022 Sample Final A, Problem 8

Find ther marginal productivity of labor and marginal productivity of capital for the following Cobb-Douglas production function:

$f(k,l)=200k^{\,0.6}l^{\,0.4}.$

(Note: You must simplify so your solution does not contain negative exponents.)

Foundations:
The word 'marginal' should make you immediately think of a derivative. In this case, the marginal is just the partial derivative with respect to a particular variable.
The teacher has also added the additional restriction that you should not leave your answer with negative exponents.

 Solution:

Marginal productivity of labor:
We take the partial derivative with respect to $l$:
${\begin{array}{rcl}{\frac {\partial f}{\partial l}}(k,l)&=&{\displaystyle 200k^{0.6}\left(0.4l^{\,0.4-1}\right)}\\\\&=&200k^{0.6}\left({\frac {2}{5}}l^{-0.6}\right)\\\\&=&{\displaystyle {\frac {80k^{0.6}}{l^{\,0.6}}}.}\end{array}}$

Marginal productivity of capital:
Now, we take the partial derivative with respect to $k$:
${\begin{array}{rcl}{\frac {\partial f}{\partial k}}(k,l)&=&{\displaystyle 200\left(0.6k^{0.6-1}\right)l^{0.4}}\\\\&=&200\left({\frac {3}{5}}k^{-0.4}\right)l^{\,0.4}\\\\&=&{\displaystyle {\frac {120l^{\,0.4}}{k^{0.4}}}.}\end{array}}$

Final Answer:
Marginal productivity of labor: ${\frac {\partial f}{\partial l}}(k,l)\,=\,\displaystyle {\frac {80k^{0.6}}{l^{\,0.6}}}.$
Marginal productivity of capital: ${\frac {\partial f}{\partial k}}(k,l)\,=\,\displaystyle {\frac {120l^{\,0.4}}{k^{0.4}}}.$

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