The Lucas asset-pricing model. (Lucas, 1978.) Suppose the only assets in the economy are infinitely lived trees. Output equals the fruit of the trees, which is exogenous and cannot be stored; thus Ct = Yt, where Yt is the exogenously determined output per person and Ct is consumption per person. Assume that initially each consumer owns the same number of trees. Since all consumers are assumed to be the same, this means that, in equilibrium, the behavior of the price of trees must be such that, each period, the representative consumer does not want to either increase or decrease his or her holdings of trees.

Let Pt denote the price of a tree in period t (assume that if the tree is sold, the sale occurs after the existing owner receives that period’s output). Finally, assume that the representative consumer maximizes E [

∞

t =0 ln Ct/(1 + ρ)

t ].

(a) Suppose the representative consumer reduces his or her consumption in period t by an infinitesimal amount, uses the resulting saving to increase his or her holdings of trees, and then sells these additional holdings in period t + 1.

Find the condition that Ct and expectations involving Yt+1, Pt+1, and Ct+1 must satisfy for this change not to affect expected utility. Solve this condition for Pt in terms of Yt and expectations involving Yt+1, Pt+1, and Ct+1.

(b) Assume that lims→∞ Et [(Pt+s/Yt+s)/(1 + ρ)

s

] = 0. Given this assumption, iterate your answer to part

(a) forward to solve for Pt. (Hint: Use the fact that Ct+s = Yt+s for all s.)

(c) Explain intuitively why an increase in expectations of future dividends does not affect the price of the asset.

(d ) Does consumption follow a random walk in this model?