# [OPE-L:4228] Re: Re: Re: Re: Re: Part Two of Volume III of Capital

From: Steve Keen (s.keen@uws.edu.au)
Date: Sun Oct 22 2000 - 18:53:07 EDT

```Oh well, one last time then.

Firstly, you were wrong to dismiss linear algebra. As it happens, linear
algebra is an important tool in the analysis of dynamic systems. This is
because the stability properties of equilibria of a dynamic system are
determined by the linear component of the Taylor expansion of the system.
If you really want to do dynamics--rather than simply propose numerical
examples which are "almost just right", then you will need linear algebra
(at least to second year level) as well as calculus and differential
equations.

Secondly, numerical instances prove nothing, even when they are derived
from a properly specified closed dynamic model (which yours are not). For
example, I am sure that I could take your numerical examples and rework
them to support the proposition that machines are the source of all surplus
value; my defence of this proposition, using the same techniques you
employ, would be just as effective as your proof of the opposing
contention. Such a "proof by example" would therefore be as meaningless as
any of the numerical examples you proffer for the labor theory of value.

Andrew and I had a good off-list discussion on this recently. I sent him a
paper on my interpretation of Hansen-Samuelson "trade cycle" models, which
concludes that the model should have been third order. I mentioned that
this system generated both cycles and growth, with the cycles being
proportional to the level of output. Andrew sent back what he thought was a
counter-example to my interpretation, and with a couple of changes to his
numbers, I showed that it was not.

To know the overall characteristics of a dynamic model--even the linear one
which I sent Andrew--you have to determine the system's equilibria, and do
eigenvalue/vector analysis of them, before you can properly say what the
models character is: one or two randomly chosen numerical examples simply
won't do.

The problem is massively more complex with a nonlinear system, where
eigenvalue analysis only charcterises the system's behaviour in the loci of
equilibria. Then you have to bring out big guns--Lyapunov coefficients,
that sort of thing (in all of which, by the way,, you need skills acquired
in learning linear algebra--Gram-Schmidt renormalisation, for example, is
an essential part of calculating Lyapunov coefficients).

As for my comment about definitive proofs, it is the same issue as you
describe--the n-body problem and its generalisation. Once you have three or
more nonlinear differential equations in a system, there is no closed form
solution for it--unlike a linear system, and especially unlike a linear
algebraic system.

Finally, if you fully specified what you believe to be Marx's economics as
a differential equation system, then if the labor theory of value is
correct, your system should be valid for all rates of everything including
technical change, population growth, depreciation, prices--from zero up. If
the truth of the labor theory of value was dependent on specific levels of
these variables, then it is not a general truth.

It is possible to characterise the "simultaneist" analysis of the
transformation problem as a dynamic solution in which all rates of change
are set to zero. The conclusion of all these analyses is that the labor
theory of value cannot be right. What you are alleging is that,
effectively, the labor theory of value is only correct for positive rates
of technical change, etc. If so, then it is not a general truth, and your
approach to eliminating the transformation problem is as much a rejection
of the labor theory of value as any simultaneist formulation. This is the
point I believe that Allin was trying to get through to you (and as you
know, I believe it is untrue at any rate of change).

Steve
At 10:12 22/10/00 -0700, you wrote:
>>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
>
>
>
>
>
>
>
>
>
>
Dr. Steve Keen
Senior Lecturer
Economics & Finance
University of Western Sydney Macarthur
Building 11 Room 30,
Goldsmith Avenue, Campbelltown
PO Box 555 Campbelltown NSW 2560
Australia
s.keen@uws.edu.au 61 2 4620-3016 Fax 61 2 4626-6683
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