 + .000273a;8, origin at (0, 0). 76 Economic Cycles: Their Law and Cause l"^ ^ ^ \ \ \ u 1^ ^ > v ^ ^x ^ V > — H ^\\ K, \ -ZS -to *S *SO + 3S ^50 + 65 Percenfade chende In the pn>ducfion of potatoes. FiGUHE 20. The law of demand for potatoes, y = 1.77 — 1.5062a; + .02489x2— .000197a;^ ori^ at (0, 0). The Law of Demand 77 But unlike the classical theory of demand which was limited to the simple enunciation of this one character- istic, caeteris paribus, the statistical laws that have just been derived apply to the average changes that society is actually undergoing. They summarize the changes in prices that are to be expected from changes in the supply of the commodity, thus enabling one to predict the probable variation in price that will follow upon an assigned variation in the amount of the commodity. They exhibit the connection of probable results not only in a qualitative but also in a quantitative form. The Prediction of Prices It has been said that the statistical laws of demand enable the economist to predict the probable variation in price that will follow upon an assigned variation in the quantity of commodity that is to be sold. How accurate are the results of prediction that are based upon the statistical law of demand? The accuracy of the prediction in the case of any given commodity will vary according to the degree of fit of the type of curve that is assumed to represent the relation between the relative change in price and the relative change in the quantity of the commodity. If, for example, the commodity in question is corn in the United States, and the type of demand curve is assumed to be linear, then, according to the results in foregoing pages, the correlation between the two variables is r= — .789, and the regression equation is y= — .8896a; +7.79, the origin being at {o,o). (Figure 16 will facili- 78 Economic Cycles: Their Law and Cause tate the discussion of the case.) By means of this law of demand it is possible to predict the probable change in the price that will follow upon a given change in the quantity to be sold. In 1911, in the United States, the quantity of corn produced was 2,531,488,000 bushels, and the mean farm price on December 1, 1911 was 61.8 cents. In 1912 the quantity of corn produced was 3,124,746,000 bushels ; what, then, was the probable price of corn on December 1, 1912? The percentage change in the quantity produced was 23.44. Sub- stitute this value for x in the formula for the law of demand y = — .8896a; +7.79, and solve for the value of y. It is found that the probable change in price would be a fall of 13.06 per cent., which, since the price in 1911 was 61.8 cents, would give 52.7 cents as the prob- able price for December 1, 1912, whereas the actual price was 48.7 cents. According to the theory of Unear correlation, the accuracy of the regression equation as a prediction formula is measured by S = ^^y"^ \ — r"^, where r is the coefficient of correlation between the variables, f^y is the standard deviation of the variable y about its mean value, and S is the root-mean-square devia- tion of the actual observations about the regression line; or, in other words, >S^ is the mean value of the mean-square deviations about the regression hne, of the observations in the several arrays of y's. From the Table of the Probability Integral it is known that in a symmetrical distribution of observations about their mean value, 68 per cent, of all the observa- The Law of Demand 79 tions fall within =•= the root-mean-square deviation of the observations from their mean value; 95 per cent., between =•= twice the root-mean-square deviation; and 99.7 per cent, between ± three times the root-mean- square deviation. It is therefore possible, by means of the Probability Integral, to affix the degree of prob- abihty that a deviation shall fall within any given multiples or submultiples of the root-mean-square deviation. In case of the use of the linear law of de- mand for corn in the United States as a prediction formula, the root-mean-square deviation of the ob- servations about the demand curve was S =^y'^l—r^ = 15.92 per cent. That is to say, if we assume the law of demand that was based upon observations from 1866 to 1911 to hold in 1912, then it is 95 to 5, or 19 to 1, that the percentage variation in the actual price for 1912 from the percentage variation as calculated from the law of demand will be between =•= 2 (15.92), or 31.84 per cent. The calculated percentage change in the price for 1912 was a fall of 13.06 per cent.; the actual fall was 21.20 per cent., giving a difference of 7.14 per cent. The precision with which the linear law of demand may be used for the prediction of the price of corn in the United States justifies the behef that for some pur- poses it is unnecessary to seek a greater degree of accuracy than is afforded by the simple Hnear laws. But it is well to be able to reach the maximum degree of precision, and for this reason we have fitted, to the data of the Tables in the Appendix, the more complex 80 Economic Cycles: Their Law and Cause curves y =a+bx+cx^-\-dx^, the graphs of which, in case of the representative commodities corn, hay, oats, and potatoes, are given in Figures 17, 18, 19, 20. What is the gain in precision when the more complex curve is substituted for the simple straight line? The scatter of the observations about the straight line of regression was measured, a while ago, by taking the root-mean- square deviation of the observations about the line, that is, by using S = o-^'^i—r^. In order to compare with this result the distribution of the observations about the more complex curve, y =a+bx+cx'^+dx^, the distribution about the latter curve will likewise be measured by the root-mean-square deviation of the observations. In the little table given below, the measures of scatter of the observations for the two types of demand curves are presented in a form that will make comparison easy. Scatter of Observations About the Law of Demand Root-Mean-Square Deviation of Observations Crops When the regres- sion is linear When the regres- sion is skew Corn Hay Oats Potatoes 15 . 92 per cent. 9. '53 " " 16.02 " " 21.29 " " 7 . 36 per cent. 4.65 " " 10.17 " " 9.94 " " It is clear that in all cases a gain in precision is ob- tained by using the more complex curve. Before leaving this topic a remark should be made The Law of Demand 81 that has a bearing upon the a priori theory of demand. In treatises on pure economics, particularly in those in which mathematical analysis is employed, the masters of the a priori method point out what they regard as the extreme difficulty of the actual problem of the rela- tion of price to quantity of commodity — a difficulty growing out of the interrelation of the many factors in the problem. If, to limit the illustration to a simple case, one wishes to know how the price of corn is re- lated to the quantity of corn that is produced, he is told that the problem is inextricably complex: If there is a deficiency in corn, then hay, or potatoes, or oats, or all three may be substituted in part for corn, and con- sequently the variation in the price of corn that fol- lows upon a deficiency of corn cannot be traced with- out knowing in what degree, when the price of corn varies, hay, oats, and potatoes are used as substitutes. But this is not all. The degree in which hay, oats, and potatoes are substituted for corn is dependent not only upon the price of corn but also on their own several prices, and these latter prices are, in turn, dependent upon the supply and price of corn! This statement of the problem, complex as it appears, is unduly simpU- fied; and it is presented not in order to ridicule the work of the masters who have elaborated the method of stating the problem in the form of simultaneous equa- tions, but to show how hopelessly remote from reality is the very best theoretical