{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Problem-Set-2-Solution

Problem-Set-2-Solution - Problem Set 2 Solutions TA Guannan...

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

View Full Document Right Arrow Icon
Problem Set 2 — Solutions TA: Guannan Luo Econ 310-2, Winter 2011 1 Transformation of Utility and SWFU (a) Formally, the problem we have to solve is Max SWFU = u 1 + u 2 subject to ( u 1 , u 2 ) being on the UPS. There are (at least) 2 methods to solve this problem. Method 1: Substituting the utility functions into the SWFU and using the feasibility constraint we get W ( u 1 ( x 1 ) , u 2 ( x 2 )) = u 1 ( x 1 ) + u 2 ( x 2 ) = x 1 + 2 x 2 = 8 - x 2 + 2 x 2 Setting the first order condition equal to zero gives 0 = 0 - 1 + 1 x 2 x 2 = 1 x 1 = 8 - 1 = 7 Hence the allocation that maximizes the utilitarian SWFU is given by x 1 = 7 and x 2 = 1, which corresponds to u 1 = 7 , u 2 = 2. See figure 1. Method 2 : We know that when the solution is interior (which is true in this exercise), welfare maximization occurs where the UPF and the social indifference curves are tangent (or, in other words, where the slope of the SWFU equals the slope of the UPF). Therefore slope of SWFU = slope of UPF - 1 = - 1 8 - u 1 u 1 = 7 Using the equation for the UPF we get u 2 = 2 8 - 7 u 2 = 2 Notice that this is the same answer ( u 1 = 7 and u 2 = 2 , which corresponds to x 1 = 7 and x 2 = 1) as before. (b) Formally, the problem we have to solve is Max SWFU = min { u 1 , u 2 } subject to ( u 1 , u 2 ) being on the UPS. You should remember from previous courses that we can’t solve this problem 1
Image of page 1

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

View Full Document Right Arrow Icon
using calculus (why?). Instead, we get the solution by equating the utilities of the individuals (why? If they are not equal, we can increase the Rawlsian SWFU by transferring some goods from the individual with higher utility to the individual with lower utility). Method 1 : Using the utility functions and the feasibility constraint we get u 1 ( x 1 ) = u 2 ( x 2 ) x 1 = 2 x 2 x 1 = 2 8 - x 1 x 2 1 = 4(8 - x 1 ) x 2 1 + 4 x 2 - 32 = 0 ( x 1 - 4) ( x 1 + 8) = 0 x 1 = 4 (solution) , x 1 = - 8 (not feasible) x 2 = 8 - 4 = 4 The allocation that maximizes the Rawlsian SWFU is given by x 1 = 4 and x 2 = 4, which corresponds to u 1 = 4 , u 2 = 4 . See figure 2. Method 2 : Using the equation for the UPF we get u 1 = u 2 u 1 = 2 8 - u 1 u 2 1 + 4 u 1 - 32 = 0 ( u 1 - 4) ( u 1 + 8) = 0 u 1 = 4 (solution) , u 1 = - 8 (not feasible) u 2 = 2 8 - 4 u 2 = 4 Notice that this is the same answer as before ( u 1 = 4 , u 2 = 4 , which corresponds to x 1 = 4 and x 2 = 4) .
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern