Consider the following modification to the stochastic growth model presented in chapter 12 of Ljungqvist and Sargent "Recursive Macroeconomic Theory" (2nd edition): The TFP that firms use to produce at t,At is a discrete random variable with support A={A1,A2,,AM}. The sequence {At}t=0 is a Markov process with transition probabilities given by: ij=P(At=AjAt1=Ai) Note that with this modification the aggregate state vector now corresponds to X[KA] as opposed to X[KA] as we have defined directly the transition probabilities of technology At. As opposed to the original setup where At is expressed as a function of st and At1. Recursive Planner Problem 1. Present the recursive formulation of the planner problem. 2. Given the Metric space (X,d) of bounded and continuos functions with the supremum distance. (a) Use the recursive formulation to define the mapping T(v(K,A)) and show that it is an operator. (b) Show that T(v(K,A)) is a contraction mapping. 3. Define the corresponding optimal policy functions. 4. Find the first order conditions of the planner problem. Recursive Competitive Equilibrium 1. Households: (a) Present the recursive version of the household problem. (b) Define the policy functions of the household (c) Find the first order conditions Consider the following modification to the stochastic growth model presented in chapter 12 of Ljungqvist and Sargent "Recursive Macroeconomic Theory" (2nd edition): The TFP that firms use to produce at t,At is a discrete random variable with support A={A1,A2,,AM}. The sequence {At}t=0 is a Markov process with transition probabilities given by: ij=P(At=AjAt1=Ai) Note that with this modification the aggregate state vector now corresponds to X[KA] as opposed to X[KA] as we have defined directly the transition probabilities of technology At. As opposed to the original setup where At is expressed as a function of st and At1. Recursive Planner Problem 1. Present the recursive formulation of the planner problem. 2. Given the Metric space (X,d) of bounded and continuos functions with the supremum distance. (a) Use the recursive formulation to define the mapping T(v(K,A)) and show that it is an operator. (b) Show that T(v(K,A)) is a contraction mapping. 3. Define the corresponding optimal policy functions. 4. Find the first order conditions of the planner problem. Recursive Competitive Equilibrium 1. Households: (a) Present the recursive version of the household problem. (b) Define the policy functions of the household (c) Find the first order conditions