Question

Suppose that a monopolist's inverse demand curve can be expressed as:

P= 10,000 +100Q - 10Q^{2} The monopolist's total cost
curve is TC= 5,000Q

a. Use Calculus to determine the monopolist's marginal revenue curve

b. Use calcuus to determine the monopolist's marginal cost curve

c. What is monopolist's profit-maximizing level of output?

d. What price should the monopolist charge to maximize its profit?

e. What is the profit that the monopolist makes?

Answer #1

(a)

Total revenue (TR) = P x Q = 10,000Q + 100Q^{2} -
10Q^{3}

Marginal revenue (MR) = dTR/dQ = 10,000 + 200Q -
30Q^{2}

(b)

Marginal cost (MC) = dTC/dQ = 5,000

(c)

Profit is maximized by equating MR and MC.

10,000 + 200Q - 30Q^{2} = 5,000

30Q^{2} - 200Q - 5,000 = 0

3Q^{2} - 20Q - 500 = 0 [Dividing by 10]

Solving this quadratic equation using online solver,

Q = 16.67 or Q = - 10 (This s inadmissible since Q >= 0)

(d)

P = 10,000 + (100 x 16.67) - (10 x 16.67 x 16.67) = 10,000 + 1,667 - 2,778.89 = 8,888.11

(e)

TR = P x Q = 8,888.11 x 16.67 = 148,164.79

TC = 5,000 x 16.67 = 83,350

Profit = TR - TC = 148,164.79 - 83,350 = 64,814.79

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