Optimal Unemployment Insurance over the Business Cycle

In this paper I study the optimal provision of unemployment insurance over the business cycle. A novel feature of the paper is that, instead of performing the analysis in an environment with exogenous restrictions to risk sharing such as borrowing constraints, the paper takes a more primitive mechanism design approach. In particular, the restrictions to risk sharing arise endogenously as a consequence of search intensities being private information of the individual agents. The economy is essentially a standard real business cycle model that incorporates unemployed agents similar to those in Hopenhayn and Nicolini (1997). Output, which can be consumed or invested, is produced with a Cobb-Douglas production function that uses capital and labor subject to an aggregate productivity shock that follows an AR(1) process. This production technology is located in a single "production island" and workers must be located in this island in order to provide their labor services. Workers get separated from the production island at the beginning of the following period with a constant separation probability. Once outside the production island agents must search in order to get back to it. The probability that a worker arrives to the production island at the beginning of the following period depends on her own search intensity level. A crucial assumption is that this search intensity level is private information of the agent. Only the location of the agent at the beginning of the period (inside or outside the production island) is observed. Agents value consumption and leisure (which is obtained outside the production island) and dislike to search. In order to guarantee stationarity I assume that agents have a constant probability of surviving between consecutive time periods. When an agent dies he is immediately replaced by an offspring from which the agent derives no utility. Newborns start their life as unemployed agents (i.e. outside the production island). In this framework the social planner offers dynamic insurance contracts to the agents under full commitment. The state of the contract is given by the location of the agent at the beginning of the period and by the value that the contract promises to the agent at the beginning of the period. Given this state, the contract determines the consumption level of the agent during the current period and the contingent promised values at the beginning of the following period. These contingent promised values depend both on the realized employment status of the agent at the beginning of the following period and on the realized aggregate productivity shock at the beginning of the following period. The contract must deliver an expected lifetime discounted utility equal to the value promised at the beginning of the period (promise keeping constraint). Although search intensities are not directly observed by the social planner he takes as given the optimal choice of individual search intensities of unemployed agents as a function of the difference that they face between the expected value of becoming employed at the beginning of the following period and the expected value of continuing unemployed (incentive compatibility constraint). The state of the economy for the social planner is the aggregate productivity level, the aggregate stock of capital, and the joint distribution of old agents (i.e. those that are not newborns at the beginning of the current period) across promised values and employment states. Given this aggregate state the social planner chooses investment and the dynamic insurance contracts to maximize the weighted expected lifetime utility of the current newborns and of all future newborns (with constant relative Pareto weights), subject to promise keeping, incentive compatibility and aggregate consumption feasibility constraints. Observe that the social planner does not seek to maximize the lifetime utilities of the current old agents since these are predetermined by their dynamic insurance contracts. Computing a solution to this mechanism design problem is a complex task given the high dimensionality of the state space. I use a method that I introduced in a previous paper (Veracierto 2017) which has the important advantage of not imposing an approximation to the law of motion for the distribution of agents across individual states. The method requires carrying as a state variable a long history of spline coefficients for the decision rules that have been chosen in the past. A (large) linear rational expectations model is then obtained by linearizing all first order conditions and aggregate feasibility constraints with respect to those spline coefficients. In that paper the method was shown to reproduce some key analytical properties that could be derived in the case of logarithmic preferences, even though the computational method did not exploit any particular feature of that case. While this provided considerable confidence about the accuracy of the computational method, the logarithmic preferences corresponded to a case in which the cross sectional heterogeneity did not play an important role in aggregate fluctuations. For other preferences, the computational method still delivered numerical solutions in which the cross sectional heterogeneity did not play a crucial role. Given those results the computational method had to wait to be applied to an environment in which the cross sectional heterogeneity plays an important role for aggregate fluctuations. I expect that the model in this paper will provide such environment. The reason is that when a positive aggregate productivity shock hits the economy and the planner wants to bring people quickly out of unemployment, the only way that he can induce individual agents to increase their search intensity is by increasing the difference between the expected value of becoming employed and the expected value of continuing unemployed. That is, the planner needs to worsen the insurance that it provides to unemployed agents in order to induce them to search more. This will introduce interesting interactions between social insurance (and, therefore, inequality) and properties of the aggregate business cycle since the social planner will consequently be induced to respond less to the aggregate shocks given the negative insurance effects that such response entails. I am currently in the process of computing a solution to the optimal business cycle fluctuations. I will provide a draft of the paper with the results as soon as I have them ready.