A marriage matching function with flexible spillover and substitution patterns

Abstract

This paper proposes a new and easy-to-estimate marriage matching function (MMF). I show under minimal conditions the existence of the equilibrium marriage matching distribution associated with the proposed MMF and provide testable conditions under which the equilibrium is unique. This MMF allows more flexible spillover and substitution patterns than the existing MMFs. I show that the static frictionless transferable utility (TU) matching model with peer effects and the dynamic (imperfect) TU marriage matching model both generate MMFs that are each a special case of this proposed MMF. Moreover, I show that the MMF generated by the dynamic TU marriage matching model $\grave{a}$ la Choo (Econometrica 83(4): 1373–1423, 2015) can be rationalized by a static frictionless TU marriage matching model with peer effects. I show how the estimation of this MMF can be used to estimate peer effect coefficients in a marriage matching model.

Appendices

Appendix A: A McFarland’s MMF axioms

I briefly enumerate here McFarland’s Axioms as presented in Pollard (1975).

• Axiom 1: The MMF should be defined for all vectors M and F whose elements are nonnegative integers.

• Axiom 2: $\mu _{} \ge 0$ for all (i, j). (The number of marriages occurring cannot be negative.)

• Axiom 3: $\sum _{i=1} \mu _{ij} \le f_j$, and $\sum _{j=1} \mu _{ij} \le m_i$. (The number of marriages cannot exceed the number of available females and males: Populations constraints.)

• Axiom 4: The number of marriages should depend heavily on the ages of the males and females. So, in partitioning the male and female components of the population into distinct categories by factors relevant to marriage analysis, at the very least, age must be recognized as an essential factor.

• Axiom 5: $\mu _{ij}$ should be nondecreasing in $m_i$ and $f_j$. (Increased availability should not decrease the number of marriages.)

• Axiom 6: $\mu _{ij}$ should be a nonincreasing function of $m_k$ for $k \ne i$, and over some interval strictly decreasing. Similarly for the other sex. (Competition.)

• Axiom 7: If the categories are ordered, the negative effect on $\mu _{ij}$ of an increase in $m_k$ should be greater in magnitude than the negative effect on $\mu _{ij}$ of an equivalent increase in $m_r$, if k is closer to i than r is. (Substitution axiom.)

Appendix B: Relaxing Galichon and Salanié (2015)’s restriction

Consider (Choo 2015’s) model with only two groups of individuals, i and $i+1$ (young and old). In this simple example, the Choo MMF takes the following forms:

$$\begin{aligned} \ln \mu _{i+1,j+1}= & {} \frac{1}{2}\ln \mu _{i+1,0} + \frac{1}{2}\ln \mu _{0,j+1} + \gamma _{i+1,j+1},\\ \ln \mu _{i+1,j}= & {} \frac{1}{2}\ln \mu _{i+1,0} + \frac{1}{2}\ln \mu _{0j} + \frac{1}{2}\beta (1-\theta ) \ln \frac{\mu _{0,j+1}}{f_{j+1}} + \gamma _{i+1,j}.\\ \ln \mu _{i,j+1}= & {} \frac{1}{2}\ln \mu _{i0} + \frac{1}{2}\ln \mu _{0,j+1} + \frac{1}{2}\beta (1-\theta )\ln \frac{\mu _{i+1,0}}{m_{i+1}} + \gamma _{i,j+1}.\\ \ln \mu _{ij}= & {} \frac{1}{2}\ln \mu _{i0} + \frac{1}{2}\ln \mu _{0j} + \frac{1}{2}\beta (1-\theta )\ln \frac{\mu _{i+1,0}}{m_{i+1}}\\&\quad + \frac{1}{2}\beta (1-\theta ) \ln \frac{\mu _{0,j+1}}{f_{j+1}} + \gamma _{ij}. \end{aligned}$$

