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 transferable 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 paradigm, 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 to create tion, we provide a detailed analysis of the a cohesive equilibrium story of frictional TU and NTU paradigms, as well as several markets. economic applications.5 Our focus is on the conditions under which assortative matching obtains. Given our interest in sorting, we 2. Frictionless Matching and Sorting

To study how search frictions and/or infor- 4 We break with tradition and use the more suggestive mation frictions shape matching outcomes in terminology in Noeldeke and Samuelson (2015) instead of the well-established one from cooperative game theory used by Legros and Newman (2007). That is, instead of transferable utility we call it perfectly transferable utility; instead of nontransferable utility we call it imperfectly 2 See, among many others, Garicano (2000); Sørensen transferable utility; and instead of strictly nontransferable (2007); Antras, Garicano, and Rossi-Hansberg (2006); utility we call it simply nontransferable utility. Grossman, Helpman, and Kircher (2013); Grossman and 5 We abuse the terminology slightly by calling the NTU Maggi (2000); Tervio (2008); Gabaix and Landier (2008); case “frictionless,” since this feature can be due to some Guadalupe et al. (2014); and Ackerberg and Botticini friction that prevents full transferability. What we mean (2002). here is that there are no search frictions (agents observe all 3 See, for example, Lise, Meghir, and Robin (2013); potential partners) and no incomplete information about Lamadon (2014); Lise and Robin (2017). partners (agents observe their characteristics). 496 Journal of Economic Literature, Vol. LV (June 2017) only discuss the NTU case results that shed TU (i.e., an assignment game, using the light on sorting patterns, leaving aside many terminology of Shapley and Shubik 1972), interesting issues in this framework that are leaving extensions for later. For definiteness, extensively covered in the book by Roth and we cast the problem in terms of a marriage Sotomayor (1990). market, but it will be obvious that other applications follow by a simple reinterpreta- 2.1 The Theory of Frictionless Sorting tion of the two sides of the market. There are with Perfectly Transferable Utility N women and N men. Each woman i has a characteristic (type) x 0, 1 and each man i ∈ [ ] The insights below encapsulate the mes- j has a characteristic y 0, 1 ; for simplic- j ∈ [ ] sage of a trio of seminal papers: Koopmans ity, assume that x x x and 1 < 2 < ⋯ < N and Beckmann (1957), Shapley and y 1 y 2 y N. If woman x i marries 6 < < ⋯ < Shubik (1972), and Becker (1973). They man y j , then they produce a positive output first analyzed the matching problem between f (x i , y j). We can thus identify each agent plants and locations and derived the prop- with his or her type. We make the innocuous erties of the optimal assignment and com- assumption that single agents produce zero petitive equilibrium as solutions to a linear output. Crucially, agents’ preferences are programming problem and its dual. The sec- linear in money (perfect TU), and thus part- ond one used as a metaphor the assignment ners can freely share the match output proof buyers and sellers in a market for hetero- duced using transfers.8 geneous houses, and provided solid game We answer the following questions: What theoretic foundations to the problem, deriv- is the optimal matching of men and women? ing the optimal assignment, core allocations, Under what conditions does this assignment and competitive equilibrium in a unified exhibit PAM or NAM? Is this allocation in way.7 None of these papers focus on sort- the core of the assignment game? Can it be ing patterns. It was Becker (1973) who, in a decentralized as a Walrasian equilibrium? marriage context, provided the fundamental 2.1.2 The Optimal Assignment Problem insight about complementarities of partners’ characteristics in the match payoff function Start with the planner’s problem. Since and the resulting positive or negative assorta- utility is transferable, efficiency demands tive matching (PAM or NAM) in the optimal/ that an optimal matching maximize the sum equilibrium assignment of men and women. of all match outputs.9 Formally, the optimal His analysis remains a cornerstone of match- matching is the solution to the following ing theory, and has also become important in maximization problem: empirical work on the subject, since it pro- N vides the theory with empirical content. (1) max f ( x i , y (i)) , π i∑1 π 2.1.1 The Basic Model = where the maximization is taken over all We derive the main insights using a simple possible permutations : 1, 2, , N π { … } → instance of the matching model with perfect 1, 2, , N . By a well-known result in { … }

6 In the mathematics literature, the Monge–Kantorovich 8 Actually, the model allows for transfers to other pairs, optimal-transport problem subsumes many frictionless but one can easily show that they are not used in the core matching problems. For an authoritative treatise of this or competitive-equilibrium allocations. problem, see Villani (2009). 9 If this were not the case, there would be a rematching 7 An excellent source for these results in both the finite of some of the agents that would increase the total size of and continuous case is Gretsky, Ostroy, and Zame (1999). the pie to be distributed, contradicting optimality. Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 497

rearrangement inequalities (e.g., see Vince the best woman is paired with the man with 1990 and the main sorting result in the the lowest type, the second best woman appendix of Becker 1973), the identity per- with the man with the second lowest type, mutation (i) i for all i solves problem (1) and so on. π = if f is supermodular10, 11 on 0, 1 2. This con- A useful alternative linear programming [ ] dition is not only sufficient but also necessary formulation of the optimal assignment prob- if the result must hold for all distributions of lem is: types for men and women. In short, PAM is optimal if and only if f is supermodular, that N N is, when men’s and women’s types are com- (2) max f (x i , y j) ij i∑1 j∑1 α plements in the match output function. In α = = this case, the planner pairs the woman and man with the best characteristics, the second N N best woman with the second best man, and subject to j 1 ij 1 for all i, i 1 ij ∑ = α ≤ ∑ = α so on. 1 for all j, and 0 for all i, j. Since ≤ αij ≥ It is easy to see why supermodularity is is not merely zero or one, the problem αij sufficient for PAM—independently of the permits fractional assignment of men and distribution of men’s and women’s types. women. Koopmans and Beckmann (1957) Under any other assignment, there are two and Shapley and Shubik (1972), however, women, say i and i with i i, respectively showed that there is an optimal solution ′ ′ > matched with two men j and j with j j . with 0, 1 . If f is supermodular, then ′ > ′ αij ∈ { } The total output of these couples f (x , y ) 1 when i j, and PAM ensues; if i j + αij = = f (x i , y j ) is lower than f (x i, y j ) f (x i , y j), not, then one can find a profitable rematch- + by ′ supermodularity.′ Hence, the′ planner′ ing. A similar analysis holds for NAM. can increase total output by assortatively rematching them. 2.1.3 Core, Stability, and Walrasian A similar argument reveals that the reverse Equilibrium permutation (i) N i 1 solves the π = − + problem if and only if f is submodular Instead of the planner’s problem, we could in (x, y). Thus, NAM is optimal when types envision men and women competing for are substitutes in production. In this case, partners in the assignment game, where they can bid for each other and sign contracts specifying how to divide the match output. 10 A real-valued function f on a lattice X n (e.g., ⊆ 핉 To fix ideas, assumef is strictly supermodular, 0, 1 2) is supermodular if f (x x ) f (x x ) f (x ) [ ] ′ ∨ ″ + ′ ∧ ″ ≥ ′ + and thus the optimal assignment is PAM. Let f (x ) for all x and x in X , where x x max x , x ″ ′ ″ ′ ∨ ″ = { ′ ″} and x x min x , x . If f is twice continuously dif- i i and j j ; by strict supermodularity, ′ ∧ ″ = { ′ ″ } > ′ > ′ ferentiable, then this is equivalent to 2 f (x)/ x x 0 f (x , y ) f (x , y ) f (x , y ) f (x , y ). ∂ ∂ i ∂ j ≥ i j i j i j i j for all i j. The function is submodular if f (x x ) + ′ ′ > ′ + ′ ≠ ′ ∨ ″ + This inequality implies f (x i , y j) f (x i , y j) f (x x ) f (x ) f (x ) for all x and x in X, and this is − ′ ′ ∧ ″ ≤ 2′ + ″ ′ ″ equivalent to f (x)/ x x 0 for all i j if f is twice f (x i , y j ) f (x i , y j ), so that the willing- ∂ ∂ i ∂ j ≤ ≠ > ′ − ′ ′ continuously differentiable. These concepts are strict if the ness to pay for the higher woman x i is higher inequalities are strict. for the higher man y than for y . So when 11 j j This is a nonlinear generalization of an inequality for competing for partners, j can outbid′ j in products of vectors in Hardy, Littlewood, and Polya (1952). ′ In the discrete case considered, it states (see Vince 1990) the quest for i. Consequently, any “stable” that if f1 , , fn are real-valued functions on an interval I, … outcome of the assignment game exhibits then i fi (bn i 1) i fi (b (i)) i fi (bi ) for all ∑ − + ≤ ∑ π ≤ ∑ sequences b b b and all if and only if PAM. Alternatively, we could explore the 1 ≤ 2 ≤ ⋯ ≤ n π fi 1 fi is increasing on I for 1 i n. performance of a competitive market where + − ≤ < 498 Journal of Economic Literature, Vol. LV (June 2017) agents from each side take “partners’ prices” maximizes f (x , y ) w . By construction i j − j as given. The same logic reveals that any of the core allocation, v f (x , y ) i = i j − Walrasian equilibrium of this market deliv- w if 1. That is, v is the wage that j αij = i ers PAM when f is strictly supermodular, and woman i obtains in the core allocation. NAM when f is strictly submodular. Also, for any other man j with ij 0, we ′ α ′ = Let v 1, , v N and w 1, , w N be the mul- have v i f (x i, y j ) w j , or f (x i , y j) w j … … ≥ ′ − ′ − tipliers associated with the constraints after f (x i , y j ) w j . Hence, when con- ≥ − (2). Then the dual problem is12 fronted ′with ′men’s wages w , , w , 1 … N woman i optimally selects the same partner N N as in the core allocation. (3) min v i w j, v, w i∑1 + j∑1 Since one can perform this analysis for = = men, the optimal matching can be decen- subject to v w f (x , y ), v 0, and tralized as a Walrasian equilibrium of the i + j ≥ i j i ≥ w 0. From the linear programming marriage market. The wage of a woman in j ≥ duality theorem, the value to this prob- this market depends only on her type, and lem is the same as that of problem (2), and not on that of the man she matches with. For thus of (1). Moreover, the s of problem these wages are formally the utility payoffs αij (2) are the multipliers of problem (3). It of each woman in the core allocation. Since follows that if ( , v, w) solves (2) and (3), these wages are the multipliers of the lin- N αN N then (i) i 1 j 1 f (x i , y j) ij i 1 v i ear programming constraints, each can be N ∑ = ∑ = α = ∑ = + j 1 w j; and (ii) v i w j f (x i , y j) for each interpreted as the shadow value of adding a ∑ = + = pair (i, j) such that 1; and (iii) v w woman to the matching market—hence the αij = i + j f (x , y ) for each pair (i, j) such that dependence only on the woman’s type. ≥ i j 0. This analysis is valid with an unequal αij = The triple ( , v, w) optimally matches number of men and women. In that case, α the two populations and provides a division some agents on the long side of the market of the match output between partners that will remain single. Similarly, if we assume exhausts output, if we interpret v i and w j as that agents can produce some output as sin- the wages of woman i and man j. That triple gles, then match surplus will be its output is also a stable matching of the assignment minus the sum of the singles outputs. In this game, for no man and woman not originally case, some agents may remain single at the matched can profitably block the assignment optimal matching. While we have assumed given (iii), since the sum of the utilities in only one agent of each type, our analysis is their original matches more than exhausts valid with different discrete distributions the match output if they rematch. Moreover, of types on each side. In this case, PAM (iii) implies that no coalition of men and matches agents from the top types down, women can improve upon ( , v, w). Hence, respecting the measure of agents of each α the solution of the dual problem character- type in the population until the populations izes the core of the assignment game. are exhausted. We now decentralize the optimal/core 2.1.4 The Large-Market Case matching as a Walrasian equilibrium. Let women take the men’s wages w , , w as As is standard in economics, the contin- 1 … N given. Then woman i chooses the man j that uum of agents idealization not only provides solid foundations for price-taking behavior, 12 See Chvatal (1983), chapter 5, for a derivation and for but also enables the use of calculus in the the proof of the duality theorem of linear programming. derivation of equilibria and their properties. Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 499

We will illustrate this convenient feature Now, if f is strictly supermodular (i.e., below with some important economic f 0), then the objective function sat- xy > applications of the frictionless matching isfies the strict single-crossing property in paradigm. (x, y).13 Hence, in any solution to this Assume an equal unit mass continuum of problem, men with higher y choose women men and women. Each female has a type with higher x.14 So if a Walrasian equilibrium x 0, 1 drawn from a strictly increasing exists, then it must exhibit PAM. This pro- ∈ [ ] and continuously differentiable cdf G with vides an alternative view of the sufficiency positive density g. Similarly, each man of supermodularity for PAM.15 From the has a type y 0, 1 , with cdf H and den- above measure-preserving (market-clearing) ∈ [ ] sity h . We can define a (pure) matching as property, the only candidate for equilib- 1 a function : 0, 1 0, 1 that is mea- rium matching is y (x) H − (G(x)) μ [ ] → [ ] 1 = μ = sure preserving, i.e., matching equal mea- or x − (y). This must satisfy the = μ sures of men and women. For instance, first-order condition v (x) f (x, (x)), and ′ = x μ PAM requires that G(x) H( (x)) for thus: 1 = μ all x. Hence, (x) H − (G(x)) is strictly μ = increasing and (x) g(x)/ h( (x)) 0. μ′ = μ > x Under NAM, G(x) 1 H( (x)) for all x, (4) v(x) v 0 fx (s, (s)) ds, =1 − μ = + 0 μ and thus (x) H − (1 G(x)), with (x) ∫ μ = − μ′ g(x)/ h( (x)) 0. = − μ < The match output of a woman with type where v 0 is a constant of integration. (One x with a man with type y is f (x, y), now can show that global optimality holds.) assumed twice continuously differentiable. Hence, if f is strictly supermodular, then 1 A continuous version of the above rearrange- (x) H − (G(x)) and (4) constitute a μ = ment inequality (e.g., see Lorentz 1953 and Walrasian equilibrium and exhibits PAM. Crowe, Zweibel, and Rosenbloom 1986) Clearly, each man y in equilibrium obtains 1 1 shows that PAM is optimal if and only if f is w(y) f ( − (y), y) v( − (y)). A similar = μ − μ supermodular, and NAM is optimal if and analysis can be done for f strictly submodular only if f is submodular. Also, the Shapley– and NAM. Shubik linear programming derivation of the optimal assignment, core allocations, and Walrasian equilibrium extends to continuous models (Gretsky, Ostroy, and Zame 1992, 1999). _ 13 A function z : X × t , t , X a lattice, satisfies We now derive the Walrasian equilibrium → 핉 the strict single-crossing [property_ ] in (x, t) if for all x x ″ > ′ and deduce the sorting pattern that ensues and t t , z(x , t ) z(x , t ) 0 implies z(x , t ) ″ > ′ ″ ′ − ′ ′ ≥ ″ ″ − when production f (x, y) is supermodular or z(x , t ) 0. ′ ″ > 14 Apply theorem 4 in Milgrom and Shannon (1994) submodular. In so doing, we draw a simple ′ (quasi-supermodularity in y trivially holds in this problem). connection between matching models and 15 In a CEO–firm assignment application, Tervio (2008) a basic monotone comparative statics result derives the Walrasian equilibrium of the model in a simi- (Topkis 1998). lar way, and points out the relationship with the incentive compatibility conditions in screening problems. What lies Consider a man of type y facing a wage at a more basic level is the monotone comparative statics profilev (x) for women x. He seeks the result alluded to above, since both the problem of each woman x that maximizes his payoff: agent in a matching setting and the incentive compatibility problem are parameterized optimization problems (in one case by an agent’s observable type, and in the other case by max f (x, y) v(x). x 0, 1 − an agent’s privately known type). ∈[ ] 500 Journal of Economic Literature, Vol. LV (June 2017)

2.2 Applications of Frictionless Sorting with positive correlation among wages of workers Perfectly Transferable Utility in different occupations within a firm.

2.2.1 The O-Ring Production Function 2.2.2 CEO–Firm Assignment Model

In an application of Becker’s marriage Tervio (2008) and Gabaix and Landier model, Kremer (1993) explores a “weakest (2008) develop a matching model of firm link” production model that naturally gen- size and CEO talent, and calibrate it using erates supermodularity. There are n tasks, US data to analyze CEO pay. They assume a each performed by a worker. Each worker i unit mass continuum of CEOs and of firms. has a type x _x , 1 , 0 _ x 1, drawn The CEO talent x has a differentiable cdf G, i ∈ [ ] < < from a continuous density g . This type is the and the firm sizey has a differentiable cdf H . probability that the worker successfully per- They (and Tervio 2008) identify each CEO forms the task. Production happens when of talent x with his quantile rank i G(x); = all n workers succeed in their tasks. That is, since G is differentiable, there is a smooth n the expected output of a firm is nB ∏ i 1 x i, relationship between i and X(i), with X (i) = ′ where B 0 is the output per worker if all 0. Likewise, associate firms with their > > perform their tasks successfully. There is a quantile rank j H(y), where firmY ( j) = unit mass of workers and a limited number smoothly increases: Y ( j) 0. They assume ′ > of identical potential firms that each hires that the revenue when CEO i is matched with n workers from a competitive labor market. firmj is C Y d( j)X(i), for constants C, d 0, > Firms take the wage function : _x , 1 from which the CEO gets paid (i). ω [ ] → ω as given. For simplicity, we assume that In a Walrasian equilibrium, firmj max- 핉+ labor is the only factor of production. imizes CY d( j)X(i) (i) over i 0, 1 . − ω ∈ [ ] In the matching problem, each firm hires Since the objective function is strictly super- n possibly heterogeneous workers to maxi- modular in (i, j), the equilibrium exhibits mize expected profits: PAM—to wit, i j, for all j. The first-order = condition (FOC) yields (i) CY d( j)X (i), n n ω′ = ′ and hence (5) max nB ∏ x i (x i). n − ω xi i 1 i 1 i∑1 { } = = = i (i) CY d(s)X (s) ds (0). 2 n ω = 0 ′ + ω Since ∏ i 1 x i / x j x k 0 for all j k, ∫ ∂ ( = ) ∂ ∂ > ≠ the expected output is strictly supermodular This wage function, along with PAM, consti- in (x , x , , x ) and the equilibrium exhib- tutes the Walrasian equilibrium of the model 1 2 … n its PAM: In other words, all the workers (modulo the constant (0)). ω employed by any given firm have the same To calibrate the model, Gabaix and type x . The first order condition for (5) in x i Landier (2008) posit a Pareto firm size dis- n 1 1/ evaluated at x x yields (x) nBx − . τ i = ω′ = tribution H(y) 1 ( _ y /y) , which yields Consequently, (x) Bx n , where the = − ω = + ω0 Y( j) _ y (1 j) −τ. Meanwhile, inspired by constant is pinned down by the firm’s zero = − ω0 extreme value theory, they posit that the profit condition, or 0. All told, each ω0 = “spacing function” for CEO talents X (i) sat- 1 ′ firm hires workers of the same skillx and pays isfies X (i) K(1 i) ν− , for constants K them the wage (x) Bx n, equally dividing ′ = − ω = and . The wage function is the expected output among its workers. ν d Kremer (1993) shows how the model sheds CK _y ( d ) (i) ______ (1 i) − τ −ν 1 (0). light on several stylized facts, such as the ω = d ( − − ) + ω τ − ν Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 501

Assuming d , Gabaix and Landier we will see in section 2.4 that moral hazard τ > ν (2008) calibrate the model and analyze sev- leads to imperfect TU. eral features of CEO pay and its increase in The type x of a principal is the variance recent years in the United States (see also of her output, while the type y of the agent Tervio 2008). They show that the model is his coefficient of absolute risk aversion. exhibits a “superstar” property (Rosen A match of principal x and any agent gen- 1981): small differences in talent can have a erates stochastic output q e , where = + ε drastic impact in pay at the top—i.e., CEOs N(0, x), and the agent’s effort e 0 ε ∼ ≥ with rank close to one, here.16 Also, the incurs disutility k e 2/2. With CARA utility increase in size of large firms in recent function, agent y’s expected utility is 1 y(I ke 2/2) − years can account for a large fraction of the e − − , where I is income. The optimal increase in CEO pay. contract is linear in output: I(q) I bq, = 0 + where I is a base wage and b the incentive 2.2.3 Matching Principals and Agents 0 power. By Holmstrom and Milgrom (1987), In the principal–agent model, the princi- the optimal contract sets b 1/(1 kyx) = + pal hires an agent to perform a task. Since and yields the principal expected profit the agent’s actions are unobservable, the con- f (x, y) 1/(2 k(1 kyx)). (The base wage = + tract is based on a stochastic signal, such as I0 is irrelevant with CARA utility.) Finally, output, that is correlated with those actions. we can check that f 0 if and only if yx x y < _ Ackerberg and Botticini (2002) convinc- 1/k, and this holds for all x _x , x and < _ _ _ ∈ [ ] ingly argue that accounting for endogenous y [y , y ] if y x 1/ k. So NAM emerges ∈ _ < matching of principals and agents is import- if this condition holds, and PAM if _ y _x ant when testing predictions of contract the- 1/ k. > ory, since it can bias many of the relevant This principal–agent model predicts that coefficients. Using data from Renaissance b is decreasing in x , namely, a negative rela- Tuscany, they find strong evidence for tionship between risk and incentives. The matching between landlords with crops of evidence on this prediction is weak: the data different riskiness and tenants with different exhibits either a positive or an insignificant levels of wealth (proxying for risk aversion), relationship. Embedding the principal–agent which affects the contract form used (share problem in a matching model can account contracts or fixed-rent contracts). for this finding if NAM is optimal: for in this Serfes (2005) explores a tractable match- case, high variance x principals are matched ing model of heterogeneous principals and with less risk averse agents y (x). Since = μ agents under moral hazard. He restricts is strictly decreasing, the incentive power μ attention to linear contracts and constant b 1/(1 k (x) x) could increase in x if = + μ absolute risk aversion (CARA) utility func- matching is endogenous. If (x) x decreases μ tion (i.e., using the standard justification of in x, which depends on the distributions G Holmstrom and Milgrom 1987), and this and H, then b increases in x, as the evidence turns the model into a matching problem shows. with perfect TU. Without this assumption, 2.3 Frictionless Sorting with Imperfectly Transferable Utility

