3.1 The models and the empirical specifications
Our policy evaluation methodology relies on a static microeconometric labor supply model. We extend the simple theoretical framework presented in the previous section in order to encompass workers’ and jobs’ heterogeneity and include labor market participation choice15. The key feature of our modeling framework is the exact representation of tax and transfer schemes when estimating the model.
It is important to underline that we will report several simulation results in relation with the marital status of the agents. That is: (i) the implied choices of unmarried men and women, separately estimated; and (ii) the simultaneous choices of spouses. For case (ii), we will show the differences in the elasticities of women’s probability of participation both when husband’s participation choice is taken as given (as in (Kleven et al. 2009)) and when it is endogenous. Note that in the case of inelastic husband’s labor supply, women solve a similar problem to that of unmarried women, where husband’s employment status and earnings are taken into account.
We build a two-stage static model of labor supply. Consider a one-person decision problem16. In the first stage, an agent decides whether to join the labor market and search for a job. If she does, she enters the second stage and receives, for each possible amount of working time h∈H⊂ℜ+, a job offer characterized by a level of gross yearly earnings w(h). She can accept one of them or reject them all and stay unemployed (h=0). The problem is discretized in the sense that the choice of hours is supposed to be made between few alternatives: part-time, full-time, or unemployment. The idea is simply that there generally are commonly agreed durations of work in the labor market, including the possibility not to work at all. This is relatively realistic and particularly appropriate in the case of Italy where social and institutional norms as well as demand-side rigidities are strong and imply concentrations around a limited number of hour choices. Thus, when modeling unmarried individuals or inelastic husband’s labor supply, the set of alternatives (h=1,…,H) corresponds to H work durations. In the case of joint spouses’ choices, participation and employment decisions are taken simultaneously by the two partners, and there will be H
2 combinations of spouses’ labor supplies.
The complex structure of the tax system implies that the disposable income is a function of total household income and household composition. Extensively, the disposable income of household i, for i=1,…,I, can be computed as
$$ {\fontsize{8.2}{12}\begin{aligned} D^{i} = \left\{ \begin{array}{ll} w_{f}\left(h_{f}\right)+w_{m}(h_{m})+y-T\left(w_{m}(h_{m})\right)-T\left(w_{f}(h_{f})\right)+F^{i}\left(w_{m}(h_{m}),w_{f}(h_{f})\right) & \text{if \textit{i} is a married couple,} \\ w_{j}(h)+y-T(w_{j}(h))+F^{i}(0,w_{j}(h)) & \text{otherwise,}\\ \end{array}\right. \end{aligned}} $$
((6))
where j={f,m}. w
f
and w
m
are the labor earnings of women and men, respectively; y is net household non-labor income; F
i(w
m
,w
f
) denotes transfers from the government; and T(w) are taxes on gross earnings. In the rest of the section, the spousal gross wage w
m
(h
m
) is taken as given if husband’s labor supply is inelastic.
We assume that the household preferences are described by a linear stochastic utility function U(·) which depends on the alternative h, household disposable income D(·), and a set of exogenous socio-demographic characteristics Z which account for observed heterogeneity across households. The utility is estimated separately for married and unmarried individuals:
$$ U^{i} = \left\{ \begin{array}{ll} \alpha_{(h_{f},h_{m})}+\beta_{(h_{f},h_{m})} D^{i}+\gamma_{(h_{f},h_{m})} Z^{i}+\epsilon_{(h_{f},h_{m})}^{i} & \text{if \textit{i} is a married couple,} \\ \alpha_{h}+\beta_{h} D^{i}+\gamma_{h} Z^{i}+{\epsilon_{h}^{i}} & \text{otherwise.}\\ \end{array} \right. $$
((7))
Notice that the effect of all variables included in Z varies with h
17. In the case of inelastic husband’s labor supply, the set of explanatory variables Z contains controls for the labor market status of the husband. The difference (α
h
−α
0)+(γ
h
−γ
0)Z
i captures the disutility of working (utility of leisure) an amount of time h. Finally, ε
i is a stochastic error component18.
