Profits Resulting From Exploitation Of Workers

1.0 Introduction
This post outlines a simple economic model. This model formalizes the claim that capitalists obtain their income from the exploitation of workers. The concept of exploitation illustrated in this model is supposed to formalize at least some of Marx's results in Capital.

I present results here without proof. Michio Morishima calls the theorem with which I conclude the "Fundamental Theorem of Marxism". In the 1970s, Ian Steedman showed the theorem does not apply to models with joint production, under a natural generalization of labor values. On the other hand, Morishima argues the theorem does apply to joint production under a different way of defining labor values in that case.

2.0 The Data and Quantity Relationships
I assume an economy in which n commodities are produced. Each commodity is produced from inputs of labor and produced commodities. Define a_i,j to be the quantity of the ith commodity used to produce one unit of the jth commodity. The nxn input-output matrix A contains these coefficients of production. By assumption, all commodity inputs are used up in a single production cycle (a year). That is, this is a circulating capital model. Assume the input-output matrix A is irreducible.

The net or gross quantities of commodity outputs of this economy are among the model's data. Let y be the column vector of net outputs (also known as the surplus), and let q be the column vector of gross outputs. Net and gross outputs are related by the following equation:

y = q - A q = (I - A)q

Assume A is such that all elements of y are strictly non-negative and at least one element is strictly positive. That is, in this economy, the circulating capital goods are reproduced, with some output left over. The assumptions allow one to invert the above relationship, thereby obtaining:

q = (I - A)^-1 y

The inverse is known as the Leontief inverse. It exists under these assumptions by a Perron-Frobenius theorem.

Assume the quantity of any type of labor can be expressed as a known ratio of simple (unskilled) labor. For example, one hour of a nurse's labor might be expressed as five hours of simple labor, to pick a ratio at random. Let a_{0, j} denote the hours of (simple) labor expended in producing one unit of the jth commodity. The n-element row vector a₀ is composed of these labor coefficients. The labor force employed to produce the observed surplus is then a₀ (I - A)^-1 y.

Finally, the real wage is among the data of the model. Suppose wages are expended on a commodity basket of known proportions. Let the column vector c be in these proportions. If c is also the numeraire, the real wage is wc, in terms of numeraire-units per labor hours.

Assume wages are advanced. Some of the fomulas below are usefully expressed in terms of the augmented input-output matrix A(w). The augmented input-output matrix is defined as:

A(w) = A + c a₀w

The coefficients of the augmented input-output matrix are the commodities advanced per unit output, where these commodities are needed both as inputs to production and to sustain the workers during a production cycle.

3.0 Labor Value Accounting
Let e_j be the jth column of the identity matrix. Suppose the net output of the economy were to consist of one unit of the jth commodity. Then e_j would be a column vector denoting the net output. The gross output of the economy would be (I - A)^-1 e_j. One could then calculate the labor time to produce at these levels of operation of each industry in the economy:

Definition: The labor value of the jth commodity, v_j is:

v_j = a₀ (I - A)^-1 e_j

In words, the labor value of a commodity is the amount of labor time that would be required, if Constant Returns to Scale prevailed, to increase the net output of the economy by one unit of that commodity. Some refer to the labor value of a commodity as the amount of labor embodied in that commodity.

The row vector of labor values of all commodities is easily seen to be:

v = a₀ (I - A)^-1

The labor value of any quantities of commodities is calculated by premultiplying the column vector representing those quantities by the row vector of labor values. This observation allows one to make sense of a further definition:

Definition: The rate of exploitation, e is defined by:

e = [v(I - A - c a₀ w)q]/(v c a₀ q w)

In words, the rate of exploitation is the ratio of the labor embodied in the surplus not paid out in wages to the labor embodied in the commodities purchased out of wages.

4.0 The Price System for the Maximum Wage
The maximum wage, w*, arises when the entire net output (also known as the surplus) is paid out to the workers and the rate of profits is zero. The stationary row vector of prices, p*, consistent with the maxiumum wage satisfies the following equation:

p* [A + c a₀ w*] = p*

Or:

p* A(w*) = p*

That is, an eigenvalue of unity exists for the augmented input-output matrix calculated for the maximum wage, and the prices comprise the corresponding left-hand eigenvector. The existence of unity as the eigenvalue follows from a Perron-Frobenius theorem. It also follows that the prices corresponding to the maximum wage are strictly positive.

These prices are proportional to labor values:

Theorem: There exists a k > 0 such that:

p* = k v

5.0 The Price System in General
Under the assumptions, a stationary price vector p satisfies the following equation:

p A(w) (1 + r) = p

where r is the rate of profits. (The question of the usefulness of these prices of production for understanding the dynamics of market prices is known as the "realization problem".) The following is an equivalent equation:

p A(w) = 1/(1 + r) p

1/(1 + r) is then an eigenvalue of the augmented input-output matrix, and the price vector is the corresponding left-hand eigenvector. Some manipulation yields an nth-degree polynomial equation for 1/(1 + r):

Determinant[ 1/(1 + r) I - A(w) ] = 0

For a wage less than the maximum wage, the maximum root of this equation is real and has a corresponding left-hand eigenvector with positive elements. (This claim also follows from a Perron-Frobenius theorem.) The price level is picked out by the condition that the price of the numeraire be unity:

p c = 1

Suppose that w < w*. That is, not all of the surplus is paid out in wages. Furthermore, assume that the organic composition varies among industries. ("Organic composition of capital" is Marxist jargon roughly equivalent to "capital-intensity".) Then there does not exist a k > 0 such that:

p = k v

That is, prices in the general case are not proportional to labor-values. In this sense, a simple labor theory of value usually does not hold.

(Have I left out any other freak cases? Is there possibly a left-hand eigenvector for the input-output matrix such that all of its elements are positive and that eigenvector is associated with some other eigenvalue that the Frobenius root of the input-output matrix? I'd look for other freak cases where the vector of direct-labor coefficients is some such eigenvector, if such exists.)

6.0 Conclusion

Definition: Labor is exploited if and only if e > 0.

In words, labor is exploited if the labor-hours embodied in the commodities workers buy is less than the labor-hours the workers expend to earn the wages with which they buy those commodities.

Theorem: r > 0 if and only if e > 0.

In words, the rate of profits is positive in a system of stationary prices consistent with smooth growth of the economy if and only if labor is exploited. In other words, the positive returns provided to ownership of capital come from paying workers less than the value they contribute to production.

As far as I am concerned, this is solid mathematics.