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    Monopolist profit maximization after tax and with price ceiling

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    Consider a monopolist facing a market demand curve given by q = 186- p(q). The monopolist's fixed costs and variable costs are equal to Cf = 2400 and C(q) = q2 /10 +10q, respectively ........(in the latter equation its actually q squared over ten plus ten q)

    Calculate:

    a) the monopolist's price-quantity combination that maximizes profits and the level of profits obtained;
    b) the monopolist's price-quantity combination that maximizes profits in case a fixed tax T = 1000 is introduced and the level of profits obtained;
    c) the monopolist's price-quantity combination that maximizes profits in case a tax for each unit of product sold tq = 11 is introduced, the amount paid in taxes and the level of profits obtained after taxes;
    d) the monopolist's price-quantity combination that maximizes profits in case a tax of the tπ = 50% on the profit is introduced; the amount paid in taxes and the level of profits obtained after taxes;
    e) the monopolist's quantity and profits in case the government fixes a p = 90.

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    https://brainmass.com/economics/monopolies/monopolist-profit-maximization-after-tax-price-ceiling-591929

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    a)
    Profit is maximized at the quantity where Marginal Revenue (MR) = Marginal Cost (MC)

    From the Demand curve:
    Q = 186 - P
    Rearranging:
    P = 186 - Q

    Total Revenue (TR) = PQ
    TR = (186 - Q)Q
    TR = 186Q - Q^2

    MR is the derivative of TR:
    MR = 186 - 2Q

    Total Cost (TC) = Variable Cost (VC) + Fixed Cost (FC)
    TC = 0.1Q^2 + 10Q + 2400

    MC is the derivative of TC:
    MC = 0.2Q + 10

    To maximize profit, let MR = MC
    186 - 2Q = 0.2Q + 10
    186 - 10 = 2Q + 0.2Q
    176 = 2.2Q
    Q = 80

    From the Demand curve:
    P ...

    Solution Summary

    This solution shows how to calculate the profit-maximizing price and quantity for a monopolist, including under scenarios where it must pay:
    - a fixed tax of 1000
    - a variable tax of 11 per unit
    - a tax of 50% of total profit

    and also where the government has imposed a price ceiling.

    $2.19