Solutions

# Solutions - ECON 5113 Advanced Microeconomics Winter 2011...

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Unformatted text preview: ECON 5113 Advanced Microeconomics Winter 2011 Answers to Selected Exercises Instructor: Kam Yu The following questions are taken from Geoffrey A. Jehle and Philip J. Reny (2001) Advanced Microeconomic The- ory , Second Edition, Boston: Addison Wesley. The up- dated version is available at the course web page: http://flash.lakeheadu.ca/ ∼ kyu/E5113/Main.html Ex. 1.14 Let U be a continuous utility function that represents % . Then for all x , y ∈ R n + , x % y if and only if U ( x ) ≥ U ( y ). First, suppose x , y ∈ R n + . Then U ( x ) ≥ U ( y ) or U ( y ) ≥ U ( x ), which means that x % y or y % x . There- fore % is complete. Second, suppose x % y and y % z . Then U ( x ) ≥ U ( y ) and U ( y ) ≥ U ( z ). This implies that U ( x ) ≥ U ( z ) and so x % z , which shows that % is transitive. Finally, let x ∈ R n + and U ( x ) = u . Then U- 1 ([ u, ∞ )) = { z ∈ R n + : U ( z ) ≥ u } = { z ∈ R n + : z % x } = % ( x ) . Since [ u, ∞ ) is closed and U is continuous, % ( x ) is closed. Similarly (I suggest you to try this),- ( x ) is also closed. This shows that % is continuous. Ex. 1.33 1 Suppose on the contrary that E is bounded above in u , that is, for some p , there exists M > such that M ≥ E ( p ,u ) for all u in the domain of E . Let u * = V ( p ,M ). Then E ( p ,u * ) = E ( p ,V ( p ,M )) = M = p T x * , where x * is the optimal bundle. Since U is continuous, there exists a bundle x in the neighbourhood of x * such that U ( x ) = u > u * . Since U strictly increasing, E is strictly increasing in u , so that E ( p ,u ) > E ( p ,u * ) = M . This contradicts the assumption that M is an upper bound. 1 It may be helpful to review the proof of Theorem 1.8. Ex. 1.45 Since d i is homogeneous of degree zero in p and y , for any α > 0 and for i = 1 ,...,n , d i ( α p ,αy ) = d i ( p ,y ) . Differentiate both sides with respect to α , we have ∇ p d i ( α p ,αy ) T p + ∂d i ( α p ,αy ) ∂y y = 0 . Put α = 1 and rewrite the dot product in summation form, the above equation becomes n X j =1 ∂d i ( p ,y ) ∂p j p j + ∂d i ( p ,y ) ∂y y = 0 . (1) Dividing each term by d i ( p ,y ) yields the result. Ex. 1.46 Suppose that U ( x ) is a linearly homogeneous utility function. (a) Then E ( p ,u ) = min x { p T x : U ( x ) ≥ u } = min x { u p T x /u : U ( x /u ) ≥ 1 } = u min x { p T x /u : U ( x /u ) ≥ 1 } = u min x /u { p T x /u : U ( x /u ) ≥ 1 } (2) = u min z { p T z : U ( z ) ≥ 1 } (3) = uE ( p , 1) = ue ( p ) In (2) above it does not matter if we choose x or x /u directly as long as the objective function and the con- straint remain the same. We can do this because of the objective function is linear in x . In (3) we simply rewrite x /u as z ....
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Solutions - ECON 5113 Advanced Microeconomics Winter 2011...

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