Exercise 16.13. Consider an economy populated by identical households each with preferences given by EP t=0 t

Exercise 16.13. Consider an economy populated by identical households each with preferences given by E£P∞ t=0 βt u (c (t))¤ , where u (·) is strictly increasing, strictly concave and twice continuously differentiable. Normalize the measure of agents in the economy to 1. Each household has a claim to a single tree, which delivers z (t) units of consumption good at time t. Assume that z (t) is a random variable taking values from the set Z ≡ {z1, ..., zN } and is distributed according to a Markov chain (all trees have exactly the same output, so there is no gain in diversification). Each household can sell any fraction of its trees or buy fractions of new trees, though cannot sell trees short (i.e., negative holdings are not allowed). Suppose that the price of a tree when the current realization of z (t) is z is given by the function p : Z → R+. There are no other assets to transfer resources across periods. (1) Show that for a given price function p (z), the flow budget constraint of a representative household can be written as c (t) + p (z (t)) x (t + 1) ≤ [z (t) + p (z (t))] x (t), where x (t) denotes the tree holdings of the household at time t. Interpret this constraint. (2) Show that for a given price function p (z), the maximization problem of the representative household subject to the flow budget constraint and the constraint that c (t) ≥ 0, x (t) ≥ 0 can be written in a recursive form as follows V (x, z) = sup y∈[0,p(z) −1(z+p(z))x] © u ((z + p (z)) x − p (z) y) + βE£ V ¡ y, z0 ¢ | z ¤ª . (3) Use the results from Section 16.1 to show that V (x, y) has a solution, is increasing in both of its arguments and strictly concave, and is differentiable in x in the interior of its domain. (4) Derive the stochastic Euler equations for this maximization problem. (5) Now impose market clearing, which implies that x (t)=1 for all t. Explain why this condition is necessary and sufficient for market clearing. (6) Under market clearing, derive p (z) the equilibrium prices of trees as a function of the current realization of z.