Suppose you are asked to model the savings decisions of a household that has an income of $100,000 this year but expects to have no income a period into the future.
A. Suppose the interest rate is 10% over this period and we consider the tradeoff between consuming now and consuming one period from now.
(a) On a graph with “Consumption now” on the horizontal and ”Future Consumption” on the vertical axis, illustrate how an increase in the interest rate to 20% over the relevant period would change the household’s choice set.
(b) Suppose that you know that the household’s tastes can accurately be modeled as perfect complements over consumption now and consumption in the future period. Can you tell whether the household will save more or less as a result of the increase in the interest rate?
(c) You are asked to advise Congress on a proposed policy of subsidizing savings in order to increase the amount of money people save. Specifically, Congress proposes to provide 5% in interest payments in addition to the interest households earn in the market. You are asked to evaluate the following statement: “Assuming that consumption is always a normal good, small substitution effects make it likely that savings will actually decline as a result of this policy, but large substitution effects make it likely that savings will increase.”
(d) True or False: If the purpose of the policy described in the previous part of the problem is to increase the amount of consumption households have in the future, then the policy will succeed so long as consumption is always a normal good.
B. Now suppose that tastes over consumption now, c1, and consumption in the future, c2, can be represented by the Constant Elasticity of Substitution utility function u(c1,c2) = (c1−ρ +c2−ρ )−1/ρ.
(a) Write down the constrained optimization problem assuming that the real interest rate is r and no government programs dealing with savings are in effect.
(b) Solve for the optimal level of c1 as a function of ρ and r. For what value of ρ is the household’s savings decision unaffected by the real interest rate?
(c) Knowing the relationship between ρ and the elasticity of substitution, can you make the statement quoted in (c) of part A more precise?