When bidders have correlated private information, optimal auction mechanisms screen bidders based on their beliefs. We show that the possibility of ex post withdrawal by bidders provides a foundation for uniform price auctions that applies even in settings with correlated information. Withdrawal rights therefore provide a realistic framework for studying optimal auction design with correlated information. Our results are not driven by ex post individual rationality constraints or their interaction with the winner's curse. Instead, our analysis relies on a strategic impact of withdrawal rights: bidders can engage in "double deviations" that involve misreporting and then (sometimes) withdrawing ex post.

A monopolist wants to sell multiple goods, accounting for the possibility of returns. For any bundle purchased, a buyer can return any part of the bundle for a partial refund specified by a return policy. The seller is constrained to offer additive return policies: the total price of a bundle must be divided into return values for each of the goods in the bundle. We show that if the buyer's values are additive and independent across goods, then selling each good separately is optimal. The result applies even if the seller can use stochastic mechanisms, and holds for general multidimensional screening problems. We show that selling separately remains approximately optimal if the buyer experiences small return costs, as well as if the return policy only has to be close to additive or the buyer's values are weakly correlated. Applying our analysis to multiproduct pricing without returns, we introduce a new concept of bundle discounts and show that obtaining revenue gains from using any complex mechanism requires discounting bundles in our sense.

We analyze the implications of financial or other budget constraints in a model of matching with contracts. We assume that agents' preferences satisfy the net substitutability condition: i.e, if a price of a good increases, then minimizing the cost of obtaining a given level of utility would lead buyers (resp. sellers) to buy (resp. sell) more (resp. less) of other goods. The net substitutability condition coincides with gross substitutability if agents' preferences are quasilinear, but is strictly weaker otherwise. If agents have sufficient incomes for hard budget constraints not to bind, stable outcomes exist and coincide with competitive equilibrium outcomes. Otherwise, competitive equilibria can fail to exist, but stable outcomes exist and coincide with quasiequilibrium outcomes. Stable outcomes are at least weakly Pareto-efficient, but do not form a lattice and do not satisfy a Lone Wolf (or Rural Hospitals) Theorem. Our results suggest a new scope for sealed-bid auctions and matching with budget constraints.

We show that, with indivisible goods, the existence of competitive equilibrium fundamentally depends on agents' substitution effects, not their income effects. Our Equilibrium Existence Duality allows us to transport results on the existence of equilibrium from transferable utility economies to settings with income effects. One consequence is that net substitutability-which is a strictly weaker condition than gross substitutability-is sufficient for the existence of equilibrium. We also extend the "demand types" classification of valuations to settings with income effects and give necessary and sufficient conditions for a pattern of substitution effects to guarantee the existence of competitive equilibrium.

We show how frictions and continuous transfers jointly affect equilibria in a model of matching in trading networks. Our model incorporates distortionary frictions such as transaction taxes, bargaining costs, and incomplete markets. When contracts are fully substitutable for firms, competitive equilibria exist and coincide with outcomes that satisfy a cooperative stability property called trail stability. In the presence of frictions, competitive equilibria might be neither stable nor (constrained) Pareto-efficient. In the absence of frictions, on the other hand, competitive equilibria are stable and in the core, even if utility is imperfectly transferable.

This paper studies a model of large trading networks with bilateral contracts. Contracts capture exchange, production, and prices, as well as frictions such as complex taxes and the absence of transfers. In our setting, under standard continuity and convexity conditions, a stable outcome exists in any acyclic network, as long as all firms regard sales as substitutes and the market is large. Thus, complementarities between inputs do not preclude the existence of stable outcomes in large markets, unlike in discrete markets. Even when sales are not substitutable, tree stable outcomes exist in our setting.

Sönmez [2013] and Sönmez and Switzer [2013] used matching theory with unilaterally substitutable preferences [Hatfield and Kojima, 2010] to propose mechanisms to match cadets to military branches. Unilaterally substitutable priorities in general exhibit complementarities. In this paper, I construct the same mechanisms in matching markets with substitutable priorities. My results show that cadet-branch matching does not require weakened substitutes conditions. The cadet-branch matching market with substitutable branch priorities constructed in this paper can be often regarded as a labor market in the model of Kelso and Crawford [1982]. The length of a contract is interpreted as the inverse of a salary, and the branches' priorities can be taken to be quasi-linear. Therefore, the substitutable branch priorities are appealing from a theoretical perspective.