16 Differentiation reveals that (i) is strictly increasing 2.3.1 Background ω and strictly convex, with (i) going to infinity asi goes to ω″ one. Thus, CEOs matched with large firms receive increas- When partners cannot transfer utility one ingly larger pay near the top. for one, we say that there is imperfect TU. 502 Journal of Economic Literature, Vol. LV (June 2017)

Many economic environments of interest fall that is, x with y and x with y . Then x and y ′ ″ ″ ′ ′ ′ into this category, such as risk-sharing prob- can block the matching and offer to rematch, lems or matching problems where moral since f (x , y ) f (x , y ) and f (x , y ) 1 ′ ′ > 1 ′ ″ 2 ′ ′ hazard is present. f (x , y ), due to the monotonicity in > 2 ″ ′ In one extreme case, NTU, partners can- partner’s type. Similarly, NAM emerges if not transfer utility at all. For instance, the one of the partial derivatives is positive and output f (x, y) from a match between x and the other one is negative. So while Becker y may be divided according to some fixed did not cite Gale and Shapley (1962), he sharing rule; more generally, assume that if x intuitively grasped their pairwise stability matches with y, then x obtains utility f1 (x, y) notion in this case. and y obtains f2 (x, y), as done in Smith Recently, Legros and Newman (2010) (1997). Actually, matching models with- have shown that PAM does not require out transfers have been extensively studied monotonicity in the partner’s type. Indeed, since Gale and Shapley (1962)—see Roth the necessary and sufficient condition for and Sotomayor (1990). In their two-sided PAM in this setting is that preferences matching model, preferences are formulated exhibit “co-ranking”: given any two men as ordinal rankings over the partners on the and women, either the top man and woman other side of the market, and an equilibrium prefer each other, or the bottom man and is defined in terms of stability. A matching woman do. In the example above, this means is stable if there exists no blocking pair of that it has to hold for any two pairs of men agents, preferring to be matched to each and women, and this condition is consis- other rather than their respective partners tent with f , i 1, 2, not being increasing i = in the candidate allocation. The fundamen- in partner’s type (see Legros and Newman tal result is that a stable matching exists. The 2010 for an example). existence proof is constructive by means of Thus far we have explored assortative the deferred acceptance algorithm. One side matching in both the perfect TU and the of the market, say women, can make offers NTU cases. Notice that the Pareto frontier of to their preferred man, who temporarily payoffs achievable by a pair of matched agents retains his best choice. Each woman who is linear in the perfect TU case (with constant has not been retained then makes an offer to slope that is independent of the agents’ types), her second most preferred man. Again, men and collapses to a point in the NTU case. retain their most preferred women, possibly What about typical intermediate cases where dropping an earlier retention. This process agents can transfer utility but not at a constant continues until no more women are left who rate, so that the Pareto frontier is decreasing prefer any man over remaining single. This but neither linear nor a single point (see fig- yields existence of a stable matching, and it ure 1)? Legros and Newman (2007) address highlights also that there may be multiple this case. We now provide a detailed summary ones. of their main insights and several illustrative Using a cardinal representation of pref- applications. erences, Becker (1973) noted that if each 2.3.2 The General Model agent strictly prefers a partner with a higher type, then PAM emerges under NTU. In our There are two populations, women and notation, we need f to be strictly increas- men, indexed by types x 0, 1 and 1 ∈ [ ] ing in y and f in x. To see this, consider two y 0, 1 . For simplicity, we assume that 2 ∈ [ ] women and two men, with types x x and they have the same size, which can be finite ′ > ″ y y , who are matched in a NAM way, or a continuum, and their autarchy payoff is ′ > ″ Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 503

Panel A. TU Panel B. NTU Panel C. Strict NTU v x v x v x ( ) ( ) ( )

x, y, w f x, y ϕ( ) 1( )

w y w w y f2 x, y w y ( ) ( ) ( ) ( ) Figure 1. Examples of Pareto Frontiers for TU and NTU Cases

Note: We depict examples of the Pareto frontiers respectively for perfect TU; the intermediate imperfectly TU case introduced in Legros and Newman (2007) (with a decreasing nonlinear frontier); and NTU.

normalized to zero. We identify agents by and w( (x)) (y, x, 0) for all x and y; μ ≤ ψ their types, so that agents of the same type and (ii) stability of with respect to v and μ behave alike and receive the same payoff in w, so that there is no pair of agents with equilibrium—the “equal-treatment” prop- x and y with w w(y) and (x, y, w) > ϕ erty deduced in Legros and Newman (2007). v(x). This subsumes the perfect TU > To capture a utility frontier for each pair model with (x, y, w) f (x, y) w, and ϕ = − of agents, let (x, y, w) be the maximum subsumes the NTU model with (x, y, w) ϕ ϕ utility that x generates when matched with f1 (x, y) 1 w f (x, y). = = 2 y, if y receives utility w. Since no agent receives less than their autarchy payoff in 2.3.3 Generalized Increasing Differences equilibrium, (x, y, 0) is the maximum that ϕ x can obtain when matched with y. We First consider supermodular production assume that (x, y, w) is strictly decreas- with perfect TU, so that PAM is optimal. If ϕ ing in w when positive. In later applica- x x and y y , then y can weakly outbid > ′ > ′ tions, we derive this Pareto frontier from y in the competition for x, or f (x, y) f (x , y ′ − ′ assumptions on technology and preferences. ) f (x, y ) f (x , y ). Rewrite this increas- ≥ ′ − ′ ′ Let (y, x, v) be the maximum utility of y ing differences condition as ψ when matched with x who receives utility v. This is the partial inverse of (x, y, ), in ϕ · (6) f (x, y) f (x , y) v the sense that (x, y, (y, x, v)) v for all − [ ′ − ] ϕ ψ = v 0, (x, y, 0) . ∈ [ ϕ ] f (x, y ) f (x , y ) v , The equilibrium concept is the core of ≥ ′ − [ ′ ′ − ] this assignment game—namely, a matching function and utility functions v for types where x obtains utility v . Inequality (6) holds μ ′ x and w for types y, that satisfy the follow- for any level of utility v. By this increasing ing properties: (i) feasibility of v and w with difference condition, higher types choose respect to , so v(x) (x, y, w( (x))) higher-matching partners, and thus PAM is μ ≤ ϕ μ 504 Journal of Economic Literature, Vol. LV (June 2017) optimal. Extending (6) to our richer class of Finally, we can write this in a standard match payoffs in the general NTU case, y single-crossing form with an inequality can weakly outbid y in the competition for premise—that for any (y, w) and (y , w ):17 ′ ′ ′ x when the outside option is a match with x ′ who earns utility v if and only if (8) (x , y, w) (x , y , w ) ϕ ′ ≥ ϕ ′ ′ ′ (7) (x, y, (y, x , v)) (x, y, w) (x, y , w ) ϕ ψ ′ ⇒ ϕ ≥ ϕ ′ ′ (x, y , (y , x , v)). for all y y and x x . ≥ ϕ ′ ψ ′ ′ > ′ > ′ This reduces to (6) with perfect TU, as We next provide a new differential version (x, y, (y, x , v)) f (x, y) (y, x , v) of (8). Let be twice continuously differen- ϕ ψ ′ = − ψ ′ ϕ f (x, y) f (x , y) v , and similarly the tiable. Then under the regularity assump- = − [ ′ − ] right sides of (6) and (7) coincide. There are tions in theorem 3 in Milgrom and Shannon generalized increasing differences if (7) holds (1994),18 the single crossing property (8) is whenever x x, y y , and v is feasible, equivalent to the Spence–Mirrlees condition, > ′ > ′ namely, v 0, (x , y, 0) . that the marginal rate of substitution / ∈ [ ϕ ′ ] −ϕy ϕw Legros and Newman (2007) prove that between y and w increases in one’s type x. when (7) holds, all equilibria are payoff Since is twice differentiable, we have ϕ equivalent to PAM. Similarly, given the reverse condition, generalized decreasing (9) (x, y, w) ϕxy differences, all equilibria are payoff equiva- (x, y, w) lent to NAM. These conditions are necessary ______y ϕ xw (x, y, w) if PAM must hold for any type distribution. ≥ w (x, y, w) ϕ To wit, as with Becker’s supermodularity ϕ and submodularity conditions, generalized since 0. In other words, a high type ϕw < increasing and decreasing differences are x is willing to pay more of w for an incre- the necessary and sufficient distribution-free ment in his partner’s type y. Consequently, conditions for PAM and NAM, respec- the indifference curves in (y, w)-space single tively. This powerful result nests perfect TU cross as x changes. and NTU as special cases, and thus greatly We are now equipped to give a simple enlarges the set of economic applications for smooth argument for why (9) leads to PAM which we can assert PAM and NAM. with a continuum of agents. Mimicking logic Condition (7) can be usefully simplified. familiar in screening models (Fudenberg Label w (y, x , v) and w (y , x , v). = ψ ′ ′ = ψ ′ ′ Then v (x , y, w) (x , y , w ), so that = ϕ ′ = ϕ ′ ′ ′ 17 type x obtains the same utility v either in a Indeed, (8) implies (7), since equality is a special case ′ of the left side of (8). To see that (7) is equivalent to (8), match with y, paying him w, or in a match recall that strictly falls in the partner’s utility. Let v ϕ 1 ≡ (x , y, w) (x , y , w ) v , so that w (y, x , v ) with y, paying him w . Then (7) asserts ϕ ′ ≥ ϕ ′ ′ ′ ≡ 2 = ψ ′ 1 ′ ′ and w (y , x , v ). Since is also decreasing in v, we (x, y, w) (x, y , w ), so that x x ′ = ψ ′ ′ 2 ψ ϕ ≥ ϕ ′ ′ > ′ have (y , x , v 2) (y , x , v 1). Along with (7), this yields obtains more utility by matching with y ψ ′ ′ ≥ ψ ′ ′ than with y , if she must pay them the same. (x, y, w) (x, y, (y, x , v 1)) ′ ϕ = ϕ ψ ′ Put differently, if x is indifferent between (x, y , (y , x , v )) (x, y , (y , x , v )) ′ ≥ ϕ ′ ψ ′ ′ 1 ≥ ϕ ′ ψ ′ ′ 2 (y, w) and (y , w ), then x x prefers ′ ′ > ′ (x, y , w ). (y, w). Formally, if x x then (x , y, w) = ϕ ′ ′ > ′ ϕ ′ 18 See also theorem 2.1 in Edlin and Shannon (1998), (x , y , w ) (x, y, w) (x, y , w ). = ϕ ′ ′ ′ ⇒ ϕ ≥ ϕ ′ ′ and theorem 3 in Athey, Milgrom, and Roberts (1998). Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 505 and Tirole 1991, chapter 7), assume a PAM We repeatedly exploit it in later applications. allocation y (x), with strictly increas- This inequality reveals a tension between = μ μ ing and smooth. Consider the problem that complementarity in one’s own type and one’s type x solves when matched with a type partner’s type ( ) or in a partner’s utility ϕxy y agent who earns w(y). If this is an equi- ( ). The latter reflects whether transfering ϕxw librium, the FOC for type x’s optimization utility to a partner becomes easier as one’s type max (x, y, w(y)) holds when evaluated at increases. In the perfect TU case, the second y ϕ y (x), or complementarity is absent, and (9) collapses = μ to Becker’s condition fxy 0, since fxy xy 1 ≥ ≡ ϕ (10) − (y), y, w (y) with perfect TU. Next, assume that ϕy (μ ) ϕ increases in one’s partner’s type ( y 0). 1 ϕ > − (y), y, w (y) w (y) 0. Since 0, inequality (9) and therefore + ϕw (μ ) ′ = ϕw < PAM ensues if 0 and 0 while ϕxy ≥ ϕxw ≥ Next, the solution to (10) is a global maximum. NAM obtains if 0 and 0. ϕxy ≤ ϕxw ≤ For consider any other type, say yˆ y. Then Intuitively, type complementarity abets sort- > ing, whereas increasing difficulty of transfer- (x, yˆ , w(yˆ )) (x, y, w(y)) ring utility to one’s partner (namely 0, ϕ − ϕ ϕxw ≤ recalling that 0) discourages sorting. ϕw < yˆ But if transferring utility becomes easier with (x, s, w(s)) = y [ϕy higher types ( 0), then the two effects ∫ ϕxw ≥ reinforce each other and PAM obtains. (x, s, w(s))w (s) ds + ϕw ′ ] 2.4 Applications of Frictionless Sorting ˆ y 1 − (y), s, w(s) with Nontransferable Utility = ∫y (−ϕw (μ )) 1 2.4.1 Matching Principals and Agents y ( − (y), s, w(s)) ______ϕ μ w (s) ds × − 1 − ′ ( w ( − (y), s, w(s)) ) We revisit this application from sec- ϕ μ tion 2.2, but now without assuming CARA ˆ and linear contracts. An agent’s character- y 1 − (y), s, w(s) ≤ y (−ϕw (μ )) istic y is her initial wealth, which affects ∫ her risk attitude. His utility is V (y I) e, + − 1 where e 0, 1 is the disutility of exerting y ( − (s), s, w(s)) ______ϕ μ w (s) ds, ∈ { } 1 effort. Per usual, we assume V 0 V , × − − (s), s, w(s) − ′ ′ > > ″ ( ϕw (μ ) ) with a decreasing coefficient of absolute risk aversion V / V . The agent’s effort is − ″ ′ where the inequality follows from / unobservable, while his output q q_ , q is −ϕy ϕw ∈ { _} increasing in x and the last line vanishes observable. If the agent exerts effort e 0, = by the FOC (10). Hence, (x, y, w(y)) then output is low q q for sure. If she ϕ = _ _ (x, yˆ , w(yˆ )), so x does not have incen- exerts effort e 1, then output q q ≥ ϕ = = tives to deviate up. A similar argument shows with probability x 0. Principals differ in x, > _ that x does not incentives to choose yˆ y. a riskiness measure. We assume that q q ≤ − _ Hence, choosing y is a global optimum for x . is large enough so that principals always want In other words, the differential inequality to implement e 1. = _ (9) is an easily-checked condition for PAM A contract is a pair (_ I , I_ ) of wages con- (NAM) in imperfect TU matching models. tingent on outputs _q and q . If principal x 506 Journal of Economic Literature, Vol. LV (June 2017) matches with agent y , whose reservation util- 2.4.2 Marriage and Risk Sharing ity is w , the resulting contracting problem is _ _ _ (x, y, w) max x q I (1 x) Consider a marriage market where men ϕ = I_ , I ( − ) + − q _ I subject to an incentive con- and women with different wealths (and thus × (_ − ) _ different risk aversion levels) marry to share straint xV y I (1 x)V(y _I ) 1 20 ( + ) + − + − risk. If a woman of wealth x marries a man V(y _I ) and a participation constraint ≥ +_ of wealth y, then they share the risk embed- xV y I (1 x)V(y I_ ) 1 w. ded in a gamble whose payoff q 0, 1 ( + ) + − + − ≥ ∈ [ ] has a continuous distribution . The utility Both constraints bind at the optimum: if Γ function of women is log (1 x I) and of either is slack, then wages can be reduced + + men is log (1 y I), where I is income. to strictly raise the principal’s expected + + profit. Solving_ the two binding con- Efficient risk sharing solves the following 1 21 straints yields I Z( w x − ) y and problem: = + 1 − _ I Z(w) y, where Z V − . So: = − ≡ (12) (x, y, w) ϕ _ (x, y, w) x q Z w __ 1 ϕ = ( − ( + x)) 1 max log (1 y q I(q)) d (q) = I( ) ∫0 + + − Γ (1 x) q Z(w) y. · + − (_ − ) − subject to Notice that is strictly concave in w, as in ϕ figure 1. Moreover, (x, y, w) 0 and 1 ϕxy = log (1 x I(q)) d (q) w, ∫0 + + Γ ≥ (11) (x, y, w) ϕxw where I(q) is the woman’s share of q and q I(q) the man’s share. __ 1 Z w __ 1 Z w __ 1 Z (w). − = x ″( + x) − ′( + x) + ′ Intuitively, the constraint binds at the optimum. Maximizing pointwise, we obtain If Z is convex,19 then the first term in (11) I(q) ( (1 x) (1 y q) )/(1 ), ′ = − + + + + ζ + ζ dominates the last two, and so 0. where is the Lagrange multiplier. Inserting ϕxw ≥ ζ Since 0, we obtain PAM by (9). I(q) into (12) and solving for yields ϕxy = ζ That is, agents with high initial wealth and thus low risk aversion matched with prin- (13) (x, y, w) ϕ cipals with safer output distributions. As 1 Legros and Newman (2010) point out, the v log(2 x y q) d (q) log(1 e −∫0 + + + Γ ) sorting pattern emerges despite the lack of = − any complementarities between x and y. 1 If instead Z is concave, then 0 and log (2 x y q) d (q). ′ ϕxw ≤ + 0 + + + Γ there is NAM. ∫

19 Since Z 1 / V , this is convex if and only if ′ = ′ V / V 3V / V . Many standard utility functions sat- 20 For a survey of the literature, see the book by − ‴ ″ ≤ − ″ ′ isfy this condition, including CARA (exponential) utility as Browning, Chiappori, and Weiss (2014). well as V(I) I ξ with 0.5. This commonly emerges 21 This is a general version of an example in Legros and = ξ ≤ in principal–agent models with moral-hazard and wealth Newman (2010), section 5.1 (see also Chiappori and Reny effects. Conversely, Z is concave, say, for V(I) I ξ with 2015 and Schulhofer-Wohl 2006). We use our differential ′ = 0.5. version of their condition to readily check for NAM. ξ > Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 507

We claim that NAM emerges, i.e., wealthy Assume the following technology for women who are less risk averse marry poor producing output: men who are more risk averse. Intuitively, f (1, ) 0, f (1, 1 ) 4, f (1, 2 ) 1, f (1, 1, 2 10 a more risk averse individual is willing to {∅} = { } = { } = { }) = f (2, ) 0, f (2, 1 ) 8, f (2, 2 ) 5, f (2, 1, 2 ) 9. pay more for insurance. To wit, a highly risk {∅} = { } = { } = { } = averse man can outbid a less risk averse one Then all possible allocations with full for a wealthy woman. employment are blocked. For instance, if To prove this result, one must check that firm 1 hires both workers, then the total ( / ) . It is easy to check that output is ten at firm 1 and zero at firm 2. ϕxy ≤ ϕy ϕw ϕxw 0 and 0, so that the quick Worker 1 must earn at least eight, namely, ϕxy < ϕxw > sufficient condition for PAM or NAM does what firm 2 is willing to offer when it hires not hold. But some algebra reveals that only him. Likewise, worker 2 must earn at ϕxy ( / ) . Thus, the optimal sorting least five. But this total-wage bill overex- < ϕy ϕw ϕxw pattern is NAM. hausts firm 1’s total output. So this allocation is not stable, and all other allocations 2.4.3 Matching in Large Firms are also blocked. The nonexistence is driven by worker The one-to-one matching paradigm misses complementarities. Specifically, if firm 1 an important feature of actual labor markets. hires worker 2 when it already employs Labor market realism demands that firms worker 1, then worker 1’s productivity rises, can hire many workers. In principle, one and she can thus command a higher wage. could reinterpret a firm as a series of inde- The gross substitutes condition in Kelso and pendent jobs that do not affect any other Crawford (1982) precludes this possibility, job’s productivity. Often, however, there are and secures for existence. It asserts that complementarities between jobs. if wages increase for some workers, then Gale and Shapley (1962) highlighted the firm will not drop from its labor force the importance of many-to-one match- any worker whose wage did not increase. ing with their college admissions problem. Additively separable production functions A more relevant setup for labor market easily obey this condition, since a work- applications is one with a large number er’s productivity does not depend on her of firms and transfers (wages). In the coworkers. Gross substitutes is sufficient O-ring technology in Kremer (1993), firms and almost necessary, as it leaves little employ a given fixed number of workers.22 room for complementarities (Hatfield and Kelso and Crawford (1982) develop a gen- Milgrom 2005 and Hatfield and Kojima eral many-to-one matching model of firms 2008). Kelso and Crawford (1982) also pro- to any number or type of workers. Existence vide an algorithm that finds the equilibrium of equilibrium, however, is not guaran- allocation and wages. It is a variation on the teed, as the following simple example illus- deferred acceptance algorithm of Gale and trates. Consider two workers x 1, 2 Shapley (1962) (see section 2.3).23 = and two firmsy 1, 2. Let f (y, X ) be Modeling firms with an endogenous size = { } the output of firm y matched with a set and labor force composition with comple- of workers X 1 , 2 , 1, 2 , . mentarities has proven difficult, given the { } ∈ {{ } { } { } {∅}} gross substitutes condition. For instance, in

