Wednesday, June 19, 2013

On "Substitutability"

"[The] validity [of the Cambridge Criticism of neoclassical theory] is unquestionable, but its importance is an empirical or an econometric matter that depends upon the amount of substitutability there is in the system. Until the econometricians have the answer for us, placing reliance upon neoclassical economic theory is a matter of faith. I personally have the faith; but at present the best I can do to convince others is to invoke the weight of Samuelson's authority." -- C. E. Ferguson (1969) [as quoted in Carter (2011)].
1.0 Introduction

In this post, I describe two different meanings of "substitutability", as used in the literature and economists' remarks on the Cambridge Capital Controversy1.

2.0 Joan Robinson's Criticism

Imagine two island capitalist economies, Alpha and Beta, each in a steady state and with access to the same technology. Suppose for some reason, the distribution of income happens to be different in the two islands. Then the capitalists on the islands will, maybe, have adopted different techniques of production and be producing a different mixture of commodities for final output. Consequentially, the structure of capital goods, both in composition and in quantities, will differ between the two islands.

An (illegitimate) thought experiment is to imagine the distribution of income slowly changing from as it is on one island to the distribution on the other. One might mistakenly consider the capital equipment slowly changing through the composition appropriate to imaginary intermediate islands. This claim ignores the reality of what Joan Robinson called historical time. One is treating a process occurring in time as if it occurring in space, ignoring that past bygones are gone, and assuming no difficulties exist in getting into equilibrium.

Neoclassical2 economists frequently ignore the structure of capital equipment and the plans of the entrepreneurs. One meaning of "substitutability" is the assumption that capital goods can be instantaneously taken apart and reassembled to be appropriate for whatever equilibrium is being considered. The tranverse from one equilibrium to another is abstracted from. Robinson satirized this meaning of substitutability by designating the capital good in, say, the Solow-Swan growth model with such names as "ectoplasm", "leets", and "mecanno sets". Post Keynesians, including Sraffians, are generally suspicious of this approach. (Any fans of Austrian school economists want to chime in in the comments?)

3.0 Substitutability and Smooth Microeconomic Production Functions

Another meaning relates to the smoothness of production functions. One might say substitutability exists when derivatives (including, second, third, etc. derivatives) exist for all production functions. That is, substitutability exists in these examples, but not in these ones. (But what would you say about this one, where the cost-minimizing technique varies continuously with the interest rate, and output and each capital good are produced with fixed-coefficients?)

As far as I know, capital-reversing, for example, is consistent with substitutability, in this sense of smooth production functions. I, too, will invoke the weight of Samuelson's authority, even though I reject it in the former case. I would like, however, to see an explicit numeric example.

4.0 Conclusion

I believe C. E. Ferguson was referring to my section 2 meaning of "substitutability". That is, when neoclassical economists claim that Sraffians rely on a lack of substitutability for their critique of neoclassical economics, they should not be objecting to a lack of differentiability of microeconomic production functions.

  1. Other usages are ignored in this post. For example, J. R. Hicks' "elasticity of substitution", as used in his mistaken Theory of Wages (1932), is not treated here.
  2. As far as I am concerned, "neoclassical" is a meaningful and appropriate word in this context.
  • Scott Carter (July 2011). C. E. Ferguson and the Neoclassical Theory of Capital: A Matter of Faith, Review of Political Economy, V. 23, N. 3: pp. 339-356


Magpie said...

"(...) neoclassical economists claim that Sraffians rely on a lack of substitutability for their critique of neoclassical economics"

Even if that were true, I am not sure why it should be a problem for Sraffians: have neoclassicals _any_ empirical evidence confirming their assumed type-2 factor substitutability (i.e. smooth and convex isoquants)?


My main concern, however, is not that. While Robinson indeed satirized the notion of capital employed by neoclassicals in their production function (the delightful mecanno sets thing!), I thought her criticism aimed precisely at the notion of smooth and convex isoquants. Am I wrong on that?


As an amusing parting comment: funny that you mentioned C.E. Ferguson. Ferguson and Gould's micro intro textbook is among the books I've been using!

I decided to use it, among other things because of the "faith" remark and the subsequent exchange between Ferguson and Robinson.

Robert Vienneau said...

