Variation in Standard Commodity with Relative Markups

by
Robert Vienneau
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I am not sure about the economic logic in this post. Maybe somebody like D'Agata or Zambelli could do something with this. These ideas were suggested to me by email with a sometime commentator.

I start out with notation for Sraffa's price system, modified in an unusual way to allow for persistent variations in the rate of profits among industries:

• a₀ is a row vector of labor coefficients in each of n industries.
• A is a Leontief input-output matrix, where a_{i, j} is the quantity of the ith commodity needed as input to produce one unit of the jth commodity.
• S is a diagonal matrix, where all off-diagonal elements are zero. s_{j, j} is the markup on non-labor costs in the jth industry.
• p is a row vector of prices.
• w is the wage.
• r is the scale factor for the rate of profits.

The coefficients of production, as expressed in the labor coefficients and the Leontief matrix are given parameters. Relative markups are also taken as given. Prices, the wage, and the scale factor for the rate of profits are the unknowns to be determined. My problem is to find a numeraire such that the wage and the scale factor for the rate of profits trade off in a straight-line relationship, at least when labor is advanced and wages are paid out of the net product:

r = R (1 - w)

I assume all elements of A are non-negative and that all elements of a₀ and all diagonal elements of S are positive. The economy is assumed to be viable, that is, as capable of producing a surplus product. For simplicity, assume that the Leontief matrix is indecomposable. More generally, I need A S to be a Sraffa matrix.

For my purposes here, I formulate price equations as so:

p A S (1 + r) + a₀w = p

Consider the case when wages are zero and the scale factor for the rate of profits is at its maximum R:

p A S (1 + R) = p

Or:

p A S = (1/(1 + R)) p

I observe that prices are a left-hand eigenvector of the matrix A S, with (1/(1 + R)) the corresponding eigenvalue. To ensure that prices are positive, of the n eigenvalues, choose the maximum. The maximum eigenvalue is also known as the Perron-Frobenius root of A S.

Let y^* be a right-hand eigenvector of A S corresponding to its Perron-Frobenius root. Let q^* be gross output such that the net output is y^*:

y^* = q^* - A q^*

These quantities flow define the standard system here, when scaled so as employ a unit quantity of labor:

a₀q^* = 1

The net output of the standard system is the desired numeraire:

p y^* = 1

With this definition of the standard system, the ratio of physical gross outputs to circulating capital inputs varies among commodities. This result contrasts with Sraffa's standard system. I suppose I could restore this property by choosing q^*, not y^*, to be an eigenvector. Either way, the ratio of net outputs to circulating capital inputs varies among industries. Either way, the relative ratios of commodities in the standard industry depends on relative markups.

Do Marx's invariants hold with the above definition of the standard system? I expect not. Nevertheless, does this mathematics provide some insight into classical or Marxist political economy?