# [OPE-L:3944] Frank Thompson's Theorem

andrew kliman (Andrew_Kliman@msn.com)
Tue, 7 Jan 1997 09:36:36 -0800 (PST)

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At the ASSA conference in New Orleans, Frank Thompson presented a paper, "The
Composition of Capital and the Rate of Profit: A Reply to Laibman." I wasn't
able to attend the session, since my own paper ("The Okishio Theorem: An
Obituary") was for some reason placed by the URPE session organizers into a
different panel at the same time, but I got a copy of the paper from Frank.
In it, he has a beautiful theorem which shows that, if the Okishio theorem
were true, then a rise in the value composition of capital cannot lower the
equilibrium rate of profit and must raise the equilibrium rate of profit if
the real wage rate is an increasing function of the demand for labor. Putting
the same thing differently, simultaneism implies that a rise in the VCC raises
the profit rate, instead of lowering it. Once more, then, simultaneism leads
to a result that contradicts Marx's value theory.

Frank examines the single-sector, circulating capital case. His own
presentation is in terms of finite changes and the math is too tedious for
e-mail, so I'll present the same (I think) thing using differential calculus.

Let K and L be the (absolute) physical amounts of the means of production and
living labor employed, and let w be the real wage per unit of living labor.
Since there's a single sector, the unit price or value of the means of
production is the same as the unit price or value of the wage good. Call it
p.

The VCC is then defined as
c = (pK)/(pwL) = K/(wL)

so that, by definition again,

w = K/(cL).

Totally differentiating, we get

w~ = K~ - c~ - L~. (1)

where w~ = (dw)/w, and similarly for K~, c~, and L~.

Now, Frank puts forth a "Minimal Assumption" that a fall in demand for labor
either leads to a fall in or does not affect, i.e., does not lead to a rise
in, the real wage rate. Thus, defining the elasticity of the real wage with
respect to labor demand as e, we have

e = w~/L~ > or = 0.

We can then rewrite (1) as

w~ = K~ - c~ - (1/e)w~, or, equivalently,

w~ = [e/(1+e)]*[K~ - c~]

so that a rise in c, the value composition, leads to a fall in the real wage
rate w if e > 0 and to no change in the real wage if e = 0, if K is held
constant. In other words, the partial derivative of w with respect to c is
negative if e > 0, and is zero if e = 0.

Now, if the Okishio theorem were true, i.e., if the simultaneist
interpretations were correct, then any fall in the real wage would lead the
equilibrium (i.e., uniform) profit rate to rise, given profit-maximizing
behavior and viable technical change. Hence, if e > 0, then a rise in the VCC
lowers the real wage and therefore raises the profit rate, and if e = 0, a
rise in the VCC has no effect on the real wage and therefore no effect on the
profit rate.

This is a beautiful example of the incompatibility of Marx's value theory and
simultaneism. Frank's theorem does not carry over to the successivist
interpretation, because for us, and Marx, the profit rate can fall when the
real wage rate falls.

Incidentally, Frank is not a proponent of the TSS interpretation and so
portrays his theorem as anti-Marx, whereas it its actual import is
anti-simultaneist. Again, as I interpret Marx, what Frank demonstrates is
wholly compatible with the logic of Marx's law of the tendential fall in the
profit rate.

Frank's paper also has some nice evidence showing that the aggregate physical
capital/output ratio in the U.S. has not risen over time, which supports the
argument John has been making. I hope to report his findings in a later post.

Nice going, Frank!

Andrew Kliman