Quasi-linear Preferences Research Articles

Implementation with Secrets

This paper studies implementation in settings where agents take strategic actions that influence preferences. We show that such settings can arise in entry auctions for markets, and that the Vickery-Clarke-Groves Mechanism is not guaranteed to be truthful because of strategic actions. We thus pursue a new approach in this paper: (i) we formalize so-called strategic private information, (ii) we characterize social choice functions that can be implemented in a way that is robust to strategic private information, and (iii) we propose a mechanism to do so. The model allows agents to have multi-dimensional types and quasi-linear preferences. We apply these results to identify social choice functions that can and cannot be implemented in entry auctions for Cournot competitions.

Consumer surplus bias and the welfare effects of price discrimination

A well-known result with important policy implications is that an output increase is a necessary condition for social welfare to increase with third-degree price discrimination. In this paper, we explore the robustness of this result to the introduction of an assumption that is different than the conventional approach, namely preferences not being quasilinear. We show that in the presence of income differences among consumers, the aggregate utility of consumers may increase with price discrimination while total output remains constant. This result questions the general policy recommendation that third-degree price discrimination should be disapproved because it reduces welfare unless output increases. Our result highlights the crucial role of the assumption of quasilinear preferences in standard welfare calculations. In the presence of income differences, consumer surplus may be a biased welfare measure, thus potentially leading to incorrect conclusions when assessing the impact of specific policies.

A General Revealed Preference Test for Quasi-Linear Preferences: Theory and Experiments

We provide a generalized revealed preference test for quasi-linear preferences. The test applies to nonlinear budget sets and nonconvex preferences and can be used in contexts such as taxation and nonlinear pricing. We use this test to evaluate the prevalence of quasi-linear preferences in a laboratory experiment on consumption. We find that half of the subjects who satisfy the Generalized Axiom of Revealed Preferences also satisfy the criteria for quasi-linear preferences.

Incentive Contracts and Efficiency in a Frictional Market

Principals seek to trade with agents by posting incentive contracts in a search environment. A contract solves the ex ante search problem, and adverse selection and moral hazard ex post. We fully characterise the equilibrium for quasi linear preferences, and derive some comparative statics. If using appropriate transfers the equilibrium allocation is constrained welfare optimal, in contrast to the one-to-one principal-agent problem. Search frictions thus correct that inefficiency because searching requires internalizing the utility of agents. Incentives are weaker than in bilateral contracting, and agents enjoy more efficient risk sharing. With a constraint on transfers search and moral hazard interact and may induce an inefficient allocation; principal competition results in over-insurance of the agents and too little effort in equilibrium.

On the manipulability of efficient exchange rules

We identify a large subdomain, D, of quasilinear economies on which any efficient exchange rule will be generically (in the Baire sense) manipulable. For generic economies outside of D, we find rules that are locally non-manipulable. The interior of the set D consists of all economies in which competitive equilibrium would prescribe that all agents consume a positive quantity of money. Since we study quasilinear preferences, this is the domain of primary interest. Our locally non-manipulable rules rely on the existence of traders who are willing to sell all of their cash and absorb the imbalances in the trading of the commodity.

Non-manipulability of uniform price auctions with a large number of objects

When agents (bidders) have multi-demand preferences, uniform price auctions are generally not immune to agents’ strategic manipulation, and they may achieve an inefficient allocation. We consider economies in which a large number of identical objects have to be allocated. Agents have quasi-linear preferences with non-increasing incremental valuations. We explore the incentives of agents in uniform price auctions. An important assumption on preferences is proposed, called “no monopoly.” It requires that preferences should be correlated in such a way that no agent’s incremental valuation for an additional object when he receives sufficiently many objects is higher than those of the other agents. We show that under no monopoly and other mild assumptions on preferences, as the number of objects goes to infinity, the payment in any uniform price auction converges to that in a Vickrey auction. We deduce that when there are sufficiently many objects, truth-telling is an approximate Bayesian Nash equilibrium in each uniform price auction.

Revisiting the foundations of dominant-strategy mechanisms

An important question in mechanism design is whether there is any theoretical foundation for the use of dominant-strategy mechanisms. This paper studies the maxmin and Bayesian foundations of dominant-strategy mechanisms in general social choice environments with quasi-linear preferences and private values. We propose a condition called the uniform shortest-path tree that, under regularity, ensures the foundations of dominant-strategy mechanisms. This exposes the underlying logic of the existence of such foundations in the single-unit auction setting, and extends the argument to cases where it was hitherto unknown. To prove this result, we adopt the linear programming approach to mechanism design. In settings in which the uniform shortest-path tree condition is violated, maxmin/Bayesian foundations might not exist. We illustrate this by two examples: bilateral trade with ex ante unidentified traders and auction with type-dependent outside option.

A note on Ramsey pricing and the structure of preferences

We represent quasi-linear preferences by the dual measure of consumer surplus, and investigate demand and the associated optimal pricing. In particular, we discuss substitutability with respect to the outside commodity, deriving a Slutsky-like decomposition of the price effects. We use our results to show that commodities with larger outside substitutability have smaller optimal Lerner indexes, and that Ramsey prices are always proportional to marginal costs only if preferences are fully homothetic.

