We consider an auction in which k identical objects of unknown value are auctioned off to n bidders. The k highest bidders get an object and pay the k + 1st bid. Bidders receive a signal that provides information about the value of the object. We characterize the unique symmetric equilibrium of this auction. We then consider a sequence of auctions A r with n r bidders and k r objects. We show that price converges in probability to the true value of the object if and only if both k r → ∞ and n r - k r → ∞, i.e., both the number of objects and the number of bidders who do not receive an object go to infinity.

THE REQUIREMENT OF NO-ENVY is at the heart of recent equity theory. An allocation is free from envy if no agent strictly prefers the bundle of goods which is assigned to another agent to the one she/he gets. An allocation rule satisfies No-Envy if it only selects envy-free allocations. In this paper, we examine the relationship between No-Envy and implementability in a general model. Our main result is that in monotonically closed domains the No-Envy property is satisfied by any allocation rule which is both horizontally equitable and Nash Implementable. The requirement of horizontal equity, called Equal Treatment of Equals, simply states that two agents having the same preferences should be treated equally, i.e., should be assigned the same welfare level. The monotonic closedness condition on the domain of admissible preferences is satisfied in many private and/or public good environments, as discussed below. Quasi-linear domains, however, are examples of nonmonotonically closed domains. Our result confirms the widespread intuition that the No-Envy requirement is justified not only from an equity point of view but also from an implementation standpoint (see Hammond (1979) and Champsaur and Laroque (1981)). Moreover, it throws some light on several previous results where specific allocation rules defined over monotonically closed domains are characterized by Nash Implementability among other axioms. As a consequence of our analysis, No-Envy can be weakened into Equal Treatment of Equals in these characterizations (see, e.g., Thomson (1990) and Nagahisa and Suh (1995)). Similar arguments apply to decentralization problems where informational efficiency is the primary concern. For instance, Calsamiglia and Kirman (1993) characterized the Equal Income Walrasian rule on the basis of informational efficiency, Pareto Optimality, and No-Envy. Again, No-Envy can be replaced by Equal Treatment of Equals in this result.2 On the other hand, our result also explains why Nash Implementable allocation rules violating No-Envy over monotonically closed domains all fail to satisfy Equal Treatment of Equals. Examples include the Lindahl solution, the ratio equilibrium solution (Kaneko (1977)) and the balanced linear cost share solution (Mas-Colell and Silvestre (1989)); see Corchon (1989) and Wilkie (1990). At the end of the paper, we show that if we restrict ourselves to allocation functions (that is, allocation rules selecting one and only one allocation per economy), then a similar result holds for Strategy-Proofness, provided the Satterthwaite-Sonnenschein (1981) property of Non-Bossiness is also imposed. That is, in monotonically closed

WE WARN OF A CLASS of problems that can occur when inverting confluent characteristic functions (CF's). The term confluence is often used in Mathematics in connection with analysis and/or dynamic (difference, differential, and integral) equations; for example, see the classic text by Whittaker and Watson (1927). A confluence (of singularities) is a joint degeneracy that occurs within a function; here, the CF. When one is dealing with the CF of a k-dimensional variate where k > 1, these joint degeneracies can distort the derivation of the marginal density of some lower-dimensional combination of the k components. The distortions are both analytical and numerical. In this note, we first express the distributional problem in the simplest bivariate case, then clarify it with examples from a simple autoregressive (AR) model. Let R, S be two continuous (for simplicity) variates based on a sample of n observations, with joint CF pn(u,v) =E[euR+ivS], i = , and Pr{S > 0} = 1. The joint density h,jr, s) of R and S is expressed by means of the inversion formula as

We derive explicitly the exact density functions of two key mixed normal variates that arise from cointegration analysis. We also plot them, and analyze their analytical features and implications.

The authors examine core convergence for economies with a large finite number of agents and an infinite number of commodities. They find a serious disconnection between economies with a large finite number of agents and economies with a continuum of agents: the authors provide examples of nonconvergence of the core for large finite economies in L[superscript 1], a commodity space for which core equivalence holds for continuum economies. In addition, they show that, if preferences exhibit uniformly vanishing marginal utility of consumption at infinity, core convergence is restored.

The authors consider a family of rank tests based on the regression rank score process introduced by C. Gutenbrunner and J. Jureckova (1992) to test the unit root hypothesis in economic time series. In contrast to tests based on least-squares methods, the rank tests are asymptotically Gaussian under the null hypothesis, and have excellent power--particularly under innovation exhibiting heavy tails. These regression rank scores arise as a vector of solutions of the dual form of the linear program required to compute the regression quantile statistics of R. W. Koenker and G. Bassett (1978). For location model, they are simple ranks of the sample observations.

In a nontransferable utility exchange economy with a continuum of agents, the Mas-Collel bargaining set coincides with the set of Walrasian equilibria. In this paper, we show that the Mas-Colell bargaining set, as well as a smaller bargaining set due to Zhou, may fail to converge to competitive outcomes in large finite NTU exchange economies.