proof_consumer_theory

proof_consumer_theory - Lecture 9 Consumer Theory I AAE 635...

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Lecture 9 Consumer Theory I AAE 635 Fall 2010 9.1 Introduction So far we have explored the production behavior of cost minimizing firms and profit maximizing firms. We have also investigated the short run production equilibrium and the long run production equilibrium. The production behavior (and its comparative statics) under optimization gives the supply side story of the market, and for the demand side story, we need to examine the consumption behavior by rational consumers. Consider a household choosing n consumption goods x = ( x 1 , …, x n ) T . The household has preferences represented by a utility function u ( x ). Also, the household faces the budget constraint: p T x y or Σ i p i x i y where y > 0 is household income, and p = ( p 1 , …, p n ) T is a n -price vector for x , p i > 0 denoting the market price of x i , i = 1, …, n . Assume that economic rationality for household decisions is represented by the following utility maximization problem V ( p , y ) = . (1) {() : , 0 , } Tn Max u y ≤≥ x xpx x xR This simply states that the households make consumption decisions so as to maximize their utility subject to a budget constraint. Denote the solution of the optimization problem (1) by x *( y , p ). Here, x* ( y , p ) are utility maximizing decision rules for household consumption. They are also called Marshallian demand functions. The function V ( y , p ) in (1) is the indirect utility function satisfying V ( y , p ) = u ( x *( y , p )). 9.2 Consumption decisions Throughout our discussion, we will make the following (intuitive) assumption. Assumption A1 : (non-satiation) For any consumption bundle x 0 and for every ε > 0, there is a bundle x ' satisfying u ( x ') > u ( x ), x ' x , x ' x , and || x - x '|| < . Non-satiation means that the household can always increase its utility by increasing the consumption of some commodity. This generates the following key result. Under (A1), the budget constraint is necessary binding at the optimum x *: p T x *( y , p ) = y . Proof : Assume that the budget constraint is not binding at x *: p T x *( y , p ) < y . Under assumption A1 and given p > 0, there exists a consumption bundle x ' such that p T x ' y and u ( x ') > u ( x *( y , p )). But this contradicts x *( y , p ) being utility maximizing. Thus, the budget constraint must be binding at x *. This means that, under non-satiation, the utility maximization problem (1) can be written as . (2) (,) : , 0 , } T VyM a x u y == x px p x x x n R Expression (2) is a standard constrained maximization problem. It can be analyzed using the Lagrangean approach. Define the associated Lagrangean L ( x , λ , y , p ) = u ( x ) + [ y - p T x ], where is a Lagrange multiplier corresponding to the budget constraint y - p T x = 0. We have the following result: Given p > 0, the constraint qualification (CQ: rank( p ) = 1) is always satisfied. Assume that the utility function u ( x ) is twice continuously differentiable. Then, given an interior solution x *, the maximization problem (2) implies the FONC 1
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Lecture 9 Consumer Theory I AAE 635 Fall 2010 L x u λ x p = 0, (3a) L y - p T x = 0. (3b) This is a system of ( n +1) equations in ( n +1) unknowns: ( x , ). Denote the solution of this system of equations by x *( y , p ) and *( y , p ). Equation (3a) implies that
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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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proof_consumer_theory - Lecture 9 Consumer Theory I AAE 635...

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