Matching Multidimensional Types: Theory and Application

TL;DR

This paper extends Becker's scalar matching model to multidimensional types, establishing new sorting patterns and complementarities, and applies the framework to analyze health and education correlations among U.S. households.

Contribution

It introduces multidimensional generalizations of sorting concepts, identifies conditions for optimal sorting patterns, and links theory with observed aggregate attributes in empirical data.

Findings

Weak positive association between health and education among spouses

Attraction effects influence household insurance risk

Multidimensional sorting patterns explain observed attribute correlations

Abstract

Becker (1973) presents a bilateral matching model in which scalar types describe agents. For this framework, he establishes the conditions under which positive sorting between agents' attributes is the unique market outcome. Becker's celebrated sorting result has been applied to address many economic questions. However, recent empirical studies in the fields of health, household, and labor economics suggest that agents have multiple outcome-relevant attributes. In this paper, I study a matching model with multidimensional types. I offer multidimensional generalizations of concordance and supermodularity to construct three multidimensional sorting patterns and two classes of multidimensional complementarities. For each of these sorting patterns, I identify the sufficient conditions which guarantee its optimality. In practice, we observe sorting patterns between observed attributes that are aggregated over unobserved characteristics. To reconcile theory with practice, I establish the link between production complementarities and the aggregated sorting patterns. Finally, I examine the relationship between agents' health status and their spouses' education levels among U.S. households within the framework for multidimensional matching markets. Preliminary analysis reveals a weak positive association between agents' health status and their spouses' education levels. This weak positive association is estimated to be a product of three factors: (a) an attraction between better-educated individuals, (b) an attraction between healthier individuals, and (c) a weak positive association between agents' health status and their education levels. The attraction channel suggests that the insurance risk associated with a two-person family plan is higher than the aggregate risk associated with two individual policies.