t-JJ3 t1qcECONOMICjl I'~ ESSAYS 1 IN HONOUR OF GUSTAV...

t-JJ3jlt1qcECONOMICI'~1 ESSAYS

IN HONOUR OF

GUSTAV CASSEL

OCTOBER 20th

1933

[K)REPRINTS OF ECONOMIC CLASSICS

Augustus M. Kelley, .BooksellerNew York 1967

PROPAGATION PROBLEMS AND IMPULSEPROBLEMS IN DYNAMIC ECONOMICS*

THE majority of the economic oscillations which we encounter seemto be explained most plausibly as free oscillations. In many cases theyseem to be produced by the fact that certain exterior impulses hit theeconomic mechanism and thereby initiate more or less regular oscil-lations. '

The most important feature of the free oscillations is that the lengthof the cycles and the tendency towards dampening are determined bythe intrinsic structure of the swinging system, while the intensity (theamplitude) of the fluctuations is determined primarily by the exteriorimpulse. An important consequence of this is that a more or less. regular fluctuation may be produced by a cause which operatesirregularly. There need not be any synchronism between the initiatingforce or forces and the movement of the swinging system. This facthas frequently been overlooked in economic cycle analysis.

If a cyclical variation is analysed from the point of view of a freeoscillation, we have to distinguish between two fundamental problems:first, the propagation problem; second, the impulse problem. Thepropagation problem is the problem of explaining by the structuralproperties of the ~winging system what the character of the swingswould be in case the system Wasstarted in some initial situation. Thismust be done by an essentially dynamic theory, that is to say, by atheory that explains how one situation grows out, of the foregoing.In this type of analysis we consider not only a set of magnitudes in agiven point of time and study the interrelations between them, butwe consider the magnitudes of certain variables in different points oftime, and we introduce certain equations which embrace at the sametime several of these magnitudes belonging to different' instants. This• The numerical results incorporated in the present study have been workedout under my direction by assistants in the University Institute of Economics,Oslo, established through generous grants from the Rockefeller Foundation,New York, and A/S Norsk Varekrig, Oslo. As directors of the Institute,Professor Wedervang and I take this opportunity of expressing our sincerethanks for the support received from these institutions.

ECONOMIC ESSAYS IN HONOUR OF GUSTAV CASSEL

is the essential characteristic of a dynamic theory. Only by a theory ofthis type can we explain how one situation. grows out of the foregoing.This type of analysis is basically different from the kind of analysisthat is represented by a system of Walrasian equations; indeed in sucha system all the variables belong to the same point of time.

In one respect, however, must the dynamic system be similar tothe Walrasian: it must be determinate. That is to say, the theory mustcontain just as many equations as there are unknowns. Only byelaborating a theory that is determinate in this sense can we explainhow one situation grows out of the foregoing. This, too, is a fact thathas frequently been overlooked in business cycle analysis. Often thebusiness cycle theorists have tried to do something which is equivalentto determining the evolution of a certain number of variables from anumber of conditions that is smaller than the number of these variables.It W-ouldnot be difficult to indicate examples of this from the literatureof business cycles.

When we approach the study of business cycle with the intentionof carrying through an analysis that is truly dynamic and determinatein the above sense, we are naturally led to distinguish between twotypes of analyses: the micro-dynamic and the macro-dynamic types.The micro-dynamic analysis is an analysis by which we try to explainin some detail the behaviour of a certain section of the huge economicmechanism, taking for granted that certain general parameters are given.Obviously it may well be that we obtain more or less cyclical fluctua-tions in such sub-systems, even though the general parameters aregiven. The essence of this type of analysis is to show the details of theevolution of a given specific market, the behaviour of a given type ofconsumers, and so on.

The macro-dynamic analysis~ on the other hand, tries to give anaccount of the fluctuations of the whole economic system taken in itsentirety. Obviously in this case it is impossible to carry through theanalysis in great detail. Of course, it is always possible to give even amacro-dynamic analysis in detail if we confine ourselves to a purelyformal theory. Indeed, it is always possible by a suitable system ofsubscripts and superscripts, etc., to introduce practically all factorswhich we may imagine: all individual commodities, all individualentrepreneurs, all individual consumers, etc., and to write out variouskinds of relationships between these magnitudes, taking care that thenumber of equations is equal to the number of variables. Such a theory,however, would only have a rather limited interest. In such a theory

172

PROPAGATION AND IMPULSE PROBLEMS

it would hardly be possible to stlldy such fundamental problems as theexact tt"me shape of the solutions, the question of whether one groupof phenomena is lagging behind or leading before another- group" thequestion of whether one part of the system will oscillate with higheramplitudes than another part, and so on. But these latter problems arejust the essential problems in business cycle analysis. In order to attackthese problems on a macro-dynamic basis so as to explain the mO'\Te-ment of the system taken in its entirety, we must deliberately disregarda considerable amount of the details of the picture. We may perhapsstart by throwing all kinds of production into one variable, all con-sumption into another, and so on, imagining that the notions "pro-duction," "consumption," and so on, can be measured by some sortof total indices.

At present certain examples of micro-dynamic analyses have beenworked out, but as far as I know no determinate macro-dynamicanalysis is yet to be found in the literature. In particular no attemptseems to have been made to show in an exact way what the relationsbetween the propagation analysis and the impulse analysis are in thisfield. ""

In the present paper I propose to offer some remarks' on theseproblems.

2. LE TABLEAU ECONOMIQUE

In order to indicate the most important variables entering into themacro-dynamic system we may use a graphical illustration as the oneexhibited in Fig. I.

The system expressed in Fig. I is a completely closed system. Alleconomic activity is here represented as a circulation in and out ofcertain sections of the system. Some of these sections may best bevisualized as receptacles (those are the ones indicated in the figure bycircles), others may be visualized as machines that receive inputs anddeliver outputs (those are the ones indicated in the figure by squares).There are three receptacles, namely, the forces of nature, the stock ofcapital goods, and the stock of consumer goods. And there are threemachines: the human machine, the machine producing capital goods,and the machine producing consumer goods.

The notation is chosen such that capital letters indicate stocks andsmall letters flows. For instance, R means that part of land (or otherforces of nature) which is engaged in the production of consumer

173


goods, ,. is the services rendered by R per unit time. Similarly V isthe stock of capital goods engaged in the production of consumer goodsand tJ the services rendered by this stock per unit time. Further, Q islabour (manual or mental) entering into the production of consumergoods, so that the total input in the production of consumer goodsis T + tJ + Q.

The complete macro-dynamic problem, as I conceive of it, consistsin describing as realistically as possible the kind of relations that existbetween the various magnitudes in the Tableau Economique exhibited

in Fig. I, and from the nature of these relations to explain the move-ments, cyclical or otherwise, of the system. This analysis, in order tobe complete, must show exactly what sort of fluctuations are to beexpected, how the length of the cycles will be determined from thenature of the dynamic connection between the variables in the TableauEconomique, how the damping exponents, if any, may be derived, etc.In the present paper I shall not make any attempt to solve this problemcompletely. I shall confine myself to systems that are still moresimplified than the one e,xhibited in Fig. I. I shall commence by asystem that represents, so to speak, the extreme limit of simplification,but which is, however, completely determinate in the sense that itcontains the same number of variables as conditions. I shall thenintroduce little by little more complications into the picture, remem-bering, however, all the time to keep the system determinate. This

17.


procedure has one interesting feature: it enables us to draw someconclusions about those properties of the system that may account forthe cyclical character of the variations. Indeed, the most simplifiedcases are characterized by monotonic evolution without oscillations,and it is only by adding certain complications to the picture that weget systems where the theoretical movement will contain oscillations.It is interesting to note at what stage in this hierarchic order oftheoretical set-ups the oscillatory movements come in.

