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Rubinstein2005-page76 - 12:18 58 master Sheet number 74...

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October 21, 2005 12:18 master Sheet number 74 Page number 58 58 Lecture Five Proof: Once again, we could use the fact that the preferences have a contin- uous utility representation and apply a standard “maximum theo- rem.” (If the function f ( x , a ) is continuous, then the function h ( a ) = argmax x f ( x , a ) is continuous.) However, I prefer to present a proof that does not use the notion of a utility function: Assume not. Then, there is a sequence of price vectors p n con- verging to p such that x ( p , w ) = x , and x ( p n , w ) does not converge to x . Thus, we can assume that ( p n ) is a sequence converging to p such that for all n the distance d ( x ( p n , w ) , x ) > ε for some positive ε . All numbers p n k are greater than some positive number m . There- fore, all vectors x ( p n , w ) belong to some compact set (the hypercube of bundles with no quantity above w / m ) and thus, without loss of generality, we can assume that x ( p n , w ) y for some y = x .
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