Suppose Jack has the utility functions:U0(c0, c1, c2) =u(c0) +u(c1) +2u(c2)

U1(c1, c2) =u(c1) +u(c2)

Above,U0is the utility function based on which Jack makes decisions in periodt= 0;U1is the utility function based on which Jack makes decisions in periodt= 1. Jack hasy >0 apples and there is no interest rate.

We want to show that Jack's behavior will be time consistent even though there is present bias int= 0 ( <1). We will do so in several ways.

a. Re-write the functionU0so as to show that when maximizingU0int= 0, Jack is implicitly also maximizingU1.

b. Write down the tangency conditions betweenc1andc2for both the period-0 and the period-1 maximization problems. What do you notice? Do the same for the budget constraints. What do you notice?

c. Assume thatu(c) = logcand solve explicitly for the optimal plan (c0, c1, c2) in periodt= 0 and the optimal plan (c, c) in periodt= 1. These optimal plans should be given as formulas in terms of, , andy.

d. Do you find the specifications ofU0andU1compelling? That is, do you think they are a good representation of actual behavior? Discuss. As part of your discussion, explain what must be true in order for present bias to produce time inconsistency.