Before continuing this exposition and explication of the evolution of the Smith/Ricardo/Marx Labor Theory of Value, let me just note that none of these luminaries made the slightest use of even the relatively elementary algebra readily available in their day. It is simply extraordinary that without the aid of correctly worked-through quantitative examples, they were able to intuit very deep formal truths. [Marx tried to carry out a numerical example in Capital, but got it wrong, even though the deeper proposition he was attempting to prove is, as it happens, true.]
With the aid of the linear algebra that I taught myself that cold Northampton January, I was able to read and grasp the mathematical reconstruction and reinterpretation of classical Political Economy carried out around the world in the 60's to 80's of the last century by a score of gifted economists. Indeed, I was even able to offer formal proofs of a few interesting propositions myself, although, alas, the most important of them was anticipated by a year by a Spanish economist [a fact pointed out in print by the most mathematically gifted of the American Marxist economists, John Roemer. So much for immortality.]
Ricardo claimed that equilibrium prices [or "natural prices," as the classicals called them] are determined by the quantities of labor directly and indirectly required for their production. He quite well understood that this explanation excluded such scarce goods as Old Masters and "wine grown on a particular side of the hill." [Ricardo had a rather comfortable lifestyle.] In mathematical terms, this means that the prices of commodities would be proportional to their labor values. Since a good deal hinges on this simple claim, let me, at the risk of totally alienating the innumerate among my readers, take a moment to explain exactly what Ricardo was asserting.
Each of the commodities bought and sold in the marketplace has a natural value, or, as modern economists would say, an equilibrium price. [The economy is equilibrated by movements of capital and of prices until there is a single set of prices and a single profit rate throughout.] Suppose we use the lowercase letter p to represent price, with subscripts to indicate which commodity we are referring to. Thus pc will stand for the natural price of corn, pl will stand for the natural price of linen, and so forth. Let us also adopt the modern convention of using the lowercase Greek letter lambda, or λ, to represent the quantity of labor directly or indirectly required to produce a unit of a commodity, which is to say the labor value of that commodity, with subscripts employed in the same fashion. If corn is measured in bushels, linen in yards, money in shillings, and labor in hours, then Ricardo's Labor Theory of Value asserts that:
The price in shillings of a bushel of corn is to the price in shillings of a yard of linen as the labor value in hours of a bushel of corn is to the labor value in hours of a yard of linen, which, using our newly defined symbols, can be stated succinctly as: pc/pl = λc/λl.
Even though Ricardo never translated his claims into convenient symbols of this sort, he grasped intuitively, and with great insight, that this proposition, which he himself had asserted, is not in general true! What is more, he understood exactly what had to be the case in order for it to be true. Briefly, the natural prices of commodities are proportional to their labor values only when the ratio of the quantity of labor directly required to the quantity of labor indirectly required is the same in all lines of production. This is the situation that Marx was later to call "equal organic composition of capital." The point is that some production techniques are what we have come to call "labor intensive" while others are "capital intensive." In an economy with production techniques of both sorts being employed, prices will in some industries lie above their labor values and in other industries below their labor values. [As is perhaps obvious by now, I find this whole subject absolutely fascinating, but I am quite well aware that not everyone, to quote the old joke about philosophers, wants to know "that much about rainbows." Readers who share my geekish enthusiasms are invited to consult the relevant chapters of Understanding Marx.]
After publishing the Principles of Political Economy and Taxation in 1817, Ricardo spent the remaining six years of his life struggling unsuccessfully with this deep problem with his Labor Theory of Value. Marx was well aware of this, and in Capital Volume Three offers a brilliant solution that is almost [but not quite] correct. And yet, despite the fact that the materials of his entire hauptwerk were worked out before 1867, Marx chose to write all of Volume One on the assumption of equal organic composition of capital! Why on earth would he make so odd a choice, considering that he had in his back pocket, as it were, a solution to Ricardo's problem? Tomorrow I shall answer that question and carry our story forward into Capital Volume One itself.