Toyboys or supergirls? An analysis of partners’ employment outcomes when she outearns him

Abstract

In this paper we study households in which the woman is the main earner, encompassing both dual-earners with the wife outearning the husband and couples in which the husband is not employed. The literature in this area is very scant. Earlier studies find that the wife outearns the husband in roughly one of every four dual-earner couples in North-American countries. According to our estimates, the wife earns a higher hourly wage than the husband in one of every six French households, including couples with an inactive partner, and, moreover, this proportion is almost the same considering partners’ monthly earnings. Economic models of marriage would predict that the wife’s earnings dominance be compensated by the husband being younger or possibly more attractive than the wife. Using a large dataset of couples, drawn from the French Labor Force surveys, we find that larger spousal age differences correlate positively with the occurrence of couples in which only the wife works but negatively with dual-earners in which she outearns the husband. Therefore, a marriage selection type of story may explain the occurrence of female solo-earner households while the emergence of “power couples” may provide a rationale for dual-earners in which the wife outearns the husband.

Appendix: Likelihood contributions

The model equations (1) and (2) and the distributional assumptions (3) and (4) are used to construct the likelihood contributions for the different types of observations.

Consider a household i with both spouses employed in year $t,\;(d_{mit}=1,\;d_{fit}=1)$, and where wages, respectively, w _mit and w _fit , are observed for both spouses. Unobserved characteristics are denoted by $(\alpha_i,\;\omega_i)^{\prime}$. We first construct the probability that both spouses are employed, conditional on the unobservables $(\alpha_i,\;\omega_i)^{\prime}$

We define the covariance matrix of the idiosyncratic errors (4) as:

$$ \left(\begin{array}{cc} \Upsigma_{\epsilon} &\Upsigma_{\epsilon u}\prime \\ \Upsigma_{\epsilon u} & \Upsigma_u \end{array}\right) \equiv \left(\begin{array}{cccc} 1 & \sigma_{mf,\epsilon} & \sigma_{m,\epsilon u} & \sigma_{mf,\epsilon u} \\ \sigma_{mf,\epsilon} & 1 & \sigma_{fm,\epsilon u} & \sigma_{f,\epsilon u} \\ \sigma_{m,\epsilon u} & \sigma_{fm,\epsilon u} & \tau_m^2 & \tau_{mf} \\ \sigma_{mf,\epsilon u} & \sigma_{f, \epsilon u } & \tau_{mf} & \tau_f^2 \end{array}\right) $$

(6)

We assume that the density distribution of the idiosyncratic errors of the employment equation, $\epsilon_{mit} = (\epsilon_{mit},\;\epsilon_{fit})^{\prime}$, conditional on the errors $u_{it}=(u_{mit},\;u_{fit})^{\prime}$ of the wage equation, is normal:

$$ \epsilon_{it}|u_{it} \sim N(\Upsigma_{\epsilon u}^{\prime}\Upsigma_u^{-1}u_{it},\Upsigma_{\epsilon} - \Upsigma_{\epsilon u}^{\prime}\Upsigma_{u}^{-1}\Upsigma_{\epsilon u}) $$

(7)

$$ \Upsigma_{\epsilon|u}:= \Upsigma_{\epsilon} - \Upsigma_{\epsilon u}^{\prime}\Upsigma_{u}^{-1}\Upsigma_{\epsilon u}:= \left(\begin{array}{cc} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{array}\right)\;\hbox{and}\;\left(\begin{array}{c} \mu_{1}(u_{it}) \\ \mu_{2}(u_{it}) \end{array}\right) = \Upsigma_{\epsilon u}^{\prime}\Upsigma_u^{-1}u_{it} $$

(8)

We write P(d _m,it  = 1, d _f,it  = 1|w _m,it , w _f,it , α _i , ω _i ).