We solve the problem by backward induction, starting from stage 2. A household i will maximize utility
$$\begin{array}{@{}rcl@{}} V^{i}(w_{f}(h_{f}),w_{m}(h_{m}),y,Z) &= \max_{\{h_{f},h_{m}\}}U^{i}(h_{f},h_{m},D(w_{f},w_{m},y),Z) & \text{if \textit{i} is a married couple,} \end{array} $$
((8))
$$\begin{array}{@{}rcl@{}} V^{i}(w_{j}(h_{j}),y,Z)\!\!\!\!\!\!\!\!\!\!\!\!\!\ &= \max_{\{h_{j}\}}U^{i}(h_{j},D(w_{j}(h_{j}),y),Z) & \text{otherwise.} \end{array} $$
((9))
The expressions for V
i(w
f
(h
f
),0,y,Z) and V
i(0,w
m
(h
m
),y,Z) are straightforward analogues of equation (8). In this stage, a household i faces a trade-off between the utility from non working (enjoying leisure and carrying out domestic work) and working, augmenting the disposable income of the household.
In stage 1, members of household i decide whether or not to enter the labor market. To make their choice, they compare the utility from not participating and the expected utility from entering the labor market. Let c be the cost of entering the labor market and E[V
i(·)] be the expected utility generated by the maximization problem in stage 2. Then,
$$ \max\!\left\{\!U^{i}(0,0,D(0,0,y),Z),\!E\!\left[V^{i}(w_{f}(h_{f}),\!w_{m}(h_{m}),y,Z)\right]\,-\,c^{i}(h_{f},\!h_{m})\right\} \text{if}\ i\ \text{is a married couple,} $$
((10))
$$ \max \left\{U^{i}(0,D(0,y),Z),E\left[V^{i}(w_{j},y,Z)\right]-c^{i}\right\}\ \text{otherwise,} $$
((11))
and two additional problems similar to (10) for V
i(w
f
(h
f
),0,y,Z) and V
i(0,w
m
(h
m
),y,Z).
We know that if ε is i.i.d. according to a type I extreme value distribution, the probability of observing an individual in the labor market, opting for a choice h
j
=k
j
or a couple choosing {h
f
=k
f
,h
m
=k
m
}, is
$$\begin{array}{@{}rcl@{}} Pr(k_{f},k_{m})&=&\frac{e^{U^{i}(k_{f},k_{m},D(w_{f}(k_{f}),w_{m}(k_{f}),y),Z)}}{\sum_{h_{f}} \sum_{h_{m}}e^{U^{i}(h_{f},h_{m},D(w_{f}(h_{f}),w_{m}(h_{m}),y),Z)}}, \end{array} $$
((12))
$$\begin{array}{@{}rcl@{}} Pr(k_{j})&=&\frac{e^{U^{i}(k,D(w_{j}(k),y),Z)}}{\sum_{h \in H} e^{U^{i}(h,D(w_{j}(h),y),Z)}} \text{for j \(=\) f,m}. \end{array} $$
((13))
Moreover, P
r(k
f
,0) and P
r(0,k
m
) can be computed in the same fashion as (12). Similarly, the probability of participating is
$$\begin{array}{@{}rcl@{}} Pr(s_{f}=1,s_{m}=1)&=&\frac{e^{E\left[V^{i}(w_{f}(h_{f}),w_{m}(h_{m}),y,Z)\right]-c^{i}(h_{f},h_{m})}}{e^{U^{i}(0,0,D(0,0,y),Z)}+\sum_{f=\{0,w_{f}(h_{f})\}}\sum_{m=\{0,w_{m}(h_{m})\}}e^{E\left[V^{i}(f,m,y,Z)\right]-c^{i}(f,m)}}, \end{array} $$
((14))
$$\begin{array}{@{}rcl@{}} Pr(s_{j}=1)& =& \frac{e^{E\left[V^{i}(w_{j},y,Z)\right]-c^{i}}}{e^{U^{i}(0,D(0,y),Z)}+ e^{E\left[V^{i}(w_{j},y,Z)\right]-c^{i}}} \text{for j=f,m}, \end{array} $$
((15))
where s indicates the labor market status.
3.2 The data and wage imputation
We use micro data from the EU-SILC, the Community Statistics on Income and Living Conditions. The survey collects information relating to a broad range of issues in relation to income and living conditions. SILC is conducted by the Statistics Offices of the European countries involved in the project on an annual basis in order to monitor changes in income and living conditions over time.
