Judo Economics: Capacity Limitation and Coupon Competition
Author(s): Judith R. Gelman and Steven C. Salop
Source: The Bell Journal of Economics, Vol. 14, No. 2 (Autumn, 1983), pp. 315-325
Published by: RAND Corporation
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Judo economics: capacity limitation
and coupon competition
Judith R. Gelman*
and
Steven C. Salop**
This article analyzes the effect of credible capacity limitation on the interaction between
an incumbent and an entrant. The prospect of price competition deters entry by a less
efficient firm with unlimited capacity. If, however, the entrant practices 'Judo economics"
and limits its capacity sufficiently, it is in the incumbent's interest to accommodate entry.
The entrant can increase its profits by selling rights (coupons) to its scarce discount output.
If the entrant is less efficient, the incumbent will choose to serve coupon holders at the
discount price. When this occurs, the entrant sells coupons, but no output, and industry
production is rationalized.
1. Introduction
* This article explores the role of precommitment in the strategic interaction between
a dominant firm and a fringe competitor. In the model examined, a fringe competitor
partially offsets its demand disadvantage by engaging in capacity limitation and discount
pricing. This strategy credibly reduces the threat posed to the dominant firm and makes
retaliation more expensive. The rational incumbent is thus induced to accommodate the
entrant. To capture the image of a small firm using its rival's large size to its own advantage,
we call this a strategy of judo economics.
The article is organized as follows. Section 2 sets out a simple capacity-limitation
model in which a less efficient entrant with unlimited capacity is deterred by the rationally
based prospect that the incumbent will not accommodate it. In contrast, following Schelling
(1960), when the entrant judiciously limits its capacity, it is in the incumbent's interest to
accommodate entry.
Section 3 examines a more sophisticated entry strategy. When the entrant both limits
its capacity and sets a low price, its scarce output must be rationed. The entrant can
increase its profits by selling coupons-transferable rights to its output. If the entrant's
coupons are transferable, the incumbent may choose to honor the coupons, serving coupon holders at the discount price. When this occurs, the entrant produces no output and
earns its profits solely from the sale of its coupons. Section 4 reviews the results of the
model and suggests a number of extensions.
* Federal Trade Commission.
** Georgetown University Law Center.
The opinions in this article do not necessarily represent the views of the Federal Trade Commission. We
would like to especially thank E. Bailey, A. Dixit, W. Dudley, J. Farrell, R. Frank, J.J. Gabszewicz, D.K. Levine,
S. Salant, R. Schmalensee, D. Scheffman, W. Schwartz, C. Shapiro, and A.M. Spence for helpful conversations
and the editors and referees for useful suggestions. The authors gratefully acknowledge research support from the
FTC's Bureau of Economics.
315
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316 / THE BELL JOURNAL OF ECONOMICS
2. Capacity and competition
* Consider the following market. Consumer demand is given by the demand function
D(p). Initially, the industry consists of an incumbent monopolist with unlimited capacity
to produce output at a constant marginal cost cl. We denote the incumbent's monopoly
price as Pim.
Suppose a single potential entrant appears. The entrant can produce up to k units
of output at a constant marginal cost c2 . We assume that the entrant has no cost advantage
so that c2 > cl. The entrant freely selects a capacity level k.' Upon entry, it incurs a
nominal sunk cost which is invariant with respect to k.2
The entrant's prospects depend on consumers' relative demand for the two brands
and the strategic interaction that follows entry. In this article we focus on the effect of the
entrant's ability to limit capacity on postentry pricing. To highlight this issue, we make a
number of unrealistic simplifying assumptions. First, we assume consumers lexicographically
prefer the incumbent's brand at equal prices; with any price differential, all consumers
prefer the less expensive brand.3 Formally, let demands xi and x2 for incumbent and entr
respectively be given as follows:
xl =I
T D(pI) if P -P2
Xi O otherwise (1
00 if PI --P2
= D(P2) otherwise.
