Competitive Many-to-One Matching: Sorting vs. Equality

Technical Summary: Competitive Many-to-One Matching: Sorting vs. Equality

Problem Statement
The paper investigates competitive equilibrium in many-to-one matching markets with transfers and peer effects. The authors model settings where heterogeneous agents on one side (e.g., firms, schools, neighborhoods) are matched with groups of heterogeneous agents on the other side (e.g., workforces, student bodies, residents). The central economic question is how the tradeoff between sorting (segregating agents by type to exploit complementarity) and equality (mixing agents to mitigate decreasing returns or peer effects) determines equilibrium matching patterns. Unlike standard one-to-one assignment models (optimal transport), this framework allows a single firm to hire a workforce with a distribution of skills, creating a tension between maximizing the match between firm productivity and workforce composition and optimizing the internal composition of the workforce itself.

Methodology
The authors employ a continuum-agent model with quasilinear utility.

1. General Framework: Firms $x$ (distributed according to $\mu$) hire workforces $\eta$ (distributions over worker types $y$ distributed according to $\nu$). The value generated is $V(x, \eta)$. A competitive equilibrium is defined as an assignment $\gamma$ and price functions $(p, q)$ such that firms maximize profit and workers maximize utility, with no blocking coalitions.
2. Duality and Existence: The authors establish that the planner's problem (maximizing total value) and its dual (minimizing total utility subject to no-entry constraints) satisfy strong duality. They prove the existence of a competitive equilibrium by showing the value function inherits Lipschitz continuity from the production function, utilizing infinite-dimensional separating hyperplane theorems.
3. Mean-Measurable Case: The core analysis focuses on the tractable case where workforce value depends only on the mean skill $z = E_\eta[y]$, i.e., $V(x, \eta) = v(x, z)$. In this setting, the problem reduces to finding a distribution of workforce means $H$ that majorizes the population distribution of worker skills $G$ (denoted $H \succeq G$), subject to a concave maximization objective.
4. Moment-Measurable Case: To resolve indeterminacy regarding individual worker assignments, the authors generalize the model to cases where output depends on a specific moment $E_\eta[m(x,y)]$, allowing for firm-specific aggregation of skills (e.g., CES aggregators).
5. Uniform Price Equilibrium (UPE): The paper contrasts competitive equilibrium (which allows personalized prices/wages) with UPE, where a single price is charged to all agents on the "many" side for a given agent on the "one" side (e.g., uniform tuition). This is relevant for applications like school choice or housing markets where personalized pricing is infeasible.

Key Results

• Equilibrium Characterization (Mean-Measurable):

  • The equilibrium matching $T$ between firm productivity $x$ and mean workforce skill $z$ is unique, continuous, and strictly increasing.
  • The equilibrium is characterized by alternating intervals of segregation and compression (mixing).
  • Segregation ($X_S$): Occurs where the majorization constraint binds ($H=G$ locally). Here, workforces are segregated by skill, and the marginal value of skill is increasing.
  • Compression ($X_C$): Occurs where the constraint does not bind. Here, workforces are mixed, and the marginal value of skill is constant across the interval.
  • Simple Patterns:
    • Positive Assortative Segregation (PAS): All workforces are segregated. This arises if the sorting force (increasing differences $v_{xz}$) dominates the pooling force (decreasing returns $v_{zz}$) sufficiently.
    • Full Compression: All workforces are mixed. This arises if decreasing returns are strong relative to sorting forces.
    • Upper/Lower Compression: Segregation occurs for low/high-skill firms, while mixing occurs for the opposite.

• Comparative Statics:

  • Segregation vs. Compression: Equilibrium becomes more segregated if the marginal rate of substitution between firm productivity and workforce skill increases (i.e., sorting forces strengthen relative to pooling forces).
  • Skill Dispersion: An increase in the heterogeneity of the worker population leads to a more heterogeneous distribution of workforce means.
  • Wages and Profits: Unlike one-to-one matching, in many-to-one matching, an increase in the supply of high-skill workers can reduce the wages of low-skill workers. This is because high-skill workers substitute for low-skill workers within a mixed workforce, altering the marginal productivity of the remaining low-skill workers.

• Moment-Measurable Refinements:

  • When production depends on moments other than the mean (e.g., CES aggregators), the equilibrium assignment between firms and individual workers is uniquely determined.
  • If the labor input function $m(x,y)$ is strictly log-supermodular, the equilibrium is strictly single-dipped: more productive firms employ more heterogeneous workforces (mixing low and high-skill workers), while less productive firms employ homogeneous workforces. Conversely, log-submodularity leads to strictly single-peaked assignments.

• Uniform Price Equilibrium (UPE):

  • UPE can be inefficiently segregated. Specifically, Positive Assortative Segregation (PAS) may be a UPE even when it is not efficient (i.e., not a competitive equilibrium).
  • Distributional Consequences: Moving from competitive equilibrium to UPE shifts the value generated by peer effects from the "many" side (students/workers) to the "one" side (schools/firms). In the context of schools, a uniform price regime (no merit scholarships) increases school profits and reduces student welfare compared to a competitive regime with personalized tuition, assuming positive peer effects.

Significance and Contributions
The paper claims to provide a tractable framework for studying equilibrium matching patterns, utilities, and comparative statics in markets with rich heterogeneity on both sides and peer effects. Its primary contributions are:

1. Generalization: It unifies the assignment model (optimal transport) and Bayesian persuasion (information design) into a single framework. The mean-measurable case is shown to be a mutual generalization of these two literatures.
2. New Matching Patterns: It characterizes equilibrium patterns that do not exist in one-to-one matching, specifically the alternation between segregation and compression intervals, and the phenomenon of "full compression" where all workforces are mixed despite increasing differences.
3. Comparative Statics on Inequality: It offers a mechanism for rising wage inequality driven by changes in production technologies (specifically the ratio of complementarity to decreasing returns) and worker heterogeneity, distinct from standard one-to-one models.
4. Policy Implications: It highlights the inefficiency and distributional consequences of uniform pricing in markets with peer effects, suggesting that agreements to forego personalized pricing (e.g., merit scholarships) may warrant regulatory scrutiny as they transfer surplus from agents to institutions.
5. Indeterminacy Resolution: It resolves the indeterminacy of individual worker assignments in the mean-measurable case by analyzing moment-measurable perturbations, revealing how firm-specific skill aggregation preferences determine workforce heterogeneity.

The authors position their work as a foundational step in understanding many-to-one matching with peer effects, providing tools to analyze whether social institutions (schools, neighborhoods, firms) are "overly sorted" or meritocratic in equilibrium.