Question 1 Consumers live for two periods and have the following lifetime utility: U = Inc, +8 Inc2 (1) Where c, and 2 are consumption in the first and second period respectively ands is the discount factor. Everyone has full-time contracts and cannot change their hours of work. They receive exogenous income Y1, profits , and pay lump sum taxes 7, in the first period and Y2, 72 and 72 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 c, and (2, and savings s in terms of the exogenous variables Y], Y2, "], #2; ], T2; B , and r. Show the optimal allocation in a graph. (10 marks) (ii) Assume that there are two consumers in this economy, Ina and Paul. They have the same stream of exogenous income Y, = 90, Yz = 78, earn the same profits #1 = 20, #2 = 20, pay the same lump-sum taxes 7, = 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 a3 = 0.8 and Paul has a8 = 0.4. Find Ina's and Paul consumption G, 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 corre- sponding graph. (10 marks) (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 , and T2- (5 marks) (iv) Find the optimal consumption G, C2, and savings s for both consumers taking into account the new taxes. Discuss the results and draw the corresponding graphs. (15 marks)