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 October 14th, 2012, 08:34 AM #1 Senior Member   Joined: Sep 2012 Posts: 112 Thanks: 0 application of differentiation in economics How to solve the following problem? A revenue maximizing monopolist requires profit of at least 1500. Its demand and cost functions are p=304-2q and c=500+4q+8q^2. Determine the output level and price. Please explain in detail.
 October 14th, 2012, 12:34 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: application of differentiation in economics Profit is revenue minus cost. Revenue is the product of price per unit times units demanded, hence: $P(q)=q\cdot p(q)-C(q)$ $P(q)=q$$304-2q$$-$$500+4q+8q^2$$$ $P(q)=304q-2q^2-500-4q-8q^2=-10q^2+300q-500=-10$$q^2-30q+50$$$ To find the maximum profit using differentiation, we find the output level for which marginal profit is zero, which may be done as follows: $\frac{d}{dq}$$q^2-30q+50$$=0$ $2q-30=0$ $q=15$ This is the output level which maximizes profit. Now to find the price at this output level, we compute: $p(15)=304-2(15)=274$ To find the output level corresponding to a profit of at least 1500, we solve the inequalilty: $1500\le-10$$q^2-30q+50$$$ $150\ge q^2-30q+50$ $q^2-30q-200\le0$ $(q-10)(q-20)\le0$ From this, we determine $10\le q\le20$. This give the range of prices as: $304-2(20)\le p\le304-2(10)$ $264\le p\le284$
 October 15th, 2012, 08:12 AM #3 Senior Member   Joined: Sep 2012 Posts: 112 Thanks: 0 Re: application of differentiation in economics But it's a revenue maximising monopolist not a profit maximising one...
October 15th, 2012, 08:47 AM   #4
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Re: application of differentiation in economics

Quote:
 Originally Posted by helloprajna But it's a revenue maximising monopolist not a profit maximising one...
Set profit equal to 1500 and find the two solutions. Take the one giving greater revenue.

October 15th, 2012, 10:11 AM   #5
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Re: application of differentiation in economics

Quote:
 Originally Posted by helloprajna But it's a revenue maximising monopolist not a profit maximising one...
Well, excuuuuuse me!

 October 16th, 2012, 01:00 AM #6 Senior Member   Joined: Sep 2012 Posts: 112 Thanks: 0 Re: application of differentiation in economics I want to solve the problem like this:- The monopolist requires a profit(P) of at least 1500. So, $P \ge1500$ or$-10q^2+300q-500 \ge1500$ or $10\le q\le20$ So the monopolist must produce his output in the range of 10 to 20 units.It's given that the monopolist maximizes his revenue. So we have to find at what value of q, revenue is maximum. At x=10, Revenue(R)= 304q-2q^2 = 2840 At x=20, Revenue(R)=5280 Clearly, as the monopolist goes on increasing his output from 10 units to 20 units, its revenue goes on increasing and becomes maximum at 20 units i.e., at x=20. So we conclude that the output level for the monopolist is x=20. Here I've combined the suggestions of you two. Please tell me if I'm right or wrong.
October 16th, 2012, 01:08 AM   #7
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Re: application of differentiation in economics

Quote:
Originally Posted by MarkFL
Quote:
 Originally Posted by helloprajna But it's a revenue maximising monopolist not a profit maximising one...
Well, excuuuuuse me!
I accidentally clicked on that "rolling eyes" smiley.
I'm sorry...
I'm really very grateful to you for all your help.

 October 16th, 2012, 01:16 AM #8 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: application of differentiation in economics Hey, no worries...I was quoting a comedian for which that expression is probably way before your time. I took it in good humor! video clip

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