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Web Supplement
to Chapter 13
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ADDITIONAL MODELS OF
OLIGOPOLY AND
MONOPOLISTIC COMPETITION
This Web supplement contains two sections: Section 13S.1 provides a more extended
treatment of the spatial model of monopolistic competition analyzed in Sections 13.12
and 13.13 of Chapter 13 in the text. Section 13S.2 provides a critical outline of the theory
of “contestable markets.”
13S.1 A Simple Spatial Model of Monopolistic Competition
In recent years, research on the theory of monopolistic competition has focused on models that incorporate the specific features of a product that make buyers choose it over all
others.1 In contrast with the Chamberlinian model, these models produce conclusions
that often differ sharply with those of the perfectly competitive model. This alternative
way of thinking about monopolistic competition was discussed in the text and is the
basis for the model that follows.
Here we explore in more detail the forces that govern behaviour in the spatial model
of monopolistic competition. Consider again our simple restaurant example, in which
there are L people living around a loop with unit circumference. Recall that each person
eats 1 meal/day, and restaurant costs are given by TC F MQ. The cost of travel is t
per person per unit of distance. Suppose we start off with N restaurants, evenly spaced
around the loop, so that each is located 1/N units from each of its nearest neighbours.
If each restaurant acts as a profit maximizer, what price will it charge and how many
meals will it serve per day? The second part of this question is easy to answer if all restaurants charge the same price. There are L consumers in total, and each restaurant will
serve L/N of them. But suppose that restaurants set their prices independently and not
all of them charge the same price. If the price charged by a more distant restaurant is
sufficiently lower than that of the one closest to home, a consumer will choose the more
distant restaurant.
1
This work is inspired by Harold Hotelling’s pioneering paper, “Stability in Competition,” The Economic
Journal, 39, 1929: pp. 41–57.
13S.1 A SIMPLE SPATIAL MODEL OF MONOPOLISTIC COMPETITION
13S-1
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What we need to do is analyze a given restaurant’s demand curve when it charges a
price different from its competitors’. Figure 13S-1 shows a portion of the island periphery that contains a restaurant and its two nearest competitors. For the sake of
convenience, the space between them is represented as a straight line rather than the arc
of a circle. The restaurant whose demand curve we are about to derive is located at 0, its
nearest competitors are at 1/N and 1/N
Let us begin by supposing that the restaurant at 0 charges a price of P, while its
competitors charge a price of P. For a person who lives d units to the right of 0, what is
the total cost, including transportation costs, of a meal at the restaurant at 0? Since the
round-trip distance for this consumer is 2d, total cost is given simply by
C0(P) P 2td
(13S.1)
For the consumer who happens to live right at 0 (that is, for someone for whom d is
0), the total cost of a meal at the restaurant at 0 is just the purchase price, P. The heavy
lines emanating from P to the right and left of 0 in Figure 13S-1 show how the total cost
of a meal at the restaurant at 0 rises with d.
Now consider the total cost to the same person of a meal in the restaurant at 1/N.
Since the person lives d units to the right of the restaurant at 0, he lives 1/N d units to
the left of the restaurant at 1/N. His total cost of a meal at the second restaurant will
therefore be
C1/N 1P¿ 2 P¿ 2ta
1
db
N
(13S.2)
The heavy line emanating from P at 1/N traces the relationship between location
and the total cost of a meal at the restaurant at 1/N.
The intersection of the total cost lines for adjacent restaurants defines the “breakeven” location, the point at which a consumer will incur the same total cost at either
restaurant. Note in Figure 13S-1 that this intersection occurs at a point that is closer to
the restaurant at 0. The reason is that this restaurant happens to be charging a higher
price than its adjacent competitors. So a person living at the halfway point (1/2N) will
find it cheaper to eat at the restaurant at 1/N than to eat at the one at 0.
Note in Figure 13S-2 that if the restaurant at 0 charges a price lower than its competitors, the breakeven point (X) will of course shift beyond the halfway point.
Once we know where a restaurant’s breakeven point is, we know exactly how
many patrons it will attract at a given price. If the restaurant at 0 again charges P
FIGURE 13S-1
Total Cost as a Function
of Both Price and
Distance
The total cost of a meal is
the price charged by the
restaurant plus the
transportation cost
incurred in getting there.
P’’ + 2td
P’ + 2t ( 1 – d )
N
P’ + 2t ( 1 – d )
N
P’’
P’
–1/N
0
–X’’
13S-2
P’
1/(2N)
1/N
X’’
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FIGURE 13S-2
The Breakeven Point of a
Low-Priced Restaurant
The restaurant at 0 is
charging a price (P)
lower than the price
charged by its competitor
at 1/N. The total cost
of a meal at the two
restaurants will therefore
be the same at X, which
lies to the right of the
halfway point.
P’ + 2t ( 1 – d )
N
P’ + 2t ( 1 – d )
N
P’’ + 2td
P’
P’
P’’
–1/N
0
1/(2N)
–X’’
1/N
X’’
and its competitors on either side charge P, the breakeven point, X, can be obtained
by solving
P– 2tX– P¿ 2t a
1
X–b
N
(13S.3)
which yields
X– 1
2t
aP¿ P– b
4t
N
(13S.4)
As a check, note that when prices charged by all restaurants are the same, we get the
expected result that X 1/(2N), the midpoint between the two restaurants.
Because the restaurant will draw patrons from both the right and the left of 0, the
total length of the arc whose residents will attend is equal to twice the breakeven distance. And since there are L consumers per unit of distance on the circle, its total
patronage will be given by
Q
1
2t
aP¿ P– b
4t
N
(13S.5)
Note that this expression behaves as a demand schedule should, in the sense that
patronage increases as the gap between P and P increases.
Equation 13S.5 tells us the restaurant’s demand curve written in the form of quantity
as a function of price. We are much more used to seeing the demand function written in
the form of price as a function of quantity. To translate Equation 13S.5 into that form, we
simply solve for P:
P– aP¿ 2t
2t
b Q
N
L
(13S.6)
We are at last in a position to solve the individual restaurant’s profit-maximization
problem. As always, we simply equate marginal revenue to marginal cost. Recall that
the marginal revenue curve that corresponds to a given straight-line demand curve is a
straight line with the same vertical intercept as and twice the slope of the demand curve.
So the marginal revenue curve that corresponds to the demand curve given by Equation
13S.6 is given by
MR aP¿ 2t
4t
b Q
N
L
13S.1 A SIMPLE SPATIAL MODEL OF MONOPOLISTIC COMPETITION
(13S.7)
13S-3
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The demand, marginal revenue, and marginal cost curves for the restaurant are
shown in Figure 13S-3, and the profit-maximizing quantity and price are given by
P* 1
2t
aP¿ Mb
2
N
(13S.8)
Q* 1
L
aP¿ Mb
2N
4t
(13S.9)
and
ˇ
respectively. The area of the shaded rectangle is the revenue in excess of variable costs.
If it exceeds F, the restaurant will earn an extranormal profit. If less, it will experience an
economic loss.
The profit-maximizing price and quantity expressions in Equations 13S.8 and 13S.9
may not seem illuminating at first glance, but on closer inspection they have a clear
logic. Note, for example, that P* increases with P, the price charged by the neighbouring restaurant, and also with t, the unit transportation cost. The more costly
transportation is, the more a restaurant can charge before it becomes worth someone’s
while to travel to a more distant location. Note also that the profit-maximizing price
increases with M, the marginal cost of producing meals. In Equation 13S.9, note that Q*
increases with the rival’s price and declines with the cost of transportation.
Things begin to look much simpler once we take into account the symmetry between
competing restaurants. Since all restaurants have the same costs and access to the market, the profit-maximizing price for one must in the end be the profit-maximizing price
for all. If we substitute P* for P into Equation 13S.8, the expression for P* simplifies to
P* 2t
M
N
(13S.10)
and substituting this expression into Equation 13S.9, the expression for Q* simplifies to
Q* FIGURE 13S-3
The Profit-Maximizing
Restaurant
The more the restaurants
on either side charge,
the higher the profitmaximizing price will be.
The profit-maximizing
price also rises with
transportation cost, t, and
with the marginal cost of
producing meals, M.
L
N
(13S.11)
$/Q
P’ +2t /N
P * = P’/2 + t /N + M/2
M
(L/2)(P’/2t + 1/N )
L(P’/2t + 1/N )
Q
Q* = (L/4t )(P’ + 2t /N – M)
13S-4
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When all restaurants charge the same price, P*, the breakeven point occurs halfway
between restaurants, and each of the N restaurants serves 1/Nth of the total market.
Economic profit for each of the N restaurants is given by the difference between its
total revenue and its total cost:
P*Q* F MQ* 2Lt
F
N2
(13S.12)
Again, this figure may be either positive or negative, depending on the relative sizes
of L, t, N, and F.
If profit for each firm was negative and expected to remain so, firms would leave the
industry until there were no longer any losses. Suppose, for the sake of discussion, that
economic profit is positive. Will free entry tend to force profit down to zero, as in both
the competitive and Chamberlinian models? The answer this time is “Not necessarily.”
To illustrate why, suppose that once a restaurant is built, its location cannot be altered
without abandoning the setup costs associated with the existing site. The term F in the
cost function is, in effect, a sunk cost. If a new restaurant tries to enter the market to take
advantage of the current excess profits, where will it locate? The best it can do is to
choose a position midway between two existing units, which means that it will have
only half as much territory to work with as the existing firms. So if existing firms hold
price constant at P*, the revenue potential for an entering firm will be only half as great
as for existing firms. And in the more likely event that existing firms cut price when a
new firm enters, the entrant’s total revenue will be even less than half that of existing
firms. Finally, costs do not fall proportionally with output (because of the fixed cost F),
so it is possible for an evenly spaced array of existing firms to have positive economic
profits, while at the same time it is not profitable for a new firm to enter the market.
Here we see a fundamental distinction between the spatial model of monopolistic
competition and the Chamberlinian model. Recall that in the Chamberlinian model,
each firm—even new entrants—received a proportional share of total market demand.
In the spatial model with fixed locations of existing firms, in contrast, an entrant’s
opportunities will be markedly less attractive than those of firms already in the market.
So we may have perfect freedom of entry in the spatial model, yet economic profit can
persist indefinitely.
The source of this difference lies, as noted, in the assumption that the locations of existing firms are fixed, but this doesn’t always need to be the case. Consider, for example, the
food-truck vendors who ply their trade on the streets of many large cities. The locations of
these vendors will matter to their customers in precisely the same way as in the restaurant
example we just considered. But in stark contrast to the restaurant example, here there is
virtually complete flexibility with respect to location. If a new food truck enters the market, existing food trucks will find it in their interests to relocate until all trucks are once
again evenly distributed throughout the territory. In such a market, the profit opportunities of entering firms are essentially the same as those of existing firms, and freedom of
entry will force economic profit down to zero, just as in the Chamberlinian model.
These examples illustrate the critical distinction between a fixed cost and a sunk cost.
A sunk cost, again, is one that is irretrievably committed. No matter what the firm does,
a sunk cost cannot be recovered. A fixed cost, by contrast, is simply one that does not
vary with output. Some fixed costs, like those we saw in the restaurant example, are
sunk. Others, like those in the food-truck example, are not.
In this simple spatial model of monopolistic competition, then, economic profit in the
long run may be either positive or zero. Consider the latter case. How many outlets
13S.1 A SIMPLE SPATIAL MODEL OF MONOPOLISTIC COMPETITION
13S-5
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(food trucks) will that imply in long-run equilibrium? Taking the expression for profit
from Equation 13S.12 and setting it equal to zero, we solve for
N** 2Lt
B F
(13S.13)
Comparing N**, the long-run equilibrium number of outlets in the spatial model
with zero profit, to N*, the optimal number of outlets (from Equation 13.10 in Chapter
13), we see that the latter is half as large as the former. Put another way, in this version
of the spatial model, we get excessive product variety. As noted earlier, however, this
result applies with respect to this particular model. It does not apply, for example, in
models that use only slightly different assumptions.2
13S.2 Contestable Markets
In a widely discussed book, economists William Baumol, John Panzer, and Robert Willig suggested that oligopolies and even monopolies sometimes behave much like
perfectly competitive firms.3 According to their theory, entry and exit must be perfectly
free for this to happen. With costless entry, a new firm will quickly enter if an incumbent
firm dares to charge a price above average cost. The term contestable markets refers to the
fact that when entry is costless, we often see a contest between potential competitors to
see which firms will serve the market.
