Question 3

1With probability p E ([1,1], an unemployed agent draws a wage offer to from the et W = {11.1, mH} , where am :3 mL 33' 0. The wage offer can be IUH or 1.21 with equal Irobability. The agent becomes employed if the wage offer is accepted. 1illith probability p, the agent does not get an offer and remains unemployed. The agent's utility is Lift], .rhere c denotes the consiunption level. Assume that It is continuously differentiable, trictly increasing, and strictly concave. Furthermore, suppose that there is a disutility rom working U 1} D, which is constant across all jobs. The goverrunent designs an unemployment policy that maximizes the agent's ex- Iected utility. If the agent accepts a job, then the wage is taxed by 11:13), so an employed gent consiunes w wa). Notice that the tax can vary with wage. If the agent remains .nemployed {either because the offer was rejected or the agent did not receive an offer}, men the agent consumes unemployment benefit I? 3 D. The policy has to satisfy the fol- Jwing budget constraint: P 1133\" + Tf'ZJHJ] : (1 _ p) f}. a. Suppose the government can observe job offers. Solve for the optimal tax 1' (to) and Imemployment benefit I? for the problem in Part a. {4 points) b. Provide a brief explanation of the consumption levels that you found for the unem- ployed 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]