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 dierent preferences over consumption and hours

worked:

u1 (c; h ) =

1/2

ln(c ) +

1/2

ln(T- h ) u2 (c; h ) =

1/24²

(c-h²)

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 = 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 rst-best allocation? Justify your

answer. You do notneed to explicitly solve for the rst-best.

For the rest of this problem, assume that agent 2's utility function is instead given by

u2 (c; h ) =

1/24²

(c -2 h²⁾:

(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.