Herbert Scarf died this 15th of November. I think of Scarf as the economist who first demonstrated that general equilibria need not be stable. Something more, some special case assumption or another approach entirely, is needed.

From his Wikipedia page, I learned that have been exposed to more of Scarf's work than I knew. Long ago I took a course in Operations Research, in which we were taught queuing theory and how to find policies for optimal inventory management. Apparently, that approach to the study of inventory policies comes from Scarf.

I did not find the *New York Times* obituary enlightening. I wish they had mentioned that his algorithm was for finding so-called Computable General Equilibrium (CGE). I have never quite got CGE models. The ones I have seen do not have the dated commodities of the Arrow-Debreu model of intertemporal equilibrium. I have never been sure that they really belong with that tradition, or, like Leontief's model, really fit with a revival of classical economics. Perhaps they are an example of temporary equilibria, as put forth by J. R. Hicks in *Value and Capital*.

Quite some time ago, Rajiv Sethi discussed Duncan Foley's appreciation of Scarf as a teacher.

**Update:** Barkley Rosser provides some comments on Scarf (hat tip to Blissex). Here is an obituary from the blog, Leisure of the Theory Class.

(Unrelated to the above, Cameron Murray recently comments on economists confusion about what is meant by "capital".)

## 7 comments:

«Scarf as the economist who first demonstrated that general equilibria need not be stable. Something more, some special case assumption or another approach entirely, is needed. [ ... ] CGE models. The ones I have seen do not have the dated commodities of the Arrow-Debreu model of intertemporal equilibrium»

The Arrow-Debreu-(Lucas) models introduce infinite markets for infinite goods at every possible time to in essence abolish time: "tatonnement" happens in effect across eternity, once and forever. It is just a cheap shallow trick to turn "dynamics" into "statics".

The deeper economic sense of that is wider: the trick is needed to get rid of the role of capital (it creates "knots" in the timeline, especially if one considers depreciation), and to avoid the "complications" like path dependency that creates. BTW Sraffa's illustration of "return of the techniques" even in comparative statics is all about path dependency especially in the accumulation of capital.

Perhaps I have been too evocative in the above...

So I had another comment ready here but I lost it, but then I did a web search for "putty capital" and I was pleased to find this:

dept.econ.yorku.ca/~avicohehttp://dept.econ.yorku.ca/~avicohen/research.htmln/research.html

This guys has done a lot of work on the Cambridges Capital Controversy and I have just read his "Whatever Happened to the Cambridge Capital Theory Controversies?" which is fairly interesting even if far too mild :-). Some relevant quotes:

«A common starting point for the neoclassical perspective on capital»

That's actually the end point too... :-)

«is a one-commodity Samuelson/Solow/Swan aggregate production function model: $Q = f(K,L)$ where the one produced good (Q) can be consumed directly or stockpiled for use as a capital good (K). With the usual assumptions, [ ... ] this simple model exhibits what Samuelson (1962) called three key “parables”: 1) The real return on capital (the rate of interest) is determined by the technical properties of the diminishing marginal productivity of capital; [ ... ]»

The real return on physical capital is not the "rate of interest" :-).

«3) the distribution of income between laborers and capitalists is explained by relative factor scarcities/supplies and marginal products. The price of capital services (the rate of interest) is determined by the relative scarcity and marginal productivity of aggregate capital, and the price of labor services (the wage rate) is determined by the relative scarcity and marginal productivity of labor (L).»

The central truthiness of Economics!

«Why do reswitching and capital-reversing occur? Samuelson (1966) provides the intuition using the Austrian conception of capital as time, so that the productivity of capital is the productivity of time itself.»

Capital is definitely related to time... :-) But "the productivity of time itself" is quite amazing handwaving :-).

«Because of Wicksell effects, in models with heterogeneous capital goods (or heterogeneous output), the rate of interest depends not only on exogenous technical properties of capital, but also on endogenously determined prices like the interest rate. The endogeneity of prices allows multiple equilibria, which complicates the one-way parable explanations of income distribution. Differences in quantities no longer yield unambiguously signed price effects. The power and simplicity of one-commodity models emanates from eliminating these endogenous price effects and measurement problems (Cohen, 1989).»

The "commodities by means of commodities" story was just a rigorously built but comically trivial neoclassical model which was not the even more ridiculous "commodity by means of commodity" one of JB Clark and others.

«Capital is fundamentally intertwined with issues of time. As Bliss (1975, p. 39) wrote: “One of the essential tasks of a theory of capital is . . . to make clear why a purely static and timeless economic theory could not be adequate.” Questions about the measurement of capital in aggregate production function growth models segued to questions about how, if at all, may dynamic processes of accumulation and distribution be analyzed within an essentially static equilibrium framework.»

All sorted out within Arrow-Debreu(-Lucas) models or DSGE models :-).

«Thus, many years ago, Robinson (1953, p. 590) put back on the agenda what we now call path-dependent equilibria: “the very process of moving has an effect upon the destination of the movement, so that there is no such thing as a position of long-run equilibrium which exists independently of the course which the econ omy is following at a particular date.” The title of her 1975 paper, “The Unimportance of Reswitching” (Robinson, 1975a), reflected her belief that while reswitching and capital-reversing were problematic for neoclassical capital theory, her methodological critique was far more important.»

Hear hear! :-)

Also very funny:

«The third theoretical neoclassical response attempted to extend the one-commodity results to more general heterogeneous commodity models. Samuelson’s (1962) attempt in the “surrogate production function” included what appeared to be a variety of physically distinct capital goods, but he also assumed equal factor proportions in all industries, making relative prices independent of changes in distribution between wages and profits. As Samuelson subsequently realized, this effectively collapsed his model back to one commodity.

