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.

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.