Approximate equilibria in markets with indivisible commodities

This paper presents results on the existence of approximate equilibria in large but finite exchange economies in which all the traders have non-convex consumption sets. As such, the paper gives a natural extension, to the finite economy, of Mas-Colell's results on existence with indivisible commodities and a continuum of agents. It also presents an approximate equilibrium theorem without any assumption of compactness or bounded non-convexity of preferences.     

Nonlinear perturbations of difference equations in population dynamics

A new method is proposed for the study of population dynamics in which the growth rate is impacted by population history, i.e., levels of one or more previous generations. In particular, nonlinear perturbations are incorporated into second-order difference equations, two fundamental time scales are assumed for the solution, and a differential system is generated to approximate the “slow” variation. Solutions of the approximating differential system are developed by computer simulation and the theory of nonlinear oscillations. The geometry of the phase planes describing the approximate “slow” variation is represented graphically and the interaction of fast and slow time scales analyzed. Relationships of the model to experimental data on small mammal populations are discussed. The experimental data was collected at the National Institute of Health under the direction of Dr. John B. Calhoun.     

Non-Additive Beliefs and Strategic Equilibria

This paper studies n-player games where players' beliefs about their opponents' behaviour are modelled as non-additive probabilities. The concept of an “equilibrium under uncertainty” which is introduced in this paper extends the equilibrium notion of Dow and Werlang (1994, J. Econom. Theory64, 305–324) to n-player games in strategic form. Existence of such an equilibrium is demonstrated under usual conditions. For low degrees of ambiguity, equilibria under uncertainty approximate Nash equilibria. At the other extreme, with a low degree of confidence, maximin equilibria appear. Finally, robustness against a lack of confidence may be viewed as a refinement for Nash equilibria. Journal of Economic Literature Classification Numbers: C72, D81.     

A note on a posterior approximation in a heteroscedastic model

This paper develops a method of obtaining approximate marginal posteriors for all parameters of interest for a heteroscedastic model. This method improves upon Box and Hill's (1974) method in suggesting a pure Bayesian estimator for a regression coefficient.     

Coping with Technological Change: The Role of Ability in Making Inequality so Persistent

This study provides an explanation to the evolution of wage inequality over the last 30 years and supports this explanation with evidence. A faster rate of technological progress introduces new unknown elements at the workplace. The need to cope with the unknown accentuates the role of ability and thus increases wage inequality within and between education groups. Inasmuch as education is an irreversible investment project the rise in within group inequality BOOSTS UP the rise of between group inequality. Guided by this theory we turn to the PSID for evidence. Using parents' education to approximate child's ability we show the following set of results: (a) Controlling for education of the child, parents' education contributed much more in the 1980s to his wage growth than in the 1970s. (b) The correlation between the parents' and the child's education increases from the 1970s to the 1980s. (c) The return to college education for an individual with no ability rents did not change—it remains steady at the 23 percent. (d) Facts (a)–(c) CANNOT be attributed to the impact of parent's income. It is parents' education and not parents' income that is more relevant for son's economic outcomes in the 1980s.     

Stochastic stability in a double auction

In a k-double auction, a buyer and a seller must simultaneously announce a bid and an ask price respectively. Exchange of the indivisible good takes place if and only if the bid is at least as high as the ask, the trading price being the bid price with probability k and the ask price with probability (1−k). We show that the stable equilibria of a complete information k-double approximate an asymmetric Nash bargaining solution with the seller's bargaining power decreasing in k.Note that ceteris paribus, the payoffs of the seller of the one-shot game increase in k. Nevertheless, as the stochastically stable equilibrium price decreases in k, choosing the seller's favorite price with a relatively higher probability in individual encounters makes him worse off in the long run.     

A decomposition algorithm for N-player games

An N-player game can be decomposed by adding a coordinator who interacts bilaterally with each player. The coordinator proposes profiles of strategies to the players, and his payoff is maximized when players’ optimal replies agree with his proposal. When the feasible set of proposals is finite, a solution of an associated linear complementarity problem yields an equilibrium of the approximate game and thus an approximate equilibrium of the original game. Computational efficiency is improved by using vertices of a triangulation of the players’ strategy space for the coordinator’s pure strategies. Computational experience is reported.     

