Consider a two-period economy where consumers 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. 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. They buy or sell any amount of bonds at the same interest rate r.

a)Derive the expressions for the optimal level of consumption c₁ and c₂, and savings s. Illustrate the solution with a graph.

b) Assume that there are two consumers in this economy, Maria and Paul. They receive the same lifetime income with Y₁ = 70, Y₂ = 100, earn the same profits ₁ = 20, ₂ = 20 and pay the same lump-sum taxes T₁ = 10, T₂ = 10 in each period respectively. Maria is patient and trustworthy and has = 0.8 and can lend and borrow at an interest rate r = 10%. On the other hand Paul is inpatient and untrustworthy and has = 0.7565 and can borrow and lend at an interest rate r = 15%.

Find Maria's and Paul's consumption c₁, c₂, and savings s assuming that can buy and sell bonds freely. Illustrate the solution with a graph and explain.