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    Calculating Profit Maximization & Cost Functions

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    Suppose there are three firms with the same individual demand function. This function is Q = 1,000 - 40P. Suppose each firm has a different cost function. These functions are:

    Firm 1: 4,000 + 5Q
    Firm 2: 3,000 + 5Q
    Firm 3: 3,000 + 7Q

    a) What price should each firm charge if it wants to maximize its profit (or minimize its loss)?
    b) Explain why the answer to the preceding question indicates that two of the firms should charge the same price and the third should charge a higher price?
    c) Which firms will be most vulnerable to a price war? Explain

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    Solution Preview

    a) The decision rule for firms to set price (to maximize their profits) is MR=MC
    First calculate the MR.
    rewriting it we get
    TR=P*Q=(1000-Q)/40 * Q = 25Q -Q^2/40
    MR is the first derivative of TR with respect to Q.
    So we have MR= dTR/dQ = 25 - 2Q/40

    Now Calculate the MC. MC=dC/dQ
    Firm 1: MC=dC/dQ=5
    Firm 2: ...

    Solution Summary

    The solution consists of displayed calculations that will guide the reader in finding the price firms should charge to maximize profit, and vulnerability in a price war. 265 words total.