The behavior of a profit-maximizing monopolist setting a single price

Basic theory

A firm is a monopolist if it has no close competitors, and hence can ignore the potential reactions of other firms when choosing its output and price.

Theory: a monopolist chooses its output to maximize its profit, given the relationship between output and price as embodied in the aggregate demand function for the good it sells.

Denote by TC the monopolist's total cost function, and by TR its total revenue function (that is, TR is the product of the firm's output and the price that output fetches, given the demand function). Then the monopolist's profit is

(y) = TR(y) TC(y).

An output y* that maximizes this profit is such that the first derivative of

is zero, or

TR'(y*) = TC'(y*)

or, denoting the derivative TR' of total revenue by MR (for marginal revenue),

MR(y*) = MC(y*).

At a maximum, rather than a minimum, the second derivative of profit is nonpositive, or

TR''(y*) TC''(y*) 0

MR'(y*) MC'(y*):

the slope of MR is at most the slope of MC at the optimal output.

Finally, the monopolist's profit at y* must be nonnegative (in the long run), otherwise it would not stay in business.

In summary:

The profit maximizing output y* of a monopolist is either 0 or is positive and satisfies the following conditions:

MR(y*) = MC(y*)
MR'(y*) MC'(y*)
(y*) 0.

Graphical representation

Given a likely form for the marginal revenue function, the conditions satisfied by a positive profit maximizing output are represented in the following figure.

The three conditions that any positive optimal output y* must satisfy correspond to the following three properties of the curves in the figure.

MR(y*) = MC(y*):: The output y* is determined by the intersection of the MR and MC curves.
MR'(y*) MC'(y*):: At the intersection of the curves the slope of the MR curve must be less than the slope of the MC curve. In the figure, the MR curve is downward sloping and the MC curve is upward sloping at y*, so the condition is satisfied. Alternatively, the condition says that "the marginal cost curve must cut the marginal revenue curve from below at y*".
(y*) 0:: At the output y* the AR curve lies above the AC curve.

Note that if there is some point at which the AR curve lies above the AC curve then the optimal output is positive. Conversely, if the AR curve lies everywhere below the AC curve then there is no positive output at which the monopolist can make a profit, so the optimal output is 0. [If the AR curve just touches the AC curve, coinciding at one point and everywhere else lying above, then the monopolist is indifferent between producing no output an producing the output where the curves touch (at both points its profit is zero). (In this case, at the output at which AR is equal to AC the slopes of AR and AC are equal, so that MR is equal to MC.)] At the optimal output y*, the monopolist's profit is

TR(y*) TC(y*) = y*(AR(y*)AC(y*)).

This profit is represented by the purple area in the figure.

The inefficiency of monopoly

In a competitive equilibrium price is equal to marginal cost; if more output were produced, marginal cost would exceed price. Thus the "gains from trade" are fully realized: no more units can be sold at a price that covers MC.

In a monopoly equilibrium the same is not true: since price exceeds MR and MR is equal to MC, we conclude that

Price exceeds MC.

Thus someone who does not buy the good values a unit more than the marginal cost of producing it. The monopolist does not produce the extra unit because the marginal revenue from doing so is less than the MC, but the average revenue---price---is still bigger. If the monopolist could produce the unit and sell it to a consumer whose reservation price exceeds MC (while not changing the price at which the other units are sold), then both the monopolist's profit and the consumer's payoff would increase.

It follows that a monopoly equilibrium is not Pareto efficient: someone can be made better off without making anyone worse off.

Consider a situation in which the market consists of a large number of buyers with various reservation prices, each of whom can either not buy the good or buy one unit of it, so that the demand curve represents the number of buyers whose reservation values exceed each possible price. Then the "surplus" in a trade between a buyer and the monopolist is the difference between the buyer's reservation price and the monopolist's cost of producing that unit---its MC. The part of this surplus that goes to the consumer is the difference between the reservation price and the price the monopolist charges, and the part that goes to the monopolist is the difference between the price it charges and its MC. Thus in this case the loss of "surplus" relative to the competitive equilibrium (in which price is equal to MC) is represented as the triangle in the following figure.

Examples and exercises on a profit-maximizing monopolist that sets a single price