“Making work pay” in a rationed labor market

Abstract

This paper empirically analyzes the labor supply effects of two “making work pay” reforms in Germany. We provide evidence in favor of policies that distinguish between low effort and low productivity by targeting individuals with low wages rather than those with low earnings. We discuss our results more generally and with comparisons to the family-based tax credits in force in the US and the UK. For the evaluation of the policies, we apply a static structural labor supply framework and explicitly account for demand-side constraints by using a double-hurdle model.

A Technical appendix

1.1 A.1 Unconstrained model

We assume the error terms ϵ _ij of the utility function to be i.i.d. according to an EV-I distribution. Then, the probability that alternative k is chosen by household i is given by McFadden (1974):

$$P_{ik}=\Pr (V_{ik}\geq V_{ij},\forall j=0,...,J)=\frac{\exp U(Lf_{ik}, C_{ik},Z_{i})}{\sum_{j=0}^{J}\exp U(Lf_{ij}, ,C_{ij},Z_{i})}.$$

(3)

Further, we model the utility function in a quadratic specification as in Blundell et al. (2000). Preferences for income and leisure coefficients vary conditional on age, number and age of children, and region of residence. We follow van Soest (1995) and introduce dummy variables for the part-time categories in order to capture specific (dis)utility from working part-time. In the estimation, we do not consider potential effects of unobserved heterogeneity, which implies that the IIA property holds. However, Haan (2006) has shown that labor supply elasticities estimated on the same data as in the present study do not differ significantly when unobserved heterogeneity is introduced.

1.2 A.2 Constrained model

Denoting d as the desired hours and p as a dummy representing nonrationing, we can summarize the situation of a single individual with three possible states,¹⁵ to be voluntarily inactive, to be rationed, and to participate without being rationed. In the present set-up, these probabilities are written as follows:¹⁶

$$P_{i0}^{VOL} =\Pr (d_{i}=0)=\frac{\exp U(L_{i0},C_{i0},Z_{i})}{ \sum_{j=0}^{J}\exp U(Li_{j},C_{ij},Z_{i})},$$

(4)

$$P_{ik}^{INVOL} =\Pr (d_{i}=k, \forall k>0, p_{i}=0)= \frac{\exp U(L_{ik},C_{ik},Z_{i})}{\sum_{j=0}^{J}\exp U(L_{ij},C_{ij},Z_{i})}[\Phi (\beta X_{i})],$$

(5)

$$P_{ik}^{EMP}= \Pr (d_{i}=k, \forall k>0, p_{i}=1)= \frac{\exp U(L_{ik},C_{ik},Z_{i})}{\sum_{j=0}^{J}\exp U(L_{ij},C_{ij},Z_{i})}[1-\Phi (\beta X_{i})].$$

(6)

As in Duncan and MacCrae (1999), we assume that the error terms of the labor supply model and the probability of rationing are independent, which allows us to estimate the unemployment risk separately. A more general approach could make use of a simulated maximum likelihood to introduce a correlation between these two terms. The assumption of independent error terms implies, in particular, that, conditional on observed characteristics, having positive desired hours is independent on the risk of rationing. In this sense, our specification rules out discouragement effects, which are unobservable. Indeed, we mix those who are “voluntarily” inactive for two different reasons: either due to high fixed cost of work (due to, e.g., childcare costs) or to high search costs (e.g., due to rationing). As mentioned above, the individual unemployment risk is conditioned on the individual employment history and, hence, partly accounts for discouragement effect related to the employment status over the last 3 years (yet the initial state is assumed random). To account for discouragement, this effect could be modeled explicitly (structurally) by introducing job search cost that would increase with the risk of unemployment (cf. Duncan and MacCrae 1999). The sample log-likelihood to be maximized can be derived by summing individuals’ probabilities for their working state, using Eq. 3 for the unconstrained model or Eqs. 4–6 for the constrained model.