[OPE-L:5388] Re: Luxuries in the New Solution

Duncan K. Foley (dkf2@columbia.edu)
Sun, 31 Aug 1997 06:47:10 -0700 (PDT)

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A reply to Andrew's OPE-L:5368:

Andrew wrote:

>Rieu and Duncan have claimed that, although the "New Solution" (NS) to the
>"transformation problem" is simultaneist, it nonetheless replicates Marx's
>result that production conditions in luxury industries affect the general
>of profit. I have shown that this cannot be the case, given that workers'
>spending on "wage goods" equals their wages. Duncan has now given what seems
>to me to be a somewhat equivocal reply, writing: "_given_ the value of
>labor-power in the NS sense, the composition of demand (including luxury
>consumption) can affect the surplus value. But if we assume that the value of
>labor-power always adjusts to maintain w = pb, the endogenous fluctuations in
>the value of labor-power offset these effects."
>If necessary, I guess I'll have to come back and debate the meaning of the
>"value of labor-power" being given.

>But let me try a simpler and more
>straightforward demonstration by abstracting from wage-related stuff
> Assume that the "value of labor-power" is zero. :-) :-) So it is given.
>Wages are also zero. :-) :-) Then it is clearly the case that the NS, like
>all other simultaneist interpretations, contradicts Marx's claim that
>production conditions in luxury industries affect the general rate of
I'm not sure Andrew and I have the same conception of the "New Solution".
Here's how I would set out the theory of the rate of profit.

For definiteness, take a one-period circulating capital model with n
commodities, each produced by other commodities and labor (the standard
Sraffa production setup), with no joint production, described by a square
matrix A of input coefficients, and a row vector l of labor inputs. Some of
the commodities may be "luxuries" in the sense that they do not appear as
inputs to themselves or any other commodities. Under these assumptions we
can calculate the row vector of embodied labor coefficients, v, since

v = vA + l, so that v = l Inverse(I-A)

Suppose workers are paid enough to consume the column vector b pure unit of
labor-power sold (assuming for the moment that there is a fixed relation
between the sale of labor power and the actual expenditure of labor time.)
The the value of labor-power is vb, which is also the wage measured in
embodied labor coefficients. If the gross output is described by the column
vector x, the total labor time is lx, the total wage bill is vblx, and the
surplus value in labor terms is lx - vblx = (1-vb)lx. The capital invested,
assuming that wages are paid at the beginning of the production period,
measured in embodied labor coefficients, is vAx + vblx. It's convenient to
define an augmented input matrix A' = A + bl, (note that the product of the
n x 1 column vector b and the 1 x n row vector l bl is an n x n matrix), so
the capital employed can be expressed in embodied labor terms as vA'x. The
rate of profit over the system as a whole is the ratio of the surplus value
to the capital invested:

r(v) = (1-vb)lx/vA'x

In general this rate of profit will depend on the output vector x, and thus
on the output of the luxury sectors. This will be true even if b = 0.

The "New Solution" considers a system in which there is row vector of money
prices p, and a scalar wage w, and proposes to define the surplus value as
px - Ax - wlx = p(I - A)x - wlx. The monetary expression of labor time, u,
is the ratio of the value-added p(I-A)x to the total labor time lx:

u = p(I-A)x/lx, so it is immediate that the surplus value is (u - w)lx =
u(1-(w/u))lx. The New Solution proposes to consider w/u to be the analogue
to the value of labor power vb (though they are not in general equal except
in the case where the prices p are proportional to the embodied labor
coefficients v.) The value of the capital invested is pAx + wlx.

The overall rate of profit is

r(p) = (u-w)lx/(pAx + wlx), which in general depends on x, u, w, p, A and
l, and thus depends on the composition of output, including the production
of luxury goods.

Notice that the above argument does not require the assumption of
equalization of the rate of profit across sectors, nor does it stipulate
any particular theory of the level of the wage, w. If, now, we add the
assumption that profit rates are equalized:

p = (1+r)(pA + wl), and the assumption that wages are determined so that
workers can buy a given vector of commodities b:

w = pb, we get by substitution,

p = (1+r)p(A + bl) = (1+r)pA'. This is an eigenvalue problem in which r is
determined independently of x, as Andrew points out, and p is determined up
to a constant of proportionality. If we then add the additional assumption
that the monetary expression of labor time, u, is given, we can determine
the level of p from

p(I-A)x = ulx.

This is what I was trying to explain in my earlier posting. But notice that
the assumptions of equalized rates of profit and the assumption w = pb are
not parts of the New Solution itself, but auxiliary hypotheses which are
consistent with it. These hypotheses are, in fact, those adopted by Seton,
Morishima, Roemer, Steedman, Samuelson, and other writers who make the
point that there is in this derivation no systematic appeal to labor time
as such, since the prices and profit rate can be determined directly purely
from the technological coefficients A and l.

An alternative would be to take the New Solution "value of labor-power",
1-e = w/u, and the monetary expression of labor time, u, as given, rather
than the real wage b. Then if we assume profit rate equalization, the full
system is:

p = (1+r)(A + wlx)
p(I-A)x = ulx
w = (1-e)u

In this system of equations, p, w, and r depend in general on u, e, and x.

This same argument goes through if e = 1 so that w = 0.

I think this line of thinking is consistent with Rieu's remarks in the thread.


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