Quantitative Macroeconomics. Thank you!

l. (26 points + 6 extra points) Competitive Equilibrium and Social Optimum In this exercise you will compute a social optimum and the competitive equilibrium for an economy without distortions. Then you will implement a policy such that the competitive equilibrium will deliver the social optimum. Consider an economy inhabited by two households indexed by 1' = 1, 2. Households can work (N,) or enjoy leisure (Li). Each household 1' is endowed with one unit of time, Li + N5 = 1. When working, households produce output according to a linear production function with K; = mNi for i = 1, 2 where fr, is the productivity of agent 5?, with 1'71 > 11'2. They can use their income to purchase consumption goods. Households derive utility from consumption Ci, and from leisure, L5. Preferences take the form 15033:. Li) = 15(05) + 0(1):) for household 32 = 1,2. (a) (6 points) Set up and derive the rst-order conditions for the household's i prob- lem for i: 1, 2. (b) (4 points) Write the system of equations that describe the competitive equilib- rium of this economy (c) (6 points) Set up and solve the social planner's problem for a planner who max- imizes the sum of the welfare of household 1 and household 2, deriving the rst order conditions with respect to (0-5, Li]. ((1) (4 points) Compare the planner's allocations with the competitive equilibrium allocations. Are they the same, i.e. do the Welfare Theorems hold? Explain your answer. (3) (6 points + 6 extra credit points) Is there any need for government intervention in this economy? What policy would you suggest to the government, if any? You cannot solve for the policy without a specication for the utility function, just describe the policy set up as well as you can. If you want to solve for it for extra credit, assume 1:105) + v(L,) = log 0,- + \"og L5