Bennett McCallum - The Instability of Kaldorian Models

7/28/2019 Bennett McCallum - The Instability of Kaldorian Models

http://slidepdf.com/reader/full/bennett-mccallum-the-instability-of-kaldorian-models 1/11

The Instability of Kaldorian ModelsAuthor(s): B. T. McCallumSource: Oxford Economic Papers, New Series, Vol. 21, No. 1 (Mar., 1969), pp. 56-65Published by: Oxford University Press

Stable URL: http://www.jstor.org/stable/2662353 .

Accessed: 24/06/2013 15:08

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

.

Oxford University Press is collaborating with JSTOR to digitize, preserve and extend access to Oxford

Economic Papers.

http://www.jstor.org

This content downloaded from 190.232.242.27 on Mon, 24 Jun 2013 15:08:45 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=oup

http://www.jstor.org/stable/2662353?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp





http://www.jstor.org/stable/2662353?origin=JSTOR-pdf

http://www.jstor.org/action/showPublisher?publisherCode=oup



THE INSTABILITY OF KALDORIAN MODELS1

By B. T. McCALLUM

DESPITE the vintage fsome of the writings, t appears that interest n

Nicholas Kaldor's macro-economicmodels of growth,distribution, ndtechnicalchangeis verymuch alive.2 Empiricalevidencefor hisproposi-tion exists in the formofseveralrecent publications: lengthydiscussionsofKaldor's modelsappear in R. G. D. Allen's Macro-EconomicTheory 1]

and in the impressive urveyarticleofHahn and Matthews 3]; reprintsof Kaldor articles appear in two new collectionsof readings;3and even

textbooks designedforintroductory ourses referto Kaldorian theories

[9, 16].Furthermllore,aldor's technical progress hypothesis has been pro-

nouncedas 'ripeforempirical esting'by Hahn and Matthews[3, p. 889];

Scitovskyhas suggestedthatKaldor's distribution heory, n conjunctionwith one of Phelps Brown and Weber, is 'the most, perhaps the only

satisfactorymacro-economicheory f ncomedistribution' 17]; andRederhas attemptedan empiricaltest [15].

Alongwiththeir dmirers, owever,Kaldor's ideas have attractedtheir

share of critics.4 Many of these have probably been impressed as haveadmirers) by the extent to which Kaldor has departed fromneoclassicalorthodoxy. Othershave criticized he writings n theirown grounds,at

least two focusingupon the distributionmechanismwhichis commonto

all thesemodels.5

In thispaper it is arguedthatinstabilitys a pervasivefeature fKaldor-ian models. While previouswritershave questionedthe stability of such

models in the sense of 'disequilibriumdynamics',we are here concerned

onlywith equilibriumdynamics'. The sortof nstabilityn question,then,

is such that (assuming forces exist to keep the model economy at full

employnentofcapital and labour) capital accumulationwill take place at

1 The author is indebted to J. K. Whitaker,John Conlisk,and Marian Krzyzaniak forimportant uggestions.

2 While Kaldor has several more papers in the field,his basic ideas are put forth n[4], [5], and [7].

3 Kamerschen's collectionofReadings in Microeconomics8] contains Kaldor's distribu-tion paper [4]. The Kaldor-Mirrlees article [7] appears in Mueller's Readings inMtacroeconomics13].

4 Modigliani [11], Tobin [18], and Weintraub [19], among others.5 Marrisfinds tsignificanthatKaldor has devoted onlya small fraction fhispublished

words on the topic to 'his account of the modus operandiof the distributivemechanism'[10, p. 309], while Moore has found it necessaryto 'reformulatethe Kaldor effect' [12].Allen also mentionsthe possibilitythat thismechanismwill not 'work' [1, p. 310].







