A sufficient condition for market demand to satisfy the Law of Demand is that the mean of all households' income effect matrices be positive definite. We show how this mean income effect matrix can be estimated from cross section data under metonymy, an assumption about the distribution of households' characteristics. The estimation procedure uses the nonparametric method of average derivatives. Income effect matrices estimated this way from U.K. family expenditure data are in fact positive definite. This result can be explained by a special form of heteroskedasticity in the data: households' demands are more dispersed at higher income levels.
This paper develops a decision-theoretic approach to normal-form refinements of Nash equilibrium and provides characterizations of (normal-form) perfect equilibrium and proper equilibrium. The approach relies on a theory of single-person decision-making that is a non-Archimedean version of subjective expected utility theory. Copyright 1991 by The Econometric Society.
IN A NOW CLASSICAL PAPER, Nash (1953) studied the following bilateral bargaining game: The two parties state their demands simultaneously. If these are compatible with a feasible agreement, each party gets the utility corresponding to his demand. Otherwise both parties get their conflict payoffs. Under quite general conditions this game has a continuum of equilibria: Any possible agreement which is both Pareto optimal and individually rational corresponds to a particular equilibrium point. Recently, the literature on sealed-bid double auctions (see, e.g., Leininger et al. (1989) and Matthews and Postlewaite (1989)) has extended Nash's model to the case where each party has incomplete information about the other party's valuation. A common feature of these models is that they lead to an even larger set of equilibria and, thus, to an aggravation of the nonuniqueness problem. In the present paper, we will show that this problem all but disappears if a different kind of uncertainty is introduced into Nash's model, viz. if one assumes that the parties make errors in choosing their actions in the bargaining process. This modification will be seen to imply the existence of an equilibrium which Pareto-dominates all other equilibria. The rationale for our assumption lies in the implausibly precise coordination needed to induce equilibrium play in Nash's original model: each (nontrivial) equilibrium consists of a pair of demands that are just compatible. Even an arbitrarily small deviation (in the wrong direction) will reduce both players to their conflict payoffs. By adding error terms to tne bids, we get rid of this discontinuity and force the players to weigh their demands against the risk of breakdown. Thus, our assumption could be seen as a way of accounting for the strategic uncertainty which seems practically unavoidable in a game where strategies are continuously variable. More fundamentally, the errors may be thought to reflect the presence of some uncertainty about the exact values of relevant parameters. Formally, the errors will be modeled by letting a player's bid result from the addition of a stochastic term to his strategy. The rules of Nash's game will also be modified by allowing a surplus to be divided between the players. While in Nash's model the players get precisely what they have demanded even when demands are more than compatible, we make the more general assumption that some fraction (ranging between zero and one) of the unclaimed surplus is divided between the parties according to a surplus partition rule. A similar assumption is made in the above-mentioned incomplete information models, but the severe indeterminacy makes it impossible to assess its influence. The above described equilibrium properties of our model hold for errors of arbitrary magnitude. Naturally, it is particularly interesting to study the properties when errors go to zero. In the special case where no surplus is divided, we find a convergence to the
A Bayesian analysis of the linear regression model with only parts of the prior distribution specified or a robust Bayesian analysis lead to sets of posterior distributions. E. E. Leamer (1978) describes the region of posterior means for conjugate priors and varying prior covariance matrices. As an extension to Bayesian confidence sets (HPD regions) the authors introduce the concept of HiFi (high fiduciary) regions. The Hifi region is a union of HPD regions and is a tool for describing the dependence of the posterior distribution on the prior covariance. The authors assume that the prior covariance matrix varies in an interval of matrices. Copyright 1991 by The Econometric Society.
A celebrated result of Black (1948a) demonstrates the existence of a simple-majority winner when preferences are single-peaked. The social choice follows the preferences of the median voter: the median voter's most-preferred outcome beats any alternative. However, this conclusion does not extend to elections in which candidates differ in more than one dimension. This paper provides a multi-dimensional analog of the median voter result. We provide conditions under which the mean voter's most preferred outcome is unbeatable according to a 64%-majority rule. The conditions supporting this result represent a significant generalization of Caplin and Nalebuff (1988). The proof of our mean voter result uses a mathematical aggregation theorem due to Prekopa (1971, 1973) and Borell (1975). This theorem has broad applications in economics. An application to the distribution of income is described at the end of this paper; results on imperfect competition are presented in the companion paper, Caplin and Nalebuff (1991).
Evolutionary games are introduced as models for repeated anonymous strategic interaction.The basic idea is that actions (or behaviors) which are more "fit," given the current distribution of behaviors, tend over time to displace less fit behaviors.Simple numerical examples motivate the key concepts of fitness function and compatible dynamics, and illustrate the relation to previous biological models.Cone fields are introduced to characterize the continuous-time dynamical processes compatible with a given fitness function.The analysis focuses on dynamic steady state equilibria and their relation to the static equilibria known as NE (Nash equilibrium) and ESS (evolutionary stable state).For large classes of dynamics it is shown that all stable dynamic steady states are NE and that all NE are dynamic steady states.The biologists' ESS condition is less closely related to the dynamic equilibria.The paper concludes with a brief survey of economic applications.