2. (32 points) There are two agents in the economy, agent 1 and agent 2. Both...

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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 1 u₁(c, h) = = ln(c) + = ln(T — h) u₂ (c, h) = 22 [c-h²] 24² Note that we have defined 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 = u₁(c, h) + u₂(c, h) (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 first-best allocation? Justify your answer. You do not need to explicitly solve for the first-best. For the rest of this problem, assume that agent 2's utility function is instead given by 1 24² u₂ (c, h) = × h²] . [c- - 2 x (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. 2 Now suppose that the government wants to raise $100 in revenue to fund an extremely valu- able public good, and can impose a lump-sum tax on each agent. Let t₁ and tå denote the lump-sum taxes on agents 1 and 2, respectively. Assume the government knows each agent's type and can require them to pay the lump-sum tax regardless of their behavior. However, each agent chooses her privately optimal hours worked given the tax. (e) (8 points) How many hours does each agent work as a function of her lump-sum tax? What is each agent's equivalent variation due to lump-sum taxes t₁ and t₂? (f) Bonus: (Only Solve if You Have Finished the Exam) What are the socially optimal lump-sum taxes t₁ and t₂? Assume that it is socially optimal to raise exactly $100 in revenue, so t₁ + t₂ = $100. Would the optimal lump-sum transfers from satisfy agent 2's incentive compatibility constraint? 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 1 u₁(c, h) = = ln(c) + = ln(T — h) u₂ (c, h) = 22 [c-h²] 24² Note that we have defined 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 = u₁(c, h) + u₂(c, h) (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 first-best allocation? Justify your answer. You do not need to explicitly solve for the first-best. For the rest of this problem, assume that agent 2's utility function is instead given by 1 24² u₂ (c, h) = × h²] . [c- - 2 x (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. 2 Now suppose that the government wants to raise $100 in revenue to fund an extremely valu- able public good, and can impose a lump-sum tax on each agent. Let t₁ and tå denote the lump-sum taxes on agents 1 and 2, respectively. Assume the government knows each agent's type and can require them to pay the lump-sum tax regardless of their behavior. However, each agent chooses her privately optimal hours worked given the tax. (e) (8 points) How many hours does each agent work as a function of her lump-sum tax? What is each agent's equivalent variation due to lump-sum taxes t₁ and t₂? (f) Bonus: (Only Solve if You Have Finished the Exam) What are the socially optimal lump-sum taxes t₁ and t₂? Assume that it is socially optimal to raise exactly $100 in revenue, so t₁ + t₂ = $100. Would the optimal lump-sum transfers from satisfy agent 2's incentive compatibility constraint?

Answer rating: 100% (QA)

ANSWER a Since each agents utility function is linear in hours worked and consumption the privately optimal hours worked for each agent is the point w... View the full answer