The marital earnings premium: an IV approach

Abstract

Numerous studies find that married men earn more than single men. However, identifying whether and why marriage affects earnings is complicated by the fact that marriage market outcomes are jointly determined with potential earnings. As such, I exploit exogenous variation in marriage induced by the introduction of no-fault divorce laws in the USA to estimate the effect of marriage on the earnings of men. I find a 38% causal increase of marriage on earnings of husbands. This increase in earnings is explained by a large increase in labor market work after marriage. My findings are robust to the possibility of unobserved heterogeneity in the effect of marriage on earnings across individuals.

A Appendix

1.1 A.1 A simple framework of marriage and divorce

This section provides a framework of marriage and divorce for analyzing the effect of divorce legislation on marriage decisions.²⁹ Consider individual utilities for partners 1 and 2:

$$\begin{aligned} U_i = u_i(q_i, Q) + \theta _i, \quad \text { for } i = 1,2, \end{aligned}$$

(A.1)

where $q_i$ is individual consumption, Q is public good consumption, and $\theta _i$ is the quality of the match which is assumed to be independent between partners and independent of their incomes. Agents live for two periods. In the first period, agents marry and consume. The quality of the match is revealed at the end of the first period and agents decide whether to remain married or divorce. In the second period, couples who remained married consume as before, while members of couples who divorced consume as newly divorced singles. Under Pareto efficiency (in a collective model), the economic gain from marriage is

$$\begin{aligned} G(t) = \max \left\{ \sum _i u_i(q_i, Q) \text { s.t. } p\cdot \sum _i q_i + P \cdot Q = t \right\} , \end{aligned}$$

(A.2)

and the total gain from marriage is $G(t) + \sum _i \theta _i$, where t is total income. In addition, let $u^D_i(q_i, Q)$ denote utility when $i=1,2$ is divorced. If and when agents divorce, they each maximize their respective divorce utilities subject to their respective budget constraints. To that end, let $y_1$ and $y_2$ denote their respective individual incomes. In the case of divorce, the woman gets $d_1(y_1,y_2) = d(y_1,y_2)$, and the man gets $d_2(y_1,y_2) = y_1+y_2 - d(y_1,y_2)$. The indirect utility function is

$$\begin{aligned} v^D_i(d_i) = \max \left\{ u^D_i(q_i, Q) \text { s.t. } p \cdot q_i + P\cdot Q = d_i \right\} . \end{aligned}$$

(A.3)

In the second period, partners divorce if the total surplus is larger when divorced than when married, that is

$$\begin{aligned} \sum _i v^D_i(d_i(y_1,y_2)) > G(y_1+y_2) + \sum _i \theta _i, \end{aligned}$$

(A.4)

and remain married otherwise.

From the above inequality, we can easily compute the probability of divorce as follows. Let F be the CDF of $\theta _1 + \theta _2$, then

$$\begin{aligned} \Pr (\text {divorce}) = F\left( \sum _i v^D_i(d_i(y_1,y_2)) - G(y_1+y_2)\right) . \end{aligned}$$

(A.5)

Notice that the probability of divorce depends on $y_1$, $y_2$, and d, so we can write $\Pr (\text {divorce}) = \Pi (y_1, y_2, d, v^D)$, where $v^D = (v^D_1, v^D_2)$.

In the first period, agents marry based on the expected surplus generated by marriage in both periods, but taking into consideration the probability of divorce. Let $\theta = \theta _1 + \theta _2$, the total gains realized from marriage in both periods is

$$\begin{aligned} \bar{G}(y_1 + y_2)&= G(y_1 + y_2 ) + \theta \nonumber \\&\quad + \beta (1 - \Pi (y_1, y_2, d, v^D)) (G(y_1 + y_2 ) + \theta ) \nonumber \\&\quad + \beta \Pi (y_1, y_2, d, v^D) \sum _i v^D_i(d_i(y_1,y_2)). \end{aligned}$$

