Question

An individual with 100, 000 dollars and must decide how much to consume now (period t) and how much to save for the future (period t + 1) to maximize the sum of utilities over the two periods: U(Ct) + U (Ct+1). Ct is the consumption amount at date-t and Ct+1 is the consumption at date-t + 1. The bank will pay interest rate r for every dollar she saves.

Can you give an example of utility function U(C) such that when the individual has such utility function, the optimal consumption Ct does not change with the interest rate r?

Can you give an example of utility function U(C) such that when the individual has such utility function, the optimal consumption Ct changes with the interest rate r?

The goal is to just maximize the person's wealth.

Answer 1

a) When a consumer has a perfect substitute utility function, the optimal consumption doesnot change with the change in the interest rate.

For example, let us consider the utility function as :

In the case of perfect substitutes, we cannot find equilibrium through tangency as the slope of indiference curve is constant. So a change in interest rate would not affect the optimal consumption bundle.

b) When a consumer has a cobb-douglas utility function, the optimal consumtion C(t) would change with the interest rate.

For example, let us consider the utility function as:

MRS=Ct+1/Ct

At equilibrium MRS=SLOPE OF BUDGET LINE=1+r

Ct+1/Ct=1+r

So as r would change, the ratio of Ct+1/Ct would change and thus there would be a change in the optimal bundle.