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

MATH105 April 2011

  •  Q1 (a)  •  Q1 (b)  •  Q2  •  Q3  •  Q4  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q6  •  Q7  •  Q8 (a)  •  Q8 (b)  •  Q8 (c)  •  Q9 (a)  •  Q9 (b)  •  Q9 (c)  •  Q9 (d)  •  Q9 (e)  •  Q9 (f)  •  Q9 (g)  •  

   Other MATH105 Exams

  •  April 2009  •  April 2011  •  April 2010  •  April 2012  •  April 2013  •  April 2014  •  April 2015  •  April 2016  •  April 2017  •  April 2018  •  

Question 08 (b)

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 government has the following two policy options, Policy I: increase labor by 1 unit without changing capital, and Policy II: increase labor by 1/2 units and capital by 1/3 units. 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!

Hint
If $g(x,y)$ is a differentiable function of two variables x and y, then near any specific point (x0, y0) we have the linear approximation $\displaystyle g(x,y)\approx g(x_{0},y_{0})+\left({\frac {\partial g}{\partial x}}(x_{0},y_{0})\right)\cdot (x-x_{0})+\left({\frac {\partial g}{\partial y}}(x_{0},y_{0})\right)\cdot (y-y_{0})$

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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. From the hint we see that a linear approximation based on a given point is $\displaystyle g(x,y)\approx f(x_{0},y_{0})+\left({\frac {\partial f}{\partial x}}(x_{0},y_{0})\right)\cdot (x-x_{0})+\left({\frac {\partial f}{\partial y}}(x_{0},y_{0})\right)\cdot (y-y_{0})$ 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 $f(x,y)>f(x_{0},y_{0})$ then the new labour/capital pairing of (x,y) is better than the old pairing $(x_{0},y_{0})=(81,16)$. If we use the first policy and only increase labour by 1 unit ($x-x_{0}=1,y-y_{0}=0$), then since the marginal productivity of labour at the current level ($f_{x}(81,16)$) is 80, the productivity will be raised by approximately 80 dollars since $\displaystyle {}f(x,y)-f(81,16)=f_{x}(81,16)(1)+f_{y}(81,16)(0)=80.$ On the other hand, if we use the second policy, an increase of labour by 1/2 unit ($x-x_{0}=1/2$) will increase productivity by $\displaystyle {}f_{x}(81,16)(x-x_{0})=80(1/2)=40$ dollars; and an increase of capital by 1/3 unit ($y-y_{0}=1/3$) will increase productivity by $\displaystyle {}f_{y}(81,16)(y-y_{0})=135(1/3)=45$ dollars. The two combined yield an increase of productivity of approximately $\displaystyle {}f(x,y)-f(81,16)=f_{x}(81,16)(1/2)+f_{y}(81,16)(1/3)=85$ dollars. So based on these approximations, the second policy should yield a higher productivity.

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Marginal productivity, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag