Assume that a monopolist faces a demand curve for its product given by: p=100?1q

Question

Assume that a monopolist faces a demand curve for its product given by: p=100?1q Further assume that the firm's cost function is: TC=470+9q Using calculus and formulas (but no tables or spreadsheets) to find a solution, what is the profit (rounded to the nearest integer) for the firm at the optimal price and quantity? Round the optimal quantity to the nearest hundredth before computing the optimal price, which you should then round to the nearest cent. Note: Non-integer quantities may make sense when each unit of q represents a bundle of many individual items.

Explanation / Answer

Demand curve of monopolist is as follows -

p = 100 - 1q

TR = p * q = (100 - 1q) * q = 100q - 1q2

MR = dTR/dq = d(100q - 1q2)/dq = 100 - 2q

TC = 470 + 9q

MC = dTC/dq = d(470 + 9q)/dq = 9

A monopolist maximize profit or is at optimal solution when it produce that level of output corresponding to which MR equals MC.

MR = MC

100 - 2q = 9

2q = 91

q = 91/2 = 45.50

p = 100 - 1q = 100 - (1*45.50) = 100 - 45.50 = 54.50

Thus,

The optimal quantity is 45.50 units and the optimal price is 54.50 per unit.

Calculate profit -

Profit = Total revenue - Total cost

Profit = (p * q) - (470 + 9q)

Profit = (54.50 * 45.50) - [470 + (9*45.50)] = 2,479.75 - 879.50 = 1,600.25

The profit for the firm is 1,600.

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