lecture_05_handout

Suppose not there exists an x 2 x p w with p x w hence

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Unformatted text preview: there exists an x 2 x (p ; w ) with p x < w . Hence, there exists some y such that ky By local non satiation, this implies y x k < " (with " > 0) and p y x contradicing x 2 x (p ; w ). w Properties of Walrasian Demand III Proposition If u is quasiconcave, then x (p ; w ) is convex. If u is strictly quasiconcave, then x (p ; w ) is unique. We proved this last class (u is (stricly) quasiconcave means % is (strictly) convex). Proof. Suppose x ; y 2 x (p ; w ) and pick 2 [0; 1]. First convexity: need to show x + (1 By de…nition of x (p ; w ), x % y . )y 2 x (p ; w ). u is quasiconcave, thus % is convex and x + (1 )y % y . By de…nition of x (p ; w ), y % z for any z 2 Bp ;w . Transitivity implies x + (1 x + (1 )y 2 x (p ; w ). )y % z for any z 2 Bp ;w ; thus Now uniqueness. x ; y 2 x (p ; w ) and x 6= y imply x + (1 )y y for any because u is strictly quasiconcave (% is strictly convex). Since Bp ;w is convex, x + (1 2 (0; 1) )y 2 Bp ;w , contradicting y 2 x (p ; w ). Properties of Walrasian Demand IV Proposition If u is continuous, then x (p ; w ) is nonempty and compact. We already proved this as well. Proof. Use A = Bp ;w = fx 2 Rn : p x + wg (a closed and bounded, i.e. compact, set) and x (p ; w ) = C% (A) = C% (Bp ;w ) where % are the preferences represented by u . Walrasian Demand: Examples How do we …nd the Walrasian Demand? Need to solve a constrained maximization problem, usually using calculus. Question 1. Problem Set 3, due next Tuesday. Consider a setting with 2 goods. For each of the following utility functions, …nd the Walrasian demand correspondence. (Hint: pictures may help) 1 u (x ) = x1 x2 for ; 2 u (x ) = minf x1 ; x2 g for ; 3 4 > 0 (Cobb-Douglas) u (x ) = x1 + x2 for ; u (x ) = 2 x1 + 2 x2 > 0 (generalized Leontief) > 0 (linear) Walrasian Demand: Examples Constant elasticity of substitution (CES) preferences are the most commonly used homothetic preferences. Many preferences are a special case of CES. Question 2. Problem Se...
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