Efficient Child Care Subsidies

Abstract

A ProofsFor ease of exposition, we report the statement in each Lemma and Proposition from the main text (in italics) before each proof. We start with a couple of preparatory results.Claim 1 In an optimal contract, we have y(z 1 ) = 0; and if h (z 1 ) > 0, then it solvesIn addition, if for some z we have y (z) = 0, then agent z gets the same allocation as type z 1 = 0.Proof. Since z 1 = 0, we must have y(z 1 ) = 0. From the first order conditions of agent z 1 = 0 with respect to c and h, we have v (h(z 1 )) ≤ ω. Since v (0) = 0 and ω > 0, we have h(z 1 ) > 0. If v (1) ≤ ω, utility can be increased strictly by increasing h whenever h(z 1 ) < 1; Hence, it must be that h(z 1 ) = 1. Consider now type z > 0 declaring σ = z 1 . When y(σ) = 0, all agents have the same preferences over c and h and get the same utility when declaring σ = z 1 . Thus, by DIC, agent z must receive at least the same utility as agent z 1 . If y(z) = 0, UIC implies the reverse inequality, so that the utility between z and z 1 must be the same.Since agent z 1 's problem is strictly concave, the allocation designed for z 1 minimizes the budget cost. Hence, we should use for all z with y(z) = 0 the allocation designed for z 1 .Claim 2 Let λ be the multiplier associated to the budget constraint (1). We haveProof. This result is shown by a simple variational exercise. Since we can increase c(z i ) by the same amount for all i without violating the incentive constraints, it must be that ∑ N i=1 π(z i )φ(z i ) ≤ λ. Since we can also decrease all c(z i ) uniformly in an incentive compatible way, it must be that ∑ N i=1 π(z i )φ(z i ) ≥ λ. Combining the two we get the desired equality.

Child Care Subsidies, Quality, and Optimal Income Taxation

“…Finally, the paper most closely related to ours is Ho and Pavoni (2016), who consider a static Mirrleesian setting and provide a rich set of results on the optimal manner in which to subsidize child care. Even though their setting is similar to ours, the two main differences are that they consider a different model of household decision-making and analyze the government's problem in a different way.…”

“…When we prespecify the policy instruments, we consider a nonlinear income tax and either linear child care subsidies (linear tax credits) or a simple opting-out public provision scheme. A final difference between our paper and Ho and Pavoni (2016) is that we restrict ourselves to a two-type setting in our theoretical analysis, whereas they consider a more general theoretical framework with an arbitrary discrete number of household types.…”

“…Another strand of the literature (see, e.g., Ho and Pavoni, 2016) assumed instead that time spent by parents with the children is a perfect substitute for time spent working in the market; also in this case, however, the longer an agent works in the market, the less time he will spend with the children. Therefore, even though it is possible that agents demand child care hours in excess of their working hours (i.e., it is possible that h c > y/w), it is still the case that center-based child care hours (and therefore child care expenditures) are increasing in the hours spent working in the market (y/w).…”

Pareto efficient income taxation without single-crossing

“…In these papers the SC condition is satisfied and the justification for negative marginal tax rates either comes from the assumption that wages are endogenous or from specific assumptions on the profile of social weights that apply to the different types of agents in the economy. 15 Recent contributions that have analyzed the optimal tax treatment of work-related expenses include Koehne and Sachs (2017), Bastani et al (2020) and Ho and Pavoni (2020), where the last two papers explicitly focus on the case of child care expenditures. A common feature of these papers is that they consider a setting where all agents are, according to our terminology, "users".…”

Nonlinear and piecewise linear income taxation, and the subsidization of work-related goods

“…2 See, e.g., Blomquist and Christiansen (1995), Boadway and Marchand (1995), Cremer and Gahvari (1997), Balestrino (2000), Pirttilä and Tuomala (2002), and Blomquist et al (2010). More recently, Koehne and Sachs (2017) analyze the desirability of providing tax breaks for work-related goods in the context of Pareto-efficient tax structures, whereas Bastani et al (2017), and Ho and Pavoni (2016) analyze optimal child care subsidies. 3 As we have already mentioned, the Mirrleesian literature provides a rationale for subjecting work-related goods/services to a relatively more lenient tax treatment (compared to other goods) based on its effects on the incentive-compatibility constraints faced by the government in the design of the nonlinear income tax.…”