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Problem-Set-2-Solution

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

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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

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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) .
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