Nash and Walras equilibrium via Brouwer

Summary. The existence of Nash and Walras equilibrium is proved via Brouwer's Fixed Point Theorem, without recourse to Kakutani's Fixed Point Theorem for correspondences. The domain of the Walras fixed point map is confined to the price simplex, even when there is production and weakly quasi-convex preferences. The key idea is to replace optimization with “satisficing improvement,” i.e., to replace the Maximum Principle with the “Satisficing Principle.” Received: July 9, 2001; revised version: February 25, 2002 RID="*" ID="*" I wish to thank Ken Arrow, Don Brown, and Andreu Mas-Colell for helpful comments. I first thought about using Brouwer's theorem without Kakutani's extension when I heard Herb Scarf's lectures on mathematical economics as an undergraduate in 1974, and then again when I read Tim Kehoe's 1980 Ph.D dissertation under Herb Scarf, but I did not resolve my confusion until I had to discuss Kehoe's presentation at the celebration for Herb Scarf's 65th birthday in September, 1995. RID="*" ID="*"Correspondence to: C. D. Aliprantis     

Evolutionary foundations of aggregate-taking behavior

Summary. I show that aggregate-taking behavior is often evolutionarily stable for finite population in symmetric games in which payoff depends only on own strategy and an aggregate. I provide economic examples exhibiting this phenomenon. Received: August 27, 2001; revised version: January 29, 2002 RID="*" ID="*" The paper has profited from the comments of Maria Montero, Burkhard Hehenkamp, Wolfgang Leininger, and Dave Furth. Financial support from the DFG via Postgraduate Programme at the University of Dortmund and via SFB 504 at the University of Mannheim is acknowledged. RID="*" ID="*" Present address: University of Mannheim, SFB 504, L 13, 15, 68131 Mannheim, Germany (e-mail: possajen@sfb504.uni-mannheim.de)     

Large economies with differential information and without free disposal

We consider exchange economies with a continuum of agents and differential information about finitely many states of nature. It was proved in Einy et al. (Econ Theory 18, 321–332, 2001) that if we allow for free disposal in the market clearing (feasibility) constraints then an irreducible economy has a competitive (or Walrasian expectations) equilibrium, and moreover, the set of competitive equilibrium allocations coincides with the private core. However when feasibility is defined with free disposal, competitive equilibrium allocations may not be incentive compatible and contracts may not be enforceable (see e.g. Glycopantis et al. in Econ Theory 21, 495–526, 2002). This is the main motivation for considering equilibrium solutions with exact feasibility. We first prove that the results in Einy et al. (Econ Theory 18, 321–332, 2001) are still valid without free-disposal. Then, motivated by the issue of contracts’ execution, we adapt the incentive compatibility property introduced in Krasa and Yannelis (Econometrica 62, 881–900, 1994) and we prove that every Pareto optimal exact feasible allocation is incentive compatible, implying that contracts of competitive or core allocations are enforceable. We would like to thank two anonymous reviewers and the Associate Editor for their valuable suggestions and remarks. This work was partially done while V.F. Martins-da-Rocha was visiting the Dipartimento di Matematica e Informatica of the Università degli Studi di Perugia. We thank the audience of the First General Equilibrium Workshop at Rio. Section 6 dealing with contract enforcement and coalitional incentive compatibility has benefited from discussions with J. Correia-da-Silva, W. Daher, F. Forges, C. Hervès-Beloso, E. Moreno-García, K. Podczeck, Y. Vailakis and N.C. Yannelis.     

An algebraic theory of portfolio allocation

Summary. Using group and majorization theory, we explore what can be established about allocation of funds among assets when asymmetries in the returns vector are carefully controlled. The key insight is that preferences over allocations can be partially ordered via majorized convex hulls that have been generated by a permutation group. Group transitivity suffices to ensure complete portfolio diversification. Point-wise stabilizer subgroups admit sectoral separability in fund allocations. We also bound the admissible allocation vector by a set of linear constraints the coefficients of which are determined by group operations on location and scale asymmetries in the rate of returns vector. For a distribution that is symmetric under a reflection group, the linear constraints may be further strengthened whenever there exists an hyperplane that separates convex sets. Received: May 15, 2001; revised version: March 20, 2002 RID="*" ID="*" Journal paper No. J-19797 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Project No. 3463, and supported by Hatch Act and State of Iowa funds. Correspondence to: D. A. Hennessy     