treatment of the problem of the relation of price to the quantity of commodity, and to suggest, from the results of the preceding pages 82 Economic Cycles: Their Law and Cause of this chapter, how imaginary, theoretical difficulties are dispelled by solving real problems. Of course it is theoretically possible when there is a deficiency in the production of corn, that oats, hay, and potatoes may be substituted in part for corn, but in- stead of conjuring up these and other possibilities that are never tested, would it not be wise to ascertain first just how closely is the variation in the price of corn related to the variation in its own supply? When the statistical investigation is made and it is found that the correlation coefficient is r = — .789, and that when a skew relation is assumed instead of the usual linear relation, the connection between the variables is still closer, one sees very clearly, if our illustration is a tjqjical case, that for most of the problems of actual life, it is unnecessary to face the complex possible in- terrelation of phenomena contemplated in the theoret- ical treatment. For the sake of economy of time and of talent, theoretical and statistical work should go hand in hand. Even the complex theoretical problem that has just been sketched may be tested as to its hypotheses and conclusion by the statistical method of multiple correlation. Elasticity of Demand The coefficient of the elasticity of demand for a commodity has been described as the ratio of the rela- tive change in the quantity of the commodity demanded to the relative change in the price, when the relative changes are infinitesimal. Starting with this descrip- The Law of Demand 83 tion, we are able, by means of the laws of demand for the several commodities, to measure their respective degrees of elasticity of demand. It will be recalled that, in the form in which the laws of demand have been presented in preceding pages, the variable x has been taken to represent the relative change in the quantity of the commodity, and the variable y, the corresponding relative change in the price. The coefficient of the dx elasticity of demand, therefore, is equal to t- when x is zero. All that is needed to obtain the measure of the degree of elasticity of demand is to differentiate y with respect to x in the equation to the law of demand, place X = zero, and then take the reciprocal of the result. The method may be illustrated in case of the four representative commodities, corn, hay, oats, and pota- toes. The law of demand for corn — see Figure 17 — is 2/ = .94- 1.0899X + .02391x2- .000234x» Therefore, ^ = - 1.0899 + 2(.02391)a;-3(.000234)x2 ax When. = 0,| = -1.0899, I =-j^=-.92 and consequently the coefficient of the elasticity of demand for corn is — .92. Since the law of demand for hay is y = 4. 17-.946a; -.0077x2 + .000385x» -^ = —.946 when x = zero, ax and the coefficient of elasticity of demand is —1.06. For similar reasons the degrees of elasticity of demand 84 Economic Cycles: Their Law and Cause for oats and for potatoes are respectively, — .84 and -.66. In obtaining these numerical values for the coefficient of elasticity, the laws of demand for the respective crops have been assumed to be parabolas of the third order. If the linear laws of demand had been taken for the piu-pose, the coefficients of elasticity would have been different. For example, the law of demand for corn — see Figure 16 — is y= — .8896x+7.79 which would give 3-=— .8896, or ~r-= — 1.12, whereas the coefficient was — .92 in case of the more complex curve. This discrepancy between the results when different types of curves are used for the demand curve shows the need of care in drawing conclusions that are based upon numerical values of the coefficient of elasticity. The discrepancy does not invafidate the method. When different measures of degrees of elasticity are afforded by different types of curves, there is a perfectly satis- factory criterion which makes it possible to decide between different coefficients of elasticity: The coeffi- cient is to be preferred which is deduced from the de- mand curve that fits the data with the highest degree of probabifity. The demand curve that fits best the data affords the best measure of the degree of elasticity of demand. The conclusions of this chapter may be briefly sum- marized. In the closing quarter of the last century great hopes were entertained by economists with regard to the capacity of economics to be made an The Law of Demand 85 "exact science." According to the view of the foremost theorists, the development of the doctrines of utiUty and value had laid the foundation of scientific economics in exact concepts, and it would soon be possible to erect upon the new foundation a firm structure of interrelated parts which, in definiteness and cogency, would be suggestive of the severe beauty of the mathematico-physical sciences. But this expectation has not been realized. On the contrary, faith in the possibihty of an adequate "exact" treatment of the science has progressively diminished, and interest in economic theory in general has decidedly lost ground. There must have been something fundamentally wrong with the traditional handling of the subject, for cer- tainly it must be admitted that the parts of a science most worthy of study are precisely those parts which are concerned with the general and the universal. Why, then, should there have been the gradual dissipation of interest in theoretical economics? The explanation is found in the prejudiced point of view from which economists regarded the possibilities of the science and in the radically wrong method which they piu-sued. It was assumed gratuitously that economics was to be modeled on the simpler mathe- matical, physical sciences, and this assumption created a prejudice at the outset both in selecting the data to be investigated and in conceiving of the types of laws that were to be the object of research. Economics was to be a "calculus of pleasure and pain," a "mechanics of utiUty," a "social mechanics," a "physique sociale." 86 Economic Cycles: Their Law and Cause The biased point of view implied in these descriptions led to an undue stressing of those aspects of the science which seemed to bear out the pretentious metaphors. One would naturally suppose from this manner of conceiving the science that the economic theorists would at once have entered upon their task with the methods that had proved themselves useful in the physical sciences. But this they did not do. They seemed to identify the method of physical sciences with experimentation, and since, as they held, scientific experimentation is impossible in social Ufe, a special method had to be devised. The invention was a dis- guised form of the classical cceteris paribus, the method of the static state. The point of view that has been exemplified in this chapter is that the facts in their full concreteness must never be lost from sight ; that the laws which are sought are of necessity, at first, proximate laws, laws that obtain in full empirical reality, and are means of arriv- ing at laws of larger generality; that the method to be followed is the method which makes progress from the data to generaUzation by a progressive synthesis — the method of statistics.