EconomicCycles Henry Ludwell Moore

8/11/2019 EconomicCycles Henry Ludwell Moore

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E C O N O M IC C Y C L ES T H E I R

LAW AND CAUSE

By H e n r y L u d w e l l M o o r e

F o r e c a s t i n g The Yie ld & The Price f C o t t o n [ 9 7]

G e n e r a t i n g E c o n o m i c C y c l e s [1923]

L a w s Of Wa g e s [1911]

S y n t h e t i c E c o n o m i c s [I9291HENRY LUDWELL MOOR

PROFE880R OF POLITICAL ECONOMY IN COLUMBIA UNIVERSITYAUTHOR OF LAWS O F WAGES

Nous croyons en eflet, pour notre part quepour avancer vraiment dans la connaissanceCconomique, il faut s'attaquer dirrctement etd'abor d, des variat ions, c'est-8-dire la formedynam ique des phEnom&nes, par la voie ex-p6rimentale. FRAN~OIS IMIAND.


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New Pork: The acmillan Company 9 4

TO

J 4NE MOOR

A C RITIC WHO NEVEII D1SHEAHTI:NS

A CO-WORKEII W H O KECI S THE ITH


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CONTENTS

CHAPTER I

Introduction

CHAPTER I1

CYCLES O F RAINFALL

The Use of Fourier s TheoremPeriodogram of Rainfa ll

The Equation to the Rainfall CurveRainfall in th e Corn Belt

CHAPTER I11

RAINFALL N D THE CROPS

The Secular Trend in th e Yield of t he CropsCritica l Periods of GrowthCycles in the Yield of the Representative Crops and the

Corresponding Cycles of RainfallCycles in th e Index of C rop Fluctuations and in th e Corre-

sponding Index of Mean Effective Rainfall

CHAPTER IV

THE I A\V OF DEM ND

The Theory of DemandStatistic al Laws of De mandThe Prediction of PricesElas tici ty of Demand

v


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ontents

THE R l E C H N I S M O F C Y C L E S

Th e Prices of Agricultural Commodities Correla ted with the

Yield of the Severa l Crops 94Rising and Falling Prices as Related to Yield-Price Curves 100Tlie Volumc of the Crops arid th e Activity of Ind ust ry 103A New Type of Demand Curves 10The Fundamenta l Persistent Cause of Economic Cycles 116

CHAPTER VI

E C O N O M I C C Y C L E S THEIR L WND CAUSE


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CHAPTER I

INTRODU TION

THERE s a considerable unanimity of opinion amongexperts tha t from the purely economic point of viewthe most general and characteristic phenomenon of a

changing society is the ebb and flow of economic life

the alternation of energetic buoyant a ctivit y with aspiritless depressed and uncertain drifting. ~ u r i n g he

creative period of the rhy thmic change each factor in

production receives an augmenting income and th e

mutual adjustment of interests in the productiveprocess is brought abou t in a natura l way primarilythrough the operation of competitive law. The period

of decline in the cycle presents a sharply contrasted

aspect of industry. With the organization of capit aland labor a t first unchanged the amount of th e product

falls; each of the interested factors seeks a t least toretain it s absolute share of the pro duct; friction and

strife ensue with a threatening of the d isruption of

industry. Wha t is the cause of this alterna tion of

periods of activity and depression? Wha t is its law?These are the fundamental problems of economic

the solution of which is offered in this Essay.

Political Economy began to make progress in a

rational way when the Physiocrats put forth their

of the dependence of all forms of economic life


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conomic Cyctes Their aw and ause Introduction

upon agriculture. Another momentous step was tak en

in the direction of theoretical development when the

English economists formu lated t he law of diminishing

returns in agriculture and traced its all-pervasive

influence in the production an d distribution of t he

product of indust ry. Th e desidera tum of economic

dynamics at t he present tim e is th e discovery of a law

that shall be to a changing society what the law of

diminishing returns in agriculture is to a society in a

comparatively static condition.

The full truth in the old Physiocratic doctrine has

not been exploited. Th e Dep art men t of Agriculture

of t he United S tates reaffirms the cent ral idea of t he

doctrine in its motto: Agriculture is the Foundation

of Man ufact ure and Commerce, and in the spirit of

this motto it publishes invaluable statistical data.

I t is proverbial tha t th e farmer is a t the mercy of t he

weather. If it be tr ue th at t he explanation of economiccycles is to be found in the law of su pply of agricultu ral

products, it is surely wise in a stu dy of rhyth mic eco-

nomic changes to inquire whether the law of the chang-

ing supply of raw mate rial is not associated with a law of

changing weather. Is ther e 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 t he crops are affected by the weather a nd how th e

cycles of t he we ather ar e reproduced in cycles of th e

yield of the principal crops.

men he changes in the physical yield of the cropsare 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

another unsolved problem i n theoretica l 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.

I t will be necessary to overcome th e difficulties of this

problem before a solution ca n be offered of th e more

funda menta l inquiry as t o th e law and cause of cycles in

economic phenomena.

When the physical yield of the crops has, on t he one

hand, been related to th e cycles of t he weather an d, 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 th e physical yield of the crops produce the

cycles in the activ ity of indu stry and the cycles of

general prices, and how, final ly, the law of th e cycles of

the crops is the law of Economic Cycles.


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Cycles of Rainfall

CHAPTERI

CYCLES O F RAINFALL

The first thing that in my opinion ought to be done towardsmaking the observations useful for scientific purposes is to performtha t kind of m ore perfect averaging which is afforded by the har-monic analysis. Ther e is a certain am oun t of averaging done, butthat is chiefly daily averages, with monthly averages, and yearlyaverages; but th e more perfect averaging of the harm onic analysiswould give the level of th e variation of th e phenomenon.

-I. oRD K E LV I N n his testimony before the Meteorological Com-mittee of the Royal Society 1876

FROM he point of view of the relation of changingweather to the varying fruitfulness of agriculture themost important factors that are usually included inthe term weather a re temperature and rainfall. Webegin our investigation with this common belief andinquire in this chapter whether the varying amountof annual rainfall is subject to a ny simple law.

In order to carry forward the inquiry as to the exist-ence of a law of annual rainfall an analysis must bemade of a long record of precipitat ion. Our choice ofa record is limited by two conditions: Firs t our objectin investigating the periodicity of rainfall is the hopeof throwing light upon the periodicity in th e yield ofthe crops and this expectation obviously makes itdesirable that the record of rainfall shall be as repre-sentat ive as possible of the conditions of precipit ation

in our leading crop area; secondly as the existingmeteorological records are of unequal lengths and of

reliability it is necessary to tak e the best long

records t hat can be found within th e limits of the croparea.

The principal region of grain production in t he UnitedStates is in the Mississippi Valley but the meteoro-logical records of t he Middle West do not extend througha long period of time. I n order to achieve the two endsof having a long record of precipitation and of havingthe record typical of the conditions in the grain areathe device has been adopte d of investigating rainfall in

the Ohio Valley-which affords the longest record ob-tainable in t he neighborhood of the centra l Mississippiregion-and of showing th at the rainfall of our lead-ing grain state Illinois follows the same law as th erainfa ll of the Ohio Valley.

The stations in the Ohio Valley with long rainfallrecords are Marietta Portsmouth and Cincinnati.Their mean annual rainfall since 839 is given inTable I of the Appendix to this chapter. The graph

of the course of rainfall in t he Ohio Valley since839

is traced with other graphs on Figures 4 5 and 6. Theproblem that must now be faced is the question as towhether the sequence of annual rainfall in the OhioValley follows a simple law and if so to give a quanti-tative formulation of the law.

The data were taken fromBul le t in W of the Weather B ureau ofthe United States and from theAnnu al Repor ts of the Chief of theWeather Bureau.


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Economic Cycles Their Law nd Cause Cycles of ainfall

The Use of Fourier's Theorem

A preliminary examination of the rainfa ll da ta of theOhio Valley leads to the conclusion that there is prob-ably no secular trend to the data, that is to say, thereis probably no tendency of t he rainfall to increase con-

tinuously or to decrease continuously with the flow oftime. I t is true th at when the amount of rainfall is

correlated wi th time, the coefficient of correlation isr 227f .075 where the coefficient is three times itsprobable error and is therefore suggestive of a decreasein the amoun t of rainfall with the flow of time. More-over, if a str aight line is fitted to the dat a, the indicatedannual decrease in the rainfall is seven hundredths ofan inch. Bu t these facts are no justification for hold-ing to a secular decrease in the amount of a nnua lrainfall. For, in the first place, if the re are cycles inthe amount of th e rainfall, t he low degree of t he ob-served correlation might be due to t he da ta of rainfallincluding incomplete cycles; in the second place, therecord is drawn from only three stations and becauseof the limited number of stations might give an acciden-ta l, low degree of correlation between amount of rain-fall and time; and in the third place, improvementsin the method of taking the observations might haveintroduced changes that would account for the ob-served small annual decrease in the amount of rain-

fall. In view of these considerations, it is probablybest to proceed with our problem on the assumptionth at there is no secular trend in the amount of a nnual

rainfall. 1 this assumption is true, it follows that, in1 probability, th e course of rai nfall in the Ohio Valley,

is cyclical, or a combination of cycles.an induc tive treatment of an y form of rhy thmic

cyclical change it is necessary that the methodadopted shall satisfy two conditions: (1) I t shall beconsistent with recognized mathematical processes;2) I t shall afford means of testing the degree of proba-

bility that the results are not chance phenomena.Unless the method rests clearly upon an approvedmathematical process, it is scarcely possible to saywhether the attained results may not be entirely formal;and unless the findings are tested for the degree of theirprobability, there is no assurance that the adducedcycle may not be a chance occurrence. The litera ture

in which rhythmic phenomena are treated in a statis-tical way t,eems with fallacies and uncertainties th atillustrate the need of observing the above conditions;for the method frequently adopte d of smoothing thedata is so arbitrary th at one is at a loss to know whether,afte r all, the alleged periodicity may not, in fact , be dueto the process of smoothing; and, in addit ion, one isleft in doubt as to whether an indefinite number ofcycles other than the particular one adduced might not,with equal or greater probability, be obtained from thesame data.

