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PS12_Solutions - Intermediate Microeconomics 2008 Problem...

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Intermediate Microeconomics, 2008 Problem Set No 12: Solutions Problems Q1) For the following utility functions and endowments calculate the contract curve. Draw the Edgeworth Box corresponding to this economy and draw the contract curve in it. Illustrate the set of allocations that are individually rational for both persons. If person A is chosen to be the dictator, which bundle will she choose? If person A is chosen to be the proposer in a simple bargaining game, which bundle will she propose? The contract curve is the set of all pareto optimal points in the Edgeworth box, or the set of all points where the indifference curves for A and B are tangent. The general steps for finding the contract curve are: 1) Find MRS A 2) Find MRS B 3) Set MRS A = MRS B 4) Substitute using the endowment constraint: w i = x A i + x B i for i = 1 , 2. So we can write x B 1 = w 1 - x A 1 and x B 2 = w 2 - x A 2 . 5) Solve for x A 2 in terms of x A 1 . This is just because we want to graph the curve and x 2 is on the y-axis. a) U A ( q 1 , q 2 ) = U B ( q 1 , q 2 ) = q 1 q 2 .e A = (20 , 20) , e B = (100 , 60) First we want to find the total endowments w 1 and w 2 . w 1 = 20 + 100 = 120. w 2 = 20 + 60 = 80. Then we need to find MRS = MU 1 MU 2 = q 2 q 1 . Since A and B have the same utility function their MRS are the same, in terms of q 1 and q 2 . So setting them equal to each other we have: q A 2 q A 1 = q B 2 q B 1 = w 2 - q A 2 w 1 - q A 1 = 80 - q A 2 120 - q A 1 q A 2 q A 1 = 80 - q A 2 120 - q A 1 120 q A 2 - q A 1 q A 2 = 80 q A 1 - q A 1 q A 2 120 q A 2 = 80 q A 1 q A 2 = 2 3 q A 1 1
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Once we have an equation for q A 2 in terms of q A 1 , we have found our contract curve. If we plug in q A 1 = 120 - q B 1 and q A 2 = 80 - q B 2 , we get the same equation: 80 - q B 2 = 2 3 (120 - q B 1 ) 80 - q B 2 = 80 - 2 3 q B 1 q B 2 = 2 3 q B 1 This contract curve is simply the straight line between the two corners of the Edgeworth box, shown below. The set of individually rational (IR) points is the set of points that make both A and B (weakly) better off than at the original endowment. So the IR set is those points that are above A’s indifference curve and below B’s indifference curve, i.e. the points that are within (and on) the lens formed by the indifference curves. The IR region is shaded in the figure below. If person A is chosen to be the dictator, A will always choose to take everything. Dictator A will propose e A = ( w 1 , w 2 ) = (120 , 80), leaving B with e B = (0 , 0). If person A is chosen to be the proposer in a simple bargaining game, i.e. A can propose any bundle and then B gets to accept or decline the new bundle, A will propose the best bundle that B will accept. The bundles that B will accept are those bundles that will make him weakly better off. The best that A can do will be the point on B’s current indifference curve that is tangent to A’s indifference curve. Since the contract curve gives us the set of points where A’s indifference curves are tangent to B’s indifference curve, the optimal point for A in a bargaining game is the point at the intersection of B’s current indifference curve and the contract curve, point B in the figure below.
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