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Unformatted text preview: T. Mitra Fall, 2008 Economics 6170 Problem Set 7 [Due on Wednesday, November 12] 1. [Converse of Euler’s 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 Euler’s 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 α 1 x 1- α 2 1 + x α 1 x 1- α 2 for all ( x 1 ,x 2 ) ∈ R 2 + where...
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- Fall '08
- Continuous function, Inverse function, X1, R++