# [OPE-L:4222] Re: Re: Re: Re: Part Two of Volume III of Capital

From: Rakesh Narpat Bhandari (rakeshb@Stanford.EDU)
Date: Sun Oct 22 2000 - 13:12:57 EDT

```>Rakesh,
>
>This may read like a "flame", but I am going to try to communicate to you
>why you do not have the knowledge (of mathematics) you need to make the
>claims you are making.

Steve,
this is not a flame but a dodge. The transformation problem has been
proven on the assumption of simple reproduction or a vector of
equilibrium prices (see for example Catephores). It is an unspoken
demand that one must be boxed in by these assumptions. Or which is
the same thing that the unit prices of production for the inputs have
to be transformed into the unit prices of production for the outputs.

Buried in your little math lecture is the invocation of Marx's
imprimatur that a solution must be possible in stable equilibrium. I
have given several reasons why this constraint is unreasonable, and a
misuse of Marx's vol 2 assumptions. None of your ramble answers any
of points here. You yourself seem to have no interest in such
conditions, save as a constraint on any solution to the
transformation problem. I may not be in the league of the
heavyweights on this list, none of you have yet to make a sensible
case for why you think the solution to the transformation problem
would have to be boxed in this way, especially since we all agree
that capital is sufficiently dynamic that unit prices of production
are indeed changing interperiodically.

You could have talked here about Alan F's simple use of difference
equations which allows for the possibility of a continuous
interperiodic decline in unit values. To allow for continuous
productivity growth, he needs a parameter to a difference equation
less than 1 of course.

For my purposes, I would simply take my modification of Allin's
simple reproduction tableau and then by determining how much unit
values would have droppped for the outputs compared to the
inputs,then determined my parameter for the difference equation for
that period.  Now it is impossible to exactly determine what the unit
input prices of production were; but it was easy to estimate that
parameter could easily be for both mp and wg above .95. That was all
I was trying to indicate, and you refuse to understand my point.

So what I showed was that Marx's transformation procedure does indeed
require that unit prices of production change from the inputs to the
outputs, but that change seems reasonable, realistic, even
statistically insignificant.

Instead you  thrown up your hands and say that once this constraint
of simple reproduction or a vector of equilibrium prices is gone, the
math becomes too difficult for me to follow. And too time consuming
for you to do. And there is no reason for you to do it because  you
reject the LTV for other reasons. This is just a dodge.

>
>For instance, you say to Allin that you introduce "one (count it: ONE)
>complicating. albeit utterly realistic, assumption to your non complex but
>utterly unrealistic simple reproduction tableau", and conclude with the
>comment that "Anybody with an 8th grade education in math could understand
>what I did to your simple reproduction scheme."...
>
>
>Your one complicating assumption--and the entire TSS endeavour--moves the
>issue of the transformation problem from the realm of linear algebra to
>that of ordinary differential equations (and strictly speaking, to
>open-dimensional nonlinear stochastic partial differential equations). That
>shift moves the subject from a realm in which, effectively, definitive
>proofs are easy, to one in which definitive proofs are not just difficult,
>THEY ARE IMPOSSIBLE.

What do you mean by a definitive proof here?

Again what I tried to show by modifying Allin's simple reproduction
tableau was that if one allows for interperiodic labor productivity
increase, then unit prices of production would only have to fall in
realistic ways for 1. the equalities to hold at the completion of the
period and 2. for the value of the inputs to determine the sum of
their prices of production.

I explicitly said that I could not determine the exact unit input
prices of production, only show that they could easily fall in a
small range and the output prices of production would only have to
decline in a small, if not statisically insignificant way, for the
the above 2 conditions to be met.

So I am not after a definitive proof or exact determination of prices
along any time path.

If one were to follow the trajectory of my modified simple
reproduction system, one could then determine the range for all the
time subscripted variables under the assumption that r remains stable
for some time (for the reasons Marx invoked).

That is, I estimated again the range of how much live labor would be
added to the system in the next period due to the greater quantity of
wage goods with which to purchase labor power.

This is what drove Allin through the roof, yet I responded to him why
such expanded reproduction was not only completely realistic but
grounded in Marx's understanding of the consequences of rising labor
productivity..