3. SIMPLIFIED SYSTEMS WITHOUT OSCILLATIONS

We shall first consider the following case. Let us assume that theyearly consumption is equal to the yearly production of consumers'goods, so that there is no stock of consumers' goods. But let us takeaccount of the stock of fixed capital goods as an essential element ofthe analysis. The depreciation on this capital stock will be made upby two terms: a term expressing the depreciation caused by the useof capital goods in the production of consumers' goods, and a termcaused by the use of capital goods in the production of other capitalgoods. For simplicity we shall assume that in both these two fields thedepreciation on the fixed capital goods employed are proportional tothe intensity with which they are used, this intensity being measuredby the volume of the output in the two fields. If h and k are thedepreciation coefficients in the capital producing industry and in theconsumer goods industry respectively, the total yearly depreciation onthe nation's capital stock will be h:c + ky, where x is the yearly pro-duction of consumers' goods and y the yearly production of capitalgoods. Our assumption amounts to saying that h and k are technicallygiven constants.

What will be the forces determining the annual production ofcapital goods y? There are two factors exerting an influence on y.First, the need to keep up the existing capital stock, replacing the partof it that is worn out. Second, the need for an increase in total capitalstock that may be caused by the fact that the annual consumption isincreasing. This latter factor is essentially a progression (or degression)factor, and does not exist when consumption is stationary. I shallconsider these two factors in turn.

First let us assume that the annual consumption is kept constantat a given level x. How much annual capital production y will thisnecessitate? This may be expressed in terms of the depreciation

175


coefficients in the following way. Let total capital stock be denoted Z.The tate of increase of this stock will obviously be

z =y - (lzx + ky)

Since the stationary case is characterized by Z = 0, the stationarylevels of x and y must obviously be connected by the relationy = hx + ky, i.e.(2) y = mx

hm=--I-k

The constant m represents the total depreciation on the capitalstock associated with the production of a unit of consumers' goods,when we take account not only of the direct depreciation due to thefact that fixed capital is used in the production of consumer goods,but also take account of the fact that fixed capital has to be used inthe production of those capital goods that must be produced forreplacement purposes. This follows from the way in which (2) wasdeduced, and it may also be verified by following the depreciationprocess for an infinite number of steps backwards. Indeed, the pro-duction of x causes a direct depreciation of lzx. In order to replacetheselzx units of capital, a further depreciation of khx is caused, andthis amount has to be added to the annual capital replacement pro~duction. But adding the amount klzx to the annual capital productionmeans that the annual depreciation is increased by k. (/We) = k2lzx,which also has to be added to the annual capital production, and so on.Continuing in this way, we find that the total annual capital productionneeded to maintain the constant consumption x (with no change inthe total capital stock) is equal to

which is formula (2). For this reason m may be called the total,hand k the partial depreciation coefficients.

Now let us consider the other factor that effects the annual capitalproduction, namely, the change x in the annual production of con-sumption goods.

Let us take a simple example. Suppose that a capital stock of1,000 units is needed in order to produce a yearly national income(i.e. a yearly national consumption) equal to 100 units, then if the

176


production of consumer goods rests stationary at a level of 100 unitsper year, it is only necessary to produce each year enough capitalgoods to replace the capitals worn out, namely, m. 100, m being theconstant in (2).

But if there is in a given year an increase, say, of 5 units, in theproduction of consumer goods, then it is necessary during that yearto increase the stock of capital goods. Indeed, in order to maintain ayearly production of consumer goods equal to 105, there is needed acapital stock equal to 1,050. During the year in question it is thereforenecessary to produce an additional 50 units of capital goods. We arethus led to assume that the yearly production of capital goods can beexpressed in a form where there occurs not only the term (2) but alsoa term that is proportional to x, i.e. y will be of the form

where m and p, are constants. The consta;nt m expresses the wear andtear on capital goods caused directly and indirectly by the productionof one unit of consumption, and p, expresses the size of capital stockthat is needed directly and indirectly in order to produce one unit ofconsumption per year. In other words, p, is the total "productioncoefficient," in the Walrasian sense, for the factor capital.

The two influences expressed by the two terms in (3) have beenthe object of a certain discussion in the literature which ought to bementioned here. Professor Wesley C. Mitchell, in one of his studies,observed that the maximum in the production of capital goods precededthe maximum in the production of consumer goods (or, which amountsto the same, the sales of consumer goods if stock variations of consumergoods are disregarded). From this he drew the conclusion that it israther in production than in consumption we ought to look for thefactors that can explain the turning-point of the cycle. Professor J. M.Clark objected to this conclusion. He said that the rate of increase ofconsumption exerts a considerable influence on the production ofcapital goods, and that the movement of this rate of increase precedesthe movement of the absolute value of the consumption. Indeed,during a cyclical movement the rate of increase will be the highest,about one-quarter period before the maximum point is reached in thequantity itself.

The effect which Clark had in mind is obviously the effect whichwe have expressed by the second term in (3). If we think only of thisterm,' disregarding the first term, we will have the situation where y is

177


simply proportional to the rate of increase of x. If the movement iscyclical, we would consequently have a situation as the one exhibitedin Fig. 2.

In Fig. 2 we notice that the peak in consumption' comes after thepeak in capital production, but if we compare production with the rateof increase of consumption, we find that there is synchronism: themaximum rate of increase in consumption occurs at the same moment

/ PE".I\ IN CAPI"rAL P"RODUCTION

/ ~ PEAK IN CONSUMPTIoN

,.,, \I \

I " ,I I ," ,I "

I ,

I 'I \,, ,,..,,~

CAPITALPRODUCTION

\ I, '~--'MAXIMUMR ••.•TE OF INCREASEIN CONSUt1PT,ON

as the peak in the actual size of capital production. The fact observed .by Mitchell can, therefore, just as well be explained-as Clark did-by saying that it is consumption which exerts an influence on pro .•duction. This is an interesting observation, and it is correct if takenonly as an expression for one of the factors infl:uencingproduction.-

In order to have a complete and correct picture we must, however,take account of both terms in (3), and we ~ust also look for someother relation between our variables. So far the problem is not yetdeterminate.

• Clark, in his discussion of this question, went further than this. He triedto prove that the fact here considered could by itself explain the turning-pointof the business cycle, bu~ this is not correct. Indeed, equation (3) is only oneequation between two variables. Consequently many types of e'V'olutionsarepossible. See the discussion between Professor Clark and the present authorin the journal of Political Economy, 1931-32.

