With probability p(0,1), an unemployed agent draws a wage offer w from the set W={wL,wH}, where wH>wL>0. The wage offer can be wH or wL with equal probability. The agent becomes employed if the wage offer is accepted. With probability 1p, the agent does not get an offer and remains unemployed. The agent's utility is u(c), where c denotes the consumption level. Assume that u is continuously differentiable, strictly increasing, and strictly concave. Furthermore, suppose that there is a disutility from working v>0, which is constant across all jobs. The government designs an unemployment policy that maximizes the agent's expected utility. If the agent accepts a job, then the wage is taxed by (w), so an employed agent consumes w(w). Notice that the tax can vary with wage. If the agent remains unemployed (either because the offer was rejected or the agent did not receive an offer), then the agent consumes unemployment benefit b0. The policy has to satisfy the following budget constraint: p[2(wL)+2(wH)]=(1p)b. a. Suppose the government can observe job offers. Solve for the optimal tax(w) and unemployment benefit b for the problem in Part a. (4 points) b. Provide a brief explanation of the consumption levels that you found for the unemployed and employed in Part a. (5 points) c. Suppose the government does not observe any job offers. Explain why the policy you found in Part a. is no longer feasible. (4 points) d. Suppose the government does not observe rejected job offers, but can observe wage offers if the job was accepted. Provide constraints on policies such that the agent would accept all job offers. ( 2 points) e. For the setting in Part d., is it optimal to fully insure against unemployment risk? Please explain. (5 points) f. For the setting in Part d., is it optimal to fully insure against wage risk so that all employed agents consume the same amount? Please explain. (5 points) With probability p(0,1), an unemployed agent draws a wage offer w from the set W={wL,wH}, where wH>wL>0. The wage offer can be wH or wL with equal probability. The agent becomes employed if the wage offer is accepted. With probability 1p, the agent does not get an offer and remains unemployed. The agent's utility is u(c), where c denotes the consumption level. Assume that u is continuously differentiable, strictly increasing, and strictly concave. Furthermore, suppose that there is a disutility from working v>0, which is constant across all jobs. The government designs an unemployment policy that maximizes the agent's expected utility. If the agent accepts a job, then the wage is taxed by (w), so an employed agent consumes w(w). Notice that the tax can vary with wage. If the agent remains unemployed (either because the offer was rejected or the agent did not receive an offer), then the agent consumes unemployment benefit b0. The policy has to satisfy the following budget constraint: p[2(wL)+2(wH)]=(1p)b. a. Suppose the government can observe job offers. Solve for the optimal tax(w) and unemployment benefit b for the problem in Part a. (4 points) b. Provide a brief explanation of the consumption levels that you found for the unemployed and employed in Part a. (5 points) c. Suppose the government does not observe any job offers. Explain why the policy you found in Part a. is no longer feasible. (4 points) d. Suppose the government does not observe rejected job offers, but can observe wage offers if the job was accepted. Provide constraints on policies such that the agent would accept all job offers. ( 2 points) e. For the setting in Part d., is it optimal to fully insure against unemployment risk? Please explain. (5 points) f. For the setting in Part d., is it optimal to fully insure against wage risk so that all employed agents consume the same amount? Please explain. (5 points)
