Stable matching in large markets with occupational choice

Carmona, Guilherme; Laohakunakorn, Krittanai Article Stable matching in large markets with occupational choice Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Carmona, Guilherme; Laohakunakorn, Krittanai (2024) : Stable matching in large markets with occupational choice, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 19, Iss. 3, pp. 1261-1304, https://doi.org/10.3982/TE5915 This Version is available at: https://hdl.handle.net/10419/320266 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 19 (2024), 1261–1304 1555-7561/20241261 Stable matching in large markets with occupational choice Guilherme Carmona School of Economics, University of Surrey Krittanai Laohakunakorn School of Economics, University of Surrey We introduce a model of large many-to-one matching markets with occupational choice where each individual can choose which side of the market to belong to. We show that stable matchings exist under mild assumptions; in particular, both complementarities and externalities can be accommodated. Our model generalizes Greinecker and Kah (2021), which focuses on one-to-one matching and did not allow for occupational choice. Applications include the roommate problem with nonatomic participants, explaining the size and distribution of firms and wage inequality. Keywords. Stable matching, large markets, occupational choice. JEL classification. C78. 1. Introduction This paper establishes the existence of many-to-one stable matchings in large markets with complementarities, externalities, and occupational choice. Stability in the presence of occupational choice differs from the standard stability notion for two-sided, many-to-one matching markets. As individuals no longer have a fixed occupation, stability requires someone being unable to find a better match even if this involves a change of occupation. Having all these features present simultaneously in the same model is important for at least the following reasons. Labor markets match a large numbers of workers to managers in a many-to-one way. Unlike standard matching markets membership in one side or the other of the market is endogenous. Complementarities and externalities are also an essential feature of labor markets. For example, firms typically want to hire workers with complementary skills and recent graduates may prefer to enter the same industry as their peers. In addition, knowledge spillovers may imply that the productivity of a manager depends on the aggregate quality of those who take managerial roles according to the matching. Guilherme Carmona: [email protected] Krittanai Laohakunakorn: [email protected] We wish to thank Michael Greinecker, Ravi Jagadeesan, Fuhito Kojima, Karolina Vocke, three anonymous referees and seminar participants at the University of Bath, the 2022 SAET conference (Canberra), the 2022 Many Player Games and Applications Workshop (Berlin), the 2023 Lisbon Meetings in Game Theory and Applications, the 2023 EWET (Naples), and the 2023 PEJ conference (Braga) for helpful comments. Any remaining errors are, of course, ours. ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE5915 1262 Carmona and Laohakunakorn Theoretical Economics 19 (2024) Prior work, summarized in Section 2, has established the existence of stable matchings in models that contain a strict subset of these elements. Our framework for large many-to-one matching markets with occupational choice subsumes several important special cases. It generalizes the two-sided, one-to-one matching setting in distributional form of Greinecker and Kah (2021) by adding manyto-one matching and occupational choice; in particular, our existence result implies existence in Greinecker and Kah’s (2021) one-to-one matching market.1 In addition, we show how several classical models that feature occupational choice, many-to-one matching and a large number of participants, such as Lucas (1978), Rosen (1982), Garicano and Rossi-Hansberg (2004), and Garicano and Rossi-Hansberg (2006), can be seen as particular cases of our framework. These models also feature a continuum of types, which can be accommodated in our framework. To illustrate the flexibility of our setting and its technical advantages, we provide a detailed analysis of Rosen’s (1982) model. We show that stable matchings exist and fully characterize them even though some of the assumptions of our general existence result do not hold. Our model is not restricted to labor markets. We illustrate this by formalizing a nonatomic version of Gale and Shapley’s (1962) roommate problem as a special case of our model—one in which individuals are indifferent between the two occupations. We show that our existence results imply the existence of stable matchings for the nonatomic roommate problem. We present our model and stability notion in Section 4after a brief literature review in Section 2and a motivating example in Section 3. Our existence results are in Section 5. In particular, we show that stable matchings exist in markets with occupational choice whenever preferences are rational and continuous and the set of feasible measures that managers can match with is bounded and rich.2Thus, we can accommodate externalities as long as preferences depend on the matching in a continuous way without any substitutability requirement— complementarities cause no problem for existence in our model. In addition, as is standard in models with a continuum of agents, preferences are not required to be convex. Section 6contains applications of our framework to the roommate problem (Section 6.1)andRosen’s (1982) model (Section 6.2), and a brief discussion of the settings of Lucas (1978), Garicano and Rossi-Hansberg (2004), and Garicano and Rossi-Hansberg (2006). Section 7contains some concluding remarks. The proofs of our results are in the Appendix. Some omitted details are in the working paper version.3 1In the working paper version, we establish formally that Greinecker and Kah’s (2021) setting can be represented as a special case of our general framework and that, specialized to this setting, our stability notion coincides with theirs. We also introduce a new two-sided, many-to-one matching model that generalizes Greinecker and Kah (2021) to allow for many-to-one matching (but not occupational choice). We show that this model is also a particular case of our framework and, specialized to this setting, our stability notion coincides with other stability concepts for two-sided markets where both sides are large, such as Azevedo and Hatfield’s (2018). 2Richness is a weak technical condition that implies that small perturbations of feasible measures are feasible. 3The working paper version is available at https://klaohakunakorn.com/ocwp.pdf. Theoretical Economics 19 (2024) Stable matching in large markets 1263 2. Literature review The study of large matching markets has commanded a great deal of recent attention; see, e.g., Azevedo and Leshno (2016), Fisher and Hafalir (2016), Ashlagi,Kanoria,and Leshno (2017), Eeckhout and Kircher (2018), Fuentes and Tohmé (2018), Nöldeke and Samuelson (2018), and Che and Tercieux (2019).4However, none of these papers allow for occupational choice. Chiappori, Galichon, and Salanié (2014), P˛eski (2017), Azevedo and Hatfield (2018) study large matching models with a restricted form of occupational choice and a large number of participants. See Section 6.1 for a more detailed discussion of these papers. Compared to these papers, we accommodate many-to-one matching and more general forms of occupational choice. Closest to this paper is Jagadeesan and Vocke (2024), which considers a many-tomany matching model where a continuum of agents of finitely many types can sign multiple contracts with each other. They do not require that the market be two-sided, and hence their existence result holds in the presence of occupational choice. However, their assumption that the set of contracts available to each agent is finite makes it less convenient to capture settings such as Rosen (1982), which was part of our motivation. While our model cannot accommodate many-to-many matching, we allow for more general type and contract spaces and we allow preferences to depend on the matching. Wu (2021) also provides a general existence result for a broad class of finite-type many-tomany matching models under a convexity condition. However, Wu’s (2021)resultdoes not apply to our setting because we allow preferences to depend on the entire matching. Externalities and complementarities cause problems for the existence of stable matchings in finite markets. Making the workers negligible allowed Che, Kim, and Kojima (2019) to obtain the existence of stable matchings in two-sided, many-to-one matching markets where managers’ preferences exhibit complementarities. This result solved a longstanding problem in matching theory since, with finitely many workers and managers, Kelso and Crawford (1982), Hatfield and Milgrom (2005), and Hatfield and Kojima (2008) have shown that managers need to have substitutable preferences to guarantee the existence of stable matchings. In contrast to Che, Kim, and Kojima (2019), we also allow for occupational choice and externalities. By assuming that all agents are negligible, we are able to show that a stable matching exists in the presence of complementarities, occupational choice, and externalities. Externalities raise some conceptual issues in finite markets. Indeed, when preferences depend on the matching, whether or not an individual gains by being part of a potential blocking coalition depends on the matching that results from such blocking. Thus, the definition of stability has to specify the (set of possible) matchings that result from each blocking coalition, and many such definitions have been proposed.5 When there are finitely many managers but a continuum of workers and only workers’ preferences depend on the matching, Cox, Fonseca, and Pakzad-Hurson (2022), Leshno 4See Greinecker and Kah (2021)forasurvey. 