178


In order to make -the problem determinate we need to introducean equation expressing the behaviour of the consumers. We shall dothis by introducing the Walrasian idea of an encaisse denree. Thisnotion will be introduced here only as a parameter by means of whichwe express a certain feature of the behaviour of the consumers. Theparameter is going to be introduced in the equations and theneliminated. Its introduction, therefore, does not mean that we areactually elaborating a monetary theory of business cycle. It is only anintermediary parameter introduced in order to enable the formulationof a certain simple hypothesis.

The encaisse desiree, the need for cash on hand, is made up of twoparts: cash needed for the transaction of consumer goods and producergoods respectively. The first of these parts may of course always bewritten as a certain factor r times the sale of consumer goods, andthe second part as a certain factor s times the production of capitalgoods, provided the factors r and s are properly defined. In otherwords, the encaisse desiree w may be written

As a first approximation one may perhaps consider rand s as constantsgiven by habits and by the nature of existing monetary institutions.

When the economic activity both in consumer goods and in producergoods production increase, as they do during a period of expansion,the encaisse desiree will increase, but the total stock of money, ormoney substitutes, cannot be expanded ad infinitum under the presenteconomic system. There are several reasons for this: limitations ofgold supply, the artificial rigidity of the monetary systems, psycho-logical factors, and so on. We do not need to discuss in detail thenature of these limiting factors. We simply assume that, as the activityand consequently the need for cash increases, there is created a tensionwhich counteracts a further expansion. This tension is measured bythe expression (4). It seems plausible that one effect of the tighteningof the cash situation, and perhaps the most important one, will be arestriction in consumption. In the boom period when consumptionhas reached a high level (in many cases it has extended to pureluxuries), consumption is one of the elastic factors in the situation.It is likely that this factor is one that will yield first to the cash pressure.To begin with this will only be expressed by the fact that the rate ofincrease of consumption is slackened. Later, consumption may perhapsactually decline. Whatever this final development it seems plausible

179


to assume that the encaisse desiree w will enter into the picture as animportant factor which, when increasing, will, after a certain point,tend to diminish the rate of increase of consumption. Assuming as afirst approximation the relationship to be linear, we have

(5) oX= c - ,\W

where c and ,\ are positive constants. The constant c expresses a ten-dency to maintain and perhaps expand consumption, while,\ expressesthe reining-in effect of the encaisse desiree.

Introducing into (5) the expression for the encaisse desiree takenfrom (4), we get(6) oX= c - '\(rx + sy)

This equation we shall call the consumption equation.The two equations (3) and (6) form a determinate system in the

two variables x and y. If the parameters p" m, '\, etc., are constants,the system may easily be solved in explicit form. By doing so we seethat the system is too simple to give oscillations. Indeed, by eliminatingx between (J) and (6) we get a linear relation between x and y.Expressing one of the variables in terms of the others by means ofthis relation and inserting in one of the two equations (3) or (6) weget a linear differential equation in a single variable. The characteristicequation is consequently of degree one, and has therefore only onesingle real root. This means that the variables will develop monotoni-cally as exponential functions. In other words, we shall have a seculartrend but no oscillations.-

The system considered above is thus too simple to be able toexplain developments which we know from observation of the economicworld. There are several directions in which one could try to generalizethe set-up so as to introduce a possibility of producing oscillations.One idea would be to distinguish between saving and investment. Thefact that, in an actual situation, there is a difference between these twofactors will tend to produce a depression or an expansion. This isKeynes' point of view. It would be exceedingly interesting to see whatsort of evolution would follow if such a set of hypotheses were subjectto a truly dynamic and determinate analysis.

Another way of generalizing the set-up would be to introduce thefact that the existence of debts exerts a profound influence on the

• Incidentally this shows that the fact pointed out by Clark does not neces-sarily lead to a development giving a turning-point.

180


behaviour of both consumers and producers. This is the leading ideaof Irving Fisher's approach to the business cycle problem.

A third direction would be to introduce the Marxian idea of a biasin the distribution of purchasing power. This idea may-with a slightchange of emphasis-be expressed by saying that under privatecapitalism production will not take place unless there is a prospect ofprofit, and the existence of profits tends to create a situation wherethose who have the consumption power do not have the purchasingpower, and vice versa. Thus, under private capitalism, production mustmore or less periodically kill itself.

A fourth direction would be the introduction of Aftalion's pointof view with regard to production. The essence of this consists inmaking a distinction between the quantity of capital goods whoseproduction is started and the activity needed in order to carry tocomplett:on the production of those capital goods whose production wasstarted at an earlier moment. The essential characteristics of the situa-tion that thus arises are that the activity at a given moment does notdepend on the decisions taken at that moment, but on decisions takenat earlier moments. By this we introduce a new element of discrepancyin the economic life that may provoke cyclical oscillations. I do notthink that AftaIion's analysis as originaUypresented by himself canbe characterized as a determinate analysis. By putting his argumentsinto equations one wiII find that he does not have as many equationsas unknowns. But his idea with regard to production is very interesting,and, if properly combined with other ideas, will lead to a determinatesystem. Not only that, but it may lead to a system giving rise toosciIIations. I now proceed to the discussion of such a system.

4. A MACRO-DYNAMIC SYSTEM GIVING RISE TO OSCILLATIONS

Let Yt be the quantity of capital goods whose production is started atthe point of time t. We shall call Yt the "capital starting" or the"production starting," and we shall assume that this magnitude isdetermined by an equation of the form (3.3). A capital object whoseproduction is started at a certain moment will necessitate a certainproduction activity during the -following time in. order to completethe object. The productive activity needed in the period following thestarting of the object will, as a rule, vary in a certain fashion whichwe may, as a first approximation, consider as given by the technicalconditions of the production. Let DT be the amount of production

181


activity needed at the point of time t + T in order to carry on theproduction of a unit of capital goods started at the point of time t.The function DT we shall call the "advancement function."

This being so, the amount of production work that will be goingon at the moment t will be

00

Zt = JDTYt-TdTT=O

The magnitude ZJ we shall call "the carry-on-activity" at the pointof time t.

In the formula of the encaisse desiree it is now Z that will occurinstead of Y, so that the consumption equation will be

where c, ~, rand s are constants.The three equations (3.3), (I) and (2) form now a determinate

system in the three variables ~, Y and z.If the carry-on function DT is given, the above system may be

solved. If DT is given only in numerical form, the system has to besolved numerically, taking for granted a certain set of initial conditions.If DT is given as a simple mathematical expression, the system mayunder certain conditions be solved in explicit form. As an example weshall assume

where E is a technicallygiven constant. This simply means that a givenelement of capital starting will cause a certain constant amount ofcarry-on-activity per unit time over the E units of time following thestarting, and that in this point the object is finished, so that nofurther carry-on-activity is needed. This is obviously a simplifiedassumption, but may perhaps be taken as a first approximation.

In this case we get from (I) by differentiating with respect to time

For certain purposes it will be convenient to differentiate also theequation (2) In order to get rid of the constant term, which gives

(5)J8a

~= - ~(rx + s%)

PROPAGAT10N AND IMPULSE PROBLEMS

The three equations to be considered now are (2), (3.3) and (4)(or possibly (5) may replace (2»). This is a mixed system of differentialand difference equations. It is therefore to be expected that the solutionwill depend, not only on the initial conditions of the system in a givenpoint of time, as initial conditions we shall have to consider the shapeof the curve over a whole interval of length €.