Second, we assume the following strategic interaction. The entrant plays first, irre-
vocably choosing a price/capacity pair (P2, k).4 The incumbent then must react by choosing
a price pi. In standard parlance, the entrant is the von Stackelberg price "leader" and the
incumbent is the price "follower." For now, we assume that the incumbent may not price
discriminate.
O Competition with unlimited capacity. Suppose the entrant's capacity is unlimited (i.e.,
k = oo). Because entry costs must be paid, entry is only profitable if the entrant's price
exceeds its marginal cost. But if the entrant has unlimited capacity and no cost advantage,
the incumbent surely maximizes its profits by meeting the entrant's price.5 This is a result
of consumers' lexicographic preferences: at any price pi > P2, the incumbent obtains no
customers; at equal or lower prices (p1 < P2) the incumbent obtains the entire market
demand D(p1).6 Thus, entry does not occur unless the entrant is strictly more efficient.
What may not be obvious is the importance of the entrant's unlimited capacity and the
incumbent's inability to price discriminate in generating these results.
' Here the entrant's choice is restricted to supply curves with marginal costs that are constant at c2 up to
a maximum capacity level k. More generally, the entrant's strategy space could include a family of cost functions.
2 Any sunk cost prevents entry unless price strictly exceeds the entrant's marginal cost. If sunk costs are
large, entry may be unprofitable even if accommodated. Presumably, the incumbent has already sunk this cost.
3Lexicographic preferences are a limiting case of differentiated products (Gelman and Salop, 1982). See
Schmalensee (1982) and Dudley (1982) for alternative specifications of the advantage of the first entrant's
"pioneer" brand.
4 The qualitative results of our model do not depend on the entrant's ability to make a credible price
commitment, only on its ability to credibly limit capacity.
5A postentry competition based on the Bertrand price equilibrium concept is more complicated, but the
results are similar. Without capacity limitations, the Bertrand equilibrium also implies zero entry.
6 Assuming a concave profit function, if the entrant sets P2 > pl,,,, the incumbent earns greater profits by
undercutting (i.e., p, = Pi,,). If the entrant sets cl < P2 < Pl,?, the incumbent earns greater profits by matching
(i.e., p, = P2).
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GELMAN AND SALOP / 317
o Competition with capacity limitations. By credibly committing itself to a limited ca-
pacity, the entrant makes itself less threatening to the incumbent and therefore improves
its strategic position. Instead of matching the entrant's price, the incumbent may allow
the entrant to sell out its limited capacity and serve the remainder of the market itself
at a higher price.
Formally, suppose the entrant irrevocably chooses a price/capacity pair (P2, k) to
which the incumbent responds by choosing a price pl. Since price discrimination is
impossible, the incumbent has two basic choices-pi > P2 (accommodating) or pi < P2
(undercutting or matching).7 The entrant obtains no customers if the incumbent matches
or undercuts its price. Hence, a profit-maximizing entrant chooses among the (P2, k)
pairs that induce an accommodation response. To derive this set of price/capacity pairs
(the accommodation set), we first examine the incumbent's problem.
If the entrant chooses P2 > pi, (the initial monopoly price), the incumbent will surely
undercut by setting pi = pim. Hence, no entrant will pick P2 > Pim. For any price
P2 < Plm, if the incumbent matches, it earns profits HM(P2) given by
HM(P2) = (P2 - cl)D(p2) for Pi = P2. (2)
Alternatively, the incumbent can accommodate the entrant by choosing p, > P2 and
allowing the entrant to sell its k units. At P2 < pl, the entrant must ration the rights to its
scarce output on some nonprice basis for k < D(p2).
Two simple rationing schemes are random rationing and reservation-price rationing.
Under random rationing, the rights are distributed randomly among all consumers willing
to pay at least P2.' This more realistic random selection scheme is presented in the
Appendix. Under reservation-price rationing, the k customers with the highest reservation
prices are given the rights to the entrant's product.9 This rationing scheme is quite simple
to analyze and, as shown in Section 3, selling the rights by means of coupons induces
exactly this latter type of rationing. For now, we assume that the rights are rationed by
reservation price but are not transferable, so that no aftermarket exists.
