The Transformation Problem

[Viva Fidel]

The so-called transformation problem, discovered by Marx himself in Chapter 9 of Capital Vol. 2 I believe, has been "contraversial" among economists. Is there a solution to this "problem," or is it something that Marxists should just accept?

http://en.wikipedia.org/wiki/Transformation_problem

[Severian]

You might find this thread interesting (http://www.revleft.com/index.php?showtopic=39750)

ComradeRed is probably the best poster on this board to ask about this stuff; not that many people really get into the 3rd volume of Capital.

[ComradeRed]

Actually, if you think about it, the labor theory of value is merely an elucidation of the "commodity-input theory of value"...this, however, is not a universally agreed upon assertion.

I would assert the first commodity that humans began with is labor-power, and from that begot the first tools. These tools encapsulated the labor power required to produce them and made new tools, and so on until today.

So really, to reduce back to values of the dated labor-power inputs ("reduction in terms of labor-power") requires an infinite sum (it's convergent, and not point-wise convergent):

\SUM^{infinity}_{t=0} L_{a}(t)w(t) (1+r(t))^{t} = output_{a}(0)price_{a}(0)

for the labor input of sector a at some t production intervals ago, the wage rate w(t) at time t, and r being the rate of profit. Note that this is in the linear framework of the Neo-Ricardians, and this is a rather "controversial" framework (because it's linear as opposed to nonlinear).

Of course, it would be feasible to replace this with an integral:
\int^{T}_{0} L_{a}(t)w(t) (1+r(t) )^(t) dt
if we notionally accept that \Delta t_{i} is set to unity (i.e. the time intervals are integer representations, i.e. the difference between production processes are 1 and thus the production processes are of equal time periods). This may not be a solvable equation (the integral representation) since L_{a}(t), w(t) and r(t) are not necessarily well enough defined.

Some Leftist Objections

It's too linear!

Yes, yes it is. However, that's not necessarily a bad thing! This is only an approximation to begin with; to make the system nonlinear, well, the second equation is tacitly nonlinear...L(t) is possibly a nonlinear function.

Supposing that L(t) were nonlinear, that would imply that w(t) and r(t) would also be nonlinear (in the Neo-Ricardian framework, the rate of profit and wages are dependent on the quantity of inputs and outputs...which in our case would directly be related to the labor inputs).

It would be possible to make this into a nonlinear system...however, since most people would jump to say "Why don't you do it then?" I don't have the time; but if you have the time: get the book "Nonlinear dynamics and Chaos" by Strigatz and read some Neo-Ricardian stuff, and do it your own damn self!

This really isn't what Marx had in Mind.

Well I'm not so sure about that. Marx seems to lay down the LTV in Capital vol. I as an elucidation of the "Input Theory of Value"...and in effect, it is in a sense.

I don't really see a problem with using the Neo-Ricardian "Reduction to Dated Labor" method, it works (and actually there has been "empirical" evidence supporting it, see Empirical Strength of the Labour Theory of Value (pdf) (http://homepage.newschool.edu/~AShaikh/labthvalue.pdf)).

There's nothing wrong with saying "Marx got a few things wrong in the third volume of Capital."

What About the Tendency of Profits to Fall?

Well, you would have to take into account one of the basic assumptions of Marxist economics (and I would assert Marxism in general): technological innovation is treated as "constant".

That means as time goes on, the amount of labor necessary to produce a fixed quantity of a given good decreases.

I haven't worked out the math to see if the Neo-Ricardian "Reduction method" would have the rate of profits fall too or not; I've been swamped lately by a number of things. It is on my "to-do list" and I'll probably do it sooner rather than later.

Brief Appendix on Neo-Ricardian Economics for Mathematically inclined people.

So the basic relations are:

p^{a}B_{ab} = (1+r)(p^{a}A_{ab} + w l^{a})
p^{a} = rp^{a}H_{a} + (1+r)w v^{a}

for the price vector p^{a}, the output matrix B_{ab}, the input matrix A_{ab}, the labor input vector l^{a}, v^{a} is the vertically integrated labor vector, the vertically integrated input vector H_{a} = SUM_{c}A_{ca}. Note that summation convention is implied.

The fundamental constraint is that:
p^{a}y_{a} = v^{a}y_{a}
where y_{a} is the net output vector, that is to say the vertically integrated output vector minus the vertically integrated input vector.

From this it is easy to find the wage rate being equal to:

w = (v^{a}y_{a}) / ( (1+r)v^{a}(I_{ab}-rH_{a})y^{b})

and the price vector:

p = (v^{a}y_{a}v^{a}(I_{ab}-rH_{a})) / (v^{a}(I_{ab}-rH_{a})y^{b}).

This math I just found in some of my notes, I'll need to verify the indices, the general idea is correct.

[Severian]

Originally posted by [email protected] 07, 2007 12:21 pm
Actually, if you think about it, the labor theory of value is merely an elucidation of the "commodity-input theory of value"
Makes sense to me. Is the commodity-input theory more widely accepted by bourgeois economists? And what's the relationship of this point to the "transformation problem"?

[ComradeRed]

Originally posted by Severian+February 07, 2007 06:57 pm--> (Severian @ February 07, 2007 06:57 pm)
[email protected] 07, 2007 12:21 pm
Actually, if you think about it, the labor theory of value is merely an elucidation of the "commodity-input theory of value"
Makes sense to me. Is the commodity-input theory more widely accepted by bourgeois economists? And what's the relationship of this point to the "transformation problem"? [/b]
It's not accepted by Marginalists, and in fact demonstrates that under the very conditions that Marginalism assumes that Marginal analysis cannot work.

So it's very controversial among vulgar economists, to the point where they just ignore it and pretend it's not there. However, two colleges that come to mind (Cambridge in the UK and the New School) that actually espouse it.

The serious problem is integrating time in a mature manner (i.e. nonlinearly). IF that can happen, then you've got a damn good paradigm on your hands.

But no, most bourgeois economists do not accept it (largely because they straw man the Neo-Ricardian school then criticize the straw man).

As far as its relation to the transformation problem, it's essentially made Capital vol. III obsolete. It has satisfactory solutions to the transformation problem; and actually, if you look into it, there is a book Marx after Sraffa by Ian Steedman which criticizes the Marxist economic theory behind Capital vol III with the Neo-Ricardian paradigm...that is to say, Steedman essentially demonstrates that there is a Commodity Input Theory of Value solution.

However, Sraffa (father of the Neo-Ricardians) went through great pains to point out that the commodity input theory of value was the same as the reduction to dated labor inputs. Taking this into account, Steedman's "critique" essentially becomes a solution to the transformation problem! It relates, undeniably, the amount of labor inputs to the current gross output of the sector in question...and from there you can get the price rather elementarily.