Economies with a finite set of equilibria
About: This article is published in Econometrica.The article was published on 1970-05-01. It has received 579 citations till now. The article focuses on the topics: Methodology of econometrics & Econometric model.
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TL;DR: In this article, the authors argue that this reduction of the behavior of a group of heterogeneous agents, even if they are all themselves utility maximizers, is both unjustified and leads to conclusions which are usually misleading and often wrong.
Abstract: A modern economy presents a picture of millions of people, either as individuals or organized into groups and firms, each pursuing their own disparate interests in a rather limited part of the environment. Somehow, these varied individual activities are more or less coordinated and some relative order emerges. Economists commonly explain that this is due to Adam Smith's "invisible hand," and that despite the conflicting interests of individuals, the result of the pursuit of their selfish ends is socially satisfactory. The market provides the mechanism which links and coordinates all the activities being pursued by individuals. Paradoxically, the sort of macroeconomic models which claim to give a picture of economic reality (albeit a simplified picture) have almost no activity which needs such coordination. This is because typically they assume that the choices of all the diverse agents in one sector-consumers for example-can be considered as the choices of one "representative" standard utility maximizing individual whose choices coincide with the aggregate choices of the heterogeneous individuals. My basic point in this paper is to explain that this reduction of the behavior of a group of heterogeneous agents even if they are all themselves utility maximizers, is not simply an analytical convenience as often explained, but is both unjustified and leads to conclusions which are usually misleading and often wrong. Why is this? First, such models are particularly ill-suited to studying macroeconomic problems like unemployment, which should be viewed as coordination
1,492 citations
Cites background or result from "Economies with a finite set of equi..."
...This suspicion is heightened by the fact that in Debreu's (1974) paper each individual always has positive excess demand for one particular good, regardless of the prices of the other goods....
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...6To state Debreu's (1974) theorem more precisely, have each agent characterized by textbook indifference curves and a positive bundle, called e(a), of initial endowment of all goods. From these is derived a well-behaved demand 4(a, p) and excess demand z(a, p) = 4(a, p) - e(a). Summing over all the individuals a, of whom we assume there are a finite number, gives aggregate excess demand Z(p). If only prices greater than some positive E are considered, which may be chosen as small as we like, three basic properties of individual excess demand carry over to Z(p); 1) Z(p) is continuous. 2) Z(p) satisfies Walras' Law, i.e. p Z(p) = 0. 3) Z(p) is homogeneous of degree 0, i.e. Z(Ap) = Z(p) for any positive A. Debreu found that given any function f(p) satisfying properties 1-3 we can find individuals with strictly convex and monotonic preferences and positive initial endowments whose aggregate excess demand Z(p) is equal to f(p) for all prices greater than E. 'I'his means that the only properties the aggregate excess demand of an economy can have are the three given above. 7The reader will observe that, for Debreu's result, prices are bounded away from zero. Balasko (1986) has argued that since we do know something, with our assumptions, about the behavior of excess demand functions when some prices go to zero, the class of such functions is much smaller than would be suggested by the Sonnenschein-Debreu results....
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...6To state Debreu's (1974) theorem more precisely, have each agent characterized by textbook indifference curves and a positive bundle, called e(a), of initial endowment of all goods....
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...However, a series of results starting with those of Sonnenschein (1972) and Debreu (1974) show unequivocally that no such conditions exist....
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TL;DR: This paper argued that rationality consists of making a decision which is justifiable by an internally consistent system of beliefs, rather than one which is optimal, post hoc, in a noncooperative game.
Abstract: THE NOTION OF EQUILIBRIUM proposed by Nash [19] has come to play a dominant role in economic applications of noncooperative games. While analyses of Nash equilibria have unquestionably contributed to our understanding of economic behavior, it would be unreasonably optimistic to maintain that Nash "solved" the problem of noncooperative strategic choice. There is a small literature (beginning with Ellsberg [6]) and a much larger oral tradition which argues that Nash behavior is neither a necessary consequence of rationality, nor a reasonable empirical proposition. In this paper I take the view that although there may be various reasons why an agent might select a Nash strategy, the notion of an equilibrium has little intrinsic appeal within a strategic context. When an agent reaches a decision in ignorance of the strategies adopted by other players, rationality consists of making a choice which is justifiable by an internally consistent system of beliefs, rather than one which is optimal, post hoc. This point of view is not original; indeed, most serious justifications of the Nash hypothesis embrace such an approach, arguing that agents will expect the game to yield a Nash outcome, and consequently will choose their equilibrium strategies. Nevertheless, when we think in terms of maximizing utility subject to expectations rather than realizations, it becomes clear that the Nash hypothesis, far from being a consequence of rationality, arises from certain restrictions on agents' expectations which may or may not be plausible, depending upon the game being played. We are then quite naturally led to ask: are there any restrictions of individuals' expectations (and hence choices) which are required by rationality alone, rather than by (subjective) plausibility? This paper is concerned with defining, justifying, characterizing, and refining a criterion for rational strategic choice, which I label "rationalizability."