22 For another application where firms hire teams with a fixed number of experts, see Chade and Eeckhout 23 Hatfield and Milgrom (2005) apply their ascending- (forthcoming). bid auction to analyze package auctions. 508 Journal of Economic Literature, Vol. LV (June 2017) a model of the span of control by manage- optimization problem reduces to maximizing ment, Lucas (1978) focused on the inten- (y, x, (x)). Since this has the NTU struc- ψ ω sive margin decision of how many workers ture analyzed in section 2.3, PAM or NAM to hire, but ignored composition. A more ensues depending on whether (y, x, ) ψ ω productive management hires a larger work has x / globally increasing or globally −ψ ψω force, increasing its span of control. This decreasing in y for all (y, x, ), i.e., ω ψxy − model shed light on the distribution of firm ( x / ) y 0 from (9). Now, x fx , ψ ψω ψω ≷ ψ = size. In reality, however, management at a , and y fy by the envelope the- ψω = −θ ψ = firm also faces an extensive-margin decision orem. Also, for fixed in (x, y, ), it fol- ω ψ ω about workers’ composition; this is precisely lows from implicitly differentiating the FOC the focus of matching models. f (x, y, ) that y fy f and θ θ = ω θ = − θ / θθ Eeckhout and Kircher (2012) develop a x fx f . Therefore, differentiating θ = − θ / θθ tractable model with size and composition yields y y, while differ- ψω = −θ ψω = −θ margins. We present a slightly simplified entiating x fx gives xy fxy fx y ψ = ψ = + θ θ version that yields their sorting condition. fxy fx fy f . Next, = − θ θ / θθ Assume match output is given by a produc- fy fx tion function F(x, y, lx , r x), where y is the _____θ θ xy fxy fy x fxy firm type,x the worker type, lx the labor ψ = + θ θ = − f θθ force size of type x , and r x the resources the and firm dedicates to workers of type x. That is, a firm of typey hires a quantity or mea- ___fx x θ . sure lx of workers of a common type x at a ψ ω = − f wage (x) per worker.24 Since a firm chooses θθ ω both the type and the number of workers, Since f 0, the condition for PAM θθ < the model embeds size and composition reduces to fxy f fy fx fx fy / 0, θθ − θ θ + ( θ θ) ≤ margins: It is inspired by Becker’s TU pair- for wise matching model, but instead allows for variable sizes of one side of the match. It is (14) 0 xy ( x / ) y ≤ ψ − ψ ψω ψω further assumed that F is strictly concave in l and r, and exhibits constant returns in fy fx fy l, r, so that F (x, y, l, r) r f (x, y, ), where f _____θ θ f / ___θ ≡ θ = xy − f + ( x θ)f f (x, y, ) F(x, y, , 1), and where l/r θθ θθ θ ≡ θ θ = is the labor resources ratio. The prob- fx fy lem of firmy reduces to the maximization ____θ fxy f fy fx / f , problem: maxx , f (x, y, ) (x). Let us = ( θθ − θ θ + ) θθ θ θ − θω θ solve first the maximization in , for each x. θ The FOC f (x, y, ) (x) yields the which is the inequality in Eeckhout and θ θ = ω unique maximizer (x, y, (x)). Recalling Kircher (2012). The reverse inequality yields θ ω that F is strictly concave in its third argu- NAM. ment, we have f 0. Define (y, x, ) We now express this in terms of F (x, y, l, r). θθ < ψ ω f (x, y, (x, y, )) (x, y, ) . Then the Since F is homogeneous of degree one in ≡ θ ω − θ ω ω (l, r), F l is homogeneous of degree zero, and so l F ll r F lr 0 by Euler’s theorem, i.e., 24 Eeckhout and Kircher (2012) provide a justifica- + = F lr F ll. Additionally, Fx is homoge- tion for focusing on a single type of worker by positing a − = θ production function that is additively separable in worker neous of degree one in (l, r), whence Fx F F , by Euler’s theorem. So the types. = θ xl + xr Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 509

inequality fxy f fy fx fx fy / 0 of workers by skill. Moreover, the optimal θθ − θ θ + ( θ θ) ≤ can be rewritten as matching need not exhibit PAM. The models analyzed above assume F F F F F __x that output f depends only on the types of xy ll − yl( xl − ) θ matched pairs. In some applications, the value of the match to a pair also depends Fyl on the entire matching, which gives rise to a F F F ___ = xy ll + xr problem with externalities. Sasaki and Toda θ (1996) and Pycia and Yenmez (2015) have F Fyl analyzed matching with externalities, intro- F ___lr F ___ 0, = − xy + xr ≤ ducing notions of stability with externalities θ θ in both the perfect TU and NTU cases, and which rearranges to the key finding of studying their implications. More recently, Eeckhout and Kircher (2012), namely, that Chade and Eeckhout (2015) analyze the F F F F delivers PAM. The left impact of externalities on the optimal and xy lr ≥ xr yl side of this inequality involves the stan- equilibrium matching patterns using a dard cross-interactive term between the two-stage model of teams where teams are worker and firm types, scaled by the cross- formed and later compete. interactive term of labor and resources of We have focused on matching by agents the firm. Meanwhile, the right side is the with scalar characteristics. The multidimen- product of the cross-interactive terms of sional problem—say where men and women firm type and labor force size, and that of differ in education, income, attractiveness, the worker type and firm resources. This etc., or in a labor market where firms have analysis highlights the importance of the many heterogeneous tasks and workers differ condition in Legros and Newman (2007) across several skill dimensions—is technically for analyzing sorting patterns in matching harder. Chiappori, McCann, and Nesheim problems. (2010) explore existence and uniqueness of equilibrium in the multidimensional perfect 2.4.4 Further Topics TU matching model. They use tools from the Many applications involve just one popula- optimal transport literature, linking match- tion of agents, such as collaboration between ing models and hedonic pricing models. partners in law firms, team members in con- Lindenlaub (2014) provides a notion of sort- sulting or sports, gay marriage, etc. Although ing for multidimensional problems and stud- the existence of stable matchings might be ies a matching model where workers have problematic (Roth and Sotomayor 1990), both manual and cognitive skills and firms sometimes one can divide agents into two have jobs demanding both skills. Using US sides and match them as if they came from data, she analyzes technological change and a two-sided problem. Kremer and Maskin its effects on the wage distribution. (1996) explore such a model in which identi- Most of the matching literature assumes cal agents might be able to perform different that agents’ characteristics are primitives tasks with different productivities. If man- of the model. A small literature explores agers and workers are drawn from the same ex ante investments followed by a matching population and they are complementary but stage. A standard question addressed in managers play a more important role, then these papers is whether the prospect of they show that technology changes can aggra- a better match induces agents to invest vate wage inequality and the segregation ex ante, thereby mitigating the hold-up 510 Journal of Economic Literature, Vol. LV (June 2017) problem. Another one is to understand go to women, but with slightly more women how imperfections at the matching stage than men, the opposite holds. 25 Or assume combine with the investment problem a world with heterogeneous people available ex ante and generate inefficiencies. Early for matches. There can be implausibly discon- work here includes Makowski and Ostroy tinuous matching allocations. Suppose that (1995), Felli and Roberts (2016), and Cole, match payoffs are f (x, y) 1 xy, with = + ε Mailath, and Postlewaite (2001). Noldeke 0 and incredibly small. Depending on | ε| > and Samuelson (2015) and Bhaskar and whether 0, we either have positive or ε ≷ Hopkins (2016) explore the efficiency of negative assortative matching, respectively. pre-matching investments, and Chade and So the frictionless predictions by Becker of Lindenlaub (2015) derive comparative stat- both the matched and unmatched agents are ics of risk on pre-matching investments. counterfactual in different ways. Additionally, Search frictions or learning, analyzed later, the Walrasian auctioneer fiction is a far less can lead to mismatch and so rematch. In accurate description of the actual matching Chade and Eeckhout (2016), agents match process, since the Walrasian fiction rings less based on observable characteristics that true for a market with a massive number of index the distributions of payoff-relevant essentially unique items for sale. Indeed, attributes that are revealed after the match. organizing this as a market is the very chal- For example, employers may sort workers by lenge facing online matching services. education, hoping this signals later produc- To fully understand how search frictions tivity, and this leads to mismatch. They pro- distort equilibrium market outcomes, we vide an empirical application of the model to first need to learn some basic decision theo- the assignment of CEOs to firms. retic tools of search theory. Below, we explore single-agent search theory and illustrate it with some economic applications. Search in 3. Foundations of Search Theory microeconomic models is usually modeled in two ways: sequential, where the decision 3.1 Why Search Frictions? maker samples options over time until she The matching paradigm as we have decides to stop, and nonsequential or simul- described it has some unrealistic economic taneous, where all the options are sampled properties. For one thing, it predicts no at once and then the best one is chosen. In unmatched agents, except due to obvi- all cases, search theory explores how option ous imbalance. For another, it says nothing value governs choices: where to search or about mismatch among those that do match. how long or how much to search. Just as in Finally, its predictions are excessively vola- finance theory, an option value is increasing tile in a counterfactual way. To see this last in the riskiness of the choices, since extreme claim, consider a standard Walrasian model. events yield the surplus. Either a small change in the supply of some 3.2 Simultaneous Search endowment, or a slight increase in the number of individuals with some preference, The seminal paper by Stigler (1961) has only a small effect on the price. But in a started the literature on search in econom- pairwise matching setting, slight imbalances ics. In this model of simultaneous search, a can sometimes have dramatic effects. For consumer samples prices from a distribution, instance, consider a marriage market with homogeneous men and women. With slightly 25 For a recent reference on this issue, see Ashlagi, more men than women, all matching rents Kanoria, and Leshno (2017). Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 511 and chooses how many searches to make. an interesting dynamic wrinkle, to which we Each search costs c 0. Specifically, sup- will return. Since the locations of prizes are > pose you are searching for a product, and known, but their realizations are not, this must buy it today. In the morning you can may be better thought of as an information call many (ex ante) identical stores, and in friction. Unlike in Stigler’s model, one must the afternoon, after searching through their choose the colleges to apply to, and not sim- stock, they will call back with a price quote. ply their number because colleges vary by Upon observing the prices sampled, the con- admission chances and career value. sumer buys the product in question from Recently, Chade and Smith (2006) the firm that quoted the lowest price. The extended the simultaneous-search paradigm optimal sample size is an easy optimization to allow for ex ante heterogeneous options. problem in one variable and, for some dis- The decision maker chooses not only the tributions, it can be obtained in closed form. number of options to sample, but also the Let the distribution of prices be given sample composition. Each option gener- by a non-degenerate distribution F(p) on ates a stochastic reward. After observing the 0, 1 . A consumer chooses a fixed sam- rewards of each option, the decision maker [ ] ple size n to minimize the expected total chooses the largest one. Specifically, imagine cost C (expected purchase cost plus search a set of colleges 1, 2, , N , with payoffs { … } cost) of purchasing it. With n independent v v v . Since better colleges 1 > 2 > ⋯ > N draws, the distribution of the lowest price is are presumably harder to secure entry to, F (p) 1 1 F(p) n. Thus, if one will assume inversely ranked admission chances n = − [ − ] purchase K units, the expected total outlay is . α1 < α2 < ⋯ < αN Chade and Smith (2006) then deduce the 1 optimal portfolio of any given size n N P(n) K p d Fn (p) ≤ = ∫0 —and thus solve the richer problem of the 1 optimal portfolio when all college applica- K 1 F (p) dp = 0 [ − n ] tion costs are c 0. (Their analysis does not ∫ > 1 help if application costs vary.) In principle, K 1 F(p) n dp. = 0 [ − ] the optimal portfolio might require search- ∫ ing through all possible n -subsets, or in the Observe that 1 F(p) n, and thus the richer problem, all 2 N portfolios. Could [ − ] expected cost P (n), falls in n , but at a dimin- college students actually be solving such a ishing rate. Thus, the second-order con- fantastically seemingly complex NP-hard dition is met, and the optimal sample size problem? The authors prove that a simple n∗ obeys the discrete first-order condition: marginal improvement algorithm (MIA) P(n ∗ 1) P(n∗ ) c P(n∗ ) P(n∗ 1). yields the optimal portfolio, and it only takes − − ≥ > − + Easily, a larger planned purchase K raises about N 2 steps to find the best portfolio with the marginal benefit of sampling, and thus N schools: At stage 1, one selects the school induces weakly more searches n ∗ . with greatest expected value. If that value Stigler’s fixed sample size search is tough exceeds c, then put college i in the tentative to motivate, as it is almost always a contrived portfolio. At any stage n 1 in the recur- + thought experiment (as above). But there is sion, one finds the school i n 1 yielding the one major occasion in life when we make greatest marginal benefit on+ the portfolio such a one-shot search experiment: applying constructed so far. Add that school to the for college. This ignores the possibility of tentative portfolio if the incremental value is early admission at one school, which adds at least the cost c. Otherwise, stop. 512 Journal of Economic Literature, Vol. LV (June 2017)

That this algorithm works is surprising, Using the algorithm, Chade and Smith since the problem is static and not amena- (2006) prove that students apply more ble to dynamic programming. One could aggressively than they would if they were easily imagine that a college optimal for unaware of how their colleges jointly inter- one portfolio size might not remain so for act in their portfolio. They should not blindly a larger portfolio. The proof that this never apply to their best expected options. For happens—a joint mathematical induction on instance, the lower-ranked colleges 2 and 3 the number of options and cardinality of the have the two highest expected payoffs. portfolio set—shows why one never wishes This is best seen as a justification for to remove a college added at an earlier stage. why students pursue “stretch schools.” For The MIA is a member of a class of “greedy assume a world with just college i and algorithms,” in which a sequence of locally many identical lower-ranked colleges j. optimal choices leads to the global optimum. Assume that even though any such college j For a minimal illustrative example, assume has a lower payoff v v , it has a higher j < i just three colleges with payoffs v 1, expected value v v . As a result, the 1 = αj j > αi i v 0.8, and v 0.6, and admission 2 = 3 = MIA starts with college j. While it may well chances 0.5, 0.8, and 1. α1 = α2 = α3 = continue to add “copies” of college j, college i The expected payoffs z v are therefore i ≡ αi i is eventually chosen by the algorithm before z 0.5, z 0.64, and z 0.6. With an 1 = 2 = 3 = exhausting all of the j colleges. Let’s see how a application fee c 0.15, the optimal port- = temptation to gamble upwards emerges. The folio includes college 2. The marginal benefit marginal benefit of adding more college j of adding college 1 to a portfolio 2 is { } copies vanishes geometrically in their number, and so eventually falls below (v v ) αi i − j − MB z (1 ) z z c 0, for any application cost c 0. By 12 = [ 1 + − α1 2 ] − 2 > > contrast, the marginal benefit of adding col- z z = 1 − α1 2 lege i to a portfolio of n colleges j is since college 2 is only relevant in the event n i v i i v j (1 ( 1 j) ) c that one is rejected at college 1. On the other α − α − − α − hand, in pondering the marginal benefit i (v i v j) c. of adding 3 to a portfolio 2 , we note that > α − − { } college 2 matters whenever one is accepted For large enough n , adding the stretch appli- there. The marginal benefit is computed cation to college i is the best course of action. therefore in a different way: By the same token, “safety schools” can only be understood if acceptances are not MB z (1 ) z z independent. For instance, suppose that a 32 = [ 2 + − α2 3 ] − 2 common unknown shock may affect all col- (1 ) z . lege evaluations. If a student is unaware, = − α2 3 e.g., that math requirements have shot up, We conclude from the MIA that college 1 then as insurance, he might wish to pursue belongs to the optimal portfolio, since a safety-school strategy. This remains a chal- z z 0.18 0.12 (1 ) z . lenging but important research avenue. 1 − α1 2 = > = − α2 3 Finally, one does not wish to add college 3 Modeling search frictions in this simul- to this portfolio since (1 )(1 ) z taneous way appears in some equilibrium − α1 − α2 3 0.06 0.15 c, and thus, the optimal search models of price dispersion such as = < = college portfolio is 1, 2 . Burdett and Judd (1983); directed search { } Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 513 models with one or multiple applications and reservation wage, and will in the future be, ex ante identical firms such as Burdett, Shi, his new optimality condition requires: and Wright (2001) and Albrecht, Gautier, _ ∞ _ and Vroman (2006); as well as search prob- (15) w max ( w , w) dF(w) c lems with multiple applications and hetero- = ∫0 − geneous options such as Chade, Lewis, and ∞ c _ 1 F(w) dw. Smith (2014) for college admissions, and ⇒ = w [ − ] Kircher (2009) and Galenianos and Kircher ∫ (2009) for labor markets. Since the problem is stationary, a wage once rejected is forever rejected. As a result, an 3.3 Sequential Search option to return to a previously declined Amusingly, while Stigler’s paper intro- option is worthless. This expression admits duced price search—deploring information some immediate predictions. For instance, as the “slum dwelling in the town of eco- the hazard rate of finding jobs is constant nomics”—his model was almost immediately through time. Also, if one interprets c as abandoned. After McCall (1965) intro- foregone unemployment benefits, then the duced sequential search to economics, the reservation wage rises in these benefits. And simultaneous-search model was essentially by standard stochastic dominance reasoning, ignored until Chade and Smith (2006). For a mean-preserving spread of the wage dis- in Stigler’s model, if the searcher could tribution F likewise raises the reservation decide sequentially on whether to continue, wage, since the max operator is convex, and he does better. Indeed, he would always have hence it encourages risk-taking behavior— available the fixed sample size commitment for instance, acquiring information about F. policy, simply by ignoring what he has seen The impact on search duration is a priori until sampling n ∗ stores, and then picking the ambiguous—for the searcher is more ambi- best so far. But if given the option to recall a tious with a mean-preserving spread, but past search, he might well wish to stop either there is also more probability weight in the earlier or later. McCall (1970) re-worked his upper tail. Choi and Smith (2016) resolve 1965 model for wage search, assuming that a this ambiguity, showing that if every pair worker samples a wage from a distribution in of percentiles of the wage distribution shift each period and decides whether to continue apart—namely, there is a right shift in the the search, or stop and work at that wage. dispersion order_—then the hazard rate of His classic model has become a fundamen- stopping 1 F( w ) falls, and consequently − tal building block for macroeconomic mod- the search duration rises. els of the labor market (see the discussion in We pursue a richer model than McCall section 3 in Rogerson, Shimer, and Wright (1970) that allows for ex ante hetero- 2005), and we will also use it extensively geneous options: the Pandora’s box of in section 4. The worker’s optimal strategy Weitzman (1979). Assume a finite number of is_ fully summarized by a reservation wage heterogeneous options, each represented by w above which the worker stops searching, a unique probability distribution Fk (w) over and below which he continues.26 Since he prizes. Opening box k costs c 0, and k > is now indifferent when faced with his incurs a time discounting factor (0, 1 , δi ∈ ] due to delay. Only one prize may ultimately be accepted. Payoffs are independent, 26 Morgan and Manning (1985) endogeneized the sample size at each stage of the search process, thereby blend- and the decision maker must sample them ing sequential and simultaneous search. sequentially. At any point in time, she can 514 Journal of Economic Literature, Vol. LV (June 2017) decide to stop the search and keep the best past options. Any such recall freedom has no reward observed thus far. So an optimal strat- value in the stationary job search setting. egy requires specifying the order to explore Keeping in mind our predictions about options and a stopping rule. how reservation wages change, we can see Uncertain options in life should be under- that the searcher will first explore options taken as long as one is sufficiently opti- with lower costs, higher means, and higher mistic. Uncited in Weitzman (1979) was variances. Weitzman gives an example where the earlier solution of the infinite horizion the first options explored are statically dom- multi-armed bandit problem in Gittins and inated, with a lower mean and a higher cost. Jones (1974) and later Gittins (1979)—the They are valuable provided they have a high so-called Gittins index.27 When arm payoffs enough variance. are independent, the index for each arm Let’s revisit the college application prob- solely reflects the uncertainty of that arm. lem of section 3.2. Assuming that one could Capturing the contingent_ decision making, it apply in sequence to colleges, one would is the fixed prize w k that leaves the decision optimally employ Weitzman’s rule. This is maker indifferent about choosing a prize, true even with college-specific application and paying to open box k , knowing that the costs. With our binary payoff distribution, the prize awaits him if he wants it. Namely: index equation (16) of college i reduces to: _ _ _ ∞ _ w (1 ) w u c (16) w max ( w , w) d F (w) c . i = − αi i + αi i − k = δi 0 k k − i ∫ _ w u c / (z c) / . ⇒ i = i − αi = i − αi Solving the problem by induction and dynamic programming, Weitzman showed So sequential decision making is governed that the optimal selection and stopping rules not by the expected net gains z i, but instead were then straightforward: at each stage, by the expected net gain divided by the the decision maker samples the option with probability of success. Chade and Smith the largest index and stops when the reward (2006) prove that individuals act more observed exceeds the reservation values of aggressively with sequential decision mak- all the remaining options.28 Notice that the ing than simultaneous choices. One should reservation wage equation (15) emerges from pursue less likely options first. For instance, (16) with homogeneous options and costs and in the example of section 3.2, college i 2 = no discounting—namely, F F, c c, is the first applied to; however, the indexes i = i = and 1. McCall (1965) intuitively feels are δi = like a special case of Weitzman (1979) when _ there is a vast number of identical options. w u c/ 1 0.15/0.5 1 = 1 − α1 = − This logic shows that the reservation wage coincides with Weitzman’s index with only 0.7 0.6125 0.8 0.15/0.8 = > = − finitely many options when there is recall of _ u c/ w . = 2 − α2 = 2

