2. Consider a pure exchange economy with two agents (let's say, Attila and Balazs) and two goods (let's say, exes and whys): . UA(TA, yA) = 2 . x4 . yA, . UB(TB, yB) = XB . yB, . wa + wp = 10, . w + Wg = 10. (a) Find mathematically and represent graphically in an Edgeworth box the set of Pareto efficient allocation of this economy. 2.5 points (b) Plot the utility possibilities frontier for this economy. Hint: Recall that, with the help of the utility possibilities frontier, we are sim- ply considering the set of Pareto efficient allocations from a different perspective. Follow the three steps below to complete this part of the exercise. i. Use the feasibility constraints and your answer to the previous question, and write uA and up (along the contract curve) as a function of x A only. ii. Given your answers to the previous point, write up as a function of uA. iii. Now you should be able to plot the utility possibilities frontier for this econ- omy. 2.5 points (c) Consider the following social-welfare maximization problem: max XA . UA(XA, yA) + AB . UB(TB, yB) TA,YA, TB, yB subject to XA + B = 10 yA + yB = 10 The function AA . UA(TA, yA) + AB . UB(B, yB) is called social welfare function. The parameters AA and AB are called Pareto weights.i. Write the first-order conditions and argue that any solution to the social- welfare maximization problem is Pareto efficient. Remember that uA (XA, yA) = 2 . x4 . yA and UB (TB, yB) = XB . yB. 2 points ii. Consider the allocation in which agent A consumes 2 units of each good, and agent B consumes the rest. A. Argue that this is a Pareto efficient allocation. 1 point B. How much should AA and AB be so that the solution of the social-welfare maximization problem is exactly the above allocation? 1 point iii. Use the graph with the utility possibilities frontier and represent the above social-welfare maximization problem. 1 point