[OPE-L:2435] response to Andrew, Part I

Fred Moseley (fmoseley@laneta.apc.org)
Thu, 30 May 1996 23:03:36 -0700

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This is a response to Andrew's (2301), Part I of our recent discussion.
Thanks very much to Andrew for continuing to clarify his arguments for me.
The version of Andrew's "proof" that my equation for the rate of profit must
equal the Sraffian equation in (2301) is much clearer than the earlier
version (and is also clearer than the version presented in 2309).

I have come to realize that Andrew's "proof" depends on a particular
intepretation of the meaning of mathematical equations. Andrew argues that
the equations in his "proof" DO NOT IMPLY ANY RELATIONS OF CAUSATION OR
DETERMINATION. They are simply equalities or identies.

I argue instead that the equations I have used to express Marx's theory DO
imply relations of causation or determination. And I argue also that the
equations used to express Sraffian theory imply DIFFERENT relations of
determination. Andrew argues that one cannot "conclude" or "infer" anything
about causal relations from these equations themselves itself. I agree. But
this is not what I am arguing. I am not arguing that these causal relations
are inferred FROM THE EQUATIONS THEMSELVES. I am arguing instead that these
causal relations are inferred FROM MARXIAN OR SRAFFIAN THEORY. These
equations by themselves do not imply causal relations. But Marxian and
Sraffian theories import (different) causal relations to these equations.
To deny these causal relations is to deny the economic theories behind them.
But the economic theories are supposed to be what we are talking about.

I argue below that the entirely different relations of determination implied
by Marxian and Sraffian theories render Andrew's "proof" invalid. Let us see.

A. Andrew's "proof" consists of four main steps:

1. My equation for the rate of profit:

(1) R = S / (C + V)

where S, C, and V are aggregate quantities. I argue that C and V are taken
as given and S and hence R are determined in the Volume 1 analysis of
capital in general, prior to the determination of prices of production.

2. An equation for prices of production for each of the three departments:

(2) Pi = (Ci + Vi) (1 + R)

where Ci, Vi, and Pi dept. money-capital totals.

3. A relation between the dept. money-capital totals in equation (2) and
the product of unit price and quantity in the three depts (assuming one type
of good produced in each dept) according to the following indenties:

(3) Ci = p1*ai*Xi
Vi = p2*bi*li*Xi
Pi = pi*Xi

4. Substitution of the indenties in (3) into the equation in (2), which
yields the following equation, in matirx terms.

(4) p = (pA + pbL) (1 + R)

Andrew argues that this is exactly the same equation as the equation for the
determination of the Sraffian rate of profit (r):

(4') p = (pA + pbL) (1 + r)

Therefore, Andrew concludes that R = r.

B. My response begins with step #2 in Andrew's "proof".

1. Within Marxian theory, equation (2) assumes or expresses the following
relations of determination:

a. the dept. totals of constant capital (Ci) and variable capital (Vi) are
TAKEN AS GIVEN as quantities of money-capital.

b. the rate of profit (R) is also taken as given, as PREDETERMINED by the
aggregate analysis of vol. 1, i.e. by equation of (1).

c. the dept price of production Pi is determined by equation (2) from the
above givens.

2. Within Sraffian theory, on the other hand, the very same equation
assumes or expresses very different relations of determination:

a. the dept. totals of constant capital (Ci) and variable capital (Vi) are
NOT taken as given as quantities of money-capital, but are instead DERIVED
FROM given technical conditions of production and the real wage (equations
(3) and (4)).

b. r is not detemined prior to prices of production and taken as given in
this equation of prices of production, but is instead determined
simultaneously with these prices of production (equation (4')).

Therefore, even though this equation is exactly the same, it expresses
entirely different relations of determination within the two different
theories.

C. Similarly, the equations in (3) also express very different relations
of determination within the two different theories.

1. In Marxian theory, these equations play no essential role. There is no
need to decompose the dept. (or industry) money-capital totals into unit
price and input and output quantities in order to explain the equalization
of profit rates and prices of production. Neither the unit prices or the
input-output quantities play a role in Marx's theory of prices of
production. However, if this decomposition is carried out, then it would be
done according to the following logic:

a. the givens in these equations are the dept. money-capital totals (Ci,
Vi, and Pi) and the input and output quantities ai and Xi.

b. unit prices could then be derived from the above givens; e.g.
p1 = Ci / ai*Xi

2. According to Sraffian theory, these equations once again play no
essential role, but for a different reason: because there is no need to
determine the department money-capital totals. These dept. money-capital
totals play no role in Sraffian theory. However, if these totals are
determined, it would be according to the following logic:

a. the givens in these equations are the input and output quantities (ai
and Xi) and the unit prices (pi) as derived from the physical quantities by
equation (4').

b. the dept. money-capital totals would then be derived as the product of
these givens as in equation (3).

Therefore, we can see that the unit prices (pi) are determined according to
very different logics. In Marxian theory, these unit prices play no role,
but they can be derived as described in (C.1.b) above. In Sraffian theory,
these unit prices are the main variables to be determined (along with the
rate of profit) and are determined according to equation (4').

D. From the above, it follows that the unit prices (pi) in equations (4)
and (4') are not the same prices. They are determined in entirely different
ways and therefore in general will not be equal. It is a mistake to use the
same letter to stand for unit prices in the two theories. If pi is used to
represent unit prices in Marx's theory, then we should use some other letter
- say pi' - to represent unit prices in Sraffian theory. Once it is
recognized that pi NOT = pi', then it does not follow that R = r.

Therefore, Andrew's "proof" that R = r in fact shows only that R and r are
both related to unit prices in ways that can be expressed with the same
mathematical equation, in which prices are determined as a profit mark-up
over costs and costs can be decomposed into unit prices and input-output
quantities. HOWEVER, (1) the precise nature of the relation (the order of
determination) between R or r and unit prices is entirely different in the
two theories, and (2) the unit-prices themselves to which R or r relate are
entirely different in the two theories. Therefore, it does not follow that
R = r.

It also seems to me that Andrew's "proof" proves too much: one could assume
that R is determined by ANYTHING and Andrew's logic would still apply, as
long as equation (2) applies and long-run equilibrium is assumed (i.e. input
prices equal to output prices). Equation (1) is really irrelevant. If one
assumes that prices are a mark-up over costs and long-run equilibrium, then
Andrew's logic "proves" that EVERY THEORY of the rate of profit leads to the
same rate of profit as in Sraffian theory. I don't think so. I think
instead that Andrew's "proof" implicitly assumes that prices in Marxian
theory (or any other theory) are the same as prices in Sraffian theory.
Once it is recognized that the prices in the different throeies are not
necessarily equal, then it follows that the rates of profit in the different
theories are also not necessarily equal.

Andrew (and others), I look forward to continuing this discussion after my
return in 10 days or so.

Fred