Rubinstein2005-page67

# Rubinstein2005-page67 - k at point x If all vectors du dx k...

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October 21, 2005 12:18 master Sheet number 65 Page number 49 Consumer Preferences 49 Figure 4.5 Differentiable preferences. Examples: The preferences represented by 2 x 1 + 3 x 2 are differentiable. At each point x , v ( x ) = ( 2, 3 ) . The preferences represented by min { x 1 , ... , x K } are differen- tiable only at points where there is a unique commodity k for which x k < x l for all l 6= k (verify). For example, at x = ( 5, 3, 8, 6 ) , v ( x ) = ( 0, 1, 0, 0 ) . Assume u is a differentiable quasi-concave utility function rep- resenting the consumer’s preferences. Let du / dx k ( x ) be the partial derivative of u with respect to the commodity
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Unformatted text preview: k at point x . If all vectors ( du / dx k ( x )) of partial derivatives are nonzero, then the in-duced preference is differentiable with v k ( x ) = du / dx k ( x ) (the partial derivative of u with respect to the commodity k at the point x ). Bibliographic Notes Recommended readings : Kreps 1990, 32–37; Mas-Colell et al. 1995, Chapter 3, A–C. The material in this lecture up to the discussion of differentiability is fairly standard and closely parallels that found in Arrow and Hahn (1971)....
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