## Microeconomics Question from Walter E. Williams:Edit

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.

## AnswerEdit

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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