Questions 2 all parts

An unemployed agent is searching for work. The agent can either invest effort in searching or relax at home. The agent incurs a cost of h > 0 when searching for work, while the cost of relaxing at home is 0. The probability of finding work is p if the agent invests search effort. The probability of finding work is 0 if the agent simply relaxes at home. Once employed, the agent receives a pre-tax income of w > 0. The agent's utility is u(c), where c denotes the consumption level. Assume that u is continuously differentiable, strictly increasing, and strictly concave. The government designs an unemployment policy that maximizes the agent's ex- pected utility. If the agent finds a job, then the income is taxed, so an employed agent consumes w - T, where r is the tax. If the agent does not find a job, then the agent con- sumes unemployment benefit b 2 0, which has to be non-negative. The policy has to satisfy the following budget constraint (if the agent invests search effort): pt = (1 - p)b. a. What is the highest possible h such that it would still be efficient to search for a job? (2 points) The rest of the question will assume that h is below the upper bound found in Part a. b. Suppose the government can observe search effort. Solve for the optimal income tax T and unemployment benefit b. (4 points) c. Provide a brief explanation of the consumption levels that you found for the unem- ployed and employed in Part b. (4 points) d. Suppose the government does not observe search effort. Write down the incentive compatibility constraint. (2 points) e. Solve for the optimal income tax T and unemployment benefit b when p = 1. (4 points) f. Suppose p E (0,1) and let A denote the multiplier on the incentive compatibility constraint. Provide an expression for the optimal risk sharing rule 10 in terms of A and p. (4 points) g. Provide a brief explanation of your result in Part f. (5 points)