Wage-Rate Of Profits Curves

1.0 Introduction
I have written about so-called factor price curves and frontiers in many posts. They are so-called because the interest rate is not a price of any factor of production. In this post, I use the more neutral expressions "Wage-Rate of Profits Curve" and "Wage-Rate of Profits Frontier". I consider the concepts denoted by these terms to be elements of mathematical economics that arise, in particular, in the analysis of steady states.

2.0 Derivation of a Wage-Rate of Profits Curve
Consider an economy in which n commodities are produced. Each commodity j is produced in a corresponding industry in which it is the sole output of a single process. This process:

• Requires inputs of labor and commodities. These inputs are represented as a_{0, j} person-years per unit output and a_{i, j} units of the ith commodity per unit output.
• Exhibits Constant Returns to Scale (CRS).
• Requires a year to complete.
• Totally uses up its commodity inputs.

A technique consists of a process for each of the n industries. The technique is represented by the row vector a₀ of direct labor coefficients and the square Leontief Input-Output matrix A. Assume:

• Each commodity enters either directly or indirectly into the production of all commodities. That is, all commodities are basic in the sense of Sraffa.
• The economy is viable. That is, there exists a level of operation of all processes such that the outputs can replace the commodities used up in their production and leave a surplus product to be paid out in the form of wages and profits.
• Wages are paid at the end of the year.
• The same rate of profits is earned on advances in all industries.

The assumptions of CRS and of all commodities being basic are made for ease of exposition.

Under these assumptions, the constant prices that allow the economy to smoothly reproduce satisfy the following system of n equations:

p A (1 + r) +w a₀ = p

where p is the row vector of prices, w is the wage, and r is the rate of profits. Given the rate of profits, this is a linear system in n + 1 variables. The last equation imposed in the model sets the value of the numeraire to unity:

p e = 1

where e is a column vector denoting the units of each commodity that comprise the numeraire. Only solutions in which all prices are positive and the wage is non-negative are considered.

The price equation can be transformed into:

w a₀ = p [I - (1 + r)A]

where I is the identity matrix. Or:

w a₀ [I - (1 + r)A]^-1 = p

where the assumption of viability guarantees the existence of the inverse for all rates of profits between zero and a maximum rate of profits. Right multiply both sides of the above equation by the numeraire:

w a₀ [I - (1 + r)A]^-1 e = p e = 1

The wage-rate of profits curve for the technique is then:

w = 1/{a₀ [I - (1 + r)A]^-1 e}

3.0 Properties of Wage-Rate of Profits Curves
The Wage-Rate of Profits Curve for a technique, under the assumptions above, has the following properties:

• There is a finite maximum rate of profits for which the wage is zero. (If no commodity were basic, this maximum would not be finite.)
• There is a maximum wage for which the rate of profits is zero.
• The wage-rate of profits curve is strictly decreasing between the rate of profits of zero and the maximum rate of profits.
• The wage rate of profits curve can be both convex to the origin and concave to the origin. (If the number of commodities n is greater than 2, the convexity can vary throughout the curve.)
• If the vector of direct labor coeffients is a left-hand eigenvector of the Leontief Input-Output matrix, the wage-rate of profits curve is a straight line, that is, affine. (This is Marx's case of equal organic composition of capitals.)
• If the numeraire is a right-hand eigenvector of the Leontief Input-Output matrix, the wage-rate of profits curve is affine. (This is the case of Sraffa's standard commodity.)

Figure 1 illustrates the wage-rate of profits curve for five techniques (α, β, δ, ε, and τ). Pasinetti uses π, not r, to denote the rate of profits. These curves are drawn under the assumption that the organic composition of capitals is not constant for any technique, and the numeraire is not the standard commodity for any of the techniques. Figure 1 also shows the wage-rate of profits frontier, formed from the outer envelope of all the wage-rate of profits curves for the individual techniques. This frontier is used to analyze the choice of technique for long-period, circulating capital models with single production.

Figure 1: The Frontier Formed From Factor-Price Curves (from Pasinetti (1977), p. 157)

Selected References

• Heinz D. Kurz and Neri Salvadori (1995) Theory of Production: A Long-Period Analysis, Cambridge University Press
• Heinz D. Kurz and Neri Salvadori "Production Theory: An Introduction"
• Luigi L. Pasinetti (1977) Lectures on the Theory of Production, Columbia University Press