R. Goodwin - A Non-linear Theory of the Cycle

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A Non-Linear Theory of the CycleAuthor(s): R. M. GoodwinSource: The Review of Economics and Statistics, Vol. 32, No. 4 (Nov., 1950), pp. 316-320Published by: The MIT Press

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A NON-LINEAR THEORY OF THE CYCLE

R. M. Goodwin

W HEN a leadingtheoristhas his atten-tion diverted to business cycle analysis,

the result is usually a happy one, as witness theexamples provided by Schumpeter, Pigou,Frisch, J. M. Clark, Harrod, and Haberler.The most recentof ProfessorHicks' works con-tinues this impressiverecord.' It is a book thatno one interested in the cycle - and what econ-omist is not? - can afford to miss reading. Thefollowing bare enumeration of its accomplish-ments should suffice to make this clear: (i) anelegant restatement of the General Theory ondynamic lines, (2) a basic elaboration of Mr.

Harrod's multiplier-accelerator version ofKeynes, (3) an alterationof acceleratortheorywhich removesits majorflaw (as distinguishedfrom numerous lesser ones), (4) the use ofthis alterationto suggest a solution to a central,unsolvedproblem,i.e., the persistence of cycles,(5) a reaffirmationand restatement of the roleof money in the cycle in such a way as to giveit an ancillary, though important, role. Such atheoreticaloffering,so well written, in such nar-row compass (I68 pages), for so little ($2.25),

is most welcome, especially if we compare it

with the ponderous,uninspired,expensive textswhich are continually being hurled at us. Itwill speak most directly to the professionaleconomist, for it is essentially an experimentaleffort, a true "model," but my guess is thatit was also aimed at the better undergraduateand graduate students as well. A preliminarytrial has convinced me that, while it has to betaught, the book's central thesis can be under-stood by honorsundergraduates.

Professor Hicks launches his model with a

humility rare aniongst economists: "I do be-lieve that the argumentwhich I am going to setout is quite likely to be the main part of theanswer to the great question with which I amconcerned. . . . I am not by any means posi-tive that the answer which I have found'is theright answer; one cannot begin to be sure ofthat until one has tested one's theory against

the facts, and I am well aware that any testingwhichI have been able to do has been extremely

superficial. If the theory which is here offeredstands up to theoretical criticism, the nextstage will be the concern of statisticians, econ-ometrists, and (most of all) economic his-torians, . . ." (Preface.) In this short note,I propose to restrict myself to the theoreticalissues and to leave - with some misgivings- the empirical aspects to others betterequipped for carrying out the arduous task oftesting.

With an engaging frankness,the author spells

out in detail his trinitarian sources Keynesfor the multiplier,Clarkfor the accelerator,andHarrod for growth phenomena so clearlythat one might almost miss the,element of orig-inality involved in erecting a quite new build-ing out of old and familiar building blocks. Hethen proceeds to lay out the nature of thedynamical multiplier in such a way as to undothe damage done by Keynes in telescoping theKahn process. If only this lucid exposition hadbeen available fifteenyears ago, what mountainsof confusion and quibblingwe might have been

spared. The chief new element is contained inhis careful discussion of the difficultproblem ofdistributed lags and in his demonstrationthatthey do not alter fundamentallythe conclusionsderived from a simple lag hypothesis.

In ChapterIV is the second pillar of the edi-fice, for "The theory of the multiplier and thetheory of the accelerator are the two sides ofthe theory of fluctuations, just as the theoryof demand and the theoryof supply are the twosides of the thory of value" (p. 38). Here ap-

pears one of the two non-linearitieswhich dis-tinguish Hicks' theory from its predecessors:"The expansion in investment, induced by arise in output, can be as large as it likes, pro-vided that the necessary resources are avail-able; but the corresponding contraction isslowed up by the condition that Gross Invest-ment in fixed capital cannot be negative"(p. 66). There is also a discussion of the ac-celeration theory of liquid and working capital

' J. R. Hicks, A Contribution to the Theory of the TradeCycle (London,I950).