(A.6)

Since $\theta$ is not observed prior to marriage, we can take the expectation of $\bar{G}$ to obtain the expected gains from marriage (but prior to marriage):

$$\begin{aligned}&{\mathbb {E}}[\bar{G}(y_1 + y_2)] \nonumber \\&\quad =G(y_1 + y_2 ) + {\mathbb {E}}[\theta ] \nonumber \\&\qquad + \beta (1 - \Pi (y_1, y_2, d, v^D)) \nonumber \\&\qquad \left( G(y_1 + y_2 ) + {\mathbb {E}}\left[ \theta \mid \theta \ge \sum _i v^D_i(d_i(y_1,y_2)) - G(y_1+y_2) \right] \right) \nonumber \\&\qquad + \beta \Pi (y_1, y_2, d, v^D) \sum _i v^D_i(d_i(y_1,y_2)). \end{aligned}$$

(A.7)

We are interested in analyzing the effect of divorce legislation on marriage decisions. Marriage decisions are embodied in ${\mathbb {E}}[\bar{G}]$, the expected gains from marriage. Larger gains induce higher propensity for marriage. Divorce legislation is harder to pin in this general framework; however, we can think about divorce legislation affecting the utility of parters after divorce that is $v^D_i$. In that regard, we can take the derivative of the expected marriage gains, which determine marriage decisions, with respect to utilities after divorce, which act as the channel through which divorce legislation affects marriage decisions and represents welfare after divorce. The total effect of $v^D_i$ on ${\mathbb {E}}[\bar{G}(\cdot )]$ can be decomposed as

$$\begin{aligned} \partial {{\mathbb {E}}[\bar{G}(\cdot )] }{v^D_i(\cdot )} = \partial {{\mathbb {E}}[\bar{G}(\cdot )] }{\sum _i v^D_i(\cdot )} \partial {\sum _i v^D_i(\cdot )}{v^D_i(\cdot )}. \end{aligned}$$

(A.8)

The derivative in the first term has a simple expression: $\partial {{\mathbb {E}}[\bar{G}(\cdot )] }{\sum _i v^D_i(\cdot )} = \beta \Pi (\cdot )$.³⁰ Then we have:

$$\begin{aligned} \partial {{\mathbb {E}}[\bar{G}(\cdot )] }{v^D_i(\cdot )} = \beta \Pi (\cdot ) \partial {\sum _i v^D_i(\cdot )}{v^D_i(\cdot )}. \end{aligned}$$

(A.9)

Equation (A.9) says that the effect of divorce legislation on (the gains of) marriage depends on how the legislation changes total welfare after divorce, and that change should be weighted by the probability of divorce and discounted over time. The second term of Eq. (A.9) will depend on the particular divorce legislation under analysis. The change from mutual consent to no-fault divorce clearly increases the post-divorce utility for the spouse that initially has less bargaining power, but the other spouse can experience a decrease in their post-divorce utility. In general, divorce legislation will make $v^D_1(\cdot )$ and $v^D_2(\cdot )$ interdependent, for example increasing one spouse’s utility at the expense of the other, which results in a non-trivial calculation of the derivative that could result in a positive or negative overall effect on the expected gains from marriage and therefore on the marriage decision.³¹

In particular, Chiappori et al. (2015) show that there are cases where mutual consent induces couples to divorce when no-fault would sustain the marriage, and vice versa. We can then conclude that the effect of divorce on marriage depends on the utilities at divorce of both partners, and that effect can go in one direction or another. Moreover, since the only source of heterogeneity in this simple model is individual incomes, a richer model would induce different types of ambiguity in the effects of divorce on marriage.

