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Problem-Set-5-Solutions

Problem-Set-5-Solutions - Problem Set 5 Solutions Econ...

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Problem Set 5 — Solutions Econ 310-2, Winter 2011 1 PE in an Exchange Economy For the purpose of taking derivatives, it may be useful to write u 1 as follows: u 1 ( x 1 , x 2 ) = x 1 y 1 = x 1 · y 1 = x 1 / 2 1 y 1 / 2 1 First, solve for the marginal utilities: MU x 1 = ∂u 1 ∂x 1 = y 1 / 2 1 · 1 2 · x - 1 / 2 1 = y 1 2 x 1 MU x 2 = ∂u 2 ∂x 2 = 1 MU y 1 = ∂u 1 ∂y 1 = x 1 / 2 1 · 1 2 · y - 1 / 2 1 = x 1 2 y 1 MU y 2 = ∂u 2 ∂y 2 = 1 We can now solve for the marginal rates of substitution: MRS 1 = MU x 1 MU y 1 = y 1 2 x 1 x 1 2 y 1 = y 1 2 x 1 · 2 y 1 x 1 = y 1 x 1 MRS 2 = MU x 2 MU y 2 = 1 The following equality identifies the interior PE allocations: MRS 1 = y 1 x 1 = 1 = MRS 2 This simplifies to y 1 = x 1 . Therefore, an interior allocation is Pareto efficient if and only if it lies on the line y 1 = x 1 and satisfies the feasibility conditions 0 x 1 12 and 0 y 1 8. Note that including both feasibility constraints is a bit redundant because y 1 = x 1 . It suffices to just put bounds on either x 1 or y 1 . However, since e x = 12 > 8 = e y , the feasibility constraint for good y is binding before that of good x . Thus, the set of interior PE allocations are those for which 0 x 1 8, y 1 = x 1 , x 2 = 12 - x 1 , and y 2 = 8 - y 1 . Graphically, the set of interior PE allocations is the line starting in the bottom-left corner of the Edgeworth box and increasing with a slope of 1 until it reaches the upper boundary of the box. That is, it is the line connecting the allocation ((0 , 0) , (12 , 8)) to the allocation ((8 , 8) , (4 , 0)). Figure 1 illustrates the Edgeworth box and this part of the contract curve. It also includes some indifference curves for each consumer. There are several ways to solve for the boundary PE allocations. I think the simplest method is to divide the Edgeworth box into two regions, (i) allocations with y 1 > x 1 , and (ii) allocations with y 1 < x 1 . Let’s consider which, if any, allocation in these regions could be PE: 1
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(a) Case (i) — y 1 > x 1 : In this region, note that MRS 1 = y 1 x 1 > 1 = MRS 2 . Therefore, at such an allocation, both individuals would benefit if consumer 1 trades some y to consumer 2 in exchange for some x (at a rate of E units of y per unit of x for some MRS 1 > E > MRS 2 ).
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