The method that was employed to reach the results

of this chapter rests upon the analysis invented byJoseph Fourier,' which is called, in English trea tises ,

l h e most philosophic exposition of Fourier s theorem is in


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8 Economic Cycles: Their Law and Cause Cycles of Rainfa 9

harmonic analysis. The perfection of the method

whereby the findings may be subjected to the test ofprobability is the work of Professor Arthur Schuste r

of Manchester.We may begin the presentation of the method with a

definition of a series of terms t ha t constantly recur inthe trea tment of periodic phe-

nomena. Figure 1 will facili-ta te the exposition by affording

a graphic description of the

terms dealt with.

Suppose that the point Q

moves uniformly in the circleof Figure 1 tha t is to say, sup-

FIGURE .pose that the point Q describes

equal arcs in equal times and, therefore, proportional

arcs in different times. Then, if the measurements ofthe arcs of the circle are made from the point andthe reckoning of time is begun when Q is at E, theangle E is called the angle a t epoch, or simply

Fourier's own work: ThBorie analytique cle la chuleur. In Freeman'sEnglish translation the treatment is found on pp. 137-212.

The funda mental memoirs of Professor Schuster ar e"On t he Invest igation of Hidden Periodicities with Application

to a Supposed 26 Day Period of Meteorological Phenomena."Terrestl-iul Afagnetisrn for March, 1898.

"The Pcriodogram of Magnetic Declination as obtained from therecords of the Green vich Observatory during t he year s 1871-1895."Camlwidge Philosophical Society Transactions, Vol. 18, 1899.

On the Periodicity of Sunspots." Philosophical Transactions ofthe Royal Societu of London, A, Vol. 206, 1906.

"The Periodogram and its Optical Analogy.'' Proceedings ofthe Royal Society of Lorltlol~, A Val 77, 1906.

the epoch of the uniform circular motion. The

radius of the circle is the amplitud e of the motion ;the time of going once around the circle is the

period of the motion; the ratio of Q to theof t he circle is the phase of t he mo-

tion.1f from each position of Q a perpendicular is dropped

upon the diameter of the circle, G H, t he foot of t heperpendicuhr will describe a simple harmonic motion.

The amplitude of th e simple harmonic mot ion is one-

half of the range of th e motion, t ha t is, one-half of G H,or the radius of th e circle. Th e period of th e simple

harmonic motion is the interval between the passingof the point P twice through the same position in thesame direction. Th e distance of the point P from the

middle of its range, 0 is a simple harmonic functionof the time, y = a sin ntf e), where a is the radiusof the circle-or th e ampl itude of the simple harmonicmotion-e is the angle of epoch, and n is the angle de-scribed by the moving point Q in the unit of time. Theperiod of the simple harmonic motion is, in the above

rr nt e

case,.

Its phase is w

Figure presents a graph of simple harmonic mo-tion. As in Figure 1 the point Q moves uniformly inthe circle; the po int P pe rfo ms simple harmonic motionaccording to the formula y a sin nt+ e), where a isthe amplitude of the motion, or radius of the circle, e

is the angle of the epoch, namely, 0 E, and n is thearc described by Q in the unit of time. If time is meas-


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1 Economic Cycles: Their Law and Cause

ured upon the line B C, the sinuous curve of Figure 2is the graph of the function, y = a sin nt+e) .

The importance of simple harmonic functions in

the stu dy of periodic phenomena grows out of the fact

th at an y periodic curve however complex can be ex-

pressed mathematic ally by a series of simple harmonic

functions. By the help of Fourier's analysis a periodic

function may be put in th e form

1 ) = A , a, cos kt a, cos 2 kt a, cos kt .b sin kt b, sin 2 kt b3 sin 3 kt

If in (I ), we put,

a , A , s in e l ; a , A , s in e, ; a , A , s in e ,; c.,b1 A 1 cos el; b, A cos e,; b, A , cos e,; c.,

We get,2 ) A , + A 1 s in k t e l ) + A , si n 2 kt e,)

+ A 3 i n 3 kt e,) .

where y is expressed as a series of sines. In a similarmanner, equ ation (I ) may be expressed as a series of

cosines,The few exceptions to the general rule are discussed in t he

mathematical te xts tha t develop Fourier s theorem.

Cycles of Rainfall 11

th e use of Fourier 's theorem for the purpose of

periodic phenomena, we follow a processto the use of Taylor's theorem in the simpler

demonstrations of mathe matic al economics. By far

the greater part of Cournot's pioneer treatise and ofsubsequent work of his school is based upon the as-

sumption tha t, if th e economic function under investi-

gation is y x ) , then f x+h) may be expanded byTaylor's theorem, and t he first terms of the series may

be used as an approximation to the form off x ) . Simi-

larly, in our use of Fourier's series, the a tten tion will befocussed upon a few harmonics as a first approximationto the solution of the problem in hand of expressing

in mathematical for m the periodicity of a nnual rainfall.Assuming that any periodic function may be ex-

pressed as a Fourie r series, the problem is presented of

determining the values of the coefficients. Th e series,as we know, is of th e form

Y = f t ) = A , + a l c o s k t + a , c o s 2 k t +

b1 sin kt b, sin 2 ktWhat a re the values of the first term a nd of th e co-

efficients of the sines and cosines? In order to deducethe necessary values, we shall have need of the follow-ing lemma:

If m and n are two unequal integers and k is put equal?r

to T hen


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2 conomic Cycles Their Law and auseCycles of Rainfall 3

Taosm k t cos nkt dt 0,

sin m k t sin nkt dt = 0,

Tin rnkt cos nk t d t 0.The lemma may be proved to be true by evaluating thethree integrals according to th e usual methods. The

first integral, for example, becomes

lT osk t cos n t dt = cos m - n ) kt cos m + n ) k t dtsin m - n ) k t sin m n k t

- [ 2 m - n ) k 2 m + n ) k 12 rBut k -, and, cos m k t cos nk t d t = 0.T

With t he ai d of this lemma we may proceed to evalu-

at e th e coefficients in F ourier s series. If we integratethe series between th e limits o and T, we get,

But all of the terms except the first on the right-handside of the e quation will vanish, a nd consequently

Since j j ( t ) d t is the a rea of t he original curve for one

whole period T, the constant term in Fourier s series isequal to th e value of the mean ordinate of the original

curve.

To determine the value of all multiply throughoutby cos k t and integra te between limits o and T.

b,J;in kt cos kt dt .

Or rf t ) oos kt dt = a , l r c o s 2 kt dt , sinceJsin kt oos kt dt are bo th equ al to zero and a11 the other

terms on the right-hand side of the equation , according

to our lemma, disappear. But

sin 2 kt]:=cosZ t dl = T cos kt dt = t 2k

and a s a result, we havef f t ) COS kt dt

Therefore al is equal to twice the mean value of t heproduct f t)cos kt.

In a similar manner t he value of any other coefficient

may be determined. Tak e, for example, b . Multiplythroughout by sin nlct and integrate between o and T,

fi t ) sin nk t d t b,, dt =T-cos nkt

sin 2 nlctb { ~ [ t 2 nk o = n 5

j f in nkt Therefore bnd, consequently, b 2


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14 Economic Cycles: Their Law and Cause Cycles of Rainfall 5

is equal to twice the mean value of the productt ) sin nk t .Having found the algebraic values of the coefficients

in Fourier s series, we may now proceed t o deter minetheir sta tistic al equivalents in the case of annual rainfall.

Th e Periodogram of R ainfa ll

If the length of a cycle of rainfall were known before-hand , the preceding exposition of Fourie r s theorem

would suffice to determine, from the dat a of precipita -

tion, the amplitudes and phases of the harmonic con-

stit uent s of the Fourier series descriptive of t he rainf allcycle. Bu t 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 a t all. And there is no

shor t method of solving the problem.

Suppose, for example, it were assumed from a prioriconsiderations tha t t he amou nt of rainfall is affected

by sunspots, and, as sunspots are known to occur in

periods of a bout eleven years, suppose it should be in-ferred th at the a nnual r ainfall will likewise show a period

of eleven years . If the rainfall da ta of th e Ohio Valleyare examined f or an eleven years period, i t 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

warr ant f or holding tha t ther e is a true rainfall period of

eleven years. Every other grouping of t he seventy-twoyears record will likewise show a definite amplitude

and a definite phase. The questions that one is in-

terested to have answered are: (I) What is the law ofthe distribution of Fourier coefficients when the d ata

are analyzed for all possible periods; and (2) how maythe true cycles be separated from the accidental,

spurious cycles that are obtained when the data are

exhaustively analyzed?

