Suppose the demand and total cost functions for a monopoly firm are as follows : \(Q = 100 - 0.2P\) \(P = 500 - 5Q\) \(TC = 50 + 20Q + Q^2\) What will be the profit maximization output ?
40
A monopoly, being the sole seller, chooses the output that maximises profit, which occurs where marginal revenue equals marginal cost. To apply this we derive marginal revenue from the demand relationship and marginal cost from the total cost function.
The demand is given as \(Q = 100 - 0.2P\), which rearranges to the inverse demand \(P = 500 - 5Q\). Total revenue is price times quantity: \(TR = PQ = (500 - 5Q)Q = 500Q - 5Q^2\). Differentiating gives marginal revenue \(MR = 500 - 10Q\) - note that MR falls twice as fast as the inverse demand curve, a standard result for a linear demand.
From the total cost \(TC = 50 + 100Q\), marginal cost is the derivative, \(MC = 100\) (the fixed 50 drops out).
Setting \(MR = MC\): \(500 - 10Q = 100\), so \(10Q = 400\) and \(Q = 40\). At this output the extra revenue from one more unit just equals its extra cost, so profit cannot be increased further.
Hence the profit maximization output is 40 units.
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