Problem_Set_7 - T. Mitra Fall, 2008 Economics 6170 Problem...

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Unformatted text preview: T. Mitra Fall, 2008 Economics 6170 Problem Set 7 [Due on Wednesday, November 12] 1. [Converse of Eulers Theorem] (a) Let f : R n ++ R ++ be a continuously differentiable function on its domain, which satisfies x f ( x ) = f ( x ) for all x R n ++ . Show that f is homogeneous of degree one on its domain. (b) Generalize the result in (a) to provide an appropriate converse of Eulers theorem. 2. [Homothetic Functions] A function F : R n + R + is called a homothetic function on R n + if there exists a function f : R n + R + which is homogeneous of degree one on R n + , and there exists a function g : R + R + which is an increasing function on R + , such that F ( x ) = g ( f ( x )) for all x R n + . (a) Let F : R n + R + be a function which is homogeneous of degree r > on R n + . Show that F is a homothetic function on R n + . (b) Let F : R 2 + R + be defined by: F ( x 1 ,x 2 ) = x 1 x 1- 2 1 + x 1 x 1- 2 for all ( x 1 ,x 2 ) R 2 + where...
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Problem_Set_7 - T. Mitra Fall, 2008 Economics 6170 Problem...

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