>
>To repeat, it is not just that the solutions of ordinary differential
>equations are difficult: even for the vast majority of one-dimensional
>systems, they are technically impossible (this is proven to most
>mathematics students in week 3 or 4 of a subject on ordinary differential
>equations).

I recall working out the technical development of the calculus in the
nbody problem. Is this what you are referring to? It's been a while.
Can you explain what this has to do with the problem at hand? Be as
brief as you want. I will restudy and try to figure out what you are
saying.

>
>Those who go on to higher level subjects (some third year subjects) learn
>that three or higher dimensional systems of nonlinear differential
>equations are also insoluble--a proof which dates back to Poincare in 1899.

True, I did not do such math. Only one year of calculus in which by
the way I did  quite well. So if I must relearn and learn more, I
have no problem Ignorance never helped anyone, but you have to some
case for why this math is needed to determine what there is fatal
logical defect in Marx's transformation procedure.

>
>Now I'm not saying that the move is not in some senses justified. I am
>willing to concede the TSS point that the analysis of Marx on this front
>should be dynamic, and I also know that the "simultaneist" equilibrium long
>run solutions will only apply if the eigenvalues of the Jacobian of the
>resulting dynamic model have negative dominant real parts (I'm using the
>jargon here deliberately--anyone who's been trained in this area knows what
>I'm saying, those that don't won't have a clue--and that is my point).

That's ok. But I am not interested in the post Waldian conditions for
equilibrium to hold. It may allow an interesting application of
advanced mathematics which are intrinsically beautiful but it has no
relevance to real world dynamics.

>
>But I also concur with Allin (and, if I'm reading him correctly in recent
>posts, also Duncan--though I could be wrong there) that Marx expected his
>system would work even with the constraint of a stable equilibrium.

Ha! Here it is buried. I have provided numerous reasons why this is not true.
>
>
>
>There is also a "burden of proof" issue here. You might give academic
>conspiracy theories as the main reason for why Marxist economics has been
>marginalised (and I'm not about to deny that they have any influence: they
>most certainly have!). But there are also many genuine radicals who have
>expressed disquiet with how the TSS approach has attempted to eliminate the
>transformation problem.

>
>In my not so humble opinion, the burden of proof of its claims lies with
>you and its adherents in general. If you really want to prove that the TSS
>approach is correct--rather than simply provide numerical couter-examples
>to the "simultaneist" approach--then you have to equip yourself with the
>technical knowledge needed.

I have expressed a couple of perhaps minor disagreements with TSS
(over how constant capital is treated and how replacement costs are
handled).

But I think that a purely logical dismissal of theory which has made
so many accurate predictions should meet a high burden of proof. And
unless you show me that there is necessarily a transformation problem
after one drops the COMPLETELY UNREALISTIC assumptions of simple
reproduction or equilibrium, then no reasonable person should write
Marx off due what is nothing more than a curio.

>
>
>This includes at least introductory first-year courses in linear algebra,
>calculus and ordinary differential equations.

Never was taught linear algebra. read the first 150 pages of the
Meek and Bradley book and then concluded that matrix algebra was just
an optimizing technique which was too blunt to allow for dynamic
reality.

>To really do what you need to
>do, you should also consider at least 2nd year courses in the same, as well
>as a subject or two in dynamics (preferably treating both difference and
>differential equations, as well as an introduction to chaos).
>
>Once you have done all that, then try to build a full TSS model, and see
>whether its specifications are internally consistent.
>
>Until you have that level of knowledge, you are fighting well outside your
>weight class in trying to engage in these arguments with myself, Allin,
>Ajit, Paul, Gil, Duncan and others.

Well such smart people should be able to explain to someone like me
why you have all thought it necessary that the transformation problem
be solved in terms of the uncompletely unrealistic constraints of
simple reproduction or equiibrium prices.

If you say that Marx did the same thing, then you have to address my
concern out of total incomprehension of Marx's method  you have taken
out of its context simplfying assumptions for the study of
circulation and turned them into controlling methodological
postulates for the analysis of all economic reality.

All the best, Rakesh
```

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