Sigh.
There is so much to explain that I am having trouble keeping this simple,
so I am just going to take things one at a time and trust to your
patience. Thus far I have been talking
about labor values, but of course no solid, upstanding English businessman
cared one whit for labor values. His eye
was fixed on

*prices*-- the prices he paid in the market for his production inputs, the price, or wage, he paid his workers, and the prices he could get in the market for his finished goods. He was a ruthlessly single-minded cheeseparer, our Entrepreneur, paying as little as he could manage for his inputs and wages and charging as much as he could get away with for his finished goods. And, of course, all the while, keeping careful track of it all so that he could calculate what rate of profit he was making on his invested capital. If Marx is correct that what is really taking place under the surface, as it were, is the extraction of surplus value from the workers, then what relation does this bear to what seems on the surface to be exchanges of money for goods or labor? How do labor values come to manifest themselves as prices? Old lefties and keepers of the sacred flame call this The Transformation Problem.
We can make a start toward solving this problem
by looking at the little corn/iron/books model with which we have been dealing,
but first, we must make a number of very powerful simplifying historical,
systemic, behavioral and knowledge assumptions of the sort that students of
modern economic theory will find familiar.
Let us focus on just four.

Assumption One: By a long process of historical evolution,
the goods offered in the marketplace have become standardized, so that one yard
of linen is interchangeable with another, one bushel of corn indistinguishable
from the next, one pound of iron as good as any other. This is not at all inevitably or universally
true, as the purveyors these days of "artisanal bread" will assure
you, but the capitalist market that Marx and the classicals are looking at has
driven all such pre-capitalist anachronisms into specialty shops and craft
fairs, concerning itself only with output that, in the ugly but accurate
terminology of modern free enterprise, is referred to simply as
"product."

Assumption Two: Exactly the same standardization and routinization has reduced the rich variety of productive work activities to "labor," measured simply in units of time. The ancient crafts to which young workers were apprenticed for as much as seven years have been replaced by the homogenized semi-skilled labor of machine operatives who can, in days or weeks, be transferred not merely from one factory to another but from one industry to another. I shall have a good deal to say about this process later, but those who would like to follow the idea up right now might usefully take a look at a fine old book,

*Labor and Monopoly Capital*, by Harry Braverman.
Assumption Three: Everyone in the economy, including workers, has a clear, unambiguous set of preferences for the available goods, and makes choices based solely on considerations of price, always opting for the lowest price. Workers are not bound, by law, custom, or desire, to any particular locale, and will go wherever there are jobs, choosing which jobs to accept, if there is a choice, on the basis of the wage offered, and not from a preference for one sort of work rather than another. In a remarkable little chapter of his widely read textbook,

*Principles of Political Economy*entitled "Of Competition and Custom" [Book 2, Chapter 4], John Stuart Mill noted that this assumption is in fact false, a fact that undermines the entire theoretical enterprise, but that is a bit too squirrely a matter to go into here.
Assumption Four: Everyone in the capitalist system has perfect knowledge of prices, wages, and products, and capitalists know, or very quickly learn, about innovations by their competitors, about demand not being met in the market, and about any other facts that might affect their returns. Workers are equally well-informed about wage rates and available jobs anywhere in the economy.

Well, with all of these assumptions in place, it is not too difficult to show that in short order an economy will exhibit a single set of consistent prices, a single wage rate, and a single rate of profit. The assumption about consistent prices simply means that if one bushel of corn will trade for five pounds of iron, and five pounds of iron will trade for one theology book, then one bushel of corn will trade for one theology book. If this were not the case, someone could get very rich very quickly by just swapping corn for iron for theology books, without ever actually producing anything. [Hmm. That does sound very familiar indeed, but never mind.]

What happens in our little corn/iron/books economy? To answer that question, let us set up some price equations. As already indicted, we will use the letter p with subscripts to indicate prices. Let us use w to stand for the wage rate, and the Greek letter π to represent the equilibrium economy-wide rate of return, or profit rate. Our table of inputs then can be translated into the following three price equations:

Corn Sector: (100w + 2p

Iron Sector: ( 90w + 9p_{c}+ 16p_{i}+ 0p_{b})(1 + π) = 300 p_{c}_{c}+ 12p

_{i}+ 0p

_{b})(1 + π) = 90p

_{i}

Books Sector: ( 20w + 1p

_{c}+ 2p

_{i}+ 2p

_{b})(1 + π) = 40p

_{b}

Why (1 + π) in each equation? Because the capitalist must make enough from the sale of his products to earn back what he laid out for inputs and labor [that is the 1] and also he must make enough to earn a rate of π on that investment. Hence 1 + π.

