Q2. In one economy there are two workers: Amy and Bill. They are living 1 period. They both consume all their income immediately and they have the following utility functions

U_A = 2C^0.5+4L^0.5 and U_B = 2C^0.5+2L^0.5

L represents hours of leisure and C is the amount of consumption. Suppose that they both work maximum 50 hours in a period. The market wage is 10 and prices are fixed at 1.

1. Find the budget constraint for the two workers
2. Find the optimal amount of consumption, leisure and hours worked for Amy and Bill
3. Find the individual supply function for A and B (meaning N as a function of the wage).

Tip: instead of putting a number for the wage, both in the budget constraint and in the condition for marginal costs and benefits, let it be an unknown variable w. Then, find L as a function of that w for both A and B. Remember that w>0, so, if necessary, don't be afraid to divide your equation by w! Finally, remember that hours worked are the maximum hours minus the hours of leisure.

1. Suppose that government introduce a labour tax of 20% on labour income. Find the new optimal choice for Amy and Bill.
2. Suppose now that the government gives back the total amount of taxes collected, equally shared between the two workers. The workers spend immediately everything in consumption (they have already made their work decision; they cannot increase leisure). Compare workers in point b and now. Are they better off? Worse off? Same?

Tip: to understand if an agent is better off you need to compare their previous level of utility (substituting C and L in the utility function) with the current one (in which C is increased by the government transfer).