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# demand1 - 8 Demand 8.2 Separability(aggregating goods 8.1...

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8 Demand 8.1 Homothethic Preferences Remember: Utility functions describe preferences up to a monotonic transformation. Thus, homotheticity homogeneity of degree 1. Properties: e ( p , u ) = e ( p ) u, v ( p , m ) = v ( p ) m, x i ( p , m ) = x i ( p ) m. Examples: Cobb-Douglas (?), Leontief (?), Linear (?), ... Is it good or bad to have all our examples homo- thetic? 8.2 Separability (aggregating goods) Functional separability: for all x , z , x 0 , z 0 ( x , z ) Â ( x 0 , z ) ( x , z 0 ) Â ( x 0 , z 0 ) . Then (weak separability) u ( x , z ) = U ( v ( x ) , z ) . Problem 1: max v ( x ) , s.t. px = m x . Problem 2: max U ( v, z ) , s.t. e ( p , v ) + qz = m. Also: Strong and Hicksian separbilities, price and quan- tity indeces.

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8.3 Aggregating consumers Aggregate demand: X ( p , m 1 , . . . , m n ) = n X i =1 x i ( p , m i ) . Properties: Continuity, Homogeneity. NO SARP, NO Slutsky, NO corresponding pref- erences. 8.3.1 “Representative” consumer Gorman indirect utility: v i ( p , m ) = a i ( p ) + b ( p ) m i . If prices are fi xed, the e ff ect of income changes is the same for all consumers. Roy’s identity implies: x i j ( p , m i ) = ∂v i ( p ,m i ) ∂p j ∂v i ( p ,m i ) ∂m = ∂a i ( p ) ∂p j b ( p ) ∂b ( p ) ∂p j b ( p ) m i = α i j ( p ) + β j ( p ) m i . Then, X ( p , m 1 , . . . , m
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demand1 - 8 Demand 8.2 Separability(aggregating goods 8.1...

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