Market hours, household work, child care, and wage rates of partners: an empirical analysis

Abstract

The aim of this paper is to provide new evidence on the effect of partners’ wages on partners’ allocation of time. Earlier studies concluded that wage rates are an important determinant of partners’ hours of market and non-market work and also that house work may lower married women’s wage rates. However, the bulk of earlier literature in this area failed to account for the endogeneity of wages or the simultaneity of partners’ time allocation choices. Here we take a reduced form approach and specify a ten simultaneous equations model of wage rates, employment and hours of market work, house work and childcare of parents. Non-participants are included in the model. We exploit a rich time use dataset for France to estimate the model. We find that the own wage affects positively own market hours and negatively own house work and childcare hours. The wage of the father has a significantly negative effect on the mother’s market hours while her wage rate has a significantly positive effect on his house work hours.

Appendix: Likelihood contributions

To deal with the multidimensional integration of the likelihood contributions, we estimate the model by simulated maximum likelihood, using the GHK algorithm (see, for instance, Börsch-Supan and Hajivassiliou 1993), proceeding as follows.

We write the variance-covariance matrix $\Upsigma$ of the errors of the time-use, employment and wage equations as:

$$\Upsigma = \left( \begin{array}{ll} A & C' \\ C & \Upomega \end{array} \right)$$

(6)

with

$$\Upomega = Eu_i u_i', u_i = \left( \begin{array}{l} u_{im}\\ u_{if} \end{array} \right), C = E u_i \left( \begin{array}{l} \epsilon_{im}\\ \epsilon_{if}\\ \nu_{im}\\ \nu_{if} \end{array} \right)^{\prime},\; A = E \left( \begin{array}{l} \epsilon_{im}\\ \epsilon_{if}\\ \nu_{im}\\ \nu_{if} \end{array} \right) \left( \begin{array}{l} \epsilon_{im}\\ \epsilon_{if}\\ \nu_{im}\\ \nu_{if} \end{array} \right)^{\prime}$$

(7)

The joint density of the errors of the time-use equations (Eq. 1) and the employment equations (Eq. 3), conditional on the errors of the wage equations (Eq. 2), is normal with mean B _i and variance-covariance matrix Z, with:

$$B_i = C'\Upomega^{-1}u_i, Z = A - C'\Upomega^{-1}C$$

(8)

Let L be the lower-triangular matrix implicitly defined by:

$$L L' = Z$$

(9)

with typical element l _js , j = 1, …8, s = 1, …, j. For each household, we draw R independent random numbers u ^* _isr , i = 1, …, N, s = 1, …, 7, r = 1, …, R from the uniform distribution over the range (0, 1). These random numbers are used to recursively simulate the likelihood contributions for the time-use equations of the husband, the time-use equations of the wife, and the employment equations of husband and wife. We initially assume that wages w _im and w _if are observed. Let l _itmjr denote the simulated likelihood contribution for the j-th time use (j = 1, 2, 3) of the husband in household i, and replication r. If the husband reports no time spent on time use j  (j = 1, 2, 3), but is employed, we set ¹⁹

$$l_{itmjr} = \Upphi\left(- \frac{\alpha_{jm}^m \ln w_{im} + \alpha_{jm}^f \ln w_{if} + x_{im}'\beta_{jm} + \sum_{s=1}^{j-1}l_{js}\nu_{isr}} {l_{jj}} \right)$$

(10)

and

$$\nu_{ijr} = \Upphi^{-1}(l_{itmjr}u_{ijr}^*)$$

(11)

where $\Upphi(.)$ represents the standard normal distribution function. If the husband is not employed, there is no information about the latent amount of paid work, because we do not know whether this state is due to choice or restriction, and the likelihood contribution is the probability (10) plus its complement, which leads to l _itm1r  = 1 for paid work.