5See, e.g., Sasaki and Toda (1996), Dutta and Massó (1997), Echenique and Yenmez (2007), Hafalir (2008), Mumcu and Saglam (2010), Bando (2012), and Fisher and Hafalir (2016). 1264 Carmona and Laohakunakorn Theoretical Economics 19 (2024) (2022), and Carmona and Laohakunakorn (2023) define stability and establish existence by specifying that each worker in a blocking coalition expects the matching to remain unchanged. In contrast to these papers, we consider the case where all agents are negligible, and thus, a blocking coalition of one (prospective) manager and a measure of (prospective) workers is negligible and, indeed, has no impact on the matching. Hence, externalities cause no conceptual issue in our framework and we can accommodate them on both sides of the market. 3. Motivating example This example is a particular case of the model in Rosen (1982). There are two types of individuals, 1 and 2. Individuals have preferences that are fully described by their types and their population is described by a measure νover the type space Z={1, 2}.Let ν(1)=ν(2)=1/2. Each individual can be a manager, a worker, or self-employed (i.e., remain unmatched). For each type z∈{1, 2}, some individuals of type zcan be managers and some others can be workers; furthermore, those who are managers (if any) can be matched with workers of type zor of type z= z. Those who are managers can hire a workforce, which we represent as a measure over worker types and contracts, from the set X,where each δ∈Xis a measure over Z×Cwith Cbeing the set of contracts. For this example, let C=R+and X={n1(z,c):z∈Z,n,c∈R+}.6Specifically, each manager can be matched with a measure n1(z,c),wherez∈Zdenotes the type of workers he employs, n∈R+ denotes their number and c∈R+denotes the wage paid to them. The preferences of each individual depend on her type, her occupation, and on her match. In this example, we specify that if someone of type z∈{1, 2}chooses to be a manager and is matched with n1(z,c),thenherpayoffisUz(m,n1(z,c))=z1+αn1−α−cn, where α∈(0, 1). If she chooses to be a worker and is matched with manager zat wage c, then her payoff is the wage: Uz(w,1 (z,c))=c. An individual can also choose to be unmatched, in which case she receives a payoff of zero. The managers’ rents are obtained via a production function of the form g(z)zαn1−α, with g(z)=z, which has labor and managers’ type as inputs, the latter being interpreted as the managers’ quality. In the context of this example, a matching is a measure μover Z×Xwith μ(z,n1(z,c))describing the measure of type zwho are managers and hire nworkers of type zat wage c. Consider first the case where each individual’s occupation is fixed, with type 1 individuals being managers and type 2 individuals being workers. There is a unique stable matching in this example without occupational choice: μ(1, 1(2,1−α))=1/2. In such matching, all workers (i.e., type 2 individuals) are matched with a manager (i.e., a type 1 individual), each manager hires a workforce consisting of a measure n=1ofworkers at wage c=1−α. Since both managers and workers obtain a strictly positive utility in this matching and zero if they were unmatched, such matching is individually rational. 6If Yis a metric space and y∈Y,1 ydenotes the probability measure degenerate on y. Theoretical Economics 19 (2024) Stable matching in large markets 1265 Furthermore, no manager and group of workers can block this matching since hiring a measure one of workers is optimal given the wage; hence, the manager cannot gain by changing his workforce since at least the newly hired workers would require a wage higher than 1 −α. In the example without occupational choice, type 1 individuals can only be managers and type 2 individuals can only be workers; these restrictions are now removed by the introduction of occupational choice. The specification of our example implies that individuals of type 2 are better managers than those of type 1 since they have higher quality. This then means that the stable matching μfor the setting without occupational choice is intuitively not stable when occupational choice is allowed. For instance, any type 2 individual could choose to be a manager and attract, e.g., a measure one of workers of type 2 by paying them 1 −α+εto obtain a rent of 21+α−(1−α)−ε;for sufficiently small ε>0, such workers are willing to work for her and her payoff is higher than 1 −α, which is her payoff in the matching μ. Thus, stability in the presence of occupational choice is more demanding than the stability notion for two-sided many-to-one matching markets. The latter roughly requires that no manager can improve his well being by changing the number of workers who work for him or by employing (an optimal number of) workers that he can target, which are those who would prefer to work for him at the proposed wage rather than for the manager with whom they are currently matched.7With occupational choice, since anyone can choose to be a manager, this condition must hold not just for those who are managers in the current match but also for those who are workers and unmatched. Similarly, since anyone can be a worker, the targets of a prospective manager are no longer restricted to be the current workers but rather can include current managers and unmatched individuals. When α=1/2, the unique stable matching in the above example is for all type 2 individuals to be managers, each of them being matched with a measure one of type 1 individuals at wage w≃1.41.8At this wage, the firm size is optimal for type 2 managers. Their rent is equal to w, so that type 2 individuals are actually indifferent between being a manager or a worker. Type 1 individuals would get a rent approximately equal to 0.18 if they were to hire an optimal number of workers at wage w, and thus, they strictly prefer to be workers rather than managers. It follows from these properties that this matching is indeed stable.9 4. Matching with occupational choice The setting we introduce in this paper is that of a matching market featuring occupational choice, many-to-one matching, and a large number of participants. We frame 7Stability also requires individual rationality for the workers. 8In the working paper version, we fully characterize the stable matchings in this example for each α∈ (0, 1); in fact, there is a unique stable matching for each α. 9Our general framework allows for externalities and their presence is often natural. In the context of the above example, it might be that the production function depends on the aggregate managerial quality in an analogous way to Romer (1986), so that the rent of a manager with quality zis, e.g., (Z×Xˆ zdμ(ˆ z,δ))z1+αn1−α−cn when the matching is μ. 1266 Carmona and Laohakunakorn Theoretical Economics 19 (2024) this problem in the context of a labor market for simplicity, so that individuals have a choice of being a manager, a worker, or self-employed. 