Rubinstein2005-page79 - x  y iF there is p w so that both...

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October 21, 2005 12:18 master Sheet number 77 Page number 61 Demand: Consumer Choice 61 Figure 5.2 (a) Satisfes the weak axiom. (b) Does not satisy the weak axiom. In the general discussion oF choice Functions, we saw that the weak axiom (WA) was a necessary and suFfcient condition For a choice Function to be derived From some preFerence relation. In the prooF, we constructed a preFerence relation out oF the choices oF the decision maker From sets containing two elements. We showed (by looking into his behavior at the choice set { a , b , c } ) that WA implies that it is impossible For a to be revealed as better than b , b revealed as better than c , and c revealed as better than a . However, in the context oF a consumer, fnite sets are not within the scope oF the choice Function. In the same spirit, adjusting to the context oF the consumer, we might try to defne
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Unformatted text preview: x  y iF there is ( p , w ) so that both x and y are in B ( p , w ) and x = x ( p , w ) . In the context oF the consumer model the Weak Axiom is read: iF px ( p , w ) ≤ w and x ( p , w ) 6= x ( p , w ) , then p x ( p , w ) > w . WA guarantees that it is impossible that both x  y and y  x . However, the defned binary relation is not necessarily complete: there can be two bundles x and y such that For any B ( p , w ) containing both bundles, x ( p , w ) is neither x nor y . ±urthermore, in the general discussion, we guaranteed transitivity by looking at the union oF a set in which a was revealed to be better than b and a set in which b was revealed to be as good as c . However, when the sets are budget sets, their union is not necessarily a budget set. (See fg. 5.2.)...
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