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Unformatted text preview: d UPF UA 185 It can be shown that the Pareto optimal solution to the Production economy results in the overall utility possibility frontier which is the envelope of all of these individual UPFs. We call this envelope the Grand UPF or Grand utility possibility frontier. All right then...what about an example. Let's consider the square root economy with unit aggregate factor endowments (K = L = 1). We solve the Pareto Optimal allocation problem for this simple economy as follows: PRODUCTION EFFICIENCY The problem is to find factor allocations (KX, KY, LX, LY) such that KX + KY = K LX + LY = L MRTSX = MRTSY (= MRTS) (1) (2) (3) Applying these general conditions to the specific problem in our square root example, we have KX + KY = 1 LX + LY = 1 LX = LY KX KY which can be solved for the following production contract curve: LX = KX L Y = KY To construct the production possibility frontier, we first substitute Pareto Optimal allocations (KX, KY, LX, LY) into the production function for good X... X = (LX KX)1/2 = (KX KX)1/2 = KX (1) (2) (3) 186 Similarly for good Y... Y = (LY KY)1/2 = (KY KY)1/2 = KY Then we can extract the following relationship between X and Y from the above information: X + Y = KX + KY =K =1 The production side of the model is thus solved with linear upward sloping contract curve (slope = 1) and a linear downward sloping PPF (MRT = slope = 1). L 1
Production Contract Curve K 1 187 CONSUMPTION EFFICIENCY The problem is to find allocations (XA, XB, YA, YB) of goods such that XA + XB = X YA + YB = Y MRSA = MRSB (= MRS) (4) (5) (6) Applying these general conditions to our specific square root example, we get XA + XB = X YA + YB = Y YA = YB XA XB (4) (5) (6) for each point on the PPF above. For example, we will solve the pure exchange model at the following three points on the PPF: Point A Point B Point C X = 1/3 X = 1/2 X = 3/4 Y = 1 1/3 = 2/3 Y = 1 1/2 = 1/2 Y = 1 3/4 = 1/4 For each point, we obtain one contract curve and one individual utility possibility frontier (UPF) as follows : [a] At point A on the PPF, we have the following equations: XA + XB = 1/3 YA + YB = 2/3 YA = YB XA XB which can be solved for the equations of the contract curve YA = 2XA YB = 2XB 188 To find the...
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This note was uploaded on 05/25/2010 for the course ECON 301 taught by Professor Sning during the Spring '10 term at University of Warsaw.
- Spring '10