The Revealed Preference Theory of Stable and Extremal Stable Matchings

We investigate the testable implications of the theory of stable matchings. We provide a characterization of the matchings that are rationalizable as stable matchings when agents' preferences are unobserved. The characterization is a simple nonparametric test for stability, in the tradition of revealed preference tests. We also characterize the observed stable matchings when monetary transfers are allowed and the stable matchings that are best for one side of the market: extremal stable matchings. We find that the theory of extremal stable matchings is observationally equivalent to requiring that there be a unique stable matching or that the matching be consistent with unrestricted monetary transfers.     

Complexity and Competition

Extensive‐form market games typically have a large number of noncompetitive equilibria. In this paper, we argue that the complexity of noncompetitive behavior provides a justification for competitive equilibrium in the sense that if rational agents have an aversion to complexity (at the margin), then maximizing behavior will result in simple behavioral rules and hence in a competitive outcome. For this purpose, we use a class of extensive‐form dynamic matching and bargaining games with a finite number of agents. In particular, we consider markets with heterogeneous buyers and sellers and deterministic, exogenous, sequential matching rules, although the results can be extended to other matching processes. If the complexity costs of implementing strategies enter players’ preferences lexicographically with the standard payoff, then every equilibrium strategy profile induces a competitive outcome.     

Sorting and Decentralized Price Competition

We investigate the role of search frictions in markets with price competition and how it leads to sorting of heterogeneous agents. There are two aspects of value creation: the match value when two agents actually trade and the probability of trading governed by the search technology. We show that positive assortative matching obtains when complementarities in the former outweigh complementarities in the latter. This happens if and only if the match‐value function is root‐supermodular, that is, its nth root is supermodular, where n reflects the elasticity of substitution of the search technology. This condition is weaker than the condition required for positive assortative matching in markets with random search.     

Multilateral Contracting with Externalities

This paper proposes a model for multilateral contracting, where contracts are written and renegotiated over time, and where contracts may impose externalities on other agents. Equilibria always exist and the equilibrium value function is linear and monotonically increasing on the contracts. If the grand coalition, or contracting among all agents, is inefficient, we show that bargaining delays arise in positive‐externality games and equilibrium inefficiency may remain bounded away from zero even as bargaining frictions converge to zero. Otherwise, if the grand coalition is efficient, there are no bargaining delays, convergence to the grand coalition occurs in a finite number of contracting rounds, and the outcome becomes efficient as players become more patient.     

Large Matching Markets as Two‐Sided Demand Systems

This paper studies two‐sided matching markets with non‐transferable utility when the number of market participants grows large. We consider a model in which each agent has a random preference ordering over individual potential matching partners, and agents' types are only partially observed by the econometrician. We show that in a large market, the inclusive value is a sufficient statistic for an agent's endogenous choice set with respect to the probability of being matched to a spouse of a given observable type. Furthermore, while the number of pairwise stable matchings for a typical realization of random utilities grows at a fast rate as the number of market participants increases, the inclusive values resulting from any stable matching converge to a unique deterministic limit. We can therefore characterize the limiting distribution of the matching market as the unique solution to a fixed‐point condition on the inclusive values. Finally we analyze identification and estimation of payoff parameters from the asymptotic distribution of observable characteristics at the level of pairs resulting from a stable matching.     

Stable and Coordinating Contracts for a Supply Chain with Multiple Risk‐Averse Suppliers

We analyze a decentralized supply chain with a single risk‐averse retailer and multiple risk‐averse suppliers under a Conditional Value at Risk objective. We define coordinating contracts and show that the supply chain is coordinated only when the least risk‐averse agent bears the entire risk and the lowest‐cost supplier handles all production. However, due to competition, not all coordinating contracts are stable. Thus, we introduce the notion of contract core, which reflects the agents' “bargaining power” and restricts the set of coordinating contracts to a subset which is “credible.” We also study the concept of contract equilibrium, which helps to characterize contracts that are immune to opportunistic renegotiation. We show that, the concept of contract core imposes conditions on the share of profit among different agents, while the concept of contract equilibrium provide conditions on how the payment changes with the order quantity.     

An Efficient Dynamic Mechanism

This paper constructs an efficient, budget‐balanced, Bayesian incentive‐compatible mechanism for a general dynamic environment with quasilinear payoffs in which agents observe private information and decisions are made over countably many periods. First, under the assumption of “private values” (other agents' private information does not directly affect an agent's payoffs), we construct an efficient, ex post incentive‐compatible mechanism, which is not budget‐balanced. Second, under the assumption of “independent types” (the distribution of each agent's private information is not directly affected by other agents' private information), we show how the budget can be balanced without compromising agents' incentives. Finally, we show that the mechanism can be made self‐enforcing when agents are sufficiently patient and the induced stochastic process over types is an ergodic finite Markov chain.     

