Question

Sierra has the utility function ?(?1,?2)=2??(?1)+4??(?2) where ?1 and ?2 are her consumption in periods 1 and 2, respectively. She will earn $200 in period 1 and $220 in period 2. She can borrow or save at an interest rate of 10% and the price of the consumption good is $1 in each period.

a. If Sierra decides to keep her savings at cash, solve for her optimal level of consumption in each period. Assume that she does not have access to other sources of credit.

b. Suppose she considers borrowing and saving at the prevailing interest rate. Solve for her optimal level of consumption in each period.

c. If inflation hits 10%, what is the equation of Sierra’s budget constraint?

Answer 1

A) no borrowing/ saving is possible

At eqm, MRS _1,2 = P1/P2 = 1

2C2/4C1= 1

C2 = 2C1

From BC : C1+C2 = Y1+Y2 = 420

Put C2 = 2C1

3C1 = 420

C1* = 140

C2* = 280

B) borrowing/ saving is possible

Then , new BC :

Saving in period 1, S1 = Y1-C1

In period 2, C2 = (1+r)S1 + Y2

(1+r)C1 + C2 = (1+r)Y1 + Y2

1.1C1 + C2 = 1.1*200 + 220

1.1C1 + C2 = 440

.

At eqm, MRS = (1+r)

C2/2C1 = 1.1

C2 = 2.2C1

From New BC :

1.1C1 + 2.2C1 = 440

3.3C1 = 440

C1* = 133.33

C2*= 293.33

(C1,C2)* = (133.33, 293.33)

C) if inflation = 10%

So P2 = 1+π

So new BC :

P1*C1 + P2*C2/(1+r) = Y1 + Y2/(1+r)

C1 + 1.1C2/(1.1) = 200 + 220/(1.1)

C1 + C2 = 200 + 200

C1 + C2 = 400