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Unformatted text preview: CHAPTER 2 4 MONOPOLY In the preceding chapters we have analyzed the behavior of a competitive
industry, a market structure that is most likely when there are a large
number of small ﬁrms. In this chapter we turn to the opposite extreme
and consider an industry structure when there is only one ﬁrm in the
industry—a monopoly. When there is only one ﬁrm in a market, that ﬁrm is very unlikely to
take the market price as given. Instead, a monopoly would recognize its
inﬂuence over the market price and choose that level of price and output
that maximized its overall proﬁts. Of course, it can’t choose price and output independently; for any given
price, the monopoly will be able to sell only what the market will bear. If
it chooses a high price, it will be able to sell only a small quantity. The
demand behaviortof the consumers will constrain the monopolist’s choice
of price and quantity. We can View the monopolist as choosing the price and letting the con—
sumers choose how much they wish to buy at that price, or we can think of
the monopolist as choosing the quantity, and letting the consumers decide
what price they will pay for that quantity. The ﬁrst approach is probably
more natural, but the second turns out to be analytically more convenient.
Of course, both approaches are equivalent when done correctly. 424 MONOPOLY (Ch. 24) 24.1 Maximizing Profits We begin by studying the monopolist’s proﬁt—maximization problem. Let
us use ﬂy) to denote the market inverse demand curve and C(31) to denote
the cost function. Let My) 2 p(y)y denote the revenue function of the
monopolist. The monopolist’s proﬁt—maximization problem then takes the
form mgx Hy) — cry). The optimality condition for this problem is straightforward: at the 0p
timal choice of output we must have marginal revenue equal to marginal
cost. If marginal revenue were less than marginal cost it would pay the ﬁrm
to decrease output, since the savings in cost would more than make up for
the loss in revenue. If the marginal revenue were greater than the marginal
cost, it would pay the ﬁrm to increase output. The only point where the
ﬁrm has no incentive to change output is where marginal revenue equals
marginal cost. In terms of algebra, we can write the optimization condition as MR=MC
OI'
£E_€E
Ay—Ay The same MR : M C condition has to hold in the case of a competitive
ﬁrm; in that case, marginal revenue is equal to the price and the condition
reduces to price equals marginal cost. In the case of a monopolist, the marginal revenue term is slightly more
complicated. If the monopolist decides to increase its output by Ay, there
are two effects on revenues. First it sells more output and receives a revenue
of pAy from that. But second, the monopolist pushes the price down by
Ap and it gets this lower price on all the output it has been selling. Thus the total effect on revenues of changing output by Ay will be Ar=pAy+yAa so that the change in revenue divided by the change in output—the mar
ginal revenuewis A?" Ap
___ l: p _+_ ___ .
Ay Ay
(This is exactly the same derivation we went through in our discussion of
marginal revenue in Chapter 15. You might want to review that material before proceeding.) LINEAR DEMAND CURVE AND MONOPOLY 425 Another way to think about this is to think of the monopolist as choosing
its output and price simultaneouslyrrecognizing, of course, the constraint
imposed by the demand curve. If the monopolist wants to sell more output
it has to lower its price. But this lower price will mean a lower price for all
of the units it is selling, not just the new units. Hence the term yAp. In the competitive case, a ﬁrm that could lower its price below the price
charged by other ﬁrms would immediately capture the entire market from
its competitors. But in the monopolistic case, the monopoly already has
the entire market; when it lowers its price, it has to take into account the
effect of the price reduction on all the units it sells. Following the discussion in Chapter 15, we can also express marginal
revenue in terms of elasticity Via the formula MRW) = My) [1 + $] and write the “marginal revenue equals marginal costs” optimality condi—
tion as 1
pg 1+——]=MCy. 24.1
( > [ 6w) ( ) < )
Since elasticity is naturally negative, we could also write this expression
as 1
29(9) [1 l=MC(y)
l€(y)l From these equations it is easy to see the connection with the competitive
case: in the competitive case, the ﬁrm faces a ﬂat demand curveaan in—
ﬁnitely elastic demand curve. This means that 1/ {cl 2 1/00 : 0, so the
appropriate version of this equation for a competitive ﬁrm is simply price
equals marginal cost. Note that a monopolist will never choose to operate where the demand
curve is inelastic. For if e < 1, then 1/ '6' > 1, and the marginal revenue
is negative, so it can’t possibly equal marginal cost. The meaning of this
becomes clear when we think of what is implied by an inelastic demand
curve: if [6' < 1, then reducing output will increase revenues, and reducing
output must reduce total cost, so proﬁts will necessarily increase. Thus any
point where {cl < 1 cannot be a proﬁt maximum for a monopolist, since it
could increase its proﬁts by producing less output. It follows that a point
that yields maximum proﬁts can only occur where [El 2 1. 24.2 Linear Demand Curve and Monopoly Suppose that the monopolist faces a linear demand curve 10(9) = a — by Figure
24.1 426 MONOPOLY (Ch. 24) Then the revenue function is 719) : p(y)y = ay — b312,
and the marginal revenue function is
fl/IRW) = a f 2by. (This follows from the formula given at the end of Chapter 15. It is easy
to derive using simple calculus. If you don't know calculus, just memorize
the formula, since we will use it quite a bit.) Note that the marginal revenue function has the same vertical intercept,
a, as the demand curve, but it is twice as steep. This gives us an easy way
to draw the marginal revenue curve. We know that the vertical intercept is
a. To get the horizontal intercept, just take half of the horizontal intercept
of the demand curve. Then connect the two intercepts with a straight line.
