## 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.

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## Notes

This is done when modelling their likelihood contribution. See the earlier working paper version of this study for details.

It is tempting to follow a naive approach and model the outcomes above using a multinomial logit or multinomial probit model, in which (to circumvent overlap between the latter two categories), category 4 would be redefined as dual-earners with the wife’s wage lower than the husband’s wage. However, under this alternative set up (used in most earlier work in this area), we would not be able to classify partners with missing wages information nor to control for the possible endogeneity of wages. This last matters because partners with higher hourly wage rates are actually more likely to be employed and may also work longer hours.

Using fixed effects did not strike us as an interesting alternative. We are interested in the effects of education and age differences of partners and these do not vary much over time.

We do not include random effects, since we found that within group variation in hours for men was fairly small, making the variance of the random effect tend to zero.

The new LFS series started in 2003. The survey is now carried out every quarter, households are followed for a year and a half, and the questionnaire is not much comparable across the old and the new LFS series unfortunately.

There were five couples in all with a partner aged less than 17 and 30 couples with a partner (most of the time the woman) aged 17 years.

Earnings are gross of (before) income tax but net of (after) employers’ and employees’ social security contributions.

These figures are equal to, respectively, 12 % (10 %) for the population of households in the sample.

See earlier working paper version of this study. The full results of estimation of the four equations model are also provided in an earlier working paper version of this study.

Full estimates of the parameters of the wage equations can be found in an earlier working paper version of this study.

Full estimates of the parameters of the covariance matrix can be found in an earlier working paper version of this study.

To compute the probabilities, the values of the continuous and count covariates were set to their sample means. To compute the marginal effects of age (experience) we increased the sample mean of age (experience) by one year. Dummy variables have been set at their reference values, so the marginal effects show deviations from the reference category.

For reasons of conciseness, we do not show these associations for “both-out-of work”, and “dual-earners” partners. These can be found in an earlier working paper version of this study.

To be more specific, we distinguish here couples in which the wife is 5 years older, couples in which the husband is between 5 and 10 years older, and couples in which the husband is 10 or more years older. The default category includes all couples in which partners are less than five years apart. Couples in which the wife is more than 10 years older than the husband are very rare. Therefore, we did not create an additional dummy category for these couples. Indeed, there are more couples with a more than 10 years older husband than couples with a more than 5 years older wife.

The estimate of the ‘trophy husband’ effect for husbands aged 45 or more is not statistically different from zero. We did an additional likelihood ratio test, in which made the ‘trophy husband’ effect independent of the husband age, but the test rejected equal coefficients for different age categories (LR = 15.8, with 4 degrees of freedom). Inspection of the coefficients of the employment equation show that for this age category, employment rates husbands with an older wife is not significantly different from couple with minor age differences, while the wife is less likely to be employed. The latter is also influenced by the fact that in those couples the wife is in the age range 50 through 54.

Inspection of the standard errors of these estimated effects show that the confidence intervals in both cases do not overlap. Thus, the differences are statistically significant.

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## Acknowledgments

This research has benefited from a French National Research Agency (ANR) grant. Earlier versions of this paper were presented at the annual meeting of the Society of Labor Economists, New York, May 2008, the IAFFE annual conference, Turin, June 2008; the European Meeting of the Econometric Society, Budapest, August 2007; the AFSE Paris, and the AIEL Naples, September 2007; and at invited seminars at University San Diego State, December 2008; University Paris 2, February 2008; University Cergy, May 2007; University Paris 1, June 2007. We thank Robert A Pollak, Catherine Sofer, Anne Winkler and all seminar participants for their comments. In particular, we thank Shoshana Grossbard and Sonia Oreffice for their valuable advice. We are also grateful to two anonymous referees for their suggestions. All errors are ours.

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## Appendix: Likelihood contributions

### 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:

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:

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:

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

with

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

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 \((d_{mit}=1,\;d_{fit} = 1,\;w_{mit},\;w_{fit})\):

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:

The conditional density of \(\epsilon_{it}\) on *u*
_{
fit
} is normal

with

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:

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

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

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

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

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:

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

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

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:

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.

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Bloemen, H.G., Stancanelli, E.G.F. Toyboys or supergirls? An analysis of partners’ employment outcomes when she outearns him.
*Rev Econ Household* **13**, 501–530 (2015). https://doi.org/10.1007/s11150-013-9212-y

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DOI: https://doi.org/10.1007/s11150-013-9212-y