Purification of incentive compatible allocations

In implementation theory it is common to consider social choice functions that map to lotteries over outcomes. For example, virtual implementation has been used widely with social choice functions that involve randomization. This paper investigates (in incomplete information environments) the possibility of purification of such social choice functions, where purification means that the “replacement” social choice function maps to outcomes rather than distributions over outcomes, is incentive compatible and generates the same distribution over payoffs and outcomes as the original social choice function. Received: July 31, 2000; revised version: December 2, 2002 RID="*" ID="*" Thanks are due to Ole Nielsen, Terence Tao and Lin Zhou for helpful conversations. Thanks are also due to an anonymous referee for detailed comments.     

On extensive form implementation of contracts in differential information economies

Summary. In the context of differential information economies, with and without free disposal, we consider the concepts of Radner equilibrium, rational expectations equilibrium, private core, weak fine core and weak fine value. We look into the possible implementation of these concepts as perfect Bayesian or sequential equilibria of noncooperative dynamic formulations. We construct relevant game trees which indicate the sequence of decisions and the information sets, and explain the rules for calculating ex ante expected payoffs. The possibility of implementing an allocation is related to whether or not it is incentive compatible. Implementation through an exogenous third party or an endogenous intermediary is also considered. Received: November 19, 2001; revised version: April 17, 2002 RID="*" ID="*" This paper comes out of a visit by Nicholas Yannelis to City University, London, in December 2000. We are grateful to Dr A. Hadjiprocopis for his invaluable help with the implementation of Latex in a Unix environment. We also thank Leon Koutsougeras and a referee for several, helpful comments. Correspondence to: N.C. Yannelis     

A common trends model of UK core inflation

In this paper the long-run trend in RPI inflation (core inflation) for the UK over the 1961–1997 period is estimated within the framework of a multivariate common trends model which extends the bivariate VAR approach of Quah and Vahey (1995). In this context core inflation is directly linked to money and wage growth and interpreted as the long-run forecast of inflation from a small-scale, cointegrated macroeconomic system. First version received: September 1999/Final version received: October 2001 RID="*" ID="*"  We thank two anonymous referees for many helpful comments and suggestions. Work on this paper was partially conducted when C. Morana was at Heriot-Watt University.     

Risk aversion in the Talmud

Summary. Evidence is adduced that the sages of the ancient Babylonian Talmud, as well as some of the medieval commentators thereon, were well aware of sophisticated concepts of modern theories of risk-bearing. Received: April 10, 2002; revised version: May 7, 2002 RID="*" ID="*"Presented at the Institute for Mathematical Studies in the Social Sciences-Economics, Stanford University, August 4, 1981. Subsequent to that presentation, the author's attention was drawn to an article by Zvi Ilani, “Models in the Economics of Uncertainty: The Cost of Concluding a Conditional Contract, according to the Talmud and the Halachic Literature,” Iyunim Bekalkala (Investigations in Economics), The Israel Association for Economics, Jerusalem, Nissan 5740 (April 1980), 246–261 (in Hebrew). Inter alia, Ilani treats the Talmudic passage that forms the subject of this paper, and provides a fairly comprehensive review of the medieval commentaries thereon; undoubtedly, he was the first to recognize in print the relevance of this passage to modern economic theories of uncertainty. It is not clear, though, whether or not his understanding of the passage agrees with ours. The current paper appeared in January 2002 in the Research Bulletin Series of the Research Center on Jewish Law and Economics, Department of Economics, Bar Ilan University.     

Non-cooperative games with a continuum of players whose payoffs depend on summary statistics

Summary. We consider static non-cooperative games with a continuum of small players whose payoffs depend on their own actions and finitely many summary statistics of the aggregate strategy profile. We prove the existence of an equilibrium in pure strategies without any convexity restrictions on payoffs or the common action space. We show that this result applies to a broad class of monopolistic competition models. Received: April 13, 2001; revised version: December 18, 2001 RID="*" ID="*" The result in this paper generalizes a result in my PhD dissertation supervised by M. Ali Khan and Joe Harrington. I thank them for support and encouragement. I also thank Sung Kim, Bruce Nanney, Ashvin Rajan, Kali Rath, and an anonymous referee for comments. The usual disclaimer applies.     

