Intermediate Microeconomics: A Modern Approach, Seventh Edition

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14.04 - Problem Set 1 Due Sept 22nd in recitation 1) Start with an arbitrary utility function u x 1 , x 2 that is differentiable. Let v u be a monotonic transformation of u . a) Solve: max u x 1 , x 2 ST : p 1 x 1 p 2 x 2 m b) Solve: max v u x 1 , x 2 �� ST : p 1 x 1 p 2 x 2 m c) Discuss the relationship between these problems. What characteristics of the utility function is generating this result? 2) Consider the following problem: max x y x , y Subject to: x py 10, x 0, y 0 a) Show formally that the utility function x y is at least weakly monotonic and strongly convex for 0 . You may use ideas from problem 1 to simplify the problem. b) Find V , p , x , p , y , p 3) Solve the following: max ln x y x , y ST : 2 x y 10, x 0, y 0 4) One way to rule out the potential that the non negativity constraints aren’t binding is to look at the marginal rate of substitution (MRS) when one of the factors gets arbitrarily close to zero. Suppose that we have a function f x 1 , x 2 . The MRS 12 x 1 , x 2 is the amount of x 1 required to keep the function f the same when x 2 changes by a small amount.
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