Costless entry does not mean that it costs no money to obtain a production facility to
serve a market. It means that there are no sunk costs associated with entry and exit. The
most important piece of equipment required to provide air service in the Iqaluit market,
for example, is an aircraft, which carries a price tag of at least $50 million. This is a hefty
investment, to be sure, but it is not a sunk cost. If a firm wants to leave the market, it can
sell or lease the aircraft to another firm, or make use of it in some other market.
Contrast this case with that of a cement producer, which must spend a similar sum to
build a manufacturing facility. Once built, the cement plant has essentially no alternative use. The resources that go into it are sunk costs, beyond recovery if the firm
suddenly decides it no longer wants to participate in that market.
Why are sunk costs so important? Consider again the contrast between the air service
market and the cement market. In each case we have a local monopoly. Because of economies of scale, there is room for only one cement factory in a given area and only one
flight at a given time of day. Suppose in each case that incumbent firms are charging
prices well in excess of average costs, and that in each case a new firm enters and captures some of the excess profit. And suppose, finally, that the incumbents react by
lowering their prices, with the result that all the firms, entrants and incumbents alike,
are losing money. In the cement market case, the entrant will then be stuck with a huge
capital facility that will not cover its costs. The airline case, in contrast, carries no similar
risk. If the market becomes unprofitable, the entrant can quickly pull out and deploy its
asset elsewhere.
The contestable market theory is like other theories of market structure in saying that
cost conditions determine how many firms will end up serving a given market. Where
2
See Avinash Dixit and Joseph Stiglitz, “Monopolistic Competition and Optimal Product Diversity,” American
Economic Review, 1977: pp. 297–308; and B. C. Eaton and Myrna Wooders, “Sophisticated Entry in a Model of
Spatial Competition,” The Rand Journal of Economics, 16, 1985: pp. 282–297.
3
William Baumol, John Panzer, and Robert Willig, Contestable Markets and the Theory of Industry Structure, San
Diego, CA: Harcourt Brace Jovanovich, 1982. For an accessible summary, see Baumol, “Contestable Markets:
An Uprising in the Theory of Industry Structure,” American Economic Review, 72, March 1982: pp. 1–15.
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there are economies of scale, we expect to see only a single firm. Where there are
U-shaped LAC curves whose minimum points occur at a substantial fraction of industry output, we expect only a few firms. With constant costs, there may be many firms.
Where the contestable market theory differs from others is in saying that there is no
clear relationship between the actual number of competitors in a market and the extent
to which price and quantity resemble what we would see under perfect competition.
Where the threat of entry is credible, incumbent firms are simply not free to charge
prices that are significantly above cost.
Critics of the contestable market theory counter that there are important sunk costs
involved in participation in every market.4 Granted, in the airline case it is possible to
lease an aircraft on a short-term basis, but that alone is not sufficient to start a viable
operation. Counter space must be obtained at the airport terminal; potential passengers
must be alerted to the existence of the new service, usually with an expensive advertising campaign. Reservations, baggage handling, and check-in facilities must be arranged.
Ground service contracts for the aircraft must be signed, and so on. Each step involves
irretrievable commitments of resources, and they add up to enough to make a brief stay
in the market very costly indeed. The fiercest critics contend that as long as there are
any sunk costs involved in entry and exit, the contestable market theory breaks down.
The contestable market theory has been a fruitful source of ideas regarding the forms
competition can take in imperfectly competitive markets. The critics have raised some
formidable objections, but there do appear to be at least some settings where the insights
hold up. In the intercity bus market, for example, one or two firms usually provide all
of the scheduled service in any given city-pair market. Traditional theories of market
structure suggest that prices would be likely to rise steeply during holiday weekends,
when substantially larger numbers of people travel. What we see in some markets,
however, is that small charter bus companies offer special holiday service at fares no
higher than normal. These companies often do little more than post a few leaflets on
college campuses, stating their prices and schedules and giving a telephone number to
call for reservations. The circumstances of the intercity bus industry come very close to
the free-entry ideal contemplated by the contestable market theory, and the results are
much as it predicts. Much more can be said about just when the threat of entry will be a
significant disciplining force, a subject considered in more detail in Section 13.9 of
Chapter 13 and Mathematical Applications 13A in the text.
4
W. G. Shepherd, “Contestability vs. Competition,” American Economic Review, 74, September 1984:
pp. 572–587.
13S.2 CONTESTABLE MARKETS
13S-7