This can be rewritten as follows:

$$\begin{aligned} \gamma _{i+1,j+1}= & {} \ln \mu _{i+1,j+1}- \frac{1}{2}\ln (m_{i+1}-\mu _{i+1,j+1}-\mu _{i+1,j}) \\&- \frac{1}{2}\ln (f_{j+1}-\mu _{i+1,j+1}-\mu _{i,j+1}),\\ \gamma _{i+1,j}= & {} \ln \mu _{i+1,j} - \frac{1}{2}\ln (m_{i+1}-\mu _{i+1,j+1}-\mu _{i+1,j}) \\&- \frac{1}{2}\ln (f_j-\mu _{i+1,j}-\mu _{ij})+\frac{1}{2}\beta (1-\theta )\ln f_{j+1}\\&- \frac{1}{2}\beta (1-\theta )\ln (f_{j+1}-\mu _{i+1,j+1}-\mu _{i,j+1}).\\ \gamma _{i,j+1}= & {} \ln \mu _{i,j+1} - \frac{1}{2}\ln (m_i-\mu _{i,j+1}-\mu _{ij}) \\&- \frac{1}{2}\ln (f_{j+1}-\mu _{i+1,j+1}-\mu _{i,j+1})+\frac{1}{2}\beta (1-\theta )\ln m_{i+1} \\&- \frac{1}{2}\beta (1-\theta )\ln (m_{i+1}-\mu _{i+1,j+1}-\mu _{i+1,j}).\\ \gamma _{ij}= & {} \ln \mu _{i,j} - \frac{1}{2}\ln (m_i-\mu _{i,j+1}-\mu _{ij}) \\&- \frac{1}{2}\ln (f_j-\mu _{i+1,j}-\mu _{ij})+\frac{1}{2}\beta (1-\theta )\ln (m_{i+1}f_{j+1}) \\&- \frac{1}{2}\beta (1-\theta )\ln \Big ((m_{i+1}-\mu _{i+1,j+1}-\mu _{i+1,j}) (f_{j+1}-\mu _{i+1,j+1}-\mu _{i,j+1}) \Big ). \end{aligned}$$

We then have the following:

$$\begin{aligned} \frac{\partial \gamma _{i+1,j+1}}{\partial \mu _{i+1,j}}= & {} \frac{1}{2}\frac{1}{\mu _{i+1,0}}; \; \; \;\; \frac{\partial \gamma _{i+1,j}}{\partial \mu _{i+1,j+1}}=\frac{1}{2}\frac{1}{\mu _{i+1,0}} + \frac{1}{2}\beta (1-\theta )\frac{1}{\mu _{i+1,0}}.\\ \frac{\partial \gamma _{i+1,j}}{\partial \mu _{i,j}}= & {} \frac{1}{2}\frac{1}{\mu _{0j}}; \; \; \;\; \frac{\partial \gamma _{i,j}}{\partial \mu _{i+1,j}}=\frac{1}{2}\frac{1}{\mu _{0j}} + \frac{1}{2}\beta (1-\theta )\frac{1}{\mu _{i+1,0}}. \end{aligned}$$

Because the Galichon and Salanié (2015) restriction can equivalently be written as $\frac{\partial \gamma _{kl}}{ \partial \mu _{ij}}= \frac{\partial \gamma _{ij} }{\partial \mu _{kl}}$, the above partial derivatives show that the Galichon and Salanié (2015) restriction is relaxed by the Choo (2015) MMF and therefore by the generalized Cobb–Douglas MMF as well. This implies that the Galichon and Salanié (2015) class do not cover the classes of behavioral matching models that can be rationalized by the generalized Cobb–Douglas.

Appendix C: Proof of the equilibrium existence

1.1 C.1 Proof of Theorem 1

Fixed-point representation of the existence of an equilibrium.

