Groucho Marxism

Value, Price and Profit is a transcript of a lecture series delivered in 1865 by Karl Marx. Having just finished reading it I thought I would provide a short summary. In this text, Marx sought to refute the theoretical basis for the economic policy of his contemporary John Weston, who argued: “(1) that a general rise in the rate of wages would be of no use to the workers; (2) that therefore … the trade unions have a harmful effect”. Marx begins by pointing out that because there is an economic law governing the value of commodities – namely, the labour theory of value – capitalists cannot raise prices at will. Nor can they lower wages at will, as wages represent the price of labour power, which is also a commodity under capitalism.

Marx argues that profit is derived not by selling commodities above their value (in which case capitalists could raise prices at will), but by selling commodities at or near their natural value, because workers are only paid for that portion of their labour which pays for their own labour power. The distinction between labour and labour power is crucial here. Recall that labour refers to work done, whereas labour power refers to the capacity to do work. In physical terms, labour has units of energy, whereas labour power has units of energy / time. Workers are paid for their labour power but the value they produce is based on their labour. In general, the value produced by a worker’s labour is greater than the value of their labour power; and this difference, according to Marx, is the source of profit.

To make this a bit more rigorous, consider an economy which produces m commodities using n types of labour. For such an economy we can define a commodity vector as a nonnegative mx1 vector, and a labour vector as a nonnegative nx1 vector. The economy is defined by an activity set, with the interpretation that an element (x,u,y) of this set represents a possible configuration of commodity inputs x, labour inputs u, and commodity outputs y. We may also define a price vector as a nonnegative 1xm vector, and a wage vector as a nonnegative 1xn vector. As noted above, Marx begins Value, Price and Profit by arguing that wages and prices are determined by the labour theory of value. How can we represent this mathematically?

For simplicity, let us assume that there is only one type of labour, so that n = 1. This can be thought of as representing Marx’s concept of ‘abstract labour’. Saying that wages are determined by the labour theory of value can be represented by stating that w = pb, where p is the price vector and b_­­­ is a ‘subsistence bundle’ of commodities, a nonnegative mx1 vector. This represents the idea that wages are precisely sufficient for the subsistence bundle b_­­­ to be affordable. Or to put it another way, that wages are determined by the value of commodities, expressed in monetary terms, required to produce the labour power that workers are then forced to sell to capitalists. Given an element (x,u,y) of the activity set and a profit rate r, the price vector is determined by the equation: py = (1+r)(px+wu) = (1+r)p(x+ub).

To make things more concrete, suppose that the activity set is such that x = Ay and u = Ly for some mxm matrix A and 1xm vector L. Such an economy is referred to as a ‘Leontief economy’ after the Russian economist Wassily Leontief. Then from the above, we have: py = (1+r)p(A+bL)y; and since y was left undetermined, we must therefore have: p = (1+r)p(A+bL). This is equivalent to saying that p is an eigenvector or the matrix A+bL with eigenvalue 1/(1+r). In order to find conditions under which a nonnegative p and r exist that satisfy this equation, we need to invoke something called the Perron–Frobenius theorem, named after the German mathematicians Oskar Perron and Georg Frobenius. This implies that under a fairly weak condition on the matrix A+bL, such a p and r exist and are unique.

Thus, under this weak assumption, the price vector p and profit rate r are uniquely determined in a Leontief economy, which validates Marx’s claim that capitalists cannot simply raise prices at will. Furthermore, as w = pb, if we assume that the subsistence bundle b is uniquely determined, then wages are uniquely determined too. We can ensure that b is uniquely determined by assuming that if there is more than one subsistence bundle, wages will be determined by using the one with the lowest price. The total value of output is given by v = C+V+S, where C = vA is the value of constant capital, V = vbL is the value of variable capital, and S = (1-vb)L is surplus value. Thus, v = vA+L. Assuming that the matrix I-A is invertible, we can write: v = L(I-A)^-1.

The labour embodied in the subsistence bundle b is then given by: vb = L(I-A)^-1b. Furthermore, the rate of exploitation is given by: e = Sb/Vb = (1-vb)Lb/vbLb = (1-vb)/vb. It can be shown that r is positive if and only if e is positive; in other words, positive profits implies exploitation and vice-versa. This is known as the ‘Fundamental Marxian Theorem’. The key step is the identification of surplus value with the quantity (1-vb)L. This can be understood by noting that for a given output level y, Ly represents the labour required to produce this level of output; and for a given amount of labour u, vbu represents the value received by the worker; hence vbLy represents the value received by the worker for the when the output level is y. The difference Ly-vbLy is therefore the surplus value received by the capitalist.

As I mentioned in a previous blog post, a critic might argue that the definition of ‘value’ used by Marxian economists is contrived and chosen specifically to produce a desired result – namely, the Fundamental Marxian Theorem. I will return to this point in a future blog post.