^ ' With regard to the methodology of the social sciences, the writings of Cournot are always helpful. The following quotation is taken from a treatise published thirteen years after his epoch making Recherches sur les principes mathknatiques de la theorie des richesses. Si nous restons dans I'ordre des causes secondaires et des faits observables, le seul auquel la science puisse atteindre, la th6orie math^matique du hasard . . . nous apparait comme I'application la plus vaste de la science des nombres, et celle qui justifie le mieux The Law of Demand 87 Starting with this point of view and pursuing the method that has just been described, we have attacked the old problem of the form of the law of demand. We have obtained the concrete laws of demand for repre- sentative commodities, have affixed the degree of preci- sion with which the laws may be used as formulae for predicting prices, and have measured the elasticity of demand for the respective commodities. In all Ukelihood it will be said that what we have achieved is not exactly what the partisans of the method of costeris -paribus proposed. To this criticism we reply that their immediate problem of the relation of price and quantity of commodity, cwteris paribus, was vaguely conceived and actually abandoned by those who sought to give it definiteness, as being incapable of concrete I'adage: Mundum regunl numeri. En effet, quoiqu'en aient pensd certains phiiosophes, rien ne nous autorise h, croire qu'on puisse rendre raison de tous les ph6nom6nes avec les notions d'6tendue, de temps, de mouvement, en un mot, avec les seules notions des grand- eurs continues sur lesquelles portent les mesures et les calculs du gfeomtoe. Les actes des Stres vivants, intelligents et moraux ne s'expliquent nullement, dans I'etat de nos connaissances, et il y a de bonnes raisons de croire qu'ils ne s'expliqueront jamais par la m6canique et la g^om^trie. lis ne tombent done point, par le c6t6 g^omdtrique ou m^canique dans le domaine des nombres, mais Us e'y retrouvent places, en tant que les notions de combinaison et de chance, de cause et de hasard, sont sup^rieures, dans I'ordre des abstractions, h, la g6om6trie et &, la m^canique, et s'appliquent aux ph^nomlnes de la nature wante comme ^ ceux que produisent les forces qui sollicitent la mati^re inorganique; aux actes r^fl^chis des toes libres, comme aux determinations fatales de I'app^tit et de I'instinct. Essai sur les fondements de nos connaissances et sur les caractires de la critique philosophique, vol. 1, pp. 64^65. 88 Economic Cyces: Th eir Law and Cause solution; that when the problem is clearly stated, it admits of solution by means of a method which we have indicated, the method of multiple correlation ; and that what we have achieved is the solution of their ultimate problem of the relation of price and quantity of com- modity in a dynamic society. APPENDIX TABLE I. — ^The Peoduction and the Price of CoRir in the United States Average Prodttction of Farm Price Percentage Percentage Yeah Corn in Thou- Per Bushel Change in Change in sands OF Bushels December 1, IN Cents Production Price 1866 867,946 47.4 1867 768,320 57.0 —11.48 + 19.41 1868 906,527 46.8 + 17.99 —17.89 1869 874,320 59.8 — 3.55 +27.78 1870 1,094,255 49.4 +25.15 —17.39 1871 991,898 43.4 — 9.35 —12.15 1872 1,092,719 35.3 + 10.17 —18.66 1873 932,274 44.2 —14.68 +25.21 1874 850,148 58.4 — 8.81 +32.13 1875 1,321,069 36.7 +55.39 —37.16 1876 1,283,828 34.0 — 2.82 — 7.36 1877 1,342,558 34.8 + 4.57 + 2.35 1878 1,388,219 31.7 + 3.40 — 8.91 1879 1,547,902 37.5 +11.50 +18.30 1880 1,717,435 39.6 +10.95 + 5.60 1881 1,194,916 63.6 —30.42 +60.61 1882 1,617,025 48.5 +35.33 —23.74 1883 1,551,067 42.4 — 4.08 —12.58 1884 1,795,528 35.7 +15.76 —15.80 1885 1,936,176 32.8 + 7.83 — 8.12 1886 1,665,441 36.6 —13.98 + 11.59- 1887 1,456,161 44.4 —12.57 +21.31 1888 1,987,790 34.1 +36.51 —23.20 1889 2,112,892 28.3 . + 6.29 —17.01 1890 1,489,970 50.6 —29.48 +78.80 1891 2,060,154 40.6 : +38.27 —19.76 1892 1,628,464 39.4 —20.95 — 2.96 1893 1,619,496 36.5 — .55 — 7.36 1894 1,212,770 45.7 —25.11 +25.21 1895 2,151,139 25.3 +77.37 —44.64 1896 2,283,875 21.5 + 6.17 —15.02 1897 1,902,968 26.3 —16.68 +22.33 1898 1,924,185 28.7 ; + 1.11 + 9.13 1899 2,078,144 30.3 + 8.00 + 5.57 1900 2,105,103 35.7 + 1.30 + 17.82 1901 1,522,520 60.5 —27.67 +69.47 1902 2,523,648 40.3 +65.75 —33.39 1903 2,244,177 42.5 —11.07 + 5.46 1904 2,467,481 44.1 + 9.95 + 3.76 1905 2,707,994 41.2 ' + 9.75 — 6.58 1906 2,927,416 39.9 + 8.10 — 3.16 1907 2,592,320 51.6 —11.45 +29.32 1908 2,668,651 60.6 + 2.94 +17.44 1909 2,772,376 59.6 + 3.89 — 1.65 1910 2,886,260 48.0 + 4.11 —19.46 1911 2,531,488 61.8 —12.29 +28.75 89 90 Economic Cycles: Their Law and Cause TABLE II. — The Production and the Price of Hay in the United States Year Production of Hay in Thou- Averaok Farm Price Per Ton Percentaqb Change in Percentage Change in sands OF Tons December 1, Production Price (Ton = 2000 lbs.) in Dollars 1866 21,779 10.14 1867 26,277 10.21 +20.65 + .69 1868 26,142 10.08 — 2.42 — 1.27 1869 26,420 10.18 + 1.06 + .99 1870 24,525 12.47 — 7.17 +22.50 1871 22,239 14.30 — 9.32 + 14.68 1872 23,813 12.94 + 7.08 — 9.51 1873 25,085 12.53 + 5.34 — 3.17 1874 25,134 11.94 + .20 — 4.71 1875 27,874 10.78 + 10,90 — 9.72 1876 30,867 8.97 +10,74 —16.79 1877 31,629 8.37 + 2.47 — 6.69 1878 39,608 7.20 +25.23 —13,98 1879 35,493 9.32 —10,39 +29.44 1880 31,925 11.65 —10,05 +25.00 1881 35,135 11.82 + 10,05 + 1.46 1882 38,138 9.73 + 8,55 —17.68 1883 46,864 8.19 +22.88 —15.83 1884 48,470 8.17 + 3.43 — .24 1885 44,732 8.71 — 7.71 + 6,61 1886 41,796 8.46 — 6.56 — 2,87 1887 41,454 9.97 — .82 + 17,85 1888 46,643 8.76 + 12.52 —12,14 1889 66,831 7.04 +43,27 —19,63 1890 60,198 7.87 — 9,93 + 11,79 1891 60,818 8.12 + 1.03 + 3,18 1892 59,824 8.20 — 1.63 + .99 1893 65,766 8.68 + 9.93 + 5.85 1894 54,874 8.54 —16,56 — 1,61 1895 47,079 8.35 —14,21 — 2,22 1896 59,282 6.55 +25,92 —21,56 1897 60,665 6.62 + 2,33 + 1.07 1898 66,377 6.00 + 9,42 — 9.37 1899 56,656 7.27 —14,65 +21.17 1900 50,111 8.89 —11.55 +22.28 1901 50,591 10.01 + .96 + 12.60 1902 59,858 9,06 + 18.32 — 9.50 1903 61,306 9,07 + 2,42 + .11 1904 60,696 8,72 — 1.00 — 3.86 1905 60,532 8,52 — ,27 — 2.29 1906 57,146 10,37 — 5,59 +21.71 1907 63,677 11,68 + 11.43 + 12.63 1908 70,798 8,98 +11.18 —23.12 1909 64,938 10,62 — 8.28 +18.26 1910 60,978 12,26 — 6.10 +15.44 1911 47,444 14,64 —22.19 + 19.41 The Law of Demand 91 TABLE III. — ^The Pboduction and the Price of Oats in the United States Average Phodtjction op Fabm Price Percentage Percentage Year Oats in Thou- Per Bushel Change in Change in sands OF Bushels December I. IN Cents Production Price 1866 268,141 35.1 1867 278,698 44.5 + 3.94 +26.78 1868 254,961 41.7 — 8.52 — 6.29 1869 288,334 38.0 + 13.09 — 8.87 1870 247,277 39.0 —14.24 + 2.63 1871 255,743 36.2 + 3.42 — 9.66 1872 271,747 29.9 + 6.26 —17.40 1873 270,340 34.6 — .52 +15.72 1874 240,369 47.1 —11.09 +36.13 1875 354,318 32.0 +47.41 —32.06 1876 320,884 32.4 — 9.44 + 1.25 1877 406,394 28.4 +26.65 —12.35 1878 413,579 24.6 + 1.77 —13.38 1879 363,761 33.1 —12.05 +34.55 1880 417,885 36.0 +14.88 + 8.76 1881 416,481 46.4 — .34 +28.89 1882 488,251 37.5 +12.43 —19.18 1883 571,302 32.7 +17.01 —12.80 1884 583,628 27.7 + 2.16 —15.29 1885 629,409 28.5 + 7.84 .+ 2.89 1886 624,134 29.8 — .84 + 4.56 1887 659,618 30.4 + 5.68 + 2.01 1888 701,735 27.8 + 6.39 — 8.55 1889 751,515 22.9 + 7.09 —17.63 1890 523,621 42.4 —30.32 +85.15 1891 738,394 31.5 +41.02 —25.71 1892 661,035 31.7 —10.48 + .63 1893 638,855 29.4 — 3.36 — 7.26 1894 662,037 32.4 + 3.63 + 10.20 1895 824,444 19.9 +24.53 —38.58 1896 707,346 18.7 —14.20 — 6.03 1897 698,768 21.2 — 1.21 + 13.37 1898 730,907 25.5 + 4.60 +20.28 1899 796,178 24.9 + 8.93 — 2.35 1900 809,126 25.8 + 1.63 + 3.61 1901 736,809 39.9 — 8.94 +54.65 1902 987,843 30.7 +34.07 —23.06 1903 784,094 34.1 —20.52 + 11.07 1904 894,596 31.3 +14.09 — 8.21 1905 953,216 29.1 + 6.55 — 7.03 1908 964,905 31.7 + 1.23 + 8.93 1907 754,443 44.3 —21.81 +39.75 1908 807,156 47.2 + 6.99 + 6.55 1909 1,007,353 40.5 +24.80 —14.19 1910 1,186,341 34.4 +17.77 —15.06 1911 922,298 45.0 —22.26 +30.81 92 Economic Cycles :\ Their Law and Cause TABLE IV. — The Production and the Price of Potatoes in THE United Stages ~'^' Production of 1 ' Average T'arm Price Percentage Percentage Yeas t^OTATOES IN Thousands of Bushels Per Bushel December 1, IN Cents, , Change in Production Change in Price 1866 107,201 47.3 1867 97,783 65.9 — 8.79 +39.32 1868 106,090 59.3 4- 8.50 —10.02 1869 133,886 