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The Columbia University Libraries reserve the right to refuse to accept a copying order if, in its judgement, fulfillment of the order would involve violation of the copyright law. Author: Pearl, Raymond Title Predicted growth of population of New York Place New York Date: 1923 MASTER NEGATIVE * COLUMBIA UNIVERSITY LIBRARIES PRESERVATION DIVISION BIBLIOGRAPHIC MICROFORM TARGET ORIGINAL MATERIAL AS FILMED - EXISTING BIBLIOGRAPHIC RECORD RESTRICTIONS ON USE: FILM SIZE : "S CDrr\ t mtm Basines5 120 P31 I DATE FILMED: TRACKING # : Pearl, Raymond, 1879- ... Predicted growth of population of New York and its environs, by Raymond Pearl and Lowell J. Eeed ... New York city, Plan of New York and its environs, 1923. 42 p. incl. illus. (map) tables, diagrs. 23*=". At head of title: P. N. Y. 4. X New York (City)— Population. i. Reed, Lowell Jacob, joint author. II. Plan of New York and its environs, in. Title. Library of Congress n 24—4436 HB3527.N7P4 is24bl| TECHNICAL MICROFORM DATA REDUCTION RATIO: l^X IMAGE PLACEMENT: lA ® IB IIB v\A^ -c\5 INITIALS: VO\a/ H15U DS- s o o 3 '*.* an 3 > 0,0 erg o m ^|0 o ?"C/3 X M X ISI ^ > ^: -^ i 3 > Ul o 3 3 A? > Ul ^, '§ *<^^ o o 3 3 O - i^ 1^ llllg a 00 K3 b ro to II ro In Ol t-j> tS^ ^t3 •** Wb ?o^ ^c? fp ^o V :.«« ^ Ik' ir i' ^o ¥cP ?^ ^l5r m H O o -o m "o > C c*> X TJ ^ 0 Sr>' '. ^,3 ',.v S* CJl 3 3 U? fo •«P ^fp ^^ IN) ■ o 3 3 o > Too f^ HIJKLM n nopqn IJKLM nopqr iilz OPQR uvwxy OPQ uvw> *< ::□ ^^c r^J c/> *•< cnx -vl-< OOM s x >>!•< OOM ^O o / ;■ PREDICTED oaom OF BOPUiATicN 'i'i H p^< •i / I "^vao Columbia (Mnitiettitp itttfteCitpofl^rtogork LIBRARY School of Business 4 ♦ I ■in ^ )v fC^ / \l LIBRARY SCHOOL OF BUSINESS P.N.Y.4 PREDICTED GROWTH OF POPULATION OF NEW YORK AND ITS ENVIRONS ' il r BY RAYMOND PEARL AND LOWELL J. REED SCHOOL OF HYGIENE AND PUBLIC HEALTH THE JOHNS HOPKINS UNIVERSITY I ! ».-» *t> K Price 25 Cents PLAN OF NEW YORK AND ITS ENVIRONS 130 East Twenty-second Street New York City 1923 A' COMMITTEE ON PLAN of NEW YORK AND ITS ENVIRONS Frederic A. Delano, Chairman Robert W. de Forest Dwight W. Morrow John M. Glenn Frank L. Polk Frederick P. Keppel, Secretary Flavel Shurtleff, Assistant Secretary 130 East 22d Street, New York City SOCIAL SURVEY Shelby M. Harrison, Director \ T3\ i J ' "I c<3 <' ^ ^ » \^ cd-S » "^ « & S CONTENTS PAGE Introduction 5 Predicted Growth of Population of New York and Its Environs 9 Mathematical Theory 10 Prediction of Total Population within the Area and in Certain of Its Subdivisions 19 Population Densities 24 Population Predictions for New York City K?*^ Trends of Certain Elements of the Population of New York City . . (^8) Distribution of Population by Age Groups 30 Negro Population of New York City 31 Foreign-Born Population of New York City 32 Summary 34 Tables of Predicted Population for the New York Region \^ Appendix. Comparison of Population Predictions made by Nelson P. Lewis, of the Committee's StaflF, with those of this Study 41 ip^: O \ \^.' t \ ' 1, „'■' 525 O o O > o « 62^3 ^11 .S c« c c H O < u c o •O 3 0) (0 r; ^ > c o c « .iie N -A; INTRODUCTION SINCE city planning has to do, among other things, with the economic and socially desirable uses of land areas, it is important to know, at least approximately, what are likely to be the future demands of the people of a region for such areas. And this demand is partly represented in the population figures— in the number of people who are likely to live in a given region at particular times in the future. The Committee on Plan of New York and Its Environs was thus early faced with the problem of predicting the growth of population in the New York Region. The Committee realized that this problem is one that will not be disposed of at a single stroke, but will need to be grappled with in one form or another through all phases of its planning. The beginning point appeared to be to compute for a certain period into the future the approximate numbers of people who will live in the New York Region as a whole and in certain large divisions of it. Without waiting, therefore, until its later studies into in- dustrial, economic, housing, and other trends, either general or specific for localities, might indicate factors which will undoubt- edly modify the population aggregates for particular sections of the Region, it was decided to see what could be ascertained at once. There are a number of methods of forecasting the populations of the future which are familiar to students of the subject, and which could be employed immediately with data already avail- able. In these the future is predicted on the basis of population figures of the past. That is, the trend of past population changes is first determined, and this trend is then extended into the future. These methods come down in general to two types, and the difference between them is found in the procedures followed in extending the curve representing the future aggregates. The more recent of them is one developed by Professors Ray- mond Pearl and Lowell J. Reed, of Johns Hopkins University. It begins by indicating several factors which should be taken into account in developing a mathematical formula for predicting the trend of future population growth in a fixed area: First, the area upon which the population grows is finite— the area has a definite size or upper limit. Second, since population lives upon limited areas there must be a definite upper limit to the number of persons who can live on that area; that is, it is inconceivable that pop- ulations on particular areas can increase without limit. Third, there is also a lower limit to population which is zero— population obviously cannot go below that. Fourth, each epoch marking an advance in human culture and economy has made it possible for a given area to support more people. And fifth, the rate of growth during each epoch, in so far as it has been observed, varies, being slow at first, then increasing in rate to a maximum, and then decreasing until almost a stationary aggregate of popula- tion is maintained. With these factors in mind Professors Pearl and Reed have developed a mathematical equation, aimed to make the known quantities of the past indicate what may be expected in the future. In other words, assuming that populations cannot grow on for- ever and that they grow at differing rates at different periods of each epoch, a formula has been developed which is believed to express the fundamental law of normal population growth. The Committee, desiring to bring as much light as possible to bear upon the local problem of population growth, asked Pro- fessors Pearl and Reed to apply this new method to the New York Region. The results obtained are presented in the accompanying report. In the available space Professors Pearl and Reed have confined themselves to a description of their process and results without