Question

Show the mathematical argument that for a monopolist who faces a downward-sloping demand curve, marginal revenue is less than price whenever quantity sold is positive.

Answer 1

A price effect is there in the monopoly. Therefore, it must reduce price to sell more output. As it gets less revenue for previous sold units, the marginal revenue on the additional unit sold is less than the price.

As the firm faces a downward sloping demand, P(q - 1) > P(q). .....(i)

P - price

q - quantity

Marginal revenue (MR) = P(q)×q − P(q−1)×(q−1)

Price = P(q)×q − P(q)×(q−1)

For, MR to be less than price, MR - Price < 0

MR - Price = - P(q-1)(q-1) + P(q)(q-1) which is less than 0 as condition (i) holds true.

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