Rubinstein2005-page70

Rubinstein2005-page70 - Fned by weak inequalities and...

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October 21, 2005 12:18 master Sheet number 68 Page number 52 LECTURE 5 Demand: Consumer Choice The Rational Consumer’s Choice from a Budget Set In Lecture 4 we discussed the consumer’s preferences. In this lec- ture we adopt the “rational man” paradigm in discussing consumer choice. Given a consumer’s preference relation on X = K + , we can talk about his choice from any set of bundles. However, since we are laying the foundation for “price models,” we are interested in the consumer’s choice in a particular class of choice problems called budget sets. A budget set is a set of bundles that can be represented as B ( p , w ) = { x X | px w } , where p is a vector of positive numbers (interpreted as prices) and w is a positive number (interpreted as the consumer’s wealth). Obviously, any set B ( p , w ) is compact (it is closed since it is de-
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Unformatted text preview: Fned by weak inequalities, and bounded since for any x ∈ B ( p , w ) and for all k , 0 ≤ x k ≤ w / p k ). It is also convex since if x , y ∈ B ( p , w ) , then px ≤ w , py ≤ w , x k ≥ 0, and y k ≥ 0 for all k . Thus, for all α ∈ [ 0, 1 ] , p [ α x + ( 1 − α) y ] = α px + ( 1 − α) py ≤ w and α x k + ( 1 − α) y k ≥ 0 for all k , that is, α x + ( 1 − α) y ∈ B ( p , w ) . We will refer to the problem of Fnding the %-best bundle in B ( p , w ) as the consumer’s problem . Claim: If % is a continuous relation, then all consumer problems have a solution. Proof: If % is continuous, then it can be represented by a continuous utility function u . By the deFnition of the term “utility representation,”...
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