solve the following questions please
6. (10) Consider the standard "cash-goods vs credit goods" model discussed in class. The representative agent maximizes life-time discounted utility S Btu(CIt, Cat). 1= 0 Assume that u11, 122 Btu(c). Each household has the following endowments: a) an initial capital stock ro at time 0, b) one unit of productive time per period that can be devoted to work, and c) one unit of land. Final output is produced according to yt = F(k, mt, 4). where F is a CRS production function, and ki, ne, It denote capital, labor, and land services, respec- tively. This technology is owned by firms whose number will be determined in equilibrium. Output can be consumed (c) or invested (it). We assume that households own the capital stock (so they make the investment decision) and the land, and they rent out capital, labor, and land services to the firms. The depreciation rate of the capital stock (r;) is denoted by 6.' The land does not depreciate, i.e. it is in fixed supply. Finally, we assume that households own the firms, i.e. they are claimants to the firms' profits. The functions u and F have the usual properties that we have described in class.? a) Consider an Arrow-Debreu world. Describe the households' and firms' problems and carefully define an AD equilibrium. How many firms operate in this equilibrium? b) In this economy, why is it a good idea to describe the AD equilibrium capital stock allocation by solving the (easier) Social Planner's Problem? From now on assume that F(k, n, 4) = Akfin;? nazi-1 02 and 6 = 1, with A > 0 and a1, 02 E (0, 1). The household's instantaneous utility function is given by u(c) = In(c). c) Fully characterize (i.e. find a closed form solution for) the equilibrium allocation of the capital stock. (Hint: Guess and verify a "policy rule" of the form ki+1 = gly, where g is an unknown to be determined.) In the remaining questions, for full credit, your answers should be functions only of the parameters of the model, i.e. 01, 02, 8, A, To etc. d) What is the ADE value of the rental rate of capital as t -+ co? e) What is the ADE price of the consumption good in t = 1? f) What is the ADE price of land services in t = 2?1. (10) Demonstrate that the Keynes-Ramsey condition (derived within the continuous time optimal growth model) and the Euler equation associated with capital (in the discrete time optimal growth model) are equivalent. (Note: To keep things simple assume no technological progress, no population growth and use constant relative risk aversion preferences.) 2. (20) Consider an optimal growth model with no technological progress and a constant population (normalized to 1). Assume that the lifetime utility function for the representative household is given by (1) where c(t) is per-capita consumption and it is assumed that 0 > 0 and 0 # 1. The representative agent is endowed with one unit of labor and supplies this inelastically. Output at each point in time is given by the constant returns to scale production function: Y (t) = F(K (t), L (t)) (2) The depreciation rate of capital is assumed to be 0. Given this environment, do the following: (a) Solve the model as a social planner problem in which the planner maximizes household utility given by eq. (1) subject to the technology given by eq. (2) and the law of motion for capital. To do this, express the social planner problem as a Hamiltonian and derive the necessary conditions. Also, present the transversality condition and its interpretation. (b) Linearize the system around the steady-state and prove that the trajectory to the steady-state is unique. (Hint: Recall that the determinant of a square matrix is equal to the product of its eigenvalues.) 3. (20) 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 is dy = dy = - where dy 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: where a is consumption and sy denotes the stock of beautiful trees owned at the beginning of period t and y > 0. (Let sy 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 pu and pat)- (c) Define the gross realized rate of return on trees of type i held from t to t + 1 as Ra. Prove that, under the assumption that y > 0, Ru > Ry. Explain
