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Unformatted text preview: int S on the diagram below). Both points S and N are Pareto efficient because they are both on the Grand UPF. However, point S is more socially desirable because it corresponds to a higher SWF contour than point N. Point S provides both economic efficiency (on the Grand UPF) and social equity (on the highest attainable SWF contour). UB N S SWF1 SWF2 Grand UPF UA HOMEWORK Construct the individual UPFs for the following additional points on the PPF of the square root economy example we did in today's lecture: Point D Point E Point F X = 0.4 X = 0.2 X = 0.1 196 Y = 1 0.4 = 0.6 Y = 1 0.2 = 0.8 Y = 1 0.1 = 0.9 ECON 301 LECTURE #12 WELFARE THEOREMS STATEMENT: "A competitive equilibrium is Pareto Optimal". PROOF: (by contradiction) Suppose that an allocation bundle x (i.e. x = (x1 A, x2A, x1B, x2 B)) is a competitive equilibrium that is not Pareto Optimal. Thus, there exists an allocation bundle y (i.e. y = (y1 A, y2A, y1B, y2 B)) such that y is feasible... y1 A + y1 B = 1 A + 1 B y2 A + y2 B = 2 A + 2 B (1) (2) and UA (y) UA (x) but UB (y) > UB (x) However, since x was the allocation bundle chosen for utility maximization at the equilibrium price vector, we have... P1x1 A + P2x2A = P11A + P22A P1x1 + P2x2 = P11 + P22 B B B B Budget Constraints are satisfied. meaning that for allocation bundle y to be "better" than allocation bundle x the following must be true... P1y1 A + P2y2A P11A + P22A P1y1B + P2y2 B > P11B + P22 B which implies, P1 (y1 A + y1B ) + P2 (y2A + y2B) > P1 (1 A + 1B ) + P2 (2A + 2B) (3) but the allocation bundle y must be feasible...subbing (1) and (2) into (3), we get P1 (1 A + 1B) + P2 (2A + 2B) > P1 (1 A + 1B ) + P2 (2A + 2B) the above says that, at allocation bundle y, the value of the individual endowments for good 1 and good 2 exceeds the value of the individual endowments of goods 1 and 2 in the economy. So at this equilibrium price vector the allocation bundle y is not feasible...we have our contradiction! 197 We derived this contradiction by assuming that the competitive equilibrium solution (allocation bundle x) was n...
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