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October 14th, 2012, 07:34 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=3042q and c=500+4q+8q^2. Determine the output level and price. Please explain in detail. 
October 14th, 2012, 11:34 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 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: To find the maximum profit using differentiation, we find the output level for which marginal profit is zero, which may be done as follows: This is the output level which maximizes profit. Now to find the price at this output level, we compute: To find the output level corresponding to a profit of at least 1500, we solve the inequalilty: From this, we determine . This give the range of prices as: 
October 15th, 2012, 07:12 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, 07:47 AM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: application of differentiation in economics Quote:
 
October 15th, 2012, 09:11 AM  #5  
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: application of differentiation in economics Quote:
 
October 16th, 2012, 12:00 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, or or 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)= 304q2q^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, 12:08 AM  #7  
Senior Member Joined: Sep 2012 Posts: 112 Thanks: 0  Re: application of differentiation in economics Quote:
My expression to your answer was not like that. I'm sorry... I'm really very grateful to you for all your help.  
October 16th, 2012, 12:16 AM  #8 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 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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