A Model of Final Offer Arbitration in Regulation

I study a regulatory process in which both the regulator and the regulated firm propose prices that, in case of disagreement, are settled through final-offer arbitration (FOA)—a practice currently used in Chile for setting prices in the water sector. Rather than submitting a single offer, each party simultaneously submits an offer for each of the firms cost units (e.g., cost of raw water, capital cost). While a multiple-offers scheme allow the arbitrator to better approximate her ideal settlement, it may induce parties to submit widely divergent offers. This divergence, however, does not affect the arbitrators ability to learn from the offers.     

Dissertation Abstract: Decision-Theoretic Models of Market Equilibration

There has been a surprising dearth of decision-theoretic approaches to market equilibration within the extended Arrow-Debreu-McKenzie framework. While game theory is replete with learning and evolutionary models in which Nash equilibria are not necessarily the only stable states of a game, general equilibrium theorists have generally been content to accept the attainment of competitive equilibrium as a matter of faith. The first two chapters of this dissertation advance the study of behaviorally plausible models of market equilibration. First, we develop the epsilon-intelligent competitive equilibrium algorithm. According to this model, the actions of minimally sophisticated agents based on local information will lead an exchange economy to approximate competitive equilibrium in a larger set of economics than Walras’ tatonnement. The algorithm also supports a behavioral interpretation of Negishi’s existence proof of competitive equilibrium. The second chapter is an empirical analysis of laboratory markets designed to test the extent to which human behavior is consistent with the algorithm’s behavioral restrictions, and suggest alternative hypotheses. The chief finding is that while subjects sufficiently sophisticated to consistently secure competitive utility for themselves exist, the majority are satisficers who follow small modifications of a simple utility-improvement rule.While general equilibrium theory has lacked a behaviorally plausible foundation of price equilibration, behavioral economic models typically ignore general equilibrium implications. In the third chapter of this dissertation, I present the surprising result that loss aversion and diminishing sensitivity for gains imply a greater redistribution of endowments than when otherwise identical preferences are reference-independent. The result is surprising because loss aversion was developed in part to account for the status quo bias, whereby people tend to value a good more when it is in their possession than when it is not. One might reasonably suspect trade to be inhibited by this bias. The counter-intuitive result is driven by the fact that, given the axiomatization of loss aversion and diminishing sensitivity for gains, the acquisition of some quantity of a good increases one’s taste for that good, thus perpetuating a taste for more trade.     

On equilibrium existence in payoff secure games

We propose a single framework for studying the existence of approximate and exact pure strategy equilibria in payoff secure games. Central to the framework is the notion of a multivalued mapping with the local intersection property. By means of the Fan-Browder collective fixed point theorem, we first show an approximate equilibrium existence theorem that covers a number of known games. Then a short proof of Reny’s (Econometrica 67:1029–1056, 1999) equilibrium existence theorem is provided for payoff secure games with metrizable strategy spaces. We also give a simple proof of Reny’s theorem in its general form for metric games in an appendix for the sake of completeness.     

Welfare Measurement and Measurement Error

The approximate effects of measurement error on a variety of measures of inequality and poverty are derived. They are shown to depend on the measurement error variance and functionals of the error–contaminated income distribution, but not on the form of the measurement error distribution, and to be accurate within a rich class of error–free income distributions and measurement error distributions. The functionals of the error–contaminated income distribution that approximate the measurement error induced distortions can be estimated. So it is possible to investigate the sensitivity of welfare measures to alternative amounts of measurement error and, when an estimate of the measurement error variance is available, to calculate corrected welfare measures. The methods are illustrated in an application using Indonesian household expenditure data.     

On the relation between regular variation and the asymptotic elasticity of substitution

This paper characterizes a class of regularly varying production functions with an asymptotic elasticity of substitution equal to one. In particular, it is shown that these functions asymptotically approximate the Cobb–Douglas form. The results generalize and unify existing results in the literature.     

