Bubbles. Consider the setup of the previous problem without the assumption that lims→∞ Et [Pt +s/(1 + r)

s ] = 0.

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

(b) of Problem 8.8 plus (1 + r)

t b , b > 0.

(i) Is consumers’ first-order condition derived in part

(a) of Problem 8.8 still satisfied?

(ii) Can b be negative? (Hint: Consider the strategy of never selling the stock.)

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

(b) of Problem 8.8 plus qt, where qt equals (1 + r)qt −1/α with probability α and equals 0 with probability 1 − α.

(i) Is consumers’ first-order condition derived in part

(a) of Problem 8.8 still satisfied?

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

(c) Intrinsic bubbles. (Froot and Obstfeld, 1991.) Suppose that dividends follow a random walk: Dt = Dt−1 + et, where e is white noise.

(i) In the absence of bubbles, what is the price of the stock in period t?

(ii) Suppose that Pt equals the expression derived in (i) plus bt, where bt =

(1+r)bt−1+cet, c > 0. Is consumers’ first-order condition derived in part (a)

of Problem 8.8 still satisfied? In what sense do stock prices overreact to changes in dividends?