I n Figure the results of a detailed, laborious ex-

amination of the d at a of annual ra infall in the OhioValley are presented in graphic form. On t he axis of

are measured, within assigned limits, the

possible lengths of cycles in the 72 years of rainfall.

By extending the calculations to 36 years, we obtainfor 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 years. On theaxis of o rdina tes are measured the squares of th e co-

efficients of t he first harmonic in th e Fourier series

corresponding to the lengths of periods recorded on

the axis of abscissas. Th e numerical values of thesesquares are given in the fourth and eighth columns of

Table I1 in the Appendix to this chapter. The methodof deriving the values may be illustrated by taking thecycle of 8 years. Suppose, as a first approximation,tha t th e equation to Fourier s series is put i n the alge-braic form

F t ) A , a , cos kt b, sin kt] A, + A, sin (kt e ) .

Then the corresponding arithmetical values derivedfrom the Ohio rainfall data ar e


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16 Economic Cycles Their Law and Cause Cgcles of Rainfall 7

r 2F t ) 41.19-3.13 cos 2.69 sin

8

41.19 4.13 sin

The values of the terms a: bi, A: are respectively3.1339) 2.6938) 4.1325) , and these values are

given in the proper columns of Table I1 in the Ap-pendix. I n Figure 3, the values of A 2 for the severalperiods are measured on the axis of ordinates.

An examination of Figure 3 will illustrate the truth ofa statement advanced a moment ago. It is clear fromthe course of th e periodograph th at if one were to

take a ny period at random between the limits of 3years and 36 years, he would in every case obtain afinite value for t he amplitude of the selected cycle; andif, by chance, selection should fall upon, say, 18, or 21, or29, or 36 years, a n argument might be made with somedegree of plausibility th at a real cycle had been dis-

covered, But, in trut h, the real significance of no onecycle taken a t random can be judged a par t from itsplace in th e distribution of all the cycles th at can be

derived from the data.Thi s last point is of fundamental importance. The

only object of invest igating cycles of rainfall or cycles ofeconomic phenomena is th at the knowledge of the

Th e terms periodograph and periodogram were coined y Pro-fessor Schuster.

The periodograph is the curve tracing the values of Az; theperiodogram is the surface between the periodograph and the baseline giving th e lengths of the periods. Schuster : Th e P eriod-ogram of Magnetic Declination, p. 108.


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18 Economic Cycles Their Law and Cause Cycles of Rainfall 19

cons tant recurrence of th e cycles may place one in a

position t o foresee an d utilize the dep endent pheno mena.Bu t the contro l of phenomena d ependen t upon a cycle

presupposes that the cycle is itself a real phenomenon

with a natural cause, and that consequently ,it ~ersistswith an increase in the number of observations. If,

however, an apparen t cycle of any length taken a t

random is obtained from the given data, one would

surely misspend his time if h e were to se t abo ut th e

search for its cause, and were to d erive conclusions based

upon t he hypothe sis of t he persistence of t he cause.

The cycles due to formal, accidental causes must be

discriminated from th e cycles with na tur al causes.

The sepa ration of t rue cycles from spurio us or

accidenta l cycles is facilitated by the periodogram of

observations. If, following Professor Schuster, we callthe squar e of th e amplitud e of an y given period the

intensity of the period, then it may be said tha t the

probab ility of the realit y 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 inten sity of the periodogram the

expectancy, the n the reality of a period is depend ent

upon th e ratio of its intensity to t he expectancy of th e

periodogram. For instanc e, if in case of a given perio dthe ratio of intensity to expectancy is, say, 3 to 1, thenin about one case in twenty we should expect to obtain

by chance a greater amplitude tha n t he amplitu de of

the particular period in question. If, on the other hand,See the preceding note.

the ratio were say, 7 to 1, a greater ratio would notoccur by chance once in a thousa nd times.'

With these facts in mind, let us again examine Fig-

ure 3. I t is clear t hat the principal periods needingatte nti on ar e those respectively of 8 , 29, 33, 36 years.

In case of the 8 year cycle there can be very lit tle

doubt a s to th e existence of a tr ue periodicity approx-

imating 8 years in length. Th e ratio of t he square of

its amplitude to th e mean square amplitude of the

periodogram is 6 71 to 1. We ma y accordingly acceptwith considerable confidence the existence of a nat ura l

period of rain fall in the Ohio Valley approximating

8 years in length.

Th e cycle of 33 years, inasmuch as th e ratio of th e

square of its amplitude to t he mean square amplitude

of th e periodogram is 3.27 to 1 is in all probability atrue cycle. The doubt t hat exists is due to the smallness

of the ratio and the few recurrences-only two 2

Schuster: The Periodogram of Magnetic Declination, pp.124125.

Those who deprecate th e use of such meager da ta should con-sider well the testim ony of Lord K elvin before the M eteorologicalCommittee of the Royal Society, 1876.

Question 1710. The sum which parliament will give for thispurpose being a limited sum, do you think t ha t it would be well toreduce the num ber of observations in order to have more money tospend upon the reduction of observations? hink at all eventsun t lone eleven years period, the sun spot period, is completed, it would ewrong to reduce the number of observation^.'^

Question 1735. Supposing that you had one of these analysescalculated for a period of 11 years, would each year's observationsand still more each period of 11 years observations, require to beintroduced into this analysis so tha t you would have an analysis of22 years, and an analysis of 33 years, and so on from time t o time,


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Economic Cycles: Their Law and CauseCycles of Rainfall

th at our data afford. A greater confidence in th e exist-ence of a real period of 33 years is given by th e fact tha t

Briickner claims to have found a tru e period of abo ut3 5 years in an examinati on of a v ast mass of rainfallmater ial all over the world. Accordingly the existence

in th e Ohio Valley of a real 33 years period of rain fall we

shall assume to be very probable.

The other two periods of 29 years and 36 years are

not easily disposed of. Bu t in the first place the ratios

of th e squares of the respective amplitudes t o the mea n

square amplitude of the periodogram are not such as to

justify the accep tance with an y degree of confidence of

th e existence of true cycles of29

years and36

years.In the second place they are both so close to th e period

of 33 years as to cause a doubt a s to whether the y may

not be spurious periods th at are likely to ap pear in the

neighborhood of a real p e r i ~ d . ~

Considering the short range of our da ta i t would not

be properly cautious to press the po int of t he existence

of an y definite real cycle. Bu t this much is certain:If there are true cycles in the dat a of t he 72 years of

rainfall in the Ohio Valley there is far greater prob-ability 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 whetheranythi ng wi th reference to Terrestrial Meteorology is done once for all.I think probably the work will never be done.

Edward Briickner: Klimaschwankungen seit 1700. Briickner'speriod fluctuates greatly in length and has a n average value of35years.

Schu ster: Th e Periodogram of Magne tic Declination, p. 130.

36 years. Moreover th e periods of 8 years and 3 3

years afford the most probable basis derivable from the

data upon which to reason both as t o the futu re course

of rainfall in the Ohio Valley and as t o the course of th e

phenomena dependent upon rainfall.Assuming then th at for the purpose in hand the 3 3

years and 8 years periods are the most probable and

valuable we tu rn to th e consideration of th e equation

to th e graph giving the course of rainfall in th e Ohio

Valley.

The Equation to the Rainfall Curve

It will be helpful to approach the algebraic descrip-tion of t he cyclical movement of rainfall in th e Ohio

Valley by observing how we obtai n an increasingly

accurate account of the actual rainfall by superposingthe constituent cycles. We shall use as an index of the

relative fit of t he several curves th e root-mean-square

deviation of the observations from each curve.

If as a preliminary step the raw da ta of th e courseof annual rainfall are examined it is found tha t t he

mean annual rainfall in the Ohio Valley is 41 19 inches

and th e root-mean-square deviation about the mean is

S 6 7 0 inches.If the long 33 years cycle is considered ry itself it

appears that the root-mean-square deviation about the3 3 years curve is S 6 39 inches. The graph of the3 3 years cycle is given in Figure 4 Its equation is


16/79

22 Economic Cycles Their Law and Cause Cycles of Rainfall 3

the origin being a t 1839 This curve traces in bold

outline th e general course of rai nfall. I t gives theground-swell of th e rainfall movement.

If the 8 years cycle is superposed upon th e 33 years

cycle th e root-mean-square deviation about t he curvebecomes =5 66 inches. The graph of the combinationof these two curves is traced in Figure 5 It s equation is

the origin being at 1839 A point of intere st with regardto the flow of the curve is th e rapidi ty with which i t

rises from the least minimum t o th e greatest maximum

and the slowness with which it then descends to thesubsequent least minimum.