The employment probability of spouse k [see Eq. (1)] is as follows:

$$ d_{kit} = 1\;\hbox{if}\;d_{kit}^* = \gamma_k^{\prime} x_{kit} + \alpha_{ki} + \epsilon_{kit}> 0\;\hbox{or}\;\epsilon_{kit} > -\gamma_k^{\prime} x_{kit} - \alpha_{ki} $$

(9)

Given (9), (7) and (8), we can write:

$$ \begin{aligned} &P(d_{m,it}=1,\;d_{f,it} = 1|w_{m,it},w_{f,it},\alpha_i,\omega_i) \\ &\quad=\int\limits_{-(x_{fit}^{\prime}\gamma_f + \mu_2(u_{it})+ \alpha_{fi})/\sigma_2}^{\infty} \Upphi \left(\frac{ x_{mit}^{\prime}\gamma_m + \alpha_{mi} + \mu_1(u_{it})+ \frac{\sigma_{12}}{\sigma_2}\nu}{\sqrt{\sigma_{1}^2- \frac{\sigma_{12}^2}{\sigma_2^2}}}\right) \frac{1}{\sqrt{2\pi}}\;\hbox{exp}\;\left\{-\frac{1}{2}\nu^2\right\}d\nu \end{aligned} $$

(10)

with

$$ u_{kit}=\;\hbox{ln}\;w_{kit} - \eta_{k}^{\prime}n_{kit} - \omega_{ki},\;k = m,f $$

The joint density of wages, conditional on $(\alpha_i,\;\omega_i)^{\prime}$ is then:

$$ f(w_{mit},w_{fit}|\omega_i,\alpha_i) = \frac{1}{w_{mit},w_{fit}2\pi |\Upsigma_u|^{1/2}}\;\hbox{exp}\left\{-\frac{1}{2}(\hbox{ln}\;w_{it} - \eta^{\prime}n_{it} - \omega_i)^{\prime}\Upsigma_u^{-1}(\hbox{ln}\;w_{it} - \eta^{\prime}n_{it} - \omega_i)\right\} $$

(11)

with $\eta^{\prime}n_{it} \equiv (\eta_m^{\prime}n_{mit}, \eta_f^{\prime}n_{fit})^{\prime}$ and $\hbox{ln}\;w_{it} \equiv (\hbox{ln}\;w_{mit},\;\hbox{ln}\;w_{fit})^{\prime}$. Finally, let l _it (α _i , ω _i ) be the joint probability density of this household with $(d_{mit}=1,\;d_{fit} = 1,\;w_{mit},\;w_{fit})$:

$$ l_{it}(\alpha_i,\omega_i) = P(d_{mit}=1,\;d_{fit} = 1|w_{mit},w_{fit},\alpha_i,\omega_i) \times f(w_{mit},w_{fit}|\omega_i,\alpha_i) $$

(12)

Second, we consider households in which we observe the employment status of the spouses, but not the wage rate (for either one or both spouses). This occurs if either monthly earnings or usual hours of work are missing. The wage rate is also set to missing if it is less than half of the minimum wage (see the data section). Take first the case of dual-earners, $(d_{mit}=1,\;d_{fit}=1,\;w_{fit})$, where we do not observe the husband’s wage rate. From (4) we know the joint distribution of the idiosyncratic errors of the employment equation and the error of the wife’s wage equation:

$$ \left(\begin{array}{c} \epsilon_{m,it} \\ \epsilon_{f,it} \\ u_{f,it} \end{array}\right) \sim N \left(\left(\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right),\;\left(\begin{array}{ccc} 1& \sigma_{mf,\epsilon} & \sigma_{mf,\epsilon u} \\ \sigma_{mf,\epsilon} & 1 & \sigma_{f,\epsilon u} \\ \sigma_{mf,\epsilon u} & \sigma_{f, \epsilon u } & \tau_f^2 \end{array}\right)\right) $$

(13)

The conditional density of $\epsilon_{it}$ on u _fit is normal

$$ \left(\begin{array}{c} \epsilon_{mit} \\ \epsilon_{fit} \end{array} |u_{fit}\right) \sim N \left(\left(\begin{array}{c} \mu_{1}(u_{it}) \\ \mu_{2}(u_{it}) \end{array}\right),\;\left(\begin{array}{cc} \sigma_1^2 &\sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{array}\right)\right) $$

