Consider the problem facing an individual in the Lucas model when Pi/P is unknown. The individual chooses Li to maximize the expectation of Ui ; Ui continues to be given by equation (6.72).

(a) Find the first-order condition for Yi , and rearrange it to obtain an expression for Yi in terms of E [Pi/P]. Take logs of this expression to obtain an expression for y_i.

(b) How does the amount of labor the individual supplies if he or she follows the certainty-equivalence rule in (6.81) compare with the optimal amount derived in part (a)? (Hint: How does E [ln(Pi/P)] compare with ln(E [Pi/P])?)

(c) Suppose that (as in the Lucas model) ln(Pi/P) = E [ln(Pi/P) |Pi]+ui , where ui is normal with a mean of 0 and a variance that is independent of Pi . Show that this implies that ln{E [(Pi/P) |Pi]} = E [ln(Pi/P) |Pi] + C, where C is a constant whose value is independent of Pi . (Hint: Note that Pi/P = exp{E [ln(Pi/P) |Pi]}exp(ui), and show that this implies that the yi that maximizes expected utility differs from the certainty-equivalence rule in (6.81) only by a constant.)