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# Demand - Consumer theory Demand functions Maria Saez Marti...

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Consumer theory: Demand functions Maria Saez Marti O ce 210, IEW tel. 044 634 37 13 e-mail: [email protected]

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Summary x B such that x % x for all x B (1) max x R n + u ( x ) s.t. p · x y. (2) Continuous preferences can be represented by utility functions For such preferences a demand correspondence exists if all prices are strictly positive If, in addition preferences are strictly convex, demand function exists If, in addition, utility functions are strictly monotone, the marginal rate of substitution equals the price ratio.
1.5 Indirect Utility and Expenditure The Indirect Utility Function The relationship among prices, income, and the maximized value of utility can be summarized by a real-valued function v : R n + × R + R de fi ned by the indirect utility function: v ( p , y ) = max x R n + u ( x ) s.t. p · x y. (3) When u ( x ) is continuous, v ( p , y ) is well-de fi ned for all p >> 0 and y 0 If, in addition, u ( x ) is strictly quasiconcave, then the solution is unique and we write it as x ( p , y ), the consumer’s demand function. v ( p , y ) = u ( x ( p , y )) . (4)

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Figure 1: Indirect utility at prices p and income y
Theorem 6: If u ( x ) is continuous and strictly increasing on R n + , then v ( p , y ) de fi ned in (3) is 1. Continuous on R n ++ × R + 2. Homogeneous of degree zero in ( p , y ) 3. Strictly increasing in y 4. Decreasing in p. 5. Quasiconvex in ( p , y ) 6. Roy’s identity holds: If v ( p , y ) is di ff erentiable at ³ p 0 , y 0 ´ and ∂v ³ p 0 , y 0 ´ /∂y 6 = 0 , then x i ³ p 0 , y 0 ´ = ∂v ³ p 0 , y 0 ´ /∂p i ∂v ³ p 0 , y 0 ´ /∂y , i = 1 , ..., n.

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CES Functions Example 2 u ( x 1 , x 2 ) = ³ x ρ 1 + x ρ 2 ´ 1 , where 0 6 = p < 1 By Example 1, Marshallian demands are: x 1 ( p , y ) = p r 1 1 y p r 1 + p r 2 , (5) x 2 ( p , y ) = p r 1 2 y p r 1 + p r 2 for r ρ/ ( ρ 1)
Calculation of indirect utility: v ( p , y ) = [( x 1 ( p , y )) ρ + ( x 2 ( p , y )) ρ ] 1 (6) = p r 1 1 y p r 1 + p r 2 ρ + p r 1 2 y p r 1 + p r 2 ρ 1 = y p r 1 + p r 2 ³ p r 1 + p r 2 ´ ρ 1 = y ( p r 1 + p r 2 ) 1 /r

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The Expenditure Function To construct the indirect utility function, we fi xed prices and in- come , and sought the maximum utility the consumer could achieve.
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