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    Short run and long run cost functions: Profit maximization

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    1. The demand and cost function for a company are estimated to be as follows: P(Q)=100-8Q; C(Q)=50+80Q-10Q^2+0.6Q^3

    (a) What price should the company charge if it wants to maximize profits in the short-tun?
    (b) What price should ti charge it it wants to maximize it's revenue?
    (c) Assume the company is a monopoly - what would expect their profits to be in the long-run, and why?
    (d) Now, assume the company exists under monopolistic conditions - what you you expect their profits to be in the long-run, and why?

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    Solution Preview

    Profit is maximized when Marginal Revenue (MR) = Marginal Cost (MC)

    To find MR, first find Total Revenue (TR)
    TR = PQ
    TR = (100 - 8Q)Q
    TR = 100Q - 8Q^2

    MR is the derivative of TR
    MR = 100 - 16Q

    MC is the derivative of Total Cost (TC)
    TC = 50 + 80Q - 10Q^2 + 0.6Q^3
    MC = 80 - 20Q + 1.8Q^2

    To maximize profit, let MR = MC
    100 - 16Q = 80 - 20Q + 1.8Q^2
    0 = 1.8Q^2 - 4Q - 20

    Solving ...

    Solution Summary

    This solution gives a detailed, step-by-step solution of the math required to calculate a firm's profit-maximizing and revenue-maximizing price and quantity when its demand and cost functions are known. Solutions are given for a firm in a monopoly market and a monopolistic competition market.