If the husband reports a positive amount of time spent on activity j, we set

$$l_{itmjr} = \frac{1}{l_{jj}} \phi\left( \frac{t_{ijm} - [\alpha_{jm}^m \ln w_{im} + \alpha_{jm}^f \ln w_{if} + x_{im}'\beta_{jm} + \sum_{s=1}^{j-1}l_{js}\nu_{isr}]} {l_{jj}} \right)$$

(12)

where ϕ(.) is the standard normal density function, and

$$\nu_{ijr} = \frac{t_{ijm} - [\alpha_{jm}^m \ln w_{im} + \alpha_{jm}^f \ln w_{if} + x_{im}'\beta_{jm} + \sum_{s=1}^{j-1}l_{js}\nu_{isr}]} {l_{jj}}$$

(13)

We take a similar approach for the time uses of the wife. If the wife reports no time spent on activity j, but she reports to be employed, we determine:

$$l_{itfjr} = \Upphi\left(- \frac{\alpha_{jf}^m \ln w_{im} + \alpha_{jf}^f \ln w_{if} + x_{if}'\beta_{jf} + \sum_{s=1}^{j+3-1}l_{js}\nu_{isr}} {l_{j+3,j+3}} \right)$$

(14)

and

$$\nu_{i,j+3,r} = \Upphi^{-1}(l_{itfjr}u_{i,j+3,r}^*)$$

(15)

If she is not employed, we have l _itf1r  = 1 for her paid work.

If the wife reports a positive amount of time spent on activity j, we have l _itfjr  = 

$$\frac{1}{l_{j+3,j+3}} \phi\left( \frac{t_{ijf} - [\alpha_{jf}^m \ln w_{im} + \alpha_{jf}^f \ln w_{if} + x_{if}'\beta_{jf} + \sum_{s=1}^{j+3-1}l_{js}\nu_{isr}]} {l_{j+3,j+3}} \right)$$

(16)

and

$$\nu_{i,j+3,r} = \frac{t_{ijf} - [\alpha_{jf}^m \ln w_{im} + \alpha_{jf}^f \ln w_{if} + x_{if}'\beta_{jf} + \sum_{s=1}^{j+3-1}l_{js}\nu_{isr}]} {l_{j+3,j+3}}$$

(17)

The likelihood contribution for the employment status of the husband, denoted by l _iemr , is equal to, for a non-employed husband:

$$l_{iemr} = \Upphi\left( -\frac{q_{im}'\gamma_m + \sum_{s=1}^6 l_{7s}\nu_{isr}} {l_{77}} \right)$$

(18)

and

$$\nu_{i7r} = \Upphi^{-1}(l_{iemr}u_{i7r}^*)$$

(19)

For an employed husband, it is equal to:

$$l_{iemr} = 1 - \Upphi\left( -\frac{q_{im}'\gamma_m + \sum_{s=1}^6 l_{7s}\nu_{isr}} {l_{77}} \right)$$

(20)

$$\nu_{i7r} = \Upphi^{-1}((1 - l_{iemr}) + l_{iemr}u_{i7r}^*)$$

(21)

For the employment of the wife, we set the likelihood contribution l _iefr if she is nonemployed equal to:

$$l_{iefr} = \Upphi\left( -\frac{q_{if}'\gamma_f + \sum_{s=1}^7 l_{8s}\nu_{isr}} {l_{88}} \right)$$

(22)

If she is employed, we set this equal to:

$$l_{iefr} = 1 - \Upphi\left( -\frac{q_{if}'\gamma_f + \sum_{s=1}^7 l_{8s}\nu_{isr}} {l_{88}} \right)$$

(23)

Next, we set the simulated likelihood contribution of household i, for replication r of the time-use equations and employment status equal to l _ir , where

$$l_{ir} = \prod_{j=1}^3 l_{itmjr} l_{itfjr} l_{iemr} l_{iefr}$$

(24)

This is then averaged over replications to yield:

$$l_i = \frac{1}{R}\sum_{r=1}^R l_{ir}$$

(25)

In the empirical application we set R = 60.

Were neither the wage rates of the husband or the wife to be observed, before computing the above likelihood contributions we simulated their wages w _imr and w _ifr , by drawing them from their joint distribution, (defined by Eq. (2) and $u_i \sim N(0,\Upomega)$), and then plugged them into the simulated likelihood contributions listed above. If the wage rate of the husband is observed, but that of his wife is not, we draw the wife’s wage rate, w _ifr , from the distribution of w _if , conditional on w _im , and plug it into the simulated likelihood contribution, as above. The likelihood contribution was completed by multiplying Eq. 25 by the marginal density of the husband’s wage rate. For households, where, on the contrary, the wife’s wage rate was observed but the husband’s was not, we proceed similarly. If both wage rates are observed, we multiply the simulated likelihood contribution Eq. 25 by the joint density function of the wife’s and husband’s wage rates.