Implementing Pareto Optimal and Individually Rational Outcomes by Veto

We introduce a simple veto mechanism where each agent can veto any subset of alternatives, by paying a veto cost for each vetoed alternative. The outcome is the set of non-vetoed alternatives or, if this set is empty, some previously fixed alternative which is declared the disagreement outcome. Under fairly mild axioms to extend individual preferences over alternatives to sets of alternatives and assuming quasi-linear preferences over outcome-money bundles, we show that the Nash equilibrium outcomes of the veto mechanism coincide with the Pareto optimal outcomes which are individually rational according to the disagreement outcome. The positive result prevails when individual preferences admit indifferences and even for the case of two agents. We also show that under stronger axioms to extend preferences over alternatives to sets, strong Nash implementation (hence double implementation) is also possible with the same veto mechanism.

Dynamic relational contracts under complete information

This paper considers a long-term relationship between two agents who both undertake an action or investment that produces a joint benefit. Agents have an opportunity to expropriate some of the joint benefit for their own use. Agents have quasi-linear preferences. Two cases are considered: where agents are risk averse but where limited liability constraints do not bind, and where agents are risk neutral and subject to limited liability constraints. We ask how to structure the investments and division of the surplus over time to avoid expropriation. In the risk-averse case, the dynamics of actions and surplus may or may not be monotonic depending on whether or not a first-best allocation can be sustained. Agents may underinvest but never overinvest. If the first-best allocation is not sustainable, there is a trade-off between risk sharing and surplus maximization; surplus may not be at its constrained maximum even in the long run and the “amnesia” property of pure risk-sharing models fails to hold. In contrast, in the risk-neutral case there may be an initial phase in which one agent overinvests and the other underinvests. Both actions and surplus converge monotonically to a stationary state, where surplus is maximized subject to the self-enforcing constraints.

Multiproduct-Firm Oligopoly: An Aggregative Games Approach

This paper introduces a new class of demand systems to study oligopolistic pricing games with multiproduct rms. We focus on the class of demand systems that can be derived from discrete/continuous choice with iid type 1 extreme-value taste shocks. We also show that these demand systems are integrable with quasi-linear preferences. The pricing game is aggregative and payo functions are uni-modal, although not necessarily quasi-concave. Firms’ tting-in and best-response functions can be entirely summarized by a uni-dimensional sucient statistic, called the iota-markup. This allows us to show that, under fairly weak conditions, the pricing game has a Nash equilibrium. Under stronger conditions, this equilibrium is unique. We also provide an algorithm which exploits the aggregative nature of the game to compute the pricing equilibrium with multiproduct rms and CES demands. The algorithm always converges. As an application, we derive a number of results on the dynamic optimality of myopic merger policy under dierentiated Bertrand competition.

On the implications of declining population growth for regional migration

Although many advanced economies nowadays experience decreasing populations, migration in models of economic growth has so far been almost exclusively analyzed for the case of non-negative population growth rates. This paper considers decreasing and possibly negative population growth rates in two two-sector growth models. As long as preferences are homothetic, neither a decrease in population growth rates nor an actual population decline does induce migration in either direction. Introducing quasi-linear preferences implies that a decline in population growth leads to migration from the rural to the industrial region. A complete depopulation of the rural region takes place if the population growth rate falls short of minus the rate of physical capital depreciation. These results reinforce pessimistic expectations about a rural exodus.

Characterizing incentive compatible, Pareto optimal and sufficiently anonymous constrained combinatorial mechanisms – Two players case

We characterize the possibility space of deterministic, dominant-strategy incentive compatible, individually rational, and Pareto-optimal combinatorial auctions in a model with two players and two nonidentical items. Our model has multidi- mensional types, private values, quasilinear preferences for the players with one relaxation - one of the players is subject to a publicly-known budget constraint. We show that the space includes two types of mechanisms: VCG and dictatorial mechanisms. Furthermore when it is publicly known that the budgeted player is not constrained by his budget, VCG uniquely fulfills the basic properties of deterministic, dominant-strategy incentive compatible, individually rational, and Pareto-optimal. When it is pub- licly known that the budgeted player is constrained on all bundles then only a dictatorial solution will fulfill the above properties. Moreover when it is publicly known that the budgeted player is constrained on the largest bundle there are preferences under which the VCG mechanism uniquely fulfills these properties.

Monotonicity and revenue equivalence domains by monotonic transformations in differences

In a mechanism design setting with quasilinear preferences, a domain D of admissible valuations of an agent is called a monotonicity domain if every 2-cycle monotone allocation rule is truthfully implementable (in dominant strategies). D is called a revenue equivalence domain if every implementable allocation rule satisfies revenue equivalence. Carbajal and Müller (2015) introduced the notions of monotonic transformations in differences and showed that if D admits these transformations then it is a revenue equivalence and monotonicity domain. Here, we show that various economic domains, with countable or uncountable allocation sets, admit monotonic transformations in differences. Our applications include public and private supply of divisible public goods, multi-unit auction-like environments with increasing valuations, and allocation problems with externalities. Single-peaked domains admit only a modified version of monotonic transformations in differences. We show that this property implies too that single-peaked domains are revenue and monotonicity domains.

