Solutions - ECON 5113 Advanced Microeconomics Winter 2011...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

This note was uploaded on 06/20/2011 for the course OPR 201 taught by Professor Pp during the Spring '11 term at Thammasat University.

Page1 / 5

Solutions - ECON 5113 Advanced Microeconomics Winter 2011...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online