Collective Models of Labor Supply with Nonconvex Budget Sets and Nonparticipation: A Calibration Approach

Abstract

We suggest a methodology to calibrate a collective model with household-specific bargaining rules and marriage-specific preferences that incorporate leisure externalities. The empirical identification relies on the assumption that some aspects of individual preferences remain the same after marriage, so that estimation on single individuals can be used. The procedure maps the complete Pareto frontier of each household in the dataset and we define alternative measures of a power index. The latter is then regressed on relevant bargaining factors, including a set of variables retracing the potential relative contributions of the spouses to household disposable income. In its capacity to handle complex budget sets and labor force participation decisions of both spouses, this framework allows the comparison of unitary and collective predictions of labor supply reactions and welfare changes entailed by fiscal reforms in a realistic setting (see Michal Myck et al., 2006; Denis Beninger et al., 2006).

Appendices

Appendix

For the sake of completeness we give here some details on the estimation procedures used.

A.1 Wage equations

In order to obtain wage rates for the whole population, including those not working, we estimate wage equations.

For singles we posit a linear normal selection model and use either the maximum likelihood method or the two-steps Heckman procedure. We tried a number of different estimation methods, including also two step methods with other regressors than the predicted normal hazard used in the Heckman approach, but the choices mentioned above gave the most accurate predictions for working singles.

For couples we also estimate wage equations separately for wives and husbands. The following conceptual difficulty arises here due to selectivity: a participation model would need to be based on the collective framework, which is difficult. However, Lewbel (2000) proposes an estimation method for the selection model which does not require the specification of the selection mechanism. ¹⁶ The method relies on the existence of a variable which is monotonically related to the selection variable: in the case of participation, household unearned income is a plausible candidate—though admittedly less so in the collective than in the unitary approach, because of the effect of household unearned income on intrahousehold allocation. We use the simplest of the estimators proposed by Lewbel for wives. For men we apply OLS, as the selectivity problem is much less severe for them, and the OLS predictions are more accurate than those based on the Lewbel estimator.

A.2 Preferences of single women and men

We estimate preferences separately for women and men, assuming LES-type preferences:

$$v_{i}\left(c_{i},l_{i}\right) = \beta_{c}^{i}\ln\left(c_{i} - \overline{c}_{i}\right) + \beta_{l}^{i}\ln\left(l_{i} - \overline{l}_{i}\right)\quad i=f,m,$$

(17)

where c _i represents consumption (i.e., disposable income in this static model) and l _i demand for leisure. $\bar{c}_{i}$ and $\overline{l}_{i}$ are, respectively the “minimum” requirements in consumption and leisure. Instead of estimating these, which proved difficult, we chose to calibrate them.

The budget constraint is defined as:

$$c_{i} = g\left(l_{i},w_{i},y_{i},\phi_{i}\right)\quad i=f,m,$$

(18)

where w _i and y _i are, respectively i’s gross wage rate and i’s unearned income, ϕ _i represents a vector of characteristics relevant to the tax system, and the function g expresses the tax-benefit schedule.

For the estimation, we use a mixed multinomial logit model (MMNL) with mass points on the consumption and leisure coefficients in order to account for unobserved heterogeneity (see Heckman and Singer, 1984; Hoynes, 1996). ¹⁷ We suppose that each person has J alternative values ℓ^j for his/her weekly labor supply, leading to leisure choices l^j=T−ℓ^j, where T is the total time available in a week: 168 h a week. The contribution to the likelihood for person i choosing combination (c ^j _i , l^j) is:

$$L =\sum_{r=1}^{R}p_{r}\frac{\exp\left[\beta_{cr}^{i}\ln\left(c_{i}^{j}- \overline{c}_{i}\right) + \beta_{lr}^{i}\ln\left(l^{j} - \overline{l}_{i}\right)\right]}{\sum_{j{\prime} = 1}^{J}\exp\left[\beta_{cr}^{i}\ln\left(c_{i}^{j^{\prime}} - \overline{c}_{i}\right) + \beta_{lr}^{i}\log\left(l^{j^{\prime}} - \overline{l}_{i}\right)\right]}$$

(19)

where R denotes the number of mass points (or regimes), and p _r the probability associated with mass point r in the mixture. Three mass points appear sufficient, given the heavy dominance of one of the regimes for both preference estimations, single women as well as single men. We do not impose the constraint β ⁱ _c +β ⁱ _l =1 in the estimation, but check that the estimates are positive, which allows to rescale the utility function by β ⁱ _c +β ⁱ _l afterwards. An alternative specification with β ⁱ _c =F(z _i γ), where F denotes the logistic cumulative distribution function, and z _i γ a linear index depending on characteristics for individual i, used for instance by Hoynes (1996), led to much lower likelihood values. The reason for the superiority of not imposing the restriction β ⁱ _cr +β ⁱ _lr =1 is that it amounts to fixing the scale of utility. The MMNL model results from adding iid-error terms to (17) for each possible choice, with an extreme value distribution. This entails a fixed variance, which in turns identifies the scale of utility (and thus the sum of the marginal valuations of leisure and consumption).

In order to ensure that the probabilities p _r do lie between 0 and 1, we adopt the following logit parameterization:

$$\begin{array}{ll} p_{r} = & \exp(e_{r})/\left[1 + \sum \limits_{s=1}^{R-1}\exp(e_{s})\right]; \quad r = 1,\ldots,R-1, \\ & p_{R} = 1-\sum \limits_{s=1}^{R-1}p_{s}. \end{array}$$

(20)

After estimation, we allocate each observation to the regime yielding the best hours prediction.