By the late 1960s, Samuelson’s (1966, p. 568) judicious “Summing Up” article admitted that outside of one-commodity models, reswitching and capital-reversing may be usual, rather than anomalous, theoretical results and that the three neo-classical parables “cannot be universally valid.” On a theoretical level, the “English” Cantabrigians won the round over aggregate production functions. Even neoclassicals like Hahn (1972, p. 8) showed no mercy for aggregate production functions, which “cannot be shown to follow from proper [general equilibrium] theory and in general [are] therefore open to severe logical objections.” They fell out of favor in the 1970s and early 1980s until their revival with endogenous growth and real business cycle theories.»

Reswitching of the techniques! :-)

«Moreover, the general equilibrium approach revitalized Robinson’s concerns about equilibrium. Theoretical work, speci cally, the disappointing Sonnenschein-Mantel-Debreu stability results, found no particular reason to believe in the stability of the general equilibrium outcome. In discussing these results, Hahn (1984, p. 53) wrote: “[T]he Arrow-Debreu construction . . . must relinquish the claim of providing necessary descriptions of terminal states of economic processes.” The lack of adequate stability results raised questions about the conception of equilibrium as the end of an economic process and the adequacy of comparative statics as explanations of the process of change following a parameter shift (Fisher, 1989; Ingrao and Israel, 1990).»

Oh noes! :-)

People like G Mankiw and L Hubbard have accumulated lots of capital which must mean that neoclassical capital theory works! :-)

«So I had another comment ready here but I lost it, but then I did a web search for "putty capital" and I was pleased to find this:»

So I garbled the URLs and they are:

dept.econ.yorku.ca/~avicohehttp://dept.econ.yorku.ca/~avicohen/research.htmln/research.html

dept.econ.yorku.ca/~avicohen/Linked_Documents/JEP_Cohen_Harcourt.pdf

Well again:

http://dept.econ.yorku.ca/~avicohen/research.html

http://dept.econ.yorku.ca/~avicohen/Linked_Documents/JEP_Cohen_Harcourt.pdf

«The ones I have seen do not have the dated commodities of the Arrow-Debreu model of intertemporal equilibrium»

To summarize some of the assumptions, "core" neoclassical models "work best" when:

* There is only one commodity that is at the same time capital and consumption goods.

* There is only one agent that is at the same time producer and consumers.

* Regardless of the one commodity and one agent, there are infinite markets for every possible future instant, each perfectly competitive because of infinite sellers and buyers competing.

* The one agent knows the probability distributions of all future events.

* There is an "auctioneer" that ensures all infinite markets for the same one commodity over infinite times with infinite buyers and sellers who are all impersonated by the same agent do clear simultaneously.

Which talk of infinities and miraculous auctioneers able to deal with them reminds me of the topic of the lost comment:

http://econospeak.blogspot.co.uk/2015/11/what-has-not-been-said-about-later.html

«In particular he was more than any of the other general equilibrium theorists aware of the deep mathematical issues involved in actually precisely computing equilibria, issues related to constructivist critiques of classical mathematics over such things as assuming the Axiom of Choice and the Law of the Excluded Middle, issues raised in recent years by K. Vela Vellupillai, although with most economists ignoring him. Maybe they are right to do so, but I happen to know from talking with both of them at the same time a few years ago that Herb Scarf took these issues seriously, and Velupillai has credited him with being the only serious general equilibrium theorist to appear to be aware of them and to have discussed them, which he did in his famous book on computing general equilibria.»

Interesting. I personally regard this as a bit weak point, especially compared to the above, because Zermelo-Frankel foundations of mathematics with the Axiom of Choice are widely accepted by mathematicians, even if constructivist ones would probably be better.

Hard to resist being drawn back to this topic because in a sense it is the vortex of doom of both Economics and political economy studies :-). In part because I put some comments on a page on a related topic here and the they are not appearing:

nakedkeynesianism.blogspot.co.uk/2015/11/misconceptions-about-heterodox.html

It is still about the significance of Sraffa's work and "proofs of general equilibrium". His "commodities by means of commodities" neoclassical model was designed to demonstrate that even a trivial multi-commodity neoclassical model would have "anomalies" that implied multiple potential equilibria with multiple distributions of income, depending in effect on government policy.

The retort by *some* neoclassical was that the model with a single "commodity" was sufficient as long as the commodity was "money": because putting in market values of K and L would "homogenize" capital, and labor too.

The problem with that as the other side pointed out (and it had been known for a long time) is that market prices themselves depend on the amount of capital and the distribution of income, therefore using "dollars" as the "commodity" is nonsensical.

Therefore a problem arose that had vexed political economists for centuries: is it possible to create a unit of account ("money") that does not depend on the distribution of income?

The really interesting part of "commodities by means of commodities" was not showing conclusively that multi-commodity neoclassical models suffer from Wicksell-effects, which was probably just an entertainment for P Sraffa.

The really interesting part was that he found a somewhat abstruse method that allowed putting together in a the same simple neoclassical model a basket of commodities with an exchange value invariant to changes in the profit rate and the distribution of income.

The proof that this was possible had been sought as the philosopher stone's of political economy studies for centuries, it was a *constructive* proof nonetheless, the crowning of many investigations in the theory of value. Sraffa for that alone belongs in the class of Great Political Economists. I am not sure that our blogger has mentioned this aspect of Sraffa's work as often as the "Wicksell effects" one.

But it all disappeared as the Cambridge UK side won conclusively the science argument but lost completely the propaganda war...

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