We shall in particular investigate whether the system is satisfiedif each of the variables is assumed to be made up of a number ofcomponents, each component being either an exponential or a dampedoscillation, Le. a damped sine curve. It is easier to handle theformulae if each such term is written in the complex exponential form,that is to say in the form

a e(-p+ia)1 + a e(-p-ia)tI 2 i= v=-;where al and a2 are constants. For brevity we may write (6)

where PI and P2 are complex numbers.This applies to a single component. Considering now several

components in each variable, we may express the above assumptionby saying that the variables x, y and z considered as time series areof the form

{

X = a. + EkakeP-.t

y = b. + EkbkeP"t

Z = c. + EkckePf&I

where PII are complex or real constants, and where a, b and c are alsoconstants. By convention we let the numbering in (8) run k =0, I, 2. ,;•Does there exist a set of functions of the form (8) which satisfy thesystem consisting of the equations (3.3), (2) and (4)? Such functionsdo exist. The exponential characteristics Ph of these functions aredetermined by the structural constants €, Asm, Asp., M that enter intothe system considered. On the contrary, the coefficients a, band c in (8)will depend on the initial conditions.

It is easy to verify this and to determine how the exponentialcharacteristics Pk depend on the structural constants. This is done bydifferentiating the various expressions in (8) md inserting the results

183


obtained into the equations of the system. If this is done, one will findthat the coefficients a, b and c must satisfy the relations

Ck AT + Pk-=- As

Ck I - e-ePk

bk = £Pk

(k = 0, 1,2 ... )

This condition entails that all the Ph must be roots of the following.. characteristic equation:

£p = _ Asm + /-,P1- e-ep TA + P

This equation may have complex or real roots. For the numericalcomputation it is therefore conveni"entto insert into (10)

and to separate the real and imaginary parts of the equation afterhaving cleared the equation of fractions. Doing this, we get the followingtwo equations to determine a and fJ (assuming £ and As/-' "" 0).

The terms of these equations have been ordered in the particularform indicated in order to facilitate the numerical solution.

The roots (fJ, a) of these two equations will determine the shape ofthe time curves x, y and z. It is obvious from (12) and (13) that if(fJ, a) is a root, then (fJ, - a) will also be a root. In other words, ifcomplex roots occur they will be conjugate, which means that the

184


corresponding component of x, y and z will actually be a damped sinecurve. If there exists a magnitude f3 such that (f3, 0) is a root, then thecorresponding component will be a secular trend in the form of anexponential.

In order to study the nature of the solutions, I shall now insertfor the structural coefficients £, /L, m, etc., numerical values that mayin a rough way express the magnitudes which we would expect to findin actual economic life. At present I am only guessing very roughlyat these parameters, but I believe that it will be possible by appropriatestatistical methods to obtain more exact information about them.I think, indeed, that the statistical determination of such structuralparameters will be one of the main objectives of the economic cycleanalysis of the future. If we ask for a real explanation of the movements,this type of work seems to be the indispensable complement neededin order to co-ordinate and give a significant interpretation to the hugemass of empirical descriptive facts that have been accumulated in cycleanalysis for the past ten or twenty years.

Let us first consider the constant £. It expresses the total length oftime needed for the completion of big units of fixed capital: bigindustrial plants, water-power plants, railways, big steamers, etc. Thisspan of time includes not only the actual time needed for the technicalconstruction (the erection of the buildings, etc.) but also time neededfor the planning and organization of the work. Indeed, the variable T

in (I) is measured from the moment when the initiative was taken.In many cases the items planning and organization takes more timethan the actual technical construction.

It seems that we would strike a fair average if we say that the actualproduction activity needed in order to complete a typical plant of theabove-mentioned kind will be distributed over time in such a waythat in general it takes place around three years after the planningbeg~n. Some work will of course frequently be done before and someafter this time, but three years can, I beli~ve, tentatively be taken asan average. In making this guess I have taken account of an importantfactor that tends to pull the average up, namely, the fact that in agiven individual case the activity will as a rule not be distributed evenlyover the period (as assumed in the simplified theoretical set-up) butthe peak activity will be concentrated near the end of the period. Ifthree years is taken as the average lag of the various elements of pro-duction activity after the beginning of the planning, we shall have toput £ = 6 in (3), and consequently in (4), indeed in the case of equal

185


distribution, as assumed in (3), the average lag will be half themax£mum Jag.

Furthermore, Jet us put JL = 10, which means that the total capitalstock is ten times as large as the annual production. Further, let usput m = o· 5, which means that the direct and indirect yearly depre-ciation on the capital stock caused by its use in the production of thenational income is one-half of that income, i.e. 20 % of the capitalstock. Finally, let us put A= 0·05, r = 2, and s = I. These latterconstants, which represent the effect of the encaUse desiree on theacceleration of consumption, are of course inserted here by a stillrougher estimate than the first constants. There is, however, reasonto believe that these latter constants will not affect very strongly thelength of the cycles obtained (see the computations below).

Inserting these values in the two characteristic equations (12) and (13)we get a numerical determination of the roots. In the actual computa-tion it was found practical to introduce E'a and E'fJ as the unknownslooked for. By so doing, and utilizing an appropriate system of graphicaland numerical approximation procedures, the roots may be determinedwithout too much trouble. A good guidance in the search for rootsis the fact· that the solutions in a are approximately the minimumpoints of the function

that is to say, a first approximation to the frequencies a will be everyother of the roots of the equation

The roots of this equation are well known and tabulated. - The resultsof the computations are given in the first columns of Table I.

• The. above characteristic equation was worked out and the roots numericallydetermined by Mr. Harald Holme and Mr. Chr. Thorbj6rnsen, assistants atthe University Institute of Economics. In brief the following procedure wasused: The right member of equations (12) and (13) was put equal to aparameter q, and (12) solved with respect to e{J, and (13) with respect toC03 ea. From (IS) a first approximation to a was determined and the corre-sponding p taken as the value determined by putting q = 0 in (12). Thisvalue of P was as a rule immediately corrected by using (12) with the valueof q that followed from the above preliminary determination of a and p.This gave a new value of q that was inserted in (13), thus determining a newvalue of a. Starting from this new value of a the whole process was iterated.This method gave good results except for very small values of A, in whichcase it was found better to start by guessing at the value of p.

186

Trend Primary Cycle Secondary Cycle Tertiary Cycle(j- 0) (j= I) (i- 2) (j= 3)

Frequency ,, ,, , , , , a. Po = - 0'08045 0'73355 1'79775 2,8533Period 211 8'5654' , ,, ,, ,, ,, p=- 3'4950 2'2021a.Damping exponent ,, , , , , f1 0'371335 0'5157 0'59105Damping factor per period , , , , e-2TrfJ1tJ. 0'0416 0' 1649 0'2721

{ Amplitude , , A - 0'32 0,6816 0'27813 0' 17524Consumption x

Phase " , , c/> 0 0 0

Production {Amplitude , , B 0'09744 5'4585 5' 1648 5'0893starting :y Phase " tfs 1'9837 1'8243 1 '7582,,

Carry-on- {Amplitude ,, C 0'12512 - 10'662 - 10'264 - 10'147activity :I

Phase " 8 1'9251 1'7980 1'7412,,


The first component (j= 0) is a trend, which in all the threevariables x, y and z is composed of an additive constant and a dampedexponential term. We may write these trends in the form

r xo(t) = a. + aoePot

1yo(t) = b. + boePJ

L zo(t) = c. + coePot

These expressions are nothing but the first terms in the compositeexpressions (8). The damping exponent Po is the first root of thecharacteristic equation, it is real and negative Po = - 0.08045 (seeTable I). The additive constants a., b. and c. are also determined bythe structural coefficients €, >..sm, etc. Indeed, if t -+ 00 the func-tions (16) will approach the stationary levels a., b. and c.. Since thederivatives will vanish in this stationary situation, we get from (3.3),(I) and (2)

c = b• •putting as an example c = 0.165 this determines uniquely the threeconstants a = 1032, b = 0,66, and c = 0·66.• • •The coefficients ao' bo and Coare not determined uniquely by thestructural coefficients, but one initial condition may be imposed onthem-for instance, a condition that d~termines ao' When ao is deter-mined, bo and Co follow from (9). If, as a numerical example, we imposethe condition that Xo shall be unity at origin, we get the functionsxo' Yo' Zo in (23a).

Besides the secular trend, there will be a primary cycle with aperiod of 8· 57 years, a secondary cycle with a period of 3' 50 years,and a tertiary cycle with a period of 2'20 years (see Table I). Thesecycles are determined by the first, second and third set of conjugatecomplex roots of (10). These sets are denotedj = I, 2, 3 in Table 2.There will also be shorter cycles corresponding to further roots of (10),but I shall not discuss them here.

The presence of these cycles in the solution of our theoreticalsystem is of considerable interest. The primary cycle of 8057 yearscorresponds nearly exactly to the well-known long business cycle. Thiscycle is seen most distinctly in statistical data from the nineteenthcent'ury, but it is present also in certain data from the present

188


century; in the most recent data it actually seems to come back withgreater strength.

Furthermore, the secondary cycle obtained is 3'50 years, whichcorresponds nearly exactly to the short business cycle. This cycle is seenmost distinctly in statistical data from this century, but it is presentalso in older series. As better monthly data become available back intothe nineteenth century the short cycle will become quite evident alsohere I believe.

The lengths of the cycles here considered depend, of course, onall the structural coefficients; but it is only € that is of great importance.The other coefficients only exert a very small influence on the lengthof the cycles. Choosing, for instance, ;\ = O· I, S = I, r = 2, I-' = 5,and different values for m, we find

Period Damping Factorm p e-21rfJ1a.

0-7 8-53 0-0420-5 8-43 0-0430-0 8-20 0-048

In other words, even an extreme variation in the total depreciationfactor m leaves both the period and the damping factor nearlyunchanged_

A change in the constant;\ that expresses the "reining-in" effectof the encaisse des;ree exerts a considerable influence on the dampingfactor, but a relatively small influence on the length of the period.For ;\ = 0-001, S = I, r = 2, I-' = 5 and m = 0-5, we find, forinstance, p = 10-6 years, e-21TP/a = .OO02סס-0 In other words, theperiod is still of the same order of magnitude, but the damping is nowenormous, the amplitude being brought down to two-millionthin the course of one period. This means that the cycle in question isvirtually non-existing_

It is interesting to interpret the last result in the light of the limitingcase where ;\ = 0, Le. 'where the need for cash as a .brake on thedevelopment of production is eliminated. In this case it follows imme-diately from (2) that x will evolve as a straight line with positiveinclination (c being assumed positive). Hence by (3.3) y must also bea straight line, and by (4) z must be a constant, hence z linear. Themovement of the system will consequently be a steady evolutiontowards higher levels of consumption and production without the set-backs caused by depressions.


Of course, the results here obtained with regard to the length ofthe periods and the intensity of the damping must not be interpretedas giving a final explanation of business cycles; in particular it mustbe investigated if the same types of cycles can be explained also byother sets of assumption, for instance, by assumptions about thesaving-investment discrepancy, or by the indebtedness effect, etc.Anyhow, I believe that the results here obtained, in particular thoseregarding the length of the primary cycle of 81 years and thesecondary cycle of 3i years, are not entirely due to coincidencebut have a real significance.

I want to go one step further: I want to formulate the hypo-thesis that if the various statistical production or monetary seriesthat are now usually studied in connection with business cyclesare scrutinized more thoroughly, using a more powerful techniqueof time series analysis, then we shall probably discooer evidencealso of the tertiary cycle, i.e. a cycle of a little more than twoyears.

Now let us consider the other features of the cycles: phase, etc.We write the various cyclical components

Xj(t) = Aje- Pjt sin (1)j + ajt)Yj( t) = Bje- Pit sin (.pj + ajt).a'j(t)= Cje- Pjt sin (OJ+ ajt)

(j=1,2 ... )

j = I means the primary cycle, i = 2 the secondary cycle, etc. Thefrequencies a and the damping coefficients fJ are uniquely determinedby the characteristic equation, but the phases 1>, .p, 0 and the ampli-tudes A, B, C are influenced by the initial conditions. For the primarycycle (j= I) two such conditions may be imposed. We may, forinstance, require that x1(o) = 0 and x1(o) = i. This leads to 1>1= 0,

IAl = -. And when the phase and amplitude for the primary cycle

2a1in x is thus determined, the phases and amplitudes of the primarycycles in y and .a'follow by virtue of (9). Similarly, if 1>" and Aa aredetermined by two initial conditions imposed on the secondary cyclein x, for instance, by the conditions xa(o)= 0 and x,,(o) = i, thephases and amplitudes of the secondary cycles in y and .a'are alsodetermined by (9).

190


When the conditions (9) are formulated in terms of phases andamplitudes, we get

{B sin (.p - q,) = Ap,aB cos (.p- q,) = A(m - p.p)

AaCsin (fJ - q,) = -1;

A(fJ- M)C cos (fJ - q,) = . M

These equations hold good for all the cycles, that is, for j = I, 2, 3 ...They show that the lag between :Je, y and :¥ are independent of theinitial conditions and depend only on the structural coefficients of thesystem. From (18) and (19) we get indeed

p,atg(.p - q,) = -m---~-p

a.tg(fJ - q,) = -- -fJ --M-

Similarly the relations between the amplitudes may be reduced to

where the square roots are taken positive. If the amplitudes aretaken positive, sin (.p - q,) has the same sign as p,a and sin (fJ - q,)the same sign as - aIM.

