[OPE-L:5006] numerical example

From: Fred B. Moseley (fmoseley@mtholyoke.edu)
Date: Wed Feb 21 2001 - 08:53:53 EST

```Andrew, here is the numerical example that you have requested for so
long, including several times in the last week. It demonstrates that my
interpretation of Marx's determination of the rate of profit and prices of
production leads to DIFFERENT QUANTITATIVE RESULTS than the Sraffian
interpretation.

1.  Let's take Steedman's famous example of physical quantities (iron,
gold, and corn) (Table 1, p. 38 of Marx after Sraffa).

Steedman takes the real wage and one price (the price of gold = 1) as
given, and then solves for the rate of profit and the other two prices
(see column 1 below).  Notice that Steedman's rate of profit = 0.52.

However, we could also solve Steedman's system of equations in a different
way.  We could instead assume that the rate of profit is determined
independently of this system of equations.  I argue that Marx determined
the rate of profit by the analysis of capital in general in Volume
1.  Then this independently determined rate of profit can be taken as
given, along with the money wage (=1), and then solve for the three
prices.  For example, if the rate of profit is assumed to be = 0.40
(i.e. NOT = Steedman's 0.52), then the resulting three prices are shown in
column 2 below.  Indeed the same system of equations is compatible with an
infinite number of rates of profit.  For example, one could also assume r
= 0.25, and then determine a different set of the three prices.  These
results are shown in column 3 below.  And so on.

(1)		    (2)	    	    (3)
Steedman	Moseley - 1	Moseley - 2

r: rate of profit 	  0.52		    0.40	    0.25

p1: price of iron	  1.705		    4.667	    3.333

p2: price of gold	  1.000		    2.645	    2.022

p3: price of corn	  4.296		   11.201	    8.399

Andrew, you have said in the past that if I produced just one such
numerical example, then you would agree that my quantitative results are
different from the Sraffian results.  I have produced two such examples
and could of course produce more.  Is that enough?

2.  Andrew has argued that a given physical structure of inputs and
outputs (as in Steedman's example), along with the assumption that input
prices = output prices, UNIQUELY DETERMINES the rate of profit.  In other
words, there is only one rate of profit (and one set of prices of
production) that is compatible with a given physical structure and the
assumption that input prices = output prices.  He based his argument on
one of the Perron- Frobenius theorems (the one about the maximum
eigenvector being the only positive solution).

However, the Perron- Frobenius theorems apply only to the case in which
the system of equations is solved by taking the real wage and one price as
given, and then solving for the rate of profit and the other two prices
(i.e. the standard Sraffian method).  In this case (and this case ONLY),
there is a unique rate of profit.  When the rate of profit is solved
within the system of equation, then there is only one ("economically
meaningful") rate of profit.

If, on the other hand, the rate of profit is determined outside this
system of equations, and then taken as given in this system of equations,
then obviously there is no unique rate of profit (see Pasinetti, Lectures
on the Theory of Production, pp. 71-89 and 767-74).  The physical
structure and the assumption of "stationary prices" do not, by themselves,
uniquely determine the rate of profit.  Andrew's argument is wrong.  It is
based on an erroneous application of the Perron-Frobenius theorem.

Andrew has argued repeatedly that "method doesn't matter" - i.e. that even
though I argue that my rate of profit and prices of production are
determined by a different method than the Sraffian method, my rate of
profit and the prices of production turn out to be the same.

I think I have shown that this is not true.  Rather, I would say that
"method makes all the difference in the world."  The logical method used
to determine the rate of profit and prices of production (whether Marxian
or Sraffian) do indeed affect the quantitative results.

I look forward to further discussion.