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

MATH105 April 2011

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Question 08 (c)

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. Find all vectors $|{\textbf {v}}|=|\nabla f(81,16)|$ that point in the direction of no change in the productivity from its current value.

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
The gradient vector $\displaystyle \nabla f(81,16)$ points to the direction of maximal increase of productivity. Any vector perpendicular to that one will have no change of productivity (it's the direction of a level curve, that is a curve where f is constant).

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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. In part (a) we computed the partial derivatives, allowing us to write the gradient vector $\displaystyle \nabla f(x,y)=(120x^{-1/4}y^{1/4},40x^{3/4}y^{-3/4})$ hence (as calculated previously) $\displaystyle \nabla f(81,16)=(80,135)$ Recall that a perpendicular vector to (a,b) with the same length is is (b,-a) and (-b,a). Therefore, the perpendicular vectors (of the same length as the gradient itself) are the vectors $\displaystyle (-135,80)\quad {\text{and}}\quad (135,-80)$

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