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

MATH105 April 2011

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[hide] 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?

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[show] 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 $\displaystyle g(x,y)\approx g(x_{0},y_{0})+\left({\frac {\partial g}{\partial x}}(x_{0},y_{0})\right)\cdot (x-x_{0})+\left({\frac {\partial g}{\partial y}}(x_{0},y_{0})\right)\cdot (y-y_{0})$

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[show] 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.