See the attached file.
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
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
In the left hand side we use the chain rule to obtain:
that is the derivative of ...
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.