Find ther marginal productivity of labor and marginal productivity of capital for the following Cobb-Douglas production function:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(k, l) = 200k^{\,0.6}l^{\,0.4}.}
(Note: You must simplify so your solution does not contain negative exponents.)
| Foundations:
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| 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.
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| The teacher has also added the additional restriction that you should not leave your answer with negative exponents.
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Solution:
| Marginal productivity of labor:
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we take the partial derivative with respect to  :
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- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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}}}
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| Marginal productivity of capital:
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Now, we take the partial derivative with respect to  :
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- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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}}}
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| Final Answer:
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Marginal productivity of labor:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial f}{\partial l}(k,l)\,=\,\displaystyle{\frac{80k^{0.6}}{l^{\,0.6}}}.}
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Marginal productivity of capital:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial f}{\partial k}(k,l)\,=\,\displaystyle{\frac{120l^{\,0.4}}{k^{0.4}}}.}
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