Science:Math Exam Resources/Courses/MATH105/April 2011/Question 08 (a)

MATH105 April 2011

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Question 08 (a)

The productivity (measured in the dollar value of goods produced) of a certain country is given by the function $\displaystyle f(x,y)=160x^{3/4}y^{1/4},$ where $x$ denotes the amount (in dollars) invested in labor, and $y$ is the amount invested in capital. Recall that the marginal productivity of labor (respectively capital) is the rate of change of $f$ with respect to $x$ (respectively $y$), holding $y$ (respectively $x$) fixed. Suppose that the country is currently utilizing 81 units of labor and 16 units of capital. Find the current marginal productivity of labor and also the current marginal productivity of capital.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Can you formulate the two marginal rates in terms of partial derivatives of the productivity?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The marginal productivity of labour is the rate of change of productivity with respect to the labour investment, so in this case, it corresponds to the partial derivative $\displaystyle {\begin{aligned}{\frac {\partial f}{\partial x}}(x,y)&={\frac {\partial }{\partial x}}(160x^{3/4}y^{1/4})\\&=160{\frac {3}{4}}x^{-1/4}y^{1/4}\\&=120x^{-1/4}y^{1/4}\end{aligned}}$ And the marginal productivity of capital is the rate of change of productivity with respect to the capital investment, so in this case, it corresponds to the partial derivative $\displaystyle {\begin{aligned}{\frac {\partial f}{\partial y}}(x,y)&={\frac {\partial }{\partial y}}(160x^{3/4}y^{1/4})\\&=160x^{3/4}{\frac {1}{4}}y^{-3/4}\\&=40x^{3/4}y^{-3/4}\end{aligned}}$ The question asks us to compute those marginal productivities for 81 units of labour and 16 units of capital, that is to compute $\displaystyle {\frac {\partial f}{\partial x}}(81,16)\quad {\text{and}}\quad {\frac {\partial f}{\partial y}}(81,16)$ which yields $\displaystyle {\begin{aligned}{\frac {\partial f}{\partial x}}(81,16)&=120\cdot 81^{-1/4}16^{1/4}\\&=120\cdot {\frac {1}{3}}\cdot 2\\&=80\end{aligned}}$ and $\displaystyle {\begin{aligned}{\frac {\partial f}{\partial y}}(81,16)&=40\cdot 81^{3/4}16^{-3/4}\\&=40\cdot 27\cdot {\frac {1}{8}}\\&=135\end{aligned}}$ So the marginal productivity of labour is 80 (dollars of productivity per dollar of labour investment) and the marginal productivity of capital is 135 (dollars of productivity per dollar of capital investment).

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