Having a two-period economy where consumers have the following lifetime utility:

U =lnc1 +lnc2 (3)

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.

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

2)Assume that there are two consumers in this economy, Gabriel and Charles. 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. Charles is inpatient and untrustworthy and has = 0.7565 and can borrow and lend at an interest rate r = 15%. Gabriel is patient and trustworthy and has = 0.8 and can lend and borrow at an interest rate r = 10%. Find Gabriel's and Charles consumption c1, c2, and savings assuming that they can buy and sell bonds freely. Illustrate the solution with a graph + explain