Microeconomics Question from Walter E. Williams:
Prove that for a monopolist faced with a straight lined demand curve and forced to charge a uniform price to all buyers, total revenue will be at a maximum if the quantity sold is exactly half the quantity which buyers would take at a price of zero. Nota bene: Mathematical exposition will help you here thought not necessary for the answer.
Answer
In order for this to be true, we must assume that marginal cost (MC) is equal to zero. If MC is zero, the monopolist will produce where MR = MC. For a monopolist this will be exactly half way between zero quantity and the quantity that would be produced if the price was zero. This occurs, because a monopolist with a straight-line demand curve has a marginal revenue curve whose slope is exactly twice that of the demand curve. This leads to the conclusion that total revenue is at a maximum at the midpoint of straight-line demand curve, where elasticity is unity.
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