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# Returns to scale and marginal product of capital and labor

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We give step by step solution to the following question.

Let f(K,L) be a production function with constants returns to scale, where K denotes capital and L denotes labor.

(a) Show that if we scale both input factors up or down by t>0, the marginal products of labor and capital remain the same.

(b) Show that

f_(11)(K,L)K+f_(12) (K,L)L=0

for all K and L. Here f_(11)=d^2f/dK^2, the second order partial derivative of f with respect to K, and f_(12)=d^2f/dKdL, the second order mixed partial derivative of f with respect to K and L.

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#### Solution Preview

Recall that a function f(K,L) is called a production function with constant returns to scale if

f(tK,tL)=t f(K,L) for all t>0 and K,L --------------(1)

(a) The marginal product of labor (MPL) is defined by MKL=df/dL and the marginal product of capital is MPK=df/dK.Here d/dL and d/dK denote the partial derivatives with respect to L and K, respectively.

In (a) we are asked to show that df/dL(tK,tL)= df/dL(K,L) and df/dK(tK,tL)= df/dK(K,L) for all K,L and any t>0.

Use (1) and differentiate both sides with respect to K

d/dK(f(tK,tL))=d/dK(t f(K,L)).

In the left hand side we use the chain rule to obtain:

d/dK(f(tK,tL))=tdf/dK(tK,tL)

that is the derivative of ...

#### Solution Summary

It is shown that for a production function f(K,L) with constants returns to scale, the marginal products of labour and capital remain the same if we scale both input factors, K and L, by t>0. Also a relation between the second derivatives of f is established.

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