[ 3I6 ]

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A NON-LINEAR THEORY OF THE CYCLE 3 7

where the asymmetry is not necessarilypresent.The simple income-expenditure ag is here (p.52 ff.) reinterpreted n terms of stocks of goodsand entrepreneurialreactions. While this stepseems desirable, it does involve non-clock time,

whichpresents treacherousproblems and leavesa certain vagueness in the length of the cycle.There is in my mind some doubt as to whetherProfessor Hicks has really succeeded in side-stepping the difficultiesof a full-dress, dynam-ical treatment along the lines of the Frischcarry-on function (an integral rather than adifferenceequation).

Mr. Harrod's Regularly Progressive Econ-omy is accepted as possible but not probable,for the reason that it is an unstableequilibriummotion, so that a small deviation does not set

up a tendency to return to the steady motion.After the rejectionof smoothgrowth,ProfessorHicks quite naturally turns to cyclical theory,for which he chooses a variant of the Hansen-Samuelson formulationof the multiplier-accel-erator interaction. Since he utilizes incomeandnot consumptionfor the acceleration effect, heloses the second difference n his equation (andhence the possibility of oscillation). This sec-ond lag he unobtrusively, and without muchattempt at justification, reinserts as a single

period lag in investment outlays, although sub-sequently he considers the more realistic andimmensely more difficult case of distributedlags. From this, purely linear, model we getthe familar four behavior types - exponential(compound nterest) growth,exponentialdecay,explosive oscillation (complex exponential),and attenuated oscillation. Beyond this thereare murky waters, with the result that the realargument is transferred to the mathematicalappendix, the text being used to report a few,rather tentative, conclusions. The reason isthat even this very simple model is too rich inpossibilities to allow one to take explicit ac-count of the complications of introducing therealistic assumptionof distributed lags in bothconsumptionand investment outlays. While itis not possible to be rigorousor precise, Profes-sor Hicks does make a pretty convincing argu-ment that these complicationsare indeed com-plications and not mattersof basic importance,so that they may be substantially ignored inanalyzing the major cycle. This very useful

result is typical of the many subsidiary pieceswhich adorn the book.

Finally, the central point of the book isreached: how do cycles persist? For all linearmodels there are three possibilities, the first of

which - explosive oscillations - may be sum-marily rejected. Secondly, we know that anacceleration coefficientof one exactly will leadto a persisting, undamped cycle. ProfessorFrisch long ago argued quite conclusively,against the early models of Tinbergen andKalecki, that this will not do.2 Since that time,it has been fashionable to assume a dampedsystem with renewal of amplitude through er-ratic shocks. Professor Hicks rejects this solu-tion in an elegant, but rather too compact,passage (pp. 90-9I), for the full understand-

ing of which the conscientious readerwill haveto wrestle with paragraph27 on pages I93-95

in the appendix. The gist of the argument isthat since random shocks will work both withand against the cycle, the net effect is too smallto be relied on to keep the cycle going with anyappreciableamplitude.

Having disposed, correctly I think, of theexisting hypotheses that will explain the per-sistence of the cycle, he is then free to developa new one. Since the basic importanceof the

problem, and its difficulties,are not likely to befamiliarto the economist (indeed, it is not clearthat they are even to the author), I should liketo indicate briefly somethingof its history andpresent situation. The simple linearized pen-dulum theory, which we all learn, fails to ex-plain most observed oscillations, i.e., the onesthat neither die away nor explode as in thecases of clocks, steam engines, electric bellsand buzzers, wind and string instruments,vacuum tube oscillators, the human heart, anda great variety of other similar phenomena.The first reasonably successful assault on theanalysis of the problem was made late in thenineteenthcentury in acoustic theory (a tuningfork oscillator) by Lord Rayleigh. At aboutthe same time the great mathematicianHenriPoincaresucceededin blocking out a large partof the basic theory and applied it to certain

2It is interesting to note that radio engineers originallymade essentially the same assumption and have been gradu-ally forced to give it up as an unsatisfactory way to explainthe maintenance of oscillation.