The numerate among you will immediately recognize that we have a problem. This is a system of three equations, but it has five unknowns: w, p

_{c}, p_{i, }p_{b}, and π. The system is, as mathematicians like to say, underdetermined by two degrees of freedom. We can take a step toward making the equations solvable ["solvable" means figuring out what the numerical values of the variables are, by the way] if we simply choose one of our three commodities and use it as our money, measuring everything else in terms of it. If we choose corn as our money, or "numeraire," as the jargon has it, then one bushel of corn will be our "dollar," and everything will be priced in bushels of corn.
This has a rather strange sound nowadays, but
it would not have struck mid-nineteenth century Englishmen as strange at all. Their money was the Pound Sterling, which is
simply one pound of one of the available commodities, namely silver. For the Masai, the unit of currency would be
a cow. For the Uplands Ndani of New
Guinea, it would be a cowrie shell. This
is known as "commodity money," and it raises as whole raft of
interesting questions with which I shan't trouble you.

So, now we can set the price of a bushel of
corn equal to 1 [i.e., a bushel of corn is worth exactly one bushel of corn],
and that reduces the number of variables to four. better, but still not good enough, we might
suppose. However, things now get rather
nifty. If we try to solve the three
equations for the prices of iron and books, letting the price of corn equal 1,
the oddest thing happens. As we
manipulate the equations, w and π drop out, and we can actually arrive at
values for p_{i}and p

_{b}! The values, in fact, are:

p

p_{i}= 3 bushels of corn_{b}= 1.5 bushels of corn

We are now in a position to test the thesis that Ricardo advanced and Marx endorsed, at least for economies exhibiting equal organic composition of capital, namely that prices are proportional to labor values. Let us see.

p

p_{c}/λ_{c}= 1/.4 = 2.5_{i}/λ

_{i}= 3/1.2 = 2.5

p

_{b}/λ

_{i}= 1.5/.6 = 2.5

**Ta da!**

What about w and π? Well, it is easy enough to show that the wage
and the rate of profit are inversely related to one another. As one rises, the other falls. Prices, being determined by labor values,
remain unaffected by changes in the wage and the concomitant change in the rate
of profit. Thus the division of the
social product, which is regulated by wages and profits [and rents, but never
mind that -- it doesn't matter here], is entirely separate from the
determination of commodity prices, which depend upon technology, which is to
say the quantities of labor directly and indirectly required for their
production.

It is truly remarkable that Ricardo and Marx
understood this perfectly, despite the fact that neither of them employed the
analytical device of simultaneous equations that I have been laying before you
here.

But alas, all of this holds true only in the
fantasy world of equal organic composition of capital. Tomorrow I shall complicate things, and
introduce the theoretical innovation which, Marx thought, would lay bare the
heartless soul of capitalism: the
distinction between labor and labor power.

## 5 comments:

I'm enjoying anew the inclusion of theology books as one of the three commodities produced, every time I read it. Witty choice.

Thank you. I kind of liked it. It raises some interesting questions, such as: How do two theology boooks get used up in the production of new ones? Are they read to death, or perhaps cut and pasted. :) I also love the fact that this is a pain free way of slipping in the notion of partially decomposable non-negative square matrices.

I still think you should consider Andrew Kliman's answer to the transformation problem. Hell I'd be willing to buy the book for you :)

http://www.amazon.com/Reclaiming-Marxs-Capital-Inconsistency-Dunayevskaya/dp/0739118528/ref=sr_1_2?ie=UTF8&qid=1360624159&sr=8-2&keywords=andrew+kliman

Prof.

The solution I found to the vector equation was the vector

-(2*w)/(pi-5),

-(6*w)/(pi-5),

-(3*w)/(pi-5)

You'll notice that the values keep the same proportionality you found: 1:3:1.5.

But I am a bit perplexed by the minus signs. I suppose I must have done something wrong, but I couldn't find any obvious mistake in my procedure.

So, my question is: are those signs important?

This was the equation:

w*l+(1+pi)*A.p=O.p

where pi is an scalar, p is the 3 commodity price column vector, A is the 3x3 matrix you used in part eight; and O is a 3x3 matrix with the gross outputs in the main diagonal

A suggestion for the future:

Speaking of the use of a numeraire, you said: "This is known as 'commodity money', and it raises a whole raft of interesting questions with which I shan't trouble you."

I wouldn't mind being troubled with that! I suspect the use of a commodity money, as opposed to fiat money, has interesting consequences in relation to Say's Law and Keynesian policies.

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