4.1 Environment and matching Individuals are (potentially) heterogenous in, e.g., their talent or knowledge. This is captured by a (nonempty, Polish) set Zof types. The population of individuals is described by a nonzero, finite, Borel measure νon Z;νis the type distribution.Adummy type ∅/∈Z is used to represent unmatched, i.e., self-employed individuals, and we let Z∅=Z∪{∅}, with the assumption that ∅is an isolated point in Z∅. A manager of type zmay be matched with a worker of type zunder some contract c. In particular, there is a (nonempty, Polish) set Cof contracts and a contract correspondence C:Z×Z∅⇒Cdescribing the set C(z,z)of contracts that are feasible for a manager of type zand a worker of type z(when z=∅the manager is, in fact, selfemployed and C(z,∅)describes the feasible contracts for a self-employed individual of type z). A manager is allowed to hire as many workers as he likes; to capture the many-to-one aspect of matching, a manager is matched with a measure of workers and contracts δ∈ M(Z×C).10 The definition of a matching below will impose feasibility constraints on δ via the contract correspondence C, and thus, constrain the contracts that the manager can offer to each of his employees. These constraints are of the form c∈C(z,z)and are therefore independent across workers. To capture interdependent and other feasibility constraints, we let Xbe a subset of M(Z×C)and require that managers be matched with δ∈X. Self-employed (or unmatched) managers are those matched with the dummy type ∅. To specify his contract (e.g., the number of hours worked as self-employed), we use matches of the form (z,1 (∅,c))to describe a self-employed individual of type zwith contract c. To unify the two cases, we let X∅=X∪{1(∅,c):c∈C}be the set of possible matches of managers and self-employed individuals. The set of occupations is A={w,s,m},wherewstands for worker, sfor selfemployed, and mfor manager. The choice set of each individual depends on his occupation; namely, a worker chooses among managers’ types and contracts, a selfemployed individual among contracts, and a manager among measures δ∈Xdescribing whom to hire and the contracts offered. To capture these differences, let Xm=X, Xs={1(∅,c):c∈C},Xw={1(z,c):(z,c)∈Z×C},and={(a,δ):δ∈Xa}.11 The set is the choice set of each individual as she can choose her occupation and a match feasible for the chosen occupation. We allow for externalities, and thus, preferences are allowed to depend on the matching. Matchings with occupational choice are elements of M(Z×X∅)satisfying certain 10Whenever Yis a metric space, M(Y)denotes the set of finite, Borel measures on Yendowed with the weak (narrow) topology (see Varadarajan (1958) for details). We often focus on MR(Y)where, for each R>0, MR(Y)={δ∈M(Y):δ(Y)≤R}. 11We do not distinguish between (z,c)and 1(z,c)for each (z,c)∈Z∅×C; hence, it would be simpler to replace the latter with the former in the definition of Xsand Xw. The formalization we use above provides an unified notation which simplifies the exposition elsewhere. Theoretical Economics 19 (2024) Stable matching in large markets 1267 properties described below. The preferences of an individual of type zare then described by a relation zdefined on ×M(Z×X∅)for each z∈Z. In summary, a matching market with occupational choice (a market, henceforth) is E=(Z,ν,C,C,X,(z)z∈Z). Amatching with occupational choice (a matching, henceforth) is a Borel measure μ∈M(Z×X∅)such that: (i) {z}×supp(δ)⊆graph(C)for each (z,δ)∈supp(μ),and (ii) νM+νS+νW=ν where, for each Borel subset Bof Z,νM(B)=μ(B×X),νS(B)=μ(B×(X∅\X)),and νW(B)=Z×Xδ(B×C)dμ(z,δ). The interpretation of μis as follows. First, μdescribes the occupational choices by the place in the match (z,δ); namely, the first coordinate refers to managers and the second to workers (as part of a firm) when δ∈X,andwhenδ∈X∅\X, the first coordinate refers to a self-employed individual and the second, which is equal to 1(∅,c)for some c∈C, describes the individual’s contract. Condition (i) requires that the contract is feasible according to the contract correspondence. Condition (ii) requires that everyone in the market is accounted for as follows: For each Borel subset Bof Z,μ(B×X)is the measure of managers whose type belongs to Band we call it νM(B). Similarly, μ(B×(X∅\X)) is the measure of self-employed individuals whose type belongs to Band we call it νS(B). Finally, Z×Xδ(B×C)dμ(z,δ)is the measure of workers whose type belongs to B,and thus, we call it νW(B).12 Since an individual must be either a manager, or a worker, or self-employed, condition (ii) must hold if everyone in the market is accounted for. 4.2 Stability Heading toward the definition of stable matchings, we start by defining the targets of individuals at a given matching and then define the stability set of a matching. Targets at a given matching μdepend on the type zand on the occupational choice a, and are denoted by Ta z(μ). Because one’s occupation is a choice and not a fixed characteristic, these targets are for someone planning to choose occupation a, i.e., if someone chooses occupation a,thenhistargetsareTa z(μ). The targets for the prospective self-employed are simply the contracts that are feasible when someone is unmatched: For each z∈Z,letTs z(μ)={∅}×C(z,∅). The targets of prospective managers and workers are more complicated as they consist of contracts and types of people on the other side of the market that managers or workers can attract. But with occupational choice, there is not a fixed “other side of the market” since anyone can change his occupation. In more detail, even if all individuals of type z∗are managers in the matching μ,anytypez∗person can choose to became a worker. In particular, if such z∗person gains by becoming a worker and by working for a 12For each Borel subset Eof a metric space Y, the function δ→ δ(E):M(Y)→Ris Borel measurable. This follows by the argument in Aliprantis and Border (2006, Theorem 15.13, p. 514) together with Varadarajan (1958, Theorem 3.1). 1268 Carmona and Laohakunakorn Theoretical Economics 19 (2024) manager of type zat some contract c,then(z∗,c)is a target for those of type zplanning to be a manager, i.e., it belongs to Tm z(μ). We then let, for each z∈Z,Tm z(μ)be the set of (z∗,c)∈Z×Csuch that c∈C(z,z∗)and there exists (a) (z,c,δ)∈Z×C×Xsuch that (z,δ)∈supp(μ),(z∗,c)∈supp(δ)and (w, 1(z,c),μ)z∗(w,1 (z,c),μ),or (b) δ∈X∅\Xsuch that (z∗,δ)∈supp(μ)and (w,1 (z,c),μ)z∗(s,δ,μ),or (c) δ∈Xsuch that (z∗,δ)∈supp(μ)and (w,1 (z,c),μ)z∗(m,δ,μ). Anyone of type zcan be a manager if he finds workers, here of type z∗, who prefer to work for him than to be in their current occupation. Each of these workers can be someone who was already a worker in μas described in condition (a), or self-employed as described by condition (b), or even a manager as described by condition (c). The targets of prospective workers are defined analogously. Thus, for each z∈Z,let Tw z(μ)be the set of (z∗,c)∈Z×Csuch that c∈C(z∗,z)and there is δ∈Xsuch that (z,c)∈supp(δ)and (a) supp(δ)\{(z,c)}⊆Tm z∗(μ)and there is (z,c,δ)∈Z×C×Xsuch that (z,δ)∈ supp(μ),(z∗,c)∈supp(δ)and (m,δ,μ)z∗(w,1 (z,c),μ),or (b) supp(δ)\{(z,c)}⊆Tm z∗(μ)and there is δ∈X∅\Xsuch that (z∗,δ)∈supp(μ)and (m,δ,μ)z∗(s,δ,μ),or (c) there is δ∈Xsuch that supp(δ)\{(z,c)}⊆Tm z∗(μ)∪supp(δ),(z∗,δ)∈supp(μ) and (m,δ,μ)z∗(m,δ,μ). As above, anyone of type zcan be a worker if she finds a manager, here of type z∗,that hires her, possibly alongside other workers as described by δ∈X,andbothagreeona feasible contract c∈C(z∗,z). This manager can be someone who was already a manager in μas described in condition (c), or self-employed as described by condition (b), or even a worker as described by condition (a). The stability set S(μ)of matching μis the set of (z,δ)∈Z×X∅such that, if δ∈X, then: (i) there does not exist (a,δ)∈such that supp(δ)⊆Ta z(μ)∪supp(δ)if a=m, supp(δ)⊆Ta z(μ)if a= m,and(a,δ,μ)z(m,δ,μ), (ii) for each (z,c)∈supp(δ),theredoesnotexist(a,δ)∈such that supp(δ)⊆ Ta z(μ)and (a,δ,μ)z(w,1 (z,c),μ), and, if δ∈X∅\X,then (iii) there does not exist (a,δ)∈such that supp(δ)⊆Ta z(μ)and (a,δ,μ)z (s,δ,μ). The set S(μ)describes matches (z,δ)that do not suffer from instability. Instability could come from those who are managers in μif a manager of type zcan find a match δthat is better than his current one δby employing workers of the types currently employed Theoretical Economics 19 (2024) Stable matching in large markets 1275 satisfied. Type βprefers γto αbut γ/∈Tβ(μ); thus, condition (ii) is satisfied. Analogous arguments establish that supp(μ)⊆S(μ), and hence μis stable. A stable matching exists in this example with a continuum of individuals because it is possible for individuals of type α,β,andγall to be matched with each other, leaving individuals of type δunmatched. More generally, our results imply that the large market version of the roommate problem admits a stable solution with or without transfers and even in the presence of externalities as long as the market is rational and continuous. 6.2 Rosen (1982) In this section, we consider the setting in Rosen (1982,Section3). Individuals are characterized by their general ability, with Z⊆Rdenoting the set of possible abilities and νdenoting its (nonzero, finite) distribution. Individuals can be workers, managers, or self-employed (here more correctly interpreted as unemployed as it will be clear from the individuals’ payoffs) and their productivity is determined both by this choice and their ability, with q=q(z)denoting the productivity of someone of ability zwho chooses to be a worker and r=r(z)his productivity if he chooses to be a manager; both rand qare nondecreasing functions of the ability z. A firm consists of one manager and several workers of the same type, i.e., there is many-to-one matching. Managers have one unit of time and need to supervise workers: The output produced by a worker with productivity qin a firm with a manager with productivity ris g(r)f(tr,q),wheretis the time spent by the manager supervising the worker, g(r)represents the quality of management decisions of a manager of productivity r,g:R+→R+is increasing, and f:R2 +→R+is continuously differentiable, homogeneous of degree 1, strictly increasing and strictly concave in each coordinate in the interior of its domain26 and satisfies f(0, y)=f(x,0 )=0 for each x,y∈R+.Forconvenience, we define θ:R+→R+as θ(x)=f(x,1 )for each x∈R+;notethatθis strictly increasing and strictly concave. The output of a firm with a manager of ability rand a measure nof workers with productivity qis ng(r)fr n,q=g(r)f(r,nq)=g(r)nqθr nq since the time spent in each worker is t=1/n.27 The manager’s rent is g(r)f(r,nq)−cn =g(r)nqθr nq−cn, where cis the wage paid by the manager to the workers. 