Manipulating the population constraints (9), (10) we have the following:

$$\begin{aligned} \mu _{i0}= & {} \frac{m_{i}}{1+\frac{1}{\mu _{i0}} \sum ^{J}_{j=1}g_{ij}(\mu _0;m,f)} \equiv B_{i0}, \quad 1\le i \le I \end{aligned}$$

(37)

$$\begin{aligned} \mu _{0j}= & {} \frac{f_{j}}{1+ \frac{1}{\mu _{0j}} \sum ^{I}_{i=1}g_{ij}(\mu _0;m,f)}\equiv B_{0j}, \quad 1\le j \le J. \end{aligned}$$

(38)

Let $B(\mu _0; m, f) \equiv (B_{10}(.),\ldots ,B_{I0}(.),B_{01}(.),\ldots ,B_{0J} (.))^{\prime }$ where $m \equiv (m_{1},\ldots ,m_{I})^{\prime }$, $f \equiv (f_{1},\ldots ,f_{J})^{\prime }$. With this representation, showing the existence of an equilibrium matching distribution associated with the generalized MMF is equivalent to show that the mapping $B(\mu _0; m, f)$ admits a fixed point. In other terms, there exists $0<\mu _0^{eq} < (m',f')$ such that

$$\begin{aligned} B(\mu _0^{eq}; m, f)=\mu _0^{eq}. \end{aligned}$$

(39)

Step 0: :

Let $\underline{\xi }^t=(\underline{\xi }_1^t,\ldots ,\underline{\xi }_{I+J}^t)$ and ${\overline{\xi }}^t=(\overline{\xi }_1^t,\ldots ,\overline{\xi }^t_{I+J})$ be vectors of arbitrarily small positive constants such that $\underline{\xi }_i^t \le \mu _{i0} \le m_i-\overline{\xi }_i^t$ for $1\le i \le I$ and $\underline{\xi }_{I+j}^t \le \mu _{0j} \le f_j-\overline{\xi }_{I+j}^t$ for $1\le j \le J$. And define, $\mathbb {T}^t_{\xi }=\{\underline{\xi }_1\le \mu _{10}\le m_{1}-\overline{\xi }_1,\ldots ,\underline{\xi }_I \le \mu _{I0} \le m_{I}-\overline{\xi }_I, \underline{\xi }_{I+1} \le \mu _{01} \le f_{1}-\overline{\xi }_{I+1},\ldots ,\underline{\xi }_{I+J}\le \mu _{0J} \le f_{J} -\overline{\xi }_{I+J} \}$. Because $g_{ij}(\mu _0;m,f)$ is a positive and continuous function, for all $\mu \in \mathbb {T}^t_{\xi }$, there exists K a positive constant such that we have $0<\frac{1}{\mu _{i0}} \sum ^{J}_{j=1}g_{ij}(\mu _0;m,f)<K <\infty$, and $0<\frac{1}{\mu _{0j}} \sum ^{I}_{i=1}g_{ij}(\mu _0;m,f)<K <\infty$. Therefore, we have $0< B_{i0}(\mu _0) <m_i$ for $1\le i \le I$ and $0< B_{0j}(\mu _0) <f_j$ for $1\le i \le I$. Moreover, because $B_{i0}(\mu _0)$, $B_{0j}(\mu _0)$ are also continuous functions on a bounded set, there exist vectors of positive constants $\underline{\eta }^t=(\underline{\eta }_1^t,\ldots ,\underline{\eta }_{I+J}^t)$ and $\overline{\eta }^t=(\overline{\eta }_1^t,\ldots ,\overline{\eta }_{I+J}^t)$ such that $\underline{\eta }_i^t \le B_{i0}(\mu _0) \le m_i-\overline{\eta }_i^t$ for $1\le i \le I$ and $\underline{\eta }_{I+j}^t \le B_{0j}(\mu _0) \le f_j-\overline{\eta }_{I+j}^t$ for $1\le j \le J$. More precisely, just take $\underline{\eta }_i^t=\inf _{\mu \in \mathbb {T}^t_{\xi }} B_{i0}(\mu _0)$, $\underline{\eta }_{I+j}^t=\inf _{\mu \in \mathbb {T}^t_{\xi }} B_{0j}(\mu _0)$, and $\overline{\eta }_i^t=m_i-\sup _{\mu \in \mathbb {T}^t_{\xi }} B_{i0}(\mu _0)$, $\overline{\eta }_{I+j}^t=f_j-\sup _{\mu \in \mathbb {T}^t_{\xi }} B_{0j}(\mu _0)$. Notice that with this construction, $\underline{\eta }_i^t>0, \underline{\eta }_{I+j}^t>0, \overline{\eta }_i^t>0, \overline{\eta }_{I+j}^t >0$, since $0< B_{i0}(\mu _0) <m_i$ and $0< B_{0j}(\mu _0) <f_j$ for all $\mu \in \mathbb {T}^t_{\xi }$ where $\mathbb {T}^t_{\xi }$ is a compact set.

