Practical Homogeneous and homothetic functions PDF

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Homogeneous and homothetic functions...

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Excursus: Homogeneous and homothetic functions Homogeneous function Let F be a function of arguments x = (x1 , x2 , ..., xn ). Then, function F is homogeneous of degree k if F (λx) = λk F (x)

for all λ > 0.

(1)

Euler’s theorem furthermore states that n X

xi Fi′ (x) = kF (x).

(2)

i=1

P To prove this theorem, we can differentiate Eq. (1) with respect to λ. This gives ni=1 xi Fi′ (λx) = kλk−1 F (x). Setting λ = 1 establishes Eq. (2). As an application of the Euler theorem, we can consider a linear homogeneous production technology (k = 1) with two inputs (labor, L and capital, K): y = F (L, K). We know from the lecture notes that profit maximization of price-taking firms establishes the marginal value product rule: pF2′ (L, K) = r,

pF1′ (L, K) = w,

(3)

with w, r referring to wages and the interest rate. Eqs. (2) and (3) then establish LF 1′ (L, K) + LF2′ (L, K) = F (L, K) =⇒ wL + rK = pF (L, K)

(4)

where the latter is the zero profit condition. There is one further notable feature of homogenous functions: If F (x) is homogeneous of degree k, then F i′ (x), i = 1, 2, ..., n, is homogeneous of degree k − 1. To see this, we can partially differentiate Eq. (1) w.r.t. xi . This gives λFi′(tx) = λk F i′ (x) and thus Fi′ (λx) = λk−1 F (x).

Homothetic functions We call a function homothetic if it can be represented by a monotonic transformation of a homogeneous function. To be more specific, we cal function G(x) homothetic, if it can be written in the following way: G(x) = H (F (x)) ,

(5)

where H(·) is strictly increasing and F is homogeneous of degree k (see above). We complete the excursus by interpreting G(x) as a homothetic utility function and show that in this case the marginal rate of substitution of any two goods is homogeneous of degree zero. For this purpose, we first partially differentiate G(x) w.r.t xi and xj . This 1

gives Gi′ (x) = H ′ (·) F i′ (x) and G′j (x) = H ′ (·) Fj′ (x). This gives for the marginal rate of substitution between i and j (M RSij (x) = −dxi /dxj ): M RSij (x) =

H ′ (·) Fj′ (x) Fj′ (x) Gj (x) = ′ = Gj (y) H (·) Fi′ (x) Fi′ (x)

(6)

Since F (x) is homogeneous of degree k, we know that F ′i (x) and Fj′ (x) are homogeneous of degree k − 1. This implies M RSij (λx) =

λk−1 F j′ (x) Fj′ (λx) F j′ (x) = = λk−1 Fi′ (x) Fi′ (x) Fi′ (λx)

(7)

and proves that M RSij (x) is homogeneous of degree zero. An immediate consequence of this is that the consumptions expansion paths for homothetic utility functions are linear rays from the origin.

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