Two-sided search and perfect segregation with fixed search costs

doi:10.1016/S0165-4896(01)00068-3

Mathematical Social Sciences

Volume 42, Issue 1, July 2001, Pages 31-51

https://doi.org/10.1016/S0165-4896(01)00068-3 Get rights and content

Abstract

This paper studies a two-sided search model with the following characteristics: there is a continuum of agents with different types in each population, match utility is nontransferable, and agents incur a fixed search cost in each period. When utility functions are additively separable in types and strictly increasing in the partner’s type, there is a unique matching equilibrium. It exhibits perfect segregation as in Smith [Smith, The marriage model with search frictions, Working paper (1997) Department of Economics, MIT] and Burdett and Coles [Quarterly Journal of Economics, 112 (1997) 141]; i.e. agents form clusters and mate only within them. A simple sufficient condition on the match utility function and the density of types characterizes the duration of the search for each type of agent. The sufficiency of additive separability in the fixed search cost case is explained and contrasted with the discounted case; moreover, the results are generalized to a broader class of search cost functions that subsumes discounting and fixed search costs as special cases.

Introduction

The class of problems that this paper studies can be illustrated as follows: suppose there is a large number of agents that belong from the outset to one of two disjoint populations. Also, suppose that in each population agents differ in their characteristics. Although it is costly for agents to locate potential mates from the population to which they do not belong, they prefer this search process to the prospect of remaining lonely forever. During each period they randomly meet in pairs with agents belonging to the other population, and they must decide if they will match forever or continue their searches. The goal of each agent is to find the best possible mate. Since agents differ, the value of the search is not the same for all, nor are their sets of acceptable matches.

In this decentralized setting with costly search and heterogeneity, I investigate the following questions: In equilibrium, who matches with whom? Do people with better characteristics spend more or less time in the market before finding a suitable match? Does the way in which search frictions are modeled matter?

This paper seeks to address these questions in a framework with a continuum of agents with different types in each population, with nontransferable utility that is additively separable in types, and where agents incur a fixed search cost in each period in order to locate potential mates.

Under these assumptions, I show that if the utility from a partner is increasing in his or her type, there exists a unique matching equilibrium in which agents partition into subpopulations and marriages occur only among agents from different populations who belong to the same element in the partition. In other words, the equilibrium exhibits the cross-sectional property that Smith (1997) calls perfect segregation, or classes as in Burdett and Coles (1997).

I then proceed to characterize the measure of agents in equilibrium subpopulations in terms of the properties of the density of types and the marginal utility of partners. I derive a simple sufficient condition that allows ordering of the sizes of these subpopulations as a function of the types they contain; this provides an easy way to determine the duration of the search as a function of the characteristic of each agent.

To illustrate these results, I fully solve the model for the case where types are uniformly distributed and match utility is simply the type of the potential mate. I then contrast the results with the discounted case.

Since I model search costs and match payoffs in a different manner from the approaches used in the aforementioned papers, I provide an intuitive explanation that clarifies why additive separability and multiplicative separability are sufficient conditions for perfect segregation in the fixed search cost and discounted cases, respectively. Moreover, I present a set of sufficient conditions that generalizes the perfect segregation result to a broader class of search cost functions that subsumes discounting and fixed search costs as special cases.

With a simple change of terminology, matching problems with search frictions and heterogeneity of the sort described above commonly arise in labor and marriage markets, as well as in the formation of partnerships in general. Therefore, the analysis conducted in this paper will shed light on the role that search and heterogeneity play in these situations.

This paper is a contribution to a recent literature that deals with matching problems with heterogeneous agents and costly search. In independent work, McNamara and Collins (1990), Burdett and Coles (1997), Smith (1997), and Bloch and Ryder (1999) derive perfect segregation results when the only search friction is discounting and except for Smith (1997), each agent’s utility is linear and depends solely on the type of the partner.¹ Morgan, 1995, Morgan, 1996 also assumes fixed search costs but with strictly supermodular match payoffs; he shows that the existence of complementarities leads to positive sorting in equilibrium, and he also analyzes the welfare properties of the model as well as the formation of clubs. However, his assumptions rule out the perfect segregation case, and he does not examine alternative search cost structures.

