Consider a model in which individuals live for two periods. There are two individuals, John and Jules, and both have utility functions of the form U = ln(C1) + ln(C2). John earns $100 in the first period and saves S to finance consumption in the second period. Jules will receive $110 in the second period, and she borrows B to finance consumption in the first period. The interest rate r is 10%.
a. Set up each individual’s lifetime utility maximization problem. Solve for the optimal C1, C2, and S (or B) for Jules and John.
b. The government now imposes a 20% tax on interest income. Solve for John’s new optimum level of S. Explain how your answer relates to the saving you found for John in a, paying attention to any income and substitution effects.
c. Suppose that the government also provides a 20% tax credit on interest, so if Jules borrows $10—and consequently owes $1 in interest—the government will give her $0.20 back. Solve for Jules’s now-optimum level of B. Explain how your answer related to the borrowing you found in a, paying attention to any income and substitution effects.