Journal Article Quality in Contestable Markets: A Historical Problem? Get access Salim Rashid Salim Rashid Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Economics, Volume 103, Issue 1, February 1988, Pages 245–249, https://doi.org/10.2307/1882653 Published: 01 February 1988

THE MAJOR INTEREST in the recent literature on economies has lain in the results about but finite economies that have been derived from the results proved for economies. It is thus important to find simple yet general proofs for economies. In this article we wish to provide a simple yet general proof of the existence of a competitive equilibrium in an infinite, nonstandard economy with production. The simplicity of our proof comes from the fact that nonstandard analysis can deal with large and small quantities very much as ordinary analysis deals with finite quantities.2 As a result, it is possible to follow very closely the proof of existence for an economy with a finite number of traders, such as that in G. Debreu's classic Theory of Value [6]. In fact, if one is willing to believe that nonstandard analysis permits us to manipulate quantities as claimed above, then no further knowledge of nonstandard analysis is required in order to follow the proof. As examples of the simplicity of nonstandard analysis, it may be pointed out that no analogue of the Fatou-Schmeidler lemma [7, p. 69], a fairly difficult mathematical theorem, is required; nor is it necessary to prove separately that preserves upper-semicontinuity, a proof that Aumann [1] has recently simplified, because integration in the nonstandard model consists of an infinite summation, hence an appeal to 1.9.4 of Debreu [6] suffices to establish this point. As our main objective is to obtain results about but finite economies, it is a welcome bonus to find out that no further effort is needed to obtain these desired theorems. This arises because of the following property of nonstandard analysis. Consider a sequence of real numbers {an} which tends to zero. If we could extend this sequence to the integers, it would surely be a necessary property of the values of {an} at the integers that they are all infinitely close to zero. What makes nonstandard analysis powerful is that the above line of reasoning can be reversed, so to speak. Suppose we have a sequence which

In a recent article A. Y. C. Koo [1973] sought to advance a “more general theory of land tenancy” and draw implications for land reform. While his contention that different models of share-tenancy (with conflicting efficiency properties) have different implications for land reform is correct, unfortunately, his mathematical analysis of oligopoly land market is both misleading and erroneous. It is misleading in the sense that it is not (as a close look at the model would verify) an oligopoly model of share-rental determination, which it purports to be. Thus, on this account, it is not really a critique of either Bardhan and Srinivasan [1971] or of Cheung [1974], which it claims to be.1 Rather it is an oligopoly model of determination of fixed-rental in the landlease market. If so, it loses much of its importance, at least empirically, since the fixed-rental system is of limited empirical significance. And the efficiency arguments for and against the share-rental lease, which he advances in the beginning, and also the land-reform question of reducing the share-rental become irrelevant for the analysis that follows. Second, even if we consider his mathematical argument on land renting on its own grounds, it is unfortunately marred by a mathematical flaw that prevails throughout his analysis and leads him to counterintuitive results. Section I gives a restatement of the Koo model and points to its errors and the resulting counterintuitive results. Section II reformulates the model in correct terms and shows how intuitively and empirically plausible results may be derived therefrom.

We prove the existence of approximate equilibria in exchange economies, giving bounds on the excess demand in terms of the number of traders and norms of the endowments, but independent of the preferences.