26Meaning that for (x,y)∈R2 ++,∂f (x,y)/∂x > 0, ∂f (x,y)/∂y > 0, and x→ ∂f (x,y)/∂x and y→ ∂f (x,y)/∂y are strictly decreasing over R++. 27This claim follows from the Jensen’s integral inequality as follows. Let μ∈M([0, 1]) be a probability distribution of time spent on workers so that μ(B)is the fraction of workers who get supervision time in B,foreachBorelsubsetBof [0, 1], and 11/n ∈M([0, 1]) be the probability distribution degenerate on 1/n.Thenntdμ(t)=1 and g(r)nqθ(rt/q)dμ(t)=nqg(r)θ(rt/q)dμ(t)≤nqg(r)θ(rtdμ(t)/q)= nqg(r)qθ(r/nq)=g(r)nqθ(rt/q)d11/n(t). 1276 Carmona and Laohakunakorn Theoretical Economics 19 (2024) To represent the above setting in the general framework of Section 4, let in addition to Zand νas above, the set of contracts be C=R+, interpreted as the set of possible wages, and the contract correspondence be C≡C. The set of feasible matches for managers is X={n1(z,c):(z,c)∈Z×Cand n∈R+}since managers can hire several workers all of the same type. Occupations are the same as in the general framework: A={w,s,m}. Finally, preferences are defined by specifying payoff functions as follows: Uz(w,1 (z,c))=cfor each 1(z,c)∈Xw, Uz(s,1 (∅,c))=0 for each 1(∅,c)∈Xs,and Uz(m,n1(z,c))=gr(z)fr(z),nqz−cn for each n1(z,c)∈Xm. We will establish existence and obtain a characterization of stable matchings for the setting of this section under the following simplifying assumptions. We let Z=[z,¯ z] with 0 ≤z<¯ z<∞and assume that q(z)>0, r(z)>0andg(r)>0 for each r>0; thus, g(r(z)) >0. A market satisfying these assumptions as well as the additional specifications described above is a Rosen market and denoted by Erosen. Concerning the existence of stable matchings, note that a Rosen market fails to satisfy two assumptions of our existence result; namely, the contract correspondence fails to be compact-valued and the market fails to be bounded. Nevertheless, by considering a sequence of truncated Rosen markets that satisfy our assumptions, we show that stable matchings exist. Corollary 3. Every Rosen market has a stable matching. We next provide a characterization of stable matchings in Rosen markets that is analogous to the formulation in Rosen (1982). The following concepts are needed. Let r∈r(Z),q∈q(Z)and w>0. If nsolves maxn∈R+[g(r)f(r,nq)−wnq],then w=g(r)∂f (r,nq) ∂y =g(r) ∂f r nq,1  ∂y since ∂f /∂y is homogeneous of degree zero. Thus, there is a continuous function φ: r(Z)×R++ →R++ such that nq =φ(r,w). The manager’s rent is then g(r) ∂f r nq,1  ∂x r=g(r) ∂f r φ(r,w),1  ∂x r. The above functions and formulas are used to define, for each manager of type z, the optimal number of workers of type zhe wants to hire at wage wq(z)and the corresponding rent. Define n:Z2×R++ →R++ by setting, for each (z,z,w)∈Z2×R++, nz,z,w=φr(z),w qz. Theoretical Economics 19 (2024) Stable matching in large markets 1277 Moreover, define R:Z×R++ →R+by setting, for each (z,w)∈Z×R++, R(z,w)=gr(z) ∂f r(z) φr(z),w,1  ∂x r(z). Theorem 3. A matching μof a Rosen market is stable if and only if there exists λ∈ M(Z2)and w>0such that λ(B×Z)+Z×B nz,z,wdλz,z=ν(B)for each measurable B⊆Z,(1) supp(λ)⊆z∈Z:R(z,w)≥wq(z)×z∈Z:wq(z)≥R(z,w),and (2) μ=λ◦h−1,(3) where h:Z2→Z×Xis defined by setting, for each (z,z)∈Z2, hz,z=z,nz,z,w1(z,wq(z)). As Theorem 3illustrates, our framework is tractable and our stability notion admits a simple characterization in applied settings; they can therefore be used to clarify important economic questions and highlight what forces might explain them. We give one such example when q(z)=r(z)=zand the technology takes the form g(z)zα(nz)1−α with α=1/2. If g≡1, then each individual is indifferent between being a manager or a worker and each individual of type zhas an income (wage or rent) equal to z/2.28 In contrast, if g(z)=z, then individual income is no longer necessarily linear in the type. For example, when Z={z1,,z4}, it is possible to construct a stable matching where individuals of type z1and z2are workers, individuals of type z3and z4are managers, each person strictly prefers his occupation to the alternative one, and for some w>0, workers’ income is wz while managers’ income is z3/4w.29 In this latter example, any change that leads to a decrease in wcauses an increase in the income of those in the top and a decrease in the income of those in the bottom of the income distribution.30 In addition, as a result of decrease in w, there is less inequality at the bottom (since the function z→ wz describing the income of those in the bottom of the distribution becomes flatter) and more at the top of the income distribution (since the function z→ z3/4w describing the income of those in the top of the distribution becomes steeper). 28Indeed, if α=1/2 and g≡1, then R(z,w)≥wq(z)if and only if 1/2≥w.Itthenmustbethatw=1/2in any stable matching since otherwise there would be no worker or no managers; thus, R(z,w)=wq(z)=z/2 for each z∈Z. 29If α=1/2, Z={z1,,z4}and gis the identity, then pick w∈(2z2,2z3), which implies that R(z,w)> wq(z)for each z∈{z3,z4}and R(z,w)<wq (z)for each z∈{z1,z2}.Letνbe such that ν(z3)=ν(z4)= 1, ν(z2)=n(z4,z2,w), and ν(z3)=n(z3,z1,w).Thenλsuch that λ(z3,z1)=λ(z4,z2)=1 yields a stable matching. Payoffs are wz for each z∈{z1,z2}and R(z,w)=z3/4wfor each z∈{z3,z4}. 30Such a decrease in wwould occur, e.g., if ν(z1)and ν(z2)increase by a small amount. 1278 Carmona and Laohakunakorn Theoretical Economics 19 (2024) 6.3 Further applications In the working paper version, we consider additional applications of our framework, which we summarize here, to illustrate its flexibility. Specifically, we show how our framework can capture the settings of Garicano and Rossi-Hansberg (2004)andGaricano and Rossi-Hansberg (2006), and how it can be extended to accommodate Lucas’s (1978) model. Both Garicano and Rossi-Hansberg (2004)andGaricano and Rossi-Hansberg (2006) require feasible matches for managers that depend on the types of the workers hired. This dependence arises because the measure of workers that a manager can hire is determined by the time constraint of the manager and is increasing in the quality of the workers. In Garicano and Rossi-Hansberg (2004), all workers have the same quality but in Garicano and Rossi-Hansberg (2006)a manager can hire workers of finitely many different qualities. In Lucas (1978), there is a capital market in addition to a labor market with occupational choice. The easiest approach to represent this setting is to consider, for each rental price of capital, the resulting market with occupational choice with the amount of capital hired by a firm being included in the contract between the manager and workers. An equilibrium is then a rental price of capital and a matching such that the matching is stable given the rental price and the capital market clears. 7. Concluding remarks In this paper, we provided a formalization of large many-to-one matching markets with occupational choice and a notion of a stable matching for them. This was done with the goal of being able to include the settings of Lucas (1978), Rosen (1982), Garicano and Rossi-Hansberg (2004), and Garicano and Rossi-Hansberg (2006) in our framework, while at the same time extending the two-sided, one-to-one matching setting of Greinecker and Kah (2021). The large matching markets we consider are, as in Greinecker and Kah (2021), formalized using a distributional approach. Thus, the set of individuals is not explicitly included, rather only the distribution of individuals’ types is present in the description of the market. This approach is tractable and this has been illustrated in Section 6.2 in the context of Rosen’s (1982) setting where stable matchings are fully characterized. The above tractability makes our setting potentially useful to address the implications of stability in large labor markets, in particular, for income inequality. We aim to do so in future work. The representation of Lucas’s (1978) setting in our framework required the introduction of capital, which proved to be a relatively easy extension. This suggests that other important elements can be added to our framework. Appendix A.1 Preliminary lemmas This section presents some lemmas on the support of a measure and on the existence of convergent subsequences for which we could not find a reference. Lemma 1shows that Theoretical Economics 19 (2024) Stable matching in large markets 1279 the support of the image μ◦h−1of a measure μunder a homeomorphism his the image of the support of μ. Lemma 1. Let Yand Ybe separable metric spaces, μ∈M(Y),h:Y→Ybe a homeomorphism and ν=μ◦h−1.Thensupp(ν)=h(supp(μ)) and supp(μ)=h−1(supp(ν)). Proof. Note first that ν(supp(ν)) =ν(Y)=μ(h−1(Y)) =μ(Y)=μ(supp(μ)) and, since supp(μ)=h−1(h(supp(μ))), νsupp(ν)≥νhsupp(μ)=μh−1hsupp(μ) =μsupp(μ)≥μh−1supp(ν)=νsupp(ν). Thus, μsupp(μ)=μh−1supp(ν)=νhsupp(μ)=νsupp(ν). Since