attempting any justification of the method itself. It may be said, however, that elsewhere they have pointed out that their formula has been tested by applying it to the figures showing actual population growth in a number of countries, with the result that the theoretical curve and the points representing the re- corded facts coincide as nearly as is found necessary for the prac- tical confirmation of most hypotheses in the physical sciences. It is realized, however, that the disturbing factors in popula- tion trends may have mugh greater effect in computing for smaller areas than for those of whole nations or countries. That is to say, the introduction of an unusually large manufacturing in- dustry to a city, like the automobile industry to Detroit, for 6 i^' I ( V example, would be likely unexpectedly to modify future popula- tion figures for a city or city region to a greater degree than such disturbance would for a whole nation. At the same time, while such a sudden change may modify the course of the population growth, the result of such a disturbance, so far as existing evidence indicates, would be merely to throw the population trend to a new curve of the same type as the old one but having a different rate. The figures presented deal not only with the growth of the whole New York Region, but with several subdivisions of it. The predictions for the Region as a whole are as follows: Predicted Population (in Round Numbers), New York Region Year Number of Persons 1930 11,500,000 1940 14,100,000 1950 16,800,000 1960 19,600,000 1970 22,300,000 1980 24,800,000 1990 27,000,000 2000 28,800,000 In this connection it is interesting and significant to note that the forecasts by Professors Pearl and Reed for a considerable period into the future— up to 1970 at least, a limit which provides ample latitude for the present regional planning project— are in quite close agreement with those worked out by Nelson P. Lewis as a part of his Physical Survey of the Region for the Committee, as well as those made by at least one important public service corporation which is also confronted with the problem of future population trends.^ There is a significant divergence, however, in the predictions after 1970. As would be expected from the description given of the method followed by Professors Pearl and Reed, their predictions for the later years are lower than those made in the other way. But even on this lower, and in this sense more conservative basis, the growth in aggregate numbers of people living in the Region promises to be very great. Planning for as large a territory as the New York Region and for a considerable distance into the future must of necessity be along broad lines. And obviously, the longer the period chosen the greater will be the necessity for flexibility and adaptability * A brief summary of the estimates made by Mr. Lewis is given in the Appen- dix, page 41. 7 in what is to be recommended— that is to say, the longer ahead we try to look the greater will be the necessity of allowing for local variations and adaptations while at the same time claiming the great benefits which will come from unified and co-ordinated plans for such a large Region. This does not minimize the great importance, however, of getting as accurate a forecast as possible as to what the future holds in store as to mere numbers of people who will be living here. Indeed, such forecasts both for the near and distant future, and the later refinements of them, constitute factors in planning which need to be continually reckoned with. And incidentally, the fact that the predictions by Professors Pearl and Reed, as well as the several others which have been made, point to such great future population aggregates for this Region constitutes in itself one of the strongest arguments for careful and comprehensive planning. Frederick P. Keppel Secretary, Committee on Plan of New York and Its Environs. 8 > f PREDICTED GROWTH OF POPULATION OF NEW YORK AND ITS ENVIRONS THE present study of the population of New York City and Its Environs may be considered as having two distinct parts. The first consists of a mathematical investigation into the growth of the population on certain areas without regard to the constitution of these populations. The second concerns certain elements of the population and the growth of these ele- ments relative to the entire population. Before considering the results of the mathematical analysis it may be well to examine the logic behind the various methods of predicting future populations. The most common way of deter- mining the probable size of the population at a future date is to determine certain facts regarding the population in its present state, and then deduce from these facts the size of the population at some future time. An illustration of the application of this method is the predicting of future populations from the present size and the rate of increase expressed either as an arithmetic or geometric rate. It is well known that by this process we may obtain good predictions for a comparatively short period of time. Another method is to correlate the growth of the population with the growth of some other variable (usually an economic variable), and then from predictions for this other variable to obtain estimates of the future population. This procedure is applied when most of the population of a community is associated with some particular industry which is known to be rapidly ex- panding. While the results obtained by this method are often accurate over short intervals of time, when applied to longer inter- vals they are usually inaccurate, for two reasons : first, the long- time prediction of the condition of any industry or group of indus- tries is itself subject to error; and second, no recognition is made of the other forces acting to increase and retard population growth. A third method of predicting future population rests on the 9 \ assumption that there exists a multiplicity of forces producing population growth, and that the future effects of these may best be determined by observing their action in the past without attempting to differentiate between the forces. This reduces the problem of predicting future populations to a mathematical-sta- tistical one. It should be noted that this method does not assume that the forces are constant in their effect at different times. It merely assumes their existence and a continuity in their action. The application of this method requires the development of a mathematical equation which will well represent the effects of the forces in the past as exhibited in the population counts. In general this method of predicting population is the most reliable one and it is the one which has been used in this paper. MATHEMATICAL THEORY The following account of the mathematical theory used in this report is based upon a paper now in press in Metron by Raymond Pearl and Lowell J. Reed.