When the entrant distributes its output to the k customers with the highest willingnessto-pay, the incumbent's residual demand when accommodating the entrant is given by
xi = D(pl) - k. The incumbent's maximized profits are therefore given by
11A(k) = max (pi - c,)[D(p,) - k]. (3)
Pi
The profit-maximizing entry-accommodating price p,(k) can be derived by differ-
entiating equation (3) and solving the following first-order condition:
(pi - c1)D'(p1) + D(p,) - k = 0. (4)
Totally differentiating equation (4), it is clear that p'1(k) < 0 as long as the profit function
is strictly concave.
7 Under certain circumstances, the incumbent may selectively match the entrant's price, offering discounts
to only those customers approached by the entrant. Such a price-discrimination strategy requires that the
entrant's actual or potential customers be distinguishable, perhaps by use of a screening device. Unless, however,
customers whom the incumbent serves have exclusive "rights" to the entrant's capacity, the entrant can offer
to sell its discount output to other customers. As discussed in Section 3 below, selective matching is facilitated
if the entrant issues coupons.
8 Of course, if the rights were transferable, then an aftermarket for these rights could exist and rights could
have resale value. If so, even consumers with reservation prices below P2 would desire them.
9 These customers must be selected on a nonprice basis. If the high-reservation-price customers are identified by using price (i.e., if the entrant offers its units at a price p,), the incumbent will match.
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318 / THE BELL JOURNAL OF ECONOMICS
We now return to the entrant's choice of (P2, k), which temporally precedes the
incumbent's choice of p,(k). Because the entrant only obtains sales when the incumbent
accommodates it, the profit-maximizing entrant chooses a (P2, k) pair such that'"
HIA(k)
>
HIM(P2).
(5)
We denote the set of (P2, k) pairs that satisfy equation (5) with equality as X(k) so
that the accommodation set satisfies the constraint P2 < X(k)."
The frontier 4(k) represents the entrant's demand curve. If the entrant chooses a
price/capacity pair along ?(k), the incumbent accommodates its entry and the entrant
sells its entire capacity k. The p,(k) function and the (P2, k) accommodation set are
illustrated in Figure 1 below.
For the entrant's profit-maximization problem, we now have
H2 = max (P2 - c2)k (6)
p2ik
subject to
P2 X(k).
At the optimum, the constraint clearly holds with equality. Substituting the constraint
into the entrant's profit function and differentiating, we have the first-order condition that
defines optimal capacity k*:
1'(k*)k* + o(k*) - C2 = 0. (7)
FIGURE 1
ACCOMMODATION SET
P\
D(p)
k
'1 Equation (5) reflects the convention that if the incumbent is indifferent between the two strategies, it
chooses to accommodate.
" It must also be shown that all (P2, k) satisfying equation (5) have the entrant charging a lower price
than the incumbent (i.e., P2 < p,(k)). We prove that P2 < p,(k) for all k in the relevant open interval (0, D(c,))
as follows. Consider a (P2, k) pair that satisfies equation (5). Substituting equations (2) and (3) into (5), we have
(p,(k) - cl)D(p1(k)) - (P2 - cl)D(P2) > k(p,(k) - cl) > 0.
Since the profit function 11(p) is concave, P2 < P,tn and p,(k) < p,n, the inequality implies P2 <p,(k).
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GELMAN AND SALOP / 319
This, in turn, defines the entrant's optimal price p' = k(k*) and the incumbent's best
response p' = pl(k*).
When the incumbent accommodates the entrant, it reduces its price below the initial
monopoly level. The incumbent does not, however, match the entrant's price. Rather,
it maintains an "umbrella" under which the entrant can prosper, as long as it remains
satisfied with its modest market share.
