Choosing Sides in a Two-Sided Matching Market

1 Introduction

A common assumption in the two-sided matching literature is that if the surplus generated by matching satisfies strict supermodularity, then the resulting stable matching will be positive assortative. This condition is often used in marriage market models as well as several labor matching models, including Kremer (1993)’s O-Ring theory. However, these models do not accurately capture labor markets with two distinct roles in which agents may be able to choose which role they prefer. For example, some doctors open their own practices, some financial analysts will enter their firm’s management track, some professors will chair their department, and some entrepreneurs will start their own small business—but all such people likely have similarly qualified peers in their field who prefer to follow the lead of others. In this paper, I model many-to-one labor markets that have a “lead” role and “support” role (which are filled by a manager and worker(s) respectively) that together generate output, in which agents have a pre-matching strategic choice over role.

The main contribution this paper makes to the literature is adding a role choice to a many-to-one labor market with two sides—agents decide if they prefer to lead or support before the matching market is realized. I find that there exists a unique rational expectations equilibrium that induces a stable matching, and that the matching pattern is socially efficient. In equilibrium, both the matching pattern and the wage structure are unique. The latter is generally not the case when the solution concept requires only stability; the unique wage structure is driven by the pre-matching role choice. A condition stronger than strict supermodularity that I call role supermodularity determines the equilibrium matching pattern and wage structure, as positive assortative matching occurs if and only if the production function satisfies role supermodularity. A stronger condition is necessary because a high skilled agent is only willing to enter the market as a worker if she expects that she can profitably cluster with other high skilled agents. The model predicts two kinds of wage differentials—differences in wages between agents of the same type in different roles, and differences in wages between agents of different types.

After the literature review in Section 2, I set up the model in Section 3 and solve for equilibrium outcomes. In Section 4, I provide comparative statics and discuss how wage differentials change in response to changes in underlying productivity. I show how the wage structure relates to observed trends in U.S. wage inequality, and discuss possible policy implications. I conclude in Section 5 by discussing directions for future work.

2 Literature Review

This paper combines elements from the two-sided assignment model and the role assignment model.

In a two-sided assignment problem as in Shapley and Shubik (1971), agents are divided into two disjoint sets and match surplus is generated if agents belonging to different sets match with each other (e.g. men and women in the marriage market, firms and employees in the labor market, managers and workers in this paper). I assume that match surplus is transferable via wages. See Chiappori (2020) for a full review of matching models with transfers, and Chapter 6 of Roth and Sotomayor (1992) for an overview of the literature on many-to-one matching. The paper reassesses the assumption that strict supermodularity in inputs induces positive assortative matching, as introduced by Becker (1973) and often used in the two-sided matching literature.

The pre-matching role choice parallels the decisions agents face in investment and matching models with transferable utility (Chiappori, Iyigun, and Weiss 2009; Nöldeke and Samuelson 2015; Zhang 2021). Structurally, they are similar: a decision is made before matching (choice of role vs. an investment choice), interim outcomes are realized (each agent’s role vs. changes in skill type), and then matching occurs. The first stage is non-cooperative, as agents strategize in anticipation of the matching market that they will face, while the second stage is a cooperative assignment problem. In the investment and matching framework, though, sides of the matching market are fixed and investment decisions change agents’ skill levels; in contrast, I allow strategic choices to change the supply and demand of agents on both sides of the market while skill levels are fixed.

The model is also similar to social games (Jackson and Watts 2008; Jackson and Watts 2010), which generalize the discrete marriage market problem. In a social game, players of different roles choose strategies and partners simultaneously. Unlike this paper, in a social game, a player’s role cannot change (e.g. “men” and “women” in the marriage market, “firms” and “employees” in the labor market), but role in the social game is similar to type in this model in that it is fixed and under loose conditions, players in the scarcer role (in the social game) or type (in this paper) are able to coordinate on an advantageous outcome for the entire group. Both games also feature cooperative and non-cooperative elements that interact—individuals are strategic over who they are willing to match with to maximize their payoffs, but matching itself is cooperative.

