Child Benefits Consider an economy with two agents who differ in...

Child Benefits

Consider an economy with two agents who differ in their earning ability, denoted by w2 (w1), for the high (low) ability type, respectively, where w2> w1> 0. Each agent is endowed with a given amount of time (normalized to unity) that can be allocated between labor and leisure (caring of children). The utility of a typical agent is given by:

U (c, n) = ( n=1c+b&n=0c )

where c denotes consumption, n denotes the number of children and b>0, the imputed value of leisure. It is further assumed that the time required for nurturing is given by 0 < a < 1, where w1< b/a < w2 .

1) Formulate the typical agent's decision problem and characterize the demand for children as a function of w.

2) Denote the level of utility derived, in equilibrium, by the high (low) ability agent by u2(u1) , respectively, with u2 > u1 > 0. Suppose the government is seeking to guarantee a minimal level of well-being given by ui, where u1 < ui < u2, for all agents, at the minimal cost, by offering a universal lump-sum transfer. Formulate the government problem and characterize the optimal level of transfer.

3) The Council of Economic Advisors has recently proposed to replace the universal transfer system with a child-benefits system, that will selectively grant a transfer to a household with children. Provide a verbal explanation for the rationale underlying the suggested reform.

4) Characterize the dead-weight loss of the child benefit system as a function of the transfer level, T.

5) Show that the suggested reform will be cost-saving when ui aw2 + (1 - a)w1. Is the First-Best allocation attained in this case?

6) Suppose now that ui > aw2 + (1-a)w1. The Council of Economic Advisors has proposed to replace the child benefits system with a paid parental leave system, in which eligibility for a transfer is conditioned on the duration of parental leave, denoted by ai > a. Reformulate the government problem in light of the suggested reform (notice that there are two incentive compatibility constraints!).

7) Draw the government optimization problem in the duration-transfer plane, where the duration (ai) is on the horizontal axis and the transfer (T) is on the vertical axis. Show, graphically, that the social optimum is obtained as a solution for a system of two equations (solved for two unknowns).

8) Prove that the paid parental leave system is cost-saving when 2 is sufficiently small. Explain the economic rationale. (Hint: consider the limiting case of 2=0 and apply continuity. Explain the trade-off b/w universal and selective systems in light of our discussion of workfare versus welfare).

Remark: Intermediate Public Economics, Jean Hindricds and Gareth Myles, MIT Press, 2006 (Main Textbook)

Guidline to the Assignment:

In assignment you will examine the potential role of using demographic criteria to determine welfare-transfers eligibility as an efficient screening device. You will provide a normative justification, dwelling on re-distributive considerations, for the use of common policy tools such as child benefits and mandated paid parental leave.

There are two households in the economy that differ in their earning abilities (assumed unobserved by the government). Both households share the same preferences and derive utility from consumption and leisure (caring of children). Each household is faced with a family-career dilemma: whether or not to have kids. Nurturing is time consuming and entails an opportunity cost (measured in forgone consumption terms) which differs b/w the two households.

In part 1 you are required to formulate and solve the household's problem: a choice b/w career (working full time) and family (having a single child and allocating a fraction alpha of the time to nurturing). You will derive the household's demand for children as a function of its earning ability,w.

In part 2 we introduce the government. We assume that the government is seeking to guarantee a minimal level of utility for all households at the minimal cost. There is a well-defined screening problem, as the low-skilled household is the only one falling below the poverty threshold, by presumption. Ideally the government would offer a lump-sum transfer only to the low skilled household. However, such a policy is infeasible due to asymmetric information. In part 2 we consider one option to overcome the screening problem - offering a universal lump-sum transfer, namely an identical transfer to both households. This is very similar to the benchmark case in the Workfare Model. You should formulate the government problem and derive the optimal level of transfer (the minimum necessary to bring the low-skilled household to the poverty threshold).

The universal system is clearly wasteful - the government offers a transfer to the high-skilled household that does not need it. In part 3 you are asked to discuss the merits of a potentially cheaper alternative: offering child benefits - a transfer which is conditioned on having kids. Provide an informal explanation for the rationale underlying the suggested reform (think about the variation in the costs of nurturing b/w the two households).

In part 4 you are asked to characterize the deadweight loss of the child benefits system. Conditioning the transfer on the family status (with or w/o children) may induce the household to change its choice (favoring family over career). You can verify that, given the parametric assumptions, the low-skilled household would choose to have children even in the absence of child benefits. The potential source of distortion would therefore come from the high-skilled household that is choosing to have no kids in the benchmark case but may change its choice when child benefits are sufficiently generous (a substitution effect!). You should demonstrate that there exists a threshold level of transfer above which the high-skilled household will change its choice and decide to have children. Use the standard methodology to calculate the deadweight loss - measuring the difference in the costs entailed by the government b/w the child benefits system and a (calibrated) lump sum transfer regime (which by definition would be cheaper). You should focus on the high-skilled household only. There is no substitution effect (and hence no deadweight loss) associated with the low-skilled household's choice in the presence of child benefits (regardless of how generous the transfer is). Explain it! To facilitate the calculations, you can use the following parameters: b=1, w_low=1, w_high=3, alpha=0.5. Show that the threshold level of transfer is given by T=0.5.

In part 5 you should demonstrate that when the poverty threshold is low enough (and hence the level of transfer required to attain the poverty threshold is relatively low) then the suggested reform will be cost saving. You should demonstrate that in this case the child benefits system attains the first best allocation (a lump sum transfer offered exclusively to the low-skilled household). Show that when u_hat is lower than 2 the child benefits system attains the first best allocation and is clearly cost-saving.

Now, suppose that u_hat is higher than 2. Verify first that given this parametric assumption there will be no difference b/w the child benefits system and the universal lump-sum regime (why?). We consider next, replacing the child benefits system (which is inefficient) with a mandated paid parental leave system: in order to be eligible for a transfer, a parent has to stay at home with the kids longer than alpha. You are asked to formulate the government problem. This problem is similar to the constrained minimization problem associated with the workfare program (in lecture 5). There are three constraints: (i) a poverty threshold condition (ensuring that the low-skilled household will meet the criterion); (ii) an incentive compatibility condition for the high skilled household inducing it to opt out of the system; and, (iii) an incentive compatibility condition for the low-skilled household that is induced to claim for the paid parental leave.

You can verify that the third constraint is redundant (it is implied by the first constraint).

In part 7 you are asked to provide a graphical illustration (similar to the figure in lecture 5). You should draw the two relevant constraints and the government's indifference curves, emphasize the feasible range (satisfying both constraints) and demonstrate graphically that the optimal solution is obtained when both constraints are binding.

Now you can solve for alpha_hat (the duration of parental leave) and T (the level of transfer) obtained as a solution for a system of two equations (the binding constraints) solved for two unknowns. Compare the parental leave regime with the universal lump sum system and show that the parental leave system is the socially preferred one (less costly to implement). Provide an informal explanation for the result. Discuss (informally) the case in which w_low would be relatively small and explain why the parental leave system would be socially desirable in this case (follow the hint!).