If the 8 years cycle and its semiharmonic of 4 yearsare combined with the 33 years cycle and its semi-harmonic of 16 5 years th e root-mean-square deviation

about the compound curve becomes S =5 29 inches.The graph of th e curve is given in Figure 6. It s equa-tion is

yx41 19 2 88 sin

the origin being a t 1839 I n this closer approximationthe characteristic rapid rise to a general maximum an dslow fall to a general minimum is reproduced. Anothercharacteristic is the longer interval that the curve


17/79

4 Economic Cycles Their aw and Cause


18/79

6 Economic Cycles The ir Law and Cause Cycles of Rainf all 2

lingers at the minima and the short period during which

it flows in the neighborhood of the maxima.

Rainfall in the Corn elt

Thus fa r we have dealt with the law of rainfall onlyin the Ohio Valley. The object in taking the Ohio

data , rather tha n the da ta of a state more representa-

tive of the leading cereal area , was to make an investiga-tion of a longer meteorological record than is affordedby the dat a of the central Mississippi Valley. But our

purpose in dealing with met~eorological ecords a t all isto show the dependence of crops upon the cyclicalmovement of the elements of the weather. We must,

therefore, prove th at the cycles of rainfa ll which we haveI should like to m ake clear the methodI have followed in th e

derivation of the equations to th e curves. My object was to obtaina summ ary description of the general course of rainfall in order tha tI might discover, later on, whether the characteristic general fea-ture s of th e movement of rainfall are reproduced in th e changingyield per acre of the crops. As a first stepI tried to detect th e realcycles in rainfall and I believeI have shown that , if the 72 yearsrecord is sufficiently long to reveal the true cycles, then the mostprobable lengths of th e cycles are, in round num bers, 33 years and8 years respectively. With so short a range of da ta I regarded it a s

useless to atte mp t to calculate the lengths of the periods to a greaterdegree of precision. I next had t b derive the equations to the curvesshowing th e characteristic general course of rainfall, and it seemedto me that, for this purpose, the method describedin the text forevaluating th e coefficients in a Fou rier series might prop erly beused. If the 33 years cycle were taken as the fundam ental cycle,then the 8 years cycle would be approximately th e fourth harmonicin the series, and th e 4 years cycle would be th e eighth harmonic.

The arithmetical process for computing the coefficients is indi-cate d by Professor Schuster in Hidden Periodicities, pp. 13, 14 an d isbriefly described by Professor Perry in an article on Harm onicAnalysis in The Electrician, for Febr uary5, 1892.

discovered for the Ohio Valley are likewise the cyclesthat exist in the heart of the grain producing area.

Among the states of the Middle West, Illinois is

probably the most highly representative of Americancereal production. It produces the largest crop ofcorn, which is the leading American cereal, and itranks second in the production of oats. Most of theother cereals tha t a re produced in the upper MississippiValley are likewise cultivated with success in Illinois.Another fact that makes Illinois a desirable state for

our purpose is tha t i ts meteorological records are fairlylong and a re obtainable from so many stations a s to berepresentative of the weather conditions in the entire

state. This last fact is all-important if the stati sticsfor crop production of the whole stat e are to be con-sidered in relation to t he weather cycles of t he state.

In Table I11 of the Appendix to this chapter therecord of the annual rainfall in Illinois is given for aperiod of 41 years.2 The ideal direct method with

This statemen t was accurate when it was first written. Bu t in1912 Iowa gained by a narrow m argin t he first place among the cornproducing states.

The raw data were taken from Bulletin W of the W eather Bu-reau of the United S tates and from the A nnual Reports of t he Chiefof the Weather Bureau. The stations used in computing theme nann ual rainfall were:-In Nor thern Illinois: Aurora, Cambridge,Chicago, Tiskilwa, Galva, Kishwaukee, Ottawa, Winnebago, andHenry . In Cent ral Illinois: Charleston, Carlinville, Coatsburg ,Decatur, Griggsville, Knoxville, Havana, LaHarpe, Pana, Peoria,and Springfield. In Sou thern Illinois: Cairo, Cobden, Carlyle,Golconda, Flora, Greenville, McLeansboro, Mascoutah, MtCam el, and Palest ine.

All of these stations do not present full records for the41 years,


19/79

8 Economic Cycles Their Law and Cause Cycleso Rainfall 9

reference to these data would be to compute theperiodogram in th e same manner in which it was com-puted in the case of the Ohio Valley dat a and then com-pare the periodograms. Bu t this method has not beenfollowed. A less direct and far less laborious processhas been adopted. We know from th e Ohio da ta tha tthere a re two cycles of rainfall a years cycle and a nyears cycle and we know furthermore th at when thecurve for rainfall in t he Ohio Valley is computed for th e

years and years periods and their semiharmonics a

good fit to the dat a is obtained. The questions that areasked with reference to the Illinois data are these:

f

we assume the existence ofa

years period and anyears period i n th e Illinois rainfall data will therainfall curve fit the Illinois data as well as the Ohiocurve fits the Ohio dat a? Will the Illinois curve re-produce the characteristi c features of t he Ohio curve?A presumption in favor of a n affirmative answer tothese questions is suggested by the f act t ha t t he correla-tion between th e annua l rainfall in t he Ohio Valley a ndthe annual rainfall in the state of Illinois is r =6.00.

Th e graph of the curve of ra infall in Illinois is givenin Figure 7 It s equation is

y=38.53 3.03 sin

th e origin being a t 1870. Th e root-mean-square devia-

but in no year w ere fewerthan seven records obtainable while foralarge proportion of the years the thirty records were complete


20/79


tion of the observations from this curve is =4 20 Incase of the Ohio curve the root-mean-square deviationwas S =5.29. But this is a better relative fit for theIllinois curve than we have a right to claim because inOhio the mean annual rainfall is 41.19 while in Illinoisthe mean is 38.53. f we express the relative scatter ofthe observations about the curve as the ratio of theroot-mean-square deviation of the observations to themean rainfall we get for Ohio and Illinois respectivelyS1=.128; S =.log.

In Figure 8 the Ohio curve for 1870-1910 is placedupon th e same chart as the Illinois curve for the sameflow of time an d the degree of correspondence of th etwo curves is seen to be so close th at with due allowancefor the difference in their mean annual rainfall theyseem to be almost congruent.

We may say therefore th at the two curves fit theirrespective dat a equally well.

Our problem has now received its solution. Annualrainfall in the chief grain-producing area of the UnitedStates has no secular trend but its mean course is theresu ltant of causes producing two cycles of 33 yearsand 8 years respectively. The manner in which thesecycles of rainfall produce a rhythmical expansion andcontraction in th e yield of the crops we shall examine inthe next chapter.

Cycles of Rainfa ll 3


21/79


PPENDIX

T BLE . A N N U A L R A I N FA L L IN THE OH IO VALLEY

Cycles of Rainfall

T BLE 11. THE PERIODOGR M RAINFALL IN THE O H I OVALLEY

y F t ) A, a cos kt bl sin kt = A, A1 sin kt )

.7449 21 .0046 4.4260 4.43064 .5692 22 ,2454 2 .4237 2 .6691

,5417 2 3 ,8471 ,8714 1.7185.3623 24 .3551 .0678 .4229

5.9707 25 .2755 ,1327 ,408217.0778 26 ,0566 .OW2 ,0568

,3447 27 ,9692 ,0019 ,9711.6168 28 .6227 ,0300 .6527

1.7547 29 4.2657 1.1153 5.3810,5514 30 ,6464 ,4767 1.12:?1,0051 31 .6112 .5923 1.2035,2748 32 ,5776 1 .1168 1 .6 941,2737 33 2.3199 5.997 4 8.3173,8115 34 .2017 1.765 2 1.9669

1.0421 35 ,0456 1.7914 1.83 703.8602 36 ,0036 6.856 7 6.86033 .4199

6255 ean v lue of A 2 = 2..5459


22/79

4 Economic Cycles The ir La w and Cause

CHAPTER I

RAINFALL AND THE CROPS

It is mere weather doing and undoing without end.-WILLIAM JAMES.

TABLE 111.-ANNUAL RAINFALL N ILLINOIS

IN he preceding chap her the course of annual rainfal lin the grea t cereal-producing area of the United Stateshas been shown to move in cycles: There is a ground-swell of thirty -three years in leng th upon which cyclesof eight years in duration are superposed. Our objectin studyin g the rhyt hmic changes in the volume of rain-fall was to bring these changes into relat ion with thevariations in the yield per acre of t he crops and in th epresent ch apter we shall be able to realize our purpose.The actual course of th e varying yield per acre of t hecrops will be shown t o have both a secular and a cyclicalmovement; these two movements will be separated forrepresentative crops; and the cyclical movements willbe shown to be dependent upon th e cyclical movementsin the weather represented by th e cycies of rainfall.

The Secular Trend n the Yield of the Crops

RAINFALLN INCHES

3 4 1 1

44 17

3 5 8 9

28 99

32 92

38 27

3 7 4 4

49 09

3 4 9 5

3 6 1 9

27 17

YEAR

1870

1871

1872

1873

1874

1875

1876

1877

1878

1879

1880

The s tat e of Illinois was chosen in the precedingchapter t o illust rate the general conditions of rainfallin the Corn Belt of the Middle West and we shall nowexamine the stat istics of the yield of i ts most impo rtan tcrops.

