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The Review of Economic Studies Ltd. The Pasinetti Paradox in Neoclassical and More General Models Author(s): Paul A. Samuelson and Franco Modigliani Source: The Review of Economic Studies, Vol. 33, No. 4 (Oct., 1966), pp. 269-301 Published by: The Review of Economic Studies Ltd. Stable URL: http://www.jstor.org/stable/2974425 Accessed: 22/03/2010 09:23 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=resl. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. 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]. The Review of Economic Studies Ltd. is collaborating with JSTOR to digitize, preserve and extend access to The Review of Economic Studies. http://www.jstor.org The Pasinetti Paradox in Neoclassical and More General Models' I. INTRODUCTION In 1962Dr Pasinetti2 enunciateda remarkableresult. Pasinetti's Theorem: Consider a system in which the labour force grows at some exponentialrate n', technologicalprogress is Harrod-neutraland occurs at a constant rate n", and therefore the " effective" laboursupply, measured in " efficiency" units,rises at the raten = n'+ n'.' Suppose further that one can meaningfullyidentify a class of income receivers-the " capitalists"-whose sole source of income is earningsfrom capital, and suppose that this grouphas a propensityto save (averageand marginal)equal to s,. Call the remainder of the community" workers" and assumethey save the fractions, of theirwage or interest income. Then, if the systemis capableof generatinga " golden-age" growthpath along which income, consumptionand capital all grow exponentiallyat the " natural rate " n, this equilibriumgrowth path has the followingremarkable properties: 1. The steady-staterate of returnto capital or rate of interest,r*, dependsonly on the rate of growth n and on sc according to the simple formula r* = n/se; it is therefore completelyindependent of the workers'saving propensity s, or of the form of the production function. 2. The steady-statecapital output ratio (KIY)*, the capital labour ratio (K/L)* and the share of income going to capital a* = (rKIY)* are also independentof s,; again, they dependon n/sc, but also on theform of the productionfunction. From this surprisingtheorem come other remarkablecorollaries such as: if sc is unity, a situationthat Pasinettiassociates (pp. 277-278)in particularwith a socialist state -although the associationis questionable,as indicatedbelow-then it follows from (1) that r will tend to the naturalrate of growthn, and savingwill approachthe Swan-Phelps4 golden rule of accumulation S = I = P, where P = rK is total income from capital. A result of this generalityputs us all in debt to Dr Pasinetti,and is worthyof further study and elucidation,which is what we offerin this paper. 1 ,We should like to acknowledgehelful researchassistance from Felicity Skidmore,and financialaid from the Ford Foundation and the Carnegie Foundation. Helpful comments were received from the Harvard-M.I.T.mathematical economics colloquium. In' particularwe are indebted to R. Solow and G. LaMalfa for helpful criticism. 2 Luigi L. Pasinetti, " Rate of Profit and Income Distribution in Relation to the Rate of Economic Growth ", Reviewof EconomicStudies, 29 (1962), 267-279, a seminalpaper. 3 To sidestep the complicationsthat come in when KIL goes asymptoticallyto zero or infinity (when capital is incapable of being widened fast enough or slow enough), we first posit that positive labour is needed to produce positive output, and that n is positive but not too large. We remove this simplifying restrictionin the Appendix which gives an exhaustiveanalysis of possible pathologicaldivergences. 4 T. W. Swan, " Of Golden Ages and Production Functions ", presented at the Round Table on Economic Development in East Asia (InternationalEconomic Association), Gamagori, Japan, April 1960 (revised 1962), mimeo., 18 pp.; E. S. Phelps, "The Golden Rule of Accumulation", American EconomicReview, 51 (1961) 628-643; J. Robinson, " A Neoclassical Theorem", Review of Economic Studies,29 (1962), 219-226, with Commentsby R. M. Solow, P. A. Samuelson,J. E. Meade et al.; C. C. von Weizsacker,Wachstum, Zins und OptimaleInvestitionsquote (Kyklos, Verlag,Basel, 1962), 96 pp. T 269 270 REVIEW OF ECONOMIC STUDIES (a) First, we shall show the limited range of the