Consider a two-period economy with a representative household who has an endowment of y = 1 in the first period but no endowment in the second period. Preferences over consumption are given by:

c1 c1 u(c1,c2)= 1 + 2

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where > 1 and 0 < < 1. The household can invest in capital in period one to obtain a gross return of 1 + r in period 2.

A representative firm uses capital to produce output in period two according to the production function:

k with probability 0.5 f(k) = 2k with probability 0.5

Capital depreciates fully after use. The household owns the firm and hence receives any profits.

(a) Set up the households decision problem.

(b) Set up the firms decision problem.

(c) What market clearing conditions must hold in a competitive equilibrium?

(d) Use your answers from (a), (b), and (c) to set up and solve the social planner problem. (ie. find the equilibrium level of capital, k, in terms of the model parameters.)

(e) Show that dk > 0. Provide an economic interpretation for this result. d

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