David Gale

ON OPTIMAL DEVELOPMENT IN A MULTI-SECTOR ECONOMY.

Abstract : An economy is considered that has n goods and k types of labor, each of which is growing at the same constant rate. Goods are produced from labor and other goods by a set of specified activities. Given an initial supply of goods and amounts of labor, all possible production programs running from the present time to infinity through discrete time periods are considered. With each program is associated a utility sequence measuring the satisfaction achieved by the program at each period of time. A program is optimal if its utility sequence overtakes all other such sequences. The paper is devoted to proving the existence of optimal programs for a wide class of economies and to deriving the properties of such programs. In particular it is shown that the optimal program approaches a certain balanced program. Essential use is made of the existence of an infinite sequence of optimal prices with respect to which the optimal program is one which maximizes the sum of profit and utility at each time period. (Author)

Pure exchange equilibrium of dynamic economic models

This paper studies competitive equilibrium over time of a one good model in which the agents are members of a population which grows at a constant rate. Each agent lives for n periods and in the i-th period of his life receives an endowment of ei units of goods. Goods can neither be produced nor stored. The model is thus the n-period generalization of the two- and three-period models studied by Samuelson in [4]. We seek to ascertain the structure of the time paths of consumption in these models. Our results can be summarized roughly as follows: In general, there will exist two kinds of steady state paths, (i) golden rule paths in which the rate of interest equals the growth rate of population and (ii) “balanced” paths in which the aggregate assets or indebtedness of the society as a whole is zero (a fundamental fact about dynamic models is that it is possible for aggregate debt not to equal aggregate credit as it must in the static case). A model is termed classical if in the golden rule state aggregate assets are negative (or debt positive) and Samuelson (following [4]) in the opposite case. It is conjectured that the golden rule program is globally stable in the classical case and the balanced program is stable in the Samuelson case. This is established for the special case n = 2.

Some remarks on the stable matching problem

Abstract The stable matching problem is that of matching two sets of agents in such a manner that no two unmatched agents prefer each other to their mates. We establish three results on properties of these matchings and present two short proofs of a recent theorem of Dubins and Freedman.

An equilibrium existence theorem for a general model without ordered preferences

In a recent paper the second author has shown that some of the usual hypotheses on consumers’ preferences are not needed for the proof of existence of a Walrasian General Equilibrium [Mas-Cole11 (1974)]. Specifically, it is not necessary that preferences come from a preference ordering. The only order property required is irreflexivity (meaning that a given commodity bundle is not preferred to itself). The properties of non-satiation, continuity and convexity of preferred sets turn out to be sufficient to obtain the existence result. The main purpose of the present note is to give a second proof of this fact which seems simpler than that of Mas-Cole11 (1974), and no more lengthy or complicated than the known equilibrium existence proofs which use ordered preferences. In two additional respects the model studied here generalizes the usual equilibrium model. The standard Walras, Arrow-Debreu theory treats what might be called the laissez-faire model in which each agent’s income is whatever he gets from selling goods plus his share of the profits of any firm in which he may own stock. In the present model the income of a consumer may be any continuous function of the prices, so the laissez-faire income function is included, but so also would any rule for assigning income to consumers (e.g., according to his ability or his need or the color of his eyes). Another way of saying this is that the model includes the possibility of arbitrary lump sum transfers of income among consumers, as might be achieved, for example, by a program of income taxes and subsidies. This substantial economic generalization requires no change whatever in the mathematical argument. The second generalization concerns production. The only requirement on our production set, besides the usual closure, convexity, and free disposal, is that it intersect the positive orthant in a boundedset.This means that the usual assumption

FAIR ALLOCATION OF INDIVISIBLE GOODS AND CRITERIA OF JUSTICE

A set of n objects and an amount M of money is to be distributed among m people. Example: the objects are tasks and the money is compensation from a fixed budget. An elementary argument via constrained optimization shows that for M sufficiently large the set of efficient, envy free allocations is nonempty and has a nice structure. In particular, various criteria of justice lead to unique best fair allocations that are well behaved with respect to changes of M. This is in sharp contrast to the usual fair division theory with divisible goods. Copyright 1991 by The Econometric Society.

Stable schedule matching under revealed preference

In a recent study Baiou and Balinski [3] generalized the notion of two-sided matching to that of schedule matching which determines not only what partnerships will form but also how much time the partners will spend together. In particular, it is assumed that each agent has a ranking of the agents on the other side of the market. In this paper we treat the scheduling problem using the more general preference structure introduced by Blair [5] and recently refined by Alkan [1, 2], which allows among other things for diversity to be a motivating factor in the choice of partners. The set of stable matchings for this model turns out to be a lattice with other interesting structural properties.

Corrections to an equilibrium existence theorem for a general model without ordered preferences

It has been brought to our attention that the proof of the equilibrium theorem in Gale and Mas-Cole11 (1975) has two gaps. First: The particular bound used to truncate consumption sets may be too small to guarantee the non-empty valuedness of the modified budget sets defined at the bottom of page 13 (we are indebted to H. Cheng for pointing this out to us). Second: The augmented preference mappings may not, as claimed, have an open graph [we are indebted to J. Foster (1978) for pointing this out to us]. The theorem and the main lines of the proof remain valid. The first problem can be repaired by taking a sufficiently large truncation of the consumption sets; the second, either by noticing that the weaker property of lowersemicontinuity of the augmented preference maps suffices, or by a slight change of the definition of the augmented preference maps. Details follow.

Optimal growth under factor augmenting progress

Abstract Optimal growth is considered in the usual neoclassical one-good model of economic growth with factor augmenting technical progress. If a program is optimal (in the sense of maximizing a discounted sum of one-period utilities), then the asymptotic rate of growth of per capita output is given as a function of the capital and labor augmentation factors and the asymptotic elasticity of the production function in per capita units. Furthermore, the intuitive notion of “critical discount factor,” δ, is put on a rigorous footing, and δ is given as a function of the asymptotic growth rate and the asymptotic elasticity of the utility function.

A further note on the stable matching problem

Using a lemma of J.S. Hwang we obtain a generalization of a theorem of Dubins and Freedman. It is shown that the core of the matching game is non-manipulable in a suitable sense by coalitions consisting of both men and women. A further strong stability property of the core is derived.

Convex functions on convex polytopes

Abstract : The behavior of convex functions is of interest in connection with a wide variety of optimization problems. It is shown here that this behavior is especially simple, in certain respects, when the domain is a polytope or belongs to certain classes of sets closely related to polytopes; moreover, the polytopes and related classes are actually characterized by this simplicity of behavior.