Rubinstein2005-page69

# Rubinstein2005-page69 - ferentiable utility function u •...

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October 21, 2005 12:18 master Sheet number 67 Page number 51 Consumer Preferences 51 Problem 7. ( Easy ) We say that a preference relation satisFes separability if it can be represented by an additive utility function, that is, a function of the type u ( x ) = 6 k v k ( x k ) . Show that such preferences satisfy that for any subset of commodities J , and for any bundles a , b , c , d , we have ( a J , c J ) % ( b J , c J ) ( a J , d J ) % ( b J , d J ) , where ( x J , y J ) is the vector that takes the components of x for any k J and takes the components of y for any k / J . Demonstrate this condition geometrically for K = 2. Problem 8. ( Moderate ) Let % be monotonic and convex preferences that are represented by a dif-
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Unformatted text preview: ferentiable utility function u . • Show that for every x there is a vector v ( x ) of K nonnegative numbers so that d is an improvement at x iff dv ( x ) > 0 ( dv ( x ) is the inner product of v ( x ) ). • Show that the preferences represented by the function min { x 1 , . . . , x K } cannot be represented by a differentiable utility function. • Check the differentiability of the lexicographic preferences in < 2 . • Assume that for any x and for any d ∈ D ( x ) , ( x + d ) Â x . What can you say about % ?...
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