42.9 +26.20 —27.66 1870 114,775 65.0 —14.27 +51.52 1871 120,462 53.9 + 4.95 —17.08 1872 113,516 53.5 : — 5.77 — .74 1873 106,089 65.2 — 6.54 +21.87 1874 105,981 61.5 — .10 — 5.67 1875 166,877 34.4 +57.46 —44.07 1876 124,827 61.9 —25.20 +79.94 1877 170,092 43.7 +36.26 —29.40 1878 124,127 58.7 —27.02 +34.32 1879 181,626 43:6 +46.32 —25.72 1880 167,660 48.3 — 7.69 + 10.78 1881 109,145 91.0 —34.90 +88.41 1882 170,973 55.7 +56.65 —38.79 1883 208,164 42.2 +21.75 —24.24 1884 190,642 39.6 — 8.42 — 6.16 1885 175,029 44.7 — 8.19 + 12.88 1886 168,051 46.7 — 3.99 + 4.47 1887 134,103 68.2 —20.20 +46.04 1888 202,365 40.2 +50.90 ^1.06 1889 204,881 35.4 + 1.24 —11.94 1890 148,290 75.8 —27.62 + 114.12 1891 254,424 35.8 +71.57 —52.77 1892 156,655 66.1 —38.43 +84.64 1893 183,034 59.4 + 16.84 —10.14 1894 170,787 53.6 — 6.69 — 9.76 1895 297,237 26.6 +74.04 —50.37 1896 252,235 28.6 —15.14 + 7.52 1897 164,016 54.7 —34.97 +91.26 1898 192,306 41.4 + 17.25 —24.31 1899 228,783 39.0 + 18.97 — 5.80 1900 210,927 43.1 — 7.80 +10.51 1901 187,598 76.7 —11.06 +77.96 1902 284,633 47.1 +51.72 —38.59 1903 247,128 61.4 —13.18 +30.36 1904 332,830 45.3 +34.68 —26.22 1905 260,741 61.7 —21.66 +36.20 1906 308,038 51.1 + 18.14 —17.18 1907 298,262 61.8 — 3.17 +20.94 1908 278,985 70.6 — 6.46 + 14.24 1909 376,537 54.9 +34.97 —22.24 1910 349,032 55.7 — 7.30 + 1.46 1911 292,737 79.9 —16.13 +43.45 CHAPTER V THE MECHANISM OF CYCLES "Agriculture is the Foundation of Manufacture and Commerce." — Motto of the United States Department of Agriculture. Thus far in our investigation of the cause and law of economic cycles, we have shown that the annual rainfall in the principal grain-producing area of the United States passes through definite, well-defined cycles; and that the yield of typical, leading crops is so closely related to the rainfall of their respective critical seasons that the cyclical movement of the rainfall of the critical seasons is approximately reproduced in the yield per acre of the corresponding crops. These cycles of crops constitute the natural, material current which drags upon its surface the lagging, rhythmically changing values and prices with which the economist is more immediately concerned. In order to understand the connection between the flow of the undercurrent of agricultural yield and the surface changes of values and prices, we have taken the necessary first step of con- necting the prices of agricultural commodities with their supply. But the supply varies with the acreage as well as with the yield, and consequently to carry further our investigation we must know how closely the prices of crops are related to their yield. 93 94 Economic Cycles: Their Law and Cause The Prices of Agricultural Commodities Correlated with the Yield of the Several Crops The method employed in the preceding chapter to derive the law of demand of the several crops contained two stages : As a first stage, the correlation between the relative change in the total supply and the correspond- ing relative change in price was assumed to be linear, and upon the hypothesis of Unearity of regression, the demand curve was computed and the degree of accuracy with which prices might be predicted from such linear demand curves we showed how to measure. The second stage in the theory of demand curves was to assume a skew relation between relative changes in price and supply, and we found that the degree of accuracy with which prices might be predicted from the skew demand curves was greater than when the law of demand was assumed to be linear. We shall follow these two stages in treating the relation between the yield per acre and the price of the crops. If the correlation between the relative change in yield per acre and the relative change in price is as- sumed to be linear, we obtain for the coefficients of correlation in case of the four typical crops, the values placed in the first row of the accompanying Table, which, for purpose of comparison, also presents the corresponding coefficients in case of the linear demand curves. The Mechanism of Cycles 95 A Comparison of the Coefficients of Correlation in Case of Linear Yield-Price Curves and of Linear Demand Curves Corn Hay Oats Potatoes Relative change in yield per acre and relative change in price — .815 — .656 — .718 — .873 Relative change in total supply and relative change in price — .789 — .715 — .722 — .856 The data used in the above computation were, in case of the yield-price curve, the average yield per acre of the respective crops in the whole of the United States and the corresponding average prices for the United States, on the first of December of the years in which the crops were produced. The data for the demand curves, it will be recalled from the preceding chapter, were the total supply of the respective crops in the United States and the corresponding prices on December 1. The period covered in both cases was from 1866 to 1911, inclusively. The data were ob- tained from recent Yearbooks of the United States Department of Agriculture. It appears, from the coefficients of correlation given in the above Table, that it is possible to predict the prices of the crops from the yield per acre with the same 96 Economic Cycles: Their Law and Cause precision with which prices may be predicted from the demand curves. Or, to put the idea in another form, the productivity of the soil is as closely related to the prices of crops as the supply of the commodity is related to the same prices. In the chapter on the "Law of Demand," we found that, when the relative change in the supply is given, the mean shift in the corresponding change of price may be obtained from the regression equation, and that, furthermore, the root-mean-square deviation of the observations may be computed by the formula S = (^y'^l—r^. This same formula may be used for a similar purpose in case of the yield-price cm-ves. We come now to the second stage in the derivation of the relation between price and the yield per acre of crops. We assume that the relation between the yield per acre and the price of a crop is skew, and that the relation between the two may be expressed by an equation of the form y =a-\-bx-\-cx^+dx^. In Figure 21, the skew yield-price curves of our four representative comjnodities are drawn to a percentage scale. The equations to the curves, which were com- puted by the Method of Least Squares, are given upon the Figure. The root-mean-square deviation of the observations from their respective yield-price curves are given in the following Table which, for purposes of comparison, reproduces the coefficients that were found, in the preceding chapter, to measure the devia- tion of the observations about the skew laws of de- mand. The Mechanism of Cycles 97 Percenf^^e change in the yield per acre of hay. Percentage chande in the yield per acre of oats. Percentage change m the yield per acre of potatoes FiGTJBB 21. The relation between the price and the yield per acre of the several crops. When the origin is at (0, 0), the equations are For corn, y = XJ — 1.2989x + .01892a;2 — .000137^'. For hay, y = 1.17 — 1.0215x + .01549a;2 + .00009x'. For o^ts, y = — 1.49 — 1.1346x + .02324x2 _ .00O238x'. For potatoes, ^ = .49 — 1.4863x + .01993x2 — .000141x3. 