Role assignment models (Anderson 2020; Kremer and Maskin 1997; Li and Suen 2001; McCann and Trokhimtchouk 2008) are one-sided matching frameworks that analyze changes in worker sorting and their downstream effects on wage dispersion.^[1] In role assignment models, the firm (acting as a social planner) first determines who is matched with whom, and then assigns roles within each match; in this paper, the timing is reversed. Kremer and Maskin (1997) show that in the role assignment model, matching patterns become positive assortative as skill levels become dispersed, and mean skill level correlates positively with wage inequality. Li and Suen (2001) show that for sufficiently dispersed skill distributions, segregation by type and wage inequality depends on how the social planner chooses to sort the agents who are indifferent between managing a lower-skilled worker or working for a higher-skilled manager. My model captures a similar tension without requiring a social planner; the equilibrium matching pattern depends on whether high-skilled agents can profitably become a worker, or whether high-skilled agents always prefer to become a manager.

In role assignment, there are two standard assumptions that pull the matching pattern in opposite directions: (1) managers and workers are complements (i.e. a highly-qualified accountant should work in a junior position at a top firm), and (2) output is more sensitive to managerial skill (i.e. a highly-qualified accountant should work in a senior position at a lower-tier firm). I similarly impose an assumption that the manager role is more sensitive to skill type, but unlike the standard role assignment environment, I do not restrict attention to supermodular production functions. Empirical justification for these assumptions is unclear. Due to data limitations and difficulty in quantifying productivity, empirical research on managerial impact is modest. There are multiple data issues: it is not clear how to choose the best measure of productivity, some types of productivity may be unobservable, and one would need rich data across many firms. However, existing papers validate the main assumptions. Lazear, Shaw, and Stanton (2015) show that in a technology-based service workplace, the average manager contributes more to output than the average worker. Bertrand and Schoar (2003) find that differences in corporate managerial practices are systematically and significantly related to differences in performance. Finally, Bloom and Reenen (2007) find that better managerial practices are significantly and positively related to higher productivity in manufacturing firms.

That said, assumptions on the production function are not innocuous and I do not claim results generalize to all labor markets. Unlike Eekhout and Kircher (2018), present a model of assortative matching in large firms, here I consider a setting that more closely aligns with small business ownership (e.g. a single owner employs a small number of workers and all agents perform a variety of tasks). However, results may also apply to larger firms in which internal distribution of human capital is important, such as technology-based service and/or innovation sectors, fields in which a high level of qualification is necessary to enter the market (e.g. law or academia), start-up companies, or sectors with a significant freelance presence. Finally, while a different application of matching models, Reynoso (2021) show conditions such that positive assortative matching among wives may emerge in marriage markets with polygamy which are comparable to the results I show in this paper on labor market matching.

3 Model

The labor market is competitive with two employment roles, r ∈ {m, w} such that a manager m must match with exactly n∈N workers for production to occur, where n is relatively small.^[2] Worker skill is additive. There is a unit mass of agents, all risk neutral, who are of a skill type θ ∈ {H, L} such that H,L∈R++ and H > L.^[3] All agents have an outside option of 0. The measure of H-type agents is MH∈(0,1n+1). Denote θ _m the type of an arbitrary manager, and denote ∑i=1nθwi an arbitrary worker composition.

The production technology is f(θm,∑i=1nθwi):R+2→R+ that satisfies three assumptions: monotonicity, inefficiency of mixed worker compositions, and managerial impact. The first assumption, monotonicity, imposes that increasing the number of H-type agents in the group increases productivity.

The second assumption rules out the possibility that strictly mixed worker compositions are efficient and allows me to narrow down the possible worker compositions under consideration. For any manager, it is always more productive to either hire all H-type or all L-type workers rather than take a strictly mixed worker composition—if a manager is willing to hire one H-type worker, then she must also be willing to hire a second H-type worker, and so on. This is because marginal productivity is increasing in the number of H-type workers hired. In practice, therefore, I only need to consider four matching patterns: H-type manager with all H-type workers, H-type manager with all L-type workers, L-type manager with all H-type workers, and L-type manager with all L-type workers.

