Rate Of Profits As A Weighted Sum Of Value Rate Of Profits

1.0 Introduction

Stefano Perri has a working paper, "Sraffa's response to Eaton's review: a note on the standard commodity and Marx's general profit rate". In this post, I present the analysis rewritten in matrix notation. The claim is, more or less, that the rate of profits in the system of prices of production is the weighted average of the rates of profits, by industry, in the system of labor values. Each weight is the product of the labor value of capital advanced in that industry and the quantity of the gross output of that industry in the standard system.

This post is unsatisfying. I end up with a claim that I do not see how to prove offhand.

2.0 Parameters and Assumptions

The setting is a model of circulating capital, with wages advanced and specified as a given commodity basket. Let A be the n x n Leontief input-output matrix, with a_i,j being the quantity of the ith commodity used as an input in producing the jth commodity. Each row of A represents the inputs of a commodity in the various industries, while each column represents an industry. Let a₀ be the n-element row vector of direct labor coefficients. a_0,j is the quantity of labor employed in manufacturing the jth commodity

The units of measurement for the commodities are chosen such that one unit is produced in each industry. That is, iron ore might be measured in units of 253 million tons, while red winter wheat might be measured in units of 1.6 billion bushels. (I am making such numbers up.) The gross output is then u, a unit column vector. Employment is measured such that total employment is one unit:

a₀ u = 1

Net output, y, is a column vector:

y = u - A u

Assume the Leontief matrix is indecomposable and productive. These assumptions assure the existence of the Leontief inverse (I - A)^-1.

Net output is divided between wages and profits here. Assume wages are advanced and specified in physical terms by the column vector d. Wages are bounded above by net output:

d ≤ u - A u

Since both commodities that function as capital goods and wage goods are advanced, one can form a new matrix:

A* = A + d a₀

Each column of A* is the total commodities advanced in an industry.

3.0 Labor Values

The data determine the labor value embodied in each commodity. Consider the jth commodity. Let e_j be the jth column in the identity matrix. Suppose this is the net output of the economy produced with this technique, where q is now the gross outputs of the economy:

e_j = q - A q

The labor value embodied in this commodity is found from the vector of direct labor coefficients and the Leontief inverse:

ν_j = a₀ q = a₀ (I - A)^-1 e_j

The row vector of labor values is then:

ν = a₀ (I - A)^-1

The reproduction of the net output of the jth industry requires workers distributed across all industries to be reproducing the capital goods used up in making this net output, the capital goods used up in making those capital goods, and so on. Labor values reflect a notional vertical integration of the observed quantity flows, with the observed technique. One can ask how much employment would be increased by a sustained increase in a given industry. So-called employment multipliers answer this question.

One can evaluate the commodities advanced in production with labor values. The labor value of the constant capitals is a row matrix, C:

C = ν A

One can also evaluate the labor value of the commodities which the workers purchase with their wages. The labor value of variable capital is a row matrix, V:

V = ν d a₀

Let T be a diagonal matrix with

t_j,j = C_j + V_j

Each entry along the diagonal is the sum to the constant and variable capital advanced in the corresponding industry.

Let Π be a diagonal matrix in which the elements along the diagonal are the value rate of profits for the industries. Postulate that workers add the same value in each industry, despite the varying amounts of capital equipment with which they work. One then has, as a matter of accounting, the following equality in the system of labor values:

(C + V)(I + Π) = ν

Multiply on the right by the inverse of T:

(C + V)(I + Π) T^-1 = ν T^-1

Multiplication, when restricted to diagonal matrices, is commutative. Hence, one has:

(C + V) T^-1 (I + Π) = u^T (I + Π) = ν T^-1

Or:

[(1 + π₁), ..., (1 + π_n)] = [ν₁/(V₁ + C₁), ..., ν₁/(V_n + C_n)]

The sum of unity and the value rate of profits in each industry is the labor value of the output of that industry, normalized by the capital advanced.

4.0 Standard System

Here, as I understand it, Sraffa defines the standard system with the Leontief matrix including the commodities the advanced wage goods. The gross outputs of the standard system, q*, are defined as follows:

A* q* (1 + r) = q*

Employment in the standard system is unity.

a₀ q* = 1

Gross outputs are an eigenvector, with 1/(1 + r) as the Perron-Frobenius root of the matrix A*.

The net output of the standard system, y*, is defined as:

y* = q* - A* q*

Net output is then:

y* = (r/(1 + r) q*

Net output of the standard system, being proportional to gross outputs, is also an eigenvector.

5.0 Prices of Production

Prices of production are defined as:

p A* (1 + r) = p

Prices of production are a left-hand eigenvector of A* for the same eigenvalue, the Perron-Frobenius root of A*. Take the net output of the standard system as numeraire:

p y* = 1

At this point, the gross output of the standard system and the rate of profits for the system of prices of production have been defined.

One can multiply the row vector defined above, of the sum of unity and the value rate of profits by the column vector T q*:

u^T (I + Π) T q* = ν T^-1 T q*

Or:

u^T (I + Π) T q* = ν q*

I claim that the labor value of the gross outputs of the standard system is the sum of unity and the rate of profits in the system of prices of production:

ν q* = 1 + r

One should be able to find a proof of a few lines for this claim.

6.0 Conclusion

The weighted sum of unity added to the value rate of profits, by industry, is unity added to the rate of profits in the system of prices of production. These weights are the element-by-element product of the labor values of the capital advanced in each industry and the gross outputs from the standard system.