Consider the following economy. Time is discrete and runs forever, t = 0, 1, 2, . . . The economy is populated by two types of agents (a measure one of each): farmers and workers. Farmers own a piece of land that pays a stochastic income yt every period. We assume that yt is i.i.d. across farmers and time, and that yt N(y, 2 ). Farmers use all their time working their land. Theirpreferences are given by u(c) = exp(c)

for some > 0. For simplicity, we assume that consumption of farmers can be negative, that is c R. Moreover, farmers can save (or borrow) in a non-state contingent and non-defaultable asset a, which has a rate of return r. Farmers face the following "NoPonzi" condition on assets lim t at (1 + r) t 0. Moreover, farmers can produce and hold capital, k, which is rented to the representative firm in competitive markets (1 unit of final consumption good produces 1 unit of capital). Let r K be the rental rate of capital and the depreciation rate.

Unlike farmers, workers don't own land, and they use their available time to work in the representative firm. Assume each worker is endowed with one unit of time. They cannot trade the asset a, but they can produce and hold capital, k (with the same technology as farmers). Their per-period utility is given by u(c), with u 0 (c) > 0, u 00(c) < 0, limc0 u 0 (c) = , limc u 0 (c) = 0. Finally, there is a representative firm that combines capital and labor to produce final good according to f(K, L), where K is the capital they operate, and L the hours/workers they hire. Assume f() satisfies the standard Inada conditions. Thus, the only financial market available in this economy is the market for assets a, where only farmers can trade. Moreover, there is a market for k, where all agents can trade. Both type of agents have the same discount factor (0, 1)

a) Given a constant path for r and r K, state the problem of a farmer. Argue that the equilibrium price of capital is equal to 1. Characterize the farmer's problem with the necessary FOCs. Show that in any equilibrium with positive capital it must hold that r = r K .

b) Let c F (a, k, y) denote the consumption policy function of the farmers, which depends on asset holdings a, capital holdings k, and land income y. Prove that: c(a, k, y) = c + r(a + k) + r 1 + r y, 5 where c is a constant. Find an expression for c in terms of the parameters of the model. Hint: If x N(, 2 ), then E[e x ] = e + 1 2 2 .

c) Let C F and KF denote the aggregate consumption and capital holdings of farmers. Show that in an equilibrium with constant C F and KF , it must hold that (1 + r) < 1. Hint: Remember to check the budget constraint.

d) Let's turn to the workers. State their problem. Hint: Remember that workers cannot save in asset a; they can only hold capital.

e) Let c W and k W denote the consumption and capital holdings of an individual worker. Show that in an equilibrium with constant c W , it must be that k W = 0. Conclude that an equilibrium with constant C F , KF , c W , and k W = 0 exists Hint: Use the result that in equilibrium (1 + r) < 1

f) Show that in such an equilibrium, an increase in 2 increases the welfare of workers. Explain.