If The Workers Were Able To Live On Air

"In so far as the development of productivity reduces the paid portion of the labour applied, it increases the surplus-value by lifting its rate; but in so far as it reduces the total quantity of labour applied by a given capital, it reduces the number by which the rate of surplus-value has to be multiplied in order to arrive at its mass. Two workers working for 12 hours a day could not supply the same surplus-value as 24 workers each working 2 hours, even if they were able to live on air and hence scarcely needed to work at all for themselves. In this connection, therefore, the compensation for the reduced number of workers provided by a rise in the level of exploitation of labour has certain limits that cannot be overstepped..." -- Karl Marx, Capital, Vol. 3, Part 3, Chap. 15, Sect. 2

Introduction
A maximum rate of profits arises in a model of the production of commodities by means of commodities. This maximum rate of profits is an upper limit on the rate of profits in any sublunary capitalist economy, where the workers produce commodities to consume and thereby reproduce their labor power.

This maximum rate of profits would be easily seen if the economy were a giant farm producing one commodity, corn, from inputs of labor and seed corn. The surplus would be the difference between harvested corn and the quantity of seed corn which needs to be set aside to continue production next year. The ratio of this surplus to the quantity of seed corn is the maximum rate of profits. The maximum rate of profits cannot be achieved because of the need to pay wages to the workers eats into this surplus.

Some of the simple lessons of the corn economy generalize to actual more-or-less capitalist economies, such as in the United States of America (USA). One can use the mathematics of eigenvalues and eigenvectors to set out the theory in this case.

2.0 The Standard System
Consider an economy in which each of n commodities are produced from labor and inputs of those n commodities. Let a_{0, j} be the person-years of labor used in producing one unit of the jth commodity. Let a_{i, j} be the physical units of the ith commodity used in producing one unit of the jth commodity. The direct labor coefficients are elements of the n-element row vector a₀. The remaining input-output coefficients are the entries in the nxn Leontief input-output matrix A, which is assumed to satisfy the Hawkins-Simon condition.

This data determines Sraffa's standard system, in which the gross output, the net output, and capital goods have specific properties. Let q* be an n-element column vector denoting the gross quantities output in each industry, that is to say, the gross output in the standard system. The column vector A q* represents the physical quantities of capital goods needed to produce the gross output in the standard system. Let y* be an n-element column vector denoting the net quantities output in each industry in the standard commodity. The net output is available to be divided up between wages and profits after replacing the capital goods needed to reproduce the gross output. Net output and gross output, in any proportions, are related as follows:

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

In the standard system, the ratio between gross output and the quantity of capital goods needed to produce the gross output is invariant among commodities:

q* = (1 + R) A q*

Or:

A q* = [1/(1 + R)] q*

As a matter of fact, [1/(1 + R)] is the maximum eigenvalue of A. The standard system is scaled such that the amount of labor employed in the standard system is unity:

a₀ q* = a₀ (I - A)^-1 y* = 1

y* is the standard commodity.

One chooses the maximum eigenvalue to ensure, under the Hawkins-Simon condition, the existence of a standard commodity in which all components are non-negative and at least some components are strictly positive. The commodities which enter the standard commodities are called "basic". They enter directly or indirectly into the production of all commodities. Those commodities with zero entries in the standard commodity are called "non-basic". Either non-basic commodities do not enter into the production of any other commodity. Or they enter into the production only of non-basic commodities. For each non-basic commodity, there exist some commodity such that the non-basic commodity does not enter, either directly or indirectly, into the production of that commodity.

To explicate the concept of a commodity entering indirectly into the production of another commodity, consider the output of the Motor Vehicles, Bodies And Trailers, And Parts industry, one of the 65 industries in the 2005 Use Table for the USA available from the Bureau of Economic Analysis (BEA). 0.18 units of the output of the Primary Metals industry enter (directly) into the production of each unit produced by the Motor Vehicles, etc. industry. (A quantity unit of any industry is one hundredth of the quantity output of each industry in the year 2000, where the quantity unit in each year is a chain index.) 0.15 units of the output of the Mining, Except Oil And Gas, industry is an input into each unit produced by the Primary Metals industry. Since Mining, Except Oil And Gas, enters into Primary Metals, and Primary Metals enters into Motor Vehicles, etc., then Mining, Except Oil And Gas, enters indirectly into the production of Motor Vehicles, etc. (0.040 units of Mining, Except Oil And Gas, also enter directly into each unit output of Motor Vehicles, etc..) Any number of steps can separate the indirect production of one commodity by another.

Summary of Some Empirical Results
I've implemented the above mathematics with 2005 data for the USA. Sixty two industries in the USA in 2005 are basic and enter into the standard commodity with positive components. The three non-basic industries are

• Hospitals and Nursing and Residential Care Facilities
• Federal General Government
• State and Local General Government

I think the non-basic property of the general government industries is an accounting convention. The industries Federal Government Enterprises and State and Local Government Enterprises are basic and enter into the standard commodity with positive values.

The maximum rate of profits in the USA in 2005 was approximately 106.4%.