1,193 citations
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TL;DR: In this article, a Ricardian trade and payments analysis in the case of a continuum of goods is presented, where tariffs and transport costs establish a range of commodities that are not traded, and the price-specie flow mechanism does or does not give rise to movements in relative cost and price levels.
Abstract: This paper discusses Ricardian trade and payments theory in the case of a continuum of goods. The analysis thus extends the development of many-commodity, two-country comparative advantage analysis as presented, for example, in Gottfried Haberler (1937), Frank Graham (1923), Paul Samuelson (1964), and Frank W. Taussig (1927). The literature is historically reviewed by John Chipman (1965). Perhaps surprisingly, the continuum assumption simplifies the analysis neatly in comparison with the discrete many-commodity case. The distinguishing feature of the Ricardian approach emphasized in this paper is the determination of the competitive margin in production between imported and exported goods. The analysis advances the existing literature by formally showing precisely how tariffs and transport costs establish a range of commodities that are not traded, and how the price-specie flow mechanism does or does not give rise to movements in relative cost and price levels. The formal real model is introduced in Section 1. Its equilibrium determines the relative wage and price structure and the efficient international specialization pattern. Section II considers standard comparative static questions of growth, demand shifts,
1,137 citations
Additional excerpts
...See Debreu (1970) and Smale (1966) ....
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TL;DR: In this paper, it was shown that mixed-strategy equilibrium points are stable even though players may make no deliberate effort to use their pure strategies with the probability weights prescribed by their mixed equilibrium strategies.
Abstract: Equilibrium points in mixed strategies seem to be unstable, because any player can deviate without penalty from his equilibrium strategy even if he expects all other players to stick to theirs. This paper proposes a model under which most mixed-strategy equilibrium points have full stability. It is argued that for any game Г the players’ uncertainty about the other players’ exact payoffs can be modeled as a disturbed game Г*, i.e., as a game with small random fluctuations in the payoffs. Any equilibrium point in Г, whether it is in pure or in mixed strategies, can “almost always” be obtained as a limit of a pure-strategy equilibrium point in the corresponding disturbed game Г* when all disturbances go to zero. Accordingly, mixed-strategy equilibrium points are stable — even though the players may make no deliberate effort to use their pure strategies with the probability weights prescribed by their mixed equilibrium strategies — because the random fluctuations in their payoffs will make them use their pure strategies approximately with the prescribed probabilities.
916 citations
TL;DR: The Generalized Nash Equilibrium Problem is an important model that has its roots in the economic sciences but is being fruitfully used in many different fields and its main properties and solution algorithms are discussed.
Abstract: The Generalized Nash Equilibrium Problem is an important model that has its roots in the economic sciences but is being fruitfully used in many different fields. In this survey paper we aim at discussing its main properties and solution algorithms, pointing out what could be useful topics for future research in the field.
838 citations
Cites background from "Economies with a finite set of equi..."
...This issue has not been much considered in the literature, though, with the significant exception of (Debreu 1970)....
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TL;DR: In this article, it was shown that the set of critical values of a function of m variables of class C constitute a set of linear measure zero, provided that q ∈ (1.1) is of n-dimensional measure zero.
Abstract: of a region R of euclidean m-space into part of euclidean w-space. Suppose that each f unction ƒ' 0' = 1, • • • , n) is of class C in R (q^l). A critical point of the map (1.1) is a point in R at which the matrix of first derivatives 2)? = ||/*|| (i = ly • • • , m;j = l, • • • , n) is of less than maximum rank. The rank of a critical point # is the rank of 5DÎ at x. A critical value is the image under (1.1) of a critical point. If » = 1, these definitions are the usual definitions of critical point and value of a continuously differentiable function. We prove the following result: If m^n, the set of critical values of the map (1.1) is of m-dimensional measure zero without further hypothesis on q; if m>n, the set of critical values of the map (1.1) is of n-dimensional measure zero providing that q^m — n + 1. Using an example due to Hassler Whitney we show that the hypothesis on q cannot be weakened. We prove also that the critical values of (1.1) corresponding to critical points of rank zero constitute a set of (m/q)dimensional measure zero. The idea of considering the measure of the set of critical values of one function or of several functions is due to Marston Morse. The first result stated above reduces, if » = 1, to the known theorem : The critical values of a function of m variables of class C constitute a set of linear measure zero. A. P. Morse has given a proof of this theorem for all m. In the present paper we make use of one of A. P. Morse's results.
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