In my haste to bring my short series of posts on Marx’s economic theory to an end, I am afraid I dropped the ball and forgot to complete one part of the argument. You will recall that Ricardo got stuck at the end of his life on the problem of explaining the very common situation in which different lines of production exhibited different proportions of labor indirectly and labor directly required in production. Marx believed that he had a solution to this problem but, as I explained, chose to wait until volume 3 before presenting it. Only after he had posed and solved the problem that the classical political economists had not even seen, namely explaining the origin of profit itself, could Marx returned to the question of what to do with Ricardo’s puzzle.
Marx’s solution was quite interesting. He argued that in a mature capitalist system it is the system as a whole and not simply individual lines of production or even individual factories that must be examined. In the general case in which different lines of production exhibit varying organic compositions of capital, to use Marx’s term, it is the entire capitalist system to which we must turn to identify the relationship between profit and surplus labor. In any particular line of production these two quantities may not be equal (or, to be more precise, proportional) but in the system as a whole they are equal, thereby establishing that capitalism as a system rests on the exploitation – which is to say the extraction of surplus value from – the working class.
Marc seems simply to have assumed that this was true; how could it be otherwise! But the modern mathematical economists who returned to Marx and Ricardo and brought their analytical tools to bear on those theories did attempt to prove Marx’s claim and they came up with a quite fascinating result. As Piero Sraffa had made clear in his foundational monograph, it is essential to distinguish between those commodities that are required directly or indirectly in the production of all other commodities – commodities which he called basics – and luxury commodities which may be required in their own production or even in the production of other luxury commodities but which are not required directly or indirectly in the production of all the commodities in the system. In the little model that I created in my book, Understanding Marx, I included a luxury sector which somewhat facetiously I identified as a sector devoted to the production of theology books – this is a nod to the religious pretensions of the early English capitalists.
Now, it seems that if there are no luxury goods being produced in an economy – or, what is pretty much the same thing, if the luxury goods sector is only a tiny part of the entire economy – then Marx is actually correct! Total profits in the system as a whole will indeed be equal to total surplus labor, for a suitable choice of numeraire. This means that all of the profits are being poured back into the economy to expand the scale of production. “Accumulate! Accumulate! That is Moses and the prophets to the capitalists,” as Marx sardonically remarks at one point in Capital.
But how do we know that there is some balance in the size of the different lines of production that will, when the capitalists do their very best to reinvest, correctly use all of the physical surplus that is produced in each cycle of production? Well, Sraffa demonstrated that that is necessarily true. Indeed, things get a little bit more delicious than that. In the 20th century, the great Hungarian – American mathematician John Von Neumann, proved an important theorem in growth economics concerning the maximum growth path for a capitalist economy and it is precisely when an economy is on that maximum growth path that Marx’s solution to Ricardo’s problem is correct.
I do not suppose many people find this as delightful a fact as I do, but it is one more way of seeing that despite using only, as Stalin said, addition, subtraction, multiplication, division, and the taking of averages, Marx was in fact a brilliant intuitive mathematical economist.