Here are the price equations of our corn/iron/tools model, revised so as to represent workers as petty commodity producers. In order to indicate that capital cannot be transferred from the labor producing "industry" to other lines of production, I introduce a new variable, r, to stand for the rate of "profit," if there is any, in that "industry.
( 0Pl + 30Pc + 15Pi + 0Pt )( 1 + r ) = 150Pl
( 60Pl + 128Pc + 2Pi + 3Pt )( 1 + R) = 240Pc
( 20Pl + 16Pc + 1Pi + 5Pt )( 1 + R) = 60Pi
( 50Pl + 6Pc + 27Pi + 4Pt )( 1 + R) = 16Pt
It is not necessary to specify the real wage because that is implied by the input quantities of the first line of production. This is now a system of four equations in six unknowns: Pl, Pc, Pi, Pt, r, and R. We reduce the number of unknowns to five by stipulating that corn is the money commodity, as before, setting Pc = 1. That still leaves one degree of freedom.
It is not difficult to demonstrate [see the essay referenced above] that r and R vary inversely to one another. That, note, is different, formally speaking, from showing that the wage and the profit rate in the old system vary inversely to one another, although the underlying facts are much the same. A non-zero r means that the workers have somehow managed to extract from the capitalists some portion of the surplus. But this way of representing things, besides making no reference to the Labor Theory of Value, also opens the way to economists like Gary Becker to introduce the notion of "human capital." Workers who, in effect, drive the wage above bare subsistence can either consume the extra in the form of a somewhat less miserable standard of living, or they can -- as Becker might say -- invest in the improvement of their capital stock [themselves] by paying for education, which will allow them to produce a new and higher priced commodity -- namely, skilled labor. This way of thinking, crazy as it is [Marx is brilliant on this], leads naturally to the countless discussions of the "return on investment in education' in the form of higher lifetime earnings.
Notice, by the way, that if we wish to continue with this charade of labor as a commodity produced by a petty bourgeois entrepreneur, we must allow for several labor sectors, each with its own particular "product" [some form of unskilled or skilled labor or educated labor] and with its own rate of return. We can even analyze a failure of some children of the educated workforce to take advantage of the opportunities provided by their parents as their choosing to consume the additional portion of the surplus made available to them instead of investing it, thus reproducing the fairytale about prudent and imprudent capitalists [ants and grasshoppers.] And, of course, the variation in the rates of return in the several labor sectors permits us to talk about the relative exploitation of unskilled workers by much better compensated skilled workers.
The point of all this is to elaborate on the fundamental verruckheit, or craziness, of the manner in which bourgeois ideology [law, economic theory, philosophy, sociology, etc.] mystifies and conceals the exploitative nature of capitalism, making it appear that the failure of the workers to improve themselves is attributable to their imprudence rather than to the disadvantageous position in which they find themselves vis-a-vis capital.
[A propos, Susie and I saw "Company Men" last night, an interesting movie about the effects of the economic crisis on three privileged and advantaged corporate executives -- Ben Affleck, Chris Cooper, and Tommy Lee Jones, with a nice turn by Kevin Costner. It is worth seeing, I think.]
It is not difficult to show, formally, that the total profit appropriated by the corn, iron, and tools sectors is equal to what is lost to the labor sector by its inability to shift its capital to a more profitable line investment.
I have now come to the end of my story, even though there is a very great deal more to be said about Marx's theories, some of which I have said at length in my two books and several articles on him. Let me explain why I have taken so very much time and expended so much effort expounding and analyzing the Labor Theory of Value, only in the end to show [with arguments original to Vegara and myself] that the theory is incorrect.
Part of the reason, to be totally honest, is that I simply find the analytics of the theory fascinating and elegant. My entire life has been spent taking impossibly difficult theories [such as those in Kant's First Critique] and clarifying them in my own mind until I can present them to others as clear, simple, elegant, and beautiful. My second reason is to rescue Marx from the unfounded criticism that he is -- in the infamous words of Paul Samuelson -- a "minor post-Ricardian autodidact." I was so offended by those words when I first read them that I felt a need to show, clearly and simply, that they were just not true. My third reason is that I believe one cannot really come to grips with the profound truth at the heart of Marx's critique of capitalism without first working through this theoretical story and becoming completely clear on its precise status and limits. Speaking simply for myself, I could not have been led to an understanding of the ironic structure of Marx's critique, and the core irrationality of treating workers as though they are participants in a free market of exchange among producers and consumers, had I not first made my peace, intellectually, mathematically, and theoretically with Marx's own attempts to capture that craziness.
I hope this tutorial has been useful to some of you. as I feared, it went on for a very long time, so long, I imagine, that I lost a number of readers along the way. But I have enjoyed writing it, and it will live forever in cyberspace, in the weird way things have of surviving in that realm.