Efficient allocation of scarce resources is one of the central themes in economics. The essays in this thesis approach two aspects of this problem. In the first essay I consider the problem of the efficient provision of a public good with congestion in a setting with asymmetric information. Mailath and Postlewaite (1990) and Rob (1989) show that in such a setting an efficient, budget balanced and individually rational mechanism for provision of pure public goods does not exist. Many public goods are not pure public goods, that is, they are excludable and to some extent congested. I show that when congestion is taken into account, in a wide class of economies it is possible to construct a mechanism that produces the public good efficiently, balances the budget ex-post and satisfies individual rationality constraints.It is known, see Krishna and Perry (2000), that a desired mechanism exists if and only if the Vickrey-Clarke-Groves (VCG) mechanism results in an expected budget surplus. In the VCG mechanism each consumer is levied a tax equal to the externality that she exerts on the rest of the society. When a good is pure public a consumer pays only if she is pivotal. When a public good is congested even a consumer that is not pivotal exerts an externality on the others. This externality is stronger when congestion is higher, hence the payment from such a consumer "increases with the level of congestion." On the other hand, when congestion is higher, fewer consumers are given access to the good, hence fewer consumers pay at all. Due to this trade off, it is too simplistic to say that the VCG mechanism runs an expected budget surplus if the good is congested enough. There are simple examples where the VCG mechanism runs an expected budget surplus with less congested good and a deficit with more congested good.I show that it is most likely that the VCG mechanism runs a surplus if the level of congestion is neither too high nor too low. I present a simple and intuitive condition on the parameters of the economy that guarantees that the VCG mechanism results in an expected budget surplus. As a result, in these economies, there also exists an efficient, incentive compatible, ex post budget balanced and individually rational mechanism.The second and third essays deal with the problem of efficient allocation of private goods. In the second essay, written jointly with Serguei Izmalkov, we study efficiency properties of a single-object irrevocable exit English auction in the setting with interdependent and asymmetric values. Maskin (1992) shows that the pairwise single-crossing condition is both necessary and sufficient for efficiency of the English auction with two bidders. Our paper extends both Maskin's result and the single-crossing condition to the English auction with N-bidders. The pairwise single-crossing imposes the following: if starting from a signal profile where the values of two bidders are equal and maximal we slightly increase the signal of one of the bidders, her value becomes the highest.We introduce the generalized single-crossing condition (GSC) a fairly intuitive extension of the pairwise single-crossing. GSC requires the following: if starting from a signal profile where the values of a group of bidders are equal and maximal we slightly increase the signals of a subset of the group, no bidder outside of the subset can attain the value higher than the maximal value attained among the bidders from the subset. Generalized single-crossing both implies the pairwise single-crossing and reduces to it in the case of two bidders. We show that GSC is both necessary and sufficient condition for existence of an efficient equilibrium in the English auction with N-bidders.In the third essay I deal with the setting of interdependent and asymmetric values and consider the class of auctions that possess an efficient ex-post equilibrium. An equilibrium is ex-post if none of the bidders wants to change his strategy after all the information that was private before the auction is revealed to her. Intuitively, this seems to be a significant restriction on the strategies, so that, hopefully, auctions that have an efficient ex-post equilibrium have few other ex-post equilibria. This intuition is, in fact, incorrect. I show that every auction that implements efficient allocation in ex-post equilibrium also has a continuum of inefficient ex-post equilibria in undominated strategies.
