Morty lives for two periods and has the following utility function: U(C1, C2) = (1 + r)
Question:
Morty lives for two periods and has the following utility function: U(C1, C2) = (1 + r) × ln C1 + ln C2 where C1 and C2 are his consumption in period 1 and period 2, respectively, and r is a parameter that you should take as given. All we know is that r > 0. Morty has a labor income of $100 in period 1 and no labor income in period 2. If he consumes less than $100 in period 1, he can save the remainder in the bank and earn some interest. The interest rate is also r.
(a) (10 points) Write down Morty’s constrained utility maximization problem. How much will he choose to consume in period 1 and period 2, respectively? Denote your solution as C ∗ 1 and C ∗ 2 . Be extra careful with your algebra.
(b) (6 points) Answer the following questions: • Look at your budget constraint. What is the price (i.e., opportunity cost) of C1? Which has a higher price, C1 or C2? • Look at the utility function. Which does Morty value more, C1 or C2? • Now look at the values of C ∗ 1 and C ∗ 2 that you found in (a). Which is larger?
(c) (10 points) Now the government imposes a 10% tax on people’s interest income. That is, only the interest Morty earns on his savings is taxed, the principal is NOT taxed. Solve for the new optimal levels of consumption, C ∗∗ 1 and C ∗∗ 2 . What are the substitution effect (SE) and the income effect (IE) of the tax on C1, respectively? Which effect dominates?
