We consider an infinitely repeated Bertrand game, in which prices are publicly observed and each firm receives a privately observed, i.i.d. cost shock in each period. We focus on symmetric perfect public equilibria (SPPE), wherein any "punishments" are borne equally by all firms. We identify a tradeoff that is associated with collusive pricing schemes in which the price to be charged by each firm is strictly increasing in its cost level: such "fully sorting" schemes offer efficiency benefits, as they ensure that the lowest-cost firm makes the current sale, but they also imply an informational cost (distorted pricing and/or equilibrium-path price wars), since a higher-cost ¯rm must be deterred from mimicking a lower-cost firm by charging a lower price. A rigid-pricing scheme, where a firm's collusive price is independent of its current cost position, sacrifices efficiency benefits but also diminishes the informational cost. For a wide range of settings, the optimal symmetric collusive scheme requires (i). the absence of equilibrium-path price wars and (ii). a rigid price. If firms are sufficiently impatient, however, the rigid-pricing scheme cannot be enforced, and the collusive price of lower-cost firms may be distorted downward, in order to diminish the incentive to cheat. When the

Abstract We study multi-unit auctions where the bidders have a budget constraint, a situation very common in practice that has received relatively little attention in the auction theory literature. Our main result is an impossibility: there are no incentive-compatible auctions that always produce a Pareto-optimal allocation. We also obtain some surprising positive results for certain special cases. JEL Classification Numbers: C70, D44, D82 Keywords: Multi-Unit Auctions, Budget Constraints, Pareto Optimality. * A preliminary version appeared in FOCS'08. We thank Peerapong Dhangwatnotai for pointing out an error in a previous version.

We explore the striking mathematical connections that exist between market scoring rules, cost function based prediction markets, and no-regret learning. We first show that any cost function based prediction market can be interpreted as an algorithm for the commonly studied problem of learning from expert advice by equating the set of outcomes on which bets are placed in the market with the set of experts in the learning setting, and equating trades made in the market with losses observed by the learning algorithm. If the loss of the market organizer is bounded, this bound can be used to derive an O ( √ T) regret bound for the corresponding learning algorithm. We then show that the class of markets with convex cost functions exactly corresponds to the class of Follow the Regularized Leader learning algorithms, with the choice of a cost function in the market corresponding to the choice of a regularizer in the learning problem. Finally, we show an equivalence between market scoring rules and prediction markets with convex cost functions. This implies both that any market scoring rule can be implemented as a cost function based market maker, and that market scoring rules can be interpreted naturally as Follow the Regularized Leader algorithms. These connections provide new insight into how it is that commonly studied markets, such as the Logarithmic Market Scoring Rule, can aggregate opinions into accurate estimates of the likelihood of future events.

The paper studies the implementation problem, first analyzed by Maskin and Moore (1999), in which two agents observe an unverifiable state of nature and may renegotiate inefficient outcomes following play of the mechanism. We develop a first-order approach to characterizing the set of implementable utility mappings in this problem, paralleling Mirrlees’s (1971) first-order analysis of standard mechanism design problems. We use this characterization to study optimal contracting in hold-up and risk-sharing models. In particular, we examine when the contracting parties can optimally restrict attention to simple contracts, such as noncontingent contracts and option contracts (where only one agent sends a message).

We study the optimal inflation tax in an economy with heterogeneous agents subject to nonlinear taxation of labor income. We find that the Friedman rule is Pareto efficient when combined with a nondecreasing labor income tax. In addition, the optimum for a Utilitarian social welfare function lies on this region of the Pareto frontier. The welfare costs from inflation are bounded below by the area under the demand curve. 1

We consider a dynamic Bertrand game, in which prices are publicly observed and each firm receives a privately observed cost shock in each period. Although cost shocks are independent across firms, within a firm costs follow a first-order Markov process. We analyze the set of collusive equilibria available to firms, emphasizing the best collusive scheme for the firms at the start of the game. In general, there is a tradeoff between productive efficiency, whereby the low-cost firm serves the market in a given period, and high prices. We show that when costs are perfectly correlated over time within a firm, if the distribution of costs is log concave and firms are sufficiently patient, then the optimal collusive scheme entails price rigidity: firms set the same price and share the market equally, regardless of their respective costs. Productive efficiency can be achieved in equilibrium under some circumstances, but such equilibria are not optimal. When serial correlation of costs is imperfect, partial productive efficiency is optimal. For the case of two cost types, first-best collusion is possible if the firms are patient relative to the persistence of cost shocks, but not otherwise. We present numerical examples of rfist-best collusive schemes.

We develop a general stochastic model of directed search on the job. Directed search allows us to focus on a Block Recursive Equilibrium (BRE) where agents ’ value functions, policy functions and market tightness do not depend on the distribution of workers over wages and unemployment. We formally prove existence of a BRE under various specifications of workers ’ preferences and contractual environments, including dynamic contracts and fixedwage contracts. Solving a BRE is as easy as solving a representative agent model, in contrast to the analytical and computational difficulties in models of random search on the job.

This paper studies experimentally how information about rivals’ types affects bidding behavior in first- and second-price auctions. The comparative static hypotheses associated with information about rivals enables us to test the relevance of such information as well as of the auction theory in general, by providing an effective means to control for risk aversion and other behavioral motives that were difficult to control for in previous experiments. Our experimental evidence provides strong support for the theory, and sheds light on the roles of risk aversion and the spite motive in first- and second-price auctions, respectively.

The paper studies bilateral contracting between one principal and N agents when each agent’s utility depends on the principal’s unobservable contracts with other agents. We show that allowing deviations to menu contracts from which the principal chooses bounds equilibrium outcomes in a wide class of bilateral contracting games without imposing ad hoc restrictions on the agents ’ beliefs. This bound yields, for example, competitive convergence as N →�in environments in which an appropriately-defined notion of competitive equilibrium exists. We also examine the additional restrictions arising in two common bilateral contracting games: the “offer game ” in which the principal makes simultaneous offers to the agents, and the “bidding game ” in which the agents make simultaneous offers to the principal.

This paper studies the effect of human capital depreciation and duration dependence on the design of an optimal unemployment insurance (UI) scheme. Our results partially confirm those obtained in most previous studies: benefits should decrease with unemployment duration. The optimal program also generates two main novel features, which are not present in stationary models. First, if human capital depreciates rapidly enough during unemployment, UI transfers are bounded below by a minimal “assistance” level that arises endogenously in the efficient program. Second, we study the optimality of imposing a history contingent wage tax after reemployment. Our numerical simulations based on the Spanish and US economies show that the wage tax should decrease with the length of worker’s previous unemployment spell, and become a wage subsidy for long-term unemployed workers. As a by-product of our study, we develop a systematic approach suitable for studying recursively a wide range of dynamic moral-hazard problems, and other models with similar characteristics.