Jane\'s utility function defined over consumption ci and c2 in two different per

Question

Jane's utility function defined over consumption ci and c2 in two different periods t 1,2 can be represented by Uc1c2)-C1C2. Her income is m and m2. The interest rate at which she can borro w 1Sr Find the Marshallian demand functions for ci and c2. Jane a borrower or a lender? How much does she borrow or lend per period? a) b) Find Jane's optimal consumption bundle if mi = 90, m2-110, and r = .1. Is c) Find Jane's optimal consumption bundle if mi = 90, m2-1 10, and r = .22. Is d) Find Jane's optimal consumption bundle if mi = 90, m12-110, and r= .375. Is e Show b) -d) graphically. Jane a borrower or a lender? How much does she borrow or lend per period'? she a borrower or a lender? How much does she borrow or lend per period? f)Is Jane better of as the interest rate increases? Explain.

Explanation / Answer

a) Intertemporal budget equation is c1 + c2/(1 + r) = m1 + m2/(1 + r).

(1 + r)c1 + c2 = m1(1 + r) + m2.

Here slope of the budget is -(1 + r). From the utility function we have |MRS| = c2/c1. Now at the optimum choice, MRS = slope of the budget equation or c2/c1 = (1 + r) or c2 = c1(1 + r). The new budget equation is

(1 + r)c1 + (1 + r)c1 = m1(1 + r) + m2.

2c1(1 + r) =  m1(1 + r) + m2

This gives c1* = (m1(1 + r) + m2)/2(1 + r) = (1/2)*(m1 + m2/(1 + r)) and c2* = (1 + r) x (1/2)*(m1 + m2/(1 + r))

c2* = (1/2)*(m1(1 + r) + m2)

b) For m1 = 90, m2 = 110 and r = 10%. we have c1 =  (1/2)*(90 + 110/(1 + 10%)) = 95 and c2 =  95(1 + 10%) = 104.5. Jane is a borrower because m1 < c1. She borrows 95 - 90 = $5

c) For m1 = 90, m2 = 110 and r = 22%. we have c1 =  (1/2)*(90 + 110/(1 + 22%)) = 90.08197 and c2 = 90.08197(1 + 22%) = 109.9. Jane is a borrower because m1 < c1. She borrows 90.08197 - 90 = -0.08197

d) For m1 = 90, m2 = 110 and r = 37.5%. we have c1 =  (1/2)*(90 + 110/(1 + 37.5%)) = 85 and c2 = 85(1 + 37.5%) = 116.875. Jane is a lender because m1 > c1. She lends 90 - 85 = $5

Jane is better off as interest rate is increased because she is turning saver from borrower.

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