Even given our one-entrant assumption, this alternative formalization captures some
features of dominant-firm/fringe-competitor interaction that are absent in the usual
model. These features have been discussed informally in the literature and are prominent
in the antitrust oral tradition. For example, Scherer (1980, p. 233) observes that in many
industries the fringe firms choose to remain small rather than risk retaliation by the
dominant firm.'2 Our model explicitly incorporates these concepts of punishment and
industry discipline. If the entrant were to increase capacity slightly beyond the ?(k) fron-
tier, the incumbent would cut its price discreetly from pi(k*) to P2, thereby driving the
entrant's sales to zero.
Our model applies to only those situations in which the entrant can make credible
capacity-limitation commitments. Credibility is enhanced by the use of contracts, strategic
product design, and other strategies.'3 Irreversibly adopting a technology with sharply
increasing marginal costs can represent a credible commitment to limit capacity. If prod-
ucts are differentiated, an entrant can effectively commit to a low capacity strategy by
designing a specialty product or one of substantially inferior (or superior) quality. Producing a product with limited consumer appeal is analogous to capacity limitation.'4 On
the other hand, if the entrant can secretly produce and sell additional units at low cost,
the judo economics model does not apply.
3. Coupon competition
* Coupons-transferable rights to discount output-provide a twist on standard price
competition. Traditionally used to compete for price-sensitive consumers of "convenience" goods, coupons have recently been introduced into other markets. Airline coupons
were first introduced by United in May 1979, after a long strike. United distributed
transferable coupons that entitled recipients to a 50% discount on a future flight. Within
days, American also began distributing coupons. An aftermarket for the coupons developed and considerable speculation ensued.'5 Each airline subsequently chose to accept
the others' coupons. The demand for flights on which coupons were issued increased
substantially.
In the Summer of 1980, Eastern Airlines used coupons to deter entry as well as to
induce accommodation. Eastern distributed coupons on its Air Shuttle routes which New
2 That incumbents only respond fully to low-priced entrants when they grow large is nicely illustrated
by experience in the airline industry. For details, see Gelman and Salop (1982).
" A credible commitment can also sometimes serve as an incumbent's counterstrategy. For example, a
meeting-competition contractual clause can commit a firm to match its rivals' bids and thus prevent the incumbent from accommodating entry, even if accommodation is the most profitable response. Delta Airlines
may have attempted such a strategy with its "We will not be undersold" advertising campaign in the Spring
of 1982. See Salop (1982) for application to cartel stabilization.
" See Gelman and Salop (1982) and d'Aspremont et al. (1979). See Yip (1982) for a practical discussion
of how product differentiation and other structural characteristics traditionally thought to be entry barriers may
be used to facilitate entry.
5 This aftermarket was far more complicated than the simple one we analyze below. First, the coupons
had an expiration date, introducing dynamic considerations into the market. Second, carriers who accepted
rivals' coupons had the option of reselling them. Third, coupon values depended on the price of restrictive
discount fares as well as on the coach fare to which they were pegged. Finally, the airlines distributed coupons
on other flights rather than selling them directly.
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320 / THE BELL JOURNAL OF ECONOMICS
York Air had recently entered. The coupons were good for a 50% discount on Eastern's
new transcontinental service. After United and American decided to accept these coupons,
Eastern cut back its transcontinental flights. By distributing coupons, Eastern effectively
cut its Shuttle fare without reducing its revenue.
Coupons can be analyzed in our incumbent/entrant framework. So far, we have
assumed that the entrant rations its scarce output on a nonprice basis. We now analyze
the market for transferable rights and its natural extension-coupon competition.
o The market for rights. Suppose that the entrant randomly distributes transferable
rights, each of which represents an option to purchase one unit of the entrant's output
at the price P2. Assume that the incumbent accommodates entry by choosing Pi > P2.
Consider a resale market for these rights, where q denotes the transfer price of a right.
Consumers compare the "full price" of the entrant's output (q + P2) to the incumbent's
price pl. Given their lexicographic preference for the incumbent's product, consumers
will demand rights only if q < P, - P2. This leads to an equilibrium price in the rights
market just below Pi - P2, which can be approximated by'6
q
=
p-
P2*
(8)
When Pi > P2, all the rights to the entrant's k units are ultimately purchased by
consumers with reservation prices no less than Pi. These high-reservation-price consumers
obtain the entrant's output, thereby leaving the incumbent with a residual demand given
by xi = D(pl) - k. Not surprisingly, this is identical to the incumbent's residual demand
in the case of reservation-price rationing.