Finally, the manager’s role is more important to overall productivity than the worker composition. This implies that the “L-type manager with all H-type workers” matching pattern is always inefficient compared to the “H-type manager with all L-type workers” matching pattern.

Assumptions 2 and 3 are strong, especially as n becomes large. I discuss how to loosen these assumptions in Appendix B and show that dropping them does not affect the structure of the main results Table 1.

The game takes place over two stages. In the first stage, agents make strategic role choices. In the second stage, these choices resolve into a matching market P and a standard assignment game occurs.

1. Strategic Stage: Agents simultaneously make pre-matching strategic decisions over what role to enter into the market as. Becoming a manager has a known, type-dependent cost c(θ)∈R+, and costs are relatively small compared to productivity.

   Let σ _θ be a type symmetric strategy, such that σ _H gives the fraction of H-type agents who chose to become a manager (and analogously for σ _L ). Denote σ = (σ _H , σ _L ) an arbitrary strategy profile.

2. Outcome Stage: σ induces a matching market P=(PmH,PwH,PmL,PwL)∈R+4 with transfers; in this setting, transfers take the form of wage determination. Denote P _rθ the measure of θ-type agents in role r (e.g. the measure of L-type workers in the matching market induced by σ is P _wL = (1 − M _H ) (1 − σ _L )).

   Once P has formed, a cooperative, non-strategic assignment game occurs. A market outcome in the assignment game is a matching μ along with a wage vector V. Since worker composition can be expressed as a linear combination of skill types, I denote a matching μ as a function μ : {L, H} × {0, 1, …, n} → [0, 1] such that μ(θ, ℓ) is the measure of θ-type managers matched with ℓ H-type workers and (n − ℓ) L-type workers. Denote V as a wage vector of up to four components, such that vrθ∈R+ is the wage of a θ-type agent in the role r whenever μ _rθ > 0.

Consider the assignment game that happens in the second stage. Given an arbitrary P, solutions to the assignment game are stable market outcomes, which is a pair (μ, V) that satisfies feasibility of the matching μ, consistency between μ and the corresponding wage vector V, and does not allow for blocking coalitions to form. First, I define a feasible matching.

Feasibility guarantees that (1) there are no unmatched agents who could find another unmatched agent on the other side of the market, (2) the measure of any manager-worker composition is strictly non-negative, (3) the total measure of θ-type managers matched not exceed the measure of θ-type managers available in the market, (4) the total measure of H-type workers matched does not exceed the measure of H-type workers available in the market, and (5) the total measure of L-type workers matched does not exceed the measure of L-type workers available in the market.

In addition to feasibility, stability additionally imposes structure on the relationship between μ and V.

Individual rationality guarantees that all agents are willing to participate in the labor market. Pairwise efficiency ensures that wages are split so no productivity is wasted. Pareto efficiency gives that no blocking coalitions exist, as any coalitions are either no better off or are unsustainable given the wage demands that members of the coalition have. Put altogether, stability imposes two features: V is feasible and compatible with μ, and there do not exist any managers and groups of workers who all prefer to be matched with each other over their current assignment.

It has already been shown that for an arbitrary matching market P with wages (transfers), stable outcomes to the assignment game exist; while the stable matching is generally unique, it can be supported by a continuum of wage vectors (see Chiappori, Pass, and McCann (2016) and Chiappori (2020)). If f satisfies strict supermodularity, then the unique stable matching is positive assortative; otherwise, it is negative assortative. Because wage determination isn’t unique, stability alone cannot generate unique predictions on what stable outcome(s) will occur in a competitive market setting. However, in this setting, P isn’t fixed until agents have made their role choice; I show later in this section that this pre-matching role choice gives more structure to the potential wage vectors that can emerge.

The solution concept for the full game follows. I use a rational expectations equilibrium, as agents must have correct expectations about how their first stage role choices affect the stable market outcomes that occur in the second stage.

