Midterm1 Solutions-W09

# Midterm1 Solutions-W09 - First Midterm Exam Solutions Econ...

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First Midterm Exam — Solutions Econ 382, Professor Platt, BYU February 11th through 14th, 2009 1. Consider an exchange economy with two goods, cookies and milk. Isaac has an en- dowment of 6 cookies and 0 glasses of milk, while John has an endowment of 2 cookies and 8 glasses of milk. Both men have a utility function is U i ( c i ,m i ) = c i + m 1 2 i . (a) (4 pts) — Solve for the competitive equilibrium in this economy. Solution: The setup: max c i ,m i c i + m 1 2 i s.t. p c c i + p m m i 6 p c + 0 p m max c j ,m j c j + m 1 2 j s.t. p c c j + p m m j 2 p c + 8 p m c i + c j = 8 ,m i + m j = 8 Solving: 1 = λ i p c and 1 2 m - 1 2 i = λp m = m i = ± p c 2 p m ² 2 Combine with budget to get c i = 6 p c + 0 p m - p m ³ p c 2 p m ´ 2 p c = 6 - p c 4 p m The solution for John is similar, only changing endowments: m j = ³ p c 2 p m ´ 2 , and c j = 2 p c +8 p m - p m pc 2 pm 2 p c = 2 + 8 p m p c - p c 4 p m . Either price can be normalized. Let p * m = 1. Either resource constraint can be used due to Walras’ law, but Milk is the easier one: ³ p c 2 ´ 2 + ³ p c 2 ´ 2 = 8 = ³ p c 2 ´ 2 = 4 = p * c = 4 Then they should also list the demand for each good: m * i = 4 ,m * j = 4 ,c * i = 5 ,c * j = 3 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 8 units wide; the m dimension should be 8 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. 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:
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Midterm1 Solutions-W09 - First Midterm Exam Solutions Econ...

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