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lecture_05_handout - Walrasian Demand Econ 2100 Fall 2012...

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Walrasian Demand Econ 2100, Fall 2012 Lecture 5, 11 September Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions
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Walrasian Demand The consumer can use her non negative income to purchase weakly positive amounts of the n available goods (commodities) at the exogenously given stricyly positive prices. De°nition Given a utility function u : R n + ! R , the Walrasian demand correspondence x ° : R n ++ ° R + ! R n + is de°ned by x ° ( p ; w ) = arg max x 2 B p ; w u ( x ) where B p ; w = f x 2 R n + : p ± x ² w g : The Walrasian demand correspondence is given by the choices induced by the % represented by u , given the budget set: x ° ( p ; w ) = C % ( B p ; w ) : Therefore x ° ( p ; w ) % x for any x 2 B p ; w :
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Walrasian Demand De°nition Given a utility function u : R n + ! R , the Walrasian demand correspondence x ° : R n ++ ° R + ! R n + is de°ned by x ° ( p ; w ) = arg max x 2 B p ; w u ( x ) where B p ; w = f x 2 R n + : p ± x ² w g : Hidden Assumptions: goods are perfectly divisible; consumption is non negative; prices are linear in consumption prices are strictly positive; income is non negative; the total price of consumption cannot exceed income. Think of possible violations. Example Suppose w = $ 100. There are two commodities, electricity and food. Each unit of food costs $ 1. The °rst 20 Kwh electricity cost $ 1, but the price of each incremetal unit of electricity is $ 1 : 50. Write the consumer±s budget set formally and draw it.
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Properties of Walrasian Demand I We next derive some properties of Walrasian demand directly from assumptions on preferences. Proposition Walrasian demand is homogeneous of degree zero: for any ° > 0 x ° ( ° p ; ° w ) = x ° ( p ; w ) Proof. For any ° > 0, B ° p w = f x 2 R n + : ° p ± x ² ° w g Since ° is a scalar, it simpli°es and B ° p w = f x 2 R n + : p ± x ² w g = B p ; w : Since the constraints are the same, the utility maximizing choices must also be the same.
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Properties of Walrasian Demand II Proposition (Walras±s Law) If u represents a locally nonsatiated % , then p ± x = w for any x 2 x ° ( p ; w ) Proof. Suppose not: there exists an x 2 x ° ( p ; w ) with p ± x < w . Hence, there exists some y such that k y ³ x k < " (with " > 0) and p ± y ² w By local non satiation, this implies y ´ x contradicing x 2 x ° ( p ; w ) .
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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 ³ ° ) y 2 x ° ( p ; w ) . By de°nition of x ° ( p ; w ) , x % y . u is quasiconcave, thus % is convex and ° x + ( 1 ³ ° ) y % y . By de°nition of x ° ( p ; w ) , y % z for any z 2 B p ; w . Transitivity implies ° x + ( 1 ³ ° ) y % z for any z 2 B p ; w ; thus ° x + ( 1 ³ ° ) y 2 x ° ( p ; w ) .
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