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Question 1: Consider an individual that must decide how much to consume in a two- period model. Let us suppose that her preferences for present consumption (c1) and future consumption (c2) can be characterized by the following utility function:
Assume that the price index of present consumption is p1, the price index of future consumption is p2, and the nominal interest rate is i. In addition to deciding how much to consume in both periods, this individual must also decide on how much to invest in her University education. This investment can only be made in the present period. Let the cost of her University education, E, be E = ?M1
where 0 < ? ? 1 is her investment intensity and M1 is her income in the present period. Thus, E represents the part of her present income that she invests in her University education. This means that, for a given ?, this individual will have (1 ? ?)M1 of her present income left over for consumption. Further, for a given ?, her income in the future period, M2, will be M2 = 1 + ? ? M11 (a) Explain in what sense a dollar spent on University education has a different effect on this individual’s utility compared to a dollar spent on c1 or c2. Use this to identify the benefit and cost associated with investing in University education. (b) What is her optimal University investment intensity, ?? Is ? negatively or positively related to the nominal interest rate? Provide some intuition for this relationship. (c) Suppose that M1 = 100, p1 = 1, p2 = 1.05, and i = 0.05. Use this along with your answer in (b) to write down the expression for this individual’s inter-temporal budget constraint. Fully illustrate this constraint with her present consumption on the horizontal axis. Explain how this inter-temporal budget constraint is different from the standard inter-temporal budget constraint discussed in the lectures. (d) Solve for this individual’s optimal values of c1 and c2 respectively. Fully illustrate this optimal choice in your diagram in part (c). Is this individual a borrower or saver? How much does she borrow or save? Now suppose that this individual is credit constrained. That is, her borrowing in the present period is constrained. Consider two ways of modelling such a constraint. (e) First, suppose that this individual is not allowed to borrow at all in the present period. What is her optimal University investment intensity now? What about her optimal values of c1 and c2 respectively? Use a new diagram to fully illustrate her optimal choice with the zero borrowing constraint. How are her choices of ?, c1, and c2 different from her choices in (b) and (d) above? Instead of preventing any borrowing, we can also model her credit constraint by adjusting the nominal interest rate that she has to pay when she borrows. For instance, suppose that if she were to save in the present period, she will continue to earn the nominal interest rate, i, on her savings. However, if she chooses to borrow in the present period, she must pay a nominal interest rate, r > i, per dollar borrowed. Assume that r = 0.15 while i = 0.05.
(f) What is her optimal University investment intensity now? What about her optimal values of c1 and c2 respectively? How are her choices of ?, c1, and c2 as well as her amount borrowed different from her choices in (b) and (d) above?
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