1.1.1 A.1.1 Proof of equation (A.9)

We take the derivative of ${\mathbb {E}}[\bar{G}(\cdot )]$ with respect to $\sum _i v^D_i(\cdot )$:

$$\begin{aligned} \partial {{\mathbb {E}}[\bar{G}(\cdot )] }{\sum _i v^D_i(\cdot )}&= \beta (1 - \Pi (\cdot )) \partial {\sum _i v^D_i(\cdot )} {\mathbb {E}}\left[ \theta \mid \theta \ge \sum _i v^D_i(\cdot ) - G(\cdot ) \right] \nonumber \\&\quad - \beta \Pi '(\cdot ) \left( G(\cdot ) + {\mathbb {E}}\left[ \theta \mid \theta \ge \sum _i v^D_i(\cdot ) - G(\cdot ) \right] \right) \nonumber \\&\quad + \beta \Pi (\cdot ) + \beta \Pi '(\cdot ) \sum _i v^D_i(\cdot ) \nonumber \\&= \beta (1 - \Pi (\cdot )) \partial {\sum _i v^D_i(\cdot )} {\mathbb {E}}\left[ \theta \mid \theta \ge \sum _i v^D_i(\cdot ) - G(\cdot ) \right] \nonumber \\&\quad - \beta \Pi '(\cdot ) \left( G(\cdot ) - \sum _i v^D_i(\cdot ) + {\mathbb {E}}\left[ \theta \mid \theta \ge \sum _i v^D_i(\cdot ) - G(\cdot ) \right] \right) \nonumber \\&\quad + \beta \Pi (\cdot ). \end{aligned}$$

(A.10)

Let $A(v^D) = \{ \theta : \theta \ge \sum _i v^D_i(\cdot ) - G(\cdot ) \}$. Then the conditional expectation in equation (A.7) can be simplified.

$$\begin{aligned}&{\mathbb {E}}\left[ \theta \mid \theta \ge \sum _i v^D_i(d_i(y_1,y_2)) - G(y_1+y_2) \right] \nonumber \\&\quad = \frac{{\mathbb {E}}[\theta \mathbb {1}_{A(v^D)}]}{\Pr (A(v^D))} \nonumber \\&\quad = \frac{{\mathbb {E}}[\theta \mathbb {1}_{A(v^D)}]}{1 - \Pi (y_1,y_2,d,v^D)} \nonumber \\&\quad = \frac{\int _{\sum _i v^D_i(\cdot ) - G(\cdot )}^\infty \theta \mathrm{d}{F(\theta )}}{1 - \Pi (y_1,y_2,d,v^D)}. \end{aligned}$$

(A.11)

Taking the derivative of the last expression, we obtain

$$\begin{aligned}&\partial {\sum _i v^D_i(\cdot )} {\mathbb {E}}\left[ \theta \mid \theta \ge \sum _i v^D_i(\cdot ) - G(\cdot ) \right] \nonumber \\&\quad = \frac{-(1 - \Pi (\cdot )) (\sum _i v^D_i(\cdot ) -G(\cdot ))F'(\sum _i v^D_i(\cdot ) - G(\cdot )) +\int _{\sum _i v^D_i(\cdot ) - G(\cdot )}^\infty \theta \mathrm{d}{F(\theta )} \Pi '(\cdot ) }{(1 - \Pi (\cdot ))^2} \nonumber \\&\quad = \frac{-(1 - \Pi (\cdot )) (\sum _i v^D_i(\cdot ) -G(\cdot )) \Pi '(\cdot ) + {\mathbb {E}}[\theta \mathbb {1}_{A(v^D)}] \Pi '(\cdot ) }{(1 - \Pi (\cdot ))^2} \nonumber \\&\quad = \frac{\Pi '(\cdot )}{1 - \Pi (\cdot )} \left( {\mathbb {E}}[\theta \mid A(v^D)] -\sum _i v^D_i(\cdot ) + G(\cdot ) \right) . \end{aligned}$$

(A.12)

Plugging that last expression into equation (A.10), we obtain

$$\begin{aligned} \partial {{\mathbb {E}}[\bar{G}(\cdot )] }{\sum _i v^D_i(\cdot )} = \beta \Pi (\cdot ). \end{aligned}$$

