An alternative proof for the linear utility representation theorem

Summary The paper presents an alternative short proof for the linear utility representation theorem. In particular a generalization of the theorem of Blackwell and Girshick (1954) and a special case of the theorem of Herstein and Milnor (1953) are proved by exploiting the topological group structure of finite-dimensional Euclidean vector space.I thank two careful referees for their helpful remarks.     

Comments on John Roemer’s first welfare theorem of market socialism*

Abstract

In this comment on John Roemer’s “theory of cooperation with an application to market socialism”, I extend Roemer’s first welfare theorem of market socialism in two directions. First, I prove a version of the theorem that deals with non-linear taxation. Second, I offer a connection between the theorem and welfare equality. I then argue that the models and questions that Roemer contribute to bring to welfare economics raise questions that go much beyond the research on socialist ethics. In particular, I introduce a positive model of moral behavior that yields different predictions from Roemer’s Kantian model. I conclude that individual morality should become a central concern of welfare economists.     

Infinite-dimensional Gale-Nikaido-Debreu theorem and a fixed-point theorem of Tarafdar

This paper proves the equivalence of five fixed-point theorems in topological vector spaces and then uses them to deduce an infinite-dimensional generalization of the classic Gale-Nikaido-Debreu theorem. The proof is based on the Hahn-Banach theorem and the Tarafdar fixed point theorem. There is also a discussion of the relationship of this generalization to the existence theorems of Aliprantis and Brown, Border, Yannelis and others.     

Boundedness of economic variables and Olech's theorem

On the basis of Olech's theorem we derive conditions on which any solution path starting from an arbitrary initial point in a two-dimensional bounded region ultimately converges to a unique equilibrium point without escaping from the region during a transition period. Then, using the result, we render one proposition more exact from the economic point of view.I would like to thank two anonymous referees for valuable comments and advices. Remaining errors are, of course, wholly due to myself.     

Folk theorem with communication

This paper proves a new folk theorem for repeated games with private monitoring and communication, extending the idea of delayed communication in Compte [O. Compte, Communication in repeated games with imperfect private monitoring, Econometrica 66 (1998) 597-626] to the case where private signals are correlated.The sufficient condition for the folk theorem is generically satisfied with more than two players, even when other well-known conditions are not. The folk theorem also applies to some two-players repeated games.     

A simple proof of K-K-M-S theorem

Summary We give a simple proof of the K-K-M-S theorem based on the Kakutani fixed point theorem, the separation theorem for convex sets and the Berge maximum theorem.The author is grateful to Gerard Debreu for his advice and to Tatsuro Ichiishi, Michael Todd, Rajiv Vohra, Nicholas Yannelis, Lin Zhou and two anonymous referees for their comments. The original version of this note was written in the fall 1992 while the author was visiting the Department of Economics, University of California at Berkeley.     

On Kakutani's fixed point theorem,the K-K-M-S theorem and the core of a balanced game

Summary We provide elementary proofs of Scarf's theorem on the non-emptiness of the core and of the K-K-M-S thoerem, based on Kakutani's fixed point theorem. We also show how these proofs can be modified to apply a coincidence theorem of Fan instead of Kakutani's fixed point theorem, for some additional simplicity.The results presented here were first reported in Shapley (1987) and Vohra (1987). A version of our proof of Theorem 1 has also been presented in a recent book by C.D. Aliprantis, D.J. Brown and O. Burkinshaw,Existence and Optimality of Competitive Equilibria (1989) Springer-Verlag. We are grateful to Ky Fan, Wanda Gorgol, Tatsuro Ichiishi and Ali Khan for comments on earlier drafts. Vohra's research has been supported in part by NSF grant SES-8605630.     

A folk theorem for repeated games played on a network

I consider repeated games on a network where players interact and communicate with their neighbors. At each stage, players choose actions and exchange private messages with their neighbors. The payoff of a player depends only on his own action and on the actions of his neighbors. At the end of each stage, a player is only informed of his payoff and of the messages he received from his neighbors. Payoffs are assumed to be sensitive to unilateral deviations. The main result is to establish a necessary and sufficient condition on the network for a Nash folk theorem to hold, for any such payoff function.     

A Lagrangean approach to the limit theorem on the core of an economy

Conclusion It is shown that an allocation which remains in the core under replication gives rise to an utility allocation with the following property: The utility level of no class can be increased without decreasing the utility level of at least one other class even if the number of consumers is allowed to vary continuously. An examination of the first order conditions the corresponding maximum problem with constraints leads to the conclusion that the allocation can be sustained by a price vector as a competitive equilibrium. This is the limit theorem on the core, proved in this paper with a simple Lagrangean technique.The notion of Pareto efficiency with a variable number of consumers proves useful for other purposes as well. For an extensive discussion of that point the reader is referred to Schweizer (1981).Ours, of course, is not the first calculus approach to the limit theorem [see e. g. L. Johansen (1978)]. But, as far as I know, the method of differentiating the Lagrangean simply with respect to the number of consumers is novel and leads to a straightforward proof of the theorem which can be reproduced in class-rooms even at the undergraduate level.The author wants to thank Joseph Greenberg and Werner Hildenbrand as well as an anonymous referee for helpful comments.     