The first part of the definition is a consistency condition. If agents optimally play σ* because they expect to face the stable market outcome (μ*, V*), then σ* induces a matching market that not only can, but actually does sustain (μ*, V*) as a stable market outcome. This prevents cases where, for example, σ _H = σ _L = 1, but agents incorrectly expect to be matched in the second period. The second part is a utility maximizing condition. V* along with Equations (1) and (2) together allow agents to have rational expectations on all wages they could possibly face, even if some type-role combinations are not in the market induced by σ*. If P _rθ > 0, then V* explicitly defines v _rθ ; if not, then Equations (1) and (2) impose that θ-type agents determine what their wages from deviating to r would be by using pairwise efficiency.

Before solving for the equilibrium, consider the social planner’s problem of assigning first roles and then matches.^[4] f(H, nH) is the most productive arrangement (f satisfies monotonicity and inefficiency of mixed worker compositions), but the opportunity cost of grouping H-type agents together is that the n H-type workers could have instead been managers to groups of L-type workers (f satisfies managerial impact). Hence, after taking into account the cost of becoming a manager, the social planner groups agents of the same type together if and only if

I call this matching pattern clustering. If Equation (3) is not satisfied, then the social planner has all H-type managers become managers and matches them with n L-type workers, and assigns both roles to L-type agents such that the L-type workers who are unable to match with the relatively scarce H-type managers form clusters with an L-type manager. I call this matching pattern specialization.

The same condition determines the equilibrium outcome. To provide intuition for why this is the case before I formally state the result, consider a discrete example with n = 1, two H-type agents, four L-type agents, and no cost of entering as a manager. Consider the perspective of the H-type agents, who have three strategies available: σH={0,12,1}. Table 2 shows the possible matching patterns that each strategy can induce, excluding cases such that agents end up unmatched. I show formally when proving the main result that rational agents indeed prevent “unbalanced” markets from emerging in equilibrium. For each pair, the first is the manager type and second is the worker type (e.g. “HL means a H-type manager matches with an L-type worker”).

L-type agents always mix strategies, implying that they are equally well off on both sides of the market, so by pairwise efficiency vmL=vwL=12f(L,L) for all matching patterns. Given L-type wages, (1) σ _H = 0 is strictly dominated by σ _H = 1 since f(H, L) > f(L, H), and (2) the HL LH LL matching is unstable as pairwise efficiency, Pareto efficiency, and the L-type wage condition cannot be satisfied simultaneously.

Hence the H-type agents face two possible outcomes: play σH=12 to induce clustering or σ _H = 1 to induce specialization. Their wages under clustering are, by the same logic as the L-type agent wage determination, vmH=vwH=12f(H,H). By pairwise efficiency and the L-type wage determination, their wages under specialization are vmH=f(H,L)−12f(L,L). So H-type agents prefer to induce clustering if and only if

which is equivalent to the SPP’s cutoff condition in Equation (3) with n = 2, c(H) = c(L) = 0.

This intuition extrapolates into the general model: H-type agents’ incentives align with the social planner’s problem, so the same condition determines both the efficient and competitive, decentralized market outcomes.

Appendix A has the full proof; I outline the argument here. Existence can be proven by construction. I first show the intuitive result that an equilibrium shouldn’t induce an unbalanced market in which there are excess agents in one of the roles, as those agents can profitably deviate to the other side of the market. Given that the market is balanced, θ-type agents mix strategies if and only if expected wages net of costs on both sides of the market are the same. At least some L-type agents must always cluster for the market to clear, so v _mL − c(L) = v _wL regardless of the matching pattern. Given that, utility-maximizing H-type agents consider whether to manage L-type agents (who demand lower wages) or to cluster with other H-type agents (a more productive arrangement). This trade-off makes the H-type agents’ optimization problem equivalent to the social planner’s problem, hence Equation (3) determines both the unique, stable matching pattern as well as the accompanying wage vector in equilibrium. Uniqueness follows by ruling out all other possibilities: H-type agents prevent inefficient matching patterns from emerging because they can always do better by disregarding what L-type agents do and clustering together.