A given set (18) (for a given J) does not-taken by itself-satisfythe dynamic system consisting of (3.3), (2) and (4). It will do so onlyif the structural constant c = o. If c ~ 0 the constant terms Q., b.and c. must be added to (18) in order to get a correct solution. If theseconstant terms are added, we get functions that satisfy the dynamicsystem, and that have the property that any linear combination ofthem (with constant coefficients) satisfy the dynamic system providedonly that the sum of the coefficients by which they are linearly com-bined is equal to unity. This proviso is necessary because any sets offunctions that shall satisfy the dynamic system must have the uniquelydetermined constants Q.,b. and c•.


The sets (18), with no constants a., b. and c. added, are solutionsof the system obtained by leaving out c in (2). Or, again, (18) may belooked upon as solutions of the system obtained by letting (5) replace (2).

If we impose on the trends the initial condition that Xo shall beunity at origin, and on each cycle in x the condition that it shall bezero at origin and with velocity = i,we get the functions in (23a, b, c, d).The corresponding cycles are represented in Figs. 3-5.

{

Xo - I' 32 - O' 32e- 0'08045t

(23a) Yo = 0·66 + 0·097404£-0·0804,5tZo = 0·66 + o· I2SI2e-0 08045t

{

Xl = 0'6816e-P1t sin alt

(23b) YI = S'4s8se-Pltsin (1'9837 + alt)Zl = - Io·662e-P1t sin (I '92SI + alt)

{

X2 = 0'27813e-Pae sin a2t(23c) Y.= 5'16480- II" sin (I ·8243 + a,I) .

Z2 - - 10' 264e-P.t sin (I '7980 + a2t)

{

Xs = o· 17S2¥-Pat sin ast

(23d) Ys= S'0893e-Pae sin (.1 '7S82 + ast)Zs = - 10' 147e-Pat sm (I '7412 + ast)

(The a and fJ are given in Table I.)From the constants given in Table I, and from the shape of the

curves in Figs. 3-5, we see that the shorter cycles are not so heavilydamped as the long cycle. Furthermore, we see that the lead or lagbetween the variables x, Y and Z is, roughly speaking, the same in theprimary, secondary and tertiary cycles. To study the lag it is thereforesufficient to consider only one of these types-for instance, the tertiarycycle (Fig. S).

Let us first compare consumption with production starting. Apartfrom the fact that the cycles in Fig. S are damped, the relation betweenX and y is very much the same as in Fig. 2, i.e. production has itspeak nearly at the same time as the rate of change of consumption isat its highest. The reason for this is that the constant p. in our exampleis chose.nrather large in comparison to m. This means that our examplerefers 10 a highly capitalistic society where the .annual depreciation isrelatively small. By reducing the size of the capital stock in relation to

192


the output (i.e. reducing p,) and increasing the annual depreciation(i.e. m) the peak in production starting will advance so as to arrivenearer the peak in consumption.

Next, comparing consumption with the carry-on-activity, we see

I

DI II II II I .~iI II I

I II

I I ,I I

I I fI I

:: ~: I -----~I I

coNSUMItTION

X1

III

I It-----,I I

(rleURES ON seAL': I ND'S ATE AMPL, TUD~

FIG. 3

that the former is leading by a considerable span of time, i.e. in thedepression the carry-on-activity starts to increase only when theupswing in consumption is well under way, and the carry-on-activitycontinues to increase even after consumption has started to decline.

The way in which the structural relations determine the time shapeof the solutions may perhaps be rendered more intuitively by a methodof successive numerical approximation.

CONSUMPTION

0 X2III

5.1~ 1IIIIII

I IL_~I ,I PRODUCTION STARTINGI

0 Y2

(FIGlIRES ON SCAL£ INDICATE AMPLITUO~

oI---tIIIIIIIIII

IIIIIII

I I1-...1I II II I

I

- y,

z,

(FIGURES ON SCALE INOIC:~TE AMP1.ITUD~

FIG. 5


In order to show this, we take for granted that the time shape of oneof the curv:es-for instance, Yl-is known in the interval -6 < t < o.That is to say,in this interval we simply consider the values of Yl as

TABLE2STEP-BY-STEP COMPUTATION OF THE PRIMARY CYCLE

t x y z X Z yt-6

0 0 S·OOOO - 10·000 o·sooo 6.4264 - 33·SS81o· 16667 0.08333 4.4229 8.929 0.4381 6·6811 - 3S·66340·33333 o· IS63S 3.8297 7·81SS 0·3752 6·7865 - 36.88940.50000 0·21887 3·2329 6.6844 0.3124 6.7592 - 37.32210·66667 0.27093 2·6435 5·5579 0·25°8 6.6153 - 37.°480

TABLE3

PRIMARY CYCLE COMPUTED DIRECTLY BY FORMULA (23b)

t x Y z

0 0 5.0000 - 10·0000o· 16667 0.07814 4.4138 - 8.9°580·33333 o· 14581 3.8179 - 7.7817

given by the expression (23b). Then we want by the dynamic equationsto determine the solutions numerically from the point t = 0 andonwards. "

With the numerical constants ~, p" m, etc., inserted, the dynamicsystem (where c in (2) is left out) will now be

{

Y = o·SX + lOX

oX = - O·IX - o·oSz- 6z,= Yt-6 + o· S(Xt + Zt)

We .shall use (24) for a step-by-step computation. Since x = 0 andx = o· 5 are given at the origin, Z = - 10 may be determined fromthe second equation in (24). Furthermore, since Yt-6 is given, we maycompute z in origin by means of the third equation in (24), and finally

196


Y may be computed by- the first equation in (24). Thus we have allthe items in the first line of Table 2. Since we know x and x in origin,we may by a straight linear extrapolation determine x and z in the nextpoint of time, that is to say, in the second line of Table 2. And knowingx and z in this point, we may from the second equation of (24)compute x. Further, taking the value of Yt-6 as given also in the nextline we can compute 14, etc. In this way we may continue from line toline and determine the development of all the three variables x,y and z.A comparison between the values in Table 2 for x, y and z with thevalues in Table 3 (determined by the explicit formulae (23b) andrepresented in Fig. 3) will give an idea of the closeness of the approxi-mation obtained by the numerical step-by-step solution.

S. ERRATIC SHOCKS AS A SOURCE OF ENERGY IN MAINTAINING

OSCILLATIONS

The examples we have discussed in the preceding sections, and manyother examples of a similar sort that may be constructed, show thatwhen an economic system gives rise to oscillations, these will mostfrequently be damped. But in reality the cycles we have occasion toobserve are generally not damped. How can the maintenance of theswings be explained? Have these dynamic laws deduced from theoryand showing damped oscillations no value in explaining the realphenomena, or in what respect do the dynamic laws need to be com-pleted in order to explain the real happenings? I believe that thetheoretical dynamic laws do have a meaning-much of the reasoningon which they are based are on a priori grounds so plausible that it istoo improbable that they will have no significance. But they must notbe taken as an immediate explanation of the oscillating phenomenawe observe. They only form one element of the explanation: they solvethe propagation problem. But the impulse problem remains.

There are several alternative ways in which one may approach theimpulse problem and try to reconcile the results of the determinatedynamic analysis with the facts. One way which I believe is particularlyfruitful and promising is to study what would become of the solutionof a determinate dynamic system if it were exposed to a stream oferratic shocks that constantly upsets the continuous evolution, and byso doing introduces into the system the energy necessary to maintainthe swings. If fully worked out, I believe that this idea will give aninteresting synthesis between the stochastical point of view and the

197


point of view of rigidly determined dynamical laws. In the presentsection I shall discuss this type of impulse phenomena. In the nextI shall consider another type which exhibits another-and perhapsequally important-aspect of the swings we observe in reality.

