1 Competitive Equilibrium (120 points) Consider an economy with a representative household (HH), a representative rm, and a govern ment. 1. The HH can work up to h hours at an hourly rate of to. Its preferences are dened by the GHH utility function w _1+ The HH owns an exogenously given amount of capital E, which can be rent to the represen tative rm at a price 7'. Taxes paid to the government, T, are lump sum. Denote by n3 and 1:5 supply of labor and capital, respectively. Let 7r be the prot of the rm distributed to the HH, who owns the rm. u(c, n) = c (n5)1+ 2. The rm operates a CRS production function y = 3031):): (nd)1a The rm pays the wage rate w for each unit of labor demanded and "r for each unit of capital demanded. 3. The government consumes an exogenous given amount 9 that is nanced via lump sum taxes, T. Answer the following questions: 1. (20 pts.) Dene a competitive equilibrium for this economy. Make sure to specify all the relevant problems and conditions. Pay attention to details. 2. (20 pts.) Pose the household's problem and define the Lagrangian. Take FOC and use them to find capital and labor supply. Explain and justify all your steps. 3. (20 pts.) Pose the firm's problem. Take FOC and interpret them in terms of marginal cost and marginal benefit. What are firm's profits in equilibrium? 4. (20 pts.) Construct the system of equations you need to solve to find the CE. Clearly state all the equations and all the unknowns. Do not solve it. 5. (20 pts.) Pose the corresponding planner's problem of this economy. Construct the La- grangian, take FOC, and construct the system of equations one need to solve to find the planners's solution. Find it. In particular, find (CSP, nSP, ISP, K.SP, ySP). 6. (20 pts.) Show that the solution to the planner's problem is the same as the competitive equilibrium. Find prices (w*, r*)