مشخص کردن اثر پس انداز های مخفی در بیمه بیکاری بهینه

In this paper, I consider the problem of optimal unemployment insurance in a world in which the unemployed agent's job-finding effort is unobservable and his level of savings is unobservable. I show that the first-order approach is not always valid for this problem, and I argue that the available recursive procedures are not currently computationally feasible. Nonetheless, for the case in which the disutility of effort is linear, I am able to provide a complete characterization of the optimal contract: the agent's consumption is constant while he is unemployed, and jumps up to a higher constant and history-independent level of consumption when he finds a job.

In a recent paper, Hopenhayn and Nicolini (1997) study the properties of an optimal insurance arrangement between a risk-neutral insurer (principal) and a risk-averse worker (agent). They assume that the agent begins life unemployed and expends a hidden amount of effort to find a job in each period. His probability of finding a job is increasing in the amount of effort exerted; once he finds a job, he keeps it forever. Importantly, the insurer has complete control over the agent’ s consumption, because the agent cannot secretly transfer consumption from one period to the next. They find that in an optimal contract between the principal and the agent, the agent’s consumption is a decreasing function of his time spent unemployed. This general result has two consequences. First, an agent who has been unemployed for t periods has a lower consumption than an agent who has been unemployed for (t − 1 ) periods. Second, an agent who finds a job after a long period of unemployment must make a higher payment to the insurer than an agent who finds a job after a short period of unemployment. As stated above, Hopenhayn and Nicolini a ssume that the principal can costlessly monitor the agent’s savings and condition contractual payments on this variable. One can show that the optimal contract in Hopenhayn and Nicolini’s setting has the property that the agent is savings-constrained when unemployed: the agent’s shadow interest rate is lower than the principal’s shadow interest rate. Nor is this feature of the Hopenhayn–Nicolini contract unique to the unemployment insurance problem. Rogerson (1985a) shows that in settings with repeated moral hazard, it is gene rally optimal to impose a sufficiently severe punishment for poor output performance that the a gent ends up being savings-constrained. Intuitively, the agent would like to save so as to mitigate next period’s punishment. 1 It follows that with moral hazard, the op timal dynamic contract is only incentive- compatible under the assumption that the principal is able to costlessly monitor the agent’s asset levels. This assumption is somewhat restrictive. After all, there are a number of ways that a person can transfer resources to the future (like foreign bank accounts or by accumulating durables) that may be hard for out siders to observe. It is therefore important to understand the intertemporal structure of optimal contracts when the agent is allowed to engage in secret asset accumulation. This paper is a contribution to this general research agenda. I relax the assumption that savings can be monitored by the principal in the Hopenhayn–Nicolini unemployment insurance model, and assume instead that the a gent can secretly save at the same rate as the principal. I then look to solve for the optimal insurance contract. 2 Not surprisingly, this problem is generally impossible to solve analytically. Unfortunately, it is also difficult to solve numerically. In a recent paper, Fernandes and Phelan (2000) have described a recursive formulation for a related class of problems. It is not known, though, how to translate their recursive formulation into a practical computational procedure when savings can take on a continuum of values. Werning (2002) and Abraham and Pavoni (2003) attack the problem by using a computationally feas ible first-order appr oach that replaces the agent’s incentive constraint s with the corresponding first order conditions. However, I show that even in simple examples, the first-order approach may not be valid because the agent’s decision problem is intrinsically non-concave in effort and savings. It is possible, though, to obtain an analytical solution in a particular case, even when the first-order approach is known to be invalid. I assume that the agent’s disutility from effort is linear in the probability of his finding a job, and that the principal wants the agent to exert an interior amount of effort while unemployed. Under these assumptions,I prove that the optimal unemployment insurance contract takes an extremely simple form. During the period that an agent is unemployed, his consumption is constant. When he becomes employed, his consumption jumps up to a new constant level that is independent of the duration of the unemployment spell. This structure implies that once the agent’s savings level is unobservable, it is optimal for the agent to be borrowing-constrained when unemployed. The intuition behind this res ult is as follows. The contract has to be designed to punish the agent as severely as possible, given that it must deter the agent from saving. This intuition would seem to lead to the optimal contract’s featuring consumption-smoothing, so that the principal and agent have the same shadow interest rate. 3 However, the very fact that the first-order approach fails is a sign that thi s intuition is wrong. The binding intertemporal incentive constraint is one in which the agent jointly deviates from the optimal contract by simultaneously saving more and working less. When the contract is designed to prevent this joint deviation, the agent ends up being borrowing-constrained given that he does work the amount specified by the contract. In this paper, I assume that the unemployed agent cannot borrow secretly. I have two reasons for this restriction. The first is technical: in the linear disutility case, there are no incentive-compatible contracts (including repetition of any static contract) if agents can engage in both hidden borrowing and lending. The second is more substantive. It is much more difficult for individuals to engage in hi dden borrowing than hidden saving, because their loans have to be enforced. In contrast , as Cole and Kocherlakota (2001) explicitly model, hidden saving can take the form of physical investment. Physical investment requires no outside enforcement and so is intrinsically more difficult to monitor.

This paper considers the optimal provision of unemployment insurance for an agent who can secretly exert effort to find a job and who can secretly save. The paper argues that it is not practical to compute an approximate solution to the contracting problem using currently available recursive methods. As well, the first-order approach is not generally valid: the complementary nature of shirking and saving makes the agent’s problem non- concave. Despite these difficulties, it is possible to completely and analytically characterize the optimal contract when the agent’s disutility of effort is a linear function of his probability of finding a job. The paper uses this characterization to show that the nature of optimal unemployment insurance is cons iderably changed if the agent can engage in secret saving. In particular, the agent’s compensation when he is unemployed or when he gets a job is independent of his history, instead of depending in complicated ways on the duration of unemployment. As well, rather than being savings-constrai ned, the agent faces binding borrowing constraints at each date. It is natural to ask whether these findings are robust to introducing small amounts of curvature in v . I suspect that the exact history independence result will collapse—although my guess is that even in those cases, there will not be much loss in welfare in restricting the contract to be history independent. As well, I suspect too that the optimal contract will continue to leave the agent borrowing-constrained (which also means that the first-order approach will not work). The challenge that re mains is to develop robust and practical numerical methods to assess these, and other, conjectures. The continuous-time approach of Williams (2003) may be a promising step in this direction.