Answer to Question #222729 in Differential Equations for frank

Question

If the productivity of s company is represented by the function

f(x,y)=40x^3/4y^1/4 with the utilisation of x units of the capital. Find

i) F_x and F_y and interpret

ii)If today the company is using 400 units and 250 units of labour and capital respectively.

determine the change in productivity if the labour is increased y 12 units and the capital is decreased by 8 units.

Answer 1

$F(x,y)=40x^{3/4}y^{1/4}$ - the productivity of a company,

$x$ - number of units of the capital, $y$ - number of units of labour

(i)

$F_x(x,y)=40\cdot \frac{3}{4}x^{3/4-1}y^{1/4}=30(y/x)^{1/4}$

$F_y(x,y)=40x^{3/4}\cdot \frac{1}{4}y^{1/4-1}=10(x/y)^{3/4}$

These formulae mean that the productivity of a company changes approximately with linear rate $F_x(x,y)=30(y/x)^{1/4}$ per each new unit of the capital and $F_y(x,y)=10(x/y)^{3/4}$ per each new unit of labour, when the current values of the capital and labour equal to $x$ and $y$ correspondently.

(ii)

If $x=400$, $y=250$ then

$F_x(x,y)=30(y/x)^{1/4}=30(250/400)^{1/4}=26.7$

$F_y(x,y)=10(x/y)^{3/4}=10(400/250)^{3/4}=14.2$

With these $x$ and $y$, if $\Delta y=12$ and $\Delta x=-8$, then the value of change will be approximately

$\Delta F\approx F_x\Delta x+F_y \Delta y=26.7\cdot(-8)+14.2\cdot12=-43.2$