lecture_05_handout

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online