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 (6.74).

(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 yi .

(b) How does the amount of labor the individual supplies if he or she follows the certainty-equivalence rule in (6.83) 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 (6.83)
only by a constant.)