35

1881

1882

1883

1884

1885

1886

1887

1888

1889

1890

RAINFALL 1N INCHES/

2 9 6 5

3 6 5 3

33 98

41 62

3 2 9 1

40 34

45 50

42 76

3 7 6 1

3 6 1 0

4 2 3 1

Y E A B

1891

1892

1893

1894

1895

1896

1897

1898

1899

1900

1901

42 32

49 04

47 81

45 83

40 80

36 16

3 3 4 0

3 9 4 1

36 27

4 0 3 4

42 65

35 97

1904 39 33

1905

1906

1907

1908

1909

1910

Mean

..

3 7 3 3

38 10

4 0 6 1

3 6 7 6

4 4 7 4

34 34

38 53


23/79

6 Economic Cycles Their Law and CauseRainfall and the Crops 7

According to the Yearbook of t he Department ofAgriculture for 1912, we find the acreage and value ofthe leading Illinois crops as they are given in thesubjoined Table

Crop(1) Corn2) Oats

(3) Hay(4) Wheat(5) Potatoes(6) Barley(7) Rye(8) Buckwheat(9) Tobacco

Acreage10,658,0004,220,0002,5 12,0001,183,000

137,00057,00048,0004,000

900

Value of Crop174,791,00054,818,0004 1,152,0008,641,0008,302,000

952,000538,00070,00062,000

I t is clear, from this Table, th at five crops-corn, oats,hay, wheat, and potatoes-make up the bulk of thecrops of Il linois, and one could not go far wrong if hebased his generalizations as to the conditions of agricul-ture in the stat e upon these five crops. But for thepurposes we have in view, in this and other chapters, itis not possible to utilize the stat istics of wheat produc-tion because both spring and winter wheat ar e grown in

the state, and t he statistics of their relative yield andprice are not given in the published material for thelong record covered in our investigation. Accordingly,the crops that have been actually used in our inquiryare corn, oats, hay, and potatoes. These crops total93.13 per cent. of the crop acreage and 96.45 per cent. ofthe crop value as these quantities are given in the aboveTable.

As the yield per acre of the various crops may show asecular as well as a complex cyclical change, it will benecessary, before their cyclical elements can be broughtinto relation with the corresponding cyclical changes ofrainfall, to eliminate from the recorded course of theyield per acre of the several crops the element of changethat is secular in character.

The method that has been adopted here to effect theelimination of the secular change is simple, bu t to securea first approximation, it is adequate . For a period oftime covered by t he statistics, a change is regarded as asecular change if, for the period of time taken as awhole, the yield per acre of t he crop shows a tendencyeither to increase or to decrease. In order to determinewhether there is a secular change in the yield per acre,for a certain period of time, th e yield data a re correlatedwith time, and the existence or non-existence of asecular change is inferred from the relative magnitudesof the coefficient of correlation and its probable error.If there be a secular change, the calculation of thecoefficient of correlation of the yield with time is then afirst-step toward the elimination of the secular elementby means of a regression equation in which the co-efficient of correlation is a factor.

The method may be illustrated by taking the historyof the yield per acre of corn. I n Figure 9 the actualyield per acre in Illinois is plotted for the period 18701910. The straight line showing the secular trend of t heyield is the graph of t he regression equation betweenthe yield per acre and time. The correlation of the


24/79

38 Economic Cycles Their Law and Cause Rainfall and the Crops 39

yield per acre and time is r . 3 8 2 ~090, an d t he regres-sion equation is, y .204x+26.93, where y yield peracre, x =time, a nd the origin is a t 1870. The seculartrend is eliminated by means of the fa cts summarized

in the regression equation: Beginning with the year1870, as many times .204 are subtracted from the

yield per acre for the several years, as the respective

years differ from 1870. For example, the yield for the

year 1872 was 39.8 bushels per acre; consequently t he

reduced yield for th at year was 39.8 2 .204) 39.8-

.408 =39.39. Figure 10 traces th e yield per acre of corn

freed from the secular trend.

Of th e four leading crops of Illinois th at form the

basis of our i nvestigation, only two, corn and potatoes ,

show a significant tendency to secular change. The

correlation between the yield per acre and time is,

for hay, r .013* .I05 and, for oats, r .043* .105;consequently the figures for the yield per acre of these

two crops have not been reduced. I n the case of

potatoes, r .122*.104, and the regression equat ion

is y .233x+70.51, where the origin is a t 1870. Thefigures for the actual yield per acre and the reduced

yield per acre for corn and potatoes, as well as the

figures for the yield of hay an d of oats , are given in

Table I of t he Appendix to this chapter.

The indicated secular trend in potatoes is not significant nthe m athem atical sense because the probable error of th e coefficientof correlation is nearly as large as the coefficient itself. I havenevertheless eliminated the indicated secular trend before using thedata.

The raw data were taken from Bulletins 56 58, 62 63 of the


25/79

4 Economic Cycles Their Law and CauseRainfall and th Crops

ritical Periods Of Growth

If th e rhythmical changes in rainfall are to give the

clue to th e changes in the yield of the crops the varia-

tions in the rainfall must be closely related with thevariations in the yield of the crops. But d ifferen t crops

have different times of planting an d of harv esting

different periods o growth and different requirements

of moisture at the various stages of growth. The direct

way to find whether t he course of rainfall determines

the course of the varying yield of the c rops is first to

ascertain the critical season for every crop; and then to

compare the course of t he yield of each crop with th e

course of t he rainfall of i ts critical season.The method of discovering the critical period of

crop may be illustrated in the treat ment of corn. InTable I1 of th e Appendix to this chapter the meanmonthly rainfall for Illinois is tabulated for seven

months March April May June July August and

September. Table I of the Appendix records the da ta

Bureau of Statistics of the United St ates Departm ent of Agricultureand from recent Yearbooks of the United States Department of

Agriculture.The raw data were taken from Bulletin W of the WeatherBureau of the United States and from the nnz~ai eports of theChief of the Weather Bureau. The stations used in computing themean monthly rainfall Tere in Northern Illinois: Aurora Cam-bridge Chicago Tiskilwa Galva Kishwaukee Ottawa Winnebagoand Henr y. In Centr al Illinois: Charleston Carlinville C oatsburgDecatur Griggsville Knoxville Havana LaHarpe Pana Peoriaand Springfield. In Southern Illinois: Cairo Cobden CarlyleGolconda Flora Greenville McLeansboro hlascoutah M t.Camel and Palest ine.


26/79

42 Economic Cycles Their Law and Cause Rainfal l and the Crops 4

referring to the yield per acre of the several crops afte rthe secular trends have been eliminated. These twoTables furnish the statistical material for ascertainingthe critical periods of the respective crops. The fact s

as to the times of planting and harvesting may beobtained from an article in the Yearbook of the UnitedSta tes Department of Agriculture, 1910, pp. 488-494,on Seedtime and Harves t: Average Date s of Plantingand Harvesting in the United States. The method of

detecting the period of critical relation between yieldand rainfall consists in ascertaining, for each crop, themont h or combination of months, within the intervalbetween planting and harvestingll whose rainfall givesthe highest correlation with the ultimate yield peracre of the crop. The time for planting corn in Illinois,according to t he official publication cited above, beginsabout April 30, it is general about M ay 13, and i t endsabout June 2. The average time for harvesting, accord-ing to t he same publication, begins about September 26,is general by October 29, and ends about December 10.The correlation between the yield of corn per acre(secular trend eliminated), and the rainfall for June is,r =.069; for July, r =.496; for August, r=. 29 3; forSeptember, r .087; for July and August combined,r =.589. The critical period of growth for corn has,therefore, been assumed to be the interval of twomonths-July and A u g ~ s t . ~

1 For some purposes it would be desirable to test the correlationbeyond these limits.

Of course all possible combinations of m onths h ave no t been

The critical periods for the other crops are, for oats-May, June , July , r .290; for hay-March, April,May, June, r .620; for potatoes-July and August,r .666. The critical season for corn, as we found

while ago, is July a nd August, r 589.The high correlation between th e yieId of t he crops

and the rainfall of their respective critical seasonspromises well for the theory as to the relation of t hecycles of rainfall and cycles of crops. In the last chapterwe found that by examining the periodogram of an nualrainfall in th e Ohio Valley, cycles of eight years a nd ofthirty-three years were discovered; and that by takingperiods of thir ty-three years and eight years with theirsemiharmonics, a good fit to the annual rainfall curvewas obtained. I t was then shown that th e annual rain-fall in Illinois is correlated with the annual rainfallin the Ohio Valley, the correlation coefficient beingr .600. Upon the basis of this relatively high correla-tion, it was assumed th at the annual rainfall in Illinoispassed through similar cycles to the rainfall in theOhio Valley, and we found that this assumption wasjustified by the facts inasmuch as the harmonic analysisapplied in the same way to the Illinois dat a afforded a sgood a fit as when it was applied to th e da ta of t heOhio Valley. Since in two of the four representativecrops the correlation between the yield and the rainfallexhausted in the above case nor have we made any at tem pt toplace the critical period fora smaller interval of time t han a m onth.f for any other period a closer relation could be found t han r 589

the conclusions tha t we dra w from o ur investigation would onlybestrengthened.