^ Careful study of the matter will convince one that at least the factors listed below must be taken account of in any mathema- tical theory of population growth which aims at completeness. The necessity for a part of these factors is evident on purely a priori grounds. The remainder are equally obvious and certain deductions from observed facts as to how populations do actually grow. 1. If any discussion of the growth of human population is to be profitable in any real or practical sense, ^e area upon which the population grows ^ust be taken as a finite one, however large its limits. For the growth of human populations the upper limit of finite areas possible of consideration must plainly be the habit- able area of the earth. Smaller areas, as politically defined coun- tries, may be treated each by itself. But whether this is done or not, there clearly is a finite upper limit of area on which human population can grow. 2. If there is a finite upper limit to the area upon which popula- tion may grow, then with equal clearness there must be a finite » See Pearl, R., and Reed, L. J.: On the Rate of Growth of the Population of the United States since 1790 and its Mathematical Representation. Proceed- ings of the National Academy of Sciences, Vol. VI, No. 6, June, 1920, p. 275. Also Pearl, R., and Reed, L. J.: A Further Note on the Mathematical Theory of Population Growth. Proceedings of the National Academy of Sciences, Vol. Vni, pp. 365-368. 10 N > upper limit to population itself, or, in other words, to the number of persons who can live upon that area. It is evident, for example, that it is a biological impossibility for so many as 50,000 human beings to live, and derive support for living, upon one acre of ground, provided every other acre of the possibly habitable area of the earth is at the same time inhabited to the same degree of density. This is obviously true whatever the future may hold in store for us in the way of agricultural discoveries, improvements, or advancements. That there is a finite upper limit to the popu- lation which can live upon a finite area (as of the earth) is really as much a physical as a biological fact. The amount of water w^hich can be obtained in a pint measure is strictly limited to a pint. It cannot by any chance be ten gallons. And this conclu- sion is in no way determined or limited by the present limitations of our knowledge of physics. Nor can it be upset by any future discoveries to be made in the realm of physics. It is this point which is so usually overlooked. From Mai thus to the present time, everyone who has pointed out that there must be some upper limit-iflLjiunian_population upon this globe has been met by the contention that he has neglected the p ossibi lities inherent in the future development o ( scienc e. Of course, mture scientific dis- coveries can have no bearing upon the bald fact that there must somewhere be an upper limit to population. They can influence only the precise location (or magnitude) of that upper limit. But the discrimination between these two ideas appears not to be sufficiently appreciated. Mathematical theories of population, even, have been more or less seriously advanced which really postulated that with the passage of time the curve of population would run off to infinity. Of course, attention was not drawn to this feature of such theories, but nevertheless it was inherent and implicit in them. 3. The lower limit to population is zero. Negative populations are in any common, practical sense, unthinkable. 4. History tells us what common-sense indicates a priori; namely, that each advancement in cultural level has brought with it the possibility of additional population growth within any de- fined area. In the hunting stage of human culture the number of persons who can be supported upon a given area is small. In the pastoral stage of culture more persons can subsist upon a given area, though the absolute number is still small. In the general agricultural stage of civilization the possibilities of population II ♦r I per unit of area become again enhanced. The commercial and industrial stages of culture permit great increases of population, provided, of course (and only under this condition), that there still remain somewhere else less densely populated areas where the means of subsistence can be produced in excess of local needs. In other words, each geographical unit which has been inhabited for any long time has, so far as the evidence available indicates, had a succession of waves of eras of population growth, each su- perposed upon the last, and each marking the duration of a more or less definite cultural epoch. 5. Within each cultural epoch or cycle of population growth the rate of growth of population has not been constant in time. Instead, the following course of events has apparently occurred generally, and indeed almost universally. At first the population grows slowly, but the rate constantly increases to a certain point where it, the rate of growth, reaches a maximum. This point may presumably be taken to represent the optimum relation between numbers of people and the subsistence resources of the defined area. This point of maximum rate of growth is the point of inflec- tion of the population growth curve. After that point is passed the rate of growth becomes progressively slower, until finally the curve stretches along nearly horizontal, in close approach to the upper asymptote, or limit, which belongs to the particular cultural epoch and area involved. All these factors must certainly be taken account of in a mathe- matical theory of population growth. For convenience they may be recapitulated in brief as follows: 1. Finite limit of area. 2. Upper limiting asymptote of population. 3. Lower limiting asymptote of population at 0. 4. Epochal or cyclical character of growth, successive cycles being additive. 