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. Assume that h = 16

The firm produces output Y using capital K and labor N according to

Y = zK^(a)N^(1a)

where z is the total factor productivity and a is the Cobb-Douglas parameter. The firm maximizes profits which are then transferred to the representative consumer. The government balances the budget using lump-sum taxes T on the representative consumer to finance government spending G. The hourly wage in this economy is w and the consumer has h hours to divide between leisure and labor.

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

(ii) 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.