A generalized additive Tobit model

This paper proposes a semi-parametric approach to estimation in Tobit models. A generalized additive Tobit model of residential local long distance (intra-LATA) telephone demand is estimated on a cross-section of residential telephone consumers across twenty-eight states. While past studies of telecommunications demand have used fully parametric models, the model presented here is non-parametric in two dimensions: first no distributional assumption is made for the error distribution, and second, the demand equation is non-parametric with respect to price. We find that the elasticity of demand is substantially lower (in absolute value) that found in previous studies for a 40% cut in tariffs. First version received: July 2000/Final version received: March 2001 RID="*" ID="*"  I thank the referee and Associate Editor for suggestions which improved the paper. The views expressed here are of the author and not Analysis Group | Economics.     

Exact arbitrage and portfolio analysis in large asset markets

Summary. We provide a detailed portfolio analysis for a financial market with an atomless continuum of assets. In the context of an exact arbitrage pricing theory (EAPT), we go beyond the characterization of the existence of important portfolios (normalized riskless, mean, cost, factor and mean-variance efficient portfolios) to furnish exact portfolio compositions in terms of explicit portfolio weights. Such an analysis has not been furnished before in the context of the asymptotic arbitrage pricing theory (APT). We also characterize conditions under which a mean-variance efficient portfolio is a benchmark portfolio used in the EAPT to proxy essential risk. We illustrate our results with several examples of specific financial markets. Received: May 30, 2002; revised version: August 15, 2002 RID="*" ID="*"Some of the results reported here constituted part of Cowles Foundation Discussion Paper– No. 1139 circulated under the title “Hyperfinite Asset Pricing Theory”; additional results were obtained when Sun visited the Department of Economics at Johns Hopkins University during March 2002. This paper was presented at the Conference on Economic Design held at NYU on July 6–9, 2002 Correspondence to: M. A. Khan     

Similarity of endowments and the factor price equalization condition

Summary. We give a geometric interpretation of the lens condition, proposed by Deardorff as a shortcut for checking the factor price equalization (FPE) condition. We identify the conditions under which the lens condition implies the FPE condition. If the FPE zone is not a neighborhood of the diagonal allocations, however, the lens condition is irrelevant despite the implication since the FPE condition (hence the lens condition) is unlikely to be satisfied in that case. We give precise conditions under which the lens condition is equivalent to the FPE condition and simultaneously, the FPE zone is a neighborhood of the diagonal allocations. Received: January 2, 2002; revised version: July 1, 2002 RID="*" ID="*" The author would like to thank an anonymous referee, Bruce Dieffenbach and Michael Jerison for helpful comments.     

A class of fair distribution rules à la Rawls and Sen

Summary. This paper discusses and develops “non-welfaristic” arguments on distributive justice à la J. Rawls and A. K. Sen, and formalizes, in cooperative production economies, “non-welfaristic” distribution rules as game form types of resource allocation schemes. First, it conceptualizes Needs Principle which the distribution rule should satisfy if this takes individuals' needs into account. Second, one class of distribution rules which satisfy Needs Principle, a class of J-based Capability Maximum Rules, is proposed. Third, axiomatic characterizations of the class of J-based Capability Maximum Rules are provided. Received: July 30, 1999; revised version: March 11, 2002 RID="*" ID="*" We are grateful to an anonymous referee of this journal, Professors Marc Fleurbaey, Nicolas Gravel, Ryo-ichi Nagahisa, Prasanta Pattanaik, Kotaro Suzumura, Koich Tadenuma, and Yongsheng Xu for their fruitful comments. An earlier version of this paper was published with the title name, “A Game Form Approach to Theories of Distributive Justice: Formalizing Needs Principle” as the Discussion Paper No. 407 of the Institute of Social and Economic Research, Osaka University, and in the proceedings of the International Conference on Logic, Game, and Social Choice held at Oisterwijk in May 1999. That version was also presented at the 3rd Decentralization Conference in Japan held at Hitotsubashi University in September 1997, at the annual meeting of the Japan Association of Economics and Econometrics held at Waseda University in September 1997, and the 4th International Conference of Social Choice and Welfare held at University of British Colombia in July 1998. This research was partially supported by the Japanese Ministry of Education and the Ministry of Health and Welfare. Correspondence to: N. Yoshihara     

Bayesian estimation of social welfare and tax progressivity measures

This paper examines Bayesian methods of examining posterior distributions of inequality, concentration, tax progressivity and social welfare measures. Use is made of an explicit income distribution assumption and two alternative assumptions regarding the distribution of pre-tax mean incomes within each income group. The methods are applied to a simulated distribution of individual incomes and tax payments. It is possible to identify a minimum acceptable number of income classes to be used. The results suggest support for the use of group means in practical applications, particularly where large sample sizes are available. First version received: August 2000/Final version received: July 2001 RID="*" ID="*"  This research was supported by a Melbourne University Faculty of Economics and Commerce Research Grant. We should like to thank Bill Griffiths and two referees for comments on an earlier draft.     