27 A multi-armed bandit is a finite action, infinite hori- A general lesson is that sequential deci- zon Bayesian experimentation problem. When the payoff sion making pushes toward more risk- of each “arm” is independent of all others, the optimal taking behavior. For example, early admis- strategy is given by Gittins indices. 28 Olszewski and Weber (2015) explore the limits of this sion might be unwise for the most elite class of index rules for different payoff functions. schools because it encourages more Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 515

aggressive “stretch” applications by weaker This admits the intuitive statement that the students who otherwise would not apply. return on the value equals the sum of the These optimal stopping problems have dividend c and the expected capital gains, − i long been explored in operations research, namely, the_ expected surplus of prizes x over where it is assumed that a decision maker the value w k. chooses a sequentially optimal time at which he takes an action to maximize his 3.4 Sequential Search with Hidden expected payoff. As expected, their solution and Known Components typically involves heavy use of dynamic- programming tools. There are excellent ref- Choi and Smith (2016) explore another erences on the subject, ranging from elemen- specialization of Weitzman (1979) in which tary to advanced, such as DeGroot (1970); they assume that the prize distributions Fk Chow, Robbins, and Siegmund (1971); reflect the sum of hidden and known com- Shiryaev (1978); Ross (1983); Ferguson ponents, each with a common distribution. (2016); and Peskir and Shiryaev (2006). For simplicity, we touch on their applica- Modeling search as a sequential process tion to web search, since it is a major new is standard in much of economics. For way that sorting and matching are proceed- instance, it is widely used in macroeco- ing now.29 nomic models of the labor market (e.g., In this case, assume that all payoffs W Rogerson, Shimer, and Wright 2005), in the hail from a Gaussian distribution. Since the literature of matching with vanishing fric- Gaussian distribution is stable, it can be tions (e.g., Osborne and Rubinstein 1990), parsed as: and in assortative matching models with _____ search (see section 4). (18) W X 1 2 Z , = α + √ − α One might view search theory as optimal stopping when one knows the payoff dis- in which X and Z are each standard normal tribution, but not the realizations. In a key random variables. For instance, after enter- extension of the sequential-search problem, ing the keyword, the search engine ranks the Rothschild (1974) explored the implications search outcomes by the realized known com- of learning the distribution while search- ponents X . To learn the idiosyncratic compo- ing. This has been revisited in many guises, nent Z, the user must click on the website recently by Adam (2001), and Gershkov and and read it. Moldovanu (2012). The search engine accuracy represents α Finally, in equilibrium applications that how effective the search engine is in reduc- we will study in section 4, a continuous- ing the idiosyncratic noise, rendering more time search model with exponential arrivals predictable web searches. When 0, α = replaces the discrete-time model. Assume the problem reduces to a stationary search that the arrival rate is 0. Subtract problem. In that case, the user employs the ρ > w from both sides of (16), and think of same cutoff for all periods, and will never δi k 1 r dt, and c as a flow search cost. use the recall option unless the last period δi = − i Then (16) becomes is reached. When 1, the websites are α = _ (17) r w k 29 Varian (1999) mused on practical advice for offering _ books to a rushed consumer in an airport bookstore. He ∞ max (w w , 0) d F (w) c . identified Weitzman (1979) as a parable for web search = ρ ∫0 − k k − i engines. 516 Journal of Economic Literature, Vol. LV (June 2017)

perfectly sorted, and the user will stop at the environment. For instance, search intensi- first result. In this case, the recall option is like- fies, with recall rates and quitting rates rising wise unused. For intermediate 0 1, over time. < α < the user faces a nonstationary search prob- 3.5 Sequential Search by Committee lem with decreasing cutoffs, and so might well recall an earlier draw. This is intuitively We finally explore an intriguing appli- the world most of us find ourselves in while cation of search theory as it applies to the searching the internet. search for job candidates. Of the two sem- Even though the search problem is inal papers here, Albrecht, Anderson, and highly nonstationary, the options stochas- Vroman (2010) and Compte and Jehiel tically worsen as one proceeds through (2010), we focus on the former (AAV), the list: x x . In this model, a since it shows how search costs skew 1 > 2 > ⋯ user clicks on the k th web site if and_ only the partner search process. Consider a if the best draw so far lies below w department seeking to hire job candidates k = α x ( ), where x reflects the common that arrive sequentially, one per period. k + ζ α α k component, and ( ) measures the search The search cost is impatience: the payoff is ζ α optionality—namely, the net benefits of the discounted by 0 1. Each member i < δ < idiosyncratic randomness. Given (18), the of hiring committee of N members observes threshold ( ) obeys a reservation equation a random private value W 0, 1 , ζ α ∈ [ ] akin to (15), conveniently invariant to the web independently drawn from the common site rank k: cdf F . After seeing her private value, each committee member casts a yea or nay vote ∞ y c 1 ______ dy. to hire the current candidate or continue = − Φ _____2 the search. Search ends with the current ( ) [ (√ 1 )] ζ α − α option if at least M N members vote to ∫ ≤ One can compute that the implied “option- hire the current candidate, and continues ality measure” ( ) monotonically falls in otherwise. ζ α accuracy —as a concave and then con- For a flavor of the theory, consider first a α vex function of , with extreme values committee of one. It votes to hire a candi- α _ (0) c (1). date when W w , given the reservation ζ > − = ζ _ ≥_ Choi and Smith (2016) assume that indi- value w V( w ), where V is a fixed point = δ viduals can quit any search and exercise an V T V of the Bellman operator T V(w) = outside quitting option u , or continue search- (1 F(w))E W W w F(w)V(w). = − [ | ≥ ] + δ ing. Search engines are keenly interested in Next, assume a size N 2 committee. In = the chance that one never quits searching, so the symmetric equilibrium, each player still that the search engine secures a successful employs a reservation value w, and secures match. This chance increases in accuracy Bellman value V (w). For M 1, the value M = if and only if u ( ). In other words, solves the Bellman recursion T V V , < ζ α 1 1 = 1 there may be a conflict of interest between where online shopping sites (for, say, Amazon) and consumers. In particular, when the outside T1 V(w) (1 F(w)) E W W w options are high or the price is low, a shop- = − [ | ≥ ] ping website secures higher sales with a nois- F(w)(1 F(w)) E W W w ier search engine. + − [ | ≤ ] Choi and Smith (2016) derive an array F (w) 2 V(w). of results for this nonstationary search + δ Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 517

For any player j’s payoff exceeds the thresh- In the single agent problem, mean- old w with chance 1 F(w), whereupon preserving spreads are unambiguously − search ends with that payoff E W W w . beneficial, since one can always dis- [ | ≥ ] Next, j’s payoff is below w with chance F (w). card low draws. In the committee search In this case, if the other player k’s payoff problem, AAV show via example that exceeds w (chance 1 F(w)), then j earns mean-preserving spreads can lower welfare. − his low payoff E W W w . Otherwise, j For a mean-preserving spread can increase [ | ≤ ] earns the discounted continuation value. either (a) the stopping externality, by mak- Since the operators are ordered T1 V ing low draws more costly, or (b) the contin- TV for all v , their fixed points are ranked uation externality, by increasing the chance < V V, and their reservation values like- that another member of the committee 1 < _ _ wise so: w w . In other words, commit- blocks you, or both. 1 < tee members are individually less picky: the This literature adding strategic elements single agent i rejects any candidate that com- to the search problem is an inviting future mittee member i rejects. The value reflects direction, in light of the important of col- a stopping externality, the bad event that lective decision making in resolving search Ms. k votes to stop when Mr. j has a low draw. frictions. Next assume unanimity is needed: M 2. = The expected value V (w) is then a fixed 2 4. Search and Matching point of: 4.1 An Introduction to Sorting in Search T V(w) (1 F(w)) 2 E W W w 2 ≡ − [ | ≥ ] and Matching Models F(w) (2 F(w)) V(w) In the Walrasian matching model, it is + − δ _ _ costless for agents to find potential partners, with reservation wage w V ( w ). be they women searching for men, work- 2 = δ 2 2 By the same logic, committee members ers searching for jobs, or buyers searching are less demanding than _solo searchers_ for sellers. Building on the basic matching in equilibrium—namely, w w —now model, we now introduce search frictions. 2 < because of a continuation externality, in This twist is important, since it eliminates the bad event that Ms. k votes to continue discontinuous matching sets and wage pro- when Mr. j has a high draw. Since the stop- files (in section 3.1) predicted by the fric- ping or continuation externalities obtain tionless model, and explains important on any search committee, the committee is phenomena like equilibrium unemployment always less choosy than the solo searcher in and imperfect sorting (mismatch), and ratio- any symmetric equilibrium. When M 1, nalizes price and wage dispersion. = this reservation value ordering implies that We distinguish between time intensive the committee concludes search faster on random search, and directed or competi- average, since_ the continuation_ probability_ tive search, where markets clear by queues is lower: F ( w ) N F( w ) F( w ). But and stockouts. Some of the questions we 1 < 1 ≤ when M 1, there is a trade-off—a candi- address are: How do market frictions affect > date must independently pass several lower match formation and sorting in marriage thresholds. AAV show that when M N, and labor markets as well as in models of < the committee concludes search faster then bilateral trade? Who matches with whom a solo searcher with enough patience or in equilibrium? The literature frontier impatience: for ( , ). assumes a common evaluation of agents, δ ∉ δL δH 518 Journal of Economic Literature, Vol. LV (June 2017) without a hint that beauty is in the eye of of unmatched agents. On the other hand, in the beholder,30 and this remains a major the quadratic-search technology, unmatched direction of future research. individuals face a constant arrival rate of poten- But the quest to enrich Becker’s framework tial partners times the mass of unmatched ρ and account for search frictions has received agents; here, the mass of new matches is pro- an enormous amount of attention in recent portional to the squared mass of unmatched years. Among the earliest such models of agents. We will refer to as a meeting rate, ρ heterogeneous agent search were Bergstrom or possibly, the rendezvous rate. For intuition, and Bagnoli (1993), which we explore in this arises if invitations to meetings arrive at section 5.2, and Smith (1992). These papers fixed rate to everyone, but when either party ρ assumed NTU, and were followed up by is already matched, he misses the meeting. Burdett and Coles (1997) and Smith (2006). The quadratic-search technology embeds But the proper extension of Becker’s model a crucial analytic advantage: players are unaf- required perfect TU, as was later assumed fected by the matching decisions of those in Shimer and Smith (2000). Recently, the unwilling to match with them. This strategic literature has explored the implications of independence greatly simplifies equilibrium replacing the assumption of anonymous analysis. By contrast, in order to hold the search by that of directed search (e.g., matching rate constant with a linear-search Eeckhout and Kircher 2010a, Shi 2001), technology, new individuals that enter one’s where agents can identify where they send matching set crowd out previous individu- their applications to find potential partners. als. This complicates and sometimes renders With random search, there are several impossible equilibrium analysis, especially in other modeling assumptions besides perfect a nonstationary environment. We now illus- TU or NTU. First, it is standard to assume trate this with Smith (1992), the first hetero- continuous time with an exponential arrivals geneous agent search model that properly of matching opportunities. Next, for models tracked the demographics. This leads us into of partner search, it is common to model the sorting results. search cost as impatience, rather than an 4.2 Sorting with Random Search and explicit search cost. Also, with a few salient Nontransferable Utility exceptions, the unisex model is assumed for simplicity, but a similar analysis can be done 4.2.1 Block Assortative Matching with two distinct sides, as in the frictionless case. When individuals sort into matches by Crucially, one must take a stand on the anonymous random search, an intriguing nature of the search technology (Diamond equilibrium matching pattern emerges. To see and Maskin, 1979). With anonymous search, this, assume that everyone is summarized by a unmatched individuals meet one another scalar type in 0, 1 , and posit a uniform den- [ ] in direct proportion to their mass in the sity on types. Assume no match complemen- unmatched pool. But what then is the propor- tarities,31 and posit that anyone matching with tionality constant? In a linear search technol- type x earns payoff x . A pairwise matching mar- ogy, potential partners arrive with constant ket is newly opening, with everyone initially rate 0. To wit, the density mass of new unmatched, and meeting potential partners ρ > matches is linearly proportional to the mass according to a quadratic-search technology

30 This assumption has already been explored in a 31 Some introspection reveals that this analysis also search and trading model in Smith (1995). works when the payoff to the match of types x and y is xy. Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 519 with meeting rate . We assume that match- Since types in v_ , 1 match just among them- ρ [ ] _ _ ing is irreversible, but this will be optimal ex selves and they have mass u v , the qua- − post anyhow. Everyone wishes to enter a per- dratic-search technology implies that_ their_ manent match with the highest discounted unmatched density u (x) falls at rate ( u v ). ρ − expected present value. In the equilibrium Because the mass and first moment of agents _ _ _2 that transpires, intuitively everyone will start below v is respectively_ v and v /2, the laws off wanting to match with the highest types, of motion for u and are thus: χ but then these types gradually vanish, and _ _ _ the cutoff monotonically falls. By the logic of u ( u v ) 2 ′ = −ρ − section 3.3, the marginal type that leaves one indifferent between matching_ and continuing and is the unmatched value v . If this marginal _ _ _ type monotonically vanishes, then the equi- ( u v )( v 2/2). χ′ = −ρ − χ − librium assumes the form: any types in v_ , 1 [ ] agree to a match when they meet, and every_ - All told, this nonstationary “rush” equilib- one declines to match with anyone in 0, v ). rium is captured by this three-dimensional [ _ _ _ To verify this equilibrium, we adapt the state ( v , u , ). Naturally, the threshold v χ reservation wage equation (17) for a non- vanishes; Smith (1992) argues that it does so stationary world, in which there is a further at rate O(1/t).33 _ capital loss reflecting the falling value v 0 Assume now a flow entry with uniform ′ < _ (suppressing the time subscripts). For any density e 0 on 0, 1), the threshold v no > [ interest rate r 0, the return is the sum of longer vanishes. For a constant inflow of ≥ the “capital gains” and “dividends,” namely: high types, there is now a strictly positive lower bound on the option value of waiting. _ _ 1 _ This equilibrium offers a unique approach (19) r v v _ (x v ) u (x) dx, = ′ + ρ ∫ v − to thinking about steady-state analysis. In the long-run limit equilibrium, we should where u(x) is the unemployment mass approach_ steady state, in which the thresh- density of type x 0, 1 . Tracking the old v is intuitively constant and strictly pos- ∈ [ ] 1 unmatched mass density u(x) is essential in itive. For the matching threshold need not heterogeneous agent search models. But we vanish in order to satisfy dynamic optimality. can greatly simplify the problem, and capture The logic of steady state requires that entry the evolution of this threshold simply using of new unmatched agents balance the flow two state variables—the_ total unmatched of agents from the unmatched pool into_ measure mass u 1 u(x) dx, and the first matches. This yieldes e u (1 v ). = ∫0 = ρ 1 − 1 moment 1 xu(x) dx of unmatched Meanwhile, the steady-state condition χ = ∫0 _ agents. So the average type of an unmatched v 0 in (20) intuitively yields an optimal- _ 1 = agent is / u . Rewrite the law of motion (19) ity′ condition: χ for the unmatched value as:32 _ 1 ______2 r v 1 u 1 _ (x v 1 ) dx (20) v r v ( v u v /2). = ρ v 1 − ′ = + ρ − χ + ∫ _ u (1 v ) 2/ 2. _ _ _ = ρ 1 − 1 32 Write (19) as v r v _1 (x v ) dx. Since types _ ′ = − ρ ∫ v − below v have not matched, they_ still have a uniform_ density, so that u(x) 1 for all x v . Hence, _1 (x v )u(x) dx 33 In pursuant work, Damiano, Li, and Suen (2005) = _ _≤_ _ ∫ v __− _ 1 xu(x) dx v x dx v ( u v ) v u v 2 / 2. develop a general theory of how matching unravels. = ∫0 − ∫0 − − = χ − + 520 Journal of Economic Literature, Vol. LV (June 2017)

_ Jointly, we can solve for the pair (u 1, v 1 ). types do not typically have entirely dis- But _then to understand how types jointed sets of match partners. Burdett and x v match, the logic starts anew. Coles (1997) instead took it as a parable of < 1 Recursively, _there _is a sequence of thresh- “marriage and class” in Britain. It is a very olds 1 v v and associ- stark form of PAM.35 The question then > 1 > 2 > ⋯ ated unmatched rates u 2, u 3, computed arose as to when matching sets were contin- 34 … inductively. In other words, whenever_ uous in types. Relatedly, with a continuum of two types in the same interval v_ k, v k 1) types, it is not even clear what we should call [ − meet, for some k 1, 2, , they agree to positive sorting. For since every type must = … match; otherwise, the higher type declines match with a positive mass of types, it is no the match. Smith (2006) suggestively called longer possible to assert that the percen- this equilibrium block segregation. This tiles of matched men and women coincide. balanced-flow approach was formally devel- Shimer and Smith (2000) offered a formula- oped for any matching sets in (the 1997 tion of PAM and NAM that simultaneously working paper version of) Smith (2006), with applies to singleton matching sets or sets of a general existence proof. positive measure. This definition asks that The earliest heterogeneous agent sort- the matching set as a subset of 2 be a lat- 핉 ing paper, McNamara and Collins (1990) tice. In other words, if (x 1, y 2) and (x 2, y 1) assumed a stationary equilibrium, without are willing to match, and x x , y y , 1 < 2 1 < 2 clarification. Later, Bloch and Ryder (2000) then so are (x 1, y 1) and (x 2, y 2). Intuitively, made an endogenous “cloning assumption,” any mismatches are explained by the thick- positing that agents who leave the matching ness of the matching set. So when matching market are somehow magically replaced by sets collapse to a singleton for each x, as in clones. This simply requires solving a set Becker’s marriage model, this notion reduces of difference equations, as does the later to an increasing or decreasing function under approach in Burdett and Coles (1997), PAM and NAM, respectively. who assumed a fixed flow entry of all types By drawing a suitably small rectangle at and no match dissolution. Finally, if match the edge of the matching set, this definition dissolution is indeed economically central, implies that each type x matches with types then one can assume a constant stock of in an interval a(x), b(x) , with both a(x) and [ ] agents, and just ask that severances balance b(x) weakly increasing in x. With NAM, the new matches for each type; Shimer and functions a and b are decreasing in x. This Smith (2000) take this tack, as their primary definition has compelling economic impli- application is labor. cations. Most easily, the distribution of part- We see how the first taste of sorting mod- ners with whom x matches is increasing in x els with search frictions entailed jumps: in the sense of first-order stochastic domi- Individuals match in ranked-equivalence nance; as a result, the expected value of the classes. Smith (2006) highlights the coun- partner with whom x matches is increas- terfactual discontinuous aspects of this block ing in x under PAM, and decreasing under segregation—extremely close individual

35 This early literature on sorting with search fric- 34 Further, by strategic independence of the qua- tions has many other participants, some with overlapping dratic-search technology (see section 4.1), these_ results, with varying insights into search costs, intermedia- can be computed_ in isolation: for we have r vk 1 tion, and the block segregation logic. See McNamara and _ _ 2 + _ vk uk 1 vk 1 (x vk 1) dx u k 1 v_k vk 1 / 2, as Collins (1990), Morgan (1996), Bloch and Ryder (2000), = ρ + ∫ + − + = ρ + [ − + ] k 1, 2, . Chade (2001), and Eeckhout (1999). = … Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 521