We have illustrated the demand curve and the marginal revenue curve in Figure 24.1. PRICE
AC Profits = In Demand (slope = —b) (slope = «213) y* OUTPUT Monopoly with a linear demand curve. The monopolist’s
proﬁtmaximizing output occurs where marginal revenue equals
marginal cost. MARKUP PRICING 427 The optimal output, 3/“, is where the marginal revenue curve intersects
the marginal cost curve. The monopolist will then charge the maximum
price it can get at this output, p(y*). This gives the monopolist a revenue
of p(y*)y* from which we subtract the total cost c(y*) 2 AC(y*)y*, leaving
a proﬁt area as illustrated. 24.3 Markup Pricing We can use the elasticity formula for the monopolist to express its optimal
pricing policy in another way. Rearranging equation (24.1) we have [email protected] [email protected])=fjjmﬁma mam This formulation indicates that the market price is a markup over marginal
cost, where the amount of the markup depends on the elasticity of demand. The markup is given by
1 1 — 1/ €(y)l‘
Since the monopolist always operates where the demand curve is elastic,
we are assured that [6 > 1, and thus the markup is greater than 1. In the case of a constantelasticity demand curve, this formula is espe
cially simple since 6(y) is a constant. A monopolist who faces a constant
elasticity demand curve will charge a price that is a constant markup
on marginal cost. This is illustrated in Figure 24.2. The curve labeled M C/ (1 — 1/ e[) is a constant fraction higher than the marginal cost curve;
the optimal level of output occurs where p : MC/(l — 1/€l) EXAMPLE: The Impact of Taxes on a Monopolist Let us consider a ﬁrm with constant marginal costs and ask what happens
to the price charged when a quantity tax is imposed. Clearly the marginal
costs go up by the amOunt of the tax, but what happens to the market
price? Let’s ﬁrst consider the case of a linear demand curve, as depicted in
Figure 24.3. When the marginal cost curve, M C , shifts up by the amount
of the tax to M C +15, the intersection of marginal revenue and marginal cost
moves to the left. Since the demand curve is half as steep as the marginal
revenue curve, the price goes up by half the amount of the tax. This is easy to see algebraically. The marginal revenue equals marginal
cost plus the tax condition is a—2by=c+t. 428 MONOPOLY (Ch. 24) PRICE yak OUTPUT Monopoly with constant elasticity demand. To locate the
proﬁtmaximizing output level we ﬁnd the output level where
the curve M C/ (1 — 1/ Id) crosses the demand curve. Solving for y yields ﬁ (1 — 0 ~ t
y ‘ 2b '
Thus the change in output is given by
Ag __ __ i
At T 2b'
The demand curve is
p(y) = a — by,
so price will change by —b times the change in output:
Ap 1 1
—— = —b —~—— 2 —.
At X 25 2 In this calculation the factor 1/2 occurs because of the assumptions of
the linear demand curve and constant marginal costs. Together these as—
sumptions imply that the price rises by less than the tax increase. Is this
likely to be true in general? The answer is no—in general a tax may increase the price by more or
less than the amount of the tax. For an easy example, consider the case of
a monopolist facing a constantelasticity demand curve. Then we have _ c—f—t
19‘ 1—1/e’ INEFFICIENCY OF MONOPOLY 429 PRICE After tax Before tax MC+t MC Demand y’ y" OUTPUT Linear demand and taxation. Imposition of a tax on a
monopolist facing a linear demand. Note that the price will rise
by half the amount of the tax. so that
Ap _ 1 EUq/W which is certainly bigger than 1. In this case, the monopolist passes on
more than the amount of the tax. Another kind of tax that we might consider is the case of a proﬁts tax.