3I8 THE REVIEW OF ECONOMICS AND STATISTICS

problems in celestial mechanics. Neither efforthad much effect on the general theory of cycles,however, until the advent of vacuum tube cir-cuits made a non-linear oscillator theory im-perative. About two 'decades ago B. van der

Pol gave a satisfactory account of what hecalled relaxation oscillations, and this work hasbeen extended by Ph. Le Corbeillerand others.Research is being actively pursued in manyplaces, especially the United States and SovietRussia, where notable results have beenachieved. A superior account of the whole fieldis to be found in the Theory of Oscillations, byA. A. Andronow and C. E. Chaikin.3 As a re-sult of all this we have a reasonably all-embrac-ing and muchprofoundertheory of oscillations.It has become abundantly clear that self-main-

taining cycles occur because of essential non-linearitiesin the structure of the system.

Since Professor Hicks proposes a theorywhich will explain the maintenance of oscilla-tion, we can be sure, on formal grounds, thatthis impliesa non-linearity. In fact, he assumestwo - the lower limit of zero in gross invest-ment and the upper limit of full employmentin real income. From the point of view of sys-tematic vibration theory, his model is probablybest regardedas an over-energeticclock. The

ordinary clock maintains its oscillation by de-livering to its pendulum in each swing a blowwhich is timed by the escapement. The econ-omy seems to have a strong tendency to in-stability, and hence wouldswing in ever greaterarcs were it not for the inevitable limit of fullemployment, which thus provides an essentialnon-linearity. By analogy, we may visualize aclock placed next to a wall; it is so constructedthat each swing of the pendulumis greater un-til it begins to hit the wall; thereafter it willoscillate perpetually with this uniquely deter-mined amplitude. Even if it is slowed down orspeeded up or nearly stopped, it will alwayswork back to its proper limit cycle.

Actually Professor Hicks considers two quitedistinct "constraints." It is evident that full

employment constitutes a barrier because inorder to get there we must have a rate of ex-pansion, and hence a level of investment, ofreal income which cannot be maintained whenonce we have attained it. Hence induced in-

vestment must fall with consequent drops inincome and employment.5 This one barrierwould suffice to maintain a steady oscillation,but in fact there is a second barrierof a quitedifferent sort from the first. There is no effec-tive lower limit to income short of zero, butthere is one to investment. Net investment can-not drop below a negative value equal to de-preciation. The economy cannot, however,remainpermanently at this depressedlevel, be-cause replacementoutlays will have eventuallyto be undertaken and then income increases,positive investment takes place, and the boomis on again. Between the floor and the ceilingthe economy bounces back and forth. Indeedthe economyneed have no tendency to oscillateat all, apart from the two thresholds, but needonly be explosive in both upward and down-ward directions.

Abstracting from trend and autonomous in-vestment, we may state the theory as follows:today's income is greater than yesterday's in-come by the differencebetween today's invest-

ment and the saving, s(y), out of yesterday'sincome. If to this we add the accelerator assole determinant of investment we have a com-plete theory. Today's investment is determinedas some multiple, say two, of the difference

3English translation published by the Princeton Uni-

versity Press, I949.

His choice of word "constraint" to describe a non-linearity seems most unfortunate. It has in dynamics awell-defined meaning which has nothing to do with non-linearities. A much better word would have been threshold,

for threshold oscillators are a clearly-defined class, e.g.,household heating systems with thermostatic controls.

