Question 2: Decentralizing the social planner's problem [25 points| C.'mmdr a two good, two individual endowment ecnnumy E:Llp'pl:bE that individual A's endowment is given by (el e \"'} and individual B's endowment is given by (e ). Also assume individual A's preferences are given b}r =16y UMz a,ya) = alog(za) + (1 - a) log(ya) and individual s preferences are given by: UB(zp,yp) = 7log(xs) + (1 - 7)log(ys). Consider the following social welfare function: BaU (xa,ya) + 08U (x5.y8). We can interpret this as a social planner that wishes to alloeate the economy's endowment between the two individuals. #; is deseriptive of how important individual is to the social planner, {A, B}. Therefore, a large value of #; indicates to us that the social planner gives a lot of weight to individual 4, while the reverse is true if #; is small. The social planner's problem is to maximize society's welfare by allocating consumption among the two individuals. The solution will be a Pareto optinnim. We can obtain all possible Pareto optima by varving the & weights of the problem. 1. Write down the Planner's problem, as well as the Lagrangian. (Hint: There should be two constraints because aggregate consumption of each good cannot excoed the economy's endowment.) Label the multipliers , and ,. |5 points] 2. What are the first order conditions of the Planner's problem? (Hint: There should be 6.). |5 points] 3. Find the solution to the Planner's problem and label the quantities(z%, %) and (25, 4f). |5 points] 4. If we compare the .mt order mnd.ltu:um of the Planner's problem to the first order conditions of a i ; ilibrium)(CE), we can obtain a relationship betweeng, , A, p,, and Py so that the CE and m Planner's problem give us the same solution. This is callediecentralizing the Planner's problem. Decentralize the Planner's problem. (That is, find the [equilibrium) prices and the value ofh so that the two problems have the same solution.) |5 points| 5. Can you relate this to the two welfare theorems? |5 points|