In this case the monopolist is required to pay some fraction T of its proﬁts
to the government. The maximization problem that it faces is then m5): (1 — mph/)9  C(11)] But the value of 3; that maximizes proﬁts will also maximize (1 — 7') times
proﬁts. Thus a pure proﬁts tax will have no effect on a monopolist’s choice
of output. 24.4 Inefficiency of Monopoly A competitive industry operates at a point where price equals marginal
cost. A monopolized industry operates where price is greater than mar
ginal cost. Thus in general the price will be higher and the output lower Figure
24.4 430 MONOPOLY (Ch. 24) if a ﬁrm behaves monopolistically rather than competitively. For this rea—
son, consumers will typically be worse off in an industry organized as a
monopoly than in one organized competitively. But, by the same token, the ﬁrm will be better off! Counting both the
ﬁrm and the consumer, it is not clear whether competition or monopoly
will be a “better” arrangement. It appears that one must make a value
judgment about the relative welfare of consumers and the owners of ﬁrms.
However, we will see that one can argue against monopoly on grounds of
efﬁciency alone. Consider a monopoly situation, as depicted in Figure 24.4. Suppose that
we could somehow costlessly force this ﬁrm to behave as a competitor and
take the market price as being set exogenously. Then we would have (pc, ya)
for the competitive price and output. Alternatively, if the ﬁrm recognized
its inﬂuence on the market price and chose its level of output so as to
maximize proﬁts, we would see the monopoly price and output (pm, ym). PRICE ym y}: OUTPUT Inefﬁcienoy of monopoly. A monopolist produces less than
the competitive amount of output and is therefore Pareto inef
ﬁcient. Recall that an economic arrangement is Pareto efﬁcient if there is no way
to make anyone better off without making somebody else worse off. Is the
monopoly level of output Pareto efﬁcient? DEADWEIGHT LOSS OF MONOPOLY 431 Remember the deﬁnition of the inverse demand curve. At each level of
output, p(y) measures how much people are willing to pay for an additional
unit of the good. Since p(y) is greater than M C (y) for all the output levels
between ym and ya, there is a whole range of output where people are
willing to pay more for a unit of output than it costs to produce it. Clearly
there is a potential for Pareto improvement here! For example, consider the situation at the monopoly level of output ym.
Since p(ym) > M C (ym) we know that there is someone who is willing to
pay more for an extra unit of output than it costs to produce that extra
unit. Suppose that the ﬁrm produces this extra output and sells it to this
person at any price p where p(ym) > p > M C (ym). Then this consumer
is made better off because he or she was just willing to pay p(ym) for that
unit of consumption, and it was sold for p <: p(ym). Similarly, it cost the
monopolist M C (ym) to produce that extra unit of output and it sold it for
p > M C(ym). All the other units of output are being sold for the same
price as before, so nothing has changed there. But in the sale of the extra
unit of output, each side of the market gets some extra surplus—each side
of the market is made better off and no one else is made worse off We
have found a Pareto improvement. It is worthwhile considering the reason for this inefﬁciency. The efﬁcient
level of output is when the willingness to pay for an extra unit of output
just equals the cost of producing this extra unit. A competitive ﬁrm makes
this comparison. But a monopolist also looks at the effect of increasing
output on the revenue received from the inframarginal units, and these
inframarginal units have nothing to do with efﬁciency. A monopolist would
always be ready to sell an additional unit at a lower price than it is currently
charging if it did not have to lower the price of all the other inframarginal
units that it is currently selling. 24.5 Deadweight Loss of Monopoly Now that we know that a monopoly is inefﬁcient, we might want to know
just how inefﬁcient it is. Is there a way to measure the total loss in efﬁciency
due to a monopoly? We know how to measure the loss to the consumers
from having to pay pm rather than pciwe just look at the change in
consumers’ surplus. Similarly, for the ﬁrm we know how to measure the
gain in proﬁts from charging pm rather than pc~we just use the change in
producer’s surplus. The most natural way to combine these two numbers is to treat the
ﬁrmior, more properly, the owners of the ﬁrmﬁand the censumers of
the ﬁrm’s output symmetrically and add together the proﬁts of the ﬁrm
and the consumers’ surplus. The change in the proﬁts of the firmethe
change in producer’s surplus—measures how much the owners would be
willing to pay to get the higher price under monopoly, and the change in 432 MONOPOLY (Ch. 24) consumers” surplus measures how much the consumers would have to be
paid to compensate them for the higher price. Thus the difference between
these two numbers should give a sensible measure of the net beneﬁt or cost
of the monopoly. The changes in the producer’s and consumers” surplus from a movement
from monopolistic to competitive output are illustrated in Figure 24.5. The
monopolist’s surplus goes down by A due to the lower price on the units he
was already selling. It goes up by C dUe to the proﬁts on the extra units
it is now selling. PRICE
MC P“ =
monopoly price \ Competitive
price —/ *
y OUTPUT Deadweight loss of monopoly. The deadweight loss due to
the monopoly is given by the area B + C. The consumers’ surplus goes up by A, since the consumers are now get
ting all the units they were buying before at a cheaper price; and it goes
up by B, since they get some surplus on the extra units that are being
sold. The area A is just a transfer from the monopolist to the consumer;
one side of the market is made better off and one side is made worse off,
but the total surplus doesn’t change. The area B + C represents a true
increase in surplus—this area measures the value that the consumers and
the producers place on the extra output that has been produced. The area B + C is known as the deadweight loss due to the monopoly.