Robustness and Separation in Multidimensional Screening

A principal wishes to screen an agent along several dimensions of private information simultaneously. The agent has quasilinear preferences that are additively separable across the various components. We consider a robust version of the principal's problem, in which she knows the marginal distribution of each component of the agent's type, but does not know the joint distribution. Any mechanism is evaluated by its worst-case expected profit, over all joint distributions consistent with the known marginals. We show that the optimum for the principal is simply to screen along each component separately. This result does not require any assumptions (such as single crossing) on the structure of preferences within each component. The proof technique involves a generalization of the concept of virtual values to arbitrary screening problems. Sample applications include monopoly pricing and a stylized dynamic taxation model.

Auction design without quasilinear preferences

I study the canonical private value auction model for a single good without the quasilinearity restriction. I assume only that bidders are risk averse and the indivisible good for sale is a normal good. I show that removing quasilinearity leads to qualitatively different solutions to the auction design problem. Expected revenue is no longer maximized using standard auctions that allocate the good to the highest bidder. Instead, the auctioneer better exploits bidder preferences by using a mechanism that allocates the good to one of many different bidders, each with strictly positive probability. I introduce a probability demand mechanism that treats probabilities of winning the indivisible good like a divisible good in net supply 1. With enough bidders, it has greater expected revenues than any standard auction, and under complete information, it implements a Pareto efficient allocation.

Mechanism Design Without Quasilinearity

This paper studies a model of mechanism design with transfers where agents' preferences need not be quasilinear. In such a model: (1) we characterize dominant strategy incentive compatible mechanisms using a monotonicity property; (2) we establish a revenue uniqueness result: for every dominant strategy implementable allocation rule, there is a unique payment rule that can implement it; and (3) we show that every dominant strategy incentive compatible, individually rational, and revenue-maximizing mechanism must charge zero transfer for the worst alternative (outside option). These results are applicable in a wide variety of problems (single object auction, multiple object auction, public good provision etc.) under suitable richness of type space. In particular, our results can be applied to models where preferences of agents are arbitrarily small perturbations of quasilinear preferences and illustrate the (non)-robustness of some of the classic results in mechanism design with quasilinearity. We show various applications of our results.

Characterizing implementable allocation rules in multi-dimensional environments

We study characterizations of implementable allocation rules when types are multi-dimensional, monetary transfers are allowed, and agents have quasi-linear preferences over outcomes and transfers. Every outcome is associated with a valuation function that maps an agent’s type to his value for this outcome. The set of types are assumed to be convex. Our main characterization theorem shows that allocation rules are implementable if and only if they are implementable on any two-dimensional convex subset of the type set. For finite sets of outcomes and continuous valuation functions, they are implementable if and only if they are implementable on every one-dimensional subset of the type set. This extends a characterization result by Saks and Yu (Weak monotonicity suffices for truthfulness on convex domains, pp 286–293, 2005) from models with linear valuation functions to arbitrary continuous valuation functions, and provides a simple proof of their result. Modeling multi-dimensional mechanism design the way we propose it here is of relevance whenever types are given by few parameters, while the set of possible outcomes is large, and when values for outcomes are non-linear functions in types.

Efficient Multi-Unit Auctions for Normal Goods

I study efficient multi-unit auction design when bidders have private values, multi-unit demands, and non-quasilinear preferences. Without quasilinearity, the Vickrey auction loses its desired incentive and efficiency properties. Instead of assuming that bidders have quasilinear preferences, I assume that bidders have positive wealth effects. This nests cases where bidders are risk averse, face financial constraints, or have budgets.With two bidders, I show that there is a mechanism that retains the desirable properties of the Vickrey auction if and only if bidders have single dimensional types. If bidders have multi-dimensional types, there is no mechanism that satisfies (1) individual rationality, (2) dominant strategy incentive compatibility, (3) ex-post Pareto efficiency, and (4) weak budget balance. When there are more than two bidders, I show that there is no mechanism with desirable incentive and efficiency properties, even if bidders have single dimensional types.

Non-Manipulability of Walrasian Mechanisms in Economies with a Large Number of Objects

We consider a problem of allocating multiple identical objects to a group of agents and collecting payments. Each agent may receive several objects and has quasi-linear preferences with a submodular valuation function. It is known that Walrasian mechanisms are manipulable. We investigate the incentive property of Walrasian mechanisms in economies with a large number of objects. Given a set of agents and a preference profile, an agent i asymptotically dominates an agent j if at sufficiently many objects, i's incremental valuation is higher than j's incremental valuation. We show that for each economy, if there is no agent asymptotically dominating the other agents, and if there are sufficiently many objects, any Walrasian mechanism is non-manipulable at the economy. We also consider replica economies, and show that for each economy, if it is replicated sufficiently many times, the minimum price Walrasian mechanisms are non-manipulable at the replica economy.