Question 08 (b)
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.
Suppose that the government has the following two policy options,
Using the marginal productivities obtained in part (a), find the approximations for the changes in productivity under each of these policies. Based on these approximations, which policy should the government encourage for higher productivity?
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!
If is a differentiable function of two variables x and y, then near any specific point (x0, y0) we have the linear approximation
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.
From the hint we see that a linear approximation based on a given point is
but how does this relate to the problem at hand? We want to know how small changes from our current labour/capital setup of (81,16) will affect the overall productivity. If we used our linear approximation formula above then if then the new labour/capital pairing of (x,y) is better than the old pairing .
If we use the first policy and only increase labour by 1 unit (), then since the marginal productivity of labour at the current level () is 80, the productivity will be raised by approximately 80 dollars since
On the other hand, if we use the second policy, an increase of labour by 1/2 unit () will increase productivity by
dollars; and an increase of capital by 1/3 unit () will increase productivity by
dollars. The two combined yield an increase of productivity of approximately
So based on these approximations, the second policy should yield a higher productivity.