ChuckWilsonhomogeneousfunctions

ChuckWilsonhomogeneousfunctions - New York University...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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 à n X i =1 α i ( λx i ) ρ ! β = λ ρβ à 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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 7

ChuckWilsonhomogeneousfunctions - New York University...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online