Asymptotical behavior of finite and possible discontinuous economies

We consider a finite economy in which the data are depending on an exogenous parameter and the utilities satisfy a condition, previously introduced by the authors and called sequential pseudocontinuity, weaker than sequential lower semicontinuity and than sequential upper semicontinuity. We show that the economy has a nice asymptotical behavior, that is: for any sequence (x _n ) _n of exogenous parameters converging to x _o , any convergent subsequence of a sequence of approximate competitive equilibria of converges to an exact equilibrium of .      

The price of anarchy of serial, average and incremental cost sharing

We compute the price of anarchy (PoA) of three familiar demand games, i.e., the smallest ratio of the equilibrium to efficient surplus, over all convex preferences quasi-linear in money. For any convex cost, the PoA is at least in the average and serial games, where n is the number of users. It is zero in the incremental game for piecewise linear cost functions. With quadratic costs, the PoA of the serial game is , and for the average and incremental games. This generalizes if the marginal cost is convex or concave, and its elasticity is bounded.      

First-order risk aversion and non-differentiability

Summary First-order risk aversion happens when the risk premium a decision maker is willing to pay to avoid the lottery , is proportional, for smallt, tot. Equivalently, 0$$ " align="middle" border="0"> . We show that first-order risk aversion is equivalent to a certain non-differentiability of some of the local utility functions (Machina [7]).We are grateful to the Social Sciences and Humanities Research Council of Canada for financial support and to Kim Border, Larry Epstein, Mark Machina and Joe Ostroy for helpful discussions and suggestions.     

Stationary measures for some Markov chain models in ecology and economics

Summary. Let continuous, exists in for x in . Let be an i.i.d. sequence from F and X₀ be a nonnegative random variable independent of . Let be the Markov chain generated by the iteration of random maps by . Such Markov chains arise in population ecology and growth models in economics. This paper studies the existence of nondegenerate stationary measures for {X_n}. A set of necessary conditions and two sets of sufficient conditions are provided. There are some convergence results also. The present paper is a generalization of the work on random logistics maps by Athreya and Dai (2000).Received: 20 March 2002, Revised: 4 December 2002, JEL Classification Numbers: C22, D9.The author wishes to thank Professor Mukul Majumdar and the referees for several useful suggestions.     

The (S,s) Policy is an Optimal Trading Strategy in a Class of Commodity Price Speculation Problems

This paper introduces a model of commodity price speculation and proves that the optimal trading strategy is of the (S,s) form when a no expected loss condition holds. A strong form of this condition is that the retail price charged to consumers at time t exceeds the expected wholesale price of the commodity at time t+1, i.e. , where β ∈(0,1) is the speculator’s discount factor. We are extremely grateful to Herbert Scarf for pointing out an important error in a previous draft of this paper and for suggesting the key argument in a revised proof that fixed the problem. We also benefited from helpful feedback from an anonymous referee, William Brainard, Zvi Eckstein, participants of seminars at Yale, the Operations Research Center at MIT, and the Econometric Society Winter School at the Indian Statistical Institute, New Delhi.     

The 2001-3 electricity distribution price control review in the Netherlands: regulatory process and consumer welfare

The outcome of the first electricity distribution price control review in The Netherlands did not deliver the savings initially suggested by the regulator (DTe). During the course of the first 3-year regulatory period, DTe revised the X-Factors four times. The impact on tariffs has been substantial. DTe initially announced in 2000 that savings would be equivalent to 25% of electric distribution revenues ( 2bn). However, final X-Factors in May 2003 resulted in savings of 10% of revenues. The total cost to consumers—when compared to the most probable outcome—has been 140 mln (7% of total revenues).      

