Consider a consumer with utility function u(xt) = ln xt for the income xt he enjoys in period t. Assume that he receives no income in the first two periods, but receives an income w > 0 in the third period. In addition, this consumer discounts his income stream (x0, x1, x2) according to u(x0, x1, x2)= ln x0 + bd ln x1 + bd2 ln x2 where d Î (0, 1) denotes his discount factor, and b £ 1. We will note that this type of utility function (commonly known as the (b, d)model) exhibits present bias when b ¹ 1. For parts (a) and (b) assume that, once the individual makes plans in period 0, he does not revise these plans in the future.

a. Assume that he borrows during periods 0 and 1, and in period 2 he uses w to pay his debt. For simplicity, assume that he pays no interest during periods 0 and 1, but in period 2 he pays (1 + r)(x0 + x1). Find his optimal consumption plan for (x0, x1, x2).

b. Compare the total debt that, at period 0, a present-biased individual (with b ¹ 1) and an individual without present bias (b = 1) plan to have at period 2.

c. While part (b) of the exercise focused on the debts that both types of individuals plan to have at period 2, the present-biased individual still has an opportunity to further increase his debt during period 1 given his time inconsistent preferences. Find his optimal consumption plan (x1, x2) at period 1.

d. Evaluate again the total debt of the present-biased individual against that of the decision maker who does not exhibit present bias.