The organic composition of capital

In Capital, Marx introduces the idea of the organic composition of capital. In the LTV, a capitalist can achieve profit only by expropriating surplus labor from the workers who actually work for him. The labor embodied in machines and raw materials is "dead" labor: the surplus value has already been expropriated by the capitalist who sold him the machines or raw materials. A capitalist has to pay for capital, raw materials and directly employed labor up front*; he realizes a profit only after he has created, marketed, delivered, and sold his commodity. From the capitalists perspective, his rate of return $r={l_s}/{l_p+c}$, where $l_p$ is the cost of labor power, $l_s$ is the surplus value of labor (total labor minus cost of labor power), and $c$ is the constant capital, the cost of raw materials and the amortized cost of capital equipment.

*Wages are usually paid for in arrears, but labor is still usually paid for before the product is sold.

This model leads to two problems, one which Marx addresses and the other a puzzle Marx was not able to solve.

The first problem is why a capitalist would invest in capital: the more value a capitalist invests in capital, the lower her overall rate of return ($c$ is in the denominator, so as $c$ goes up, $r$ goes down). Marx answers this problem in two ways. First, as the value of wage goods falls, $l_p\/l_t$ falls, which means that more surplus labor can be extracted from workers, increasing the rate of profit. Second, Marx invokes competition. The value of a commodity is the socially necessary labor time. If a firm can produce a commodity for less that the socially necessary labor time, it can capture a producer surplus in addition to the surplus value of the labor it employs. (Essentially, the firm is exploiting its customers, getting them to give more embodied labor than they receive.) In the short term, a firm can make more money by investing in capital and become more efficient than its competitors; in the long term, absent monopoly, what one capitalist can figure out another can copy, and efficiency just leads to a lower rate of profit overall. Thus capitalism must constantly innovate to keep going in the short term, but this constant innovation leads to the constantly falling rate of profit.

The second problem is more severe. Firms in an economy create many different commodities. Some of these commodities will use more capital relative to labor, some less: they will have different organic compositions of capital. However, capitalists want to equalize profit. To make the typing simpler, I'll use $w$ to represent labor power ($l_p$) and $s$ to represent surplus labor ($l_s$). We'll keep the corn-corn model from my earlier post.

$v={w+s}/{1-a}$
$p=(1+r){w}/{1-(1+r)a}$

If the labor theory of value holds, we would expect that the values and prices of two commodities with different organic compositions of capital would be in the same proportion: $v_1/v_2=p_1/p_2$. However, unlike the previous model, adding an extra degree of freedom by separating labor into cost of labor power and surplus labor doesn't help us. We see that, holding the quantity of labor and the rate of profit constant, and letting the proportion of constant capital ($a$) vary,

$v_1/v_2=p_1/p_2→{{w+s}/{1-a_1}}/{{w+s}/{1-a_2}}={(1+r){w}/{1-(1+r)a_1}}/{(1+r){w}/{1-(1+r)a_2}}$

therefore,

${1-a_2}/{1-a_1}={1-(1+r)a_2}/{1-(1+r)a_1}$

which can be true if and only if $r=0$. Labor just divides out, no matter how it's expressed.

There are several ways of interpreting this equation. One way, which is not illegitimate, is that the LTV is just internally contradictory. But there are other ways. The LTV is, like most traditional economic theories, an equilibrium model; it does not talk about how or why an economy might not be in equilibrium, or how it achieves equilibrium. Thus we might conclude that in equilibrium all firms must have the same organic composition of capital; alternatively, we might conclude that in equilibrium the rate of profit is zero.

Another way of looking at this equation is that differences in the organic composition of capital is a source of fundamental instability in a capitalist economy. (A growing economy, protestations of traditional economists notwithstanding, is not in equilibrium.) It does not have an economic solution; instead, it requires an exogenous political solution: to stabilize the economy, capitalists can redistribute profits among themselves to equalize the rate of profit. It's not common, bu capitalists can work together, especially when their main tool, control of capital, cannot be used to gain short-term advantage.