Help me solve this, please.

Consumers live for two periods and have the following lifetime utility: U=lncl+lncz (1) Where cl and 62 are consumption in the rst and second period respectively and ,8 is the discount factor. Everyone has full-time contracts and cannot change their hours of work. They receive exogenous income Y1, prots 771 and pay lump sum taxes T1 in the rst period and Y2, 1T2 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 01 and (:2, and savings 8 in terms of the exogenous variables Y1, Y2, 771, 71-2, T1, T2, {3, and T. 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 Y1 = 90, Yg = 78, earn the same prots 7n = 20, 172 = 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 ,6 = 0.8 and Paul has a [3 = 0.4. Find Ina's and Paul consumption cl, Cg, and savings 3 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 rst period without changing the overall government spending. Given that the government needs to balance the lifetime budget over the two periods, nd the new T1 and T2. (5 marks) (iv) Find the optimal consumption Cl, 02, and savings 8 for both consumers taking into account the new taxes. Discuss the results and draw the corresponding graphs. (15 marks) Total: 40 marks