ChuckWilsonhomogeneousfunctions

# ChuckWilsonhomogeneousfunctions - New York University...

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New York University Department of Economics V31.0006 C. Wilson Mathematics for Economists May 7, 2008 Homogeneous Functions For any α R , a function f : R n ++ R is homogeneous of degree α if f ( λx )= λ α f ( x ) for all λ> 0 and x R n ++ . A function is homogeneous if it is homogeneous of degree α for some α R . Afunct ion f is linearly homogenous if it is homogeneous of degree 1. Along any ray from the origin, a homogeneous function de f nes a power function. If f is linearly homogeneous, then the function de f ned along any ray from the origin is a linear function. Example: Consider a Cobb-Douglas production, f ( x Q n j =1 x α j j , where each α j > 0 , and let β = P n j =1 α j . Then f ( λx n Y j =1 ( λx j ) α j = λ β n Y j =1 ( x j ) α j = λ β f ( x ) . So f is homogeneous of degree β. Example: Consider a CES production function f ( x )=( P n i =1 α i x ρ i ) β , where β,ρ > 0 . Then f ( λx Ã n X i =1 α i ( λx i ) ρ ! β = λ ρβ Ã n X i =1 α i x ρ i ! β = λ ρβ f ( x ) . So f is homogeneous of degree ρβ. If f is homogeneous of degree 0 , then f ( λx f ( x ) . Why? If f is homogeneous of degree α 6 =0 , then f 1 α is homogenenous of degree 1. Why? Let f ( x ³ Q n j =1 x α j j ´ 1 β , where each α j > 0 and β = P n j =1 α j . Then f is linearly homogeneous. Why? f ( x )=min { x i : i =1 ,...,n } is linearly homogeneous. Why? The following theorem relates the value of a homogeneous function to its derivative. Theorem 1: If f : R n ++ R is continuously di f erentiable and homogeneous of degree α, then Df ( x ) · x = n X i =1 f i ( x ) x i = αf ( x ) . (Euler’s theorem) Proof. If f is homogeneous of degree α, then for any x R n ++ and any 0 , we have f ( λx λ α f ( x ) . http://www.wilsonc.econ.nyu.edu

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V31.0006: Homogeneous Functions May 7, 2008 Page 2 Then holding x f xed and di f erentiating both sides with repect to λ, we obtain df ( λx ) = Df ( λx ) · x = n X i =1 f i ( λx ) x i = d ( λ α f ( x )) = αλ α 1 f ( x ) Letting λ =1 , yields the statement to be proved. The next theorem relates the homogeneity of a function to the homogeneity of its partial derivatives. Theorem 2: If f : R n ++ R is continuously di f erentiable and homogeneous of degree α, then each partial derivative f i is homogeneous of degree α 1 . Proof. For f xed x R n ++ and λ> 0 , de f ne each g i ,h i :( x i , ) R by g i ( t )= f ( λ ( x + e i t )) and h i ( t λ α f ( x + e i t ) Then the homogeneity of f implies g i ( t f ( λ ( x + te i )) = λ α f ( x + te i h i ( t ) and therefore g 0 i ( t h 0 i ( t ) for all t ( x i , ) But g 0 i (0) = Df ( λx ) · λe i = λf i ( λx ) h 0 i (0) = λ α Df ( x ) · e i = λ α f i ( x ) So f i ( λx λ α 1 f i ( x ) . Example: In the example above, we showed that f ( x Q n i =1 x α i i is homogeneous of degree β = P n i =1 α i . To verify Euler’s theorem, observe that, for each j ,...,n, we have f j ( x α j x α j 1 j Y i 6 = j x α i i = α j x j n Y i =1 x α i i = α j f ( x ) x j .
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ChuckWilsonhomogeneousfunctions - New York University...

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