NAM. This is a testable implication for the We now set up and solve a contin- data. uum of heterogeneous but interlaced dynamic-programming problems. Let v(x) 4.2.2 Strict Assortative Matching be type x’s expected present discounted Under NTU, a sufficient condition for unmatched value, and v(x y) the analo- | PAM in the frictionless case is that the match gous value from being matched with y. output function is increasing in partner’s In the Bellman equation, there is no divi- type (Becker 1973). We now explore under dend (zero payoff while unmatched), and what conditions PAM ensues in this case an arrival rate of a capital gain equal to the with search frictions. expected match surplus: In Smith (2006), time is continuous on 0, ) and search is a time cost: unmatched [ ∞ (21) r v(x) agents discount the future at rate r 0. > There is a continuum of types x 0, 1 with ∈ [ ] cdf G and a positive density g. Unmatched (x) max (v(x y) v(x), 0) u(y) dy agents earn zero flow payoffs, while a = ρ ∫Ω | − match with a type x agent yields flow payoff f (x, y) 0. Here, f is strictly increasing in > A(x) (x) (v(x y) v(x)) u(y) dy. partner’s type, so that f 0 everywhere. = ρ ∫ ∩Ω | − y > No side payments are allowed (NTU). Already, Gale and Shapley (1962) predict Similarly, the matched value solves rv(x y) | that PAM is the unique stable matching. Is f (x, y) v(x) v(x y) . Naturally, v (x y) = + κ[ − | ] | that still true in a model with frictions? v(x) since f (x, y) r v(x). Since f 0, > > y > Let unmatched agents randomly meet, type x accepts all types in an upper set according to a quadratic-search technol- A(x) a(x), 1 . Also, the threshold part- = [ ] ogy with meeting rate 0. When agents ner a(x) 0 acts like a reservation wage, ρ > > meet, they approve a match if both earn non- and solves the indifference condition negative surplus. Matches vanish at match f (x, a(x)) r v(x). So the opportunity set = dissolution rate 0, i.e., the match lasts (x) y x a(y) is increasing in x. κ > t Ω = { | ≥ } past time t with chance e −κ . Substituting the expression for v(x y) and | Each type x agent chooses an acceptance A(x) a(x), 1 into (21) yields the explicit = [ ] set A(x) 0, 1 with whom she is will- recursion equation: ⊆ [ ] ing to match. In turn, x is deemed accept- able by the types in the opportunity set (x) y x A(y) . Hence, the match- (22) r v(x) Ω = { | ∈ } ing set of an agent with type x is A(x) (x). ∩ Ω Observe how NTU captures the classic dou- _____ρ f (x, y) r v(x) u(y) dy. ble coincidence of wants that money solves. = r A (x) (x) [ − ] + κ ∫ ∩Ω Of course, in the TU model, the matching decision is mutual, and so A(x) (x), = Ω whereas in the NTU model, A(x) should be This expression reveals how the return on a higher set than (x), since one’s prefer- the unmatched value reflects how matches Ω ences invariably surpass one’s opportunities. dissolve at rate . Finally, an equilibrium is κ The model is in steady state with a constant a triple (v, a, u), such that v(x) obeys (22) unmatched density function u, satisfying for the acceptance set A(x) a(x), 1 , a(x) = [ ] 0 u(x) g(x) for all x. solves f (x, a(x)) r v(x) given u ( ), and u (x) ≤ ≤ = · 522 Journal of Economic Literature, Vol. LV (June 2017) obeys the balanced flow condition (23) at Since y a, the integrand on the right > every type x:36 side increases in x if f (x, y)/ f (x, a) is strictly increasing in x—namely, f (x, y) is strictly (23) g(x) u(x) log supermodular (that is, log f is super- κ − ( ) modular). As a result, a(x) is increasing,

and so PAM ensues for the highest agents.

u(x) A(x) (x) u(y) dy. = ρ ∫ ∩Ω But then the inverse b(x) increases, and this logic works for all types x. To follow Smith (2006) and Chade (2001) suggest a the key observation that led to this general unified approach to exploring sorting under log-supermodularity condition for PAM, NTU; it applies to search with impatience, note that block segregation arises in Smith but extends to the case of fixed search costs. (1992) for any multiplicative payoffs f (x, y) If a(x) is weakly increasing in x , then so too is f (x) f (y), where each f is positive and = 1 2 i its inverse b (x).37 Graphically, matching then increasing. Next observe that such multi- engulfs all types between two increasing plicative functions are obviously log mod- bands a(x), b(x) , whereupon PAM obtains. ular. Not surprisingly, a(x) is constant if [ ] Assume for now that (x) 0, b(x) for f (x, y)/ f (x, a) is always constant. This also Ω = [ ] all x, and with b(x) weakly increasing. For offers another way of understanding why the highest types, this is true since b (x) 1. block segregation emerges for the assumed = Since a (x) is an optimal lower threshold, (22) payoffs when one cares about one’s partner’s becomes type, namely, f (x, y) y. = Although we have not solved the fixed b(x) search cost search, a similar analysis yields a(x) f (x, y)u(y) dy (24) r v(x) ______∫ the first-order condition: = b(x) u(y) dy ψ + a (x) ∫ b(x) (26) c ( f (x, y) f (x, a))u(y) dy. b(x) = a − f (x, y)u(y) dy ∫ a max ______∫ , = a b(x) The integrand here increases in x if f is u(y) dy ψ + ∫a strictly supermodular in (x, y), for then f (x, y) f (x, a) is strictly increasing in x for − where (r )/ encapsulates search all y a. Once more, the optimal threshold ψ = + κ ρ > frictions in a scalar constant. Assume an a(x) increases. But in this fixed-search case, interior solution. The first-order condition block-segregation PAM arises with a modular that determines the optimal threshold a(x) production function f (x, y) f (x) f (y) = 1 + 2 satisfies (Chade 2001). Morgan (1996) also studies NTU matching with fixed-search costs. b(x) f (x, y) All told, the productive conditions for (25) ______ 1 u(y) dy. ψ = f (x, a) − PAM are harder to satisfy in the presence of a ( ) ∫ time cost search frictions. In the frictionless setting, supermodularity gives higher types 36 Smith (1997, 2006) analyzes this equation. Smith (2011) shows that one essentially applies the implicit func- greater gains from matching up. But since tion theorem to deduce a continuous map from a( ) to u ( ). the search costs rise in proportion to the · · 37 The logic for PAM holds with two-sided matching, value, a stronger assumption is required—log with two populations, like men and women. In that case, the upper bound function b(x) is derived from the accep- supermodularity rather than just supermod- tance threshold of the other population. ularity. This ensures that higher types have Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 523 proportionately greater gains from matching Easily, match surplus equals match output up. While log-supermodular payoffs are suf- less the sum of the returns on the unmatched ficient for PAM, it is also necessary to ensure values: s(x, y) f (x, y) r v(x) r v(y). In ≡ − − PAM for all unmatched distributions—oth- a unisex model, it is natural to split surplus erwise, one could put a large mass on the equally.38 Since match surplus is nonneg- failure type set, and violate PAM. ative, we need no longer keep track of an The outlined argument yields a unique acceptance set and opportunity set, but a sin- equilibrium, given the unmatched density gle matching set M (x) y s(x, y) 0 . = { | ≥ } function u (x). But an equilibrium is really Analogizing (22): a triple (v, a, g). Moving outside our model with a differentiable type distribution, (27) r v(x) __1 _____ ρ = r Burdett and Coles (1997) provide a simple 2 + κ example with just two types, low and high, that exhibits multiple equilibria once the f (x, y) r v(x) r v(y) u(y) dy. unmatched density is accounted for. In the × ∫M (x) [ − − ] nonselective equilibrium, high-type agents accept both high and low types. In the The sufficient condition for PAM or NAM selective one, high-type agents only accept is much less obvious than it was with NTU, matches with other high types. If less than for the integral includes endogenous value half of types are high, then high types match functions. Shimer and Smith (2000) simpli- with lower probability than the low types, fied matters and restricted focus to increasing and so comprise most of the unmatched payoffs f , f 0. As in Becker (1973), they x y > pool; this raises the option value of wait- first argued that the locus of zero-surplus ing, and thereby induces them to choose matches is increasing in one’s type provided a higher reservation type. The selection of f 0. PAM then obtains if matching sets xy > types in the pool leads to multiplicity. are convex and a(0) 0, as a(x) and b(x) are = then weakly increasing. 4.3 Sorting with Random Search Now, if we assume f (x, 0) 0, then and Perfectly Transferable Utility x ≡ a(0) 0. Next, to deduce convex match- = We now turn to the other benchmark: ing sets, differentiate the Bellman equation matching with search frictions under TU. (27). Since match surplus s(x, y) vanishes It is essential to consider transfers between along the edge of the matching set M (x), an matched partners in order to broaden the intuitive application of the fundamental the- applicability of the search and matching orem of calculus yields: models. For instance, while there are cer- tainly nontransferable aspects to an employ- (28) r v (x) __ 1 _____ ρ ′ = r 2 + κ ment relation, the wage is the central part that determines the terms of trade. Even in f (x, y) r v (x) u(y) dy the marriage market, there are many trans- × ∫M (x) [ x − ′ ] fers between partners, both monetary (like shared income or joint mortgage payment) M (x) fx (x, y)u(y) dy ______ . and nonmonetary (such as division of child = ∫ 2 u(y) dy care or household chores). ψ + ∫M (x) The present value to any two matched types x, y 0, 1 is no longer exogenously ∈ [ ] 38 Called the Nash bargaining solution, this uses none fixed, since the surplus split is endogenous. of that concept’s richness, as the surplus frontier is linear. 524 Journal of Economic Literature, Vol. LV (June 2017)

This expression obviously parallels (22) and supermodularity and even more, depend- (24), and so the proof takes inspiration from ing on the precise form of frictions. An Smith (1997, 2006). Now, a simple suffi- important question is whether either model cient condition for the matching set of type just described can shed light on actual labor x to be convex is that his surplus s(x, y) be markets or if a new one is needed. The form quasi-concave in his partner y . To argue this, of search frictions is critical when identify- Shimer and Smith (2000) use f 0 and ing the complementarities between worker xy > the right v (x) formula in (28) to argue that ability and firm productivity. There are now ′ s (x, y) 0 for all large enough y 1. many papers bringing macro and micro x > < In this case, s(x, y) is quasi concave in y. search models with heterogeneous agents Next, using the same v (x) expression, and f to the data on labor markets. This exercise, ′ x log supermodular, Shimer and Smith (2000) even for applying Becker (1973) to the data deduce that r v (x)/ f (x, y) is increasing for on labor markets, is futile without possibly ′ x all small y 0. So in this case, s (x, y) profound modifications for frictions. Finding > x f (x, y) r v (x) is reverse single-crossing the conditions for sorting with more general ≡ x − ′ in y, i.e., positive and then negative as y rises. search costs is an important agenda. This implies that s(x, y) is quasi concave in The effects of frictional mismatch is y. Finally, a third single crossing property another frontier of this literature. For exam- argues that y is large enough for the first case, ple, Burdett and Mortensen (1998) intro- or small enough for the second, provided duced on-the-job search and Postel-Vinay fxy is log supermodular. Smith (2011) care- and Robin (2002) used it to capture import- fully distills this argument. ant aspects of the data. This creates an ini- Besides providing sufficient conditions tial mismatch among firms and workers, and for PAM, Shimer and Smith (2000) also then a career ladder for workers—with or prove existence of equilibrium. Their proof without complementarities. assumes the quadratic-meeting technology; Another line of research beginning with later, Noldeke and Troger (2009) showed Shimer and Smith (2000) questions the existence under the technically harder lin- correlation between worker and firm fixed ear-meeting technology. Manea (2017) effects. Using a simple two-period model recently dispensed with the quadratic with heterogeneous workers and firms, assumption, deducing existence in a larger Eeckhout and Kircher (2011) find that class of stationary search and matching mod- wages are non-monotonic in job productiv- els when there is a finite number of types. ity. While output rises in job productivity, Instead of time discounting, one could given the mismatch from search, a high assume a fixed search cost. In environ- productivity job has a high option value of ments where search resolves swiftly, a direct continuing vacancy. That job will match with search cost in monetary terms may be a a lower-skilled worker if it receives a high more appropriate measure of search costs share of the output. Wages then have an than the opportunity cost of time. Atakan inverted U-shape. (2006) shows that a sufficient condition for The presence of equilibrium mismatch PAM with TU is that production f (x, y) be offers sufficient variation to identify the supermodular. technology underlying the match value out- In summary, PAM obtains under super- put function f (x, y). In a two-step procedure, modularity in the benchmark TU model. Eeckhout and Kircher (2011) first derive But with search frictions, greater comple- the search cost from the wage distribution mentarity in types is needed, such as log earns across jobs, and then use the range Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 525 of job types that the worker matches with So the stable matchings in Gale and Shapley to derive the degree of complementarity (1962) are the limits of equilbria of the between workers and jobs. Hagedorn, Law, decentralized market with search frictions. and Manovskii (2012) extend this model to Without cloning, Lauermann and Noldeke a general setting akin to Shimer and Smith (2014) show that convergence of equilibrium (2000), and provide a methodology to iden- matchings is guaranteed if and only if there is tify the model.39 a unique stable matching. Otherwise, there It has long been recognized that labor mar- exists a sequence of equilibria converging to kets can generate multiplicity, and therefore unstable allocations. cyclical outcomes. Diamond (1982) shows 4.4 Directed Search and Sorting that multiple steady states can arise in a simple exchange economy, but his logic exploits 4.4.1 Background the increasing return-to-scale property of the quadratic-search technology. Pissarides and Anonymous random search takes the Petrongolo (2001) find evidence of constant dynamic process of partner quest in labor returns in the observed matching technol- and marriage markets seriously. Yet, when ogy in the labor market. Burdett and Coles agents are heterogeneous, the purely ran- (1997) deduce that heterogeneous types and dom meetings process perforce assumes an endogenous searching pool creates local that high types meet low types, even though increasing returns for subsets of types. In a they know they will never form a match. This related two-type model with transferable is particularly costly and, not surprisingly, utility, Shimer and Smith (2001) analyze quasi-market structures or coordination optimal policy and characterize the planner’s games have emerged that seek to mitigate assignment constrained by the search tech- these losses. nology. They find that even absent any intrin- In a competitive economy, prices play an sic uncertainty, optimal allocations may be informative role, since they signal willing- nonstationary. This draws into question the ness to buy and sell. With anonymous ran- focus on steady-state models in search, and dom search, prices simply determine the suggests that it may prove an intrinsic source output split between buyers and sellers, but of volatility. This is a difficult and inviting in no way influence the actual meeting pro- frontier of the search literature. cess. Directed search instead assumes that We have explored models with nonvanish- prices influence the meeting process rather ing search frictions. The literature on search than solely the surplus split. Whereas trading and trade, by contrast, focuses on the mini- partners meet and then determine the price mal frictions case and seeks a foundation for through bargaining with random search, the the Walrasian outcome. To briefly touch on order is reversed under directed search. the analogous question here, Adachi (2003) There, sellers commit to a price and post it, assumes cloning and impatience and shows and after observing the price, buyers choose that, for a general match output function f, with whom to trade. This allows buyers to the set of stationary equilibria converges to direct their search towards sellers that offer the set of stable matches as frictions vanish. better prices.40 But it also allows coordination

39 See also the related work by Teulings and Gautier 40 The directed-search model, mostly for homogeneous (2004), De Melo (2009), Bagger and Lentz (2014), agents or heterogeneous agents without complementar- Lamadon et al. (2013), and Bartolucci and Devicienti ities, has extensively been analyzed since the late 1970s: (2013). to name a few, see Butters (1977), Peters (1984), Moen 526 Journal of Economic Literature, Vol. LV (June 2017) failures, in which either multiple workers buyer x pays price p for a good bought from turn up for one job and some workers remain seller y , then her payoff is f (x, y) p and the − unemployed in equilibrium, or in which no seller’s is p. The seller’s characteristics are workers show up, and some vacancies are observable. The characteristic distributions thus left unfilled. This yields wholly different G(x) and H (y) are continuous, with positive predictions than the Walrasian one. densities g(x) and h(y). There are two formulations of directed With anonymous random matching, the search in the literature. In one, the frictions expected number of newly forming matches are captured by queues, which determine depends on the mass of buyers and sellers a trading probability. This is often referred in the market. Under a standard assumption to as competitive search, following Moen that the “matching function” exhibits constant (1997), who analyzes stationary equilibria in returns to scale—e.g., twice as many buyers a continuous-time setting. In another formu- and sellers leads to twice as many matches— lation, the friction is embodied by the chance the matching function is linear in the ratio of of a stockout. Here, if several workers turn up buyers to sellers . This same matching-func- θ for a job, it is probabilistically rationed. Both tion logic applies to the directed-search approaches generate a trading probability as environment, where now coincides with θ a function of the ratio of applicants to jobs, the expected queue length in each submar- and a wage (transfer).41 ket at each seller. The interaction here takes Because of the search frictions, traders two stages. First, each seller y posts a price now value both the price and the probability p at which she is willing to sell the good. with which trade occurs. This trade-off gov- Second, buyers choose which seller (y, p) to erns agent’s optimal strategies: sellers that visit. We illustrate this idea in the model of post lower prices will attract more poten- Eeckhout and Kircher (2010a).42 tial buyers and will therefore sell with a With queue length , buyers meet sellers θ higher probability. Buyers who pursue low- with chance q( ) and sellers meet buyers θ er-priced goods must accept lower trade with chance m ( ) q( ). Buyer x thus has θ = θ θ probabilities since there are more compet- expected payoff q( ) f (x, y) p in pursuing θ [ − ] ing buyers. With two-sided heterogeneity seller y with price p and queue , and that θ and complementarity, this trade-off plays an seller has expected payoff m ( )p. Here, m is θ important role in the determination of the assumed twice continuously differentiable, equilibrium sorting patterns. strictly increasing, and strictly concave. Seller y could choose different pairs (p, ), 4.4.2 Sorting and Directed Search θ but buyer x would have to enjoy the same There is a continuum mass of buyers each reservation utility, say v (x). All told, seller y with a characteristic x 0, 1 and unit solves ∈ [ ] demands, and likewise of sellers with characteristic y 0, 1 , each holding a unit. For max m( ) p ∈ [ ] x, , p θ instance, this might be the housing-exchange θ model of Shapley and Shubik (1972). If the subject to

q( ) f (x, y) p v(x) θ ( − ) = (1997), Acemoglu and Shimer (1999), and Burdett, Shi, and Wright (2001). 41 The difference between the two interpretations 42 Versions of this model have been analyzed by Shi, becomes real for more general mechanisms than mere (2001), Mortensen and Wright (2002), Eeckhout and price posting (Eeckhout and Kircher 2010b). Kircher (2010a), and Jerez (2012). Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 527 where v(x) is the reservation utility of x. We vacancy; therefore, they prefer to trade with will show that this optimization is of the higher probability, and so opt for a shorter imperfect TU form in section 2.3 and, as a queue . As a result, many high-type workers θ result, we derive the sorting conditions by match with fewer low-type firms, and vice applying the differential inequality (9). versa. PAM only emerges if there are suffi- Substituting for p from the constraint yields ciently strong match complementarities. the optimization maxx, m( ) f (x, y) v(x). If the elasticity ( ) is identically zero, then θ θ − θ ξ θ Equivalently, each type y must maximize supermodular production f suffices for PAM, (y, x, v(x)) m( (x, y, v(x))) f (x, y) by the criterion. For in this case, m ( ) , ψ ≡ θ − θ = θ (x, y, v(x)) v(x) in x, where the optimal and so q( ) 1, which is to say that the θ θ = queue length (x, y, v(x)) solves the FOC arrival rate of sellers for each buyer is con- θ m ( ) f (x, y) v(x). Despite having lin- stant. At the opposite extreme, if the elastic- ′ θ = ear preferences over money, the objec- ity ( ) is identically one, then the criterion ξ θ tive function is nonlinear in v, reflecting asserts that log-supermodular production f the imperfect TU structure of match- gives PAM. So unless production f is “suffi- ing. PAM or NAM arises in equilib- ciently supermodular,” the search frictions rium depending on whether (y, x, v) lead to NAM. ψ m( (x, y, v)) f (x, y) (x, y, v) v satisfies The required condition on production is = θ − θ ( / ) 0. This allows us a stronger than supermodularity and weaker ψxy − ψx ψv ψvy ≶ quick derivation of the sorting condition (14) than log supermodularity for intermediate of Eeckhout and Kircher (2010a). For one elasticities 0 ( ) 1. For instance, if < ξ θ < can verify that m f / m f, as well as 0 ( ) 1 1, then PAM obtains θy = − ′ y ″ < ξ θ < − ϒ < m f and by the envelope the- if ψx = x ψv = −θ orem. Also, differentiating m f and ψx = x respectively yield m f fxy (x, y) f (x, y) ψv = −θ ψxy = xy ______ 1 . m f and . Hence: f (x, y) f (x, y) ≥ − ϒ + ′θy x ψvy = −θy x y