On the local convergence of economic mechanisms

For certain informationally decentralized economic models, it is known that the associated dynamical process need not be locally stable. In this paper, the causes of the conflict between the local stability of the process and the decentralization of information are identified. This motivates a resolution for this problem. The method describes how approximate, local information about the economy can be used to modify a decentralized, dynamical procedure to achieve local stability. Explicit procedures are given for constructing the decentralized systems with which the agents communicate this approximate information. As an illustration, this approach provides a partial response to the well-known convergence problems with the tâtonnement model.     

Interlinkage and Generous Tit‐for‐Tat Strategy

We investigate an infinitely‐repeated prisoners' dilemma with imperfect monitoring and consider the possibility that the interlinkage of the players' distinct activities enhances implicit collusion. We show a necessary and sufficient condition for the existence of a generous tit‐for‐tat Nash equilibrium. Such an equilibrium, if it exists, is unique. This equilibrium achieves approximate efficiency when monitoring is almost perfect, where the discount factors are fixed.     

Approximate efficiency in repeated games with correlated private signals

This paper presents repeated games with hidden moves, in which players receive imperfect private signals and are able to communicate. We propose a conditional probability approach to solve the learning problem in repeated games with correlated private signals and delayed communication. We then apply this approach to symmetric n-player games to obtain an approximate efficiency result.     

Methods for estimating optimal Dickson and Waters modification dividend barrier

In applications of collective risk theory, complete information for the distribution of individual claims amount is often unknown, but reliable estimates of its first few moments may be available. Dickson and Waters [Dickson, D.C.M. and Waters, H.R., (2004) Some optimal dividends problems, Astin Bulletin, 34, 49–74.] pointed out that shareholders should be liable to cover the deficit at ruin. Thus, they considered b the level of the barrier that maximizes the expectation of the difference between the discounted dividends until ruin and the discounted deficit at ruin. For such a situation, this paper develops methods for estimating the Dickson–Waters modification for the optimal dividend barrier b with the expectation of discounted penalty at ruin. In particular, two De Vylder approximations are explained, and the diffusion approximation for the expectation of discounted penalty at ruin is examined. For several claim amount distributions, the approximate values are compared numerically with exact values.     

Exact arbitrage, well-diversified portfolios and asset pricing in large markets

For a market with an atomless continuum of assets, we formulate the intuitive idea of a “well-diversified” portfolio, and present a notion of “exact arbitrage”, strictly weaker than the more conventional notion of “asymptotic arbitrage”, and necessary and sufficient for the validity of an APT pricing formula. Our formula involves “essential” risk, one based on a specific index portfolio constructed from factors and factor loadings that are endogenously extracted to satisfy an optimality property involving a finite number of factors. We illustrate how our results can be translated to markets with a large but finite number of assets.     

Non-myopic learning in differential information economies: the core

Summary. We study the process of learning in a differential information economy, with a continuum of states of nature that follow a Markov process. The economy extends over an infinite number of periods and we assume that the agents behave non-myopically, i.e., they discount the future. We adopt a new equilibrium concept, the non-myopic core. A realized agreement in each period generates information that changes the underlying structure in the economy. The results we obtain serve as an extension to the results in Koutsougeras and Yannelis (1999) in a setting where agents behave non-myopically. In particular, we examine the following two questions: 1) If we have a sequence of allocations that are in an approximate non-myopic core (we allow for bounded rationality), is it possible to find a subsequence that converges to a non-myopic core allocation in a limit full information economy? 2) Given a non-myopic core allocation in a limit full information economy can we find a sequence of approximate non-myopic core allocations that converges to that allocation? Received: May 25, 1999; revised version: August 9, 1999     

Incentives for motivated agents under an administrative constraint

Consider an agent who has an expertise in producing a non-marketable good. This good is valued by a single principal, and there is a verifiable measure of the agent’s performance. Crucially, the agent is intrinsically motivated due to ‘warm glow altruism’. In addition, the agent’s budget, which is controlled by the principal, must not be less than the monetary performance-cost faced by the agent. This gives rise to a limited-liability constraint. It also restricts the agent’s ability to under-report costs. In such environment, we determine the link between the agent’s budget and performance. Our results come in contrast to the received solution of the principal-agent problem and to most in the literature on mission-motivated organisations and public services provision.