Assume utility is given by U(c, c' ) = lnc + lnc' with (0, 1]. Household net income today is y t and household net income tomorrow is y' - t' . The household takes the interest rate r as given.

1. Write down the household problem in this two-period model.

2. Solve for optimal c and c' .

3. Now suppose the interest rate rises. Does the choice of c change? What does this imply about substitution and income effects?

4. Now assume that the discount factor and the interest rate r are such that = 1/(1 + r). Explain the relationship between current and future consumption at the optimal level.

5. Using your results from part 4, solve for the optimal choice of savings of the consumer. Suppose y t > y' - t' . Is savings positive, negative or zero? What happens to savings if

y t < y' - t' ? Or if y t = y' - t' .