h−1(supp(ν)) is closed, supp(μ)⊆h−1(supp(ν)), and hence h(supp(μ)) ⊆h(h−1(supp(ν))) =supp(ν). Letting fdenote the inverse of h,wehavethath(F)=f−1(F)is closed for each closed subset Fof Y. Thus, it follows that supp(ν)⊆h(supp(μ)). It follows from supp(ν)=h(supp(μ)) that h−1(supp(ν)) =h−1(h(supp(μ))) = supp(μ). Lemma 2shows that the support correspondence is lower hemicontinuous. Lemma 2. If Yis a separable metric space, then the correspondence μ→ supp(μ),from M(Y)to Y, is lower hemicontinuous. Proof. WehavethatM(Y)is a separable metrizable space by Varadarajan (1958,Theorem 3.1). The conclusion then follows from (the proof of) Aliprantis and Border (2006, Theorem 17.14, p. 563). Lemma 3provides conditions for the existence of a convergent subsequence. Lemma 3. If Yis a separable metrizable space and {μk}∞ k=1is a tight sequence in M(Y) such that, for some R>0,μk(Y)≤Rfor all k∈N,then{μk}∞ k=1has a convergent subsequence. Proof. The proof reduces to the case of probability measures as follows: Suppose first that there is a subsequence {μkj}∞ j=1such that μkj(Y)→0. Then this subsequence converges to the zero measure. Thus, we may assume that there is ε>0suchthatμk(Y)≥ε for all but finitely many k. The sequence {μk(Y)}kis bounded, thus we may assume that it converges; let θ=limkμk(Y).Consider {νk}∞ k=1with νk(B)=μk(B)/μk(Y)for each Borel B. This is a tight family of probability measures, so it has a convergent subsequence {νkj}∞ j=1;letν=limjνkj,μ=θν,andBhas μ-null boundary, which happens if and only if it has ν-null boundary since θ≥ε.Thenμkj(B)=μkj(Y)μkj(B)/μkj(Y)→ θν(B), and hence μkj→μ. 1280 Carmona and Laohakunakorn Theoretical Economics 19 (2024) A.2 Proof of Theorem 1 InastablematchingofGale and Shapley’s (1962) marriage market, (i) each woman cannot find a man (including the empty man) that she prefers to her husband and who would prefer her to his wife, i.e., each woman cannot find a man in her targets that she prefers to her husband, and (ii) each man cannot find a woman in his targets that he prefers to his wife. It turns out that (ii) implies (i) and Theorem 1is the analog of this in our setting. We now turn to the proof of Theorem 1. Note first that supp(μ)⊆S(μ)implies that supp(μ)⊆SM(μ)∩IR(μ)since S(μ)⊆SM(μ)∩IR(μ). Conversely, suppose that supp(μ)⊆SM(μ)∩IR(μ).Let (z,δ)∈supp(μ)and assume, to reach a contradiction, that (z,δ)/∈S(μ). Since (z,δ)∈supp(μ)⊆SM(μ)∩ IR(μ), it follows that there is (z∗,c)∈Z×Cand ¯ z∈Zsuch that (z∗,c)∈Tw ¯ z(μ),¯ z=z or (¯ z,¯ c)∈supp(δ)for some ¯ c∈C,(1)(w,1 (z∗,c),μ)¯ z(m,δ,μ)if ¯ z=zand δ∈X,(2) (w,1 (z∗,c),μ)¯ z(w,1 (z,¯ c),μ)if (¯ z,¯ c)∈supp(δ),and(3)(w,1 (z∗,c),μ)¯ z(s,δ,μ)if ¯ z=z and δ∈X∅\X. Since (z∗,c)∈Tw ¯ z(μ), it follows that c∈C(z∗,¯ z). We now show that (¯ z,c)∈Tm z∗(μ). Indeed, we have that c∈C(z∗,¯ z)and (z,δ)∈ supp(μ). Thus, in case (1), the conclusion follows by condition (c) in the definition of Tm z∗(μ)since ¯ z=zand (w,1 (z∗,c),μ)z(m,δ,μ); in case (2), the conclusion follows by condition (a) in the definition of Tm z∗(μ)since (¯ z,¯ c)∈supp(δ)and (w,1 (z∗,c),μ)¯ z (w,1 (z,¯ c),μ); and, in case (3), the conclusion follows by condition (b) in the definition of Tm z∗(μ)since ¯ z=zand (w,1 (z∗,c),μ)z(s,δ,μ). Since (z∗,c)∈Tw ¯ z(μ),thereis ˜ δ∈Xsuch that (¯ z,c)∈supp(˜ δ)and (a) or (b) or (c) in the definition of Tw ¯ z(μ)holds. In either case, we will show that supp(μ)⊆SM(μ)fails, which is a contradiction to supp(μ)⊆SM(μ)∩IR(μ). Suppose that condition (a) in the definition of Tw ¯ z(μ)holds. Then, in addition, supp(˜ δ)\{(¯ z,c)}⊆Tm z∗(μ),andthereis(z,c,δ)∈Z×C×Xsuch that (z,δ)∈ supp(μ),(z∗,c)∈supp(δ),and(m,˜ δ,μ)z∗(w,1 (z,c),μ). Since (¯ z,c)∈Tm z∗(μ),itfollows that (z,δ)∈supp(μ)\SM(μ)since (ii) of the definition of SM(μ)fails. Indeed, (z,δ)∈supp(μ),(z∗,c)∈supp(δ),supp (˜ δ)⊆Tm z∗(μ),and(m,˜ δ,μ)z∗(w,1 (z,c),μ). Suppose next that condition (b) in the definition of Tw ¯ z(μ)holds. Then, in addition, supp(˜ δ)\{(¯ z,c)}⊆Tm z∗(μ),andthereisδ∈X∅\Xsuch that (z∗,δ)∈supp(μ)and (m,˜ δ,μ)z∗(s,δ,μ). Since (¯ z,c)∈Tm z∗(μ), it follows that (z∗,δ)∈supp(μ)\SM(μ) since (iii) of the definition of SM(μ)fails. Indeed, (z∗,δ)∈supp(μ),supp (˜ δ)⊆Tm z∗(μ) and (m,˜ δ,μ)z∗(s,δ,μ). Finally, suppose that condition (c) in the definition of Tw ¯ z(μ)holds. Then, in addition, there is δ∈X∅\Xsuch that (z∗,δ)∈supp(μ),supp (˜ δ)\{(¯ z,c)}⊆Tm z∗(μ)∪ supp(δ),and(m,˜ δ,μ)z∗(m,δ,μ). Since (¯ z,c)∈Tm z∗(μ), it follows that (z∗,δ)∈ supp(μ)\SM(μ)since (i) of the definition of SM(μ)fails. Indeed, (z∗,δ)∈supp(μ), supp(˜ δ)⊆Tm z∗(μ)∪supp(δ),and(m,˜ δ,μ)z∗(m,δ,μ). A.3 Proof of Theorem 2 The first step in the proof of our existence result consists in the following lemma, which considers the special case where Z,X,andCare finite. Our approach in this special Theoretical Economics 19 (2024) Stable matching in large markets 1281 case builds on ideas from Section S.10 in Che, Kim, and Kojima (2019) but requires many changes since there are externalities in preferences, workers’ preferences are not strict, and there is occupational choice. There are three main changes, which we now briefly describe.31 Our approach in the special case where Z,X,andCare finite is similar to the one in Che, Kim, and Kojima (2019) to the extent that we use a fixed-point argument. In their paper, stable matchings are fixed points of a correspondence whose domain consists of pairs of matchings and measures of available workers. In our case, (i) we consider a sequence of correspondences, each of which has a fixed point, but only limit points of the sequence of fixed points will yield a stable matching, (ii) the domain of each correspondence consists of pairs of allocations of types to occupations and matches and measures of available workers and contracts, and (iii) the measure of available workers and contracts depends on the allocations of types to occupations and matches in a discontinuous way, and thus, needs to be suitably approximated. Lemma 4. If Eis a rational and continuous market such that Z,X,andCare finite, then Ehas a stable matching. Proof. Note first that Z∅,X∅,andare also finite. Define ¯τ∈RZ×by setting, for each (z,a,δ)∈Z×, ¯τ(z,a,δ)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0if {z}×supp(δ)graph(C)and a= w, 0ifsupp (δZ)×{z}×supp(δC)graph(C)and a=w, ν(z)otherwise. Let ¯κ∈RZ×Z×Cbe such that ¯κ(z,z,c)=ν(z)if (z,z,c)∈graph(C),and¯κ(z,z,c)= 0otherwise. Define T=τ∈RZ× +:τ(z,a,δ)≤¯τ(z,a,δ)and  (a,δ)∈ τ(z,a,δ)≤ν(z) for each (z,a,δ)∈Z×and K=κ∈RZ×Z×C +:κz,z,c≤¯κz,z,cfor each z,z,c∈Z×Z×C. Note that Tand Kare nonempty, convex, and compact subsets of Euclidean spaces. Each τ∈Tis an allocation of types to occupations and matches and, for each κ∈K,we interpret κ(z,z,c)as the measure of workers of type zand contract cthat are available to z. Below, we will consider allocations τthat maximize preferences subject to the constraints that each manager type does not hire more workers than available to him (given by κ) and that the measure of each worker type allocated to a manager does not exceed the manager’s demand (given by some reference μ). 31See the working paper version for a more detailed outline of the proof of Theorem 2. 1282 Carmona and Laohakunakorn Theoretical Economics 19 (2024) Let u:Z××M(Z×X∅)→Rbe a continuous utility function that represent preferences. We normalize so that u≥1. For each n∈N,letun=un. Since x→ xnis strictly increasing on [1, ∞),unand urepresent the same preferences. Define d:T→RZ×X∅ +by setting, for each τ∈Tand (z,δ)∈Z×X∅, d(τ)(z,δ)=τ(z,m,δ)if δ∈X, τ(z,s,δ)if δ∈X∅\X. The function dis continuous. We abuse notation and, for each (z,a,δ,τ)∈Z××T, write u(z,a,δ,τ)for u(z,a,δ,d(τ)) and analogously for un.Wealsowrite (a,δ,τ)z (a,δ,τ)for (a,δ,d(τ)) z(a,δ,d(τ)),where(a,δ)∈. For each n∈N,letDn:T×K⇒Tbe defined by setting, for each (μ,κ)∈T×K, Dn(μ,κ)=τ∈T:τ∈argmax τ∈T z∈Z,(a,δ)∈ un(z,a,δ,μ)τ(z,a,δ) subject to  (a,δ)∈ τ(z,a,δ)=ν(z)for all z∈Z,  δ∈X τ(z,m,δ)δz,c≤κz,z,cfor all z,z,c∈Z×Z×C,and τ(z,w,1 (z,c))≤ δ∈X μz,m,δδ(z,c)for all z,z,c∈Z×Z×C. Claim 1. Dnis upper hemicontinuous with nonempty, compact, and convex values. Proof. It follows by the linearity of the objective function together with the convexity of the constraint set that Dnhas convex values. It follows from Berge’s maximum theorem that Dnis upper hemicontinuous with nonempty and compact values. To see this, first note that the objective function is continuous and that the constraint set, denoted by (μ,κ), is contained in the compact set T. It is clear that is upper hemicontinuous with compact values. To see that has nonempty values, define ˜τ∈Tas follows. For each z∈Z,letcz∈C(z,∅),˜τ(z,s,1 (∅,cz))=ν(z)and ˜τ(z,a,δ)=0 for each (a,δ)∈ \{(s,1 (∅,cz))}. We then have that ˜τ∈(μ,κ)for each (μ,κ)∈T×K. Finally, to see that is lower hemicontinuous, let (μ,κ)∈T×K,O⊆Tbe an open set such that (μ,κ)∩ O= ∅,andτ∈(μ,κ)∩O.Letˆτ=λτ +(1−λ)˜τ∈Ofor some λ∈(0, 1).Notethat for each z∈Z,(a,δ)∈ˆτ(z,a,δ)=ν(z),δ∈Xˆτ(z,m,δ)δ(z,c)<κ (z,z,c)for each (z,z,c)∈Z×Z×Csuch that κ(z,z,c)>0andˆτ(z,w,1 (z,c))<δ∈Xμ(z,m,δ)δ(z,c) for each (z,z,c)∈Z×Z×Csuch that δ∈Xμ(z,m,δ)δ(z,c)>0. Thus, there is an open neighborhood Vof (μ,κ)such that ˆτ∈(μ,κ)∩Ofor each (μ,κ)∈V. Claim 2. If (μ,κ)∈T×K,τ∈Dn(μ,κ),and(z,a,δ)∈Z×is such that τ(z,a,δ)>0, then τ(z,w,1 (ˆ z,ˆ c))=δ∈Xμ(ˆ z,m,δ)δ(z,ˆ c)for each (ˆ z,ˆ c)∈Z×Csuch that (w,1 (ˆ z,ˆ c), μ)z(a,δ,μ). Theoretical Economics 19 (2024) Stable matching in large markets 1283 Proof.Ifnot,thenτ(z,w,1 (ˆ z,ˆ c))<δ∈Xμ(ˆ z,m,δ)δ(z,ˆ c)for some (ˆ z,ˆ c)∈Z×C such that (w,1 (ˆ z,ˆ c),μ)z(w,1 (z,c),μ).Then δ∈Xμ(ˆ z,m,δ)δ(z,ˆ c)>0, and hence, (ˆ z,z,ˆ c)∈graph(C). Thus, increase τ(z,w,1 (ˆ z,ˆ c))and decrease τ(z,a,δ)by the same amount ε∈(0, τ(z,a,δ)) such that τ(z,w,1 (ˆ z,ˆ c))+ε<δ∈Xμ(ˆ z,m,δ)δ(z,ˆ c). This increases the objective function in Dn(μ,κ)while satisfying the constraints, thus contradicting τ∈Dn(μ,κ). For each μ∈Tand (z,z,c)∈Z×Z×C,let Wz,z,c,μ=(a,δ)∈:uz,w,1 (z,c),μ>u z,a,δ,μ. Let g:T→Kbe defined by setting, for each μ∈Tand (z,z,c)∈Z×Z×C, g(μ)z,z,c=⎧ ⎪ ⎨ ⎪ ⎩  (a,δ)∈W(z,z,c,μ) μz,a,δif z,z,c∈graph(C), 0otherwise. To see that g(μ)∈K, first note that if (z,z,c)/∈graph(C),g(μ)(z,z,c)=0. If (z,z,c)∈ graph(C), then since μ∈T,0≤g(μ)(z,z,c)≤ν(z)=¯κ(z,z,c). The function gmay fail to be continuous, and thus, we will consider a continuous approximation to it. For each n∈Nand (z,z,c)∈Z×Z×C,letαn,(z,z,c):×T→[0, 1] be defined by setting, for each (a,δ,μ)∈×T, αn,(z,z,c)(a,δ,μ)=nmax0, minuz,w,1 (z,c),μ−uz,a,δ,μ,1 n. Let gn:T→Kbe defined by setting, for each μ∈Tand (z,z,c)∈Z×Z×C, gn(μ)z,z,c=⎧ ⎪ ⎨ ⎪ ⎩  (a,δ)∈ αn,(z,z,c)(a,δ,μ)μz,a,δif z,z,c∈graph(C), 0otherwise. We have that gnis continuous since αn,(z,z,c)is continuous for each (z,z,c)∈Z×Z×C. Note that αn,(z,z,c)(a,δ,μ)∈⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ {0}if uz,a,δ,μ≥uz,w,1 (z,c),μ, (0, 1)if uz,w,1 (z,c),μ−1 n<u z,a,δ,μ<u z,w,1 (z,c),μ, {1}if uz,a,δ,μ≤uz,w,1 (z,c),μ−1 n. Hence, it follows that gn(μ)z,z,c≤g(μ)z,z,c(4) for each μ∈Tand (z,z,c)∈Z×Z×Csince g(μ)z,z,c=⎧ ⎪ ⎨ ⎪ ⎩  (a,δ)∈ α(z,z,c)(a,δ,μ)μz,a,δif (m,w,c)∈graph(C), 0otherwise 1284 Carmona and Laohakunakorn Theoretical Economics 19 (2024) with α(z,z,c)(a,δ,μ)=1ifuz,a,δ,μ<u z,w,1 (z,c),μ, 0otherwise. To see that gn(μ)∈K,notethat0≤gn(μ)≤g(μ)≤¯κ. Let fn:T→Kbe defined by setting, for each μ∈Tand (z,z,c)∈Z×Z×C, fn(μ)z,z,c=μz,w,1 (z,c)+1 ngn(μ)z,z,c. To see that fn(μ)∈K,notethatif(z,z,c)/∈graph(C), μz,w,1 (z,c)=gn(μ)z,z,c=0, and hence fn(μ)(z,z,c)=0. If (z,z,c)∈graph(C),then 0≤μz,w,1 (z,c)+1 ngn(μ)z,z,c ≤μz,w,1 (z,c)+g(μ)z,z,c≤νz=¯κz,z,c. We have that fnis continuous since so is gn. Let n:T×K⇒T×Kbe defined by setting, for each (μ,κ)∈T×K, n(μ,κ)=Dn(μ,κ)×fn(μ). It follows from the continuity of fnand from Claim 1that nis upper hemicontinuous with nonempty, compact, and convex values. Hence, by Kakutani fixed-point theorem, let (μn,κn)be a fixed point of n.Thus,μn∈Dn(μn,κn)and κn=fn(μn). Since T×Kis compact, taking a subsequence if necessary, we may assume that {(μn,κn)}∞ n=1converges; let (μ,κ)=limn→∞(μn,κn). For each n,wehaveκn=fn(μn), and so κz,z,c=lim n→∞ fn(μn)z,z,c=μz,w,1 (z,c)(5) for each (z,z,c)∈Z×Z×C.Let μ∗=d(μ) and μ∗ n=d(μn)for each n∈N. For each z∈Zand n∈N, it follows from μn∈Dn(μn,κn)that  (z,c)∈Z×C μn(z,w,1 (z,c))≤ (z,c)∈Z×C δ∈X μnz,m,δδ(z,c)≤ (z,c)∈Z×C κnz,z,c. By (5), limn(z,c)∈Z×Cκn(z,z,c)=(z,c)∈Z×Cμ(z,w,1 (z,c)), and hence,  (z,c)∈Z×C μ(z,w,1 (z,c))= (z,c)∈Z×C δ∈X μz,m,δδ(z,c)for each z∈Z. Theoretical Economics 19 (2024) Stable matching in large markets 1291 Condition (i) holds because, by Carmona and Podczeck (2009, Lemma 12), for each (z,δ)∈supp(μ)and (z,c)∈supp(δ), there exists a subsequence {μkj}∞ j=1of {μk}∞ k=1and corresponding {(zkj,δkj,z kj,ckj)}∞ j=1such that (zkj,δkj,z kj,ckj)→(z,δ,z,c),andfor each j∈N,(zkj,δkj)∈supp(μkj)and (z kj,ckj)∈supp(δkj).Hence,ckj∈Ckj(zkj,z kj)⊆ C(zkj,z kj), and since Cis continuous, c∈C(z,z). Part 3:Let(z,δ)∈supp(μ)and suppose that (z,δ)/∈IR(μ). Then either (i) there exists c∈C(z,∅)such that (s,1 (∅,c),μ)z(a(δ),δ,μ)where a(δ)=mif δ∈Xand a(δ)=sif δ∈X∅\X, or (ii) there exists (z,c)∈supp(δ)and c∈C(z,∅)such that (s,1 (∅,c),μ)z(w,1 (z,c),μ). Consider case (i) first. The continuity of (z)z∈Zand Cimplies that there are open neighborhoods Vc,Vz,Vδ,andVμof c,z,δ,andμ, respectively, such that (s,1 (∅,ˆ c),ˆμ)ˆ z (a(δ),ˆ δ,ˆμ)and C(ˆ z,∅)∩Vc= ∅ for each ˆ c∈Vc,ˆ z∈Vz,ˆ δ∈Vδ,and ˆμ∈Vμ. Since (z,δ)∈ supp(μ), it follows that 0 <μ (Vz×Vδ)≤liminfkμk(Vz×Vδ); hence, for each ksufficiently large, μk(Vz×Vδ)>0andμk∈Vμ. This means that, for any such k,thereexist (ˆ z,ˆ δ)∈supp(μk)∩(Vz×Vδ)and ˆ c∈C(ˆ z,∅)∩Vc.Butthen(s,1 (∅,ˆ c),μk)ˆ z(a(δ),ˆ δ,μk), and hence (s,1 (∅,ˆ c),μk)ˆ z,k(a(δ),ˆ δ,μk), contradicting the individual rationality of μk. Consider next case (ii). The continuity of (z)z∈Zand Cimplies that there are open neighborhoods Vc,Vc,Vz,Vz,andVμof c,c,z,z,andμ, respectively, such that (s,1 (∅,˜ c),ˆμ)˜ z(w,1 (ˆ z,ˆ c),ˆμ)and C(∅,˜ z)∩Vc= ∅ for each ˜ c∈Vc,ˆ c∈Vc,ˆ z∈Vz, ˜ z∈Vz,and ˆμ∈Vμ. Since (z,c)∈supp(δ), there is an open neighborhood Vδof δ such that supp(ˆ δ)∩(Vz×Vc)= ∅ for each ˆ δ∈Vδby Lemma 2. Since μk→μand (z,δ)∈supp(μ), it follows that 0 <μ (Vz×Vδ)≤liminfkμk(Vz×Vδ); hence, for all k sufficiently large, μk(Vz×Vδ)>0andμk∈Vμ. This means that, for any such k,there exists (ˆ z,ˆ δ)∈supp(μk)∩(Vz×Vδ),(˜ z,ˆ c)∈supp(ˆ δ)∩(Vz×Vc)and ˜ c∈C(∅,˜ z)∩Vc.But then (s,1 (∅,˜ c),μk)˜ z(w,1 (ˆ z,ˆ c),μk), and hence (s,1 (∅,˜ c),μk)˜ z,k(w,1 (ˆ z,ˆ c),μk),contradicting the individual rationality of μk. Part 4: In this proof, to avoid confusion, we write Tm z(μ;E)for Tm z(μ)in a market E. Let (z,δ)∈supp(μ)and suppose that (z,δ)/∈SM(μ). Then there exists δ∈Xsuch that either (i) supp(δ)⊆Tm z(μ)∪supp(δ)and (m,δ,μ)z(a(δ),δ,μ),wherea(δ)= mif δ∈Xand a(δ)=sif δ∈X∅\X,32 or (ii) there exists (z,c)∈supp(δ)such that supp(δ)⊆Tm z(μ)and (m,δ,μ)z(w,1 (z,c),μ). Consider case (i) first. Let Vz,Vδ,Vδ,andVμbe open neighborhoods of z,δ,δ,and μ, respectively, such that (m,γ,¯μ)¯ z(a(δ),γ,¯μ)for each ¯ z∈Vz,γ∈Vδ,γ∈Vδ,and ¯μ∈Vμ.Let,bytherichnessofE,˜ Vz,˜ Vδ,and ˜ Vμbe open neighborhoods of z,δ,andμ, respectively, such that (¯ z,γ,¯μ)∩Vδ= ∅ for each (¯ z,γ,¯μ)∈˜ Vzט Vδט Vμ. By Carmona and Podczeck (2009, Lemma 12), there is a subsequence {μkj}∞ j=1of {μk}∞ k=1and corresponding sequence {(zkj,δkj)}∞ j=1such that (zkj,δkj)→(z,δ)and (zkj,δkj)∈supp(μkj)for each j∈N. Let J∈Nbe such that μkj∈Vμ∩˜ Vμ,zkj∈Vz∩˜ Vz,δkj∈Vδ∩˜ Vδ,and{γ∈Xkj: {zkj}×supp(γ)⊆graph(Ckj)}∩(zkj,δkj,μkj)∩Vδ= ∅ for all j≥J.Letj≥Jand let δ kj∈{γ∈Xkj:{zkj}×supp(γ)⊆graph(Ckj)}∩(zkj,δkj,μkj)∩Vδ.Thensupp (δ kj)⊆ 32Note that when δ∈X∅\Xand δ∈X,supp (δ)⊆Tm z(μ)∪supp(δ)if and only if supp(δ)⊆Tm z(μ). 1292 Carmona and Laohakunakorn Theoretical Economics 19 (2024) Tm zkj(μkj;E)∪supp(δkj)and (m,δ kj,μkj)zkj(a(δ),δkj,μkj). It then follows that supp(δ kj)⊆Tm zkj(μkj;Ekj)∪supp(δkj)and (m,δ kj,μkj)zkj,kj(a(δ),δkj,μkj).Butthis contradicts the stability of μkj. Consider next case (ii). Let Vz,Vz,Vδ,Vc,andVμbe open neighborhoods of z,z, δ,c,andμ, respectively, such that (m,ˆ δ,ˆμ)ˆ z(w,1 (ˆ z,ˆ c),ˆμ)for each ˆ z∈Vz,ˆ z∈Vz, ˆ δ∈Vδ,ˆ c∈Vc,and ˆμ∈Vμ.Let,bytherichnessofE,˜ Vz,and ˜ Vμbe open neighborhoods of zand μ, respectively, such that 0(ˆ z,ˆμ)∩Vδ= ∅ for each (ˆ z,ˆμ)∈˜ Vzט Vμ. By Carmona and Podczeck (2009, Lemma 12), there is a subsequence {μkj}∞ j=1 of {μk}∞ k=1and corresponding sequence {(zkj,δkj,z kj,ckj)}∞ j=1such that (zkj,δkj)∈ supp(μkj)and (z kj,ckj)∈supp(δkj)for each j∈Nand (zkj,δkj,z kj,ckj)→(z,δ,z,c). Let J∈Nbe such that δkj∈Vδ,zkj∈Vz,z kj∈Vz∩˜ Vz,ckj∈Vc,μkj∈Vμ∩˜ Vμ, and {γ∈Xkj:{z kj}×supp(γ)⊆graph(Ckj)}∩0(z kj,μkj)∩Vδ= ∅ for all j≥J.Let j≥Jand let δ kj∈{γ∈Xkj:{z kj}×supp(γ)⊆graph(Ckj)}∩0(z kj,μkj)∩Vδ.Then supp(δ kj)⊆Tm z kj (μkj;E)and (m,δ kj,μkj)z kj (w,1 (zkj,ckj),μkj). It then follows that supp(δ kj)⊆Tm z kj (μkj;Ekj)and (m,δ kj,μkj)z kj,kj(w,1 (zkj,ckj),μkj). But this contradicts the stability of μkj. The second step in the proof of our existence result consists in the following lemma, which considers the special case where Zis finite and X=MR(Z×C)for some R>0. Lemma 6. If Eis a rational and continuous market such that Zis finite and X=MR(Z× C)for some R>0,thenEhas a stable matching. Proof. For each (z,z)∈Z×Z∅,let{cn z,z}∞ n=1be a dense subset of C(z,z). For each k∈N, define Ck(z,z)={cn z,z:n≤k}and Ck=(z,z)∈Z×Z∅Ck(z,z). In addition, enumerate Q={q1,q2,}and, for each k∈N,letXkbe the set of δ∈ MR(Z×C)such that supp(δ)is a subset of Z×Ckand, for each (z,c)∈Z×Ck,δ(z,c)∈ {qn:n≤k}.LetX∅,k=Xk∪{1(∅,c):c∈Ck},Xm,k=Xk,Xs,k={1(∅,c):c∈Ck},Xw,k= {1(z,c):(z,c)∈Z×Ck},andk={(a,δ):δ∈Xa,k}. For each k∈N,letEk=(Z,ν,Ck,Ck,Xk,(z)z∈Z)be a market where zis restricted to k×M(Z×X∅,k)for each z∈Z.Letμk∈M(Z×X∅,k)be a stable matching in Ek, which exists by Lemma 4since Z,Ck,andXkare finite. It follows by part 1 of Lemma 5that we may assume that {μk}∞ k=1converges; let μ=limkμk. It then follows by parts 2 and 3 of Lemma 5that μis a matching and that supp(μ)⊆IR(μ). The following claim will be used to show that condition (a) of part 4 of Lemma 5 holds. Claim 8. Let (˜ z,˜ c)∈Tm z(μ)and V˜ cbe an open neighborhood of ˜ c. Then, for all ksufficiently large, there exists ck∈Ck(z,˜ z)such that (˜ z,ck)∈Tm z(μk)∩({˜ z}×V˜ c). Proof.Let(˜ z,˜ c)∈Tm z(μ)and V˜ cbe an open neighborhood of ˜ c.Then ˜ c∈C(z,˜ z) and either (i) there exists (ˆ z,ˆ δ,ˆ c)such that (ˆ z,ˆ δ)∈supp(μ),(˜ z,ˆ c)∈supp(ˆ δ),and Theoretical Economics 19 (2024) Stable matching in large markets 1293 (w,1 (z,˜ c),μ)˜ z(w,1 (ˆ z,ˆ c),μ), or (ii) there exists ˜ δ∈X∅such that (˜ z,˜ δ)∈supp(μ)and (w,1 (z,˜ c),μ)˜ z(a(˜ δ),˜ δ,μ),wherea(˜ δ)=sif ˜ δ∈X∅\Xand a(˜ δ)=mif ˜ δ∈X. Consider case (i) first. Let O˜ c,Oˆ c,Oˆ δ,andOμbe open neighborhoods of ˜ c,ˆ c,ˆ δ,and μ, respectively, such that (w,1 (z,˜ c),μ)˜ z(w,1 (ˆ z,ˆ c),μ)and supp(ˆ δ)∩({˜ z}×Oˆ c)= ∅ for each ˜ c∈O˜ c,ˆ c∈Oˆ c,ˆ δ∈Oˆ δ,andμ∈Oμ. Since 0 <μ ({ˆ z}×Oˆ δ)≤liminfkμ({ˆ z}×Oˆ δ),it follows that, for each ksufficiently large, there is ˆ δk∈Oˆ δsuch that (ˆ z,ˆ δk)∈supp(μk), and for some ˆ ck∈Oˆ c,(˜ z,ˆ ck)∈supp(ˆ δk). In addition, μk∈Oμand there exists ck∈ Ck(z,˜ z)∩O˜ c∩V˜ csince, respectively, μk→μand Ck(z,˜ z)increases to a dense subset of C(z,˜ z).Then(w,1 (z,ck),μk)˜ z(w,1 (ˆ z,ˆ ck),μk), and hence (˜ z,ck)∈Tm z(μk)∩({˜ z}×V˜ c) for all ksufficiently large. Consider next case (ii). Let O˜ c,O˜ δ,andOμbe open neighborhoods of ˜ c,˜ δ,and μ, respectively, such that (w,1 (z,˜ c),μ)˜ z(a(˜ δ),˜ δ,μ)for each ˜ c∈O˜ c,˜ δ∈O˜ δ,and μ∈Oμ. Since 0 <μ ({˜ z}×O˜ δ)≤liminfkμk({˜ z}×O˜ δ), it follows that, for each ksufficiently large, there is ˜ δk∈O˜ δsuch that (˜ z,˜ δk)∈supp(μk). In addition, μk∈Oμ and there exists ck∈Ck(z,˜ z)∩O˜ c∩V˜ csince, respectively, μk→μand Ck(z,˜ z)increases to a dense subset of C(z,˜ z).Then (w,1 (z,ck),μk)˜ z(a(˜ δ),˜ δk,μk), and hence (˜ z,ck)∈Tm z(μk)∩({˜ z}×V˜ c)for all ksufficiently large. We now show that condition (a) of part 4 of Lemma 5holds. Let (z,δ,μ)∈Z×X× M(Z×X∅),δ∈(z,δ,μ),Vδbe an open neighborhood of δand {(zkj,δkj,μkj)}∞ j=1be a sequence such that (zkj,δkj,μkj)→(z,δ,μ)and (zkj,δkj,μkj)∈Zkj×Xkj×M(Zkj× X∅,kj)for each j∈N. In particular, supp(δ)⊆Tm z(μ)∪supp(δ)and we may assume that supp(δ)is finite, i.e., δ=(˜ z,˜ c)∈supp(δ)a(˜ z,˜ c)1(˜ z,˜ c)for some a=(a(˜ z,˜ c))(˜ z,˜ c)∈supp(δ).LetVabe an open neighborhood of a, and for each (˜ z,˜ c)∈supp(δ),V(˜ z,˜ c)be an open neighborhood of (˜ z,˜ c)be such that  (˜ z,˜ c)∈supp(δ) ˆ a(˜ z,˜ c)1(z(˜ z,˜ c),c(˜ z,˜ c)) ∈Vδ whenever (z(˜ z,˜ c),c(˜ z,˜ c)) ∈V(˜ z,˜ c)for each (˜ z,˜ c)∈supp(δ)and ˆ a∈Va.Let ˆ a= (ˆ a(˜ z,˜ c))(˜ z,˜ c)∈supp(δ)∈Q|supp(δ)| +∩Vaand V˜ cbe an open neighborhood of ˜ csuch that {˜ z}×V˜ c⊆V(˜ z,˜ c). For each (˜ z,˜ c)∈supp(δ)∩Tm z(μ), and for each ksufficiently large, let ck(˜ z,˜ c)∈ Ck(z,˜ z)be such that (˜ z,ck(˜ z,˜ c)) ∈Tm z(μk)∩({˜ z}×V˜ c), which exists by Claim 8. If (˜ z,˜ c)∈supp(δ)\Tm z(μ),thenδ∈X,(˜ z,˜ c)∈supp(δ),and0<δ ({˜ z}×V˜ c)≤ liminfjδkj({˜ z}×V˜ c). Hence, for each jsufficiently large, let ckj(˜ z,˜ c)∈V˜ cbe such that (˜ z,ckj(˜ z,˜ c)) ∈supp(δkj). Let J∈Nbe such that, for each j≥J,(˜ z,ckj(˜ z,˜ c)) ∈Tm z(μkj)∩({˜ z}×V˜ c)if (˜ z,˜ c)∈ supp(δ)∩Tm z(μ)and (˜ z,ckj(˜ z,˜ c)) ∈supp(δkj)∩({˜ z}×V˜ c)if (˜ z,˜ c)∈supp(δ)\Tm z(μ). Thus, letting δ kj=(˜ z,˜ c)∈supp(δ)ˆ a(˜ z,˜ c)1(˜ z,ckj(˜ z,˜ c)) for each j≥J,wehavethatδ kj∈Vδ and supp(δ kj)⊆Tm z(μkj)∪supp(δkj). Since {ˆ a(˜ z,˜ c):(˜ z,˜ c)∈supp(δ)}is finite, it follows that there is J>J such that δ kj∈Xkjfor each j≥J. 1294 Carmona and Laohakunakorn Theoretical Economics 19 (2024) An analogous argument shows that condition (b) of part 4 of Lemma 5also holds. Hence, it follows that supp(μ)⊆SM(μ). This together with the fact that μis a matching and supp(μ)⊆IR(μ)shows that μis stable. The next step of the proof of Theorem 2extends Lemma 6by requiring only that E be rich. Lemma 7. If Eis a rational, continuous, bounded, and rich market such that Zis finite, then Ehas a stable matching. Proof. It follows by Debreu (1964, Proposition 3) and by the finiteness of Zthat there exists a continuous function u:Z××M(Z×X∅)→[1, 2]such that (a,δ,μ)→ u(z,a,δ,μ)represents zfor each z∈Z, using the fact that [1, 2]and the extended reals are homeomorphic. Let R>0besuchthatX⊆MR(Z×C),∗=({m}×MR(Z×C)) ∪({w}×Xw)∪ ({s}×Xs),andX∗=MR(Z×C)∪{1(∅,c):c∈C}. By the Tietze extension theorem, let U:Z×∗×M(Z×X∗)→[1, 2]be a continuous extension of u. Let ρbe a metric on MR(Z×C). For each k∈N,let k={m}×δ∈MR(Z×C):ρ(δ,X)≥k−1. Let, by Urysohn’s lemma, gk:∗→[0, 1]be a continuous function such that g−1 k(1)= and g−1 k(0)=k. Then define Uk:Z×∗×M(Z×X∗)→Rby setting, for each (z,a,δ,μ)∈Z×∗×M(Z×X∗),Uk(z,a,δ,μ)=gk(a,δ)U(z,a,δ,μ). Consider the market Ek=(Z,ν,C,C,MR(Z×C),Uk), i.e., Ekis equal to Eexcept that Xis replaced with MR(Z×C)and uwith Uk. Since Ekis rational and continuous with Zfinite and X=MR(Z×C),thenEkhas a stable matching μkby Lemma 6. Let E∗=(Z,ν,C,C,MR(Z×C),U). To avoid confusion, we write IR(μ;E)for IR(μ)and SM(μ;E)for SM(μ)whenever μis a matching of a market E. It follows by part 1 of Lemma 5that we may assume that {μk}∞ k=1converges; let μ=limkμk.Itthen follows by part 2 of Lemma 5that μis a matching of E∗. The proof of part 3 of Lemma 5implies that supp(μ)⊆IR(μ;E∗)since the requirement that z,kis the restriction of zto k×M(Zk×X∅,k)for each z∈Zkcan be replaced with the following condition: (s,δ,ˆμ)z,k(a,δ,ˆμ)for each k∈N,z∈Zk,δ∈ Xs,k,(a,δ)∈k,and ˆμ∈M(Zk×X∅,k)such that (s,δ,ˆμ)z(a,δ,ˆμ). This condition holds because Uk(z,a,δ,ˆμ)≤U(z,a,δ,ˆμ)and Uk(z,s,1 (∅,ˆ c),ˆμ)=U(z,s,1 (∅,ˆ c),ˆμ)for each k∈N,z∈Z,(a,δ)∈∗,ˆ c∈C,and ˆμ∈M(Z×X∗)since (s,1 (∅,ˆ c))∈, and hence gk(s,1 (∅,ˆ c))=1. We have that μbelongs to M(Z×X∅). Indeed, let k∈Nand (z,δ)∈supp(μk)∩ M(Z×C).Ifδ∈Xand ρ(δ,X)≥k−1,thenletc∈C(z,∅)and δ=1(∅,c)to obtain that supp(δ)⊆Ts z(μk)and Uk(z,s,δ,μ)=U(z,s,δ,μ)>0=Uk(z,m,δ,μ), the latter since (s,δ)∈,andthus,gk(s,δ)=1, U(z,s,δ,μ)∈[1, 2],andgk(m,δ)=0. But this contradicts the stability of μk. Hence, it follows that ρ(δ,X)<k −1. Theoretical Economics 19 (2024) Stable matching in large markets 1295 Thus, for each k∈N, supp(μk)⊆Z×δ∈MR(Z×C):ρ(δ,X)≤k−1∪Z×{1(∅,c):c∈C}. Hence, supp(μ)⊆Z×X∅as claimed. It then follows that μis a matching of Eand that supp(μ)⊆IR(μ;E)since IR(μ;E∗)∩(Z×X∅)⊆IR(μ;E).Claim9, which is analogous to part 4 of Lemma 5, shows that supp(μ)⊆SM(μ;E). Claim 9. supp(μ)⊆SM(μ;E). Proof.Let (z,δ)∈supp(μ)and suppose that (z,δ)/∈SM(μ;E). Then there exists δ∈ Xsuch that either (i) supp(δ)⊆Tm z(μ)∪supp(δ)and U(z,m,δ,μ)>U(z,a(δ),δ,μ), where a(δ)=mif δ∈Xand a(δ)=sif δ∈X∅\X(see footnote 32), or (ii) there exists (z,c)∈supp(δ)such that supp(δ)⊆Tm z(μ)and U(z,m,δ,μ)>U(z,w,1 (z,c),μ). Consider case (i) first. Let Vδ,Vδ,andVμbe open neighborhoods of δ,δ,andμ, respectively, such that U(z,m,γ,¯μ)>U(z,a(δ),γ,¯μ)for each γ∈Vδ,γ∈Vδ,and ¯μ∈ Vμ.Let,bytherichnessofE,˜ Vδ,and ˜ Vμbe open neighborhoods of δand μ, respectively, such that (z,γ,¯μ)∩Vδ= ∅ for each (γ,¯μ)∈˜ Vδט Vμ. By Carmona and Podczeck (2009, Lemma 12), there is a subsequence {μkj}∞ j=1of {μk}∞ k=1and corresponding sequence {δkj}∞ j=1such that δkj→δand (z,δkj)∈supp(μkj) for each j∈N. Let J∈Nbe such that μkj∈Vμ∩˜ Vμand δkj∈Vδ∩˜ Vδfor all j≥J, and for each j≥J, let δ kj∈(z,δkj,μkj)∩Vδ. Then, for each j≥J, Ukjz,m,δ kj,μkj=Uz,m,δ kj,μkj>Uz,a(δ),δkj,μkj≥Ukjz,a(δ),δkj,μkj since δ kj∈Xby the definition of , and supp(δ kj)⊆Tm z(μkj)∪supp(δkj). But this contradicts the stability of μkj. Now assume there exists (z,c)∈supp(δ)and δ∈Xsuch that supp(δ)⊆Tm z(μ) and U(z,m,δ,μ)>U (z,w,1 (z,c),μ).LetVδ,Vc,andVμbe open neighborhoods of δ,c,andμ, respectively, such that U(z,m,ˆ δ,ˆμ)>U (z,w,1 (z,ˆ c),ˆμ)for each ˆ δ∈Vδ, ˆ c∈Vc,and ˆμ∈Vμ.Let,bytherichnessofE,˜ Vμbe an open neighborhood of μsuch that 0(z,ˆμ)∩Vδ= ∅ for each ˆμ∈˜ Vμ. By Carmona and Podczeck (2009, Lemma 12), there is a subsequence {μkj}∞ j=1 of {μk}∞ k=1and corresponding sequence {(δkj,ckj)}∞ j=1such that (δkj,ckj)→(δ,c), (z,δkj)∈supp(μkj),and(z,ckj)∈supp(δkj)for each j∈N. Let J∈Nbe such that δkj∈Vδ,ckj∈Vc,andμkj∈Vμ∩˜ Vμfor all j≥J,and for each j≥J,letδ kj∈0(z,μkj)∩Vδ. Then, for each j≥J,Ukj(z,m,δ kj,μkj)= U(z,m,δ kj,μkj)>U (z,w,1 (z,ckj),μkj)≥Ukj(z,w,1 (z,ckj),μkj)since δ kj∈Xby the definition of 0, and supp(δ kj)⊆Tm z(μkj). But this contradicts the stability of μkj. It follows by supp(μ)⊆IR(μ;E)and by Claim 9that supp(μ)⊆SM(μ;E)∩IR(μ;E). Thus, μis stable. We now complete the proof of our existence result. 