[OPE-L:2309] Re: response to Andrew - Part 1

akliman@acl.nyit.edu (akliman@acl.nyit.edu)
Tue, 21 May 1996 13:38:04 -0700

[ show plain text ]

Fred didn't like my 1st proof that "his" profit rate is quantitiatively
identical to the "Sraffian" profit rate, and that value in Fred's
interpretation of Marx is quantitatively redundant. I provided a 2d
proof yesterday, likewise valid IMHO, but it might not be so accessible
to everyone on the list.

Here, therefore, is a 3d proof. Like the 2d proof, it begins the way
Fred wishes, from "his" profit rate, R, determined (logically) prior
to prices of production, in Vol. I's analysis of capital in general:

R = (S1 + S2 + S3)/(C1 + C2 + C3 + V1 + V2 + V3) [1]

Hereafter, I'll use R instead of the right-hand side expression, but
keep in mind that R depends on S3, C3, and V3. Now, under the
assumptions we've been employing in our discussion, Fred's production
price equation for Dept. I is

P1 = (C1 + V1)(1+R) [2]

and, since Ci = p1*ai*Xi, Vi = p2*bi*li*Xi, and Pi = pi*Xi, we can
plug them into [2] and, after cancelling out X1, rewrite it as

p1 = (p1*a1 + p2*b1*l1)(1+R) [2']

or, letting p1/p2 = z,

z = (z*a1 + b1*l1)(1+R) [2'']

We can solve [2''] for z:

z = (b1*l1[1+R])/(1 - a1*[1+R]) [3]

The production price equation for Dept. II is

P2 = (C2 + V2)(1+R) [4]

which, in a manner analogous to the rewriting of [2] as [2''], can also
be written as

1 = (z*a2 + b2*l2)(1+R) [4'']

Substituting the right-hand side of [3] for z in [4''], we have

(a1*b2*l2 - a2*b1*l1)(1+R)^2 - (a1 + b2*l2)(1+R) + 1 = 0 [5]

after a little manipulation.

Note that *each and every* equation given a number in square brackets
includes R, and thus includes S3, C3, and V3. But the "Sraffian"
profit rate, r, is given by

(a1*b2*l2 - a2*b1*l1)(1+r)^2 - (a1 + b2*l2)(1+r) + 1 = 0 [6]

So that a comparison of [5] and [6] shows immediately that R = r.

Andrew Kliman