27/79

Economic Cycles: TheirLaw and CauseRainfall and the Crops 5

of the critical season of growth is greater tjhan the

correlation between t2he annual rainfall in Illinois and

the annual rainfall in the Ohio Valley, there would

seem to be excellent ground for believing t ha t the cycles

of the yield of the crops would flow congruently wi ththe cycles of rainfall during their respective critical

periods.

Cycles n the Yield o the Representative Crops and theCorrespo nding Cycles of Rai nfa ll

Th e method of bringing the cycles of rainfal l for the

crit ical period of growth of the several crops into rela-

tion with the cycles of the respective crops is similar to

the method that was employed in passing from thecycles of annu al rainfall in the Ohio Valley to the

corresponding cycles in the sta te of Illinois. The

laborious bu t direct way of treati ng th e problem would

be t o compute the periodogram of rainfall for the

crit ical period of growt,h of each crop, an d then t o com-

pare the results with the corresponding periodograms of

the respective crops. I t may be that this laborious

process may eventually have to be followed. The

process t ha t ha s been adopt,ed in the present investiga-tion makes several assumptions which it is highly

desirable to have clear in mind. I t is assumed1) Th at the course of the annual rainfall is the mean

course of the rainfall of the pa rts of the year

and that, consequently, by computing the

periodogram of annual rainfall, we obtain a

general type of curve for describing not only

the annual rainfall but also the rainfall of

an y considerable pa rt of th e year. Or, moreconcretely, t ha t the annu al rainfall and th e

rainfall of a ny considerable pa rt of th e yearmay be described by a n equation of the form

A , +A , sin g + e , +A , s n (&+ e2+ l n g q l + s n ( q 2 ,

where the constants in the series may be differ-

ent for th e several par ts of the year.2) That where the correlation between the yield per

acre of a particular crop and t he rainfall of itscritical season is high, the same general t8ypeof equation will fit both groups of da ta, the

dat a of rainfall and the da ta of th e yield per

acre of t he crops.3) Th at bot h of the preceding assumptions are

great ly fortified if t he compound curves de-

duced from the actual data of rainfall and

yield satisfy a reasonable test of fit to th e data .

Th e working out of the consequences of these as-

sumptions is exhibited in Figures 11, 12, 13, and the

equations descriptive of the several curves appear on

the corresponding Figures.

To measure the degree of fit of the curves to t heir

respective data, we shall employ a coefficient K, whichmay be described as the ratio of th e arithmetic al sumo the deviations of the observa tions from th e curve


28/79

6 Economic Cycles: Their La w nd Cause Rain fall and the Crops 7

o n s o dY p r cre

P

r1


29/79

8 conomic Cycles Their Law and ause

ushels o f orn per acreV

I

I I I I I

Rainfa l l and the rops 9

divided by th e area included between th e curve and thestraight line indicating the mean va lue of t he observa-tions. I n the equation to the compound cycle describ-ing the typical curve with which we shall have to dealthe first te rm gives the mean value of the observationsand the remaining four harmonic terms trace the areaabout the horizontal line drawn at a distance from thebase line equal to the mean value of the observations.The reason for adopting this complex coefficient K isth at the curves whose relative degrees of fit are inquestion apply to qualitatively different things. Fromthe method of calculating K it follows tha t th e smallerthe value of K the bette r is the degree of fit of the curveto the observations.

Passing now to the calculations referring to therepresentative crops we find

For potatoes the correlation of th e yield per acrewith the rainfall of its critical period-July and Au-gust-is r 666 The measure of t he fit of t he com-pound cycle of th irty-t hree years and eight years withtheir semiharmonics is in case of the yield per acreK = 1.97 and in case of the rainfal l of the critical periodof growth K = 1 30

For hay the correlation of the yield per acre withthe rainfall of its critical period-March April MayJune-is = 620 The measure of the fit of t he com-pound cycle to the da ta is in case of the yield per acreK = 1.57 an d in case of the rainfall of the critical seasonK = 1 63

For corn the correlation of the yield per acre with


30/79

5 Economic Cycles: Their Law and Cause Rainfall and the Crops 51

the rainfall of th e critical season-July an d August-is r .589. The measure of the fit of the compoundcycle to the data is, for the yield per acre, K 1.52,and, for the rainfall of the critical season, K 1.30.

For oats, the computation of the equation has notbeen carried ou t because no critical period of growthcould be found in which the correlation between yieldand rainfall was higher than r .3. The correlationswere, for March, r = -.181; for April, r = -.147; forMay, r 120; for June, r .297; for July, r .140; forMay, June, and July, r .290.

Referring now to the Figures 11 12, 13 and to thecalculations that have just been reviewed, we observeth at the compound cycles of yield per acre and of therainfall of the critical seasons flow almost congruently,and th at the compound cycle of th irty-three yearsand eight years with their semiharmonics fits the yielddata nearly as well as it fits the rainfall da ta.

Cycles n the Ind ex of Crop Fluctuatio ns and n the Corresponding Index of Mean Eflective Rainfall

Does the cyclical movement of rainfall give a rhy th-mic movement to t he fluctua tions in the yield of th ecrops taken all together? The preceding section hastreate d the relation of the yield of the separate cropsto the rainfall of their respective critical seasons; wenow inquire whether the yield of al l of the crops takentogether shows a tendency to conform to the cyclicalmovement of rainfall. In order to answer this questiontwo preliminary steps must be taken : 1) A method

must be devised for measuring the fluctuation in theyield of t he crops when th e crops are taken all together;and 2) a method must be devised for combining therainfall of th e critical periods of the growth of t he

several crops. These two steps we shall now consider.I n regard to the first of these desiderata, it is clear

tha t t he measure of the fluctuation of crops taken a s awhole should be based upon the best measure of th efluctuat ion of the yield of th e crops taken singly. More-over, there is a general agreement that the standarddeviation of a frequency scheme is a good measure ofthe sca tter of th e observations abo ut their mea n value.

natural step, therefore, would be to assume that if

the observations form a series in time, a good rela-tive measure of their fluc tuations a t different epochs isafforded by the rat io of the deviations of t he observa-tions from their mean divided by the standard devia-tion. For example, the mean yield of oa ts in Illinois,for th e period 1870 to 1910, was 31.4 bushels per acre,and t he stand ard deviation of the yield for the sameperiod of time was =5.2 bushels. The yield per acrefor the year 1910 was 38.0 bushels. If be taken to rep-resent the dev iation of the yield of any year from themean yield of the whole period, then the for 1910 was38.0-31.4=6.6, an d the fluctuation for 1910 was

1.27. Similarly, for the year 1908, when the5 2

yield was 23.0 bushels, th e fluctuation was, 1.62.

It happens th at in th e case of oats, there is no seculartrend to the yield, but when the secular trend exists,


31/79

Rainfall and the Crops 552 Economic Cycles: Their Law and Cause

it must be eliminated before the fluctuation is com-

puted.

In Table I11 of the Appendix to this chapter thefluctuation for each of t he forty-one years 1870-1910is given for corn oats hay and potatoes. By taking

the algebraic sum of the flu ctuations for all the crops

for an y given year and dividing by four-the number of

th e crops-a measure of th e fluctuation of th e cropstaken all together is obtained. This measure we shall

refer to as the index of th e fluctuation of crops. Th e

index for each of the yea rs 1870-1910 is recorded in t he

last column of Ta ble 111.Th e index of c rop fluctuation computed in th e man-

ner that has just been described is regarded as a more

accu rate measure of t he fluctuation of crops than

would be obtained from an index formed by taking as

the fluctuation for each year in case of each crop the

ratio of the deviation from the mean divided by th e

mean. If the crops differ in their coefficients of varia-

tion th at is to say if t he ratio 5 here M is the meanyield and is the standa rd deviation is not th e same

for all crops the n th e crop with t he largest coefficient ofvariation would receive the largest weight in t he general

index. Th e coefficients of varia tion for th e crops in5 84

our Table are for corn5 17

26 93 217; for oats 3 1 4418

1 3123 25 330. If164; for hay = .137; for potatoes 70 51

the usual method of forming index numbers were em-

ployed in this case to nieasure crop fluctuations the

several crops would in consequence of their different

receive disproportiona te weights. The

method of calculating the index which we have em-

ployed obviates this difficulty.

Having now obtained an index of the fluctuation of

crops we next consider the method of combining the rain-

fall of the critical periods of growth for th e several crops.

The method will be clear if we bear in mind tha t the

critical period of growth of a crop is t he combination of

months whose rainfall gives the highest correlation wi th

the yield. The mean effective month ly rainfall for th e

critical period of a crop is the t ota l rainfall of th e critical

period of growth divided by t he number of mont hs mak-

ing up the critical period. In case of hay for examplethe critical period of growth is March April May

June. The mean effective rainfall for an y given year

would be the tot al rainfall for the four months March

April Ma y Jun e divided by the number of the months.

If the mean effective monthly rainfall for the severalcrops is summed for each year and divided by the num-

ber of crops a measure is obtained of th e mean effective

monthly rainfall for the crops tak en all together. In

TableIV

of the Appendix to this chapter the meaneffective rainfall of t he several crops an d of t he crops

taken all together is tabula ted for each of the years1870-1910.