(14)

with

$$ \begin{aligned} \mu_{1}(u_{it}) &= \frac{\sigma_{mf,\epsilon u}}{\tau_f^2}u_{fit} \\ \mu_{2}(u_{it}) &= \frac{\sigma_{f,\epsilon u}}{\tau_f^2}u_{fit} \\ \sigma_1^2 &= 1 - \frac{\sigma_{mf,\epsilon u}^2}{\tau_f^2} \\ \sigma_{12} &= \sigma_{mf,\epsilon u} - \frac{\sigma_{mf,\epsilon u} \sigma_{f,\epsilon u} }{\tau_f^2} \\ \sigma_{2}^2 &= 1 - \frac{\sigma_{f,\epsilon u}^2}{\tau_f^2} \end{aligned} $$

(15)

We can then compute P(d _mit  = 1, d _fit  = 1|w _fit , α _i , ω _i ) as in expression (10), but applying the conditional means and variances specified in the block (15). The complete likelihood contribution, $l(\alpha_{it}, \omega_{it})^{\prime}$ for this household in year t, is obtained by multiplying this probability by the marginal density of the wife’s wage, conditional on the unobservables.

Similarly, we can construct the likelihood contribution of dual-earner households where the wife’s wage is missing. The relevant conditional means and variances are:

$$ \begin{aligned} \mu_{1}(u_{it}) &= \frac{\sigma_{m,\epsilon u}}{\tau_m^2}u_{mit} \\ \mu_{2}(u_{it})&= \frac{\sigma_{fm,\epsilon u}}{\tau_m^2}u_{fit} \\ \sigma_1^2 &= 1 - \frac{\sigma_{m,\epsilon u}^2}{\tau_m^2} \\ \sigma_{12} &= \sigma_{mf,\epsilon u} - \frac{\sigma_{fm,\epsilon u} \sigma_{m,\epsilon u} }{\tau_m^2} \\ \sigma_{2}^2 &= 1 - \frac{\sigma_{fm,\epsilon u}^2}{\tau_m^2} \end{aligned} $$

(16)

For dual-earner households with missing wages for both spouses, we write:

$$ \begin{aligned} &P(d_{m,it}=1, d_{f,it} = 1|\alpha_i,\omega_i) \\ &\quad=\int\limits_{-(x_{fit}^{\prime}\gamma_f + \alpha_{fi})}^{\infty} \Upphi \left(\frac{ x_{mit}^{\prime}\gamma_m + \alpha_{mi} + \sigma_{mf,\epsilon}\nu}{\sqrt{1 - \sigma_{mf,\epsilon}^2} }\right) \frac{1}{\sqrt{2\pi}}\;\hbox{exp}\left\{-\frac{1}{2}\nu^2\right\}d\nu \end{aligned} $$

(17)

Third, we construct the likelihood contribution of wife-sole-earner households when we observe the wage:

$$ \begin{aligned} &P(d_{m,it}=0, d_{f,it} = 1|w_{f,it},\alpha_i,\omega_i) \\ &\quad=\int\limits_{-(x_{fit}^{\prime}\gamma_f + \mu_2(u_{it}) + \alpha_{fi})/\sigma_2}^{\infty} \left[1-\Upphi \left(\frac{ x_{mit}^{\prime}\gamma_m + \alpha_{mi} + \mu_1(u_{it}) + \frac{\sigma_{12}}{\sigma_2}\nu}{\sqrt{\sigma_{1}^2 - \frac{\sigma_{12}^2}{\sigma_2^2}}}\right)\right] \frac{1}{\sqrt{2\pi}}\;\hbox{exp}\left\{-\frac{1}{2}\nu^2\right\}d\nu \end{aligned} $$

(18)

with the conditional means and variances defined as in block (15). The likelihood contribution for this household in year t, conditional on the random effects, l _it (α _i , ω _i ), is obtained by multiplying this probability by the marginal distribution of the wife’s wage.

