Consider an economy where individuals live for two periods only. They have the utility function over consumption in period 1 (C₁) and period 2 (C₂) given by U = 2 log(C₁) + 2 log(C₂). The labor income of each individual in period 1 is fixed at $10, and there is no labor income in period 2. They can save as much of their income in period 1 as they like in bank accounts, earning interest rates of 200 percent per period (recall, a period is the entire active life). The income tax rate is 50 percent, which is used to pay back the public debt inherited from the past generation.

a. Derive the optimal lifetime consumption profile of this consumer. What would be the consumption profile without income tax?

b. Suppose that a ‘‘retirement saving program’’ is introduced allowing each consumer to save up to 20 percent in the first period in a tax-free account. Compare the lifetime budget constraints with and without the retirement savings program.

c. Derive the optimal lifetime consumption profile with the retirement savings program. Explain the impact of this program on private savings.

d. Now suppose that the retirement savings program in part b is replaced by a new savings program taxing investment income on the first 50 percent of savings and exempting any savings in excess of 50 percent from taxation. Draw the budget set associated with this program, and find the optimal lifetime consumption profile. Explain the difference with the program in part b.

e. If the threshold for tax-exempt savings in part b is increased from 50 to 51 percent, how would this affect private savings? How does this affect total savings in society?