Every person aged 16 years and over in a household is required to participate in the survey. Two different types of questions are asked in the household survey: household questions, and personal questions. The former covers details of accommodation and facilities together with regular household expenses (mortgage repayments, etc.). This information is supplied by the head of the household. The latter covers details on variables such as work, income, and health and are obtained from every household member aged 16 years and over. We combine household and personal information to construct a data set which contains information on the spouse of the interviewed household member.
We focus on the cross-sectional information of the years 2007-201119. We restrict the sample to women aged 26-54 years to avoid the modeling of schooling and retirement decisions. Moreover, we exclude self-employed men and women and individuals that are coded as disabled or unfitted to work. Descriptive statistics are in Table 13, Additional file 1: Appendix B.
The data set provides information on gross labor income of all members of the household (w
m
,w
f
), and total household income. By difference, it is possible to compute the non-labor income y. Nevertheless, it is necessary to compute the potential income for all possible labor supply choices h∈H, including the non-employed. To correct for selection bias, a two-stage non-linear procedure is adopted which departs in few features from the standard Heckman correction.
We consider a model in which individuals i are sorted into 4 categories 0,…,3 on the basis of an ordered-probit selection rule:
$$ z^{*}_{i}=\gamma't_{i} + u_{i}; $$
((16))
$$z_{i} = \left\{ \begin{array}{ll} 0 \text{\textquotedblleft out of the labor force}^{\prime\prime} & \text{if} -\infty < z^{*}_{i} \leq \mu_{1}, \\ 1 \text{\textquotedblleft unemployed}^{\prime\prime} & \text{if}\, \,\mu_{1} < z^{*}_{i} \leq \mu_{2}, \\ 2 \text{\textquotedblleft part-time employed}^{\prime\prime} & \text{if}\, \,\mu_{2} < z^{*}_{i} \leq \mu_{3}, \\ 3 \text{\textquotedblleft full-time employed}^{\prime\prime} & \text{if} \,\,\mu_{3} < z^{*}_{i} < \infty, \\ \end{array} \right. $$
where γ is an unknown vector of parameters, u
i
is a standard normal shock, and the unknown cutoffs μ
1,μ
2,μ
3 satisfy μ
1<μ
2<μ
3. We assume that the independent variables t
i
, vectors of demographic characteristics of agent i and her spouse, and the categorial variable z
i
are observed, but the latent selection variable \(z_{i}^{*}\) is unobserved.
We also consider an observed dependent variable w
i
, the wage, that is a linear function of some observed demographic characteristics of agent i and her spouse x
i
, but the coefficients of x
i
depend on the category z
i
:
$$w_{i} = \left\{ \begin{array}{rl} \beta_{0}'x_{i} + \nu_{i0} & \text{if} \,\,z_{i} = 0, \\ \beta_{1}'x_{i} + \nu_{i1} & \text{if} \,\,z_{i} = 1, \\ \beta_{2}'x_{i} + \nu_{i2} & \text{if} \,\,z_{i} = 2, \\ \beta_{3}'x_{i} + \nu_{i3} & \text{if} \,\,z_{i} = 3, \\ \end{array} \right. $$
where for each j∈{0,1,2,3}, ν
ij
has a mean 0, a variance \({\sigma ^{2}_{j}}\), and is bivariate normal with u
i
, with correlation ρ
j
. We assume that the shocks ν
ij
and u
i
are independently and identically distributed across observations. Our goal is to estimate the parameter vectors β
0,…,β
3. w
i
could also be missing for certain categories j, in which case β
j
, ρ
j
, and σ
j
do not exist.
Since only one category j is observed for each individual and the observations are independent, the correlations between ν
ij
and ν
ik
for j≠k cannot be identified, so we do not model or estimate them.
We proceed with a two-step estimation procedure that is a generalization of Heckman’s (1979) estimator for the binary case. In the first step, we estimate (16) by an ordered probit of z on t, yielding the consistent estimates \(\hat {\gamma }, \hat {\mu }_{1}, \hat {\mu }_{2}, \hat {\mu }_{3}\). Define \(\hat {z}_{i}^{*}\equiv t_{i}\hat {\gamma }\) and λ
i
≡E[u
i
|z
i
,t
i
]. Then, a consistent estimator of λ
i
is
$$ \hat{\lambda_{i}} \equiv \frac{\phi \left(\hat{\mu}_{j} - \hat{z}_{i}^{*}\right) - \phi \left(\hat{\mu}_{j+1} - \hat{z}_{i}^{*}\right)}{\Phi \left(\hat{\mu}_{j+1} - \hat{z}_{i}^{*}\right) - \Phi \left(\hat{\mu}_{j} - \hat{z}_{i}^{*}\right)}, $$
((17))
where j=z
i
. Using E[w
i
|z
i
,t
i
,x
i
]=β
j′x
i
+ρ
j
σ
j
λ
i
, we can consistently estimate β
j
with OLS regressions of w on x and \(\hat {\lambda }\) by using only the observations i for which z
i
=j
20.
