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Sorting Through Search and Matching Models in Economics† Journal of Economic Literature 2017, 55(2), 493–544 https://doi.org/10.1257/jel.20150777 Sorting through Search and Matching Models in Economics† Hector Chade, Jan Eeckhout, and Lones Smith* Toward understanding assortative matching, this is a self-contained introduction to research on search and matching. We first explore the nontransferable and perfectly transferable utility matching paradigms, and then a unifying imperfectly transfer- able utility matching model. Motivated by some unrealistic predictions of frictionless matching, we flesh out the foundational economics of search theory. We then revisit the original matching paradigms with search frictions. We finally allow informational frictions that often arise, such as in college-student sorting. ( JEL C78, D82, D83, I23, J12) 1. Introduction Matching models enrich the Walrasian par- adigm, capturing person-specific goods conomics is built on the Walrasian sup- and relationships. The frictional matching Eply and demand cornerstone, with trade literature replaces the auctioneer’s gavel by anonymously guided by a fictitious impar- a mixture of dynamic choice and chance. It tial auctioneer. This survey article explores thus impedes the invisible hand with costs or the literature that has largely emerged in imperfect information. the last quarter century on decentralized In this research thread of the assignment matching models with and without frictions. and matching literature, a dominant theme is positive sorting—the best are matched with one another, as are the next best, and so on. * Chade: Department of Economics, Arizona State Indeed, we observe firms spending signifi- University. Eeckhout: Department of Economics UPF– ICREA–Barcelona GSE and University College London. cant resources to hire the right employee; the Smith: Department of Economics, University of Wis- government spends large sums on unemploy- consin–Madison. We are grateful to the editor and five ment benefits to provide incentives for work- anonymous referees for their helpful comments and sug- gestions. We also benefited from comments from Manolis ers to search for the right jobs; people invest Galenianos, Shoshana Grossbard, Belen Jerez, Stephan great time resources searching for the right Lauermann, Patrick Legros, Ilse Lindenlaub, Jeremy mates—including the use of online dating Lise, Qingmin Liu, Benny Moldovanu, Sam Schulhofer- Wohl, and Randy Wright. An early version of parts of this markets to find more and better partners; and draft was developed for lectures at the Centre for Market house buyers generally hire agents to help Design in Melbourne, Western Ontario, and the Becker find their ideal property matching their tastes. Friedman Center in Chicago. † Go to https://doi.org/10.1257/jel.20150777 to visit the Even in the Walrasian setting with cen- article page and view author disclosure statement(s). tralized trade, frictionless matching of 493 494 Journal of Economic Literature, Vol. LV (June 2017) heterogeneous agents makes explicit the sort- frictions or because of differences in private ing patterns between agents. The theoretical valuations, they would need costly inspection. literature on frictionless matching has largely Until the 1990s, the matching and search pursued two main lines of thought. In one, theory literatures largely proceeded in iso- match payoffs are nontransferable, and equi- lation.1 But the development of frictional librium (stability) requires checking pair- matching models that began in the early wise double coincidence of wants. This work 1990s has sparked renewed interest in both began with the path-breaking math article the frictionless matching and search para- by Gale and Shapley (1962) that developed digms. This has been driven by the impor- an intuitive algorithm for generating stable tance of heterogeneity and sorting in many matchings; this is the cornerstone of the large economic environments where search centralized matching literature. Meanwhile, frictions are significant. Since then, the a parallel model allowing transferable payoffs two literatures have been bedfellows. For emerged, closer in spirit to market econom- search frictions create equilibrium feedback ics, in which a welfare theorem held. This between types who would otherwise remain social planner’s problem for the matching lit- unmatched. For instance, when low-produc- erature dates back to the early work by Monge tivity jobs are filled by high-ability workers, (1781) and Kantorovic (1942) on the mass this affects the labor market prospects of the transportation problem, and to Koopmans low-ability workers. This has been a major and Beckmann (1957) who introduced a pric- theme that has emerged. ing system to solve the problem. This liter- Pursuant to this merger of the search and ature saw its fruition in Shapley and Shubik matching literatures, an alternative approach (1972). Whereas Gale and Shapley allowed to search theory developed. Rather than heterogeneous preferences, the seminal mar- explicitly and separately model the dynamic riage-model paper by Becker (1973) assumed matching and price negotiation processes, common ordinal preferences over partners. in directed search, firms first set prices and He found that matching was assortative when then buyers direct their search, and finally the match-payoff function was supermodu- meetings materialize. One slice of the liter- lar. This literature was naturally drawn to this ature here captured market-clearing failures pivotal sorting question in a variety of eco- in two-sided matching models in the spirit of nomic contexts like marriage markets, labor Gale and Shapley (1962) by explicitly mod- markets, housing markets, industrial organi- eling the queues that form. Buyers arrive at zation, and international trade. sellers, and the queue length acts like a price, Concurrent with the matching literature, as it does at an amusement park. Another the economics of search theory was devel- approach instead explicitly models the oping. Motivated by the failure of the law of stockouts that emerge—students apply for one price, Stigler (1961) had formulated the slots at colleges, and are generally rejected. first search optimization in economics. This Formally, they are told that no slot is avail- has since proven useful for understanding able. Directed search often also exploits the wage formation and unemployment in the role of prices: sellers post prices first, upon labor market. Search offers a way to formalize decentralized trade. In many formulations, 1 Sattinger (1993) nicely surveyed the matching models the Walrasian assumptions on price setting that were standard in labor-market applications until the are too strong. For example, agents do not 1990s. Search and information frictions play only a minor role in that survey. The large literature that we cover in see all prices of houses transacted, and even this paper illustrates how much it has progressed in the last if they did, in the presence of information twenty-five years. Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 495 which buyers make their purchase decisions, economic environments, we first analyze the taking matching frictions into account. benchmark case without frictions. We offer a self-contained review of Many important problems can be thought this literature. We introduce the bench- of as pairwise matching or an assignment mark matching models without frictions. of two groups of heterogeneous elements, Motivated by some unrealistic implications, either individuals or goods. This can be we then explore the search models that have accomplished by a benevolent planner or can emerged that best address these failings. We take place in a decentralized setting where finally assemble these pieces, fleshing out there is competition for agents or objects. matching models beset by search and infor- Examples abound: sorting men and women mation frictions. By focusing on its main into marriages, assigning workers to firms, analytic idea, possibly by way of example, we locations to plants, buyers to sellers, coun- present each as a teachable unit. We then tries to goods, etc. A distinctive feature is touch on salient applications, for in recent that agents or objects on each side are indi- years matching models have been applied visible and frequently heterogenous. broadly in economics. Examples without An important modeling choice in this frictions include marriage markets, hier- framework is how payoffs are shared within archies, international trade, finance, CEO a match. Two polar choices are perfectly compensation, foreign direct investment, transferable utility (perfect TU), where and development.2 Matching models with agents can freely transfer payoffs between frictions afford analyses of unemployment them at a constant rate, and nontransferable in the presence of sorting such as mismatch, utility (NTU), where either no transfers are the transmission of labor market risk, and possible or the division of the match sur- the impact of macroeconomic fluctuations.3 plus is exogenously given and preferences Throughout the review, we refer to some of over mates can be fully expressed in ordinal these papers, and highlight open-research terms. A blend of both cases is imperfectly agendas as a roadmap for future work. transferable utility (imperfect TU), where Overall, this survey explores how two eco- payoffs are neither fully transferable nor nomic literatures, one in optimization and exogenously given.4 In the rest of the sec- another in equilibrium, merged
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