This point has not, of course, by any means gone un-noticed before. For example in Monetary Theory and FiscalPolicy, Professor Hansen writes on page I99, "In the firstplace, when the rate of increase of output has begun todecline, as it must as full employment is approached, theinduced investment in inventories and in fixed plant and

equipment will fall." It is in this form that the Hansen-Frisch-Clark controversy (Journal of Political Economy,October I93I and subsequent issues) should have been posed.Then it would have been clear that Professor Clark wasreally right, although formally Professor Frisch (and Mr.Harrod and Professor Domar after him) was right in sayingthat a smooth exponential growth will always satisfy afirst order differential equation. The point is that if byinduced investment we rise from unemployment to fullemployment, there must at that point occur an impasseand the smooth development breaks down. Even if thestructural parameters were such as to permit a steady 3 percent growth rate - which seems to me quite unrealistic -

once unemployment existed it would never be reduced andhence we would never reach boom and full employment.








320 THE REVIEW OF ECONOMICS AND STATISTICS

in autonomousinvestment and in the full em-ployment ceiling, quite independently of thecycle. Significantlyenough Schumpeter'snameis never mentioned, and yet surely some formof his innovational theory is essential to the

descriptionof the contoursof capitalistic evolu-tion, particularly in relating trend to cycle.This leads into another problem, peculiar toKeynesian theory, on which Professor Hicks isless than adequate. Even if we include secularprogress in the form of the accumulation ofcapital and labor, whethersteadily or in spurts,we are still left with an awkwardquestionaboutthe savings function. If s(y) is as drawn inChart I, it is adequate for cycle analysis, butwe get into trouble over the long run, for wewould always return to the same trough level,e.g., a depressionin I920 would carry us downto an I820 level and back up again but onlywith fantastically high levels of savings andinvestments. We escape this difficulty (bydrawing a flatter s(y) curve passing throughthe origin) only to fall into another (incomewould go to zero or lower in each cycle).

This dilemmamay be resolved by a resoluteapplicationof ProfessorDuesenberry's"ratcheteffect," whereby savings depend not only onincomebut on the previoushighest income.8As

a resultof secularprogressthe full employmentlimit moves to the right, and once we pass theprevious highest income we proceed along thesecular savings function as distinguished fromthe cyclical one. Yet the threshold does existand, being encountered,reversesthe boom. Butnow we descend along a new cyclical savingsfunction, which is simply the old one translatedto the right by the differencebetween the oldpeak and the new. In this way, without toogreat a strain on our geometrical intuition, wecan fuse trend and cycle in a more satisfactorymanner that is strongly reminiscent of, andparallel to, the Schumpeterian innovationstheory. An example is given in Chart 2, with

no particular attention to accuracy or realism,of such a cyclical-secular process. By joiningcogetherwith a solid line the points represent-ing successive combinationsof investment andincome, we get a vivid picture of how the econ-

omy may slip and slide along. Each forwardlurch along the secular savings function (thestraight line) generates consumption habits

which produce a new cyclical curve, which wefollow down and back up again. No accounthas been taken of autonomousinvestment, butit is easily inserted by simply adding it to in-duced investment, and, since it is likely neverto be of the same magnitude, it imparts to eachcycle an historical individuality.

In this short compass I have tried only tosuggest to the reader the main outlines of amost impressiveperformance. My purposewillhave been served if I have sent him hurryingtothe original, to which he must repair for aproper view of the rich embroidery of theoret-ical detail. There he will find illuminating dis-cussions of inflation and the monetary theoryof the cycle, of a hitherto unnoticed cobweb inthe Keynesian theory (but surely not as a re-sult of disuse!), of distributed lags, of differ-

ence equations (the mathematical appendixprovides perhaps the best available introduc-tion to that importantsubject), of the dynamicsof the pure theoryof investment and of the dis-tinction between its fixed and liquid forms, ofthe theory of the secular steady state, andfinally of the many problemsof saving and in-vestment in a dynamical multiplierprocess.

i s

(y)

ty)~~~~~~~~~~~Ym,YM Ynm3 Ymn4 YMi5

CHART 2

8 James Duesenberry, Income, Saving and the Theory ofConsumer Behavior (I949), Chapter vii. Unfortunately, theimportant results obtained by Professors Duesenberry andModigliani appeared too late for incorporation in the book(cf. page vi of the Preface).






R. Goodwin - A Non-linear Theory of the Cycle

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