[OPE-L:5561] RE: Luxury goods and profit rate

Duncan K. Foley (dkf2@columbia.edu)
Thu, 2 Oct 1997 05:56:47 -0700 (PDT)

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In reply to Ajit's OPE-L:5478:

>At 07:34 15/09/97 -0700, Duncan wrote:
Ajit continues the argument that the NS does not hold constant the rate of
surplus value and the value of labor power under a change of numeraire, by
considering a particular case:

>I'm not convienced. Let us take a simple case of two commodity model:
>[a(11)p(1) + a(12)p(2)] (1 + r) ....(1)
>[a(21)p(1) + a(22)p(2)] (1 + r) ....(2)
>Where a(ij) refers to the amount of j needed to produce one unit of i; p(i)
>refers to price of i, and r is the uniform rate of profit.

Equations (1) and (2) , once the ellipses are completed, read:

[a(11)p(1) + a(12)p(2)] (1 + r) = p1 (1)
[a(21)p(1) + a(22)p(2)] (1 + r) = p2 (2)

Sector 1 produces means of production, and sector 2 produces means of
subsistence, or consumption goods. Since there is not explicit
representation of labor input, I take this to be the _augmented_ A' matrix
(A' = A + bl), where the real wage bundle b = (0,b2), so that a12 = b2l1
and a22 = b2l2 in the notation I used in an earlier post. Thus the basic
input coefficients matrix A has A11 = a11, A12 = 0, A21 = a21 and A22 = 0.

>[a(12) + a(22)] is the total consumption of the working class, and treated
>as variable capital advanced.

Here I have two problems. First, there is not explicit representation of
the gross output bundle x. In my notation the total consumption of the
workers is blx. If the gross output x = (1,1), then I can make sense of
this remark. So I will proceed on that assumption.

>The net output = [1 - {a(11) + a(21)}] of good(1) + [1 - {a(12) + a(22)}]
>of good(2)

Here I think Ajit parts company with the usual NS treatment. The NS treats
wages and workers consumption as part of the net product, as does modern
national income accounting. (Occasionally Smith and Ricardo refer to "net
income" in the sense here defined, but hardly anybody does afterward.) Thus
I would measure the net output vector as y = (I-A)x, _not_ (I-A')x. In
terms of Ajit's notation this would be (1-a11,-a21) rather than the
expression he proposes.

>The two price equations give us:
>{a(11)p(1) + a(12)p(2)}/{a(21)p(1) + a(22)p(2)} = p(1)/p(2) ....(3)

I agree that this is a consequence of the price equations.

>First case. Take p(1) = 1
>--> p(2) = [a(22) - a(11) +- square root of {a(11)square + a(22)square -
>2a(11)a(22) +
>4a(12)a(21)}]/-2a(12) ... (4)
>Let's call the positive value of equation 4 = A; i.e. p(2) = A
>Now, the total money value of net output, when p(1) = 1, would be:
>1 - a(11) - a(21) + A - Aa(12) - Aa(22) ....(5)

Here I disagree, on the ground that the net output vector is not defined
according to the NS definitions. I think the value of net output is
p(I-A)x, not p(I-(A+bl))x.

>Suppose direct labor-time spent in production is equal to L
>Then, the value of money = L/expression (5) ... (6)
>Therefore, value of variable capital = {a(12) + a(22)}A multiplied by
>expression (6)....(7); and the surplus value would be L - expression (7)

If the concept of net output were correct, these expressions would
correspond to the NS definitions.

>The Second Case. Put p(2) = 1 in equation (3)
>--> p(1) = [a(11) - a(22) +- square root of {a(22)square + a(11)square
>-2a(11)a(22) + 4a(21)a(12)}]/-2a(21) ...... (9)
>Clearly (9) is different from (4). Let's call it B, i.e. p(1) = B
>Now in the regime when commodity (2) is the money commodity,
>The money value of Net output would be = {1 - a(11) - a(21)}B + 1 - a(12) -
>a(22) ....(10)
>The value of money would be = L/expression in (10)... (11)
>The value of variable capital would be = {a(12) + a(22)} multiplied by
>expression in (11) ....(12)
>Since (10), (11), and (12) have all B element instead of A, and so all the
>value expressions in the two regimes are different.

The effect of solving equation (3) is to discover the lefthand eigenvector
of the augmented production matrix A + bl corresponding to the largest
(Perron-Frobenius) eigenvalue, which are the usual "prices of production."
Eigenvectors are determined only up to a scalar multiple, so (1,A) must be
proportional to (B,1) in Ajit's terminology. (You can check this in terms
of the quadratic formulae, as well.)

Thus the NS monetary expression of value will change between the two
numeraires, as it should, but the NS rate of surplus value
p(I-(A+bl))x/pblx will remain invariant with the change in numeraire, since
the price vector appears in both the numerator and the denominator.

>I think I have followed the New Interpretation method faithfully, and have
>only kept the real wages constant. Tell me where did I go wrong.

I'm not sure you went "wrong," but you deviate from the NS definitions by
deducting workers' consumption from gross output in calculating net output.


Ajit continues:

>If my above example is not flawed, then given real wages proposition cannot
>be maintained by the new interpretation. And this would be a serious issue.
>Moreover, I think the critique I have presented in the latest issue of
>RRPE, which I think must be in everybody's mail box by now, is also
>important. The rate of exploitation as a concept seem to lose consistency
>in the new interpretation. Some rethinking is required on this point, I think.

I think the example is flawed, and that the NS measure of the rate of
surplus value is invariant to a change in the numeraire. I wish your paper
had addressed more directly the converse aspect of the propositions: if you
believe that the money value added represents the living labor time, and
that gross profits represent unpaid labor time, as Marx argues, then you
have to define the value of labor-power in the NS fashion.


Duncan K. Foley
Department of Economics
Barnard College
New York, NY 10027
fax: (212)-854-8947
e-mail: dkf2@columbia.edu