San José State University
Department of Economics
Thayer Watkins
Silicon Valley
& Tornado Alley

Monopoly Pricing

Economist have known the rule, since Methuselah was a child, that a protected monopolist sets a level of output such that marginal revenue is equal to marginal cost. This level of output determines the price. This is correct but there is more insights to be gained by making price rather than output the operative variable.

Let P be the price and the demand function be Q(P). Let C(Q) be the cost function. Then in the absence of taxes the profit Π is given as

Π = PQ(P) − C(Q(P))

The effect of an increase in price on profit is given by:

dΠ/dP = Q(P) + P(dQ/dP) − C'(Q)(dQ/dP)
or, equivalently
dΠ/dP = Q(P) + [P − C'(Q)](dQ/dP)

The quantity demanded Q(P) is positive, [P − C'(Q)] is positive so unless dQ/dP is negative, a monopolist will not stop raising the price. Thus if the government keeps creating programs, as in the case of health care, to prevent price from affecting the quantity demanded the price will keep rising indefinitely.

In general the above condition means that (dΠ/dP)>0 if

Q(P) + [P − C'(Q)](dQ/dP) > 0
or, equivalently
−[P − C'(Q)](dQ/dP) < Q(P)
−(1/Q)(dQ/dP) < 1/[P−C'(Q)]
|ε| < P/[P−C'(Q)]
[P−C'(Q)]/P < 1/|ε|

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