The nature of the steady state in models of optimal growth under uncertainty

Summary. We study a one-sector stochastic optimal growth model with a representative agent. Utility is logarithmic and the production function is of the Cobb-Douglas form with capital exponent . Production is affected by a multiplicative shock taking one of two values with positive probabilities p and 1-p. It is well known that for this economy, optimal paths converge to a unique steady state, which is an invariant distribution. We are concerned with properties of this distribution. By using the theory of Iterated Function Systems, we are able to characterize such a distribution in terms of singularity versus absolute continuity as parameters and p change. We establish mutual singularity of the invariant distributions as p varies between 0 and 1 whenever . More delicate is the case . Singularity with respect to Lebesgue measure also appears for values such that . For and Peres and Solomyak (1998) have shown that the distribution is a.e. absolutely continuous. Characterization of the invariant distribution in the remaining cases is still an open question. The entire analysis is summarized through a bifurcation diagram, drawn in terms of pairs .Received: 9 April 2002, Revised: 29 October 2002, JEL Classification Numbers: C61, O41.Correspondence to: Tapan MitraThis research was partially supported by CNR (Italy) under the "Short-term mobility" program and by M.U.R.S.T. (Italy) National Group on "Nonlinear Dynamics and Stochastic Models in Economics and Finance" . We are indebted to Rabi Bhattacharya for providing us with the reference to Solomyak's (1995) paper. The present version has benefitted from comments by Mukul Majumdar and two anonymous referees.     

More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model

Summary. This paper studies monotone risk aversion, the aversion to monotone, mean-preserving increase in risk (Quiggin [21]), in the Rank Dependent Expected Utility (RDEU) model. This model replaces expected utility by another functional, characterized by two functions, a utility function u in conjunction with a probability-perception function f. Monotone mean-preserving increases in risk are closely related to the notion of comparative dispersion introduced by Bickel and Lehmann [3,4] in Non-parametric Statistics. We present a characterization of the pairs (u,f) of monotone risk averse decision makers, based on an index of greediness G _u of the utility function u and an index of pessimism P _f of the probability perception function f: the decision maker is monotone risk averse if and only if . The index of greediness (non-concavity) of u is the supremum of taken over . The index of pessimism of f is the infimum of taken over 0 < v < 1. Thus, , with G _u = 1 iff u is concave. If then , i.e., f is majorized by the identity function. Since P _f = 1 for Expected Utility maximizers, forces u to be concave in this case; thus, the characterization of risk aversion as is a direct generalization from EU to RDEU. A novel element is that concavity of u is not necessary. In fact, u must be concave only if P _f = 1.Received: 10 April 2001, Revised: 18 November 2003, JEL Classification Numbers: D81. Correspondence to: Michéle CohenAlain Chateauneuf, Michéle Cohen, Isaac Meilijson: We are most grateful to Mark Machina, Peter Wakker and two anonymous referees for very helpful suggestions and comments.     

Optimal Control in Infinite Horizon Problems: A Sobolev Space Approach

In this paper, we make use of the Sobolev space to derive at once the Pontryagin conditions for the standard optimal growth model in continuous time, including a necessary and sufficient transversality condition. An application to the Ramsey model is given. We use an order ideal argument to solve the problem inherent to the fact that L ¹ spaces have natural positive cones with no interior points. The paper was written when Cuong Le Van was visiting CORE, Université catholique de Louvain, and Cagri Saglam was a research fellow of the Economics Department of the same university. The authors are indebted to an anonymous referee and Takashi Kamihigashi for very useful comments. R. Boucekkine acknowledges the support of the Belgian research programmes PAI P4/01 and ARC 03/08-302.     

Voting in assemblies of shareholders and incomplete markets

Summary. An economy with two dates is considered, one state at the first date and a finite number of states at the last date. Shareholders determine production plans by voting - one share, one vote - and at -majority stable stock market equilibria, alternative production plans are supported by at most percent of the shareholders. It is shown that a -majority stable stock market equilibrium exists if where S is the number of states at the last date and J is the number of firms. Moreover, an example shows that -majority stable stock market equilibria need not exist for smaller s.Received: 23 December 2002, Revised: 14 June 2004, JEL Classification Numbers: D21, D52, D71, G39. Correspondence to: Hervé CrésThe authors are grateful to an anonymous referee for helpful comments and suggestions. Financial support from the Danish Research Councils and hospitality of HEC is gratefully acknowledged by Mich Tvede and support from Fondation HEC is gratefully acknowledged by Hervé Crés.     