98 Economic Cycles: Their Law and Cavse A Comparison of the Root-Mean-Squabe Deviation in Case op Skew Yield-Price Curves and of Skew De- mand Curves Corn Hay Oats Potatoes Yield-Price Curves 5.48 5.72 7.05 9.39 Demand Curves 7.36 4.65 10.17 9.94 From the results given in the last two Tables, it is clear that the prices of the representative crops are as closely related to the yield per acre as to the total supply of the crops. This conclusion is of importance in the task of connecting the cycles in the productivity of the soil with the cycles in values and prices. In obtaining the preceding close relations between the changes in prices and changes in yield, the figures for the whole of the United States were employed. The object of broadening the field of observation from the detailed investigation of the Middle West to the whole of the United States was two-fold: First, it seemed likely, a priori, that a more intimate relation between prices and yield would be obtained if the large market of the whole country were substituted for the local market of Illinois; secondly, because the object of this chapter is to bring the physical cycles of crops into relation with the industrial and commercial changes of the whole country, and to this end it seemed desirable that the crops of the The Mechanism of Cycles 99 whole country should be considered. We need, how- ever, to assure ourselves that, in taking this more comprehensive view of the yield of crops, we have not lost the characteristic cyclical movement of the yield which we discovered in the more limited study. We desire to know how closely the yield per acre of the whole country is correlated with the yield per acre of our representative state of Illinois. The correlations of the annual differences in the yield per acre in Ilhnois and the annual differences in the yield per acre in the United States were, in case of our four typical crops, for corn, r = .855 ; for hay, r = .745 ; for oats, r = .800; for potatoes, r = .843. The period covered in all cases was from 1866 to 1912 inclusively. The data were obtained from Bulletins, 56, 58, 62, 63 of the Bureau of Statistics of the United States Depart- ment of Agriculture and from the recent Yearbooks of the same Department. A reference to the Table given a moment ago will show that the yield per acre of crops in IlUnois is at least as closely related to the yield per acre of the same crops in the United States, as the prices of the several crops are related either to the supply of the crops or to the yield per acre of the crops. More- over, the very high values of the coefficients leave but Uttle room for doubt that the cyclical movement of the yield per acre in the Middle West is representative of the movement of the crop yield in the whole of the United States. 100 Economic Cycles: Their Law and Cause Rising and Falling Prices as Related to Yield-Price Curves Thus far it is clear that the prediction of agricul- tural prices is dependent upon a knowledge (1) of the law of the variations of price with the yield per acre, and (2) of the law of the annual change in the yield per acre of the several crops. If the relation between prices and yield per acre were constant, the theory of agricul- tural cycles would be completely elucidated; for, once having discovered the law of the relation of price to yield per acre, nothing more would be necessary then to connect the yield with the meteorological conditions of its critical season, and the resulting prices for a long term of years could be predicted with great probability. But the relation between the price of the crops and the yield per acre varies with the level of general prices, and it is of the first importance to know the manner of varia- tion. If the course of prices in the United States for the period 1866 to 1911 is examined, it will be seen that, in general terms, we may with justness characterize the period 1866 to 1890 as a period of falUng prices, and the period 1890 to 1911 as a period of rising prices. If therefore, in case of each of our representative com- modities, we construct two yield-price curves, one for the period of falliag prices and one for the pe- riod of rising prices, we shall, by comparing the two curves for the two periods, discover how the demand curves, or yield-price curves, vary in periods in which The Mechanism of Cycles 101 the movement of general prices is in opposite direc- tions. In Figure 22, the eight curves are drawn. Compar- ing the curves in the two periods for each of the four representative crops we infer that (1) the demand schedule or yield-price curve is high when the general level of prices is high; and the demand schedule is low when the general level of prices is low; (2) the general run of the curves remains nearly the same. That is to say, the principal difference between the period of falUng prices and period of rising prices is that the yield-price schedules move down or up. These are general statements in which quite obvious deviations are ignored and which, consequently, do not pretend to quantitative accuracy. The construc- tion of the curves is dependent upon too few observa- tions to admit of attaching significance to the apparent exceptions to the rule. Since the prices of the representative crops are, as we know, dependent upon the yield per acre and the law of the relation between prices and the yield per acre, and since, as we have proved, the yield-price curves move with the general level of prices, our desideratum is to discover what determines the change in the level of general prices. 102 Economic Cycles: Their Law and Cause Percenfdde change in fheyie/d peracre of oafs. ^errenia^e change infheyiftd per 3cre ofpot^foes. Figure 22. The relation between the price and the yield per acre of the several crops. When the origin is at {0,0), the equations are I yrs. 1866-1889, _ -, ?/=— 2.00— 1.0299i+.01926i2— .000312a;'. |yrs. 1890-1911, ,y= 3.06— 1.4894a; + .01737i2— .000049x'. yrs. 1866-1889, - . _, 2/ = - 5.72— 1.6435i + .07798j2— .000574x'. yrs. 1890-1911, ,y= 5.41— .7306x— .00591i2+.000075x'. yrs. 1866-1889, _-, 2/ = — 2.78— 1.6039.1;— .00546,t2+.000778x'. yrs. 1890-1911, ,y= .99— 1.0240x + .02394x2— .000383x'. yrs. 1866-1889, -._,i/=— 3.92— 1.4424x+.01684x2—.000020x». yrs. 1890-1911, ,y = — .91— 1.6068x+.03831x2— .000397i». For corn For hay For oats For potatoes The Mechanism of Cycles 103 The Volume of Crops and the Activity of Industry We shall approach the problem of the cause of the changing level of prices by considering two preliminary questions which will enter into the subsequent argu- ment: (1) Is there any relation between the changing volume of the crops and the changing volume of those producers' goods whose fluctuations are generally re- garded as indices of the activity of trade? (2) Is the law of demand for crops the type of law that is repro- duced in the demand for all commodities, or is it not rather the case that the law of demand for pure pro- ducers' goods is of a different type from the law of demand for those commodities of which our four crops are samples? The first of these two questions we shall consider in a form modified to bring its significance to bear upon the results that have already been established. The volume of crops varies with the extent of the acreage and with the average yield per acre. The question of interest to us at this point is whether the volume of producers' goods fluctuates with the yield per acre of the crops. We shall investigate this question, and, as a means of carrying forward our inquiry, we first construct an index number of the yield per acre of crops. The nine crops of the United States whose yield per acre through- out a long period is recorded in the Yearbooks of the Department of Agriculture are : corn, wheat, oats, bar- ley, rye, buckwheat, potatoes, hay, cotton.