B. T. McCALLUM 57

a rate differing rogressively romthat which can be maintained overtime.1 n otherwords,growth quilibriawilldivergefrom he steady-statepath.2

Two limitations nwhatfollows hould bementioned.First,onlythenon-vintageversionofKaldor's technicalprogress unctionwill be examined.Second,Kaldor-typemodels ratherthan the preciseversionspresentedbyKaldor will be treated owingto the analytical 'messiness' of the latter.Subject to these limitations, hispaper will argue that Kaldorian growthand distributionmodels are unstable under rathergeneral conditions.

In Section I a streamlinedversionofKaldor's basic growthmodelwillbe presented; ts instabilitywill be demonstrated n II. Sections III and

IV will presentgeneralizationson the results of II and a briefconclusionwillfollow.

I

Kaldor's basic non-vintage growth model, which incorporates hisdistribution heory,has been well summarizedby Allen [1, pp. 305-12].3His treatment nd notation will be followedratherclosely n this section.

The model assumes fullemployment fcapital and labour but, in con-trastwithneoclassicalmodels, ncorporates unctionspecifyingggregative

behaviourofbothsavingand investment.This ispossiblebecause marginalproductivity actorpricing s not followed; ncome distribution djusts tosatisfythese specifications. Aggregate saving S depends upon incomedistribution ince s,, the marginaland average propensity o save out of'profits' ,4 differs rom W,the propensity o save out ofwages Y-P:

S SWY+(s, -8.)P (o < SW < so < 1). (1)

The investmentfunctionspecifiesthat the desired capital-output ratio

K*/Y is an increasing inearfunction f the profit ate PIK:K*/Y v+P/IK (v,8 > 0). (2)

As usual in growthmodels abour growsexponentially t the raten:

L eat. (3)

The usual aggregate production functionwith labour augmentingtechnicalchange is replaced, however, by a 'technicalprogressfunction'whichsuggeststhat 'the use of morecapital perworker nevitablyentails

the introductionof superiortechniques which require inventivenessofsomekind',whilemost technical nnovationswhichare capable ofraising

1 Hahn and Matthews [3] questioned the stability n this sense ofKaldor's models butdid not undertake analysis.

2 On these concepts and distinctions ee Hahn and Matthews [3, pp. 781-2].3 One feature ofAllen's versionwhich seems to violate the spirit of Kaldor's writings s

discussed in Section IV below.'Profit' is non-wage incomein the Kaldor scheme.







58 THE INSTABILITY OF KALDORIAN MODELS

the productivityof labour require the use of more capital per man' [5,p. 595]. If we denote output per man and capital per man by y Y/Land k K/L the technicalprogressfunctionmay be written s

~ly F(k/k), (4)wherea dot designatesdifferentiationithrespectto time. The functionin (4) is taken to be 'well behaved' so that it will cut a 450 line from heorigin nd a steady-state rowth ath (with j/y ic/k mn)canbe attained.More precisely the specification s that F(O) > 0, F' > 0, F'(oo) 0,and F" < 0.

These four equations make up the basic model. Equilibrium (fullemploymentof capital and labour) prevails only ifK =K*. Using thisconditionand the S I K identitywith (1) and (2) one can obtain

K SW +[(sv- s^,)/,](K2!Y-vK) (5')

or,dividingby K and subtractingn from ach side,

k/kk (5)

If we defineR1 K/Y as the capital-outputratio prevailingat a point in

time,then its (proportionate) ate of change is related to per capita ratesofchange in capital and incomeby the identity

R k y (6)

Equations (4), (5), and (6) thenyield

W+ (8 R-8)(R)-nF(k/1k) (7)

In steady-stategrowth quilibriumF(k/k) n and the capital-outputratio is unchanging o the equation

0 - + (R-v)-n-qu (8)

can be solved forthesteady-statevalue of R. (See Allen [1, p. 309].)Since (8) is a quadratic, tyieldstwo solutions. In his 1957 articleKaldor

assertedthat for any reasonablevalues ofthe coefficients,he largerrootwill be greaterthan unityand the smaller root less than unitywhen Y isthoughtof as annual income' and concluded that 'normallythe greaterof the two [roots]onlywould be relevant' [5, p. 613]. Allen finds,underthe assumptionv > s?,/(n+rn), that the smaller root implies a solutionwithnegative profits.He concludes that 'the model has a singlesteady-state solution' [1, pp. 309-10].







