If there is only one firm in the industry because of some artificial restriction, then one possibility is to remove that restriction---if the outcome that results after the restriction is lifted is likely to be better than the monopoly outcome. But if the industry is a natural monopoly then lifting a restriction to entry will not in fact induce entry---in this case, an entrant cannot make a profit (the market is not big enough to support more than one firm).

Three ways in which the monopolist might be controlled are considered in the following sections.

Suppose the monopolist is not allowed to charge a price above *p*_{0}. Then if it sells less than is demanded at *p*_{0} it must do so at the price *p*_{0} (rather than at a higher price), and so its marginal revenue is *p*_{0}. If it sells more than is demanded
at the price *p*_{0} then the price is the same as it is in the absence of any restriction, and hence its marginal revenue is the same as it was originally. Thus its marginal revenue has a discontinuity, as in the following figure.

In the presence of the restriction, the firm's optimal output is *y*_{0} (note that its profit at this output is positive), where the marginal revenue has a discontinuity: for smaller outputs MR exceeds MC, and for larger outputs MR is less than MC. The outcome of the regulation is thus that the price falls to *p*_{0} (from its
original value *p**) and output increases from *y** to *y*_{0}.

If the regulated price ceiling is exactly the competitive price then *if the firm makes a profit*, it is induced to produce the efficient output. If it does not make a profit at this output, then a subsidy, in addition to a price ceiling, is necessary to induce the firm to produce the efficient output.

Notice that in order to set the price ceiling exactly at the level that induces the firm to produce the efficient output, the regulator has to know the firm's marginal cost curve. (And if the firm knows how the information about its MC is going to be used by the regulator, obviously it has an incentive not to reveal what it is.)

The outcome that this regulation produces is illustrated in the figure. In the left panel the output *y*_{0} under the regulation is larger than the efficient output *y**, while in the right panel it is less than the efficient output. (In both cases the curves are drawn so that there is only one output at which AC and AR are equal; in other cases there may
be more than one such output.)

To see this, suppose that a firm uses capital (input 2) and another input (1). The return on capital is

TR(wherey)w_{1}z_{1},

TR(wherey)w_{1}z_{1}rz_{2},

If *r* = *w*_{2} then rate-of-return regulation is equivalent to average cost pricing: it requires the firm to make zero profit. Assume that *r* > *w*_{2}. Then under rate-of-return regulation the firm's profit-maximization problem is:

max_{y,z}_{1}_{,z}_{2}[TR(y)w_{1}z_{1}w_{2}z_{2}] subject toF(z_{1},z_{2})yandTR(y)w_{1}z_{1}+rz_{2}.

Let (*y**,*z*_{1}*,*z*_{2}*) be a solution of this problem. Consider the *y**-isoquant. The regulation requires that the firm use a pair of inputs satisfying the regulatory constraint TR(*y*)
*w*_{1}*z*_{1} + *r**z*_{2}; that is, a pair (*z*_{1}, *z*_{2}) **above** the line TR(*y*) = *w*_{1}*z*_{1} +
*r**z*_{2}. The slope of this line is *w*_{1}/*r* (write *z*_{2} as a function of *z*_{1}), so since *r* > *w*_{2}, the line is not as steep as an isocost line. Note that
as *r* changes, the line pivots around the point it intersects the *z*_{1} axis.

The problem of the firm is represented in the following figure. The firm has to choose an input bundle (*z*_{1}, *z*_{2}) on or above the *y**-isoquant **and** on or above the regulatory constraint (i.e. in the area shaded light blue); it wants to choose the one that is on the lowest possible isocost line. The point
in the figure that satisfies this condition is (*z*_{1}*, *z*_{2}*). In the absence of regulation the firm would use the input bundle (*z*_{1}', *z*_{2}').

Thus we see that the firm uses more of input 2 than if would if it were unregulated: the output is not produced at minimal cost. If the input prices reflect their social costs (as they do in a competitive equilibrium of the input markets) then the outcome is inefficient. Of course, this inefficiency has to be balanced against any change in the output of the monopolist in the direction of the efficient output, and also against any change in the distribution of income that the policy induces.

As *r* changes, the regulatory constraint pivots around the point it intersects the *z*_{1} axis. Thus when *r* is very large, the regulatory constraint has no effect on the firm. When *r* is not so large, as in the figure, the firm uses less of input 1 and more of input 2 than it does to the output *y* at minimal cost.

Suppose there are many demanders, each of whom either buys 0 or 1 units of the good; buyer *i* has the reservation price *R*_{i}, with *R*_{1} > *R*_{2} > ... *R*_{n} > 0. Then the following mechanism induces a
monopolist to produce efficiently and requires the regulator to know only the demand function.

How should a monopolist behave when confronted with this scheme? Its revenue from sellingMechanism: If the monopolist charges the pricepand sellsmunits it gets thesubsidyR_{1}+R_{2}+ ... +R_{m}mp.

That is, its revenue is exactly the revenue that a perfectly discriminating monopolist obtains when it maximizes its profit. Since the output chosen by a profit-maximizing perfectly discriminating monopolist is Pareto efficient, this subsidy scheme induces the monopolist to produce the Pareto efficient output.mp+ (R_{1}+R_{2}+ ... +R_{m}mp) =R_{1}+R_{2}+ ... +R_{m}.

The calculation that leads the monopolist to do so is the following. Suppose it is now selling *m* units and considers selling another unit. Its additional revenue is *p* + (*R*_{m+1} *p*) = *R*_{m+1}, which is exactly the AR at *m* units. Thus
the monopolist chooses to sell exactly the number for which AR is equal to MC, which is the efficient amount.

The point is that when the monopolist decides to sell another unit, the price on the units that have "already" been sold is not lowered---the monopolist gets the full value of *R*_{m+1}.

Obviously the scheme has the disadvantage that the monopolist gets the entire surplus: the distribution of income induced by the scheme is very inequitable, unless the monopolist is particularly deserving. But as we know, a proportional tax on profit does not affect the monopolist's behavior; it would allow the surplus to be distributed to other members of the economy.

Examples and exercises on controlling a monopolist

Copyright © 1997 by Martin J. Osborne