Equilibrium Wage‐Tenure Contracts

In this study we consider a labor market matching model where firms post wage‐tenure contracts and workers, both employed and unemployed, search for new job opportunities. Given workers are risk averse, we establish there is a unique equilibrium in the environment considered. Although firms in the market make different offers in equilibrium, all post a wage‐tenure contract that implies a worker's wage increases smoothly with tenure at the firm. As firms make different offers, there is job turnover, as employed workers move jobs as the opportunity arises. This implies the increase in a worker's wage can be due to job‐to‐job movements as well as wage‐tenure effects. Further, there is a nondegenerate equilibrium distribution of initial wage offers that is differentiable on its support except for a mass point at the lowest initial wage. We also show that relevant characteristics of the equilibrium can be written as explicit functions of preferences and the other market parameters.     

A Unique Costly Contemplation Representation

We study preferences over menus which can be represented as if the individual is uncertain of her tastes, but is able to engage in costly contemplation before selecting an alternative from a menu. Since contemplation is costly, our key axiom, aversion to contingent planning, reflects the individual's preference to learn the menu from which she will be choosing prior to engaging in contemplation about her tastes for the alternatives. Our representation models contemplation strategies as subjective signals over a subjective state space. The subjectivity of the state space and the information structure in our representation makes it difficult to identify them from the preference. To overcome this issue, we show that each signal can be modeled in reduced form as a measure over ex post utility functions without reference to a state space. We show that in this reduced‐form representation, the set of measures and their costs are uniquely identified. Finally, we provide a measure of comparative contemplation costs and characterize the special case of our representation where contemplation is costless.     

Supply Chain Contracting Under Competition: Bilateral Bargaining vs. Stackelberg

We analyze contracting behaviors in a two‐tier supply chain system consisting of competing manufacturers and competing retailers. We contrast the contracting outcome of a Stackelberg game, in which the manufacturers offer take‐it‐or‐leave‐it contracts to the retailers, with that of a bargaining game, in which the firms bilaterally negotiate contract terms via a process of alternating offers. The manufacturers in the Stackelberg game possess a Stackelberg‐leader advantage in that the retailers are not entitled to make counteroffers. Our analysis suggests that whether this advantage would benefit the manufacturers depends on the contractual form. With simple contracts such as wholesale‐price contracts, which generally do not allow one party to fully extract the trade surplus, the Stackelberg game replicates the boundary case of the bargaining game with the manufacturers possessing all the bargaining power. In contrast, with sophisticated contracts such as two‐part tariffs, which enable full surplus extraction, the two games lead to distinct outcomes. We further show that the game structure being Stackelberg or bargaining critically affects firms' preferences over contract types and thus their equilibrium contract choices. These observations suggest that the Stackelberg game may not be a sufficient device to predict contracting behaviors in reality where bargaining is commonly observed.     

The Limits of ex post Implementation

The sensitivity of Bayesian implementation to agents' beliefs about others suggests the use of more robust notions of implementation such as ex post implementation, which requires that each agent's strategy be optimal for every possible realization of the types of other agents. We show that the only deterministic social choice functions that are ex post implementable in generic mechanism design frameworks with multidimensional signals, interdependent valuations, and transferable utilities are constant functions. In other words, deterministic ex post implementation requires that the same alternative must be chosen irrespective of agents' signals. The proof shows that ex post implementability of a nontrivial deterministic social choice function implies that certain rates of information substitution coincide for all agents. This condition amounts to a system of differential equations that are not satisfied by generic valuation functions.     

Matching With (Branch‐of‐Choice) Contracts at the United States Military Academy

Branch selection is a key decision in a cadet's military career. Cadets at USMA can increase their branch priorities at a fraction of slots by extending their service agreement. This real‐life matching problem fills an important gap in the market design literature, providing strong empirical legitimacy to a series of elegant theoretical works on matching with contracts. Although priorities fail a key substitutes condition, the agent‐optimal stable mechanism is well defined, and in contrast to the current USMA mechanism it is fair, stable, strategy‐proof, and respects improvements in cadet priorities. Adoption of this mechanism benefits cadets and the Army. This new application shows that the matching with contracts model is practically relevant beyond traditional domains that satisfy the substitutes condition.     

The Coalitional Nash Bargaining Solution

The coalitional Nash bargaining solution is defined to be the core allocation for which the product of players' payoffs is maximal. We consider a non‐cooperative model with discounting in which one team may form and every player is randomly selected to make a proposal in every period. The grand team, consisting of all players, generates the largest surplus. But a smaller team may form. We show that as players get more patient if an efficient and stationary equilibrium exists, it must deliver payoffs that correspond to the coalitional Nash bargaining solution. We also characterize when an efficient and stationary equilibrium exists, which requires conditions that go beyond the nonemptiness of the core.     

