Science:Math Exam Resources/Courses/MATH105/April 2011/Question 08 (b)/Solution 1

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.