To compare the equilibrium to the setting without a role choice, again set costs to 0 and let n = 1 (but continue to assume a continuum of agents). Then Equation (3) simplifies to

and positive assortative matching occurs in equilibrium if and only if role supermodularity is satisfied. I call the condition role supermodularity because the roles that an H-type agent is willing to take determines the equilibrium matching pattern. Role supermodularity is a stronger necessary condition for positive assortative matching than the standard strict supermodularity condition,

because of the role choice—H-type agents are never willing to become workers if her only choice is to match with an L-type manager. This is in contrast to the standard assignment model, where f(L, H) matters because the matching market is pre-determined and H-type agents may end up a worker because she has no role choice.^[6] Here, though, H-type agents are only willing to become workers if the high productivity of the H-type cluster outweighs the fact that L-type workers would demand a lower wage from her if she manages.

This theoretical prediction aligns with Adhvaryu et al. 2020’s empirical study of a garment facturing firm in India. They find that negative assortative matching occurs even though the underlying production function displays complementarities between managers and workers, which is consistent with the hypothesis that a stronger condition than strict supermodularity (which can be interpreted informally as “inputs behaving more like complements than substitutes”) such as role supermodularity (which informally requires that inputs behave strongly as complements) is needed to induce positive assortative matching in labor markets. In general, the model predicts that in labor markets in which agents must preemptively decide to enter as the lead role (e.g. entering the management track at a company may require extra training or external credentials), more complementarity between agents is needed to sustain positive clustering.

4 Wage Differentials and Productivity

Theorem 1 implies that two separate features describe wage inequality: (1) wage differentials between agents of different types in the same role (which are driven by differences in productivity and costs), and (2) wage differentials between agents of the same type in different roles (which are driven by the ability to pass on the cost of entering as a manager to workers). The latter feature is straightforward to pin down: in both equilibria, whenever θ-type agents are on both sides of the market, v _mθ − v _wθ = c(θ).

In a clustering equilibrium, the wage differentials between agents of different types in the same role are

Changes to f(θ, nθ) and/or c(θ) affect only wages of θ-type agents, but affect both sides of the market. Given that, the main productivity-related reason that clustering may switch to specialization is growth in f(H, nL), which shrinks marginal productivity of H-type clusters.

In a specialization equilibrium, the wage differential between H-type and L-type managers is

and c(H) does not enter into the expression because H-type managers are unable to directly pass on c(H) to workers, while managers of both types continue to pass a fraction of c(L) onto L-type workers. Changes to f(H, nL) and c(H) affect only H-type managers’ wages, while changes to f(L, nL) or c(L) affect all agents in the market: if L-type clusters become more productive, all L-type agents’ outcomes improve while H-type managers are worse off, while increasing L-type cost works in the opposite direction. This suggests there are two productivity-related reasons that specialization may switch to clustering: (1) growth in f(H, nH) or (2) growth in f(L, nL).

How do these comparative statics match up with observed wage patterns in the U.S.? Data from the U.S. Census Bureau (2022) as shown in Figure 1 shows that the mean income of college graduates has increased faster than that of high school graduates over the last four decades.^[7] I also show the time trend in standard errors of mean income in Figure 2. In my model, agents of the same type have variance in wages due to c(θ). Interestingly, the pattern of dispersion in wages across agents of the same education level is similar—suggesting that the cost of becoming a manager is increasing in general—but has consistently been higher for high school graduates. This implies that the cost of becoming a manager is larger for L-type agents, perhaps because they face a larger opportunity cost or higher risk in entering into the market as a manager.

Figure 1:

Mean income by level of education.

Figure 2:

Standard error of mean income by level of education.

That college educated adults are experiencing faster income growth and variance in both groups’ incomes is increasing over time are together consistent with the hypothesis that the overall landscape of the U.S. labor market has moved towards clustering equilibria over time. Specialization is ruled out because variance in college graduates’ incomes would be steady in this case. This may be driven by fields like technology-based start-ups or finance, in which productivity has increased the fastest among high-skilled matches.^[8] This is also consistent with Song et al. (2019), who find that between 1978 and 2013, increased within-firm positive sorting correlates with increases in between-firm wage disparity.^[9]

The model has a notable policy implication: in some cases, making it less costly for low-skilled agents to enter in the lead role can simultaneously increase productivity and decrease wage differentials. As previously noted, H-type clustering is always the most productive assignment disregarding costs; however, it may not occur in equilibrium because H-type agents prefer to manage L-type workers who demand lower wages. Suppose that firms and/or policymakers want to push the matching pattern towards clustering—for instance, if the most technically demanding projects are expected to generate positive externalities. If so, then they should subsidize L-type agents who wish to enter in the lead role: by decreasing c(L), role supermodularity is easier to attain because vwL* is always decreasing in c(L) and the H-type manager’s trade-off between higher productivity and paying higher wages to H-type workers shifts towards the former. This simultaneously pushes the matching pattern towards the most “high powered” arrangement, clustering, and decreases wage differentials.

As an example, consider web and/or mobile app development, a sector that has multiple entry points depending on experience and training. Also, cost to receive credentials is increasing in type—aspiring developers without experience may attend short-term coding bootcamps to get their foot in the door of an entry-level job, but high profile jobs may require years of university education.^[10] For example, senior mobile developers at Google require a bachelor’s degree at minimum and consider an advanced degree substitutable with professional experience.^[11] This model advises policymakers to subsidize short-term programs like coding bootcamps rather than providing scholarships for advanced degrees in computer science. By making entry-level coders better off, higher-level coders will prefer to group together.

5 Conclusion and Future Work

I present a two-sided matching model of a labor market in which agents can choose their role. The pre-matching strategic decision causes positive assortative matching to become more difficult to attain in equilibrium compared to the standard assignment model; this is primarily driven by H-type agents’ incentives. The production technology must satisfy role supermodularity, a stronger condition than the standard strict supermodularity condition, for positive assortative matching to occur.

However, these results follow from stylized assumptions on how matching works. For example, I impose that managers must match with exactly n workers. In Appendix D, I show that n can be endogenized when considering specific functional forms. I also impose two strong properties on the production function, inefficiency of mixed worker compositions, and the importance of managerial impact. However, these properties can be dropped while maintaining the overall structure of results, as I discuss in Appendix B.

A natural extension of the paper is to add skill types. To conclude, I give some informal discussion about how I anticipate results would extrapolate to a model with more skill types. To simplify discussion, I assume n = 1 and only discuss matching patterns. Suppose there are three skill types, H > M > L, and f satisfies the following: (i) if θm>θm′, then f(θm,θ̄w)>f(θm′,θ̄w), (ii) if θw>θw′, then f(θ̄m,θw)>f(θ̄m,θw′), and (iii) if θ > θ′, then f(θ, θ′) > f(θ′, θ).^[12] I anticipate five possible equilibrium matching patterns:

1. Full clustering: All types match with their own type.

2. H-clustering: H-type agents match with their own type. M-type managers match with L-type workers.

3. M-clustering: H-type managers match with L-type workers. M-type agents match with their own type.

4. L-clustering: H-type managers match with M-type workers. L-type agents match with their own type.

5. Impure specialization: H-type managers match with M-type workers. M-type managers match with L-type workers.

By adding the assumption f(H, M) − f(M, M) > f(H, L) − f(M, L), (i.e. M-type clustering is inefficient; this rules out M-clustering), the production assumptions are analogous to the setting in Anderson (2020), which generalizes Kremer and Maskin (1997). Moreover, the proposed equilibria matching patterns align with the matching pattern that Anderson (2020) derives: skill types in a connected interval form groups, and within those groups, specialization occurs.^[13] This suggests that as more and more skill types are added, equilibria matching patterns in a two-sided framework may remain efficient.