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, Sara and Jack. 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 Sara is more patient than Jack. Sara has a = 0.8 and Jack has a = 0.4. Find Sara's and Jack 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.