Consider a two good, two individual endowment economy. Suppose that individual A's endowment is given by {8A 3A) and individual B's endowment is given by [s39 :38). Also assume individual A's preferences are I! y 3' 31' given by: {Vilma v.4) = balm} + (1 - Q) 10E'lyA] and individual B's preferences are given by: UBEasyB] = 71030113) + {1 'r'} log{y:3}- Consider the following social welfare function: 3A5\": {Imam} + EBUB {mews}- We can interpret this as a social planner that wishes to allocate the economy's endowment between the two individuals. 49,- is descriptive of how important individual i is to the social planner, 1' E {A,B}. Therefore, a large value of 3,: indicates to us that the social planner gives a lot of weight to individual i, while the reverse is true if 3,: 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 optimum. we can obtain all possible Pareto optima by varying the 3 weights of the problem. . write down the Planner's problem, as well as the Lagrangian. (Hint: There should be two constraints because aggregate consumption of each good cannot exceed the economy's endowment.) Label the multipliers 51 and (g. [5 points] . YWhat are the rst order conditions of the Planner's problem? (Hint: There should be 6.}. [5 points] . Find the solution to the Planner's problem and label the quantities (mi, ya) and {$%,yp}. [5 points] . If we compare the rst order conditions of the Planner's problem to the rst order conditions of a corresponding competitive equilibrium [CE], we can obtain a relationship between $1, ($5., A, p1, and 13;, so that the CE and the Planner's problem give us the same solution. This is called decentralizing the Planner's problem. Decentralize the Planner's problem. {That is, nd the equilibrium prices and the value of A so that the two problems have the same solution.) [5 points] . Can you relate this to the two welfare theorems? |5 points]