___ x This suggests a simple sufficient condition (29) xy ψ vy ψ − v ψ for PAM is supermodularity of the power ψ function f ϒ, and in the limit as 0, we ϒ → m fxy m y fx m fx y / recover log supermodularity (to see this, = [ + ′θ ] − [ ]θ θ take the limit of ( f ϒ 1)/ ). Altogether, − ϒ Becker’s supermodularity assumption on pro- f f ___xy m fx fy ( ) , duction no longer suffices for sorting in the = [fx fy − ξ θ ] frictional world. With anonymous random search and ex post equal surplus splitting, where ( ) m ( m m)/(m m ). Then log supermodular production is required. ξ θ = ′ θ ′ − ″ θ PAM or NAM arises as f f / (f f ) ( ) for When search is directed, the reduced fric- xy x y ≷ ξ θ all . tions entail generally an intermediate level of θ To gain some intuition about this condi- supermodularity. tion, assume first no match complementari- 4.4.3 Market Segmentation ties f 0, as in the labor market model of xy = Mortensen and Wright (2002). If the elastic- An alternative to directed search assumes ity ( ) is always positive, then NAM arises, by random matching, but allows heterogeneous ξ θ this criterion. Since output is purely additive, agents to set up separate trading posts. high-type firms have the most to lose with a Jacquet and Tan (2007) find that even with 528 Journal of Economic Literature, Vol. LV (June 2017) random matching, there are gains for indi- optimal second-price auction by a monop- viduals to set up segmented markets. For olist, but here it coincides with the cost. example, starting from the block assorta- This reflects the competition among sellers tive matching equilibrium in section 4.2, all for buyers, as with Bertrand price competi- agents in the upper class prefer to meet only tion. Since the number of buyers each seller among themselves in a segregated market. attracts is inversely related to the seller’s Since the distribution of singles is trun- reserve price, this is a force toward lower cated, the acceptance threshold of each agent reserve prices. rises. This gives them incentives to continue In an endorsement of second-price auc- segregating, further refining their type parti- tions, McAfee (1993) also shows that this tion. One might expect that this leads to per- mechanism is always a best response to any fect segregation, as in the frictionless model, arbitrary set of mechanisms by the other with each agent type matching with a unique sellers. Peters and Severinov (1997) and type. Jacquet and Tan (2007) refute this intu- Peters (1997) later justified the large market ition. For no agent can commit to rejecting assumption in McAfee (1993) by considering a partner slightly below his ideal. Because the limit of equilibria with finite markets as of search frictions, it is costly to wait for the the number of agents grows large. McAfee’s ideal partner, and so if the current candidate results suggest that auctions are superior is marginally lower, the agent will accept. to posted prices when there is competition So equilibrium segmentation consists of a between auctioneers. But Eeckhout and sequence of nondegenerate intervals of types. Kircher (2010b) argue that this result depends on the ability of auctioneers to 4.4.4 Competing Mechanisms extract rents ex post from buyers by having There is a close relationship between them simultaneously participate in the auc- directed search and competing mechanism tion and then screening them. To see this, design. McAfee (1993) and Peters (1997) notice that in the queueing interpretation of argue that if one allows the set of feasible directed search (Moen 1997), a posted price mechanisms to include more than just price is optimal because a firm only faces one bid- posting, then price posting need not be opti- der at a time. Thus, the use of posted prices mal. Assume that buyers have heteroge- or auctions reflects the nature of the search neous valuations for the good sellers offer, frictions. If the auctioneer cannot round up and sellers have heterogeneous costs for the sufficient applicants to bid in the mechanism, good they sell. Sellers simultaneously choose she is better off posting a price.43 Auctions and commit to a mechanism that maximizes are optimal when meeting probabilities are their profits within a broad class, includ- unaffected by the decision of another buyer ing auctions, price posting, and bargaining. to visit a submarket.44 With posted prices, Observing the announced mechanism, buyers then visit a seller, equally randomiz- 43 Pinheiro (2012) provides a microfoundation in a ing over all sellers who announce the same model where the firm’s mechanism design problem takes mechanism. place in continuous time with random arrival of buyers. McAfee (1993) finds an equilibrium in Now firms face a trade-off between trading fast and having few bidders or waiting longer to round up more bidders. which firms choose a second-price auction He shows there is an optimal interior solution with finite with a reserve price equal to the firm’s cost time before trading and applies it to initial public offerings. for the good. Buyers visit each seller with a 44 Lester, Visschers, and Wolthoff (2015) strengthen this “non-rivalry” condition—namely, the quadratic-search probability falling in the reserve price. The technology defined in section 2.4—by deriving a suffi- reserve price strictly exceeds the cost in the cient condition that is also necessary and which they call Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 529 buyers sort across posted price-trading prob- This is useful for applied analyses of match- ability pairs, revealing their types ex ante. ing workers to firms, and where firm size But with competing auctions, buyers visit the is endogenous.45 Finally, Eeckhout and sellers randomly and are screened ex post. Kircher (2012) (in section 4.1) contains an In a labor market setting, Shi (2002) and extension with directed search and derives Shimer (2005) consider different forms of a sorting result with heterogeneous workers competition with the flavor of an auction, and large firms. where the price paid depends on the composition of the ex post demand—specifically, 5. Matching, Information, and Dynamics the number and characteristics of agents that show up. With observable worker charac- Search frictions is the story of costly coor- teristics, in Shimer’s equilibrium, high-type dination—not knowing where a counter- workers obtain a job with the highest proba- party is, or how hard it is to match with him. bility, while low types only succeed if no high Informational frictions expand the scope types show up. This allocation is similar to towards not knowing the match payoffs or that of a second-price auction. As Shi (2002) types or other costs of individuals; this richer argues, this is not only realistic in many form of frictions promises to be the next market settings, but is also important for the frontier in matching models. We touch on efficient allocation of resources. some promising highlights of the work. 4.4.5 Directed Search and Large Firms 5.1 Sorting in Static Models The directed-search model can be used 5.1.1 Stability under Incomplete to analyze large firms in the presence of Information frictions. Under random search, Smith (1999) focuses on the role of hiring in A crucial assumption in Becker’s match- large firms that have decreasing returns ing model is that agents’ types are publicly to employment. To understand how these observable. This is not the case in many firms set wages, Smith (1999) assumes marriage and labor market applications. A a reduced-form bargaining process (in natural question then is what constitutes a the spirit of an earlier paper by Stole and stable matching under incomplete informa- Zwiebel 1996) where each worker is treated tion about agents’ types because in checking as the marginal worker and wages depend whether a blocking pair exists, the agents on marginal productivity. He shows that the involved must be able to compute their outcome is inefficient and leads to overem- payoffs from rematching, and that requires ployment, with firm size larger than opti- some knowledge about their partner’s type. mal. Kaas and Kircher (2015) and Schaal There have been some attempts at formal- (forthcoming) propose a directed-search izing a workable notion of stability, the most model with large firms, finding that price recent and relevant one for our purposes posting with coordination frictions yields being the definition of stability in Liu et al. a constrained efficient surplus division. (2014), who analyze matching with one sided These search models can handle realistic incomplete information and TU.46 Their environments where firms hire multiple workers and technology is nonadditive. 45 For example, Sepahsalari (2016) analyzes cyclical variations in the presence of credit frictions. “invariance,” i.e., the action of one buyer does not affect 46 For the NTU case and centralized matching, see Roth the distribution of buyers in the submarket. (1989) and Chakraborty, Citanna, and Ostrovsky (2010). 530 Journal of Economic Literature, Vol. LV (June 2017) incomplete information stability notion is in function , then the equilibrium utility of β the spirit of rationalizability in game theory, type x equals: rather than mechanism design. An interest- 2 ing result they show is that a mild strength- (x) max x z (z) x (x). π = z − β = − β ening of supermodularity yields PAM under incomplete information. We proceed as in a first-price auction. Since every type x must optimally bid as if it had 5.1.2 Sorting with Signaling Costs type x, the envelope theorem yields (x) π′ x, and so (x) x s ds. Altogether, = π = ∫0 In some matching applications in labor and marriage markets, agents with private (x) x 2 (x) information about their characteristics try β = − π to signal them to the other side of the mar- x x x 2 s ds s ds, ket before matching takes place. Intuitively, = − 0 = 0 those signals may be costly to send, and such ∫ ∫ costs reduce the benefits of sorting. Hoppe, where the second equality follows from inte- Moldovanu, and Sela (2009) analyze this gration by parts. Hence, (x) x s ds for all β = ∫0 issue in a static model with production com- x constitutes the Bayesian equilibrium of the plementarities and incomplete information game that exhibits PAM. Total welfare is then about types. In their model, if x matches 1 1 x with y then the utility of each agent is xy 2 x 2 g(x) dx 2 s ds g(x) dx minus any signaling cost. They consider two ∫0 − ∫0 (∫0 ) populations, men and women, who engage 1 in the following contest: agents simultane- x 2 g(x) dx. = 0 ously send signals, which consist of a bid or ∫ amount of utility that they give away. After That is, equilibrium signaling costs consume observing all the signals, a planner assorta- half of output. Thus, comparing welfare under tively matches men and women by signal. PAM versus random matching, we obtain In one equilibrium, everyone bids zero 1 1 2 and the planner randomly matches the two (30) x 2 g(x) dx 2 xg(x) dx sides. The paper shows that there is another ∫0 − (∫0 ) equilibrium in strictly increasing strate- var (x) E x 2 gies, with positively sorted agents. They ask = − [ ] whether the random matching welfare dom- E x 2 CV (x) 2 1 , inates PAM, net of signaling costs. The paper = [ ] ( − ) provides conditions under which this is the case in both the case with a finite number of where CV(x) _____var (x) /E x is the coeffi- = √ [ ] agents and with a continuum of them. In the cient of variation of x. Thus, if CV (x) 1 < latter, simpler context, assume that two unit random matching outperforms PAM, and mass populations with the same type distri- the opposite holds if greater than one. bution G and density g on 0, 1 . It turns out (Barlow and Proschan 1996, [ ] In the random matching equilibrium, corollary 4.9) that if the hazard rate g(x)/ the expected total welfare is 2E x E y (1 G(x)) is increasing in x, then CV(x) 1 [ ] [ ] − < 2 1 xg(x) dx 2. Under perfect PAM, and thus random matching dominates PAM, = (∫0 ) the total expected output is 2 1 x 2 g(x) dx. while the opposite is true if it is decreasing ∫0 If both sides use the same signaling quantity in x. Hence, the class of distributions with Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 531

Panel A Panel B

1.0 1.0

0.8 0.8

B B 0.6 C2 0.6 C2 ) x ( ( x ) 2 2 α α 0.4 0.4

0.2 0.2 C C Φ 1 Φ 1 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x α1( ) α1( ) Figure 2. Students’ Portfolio Problem

Notes: In the left panel, a student in the blank region applies nowhere. He applies to college 2 only in the Φ vertical shaded region C2 ; to both in the hashed region B, and to college 1 only in the horizontal shaded region _ 2 /_ 1 C1 . The right panel depicts the acceptance function ( 1) 1σ σ , which arises with exponential signals − /x ψ α = α m( | x) (1/x)e σ . As their caliber increases, students apply to nowhere ( ), college 2 only (C ), both col- σ = Φ 2 leges (B ), and finally college 1 only (C1 ). Student behavior is monotone in this case. Source: Chade, Lewis, and Smith (2014).

monotone hazard rate functions yields strong Students uniformly prefer college 1 to col- predictions regarding the welfare under ran- lege 2: Attending college 1 yields a utility 1, dom matching compared to PAM. college 2 yields u (0, 1), and zero is the ∈ utility for not attending college.47 By fix- 5.1.3 College Student Matching ing the payoffs of colleges, one might thus Gale and Shapley (1962) ignore the wealth understand this as an accurate short-to- of search and information frictions that medium-run description of the college afflict the sorting of students into colleges. world. Students maximize expected college Chade, Lewis, and Smith (2014) explore how payoff less application costs. Colleges max- students and colleges react to these frictions imize the integral quality of their student and what happens to sorting. bodies. In their model, there are two colleges, 1 The paper largely focuses on the case when and 2, with capacities and , and a unit students know their type, but colleges only κ1 κ2 mass of students with type x whose distribu- observe a noisy conditionally independent tion has a positive density g(x) over 0, ). signal of each applicant. Signal outcomes [ ∞ σ College capacity cannot accommodate all the students, i.e., 1 2 1. Capturing κ + κ < 47 Enrollment here is obviously deterministic. Che and the search friction, students pay a separate Koh (2015) explore a different model of college admissions application cost c 0 for each college. > with stochastic enrollments, instead. 532 Journal of Economic Literature, Vol. LV (June 2017) are drawn from a continuous density m ( x) of adding college i to a portfolio of college σ| with support on an interval of (e.g., 0, 1 ), j is given by M B (1 ) u and 핉 [ ] 21 = − α1 α2 and cdf M ( x). The density has the strict MB (1 u). The plot of these two σ| 12 = α1 − α2 monotone likelihood ratio property (MLRP): curves looks like figure 2 whenc u(1 u) < − m( x)/m( x) is increasing in x if . and c u/4, i.e., with applications not too τ| σ| τ > σ < To ensure that very high types are almost costly. never rejected, and very poor ones are This optimal-decision rule neatly parti- almost always rejected, the signals must be tions the unit square into four application able to reveal extreme types: So assume that regions, corresponding to the four portfolio M( x) 0 as x and M( x) 1 choices, denoted , C , B, C , shaded in the σ| → → ∞ σ| → Φ 2 1 as x 0 for any interior . right panel of figure 2. RegionB consists of → σ Students choose a portfolio of college students who either apply to college 2 and applications S(x) , 1 , 2 , 1, 2 for send a stretch application to college 1, or ∈ {⌀ { } { } { }} each x, while colleges set admissions stan- who apply to college 1 and send a safety dards , such that college i admits students application to college 2. σ i with signal_ realizations above . An equi- Let us now endogenize the accep- _σ i librium is a triple (S ∗ ( ), ∗ , ∗ ) such that, tance chances by considering the noisy · _σ 1 _σ 2 given ( ∗ , ∗ ), S ∗(x) is an optimal portfolio admissions process. Notice that not all _σ 1 _σ 2 for each x, and given (S ∗( ), ∗ ), standard ∗ pairs of acceptance chances ( , ) are · σ j σ i α1 α2 maximizes college i’s payoff._ _ “feasible,” since these chances are pinned An equilibrium exhibits sorting if college down by the student’s type and the college and student strategies are “increasing.” This thresholds. Fix the thresholds and σ 1 σ 2 means that the better college is more selec- set by college 1 and college 2. Student_ _x’s tive ( ∗ ∗ ) and higher-type students acceptance chance at college i 1, 2 is σ 1 > σ 2 = are increasingly_ _ aggressive in their portfo- given by (x) 1 M( x). Since a αi ≡ − σ i | lio choice: the weakest apply nowhere; bet- higher-type student generates_ stochasti- ter students apply to college 2; even better cally higher signals, (x) increases in x. αi ones “gamble” by applying also to college 1; We can then invert and define the fol- α1 the next tier up applies to college 1 while lowing acceptance function that links accep- shooting an “insurance” application to col- tance chances for each type x given colleges lege 2; finally, the top students just apply to thresholds: college 1. Strategies that are monotone in this fashion ensure the intuitive result that ( , , ) α2 = ψ α1 σ 1 σ 2 the distribution of student types accepted at _ _ college 1 first order stochastically dominates 1 M( ( , )). = − σ 2 | ξ α1 σ 1 that of college 2. _ _ We will exploit a simple graphical anal- Although the acceptance function need not ysis of the student’s problem for given col- in general be concave as in figure 2, it does lege thresholds in Chade, Lewis, and Smith have a falling secant: / is a decreasing α2 α1 (2014). Consider a student with respective function. The acceptance function and the admission chances 0 , 1. Using application strategy respectively capture ≤ α1 α2 ≤ the simultaneous-search solution in sec opportunities and preferences for student tion 3.2, we obtain the student’s opti- applications. Superimposing them, figure 2 mal portfolio choice. His expected payoff (right panel) depicts a monotone application of applying to both colleges is v strategy, in which higher types apply more α1 + (1 ) u. The marginal benefitM B aggressively to college. And since − α1 α2 ij _σ 1 > _σ 2 Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 533

Panel A Panel B 1.0 1.0

0.8 0.8 C C 2 B 2 B 0.6 0.6 ) x ( ( x ) 2 2 α α 0.4 0.4 P Q 0.2 0.2

Φ C1 C1 0.0 Φ 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x α1( ) α1( ) Figure 3. Non-Monotone Behavior

Notes: In the left panel, student behavior is non-monotone, since there are both low and high types who apply to college 2 only (C 2 ), while intermediate ones insure by applying to both. In the right panel, equal thresholds at both colleges induce an acceptance function along the diagonal, . Student behavior α1 = α2 is non-monotone, as both low and high types apply to college 1 only (C 1 ), while middle types apply to both. Source: Chade, Lewis, and Smith (2014).

in the picture, it follows that this strategy pro- threshold. To see this, assume that both col- file, if it could be sustained in equilibrium, leges set the same thresholds. As seen in the would exhibit the stochastic form of PAM right panel of figure 3, the application sets described above, as casual intuition suggests. transition through , 1 , 1, 2 , 1 as the Φ { } { } { } Yet there are two possible sorting viola- student type rises. In this case, college 2 tions, both illustrated in figure 3. The first attracts only safety applications. The paper occurs when stronger students do not apply shows that this is an equilibrium outcome for more aggressively. For relatively high types a small enough capacity of college 2. For the may apply just to college 2, while some paper shows that college 2 imposes a higher lower types also send stretch applications standard than college 1 if its capacity is to college 1. This is depicted in the left small enough—thus explaining how a poorly panel of figure 3, where application sets are ranked small private college can nonetheless , 2 , 1, 2 , 2 , 1, 2 , 1 as student type impose higher standards in equilibrium than Φ { } { } { } { } { } rises. This can be an equilibrium if col- a much larger public university.48 The paper lege 1 is not “sufficiently better” than college 2, for then one can find signal densities with the strict MLRP that engenders this 48 The sorting failures can be drastic. For instance, con- non-monotone behavior. sider the right panel of figure 3. Ifg (x) concentrates most of its mass on the interval of low calibers who apply just to The second violation occurs when the college 1, then the average caliber of students enrolled at lesser college imposes a higher admissions college 1 will be strictly smaller than that at college 2. 534 Journal of Economic Literature, Vol. LV (June 2017) also shows that all equilibria exhibit sorting if their continuation payoffs in future matches. college 2 is sufficiently worse than college 1 Anderson and Smith (2010) explore the (specifically,u 0.5), and college 1 is small trade-off between these two goals. They ≤ enough in capacity relative to college 2: show that despite production complemen- Graphically, in this case the acceptance func- tarities, PAM generally fails at high discount tion traverses the unit square high enough as factors due to the importance of information. to preclude the case in the right panel of fig- They argue that it is neither an equilibrium ure 3. nor an optimum that agents with identical The paper also conducts equilibrium anal- current reputations always match. ysis in the spirit of supply and demand, where The paper presents a general matching the supply is the college capacity, and the model with evolving human capital. They demand is the derived enrollment function first show that a Pareto optimal steady state at each school. In this metaphor, the accep- and a Walrasian equilibrium exist, and prove tance thresholds act like prices that equili- the welfare theorems. We illustrate this find- brate the two college markets. Comparative ing in their simpler motivational two-period statics reflect not only a “standards effect” partnership model. by existing applicants, but also a “portfo- Anderson and Smith (2010) assume a lio effect,” as relaxed standards encourage continuum of agents of two underlying true applications. The latter yields surprising types, high or low, i.e., , h . No one θ ∈ {θℓ θ } results: for example, a capacity increase at knows his own type, but merely the proba- the worse college can reduce admission stan- bility x 0, 1 of a high true type—called ∈ [ ] dards at the better college, via portfolio real- his reputation. Output is stochastic, and can location effects triggered by the students’ assume a finite number of positive values applications. q , , q . The chance of each output q 1 … N i is h , m , and , respectively, from a match 5.2 Sorting in Dynamic Models i i ℓi between two high types, a low and high type, and two low types. Then the chance of pro- 5.2.1 Sorting with Evolving Reputations duction q i from a match between two agents Anderson and Smith (2010) ask whether with reputations x and y is Becker’s assortative matching of types extends to reputations. For in many eco- p (x, y) xy h x(1 y) y(1 x) m i = i + [ − + − ] i nomic settings, parties to a match do not know their characteristics and learn them (1 x)(1 y) . + − − ℓi over time as they observe the output pro- 49 duced in a match. Matching then solves Let H i q i hi , M i q i m i, and = ∑ = ∑ two distinct objectives. On the one hand, it L q . Then the expected output is = ∑ i i ℓi serves to exploit complementarities in production between the partners. On the other f (x, y) q i p i (x, y) hand, it provides information about agents’ = ∑i attributes that may allow them to improve xyH x(1 y) y(1 x) M = + [ − + − ] 49 Early examples of matching models with learn- (1 x)(1 y) L. ing about the match are Jovanovic (1978) and Jovanovic + − − (1984). Although these papers derive very useful insights on the dynamics of turnover, they do not include ex ante Assuming H L 2M 0, production is heterogeneity and thus they do not shed light on comple- + − > mentarities and sorting patterns. strictly supermodular in reputations, since Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 535 f H L 2M 0. In a one-shot matching exhibits PAM. To see this, con- xy = + − > model with transferable utility, PAM arises sider three pairs (0, 0), (1, 1), and (x, x), (Becker 1973). where x (0, 1). Strict convexity of (x y) ∈ ψ | Consider a two-period matching model. in y implies that either (x 0) (x x) or ψ | > ψ | Let agents discount future payoffs by a (x 1) (x x). Easily, (0 x) (0 0) ψ | > ψ | ψ | = ψ | common factor . In the second and last and (1 x) (1 1), for there is no δ ψ | = ψ | period, output is strictly supermodular, and Bayesian updating when either of these so the matching exhibits PAM. As a result, extreme types match with anyone. So either the equilibrium wage of x is half of the out- (x 0) (0 x) (x x) (0 0) or ψ | + ψ | > ψ | + ψ | put for the agent, w(x) f (x, x)/2. Easily, (x 1) (1 x) (x x) (1 1). = ψ | + ψ | > ψ | + ψ | w (x) 2 f 0, and so the wage is con- Hence, PAM fails since rematching x agents ″ = xy > vex in reputation. with either 0 or 1 raises total payoffs. Since But in the first period, matching plays both this holds for 1, by continuity PAM δ = a production and an information role. To fails for a high enough discount factor . δ pin down the expected continuation payoff Intuitively, the learning value of matching for an agent with current type x, Anderson outweighs the productive complementarities and Smith (2010) notice that if he matches in this case. Since any nonproductive vari- with y , then after observing output q i in the ability in a match with an extreme type (0 first periodx , he updates his belief that his or 1) reflects uncertainty about the uncer- type is high to z (x, y) p (1, y)x/p (x, y).50 tain x (0, 1), assortatively matching x is i = i i ∈ Since the expected continuation payoff for x intuitively informationally dominated by is (x y) p (x, y)w(z (x, y)), the pres- cross-matching them with type 0 or 1. ψ | = ∑ i i i ent value of a match between agents x and So sufficiently forward-thinking behavior y is leads to a failure of PAM. Unfortunately, as the discount factor rises to one in an v(x, y) (1 ) f (x, y) infinite-horizon model, the continuation = − δ value tends to linear. For intuitively, in the ( (x y) (y x)). perfect patience limit, almost all produc- + δ ψ | + ψ | tion arises when one perfectly knows all If were supermodular in (x, y), then v types: this means that output of type x is the ψ would be supermodular, and PAM would linear weighted average xH (1 x)L that + − ensue, per Becker (1973). We will next show results from PAM, given the true types. In that PAM fails with sufficient patience, or the infinite-horizon version, Anderson and large enough 1. Smith (2010) find a robust PAM failure: as δ < Anderson and Smith (2010) then make the number N of output levels explodes, PAM a key preliminary observation. If 1, fails near both high enough and low enough δ = then v(x, y) (x y) (y x) and only types with probability tending to one. The = ψ | + ψ | the continuation payoff matters for match- proof turns on the asymptotic behavior of the ing. They show that is strictly convex continuation value function, that the second ψ in x and in y.51 We now ask whether the derivative explodes near 0 and 1.

