Question 08 (c)
The productivity (measured in the dollar value of goods produced) of a certain country is given by the function
where denotes the amount (in dollars) invested in labor, and is the amount invested in capital. Recall that the marginal productivity of labor (respectively capital) is the rate of change of with respect to (respectively ), holding (respectively ) fixed.
Find all vectors
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!
The gradient vector
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).
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
In part (a) we computed the partial derivatives, allowing us to write the gradient vector
hence (as calculated previously)
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