Uzawa-Equivalence Theorem

I found an interesting section in a nice intermediate textbook on General Equilibrium Theory by Ross M. Starr:




In Section 18.4 titled "Uzawa-Equivalence Theorem" shows the equivalence of two existence theorems:

1. The existence of equilibrium in an economy characterized by a continuous excess demand function fulfilling Walras' Law 
2. The Brouwer Fixed-Point Theorem

Interestingly, the two apparently distinct results are mathematically equivalent, which is originally shown by Hirofumi Uzawa (1962) in his short note: "Walras' Existence Theorem and Brouwer's Fixed Point Theorem." Economic Studies Quarterly, 8: 59-62.

The importance of the theorem is stressed by Prof. Starr as follows:

What are we to make of the Uzawa Equivalence Theorem? It says that use of the Brouwer Fixed-Point Theorem is not merely one way to prove the existence of equilibrium.  In a fundamental sense, it is the only way. Any alternative proof of existence will include, inter alia, an implicit proof of the Brouwer Theorem. Hence, this mathematical method is essential; one cannot pursue this branch of economics without the Brouwer Theorem. If Walras (1874) provided an incomplete proof of existence of equilibrium, it was in part because the necessary mathematics was not yet available.

The paper is included in this volume of Uzawa's collected papers. (I thank Prof. Kawamura for the information.)

Matching and Market

I found insightful comments on the relationship between matching theory and market economy (more specifically, general equilibrium) in the following paper:
Vincent Crawford (1991), "Comparative Statics in Matching Markets" Journal of Economic Theory, 54: 389-400.

Perhaps the most important advantage of the matching approach is its robustness to heterogeneity. A traditional competitive equilibrium cannot exist in general unless the goods traded in each market are homogeneous, because all goods in the same market must sell at the same price. A traditional model of a labor market with the degree of heterogeneity normally encountered therefore has the structure of a multi-market general equilibrium model. But because the markets in such a model are very thin, the usual arguments in support of price-taking are strained. The theory of matching markets replaces this collection of thin markets with a single market game, in which the terms of partnerships are determined endogenously, along with the matching, via negotiations between prospective partners. Gale and Shapley's notion of stability(*), suitable generalized, formalizes the idea of competition, and thereby makes it possible to evaluate the robustness of traditional competitive analysis to heterogeneity. (Stable outcomes in matching markets can in fact be viewed as traditional competitive equilibria when prices are allowed to reflect the differences between matches; see, for example, Shapley and Shubik, 1972(**))

The author, Vince Crawford, who is known as a leading researcher in game theory has written a few influential papers on matching theory. Especially, the following two are of great importance since they initiated the area of (many-to-one) matching with monetary transfers.

"Job Matching with Heterogeneous Firms and Workers"

with Elsie Marie Knoer, Econometrica, Vol. 49(2): 437-450, 1981.

"Job Matching, Coalition Formation, and Gross Substitutes"

with Alexander S. Kelso, Jr., Econometrica, Vol. 50(6): 1483-1504, 1982.

* Gale and Shapley (1962) "College Admissions and the Stability of Marriage" American Mathematics Monthly, 69: 9-15.

** Shapley and Shubik (1972) "The Assignment Game. 1. The Core" International Journal of Game Theory, 1: 111-130.

Grossman (1989)

Original article (link) posted: 20/09/2005

I was taking a look at Sanford Grossman's book "The Informational Role of Prices". This is a collection of his articles focusing on informational role of prices or that of contracts. All the articles except for the first one were reprinted from journals. Here, I take a memo for the chapter one which is an introduction of the book.

The main theme of the book is to propose models which incorporate two aspects of the role of prices at the same time. He says;

I have elaborated a model of economic equilibrium that is based upon the idea that prices have a dual role: They constrain behavior by affecting the costs or benefits of acts, but they also convey information about what will be the costs and benefits of the acts.

In the framework of Marshall or Walras, people are merely constrained by prices, and so, no one learns anything from prices. That is, their models cannot capture the second role of prices. To be more precise, in the Walrasian model, the demand function specifies a desired level of holdings of the security at each particular price, irrespective of whether that price is a market clearing price. However, a trader might be induced to adjust his "demand" function to reflect the fact that the price at which market clears conveys information. To take this informational role into account, the author assumes that the consumer faces a price that is a real offer of another person, or the outcome of a market process.
In short, his idea is to redefine the "demand" as an expression of desired holdings at prices that are "market clearing", i.e., each consumer chooses his demand at "p" to maximize his expected utility conditioned on both his private information and on the information contained in the event that "p" is a market clearing price. As a consequence, there is no desire to recontract after observing that a particular price is the market clearing price because each person has already incorporated it.
In the rest of the chapter 1, the author introduces the concept of "uninformed traders" or "noise traders" who demand a security for noninformational reasons. Incorporating them into his models, he tries to explain the stock market crush in 1987. Interestingly, he claims this event supports his models with rational agents rather than irrational behavior of traders, which is described as follows.

Some have suggested that this is strong evidence against investor rationality, and point to the October 1987 episode as another example of irrational behavior. In contrast, I think that these events and the excessive volatility of stock prices relative to the volatility of expected payoffs are evidence in favor of the type of "rationality" embodied in the R.E. approach outlined above, rather than evidence for irrationality. As I argue below, once the Walrasian notion of demand is eliminated, the volatility phenomena can be seen as an expression of a sophisticated trading strategy rather than irrationality.

Interesting papers in References

Grossman and Stiglitz (1976) "Information and Competitive Price Systems" AER, 66-2
Kreps (1988) "In Honor of Sandy Grossman, Winner of the John Bates Clark Medal" J. of Economic Perspectives, 2-2
Kyle (1985) "Continuous Auctions and Insider Trading" Econometrica, 53
Roll (1984) "Orange Juice and Weather" AER, 74

The following book might be useful for those who are interested in the line of research mentioned above.

Brunnermeier (2001) "Asset Pricing under Asymmetric Information" Oxford University Press