^ If, in case 1 The figures for the yield per acre of cotton, 1870-1910, were ob- 104 Economic Cycles: Their Law and Cause of each of these crops, the mean yield per acre for the years 1890-1899 is taken as a base, and the yield per acre for each of the years 1870-1911 is expressed as a ratio of the base, comparable indices for the crops dur- ing the period of forty-two years will be obtained. In order to combine the nine series of figures into a series that shall be representative of the whole of agriculture, the several series must be properly weighted. The method of weighting that was adopted in this particular case was to assign to each crop an importance propor- tionate to its value as compared with the total value of the nine crops in 1911. The several weights were: for corn, 36; wheat, 12; oats, 9; barley, 3; rye, .7; buck- wheat, .3; potatoes, 6; hay, 16; cotton, 17. The index numbers are given in Table I of the Appendix to this chapter. Before comparing the index number for the yield per acre of the crops with the volume of producers' goods, we must make sure that we are keeping close to the results obtained from a detailed investigation of our four representative crops. If an index number of the four representative crops is constructed upon the same principle as the index for the nine crops, how closely would the indices be correlated? In computing the index of the yield per acre of the four representa- tive crops, the weights assigned were: for corn, 50; hay, 28; oats, 15; potatoes, 7. The index is given in tained from Circular 32, Bureau of Statistics, U. S. Department of Agriculture. The yield for 1911 was obtained from the Yearbook of the Department of Agriculture, 1911. The Mechanism of Cycles 105 Table I of the Appendix to this chapter. The coeffi- cient of correlation between the index for the four representative crops and the index for the nine crops, is r = .960. It is a common observation of writers on economic crises that the production of pig-iron is an unusually- good barometer of trade. The amount of pig-iron that is annually produced swells with the activity and volume of industry and trade, and it is among the first commodities to indicate the general shrinking in the ultimate demand which checks the activity of trade and causes its temporary decline. Is there any relation between the movement of this barometer of trade, the production of pig-iron, and the cycles of the crops? Can it be that the increase and decrease of the "ultimate demand" which hes back of the flow and ebb of trade has its source in the cychcal movements of the yield per acre of the crops? The data for testing whether there is a relation be- tween the yield per acre of the crops and the annual production of pig-iron are the statistics of the annual production of pig-iron and the index numbers of the yield per acre of our nine crops. The method of testing the relation presents difficul- ties, and as it will be used again to measure the relation between the cycles of crops and the cycles of general prices, we shall have a firmer grasp upon our problem if we stop now to gain a clear idea of the terms that continually occur in the argument. In any one of the 106 Economic Cycles: Their Law and Cause series of figures that we shall use there are three distinct movements which need to be discriminated, and when any two of the series are compared, another important characteristic of the series requires to be taken into account. The three movements that are combined in each series are: (a) The continuous fall or rise of the figures with the flow of time. This movement will be referred to as the secular trend of the figures; (6) The rhythmical fluctuation of the figures about their secular trend. When this movement superposed upon the secular trend is the ob- ject of investigation, the combined movement will be referred to as the general cycUcal movement of the figures. When the rhyth- mical movement unaffected by the complicat- ing trend is being considered, it will be referred to simply as the cycles of the figures; (c) The year to year temporary fluctuation about the general cycHcal movement. These fluctua- tions will be referred to as the deviations of the figures. When the cycles of any two series are compared, it will frequently happen, particularly if the one series is the cause of the other, that there is a considerable interval between the corresponding parts of the cycles in the two series. This interval will be referred to as the lag of the second series. We shall be interested throughout the rest of this chapter primarily in the interrelations of cycles of The Mechanism of Cycles 107 crops, cycles in the activity of industry, and cycles in general prices. But we approach our general problem by considering first the temporary fluctuations which we have agreed to call deviations, and we inquire whether there is a relation between the deviations of the yield of the crops and the deviations in the produc- tion of pig-iron. The method that was adopted was first to obtain the general cyclical movements of the two series by averaging, in case of each series, the figures for each year with the figures that immediately preceded and followed the given year. For example, the index number of the yield per acre for the years 1870, 1871, 1872, 1873 were respectively 108, 105, 110, 99. The smoothed figure for the yield per acre in 1871 would therefore be = = 107.7. Simi- 3 3 larly, the smoothed index for 1872 would be 104.7. In Tables II and III of the Appendix to this chapter are presented the original and the smoothed figures for the production of pig-iron and for the index number of the yield per acre of the nine crops. The statistics of the production of pig-iron were obtained from the Statis- tical Abstract of the United States for 1912, p. 774. After the general cycUcal movements of the two series were determined, the deviations of the actual figures from the smoothed figures for each of the years were calculated for both series of figures. These deviations are also given in Tables II and III of the Appendix to this chapter. The question upon which these differ- ences are to throw fight may be put in this form: Is 108 Economic Cycles: Their Law and Cause the deviation of the yield per acre of the crops from its general cychcal movement associated with the devia- tion, in the following year, of the production of pig- iron from the general cychcal movement of pig-iron? The answer is found by correlating the differences, always remembering that the difference for the yield per acre in any given year is to be taken with the dif- ference of the production of pig-iron in the following year. The coefficient of correlation is r = .254. We come now to the association between the cychcal movement of the yield per acre of the crops and the cychcal movement of the production of pig-iron. Each of these movements is superposed upon a rising secular trend, and before we can test the degree in which the cycles are related the secular trends must be eliminated. If, as a first approximation, the secular trend in each case is assumed to be linear, then by fitting a straight line ^ to the data, it is possible to calculate the fluctua- tions of the cycles of crop yield and of production of pig-iron about their respective trends, and these fluctua- tions may be correlated. In Table IV of the Appendix to this chapter, the data for the calculation of the con- nection between the cycles are given. In columns 2 and 5 are tabulated the general cyclical movements of 1 The equations to the linear secular trends are, respectively, 2/ = .1844a;+98.57, for the yield per acre of crops; and 2/ = 582.71a:+ 9525, for the production of pig-iron. The origin in the first case is at 1871 and in the latter case, at 1890. The first equation was com- puted from the data for the years 1871-1906, and the second equa- tion, from the data for 1871 to 1910. The Mechanism of Cycles 109 the yield per acre of the crops and of the production of pig-iron; in columns 3 and 6, the values of the Hnear secular trends are given; and in columns 4 and 7, the deviations of the cyclical movement from the secular trend are recorded. These last deviations are the ma- terial for calculating the connection between the cycles of the yield per acre of the crops and the cycles of the production of pig-iron. If the deviations of the cycles from their respective secular trends are correlated, the coefficient of correla- tion reaches the value, r = .625, but we must not be content to assume that even this relatively high