2. Constructing an Equilibrium Households live two periods and have preferences U() + BU(ca), where 0 0. The type I agent faces budget constraints = 0+ rs! = where consumption for the type i agent in period j is denoted. The type 2 agent faces budget constraints 0 = +2 1 +rg - The resource constraints are Nin = Nc+Nc No - Ne+N (a) State the maximization problem solved by each type of agent and derive the first- and second-order conditions. Derive the solution using the implicit function theorem (b) Determine the equilibrium conditions for the three markets using the resource constraints and the budget constraints. Provide a statement of the equilibrium (e) Assume logarithmic utility Ue) = ln(c) and derive a closed form solution for consumption in both periods and savings for both types of agents. (d) Solve the social planning problem. Compare the solution of the social planning problem with the competitive equilibrium. Demonstrate that the decentralized solution solves the social planning problem for a particular set of Pareto weights Explain how this is an example of the First Welfare Theorem. 2. Constructing an Equilibrium Households live two periods and have preferences U() + BU(ca), where 0 0. The type I agent faces budget constraints = 0+ rs! = where consumption for the type i agent in period j is denoted. The type 2 agent faces budget constraints 0 = +2 1 +rg - The resource constraints are Nin = Nc+Nc No - Ne+N (a) State the maximization problem solved by each type of agent and derive the first- and second-order conditions. Derive the solution using the implicit function theorem (b) Determine the equilibrium conditions for the three markets using the resource constraints and the budget constraints. Provide a statement of the equilibrium (e) Assume logarithmic utility Ue) = ln(c) and derive a closed form solution for consumption in both periods and savings for both types of agents. (d) Solve the social planning problem. Compare the solution of the social planning problem with the competitive equilibrium. Demonstrate that the decentralized solution solves the social planning problem for a particular set of Pareto weights Explain how this is an example of the First Welfare Theorem