Consumers live for two periods and have the following lifetime utility:

U = ln c1 + ln c2

Where c1 and c2 are consumption in the first and second period respectively and is the discount factor. Everyone has full-time contracts and cannot change their hours of work. They receive exogenous income Y1, profits 1 and pay lump sum taxes T1 in the first period and Y2, 2 and T2 respectively in the second period. Consumers can buy or sell any amount of bonds at the same interest rate r.

(i) Derive the expressions for the optimal level of consumption c1 and c2, and savings s in terms of the exogenous variables Y1, Y2, 1, 2, T1, T2, , and r. Show the optimal allocation in a graph.

(ii) Assume that there are two consumers in this economy, Ina and Paul. They have the same stream of exogenous income Y1 = 90, Y2 = 78, earn the same profits 1 = 20, 2 = 20, pay the same lump-sum taxes T1 = 10, T2 = 10 and face the same market interest rate r = 10%. However, they have a single difference as Ina is more patient than Paul. Ina has a = 0.8 and Paul has a = 0.4. Find Ina's and Paul consumption c1, c2, and savings s assuming that consumers cannot sell bonds (i.e. cannot borrow) but can buy bonds (i.e. can save) earning interest rate r = 10%. Explain why or why not they choose the same consumption and savings, and draw the corresponding graph.

(iii) Assume that the government wants to give a tax break of 5 units in the first period without changing the overall government spending. Given that the government needs to balance the lifetime budget over the two periods, find the new T1 and T2.

(iv) Find the optimal consumption c1, c2, and savings s for both consumers taking into account the new taxes. Discuss the results and draw the corresponding graphs.