midterm_301_ans_2001 - THE UNIVERSITY OF AUCKLAND 616.301...

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THE UNIVERSITY OF AUCKLAND 616.301 Advanced Microeconomics Midterm Test 2001 Answers 1. (a) Total endowment is (6 , 6). Thus in any feasible allocation that uses all the endowment x 12 = 6 - x 11 (1) x 22 = 6 - x 21 . (2) Now, notice that for any price Consumer 2 will choose x 12 = x 22 —or equivalently that if x 12 = x 22 we can take some of one of the goods away from her and give it to Consumer 1, making him better off. Thus x 12 = x 22 , and so, from equations 1 and 2, x 11 = x 21 and this is the equation for the contract curve. (b) We know that the equilibrium will be on the contract curve. (This is the First Welfare Theorem.) From part (a) the contract curve is x 11 = x 21 . Either by appealing to symmetry, or by direct calculation the slope of Consumer 1’s indifference curve on the x 11 = x 21 line is - 1. Direct calculation: The equation for Consumer 1’s indifference curve is x 21 = ¯ u x 11 and so d x 21 d x 1 u 1 u = - ¯ u x 11 2 . Now, at the point ( x, x ) we have ¯ u = xx = x 2 and so d x 21 d x 1 u 1 u ( x, x ) = - x 2 x 2 = - 1 .
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