co- efficient represents the full degree of the relation be- tween the cyclical movement of the crops and the cycHcal movement of the activity of industry as that activity is typified in the production of pig-iron. It is quite hkely that the good or bad crops may produce their maximum effect at a considerable interval after the period in which the crops are actually harvested. Time is required for the changing productivity of crops to work out its maximum effect, and this causes a lag in the adjustment of the cycles of the activity of industry to the cycles of the yield of the crops. We must therefore measure the amount of the lag. If instead of correlating the cycles of the yield of the crops and of the production of pig-iron for correspond- ing years, we correlate them for lags of various intervals, we shall find it possible to determine the lag that will give the maximum coefficient of correlation, and this particular value of the lag we may then regard as the 110 Economic Cycles: Their Law and Cause interval of time required for the cycles in the crops to produce their maximum effect upon the cycles of the activity of industry. When the calculation of the co- efficients of correlation is made according to this plan, it is found that for a lag Of zero years, r = .625 Of one year, r = .719 Of two years, r = .718 Of three years, r = .697 Of four years, r = .572. It is clear, therefore, that the cycles in the yield per acre of the crops are intimately related to the cycles in the activity of industry, and that it takes between one and two years for good or bad crops to produce the maximum effect upon the activity of the pig-iron in- dustry. Figure 23 illustrates the general congruence of the cycles of the crops and of the cycles in the produc- tion of pig-iron when a lag of two years is eliminated. As to the general question concerning the relation between the harvests and the activity of industry, we may conclude from our statistical inquiry that there is a positive, intimate connection, and very probably a direct causal relation, between the bounty or niggardh- ness of nature and the flow or ebb of trade. ^ A New Type of Demand Curve A moment ago, we saw that two prehminary problems had to be treated before we could pass to the direct 1 In a later section of the chapter the method that has been used in treating this problem wall be employed for another purpose and will then be illustrated in detail by means of graphs. The Mechanism of Cycles 111 De\/tafion of the denera/ cyc/tM/ mot^ement o^the product/on ofptd-iron fi'otn tfi secu/ar tren^. ^ s ^ 5 5 \ \ f 5 > 3 o~ --^^ ,/* ..J< ■ — 0-.^ \ ? f V V^^ > <.- S 'x, ? s ,,^ -y ^ S, + + ^ II 112 Economic Cycles: Their Law and Cause consideration of the cause and law of cycles of general prices. The first of these preliminary problems, namely, the influence of the bounty of nature upon the volume and activity of trade, we have just discussed, and we come now to the second preliminary problem, which we shall put in the form of a question: Are all demand curves in a dynamic society of the same type as the demand curves for the representative crops: corn, hay, oats, and potatoes? This question must be answered as a preliminary to the more fundamental inquiry as to the cause of cycles of general prices, because if we assume that all demand curves are of the same negative type, we are confronted with an impossibihty at the very beginning of our in- vestigation. Upon the assumption that all demand curves are of the negative type, it would be impossible for general prices to fall while the yield per acre of crops is decreasing. In consequence of the decrease in the yield per acre, the price of crops would ascend, the volume of commodities represented by pig-iron would decrease, and upon the hypothesis of the uni- versahty of the descending type of demand curves, the prices of commodities hke pig-iron would rise. In a period of dechning yield of crops, therefore, there would be a rise of prices, and in a period of increasing yield of crops there would be a fall of prices. But the facts are exactly the contrary. During the long period of falling prices from 1870 to 1890, there was a decrease in the yield per acre of the crops, and during the long period of rising prices from 1890 to 1911, there was an increas- The Mechanism of Cycles 113 ing yield of crops. It is obviously inadmissible to assume that in a dynamic society there is one law of demand for all conunodities. The dogma of the uni- formity of the law of demand is an idol of the static state. If there are differences in types of demand curves, it is quite Hkely that as one type has been illustrated by the crops, another type will be exempUfied by pure producers' goods. We shall accordingly investigate the demand curve of pig-iron, our representative pro- ducers' good. In Table V of the Appendix to this chapter is con- tained the material for the computation of the law of demand for pig-iron. The annual percentage changes in the production of pig-iron were computed from the figures of annual production, which were taken from the Statistical Abstract for 1912, p. 774. It was impos- sible to obtain directly the mean prices for which the annual production was sold, and consequently the per- centage change in the mean price could not be com- puted directly. The device that was utiUzed to ap- proximate these percentage changes is illustrated in Table V of the Appendix. As the data needed for the solution of the problem were the annual percentage changes in the mean price and not the actual mean annual prices themselves, it was regarded as sufficient for our purpose to substitute for the unobtainable an- nual percentage changes in the mean price, the mean annual percentage changes in the prices of representa- tive kinds of pig-iron. The annual prices for the lead- 114 Economic Cycles: Their Law and Cause ing four kinds of pig-iron were obtained from the Statis- tical Abstract for 1912, p. 572, and the annual percentage changes in the prices of the four kinds, together with their mean annual percentage changes, are given in Table V of the Appendix. The second and last columns of Table V were used in computing the law of demand for pig-iron in the United States. The graph of the law of demand for pig-iron is given in Figure 24. The correlation between the percentage change in the product and the percentage change in the price is r = .537. The equation to the law of demand is ?/ = .5211x— 4.58, the origin being at {o,o). Our re- presentative crops and representative producers' good exempUfy types of demand curves of contrary charac- ter. In the one case, as the product increases or de- creases the price falls or rises, while, in the other case, the price rises with an increase of the product and falls with its decrease. The two preliminary difficulties are now cleared away. We know that as the yield per acre of the crops increases the physical volume of trade for producers' goods increases; and we know, furthermore, that the law of demand for a representative producers' good is such that as the product increases the price increases. If now a third fact, which has already been estabhshed, be added to these two, an hypothesis conformable to the three facts may be made which will give a working theory for examining whether the cycles in crops pro- duce the cycles in general prices. The third fact to which reference is made is that the law of demand for The Mechanism of Cycles 115 ■UOM-fldjO 33ud SLf^ Ul sfuei/O 3p9^U93J9(j 116 Economic Cycles: Their Law and Cause the crops falls during a period of falling general prices, and rises during a period of rising general prices. With these facts in mind it is not difficult to conceive how general prices may fall during a period of diminishing yield per acre of the crops and rise during the period that the yield is increasing. The falling yield