Hi Can you please help me with this question?

6.15 Consider the problem facing an individual in the Lucas model when PP is unknown. The individual chooses _; to maximize the expectation of Uf: Uf continues to be given by equation (6.74). 308 (a) Find the first-order condition for Y; and rearrange it to obtain an expression for Y, in terms of E[PP]. 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[In(P; P)] compare with In(E[PP])?) (c) Suppose that (as in the Lucas model) In(P;/P) = E[In(P;/P) | Pi] + ui where u; is normal with a mean of 0 and a variance that is independent of Pi. Show that this implies that In (E[(P/ P) | Pi]) = E[in(P; P) | Pi] + C. where C is a constant whose value is independent of P; (Hint: Note that PP = exp (E[In(P/P) | Pillexp(u;). and show that this implies that the y; that maximizes expected utility differs from the certainty- equivalence rule in (6.83) only by a constant.)