5. General shape of curve of growth. With these fundamental postulates in mind we may now proceed to their mathematical expression and generalization. In our first paper^ we took as a first approximation to the law expressing normal population growth, equation (I). b . y e -ax -\-c 1 Pearl, R., and Reed, L. J. Loc. cit. 12 (I) ^ < l ( This satisfied perfectly postulates 1, 2, 3, and in a fair degree 5. It made no attempt to satisfy 4. Since our first paper was pub- lished we have learned that nearly three-quarters of a century ago a Belgian mathematician, Verhulst,^ in two long since forgotten papers, which appear never to have been generally recognized in the later literature on population, anticipated us in the use of equa- tion (I) to represent population growth. The only recent writer on the subject who seems to have known of Verhulst's work is DuPasquier,* who himself makes use of a slight and, as it seems to us, entirely unjustified and in practice usually incorrect modi- fication of (I). There have, of course, been many attempts at getting mathematical expressions of population growth, or of growth in general. We shall make no attempt to review all this literature, chiefly for the reason that most of the mathematical expressions brought forward have been wholly lacking in gener- ality. They have been special curves, doctored up with greater or less skill to fit a particular set of observations, often involving assumptions which could not possibly hold in any general law of growth. A recent paper by Lehfeldt* develops an idea as to the changes of a variable in time which, fundamentally, seems to be similar to that set forth in the present paper. He says : "Let q be the quantity whose changes in time t are to be studied. It is not to be expected that the changes of q itself should be symmetrical in time, for all the changes observed in the later half of the period of change refer to values of q larger — possibly many times larger — than in the earlier half. But log q may very possibly undergo symmetrical changes, so we will assume that it is a 'normal error function' of the time, i. e., log q = log qo + kF (|) where q© is the value of q at a certain moment (the 'epoch'): t is the time in years before or after the epoch : T is a constant period and F (x) = —7= 1 e dx and k is a constant." VirJo * Verhulst, P. F. : Recherches Mathematiques sur la Loi d' Accroissement de la Population. Mem. de I'Acad. Roy. de Bruxelles. T. XVIII, pp. 1-58, 1844. Idem. Deuxieme Memoire sur la Loi d 'Accroissement de la Population. Ibid. T. XX, pp. 1-52, 1846. * DuPasquier, L. G.: Esquisse d'une Nouvelle Theorie de la Population. Vierteljahrschr. der Naturforsch. Ges., Zurich, Jahrg. 63, pp. 236-249, 1918. ' Lehfeldt, R. A.: The Normal Law of Progress. Journal Royal Statistical Society, Vol. LXXIX, pp. 329-332, 1916. 13 I 't !'• II It seems to us that, mathematically, this method of approaching the problem is much less general, and much more difficult of application and of interpretation than our treatment of the prob- lem which follows. Considered generally, the curve y e -ax +c may be written k (II) y = where 1 + me ka' X 1 k = -; m = -; and ka' = —a. c c Now the rate of change of y with respect to x is given by ^ = -a'y (k-y) or dy dx y (k-y) = - a' (III) If y be the variable changing with time x (in our case popula- tion), equation (III) amounts to the assumption that the time rate of change of y varies directly as y and as (k— y), the constant k being the upper limit of growth, or in other words the value of the growing variable y at infinite time. Now since the rate of growth of y is dependent upon factors that vary with time we may generalize (III) by using f (x) in place of —a', f (x) being some as yet undefined function of time. Then whence and where dy y (k-y) = f (x) dx. y = h^Zl = e -k/f (x)dx my k k 1 + me - k/f (x) dx 1 4- me F (x) F(x) = -kjf (x)dx (IV) Then the assumption that the rate of growth of the dependent variable varies as (a) that variable, (b) a constant minus that variable, and (c) an arbitrary function of time, leads to equation 14 t i, r (IV), which is of the same form as (I), except that ax has been replaced by F (x). If now we assume that f (x) may be repre- sented by a Taylor series, we have k y = If 1 + me aix + ajx^ -f aaJ^ -h anx" a» = as = a4 = an = (V) then (V) becomes the same as (I). If m becomes negative the curve becomes discontinuous at finite time. Since this cannot occur in the case of the growth of the organism or of populations nor, indeed, so far as we are able to see, for any phenomenal changes with time, we shall restrict our further consideration of the equation to positive values only of m. Also since negative values of k would give negative values of y, which in the case of population or individual growth are unthink- able, we shall limit k to positive values. With these limitations as to values of m and k we have the following general facts as to the form of (V). y can never be negative (i. e., less than zero), nor greater than k. Thus the complete curve always falls between the x axis and a line parallel to it at a distance k above it. Further we have the following relations : If F (x) = 00 y === F (x) = - 00 y = k F (x) = - y = T-r— ^ ^ "^ 1 + m k F (x) = + y = 1 + m from below from above dy The maximum and minimum points of (V) occur where j~ = 0« But 5^ = y(k-y) F(x); therefore we have maximum and minimum points wherever F' (x) = 0. dx The fact that -r- = when either y = Oory — k = shows dy that the curve passes oflF to infinity asymptotic to the lines y = and y = k. 15 I I6 S5 O o (^ O > S o H u H Ptf O fa > U o r CO I' i| •o " o O 3W ^ Wo 0«fi 3 O 0;a rt Si « ega •"" Ml ^^3^? D o V 0) s jcj: 9 •s a> o S'o a y > a 3 •3 3 « 0) J3 — >: The points of inflection of (V) are determined by the inter- sections of (V) with the curve y = o- k k F'^Cx) 2 2 F'(x)2 (VI) Since we are seldom justified in using over five arbitrary con- stants in any practical problem, we may limit equation (V) still further by stopping at the third power of x. This gives the equa- tion y = 7-1 ^ — nr^ (VH) ^ 1 + me aix + ajx* + ajx* ^ ' If an is positive, the curve of equation (V) is reversed and be- comes asymptotic to the line A B at x = — 00 and to the x axis at X = + 00 . Thus in equation (VII) as negative is a case of growth, and as positive is a case of decay. Equation (VII) has several special forms that are of interest, among them being a form similar in shape to the autocatalytic curve (i. e., with no maximum or minimum points and only one point of inflection), except that it is free from the two restrictive features mentioned in our first paper, that is, location of the point of inflection in the middle and symmetry of the two limbs of the curve. Asymmetrical or skew curves of this sort can only arise when equation (V) has no real roots. While any odd value of n may yield this form of curve the simplest equation that will do it is that in which n = 3, so that the equation of this curve be- comes that of (VII). Having determined that the growth within any one epoch or cycle may be approximately represented by equation (I), or more accurately by (VII) the next question is that of treating several epochs or cycles. Theoretically some form of (V) may be found by sufficient labor in the adjustment of constants so that one equation, with say 5 or 7 constants, would describe a long history of growth involving several cycles. Practically, however, we have found it easier and just as satisfactory in other respects to treat each cycle by itself. Since the cycles of any case of growth are additive we may use for any single cycle the equation k y = d + or more generally y = d + 1 + me ka'x k (VIII) 1 + me aix + ajx* + aix* 17 MILLIONS 1850 MILLIONS 1850 MILLIONS 5 1850 20 15 10 ^^^ 5 ^^^.^..r-^ AREA I r— *" ' .