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

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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
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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 0 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? Because we know that this set has a best and worse lottery, we only might need to rename the lotteries for this to hold). Let L Δ and let braceleftbig α L 1 , . . . , α L N bracerightbig be the weights of equation (1.1) attached to this lottery. Now, the independence axiom and the consequentialist axiom we have that L 1 α L 1 L 1 + (1 α L 1 ) L 1 followsequal α L 1 L 1 + (1 α L 1 ) L 2
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