Unique equilibria and substitution effects in a stochastic model of the marriage market

Colin Decker; Elliott H. Lieb; Robert J. McCann; Benjamin K. Stephens

doi:10.1016/j.jet.2012.12.005

Unique equilibria and substitution effects in a stochastic model of the marriage market

Colin Decker, Elliott H. Lieb, Robert J. McCann, Benjamin K. Stephens

Mathematics

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24 Scopus citations

Abstract

Choo and Siow (2006) [7] proposed a model for the marriage market which allows for random identically distributed McFadden-type noise in the preferences of each of the participants. In this note we exhibit a strictly convex function whose derivatives vanish precisely at the equilibria of their model. This implies uniqueness of the resulting equilibrium marriage distribution, simplifies the argument for its existence, and gives a representation of it in closed form. We go on to derive smooth dependence of this distribution on exogenous preference and population parameters, and establish sign, symmetry, and size of the various substitution effects. This leads to the testable but unexpected prediction that the percentage change of type i unmarrieds with respect to fluctuations in the total number of type j men or women turns out to form a symmetric positive-definite matrix r_ij=r_ji, and thus to satisfy bounds such as |r_ij|≤(r_iir_jj)^1/2.

Original language	English (US)
Pages (from-to)	778-792
Number of pages	15
Journal	Journal of Economic Theory
Volume	148
Issue number	2
DOIs	https://doi.org/10.1016/j.jet.2012.12.005
State	Published - Mar 2013

All Science Journal Classification (ASJC) codes

Economics and Econometrics

Keywords

Choo-Siow
Comparative statics
Convex analysis
Marriage market
Matching
Random
Unique equilibrium

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10.1016/j.jet.2012.12.005

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title = "Unique equilibria and substitution effects in a stochastic model of the marriage market",

abstract = "Choo and Siow (2006) [7] proposed a model for the marriage market which allows for random identically distributed McFadden-type noise in the preferences of each of the participants. In this note we exhibit a strictly convex function whose derivatives vanish precisely at the equilibria of their model. This implies uniqueness of the resulting equilibrium marriage distribution, simplifies the argument for its existence, and gives a representation of it in closed form. We go on to derive smooth dependence of this distribution on exogenous preference and population parameters, and establish sign, symmetry, and size of the various substitution effects. This leads to the testable but unexpected prediction that the percentage change of type i unmarrieds with respect to fluctuations in the total number of type j men or women turns out to form a symmetric positive-definite matrix rij=rji, and thus to satisfy bounds such as |rij|≤(riirjj)1/2.",

keywords = "Choo-Siow, Comparative statics, Convex analysis, Marriage market, Matching, Random, Unique equilibrium",

author = "Colin Decker and Lieb, {Elliott H.} and McCann, {Robert J.} and Stephens, {Benjamin K.}",

note = "Funding Information: The authors are grateful to Aloysius Siow for attracting their attention to this question, and for many fruitful discussions. This project developed in part from the Masterʼs research of CD; however, the present manuscript is based on a new approach which greatly extends (and largely subsumes) the results of his thesis [9] , and of the 2009 preprint by three of us entitled When do systematic gains uniquely determine the number of marriages between different types in the Choo–Siow matching model? Sufficient conditions for a unique equilibrium. CD, RJM and BKS are pleased to acknowledge the support of Natural Sciences and Engineering Research Council of Canada (NSERC) grant 217006-03 RGPIN ; CD also benefited from an Undergraduate Student Research Award (URSA) held in the summer of 2009. EHL acknowledges a United States National Science Foundation grant PHY-0965859 . ",

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AU - Decker, Colin

AU - Lieb, Elliott H.

AU - McCann, Robert J.

AU - Stephens, Benjamin K.

N1 - Funding Information: The authors are grateful to Aloysius Siow for attracting their attention to this question, and for many fruitful discussions. This project developed in part from the Masterʼs research of CD; however, the present manuscript is based on a new approach which greatly extends (and largely subsumes) the results of his thesis [9] , and of the 2009 preprint by three of us entitled When do systematic gains uniquely determine the number of marriages between different types in the Choo–Siow matching model? Sufficient conditions for a unique equilibrium. CD, RJM and BKS are pleased to acknowledge the support of Natural Sciences and Engineering Research Council of Canada (NSERC) grant 217006-03 RGPIN ; CD also benefited from an Undergraduate Student Research Award (URSA) held in the summer of 2009. EHL acknowledges a United States National Science Foundation grant PHY-0965859 .

PY - 2013/3

Y1 - 2013/3

N2 - Choo and Siow (2006) [7] proposed a model for the marriage market which allows for random identically distributed McFadden-type noise in the preferences of each of the participants. In this note we exhibit a strictly convex function whose derivatives vanish precisely at the equilibria of their model. This implies uniqueness of the resulting equilibrium marriage distribution, simplifies the argument for its existence, and gives a representation of it in closed form. We go on to derive smooth dependence of this distribution on exogenous preference and population parameters, and establish sign, symmetry, and size of the various substitution effects. This leads to the testable but unexpected prediction that the percentage change of type i unmarrieds with respect to fluctuations in the total number of type j men or women turns out to form a symmetric positive-definite matrix rij=rji, and thus to satisfy bounds such as |rij|≤(riirjj)1/2.

AB - Choo and Siow (2006) [7] proposed a model for the marriage market which allows for random identically distributed McFadden-type noise in the preferences of each of the participants. In this note we exhibit a strictly convex function whose derivatives vanish precisely at the equilibria of their model. This implies uniqueness of the resulting equilibrium marriage distribution, simplifies the argument for its existence, and gives a representation of it in closed form. We go on to derive smooth dependence of this distribution on exogenous preference and population parameters, and establish sign, symmetry, and size of the various substitution effects. This leads to the testable but unexpected prediction that the percentage change of type i unmarrieds with respect to fluctuations in the total number of type j men or women turns out to form a symmetric positive-definite matrix rij=rji, and thus to satisfy bounds such as |rij|≤(riirjj)1/2.

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KW - Matching

KW - Random

KW - Unique equilibrium

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Unique equilibria and substitution effects in a stochastic model of the marriage market

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