*v. 1900 1950 2000 Z050 ZIOO 1900 1950 2000 2050 2100 2050 2100 Diagram 2. Population Curves for Areas I, II, and III Small circles indicate observed iK>pulations, by which the curves are deter- mined. Horizontal lines mark the upper asymptotes or limits of the respec- tive curves. I8 ^ > ^ I where, in both forms, d represents the total growth attained in all the previous cycles. The term d is, therefore, the lower asymp- tote of the cycle of growth under consideration and d + k is its upper asymptote. In treating any two adjacent cycles it should be noted that the lower asymptote of the second cycle is frequently below the upper asymptote of the first cycle due to the fact that the second cycle is often started before the first one has had time to reach its natural level. This, for instance, would be the case where a population entered upon an industrial era before the country had reached the limit of population possible under purely agricultural conditions. The mathematical theory herein developed has been applied by the authors to the populations of all the large countries of the world and to several of the large cities. In all of these cases the agreement between the observed values and those of the equation is of the same order as that usually found in the application of a mathematical law to measurements in the field of science. PREDICTION OF TOTAL POPULATION WITHIN THE AREA AND IN CERTAIN OF ITS SUBDIVISIONS The area under consideration is shown in the map preceding this report and is comprised of the following counties: New York: New York, Kings, Queens, Bronx, Richmond, Nassau, Suffolk, Westchester, Putnam, Dutchess (part), Rock- land, Orange (part). New Jersey: Hudson, Bergen, Essex, Union, Passaic, Morris, Somerset, Middlesex, Monmouth (part). Connecticut: Fairfield (part). This total area is subdivided as indicated on the map into three areas defined as follows : Area I. New York City, Hudson County, Newark. Area II. Nassau County, Westchester County, part of Fair- field County, Essex County (excluding Newark), part of Bergen County, part of Passaic County, Union County. Area III. The remainder of the total area. 19 20 c .1 CO <2 < 8. g o CO 0) 5 3 o u Bp » 04 O to en U z o h s a 0) >« .C CO 2 ;5 ^-^ Irt 0) J -" 4) O jj.a ^ 0*2 E g CO (O »» •v A > 1 The following table shows the population in thousands for each of the areas, as compiled from the United States census reports: Table 1.— Recorded Population of the New York Region (in Thousands) Year Area I Area II Area III Total area 1790 225 1800 290 1810 359 1820 428 1830 560 1840 764 1850 758 135 270 1.163 1860 1,311 210 314 1.835 1870 1,714 280 381 2,375 1880 2,237 349 440 3,026 1890 2,966 477 523 3,966 1900 4,074 683 628 5,385 1910 5,651 1,029 787 7.467 1920 6,664 1.383 932 8,979 The method used in predicting the future populations of these areas is that described in the preceding section. The actual pro- cedure was to fit the equation y = d H ^^ , to the popu- e ~ ^x -|-c lation counts of the four areas. In the case of the total area and of Area I the term d was found to be so small that it could be neglected. Thus the actual curve used in these areas was y = ; — . The four curves adjusted by the method of •^ e - ax -l-c least squares to the populations in question lead to the following equations, in which y represents population in thousands, and x represents time in years since 1800. 256.0527 Total area Area I Area II y e - .032300X -f .0073368 156.2838 y -" e - .0349348X + .0078578 8.77755 y = 87 + — : .0428817X -I- .0008207 30.3083 Area III y - 150 + ^ _ .029018lx + .0074311 Since the accuracy of any mathematical prediction depends in part on the degree to which the curve satisfies the original observa- 21 . ~ '*> » » H prt < 2 o e H O 5^0 n O en g U en o 1-4 Pi < O o put IS o u o a* Pi H O - H P< O to in o M to > u o H £> O fit o to O J ! u 1 A) 22 tions, it is of interest to compare the observed values of the popu- lation with the values obtained from the equations. Table 7, in the series of tables at the end of the report, shows the computed values for the populations of the different areas at ten-year inter- vals from 1790 to 2100 with the observed values for the years through 1920. The relationship of the observed population to the curves may also be seen in Diagrams 1 and 2. Examination of the table and of these diagrams shows that the curves do rep- resent the growth of the population throughout the observed period with a high degree of accuracy. As a further check on the precision of the curves the root mean squared deviation of the observed values from the curves was computed, and in no case was the root mean squared deviation over 4 per cent of the aver- age population. Diagrams 3 and 4 are presented in order that the growth of the populations of the different areas may be compared. Diagram 3 is on arithmetic scale and may be used to compare the propor- tion of the population in the different areas at any particular instant of time. Diagram 4 is on logarithmic scale and shows by the slopes of the curves the varying rates of growth of the popu- lations of the different areas. Comparison of these slopes shows that, whereas up to 1880 Area I had the most rapidly growing population, after that time Area II increased at the most rapid rate. The rate for Area II continues to be most rapid until about the year 2000, when the rate for Area III exceeds it. The percentage distributions of the total population by areas are given in Table 8 at the end of the report, and are plotted in Diagram 5. Area I, through its rapid growth in population from 1850 to 1900, reached a position where it contained three-fourths of the population of the entire area. Since 1900, however, the percentage of the total population in Area I has been declining, while that in Area II has been rising. This movement continues in the predicted population until about the year 2050, when these two areas stabilize. Area I containing about 57 per cent and Area 11,31 per cent of the total population. Area III has lost ground in comparison with the other areas throughout the observed por- tion of the curve. Starting at a value of 21 per cent in 1850, the percentage has decreased steadily since that time to a little more than 10 in 1920. It continues to decrease in the predicted por- tion of the curve to a low point of 9.53 in 1960, after which it rises slightly and stabilizes at about 12 per cent. 23 I PER CCNT 100 FCRSONS PCR SQUARE MILE 100,000 50^000 10,000 5.000 „ t /■ 1850 1900 1950 2000 2050 2100 Diagram 5. Percentage Curves Showing Change in the Propor- tion OF THE Total Population Falling in Each of the Three Subdi- visions OF THE Region POPULATION DENSITIES From the predicted population figures we may obtain values for the predicted densities of population. Table 9, following the report, shows these densities for the three sub-areas and for the total area. It should be noted that, although there is a