Robinson did not accept smooth production functions. I read somewhere that she would illustrate the input of labor with a stick figure to emphasize lack of differentiability with respect to that particular input.

But, I think, her fundamental objection had more to do with time, as in my first definition of substitutability. Even if production functions were continuously differentiable, this objection would still apply.

Magpie said...

"Robinson did not accept smooth production functions. (…) But, I think, her fundamental objection had more to do with time, as in my first definition of substitutability".

While I am not saying that I disagree with your interpretation of Robinson's views, I am not sure the matter of time should be considered the main issue in her critique.

In any case, the CCC overviews I've seen seem to place very little import on the issue of output differentiability respect to factors.

Cohen and Harcourt[*], for instance, see three main issues discussed in the CCC:
"In this article, our aim is to put into perspective (…) the latest in a series of still-unresolved controversies over THREE DEEP ISSUES. The first is the meaning and, as a corollary, the MEASUREMENT of the concept of capital in the analysis of industrial capitalist societies. The second is Joan Robinson's complaint that EQUILIBRIUM was not the outcome of an economic process (…). The third issue is the role of IDEOLOGY and vision in fuelling controversy when the results of simple models are not robust."

You'll notice that the matter of capital measurement largely confines Robinson's critique to aggregate production functions.

Additionally, in Cohen and Harcourt's views, at least as expressed in that article, there is no immediate connection between capital measurement and output differentiability respect to factors.

From a mainstream point of view, Stiglitz[#], whose review of Harcourt you commented elsewhere, also refers to three main issues, which he comments upon in three consecutive sections: "the determination of savings and the rate of interest", "reswitching of techniques", and "aggregate capital". Again, not a word about output differentiability respect to factors.

But if partial derivatives of output respect to labour are undefined, then isoquants exhibiting factor substitutability are undefined: their derivative in K-L space is the ratio of the marginal productivities of L and R.

I understand that Leontief[%] pointed to this.

[*] Avi J. Cohen and G. C. Harcourt. 2003. "Retrospectives- Whatever Happened to the Cambridge Capital Theory Controversies?" Journal of Economic Perspectives-Volume 17, Number 1-Winter 2003-Pages 199-214
[#]Joseph Stiglitz. 1974. "The Cambridge-Cambridge Controversy in the Theory of Capital; a View from New Haven: A Review Article". Cowles Foundation Paper 410, reprinted from Journal of Political Economy, 4, 1974.
[%] Wassily Leontief. 1941. "The Structure of the American Economy, 1919-1939".

Robert Vienneau said...

I do not think we disagree.

The neoclassicals who I have in mind are economists I have met on the Internet. For some unfathomable reason, they leap to the incorrect conclusion that if an example of, say, capital reversing has non-differentiable microeconomic production functions, differentiability rules out capital-reversing.

I have found that when I ask neoclassicals economists about the CCC in such informal settings, the responses I receive are incompetent and have no discernible connection to the neoclassical side of the literature on the CCC.

And I did not think Section 2 in my post was about aggregation. I tried to point out some issues around historical time and the difficulties of getting into equilibrium.

Magpie said...

I don't think we disagree, either!

What I am is very puzzled with the accounts I've seen (like Cohen & Harcourt's and Stiglitz's, for instance).

These accounts don't mention what for me is the key issue: the law of diminishing returns.

I guess what I want to know is, to the best of your knowledge, was the issue of the law of diminishing returns discussed during the CCC?

If yes, what was the conclusion? If not, why wasn't it discussed?

Robert Vienneau said...

I do not think the CCC contains much, if any, discussion of diminishing returns. Most of the controversy took places in discussions of models of long run equilibrium where the only inputs were capital goods and labor. In such models, one can always, say, double all inputs. The law of diminishing returns, as I understand it, assumes the quantity of at least one input cannot be expanded beyond some given maximum.

On the other hand, some literature does look at the evolution of Sraffa's thought from his 1925 and 1926 papers to his 1960 book.

Magpie said...

Thanks for the reply.

Try as hard as I did, I myself could not find any discussion about the law of diminishing returns, either.

Due to the lack of discussion on this issue, I suppose it legitimate to infer that both Cambridges were more or less in agreement about the validity of the law of diminishing returns.

Very strange.