Knut Wicksell seems to be the first who has been definitely awareof the two types of problems in economic cycle analysis-the propa-gation problem and the impulse problem-and also the first whohas formulated explicitly the theory that the source of energy whichmaintains the economic cycles are erratic shocks ..•.He conceived moreor less definitely of the economic system as being pushed alongirregularly, jerkingly. New innovations and exploitations do not comeregularly he says. But, on the other hand, these irregular jerks maycause more or less regular cyclical movements. He illustrate~ it by oneof those perfectly simple and yet profound illustrations: "If you hit awooden rocking-horse with a club, the movement of the horse will bevery different to that of the club."

Wicksell's idea on this matter was later taken up by ]ohanAkerman, who in his doctorial dissertationt discussed the fact thatsmall fluctuations may be able to generate larger ones. He used, amongothers, the analogy of a stream of water flowing in an uneven river bed.The irregularities of the river bed will cause waves on the surface.The irregularities of the river bed illustrate in Akerman's theorythe seasonal fluctuations; these seasonals, he maintains, create thelonger cycles. Unfortunately Akerman combined these ideas with theidea of a synchronism between the shorter fluctuations and the longerones. He tried, for instance-in my opinion in vain-to prove thatthere always goes an exact number of seasonal fluctuations to eachminor business cycle. This latter idea is, to my mind, very misleading.It is also, as one will readily recognize, in direct opposition to Wicksell'sprofound remark about the rocking-horse.

Neither Wicksell nor Akerman had taken up to a closer mathe-matical study the mechanism by which such irregular fluctuationsmay be transformed into cycles. This problem was attacked inde-pendently of each other by Eugen Slutsky: and G. Udny Yule.§• See, for instance, his address, "Krisemas gatta," delivered to the NorwegianEconomic Society, If)07, Statsekonomisk Tidsskrift, Oslo, 1907, pp. 255-86.t Det ekonomiska livets rytmik, submitted 1925, published Lund, 1928.t The Summation of Random Causes as the Source of Cyclic Processe" vol. iii,no. I, Conjuncture Institute of Moscow, 1927. (Russian with English summary.)§ On a Method of Investigating Periodicity in Disturbed Series, Trans. RoyalSociety, London, A, vol. 226, 1927.

198


In this connection may also be mentioned a paper by HaroldHoteIling.·

Slutsky studied experimentally the series obtained by performingiterated differences and summations on random drawings (lotterydrawings, etc.). Yule only used second order differences, but tried tointerpret the random impulses concretely as shocks hitting an oscillatingpendulum. By the experimental numerical work done by these authors,particularly by Slutsky, it was definitely established that some sort ofswings will be produced by the accumulation of erratic influences, butthe exact and general law telling us what sort of cycles that a givenkind of accumulation will create was not discovered.

Later certain mathematical results which are of interest in con-nection with this problem were given by Norbert Wiener.t

But still the main problem remained, both with regard to themechanism by which the time shapes of the resulting curves are deter-mined and with regard to the concrete economic interpretation. Inthe present section I shall offer some remarks on these questions.For a more detailed mathematical analysis the reader is referred to apaper to appear in one of the early numbers of Econometrica.

Consider for simplicity an oscillating pendulum· whose movementis hampered by friction. If y denotes the deviation of the pendulumfrom its vertical position, the equation governing the movement ofthe pendulum will be(I) j + 2,8y + (0.2+ ,82)y = 0

where y and j are the first and second derivatives of y with respect totime, and fJ and a are positive constants, fJ expressing the strength ofthe friction. The equation expresses the fact that the net force acting onthe pendulum (and being expressed by the acceleration 5') is made up oftwo factors. First a factor which tends to make the force proportionalto the deviation y .(and of opposite sign). This gives the gross forceexpressed by the last term of the equation. From this gross force mustbe subtracted the effect of the friction, and this effect is proportionalto the velocity y and is expressed by the second term of the equation.

It is easily verified that the solution of (I) is a function of the formHe-PI sin (</J + at)

where a and fJ are the constants occurring in (I).• Differential Equations Subject to Error, Journal of the American StatisticalAssociation, 1927, pp. 283-314.t Generalized Harmonic Analysis, Acta Mathematica, 1930.


The amplitude H and the phase ef> are determined by the initialconditions. For our present purpose it is convenient to write thesolution in such a way that we can see immediately how the initialconditions determine the curve. If Yo and Yo are the values ofy and y respectively at the point of time t = to the solution maybe written in the form

y(t) = pet - to) .Yo + Q(t - to) . Yowhere peT) and Q(T) are two functions independent of the initialconditions and defined by

Va2 + p2 .(3) peT) = ---·e-PT sm (v+ aT)

a

. asIn v = ~/ 2 C)

'Va + p"p

cos V = ~/ 2 2va +P

By convention the square root in (3) and (5) may be chosen positive.By insertion into the equation (I) it is easily verified that (2) is afunction that satisfies the equation and the specified initial condition.peT) may be looked upon as the solution of (I), which is equal tounity and whose derivative is equal to zero at the origin T = 0, andQ(T) may be looked upon as the solution which is equal to zero andwhose derivative is equal to unity at the origin. These functions satisfyindeed the equation, and we have

{~(o) = 1

P(o) = 0

Q(o) = 0

Q(o) = I

Suppose that the pendulum starts with the specified initial con-ditions at the point of time to and that it is hit at the points of timetl, t2 ••• tn by shocks which may be directed either in the positive orin the negative sense and that may have arbitrary strengths. Let Ykand Yk be the ordinate and the velocity of the pendulum immediatelybefore it is hit by the shock number k. The ordinate Yk is not changedby the shock, but the velocity is suddenly changed from Yk to Yk +ek,where ek is the strength of the shock; mechanically expressed it is thequantity of motion divided by the mass of the pendulum. The concrete

zoo

interpretation of the shock ek does not interest us for the moment.The essential thing to notice is that at the point of time tk the onlything that happens is that the velocity is increased by a constant ek.Let us consider separately the effects produced by the two termsYk and eke From (2) we see that the initial conditions enter linearly.Consequently we can consider Yk and ek as two independent contribu-tions to the later ordinates of the variable. In other words, the fact ofthe shock may simply be represented by letting the original pendulummove on undisturbed but letting a new pendulum start at the point oftime tk with an ordinate equal to zero and a velocity equal to eke Thisargument may be applied to all the points of time. We simply haveto start in each of the points of time t1, t2 ••• tn a new pendulum withan ordinate equal to zero and a velocity equal to the strength of theshock occurring at that moment, and then let all these pendulumscontinue their undisturbed motion into the future. The sum of theordinates of all these pendulums at any given point of time t will thenbe the same as the ordinate y(t) of a single pendulum which has beensubject to all the shocks. In other words, the ordinate y(t) willsimply be

n

(7) y(t) = P(t - to) . Yo + Q(t - to»)1o+LQ(t - tk)ekk=l

If the point.t is very far from the initial point to, and if fJ is positiveso that there is actually a dampening, then the influence of the initialsituation Yo and Yo on the ordinate y(t) will be negligible, that is, theordinate will be

This means that the ordinate y(t) of the pendulum at a given momentwill simply be the c~mulation of the effects of the shocks, the cumu-lation being made according to a system of weights. And these weightsare simply the shape of the function Q(-r). That is to say, y(t) is theresult of applying a linear operator to the shocks, and the system ofweights in the operator will simply be given by the shape of the timecurve that would have been the solution of the determinate dynamicsystem in case the movement had been allowed to go on undisturbed.

