# [OPE-L:4421] Re: Re: Re: Re: 2 equalities, one invariance condition

From: Rakesh Narpat Bhandari (rakeshb@Stanford.EDU)
Date: Fri Nov 03 2000 - 07:16:52 EST

```Ajit, you have not understood what I have proposed. So only a quick
response to point out where you have not understood me.

>
>___________________________
>Your first three equations will determine the relative prices of x, y, and the
>third commodity z, and the rate of profits r.

The set of equations can be solved for the absolute prices as well.

>  Before I mention the problems with
>your equations (4) and (5), let me first suggest to you that your
>equations are
>pure numbers. You consistently fail to mention the units in which
>the variables
>are measured.

One money dollar represents one hour of social labor time. The value
of money is held constant. So the total value of 875 is simply \$875.
In the initial tableau, the input means of production thus cost \$375
and the input wage goods \$300. Etc.

>  As a matter of fact, the question of the unit of measure is the
>crux of the transformation problem. So if you don't have the problem of unit
>upper most in your mind, you cannot even begin to understand the nature of the
>problem, let alone solving it. Now, my sense is that you would say,
>the numbers
>are given in money terms.

Right. This is exactly how Marx proceeds.

>Now, your world of equations have three commodities.
>It appears that the first one is something like iron, second one is something
>like wheat, and the third one could be gold.

Nope it's a luxury good, let's say porcelain.  I am not letting the
money commodity into this; the value of money is held constant. One
labor hour is represented by \$1. That's it. I have provided you with
a long quote from Grossmann justifying this theoretical choice.

>So let us say, gold is the money

Nope won't allow it.

>so the values/prices of x and y are given in terms of
>gold. In that case, your third equation turns out to be
>
>50x + 90y + r(50x + 90y) = 200   (3').
>
>Now, the system of equations (1), (2), and (3') are in well defined units, and
>they solve for x, y, and r. Given your unnecessary simple reproduction
>constraint on the system, it must follow that:
>
>r(225x+90y)+r(100x+120y)+r(50x+90y) = 200.

You just won't listen to what I am saying. The left hand is the sum
of profits. I am saying that since the mass of surplus value is
defined as total value minus cost price, the mass of surplus value
can no longer on the right hand be the same 200 it was before cost
prices were modified.

difference between me and Allin--but rather the equation which I have

875-375x-300y (total value minus modified cost price=surplus value).

I am NOT postulating the mass of surplus value (or rate of profit)
as invariant since I think that's impossible as we modify cost prices
given Marx's definition of surplus value as total value minus cost
price. Of course if the sum of surplus value changes as a result of
the modification of cost prices, so must the rate of profit which is
now modified sum of surplus value/modified cost prices.

Below you take r as invariant. But r, as well as the sum of surplus
value, is an unknown in my equations. It has to be solved for, and r
and the sum of surplus value can be solved in absolute terms!

>Here by design, total surplus value will always be
>equal to total profits.

That's absolutely correct.  I have written the set of transformation
equations in such a way that Marx's two equalities not only both hold
but also--it turns out to my surprise--are needed to determine the
system.

This is my point!

The point is that with the two equalities,as I have them,  they  no
longer overdetermine the system.

You can be assured that I did not invent my equations 4 and 5 because
I knew the system would not be overdetermined. I wrote equation 4
exactly as I understood Marx. That is, I read Marx defining surplus
value as total value minus cost price, so since you and Bortkiewicz
wanted to modify the cost price by having the inputs transformed as
well, I then wrote the left hand of the equation

875-375x-300y because that would now represent the new surplus value
as cost price is modified

and then I wrote the right hand

as the sum of the branch profits

Because that it is exactly what I understand the second equality to be.

My fourth equation has never been proposed before. But it is exactly
how I understand Marx.

I think we are agreed that any changing of the outward appearance of
the input and output prices of a system should not change the total
value/price which the commodity output embodies.

So that gave me my fifth equation which expresses both the invariance
of total value/price and the determination of total prices (the right
hand) by the invariant total value (the left hand).

875=375x+300y+r(375x+300y).

It turns out that my fourth and fifth equations do not overdetermine
the system.

So what I am saying to you, Ajit, is that when I wrote down the set
of input transformation equations for the scheme which Allin provided
me, I was left with those five equations.

And they do provide a solution not only for x/y and r but x, y, and
r. This system is determined in absolute terms.

I of course believe that I am the first to have correctly written the
input transformation equations in Marx's own terms. The innovation is
in my fourth equation, and it simply follows from my understanding of
how Marx defines surplus value and what the second equality means.

I am not trying to be cute. I am following Marx to the letter. And
that's how the equations turn out on my reading.. I would have been
disappointed if equations 4 and 5 overdetermined the system. But they
do not.

>   And if the gold sector is made of average organic
>composition of capital, then total value will also be equal to total surplus
>value. This is just one of those special cases.

But my set of equations does not require any special assumptions
about the organic composition of capital in the Div III, porcelain
production. . Another virtue of equations is that no such assumptions
are required.  In fact the tableau Allin gives us is exactly the one
Sweezy uses when he relaxes the special assumptions about Div III. So

>  Mathematically,
>your r has to be either known or unknown, they cannot be both at the
>same time.

Ajit, r is unknown in my set of equations.

>In equation (4), on the right hand side you have r as an unknown variable,
>whereas the left hand side 875 is derived by taking r = 8/27.

That is not how the \$875 is derived. \$875 is simply the direct and
indirect labor the commodity output represents. It is the value of
the means of production+the direct labor embodied in the commodity
output. This value cannot change simply by playing around with the
outside price appearances of the system, which is what the

r is not used to determine 875; in fact r is determined only by
dividing that total value by cost price.

After my transformation the sum of surplus value and r will not
remain invariant. But the  two equalities hold. In fact the two
equalities are needed to solve the system to get a new sum of surplus
value, rate of profit and prices of production.

My equations then allow for a substantiation of Marx's intuition that
if the cost prices are left unmodified, it is possible to go wrong...

>  So this is simply
>illegitimate. Same with equation (5). Whether you like it or not, you  have
>presented a simultaneous equation system with three unknowns and
>five equations.
>If all your five equations are independent ones, then your system is
>overdetermined (try solving for x, y, and r from your five
>equations, which you
>haven't done yet).

Nope I really only have 4 equations. The fifth is a mathematical
tautology (it is simply equation 3 and 4).

>
>On a general note: I would advise that a solution to the
>transformation problem
>does not lay in being cute by somehow showing that the two
>invariance conditions
>satisfy.

Again, you have not understood me. I do not have two invariance
conditions! Remember the slogan.

All the best, Rakesh
```

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