Welfare Effects of Local versus Central Wage Bargaining

The paper analyses the welfare effects of union bargaining (de)centralization in a dual labour market with a unionized and a competitive sector. We show that social welfare depends on both the structure of the union's objective function and the elasticities of labour demand in both sectors. The welfare‐maximizing employment allocation can be obtained under a high degree of centralization if the union maximizes the total wage‐bill. Otherwise, if the union is rent maximizing, welfare is higher under local bargaining. However, in that case neither central nor local wage setting yields the social optimum.     

Homo Moralis—Preference Evolution Under Incomplete Information and Assortative Matching

What preferences will prevail in a society of rational individuals when preference evolution is driven by the resulting payoffs? We show that when individuals' preferences are their private information, a convex combination of selfishness and morality stands out as evolutionarily stable. We call individuals with such preferences homo moralis. At one end of the spectrum is homo oeconomicus, who acts so as to maximize his or her own payoff. At the opposite end is homo kantiensis, who does what would be “the right thing to do,” in terms of payoffs, if all others would do likewise. We show that the stable degree of morality—the weight placed on the moral goal—is determined by the degree of assortativity in the process whereby individuals are matched to interact.     

Money in Search Equilibrium,in Competitive Equilibrium,and in Competitive Search Equilibrium

We compare three market structures for monetary economies: bargaining (search equilibrium); price taking (competitive equilibrium); and price posting (competitive search equilibrium). We also extend work on the microfoundations of money by allowing a general matching technology and entry. We study how equilibrium and the effects of policy depend on market structure. Under bargaining, trade and entry are both inefficient, and inflation implies first‐order welfare losses. Under price taking, the Friedman rule solves the first inefficiency but not the second, and inflation may actually improve welfare. Under posting, the Friedman rule yields the first best, and inflation implies second‐order welfare losses.     

Temporary Contracts,Employment Protection,and Collective Bargaining

This paper examines the effect of employment protection in a matching model with endogenous job destruction, collective bargaining, and two types of employment contracts. Using this framework, we show that (i) the impact on job creation and job destruction caused by reducing the firing costs associated with temporary jobs depends on the labour unions' bargaining strength and the gap in firing costs between contracts; (ii) reducing the firing costs associated with permanent jobs unambiguously decreases equilibrium unemployment if labour unions have strong bargaining power; and (iii) the impact caused by the firing costs differs between collective and individual bargaining.     

Unobservable Investment and the Hold‐Up Problem

We study a two‐person bargaining problem in which the buyer may invest and increase his valuation of the object before bargaining. We show that if all offers are made by the seller and the time between offers is small, then the buyer invests efficiently and the seller extracts all of the surplus. Hence, bargaining with frequently repeated offers remedies the hold‐up problem even when the agent who makes the relation‐specific investment has no bargaining power and contracting is not possible. We consider alternative formulations with uncertain gains from trade or two‐sided investment.     

Perceived Ambiguity and Relevant Measures

We axiomatize preferences that can be represented by a monotonic aggregation of subjective expected utilities generated by a utility function and some set of i.i.d. probability measures over a product state space, S^∞. For such preferences, we define relevant measures, show that they are treated as if they were the only marginals possibly governing the state space, and connect them with the measures appearing in the aforementioned representation. These results allow us to interpret relevant measures as reflecting part of perceived ambiguity, meaning subjective uncertainty about probabilities over states. Under mild conditions, we show that increases or decreases in ambiguity aversion cannot affect the relevant measures. This property, necessary for the conclusion that these measures reflect only perceived ambiguity, distinguishes the set of relevant measures from the leading alternative in the literature. We apply our findings to a number of well‐known models of ambiguity‐sensitive preferences. For each model, we identify the set of relevant measures and the implications of comparative ambiguity aversion.     

A Theory of the Firm with Non‐binding Employment Contracts

This paper examines an employment relation in which individual workers enjoy some bargaining power vis‐a‐vis the firm although they are not unionized. The main elements of the situations studied here are that the employment contracts are non‐binding across periods of production and that the firm has opportunities to replace workers. The paper analyzes a dynamic model in which the processes of contracting and recontracting between the firm and its workers are intertwined with the dynamic evolution of the firm's workforce. The analysis of the model is somewhat complicated because the employment level is a nondegenerate state variable that evolves over time and is affected by past decisions. The main analytical results characterize certain important equilibria: the profit maximizing and stationary equilibria. The unique stationary equilibrium is markedly inefficient: it exhibits inefficient over‐employment and the steady state wages coincide with the workers' reservation wage. It confirms earlier results derived by Stole and Zwiebel (1996a, b) in the context of a static model and shows that they are very robust even when the firm has nearly frictionless hiring opportunities. In contrast, in the profit maximizing equilibrium the outcome is nearly efficient and the wage exhibits a mark‐up over the reservation wage.