{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Problem-Set-5-Solutions

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(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 ).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern