The effect of meeting rates on matching outcomes

Abstract

We extend the classic matching model of Choo and Siow (J Polit Econ 114(1):175–201, 2006) to allow for the possibility that the rate at which potential partners meet affects their probability of matching. We investigate the implications for the levels and supermodularity of the estimated match surplus.

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Notes

  1. 1.

    There is, however, evidence that individuals have at least some preference for partners with similar education levels, e.g., in Bruze (2011) and Belot and Francesconi (2013).

  2. 2.

    As we discuss in Sect. 3, the outcome is still constrained-efficient.

  3. 3.

    An exception is Gihleb and Lang (2016). Papers in sociology and demography (e.g., Schwartz and Mare 2005; Kalmijn 1991) have also measured assortativeness and analyzed its trends using statistical methods such as log-linear models. Some recent work in economics has focused on the impact on between-household inequalities (Greenwood et al 2014).

  4. 4.

    Most data on marriages do not include how the couple met, but retrospective survey evidence suggests that the fraction of heterosexual couples that meet in a given year through college increased between 1940 and 2000 (Rosenfeld and Thomas 2012).

  5. 5.

    See Goussé et al (2017) for a state-of-the-art application to the marriage market.

  6. 6.

    Equivalently, assume that the idiosyncratic part of the match is infinitely negative for women of other subtypes.

  7. 7.

    Indeed, the matching \(\mu _{ij}\) is given by the matching function that appears implicitly in Eq. (2) and must satisfy the constraints \(\mu _{i0} + \sum _{j\in C^i} \sqrt{\mu _{i0}\mu _{0j}}\exp \left( \frac{\alpha _{x_{i}y_{j}}+\gamma _{x_{i}y_{j}}}{2}\right) = 1\) for all subtypes \(i\) (the same constraints must hold for all j on the other side of the market). Since for any two subtypes \(i\) and \(i'\) of the same type \(x\) the mass of these subtypes, the number of meetings and the systematic utility with any women of given type \(y\) is assumed to be the same, it must be the case that \(\mu _{i0} = \mu _{i'0}\).

  8. 8.

    It is always true that the ratio of the average number of women a man meets to the average number of men a woman meets is equal to \(\frac{M}{N}\), but in other models, the number is not the same across types of men or across types of women.

  9. 9.

    The meeting frequencies satisfy the adding-up constraint \(a_{xy}n_x=b_{xy}m_y\) because

    $$\begin{aligned} \frac{1}{a}\sum _ym_yr_y=\frac{N}{M }\frac{1}{b}\left( M - \frac{\gamma }{a}\sum _x\frac{m_xn_x}{m_x+n_x}\right) =\frac{1}{b}\left( N - \frac{\gamma }{a\frac{M}{N}}\sum _x\frac{m_xn_x}{m_x+n_x} \right) =\frac{1}{b}\sum _xn_xq_x. \end{aligned}$$
  10. 10.

    This is a sufficient, but not necessary condition.

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Correspondence to Sonia Jaffe.

Additional information

The authors thank the editor and two anonymous referees for very helpful comments. In addition, the authors are grateful for the insightful comments of Pierre-Andre Chiappori, Alfred Galichon, Scott Kominers, and workshop participants at the University of Chicago.

Appendices

Appendix

Nested Logit

Let the correlation of the random utility shocks for the subtypes within each type be \(\rho \); if \(\rho =1\) then all the subtypes within a type are equivalent and if \(\rho =0\) then the random component of utility is as different across subtypes within a type as across subtypes of different types. In this case the match probabilities for men satisfy

$$\begin{aligned} \frac{\mu _{ij}}{\mu _{i0}}&=\exp \left( \frac{\alpha _{x_iy_j}-\tau _{x_iy_j}}{1-\rho }\right) \left( \sum _{j'\in c_{y_j}^i}\exp \left( \frac{\alpha _{x_iy_j}-\tau _{x_iy_j'}}{1-\rho }\right) \right) ^{-\rho }\\&= \exp \left( \frac{\alpha _{ij}-\tau _{x_iy_j}}{1-\rho }\right) ^{1-\rho }a_{x_iy_j}^{-\rho }. \end{aligned}$$

