INSIDE BASEBALL

I don’t know whether anyone is interested in this, but I will post it in case someone is.  I said in class on Tuesday that Ricardo could not figure out what to do with the case in which there are unequal capital/labor ratios [unequal organic composition of capital] and Marx had an answer that almost worked.  It worked when the economy is on what is now called a Von Neumann balanced growth path.  A graduate student in the course asked me where he could find an exposition of that and I drew a blank, so I wrote him this email:

            We start with Ricardo, who spent some time analyzing an imaginary economy with only one commodity – corn is Ricardo’s choice.  If there is only one commodity, then the only inputs into production are corn and labor.  One unit of corn is taken as money, the wage is some amount of corn, and the profit rate is paid in corn units.  Not surprisingly, everything in this model is simple and unproblematic.  Prices are proportional to labor values, the total profit in the system is equal, in corn money units, to the surplus labor extracted from the labor inputs, and so forth.

            Now fast forward to Sraffa, who not only wrote the very important monograph Production of Commodities by Means of Commodities [1960] but also edited the splendid 10 volume edition of the complete works of David Ricardo.  In his monograph, Sraffa analyses an economy with nothing but “basic” commodities and no “luxury goods,” a basic commodity being defined as a commodity required directly or indirectly in every line of production, and a luxury good being defined as a commodity that is not required, directly or indirectly, in the production of all commodities.  [Mathematically, this means that the square matrix of input coefficients is a non-negative non-decomposable matrix, although Sraffa never uses that language – is this clear?]

            Sraffa defines a notional complex commodity which he calls a “standard commodity”, consisting of quantities of all the basic commodities so chosen that the balance of the components of the Standard Commodity exactly equals the proportions of basic commodities in the economy when it is balanced, so that there is no excess or shortfall of demand.  Sraffa then proves that every economy with no luxury goods but only basic commodities can, by the workings of competition with each producer seeking to maximize profits, be brought into balanced form.  Thus it can be thought of as though a quasi-one commodity economy, the commodity being that Standard Commodity.  For details, see Chapter Six of my book, Understanding Marx, especially the Technical Appendix. 

            A Balanced Growth Path is a growth path in which the excess of each commodity in one cycle, which is to say the excess over and above what is needed to run the economy for another cycle at the same level, is just enough to expand production in the next cycle with no shortfalls or excesses of inputs.  Von Neumann proved a famous theorem about capitalist economies on a Balanced Growth Path which essentially shows [he did this before Sraffa, by the way] that any capitalist economy without luxury goods has a balanced growth path in which all surplus output in one cycle is reinvested in expanding the scope of production in the next cycle.  [If you Google “Von Neumann balanced growth theorem” lots of results pop up.]

            Now, it is easy to prove that in a Sraffian Standard Commodity economy or alternatively in a Von Neumann economy on a maximal Balanced Growth path, Marx’s claim is correct that total profits are proportional to total surplus labor.  This is obvious because such an economy is in effect a one commodity economy in which the inputs and outputs consist of quantities of the Standard Commodity.