[OPE-L:4420] Re: Re: Re: 2 equalities, one invariance condition

From: Ajit Sinha (ajitsinha@lbsnaa.ernet.in)
Date: Fri Nov 03 2000 - 05:47:53 EST

Rakesh Narpat Bhandari wrote:

> >
> >Rakesh, I think you have found the right vocation for yourself
> >finally! Now, my
> >advise would be to put all your energy into t-shirt business. All the best!
> >Cheers, ajit sinha
> Glad you're back, Ajit. Truly. My joke was made at the end of the
> argument. Your joke is your argument.
> Remember you said that the two equalities overdetermine the system. I
> am saying that this is not true. We can have total value=total price
> and mass of surplus value=sum of profits as long as surplus value is
> defined,as Marx explicitly does,  as total value minus cost price
> (instead of value of inputs) which allows for the modification of the
> latter to change the mass of surplus value. The two equalities do not
> then overdetermine the system of transformation equations. I am the
> first to argue that the problem of overdetermination disappears once
> we use Marx's definition of surplus value. So I can understand why
> you may not have got the point.
> To see this, I'll have to copy this again:
> _______________________
> The initial value table:
>           c       v       s     value
>     I  225.00   90.00   60.00   375.00
>    II  100.00  120.00   80.00   300.00
>   III   50.00   90.00   60.00   200.00
> Tot.  375.00  300.00  200.00   875.00
> Marx's first-step transformation takes the given total s
> and distributes it in proportion to (c+v).  Thus:
>           c       v    profit   price   pvratio
>     I  225.00   90.00   93.33   408.33   1.0889
>    II  100.00  120.00   65.19   285.19   0.9506
>   III   50.00   90.00   41.48   181.48   0.9074
> Tot.  375.00  300.00  200.00   875.00   1.0000
> _________________
> I propose these input transformation equations in which total
> value/price is invariant from the original tableau (equation 5) and
> the sum of surplus value equals (determines) the sum of profits
> (equation 4).
> (1) 225x+90y+r(225x+90y)=225x+100x+50x
> (2) 100x+120y+r(100x+120y)=90y+120y+90y
> (3) 50x+90y+r(50x+90y)=r(225x+90y)+r(100x+120y)+r(50x+90y)
> (4) 875-(225x+100x+50x+90y+120y+90y)=r(225x+90y)+r(100x+90y)+r(50x+90y)
> (5) 875=375x+300y+r(225x+90y)+r(100x+90y)+r(50x+90y)

Your first three equations will determine the relative prices of x, y, and the
third commodity z, and the rate of profits r. 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. 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. 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. So let us say, gold is the money
commodity in your world, 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.

But this is nothing but one of Bortkiewicz's solution in which the gold sector
is taken as the luxury/money commodity sector with simple reproduction
constraint on the system. Here by design, total surplus value will always be
equal to total profits. 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.

All this should have already made it clear that your equations (4) and (5) must
be at best redundant. But actually they are worse than that. Mathematically,
your r has to be either known or unknown, they cannot be both at the same time.
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. 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). In a system with unique solution, the two equations should be
derivable from the first three equations.

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. One needs to first think about what is the nature of the problem,
before trying to come up with a quantitative solutions.
Cheers, ajit sinha

> Allin proposes that the transformation should keep the mass of
> surplus value invariant even as cost prices are modified :
> (6) 225x+90y+r(225x+90y)=225x+100x+50x
> (7) 100x+120y+r(100x+120y)=90y+120y+90y
> (8) 50x+90y+r(50x+90y)=875-375-300 (200)
> (9) 875-375-300 (200)=r(225x+90y)+r(100x+90y)+r(50x+90y)
> (10)875=375x+300y+r(225x+90y)+r(100x+90y)+r(50x+90y)
> My set of equations has a determinate solution for x,y and r; this
> much you will have to grant.
> Now tell me why my equation 4 is the incorrect expression for mass of
> surplus value=sum of profits. I have responded to Allin's criticism.
> What's yours? It would be truly appreciated.
> All the best, Rakesh

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