Comments on Yoshihara and Kwak on Sraffian Indeterminancy

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Robert Vienneau
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Yoshihara and Kwak (henceforth YK) presented a paper, on Sraffian indeterminacy, at the last annual meeting for the American Economic Society. I want to register my qualified disagreement.

YK are arguing against Michael Mandler. In a 1999 paper, a book, and a series of papers since, Mandler has been criticizing Sraffa and his followers. In Mandler's reading, Sraffa argues that, in neoclassical theory, the distribution of income is generically indeterminate. That is, for any long-period equilibrium solution, one can find another solution as nearby as you want. Or, in still other words, equilibrium solutions are a continuum in some space. In much simpler terms, any distribution and prices along the wage-rate of profits curve is a long-period equilibrium in a circulating capital model with a single technique. Mandler says that Sraffa is more-or-less mistaken.

YK say that Mandler is correct if one confines oneself to stationary equilibria. But, if one considers steady states, with a not-necessarily zero, endogenous rate of growth, then indeterminate equilibria are generic.

I should offer some clarifications and caveats. First, Mandler's claim is consistent with multiple, non-unique equilibria. Equilibria are not indeterminate, as long as the number of equilibrium is finite or, I guess, at most countable. So, if I produce a model in which several points on the wage frontier are neoclassical equilibria, I am not offering a counter-example or disproof of Mandler's claim that Sraffian equilibria are determinate.

Second, Mandler has a caveat. In particular, he argues indeterminancy arises in a short-run model with technology modeled as Leontief matrices. The capital goods that exist at the start of any period are the result of production in the previous period. If they were taken as given, some might be in excess supply with an equilibrium price of zero. And those not in excess supply would have a determinate price. But there is a boundary, just before capital goods are in excess supply price. There, equilibrium prices would range from zero to some upper bound. For strategic reasons, managers of firms have an incentive to produce just this quantity of capital goods.

Third, YK are arguing in the framework of a model of overlapping generations. The production technology is specified by a Leontief matrix. The model is extended to include utility maximization by households. Each household must decide how much and what to consume out of wages and what to consume, at the end of a second period, out of retirement savings. I think assumptions that households only live for two periods and that they must work the first period and be retired the second period are inessential. Their results are most likely consistent with labor supply being determined by including a preference for leisure in the utility function. And one can have households lasting more than two periods, with more than two generations being included in the demand for consumer goods at any period.

I have not read Sraffa for decades, if ever, in line with Mandler. Sraffa does not mention utility functions. And he does not model quantity equations, although I find it natural to add a system of steady state growth to Sraffa's model. Sraffa says he intends his book to provide a foundation for a critique of (neoclassical?) economics, but he certainly presents problems of interpretation for stating what that critique is.

I think of Sraffa as presenting an open model. I guess one can say his solutions are indeterminate, but I do not see him as saying that a neoclassical closure is necessarily indeterminate.

Rather Sraffa shows that one can still model prices and distribution without any reference to subjective, neoclassical theory. He presents a model that can be closed in various ways. Neoclassical theory is only one approach out of others. Furthermore, Neoclassical economists do not seem to have any theoretical foundation for their vision of prices as indices of relative scarcity, as reflecting the result of an overriding principle of substitution.

As I understand it, YK define a steady-state equilibrium by a stationary vector of relative prices, wage, rate of profits, a vector of gross outputs, and a rate of growth. The gross outputs and the rate of growth specify a time path for gross outputs and employment, all components of which grow at the steady state rate.

One can vary the rate of growth continuously in a certain range. Since the parameters of the household utility functions are given, the steady state distribution of income and, consequently, prices must vary too. Voila, steady state equilibrium are indeterminate.

I guess this is consistent with an extension of Mandler's concept of indeterminancy. But it does not seem in the spirit of neoclassical economics, which is the about the allocation of given resources. In my excursions into models of overlapping generations, I have always taken the rate of growth of the population as given, that is, exogenous. I have seen, at least, that if one varies certain parameters in the utility function, the stationary state equilibrium varies continuously. By labeling this a model of endogenous time preferences, have I proven Mandler wrong, even for stationary state equilibria? Do not these disproofs, if that is what they are, even work for aggregate Cobb-Douglas production functions?

I am not at sure these are at all questions that can be settled by mathematical modeling. Sraffa presents an open model. Why feel obligated to close it with a formal model, much less with neoclassical assumptions of utility maximizing?

Instead, one can say that Sraffa has presented a model where one can find a place for political forces to impact the distribution of income, in all runs. One can use Sraffa as a justification for, for example, looking at the impact of the policies of the Federal Reserve on income distribution, without being required to create a formal model at the level of abstraction of Sraffa's book.

Likewise, isn't the question of whether the size of the workforce varies endogenously also an empirical question? Offhand, I think of how the labor force participation rate has varied over the last decade, with the advent of the Global Financial Crisis; how estimates of the Non-Accelerating Inflation Rate of Unemployment (NAIRU) have fallen with unemployment over decades; and the increased participation of women in the workforce during World War II as cases to pursue.

• James K. Galbraith (2000). Created Unequal: The Crisis in American Pay, University of Chicago Pay
• Michael Mandler (1999). Sraffian Indeterminancy in General Equilibrium. Review of Economic Studies 66: 693-711.
• Frank Hahn (1982). The Neo-Ricardians. Cambridge Journal of Economics 6: 353-374.
• Stephen A. Marglin (1984). Growth, Distribution, and Prices. Boston: Harvard University Press.
• Naoki Yoshihara and Se Ho Kwak (2017). Sraffian Indeterminacy in General Equilibrium Revisited. Proceedings of the American Economic Society