**1.0 Introduction**Stupidity about Marx is never-ending. So I thought I would put up a post about Marx as a mathematical economist. This is exposition of unoriginal ideas. To amuse myself, I didn't review Sraffa or any other author when writing this.

**2.0 The Technology**Consider an economy in which

*n* commodities are produced, each in a separate industry. The technique in use is represented by the

*n*x

*n* Leontief matrix

**A** and the

*n*-element row vector

**a**_{0} of labor inputs. A column,

**a**_{.,j}, in

**A** and an element of

*a*_{0,j} represent an industry. The

*i*th,

*j*th element of

**A** is the quantity of the

*i*th commodity input per unit output of the

*j*th industry. Quantities are here measured in physical units (e.g., bushels corn per ton steel). The

*j*th element of the row vector of labor inputs,

*a*_{0,j}, is the person-years of labor services hired in the

*j*th industry per unit output.

By assumption, all industries require at least some positive amount of labor to produce their outputs. All industries produce their outputs in a year, and they consume all their inputs in producing their output. This is a model of circulating capital alone; no fixed capital (e.g., long-lasting machines) appears in the model.

Assume that the economy is

*viable*, that is, some levels of operation exist for the industries such that there is a surplus product available, after replacing used-up means of production, when industries are operated at that level. For simplicity, assume all commodities are

*basic*. In other words, every commodity enters either directly or indirectly into the production of every other industry. Presumably, steel enters directly into the production of automobiles. Iron would enter indirectly into the production of automobiles through its use in the production of steel.

No choice of technique occurs in the model.

**3.0 Quantity Flows****3.1 Labor Values**Let

**q** be an

*n*-element column vector, where each entry is the gross output of that industry. Each entry is measured in the corresponding physical units (tons, bushels, kilograms, etc.). Let

**y** be an

*n*-element column vector of net outputs. Gross and net outputs are related like so:

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

Or:

**q** = (**I** - **A**)^{-1} **y**

where

**I** is the identity matrix. The existence of the inverse follows from viability. If industries were operated at levels to produce the gross outputs, the net output would be available for consumption or accumulation after replacing exactly the inputs consumed in production.

The amount of labor hired to produce the net output

**y** is:

*L* = **a**_{0} **q** = **a**_{0} (**I** - **A**)^{-1} **y**

Suppose net output consisted of only one unit of the commodity produced in the

*j*th industry:

**y** = **e**_{j}

where

**e**_{j} is the

*j*th column in the identity matrix. The

*labor value* of the

*j*th commodity, that is, the amount of labor hired to produce one more unit of the

*j*th commodity net, is:

*v*_{j} = **a**_{0} (**I** - **A**)^{-1} **e**_{j}

Labor values are expressed as an

*n*-element row vector:

**v** = **a**_{0} (**I** - **A**)^{-1}

Labor values then have a sensible meaning; nothing radical is involved in defining them.

**3.2 The Standard Commodity**One might expect any arbitrary basket with a large number of commodities to have both some labor-intensive and some capital-intensive commodities, in some sense. On average, these will approximately cancel in an arbitrary basket. Accordingly, let's assume that gross outputs, net outputs, and the commodity in which wages are paid are all of average capital-intensity, in some sense.

Since both gross and net outputs are of average capital intensity, it seems sensible to assume they are composed of the same proportions, just different in amount:

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

where the asterisks indicate a basket in

*standard* proportions.

*R* is a strictly positive constant. As a normalization condition, the standard system is assumed to employ a unit amount of labor:

**a**_{0} **q*** = 1

It follows that the gross output of the standard system is a right-hand eigenvector of the Leontief input-output matrix:

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

The outputs of the standard system are guaranteed to be positive by setting [1/(1 +

*R*)] to the maximum eigenvalue of the Leontief matrix, also known as the

Perron-Frobenius root of the Leontief matrix.

The net output of the standard system,

**y***, is the

*standard commodity*.

