Topic3-Consumer - Advanced Microeconomics Topic 3: Consumer...

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Advanced Microeconomics Topic 3: Consumer Demand Primary Readings: DL – Chapter 5; JR - Chapter 3; Varian, Chapters 7-9. 3.1 Marshallian Demand Functions Let X be the consumer's consumption set and assume that the X = R m + . For a given price vector p of commodities and the level of income y , the consumer tries to solve the following problem: max u ( x ) subject to p x = y x X The function x ( p , y ) that solves the above problem is called the consumer's demand function . It is also referred as the Marshallian demand function . Other commonly known names include Walrasian demand correspondence/function , ordinary demand functions , market demand functions , and money income demands . The binding property of the budget constraint at the optimal solution, i.e., p x = y , is the Walras’ Law . It is easy to see that x ( p , y ) is homogeneous of degree 0 in p and y . Examples: (1) Cobb-Douglas Utility Function: . ,..., 1 , 0 , ) ( 1 m i x x u i m i i i = = = α From the example in the last lecture, the Marshallian demand functions are: . i i i p y x = where = = m i i 1 . (2) CES Utility Functions: ) 1 0 ( ) ( ) , ( / 1 2 1 2 1 < + = ρ x x x x u Then the Marshallian demands are: , ) , ( ; ) , ( 2 1 1 2 2 2 1 1 1 1 r r r r r r p p y p y x p p y p y x + = + = - - p p where r = /( -1). And the corresponding indirect utility function is given by r r r p p y y v / 1 2 1 ) ( ) , ( - + = p Let us derive these results. Note that the indirect utility function is the result of the utility maximization problem: y x p x p x x x x = + + 2 2 1 1 / 1 2 1 , subject to ) ( max 2 1 Define the Lagrangian function: 1
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) ( ) ( ) , , ( 2 2 1 1 / 1 2 1 2 1 y x p x p x x x x L - + - + = λ ρ The FOCs are: 0 0 ) ( 0 ) ( 2 2 1 1 2 1 2 1 ) / 1 ( 2 1 2 1 1 1 1 ) / 1 ( 2 1 1 = - + = = - + = = - + = - - - - y x p x p L p x x x x L p x x x x L Eliminating λ , we get + = = - 2 2 1 1 ) 1 /( 1 2 1 2 1 x p x p y p p x x So the Marshallian demand functions are: r r r r r r p p y p y x x p p y p y x x 2 1 1 2 2 2 2 1 1 1 1 1 ) , ( ) , ( + = = + = = - - p p with r = ρ /( ρ -1). So the corresponding indirect utility function is given by: r r r p p y y x y x u y v / 1 2 1 2 1 ) ( )) , ( ), , ( ( ) , ( - + = = p p p 3.2 Optimality Conditions for Consumer’s Problem First-Order Conditions The Lagrangian for the utility maximization problem can be written as L = u ( x ) - ( p x - y ). Then the first-order conditions for an interior solution are: y u i p x u i i = = 2200 = x p p x x ) ( i.e. ; ) ( (1) 2
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Rewriting the first set of conditions in (1) leads to , , k j p p MU MU MRS k j k j kj = = which is a direct generalization of the tangency condition for two-commodity case. Sufficiency of First-Order Conditions Proposition : Suppose that u ( x ) is continuous and quasiconcave on R m + , and that ( p , y ) > 0 . If u if differentiable at x* , and ( x* , λ *) > 0 solves (1), then x* solve the consumer's utility maximization problem at prices p and income y . Proof . We will use the following fact without a proof: For all x , x ' 0 such that u ( x' ) u ( x ), if u is quasiconcave and differentiable at x , then u ( x )( x' - x ) 0.
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This note was uploaded on 06/20/2011 for the course OPR 201 taught by Professor Pp during the Spring '11 term at Thammasat University.

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Topic3-Consumer - Advanced Microeconomics Topic 3: Consumer...

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