slight movement toward an equality of density throughout the area, this movement falls far short of realization. The plot of these density figures on logarithmic paper in Diagram 6 shows that although the rate of increase of density in Area II is much greater than that in Area I, in absolute value the density of Area II is always much less than that of Area I. 24 i 1,000 500 too 1850 1900 1950 2000 2050 2100 Diagram 6. Population Density Curves Obtained from the Predicted Populations for the Total and Component Areas As in the previous diagrams, the prominent horizontal lines indicate the upper asymptotes of the respective curves, and here mark the predicted limits of population density for each area. 25 o t u z u < a o Ul £0 OS "Jo ^o •5 o o »4 •T3 ^ I C u c o •M o (0 o 1-4 - e a> o .ti o u C rt O *** 000^ «0 C'Z! CO "-^ r« —- H so a o M Ul 1 — ^ c ; h f ■ ■ I 1 I o •H CO CO ""J ^^ 45 c (y a> v< , fi ra ^>7 26 27 Applying the methods used for the other areas we determine the following population curve for New York City: 138.017 MILLIONS 16 y = e-.034935x +.008971 In this curve as in the others x represents time in years measured from the year 1800 as an origin, and y represents population in thousands. This curve is plotted in Diagram 9, together with the values of the observed population. The accuracy with which the curve represents the observed values is obviously on a par with that of the previous cases. The values of the predicted popu- lation at ten-year intervals are given in the last column of Table 10. The proportion of the predicted population of Area I that falls within New York City is shown in the following table: Table 3.— Predicted Population of Area I and of Present New York City Year Predicted population (in thousands) Per cent of population of Area I Area I New York City in New York City 1850 1900 1950 2000 2050 2100 857 4,086 11,878 17,797 19,489 19,818 753 3,506 9,672 13,948 15,105 15.337 87.86 85.81 81.43 78.37 77.51 77.39 Asymptote 19,889 15,385 77.35 The steady decline of these percentages shows the tendency toward a uniform density of population within Area I, which fact is in keeping with the findings of the previous section with regard to density throughout the entire area. TRENDS OF CERTAIN ELEMENTS OF THE POPULATION OF NEW YORK CITY In considering the changes that take place in the size of a single element of a population, two procedures are possible. We may consider the element of the total population as a distinct universe and fit to it the same type of equation used for the total popula- 28 V '( M 1890 1900 1950 2000 2050 2100 Diagram 9. Population Curve for New York City The area included is that of the city as at present constituted. Small circles indicate observed population, and a horizontal line marks the asymptote of the curve and the predicted limit of population. tion, or we may form the ratios of the particular element to the total population ak different instants of time and then determine the trend of these ratios. The latter procedure will in general be the better of the two. These ratios for certain elements of the population remain so nearly constant that there is no evidence of a trend during the period for which we have observations. In these cases the average ratio over the known period should be obtained and applied to the predicted values of the total popula- tion. In case there is evidence of a trend in the ratios, the ques- tion arises as to what type of curve is to bie used to represent this trend. A consideration of the general forms used to represent population growth shows that the ratios of two such curves may be well represented by the equation, y = d + _ ^^ , Thus, in an actual example, we may fit the above equation to the known ratios, compute ratios at future times, and by applying these to the predicted values of the total population obtain the size of the population element. We shall now consider certain elements of population and shall determine their probable future 29 I values. It should be noted that the probable errors of the pre- dicted values are greater in the case of the elements of the popu- lation than in the case of the population as a whole. DISTRIBUTION OF POPULATION BY AGE GROUPS The following table shows the population of New York City for certain age groups, stated in actual numbers and also as per- centages of the total population. Table 4.— Recorded Population of New York City by Age Groups Age group Years Population Per cent of total population 1890 1900 1910 1920 1890 1900 1910 1920 0- 4 5- 9 10-14 15-19 20-44 45 and over 164,686 140,026 130,651 148,843 686,825 240,652 397,287 354,747 301,264 302,751 1,530,239 545,478 507,080 438,263 422,431 457,616 2,145,583 789,108 560,869 536,490 494,867 453,758 2,488,415 1,077,844 10.894 9.263 8.643 9.846 45.434 15.920 11.577 10.337 8.779 8.822 44.590 15.895 10.653 9.207 8.874 9.614 45.074 16.578 9.994 9.559 8.818 8.085 44.339 19.205 Total 1,511,683 3,431,766 4,760,081 5,612,243 100.000 100.000 100.000 100.000 Examination of the table shows no appreciable trend in the per- centages of the different age groups. This being the case we may take the average of these percentages as representing the propor- tions of the total population to be found in the different age groups. These average percentages are : Age group Per cent of Years total population 0- 4 10.779 5-9 9.592 10-14 8.778 15-19 9.092 20-44 44.859 45 and over 16.900 Applying these percentages to the values of the predicted popu- lation of New York City we find probable values for the popula- tion within each age group. These values are given in Table 10 at ten-year intervals from 1920 to 2100. Other elements of population for which the statistics available 30 show no evidence of trend may be treated by the method used for age distributions. NEGRO POPULATION OF NEW YORK CITY Many racial groups within the total population have rates of growth that are greater than that of the population taken as a whole. The Negro element within New York City is an illustra- tion of this fact. The size of the Negro population and its per- centage of the total is shown in the following table: Table 5.