**3.3 The Rate of Exploitation**The labor embodied in the gross output of the standard system is easily found. One has:

1 = **a**_{0} **q*** = **v**(**I** - **A**)**q*** = **v** **q*** - **v** **A** **q***

Or, taking advantage of the fact that the gross output of the standard system is an eigenvector of the Leontief matrix:

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

Hence, the labor value of gross output of the standard system is found as:

**v** **q*** = 1 + 1/*R*

Marx expressed the labor value of gross output as

*C* +

*V* +

*S*. Constant capital

*C* is

**v****A****q***, the labor value of the means of production used up in producing the net output. Variable capital

*V* is the labor value of the value added by labor paid out in wages. Surplus value

*S* is the labor value of the remaining net output, which is obtained by the capitalists. Since one person-year is employed, the sum of variable capital and surplus value in the standard system is unity:

*V* + *S* = 1

Let

*w* denote the proportion of the net output of the standard system (that is, the standard commodity) that is paid out in wages. Hence:

0 ≤ *w* ≤ 1

And variable capital is given as:

*V* = *w*

It follows that surplus value is now defined:

*S* = 1 - *w*

Marx denoted the ratio of surplus value to variable capital as the rate of exploitation:

*e* = *S*/*V* = (1 - *w*)/*w* = (1/*w*) - 1

where

*e* is the rate of exploitation.When the whole value of the net product is paid out to workers as wages, workers are not exploited and the rate of exploitation is zero. The rate of exploitation is otherwise positive, and increases without bound as the wage becomes a lesser proportion of the value of the net product.

**4.0 Price Equations**Let

**p** denote a row vector of prices of production. Prices of production permit smooth reproduction in a competitive capitalist economy. They are defined by the condition that the same rate of profits is obtained in each industry:

**p** **A**(1 + *r*) + **a**_{0} *w* = **p**

where

*r* is the rate of profits. Since profits are not earned on wages, the workers are paid at the end of the year. Wages are not advanced in this model. Since the same symbol for wages is used in calculating the labor value of quantities in the standard system, the standard commodity is the numeraire. Thus, the price of the standard commodity is unity:

**p** **y*** = 1

Recall that the net and gross outputs of the standard system are in proportion. One can thus calculate the price of the gross output of the standard system:

**p** **q*** = 1 + 1/*R*

Postmultiply the price equations by the gross output of the standard system:

**p** **A** **q*** (1 + *r*) + **a**_{0} **q*** *w* = **p** **q***

Or:

**p** **q*** [1/(1 + *R*)] (1 + *r*) + *w* = (1 + 1/*R*)

Or:

(1 + 1/*R*)[1/(1 + *R*)] (1 + *r*) + *w* = (1 + 1/*R*)

The rate of profits is an affine function of the wage:

*r* = *R*(1 - *w*)

The above equation can also be expressed as:

*w* = 1 - *r*/*R*

The rate of profits ranges from zero to the maximum

*R*. The maximum rate of profits is obtained when workers live on air, with a wage of zero. A higher wage is associated with a lower rate of profits, with a very simple relationship with this numeraire.

Total wages are

**a**_{0} **q*** *w*. But, since one person-year is employed in the standard system, total wages are simply

*w*.

Total profits are

**p** **A** **q*** *r*, that is:

**p** **A** **q*** *r* = [1/(1 + *R*)] **p** **q*** *R*(1 - *w*)

Or:

**p** **A** **q*** *r* = [1/(1 + *R*)](1 + 1/*R*)*R*(1 - *w*)

Or:

**p** **A** **q*** *r* = 1 - *w*

The above is hardly surprising. The ratio of the rate of profits to the maximum rate is an increasing function of the rate of exploitation:

*r*/*R* = *e*/(1 + *e*)

When the rate of exploitation is zero, the rate of profits in the system of prices of production is also zero. As the rate of exploitation increases without bound, the ratio of the rate of profits to the maximum rate monotonically increases to unity.

**5.0 Invariants**The following statements hold, whether the quantities mentioned are evaluated in labor values or in prices of production:

- The gross output of the standard system is valued at 1 + 1/
*R* - The net output of the standard system is unity
- Variable capital is valued at
*w* - Surplus value, that is, profits are (1 -
*w*)

Furthermore, the rate of profits is positive if and only if workers are

exploited.

This model certainly suggests that market phenomena are a veil over the exploitation inherent in capitalism. And calculations with labor values exhibit that exploitation.