Step 1: :

Define $\underline{\xi }_i^{t+1}=\min (\underline{\xi }_i^{t}, \underline{\eta }_i^t)$ for $1\le i \le I+J$ and $\overline{\xi }_i^{t+1}=\min (\overline{\xi }_i^t, \overline{\eta }_i^t)$ for $1\le i \le I+J$.

Step 2: :

If $\underline{\xi }_i^{t+1}= \underline{\xi }_i^t$ and $\overline{\xi }_i^{t+1}=\overline{\xi }_i^t$ then stop the iteration and define $\underline{\epsilon }_i= \underline{\xi }_i^{t+1}$, $\overline{\epsilon }_i=\overline{\xi }_i^{t+1}$.

Step 3: :

If $\underline{\xi }_i^{t+1}\ne \underline{\xi }_i^t$ or $\overline{\xi }_i^{t+1}\ne \overline{\xi }_i^t$ then $t \leftarrow t+1$ and go back to step 0.

By construction $\underline{\xi }_i^{t}$ and $\overline{\xi }_i^{t}$ are decreasing (strictly) positive sequences bounded away from below by 0 then converge. So, when the iteration will stop in Step 2, let $\mathbb {T}_{\epsilon }=\{\underline{\epsilon }_1\le \mu _{10}\le m_{1}-\overline{\epsilon }_1,\ldots ,\underline{\epsilon }_I \le \mu _{I0} \le m_{I}-\overline{\epsilon }_I, \underline{\epsilon }_{I+1} \le \mu _{01} \le f_{1}-\overline{\epsilon }_{I+1},\ldots ,\underline{\epsilon }_{I+J}\le \mu _{0J} \le f_{J} -\overline{\epsilon }_{I+J} \}$ be a closed and bounded rectangular region in $\mathbb {R}^{I+J}$.

$B(\mu _0; m, f)$ is a continuously differentiable mapping such that $B(\mu _0; m, f)$: $\mathbb {T}_{\epsilon } \rightarrow \mathbb {T}_{\epsilon }$. Thus, the existence of an equilibrium matching distribution $\mu ^{eq}$ associated with the generalized MMF exists by invoking the Brouwer fixed-point theorem.

1.2 C.2 Proof of Theorem 2

The existence of the equilibrium is already ensured by Theorem 1. Following Gale and Nikaido (1965)’s result, we know that the equilibrium is unique if the determinant of the Jacobian of ${\mathscr {G}}(.)$ is a P-matrix for all $0<\mu ^{eq} \le (m',f')'$. However, this condition is generally difficult to verify in practice. However, a direct implication of Lemma 1 is that the Jacobian of ${\mathscr {G}}(.)$ is a P-matrix if the Jacobian $J(\mu )$ or its transpose $J^t(\mu )$ is positive diagonally dominant for all $0<\mu \le (m',f')'.$ One can check that under condition (1) and (2) of Theorem 2, $J^t(\mu )$ is a positive diagonally dominant matrix by just taking the special case of the definition where $d_1=d_2=\cdots =d_n=1$. This completes the proof.

Appendix D: Tables

See Tables 1, 2, 3, 4, 5, and 6.

Table 1 First-stage regressions for each endogenous variable (columns) used in the IV regression displays in Table 3

Table 2 First-stage regressions for each endogenous variable (columns) used in the IV regression displays in Table 4

Table 3 IV estimates of the generalized Cobb–Douglas MMF coefficients with the 2 closest age groups as reference group

Table 4 IV estimates of the generalized Cobb–Douglas MMF coefficients using the nearest age groups as reference group

Table 5 Numbers of matched and unmatched males and females by age types

Table 6 Numbers of matched and unmatched males and females by education level types