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March 8th, 2018, 09:27 PM   #1
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How to solve Differential Problem?

The productivity of a certain country is given by the function
f(x, y) = 45x^(4/5)y^(1/5)
when x units of labor and y units of capital are utilized. What is the approximate change in the number of units produced if the amount expended on labor is decreased from 243 to 240 units and the amount expended on capital is increased from 30 units to 35 units? (Round your answer to the nearest whole number.)

May someone please help?

Thanks!
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March 9th, 2018, 05:52 AM   #2
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What have you tried?
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March 9th, 2018, 06:30 AM   #3
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Assuming x is multiplied by 240/243 and y is multiplied by 35/30,
f(x, y) is multiplied by (240/243)^(4/5)(35/30)^(1/5), which is about 1.0211.

What does "productivity" mean?
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March 9th, 2018, 06:55 AM   #4
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Quote:
Originally Posted by skipjack View Post
Assuming x is multiplied by 240/243 and y is multiplied by 35/30,
f(x, y) is multiplied by (240/243)^(4/5)(35/30)^(1/5), which is about 1.0211.

What does "productivity" mean?
Productivity is the number of units produced (i.e. $f$) and I'm pretty sure the problem is asking to estimate the change in productivity using the linearization of $f$ at $(243,30)$. I'm not sure why you divided these.
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March 9th, 2018, 07:44 AM   #5
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My math instructor told me to use partial derivatives and plug in given values.
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March 9th, 2018, 08:51 AM   #6
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Quote:
Originally Posted by SDK View Post
Productivity is the number of units produced . . .
Units of what? During what period? Isn't the productivity of a country usually given in financial terms, such as annual GDP per capita?

The problem refers to the amount expended on labor, whereas the equation given uses x for the number of units of labor. Those aren't the same.
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March 15th, 2018, 10:05 PM   #7
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Ask for clarification on the units of measure. Generic units of labor and capital aren't conducive to thought.

Hours of labor and dollars of capital result in a function of the form hours^dollars, which is somewhat mind-boggling.

So find out what the unit of measure is for f(x,y)
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April 7th, 2018, 03:28 PM   #8
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If $\displaystyle f(x, y) = 45x^{4/5}y^{1/5}$

Then $\displaystyle \frac{\partial f}{\partial x}= (4/5)(45)x^{4/5-1}y^{1/5}= 36x^{-1/5}y^{1/5}$
and $\displaystyle \frac{\partial f}{\partial y}= (1/5)(45)x^{4/5}y^{1/5- 1}= 9x^{4/5}y^{-4/5}$.

Now, if x and y are functions of some variable, t, $\displaystyle \frac{df}{dt}= \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}$

$\displaystyle \frac{df}{dt}= \left(36x^{-1/5}y^{1/5}\right)\frac{dx}{dt}+ \left(9x^{4/5}y^{-4/5}\right)\frac{dy}{dt}$.

"the amount expended on labor is decreased from 243 to 240 units and the amount expended on capital is increased from 30 units to 35 units"

So we can take x= 243, y= 30. Taking the time in which this took place is one unit of "t", $\displaystyle \frac{dx}{dt}= 240- 243= -3$ and $\displaystyle dy/dt= 35- 30= 5$.

Last edited by Country Boy; April 7th, 2018 at 03:31 PM.
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