50 This is just an application of Bayes’s rule: the denom- inator is the probability of qi while the numerator is the prior probability x that his type is high times the probability planner in assigning matches, and also induces mean zero of qi if his type is indeed high and he matches with y . noise in the posterior reputation. Since all zero-mean gam- 51 For any information about one’s own or one’s part- bles have positive expected value, both strict convexity ner’s type is intuitively productively valuable to the social claims follow from Pratt (1964). 536 Journal of Economic Literature, Vol. LV (June 2017)

A key implication is that partnerships of PAM first order stochasticallydominates identical types (either both or both h) the continuation distribution H (x ) (i.e., θℓ θ |μ eventually break up. Intuitively, as informa- minimizes H(x )) across all feasible |μ tion accumulates over time, the probability matchings . Lorentz (1953) argues that μ that anyone is a high type approaches 0 or this holds when (s x, y) is submodular in  | 1, and at that point, the above PAM failure (x, y) for all s. With deterministic transitions, kicks in, the match dissolves. where (x, y) matched implies x updates to (x, y), we can write (s x, y) 1 s (x, y). 5.2.2 Sorting and Evolving Types τ  | = ≥τ A salient special case in which is submod-  Inspired by the changing reputational ular is (x, y) min x, y . This is the “bad τ = { } types in Anderson and Smith (2010), apples” case in the peer effects literature, in Anderson (2015) explores the dynamics which the greater type is pulled down to the that arise when individuals are changed by lesser one. the association with their match partners. In the two-period model, the continuation Assume an initial distribution over human value is exogenous. In order to extend these capital G.52 A matching is feasible when PAM results to the infinite-horizon model, μ the measure of all matched types weakly Anderson (2015) first analyzes the planner’s below x equals G (x). For a taste of his con- preferences over human capital distributions. clusions, assume a two-period model with These analytical results require additional types changing after period one: specifically, assumptions on the transition distribution. if types (x, y) match in period one, then For example, the planner’s value rises in the type x transitions to a new type z s with increasing convex order over human capital ≤ probability (s x, y). Given any feasible distributions when f (x, y) is individually con-  | matching in period one, the distribution vex in x and in y and 1 (s x, y) ds is indi- μ ∫z  | over human capital in the final periodH (x ) vidually concave in x and in y. |μ can be naturally defined, given . Assume In a related model, Jovanovic (2014a)  symmetric, supermodular output f (x, y), so explores a dynamic matching model with that PAM is optimal in the final period.53 imperfect information, where agents do Given the final wagew (x) f (x, x)/2, the not know their types and are randomly ≡ period-one continuation value is: matched in the first period. (This precludes any matching role for information in the

first period.) They then observe the out- (31) V( ) w(x) dH(x ). μ ≡ ∫ |μ put produced, equal to the product of their true types. Finally, they decide whether to For a high enough discount factor, PAM rematch (“recombine”) in the second period. is initially optimal when it maximizes (31) He shows that if the output produced is across all feasible matchings. Since the wage publicly observed, as in Anderson and Smith w(x) is increasing, PAM maximizes (31) if and (2010), then all agents recombine in a PAM only if the continuation distribution under way in the second period. For signals enter in a complementary fashion in the expected product of the second period. But if output 52 Jovanovic (2014b) explores a related idea in an overlapping-generations setup with two-period lives to study is only observed by the pair, then only those assortative matching and growth in the presence of mis- with low output recombine in the second match due to shocks. period (adverse selection), and the over- 53 Anderson and Smith (2010) establish the welfare theorems for this dynamic matching model. In particular, all matching exhibits negative correlation PAM is optimal if and only if PAM is a market outcome. among pairs. Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 537

5.2.3 Marriage Markets and Age Gaps A woman’s type is publicly observable, while a man’s type is his private information in Bergstrom and Bagnoli (1993) may be the period 1, and publicly observable in period 2. first paper to incorporate incomplete informa- As a result, the period that a man chooses to tion into a dynamic matching market. To shed marry signals his type. Divorce is ignored, light on the empirical regularity observed in since the match payoff is one-time only. most countries and across time that women A centralized matchmaker matches agents on average marry older men, they develop as follows—which also delivers the unique an infinite-horizon overlapping-generations stable assignment. In each period, the planner marriage market model with heterogeneous positively assortatively matches men of age 2 types, incomplete information about men’s and the best women who choose to marry, types, in which men and women time their until exhausting the supply of women, or of entry into the marriage market. In equilib- men of age 2 whose types exceed the expected rium, males use their entry date into the value of the type of men of age 1. Since the matching market to signal their type, and men type of every age 2 man x is revealed, he will with higher types tend to marry later in life. be assigned to marry woman (x), where μ Assume a heterogeneous continuum of (x) 0 and (1) 1. The remaining μ′ > μ = men and women. The type of a man is x G lesser women are randomly assigned to age 1 ∼ on 0, 1 and of a woman is y H on 0, 1 . males who opt to enter the marriage market [ ] ∼ [ ] In an important novelty for the matching when young, whose true types are as yet hid- literature, this paper introduces the assump- den. Any unmatched age 1 men or women tion of log concavity, a property satisfied by remain in the marriage market when they many common distributions.54 The cdf H is reach age 2. The population has constant size, log concave in y. Utility is nontransferable: with men and women of age 1 and 2 always if man x ever marries woman y, then he present in the market and the same mass enjoys a positive flash utilityf 1 (x) f2 (y), of each entering period 2, so that everyone where f (x) 0 for all x, f (y) 0, eventually matches. We now explore which 1′ ≥ 2′ > f (y) 0, and additionally f (y) 0 for all men choose to marry when young. 2″ ≥ 2‴ ≤ y. In turn, a woman of type y enjoys a An equilibrium must specify the agents’ match utility (y) (x) if she ever mar- marrying strategies (age 1 or age 2). First of β1 β2 ries a man of type x, with (y) 0 all, observe that women have no incentive to β1′ ≥ and (x) 0. Bergstrom and Bagnoli delay. Given the demographic stationarity, β2′ > (1993) analyzed the simpler case with f1 (x) they secure the same expected payoff from (y) 1, f (y) y, and (x) x. marriage, but incur a fixed search costc only = β1 = 2 = β2 = 2 An equal number of men and women are in period 2. But men solve a timing problem: born in each period. Everyone lives for two in the spirit of a reservation wage, there is periods and their only decision is whether a cutoff value for men: high types wait until to enter the marriage market in period 1 or age 2, and low types enter at age 1. To see period 2. Delaying marriage entails a fixed this, let women of types C 0, 1 seek to ⊂ [ ] cost c 0 for men and c 0 for women. marry age 1 men. Then a type x man strictly 1 > 2 > prefers to delay marriage until age 2 when 54 In an underground classic that was published more than a decade later, they then authored the log-concavity (32) c 1 f1 (x) f2 ( (x)) encomium Bergstrom and Bagnoli (2005). The importance − + μ of this property generally in economics had previously f (x) f (s) dH(s) been introduced in proposition 1 of Heckman and Honore C 1 2 ______ . (1990). ≥ ∫ dH(s) ∫C 538 Journal of Economic Literature, Vol. LV (June 2017)

Easily, if this inequality holds for any type x , since f (z) 0, it suffices that / strictly 2″ ≥ ξ ξ′ then it also holds for any higher type, thereby increases in z. This holds when 2 ξ″ξ − ξ′ confirming the cutoff value property. Hence, 0 or, equivalently, when is strictly log _ < ξ there _is a threshold x such that men with concave in z. Since f 0, we have f is x x choose to marry at age 1 and those 2‴ ≤ 2′ ≤ _ concave and thus log concave. If we assume a with types x x choose to marry at age 2. > strictly log-concave distribution H , then f H When interior, the threshold solves indiffer- 2′ ence equation, namely (32) with equality, is strictly log concave, and so too is the inte- namely: gral, as log concavity is preserved by integration. All told, there is a unique equilibrium. _ ( x ) μ _ _ f2 (s) dH(s) 5.2.4 Matching and the Acceptance Curse (33) f ( x ) f ( ( x )) ______∫0 _ 1 ( 2 μ − H( ( x )) ) μ In many matching applications, such as the college admissions problem or mar- c . = 1 riage, the characteristics of agents on one or both sides of the market are only observed In the purported equilibrium, in every with noise prior to matching. Chade (2006) period, age 1 women with high types marry considers an NTU matching market with age 2 men with high types, assortatively, random search, where agents know their whereas age 1 women with low types marry types but they only observe a noisy signal of age 1 men with low types, but randomly. potential partners they meet. After observ- This is their story of the marriage age gap ing the signal, an agent updates his belief between men and women. about the partner’s type and then chooses Does this equilibrium exist and is it whether to accept or reject. Intuitively, unique? The answer is yes if (33) has a agents set a threshold for the signal reali- unique solution. First, _the left side of (33) zation and accept a partner when the sig- vanishes in the limit x 0 by l’Hopital’s nal observed exceeds a threshold. If both ↓ rule. Since f2 is increasing,_ the left side of accept, they marry and leave the market, (33) exceeds c at x 1, for small enough while in any other case they continue the 1 = c 0. Existence follows by continuity. search. Under the standard MLRP condi- 1 > Uniqueness follows if_ the left side of (33) is tion on the signal distribution, higher sig- strictly increasing in x . For this, Bergstrom nal realizations convey better news about a and Bagnoli (1993) introduce_ a log-concavity partner’s type. The twist here is that agents assumption. Since f ( x ) 0, it suffices to must also account for the information in the 1′ ≥ show that the term_ in parenthesis is increas- event that the partner agrees to match. And ing in z ( x ). Now, integration by parts if agents on the other side of the market = μ reveals that grow more choosy as their types increase, z then being accepted leads one to down- f (s) dH(s) ______0 2 grade the posterior estimate of the potential f2 (z) ∫ − H(z) partner’s type. Chade suggestively called z this the acceptance curse, since it is akin to f (s)H(s) ds (z) the winner’s curse effect in auction theory ______∫0 2′ f (z) ____ ξ , = H(z) = 2′ (z) (Milgrom and Weber 1982). ξ′ The model is in steady state over an where (z) z f (s) H(s) ds. So it suffices ξ = ∫0 2′ horizon infinite in discrete time, with that f (z) (z)/ (z) is increasing in z, and matched agents replaced by clones. Using 2′ ξ ξ′ Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 539 the marriage market metaphor, there are Consider a man y seeing a signal . That σ continuum populations of men and women. woman accepts with probability a(y ). | σ The density of women’s types x 0, 1 is In this event, the man decides whether to ∈ [ ] g(x), and of men’s types y 0, 1 is h(y). accept and leave the market, securing a dis- ∈ [ ] The per period utility of each agent is 0 if counted expected payoff f (y ), or reject | σ single, and the type of the spouse if matched and continue searching, and thereby earn (NTU). Every period, men and women ran- expected discounted payoff (y). If the δΨ domly meet. When a woman x meets a man woman does not accept, which occurs with y, he observes a signal 0, 1 drawn probability 1 a(y ), then the man con- σ ∈ [ ] − | σ from m( x), and she observes a signal tinues to search. His Bellman equation is σ | 0, 1 drawn from a conditional den- thus: τ ∈ [ ] sity n( y), where m and n satisfy the strict τ | MLRP. After observing the signals, both (34) v(y ) a(y ) max f(y ), (y) announce simultaneously accept or reject; |σ = |σ { |σ δΨ } if they both accept, they marry and exit the (1 a(y )) (y), market, otherwise they continue searching + − |σ δΨ next period. Agents discount the future by where (y) 1 v(y ) m( ) d is the (0, 1). Ψ = ∫0 |σ σ σ δ ∈ optimal continuation value, and solves A stationary strategy for a man of type y or _σ f y (y) (y). a woman with type x is a fixed set of signals ( |σ ) = δΨ that led either to accept. Intuitively, these ‾Similarly, the optimal strategy of a woman are upper intervals of signals, (y) of type x is a threshold (x). Thus, the σ ≥ σ τ and (x), by the MLRP. Focus on_ a search for a stationary equilibrium_ reduces τ ≥ τ man of type_ y facing a population of women. to finding a pair of functions( ( ), ( )) that σ · τ · Let m( ) 1 m( x) g(x) dx be the are mutual best responses. The_ downward_ σ = ∫0 σ| unconditional density of signal , and k (x ) recursive construction in section 4 under σ |σ m( x) g(x)/m( ) the posterior density complete information is inapplicable here, = σ| σ on x, given the signal realization . Then since any type may be accepted by any other, σ the chance a(y ) that y’s current partner owing to signal noise. Chade (2006) shows |σ accepts, conditional on , equals: that the model can be reinterpreted as a two- σ player game with incomplete information 1 1 with a continuum of types and actions, and

a(y ) (x) n( y)k(x ) d dx. |σ = ∫0 ∫_τ τ| |σ τ then one can appeal to a theorem in Athey (2001) to show that there exists a equilibrium Next, let f (y ) be the expected discounted in increasing strategies. |σ utility from marriage, given the signal real- Finally, observe that the accep- ization and the information contained in tance curse emerges: For since f (y ) σ |σ the event that he is accepted by the current E X/(1 ) , the event of being ≤ [ − δ |σ] partner. Formally, accepted is a discouraging signal for a man of type y. Nevertheless, stochastic sorting f (y ) still emerges: the distribution of types an |σ agent can end up matched is ordered in the sense of first order stochastic dominance as 1 1 a function of the agent’s type. As a result, in x ( ( x) n( y) d ) k(x ) E ______∫_τ τ| τ |σ dx . equilibrium one’s expected partner’s type is = 1 a( , y) [∫0 − δ σ ] increasing in the agent’s type. 540 Journal of Economic Literature, Vol. LV (June 2017)

6. Conclusion References

Acemoglu, Daron, and Robert Shimer. 1999. “Holdups We have reviewed the main frameworks and Efficiency with Search Frictions.” International used in micro models of search and match- Economic Review 40 (4): 827–49. ing, focusing on the conditions for sorting Ackerberg, Daniel A., and Maristella Botticini. 2002. “Endogenous Matching and the Empirical Determi- (either positive or negative) both with and nants of Contract Form.” Journal of Political Econ- without search or information frictions. We omy 110 (3): 564–91. have started from the benchmark frictionless Adachi, Hiroyuki. 2003. “A Search Model of Two-Sided Matching under Nontransferable Utility.” Journal of assignment model both with TU and NTU. Economic Theory 113 (2): 182–98. Despite its simplicity, the model explains Adam, Klaus. 2001. “Learning While Searching for the many interesting economic phenomena Best Alternative.” Journal of Economic Theory 101 (1): 252–80. ranging from the labor market to corpo- Albrecht, James, Axel Anderson, and Susan Vroman. rate finance to marriage markets. The many 2010. “Search by Committee.” Journal of Economic variations of the model allow for a simple Theory 145 (4): 1386–407. Albrecht, James, Pieter A. Gautier, and Susan Vroman. relation between technology and the result- 2006. “Equilibrium Directed Search with Multiple ing sorting pattern. Applications.” Review of Economic Studies 73 (4): We have also explored sorting with both 869–91. Anderson, Axel. 2015. “A Dynamic Generalization of search and information frictions. While Becker’s Assortative Matching Result.” Journal of this duly complicates many aspects of Economic Theory 159 (Part A): 290–310. the analysis, it renders the setting more Anderson, Axel, and Lones Smith. 2010. “Dynamic Matching and Evolving Reputations.” Review of Eco- realistic. We have carefully reviewed the nomic Studies 77 (1): 3–29. most important sorting results in this Antràs, Pol, Luis Garicano, and Esteban area, explaining in a unified way the logic Rossi-Hansberg. 2006. “Offshoring in a Knowledge Economy.” Quarterly Journal of Economics 121 (1): underlying them, and we have also dis- 31–77. cussed the emerging applied literature on Ashlagi, Itai, Yash Kanoria, and Jacob D. Leshno. 2017. the subject. “Unbalanced Random Matching Markets: The Stark Effect of Competition.” Journal of Political Economy We think these models can be a build- 125 (1): 69–98. ing block for further theory and a spark for Atakan, Alp E. 2006. “Assortative Matching with empirical work. This literature is at best Explicit Search Costs.” Econometrica 74 (3): 667–80. Athey, Susan. 2001. “Single Crossing Properties and in its infancy, and many open questions the Existence of Pure Strategy Equilibria in Games remain. These include a better understand- of Incomplete Information.” Econometrica 69 (4): ing of many-to-one matching models, the 861–89. Athey, Susan, Paul R. Milgrom, and John Roberts. role of externalities in matching, stochas- 1998. “Robust Comparative Statics.” Unpublished. tic types, nonstationary models, the role of Bagger, Jesper, and Rasmus Lentz. 2014. “An Empir- on-the-job search, and multidimensional ical Model of Wage Dispersion with Sorting.” National Bureau of Economic Research Working models. Of course, there are also trade- Paper 20031. theoretic models of search. That side is less Bagnoli, Mark, and Ted Bergstrom. 2005. “Log-Concave well-explored, and has so far focused on Probability and Its Applications.” Economic Theory 26 (2): 445–69. low levels of search frictions. The search Barlow, Richard E., and Frank Proschan. 1996. Math- and matching framework naturally captures ematical Theory of Reliability. Philadelphia: Society heterogeneity, which is the hallmark of eco- for Industrial and Applied Mathematics. Bartolucci, Cristian, and Francesco Devicienti. 2013. nomic exchange and the source of gains from “Better Workers Move to Better Firms: A Simple trade. Formally modeling the choice of with Test to Identify Sorting.” Institute for the Study of whom to trade should prove useful in all Labor Discussion Paper 7601. Becker, Gary S. 1973. “A Theory of Marriage: Part I.” fields of economics. Journal of Political Economy 81 (4): 813–46. Bergstrom, Theodore C., and Mark Bagnoli. 1993. Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 541