workersand capitalistssaving co- efficientswithin which the above formulationof the PasinettiTheorem is valid. Outside that range we formulatea theoremthat is dual to it and of the same generality. It too involves a paradox, namely that the averageproduct of capital-the reciprocalof the capitaloutput ratio-to whichthe systemsettles is, this time, equalto n/sw and completely independentof the sc propensityto save out of profit or of the form of the production function. All the other golden-agevariables of the system depend only upon n/sw and on the form of the productionfunctions. The completeduality of all this with the Pasinetti theoremis notableand we state the generalcase that covers all contingencies. (b) A second, though relativelyminor purpose, is to help dispel the notion, which seemsto have been entertainedby some readers,that Pasinetti'sanalysis has some peculiar relevanceto a Kaldorianalternative theoryof distribution,of the type presentedby Kaldor in 1955, 1957, 1961, and 1962, or to some versionof a " Cambridge"theory of distribu- tion.' The following lead sentence of Pasinetti(and indeed his whole lead paragraph) might predisposethe readerin the streetto this view. " One of the most excitingresults of the macro-economictheories which have recently been elaboratedin Cambridgeis a very simple relationconnecting the rate of profit and the distributionof income to the rate of economicgrowth, through the interactionof the differentpropensities to save"9(p. 267). Otherpassages may give the impressionthat the majoraccomplishment of his analysis is to remove " a logical slip ' (p. 270) in the Kaldorianformulation which allowed for some saving by workers but did not permit these savings to accumulateand produce income.2 Actually, as Dr Pasinettimakes clear at many places (and in particularin the long footnote on p. 276) his analysisis one of the greatestgenerality. His theoremapplies in fact to any system capable of a golden-agegrowth path. But, preciselybecause of this great generality,his analysis can in no way help us to discriminatebetween alternative theories of income distribution. In order to make this point perfectlyclear, and for its own sake, our own analysisshall deal primarilywith a neoclassicalproduction function capableof smooth factor substitutionand with the case of perfectlycompetitive markets. Later we shall providesome indicationsof how the analysismight be extendedif either of these conditionsfails. (c) Finally, we shall investigate,and prove, the stabilityof the Pasinettigolden-age in the case where it is valid. That is, we shall prove that the system will asymptotically approachthe steady-statefrom arbitraryinitial conditions,at least in a local neighbour- hood of that unique state. And where our anti-Pasinettigolden age holds, in which the workers'saving propensity is all important,we demonstrateits global asymptoticstability. Our asymptoticstability analysis, which can be extendedin considerablemeasure to certain cases of fixed-proportionsand distributiontheories differentfrom that of 1 N. Kaldor, " AlternativeTheories of Distribution", Reviewof EconomicStudies, 23 (1955), 83-100 (reprintedin Essays on Valueand Distribution,pp. 228-236); " A Model of Economic Growth ", Economic Journal,67 (1957), 591-624 (reprintedin Essays in EconomicStability and Growth,pp. 256-300); F. Lutz and D. C. Hague (eds.), The Theoryof Capital(1961), pp. 177-220,from the 1958 CorfuI.E.A. Conference; with J. A. Mirrlees," A New Model of EconomicGrowth," Review of EconomicStudies, 29 (1962), 174-192. Cambridgewritings related to Kaldorism,but not necessarilyidentical with it, are J. Robinson, Accumulation of Capital (1956) and Collected EconomicPapers, 2 (1960), particularlypp. 145-158, " The Theory of Distribution"; Exercisesin EconomicAnalysis (Macmillan, London, 1960),Essays in the Theoryof Economic Growth(Macmillan, London, 1962); D. G. Champernowne," CapitalAccumulation and the Maintenance of Full Employment", EconomicJournal, 67 (1958), 211-244; R. F. Kahn, " Exercisesin the Analysis of Growth ", OxfordEconomic Papers, New Series, 10 (1958), 143-156. 2 Actually, it might be argued that there need not be a " logical slip " in the Kaldorian model, if it will merely assume that the propensity to save out of income from capital is s, whether that income is received by capitalists