Kalecki And Sraffa: Compatible?

Two Great Economists

1.0 Introduction

Michal Kalecki set out macroeconomic models in which markup pricing was common. Economists in this tradition rarely explore the effect of inter-industry flows on prices. Sraffians, on the other hand, usually specify prices, at least, to a first approximation, in a model of full competition. Can work in the traditions of Michal Kalecki and of Piero Sraffa be usefully combined?

2.0 A Model

Consider an economy in which n commodities are produced by n (single-product) industries. Inter-industry flows are described by a nxn matrix A, where a_{i, j} is the amount of the ith commodity used as input per unit output in the jth industry, at the given level of output of the jth industry. Labor inputs are described by the row vector a₀, where a_{0, j} is the quantity of labored hired in the jth industry per unit output, at the given level of output of the jth industry.

The positive constants m₁, m₂, ..., m_n represent barriers to entry among the different industries. The going rate of profits is earned in industries in which m_j is unity. Industries in which m_j exceeds unity have high barriers to entry. Perhaps a large scale of production is needed to operate profitably in such an industry. Industries with m_j less than unity are backwards, in some sense. At any rate, they earn less than the going rate of profits. These constants lie along the principal diagonal of the diagonal matrix M. That is, m_{i, j} is m_j, for i equal to j. And m_{i, j} is zero, for i unequal to j.

The row vector p represents prices, where p_j is the price of a unit quantity of the output of the jth industry. Suppose w represents the wage, and r represents The rate of profits.

The matrix A, the row vector a₀, the diagonal matrix M, and one of the distributive variables (say, the rate of profits r) are the given data for this model. The prices p and the remaining distributive variable (for example, wages w) are the unknowns to be found. One can set out the (modified) Sraffa equations for prices:

(p A M + a₀ w)(1 + r) = p

(I think models of full cost prices typically show markups being earned on both labor and material costs.) A numeraire should be specified. For example, one can set out the following normalization:

p₁ + p₂ + ... + p_n = 1

Likewise, the markups are only specified by the model, so far, up to a scalar multiple. I suggest the following normalization condition for markups:

m₁ x m₂ x ... x m_n = 1

Presumably this model can be extended, as in Sraffa (1960) to embrace fixed capital, land, joint production in general, and an analysis of the choice of technique.

3.0 Conclusion

The above has set out a model of prices of production. This model provides a framework for analyzing both the effects of inter-industry flows on prices and of markup pricing, arising from barriers to entry and other hindrances to full competition. The compatibility of some such model with both Kaleckian macroeconomics and the larger research agenda of Sraffa remains to be argued. Likewise, I have not shown the usefulness of this sort of model in empirical explanations of actual capitalist economies. One important issue in such discussions would probably be the applicability of models of prices of production to industries in which the planned operating level is less than full capacity.

This post should really have a bibliography, since the question of the compatibility of the economics of Kalecki and of Sraffa has been raised before. I gather that Paolo Sylos Labini, in some unpublished work in the 1960s, set out and analyzed a model rather like the above.