B. T. McCALLUM 59

The implausibility f the solution corresponding o the smallerroot canbe emphasized by using 'reasonable' coefficient alues and obtainingsolutions to (8), using years as the time unit. For example, the set ofvalues s- 055, s,, 005, n+m 0 04, v 3, and /3 2 gives the

two roots0 06 and 3a . The smallervalue is quite implausible as a steady-statecapital-output ratio,when output s inper annumunits. Such resultsare not very sensitive to the choice of values for the coefficients.

The smaller root of (8) thereforeyields implausible results and thelarger oot s supposed to define he steady-state apital-output ratio whichcan be used in equations (1)-(5) to find teady-statevalues of the variablesforany point in timewhen initial conditions are specified. In the nextsection twill be shown,however, hat the steady-state quilibriumdefinedby this root is unstable.

II

Proof of the instability begins by noting that (6) could be rewritten,usingthe technical progressfunction, s

R _k F(k/k) _+(k/ (9)

Now the first erivativeof bis f' 1- ' and we know that F' < 1 inthevicinity f the solutionto (8) from hespecification fF. Thus we canbe sure that b'> 0 in thevicinity f thesteady-state olution. The secondderivative of bis sb" -F" and since F" < 0 by assumption,we knowthatO" must be positive.

From equation (5) and the R KI/Ydefinitionwe see that

= RU+(P st) (R V)-n. (10)

Thuswe can writek/k G(R) and can verify hat the second derivativeofG is G" 2s /R3.Since SW> 0, G' is positiveforall positivevalues ofR,which are the onlyones ofinterest.

Now using P/R (k/k)and k/k G(R), we can write the crucialfunction 7) as 1/R [G(R)]. Its second derivative is O'G"+b"(G')2and is positiveforpositiveR since O', s", and G" have been found to bepositive. The rate ofchange ofR is therefore convex function f R forpositivevalues of the latter. The function 7) is thusofthe generalshapeindicated by Fig. 1, where the solutions to the quadratic equation (8)arerepresented y RL and Ru. We have seenabove whyKaldor and Allenview RL as unacceptable. The graph instantlyshows that Ru gives anunstablesteady-stategrowthpath,for fP/R is positive,R increaseswithtimeand vice versa, so ifR > Ru, R will increase,while ifR < RU, thecapital-outputratio R will fall.








Instability s thusestablished n thebasic model.1Not onlythe capital-outputratio is unstable,ofcourse. The investment unctionmpliesthatmovementsnR are accompaniedbychanges nthe ncome hares, s would

be expected. In thefollowing ectionthis resultwill be shownto apply tomoregeneralmodelswhich retainthe essentialKaldorian features.

A

0

FIG. 1.

III

As a first eneralization he inear nvestment unction 2) canbe replacedwiththemoregeneralspecification

K* = h(P/K), (11)

whereh is a continuous,monotonic ncreasingfunctionwithh" < 0. Forthese conditions mplythat the inverse function

P*/-(K*IY) H(K*/Y) (11')

has positive first nd second derivatives. The counterpartof (5) thenbecomes

k8?f

+ (sp8 jH(R)-nm, (12)and accordingly

G"(R) 2s3./R3+(P- sw)H`(B), (13)

1 SinceKaldorhas stressedhespecial ase in which here s no saving utofwages, tshould e noted hat f w= 0 then he first erivativef1/R s positive or ll R > 0 sothattheres only neroot.The curve hen utstheR axisfrom elow ndthe ingle ootis thereforenstable.







B. T. McCALLUM 61

so G" > 0 and again the plot of P/R against R has the shape of Fig. Isince b s unchanged.'

