2. {32 points) There are two agents in the economy, agent 1 and agent 2. Both agents can earn a wage of $24 per hour. However, they have different preferences over consumption and hours worked : 1 chg] wish) = $111M + ln\" h) mm) = W Note that we have dened the agents' utility functions in terms of hours worked rather than leisure. Both agents can work a maximum of T = 24 hours per day. There is a government which would like to maximize the sum of the two agents' utilities: SU : \"1('31 h) + 32(61):) (a) [8 points) Show that if each agent works the privately optimal amount and consumes her own income, they will choose to work the same number of hours. (b) [8 points) Compare the two agents' marginal utilities of consumption and marginal disutilities of work in the privately optimal allocation. Would the government want to make any transfers between the two agents or adjust their hours of work in the rstbest allocation? Justify your answer. You do not need to explicitly solve for the rstbest. For the rest of this problem, assume that agent 2's utility function is instead given by 1 g(,h)= E [c2Xh2]. (c) [4 points) What is agent 2's privately optimal hours worked? (d) [4 points} The government observes that agent 2 earns less income than agent 1, and con cludes that it would be socially optimal to transfer some of agent 1's income to agent 2. Is this correct? Justify your answer. Now suppose that the government wants to raise $100 in revenue to fund an extremely valu able public good, and can impose a lumpsum tax on each agent. Let t1 and t2 denote the lumpsum taxes on agents 1 and 2, respectively. Assume the government knows each agent's type and can require them to pay the lumpsum tax regardless of their behavior. However, each agent chooses her privately optimal hours worked given the tax. {e} (3 points) Howr many hours does each agent work as a function of her lump-sum tax? Mat is each agent's equivalent variation due to lump-sum taxes t1 and t2? (1") Bonus: (Only Solve if You Have Finished the Exam) What are the socially optimal lumpsum taxes t1 and t2? Assume that it is socially optimal to raise exactly $100 in revenue, so 1 + 2 = $100. Would the optiJIlal lumpsum transfers from satisfy agent 2's incentive compatibility constraint