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Unformatted text preview: Midterm Solutions Econ 382, Professor Platt, BYU March 4, 2008 1. Consider an exchange economy with two goods, apples and bananas. Rachel has an endowment of 12 apples and 0 bananas, while Sara has an endowment of 0 apples and 15 bananas. Both women have the utility function U i ( a i ,b i ) = a 1 3 i b 2 3 i . (a) (4 pts) — Solve for the competitive equilibrium in this economy. The setup: max a r ,b r a 1 3 r b 2 3 r s.t. p a a r + p b b r ≤ 12 p a max a s ,b s a 1 3 s b 2 3 s s.t. p a a s + p b b s ≤ 15 p b a r + a s = 12 ,b r + b s = 15 Solving: 1 3 b i a i 2 3 = λ i p a and 2 3 a i b i 1 3 = λ i p b = ⇒ b i 2 a i = p a p b = ⇒ p b b i = 2 p a a i Combine with budget to get a r = 12 p a 3 p a = 4 ,b r = 8 p a p b and a s = 15 p b 3 p a = 5 p b p a ,b s = 10 . Either price can be normalized. Let p * a = 1. Either resource constraint can be used due to Walras’ law: 4 + 5 p b p a = 12 = ⇒ 5 p b = 8 = ⇒ p b = 1 . 6 Then they should also list the demand for each good: a * r = 4 ,a * s = 8 ,b * r = 5 ,b * s = 10 (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 Second Fundamental Welfare Theorem. The box must exhibit each of the following features: • The a dimension should be 12 units wide; the b dimension should be 15 units wide. It should be clearly indicated which is which, and the length must be indicated. 1 • The endowment point must be identified and agree with the problem setup (that is, it is in the corner). The equilibrium allocation should also be identified and agree with the previous 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). • The appropriate explanation would say, “The point (4,8) is efficient (since the two indifference curves are tangent). Note that it is also a competitive equilibrium, using prices (1,1.6).” The point must be communicated that prices exist which make this efficient allocation a competitive outcome. It is likely that many students will invert this, saying that this market allocation is automatically efficient. That is the 1st Welfare theorem; give half a point for it. (c) (5 pts) — Correct: For any exchange economy, if the endowment is not on the contract curve, then moving to a point on the contract curve is always a Pareto improvement. This is typically false. Only the points within the “lens of opportunity” are Pareto improving. Elsewhere, one of the two participants will be harmed. Draw- ing an Edgeworth box example is an excellent way to show an exception to this statement....
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This note was uploaded on 03/17/2010 for the course ECON 388 taught by Professor Mcdonald,j during the Spring '08 term at BYU.
- Spring '08