Alterationof the technologyspecificationrequires a somewhat moreextensiverevisionof the analysis. But the effectof Kaldor's technicalprogressfunction s of considerable interestsince it involves a severe

k

A

ZL ZU z = eAt/kFIG. 2.

departurefrom he conventionalproductionspecification.We thus wishto examinethe modelwhenthetechnicalprogressfunction 4) is replaced

with a standard neoclassical aggregate production function. Technicalchange s assumed to be of the abour-augmentingHarrodneutral)varietyso that steady-stategrowth s possible.

Instead of (4) let us then utilize a well-behavedproductionfunction,homogeneousofdegreeone:

Y f*(K, e~tL), ( 14)

where A is the rate of technical change, or

Y/K 1/R = f*(1, eA/k) _ f(eA1/k) (14')

with f(O) =0, f' > 0, f" < 0. It will be convenient to let z -e t/k and

let g(z)- 1/f(z) -- R in which case g'(z) < 0 and g"(z) > 0. Then sub-

stituting into (12) one obtains

k/k =s/g(z) +(sp -sw)H[(z)] -n. (15)

The second derivative with respect to z is

Isu~ '+(s 8sw)[H'y`'+H`'(q )2], (16)

which is almost certainly positive.2 Thus we have an equation generally

of the form depicted in Fig. 2.3 In steady-state equilibrium i/k must1 The graphmaynot be quite ofthisshape,but itis apparent from12) and theproperties

ofq thatR/R willgrowas R -? co and as R -O 0. It is possible that therewillbe more or less)than tworootsbut such a case seemsunlikely. In any case, we see that the largestrootwillbe unstable.

2 The 'reasonable' values ofp. 59, for nstance,give a magnitudeto the positive secondterm which is over 200 times that of the negative first ermwhen f * is Cobb-Douglaswith labour elasticityof output equal to 0.65.

3 Remarks similar to those of n. 1 apply here.








equal A, the rate of technical change. But if z > zu, k/k Aso k growsfaster han eAtnd z tends to fall. Conversely,f L < Z < zu, then k/k Aso that z increases. Finally,ifz < ZLthen k/k Aand z falls towardzero.In sum

zurepresents stable solution while ZL representsone which is

unstable. But since z eAt/k,he lower root ZL again corresponds o thegreater capital stock and the largercapital-outputratio.

As suggested bove, this ast result s most significant.For it showsthatthe instabilitydoes not derivefrom he technicalprogressfunction1ndmust therefore e due to the distribution chemewhich is at the heart ofKaldorian models. The instabilityresult holds, of course, in the specialCstationarytate' case inwhichn A 0.

IVThere appears to be one importantway in whichthe Allen formulation

utilized above is contrary o the spirit of Kaldor's writings. While theinvestment unction2) makes thecapital-output ratiodesiredby investorsa function f the existingprofit ate P/K, Kaldor has been explicitaboutrelating he desiredK/Y to the expectedr prospective rofit ate [5, p. 600][6, pp. 210-16]. Thus we are led to examinethe resultofreplacing 2) with

K*1Y v+13r, (17)wherer is the expected rate ofprofit. In order to complete themodelwethen need some specificationof the manner in which expectations areformed. Kaldor has asserted that 'expectations are invariablybased onpast experience' [6, p. 213], a viewwhich s consistentwithour adoptionofan expressionwhichcan be interpreteds a continuous orm fthe widelyused 'adaptive expectations' model [14] or as a continuouslydistributedlag with an exponentialweighting unction 1, pp. 86-93]. Specificallywe

assume dr/dt= o[(P/K)-r] (a > 0), (18)

whichimpliesthat expectationsadjust to wipe out, in a time intervalofunit ength, he fraction of thediscrepancy P/K) r between the actualand expected profitrates. The alternative interpretation f (18) is thatexpected profit t timet, nowdenoted r(t), s relatedto actual past valuesby the expression 00

r(t) w()K (t ) dT,K0

where theweighting unctionwv(-i-)s taken to be wi-r) =a

1 This should come as no surprise as it is well known that a linear teclhnicalprogressfunction,which was used by Kaldor in [5], can be integrated into a constant-returnsCobb-Douglas productionfunction 1, 2, 3]. Specifically,y/y a+biklk when integratedyields Y = CeatKbLlb where C is a constant of integrationwith magnitude dependentupon initial conditions.