(A.13)

1.2 A.2 A local average treatment effect with defiers

This section sketches the main result in de Chaisemartin (2017), the estimation of a local average treatment effect under the presence of defiers. Consider a binary instrument Z, let $D_z \in \{ 0, 1 \}$ be the treatment when the instrument takes a value $Z=z$ and $Y_{dz}$ denote the outcome when the instrument takes value z and treatment takes value $d \in \{ 0, 1 \}$. Only Z, $D \equiv D_Z$ and $Y \equiv Y_{DZ}$ are observed. Four subpopulations are defined:

• 1. Never takers (NT): individuals for whom $D_0 = 0$ and $D_1 = 0$.

• 2. Always takers (AT): individuals for whom $D_0 = 1$ and $D_1 = 1$.

• 3. Compliers (C): individuals for whom $D_0 = 0$ and $D_1 = 1$.

• 4. Defiers (F): individuals for whom $D_0 = 1$ and $D_1 = 0$.

Now, under the assumptions that (1) the instrument is independent of the potential values of D and Y

and (2) Z does not enter the structural equation,

$$\begin{aligned} Y_{d0} = Y_{d1} = Y_d \qquad \forall d \in \{0, 1\}, \end{aligned}$$

the Wald estimator W can be written as:

$$\begin{aligned} W = \frac{\Pr (C) {\mathbb {E}}[Y_1 - Y_0 \mid C] - \Pr (F) {\mathbb {E}}[Y_1 - Y_0 \mid F]}{\Pr (C) - \Pr (F)}. \end{aligned}$$

In addition, if either $\Pr (F) = 0$ or ${\mathbb {E}}[Y_1 - Y_0 \mid C] ={\mathbb {E}}[Y_1 - Y_0 \mid F]$, W is the average causal effect of treatment on the compliers ${\mathbb {E}}[Y_1 - Y_0 \mid C]$ and the coefficient of the first stage in a 2SLS framework is equal to the subpopulation of compliers $FS = \Pr (C)$. de Chaisemartin (2017) relaxes these conditions to:

$$\begin{aligned}&\text {(3) There exists a subpopulation of compliers } C_F \text { such that} \\&\Pr (C_F) = \Pr (F), \\&{\mathbb {E}}[Y_1 - Y_0 \mid C_F] = {\mathbb {E}}[Y_1 - Y_0 \mid F]. \end{aligned}$$

In words, it says that there exists a subpopulation of compliers (the compliers–defiers or “comfiers”) that has the same size as the subpopulation of defiers and that the average treatment effect for these two subpopulations is the same.

Theorem 2.1 in de Chaisemartin (2017) then applies:

$$\begin{aligned}&C_V = C \setminus C_F \text { satisfies} \\&FS = \Pr (C_V), \\&W = {\mathbb {E}}[Y_1 - Y_0 \mid C_V]. \end{aligned}$$

That is, the Wald estimator identifies the treatment effect on the subpopulation of compliers that “survive” elimination with the defiers (these compliers–survivors are the “comvivors”). A sufficient condition for the last theorem to hold is

$$\begin{aligned} \Pr (F \mid Y_1 - Y_0) \le \Pr (C \mid Y_1 - Y_0). \end{aligned}$$

This last expression says that at any point of the distribution of treatment effects ($Y_1 - Y_0$), there are more compliers than defiers. This condition is significantly stronger than the necessary conditions since it requires that the distribution of treatment effects for compliers and defiers to fully overlap.

1.3 A.3 Other potential threats to identification

To assess whether the relationship between the instrument and the marriage decisions is spurious, I regress the marriage decision on the instrument and dummies for up to 3 years before the introduction of the no-fault divorce laws. The results are presented in Table 8. Globally, the dummy variables are indistinguishable from zero.

