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.

## 4 comments:

Prof. Wolff,

I go through the arithmetic of a 2-commodity example here:

http://www.dreamscape.com/rvien/Economics/Essays/sraffa.html

And I vaguely refer to some John Eatwell papers from the 1970s.

I read quickly through the essay to which you link, and agree with most if not all of it [it tells you what a nerd I am that it was like a walk down memory lane.] For my critique of this mode of analysis of Marx, you can see my paper Critique and Reinterpretation of Marx's Labor Theory of Value, archived at box.net via the link at the top of my mail blog page. Thanks for the link.

"[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?]"

Unfortunately, I have no idea what that means. So it isn't quite clear to me.

The question, "Is this clear?" was originally directed at Mark Benz, a grad student who is a mathematician. I knew it would be unclear to a larger audience.

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