7.8. Bubbles. Consider the setup of the previous problem without the assumption that lim EP/(1+r)*1-0.

(a) Deterministic bubbles. Suppose that P, equals the expression derived in part

(b) of Problem 7.7 plus (1 + ryb, b> 0. (i) Is consumers' first-order condition derived in part

(a) of Problem 7.7 still satisfied? (II) Can b be negative? (Hint: consider the strategy of never selling the stock.)

(b) Bursting bubbles. (Blanchard, 1979.) Suppose that P, equals the expression derived in part

(b) of Problem 7.7 plus q, where q, equals (1+r)q-1/a with probability a and equals zero with probability 1-a.

(f) Is consumers' first-order condition derived in part

(a) of Problem 7.7 still satisfied? (ii) If there is a bubble at time t (that is, if qr > 0), what is the probability that the bubble has burst by time t+s (that is, that a 0)? What is the limit of this probability as s approaches infinity?

(c) Intrinsic bubbles. (Froot and Obtsfeld, 1991.) Suppose that dividends fol- low a random walk: DD-1 + er, where e is white noise. (i) In the absence of bubbles, what is the price of the stock in period t?

(f) Suppose that P, equals the expression derived in (1) plus b,, where by (1+r)b-1+ce,,c>0. Is consumers' first-order condition derived in part

(a) of Problem 7.7 still satisfied? In what sense do stock prices overreact to changes in dividends?