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Production function, returns to scale and Taylor expansion

Consider a production function of the form F(K,L)=(K^(-a)+L^(-a))^(-1/a).

(a) Is this function homogeneous?

(b) Does it display increasing, constant or decreasing returns to scale?

(c) Let G be a differentiable function. Find an expression for G(K+g,L+h) by taking a first-order Taylor expansion of G about (K,L).

(d) Show using Euler's theorem for homogeneous functions that this approximation is exact when G is the function F given above, g=K and h=L.

Solution Preview

(a) Let t>0. We calculate F(tK,tL),

F(tK,tL)=((tK)^(-a)+(tL)^(-a))^(-1/a)=(t^(-a)(K^(-a)+L^(-a)))^(-1/a)=(t^(-a))^(-1/a)(K^(-a)+L^(-a))^(-1/a)
=t (K^(-a)+L^(-a))^(-1/a)
=t F(K,L)

Then F is homogeneous (of degree 1).

(b) Given a production function F(K,L) we say that F has:

constant returns to scale if F(tK,tL)=tF(K,L), for any constant t greater than 0.
increasing returns to scale if F(tK,tL)>tF(K,L), for any constant a greater than 1
decreasing returns to scale if F(tK,tL)<tF(K,L), for any constant a greater than 1

where K and L are ...

Solution Summary

A homogeneous production function F(K,L) is studied. First, we review the concept of returns to scale from the formal point of view.

Then, after writing the Taylor expansion of a differentiable function G(K,L), we show that for the case of G(K,L)=F(K,L), the Taylor expansion and the function agree for a certain value of (K,L).

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