If transferable rights are allocated on a nonprice basis, the k individuals who initially
obtain the rights gain a windfall profit of q on each right. If, instead, the entrant sells the
rights, it obtains the additional revenue.'7 Finally, the incumbent may choose to honor
the rights, serving bearers at the P2 and all others at Pi > P2. We refer to this price
discrimination strategy as selective matching.'8
If the entrant sells the rights to its output, its optimal strategy may change for two
reasons. First, it receives additional revenue for every unit sold. Second, if the incumbent
honors the coupons, the entrant's profit function is altered.
o The incumbent's problem. If the entrant issues coupons, the incumbent may selectively match the entrant's price by honoring the k coupons at the price P2 and setting a
higher price p, for sales to its remaining D(pl) - k customers. Under this selective matching
strategy (denoted by the subscript S), the incumbent maximizes profits as follows:
11s = max (pi - cl)[D(pi) - k] + (P2 - ci)k. (9)
Pi
16 If q > pi - P2, consumers strictly prefer the incumbent's product. At any q < pi with reservation prices above p, prefer the entrant's brand. If there is excess demand for the rights when
q < pi - P2 holds (i.e., if D(p,) > k), the equilibrium price of the k coupons must be infinitesimally less than
the price differential. Although there is no true equilibrium, by using the approximation in equation (8), we
derive an epsilon equilibrium.
17 The timing of this sale is crucial. For the market for these rights to equilibrate as assumed, the output
prices pi and P2 must already be known when the sale occurs. Given this timing, selling coupons represents a
credible commitment by the entrant slightly to undercut the incumbent's price as long as pi > P2. Thus, coupons
act as a "beating competition" clause. See Salop (1982).
18 The entrant may prevent selective matching by keeping a secret registry of customers entitled to the
low-priced output or by replacing lost (or honored) coupons, thereby maintaining its output sales at k. We
assume throughout that the entrant precommits itself to limiting its coupon sales to the original k units.
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GELMAN AND SALOP / 321
Differentiating equation (9) with respect to pi yields a first-order cond
to that of the accommodation strategy, as given by equation (4). Hence, for any k, the
incumbent chooses the same pi whether honoring the coupons or accommodating entry
The incumbent's optimal response function is derived by comparing profits from
selective matching, accommodating, and matching. Selective matching is more profitable
than accommodating only if
P2>
Cl.
(10)
If equation (10) is satisfied, the incumbent compares its profits from matching and
selective matching. Under either strategy, the incumbent serves the entrant's k potential
customers at the price P2 and earns profit Hlc = (P2 - cl)k from these sales. Subtracting
LIc from equations (2) and (9), and recalling that p,(k) maximizes
Hl(pl) = (pi - cl)[D(p) -k],
it follows that Ils > rIM if and only if
(p,(k) - cl)(D(pl(k)) - k) > (P2 - cl)(D(p2) - k). (11)
Hence, a (P2, k) pair induces the incumbent to selectively match the entrant if and only
if p,(k) _ P2-'9
Figure 2 illustrates the (P2, k) pairs for which the incumbent's optimal response is
undercutting (U), matching (M), selectively matching to coupon holders (S), or accommodating (A).
3 The entrant's problem. Taking into account the incumbent's reaction function, the
entrant chooses the (P2, k) pair that maximizes its profits. If the entrant issues coupons,
it has two potential revenue sources: the sale of coupons and the sale of output.