Existence of Sparsely Supported Correlated Equilibria

We show that every N-player K ₁ × ... × K _N game possesses a correlated equilibrium with at least zero entries. In particular, the largest N-player K × ... × K games with unique fully supported correlated equilibrium are two-player games. We thank an anonymous referee for most useful comments. The first author acknowledges financial support from Spanish Ministry of Science and Technology, grant SEJ2004-03619, and in form of a Ramón y Cajal fellowship. The second author acknowledges support by the PASCAL Network of Excellence under EC grant no.506778, as well as from Spanish Ministry of Science and Technology and FEDER, grant BMF2003-03324. Both authors also acknowledge financial support from BBVA grant “Aprender a jugar.”     

Random continued fractions: a Markov chain approach

Summary. By allowing the numbers appearing in a continued fraction to be random, one gets what are called random continued fractions. Under fairly general conditions, including the case when the random variables are i.i.d. non-negative, random continued fractions converge with probability one. A markovian algorithm seems to play a crucial role in studying the distribution of random continued fractions. This Markov Chain on is generated by iteration of random monotone decreasing maps on S and the connection comes from the fact that the distribution of random continued fraction is obtained as an invariant probability of the Markov Chain. Using the splitting condition, it is shown that the distribution of the Markov Chain converges exponentially fast in the Kolmogorov distance to an unique invariant probability , which is shown to be non-atomic, except in the degenerate case. A sufficient condition is given for the invariant probability to have full support S. In some special cases, the invaraint probability is obtained explicitly and this includes one case when the probability turns out to be a singular non-atomic probability with full support S. Extensions of some results to higher dimensions are also discussed.Received: 22 May 2002, Revised: 9 September 2002, Subject Classification Numbers: 60F05, 60J05.Alok Goswami: I would like to thank Rabi Bhattacharya for useful discussions at the time of writing this paper. Thanks are also due to the referee for offering various suggestionsfor improvement over a previous draft of the paper.     

On the hellwig method of feasible portfolio construction

The work feasible portfolio is built into the work, that is, the k-dimensional Q column vector with components q_i where q_i 0 for i=1,...,k and q₁+...+q_k=1. We define i=1,...,k in the following way:

The extent of the market and optimal specialization

Summary. In this paper I examine how the socially optimal allocation, and specialization in particular, depends on the extent of the market. I interpret the societys ability to keep transaction records as the extent of the market and measure it by a probability with which the society can update agents past transactions into the public record. The society uses this record to detect potential defections from the optimal allocation and to punish the defectors with autarky. I show that when is small, increasing increases optimal specialization. However, when is close to 1, increasing further has no effect on the optimal allocation. I also show that optimal specialization is gradual over time when there is cost to reduce specialization. Even for small , the process converges to the unconstrained optimum that would occur under .JEL Classification Numbers: D60, C78.An earlier version of this paper was presented at the Society for Economic Dynamics meetings in Stockholm (2001). I thank a referee and the editor for useful comments and suggestions. For financial support, I gratefully acknowledge the Social Sciences and Humanities Research Council of Canada, and the Bank of Canada Fellowship. The opinion expressed here is my own and does not reflect the view of the Bank of Canada.     

An axiomatic approach to \boldsymbol{\varepsilon}-contamination

Summary.  Suppose that an economic agent is 100% certain that uncertainty she faces is characterized by a particular probability measure, but that she has a fear that, with 100% chance, her conviction is completely wrong and she is left perfectly ignorant about the true measure in the present as well as in the future. This situation is often called -contamination of confidence. The purpose of this paper is to provide a simple set of behavioral axioms under which the decision-makers preference is represented by the Choquet expected utility with the -contamination of confidence.Received: 25 November 2002, Revised: 15 November 2004, JEL Classification Numbers:   D81. Correspondence to: Hiroyuki OzakiWe are grateful to an anonymous referee. The referees comments greatly improved the exposition of the paper. The work reported here is partially supported by a grant from the Economic and Social Research Insitute, the Cabinet Office, the Government of Japan.     

