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.     

Advanced Industrial Economics,by Stephen Martin

The author presents a simple proof of a property of the method of least squares variously known as the FWL, the Frisch-Waugh-Lovell, the Frisch-Waugh, or the decomposition theorem.     

The Paper River Revisited: A Common Property Externality Exercise

The author describes a modified version of the Paper River exercise that appeared in this journal in the Spring 1999 issue. The original game was designed to illustrate the application of the Coase theorem for solving an externality problem. In the Paper River, pairs of students share a single productive resource: small pieces of paper. At question is who has the rights to the paper. The author presents a modified version of the game that retains many of the same elements from the original, but the students work in groups rather than in pairs. Doing so increases the transaction costs associated with negotiating a solution. In addition, the productive resource is common property, so property rights are difficult to establish. These difficulties make a Coasian solution more difficult but make the simulation more realistic.     

Another Graphical Proof of Arrow's Impossibility Theorem

Arrow's (1951) Impossibility Theorem is the idea that, given several well-known assumptions, the social orderings of particular alternatives that are meant to reflect individuals' preferences must match the preferences of an arbitrary individual (the dictator). A social-choice rule other than dictatorship is impossible. Following from Fountain (2000), the author presents another graphical proof of the theorem that is intended to be more accessible to students and teachers of economics. The principal strength of this approach is that the patterns of agreements and conflicts over all possible combinations of two individuals' rankings of alternatives are transparent; appreciating these patterns is the key to intuitively understanding Arrow's theorem. A self-test for readers (or a classroom exercise for students) is included.     

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.     

Parametric continuity of stationary distributions

For Markovian economic models, long-run equilibria are typically identified with the stationary (invariant) distributions generated by the model. In this paper we provide new sufficient conditions for continuity in the map from parameters to these equilibria. Several existing results are shown to be special cases of our theorem. This paper has benefitted from the helpful comments of Kevin Reffett, Rabee Tourky, an anonymous referee and participants at the 13th European Workshop on General Equilibrium Theory, Venice 2004. The second author is grateful for financial support from Australian Research Council Grant DP0557625.     

On the existence of optimal processes in non-stationary environments

We consider an aggregative model of intertemporal allocation under uncertainty, in which the utility and production functions are allowed to be time dependent, the random shocks occurring in each period are entirely arbitrary, and the production functions are permitted to be non-concave. In this framework, we provide a theorem on the existence of infinite-horizon optimal processes. In the course of establishing this result, we obtain the existence of optimal policy functions and we show that they are monotone in the stock levels.This paper has benefitted from the comments of two referees of the journal. Research of the first author was supported by a National Science Foundation Grant.     

Existence of steady – state equium in an overlapping-generations model with production

Summary This paper establishes an existence theorem of a non-trivial (positive capital stock) steady-state equilibrium in Diamond's (1965) overlapping-generations model with production by employing the steady-state consumption curve introduced in Ihori (1978). The assumptions on preferences and production technologies that ensure the existence of a nontrivial steadystate equilibrium are separated from each other, unlike in Galor and Ryder (1989). We also provide two simple examples which illustrate the importance of two conditions in the theorem.Detailed comments by Tomoichi Shinotsuka and the referees of the journal were quite helpful. We also thank Marcus Berliant, Mark Bus, John H. Boyd III, Ban Chuan Cheah, Rajat Deb, Jim Dolmas, Oded Galor, Greg Huffman, Toshihiro Ihori, Radhika Lahiri, Lionel McKenzie, Arundhati Sen, and the seminar participants at the Midwest Mathematical Economics Conference in Ann Abor and at University of Rochester. The second author gratefully acknowledges the financial supports from the European Community Human Capital Mobility Program.     

On a mixed competitive-monopolistic macrodynamic model in a monetary economy

In this paper we reconsider the dynamic stability of the mixed competitive-monopolistic system in an analytical framework of the macroeconomic growth model in a monetary economy. We construct a simple monetary growth model in the so-called Keynes-Wicksell tradition and investigate how the degree of competition affects the dynamic stability of the system. Our analysis reveals the destabilizing rather than the stabilizing forces of the monopolistic system in amonetary economy contrary to the usual analyses. We also show that the system produces the purely cyclical behavior at some intermediate degrees of competition by using the Hopf-Bifurcation theorem.This paper was written while the author was staying at the New School for Social Research in New York as a visiting research scholar. Special thanks are due to Prof. Willi Semmler for providing the comfortable research environment and to Komazawa University for the financial support. The author is also grateful to Dr. Reiner Franke and two anonymous referees of this journal for their constructive comments and helpful suggestions. Needless to say, however, only the author is responsible for possible remaining errors.     

Recycling and International Trade Theory

Recently recycling and the production of secondary materials have increased in many countries. However, there is little analysis examining the effects of recycling on comparative advantage, trade, and welfare. In a model with recycling sectors, the author examines whether the Rybczynski theorem is valid, how the price effects are modified, how a recycling subsidy changes the production structure and comparative advantage. It is found that demand has an effect on final goods production and comparative advantage since consumption goods are transformed into recycling inputs.     

Comparing random and deterministic time series

Summary This paper addresses the question of distinguishing the output of a stochastic process from that of a deterministic process. An impossibility theorem is described which states that time a series resulting from deterministic B-processes is observationally equivalent to, and hence indistinguishable from, the output of a continuous time Markov process on a finite number of states.This is an expanded version of a lecture presented at Cornell University in June, 1992 in a conference honoring Roy Radner on his 65th birthday. I would like to thank the organizers of this conference, especially Mukul Majumdar. The author is partially supported by a grant from the NSF.     

Theorems on correspondences and stability of the core

In this note two theorems strengthening Grodal's (1971) Theorem on correspondences are proved. The first drops the convexity assumption. The second strengthens that theorem further for the case when the range is the positive orthant. In this case, the conclusion of Grodal's Theorem - the intersection of the integral with the interior of the range being open- is modified to read as the integral being a relative open subset of the positive orthant. An example is provided to show that, such a strengthening is not valid in general. This allows us to dispense with the requirment of convexity of preferences in Grodal's (1971) theorems on the closedness of the set of Pareto optimal allocations, the core, and the continuity of the core correspondence for pure exchange economies. We apply this result to show that blocking coalitions in a large economy are stable. Received: September 30, 1998; revised version: September 18, 2001 RID="*" ID="*" The author is grateful to an anonymous referee for helpful comments. The usual disclaimer applies.     

Backward Induction and the Game-Theoretic Analysis of Chess

This paper scrutinizes various stylized facts related to the minmax theorem for chess. We first point out that, in contrast to the prevalent understanding, chess is actually an infinite game, so that backward induction does not apply in the strict sense. Second, we recall the original argument for the minmax theorem of chess—which is forward rather than backward looking. Then it is shown that, alternatively, the minmax theorem for the infinite version of chess can be reduced to the minmax theorem of the usually employed finite version. The paper concludes with a comment on Zermelo's (1913) nonrepetition theorem. Journal of Economic Literature Classification Number: C72.     

论科斯定理的真理性

科斯定理具有隐含的私有产权制度偏好。在逻辑意义上,对科斯定理中"交易成本"、"可交易权利的初始配置"、"最终配置"作特定的理解,科斯定理在西方主流经济学逻辑、方法论意义上是正确的,但此时科斯定理完全等价于完全竞争理论,没有任何现实意义。在经验意义上,科斯定理是谬误。科斯定理的逻辑推论必然是私有产权企业比公有产权企业更有效率。在经济实践中,存在着三类证据:完全肯定科斯定理、完全否定科斯定理以及在肯定科斯定理的同时包含着否定科斯定理的成分。"超产权论"是对科斯定理的"革命",是对科斯定理的否定。     

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.     

On separability: Functional structure

Bliss' notion of nonstrict separability is shown to be the most general yet practical formulation of separability. This paper extends the classic theory of strict separability to nonstrict separability. Gorman's basic theorem for overlapping strictly separable groups of variables is generalized. With the generalized basic theorem on hand, Gorman's procedure for deducing functional structures can be extended for deducing the internal structure of a wider class of economic systems modelled by continuous functions. The generalized basic theorem also prepares for the joint characterization of the additively separable and Leontief functional forms.     

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.     

Lerner symmetry and trilateral relations in international monetary economics

The classical Lerner symmetry theorem concerning the equivalence of import and export tariffs is extended to a monetary economy under flexible exchange rates. Under fixed exchange rates, this should be replaced by the trilateral-relation theorem. The Meade theorem is also extended.     

An extended Heckscher-Ohlin model with transaction costs and technological comparative advantage

Summary. This paper introduces technological differences and transaction costs into the Heckscher-Ohlin (HO) model and examines the HO theorem, factor price equalization theorem, the Stolper-Samuelson theorem and the Rybczynski theorem. It shows that the HO theorem can be refined, and that the factor price equalisation theorem, the Stolper-Samuelson Theorem and the Rybczynski theorem do not always hold. It also shows that transaction costs play an important role in determining the equilibrium trade pattern.Received: 26 February 2001, Revised: 27 May 2003, JEL Classification Numbers: F10, F11. Correspondence to: Wenli ChengWe are grateful for comments from the anonymous referee, Hugo Sonnenschein, Guangzhen Sun and participants of the seminar on this paper at University of Washington.     

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.