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Unformatted text preview: Microeconomics Comprehensive Examination Homotheticity Michael Powell Department of Economics, UCLA June 21st, 2005 1 Homotheticity The purpose of these notes is to clearly de&ne homotheticity, to introduce an alternative de&nition and prove its equivalence, and then to discuss its importance in aggregating preferences. De&nition 1 The utility function u ( & ) is homothetic if 8 x; x 2 X , u ( x ) ¡ u ( x ) implies that u ( &x ) ¡ u ( &x ) , & > . This de&nition is quite di¢ cult to work with, especially with respect to establishing the homotheticity of a utility function. It is much easier to work, instead, with the de&nition which is in Simon and Blume, which I will state as a proposition. Proposition 2 u ( & ) is homothetic if and only if u ( & ) is an increasing transformation of a homogeneous (of any degree) function. Proof. ( ( ) u ( & ) an increasing transformation of a homogeneous (of any degree) function implies u ( & ) is homothetic. Let v ( & ) be homogeneous of degree k (i.e., v ( &x ) = & k v ( x ) ). Let f be an increasing transformation (i.e., a ¡ b () f ( a ) ¡ f ( b ) ) De&ne u = f ¢ v ....
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This note was uploaded on 02/01/2011 for the course ECONOMY 6 taught by Professor Fallahi during the Spring '10 term at Cambridge.
- Spring '10