We have now an index of the fluctuation of crops and

an index of t he mean ef fecti ve rainfall of th e critical

Periods of th e crops. The correlation between thetwo series is r .584. In Figure 14 are traced the graphs


32/79

Rainfall and the Crops54 Economic Cycles Their Law and Cause

of the compound cycles that describe the two seriesI n d e x o f /urfuef/on of c m p ~ .

/ r ; each graph consisting of two cycles and their semi-r,-, 0m harmonics a thirty -three years cycle describing th em ps

? % Q g r o u n d - ~ ~ e l lnd the smaller cycle of eight years sum-? +

w

+* rnarizing the minor cyclical movements. The measure

5 ** k o the degree of fit to th e observations is in case of theI .i a- yield curve K =2.46 and in case of the rainfall curve3 03

h ? K = 1.68. The yield curve reproduces the generalg + +

- characteristic features of the rainfall curve.E % me

Our findings with reference to the crops taken to-h8 . ? e a

5 2 rngether are similar to what we discovered in case of t he+

.% + :2

single crops: The yield per acre and the rainfall of theC . 6 - k l o ob l c n am-d .? mu. critical season are highly correlated; the rhythmical.- .E;? O .

6movements of the yield and of the effective rainfall

s s ;C l c?.- , + may be accurately described by a compound cycle of- ; + I.?

2 3 - gthirty-three years and eight years with their semi-

g s- g~ e., B .

harmonics; and the yield curve reproduces the general0

- 3 M characteristics of th e curve of effective rainfall.0 C2 + g +

SO -+ E m - Passing now to a summary of th e contents of th is

- 2 g . 5 7 i9 4 . chapter we may collect our results in a series of prop-= %

8 a $ ositions.- 29 . s g + 3 / ?+

E Z T 01) The yield per acre of the four representative

Y,m

3 g 0 0m crops corn hay oats and potatoes is associ-%

m+ ated with th e amount of t he rainfall of their

- 2 4 + U9 - 81% respective critical periods of growth. n

& I %- .9 three out of the four cases the degree of cor-- k 3 0F? relation lies between = 589 and r = 666.

- ?9

? ++ (2) The rhythmical changes in the yield per acre ofa g4 the crops and in the rainfall of the respective

- mII

I

aa

-

y / / l ~ f u P J ) I y C u o ~ UPaW

I I I I II

-

-

-

-

-

- 2

- 0

-

-

-

-

-

-

-

-

-

-

-I I I I I I I l I

8


33/79


critical seasons may both be accurately de-scribed by a compound cycle cornposed of athirty-three years cycle with its semihar-monic, which summarizes the ground-swell of

the movement, and a superposed cycle of eightyears with its semiharmonic, describing theshorter rhythmical movements.

3) I n three of the four representative crops, thecompound cycles summarizing the changes inthe rainfall of the critical periods of growthand the changes in the yield per acre of thecrops are so nearly congruent that, consider-ing the high correlation of the yield with th e

rainfall, one may conclude, with a high degreeof probability, tha t th e rhythmical movementin the weather conditions represented byrainfall is the cause of t he cycles of the crops.

4) Th e index of the fluctuation of the crops takentogether, and the index representative of themean effective rainfall during the criticalseasons are highly correlated, r 584

5) Th e rhythmical changes in the index of the

fluctuation of the crops and in the index ofthe mean effective rainfall are accuratelydescribed by a compound cycle which is madeup of a thirty-three years cycle and an eightyears cycle with their semiharmonics, andthese two compound curves are, in their gen-eral characteristics, much alike.

6) The investigation of the crops taken singly and

Rainf all and the Crops 57

taken together leads to the general conclu-sions

a ) that there is a rhythmical movementin both the yield of the crops and in therainfall of the critical periods which issummarized in a compound cycle, inwhich the constituent elements are aground-swell of th irty-three years an dits semiharmonic, and a shorter super-posed cycle of eight years with itssemiharmonic

b) that the cyclical movement in theweather conditions represented by rain-fall is the fundamental, persistent causeof the cycles of th e crops.


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aznfall and the ropsPPENDIX

T BLE I THE CROPS F ILLINOIS

PERACRE OHAY N

TONSTon 2

bs

18

31

1 35

1 25

1 2037

40

1 6 0

1 491 2 1

1 4 5

1 30

25

45

1 401 30

1 3480

40

39

1 3 0

25

2521

1 1466

38

1 2 9

56

2927

1 0850

1 54

1 36

35

98

40

53

1 45

33

YEAR

18701871

1872

1873

1874

1875

1876

1877

1878

1879

1880

18811882

1883

18841885

1886

1887

1888

1889

1890

1891

1892

18931894

1895

18961897

1898

18991900IS01

1902

1903

19 4

1905

19061907

1908

1909

1910

PERACRE FOATS YBUSHELS

6 0

33

36 6

30 0

17 533 0

20 0

3 7 0

35 9

3 2 0

3 1 8

33 4

40 7

36

32 832 8

31 8

29 5

35 8

37 5

2 1 0

34 026 3

27 2

36

24 4

28 0

3 2 0

29 0

38 038 0

28 2

37 7

26 6

32 0

35 5

29 524 5

23 036 6

38 0

YIEID ERPOTATOES

ACTUALYIELD

81

61

75

40

55128

75

93

67

88

7548

85

92

7987

6733

80

99

30

92

52

53

50

77

97

38

70

9690

35

118

72

108

75

97

87

7 1

91

75

ACREN BUSHELS

REDUCEDYIELD

81 0

60 8

74 5

39 3

54

126 8

73 6

9 1 4

65

8 5 9

7 2 745 4

81 8

89 0

75 783 563 3

29 0

75 8

94 62 5 3

87 146 9

47 644 4

71 2

90 9

3 1 7

63 5

89 283 027 8

110 5

64 3

100 1

66 8

88 6

78 4

62 2

81 9

65 7

YIELD ERCORN N

ACTUALYIELD

PA

35 2

38 3

39 821 0

18 0

34 3

25 0

29 0

27

35 0

27 2

19 4

23 0

25 0

30 031 4

24 519 2

35 732 3

26 2

33 5

26 2

25 7

28 837 4

40 532 530 0

36 037 0

21 4

38 7

32 2

36 5

39 8

36 136 0

3 1 6

35 9

39 1

ACRE rBUSHELS

REDUCEDYIELD

35 2

38

35 4

20 4

17 2

33 3

23 8

27 6

25 5

33 2

25 2

17 2

20 6

22 4

2728 3

21 2

15 7

32 0

28 4

22

29 2

21 7

21 0

23 9

32 335 2

27 024 3

3030 915 1

32 2

25 5

29 6

32 7

28 8

28 4

2 3 8

27 9

30 9


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6 Economzc ycles Th ew Lawand auseRai nfa ll and the rops 1

TABLE INDEX OF F L U C T U ~ T I O N F CROP S A DEVIA

rION FROM THE M E N AFTER THE S E C U L ~ R RE\D A S BEENELIMINATLD ST ND RD E V I AT I O N


36/79

The Law of emand 63

CHAPTER IV

THE LAW OF DEMAND

Kann man nicht die Nachfragefunktion genauer feststellen, sogenau, dass wir nicht bloss ein eindeutiges, sondern ein konkretesResu ltat gewinnen? Ich g laube die Antwo rt xu hijren: Welch'ein phantastisches Unterfangen-Unberechenbarkeit der wirtschaft-lichen Vorgange-steter Wechsel-u. s. w

JOSEPH CHUMPETER

QUESTIONS ffecting for the most part the supply ofcommodities have th us far been the object of our in-vestigation, but the inquiry a s to the cause and law ofeconomic cycles must extend to a consideration ofcycles of values and prices. Since the rhythmicalvariation in the supply of crops produces its effect uponcrop prices in accordance with the laws of demand forthe several crops, the obvious first and necessary stepin bringing th e results of the preceding chapters to bearupon the question of tjhe cause and law of economiccycles is to solve the problem of t he relation betweenthe variations in the supply of the several crops andthe resulting variations in their respective prices. Itis required to derive from existing data the concretelaws of demand fo r the representative crops.

The Theory o emand

The mathematical treatmen t of the theory of demandfurnishes two doctrines tha t are of importance in our

62

work: The doctrine of the uniformity of thedemand function and the doctrine of the elastic ity ofdemand. The exposition of these two doctrines will befacilitated by reference to Figure 15, in which, accord-ing to the usual practice, quant ities of commodity aremeasured upon the axisof abscissas, and the cor- Yresponding prices perunit, upon the axis ofordinates.

The doctrine of t heuniformity of the demand nlfunction, which is trace-

M M

to Cournot l but is FIGURE 5. The law of demand.especially stressed byProfessor Marshall, has been put in these words:

There is then one general l w of demand viz., thatthe greater the amount to be sold, the smaller willbe the price at which it will find purchasers; or, in otherwords, that the amount demanded increases with afall in price and diminishes with a rise in price.'' Re-

Cou rnot: Recherches sur les principes ma thha tiqu es dela thdoriedes richesses, 5 21, 22. Assuming tha t th e relation between priceand the a mount demanded is represented by F (p), he says, p.54:"Si la fonc tion F(p) est continu e, elle jouira de la propri6t6 comm une

toutes les fonctions de ce tte nature, e t sur laquelle reposent t an td'applications importantes de l'analyse mathkmatique: les varia-tions de la demande seront sensiblement proportionelles aux varia-tions du prix, tant que celles-ci seront de petites fractions du prixorkinaire. D'ailleurs, ces variations seront de signes contraires,cJest-&-dire ul& ne augmentation de prix correspondra une dimi-nution de la demande."

