[OPE-L:5020] Re: RE: numerical example!!!

From: Fred B. Moseley (fmoseley@mtholyoke.edu)
Date: Wed Feb 21 2001 - 18:03:21 EST

Hi Andrew,

Thanks for your reply.  I think we are making progress.

You are right that in my example I assumed the money wage as given rather
than the real wage, as in Steedman.  I did that to make it easier to solve
the system of equations, but I realize now that that distorts the
comparison with Steedman.

However, I think what my example shows, at least, is that, if the money
wage is taken as given, then a given set of physical inputs and outputs
and the assumption of "stationery prices" does not uniquely determine the
rate of profit.  My example shows that, with a given money wage, at least
two rates of profit are compatible with the given physical structure and
the assumption of "stationery prices" (and we could compute more).  

The reason why more than one rate of profit is compatible the given
physical structure is that we have a system of 3 equations in 5 unknowns
(three prices, the wage rate, and the profit rate).  A determinate
solution is possible only if two additional variables are taken as
given.  And there are several different options for which two variables
should be taken as given.  The standard Sraffian method is to take the
real wage (or the money wage; e.g. Pasinetti) and one price as given, and
then solve for the rate of profit and the remaining (n - 1) prices.  In
this case, there is indeed a unique rate of profit, as proven by the
Perron-Frobenius theorem.  

However, one can also take the rate of profit and the money wage as given,
and then solve for the 3 prices, as I did in my example.  In this case,
there are an infinite number of rates of profit that are compatible with
the given quantities of inputs and outputs.  

Andrew, do you agree with this?  If not, why not?

If that point is agreed, then the next question is: does this result still
hold when the real wage is taken as given, rather than the money wage?  I
don't see why not, but unfortunately, the system of equations is more
difficult to solve under the assumption of a given real wage.  The
constant terms drop out and the system becomes homogeneous of degree zero,
which is more difficult to solve.  I have not been able to solve it by
hand and I don't have the software to solve systems of equations.  I will
get some software soon, and then we shall see whether the result still
holds under the assumption of a given real wage.

In the meantime, perhaps another listmember who has the software can solve
this system of equations for us and let us know the results. 

Assuming to begin with that the rate of profit = 0.4, the equations are:

	(1.4) [28 p1+ 56 (5/80 p3)]   =   56 p1

	(1.4) [16 p1+ 16 (5/80 p3)]   =   48 p2

	(1.4) [12 p1+  8 (5/80 p3)]    =   8 p3

Please solve for the three prices.

And then assume the rate of profit = 0.25, replace the 1.4 in the above
equations by 1.25, and solve for the three prices again.  Thanks in
advance for any help.  

Logically I don't understand why this system of equations would not be
solvable for different rates of profit.  This is a system of three
equations in three unknowns, which should in principle be solvable.  Does
anybody see a reason why this system might be solvable?

I can't wait to see the results!


P.S.  Andrew said:

> In your second example (column 3), the real wage rate isn't even
> equalized across sectors any more.  (Nor is the money wage rate
> equalized any more).  

Andrew, I don't understand why you think that the money wage rate is not
equal across sectors.  The money wage rate is assumed = 1 in all
sectors.  From which it follows that the real wage rate will also be equal
across sectors, since the real wage rate = 1 / (price of corn).  
Thanks for the clarification.  

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