FIGURE 2
STRATEGY REGIONS
D(p)
Pi
m
XP1
(k)
19
The
incumbent
cannot
di
cannot
be
induced
to
reveal
th
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322 / THE BELL JOURNAL OF ECONOMICS
If the entrant picks P2 < cl, the incumbent's best response is to accommodate its
entry, allowing the entrant to sell coupons and output. Given this accommodation re-
sponse, the entrant maximizes its profits as follows:
JI2A = max (P2 - c2)k + qk (12)
P2,k
subject to
q= pi -P2; pi = p,(k);
P2 < cl; P2 <P,
Substituting for q and pi, we have
max
(p,(k)
-
c2)k
(13)
P2,k
subject to
P2 < Cl.
Differentiating the maximand in (13) with respect to k and rearranging the first-order
condition, we have
p,(ka) - c2 + kapIl(ka) = 0, (14)
where ka denotes the entrant's optimal capacity choice, given that it issues coupons and
induces an accommodation response. The optimal choice of P2 is indeterminate as long
as P2 < Cl .
If the entrant picks P2 such that p,(k) > P2 > cl, the incumbent selectively matches
by honoring the coupons. This implies that the entrant sells only coupons and no output.
The entrant maximizes its revenue from coupon sales as follows:
I12S
P2,k
=
max
qk
(15)
subject to
q =pi - P2; pi = p,(k);
P2 > Cl; P2 <P,.
Substituting for q and pi, we have
max (pi(k) - p2)k (16)
p2,k
subject to
P2 > Cl.
By inspection, it is clear that the entrant maximizes its profits by setting P2 = c
Substituting this equation into equation (16) and differentiating, we have the first-order
condition
p,(ks) - cl + kspIi(ks) = 0, (17)
where kS denotes the entrant's optimal capacity choice, given that it issues coupons and
induces selective matching.
Comparing equations ( 13) and ( 16), it is clear that if C2 > cl, the entrant's profits are
higher when the incumbent honors its coupons.2' By comparing equations (14) and (17),
20 For c2 > cl, this implies P2 < C2. This introduces the possibility that the incumbent could destroy the
entrant by subsequently refusing to honor its coupons. Such a strategy violates the leader/follower equilibrium
concept used in this article.
21 If the more efficient entrant issues coupons, it sets P2 < cl, thus inducing accommodation by the
incumbent.
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GELMAN AND SALOP / 323
it is easy to show that in inducing the incumbent to honor coupons, the less efficient
entrant selects a larger k (i.e., kS > ka). Finally, by comparing the coupon strategy with
pure capacity limitation, we show that the less efficient entrant earns additional profits
from selling coupons.22
Coupons can yield a peculiar equilibrium in which the entrant sells no output and
earns revenue only from coupon sales. Although the entrant's output price (when ac-
companied by a coupon) is P2, consumers only realize the benefit of the reduction in the
incumbent's price p1(k) because they must purchase coupons at price q = pi - P2. The
entrant obtains the windfall profit. But coupons do result in a higher level of industry
output than would occur under pure capacity limitation.
The sale of coupons also facilitates the rationalization of industry production. The
entrant sets its price so that the incumbent accepts coupons and produces the k units of
output only if the incumbent is the more efficient producer. If the entrant is less efficient,
this results in an industry cost savings of (C2 - cl)k-a savings which accrues entirely to
the entrant as profits.
Finally, selling coupons represents a counterstrategy by which the entrant can par-
tially overcome the incumbent's lexicographic demand advantage. In effect, the entrant
extorts some of the incumbent's profits by credibly threatening to produce k units unless
it is bought off. Faced with this threat, the incumbent purchases the rights to the output.
Although the market mechanism by which this deal is carried out is complex in its details,
it is quite simple in its essence: the entrant blackmails the incumbent into sharing its
profits by threatening to spoil the market.
4. Conclusions
* This article has presented a new model of incumbent/entrant interaction. By setting
a low price and limiting its capacity, the entrant makes accommodation more profitable
than retaliation in two ways. The capacity limitation restricts the incumbent's loss of
market share from accommodation. The entrant's low price increases the incumbent's
loss from matching.
The entrant can further increase its own profits by coupling the capacity limitation
with sales of transferable coupons. If the incumbent has a cost advantage, optimal pricing
by the entrant leads to an industry equilibrium in which the entrant sells coupons and
no output. All output is produced and sold by the incumbent.
