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Midterm Solutions1

Midterm Solutions1 - First Midterm Exam Solutions Econ 382...

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First Midterm Exam — Solutions Econ 382, Professor Platt, BYU October 9th through 11th, 2008 1. Consider an exchange economy with two goods, coats and sweaters. David has an endowment of 6 coats and 6 sweaters, while Rachel has an endowment of 6 coats and 9 sweaters. David’s utility function is U d ( c d , s d ) = c 1 3 d s 2 3 d , while Rachel’s is U r ( c r , s r ) = c 2 3 d s 1 3 d . (a) (4 pts) — Solve for the competitive equilibrium in this economy. Solution: The setup: max c d ,s d c 1 3 d s 2 3 d s.t. p c c d + p s s d 6 p c + 6 p s max c r ,s r c 2 3 r s 1 3 r s.t. p c c r + p s s r 6 p c + 9 p s c d + c r = 12 , s d + s r = 15 Solving: 1 3 s d c d 2 3 = λp c and 2 3 c d s d 1 3 = λp s 2 p c c d = p s s d = p c c d + 2 p c c d = 6 p c + 6 p s = c d = 6 p c + 6 p s 3 p c , s d = 12 p c + 12 p s 3 p s 2 3 s r c r 1 3 = λp c and 1 3 c r s r 2 3 = λp s p c c r = 2 p s s r = 2 p s s r + p s s r = 6 p c + 9 p s = s r = 6 p c + 9 p s 3 p s , c r = 12 p c + 18 p s 3 p c Either price can be normalized. Let p * c = 1. Either resource constraint can be used due to Walras’ law: 6 p c + 6 p s 3 p c + 12 p c + 18 p s 3 p c = 12 18 + 24 p s 3 = 12 = 24 p s = 18 = p * s = 3 4 Then they should also list the demand for each good: c * d = 3 . 5 , s * d = 9 . 33 , c * r = 8 . 5 , s * r = 5 . 66 1

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(b) (4 pts) — Illustrate: Sketch an Edgeworth box representing this economy. Be sure to label the axes, indicate the endowment point and the equilibrium allocation, and sketch in the budget line and indifference curves at equilibrium. Explain how your graph agrees with the First Fundamental Welfare Theorem. Solution: The box must exhibit each of the following features: The c dimension should be 12 units wide; the s dimension should be 15 units wide. It should be clearly indicated which is which, and the length must be indicated. The endowment point must be identified and agree with the problem setup. The equilibrium allocation should also be identified and agree with the pre- vious answer. Check this carefully. The budget line must run through both points. The indifference curves must be tangent to the budget line at the equilibrium allocation, and should be smooth (no intentional kinks). An explanation must be present such as: “The two indifference curves are tangent to each other, so there is no opportunity for Pareto improvements at the competitive equilibrium.” Be strict on this; whatever they say must correctly link the graph to the 1st welfare theorem. (c) (5 pts) — Set Up the problem to find all Pareto efficient allocations. Solution: max c d ,s d ,c r ,s r c 1 3 d s 2 3 d s.t. c 2 3 r s 1 3 r ¯ u, c d + c r = 12 , and s d + s r = 15 They may also substitute to eliminate c r and s r .
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Midterm Solutions1 - First Midterm Exam Solutions Econ 382...

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