It provides a measure of how much worse off people are paying the mon— DEADWEIGHT LOSS OF MONOPOLY 433 opoly price than paying the competitive price. The deadweight loss due to
monopoly, like the deadweight loss due to a tax, measures the value of the
lost output by valuing each unit of lost output at the price that people are
willing to pay for that unit. To see that the deadweight loss measures the value of the lost output,
think about starting at the monopoly point and providing one additinal
unit of output. The value of that marginal unit of output is the market
price. The cost of producing the additional unit of output is the marginal
cost. Thus the “social value” of producing an extra unit will be simply
the price minus the marginal cost. Now consider the value of the next
unit of output; again its social value will be the gap between price and
marginal cost at that level of output. And so it goes. As we m0ve from
the monopoly level of output to the competitive level of output, we “sum
up” the distances between the demand curve and the marginal cost curve
to generate the value of the lost output due to the monopoly behavior.
The total area between the two curves from the monopoly output to the
competitive output is the deadweight loss. EXAMPLE: The Optimal Life of a Patent A patent offers inventors the exclusive right to beneﬁt from their inven—
tions for a limited period of time. Thus a patent offers a kind of limited
monopoly. The reason for offering such patent protection is to encourage
innovation. In the absence of a patent system, it is likely that individuals
and ﬁrms would be unwilling to invest much in research and development,
since any new discoveries that they would make could be copied by com
petitors. In the United States the life of a patent is 17 years. During that period,
the holders of the patent have a monopoly on the invention; after the
patent expires, anyone is free to utilize the technology described in the
patent. The longer the life of a patent, the more gains can be accrued by
the inventors, and thus the more incentive they have to invest in research
and development. However, the longer the monopoly is allowed to exist, the
more deadweight loss will be generated. The beneﬁt from a long patent life
is that it encourages innovation; the cost is that it encourages monopoly.
The “optimal” patent life is the period that balances these two conﬂicting
effects. The problem of determining the optimal patent life has been examined
by William Nordhaus of Yale University.1 As Nordhaus indicates, the prob—
lem is very complex and there are many unknown relationships involved.
Nevertheless, some simple calculations can give some insight as to whether 1 William Nordhaus, Invention, Growth, and Welfare (Cambridge, Mass: M.I.T. Press,
1969). 434 MONOPOLY (Ch. 24) the current patent life is wildly out of line with the estimated beneﬁts and
costs described above. Nordhaus found that for “run—ofthemill” inventions, a patent life of 17
years was roughly 90 percent efﬁcientAmeaning that it achieved 90 percent
of the maximum possible consumers’ surplus. On the basis of these ﬁgures,
it does not seem like there is a compelling reason to make drastic changes
in the patent system. EXAMPLE: Patent Thickets The intellectual property protection offered by patents provides incentives
to innovate, but this right can be abused. Some observers have argued
that the extensions of intellectual property rights to business processes,
software, and other domains has resulted in lower patent quality. One might think of patents as having three dimensions: length, width,
and height. The “length” is the time that the patent protection applies.
The “width” is how broadly the claims in the patent are interpreted. The
“height” is the standard of novelty applied in determining whether the
patent really represents a new idea. Unfortunately, only the length is easily
quantiﬁed. The other aspects of...
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