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posted by  unknownstudent on 7/15/2008 9:57:49 PM  |  status: Closed  

Elasticity

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Question Details:
The demand function of a company's product is:
The current demand is 3000 units.
 
What price will maximize revenues? What will be the demand at this price?
 
I've tried everything that I could to try and solve it but I can't seem to do it. I'll show everything that I've done so far.
 
 
 <---- used implicit differentiation
   <------ equation for elasticity
<----- set E =1 to try and solve for maximum revenue
<--- try and solve for p
 
 
 
 
 
I keep getting p = 0 but that doesn't really make any sense at all.  When I tried to solve it in terms of x, I kept on getting 0 = 3000.
 
Please help me, I promise i'll rate as high as possible!
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Sage
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posted by Achilles on 7/15/2008 10:35:22 PM  |  status: Live
Asker's Rating: Lifesaver   
Response Details:
The demand function is a hyperbolic function translated -3000 units
Let us look at the graph first


As price cannot be smaller than zero, as price  gets close and close to zero  demand increases more and max price occurs at p=0
You can clearly see the function increasing and max occurs at x=0 without calculating
As p approaches zero demand approaches 60000
So max at x=0
Oracle
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posted by Cdutch88 on 7/15/2008 10:43:33 PM  |  status: Live
Asker's Rating: Helpful   
unknownstudent's comment:
"Your explanation clarified a lot of things. Thank you so much!"
Response Details:
The reason you are arriving at this conclusion is because the price at which revenue will be maximized is infinitesimally larger than zero. That's why you are getting p=0.

Let me explain with some numbers:
If you plot a few points and calculate the matching revenues:
(p, x) = R
(20, 0) = 0
(10, 3000) = 30,000
(1, 57000) = 57,000
(0.5, 117,000) = 58,500
(0.25, 237,000) = 59,250

you will see that the revenue approaches a limit of 60,000 as the price approaches 0.
This reflects the equation that you derived above:
x = 60000/p - 3000
R = xp = 60000 - 3000p

Clearly from the above equation, to maximize revenue, p = 0 so that R = 60000.

I hope that helps to clarify things!
Lifesaver means I answered your question correctly and error-free. Please rate accordingly, I'd be super grateful! Thanks!
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