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Applied Maxima and Minima

  1. Dec 10, 2007 #1
    1. The problem statement, all variables and given/known data
    For a monopolist's product, the demand equation is:

    p = 42 - 4q

    and the average-cost function is

    c = 2 + (80/q)

    Find the profit-maximizing price.

    2. Relevant equations
    When i started to solve the problem, i deduced from what i needed to find that I needed to make up a function, plugging in the above functions into profit = (price-cost)quantity.
    To what extent I'm correct I'm not sure.

    3. The attempt at a solution
    I tried plugging it in like this: profit = ((42-4q-2-(80/q)) all that multiplied by q. I dont seem to know if i multiply that whole equation just by q, or by finding a formula for q from those other functions they already gave me. I tried it and it gave me q= 80/(c-2) but then, do i have to substitute c in that formula.

    I really dont know where to go from here. would appreciate any help given please.
  2. jcsd
  3. Dec 10, 2007 #2


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    Staff Emeritus
    Science Advisor

    At maximimum profit, demand= cost. I would have thought that was one of the frst thing you would have learned!
  4. Dec 10, 2007 #3
    go it solved! thanks for that little fact i did miss. that equals to 4q^2-40q+80, which its derivative is 8q-40, and its critical value is 5, with a maximum of 5. thanks. :D
  5. Dec 10, 2007 #4


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    Science Advisor
    Homework Helper

    Since this is a monopoly problem, you should be looking to equate marginal cost (MC) to marginal revenue (MR).

    Total cost is:

    C = Average Cost times q = cq = 2q + 80

    MC = C' = 2.

    Total revenue is:
    R = pq = 42q - 4q^2.

    MR = R' = 42 - 8q

    MC = MR ===> 2 = 42 - 8q ===> 8q = 40 ===> q = 5, so your answer is correct.
    Last edited: Dec 11, 2007
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