Question

Consider a model in which individuals live for two periods (1 and 2) and have utility

functions of the form U = ln(C1) + ln(C2). They earn income of $100 in the first period and save S to finance consumption in the second period. The interest rate, r, is 10%.
a. Set up the individual’s lifetime utility maximization problem. Solve for the optimal C1, C2, and S. Draw a graph showing the opportunity set.
b. The government imposes a 20% tax on labor income. Solve for the new optimal levels of C1, C2, and S. Explain any differences between the new level of savings and the level in part a, paying attention to any income and substitution effects.
c. Instead of the labor income tax, the government imposes a 20% tax on interest income. Solve for the new optimal levels of C1, C2, and S. Explain any differences between the new level of savings and the level in a, paying attention to any income and substitution effects.

Answer 1

To solve for the optimal levels of consumption and savings, one must maximize the given utility function subject to the individual's budget constraint. The impact of taxes on labor and interest income affects the optimal levels differently, demonstrating how each type of tax influences intertemporal choices through the substitution and income effects.

Step-by-step explanation:

Lifetime Utility Maximization Problem

In the model described, an individual's lifetime utility function is given by U = ln(C1) + ln(C2), where C1 and C2 represent consumption in periods 1 and 2, respectively. With an income of $100 in the first period and a savings rate of S, the second period's consumption is given by C2 = S(1+r), where r is the interest rate of 10%. The optimal values of C1, C2, and S are determined by maximizing U subject to the budget constraint C1 + S = $100 and C2 = S(1+0.10). The result will provide the levels of consumption and savings that maximize utility over the two periods. A graph of the opportunity set would show the trade-off between present and future consumption.

Impact of Taxes on Savings and Consumption

With a 20% tax on labor income, the income in the first period is reduced to $80 (after tax), which impacts the individual's budget constraint. The new optimal levels of C1, C2, and S will reflect the substitution and income effects. Similarly, a 20% tax on interest income affects only the returns on savings and thus the future consumption without changing the first period's budget constraint. The comparison of new levels of savings and consumption in each scenario will show how taxes influence intertemporal choices.