Core equivalence theorem with production

In production economies, the extent to which non-equilibria are blocked depends on the allocation of control rights among shareholders, because a blocking coalition's resources are affected by the firms it jointly owns with outsiders. We formulate a notion of blocking that takes such interdependency problem into account, and we prove an analog of the Debreu-Scarf theorem for replica production economies. Our theorem differs from theirs in using an additional assumption, which we argue is indispensable and is driven by the interdependency problem.     

On the Phelps–Koopmans theorem

We examine whether the Phelps–Koopmans theorem is valid in models with nonconvex production technologies. We argue that a nonstationary path that converges to a capital stock above the smallest golden rule may indeed be efficient. This finding has the important implication that “capital overaccumulation” need not always imply inefficiency. Under mild regularity and smoothness assumptions, we provide an almost-complete characterization of situations in which every path with limit in excess of the smallest golden rule must be inefficient, so that a version of the Phelps–Koopmans theorem can be recovered. Finally, we establish that a nonconvergent path with limiting capital stocks above (and bounded away from) the smallest golden rule can be efficient, even if the model admits a unique golden rule. Thus the Phelps–Koopmans theorem in its general form fails to be valid, and we argue that this failure is robust across nonconvex models of growth.     

Bounded rationality,neural network and folk theorem in repeated games with discounting

Summary The perfect folk theorem (Fudenberg and Maskin [1986]) need not rely on excessively complex strategies. We recover the perfect folk theorem for two person repeated games with discounting through neural networks (Hopfield [1982]) that have finitely many associative units. For any individually rational payoff vector, we need neural networks with at most 7 associative units, each of which can handle only elementary calculations such as maximum, minimum or threshold operation. The uniform upper bound of the complexity of equilibrium strategies differentiates this paer from Ben-Porath and Peleg [1987] in which we need to admit ever more complex strategies in order to expand the set of equilibrium outcomes.I would like to thank Hao Li and John Curran for excellent research assistance. Financial support from National Science Foundation (SES-9223483), Sloan Foundation and the Division of Social Sciences at the University of Chicago is gratefully acknowledged.     

On a covering theorem

Summary. A simple proof of Reny and Wooders' recent strengthening of Shapley's extension of the Knaster-Kuratowski-Mazurkiewicz lemma on a closed cover of a simplex is given. The proof uses Ky Fan's coincidence theorem. Received: May 26, 2000; revised version: January 22, 2001     

Coase theorem, complexity and transaction costs

This paper, by introducing complexity considerations, provides a dynamic foundation for the Coase theorem and highlights the role of transaction costs in generating inefficient bargaining/negotiation outcomes. We show, when the players have a preference for less complex strategies, the Coase theorem holds in negotiation models with repeated surplus and endogenous disagreement payoffs if and only if there are no transaction costs. Specifically, complexity considerations select only efficient equilibria in these models without transaction costs while every equilibrium induces perpetual disagreement and inefficiency with transaction costs. We also show the latter is true in the Rubinstein bargaining model with transaction costs.     

A general structure theorem for the Nash equilibrium correspondence

I consider n-person normal form games where the strategy set of each player is a non-empty compact convex subset of an Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in the player i's strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. I demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well-known structure theorem in [Kohlberg, E., Mertens, J.-F., 1986. On the strategic stability of equilibria. Econometrica 54, 1003–1037]. It is supplemented by an extension analogous to the unknottedness theorems in [Demichelis S., Germano, F., 2000. Some consequences of the unknottedness of the Walras correspondence. J. Math. Econ. 34, 537–545; Demichelis S., Germano, F., 2002. On (un)knots and dynamics in games. Games Econ. Behav. 41, 46–60]: the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.     

A general theorem on tariff harmonization

This paper proves a general theorem on welfare improving tariff changes that a tariff harmonization towards any arbitrary common target tariff rates among member countries can improve potential world welfare as long as international lump-sum transfers are available.     

A new turnpike theorem for discounted programs

Summary In this paper we present new results on the local and global convergence property of solutions to an optimization model where the objective function is a discounted sum of stationary one-period utilities. The asymptotic local turnpike is given without differentiability assumptions but imposing some mild curvature restrictions on the utility function. This approach allows us to get easy estimates on the range of discount factors and the size of the neighborhood for which the asymptotic property occurs. The paper concludes by providing two global turnpike theorems. The first one is an asymptotic theorem derived from a result similar to Scheinkman's visit lemma. The second one turns out to be a restatement of McKenzie's neighborhood turnpike theorem.This research was partially supported by MURST, National Group on Nonlinear Dynamics in Economics and Social Sciences.     

A short proof of Harsanyi''s purification theorem

A short proof of more general version of Harsanyi's purification theorem is provided through an application of a powerful, yet intuitive, result from algebraic topology.     

Inverse of the Berge maximum theorem

Summary We consider an inverse of the Berge maximum theorem. We also give an application of our result to fixed point theory.     

An expository note on the composite commodity theorem

Summary This note offers an alternative derivation of the composite commodity theorem using only elementary economic and mathematical tools. It offers some insight as to why constancy of relative prices induces separability in the consumers optimization problem. It should be more readily accessible to students in upper level microeconomics courses and prove useful in the classroom in presenting this central theorem of economic analysis.The author gratefully acknowledges the useful comments made by John Fountain, Leslie Young and Peyton Young.