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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 ′ ⇐⇒ ∀ α ∈ [0 , 1] , ∀ L ′′ : αL + (1 − α ) L ′′ followsequal αL ′ + (1 − α ) L ′′ 1. ( The Axiom extends to strict preference and to indifference. ) Show that the Independence Axiom implies: (i) L ≻ L ′ ⇐⇒ ∀ α ∈ (0 , 1] , ∀ L ′′ : αL + (1 − α ) L ′′ ≻ αL ′ + (1 − α ) L ′′ (ii) L ∼ L ′ ⇐⇒ ∀ α ∈ [0 , 1] , ∀ L ′′ : αL + (1 − α ) L ′′ ∼ αL ′ + (1 − α ) L ′′ Solution . (i) Let α ∈ (0 , 1], then by the definition of ≻ , we have that L ≻ L ′ ⇐⇒ ( L followsequal L ′ ) ∧ ( L ′ notfollowsoreql L ). Thus, by the independence axiom, αL +(1 − α ) L ′′ followsequal αL ′ +(1 − α ) L ′′ and αL ′ + (1 − α ) L ′′ notfollowsoreql αL + (1 − α ) L ′′ which by the definition of ≻ implies the desired result. (ii) Let α ∈ (0 , 1], then by the definition of ∼ , we have that L ≻ L ′ ⇐⇒ ( L followsequal L ′ ) ∧ ( L ′ followsequal L ). Thus, by the independence axiom, αL +(1 − α ) L ′′ followsequal αL ′ +(1 − α ) L ′′ and αL ′ + (1 − α ) L ′′ followsequal αL + (1 − α ) L ′′ which again by the definition of ∼ implies the desired result. trianglesolid ∗ Prepared by ¨ Omer ¨ Ozak ([email protected]), 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 Δ ⊂ R N + be the space of lotteries, i.e. Δ = braceleftBigg x ∈ R n  N summationdisplay i =1 x i = 1 , x i ≥ bracerightBigg . Notice that Δ is a convex polyhedron and that braceleftbig L 1 ,...,L N bracerightbig are its vertices, so that for every L ∈ Δ there exists a unique set of numbers { α 1 ,...,α N } such that L = N summationdisplay i =1 α i L i and by the conquentialist axiom we have that L ∼ N summationdisplay i =1 α 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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This note was uploaded on 07/06/2009 for the course ECON 206 taught by Professor G.loury during the Spring '07 term at Brown.
 Spring '07
 G.LOURY
 Microeconomics

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