OCC Varies Less Among Vertically-Integrated Industries (Part 1)

1.0 Introduction
Anwar Shaikh claims that one can expect the Organic Composition of Capital (OCC) to vary less among vertically integrated industries than among non-vertically integrated industries. Shaikh shows his claim holds for the United States of America in 1947. Petrovic demonstrates the claim for Yugoslavia in 1976 and 1978.

This post lays out the theory formulating this empirical claim. Results replicating Shaikh's and Petrovic's test of the theory in new data are left for Part 2. I have yet to test the theory, and Part 2 remains unwritten for now.

2.0 Vertical Integration

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. The challenge is to express an empirical claim about the variability of the OCC in terms of this empirically-observable data.

Let q be an n-element column vector denoting the gross quantities output in each industry. The column vector A q represents the physical quantities of capital goods needed to produce this gross output. Let y be an n-element column vector denoting the net quantities output in each industry. 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 are related as follows:

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

Or:

q = (I - A)^-1 y

The jth column of (I - A)^-1 represents the gross output in a vertically integrated industry producing a net output of one unit of the jth commodity. This interpretation becomes apparent when one considers a net output vector consisting of one unit of the jth commodity:

y = e_j

where e_j is the jth column of the nxn identity matrix.

The above analysis of vertically integrated industries allows one to specify the amount of labor and the capital goods employed in each vertically integrated industry. Consider the n-element row vector v defined as:

v = a₀ (I - A)^-1

The jth element of v represents the labor (in person-years) employed in a vertically integrated industry producing one unit of the jth commodity net. This element is the labor directly and indirectly embodied in one unit of the jth commodity. v is the vector of labor values for this economy. The capital goods used in producing any gross output vector is found by pre-multiplying that vector by the Leontief input-output matrix A. Define the matrix H such that each column is the product of A and the gross output of a vertically integrated industry producing a net output of one unit of the corresponding commodity:

H = A (I - A)^-1

Luigi Pasinetti calls the columns of H "the vertically integrated units of productive capacities." A column "expresses in a consolidated way the series of heterogeneous physical quantities of commodities which are directly and indirectly required as stocks, in the whole economic system, in order to obtain one physical unit of [the corresponding commodity] as a final good."

3.0 The Organic Composition of Capital
According to Karl Marx, the labor value of a commodity is the sum of the labor embodied in the capital goods used in the production of that commodity, the labor value of the labor power used in the production of that commodity, and the surplus value:

v_j = c_j + w_j + s_j

where

• c_j is constant capital expended in producing one unit of the jth commodity
• w_j is variable capital (that is, the labor value of capital spent on the wages of workers) expended in producing one unit of the jth commodity
• s_j is the surplus value obtained in producing one unit of the jth commodity.

For Marx, the labor value of constant capital appears unchanged in the product. The source of profits is the appropriation by the capitalists of surplus value produced throughout a capitalist economy.

The OCC is defined to be the ratio of constant capital and variable capital, both expressed in labor values:

occ_j = c_j/w_j

where occ_j is the OCC for the jth industry. Marxist economics would be much less problematic if the OCC were invariant across industries. The rate of exploitation e is another important parameter in Marxist economics. The rate of exploitation is the ratio of surplus value to variable capital in each industry:

e = s_j/w_j

The equality of the rate of exploitation across industries follows from an assumption of competitive labor markets, inasmuch as workers are free to transition among industries in seeking work. The OCC in each industry can be expressed as a function of the rate of exploitation and the ratio of constant capital to the remaining labor value of the product:

occ_j = (e + 1) c_j/(w_j + s_j)

The rate of exploitation can be treated as a nuisance parameter in exploring the empirical question raised in this post.

Consider non-vertically integrated industries, each producing a gross output of one unit of each commodity. The jth industry in this case employs a_{0, j} person-years of labor. That is, the labor value of the product from newly applied labor is merely the corresponding direct labor coefficient:

w_j + s_j = a_{0, j}

The columns of A represent the capital goods needed in each of these industries. The labor embodied in the capital goods for the jth industry is the dot product of the row vector expressing the labor values of a unit of each commodity and the column vector denoting the quantities of these capital goods. Thus, one has:

c = v A

On the other hand, consider vertically integrated industries, each producing a net output of one unit of each commodity. The amount of labor directly employed in the jth vertically integrated industry is v_j. The labor value c^*_j embodied in the capital goods for the jth vertically integrated industry are easily found:

c^* = v H

where the elements of c^* are the desired labor values.

The above observations can be brought together to summarize the empirical claim of interest here. The OCC in each non-vertically integrated industry is proportional to the ratio of the labor value of the capital goods used in that industry and the labor directly employed in that industry:

occ_j/(e + 1) = c_j/a_{0, j}

The OCC in each vertically integrated industry is also proportional to the ratio of the labor value of the capital goods used in that industry and the labor employed in that industry:

occ^*_j/(e + 1) = c^*_j/v_j

where occ^*_j is the OCC in the jth vertically integrated industry. The proportionality constant is the same function of the rate of exploitation in the above pair of equations. The empirical claim is that the expression on the right hand side varies less among industries for vertically integrated industries than among non-vertically industries. That is, the coefficient of variation is less among vertically integrated industries. Perhaps one should take a variance-stabilizing transformation, such as natural logarithms, before calculating the coefficient of variation.

References

• Luigi L. Pasinetti (1973) "The Notion of Vertical Integration in Economic Analysis", Metroeconomica, V. 25: pp. 1-29 (Republished in Pasinetti 1980)
• Luigi L. Pasinetti (Editor) (1980) Essays on the Theory of Joint Production, Columbia University Press
• P. Petrovic (1991) "Shape of a Wage-Profit Curve, Some Methodology and Empirical Evidence", Metroeconomica, V. 42, N. 2: pp. 93-112
• Anwar Shaikh (1984) "The Transformation from Marx to Sraffa", in Ricardo, Marx, Sraffa (Ed. by E. Mandel and A. Freeman), Verso