B. T. McCALLUM 63

We now wishtoconsiderhowthe use ofr nstead ofP/K in the nvestmentfunction ffectsthe instability conclusions of Section II. There we ex-pressedP/R s q[G(R)] and showedthat the second derivative f he compo-site functions positive. Our current xamination of themodel with profitexpectationscan be limited to the propertiesof the counterpartof G(R)sinceq is the same as in II.

First we use equations (17) and (18) with the equilibrium conditionK = K* to obtain

P 11K r+-r p (R-v) + ad. (19)

Substitution ntothe saving function eads to

A;^8+s _____ ___(R )+ n. (20)

Except for the presence of P this is analogous to equation (10), that is,to k/k G(R). We use equation (9), P - R(1(k/k),o eliminateP andwritetheresult as an implicitfunctionn R and k/k:

#(k/k, ) =-0,.+ + (R-v) + -Ro(klk) n ? (21)

Here8 is used for s -sW)/I3.At this point a demonstration hat the second derivative of k/kwithrespect to X exceeds zero would again imply instability. The situation,however, s not as clear-cut s before. It is shown nthe appendix that theoccurrenceof instability s related to the speed of adjustment of profitexpectations. Specifically hefollowing ondition, ufficient or nstability,is derived: cX &RI'. (22)

The coefficient represents he speed of adjustment of expectations,so(22) says that instabilitywill prevail if profit xpectationsadjust rapidly.

We can use the 'reasonable' numericalcoefficientsf Section I to seewhat sortofadjustmentspeed is impliedby condition 22). The derivativeb'equals 1- ', and therelationshippointedout inp. 62,n. 1 shows thatF' should be roughly qual inmagnitude o theprofitncomeshare. Thussb' s about 0-65 and with values of 8 0-25 and R1 3-1 we see thatinstabilitywill occur fa exceeds 0 5. This value impliesthatitwould take

about 3-2yearsfor nvestors o make an 80percent correctionnerroneousprofit xpectations, if the 'true' figuresbecame available immediately.Whileresponse ags of suchlengthmightbe typical n the implementationofdecisions, t seemsunlikelythat expectationswould be so sticky.

Thus, while suitably sluggish adaptation of profitexpectations couldlend stability o the Kaldor model,we findyet anotherversionto be quiteproneto instability.







64 THE INSTABILITY OF IKALDORIAN MODELS

V

Our conclusions can be summarized briefly.Two steady-state growthpaths typically exist in Kaldorian models, corresponding o different

capital-output ratios. The path for the smaller ratio has previouslybeenrejected as extremelyunrealistic. It is here shownthat the other path isunstable. This resultholds notonly n thebasic Kaldorian model summar-ized byAllen, but also in revisedversions ncorporating1) a more generalinvestment unctionnd (2) a production unction ather hanthe unortho-dox technical progressfunction. If investment s made to depend uponexpected,rather han current, rofitabilityhen nstability ersists xceptwhen expectationsadjust slowly.

Thus while somespecificationsmaysidestep thedifficulty,t is apparentthat there exists a strongtendency toward instability n Kaldor-typemodels,a resultwhich sits ratheruncomfortably ith Kaldor's beliefthat'one of themeritsof .. [his] .. model is that it showsthat the constancyin the capital/output atio, n the shareof profits nd in therate of profitcan be shown to be the consequence of endogenousforces' [5, p. 593].