Table 8 Robustness of the instrument

In addition, I explore whether other potential instruments induce the decision to marry. I restrict my search to variables that a priori may be correlated or be confounded with the introduction of no-fault divorce or that can be thought of as instruments themselves. Table 9 shows the results of using different variables as instruments for marriage in a regression of total earnings in the lhs. None of the reported coefficients is significant.

Table 9 Other potential instruments

Another potential threat to identification arises from different compositions in the population of men across different states. Specifically, I assess whether there are differences in the ages of single men in states that enacted no-fault divorce laws early and states that enacted no-fault divorce laws later. I regress age on a dummy that indicates early enactment (before 1973), while controlling for time fixed effects. The results are shown in Table 10. The coefficient on early is not statistically different from zero.

Table 10 Comparison between state early and late adopters

1.4 A.4 Testability of the identification assumptions

Here I check the validity of the implications of the assumptions necessary for the LATE with defiers. This represents a mathematically necessary condition for identification, as if the check is not satisfied then the assumptions in Sect. A.2 cannot hold, which renders invalid the identification of the treatment effect. Specifically, as de Chaisemartin (2017) points out, if Assumptions 1, 2, and 3 in Sect. A.2 are satisfied, then

$$\begin{aligned} \underline{L} \le W \le \overline{L}, \end{aligned}$$

(A.14)

where

$$\begin{aligned} \underline{L}&= {\mathbb {E}}[Y \mid D = 1, Z = 1, U_{11} \le p_1] -{\mathbb {E}}[Y \mid D = 0, Z = 0, U_{00} \ge 1 - p_0], \\ \overline{L}&= {\mathbb {E}}[Y \mid D = 1, Z = 1, U_{11} \ge 1 - p_1] -{\mathbb {E}}[Y \mid D = 0, Z = 0, U_{00} \le p_0], \end{aligned}$$

$p_d = \frac{FS}{\Pr (D = d \mid Z = d)}$, and $U_{dz}$ is the rank of an observation in the distribution of $Y|(D = d, Z = z)$. The intuition is that the LATE W is also partially identified since $C_V$ is also included in the population for whom $\{ D = 0, Z = 0 \}$, and it accounts for $p_0$% of that population. Therefore, ${\mathbb {E}}[Y_0 \mid C_V]$ cannot be larger than the mean of $Y_0$ for the $p_0$% with highest $Y_0$. Also, it cannot be smaller than the mean of $Y_0$ for the $p_0$% with lowest $Y_0$. One can establish analogous bounds for ${\mathbb {E}}[Y_1 \mid C_V]$. When one combines those two results, one can arrive at worst-case bounds for the treatment effect of comvivors ${\mathbb {E}}[Y_1 - Y_0 \mid C_V]$, which are $\underline{L}$ and $\overline{L}$. Then the point-identified treatment effect W must lie within those bounds.

Using the data in this application, I compute all the components of condition (A.14). They are displayed in the Table 11. Condition (A.14) is satisfied.

Table 11 Components of condition (A.14)

1.5 A.5 Other regressions

1.5.1 A.5.1 Specifications with state trends

See Table 12.

Table 12 Specifications with state trends

1.5.2 A.5.2 Regressions for women

See Table 13, 14, 15, 16 and 17.

Table 13 Weekly labor earnings, women

Table 14 Hourly wages, women

Table 15 Weekly hours of work, women

Table 16 Weekly hours of housework, women

Table 17 Labor force participation, women

1.6 A.6 Power of estimates for hours of housework

Table 7 shows that after marriage, comvivors reduce their hours of housework by 24%. However, I fail to reject the null hypothesis that the effect of marriage is equal to zero. Note that the sample size in that table is smaller than the sample size in Tables 3, 4, and 5. The lack of statistical significance in Table 7 can be due to low power. To explore that possibility, I reestimate the regressions in column IV FE2 of Tables 3, 4, and 5 but keeping the same sample as in Table 7. Table 18 shows the results of such exercise; it illustrates the loss of power when using a smaller sample size.

Table 18 Significance of marriage on outcome variables, restricted sample