1296 Carmona and Laohakunakorn Theoretical Economics 19 (2024) Proof of Theorem 2.Let {νk}∞ k=1be such that νk→νand supp(νk)is a finite subset of Zfor each k∈N. Define Zk=supp(νk),Z∅,k=Zk∪{∅},Xk=X∩M(Zk×C),X∅,k= Xk∪{1(∅,c):c∈C},Xm,k=Xk,Xs,k={1(∅,c):c∈C},Xw,k={1(z,c):(z,c)∈Zk×C},and k={(a,δ):δ∈Xa,k}for each k∈N.NotethatXkis closed for each k∈N. For each k∈N,let ˜ Ek=(Zk,νk,C,C,Xk,(z)z∈Zk)be a market where zis restricted to k×M(Zk×X∅,k)for each z∈Z. Furthermore, let Ekbe exactly as ˜ Ek,except with Xin place of Xkand Zin place of Zk;moreprecisely,Ek=(Z,νk,C,C,X,(z )z∈Z). Claim 10. For each k∈N,ifμis a stable matching of ˜ Ek,thenμis a stable matching of Ek. Proof. In this proof, to avoid confusion, we write IR(μ;E)for IR(μ)and SM(μ;E)for SM(μ)whenever μis a matching of a market E. Let k∈Nand μbe a stable matching of ˜ Ek. Clearly, μis a matching of Ekand supp(μ)⊆IR(μ;Ek). We show that supp(μ)⊆SM(μ;Ek). Suppose not, then let (z,δ)∈ supp(μ)\SM(μ;Ek). First, suppose that there exists δ∈Xsuch that supp(δ)⊆Tm z(μ)∪supp(δ)and (m,δ,μ)z(a(δ),δ,μ),wherea(δ)=mif δ∈Xand a(δ)=sif δ∈X∅\X.Weclaim that δ∈Xk, i.e., that supp(δ)⊆Zk×C, from which we obtain a contradiction to the stability of μin ˜ Ek. Note that supp(¯ δ)⊆Zk×Cwhenever ¯ δ∈Xand (¯ z,¯ δ)∈supp(μ)for some ¯ z∈Zk since μis stable in ˜ Ek. Thus, it follows that supp(δ)∩supp(δ)⊆Zk×Csince if supp(δ)∩supp(δ)= ∅,thenδ∈X. We also have that supp(δ)∩Tm z(μ)⊆Zk×C. Indeed, if (z,c)∈Tm z(μ),then(z,¯ c)∈supp(¯ δ)and (¯ z,¯ δ)∈supp(μ)for some ¯ c∈C, ¯ z∈Zk,and¯ δ∈Xwhenever supp(δ)∩Tm z(μ)= ∅;hence,z∈Zk. Thus, supp(δ)= (supp(δ)∩supp(δ))∪(supp(δ)∩Tm z(μ)) ⊆Zk×Cas desired. Now suppose that there exists δ∈Xand (z,c)∈supp(δ)such that supp(δ)⊆ Tm z(μ)and (m,δ,μ)z(w,1 (z,c),μ). As above, we obtain a contradiction to the stability of μin ˜ Ekby showing that δ∈Xk. To establish this claim, it suffices to show that Tm z(μ)⊆Zk×C.If(˜ z,˜ c)∈Tm z(μ),then(˜ z,¯ c)∈supp(¯ δ)and (¯ z,¯ δ)∈supp(μ)for some ¯ c∈C,¯ z∈Zk,and ¯ δ∈X;hence,(˜ z,˜ c)∈Zk×Cas required. For each k∈N,letμk∈M(Z×X∅,k)be a stable matching in Ek,whichexistsby Lemma 7(since Zkis finite and ˜ Eksatisfies its assumptions) and Claim 10. It follows by part 1 of Lemma 5that we may assume that {μk}∞ k=1converges; let μ= limkμk. It then follows by parts 2–4 of Lemma 5that μis a matching and that supp(μ)⊆ SM(μ)∩IR(μ).Hence,μis stable. A.4 Proof of Corollary 3 Let Ebe a Rosen market. For each k∈N,letCk≡[0, k],Xk={n1(z,c):(z,c)∈Z× Cand n∈[0, k]}and Ekbe equal to Eexcept for these changes to Ckand Xk. It follows by Theorem 2that there exists a stable matching μkof Ek. Theoretical Economics 19 (2024) Stable matching in large markets 1297 Claim 11. supp(μk)⊆Z×Xfor each k∈N. Proof. Suppose not, then let (z,δ)∈supp(μk)∩(Z×(X∅\X)).Letε>0besuch that g(r(z))q(z)θ(r(z)/q(z)) −ε>0. Then (z,ε)∈Tm z(μk)since (z,δ)∈supp(μ)and Uz(w,1 (z,ε))=ε>0=Uz(s,δ). Thus, letting δ=1(z,ε), it follows that supp(δ)⊆ Tm z(μk)and Uz(m,δ)=g(r(z))q(z)θ(r(z)/q(z)) −ε>0=Uz(s,δ).Hence, (z,δ)/∈ S(μk), a contradiction to the stability of μk. Claim 12. There exist K,M∈Nsuch that, for each k≥Kand (z,δ)∈supp(μk),δ(Z× C)≤Mand δ(Z×([0, 1/M)∪(M,∞))) =0. Proof. Suppose not, then for each j∈N,thereexistskj≥jand (zkj,δkj)∈supp(μkj)⊆ Z×Xsuch that δkj(Z×C)>jor δkj(Z×([0, 1/j )∪(j,∞))) >0. Suppose first that δkj(Z×C)>j holds for infinitely many js. Taking a subsequence if needed, we may assume that δkj(Z×C)>jholds for each j. Thus, for some (z kj,ckj,nkj)∈Z×C×[0, kj],δkj=nkj1(z kj,ckj)with nkj>j.Wehavethat Uzkj(m,nkj1(z kj,ckj))≤gr(¯ z)fr(¯ z),nkjq(¯ z)−ckjnkj =gr(¯ z)q(¯ z)θr(¯ z) nkjq(¯ z)−ckjnkj. Since μkjis stable, it follows that Uzkj(m,nkj1(z kj,ckj))≥0 for each j;hence, 0≤ckj≤gr(¯ z)q(¯ z)θr(¯ z) nkjq(¯ z). Since nkj→∞, it follows that g(r(¯ z))q(¯ z)θ(r(¯ z)/nkjq(¯ z)) →0, and hence ckj→0. Since g(r(z))q(z)θ(r(z)/q(z)) >0, let ε>0besuchthat gr(z)q(z)θr(z) q(z)−ε>0. We have that (z kj,ckj+ε)∈Tm z kj (μkj)for each jand that Uz kj (m,1 (z kj,ckj+ε))≥gr(z)q(z)θr(z) q(z)−ckj−ε>c kj for all jsufficiently large. But this contradicts the stability of μkj. It follows from what has been shown above that δkj(Z×([0, 1/j )∪(j,∞))) >0 holds for each jsufficiently large. Thus, for some (z kj,ckj,nkj)∈Z×C×[0, kj],δkj= nkj1(z kj,ckj)with ckj>jor ckj<1/j . First, suppose that ckj<1/j holds for infinitely many js. Note that (z kj,1/j )∈Tm z kj (μkj)and Uz kj (m,1 (z kj,1 j))≥gr(z)q(z)θr(z) q(z)−1 j>1 j>c kj for jsufficiently large, contradicting the stability of μkj. 1298 Carmona and Laohakunakorn Theoretical Economics 19 (2024) Now suppose that ckj>jfor all jsufficiently large. Since μkjis stable, we then have that 0≤Uzkj(m,nkj1(z kj,ckj))≤gr(¯ z)q(¯ z)θr(¯ z) nkjq(¯ z)−ckjnkj. Thus, nkj→0asckj→∞, and hence Uzkj(m,nkj1(z kj,ckj))≤gr(zkj)fr(zkj),nkjq(zkj)→0. Let ε>0besuchthat gr(z)q(z)θr(z) q(z)−ε>0. We have that (zkj,ε)∈Tm zkj(μkj)and that Uzkj(m,1 (zkj,ε))≥gr(z)q(z)θr(z) q(z)−ε>0 for all jsufficiently large. But this contradicts the stability of μkj. Claim 12 implies that, for each k≥K, the payoff of a manager in μkis bounded above by maxn∈[0,M]g(r(¯ z))f(r(¯ z),nq(¯ z)) =maxn∈[0,M]g(r(¯ z))nq(¯ z)θ(r(¯ z)/nq(¯ z)).Inaddition, the payoff of a manager is bounded below by (1/2)g(r(z))q(z)θ(r(z)/q(z)), since if (z,δ)∈supp(μk)and Uz(m,δ)<(1/2)g(r(z))q(z)θ(r(z)/q(z)), then letting ε>0be such that g(r(z))q(z)θ(r(z)/q(z)) −ε>2Uz(m,δ), it follows that (z,Uz(m,δ)+ε)∈ Tm z(μk)and Uz(m,1 (z,Uz(m,δ)+ε))=gr(z)fr(z),q(z)−Uz(m,δ)−ε ≥gr(z)q(z)θr(z) q(z)−Uz(m,δ)−ε>U z(m,δ), which contradicts the stability of μk. The payoff of a worker in μkis bounded below by 1/M; since by Claim 11 there is no unemployment, it follows that minUz(m,n1(z,c)),Uz(w,1 (z,c))≥min1 M,1 2gr(z)q(z)θr(z) q(z) (8) for each (z,n1(z,c))∈supp(μk)and k≥K. Let ¯ M=maxM,max n∈[0,M]gr(¯ z)nq(¯ z)θr(¯ z) nq(¯ z),2 gr(z)q(z)θr(z) q(z), n(z,z,c)be the solution of maxn∈R+[g(r(z))nq(z)θ(r(z)/nq(z)) −cn]for each z,z∈Z and c∈[1/¯ M,¯ M+1]and ¯ n=max(z,z,c)∈Z2×[1/¯ M,¯ M+1]n(z,z,c); the existence of ¯ nfollows by the compactness of Z2×[1/¯ M,¯ M+1]and the continuity of (z,z,c)→ n(z,z,c). Let k>max{K,¯ M+1, ¯ n}and μ=μk. Theoretical Economics 19 (2024) Stable matching in large markets 1299 Claim 13. μis a stable matching of E. Proof. We will explicitly indicate the market we are considering in the stability set of μ, and thus write SM(μ;E)and SM(μ;Ek). We use analogous notation for IR(μ)and Tm z(μ)for each z∈Z. We first claim that, for each (z,z,c)∈Z2×C,if(z,c)∈Tm z(μ;E),then(z,¯ M+1)∈ Tm z(μ;Ek). Indeed, (z,c)∈Tm z(μ;E)implies that c=Uz(w,1 (z,c))>U z(a,δ)for some (a,δ)∈such that (a) If a=w,thenδ=1(ˆ z,ˆ c)with (ˆ z,ˆ n1(z,ˆ c))∈supp(μ),andthus,Uz(w,δ)=ˆ c≤Mby Claim 12. (b) If a=s,thenUz(s,δ)=0. (c) If a=m,then(z,δ)∈supp(μ),andthus,Uz(m,δ)≤¯ Mby Claim 12. Hence, Uz(a,δ)≤¯ Mand it follows that (z,¯ M+1)∈Tm z(μ;Ek)since k> ¯ M+1. We now establish that μis a stable matching of E.Let (z,δ)∈supp(μ). Since μ is a stable matching of Ek,(z,δ)∈SM(μ;Ek)∩IR(μ;Ek)and δ∈Xby Claim 11.We have that Uz(s,δ)=0 for each (z,δ)∈Z×Xs,andthus,IR(μ;Ek)⊆IR(μ;E).Hence, (z,δ)∈IR(μ;E). It thus remains to show that (z,δ)∈SM(μ;E).Letδ=n1(˜ z,c)and let (i) (ˆ z,ˆ δ)=(z,δ) and a=mor (ii) (ˆ z,ˆ δ)=(˜ z,1 (z,c))and a=w.Letδ∈Xbe such that supp(δ)⊆ Tm ˆ z(μ;E)and let δ=n∗1(z∗,c∗).Notethat(z∗,c∗)∈Tm ˆ z(μ;E)implies that c∗≥1/¯ Mby (8). If c∗≤¯ M+1, then (z∗,c∗)∈Tm ˆ z(μ;Ek)and Uˆ zm,δ=Uˆ zm,n∗1(z∗,c∗)≤Uˆ zm,nˆ z,z∗,c∗1(z∗,c∗)≤Uˆ z(a,ˆ δ), where the last inequality follows from (z,δ)∈SM(μ;Ek)and k> ¯ n.Ifc∗>¯ M+1, then (z∗,¯ M+1)∈Tm ˆ z(μ;Ek)and Uˆ zm,δ=Uˆ zm,n∗1(z∗,c∗)≤Uˆ zm,n∗1(z∗,¯ M+1) ≤Uˆ zm,nˆ z,z∗,¯ M+11(z∗,¯ M+1)≤Uˆ z(a,ˆ δ), where the last inequality follows from (z,δ)∈SM(μ;Ek)and k> ¯ n. Finally, let δ∈Xbe such that supp(δ)⊆supp(δ)in case (i). Then δ=n1(˜ z,c)for some n∈R+. Since 1/M ≤c≤Mby Claim 12, it follows that Uzm,δ=Uzm,n1(˜ z,c)≤Uzm,n(z,˜ z,c)1(˜ z,c)≤Uz(m,δ), where the last inequality follows from (z,δ)∈SM(μ;Ek)and k>¯ n.Thisconcludesthe proof that (z,δ)∈SM(μ;E)and establishes the claim. 1300 Carmona and Laohakunakorn Theoretical Economics 19 (2024) A.5 Proof of Theorem 3 In this section, we show that the conditions in the statement of Theorem 3are necessary and sufficient for μto be a stable matching of the Rosen market.33 Note that the function his an homeomorphism between Z2and h(Z2). Sufficiency. Let μ=λ◦h−1for some wand λas in the statement of the theorem. To see that μis a matching, note that for each measurable B, μ(B×X)+Z×X δ(B×C)dμ(z,δ) =λ◦h−1(B×X)+Z×X δ(B×C)dλ◦h−1(z,δ) =λ(B×Z)+Z×B nz,z,wdλz,z=ν(B). Wenowshowthatμis stable by establishing that supp(μ)⊆SM(μ)∩IR(μ).Let (z,δ)∈supp(μ);thenδ=n(z,z,w)1(z,wq(z)) for some z∈Zand (z,z)∈supp(λ)by Lemma 1. To see that (z,δ)∈IR(μ),notethatUz(m,n(z,z,w)1(z,wq(z)))=R(z,w)>0 and Uz(w,1 (z,wq(z)))=wq(z)>0. Suppose that (z,n(z,z,w)1(z,wq(z)))/∈SM(μ). Then either there exists (z∗,c∗)∈ Tm z(μ)∪{(z,wq(z))}such that Uz(m,n(z,z∗,c∗/q(z∗))1(z∗,c∗))>R (z,w)or there exists (z∗,c∗)∈Tm z(μ)such that Uz(m,n(z,z∗,c∗/q(z∗))1(z∗,c∗))>wq (z).If (z∗,c∗)= (z,wq(z)),thenUz(m,n(z,z∗,c∗/q(z∗))1(z∗,c∗))=R(z,w).Thus, (z∗,c∗)∈Tm z(μ)∪ Tm z(μ), and hence c∗>wq (z∗); indeed, condition (b) of Tm z(μ)∪Tm z(μ)cannot happen since supp(μ)⊆Z×X, condition (a) implies c∗>wq (z∗)and condition (c) implies that z∗∈proj1(supp(λ)) and c∗>R (z∗,w),andthus,thatc∗>wq (z∗)since then R(z∗,w)≥wq(z∗).If(z∗,c∗)∈Tm z(μ),thenUz(m,n(z,z∗,c∗/q(z∗))1(z∗,c∗))<R (z,w)= Uz(m,n(z,z,w)1(z,wq(z))).If(z∗,c∗)∈Tm z(μ),then Uzm,nz,z∗,c∗ qz∗1(z∗,c∗)<R z,w≤wqz, the last inequality holding since z∈proj2(supp(λ)).Thus, (z,n(z,z,w)1(z,wq(z)))∈ SM(μ), and hence μis stable. Necessity. Let μbeastablematchingofaRosenmarket. Wefirstshowthat supp(μ)⊆h(Z2).Letz,z,ˆ z,˜ z∈Z,andˆ n,˜ n,c(z),c(z)∈R+be such that (ˆ z,ˆ n1(z,c(z))) and (˜ z,˜ n1(z,c(z)))belong to supp(μ). Suppose for a contradiction that c(z)/c(z)= q(z)/q(z). For concreteness, assume c(z)>(c(z)/q(z))q(z)and let w=c(z)/q(z). It follows that Uˆ z(m,ˆ n1(z,c(z)))<max nUˆ z(m,n1(z,wq(z)))=R(ˆ z,w)=max nUˆ z(m,n1(z,wq(z))). Thus, there is ε>0suchthatUˆ z(m,ˆ n1(z,c(z)))<R (ˆ z,w+ε). Since (w+ε)q(z)= c(z)+εq(z)>c (z), it follows that (z,(w+ε)q(z)) ∈Tm ˆ z(μ).Thus,δ=n(ˆ z,z,w+ 33See the working paper version for an illustration of Theorem 3and its proof in the Cobb–Douglas case.