in the crops would lead to a diminution of the volume of trade, a dechne in the demand for producers' goods, a fall in the prices of producers' goods, a decrease in employ- ment, a fall of the demand curves for crops, with the final result of a fall in general prices. Similarly, a rising yield in the crops would lead to an increase in the volume of trade, an increase in the demand for producers' goods, an increase of employment, a rise in the demand curves for crops, with the final result of a rise in general prices. Provided the interrelation of the economic factors are in accordance with this de- scription, then it would follow that the cychcal move- ments in the yield of the crops should be reproduced in cychcal movements of general prices. If the actual facts bear out this deduction, there can be no doubt that the cause and law of economic cycles have been discovered. The Fundamental, Persistent Cause of Economic Cycles To put the theory to the test of facts we require an index number of general prices throughout the period covered by most of the iavestigation in this Essay — - the period from 1870 to 1911. There is no one index number covering this period for the United States, but The Mechanism of Cycles 117 very fortunately there are two series that overlap in the middle of the period, so that it is possible to construct a series covering the whole term of years. The two series of index numbers in question are the FaUcner index for "all articles" extending from 1870 to 1890, and the index of the Bureau of Labor for "all commodities" extending from 1890 to 1911. Since these two have the year 1890 in common it is possible, by applying the simple rule of proportion, to reduce the Falkner series to the base of the series pubUshed by the Bureau of Labor. The two original series and the continuous series are given in Table VI of the Appendix to this chapter. The test of the theory that the cause and law of economic cycles are the cychcal movements of the yield per acre of the crops will be given in answer to two questions: First, are the deviations of the indices of general prices from their general cyclical movement correlated with the deviations of the indices of the yield per acre of the crops from their general cyclical move- ment? Secondly, are the cycles of prices and the cycles of crops correlated? The answers to these two questions are the substance of the following paragraphs. In Tables III and VI of the Appendix to this chapter are given the indices of the yield per acre of the crops and the indices of general prices. The Tables hke- wise contain the smoothed indices and the deviations of the actual indices from the smoothed indices. The smoothed series were obtained in the manner that was described when the relation between the yield of the crops and the production of pig-iron was being treated. 118 Economic Cycles: Their Law and Cause It will be recalled from that description that the smoothed index for any given year is the mean of three actual indices : the actual index for the given year, the actual index for the year preceding the given year, and the actual index for the year following the given year. The quantities whose correlation is in question are the deviations of the actual indices of general prices, and of yield per acre, from their respective smoothed series. The results of the computation are as follows : From 1870-1911, r = .303, From 1870-1890, r = .370, From 1890-1911, ?-=.250. In the first row the correlations were obtained from the continuous series in which the Falkner index was adjusted to the index of the Bureau of Labor. In the second row the correlations were derived from the Falkner index unaltered. In the third row the correla- tions were computed from the index of the Bureau of Labor. We infer that the deviations from their general cyclical movement of thejindices of general prices vary directly with the deviation from their general cycHcal movement of the indices of the yield per acre of the crops. The second of the two questions as to the cause and law of the cycles of general prices was stated in this form: Are the cycles of prices and the cycles of crops correlated? The preceding paragraphs have presented the results of the inquiry as to the relation between the deviations of actual prices and of yield from their The Mechanism of Cycles 119 respective general cyclical movements. The present question concerns the relation of the cyclical move- ments themselves, after their respective secular trends have been eliminated. It will be recalled that the general cyclical movements were obtained by a process of smoothing the actual series of the indices of prices and of yield per acre, the process consisting in the formation of a progressive mean of the indices for three consecutive years. These smoothed series, which are given in Tables III and VI of the Appendix to this chapter, form the data of the present investigation. The method of the investigation is presented in Fig- ures 25, 26, 27. In the first of these three Figures, the general cyclical movements of prices and of yield per acre are described according to the data of Tables III and VI. The graphs bring out clearly the rhythmical motions of both prices and yield and a comparison of the curves suggests that the price curve is a lagging reproduction of the yield curve. But before the amount of the lag and the degree of correlation between the cycles can be computed, the secular trends in the two series of values must be eliminated. From Figure 25 it is apparent that the price cycles move upon a falling secular treiid while the yield cycles move upon a rising secular trend. If it is assumed as a first approximation that these secular trends are both linear, the equation to the trend for prices is y = — .3702a; -|- 122.01, and to the trend for the yield per acre, y =. 1844a; -(-98. 57, the ori- gin, in the former case, being at 1875 and, in the latter, 120 Economic Cycles: Their Law and Cause \ / \ • /' V^ K \ \ "^ "n k / ( \ 1 1 4 I I 1 / .y 1 f y^ 1 X 1 .X d •^33uc/ /P-^JUsfjO Xjpui P3LI4DOUJS puO sa(U:>JO 3JJ9 jsd p/St/^ dU^JO Xdpu/ DdU^OOWC The Mechanism of Cycles 121 S ? '^ ? ?^ ? 122 Economic Cycles: Their Law and Cause at 1871.^ These two equations make it possible to eliminate the secular trends upon which move the cycles of prices and the cycles of yield. The results of the calculations are given in Table VII of the Appendix to this chapter. Figure 26 presents the cycles of yield per acre and the cycles of general prices after the secular trends upon which they were respectively superposed have been eliminated. It is quite evident, now, from the appear- ance of the graphs, that the cycles of yield per acre and the cycles of general prices are closely related, and that the cycles of prices lag several years behind the cycles of crops. What is the amount of the lag and how closely are the cycles correlated? Both of these questions may be answered at once by following the method that was adopted to measure the lag in the cycles of pig-iron production. If the cycles of the yield per acre are correlated ^ with the cycles of general prices we find, for a lag of three years in general prices, r = .786; for a lag of four years, r = .800; for a lag of five years, r = .710. The cycles in the yield per acre of the crops are, there- fore, intimately connected with the cycles of general prices, and the lag in the cycles of general prices is approximately four years. Figure 27 presents the two series of cycles with the lag of four years in the cycles of prices ehminated. It is » The first equation was computed from the data for 1875-1910, and the second equation, from the data for 1871-1906. ' The data for the calculation are given in columns 4 and 7 of Table VII in the Appendix to this chapter. The Mechanism of Cycles 123 \ '\ y V ^. \ ^^^v > 1 I /* ^ r/ 1 .^■° t — ^ > 9 ( ^y 5