Unlike the papers described above, this is the first paper that (i) presents a detailed analysis of the perfect segregation result under fixed search costs; (ii) investigates the dependence of the size of equilibrium subpopulations (and therefore the duration of the search) on properties of the distribution of types and utility functions; and (iii) analyzes the effects of different modeling assumptions regarding search frictions. In this sense, the present paper represents a theoretical contribution that covers an important case that has not been heretofore analyzed, and reconciles the results obtained in the literature under alternative assumptions. It is easy to think about economic examples from marriage markets or entry-level labor markets where fixed search costs are important and complementarities of types in the payoff function are not. Moreover, the simplicity and tractability of the analysis and conditions presented in this paper make my model an attractive benchmark to use in empirical applications to labor and marriage markets. In fact, a recent paper by Wong (2000) develops a method for estimating a two-sided search model that exhibits perfect segregation, and presents some results for the US; the results derived in this paper expand the range of applicability of Wong’s method.²

The present work is also related to the class of two-sided matching models, thoroughly surveyed by Roth and Sotomayor (1990), that focuses on the analysis and design of centralized matching mechanisms. Here, I focus on the effects of search frictions when matching proceeds in a decentralized fashion and there is replenishment when agents mate and exit the market. I present an example that illustrates the relationship between the two approaches.

The paper is organized as follows. Section 2 describes the basic model. The main results are presented in Section 3. Section 4 concludes. All proofs are collected in Appendix A.

Section snippets

The model

In this section I describe the main characteristics of the two-sided search model that will be analyzed in the next section. To fix the terminology, I cast the model in terms of a marriage market where women and men are (costly) searching for spouses. It is important to keep in mind that the problem analyzed in this paper arises in other markets as well.

Time

Time is divided into discrete periods of length equal to one.

Agents

There are two populations of agents, one with a continuum of males and one with a

Reservation-type strategies

Consider a woman with type y who is accepted by men whose types belong to $Ω_{f} (y)={x:y∈A_{m} (x)}$ , with A_m(x) stationary; her problem is to find an acceptance rule that maximizes her expected utility.

Let Φ_f*(y) be the expected value of her search under an optimal strategy. It follows from standard results in optimal stopping theory (Shiryayev, 1978, Theorem 23, p. 94) that there exists an optimal acceptance rule described as follows: ‘Accept if α₁(x)+α₂(y)≥Φ*(y); otherwise, Reject and continue the

Concluding remarks

In this paper, I present a market containing two large populations of heterogeneous agents who costly and randomly search for mates belonging to the other population; mating is voluntary and requires the agreement of both parties. I prove that there is a unique matching equilibrium where each population partitions into intervals; mating occurs only among men and women belonging to corresponding intervals who segregate themselves from other agents. In other words, if we take a cross-section

Acknowledgements

This paper is based on Chapter 2 of my Ph.D. dissertation, Chade (1997). I would like to thank Bart Taub for his guidance and encouragement, and Mark Huggett, Steve Williams, Alejandro Manelli, Ed Schlee, Gustavo Ventura, and two anonymous referees for their detailed and helpful comments and suggestions. I have also benefited from seminar participants at Arizona State University, University of Western Ontario, University of Bristol, University of Aarhus, ILADES/Georgetown University,

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Two-sided search and perfect segregation with fixed search costs

Abstract

Introduction

Section snippets

The model

Reservation-type strategies

Concluding remarks

Acknowledgements

Review of Economic Dynamics

Set-valued Analysis

A theory of marriage: part I

Journal of Political Economy

A theory of marriage: part II

Journal of Political Economy

Two-sided search, marriages and matchmakers

International Economic Review

Marriage and class

Quarterly Journal of Economics

Two-sided search with heterogeneous agents

Essays on two-sided search with heterogeneous agents

Two-sided search, heterogeneity, and noisy observable types

Modeling marital connections among family lines

Journal of Political Economy