Many questions are left unanswered. The structure of the model is special, and many
properties of the equilibrium remain unanalyzed. Extension of the model to consider
multiple entrants in a free-entry equilibrium is needed. Even within the context of the
present model, we have not explored the relative profits nor the share gained by an equally
efficient entrant. In addition, we have not fully analyzed alternative commitment strategies
such as meeting-competition, beating-competition, and most-favored-nation provisions.
Appendix
Random rationing
* In Section 2 we assumed that the consumers with the highest reservation prices obtain
the entrant's scarce output. In the absence of transferable rights and an aftermarket, this
22 By comparing equations ( 13) and (6), it is easy to show that a more efficient entrant also prefers selling
coupons to limiting capacity. A more efficient entrant has, however, an alternative strategy: it can undercut the
incumbent and supply the entire market. For some range of c2 < cl, the entrant prefers undercutting the
incumbent to limiting capacity, but prefers selling coupons to undercutting. For such an entrant, the ability to
sell coupons results in lower output by the entrant, a higher market price pi, and more production by the highcost incumbent. See Gelman and Salop (1982) for details.
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324 / THE BELL JOURNAL OF ECONOMICS
class of consumers would not necessarily obtain the product. We now consider the perhaps
more realistic case in which nontransferable rights to the scarce output are distributed
randomly among the willing purchasers.
Let the density of consumers' reservation prices be given by h(w). Total demand is
thus the integral of this density, or
D(p)=
f
h(w)dw.
(A1)
Suppose the incumbent accommodates the entrant's price/capacity choice (P2, k) by
setting pi > P2. Each of the D(p2) consumers with reservation prices greater than or equal
to P2 prefers to buy from the entrant. Under random rationing, the rights to the k units
are apportioned as follows. Let r be the proportion of consumers served at each reservation
price, where r is defined according to
k
=
f
rh(w)dw.
(A2)
2
Substituting equation (A1) into (A2) and rewriting, we have the rationing function
r(p2,
k),
k
r = r(p2, k) = D(P) (A3)
When the entrant's units are rationed to k lucky customers in this way, the incumbent
is left with a residual demand of disappointed customers xi who constitute a fraction
(1 - r) of the potential market, or
xi = (1 -r) h(w)dw. (A4)
Substituting equation (Al) into (A4), we have
X=
(1
-
r)D(p1).
(A5)
Under the optimal accommodation strategy, the incumbent's profits are given by
HA = max (pi - cl)(1 - r)D(pl) (A6)
Pi
subject to
Pi > P2.
Differentiating (A6), we have the first-order condition
(1 - r)[(p1 - cl)D'(pl) + D(pl)] = 0. (A7)
Inspecting equation (A7), it is obvious that for the case of constant marginal cost,
the accommodation price is invariant with respect to the entrant's (P2, k) choice. In fact,
the accommodation price equals the (preentry) monopoly price (i.e., Pi = pim). Defining
the incumbent's maximum accommodation profits as IIA(P2, k), we have
11A(P2, k) = [1 - r(p2, k)](pim - cl)D(plm). (A8)
Of course, the incumbent still has the choice between accommodating and matching.
Following equation (5) in the text, we define the accommodation set as the (P2, k) pairs
such that HA > HM. Upon rewriting this constraint, we have
(1 - r)H1(pim) > H(P2), (A9)
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GELMAN AND SALOP / 325
where 11(p) = (p - c)D(p). Equation (A9) can be inverted to derive the demand curve
P2 1< kr(k), and the entrant's optimal strategy can be calculated as in equation (7) in
the text.
This uniform rationing case captures an often-asserted feature of dominant-firm
behavior. As long as the fringe (entrant) stays within the accommodation set, the incumbent does not react at all, but continues to price at the monopoly level. Only if the fringe
violates the industry "discipline" does the incumbent respond. When the punishment
comes, it is discontinuous and extreme-the incumbent matches the entrant's price and
captures all its customers.
References
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All use subject to http://about.jstor.org/terms