Question

A consumer is making multi-period consumption decisions. Their income is higher in period 2 than in period 1: I1=50; I2=55. Their utility is multiplicative: U(C1,C2)=C1C2. (a) Calculate whether the consumer will borrow, save or do neither for each interest rate below:

(1) r = 5%

(2) r = 10%

(3) r = 20%

(b) What interest rate would cause the consumer neither to save or borrow?

Answer 1

We have intertemporal budget using I1=50; I2=55 as

C1 + C2/(1 + r) = 50 + 55/(1 + r). Here MRS = -C2/C1.

(1) r = 5%

At this level, budget constraint becomes C1 + C2/(1 + 5%) = 50 + 55/(1 + 5%)

C1 + 0.952C2 = 102.381

Optimal choice has MRS = slope of budget line

-C2/C1 = -1/0.952 or C1 = 0.952C2

Use this rule in the budget 0.952C2 + 0.952C2 = 102.381

This gives C2 = 53.75 and C1 = 0.952*53.75 = 51.17

Note that C1 > I1 and C2 < I2, the individual is a borrower.

(2) r = 10%

At this level, budget constraint becomes C1 + C2/(1 + 10%) = 50 + 55/(1 + 10%)

C1 + 0.909C2 = 100

Optimal choice has MRS = slope of budget line

-C2/C1 = -1/0.909 or C1 = 0.909C2

Use this rule in the budget 0.909C2 + 0.909C2 = 100

This gives C2 = 55 and C1 = 0.909*55 = 50

Note that C1 = I1 and C2 = I2, the individual is neither a borrower nor a saver

(3) r = 20%

At this level, budget constraint becomes C1 + C2/(1 + 20%) = 50 + 55/(1 + 20%)

C1 + 0.833C2 = 95.83

Optimal choice has MRS = slope of budget line

-C2/C1 = -1/0.833 or C1 = 0.833C2

Use this rule in the budget 0.833C2 + 0.833C2 = 95.83

This gives C2 = 57.5 and C1 = 47.2

Note that C1 < I1 and C2 > I2, the individual is a saver

(b) The interest rate that would cause the consumer neither to save or borrow is 10%