Saturday, January 14, 2017

A Model Of Oligopoly

1.0 Introduction

Suppose barriers to entry exist in an economy. Entrepreneurs and capitalists find that they cannot freely enter or exit some industries. And these barriers are manifested by stable ratios of rates of profits among industries. This post presents equations for prices of production under these assumptions.

I suggest that the model presented here fits into the tradition of Old Industrial Organization, as formulated by Joe Bain and Paolo Sylos Labini. As I understand it, Sylos Labini may have once written down equations like these, but never presented them or published them. I suppose this model is also related to work Piero Sraffa published in the 1920s.

2.0 The Model

Consider an economy consisting of n industries. Suppose the rate of profits in the jth industry is (sj r), where r is the base rate of profits, sj is positive, and:

s1 + s2 + ... + sn = 1

For simplicity, I limit my attention to a circulating capital model of the production of commodities by means of commodities. For the technique in use, let ai, j be the quantity of the ith commodity used to produce a unit of output in the jth industry. Homogeneous labor is the only unproduced input in each industry. Let a0, j be the person years of labor used to produce a unit output in the jth industry. I assume labor is advanced, and wages are paid out of the surplus at the end of production period, say, a year. Then prices of production, which ensure a smooth reproduction of the economy, satisfy the following system of equations:

(a1, 1 p1 + a2, 1 p2 + ... + an, 1 pn)(1 + s1 r) + w a0, 1 = p1
(a1, 2 p1 + a2, 2 p2 + ... + an, 2 pn)(1 + s2 r) + w a0, 2 = p2
. . .
(a1, n p1 + a2, n p2 + ... + an, n pn)(1 + sn r) + w a0, n = pn

The coefficients of production, including labor coefficients, and the ratios of the rate of profits are given parameters in the above system of equations. The unknowns are the prices, the wage, and the base rate of profits. Since only relative prices matter in this model, one degree of freedom is eliminated by choosing a numeraire:

p1 q*1 + ... + pn q*n = 1

Since there are n price equations, appending the above equation for the specified numeraire yields a model with (n + 1) equations and (n + 2) unknowns. One degree of freedom remains.

3.0 In Matrix Form

The above model can be expressed more concisely in matrix form. Define:

  • I is the identity matrix.
  • e is a column vector in which each element is 1.
  • S is a diagonal matrix, with s1, s2, ..., sn along the principal diagonal.
  • p is a row vector of prices.
  • q* is the column vector representing the numeraire.
  • A is the Leontief input-output matrix, representing the technique in use.
  • a0 is the row vector of labor coefficients for the technique.

The model consists of the following equations:

eT S e = 1
p A (I + r S) + w a0 = p
p q* = 1

4.0 Conclusion

One could develop the above model in various directions. For example, one could plot the wage-base rate of profits curve for the technique in use. Of interest to me would be presenting examples of the choice of technique, including reswitching and capital-reversing. The Sraffian critique of neoclassical economics is not confined to the theory of perfect competition.

Update (16 January 2017): I find I have outlined this model before.

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