# Thus n x v x v x 0 therefore v x v x 0 v x v x 0

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Thus n x = v ( x ) v ( x 0 ). Therefore v ( x ) = v ( x 0 ) + v ( x ) v ( x 0 ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright = n x Hence, for all x R n 1 , v ( x ) v ( x ) = v ( x 0 ) v ( x 0 ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright a constant c = v ( x ) = v ( x ) c. Exercise 3.19 (5.2). Prove that if u, v : R n + R represent the same preference relation on R n + , then their Walrasian demand correspondences are equal. Proof. Suppose that u and v represent the same preference relation on R n + . Then v = h u , where h is strictly increasing. Hence x v ( p, w ) arg max x R n + v ( x ) subject to p · x w = arg max x R n + ( h u )( x ) subject to p · x w = arg max x R n + u ( x ) subject to p · x w x u ( p, w ) for each p R n ++ , w R + . Exercise 3.20 (5.3). Suppose there are two commodities, electricity and food. The price of food is linear: each unit costs $1. The price of electricity is nonlinear: the price for the first M units of electricity is$1, but the price for each unit of electricity over M is $1.50. Suppose w > M . Express the consumer’s budget set formally and draw it graphically. Note that a consumer can consume up to M units of electricity at$1, so we need to break up his budget set for when she consumes M or less electricity and for when he consumes more than M units of electricity. B ( p F , p E , w ) = braceleftbigg F + E w if E M F + M + 1 . 5( E M ) w if E > M
Section 3: More on Utility and Walrasian Demand 3-17 An alternate definition is to note that a consumer can consume at most w units of F , and at most M + 2( w M ) 3 units of E . So we can assume linear budget sets, one with F -intercept w , and prices (1 , 1), and one with E -intercept M + 2( w M ) 3 , and prices (1 , 1 . 5), and take their intersection. So in this case the budget set would be: B ( p F , p E , w ) = { x R 2 : F + E w } ∩ { x R 2 : F + 3 2 E w + 1 2 M } One last characterization of the budget set is as the convex hull of (0 , 0) , ( M, M ) , ( w, 0) , (0 , M + 2 3 ( w M )). Exercise 3.21 (5.4). Suppose there are two commodities. Compute the Walrasian demand correspondence x ( p, w ) given the following utility functions: 1. u ( x 1 , x 2 ) = x α 1 x 1 α 2 (Cobb-Douglas). 2. u ( x 1 , x 2 ) = min { x 1 , x 2 } (Leontief). 3. u ( x 1 , x 2 ) = αx 1 + βx 2 (Linear).