3.3 Estimation results
To assess the properties of the model, we first examine its ability to reproduce the basic features of labor force participation and employment observed in our sample by comparing model simulations to observed data.
Table 4 and Table 16 in Additional file 1: Appendix C summarize the results of the estimations of the labor force participation and employment rates (part-time and full-time). The models replicate the percentage of men and women in the labor force and the percentage of those who are employed (in part-time and full-time jobs) for both gender and marital status.
Table 4
Estimation results (
%
)
Figure 4 plots the realized and predicted labor force participation rates of married women by percentile of husbands’ incomes. The model with inelastic husband’s labor supply slightly underestimates the participation rates of women married to husbands in the lowest percentile and overestimates the participation of higher income wives. The model with joint decisions delivers a more precise match.
A similar trend for married women is confirmed in panel (a) of Figure 5, where the realized and predicted participation rates are plotted against the household’s disposable income. In panel (b) of the same figure, we plot the participation rates of husbands, showing that it is inelastic to the household’s disposable income. Interestingly, a similar trend is obtained if participation rates are plotted against wife’s income (red lines).
As a final validation exercise, we plot the estimated participation rates by age and marital status in Figure 6. We can observe that both models generate the levels and the decreasing trend of the participation rate of the different subgroups of women. Even though the taxation system is not age-dependent, the age of women is correlated with their own potential earnings, their husband’s earnings, and the number of children. As we described above, all of these elements affect the tax burden, and hence, the labor decision of second earners21.
3.4 Uncompensated labor supply elasticities
The parameter estimates of our models do not directly reveal the sensitivity of labor supply to financial incentives. Hence, in this section, we provide labor supply elasticities for several subgroups of the sample. Uncompensated elasticities for both spouses’ wage rates are relevant to analyze because they are the main driving force behind the tax policy effects.
The elasticities are derived by predicting labor supply for each individual when the gross wages are increased by 1 percent. The estimates are based on 100 sets of simulations, where the coefficients of the preferences are kept fixed to those estimated before the increase in the wages. Individual responses are averaged across individuals to yield aggregate labor supply elasticities. Note that the elasticities depend on preferences, demographic and educational structure, and tax functions. The results are in Table 5.
Table 5
Uncompensated labor supply elasticities
The own wage elasticity of married men is numerically smaller than the wage elasticities of married women, in accordance with the findings in labor supply studies. In the rest of the table, we illustrate the variation of the labor supply elasticities with disposable household income, geographical area of residence, and presence of children.
The own wage elasticities of married men are decreasing with household disposable income, while the opposite is true for married women (in both models). This is in line with the system of negative incentives provided to wives living in low-income households.
The cross wage elasticity of married women (in the joint decision model) is negative and increasing in household disposable income. By contrast, the cross wage elasticity of married men is positive and increasing in household disposable income. It is estimated to be 0.134, on average, for men (about a fourth of the corresponding own wage elasticity) and -0.055 for women. This means that a 10 percent increase in spousal wage causes a 1.34 percent increase in husband’s participation rate and a 0.55 percent decrease in wife’s participation rate. Hence, changes in wage differences in the family may lead to within-household divergence in participation.
An important feature distinguishes the joint decision model from the model with inelastic husband’s labor supply: In the latter, the cross wage elasticity of wives is positive for all levels of household income. This suggests that assuming a perfectly inelastic labor supply of husbands may deliver an inaccurate analysis of spousal responses to wage increases.
Single women have an own wage elasticity which is lower than the elasticity of men. In general, unmarried agents have an elasticity which is increasing with household’s disposable income.
Geographically, own wage elasticity is increasing from north to south for all but married women. The opposite is true for the cross wage elasticity, which numerically decreases for both husbands and wives. Lastly, households with children are more sensitive to changes in own wages, regardless of the marital status.