“Courtship as a Waiting Game.” Journal of Political “Matching to Share Risk.” Theoretical Economics 11 Economy 101 (1): 185–202. (1): 227–51. Bhaskar, V., and Ed Hopkins. 2016. “Marriage as a Rat Choi, Michael, and Lones Smith. 2016. “Optimal Race: Noisy Premarital Investments with Assortative Sequential Search among Alternatives.” University of Matching.” Journal of Political Economy 124 (4): Wisconsin PhD. Thesis. 992–1045. Chow, Yuan Shih, Herbert Robbins, and David Sieg- Bloch, Francis, and Harl Ryder. 2000. “Two-Sided mund. 1971. Great Expectations: The Theory of Search, Marriages, and Matchmakers.” International Optimal Stopping. Boston: Houghton Mifflin. Economic Review 41 (1): 93–115. Chvátal, Vašek. 1983. Linear Programming. New York: Browning, Martin, Pierre-André Chiappori, and Yoram W. H. Freeman and Company. Weiss. 2014. Economics of the Family. Cambridge Cole, Harold L., George J. Mailath, and Andrew and New York: Cambridge University Press. Postlewaite. 2001. “Efficient Non-contractible Burdett, Kenneth, and Melvyn G. Coles. 1997. “Mar- Investments in Large Economies.” Journal of Eco- riage and Class.” Quarterly Journal of Economics 112 nomic Theory 101 (2): 333–73. (1): 141–68. Compte, Olivier, and Philippe Jehiel. 2010. “Bargain- Burdett, Kenneth, and Kenneth L. Judd. 1983. “Equi- ing and Majority Rules: A Collective Search Perspec- librium Price Dispersion.” Econometrica 51 (4): tive.” Journal of Political Economy 118 (2): 189–221. 955–69. Crowe, J. A., J. A. Zweibel, and P. C. Rosenbloom. Burdett, Kenneth, and Dale T. Mortensen. 1998. 1986. “Rearrangements of Functions.” Journal of “Wage Differentials, Employer Size, and Unem- Functional Analysis 66 (3): 432–38. ployment.” International Economic Review 39 (2): Damiano, Ettore, Hao Li, and Wing Suen. 2005. 257–73. “Unraveling of Dynamic Sorting.” Review of Eco- Burdett, Kenneth, Shouyong Shi, and Randall Wright. nomic Studies 72 (4): 1057–76. 2001. “Pricing and Matching with Frictions.” Journal DeGroot, Morris H. 1970. Optimal Statistical Deci- of Political Economy 109 (5): 1060–85. sions. Hoboken, NJ: Wiley. Butters, Gerard R. 1977. “Equilibrium Distributions of Diamond, Peter A. 1982. “Aggregate Demand Man- Sales and Advertising Prices.” Review of Economic agement in Search Equilibrium.” Journal of Political Studies 44 (3): 465–91. Economy 90 (5): 881–94. Chade, Hector. 2001. “Two-Sided Search and Perfect Diamond, Peter A., and Eric Maskin. 1979. “An Equi- Segregation with Fixed Search Costs.” Mathematical librium Analysis of Search and Breach of Contract, Social Sciences 42 (1): 31–51. I: Steady States.” Bell Journal of Economics 10 (1): Chade, Hector. 2006. “Matching with Noise and the 282–316. Acceptance Curse.” Journal of Economic Theory 129 Edlin, Aaron S., and Chris Shannon. 1998. “Strict (1): 81–113. Single Crossing and the Strict Spence–Mirrlees Chade, Hector, and Jan Eeckhout. 2016. “Stochastic Condition: A Comment on Monotone Comparative Sorting.” Unpublished. Statics.” Econometrica 66 (6): 1417–25. Chade, Hector, and Jan Eeckhout. 2015. “Competing Eeckhout, Jan. 1999. “Bilateral Search and Vertical Teams.” Unpublished. Heterogeneity.” International Economic Review 40 Chade, Hector, and Jan Eeckhout. Forthcoming. (4): 869–87. “Matching Information.” Theoretical Economics. Eeckhout, Jan, and Philipp Kircher. 2010a. “Sorting Chade, Hector, Gregory Lewis, and Lones Smith. and Decentralized Price Competition.” Economet- 2014. “Student Portfolios and the College Admis- rica 78 (2): 539–74. sions Problem.” Review of Economic Studies 81 (3): Eeckhout, Jan, and Philipp Kircher. 2010b. “Sorting 971–1002. versus Screening: Search Frictions and Competing Chade, Hector, and Isle Lindenlaub. 2015. “Risky Mechanisms.” Journal of Economic Theory 145 (4): Matching.” Unpublished. 1354–85. Chade, Hector, and Lones Smith. 2006. “Simultaneous Eeckhout, Jan, and Philipp Kircher. 2011. “Identifying Search.” Econometrica 74 (5): 1293–307. Sorting-In Theory.” Review of Economic Studies 78 Chakraborty, Archisman, Alessandro Citanna, and (3): 872–906. Michael Ostrovsky. 2010. “Two-Sided Matching with Eeckhout, Jan, and Philipp Kircher. 2012. “Assortative Interdependent Values.” Journal of Economic The- Matching with Large Firms: Span of Control over ory 145 (1): 85–105. More versus Better Workers.” Unpublished. Che, Yeon-Koo, and Youngwoo Koh. 2016. “Decentral- Felli, Leonardo, and Kevin Roberts. 2016. “Does Com- ized College Admissions.” Journal of Political Econ- petition Solve the Hold-Up Problem?” Economica 83 omy 124 (5): 1295–338. (329): 172–200. Chiappori, Pierre-André, Robert J. McCann, and Ferguson, Thomas S. 2016. “Optimal Stopping Lars P. Nesheim. 2010. “Hedonic Price Equilibria, and Applications.” http://www.math.ucla.edu/~tom/ Stable Matching, and Optimal Transport: Equiva- Stopping/Contents.html. lence, Topology, and Uniqueness.” Economic Theory Fudenberg, Drew, and Jean Tirole. 1991. Game The- 42 (2): 317–54. ory. Cambridge, MA and London: MIT Press. Chiappori, Pierre-André, and Philip J. Reny. 2016. Gabaix, Xavier, and Augustin Landier. 2008. “Why Has 542 Journal of Economic Literature, Vol. LV (June 2017)

CEO Pay Increased So Much?” Quarterly Journal of Jacquet, Nicolas L., and Serene Tan. 2007. “On the Economics 123 (1): 49–100. Segmentation of Markets.” Journal of Political Econ- Gale, D., and L. S. Shapley. 1962. “College Admissions omy 115 (4): 639–64. and the Stability of Marriage.” American Mathemati- Jerez, Belen. 2014. “Competitive Equilibrium with cal Monthly 69 (1): 9–15. Search Frictions: A General Equilibrium Approach.” Galenianos, Manolis, and Philipp Kircher. 2009. Journal of Economic Theory 153: 252–86. “Directed Search with Multiple Job Applications.” Jovanovic, Boyan. 1979. “Job Matching and the The- Journal of Economic Theory 144 (2): 445–71. ory of Turnover.” Journal of Political Economy 87 Garicano, Luis. 2000. “Hierarchies and the Organiza- (5 Part 1): 972–90. tion of Knowledge in Production.” Journal of Politi- Jovanovic, Boyan. 1984. “Matching, Turnover, and cal Economy 108 (5): 874–904. Unemployment.” Journal of Political Economy 92 Gershkov, Alex, and Benny Moldovanu. 2012. “Opti- (1): 108–22. mal Search, Learning and Implementation.” Jour- Jovanovic, Boyan. 2014a. “Learning and Recombina- nal of Economic Theory 147 (3): 881–909. tion.” Unpublished. Gittins, J. C. 1979. “Bandit Processes and Dynamic Jovanovic, Boyan. 2014b. “Misallocation and Growth.” Allocation Indices.” Journal of the Royal Statistical American Economic Review 104 (4): 1149–71. Society, Series B 41 (2): 148–77. Kaas, Leo, and Philipp Kircher. 2015. “Efficient Firm Gittins, J. C., and D. M. Jones. 1974. “A Dynamical Dynamics in a Frictional Labor Market.” American Allocation Index for the Sequential Design of Exper- Economic Review 105 (10): 3030–60. iments.” In Progress in Statistics, edited by Joseph Kantorovich, L. V. 1942. “On the Translocation of Mark Gani, et al. Amsterdam: North-Holland. Masses.” Dokl. Akad. Nauk SSSR 37 (7–8): 227–29. Gretsky, Neil E., Joseph M. Ostroy, and William R. Kelso, Alexander S, Jr., and Vincent P. Crawford. 1982. Zame. 1992. “The Nonatomic Assignment Model.” “Job Matching, Coalition Formation, and Gross Sub- Economic Theory 2 (1): 103–27. stitutes.” Econometrica 50 (6): 1483–504. Gretsky, Neil E., Joseph M. Ostroy, and William R. Kircher, Philipp. 2009. “Efficiency of Simultane- Zame. 1999. “Perfect Competition in the Continuous ous Search.” Journal of Political Economy 117 (5): Assignment Model.” Journal of Economic Theory 88 861–913. (1): 60–118. Koopmans, Tjalling C., and Martin Beckmann. 1957. Grossman, Gene M., Elhanan Helpman, and Philipp “Assignment Problems and the Location of Eco- Kircher. 2013. “Matching and Sorting in a Global nomic Activities.” Econometrica 25 (1): 53–76. Economy.” National Bureau of Economic Research Kremer, Michael. 1993. “The O-Ring Theory of Working Paper 19513. Economic Development.” Quarterly Journal of Grossman, Gene M., and Giovanni Maggi. 2000. Economics 108 (3): 551–75. “Diversity and Trade.” American Economic Review Kremer, Michael, and Eric Maskin. 1996. “Wage 90 (5): 1255–75. Inequality and Segregation by Skill.” Unpublished. Guadalupe, Maria, Veronica Rappoport, Bernard Sal- Lamadon, Thibaut. 2014. “Productivity Shocks, anie, and Catherine Thomas. 2014. “The Perfect Dynamic Contracts and Income Uncertainty.” Match: Assortative Matching in International Acqui- Unpublished. sitions.” Unpublished. Lamadon, Thibaut, Jeremy Lise, Costas Meghir, and Hagedorn, Marcus, Tzuo Hann Law, and Iourii Jean-Marc Robin. 2013. “Matching, Sorting, and Manovskii. 2012. “Identifying Equilibrium Models Wages.” Unpublished. of Labor Market Sorting.” Unpublished. Lauermann, Stephan, and Georg Noldeke. 2014. “Sta- Hardy, G. H., J. E. Littlewood, and G. Polya. 1952. ble Marriages and Search Frictions.” Journal of Eco- Inequalities, Second edition. Cambridge and New nomic Theory 151: 163–95. York: Cambridge University Press. Legros, Patrick, and Andrew F. Newman. 2007. Hatfield, John William, and Fuhito Kojima. 2008. “Beauty Is a Beast, Frog Is a Prince: Assortative “Matching with Contracts: Comment.” American Matching with Nontransferabilities.” Econometrica Economic Review 98 (3): 1189–94. 75 (4): 1073–102. Hatfield, John William, and Paul R. Milgrom. 2005. Legros, Patrick, and Andrew F. Newman. 2010. “Matching with Contracts.” American Economic “Co-ranking Mates: Assortative Matching in Mar- Review 95 (4): 913–35. riage Markets.” Economics Letters 106 (3): 177–79. Heckman, James J., and Bo E. Honore. 1990. “The Lester, Benjamin, Ludo Visschers, and Ronald Empirical Content of the Roy Model.” Economet- Wolthoff. 2015. “Meeting Technologies and Optimal rica 58 (5): 1121–49. Trading Mechanisms in Competitive Search Mar- Holmstrom, Bengt, and Paul R. Milgrom. 1987. kets.” Journal of Economic Theory 155: 1–15. “Aggregation and Linearity in the Provision of Inter- Lindenlaub, Ilse. 2017. “Sorting Multidimensional temporal Incentives.” Econometrica 55 (2): 303–28. Types: Theory and Application.” Review of Economic Hoppe, Heidrun C., Benny Moldovanu, and Aner Sela. Studies 84 (2): 718–89. 2009. “The Theory of Assortative Matching Based on Lise, Jeremy, Costas Meghir, and Jean-Marc Robin. Costly Signals.” Review of Economic Studies 76 (1): 2013. “Mismatch, Sorting and Wage Dynamics.” 253–81. National Bureau of Economic Research Working Chade, Eeckhout, and Smith: Sorting through Search and Matching Models 543

Paper 18719. Theory 160: 429–37. Lise, Jeremy, and Jean-Marc Robin. 2017. “The Mac- Osborne, Martin J., and Ariel Rubinstein. 1990. Bar- rodynamics of Sorting between Workers and Firms.” gaining and Markets. Bingley, UK: Emerald. American Economic Review 107 (4): 1104–35. Peskir, Goran, and Albert Shiryaev. 2006. Optimal Liu, Qingmin, George J. Mailath, Andrew Postlewaite, Stopping and Free-Boundary Problems. Basel and and Larry Samuelson. 2014. “Stable Matching Boston: Birkhauser Verlag. with Incomplete Information.” Econometrica 82 Peters, Michael. 1984. “Bertrand Equilibrium with (2): 541–87. Capacity Constraints and Restricted Mobility.” Lopes de Melo, Rafael. 2009. “Sorting in the Labor Econometrica 52 (5): 1117–27. Market: Theory and Measurement.” Yale University Peters, Michael. 1997. “A Competitive Distribution of PhD. Thesis. Auctions.” Review of Economic Studies 64 (1): 97–123. Lorentz, G. G. 1953. “An Inequality for Rearranage- Peters, Michael, and Sergei Severinov. 1997. “Compe- ments.” American Mathematical Monthly 60 (3): tition among Sellers Who Offer Auctions Instead of 176–79. Prices.” Journal of Economic Theory 75 (1): 141–79. Lucas, Robert E., Jr. 1978. “On the Size Distribution Petrongolo, Barbara, and Christopher A. Pissarides. of Business Firms.” Bell Journal of Economics 9 (2): 2001. “Looking into the Black Box: A Survey of the 508–23. Matching Function.” Journal of Economic Literature Makowski, Louis, and Joseph M. Ostroy. 1995. “Appro- 39 (2): 390–431. priation and Efficiency: A Revision of the First The- Pinheiro, Roberto. 2012. “Venture Capital and Under- orem of Welfare Economics.” American Economic pricing: Capacity Constraints and Early Sales.” Review 85 (4): 808–27. Unpublished. Manea, Mihai. 2017. “Steady States in Matching and Postel-Vinay, Fabien, and Jean-Marc Robin. 2002. Bargaining.” Journal of Economic Theory 167: “Equilibrium Wage Dispersion with Worker and 206–28. Employer Heterogeneity.” Econometrica 70 (6): McAfee, R. Preston. 1993. “Mechanism Design By 2295–350. Competing Sellers.” Econometrica 61 (6): 1281–312. Pratt, John W. 1964. “Risk Aversion in the Small and in McCall, John J. 1965. “The Economics of Information the Large.” Econometrica 32 (1–2): 122–36. and Optimal Stopping Rules.” Journal of Business 38 Pycia, Marek, and M. Bumin Yenmez. 2015. “Matching (3): 300–317. with Externalities.” Unpublished. McCall, John J. 1970. “Economics of Information and Rogerson, Richard, Robert Shimer, and Randall Job Search.” Quarterly Journal of Economics 84 (1): Wright. 2005. “Search-Theoretic Models of the 113–26. Labor Market: A Survey.” Journal of Economic Lit- McNamara, J. M., and E. J. Collins. 1990. “The Job erature 43 (4): 959–88. Search Problem as an Employer–Candidate Game.” Rosen, Sherwin. 1981. “The Economics of Superstars.” Journal of Applied Probability 28: 815–27. American Economic Review 71 (5): 845–58. Milgrom, Paul R., and Chris Shannon. 1994. “Mono- Ross, Sheldon M. 1983. Introduction to Stochastic tone Comparative Statics.” Econometrica 62 (1): Dynamic Programming. London: Academic Press. 157–80. Roth, Alvin E. 1989. “Two-Sided Matching with Milgrom, Paul R., and Robert J. Weber. 1982. “A The- Incomplete Information about Others’ Preferences.” ory of Auctions and Competitive Bidding.” Econo- Games and Economic Behavior 1 (2): 191–209. metrica 50 (5): 1089–122. Roth, Alvin E., and Marilda A. Oliveira Sotomayor. 1990. Moen, Espen R. 1997. “Competitive Search Equilib- Two-Sided Matching: A Study in Game-Theoretic rium.” Journal of Political Economy 105 (2): 385–411. Modeling and Analysis. Cambridge and New York: Monge, Gaspard. 1781. “Memoire sur la Theorie des Cambridge University Press. Deblais et des Remblais.” Histoire de l’Academie Rothschild, Michael. 1974. “Searching for the Lowest Royale des Sciences 666–704. Price When the Distribution of Prices Is Unknown.” Morgan, Peter. 1996. “Two-Sided Search and Match- Journal of Political Economy 82 (4): 689–711. ing.” Unpublished. Sasaki, Hiroo, and Manabu Toda. 1996. “Two-Sided Morgan, Peter, and Richard Manning. 1985. “Optimal Matching Problems with Externalities.” Journal of Search.” Econometrica 53 (4): 923–44. Economic Theory 70 (1): 93–108. Mortensen, Dale T., and Randall Wright. 2002. “Com- Sattinger, Michael. 1993. “Assignment Models of the petitive Pricing and Efficiency in Search Equilib- Distribution of Earnings.” Journal of Economic Lit- rium.” International Economic Review 43 (1): 1–20. erature 31 (2): 831–80. Noldeke, Georg, and Larry Samuelson. 2015. “Invest- Schaal, Edouard. Forthcoming. “Uncertainty and ment and Competitive Matching.” Econometrica 83 Unemployment.” Econometrica. (3): 835–96. Schulhofer-Wohl, Sam. 2006. “Negative Assortative Noldeke, Georg, and T. Troger. 2009. “Matching Het- Matching of Risk-Averse Agents with Transfer- erogeneous Agents with a Linear Search Technol- able Expected Utility.” Economics Letters 92 (3): ogy.” Unpublished. 383–88. Olszewski, Wojciech, and Richard Weber. 2015. “A Sepahsalari, Alireza. 2016. “Financial Market Imperfec- More General Pandora Rule?” Journal of Economic tions and Labour Market Outcomes.” Unpublished. 544 Journal of Economic Literature, Vol. LV (June 2017)

Serfes, Konstantinos. 2005. “Risk Sharing vs. Incen- Smith, Lones. 2006. “The Marriage Model with Search tives: Contract Design under Two-Sided Heteroge- Frictions.” Journal of Political Economy 114 (6): neity.” Economics Letters 88 (3): 343–49. 1124–44. Shapley, L. S., and M. Shubik. 1971. “The Assignment Smith, Lones. 2011. “Frictional Matching Models.” Game I: The Core.” International Journal of Game Annual Review of Economics 3: 319–38. Theory 1 (1): 111–30. Sørensen, Morten. 2007. “How Smart Is Smart Money? Shi, Shouyong. 2001. “Frictional Assignment I: Effi- A Two-Sided Matching Model of Venture Capital.” ciency.” Journal of Economic Theory 98 (2): 232–60. Journal of Finance 62 (6): 2725–62. Shi, Shouyong. 2002. “A Directed Search Model Stigler, George J. 1961. “The Economics of Informa- of Inequality with Heterogeneous Skills and tion.” Journal of Political Economy 69 (3): 213–25. Skill-Biased Technology.” Review of Economic Stud- Stole, Lars A., and Jeffrey Zwiebel. 1996. “Intra-firm ies 69 (2): 467–91. Bargaining under Non-binding Contacts.” Review of Shimer, Robert. 2005. “The Assignment of Workers to Economic Studies 63 (3): 375–410. Jobs in an Economy with Coordination Frictions.” Tervio, Marko. 2008. “The Difference that CEOs Journal of Political Economy 113 (5): 996–1025. Make: An Assignment Model Approach.” American Shimer, Robert, and Lones Smith. 2000. “Assorta- Economic Review 98 (3): 642–68. tive Matching and Search.” Econometrica 68 (2): Teulings, Coen N., and Pieter A. Gautier. 2004. “The 343–69. Right Man for the Job.” Review of Economic Studies Shimer, Robert, and Lones Smith. 2001. “Nonstation- 71 (2): 553–80. ary Search.” Unpublished. Topkis, Donald M. 1998. Supermodularity and Com- Shiryaev, A. N. 1978. Optimal Stopping Rules. New plementarity. Princeton and Oxford: Princeton Uni- York and Berlin: Springer. versity Press. Smith, Eric. 1999. “Search, Concave Production, and Varian, Hal R. 1999. “Economics and Search.” Optimal Firm Size.” Review of Economic Dynamics Unpublished. 2 (2): 456–71. Villani, Cédric. 2009. Optimal Transport: Old and New. Smith, Lones. 1992. “Cross-Sectional Dynamics in a Berlin: Springer. Two-Sided Matching Model.” Unpublished. Vince, A. 1990. “A Rearrangement Inequality and the Smith, Lones. 1995. “A Model of Exchange Where Permutahedron.” American Mathematical Monthly Beauty Is in the Eye of the Beholder.” Unpublished. 97 (4): 319–23. Smith, Lones. 1997. “The Marriage Model with Search Weitzman, Martin L. 1979. “Optimal Search for the Frictions.” Unpublished. Best Alternative.” Econometrica 47 (3): 641–54.