-Recorded Total and Negro Population of New York City Year 1860 1870 1880 1890 1900 1910 1920 Total population 813,669 942,292 1,206,299 1,515,301 3,437,202 4,766,883 5,620,048 Negro population 12,574 13,072 19,663 23,601 60,666 91,709 152,467 Per cent of total 1.55 1.39 1.63 1.56 1.76 1.92 2.71 The figures for the first four years are based on the old area of New York City, since for these years the numbers of Negroes within the present city limits are not available. However, as the curve is to be based on percentages, this difference can have little or no effect. Applying to these percentages the equation y = d + e-ax + c we obtain the following expression for the percentage of Negroes in the total population .0003567 y = 1.409 + g _ .07094X + .00009641 In this equation y represents the percentage of the total popu- lation that is Negro, and x represents the time measured in years from 1800 as an origin. From this equation we derive the values of the future Negro population given in Table 11 and plotted in Diagram 10. It should be noted that the observed values for the years 1860, 1870, 1880, and 1890 have been adjusted to the area within the present city limits. The equation shows that the 31 THOUSANDS 800 «00 200 i 400 — 1850 1900 1950 2000 2050 2100 Diagram 10. Prediction Curve for Negro Population of New York City Small circles indicate observed population, and a horizontal line marks the asymptote of the curve and the predicted limit of Negro population. Negro population is increasing toward a position where in the asymptotic conditions it will comprise 5 per cent of the total population. FOREIGN-BORN POPULATION OF NEW YORK CITY The foreign-born population of New York City is an illustra- tion of a population element whose rate of growth is less than that of the total population. The actual numbers of foreign born are increasing steadily, but, as we see from Table 6, the pro- Table 6.— Recorded Total and Foreign-Born Population of New York City Year Total Foreign-born Per cent population population of total 1870 942,292 419,094 44.48 1880 1.206,299 478,670 39.68 1890 1,515,301 639,943 42.23 1900 3,437,202 1,260,918 36.68 1910 4,766,883 1,927,703 40.44 1920 5,620,048 1,991,547 35.44 32 Y ( > portion of foreign born in the total population of the city is de- clining. As in the case of the Negroes, the percentages from 1870 to 1890, inclusive, are computed from the populations of the old city. While the movement of these percentages has not been as steady as in the case of the Negroes, there has evidently been within the past fifty years a decrease in the proportion of the population that is foreign born. The curve obtained from these percentages is 88.5143 y = 9.91 + g _ .0071962X + .982510 ']S This curve leads to the predicted values for the foreign-born popu- lation that are given in Table 1 1 and plotted in Diagram 1 1 . This curve brings out very clearly the fact that continuation of the present decrease in the percentage of foreign born in the popula- MILLIONS 4 1850 1900 1950 2000 2050 2100 2150 2200 2250 Diagram 11. Prediction Curve for Foreign-born Population of New York City The circles represent observed population, and the horizontal line which the curve approaches indicates the upper asymptote of the curve. Ip thiscase the rate of ^owth becomes negative after the year 2000, and the limit of popu- lation is predicted at about three-quarters of the present number of foreign born in the city. 33 tion will lead to a time when the actual numbers of foreign born will begin to decrease. This fact may seem strange in the light of the large increase in our foreign-born population at the present time, but it is quite in harmony with the present immigration policies of the country. The prediction furnished by the curve is that about eighty years from now New York City will contain its maximum num- bers of foreign born, after which time the foreign-born popula- tion will decrease until it reaches a limiting position at about a million and a half. In considering this forecast it should be real- ized that any prediction of the actual number of foreign-born population of the city must possess much less reliability than the earlier predictions of this report. On the one hand the observed values are too few to be fully satisfactory for the purpose of pre- diction and, as this element of the population has undergone less regular increase, the observed points show greater variation from the predicted curve than in the other cases. In the case of the foreign-born population it is evident that single incidental fac- tors have much greater effect upon the rate of growth than in any of the other cases considered. SUMMARY Perhaps the best way to appreciate the great increase in popu- lation that may be expected to take place in the area under dis- cussion is to take a cross-section view of the population as pre- dicted for some specific date, for example, the year 2000. At that time the population of the total area will have increased from its present value of 9,000,000 to about 29,000,000; that is, the area will contain more than three times its present popula- tion. This increase is not evenly distributed throughout the area, however, for we find Area I with 2.62 times its present popula- tion, while the present populations of Area II and Area III are multiplied by 6.24 and 3.23, respectively. It is therefore evident that Area II makes a far greater relative gain than either of the other two areas. In density. Area I will have reached by the year 2000 the value of 48,759 persons per square mile, which is about the present density of Manhattan and Bronx boroughs. Area II will contain 7,507 persons per square mile. Its density will be not quite half the present density of Area I. Area III will not be very densely 34 ,>- f populated by the year 2000, its density at that time being 764 persons per square mile, or about that of Area II in 1910. The distribution of population by age will be about the same in the year 2000 as at present, so that such groups as children of school age, persons of voting age, etc., will be increased over their present value by the same factors as those previously given for the general population. The Negro population will have tripled and will constitute 5 per cent of the total population, whereas at present they form but 2.6 per cent of the total. The foreign-born population will have increased from its present value of 2,080,000 persons to approximately 3,750,000, which will be about the peak for this element of the population. Expressed as a percentage of the total population, the foreign born in the year 2000 will be less than at present, the percentage at that time being 26.9 as against 36.3 at present. According to the prediction equations the population situation in the year 2000 will be very near that at which the population will tend to stabilize, the one outstanding exception being that the number of foreign born in the population will ultimately tend to stabilize at about 1,525,000, which is far below the number predicted for the year 2000. i 35 TABLES OF PREDICTED POPULATION FOR THE NEW YORK REGION .M^l Table 7.— Predicted and Observed Populations (in Thousands) of Total Area and Three Subordinate Areas * Area I ^ Area II Area III Total area Year Pre- Ob- Pre- Ob- Pre- Ob- Pre- Ob- dicted served dicted served dicted served dicted served 1790 • • • • • • 184 225 1800 254 290 1810 350 359 1820 482 428 1830 662 560 1840 908 764 1850 857 758 161 135 275 270 1.242 1,163 1860 1,195 1,311 201 210 316 314 1,692 1,835 1870 1,652 1,714 261 280 369 381 2,295 2,375 1880 2,265 2,237 351 349 437 440 3,092 3,026 1890 3,067 2,966 488 477 525 523 4,131 3,966 1900 4,086 4,074 690 683 636 628 5,460 5,385 1910 5,336 5,651 986 1,029 775 787 7,li7 7,467 1920 6,803 6,664 1,408 1,383 944 932 9,122 8,979 1930 8,441 1,990 1,146 , , 11,458 1940 10,166 2,754 1,380 14,066 1950 11,878 3,700 1,643 16,841 1960 13,479 4,785 1,924 19,647 1970 14,895 5,927 2,221 22,342 1980 16,086 7,025 2.514 24,806 1990 17,047 7.993 2,794 26,958 .. 