The fundamental question which arises is, therefore: If we performa cumulation where the weights have the form Q(T), what sort of time

201

IN HONOUR OF GUSTAV CASSEL

.• shape will the function yet) get?:l The answer to this question is

given by studying the effects oflinear operators on erratic shocks.The result of this analysis is thatthe time shape of the curve willbe a. changing harmonic with thesame frequency a as the oneoccurring in QC'r). By a changingharmonic I understand a curvethat is moving more or less regu-larly in cycles, the length of theperiod and also the amplitudebeing to some extent variable,these variations taking place,however, within such limits thatit is reasonable to speak of anaverage period and an averageamplitude. In other words, thereis created just the kind of curves

\0 which we know from actual stat-.o istical observation. I shall not: ~ attempt to give any formal proof

of these facts here. A detailedproof, together with extensivenumerical computations, will begIven m the above-mentionedpaper to appear in Econometrica.Here I shall confine myself to re-producing the graph (see Fig. 6)of a changing harmonic producedexperimentally as the cumulationof erratic impulses, the weightfunction being of the form (4).

Thus, by connecting the twoideas: (I) the continuous solutionof a determinate dynamic systemand (2) the discontinuous shocksintervening and supplying theenergy that may maintain theswings-we get a theoretical set-up which seems to furnish a


rational interpretation of those movements which we have beenaccustomed to see in our statistical time data. The solution of thedeterminate dynamic system only furnishes a part of the explanation:it determines the weight system to be used in the cumulation of theerratic shocks. The other and equally important part of the explana-tion lies in the elucidation of the general laws governing the effectproduced by linear operations performed on erratic shocks.

The idea of erratic shocks represents one very essential aspect of theimpulse problem in economic cycle analysis, but probably it does notcontain the whole explanation. There is also present another sourceof energy operating in a more continuous fashion and being moreintimately connected with the permanent evolution in human societies.The nature of this influence may perhaps be best exhibited by inter-preting it in the light of Schumpeter's theory of the innovations andtheir role in the cyclical movement of economic life. Schumpeter hasemphasized the influence of new ideas, new initiatives, the discoveryof new technical procedures, new financial organizations, etc., on thecourse of the cycle. He insists in particular on the fact that these newideas accumulate in a more or less continuous fashion, but are put intopractical application on a larger scale only during certain phases ofthe cycle. It is like a force that is released during these phases, andthis force is the source of energy that maintains the oscillations. Thisidea is also very similar to an idea developed by the Norwegianeconomist, Einar Einarsen. - In mathematical language one couldperhaps say that one introduces here the idea of an auto-maintainedoscillation.

Schumpeter's idea may perhaps be best explained by a mechanicalanalogy. Personally, I have found this illustration very useful. Indeedit is only after I had constructed this analogy that I really succeededin understanding Schumpeter's idea. After long conversations andcorrespondence with Professor Schumpeter I believe the analogy maybe taken as a fair representation of his point of view.

Suppose that we have a pendulum freely suspended to a pivot.Above the pendulum is fixed a receptacle where there is water. A smallpipe descends all along the pendulum, and at the lower end of thependulum the pipe opens with a valve which has a peculiar way of

• Gode 08 daarlige tider, Oslo, 1904.


functioning. The opening of the valve points towards the left and islarger when the pendulum moves towards the right than when it movestowards the left. Concretely one may, for example, assume that thevalve is influenced by the air resistance or by some other factor thatdetermines the opening of the valve as a function of the velocity ofthe pendulum. Finally we assume that the water in the receptacle isfed from a constantly running stream which is given as a function oftime. The stream may, for instance, be a constant.

Now, if the instrument is let loose it is easy to see what will happen.The water will descend through the pipe, and the force of reactionat the lower end of the pendulum created by the fact that the wateris emptying through the valve will push the pendulum towards theright, and this movement will continue until the force of gravitationhas become large enough to pull the pendulum back again towards itsequilibrium point. During the return the opening of the valve andconsequently the force that tends to push the pendulum towards theright will diminish, and thus the movement back towards the centralposition will be accelerated. The pendulum which is now returningwith considerable speed will work up an amount of inertia that willpush it behind its equilibrium point away over to a position at theleft, but here again the gravitation will start to pull it back towardsthe centre, and now the valve will widen, and by doing so will increasethe force which accelerates the movement towards the right. In thisway the movement will continue, and it will continue even althoughfriction is present. One could even imagine that the movement wouldbe more than maintained, i.e. that the oscillations would becomewilder and wilder until the instrument breaks down. In order "toavoidsuch a catastrophe one may of course, if necessary, add a dampeningmechanism which would tend to stabilize the movement so that theamplitude did not go beyond a certain limit.

The meaning of the various features of this instrument as anillustration of economic life is obvious. The water accumulating inthe receptacle above the pendulum are the Schumpeterian innovations.To begin with they are kept a certain time without being utilized.Some of them will perhaps never be utilized, which is illustrated bythe fact that some of the atoms in the receptacle will rest thereindefinitely. But some others will descend down the pipe, whichillustrates that these new ideas are utilized in economic life. Thisutilization constitutes the new energy which maintains the oscillations.

The instrument as thus conceived will give a picture of thea04


oscillations but not of the secular or perhaps supersecular tendency ofevolutions. This tendency seems to us to be irreversible because wehave not yet lived long enough to see the decline. It is not difficult tocomplete the instrument in such a way that it will express also thissecular or supersecular movement. We may, for instance, imaginethat the pivot to which the pendulum is suspended is not fixed butslides in a crack in the wall, the crack ascending towards the right.This being so, the whole instrument will move by jumps, and eachjump will carry it to a higher position than before. We only have tofeed the instrument by a constantly running stream of water. Theimpulsion which the water creates as it leaves the valve will maintainthe oscillations, and these oscillations will constitute the jumps whichcarry the instrument steadily to higher levels. Thus there will be anintimate connection between the oscillations and the irreversibleevolution.

It would be possible to put the functioning of this whole instrumentinto equations under more or less simplified assumptions about theconstruction and functioning of the valve, etc. I even think this willbe a useful task for a further analysis of economicoscillations, but I donot intend to take up this mathematical formulation here.

t-JJ3 t1qcECONOMICjl I'~ ESSAYS 1 IN HONOUR OF GUSTAV...

Documents

Transcript of t-JJ3 t1qcECONOMICjl I'~ ESSAYS 1 IN HONOUR OF GUSTAV...