[OPE-L:5577] Re: [OPE-L:5561] RE: Luxury goods and profit rate

Ajit Sinha (ecas@cc.newcastle.edu.au)
Tue, 07 Oct 1997 18:12:02 +1000

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Continuing the discussion on the 'new solution' and its incompatibility
with given real wage. ...

In the last message [ope-l:5561] Duncan suggested that my earlier proof did
not hit the target because I counted real wages as variable capital
advanced and deducted it from the gross output to derive the net output.
The 'new solution' on the other hand counts real wages in the net output:

Duncan:
>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.

I, however, do not think that whether the real wages is taken as the
capital advanced or paid after production from the net output changes the
nature of the problem at all. Later Duncan argues that:
>
>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 you are right if you define the rate of exploitation as
p(I-(A+bl))x/pblx. However, this is nothing but the ratio of the money
value of total profit divided by the money value of total wages, and is NOT
the rate of exploitation defined by the 'new solution' as I understand it.
The rate of exploitation in the 'new solution' is given in labor units. One
needs to derive the 'value of money' with different numeraires, and then
only could one derive the value of variable capital given the real wages.
It is here, i.e. the idea of 'value of money' in the 'new solution' where
the problem, in my opinion, lies. So let us one more time go through my old
example taking real wages as part of the net output this time.

a(11)p(1) (1+r) + a(12)p(2) = p(1) ...(1)
a(21)p(1) (1+r) + a(22)p(2) = p(2) ...(2)

>From (1) and (2)

p(1)/p(2) = [a(11)-a(11)a(22)+a(12)a(21)]/a(21) ...(3)

First case, put p(1) = 1:

then, p(2) = a(21)/[a(11)-a(11)a(22)+a(12)a(21)]; for simplicity sake we
will call the denominator A, thus

p(2) = a(21)/A ... (4)

Similarly, when we put p(2) = 1, we have

p(1) = A/a(21) ... (5)

The physical net output in the system is [1-a(11)-a(21)] of good one and 1
of good two.

In the case of p(1) = 1, the total price of net output = 1-a(11)-a(21) +
a(21)/A, which = [A-Aa(11)-Aa(21)+a(21)]/A
Let's call the numerator B, Thus the total price of net output
= B/A ... (6)

Suppose the total live labor-time spent in the production process is equal
to L, then
The Value of Money = LA/B ... (7)

Thus the value of variable capital = LA/B [a(12)+a(22)] ...(8)
The value of surplus value = L-[LA/B {a(12)+a(22)}] ... (9)
The rate of surplus value = (9)/(8) ... (10)

Now, take the case when p(2) is equal to 1:

Total price of net output = [1-a(11)-a(21)] A/a(21) +1, which
= B/a(21) ... (11) [check the difference between (6) and (11)]
The value of money = La(21)/B ...(12)
The Value of Variable Capital = La(21)/B [a(12)+a(22)] ...(13)
The Value of Surplus Value = L - [La(21)/B {a(12)+a(22)}] ...(14)
The rate of surplus value = (14)/(13) ... (15)

Clearly, in all likelihood the solution to (10) would be different from the
solution to (15). What do you think?

Cheers, ajit sinha