Question

Given the following cobb-douglas production function
Q=2k2-30L+200
Calculate the marginal product of labour calculate the marginal product of capital calculate the marginal rate of technical substitution

Answer 1

For the production function given,

the marginal product of labour will be,

$\frac{\delta Q}{\delta L}=0-1\times{30}\times{L^{1-1}}+0=-30$

and the marginal product of capital will be,

$\frac{\delta Q}{\delta K}=2\times{2\times{K^{2-1}}}-0+0=4\times{K}$

now, for finding the marginal rate for technical substitution(MRTS)

we use,

$Q=2K^2-30L+300$

or,

$K={(\frac{Q+30L-300}{2})}^{1/2}$

on differentiating with respect to L, we obtain

MRTS(K,L) =$\frac{\delta K}{\delta L}=1/2\times15\times{(\frac{Q+30L-300}{2})}^{-1/2}$