2000 17,797 8,783 3,051 28,765 2010 18,366 9,390 3,278 30,232 2020 18,790 9,832 3.473 31,391 2030 19,102 10,144 3.636 32,287 2040 19,327 10,358 3.768 32,968 s 2050 19,489 10,502 3,874 33,480 2060 19,606 10,598 3,958 33,860 2070 19,688 10,661 4,022 34,141 2080 19,747 10,703 4.072 34,347 2090 19,789 10,730 4,111 34,498 2100 19,818 10,748 4.140 34,608 Asymptote 19,889 • • 10,782 ■ • 4,229 • • 34,900 « • Table 8.— Percentage Distribution of Predicted Population, by Areas Year Area I Area II Area III 1850 66.28, 12.45 21.27 1860 69.80 11.74 18.46 1870 72.39 11.44 16.17 1880 74.19 11.50 14.31 1890 75.17 11.96 12.87 1900 75.50 12.75 11.75 1910 75.19 13.89 10.92 1920 74.31 15.38 10.31 1930 72.91 17.19 9.90 1940 71.09 19.26 9.65 1950 68.97 21.49 9.54 1960 66.77 23.70 9.53 1970 64.64 25.72 9.64 1980 62.78 27.41 9.81 1990 61.24 28.72 10.04 2000 60.06 29.64 10.30 2010 59.18 30.26 10.56 2020 58.55 30.63 10.82 2030 58.09 30.85 11.06 2040 57.78 30.96 11.26 2050 57.55 31.01 11.44 2060 57.39 31.02 11.59 2070 57.28 31.02 11.70 2080 57.20 31.00 11.80 2090 57.14 30.99 11.87 2100 57.10 30.97 11.93 Asymptote 56.99 30.89 12.12 \i 36 37 >C^Q.GLy^ r rSVC, MUO(==>OnJ croofsiTV^ rJe->is/>e»Ji< w y Table 9.— Predicted Density of Population (Persons per Square Mile) OF Total Area and Three Subordinate Areas Year Area I Area II Area III Total area 1790 • • 33 1800 46 1810 • • 63 1820 « • 87 1830 • • 120 1840 • • 164 1850 2,348 138 69 225 1860 3,274 172 79 306 1870 4,526 223 92 415 1880 6,205 300 109 559 1890 8,403 417 131 747 1900 11,194 590 159 988 1910 14,619 843 194 1,287 1920 18,638 1,203 236 1,650 1930 23,126 1,701 287 2,073 1940 27,852 2,354 346 2,545 1950 32,542 3,162 411 3,046 1960 36,929 4,090 482 3,554 1970 40,808 5,066 556 4,042 1980 44,071 6,004 630 4,487 1990 46,704 6,832 700 4,877 2000 48,759 7,507 764 5,204 2010 50,318 8,026 821 5,469 2020 51,479 8,403 870 5,679 2030 52,334 8,670 911 5,841 2040 52,951 8,853 944 5,964 2050 53,395 8,976 970 6,056 2060 53,715 9,058 991 6,125 2070 53,940 9,112 1,007 6,176 2080 54,101 9,148 1,020 6,213 2090 54,216 9,171 1,030 6,241 2100 54,296 9,186 1,037 6,260 Asymptote 54,490 9,215 1,059 6,313 Table 10.— Predicted Age Distribution of Population of New York City (in Thousands) 'V Years of age Total Year 0-4 5-9 10-14 15-19 20-44 45 and over 1920 618 550 503 521 2,571 968 5,731 1930 758 675 617 639 3,155 1,188 7,032 1940 902 803 735 761 3,756 1,415 8,372 1950 1,043 928 849 879 4,339 1,634 9,672 1960 1,171 1,042 953 988 4,872 1,835 10,861 1970 1,282 1,141 1,044 1,081 5,335 2,009 11,892 1980 1,374 1,222 1,119 1,159 5,717 2,154 12,745 1990 1,447 1,288 1,178 1,220 6,022 2,269 13,424 2000 1,504 1,338 1,224 1,268 6,257 2,357 13,948 2010 1,546 1,376 1,259 1,304 6,435 2,424 14,344 2020 1,577 1,404 1,285 1,331 6,566 2,473 14,636 2030 1,601 1,424 1,304 1,350 6,661 2,509 14,849 2040 1,617 1,439 1,317 1,364 6,730 2,536 15,003 2050 1,628 1,449 1,326 1,373 6,776 2,553 15,105 2060 1,638 1,457 1,334 1,381 6,815 2,567 15,192 2070 1,644 1,463 1,338 1,386 6,841 2,577 15,249 2080 1,648 1,466 1,342 1,390 6,859 2,584 15,289 2090 1,651 1,469 1,345 1,393 6,871 2,588 15,317 2100 1,653 1,471 1,346 1,395 6,880 2,592 15,337 Asymptote 1,658 1,476 1,351 1,399 6,902 2,599 15,385 38 I 1^ 39 APPENDIX Table 11.— Predicted Negro and Foreign-Born Population OF New York City Total Negro Per cent Negro Foreign-born Per cent Year population population population foreign (in t lousands) (in thousands) (in thousands) born 1860 1,046 15 1.434 471 45.00 1870 1,443 21 1.459 627 43.47 1880 1,969 30 1.510 826 41.97 1890 2,650 43 1.609 1,073 40.50 1900 3,506 63 1.794 1,369 39.06 1910 4,539 96 2.117 1,709 37.66 1920 5,731 150 2.609 2,080 36.30 1930 7,032 228 3.236 2,460 34.98 1940 8,372 324 3.869 2,821 33.70 1950 9,672 423 4.373 3,140 32.46 1960 10,861 511 4.707 3,395 31.26 1970 11,892 583 4.899 3,581 30.11 1980 12,745 638 5.003 3,697 29.01 1990 13,424 679 5.056 3,752 27.95 2000 13,948 709 5.083 3,756 26.93 2010 14,344 731 5.096 3,724 25.96 2020 14,636 747 5.102 3,663 25.03 2030 14,849 758 5.106 3,586 24.15 2040 15,003 766 5.107 3,497 23.31 2050 15,105 772 5.108 3,400 22.51 2060 15,192 776 5.108 3,304 21.75 2070 15,249 779 5.109 3,207 21.03 2080 15,289 781 5.109 3,110 20.34 2090 15,317 783 5.109 3,017 19.70 2100 15,337 784 5.109 2,928 19.09 Asymptote 15,385 786 5.109 1,525 9.91 I COMPARISON OF POPULATION PREDICTIONS MADE BY NELSON P. LEWIS, OF THE COMMITTEE'S STAFF, WITH THOSE OF THIS STUDY IN connection with other investigations carried on by the Com- mittee, Nelson P. Lewis, Director of the Physical Survey, made a series of population predictions for the total area and for the three subdivisions of it which were later used by Pro- fessors Pearl and Reed in the present study. Mr. Lewis's esti- mates are shown in the diagram and table reproduced below. The method followed consisted in determining the rate of in- crease of population by decades from 1850 to 1920 and then, with this as a basis, estimating empirically the trend up to the year 2000. In doing this he recognized the fact that the population of the Region has been increasing by a series of de- MILLI0N5 50.0 ^ r i-. i 1 1650 1900 950 2000 40 Population Curves for the New York Region and the Three Com- ponent Areas, Showing Actual Increases from 1850 to 1920, and Esti- mated Future Growth up to the Year 2000 41 creasing percentages, and it was assumed that this would con- tinue. It will be observed, as suggested in the Introduction, that the predictions made in these ways for the whole area run very close together for the next fifty years. This is true to a considerable degree also for the smaller divisions of the Region, as will be seen by further comparison of the diagram shown here and the one presented on page 22. The difference between the figures is in no case more than 4 per cent until 1970. From 1970 upward the divergence becomes greater, the figures of Mr. Lewis being 25 per cent higher than those of Professors Pearl and Reed when the year 2000 is reached. Comparison of Population Predictions for the Total New York Region Predicted Predicted Per cent of Year by Professors Pearl and Reed by Mr. Lewis difference 1930 11,500,000 11,000,000 - 4 1940 14,100,000 13,800,000 - 2 1950 16,800,000 16,600,000 - 1 1960 19,600,000 20,000,000 -f- 2 1970 22,300,000 24,000,000 + 8 1980 24,800,000 28,000,000 4- 13 1990 27,000,000 32,000,000 -h 19 2000 28,800,000 36,000,000 + 25 T 42 /■ Date Due iEB^0.950_V MA«6'1«Sfl ifl^^^ .% ^ hs^ e?r^7;2. FEB 1 :, 1^95 il «l :i^ stP c^'^ END OF TITLE