Combining with the equivalent formula for women, we get

$$\begin{aligned} \mu _{ij}^{2}=\exp (\alpha _{x_{i}y_{j}}+\gamma _{x_{i}y_{j}})\mu _{i0}\mu _{0j}a_{x_iy_j}^{-\rho }b_{x_iy_j}^{-\rho }. \end{aligned}$$

Aggregating, as done in the text, gives

$$\begin{aligned} \mu _{xy} =\left( a_{xy}^{1-\rho }b_{xy}^{1-\rho } \exp (\alpha _{xy}+\gamma _{xy})\mu _{0y}\mu _{x0}\right) ^{\frac{1}{2}}. \end{aligned}$$

This gives the surplus formula

$$\begin{aligned} \phi _{xy}&= \log \left( \frac{\mu _{xy}^2}{\mu _{0y}\mu _{x0}}\right) - (1-\rho )\log (a_{xy}b_{xy}) = \phi _{xy}^{CS} - (1-\rho )\log (a_{xy}b_{xy}). \end{aligned}$$

Effect of population counts on the CS measure of assortativeness

The increased assortativeness due to meeting (from Eq. (7)) is

$$\begin{aligned} 2\log \left( 1+\frac{\gamma }{b}\frac{\theta }{f(n_t,m_t)}\right) +2\log \left( 1+\frac{\gamma }{b}\frac{\theta }{f(n_{t^\prime },m_{t^\prime })}\right) \end{aligned}$$

where \(f(n_t,m_t)=(n_t+m_t)q_tr_t>0\).

Consider moving a woman from \(t\) to \(t'\). This will decrease the assortativeness whenever

$$\begin{aligned} \frac{-f_2(n_{t},m_{t})}{f^2(n_{t},m_{t})+\frac{\theta \gamma }{b}f(n_{t},m_{t})} (-1) + \frac{-f_2(n_{t^\prime }, m_{t^\prime })}{f^2(n_{t'},m_{t'})+\frac{\theta \gamma }{b}f(n_{t'},m_{t'})} (1) <0, \end{aligned}$$

that is when

$$\begin{aligned} \frac{f_2(n_{t^\prime }, m_{t^\prime })}{f^2(n_{t'},m_{t'})+\frac{\theta \gamma }{b}f(n_{t'},m_{t'})} > \frac{f_2(n_{t},m_{t})}{f^2(n_{t},m_{t})+\frac{\theta \gamma }{b}f(n_{t},m_{t})}. \end{aligned}$$
(8)

Expanding \(f(\cdot ,\cdot )\) (and recalling that \(aN=bM\)), we have

$$\begin{aligned} f=(n_{t}+m_{t})- \frac{\gamma }{b}n_{t} - \frac{\gamma }{b}\frac{N}{M}m_{t} +\frac{N}{M}\left( \frac{\gamma }{b}\right) ^2\frac{n_{t}m_{t}}{n_{t}+m_{t}}, \end{aligned}$$

so

$$\begin{aligned} f_2&=1-\frac{\gamma }{b}\frac{N}{M}+\frac{N}{M}\left( \frac{\gamma }{b}\frac{n_{t}}{n_{t}+m_{t}}\right) ^2 >0,\\ f_{22}&=-2\frac{N}{M}\left( \frac{\gamma }{b}\right) ^2\frac{n_{t}^2}{(m_{t}+n_{t})^3} <0. \end{aligned}$$

The function f is increasing in the number of women m, and the derivative of f with respect to m is decreasing in m. Therefore, if \(n_{t} = n_{t'}\), moving a woman from \(t\) to \(t'\) decreases the wedge (Eq. (8) holds), for a given \(\theta \), whenever \(m_{t} > m_t'\). That is, the wedge will decrease as we split women more evenly across the two groups.

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Jaffe, S., Weber, S. The effect of meeting rates on matching outcomes. Econ Theory 67, 363–378 (2019). https://doi.org/10.1007/s00199-018-1113-0

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Keywords

  • Matching markets
  • Assortative matching
  • Matching function
  • Marriage markets

JEL Classification

  • C78
  • J12