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April 26th, 2011, 02:33 PM   #1
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Average cost differentiation

Hi i need some help to derive the AC (average cost) function and find the output which minimises the AC

I've been given TC= 24 +0.3q
TR= 297 - 13q
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April 26th, 2011, 06:41 PM   #2
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Re: Average cost differentiation

The average cost AC is found by dividing the total cost TC by the output quantity q:



Now, to find the output that minimizes AC, we must differentiate with respect to q, and equate to 0:



Since this is equal to zero when the numerator is equal to zero, we have:



Taking the positive root:



To ensure that this is a minimum, we note 8 < q < 9 and:

< 0" /> and

Now, if q is in units, then we should round to q = 9. This would have been a nicer problem if we had been given TC = 24.3 + 0.3q.

Something else we may use is "When average cost is neither rising nor falling (at a minimum or maximum), marginal cost equals average cost."

To see why this works, consider:







Now, marginal cost MC is the derivative of TC with respect to q:





Equating the two, we have:







as we found before.
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April 26th, 2011, 10:41 PM   #3
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Re: Average cost differentiation

Thanks that really helped alot
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April 26th, 2011, 10:48 PM   #4
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Re: Average cost differentiation

You're welcome and welcome to the forum!

Does this question have more to it? I'm wondering where the total revenue function comes into play.
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April 26th, 2011, 11:56 PM   #5
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Re: Average cost differentiation

oh yeah the TR is then used to; set up the firms profit function and then find the output that will maximize the profit.
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April 27th, 2011, 12:00 AM   #6
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Re: Average cost differentiation

but as far as i could see i didnt need AC to work out a profit function as

?= TR-TC

my maximizing output was 11.165 or 11 rounded i wonder if i was right
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April 27th, 2011, 12:15 AM   #7
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Re: Average cost differentiation

I've never seen ? used to represent profit, but any character can be used!



Obviously, the profit function has its maximum at q = 0...are you sure the TR function is correct?
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April 27th, 2011, 12:29 AM   #8
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Re: Average cost differentiation

Its what i got on my sheet... but the TR and TCs numbers are based on my student ID number so everyones numbers are meant to be different...
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April 27th, 2011, 12:37 AM   #9
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Re: Average cost differentiation

How did you get 11.165?
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