Sraffian Indeterminacy in General Equilibrium

Abstract

The indeterminacy claim for competitive price systems made by Sraffa (1960) is examined by placing Sraffa's work in an intertemporal general equilibrium model. We show that indeterminacy occurs at a natural type of equilibrium. Moreover, the presence of linear activities instead of a differentiable technology is crucial and the indeterminacy is constructed, as in Sraffa, by fixing some or all of the economy's aggregate quantities. On the other hand, an extra condition, that some factors have inelastic excess demand is necessary, and, unlike Sraffa's model, relative prices must be allowed to vary through time. Sraffian indeterminacy and the generic finiteness of the number of equilibria are reconciled by showing that indeterminacy occurs at a measure-zero set of endowments. We use an overlapping-generations model to show that these endowments nevertheless arise systematically and that indeterminacy does not occur when relative prices are constant through time. Near the beginning of his Production of Commodities by Means of Commodities (1960), Sraffa specifies a system of equations that prices must obey if the capital invested in each sector of an economy is to earn the same rate of profit. The system contains one more unknown than equation, leading Sraffa to conclude that prices, and wage and interest rates in particular, are indeterminate. Sraffa's book has since been read as arguing that some outside, non-economic consideration must therefore set prices and the distribution of income. Part of the controversy the book has provoked is due to Sraffa's apparent agreement with the neoclassical emphasis on optimization and competitive markets. The standard Sraffian system of equations is nothing other than the first-order conditions that hold for price-taking, profit-maximizing firms using linear activities. But if Sraffa's assumptions are in accord with neoclassical theory, how can his assertions of indeterminacy be correct? I examine Sraffa's indeterminacy argument and his critique of neoclassical economics by embedding his equations in a general equilibrium model. As we will see, the indeterminacy claim, properly interpreted, correctly describes the equilibria with strictly positive prices. Moreover, the indeterminacy demonstrated here closely duplicates the indeterminacy Sraffa claimed. First, it is prices alone that are indeterminate. I follow Sraffa's striking procedure of fixing the quantities of goods produced: an infinity of (normalized) price vectors support a single vector of aggregate quantities. Second, Sraffa seemed to claim, and his advocates (e.g. Robinson (1961)) have certainly claimed that his book is a critique of marginal productivity theory. Correspondingly, indeterminacy in the present model hinges on marginal products not being everywhere well-defined when there are linear activities; with smooth production sets, our indeterminacy arguments do not work. Third, the dimension of the indeterminacy I demonstrate matches Sraffa's accounting of the dimension of indeterminacy.