HW1 - Microeconomic Theory II Assignment 1: Choice under...

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Unformatted text preview: Microeconomic Theory II Assignment 1: Choice under Uncertainty Solutions February 20, 2007 Problem 1. Concerning the Independence Axiom: L followsequal L [0 , 1] , L : L + (1 ) L followsequal L + (1 ) L 1. ( The Axiom extends to strict preference and to indifference. ) Show that the Independence Axiom implies: (i) L L (0 , 1] , L : L + (1 ) L L + (1 ) L (ii) L L [0 , 1] , L : L + (1 ) L L + (1 ) L Solution . (i) Let (0 , 1], then by the definition of , we have that L L ( L followsequal L ) ( L notfollowsoreql L ). Thus, by the independence axiom, L +(1 ) L followsequal L +(1 ) L and L + (1 ) L notfollowsoreql L + (1 ) L which by the definition of implies the desired result. (ii) Let (0 , 1], then by the definition of , we have that L L ( L followsequal L ) ( L followsequal L ). Thus, by the independence axiom, L +(1 ) L followsequal L +(1 ) L and L + (1 ) L followsequal L + (1 ) L which again by the definition of implies the desired result. trianglesolid Prepared by Omer Ozak (ozak@brown.edu), Department of Economics, Brown University. If you find any typos or mistakes please let me know, so that it can be fixed. 1 Microeconomic Theory II Solutions 2. ( The Axiom implies there is a best and a worst lottery .) Let the set of consequences be finite: C = { c 1 ,...,c N } , and let L n be the lottery giving consequence c n with probability 1, n = 1 ,...,N . Assume, WLOG, that L 1 followsequal L 2 followsequal ... followsequal L N . Then, show that the Independence Axiom implies: lotteries L : L 1 followsequal L followsequal L N Solution . Let R N + be the space of lotteries, i.e. = braceleftBigg x R n | N summationdisplay i =1 x i = 1 , x i bracerightBigg . Notice that is a convex polyhedron and that braceleftbig L 1 ,...,L N bracerightbig are its vertices, so that for every L there exists a unique set of numbers { 1 ,..., N } such that L = N summationdisplay i =1 i L i and by the conquentialist axiom we have that L N summationdisplay i =1 i L i . (1.1) Now, since the set of degenerate lotteries is finite we can always find the best and worse lottery in this set. Assume without loss of generality that L 1 followsequal L 2 followsequal ... followsequal L N (Why?...
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HW1 - Microeconomic Theory II Assignment 1: Choice under...

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