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.     

Real exchange rate determination: Empirical observations from East-Asian countries

In this paper the models for the real exchange rate determination are re-examined between Japan and five East-Asian countries. Two important findings are reported. First, the real interest rate-bias model is valid for Korea-, Malaysia-, Indonesia-, and Philippines-Japan, and the productivity-bias model is valid for Indonesia-, and Philippines-Japan: that is, the coefficients of relative variables are stable and statistically significant. Second, there is no evidence that the political risk premium model is valid. First version received: September 2000/Final version received: April 2001 RID="*" ID="*"  This paper was presented at the Tohoku University Economics Conference April 1999. I acknowledge Yoshihiko Tsukuda, Hiroya Akiba, Tadashi Kuriyama, Jiro Akita and Hiroyuki Ozaki for their helpful comments. Also, I am very grateful to two referees of this journal for many valuable comments. The research was supported by the Nomura foundation for Social Science in 2000.     

On strategic complementarity conditions in Bertrand oligopoly

Summary. For Bertrand duopoly with linear costs, we establish via a single (counter-)example that: (i) A new monotone transformation of the firms' profit functions may lead to the supermodularity of transformed profits when the standard log and identity transformations both fail to do so, and (ii) Topkis's notion of critical sufficient condition for monotonicity of a Bertrand firm's best-reply correspondence cannot be extended to rely only on positive unit costs. Received: January 16, 2001; revised version: March 20, 2002 RID="*" ID="*" This work was completed while the first author was visiting the Institute for Industrial Economics at the University of Copenhagen during Spring 2000. Their financial support and stimulating research environment are gratefully acknowledged. The views expressed here are those of the authors and should not be attributed to the European Commission. Correspondence to: R. Amir     

Evolutionary economics and the counterfactual threat: on the nature and role of counterfactual history as an empirical tool in economics

Counterfactual conditional statements are ubiquitous in any scientific endeavour. This paper contains an analysis of the nature of counterfactual conditionals and the conditions under which they are considered assertable by scientists. The paper then applies this analysis to the use of counterfactuals in evolutionary economics, arguing that because evolutionary economics is inherently concerned with historical processes it cannot avoid the use of counterfactual history as one of its tools of empirical analysis. We discuss the strengths and pitfalls of counterfactual history. We argue that because evolutionary economics starts from the foundation that randomness may be inherent in any economic system, the very aspects of evolutionary economics that make counterfactual history a desirable empirical tool also make that tool difficult to employ. RID="*" ID="*" We thank the participants of the International Seminar on Evolutionary Economics as a Research Programme in Stockholm, May 1997, for many helpful comments. We also thank Lorri Baier for many helpful substantive and textual comments. Correspondence to: R. Cowan     

Existence of a competitive equilibrium in a one sector growth model with heterogeneous agents and irreversible investment

Summary. We prove existence of a competitive equilibrium in a version of a Ramsey (one sector) model in which agents are heterogeneous and gross investment is constrained to be non negative. We do so by converting the infinite-dimensional fixed point problem stated in terms of prices and commodities into a finite-dimensional Negishi problem involving individual weights in a social value function. This method allows us to obtain detailed results concerning the properties of competitive equilibria. Because of the simplicity of the techniques utilized our approach is amenable to be adapted by practitioners in analogous problems often studied in macroeconomics. Received: September 13, 2001; revised version: December 9, 2002 RID="*" ID="*" We are grateful to Tapan Mitra for pointing out errors as well as making very valuable suggestions. Thanks are due to Raouf Boucekkine and Jorge Duran for additional helpful discussions. We also thank an anonymous referee for his/her helpful comments. The second author acknowledges the financial support of the Belgian Ministry of Scientific Research (Grant ARC 99/04-235 “Growth and incentive design”) and of the Belgian Federal Goverment (Grant PAI P5/10, “Equilibrium theory and optimization for public policy and industry regulation”). Correspondence to: C. Le Van