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6. (10) Consider the standard "cash-goods vs credit goods" model discussed in class. The representative agent maximizes life-time discounted utility _ Blu(cut, Cat). Assume that uil, uzz 0. (Let sur 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 pur and pat). (c) Define the gross realized rate of return on trees of type i held from ? to t + 1 as Ry. Prove that, under the assumption that y > 0, Ry > Ry, Explain.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. (25) 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 1 _ge(t)- de (1) where c(t) is per-capita consumption and it is assumed that 0 > 0 and & # 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 (1) , L() ) (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 laws of motion for capital and labor. 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.)