A consumer’s income in the current period is y = 250, and her income in the future period is y′ = 300. The real interest rate r is 0.05, or 5%, per period. Assume there are no taxes. Suppose that the utility function is
U(c, c′) = √c + β√c′
and let β = 0.9
(a) Determine the consumer’s lifetime wealth we (present discounted value of life-
time income). Write down the lifetime budget constraint.
(b) Derive the Marginal Rate of Substitution between c and c′.
(c) Write down the consumer maximization problem. What is the objective function? What are exogenous variables? What are endogenous (choice) variables?
(d) Solve for c′ as a function of c using the lifetime budget constraint. Replace it into the objective function so that the problem only depends on c.
(e) Take the first order condition with respect to c and use it to and c∗ and c∗′ .
(f) Is this agent a borrower or a saver? Compute her optimal savings/borrowings
using the computations above and her budget constraint.