This exercise reviews some concepts from earlier chapters on consumer theory in preparation for exercise 19.5.

A. Consider an individual saver who earns income now but does not expect to earn income in a future period for which he must save.

(a) Draw a consumer diagram with current consumption c₁ on the horizontal axis and future consumption c₂ on the vertical. Illustrate an inter temporal budget constraint assuming an interest rate r —then draw an indifference curve that contains the optimal bundle A.

(b) Now suppose the interest rate increases to r ′. Illustrate the new budget constraint and indicate where the new optimal bundle C will lie given that the individual does not change his savings decision when interest rates change.

(c) How much, in terms of future dollars, would this person be willing to pay to get the interest rate to change from r to r ′? If he pays that amount, will he end up saving more or less?

(d) Suppose instead that the interest rate starts at r ′ and then falls to r . Illustrate how much I would have to give this individual to compensate him for the drop in the interest rate. If this is done, will he save more or less than he did at the high interest rate?

(e) On a new graph, illustrate the individual’s inelastic savings supply curve. Then illustrate the compensated savings supply curves that correspond to the utility levels the individual has at the interest rates r and r ′.

(f) True or False: Compensated savings supply curves always slope up.

B. Suppose your tastes over current consumption c₁ and future consumption c₂ can be modeled through the utility function u(c₁,c₂) = c₁^α  c₂^(1−α) , your current income is I and you will earn no income in the future. The real interest rate from this period to the future is r.

(a) Derive your demand functions c₁(r, I) and c₂(r, I ) for current and for future consumption.

(b) Define “savings” as the difference between current income and current consumption. Derive your savings—or capital supply—function k_s (r, I ). (Note: It turns out that this function is not actually a function of r .)

(c) Derive the indirect utility function V (r, I )—i.e. the function that gives us your utility for any combination of (r, I ).

(d) Next, derive your compensated demand functions c^c₁ (r,u) and c^c₂ (r,u) for current and future consumption.

(e) Define the expenditure function E(r,u) — i.e. the function that tells us the current income necessary for you to reach utility level u at interest rate r .

(f) Can you verify your answers by comparing V (r, I ) to E(r,u)?

(g) Finally, suppose that we begin with an interest rate r̅ and derive from it V (r̅ , I ). Define the compensated savings or compensated capital supply function as

(h) What is the interest rate elasticity of savings? Without deriving it precisely, can you tell whether the interest rate elasticity of compensated savings is positive or negative?

Transcribed Image Text:

k(r,7) = I – c (r, V(7, 1)).