APPENDIXDefine x = k/k. Then the object here is to examine d2x/dR2forequation (21). Thefirst nd second partial derivatives are

+R = a S+

ORR= 2s.1/R3.

Then d2x/df2is given by

d2x 0 2x~O+ X0 /3(3dR2 (ORR4 x-2+x R + x.rVJIX)/V3x23)

Since +' and +" are positive, the second partials are all positive so (23) will be positiveif (but not only if) O:c< 0 and OR > 0. These two inequalities are not inherent inthe a priori ign restrictionsbut we can use the 'reasonable' iumnericalmagnitudes ofSection I to see that the second inequality will almost surely hold. For + will be zeroon a steady-state path and the value of8 implied by those numbers is about fifty imesas great as the implied value of slvjR2: 8 = 0 25 while sj/R2 = 0-0052.

The other inequality, Ad < 0. then yields the condition (22) of Section IV. Thereis no ambiguity about the number of moots n this case. In steady-state growth k/kis a constant and equation (21) is then a quadratic in R, giving two solutions.







B. T. McCALLUM 65

REFERENCES

1. R. G. D. Allen,Macro-EconomicTheory London: Macmillan, 1967).2. J. Black, 'The technical progressfunction and the production function',

Economica,May 1962,vol. 29, 166-70.

3. F. H. Hahn and R. C. 0. Matthews,The theory f economicgrowth: survey',EconomicJournal,Dec. 1964, vol. 74, 779-902.4. N. Kaldor, 'Alternativetheoriesofdistribution',ReviewofEconomicStudies,

1955-6, vol. 23, 83-100.5. N. Kaldor, A modelof economicgrowth', conomicJournal,Dec. 1957,vol. 67,

591-624.6. N. Kaldor, Capital accumulation nd economicgrowth',nF. A. Lutz and D. C.

Hague, editors,TheTheory f Capital (NewYork: St. MartinsPress, 1961).7. N. Kaldor and J. A. Mirrlees,A new model of economicgrowth',Reviewof

EconomicStudies,June 1962,vol. 29, 174-92.8. D. R. Kamerschen, editor, Readings in MicroeconomicsCleveland: World

Pub. Co., 1967).9. R. G. Lipsey, An Introduction to Positive Economics (London: Weidenfeld and

Nicholson, 1963).10. R. Marris, The Economic Theory of Managerial' Capitalism (London: Macmillan

& Co., 1964).11. F. Modigliani, Comment', nN.B.E.R., The Behavior f ncomeShares, Studies

in Income and Wealth, vol. 27 (Princeton:PrincetonUniv. Press, 1964).12. A. M. Moore,'Areformulationf heKaldoreffect', conomicJournal,Mar. 1967,

vol. 77, 84-99.13. M. G. Mueller, ditor,Readings n MacroeconomicsNew York: Holt, Rinehart

and Winston,1966).14. M. Nerlove,Distributed Lags and Demand Analysis for Agricultural and Other

CommoditiesWashington:U.S. Dept. ofAgriculture, 958).15. M. W. Reder, Alternative heories f abor's share', inM. Abramowitz, ditor,

The Allocation of Economic Resources (Stanford: Stanford Univ. Press, 1959).16. P. A. Samuelson, Economics, An Introductorynalysis, 7th ed. (New York:

McGraw-HillBook Co., 1967).17. T. Scitovsky,A surveyof some theories f incomedistribution',nN.B.E.R.,

TheBehavior f ncome hares,Studies n ncome andWealth,vol. 27 Princeton:PrincetonUniv. Press, 1964).

18. J. Tobin, 'Towards a general Kaldorian theoryof distribution',ReviewofEconomicStudies,Feb. 1960,vol. 27, 119-20.19. S. W'eintraub,An Approach to the Theory of Income Distribution (Philadelphia:

ChiltonCo., 1958).

University fVirginia

4520.1 F

Bennett McCallum - The Instability of Kaldorian Models

Documents