Intermediate Macroeconomics Theory Question

There are two friends, Amanda and Beth. Amanda's preferences over current and future
consumption are given by the utility function UA(c; c') = ln(c) + (Beta)A ln(c'), where (BetaA) = :99.
Beth's preferences are given by UB(c; c') = ln(c) + (BetaB) ln(c0), where (Beta)B = :9. That is, Beth
is less patient than Amanda. Both Amanda and Beth have the same after-tax incomes in
the current and future periods, y - t = 11,000 and y' - t' = 10,710.


(a) Suppose that the interest rate is 2%. Write down budget constraints in
period 1, period 2, and the lifetime budget constraints for Amanda and Beth. Write
down their utility maximization problems.


(b)How much does each of them consume in the two periods? How much does
each of them save/borrow in the first period? Compare your answers for Amanda and
Beth and explain differences intuitively.




(c) Now suppose that the interest rate is 5% instead. Compute lifetime wealth
for Amanda and Beth in this case. Does it increase or decrease? How much would each of them consume in each period in this case? How much is their saving/borrowing in the first period now? Do they consume more or less in the two periods compared to before? Do they save/borrow more or less than before? What do your answers imply regarding whether an income or a substitution effect dominates for Amanda/Beth?


(d) Is Amanda better or worse off after the interest rate increase? What about
Beth?


(e) Suppose the interest rate is still 2%, and suppose that both Amanda
and Beth expect to get a tax reduction in the first period, which increases their first period's after-tax incomes by 2,000. Compute the new levels of current and future consumption, as well as saving/borrowing for Amanda and Beth.


(f) Suppose that with the new after-tax income levels, the interest rate in-
creases. Without performing calculations, can you tell whether Amanda and Beth are
better or worse o because of the interest rate increase?