Saturday, November 21, 2015

What Has Not Been Said About The Late Herbert Scarf

Economic theorist Herbert Scarf recently died.  There have been a number of blog posts about him and his work.  A good one is by Tim Taylor here.  He notes innovative work Scarf did  on inventory theory and the theory of  indivisibilities.  Other commentators have pointed out his work on the relationship between general equilibrium and the core.  I have no  complaint with any of this, although many may not  know of it or care about it as it was some time ago and has largely gotten built into the grad theory textbooks, especially the stuff on GE and core.   All of this makes him look like a sort of minor panjandrum of very orthodox theory, orthodox now almost to the point of boring.  However, I personally think that he has been under-appreciated and under-recognized.

In this regard let me mention just two items.  One is his reasonably well known demonstration of how easy it is for a general equilibrium to be unstable.  That has gotten into a lot of the textbooks, but it seems to get overlooked a lot, even though we have seen some pretty dramatic examples of major instability in the real economy, most notably involving speculative bubbles and their crashes, with one of these bringing about a global economic crisis worse than anything since the Great Depression.  People like Kenneth Arrow and others definitely recognized the importance of Scarf's work, and indeed it is indeed in the textbooks that the conditions for stability of GEs are very strong and unlikely to be met, even as many macroeconomists blithely assume not only that GE holds, but that it is unique and stable, thinking that this somehow shows how rigorous and theoretically astute they are.  Herb Scarf knew better.

The other is much more obscure, and arguably less important.  It came out of his concern for  the issues surrounding actually calculating general equilbria, which many microeconomists do all the time with CGE models, whose usefulness I am not going to totally deny, although Scarf's  work raises warning flags.  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.

Anyway, a sad loss of fine man and an under-appreciated brilliantly innovative scholar.

Barkley Rosser


Wallfly said...

Very interesting posting, thanks much.

Though I can't remember it specifically I remember reading recently that a fairly non-obscure piece of statistics relied on the Axiom of Choice for its proof (much to my surprise).

I don't think you have to be so foundational to get calculation problems. Anyplace you need to numerically compute eigen values, the process becomes chaotic for non-symmetric matrices. Since I am not versed in econ theory I am just assuming that any realistic model of an economy uses numerical linear algebra but I don't really know. said...

Yes, standard economic theory uses a lot of linear algebra.