
Figure 1: Rate of Profits Unequal to Scale Factor for Rate of Profits 
This post continues the example in the previous
post.
I modify the prices equations so that the rate of profits in producing corn
is (s_{1} r̂), and the rate of profits in producing ale
is (s_{2} r̂). The solution to the price equations are:
p_{corn} = 16 [16 + (s_{1}  s_{2}) r̂]/[204 + (3 s_{1} + 9 s_{2}) r̂]
p_{ale} = 32 [10  (s_{1}  3 s_{2}) r̂]/[204 + (3 s_{1} + 9 s_{2}) r̂]
w = 4 [51  (9 s_{1} + 5 s_{2}) r̂  s_{1} s_{2} r̂^{2}]/[204 + (3 s_{1} + 9 s_{2}) r̂]
where r̂ is what I have been calling the scale factor for the rate of profits.
I want to show that, only in exceptional cases can the the markups s_{1} and s_{2} be rescaled
such that the scale factor is equal to the economywide rate of profits, whatever the distribution of income.
For concreteness, assume that the rate of profits is four times as high in the corn industry, as compared to the ale
industry. That is, introduce a new parameter α such that:
s_{1} = α
s_{2} = (1/4) α
Suppose the wage is unity and workers receive the entire standard commodity. Then both the scale factor for the
rate of profits and the rate of profits are identically zero percent.
Next, consider the other extreme case, where the wage is zero. The maximum rate of profits is 300 percent.
For the scale factor to also be three, the numerator for the wage, in the third equation above must be
zero, when a rate of profits of 300 percent and the above scale factors are substituted in. One thereby
obtains a quadratic equation:
3 α^{2} + 41 α  68 = 0
The positive solution is:
α = (1/6)( 41 + 2,497^{1/2})
For any scale factor for the rate of profits, one can find the wage with these markups.
For any wage, one can find the economywide rate of profits:
r = 3 (1  w)
The simplicity of the above equation results from taking the standard commodity
as the numeraire. The graph at the top of this post shows the difference
between the rate of profits and the scale factor, as a function of the
proportion of the standard commodity paid out in wages. The rate of
profits and the scale factor are equal only at the two extremes.
I guess that in a model with more commodities, the difference does not
come out looking like a quadratic function.
This counterexample demonstrates that, in general, one cannot rescale
markups such that the scale factor for the rate of profits is
the economywide rate of profits, however distribution stands.
Part of Marx's point in Capital is that observers who
focus on surface phenomena will not perceive underlying
value relations.
"The relations connecting the labour of one individual with that of the rest appear,
not as direct social relations between individuals at work, but as what they really are,
material relations between persons and social relations between things."
I guess I have shown that the existence of persistent ratios between the rate
of profits in various industries, as with a corporate sector and a sector
of small firms and proprietorships, provides another layer of confusion
in economic analysis. Here is a source of another
illusion created by competition.
Aside: I have stumbled across the
International Symposium on Marxian Theory (ISMT).
They have a series of books out, with a focus on reacting to the complete works of Marx and Engels (MEGA^{2}).
MEGA^{2} is published
under the auspices of the Internationale MarxEngelsStiftung (IMES) in Amsterdam and of other groups in other countries. Apparently,
volume 1 of Capital varied among editions, and Marx had several drafts. Furthermore, MEGA^{2} apparently has
editions of volumes 2 and 3 that show exactly how Engels edited them. These facts reenforce the point of
this post, not
that I want to read all of these variants.