ConsidertheTwo-PeriodEndowmentModelofhousehold behaviour studied in class. The household derives utility from consumption today c and consumption tomorrow c′. Suppose that preferences are represented by the utility function:
(c, c′) = c + (c′)
where , (0, 1)
The household receives exogenous income y today and y′ tomorrow. For simplicity, assume that there are no lump-sum taxes (i.e. = ′ = 0, using the notation in class). The real interest rate is given by .
a)Write down the problem of the household.
b)Solve for the optimal level of current consumption (i.e. c∗).
c) Suppose that the parameters of the model take the following values:
y=500, y′ =105, =0.95, =0.08, =0.3.
Based on these values, is the household a lender or a borrower? Justify your answer.
d) Supposethatnowtheinterestratedecreasesto0.05(while the values ofall other parameters are unchanged). Compute the new level of optimal consumption in the first period. What does your finding imply about the relative magnitudes of income and substitution effects? Explain.
e) Using a diagram on the (c,c′)plane, illustrate the effect of a decrease in on the optimal consumption bundle of the household who is a lender, and identify income and substitution effects. Would the consumer be better-off or worse-off after the decrease in the interest rate?
Note: To answer Part (e), you don’t need to do any algebra or calculations.