For an economic system with given technological and resource limitations, individual needs and tastes, a valuation equilibrium with respect to a set of prices is a state where no consumer can make himself better off without spending more, and no producer can make a larger profit; a Pareto optimum is a state where no consumer can be made better off without making another consumer worse off. Theorem 1 gives conditions under which a valuation equilibrium is a Pareto optimum. Theorem 2, in conjunction with the Remark, gives conditions under which a Pareto optimum is a valuation equilibrium. The contents of both theorems (in particular that of the first one) are old beliefs in economics. Arrow and Debreu have recently treated this question with techniques permitting proofs. A synthesis of their papers is made here. Their assumptions are weakened in several respects; in particular, their results are extended from finite dimensional to general linear spaces. This extension yields as a possible immediate application a solution of the problem of infinite time horizon (see sec. 6). Its main interest, however, may be that by forcing one to a greater generality it brings out with greater clarity and simplicity the basic concepts of the analysis and its logical structure.

The core of a finite economy has been shown to converge, as the number of its agents tends to infinity, under conditions of increasing generality in a series of contributions, of which the first, by Edgeworth (1881), studied replicated exchange economies with two commodities and two types of agents, and the latest, by Hildenbrand (1974), considers sequences of finite exchange economies (with a given finite number of commodities) whose distributions on the space of agents' characteristics converge weakly.However, information on the rate of convergence of the core seems to be contained in only two articles. In Shapley and Shubik (1969, section 5) an example is given of an Edgeworth replicated economy whose core converges like the inverse of the number of agents. Recently, Shapley (1975) provided examples of Edgeworth replicated economies whose cores converge arbitrarily slowly, but concluded with the conjecture that for any fixed concave utility functions only a set of initial allocations of measure zero will yield cores that converge more slowly than the inverse of the number of agents. The theorem stated below for replicated economies with arbitrary numbers of commodities, and of types, asserts that such is indeed the case provided that preference relations are of class C2, and satisfy the conditions listed in the definition of the economy. At the same time, the theorem implies that the set of exceptional allocations is closed as well as of measure zero.

The recent introduction of differential topology into economics was brought about by the study of several basic questions that arise in any mathematical theory of a social system centered on a concept of equilibrium. The purpose of this paper is to present a detailed discussion of two of those questions, and then to make a rapid survey of some related developments of the last five years.

Let e be a complete mathematical description of the economy to be studied (e.g., for an exchange economy, e might be a list of the demand functions and of the initial endowments of the consumers). Assumptions made a priori about e (e.g., assumptions of continuity on the demand functions) define the space ℰ of economies to which the study is restricted. By a state of an economy we mean a list of specific values of all the relevant endogenous variables (e.g., prices and quantities of all the commodities consumed by the various consumers). We denote by S the set of conceivable states. Now a given equilibrium theory associates with each economy e in ℰ, the set E(e) of equilibrium states of e, a subset of S (see Figure 1).