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Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : Journal of Economic Theory, Volume 145, Issue 6, November 2010, Pages 2078–2107
This paper studies the design of unemployment insurance when neither the searching effort nor the savings of an unemployed agent can be monitored. If the principal could monitor the savings, the optimal policy would leave the agent savings-constrained. With a constant absolute risk-aversion (CARA) utility function, we obtain a closed form solution of the optimal contract. Under the optimal contract, the agent is neither saving nor borrowing constrained. Counter-intuitively, his consumption declines faster than implied by Hopenhayn and Nicolini (1997) . The efficient allocation can be implemented by an increasing benefit during unemployment and a constant tax during employment.
Unemployment insurance must balance the benefits of insurance against the concern that an over-generous program will discourage search effort. Card, Chetty and Weber  provide evidence that search effort varies according to an agent’s financial situation. They find that ex- ogenously richer agents take longer to find a job but do not find higher wages on their next job. In other words, it seems that wealth is one important determinant in the search effort decision. Since neither search effort nor wealth are readily observable to the planner, it is natural to wonder how these information frictions might affect the optimal unemployment system. We in-troduce hidden savings into an environment similar to the Hopenhayn and Nicolini  version of the model of Shavell and Weiss . We show that the addition of hidden savings leads to faster consumption declines during an unemployment spell than the declines in a model with observ- able savings. Moreover, with hidden savings, agents with relatively high initial insurance claims have the fastest rate of consumption decline, eventually having lower claims than those agents who started out receiving less. We show that the unemployment benefit rises over the course of the spell. Hidden savings is naturally relevant in repeated moral hazard models like this one. When savings can be monitored, as in Hopenhayn and Nicolini  or the repeated moral hazard model of Rogerson , the optimal policy leaves the agent savings-constrained: his marginal utility is lower today than tomorrow. By making an agent poor in the future, it encourages the agent to search harder for a job. Our paper is related to both Werning [5,6] and Abraham and Pavoni [7,8], who use the first-order approach to study models with hidden savings and borrowing. Briefly speaking, the first-order approach studies a relaxed problem, which replaces the incentive constraints in the original problem with some first-order conditions of the agent. In these papers, the first-order condition for the type that has never deviated in previous periods and thus has zero hidden wealth is imposed. Werning  acknowledges that imposing first-order conditions may not be sufficient to ensure incentive compatibility. Furthermore, Kocherlakota  shows that when the disutility function is linear, the agent’s problem is severely non-convex and the first-order condition cannot be sufficient. When the first-order approach is invalid, the number of state variables in a recursive formulation would be infinite, making even numerical computations intractable. We overcome this problem by focusing on a special case of constant absolute risk-aversion (CARA) utility from consumption and linear disutility of effort. In this case we conjecture and verify the countable set of constraints that bind. With this in hand, it is straightforward to explicitly solve for the principal’s optimum. It has the interesting feature that the incentive constraints of the searching agent never bind. Instead, the binding incentive constraint in any period is the one for the agent who has always shirked, and meanwhile saved. The basic intuition for this structure of binding incentive constraints relates to the way in which shirking and saving interact. When an agent shirks, he increases the odds of continuing to be unemployed. The unemployed state involves lower consumption, so he wants to save in preparation for the greater probability of this low outcome. Therefore, saving and shirking are complements. The agent who saves the most is the one who has always done maximum shirking, and who knew he would never become employed. Given that he has saved the most, he is best equipped to do additional shirking, which is, again, complementary with saving. This example shows, in two ways, the sense in which the first-order approach is not appropriate for this kind of problem: first, the complementarity between shirking and saving can make the first-order condition for effort insufficient for optimality, and second, since the binding incentive constraint is not for the agent who always searches, it is not enough to look at the always-searching agent’s optimality condition in the first place. The contract we study always implements, at an optimum, a one-time lottery over always searching as hard as possible, or always searching the least, which we refer to as not searching. The interesting case is when the agent searches. The reason for keeping the agent on the Euler equation in this case relates to the savings-constrained nature of the optimal contract in Hopen- hayn and Nicolini . There, making the agent poor in the future generates incentives to search today. When the agent can freely save, it is no longer possible to keep the marginal utility of consumption high tomorrow, but there is still no reason to make it fall over time. In that case, the same logic as in the case with observable savings would suggest lowering today’s marginal utility and raising tomorrow’s in order to increase search incentives today and make the contract deliver utility more efficiently. As a result, the agent is on the Euler equation. As a corollary, our solution would also be the solution to a case where the agent had access to both hidden borrowing and hidden savings. In the state-of-the-art model of optimal unemployment insurance in Shimer and Werning , the focus is on the wage-draw aspect of the Shavell and Weiss  structure. 1 The fundamental trade-off is between search time and quality of match (in terms of wages). They find that the op- timal contract has constant benefits (for CARA utility, and approximately so for constant relative risk-aversion (CRRA) utility), and keeps the agent on the Euler equation. The reason why their agents are not borrowing constrained is different from ours, however. In the CARA model, the agent’s reservation wage is independent of wealth; therefore, there is no incentive benefit from distorting the first-order condition, and there is the usual resource cost to the principal from the distortion. In that sense their model has no “wealth effects” of the kind emphasized by Card, Chetty, and Weber . Our model, on the other hand, does have wealth effects; richer agents have less incentive to give effort than poorer agents. Hopenhayn and Nicolini  show that there is an incentive benefit in making the agent savings-constrained; in our model such a constraint is impossible, so the optimal contract moves to the Euler equation. The case where search effort is a monetary cost and utility is CARA is studied in Werning . Monetary search cost implies that poor agents have no greater incentives than rich agents, since the cost and benefit of search are both proportional to wealth. The main insight of this model is similar to Shimer and Werning : agents are not borrowing constrained and a constant benefit sequence is needed to implement the optimal allocation. Our results contrast with Kocherlakota , who studies a similar model with linear disutility of effort, but assumes that the principal aims at implementing interior effort. He finds that agents are borrowing constrained. In contrast, we show that the optimal contract always implements corner solutions for effort. The optimal contract does not leave the agents borrowing constrained, even if doing so is feasible for the principal. The first-order approach remains a practical and useful method to solve hidden-savings prob- lems when the disutility function is sufficiently convex. In particular, Abraham and Pavoni  numerically verify the incentive compatibility of the solution obtained with the first-order ap- proach for a large range of convex disutility functions. Abraham and Pavoni  show analytically, in a two-period model, that the first-order approach is valid if the utility function has the non- increasing absolute risk aversion property, and the job-finding probability (as a function of effort) satisfies some concavity condition. Williams  provides sufficient conditions for the first-order approach based on the Hamiltonian in a continuous-time model. The paper proceeds as follows. In the next section we introduce the basic model. In Section 3 we show that the optimal contract either implements high effort forever or no effort forever. In Section 4 we solve for the optimal contract to implement high effort. In Section 5 we show the important characteristics of the optimal contract, and compare them to the case without hidden savings. We then conclude. All proofs are contained in Appendix A.