If information on the wife’s wage, w _fit , is missing, we write

$$ \begin{aligned} &P(d_{m,it}=0, d_{f,it} = 1|\alpha_i) \\ &\quad= \int\limits_{-(x_{fit}^{\prime}\gamma_f + \alpha_{fi})}^{\infty} \left[1 - \Upphi \left(\frac{ x_{mit}^{\prime}\gamma_m + \alpha_{mi} + \sigma_{mf,\epsilon}\nu}{\sqrt{1- \sigma_{mf,\epsilon}^2} } \right) \right] \frac{1}{\sqrt{2\pi}}\;\hbox{exp}\left\{-\frac{1}{2}\nu^2\right\}d\nu \end{aligned} $$

(19)

Fourth, the likelihood contribution of a male-breadwinner household with observed wages can be written as:

$$ \begin{aligned} &P(d_{m,it}=1, d_{f,it} = 0|w_{m,it},\alpha_i,\omega_i) \\ &\quad=\int\limits_{-\infty}^{-(x_{fit}^{\prime}\gamma_f + \mu_2(u_{it}) + \alpha_{fi})/\sigma_2} \Upphi \left(\frac{ x_{mit}^{\prime}\gamma_m + \alpha_{mi} + \mu_1(u_{it}) + \frac{\sigma_{12}}{\sigma_2}\nu}{\sqrt{\sigma_{1}^2 - \frac{\sigma_{12}^2}{\sigma_2^2}} }\right) \frac{1}{\sqrt{2\pi}}\;\hbox{exp}\left\{-\frac{1}{2}\nu^2\right\}d\nu \end{aligned} $$

(20)

where the conditional means and variances are defined by (16). The likelihood contribution, l _it (α _i , ω _i ) for this household in year t, conditional on random effects, is obtained by multiplying this probability by the marginal distribution of the husband’s wage.

If the husband’s wage is not observed, the likelihood contribution is:

$$ \begin{aligned} &P(d_{m,it}=1, d_{f,it} = 0|\alpha_i) \\ &\quad=\int\limits_{-\infty}^{-(x_{fit}^{\prime}\gamma_f + \alpha_{fi})} \Upphi \left(\frac{ x_{mit}^{\prime}\gamma_m + \alpha_{mi} + \sigma_{mf,\epsilon}\nu}{\sqrt{1 - \sigma_{mf,\epsilon}^2}}\right) \frac{1}{\sqrt{2\pi}}\;\hbox{exp}\left\{-\frac{1}{2}\nu^2\right\}d\nu \end{aligned} $$

(21)

Finally, we look at the case of spouses who are both out of work. Their likelihood contribution is determined as follows:

$$ \begin{aligned} &P(d_{m,it}=0, d_{f,it} = 0|\alpha_i) \\ &\quad= \int\limits_{-\infty}^{-(x_{fit}^{\prime}\gamma_f + \alpha_{fi})} \left[1 - \Upphi \left(\frac{ x_{mit}^{\prime}\gamma_m + \alpha_{mi} + \sigma_{mf,\epsilon}\nu}{\sqrt{1 - \sigma_{mf,\epsilon}^2} } \right) \right] \frac{1}{\sqrt{2\pi}}\;\hbox{exp}\left\{-\frac{1}{2}\nu^2\right\}d\nu \end{aligned} $$

(22)

Having constructed the likelihood contributions for different types of households in a given year, conditional on the random effects $(\alpha_{i},\omega_{i}),\;l_{it}(\alpha_{i},\omega_{i})$, we now see how these change when the household is observed for more than one year. Households stay in the sample for at most three years. If either spouse does not answer the survey, the household is dropped from the sample. If one of the spouses changes over time, then the household is also dropped (see the data section for more details). Take a household i that is observed from year T _i1 to year T _i2. Its likelihood contribution, conditional on random effects, is

$$ l_i (\alpha_{i},\omega_{i}) = \prod\limits_{t=T_{i1}}^{T_{i2}} l_{it}(\alpha_{i},\omega_{i}) $$

(23)

Finally, we complete the likelihood function by integrating over the random effects. Let f(α _i , ω _i ) denote the joint normal density of the random effects [see expression (3)]. The complete likelihood contribution for household i is then:

$$ l_i = \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty} l_i(\alpha_{i},\omega_{i}) f(\alpha_{i},\omega_{i}) d\alpha_{i}d\omega_{i} $$

(24)

where both α _i and ω _i have dimension 2. It follows that the computation of the likelihood contributions requires up to five-dimensional integration, depending on the type of household observed. We use the method of simulated maximum likelihood (SML) to estimate the model, replacing integration by simulation (see Börsch-Supan and Hajivassiliou 1993). We use 20 replications for each observation to simulate the integrals.