is dit 1. Consider the following version of Lucas's tree economy. In this economy there are two kinds of trees which produce the same quantity of "fruit" or dividends. That d2+ = where d is the total dividend at time t and each tree, denoted 1 and 2, produces half of the total. The aggregate dividend is a random variable that follows a first-order Markov process characterized by positive serial correlation. There are N identical individuals in the economy and each person is endowed with one of each kind of tree at birth. The first tree is a standard tree but the second tree is quite beautiful; consequently, ownership of the tree provides direct utility. This is represented by agents' utility function: = Eo Bt (Inc + ylns2t) where ct is consumption and s2t denotes the stock of beautiful trees owned at the beginning of period t and y> 0. (Let sit denote the stock of normal trees owned at the beginning of time t.) As in the Lucas model, the owner of a tree at the beginning of a period receives the dividends for that period. Given this environment, do the following: (a) Set up the agent's maximization problem as a dynamic programming problem and explicitly identify the state and control variables. Derive the necessary conditions. (b) Define a recursive competitive equilibrium. Solve explicitly for the prices of both trees (denote these as pit and p2t). (c) Define the gross realized rate of return on trees of type i held from t to t+1 as Rit: Prove that, under the assumption that y > 0, Rt > R2t. Explain. 2. Consider a Lucas-tree type economy in which the level of the endowment is indepen- dently and identically distributed with support x (z', x"). Agents trade one- and two-period bonds that have prices plt and p2t, respectively, and return 1 unit of con- sumption at maturity. In addition, agents can purchase a one-period futures contract for the price fl,. A futures contract purchased at time t states that the owner agrees to give up f1 units of consumption in period t+1 for the return of 1 unit of consumption in period t + 2. Note that in this economy, equity is not traded; agents receive the endowment, It, at the beginning of each period. Given this setup, do the following: (a) Assuming standard preferences (i.e. infinitely lived, risk-averse agents), set up the household's maximization problem as a dynamic programming problem. (Sugges- tion: The setup is easier if it is assumed that households sell all two period bonds after holding them for one period.) Prof. Seungduck Lee 1 Graduate Macroeconomic Theory 1 (b) Derive and interpret the associated necessary conditions. (c) Define a recursive competitive equilibrium in this economy. (d) Characterize the equilibrium behavior of bond prices and the price of the fu- tures contract. Derive an exact relationship between these prices; interpret this relationship.