The Law of Demand 65


37/79

The Law of Demand 6564 Economic Cycles: Their Law and Cause

ferring to Figure 15, this statement means that if a t

any point in the demand curve DD', say the point P

a straight line is drawn tangent t o th e curve, then th e

trigonometric tangen t of t he angle which th e line makes

with t he positive direction of t he axis of x is negative.In Professor Marshall's words: Th e one universal

rule to which the demand curve conforms is that it is

inclined negatively throughou t t he whole of i ts length. '

As we proceed we shall find tha t t he law of demand for

some commodities does indeed conform to the type of

curve which has just been described, but i t will be a part

of the work of the next ch apter to show th at the doc-

trine of t he uniformity of the demand function is an

idol of th e stati c state-of th e method of cceteris

paribus-which has stood in th e way of th e successful

treat ment of concrete dynamic problems.

Assuming th at th e law of d emand for a given com-modity is represented by the descending curve DD' inFigure 15, the elasticity of d emand for th e commoditywhen O n l units are bought is measured by the ratioMM' Q P

P ~ hat is to say, in general terms, if th e price

of the commodit,y undergoes a small change, th e amoun t

of the commodity t ha t is demanded likewise undergoes

a small change, and the degree of th e elasticity of de-

mand for the commodity, in th e given sta te of t he mar-

ket, is measured by t he ratio of th e relative change in

Marshall: Principles of Economics, 4th edit. pp. 174 174 note 2In the subsequent reasoning we shall call tk-is type of demandcurve the negative type.

the amount demanded to the small relative change in

the price. Or, more definitely, if a fall of 1 per cent.in price would cause an increase of per cent. in the

amount demanded, the elasticity of demand would be

two; if a fall of 1 per cent. in price would cause an

increase of I 3 per cent. in the amount demanded, theelasticity of d emand would be one- third; an d so on.

I t will be observed tha t th e theory of elasticity of

demand in this classical form is presented from the

point of view of in finitesimal changes in th e two va-riable~ , price and commodity demanded. I t givesthe degree of elastici ty of deman d for a poin t in time,

for a given sta te of the mark et assuming all other

things to remain the s ame; and for this reason it may be

said to treat of elast icity of dem and from a statical

point of view. Bu t this is not its most serious limitation.

I t postulates a knowledge of the demand curve, and

while it gives an exposition of t he method by which th e

degree of elas ticit y of demand might be determined

provided the demand curve were known, there have

been grave doubts as to whether the practical difficulty

of deriving the demand curve would ever be overcome.

The problem before us is to derive the demand curve

from statistics; to measure th e degree in which i t is a n

accurate description of t he changes of a ctu al industr y;

and to give t he numerical coefficients of elasticity of

demand for typical commodities.

Marshall: Principles of Economics, 4th edit. pp. 177-178note.

Economic Cycles Their aw and ause The Law of Demand 67


38/79

y

Statistical Laws of Demand

Two fundamental defects in the current theoretical

method of trea ting economic questions ar e exemplifiedin the case of the theor y of dema nd: first, the assump-

tion is made that all other things being equal (the old

cceteris paribus), an increase in th e supply of th e com-

modity will lead to a corresponding fall in the price;

secondly, it is assumed t ha t the concrete problem of

the rel ation of price and supply of commodity will be

simplified by attacking first the constituent elements

of t he question rather than by attacking directly the

problem in its full concreteness. Neither assumpti on

is satisfactor y nor indeed admissible. The "othe r

things" th at ar e supposed to remain equal are seldom

mentioned and are never completkly enumerated; and

consequently the assumption that, other unmentioned

and unenumerated factors remaining cons tant, the law

of dem and will be of a certain typ e, is really tant amo unt

to saying that under conditions which are unanalyzed

an d unknown, the law of demand will tak e the supposed

definite form. The burden of proof is upon anyone

using this method t o show tha t the assumption does not

a t least involve a physical impossibility.

Th e second of the above two assumptions is not more

satisfactory than th e first. I t reproduces the defects

of the first assumption with others superadded. Th e

movement of prices results from changes in many

factors: According to the statical method, the method

of cceteris paribus, the proper course to follow in the

explanation of th e phenomenon is to invest igate in

tur n, theoretically, t he eff ect upon price of each factor,

,-@teris paribus, an d then finally to make a synthesis

~~t if in case of t he relation of each factor to price the

assumption caeteris paribus involves large and a t least

questionable hypotheses, does one not completely lose

himself in a maze of implicit hypotheses when he speakso a final synthesis of th e several effects? We shall not

adopt this bewildering method, but shall follow the

opposite course and a tta ck th e problem of the relationo prices and suppl y in its full concreteness.

The fruitfu lness of the st atist ical theory of correlation

stands in significant contrast to the vast barrenness of

the method that has just been described, and the two

methods follow opposed courses in dealing with a

problem of multiple effects. Tak e, for example, th e

question of the effects of weather upon crops . What a

useless bit of speculation i t would be to tr y to solve, in a

hypothetical way, th e question as to the effect of rain-

fall upon the crops, other unenumerated elements of

weather remaining constant? The question as to the

effect of temperature, caeteris paribus? How, finally,would a synthes is be made of t he several individual

effects? The statist ical method of multiple correlation

formulates no such vain questions. I t inquires, di-rectly, what is the relation between crop and rainfall,

not caeteris paribus, but other things changing accord-

ing to their natural order; what is the relation between

crop and temperature, other things conforming to the

observed changes in temperature; and, finally, what is

8 Economic Cycles Their Law and Cause


39/79

The Law o Demand 9the relation between crop and rainfall for constantvalues of temperature? The problem of the effects ofthe constituent factors is solved only after the moregeneral problem has received its solution. This methodoffers promise of an answer to the question as to therelation between the effective demand price and thesupply of th e commodity.

The chief difficulties in the computation of statis ticallaws of demand are due to changes th at occur in themarket during the period to which the statistics ofprices and of quant ities of commodities refer. In orderthat the st atisti cal laws of demand shall have sufficientvalidity to serve as prediction formula, the observationsmust be numerous; and in order to obtain the requisitenumber of observations, a considerable period must becovered. This usually means tha t, during the intervalsurveyed in the statistical series, important changesoccur in the condition of the market . Bu t in caseof s taple commodities, such as the agricultural productswith which we shall have to deal, the effects of thosechanges in the condition of the market that obscure therelation between prices and amounts of commodity maybe largely eliminated. As fa r as the law of demand isconcerned, the principal dynamic effects that need tobe considered are changes in the volume of t he com-modity that arise from the increasing population, andchanges in the level of prices which are the combinedresult of causes specifically responsible for price cyclesand of causes tha t produce a secular tr end in prices.

The effects of these two fundamental changes may beeliminated approximately by a single statistica l device,namely, by deducing the law of demand from a gen-eralized trea tment of the elasticity of demand.

The degree of elasticity of demand, according to theclassic formula, is measured by the ratio of the relativechange in the amount of the commodity that is boughtto th e relative change in the price per uni t of the com-

odity. Suppose, now, that instead of restrictingthis conception to infinitesimal changes in price and inamount of commodity, we extend i t to the finite changestha t actually occur in the market. Then, the relativechange in the amount of commodity that is bought

may be correlated with the relative change in thecorresponding price, and the resulting appropriateregression equation will give the statistical law ofdemand for the commodity. By taking the relativechange in the amount of t he commodity tha t is de-manded, instead of t he absolute quantities, the effectsof increasing population are approximately eliminated;and by taking the relative change in the correspondingprices instead of th e corresponding absolute prices, the

errors due to a fluctuating general price level are par-tially removed. f the observations should cover theperiod of a major cycle of prices, and the commodityunder investigation should be a staple commodity suchas the representative agricultural products with whichwe shall have to deal, the above method of deriving thedemand curve will give an extremely accurate formulasummarizing the relation between variations in price

70 E i C l Th i d C The La of Demand 71


40/79

70 Economic Cycles Their aw and Cause The Law of Demand 71

and varia tions in the amount of the commodity th at isdemanded.

The method may be il lustrated by deriving the law ofdemand for corn. In Table I of the Appendix to this

chapter are recorded, for the period of 1866-1911, in theUnited States, the quanti ties of corn annually pro-duced, the corresponding prices per bushel, the relativechanges in the quantity produced and the relativechanges in the price per bushel. If the correlation ofthe relative change in the amount of corn th at is pro-duced and the relative change in the correspondingprice per bushel of corn is assumed to be linear, th ecoefficient of correlat ion is r = 789, and the equation

of regression is y = -.8896x+7.79, the origin being a to,o). See Figure 16.)

In Tables 11, 111 IV, of the Appendix to thischapter, similar data are given for hay, oats, andpotatoes. The coefficients of correlation are, forhay, r = -.7

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EconomicCycles Henry Ludwell Moore