Knowledge-independent Invariance Rules for Copula Duration in Dungflies

In dung flies, copula duration decreases, and the proportional rate of sperm transfer increases, as male body size becomes larger. From a marginal value approach to optimal copula duration, we show that these relationships result in the product, , remaining approximately constant across the range of male body size. The expected proportion of a female's eggs fertilized by a copulating male, equivalent to 1 – e^–c·t , is likewise invariant with male body size. (Overbars and refer to the averages over female sizes). We assume that the information or cues a male can perceive about females forms a set of discrete recognition categories, each of which is uniquely recognizable by a male, but within which he cannot discriminate. There are then likely to be different male optima for the product between categories. But the invariance rules still hold within categories, independently of exactly what the recognition categories are, provided that all males perceive the same categories. For example, suppose that males of all sizes categorise females as either 'large', 'medium', or 'small'. Then though the optimal male strategy (product ) for (say) 'large' females may differ from the corresponding optima for the other two categories, it remains constant with male size across all the 'large' females. Further, the product should remain constant for all male sizes if we take the average across all females, or across any subset of recognition categories. We believe that these conclusions have general applicability and implications for optimal foraging under the marginal value theorem, and demonstrates how we can sometimes make predictions (e.g. the relation between copula duration and male body size in dungflies) without determining exactly what a forager 'knows'.     

The asymptotic shapley value for a simple market game

We consider the game in which b buyers each seek to purchase 1 unit of an indivisible good from s sellers, each of whom has k units to sell. The good is worth 0 to each seller and 1 to each buyer. Using the central limit theorem, and implicitly convergence to tied down Brownian motion, we find a closed form solution for the limiting Shapley value as s and b increase without bound. This asymptotic value depends upon the seller size k, the limiting ratio b/ks of buyers to items for sale, and the limiting ratio of the excess supply relative to the square root of the number of market participants. This work was sponsered in part by NSF Grant DMS-03-01795.     

Infinite inequality systems and cardinal revelations

Summary. Many economics problems are maximization or minimization problems, and can be formalized as problems of solving linear difference systems of the form and r _k -r _l > c _kl , for r-unknowns, with given c-constants. They typically involve strict as well as weak inequalities, with infinitely many inequalities and unknowns. Since strict inequalities are not preserved under passage to the limit, infinite systems with strict inequalities are notoriously hard to solve. We introduce a unifying tool for solving them. Our main result (Theorem 1 for the countable case, Theorem [2] for the not-necessarily-countable case) introduces a uniform solvability criterion (the -Axiom), and our proof yields a method for solving those that are solvable. The axioms economic intuition extends the traditional ordinal notion of revealed preference to a cardinal notion. We give applications in producer theory, consumer theory, implementation theory, and constrained maximization theory.Received: 21 May 2003, Revised: 29 October 2004, JEL Classification Numbers: C69, D21, D11, D82. Correspondence to: Marcel K. RichterWe thank Professor Leonid Hurwicz for helpful discussions, and a referee for helpful comments.     

Stationary Markov equilibria for overlapping generations

Summary.  At a stationary Markov equilibrium of a Markovian economy of overlapping generations, prices at a date-event are determined by the realization of the shock, the distribution of wealth and, with production, the stock of capital. Stationary Markov equilibria may not exist; this is the case with intra-generational heterogeneity and multiple commodities or long life spans. Generalized Markov equilibria exist if prices are allowed to vary also with the realization of the shock, prices and the allocation of consumption and production at the predecessor date-event. (Stationary) Markov -equilibria always exist; as allocations and prices converge to equilibrium prices and allocations that, however, need not be stationary.Received: 2 March 2004, Revised: 2 April 2004, JEL Classification Numbers:   D50, D52, D60, D80, D90.Correspondence to: Felix KublerWe thank participants in seminars in Athens and Lund, at Penn, at IMPA and at Stanford, the 2002 CEME (NBER) General Equilibrium Conference and the 2002 SED meetings, and especially Martin Hellwig, George Mailath and an anonymous referee for very helpful comments.