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Product differentiation under congestion:Hotelling was right
Christian Ahlin and Peter D. Ahlin ∗
May 2012
Abstract
We introduce negative network externalities – “congestion costs” – into Hotelling’s(1929) model of spatial competition with linear transportation costs. For any firm lo-cations on opposite sides of the midpoint, a pure strategy price equilibrium exists andis unique if congestion costs are strong enough relative to transportation costs. Weanalyze product differentiation and find that Hotelling’s Principle of Minimum Differ-entiation comes closer to holding in the presence of congestion costs. The greater arecongestion costs, the less differentiated products can be in (locationally-symmetric)equilibrium. In fact, minimum differentiation comes arbitrarily close to holding de-pending on the magnitude of these costs relative to transportation costs. Intuitively,greater congestion effects stabilize competition at closer quarters, eliminating aggres-sive pricing equilibria. Thus, negative network externalities can play a significant rolein product differentiation.
1 Introduction
“Nobody goes there anymore; it’s too crowded.” — attributed to Yogi Berra
In the seminal Hotelling (1929) model of spatial competition, two firms compete to sell
a product. In stage one, they locate along a unit line; in stage two, they set their prices.
Consumers spread evenly along the line choose the best deal based on the posted price and
linear transportation costs.
∗We thank especially Herve Moulin for initiating a preliminary version of this project with the secondauthor. We also thank two referees and the editor Tim Brennan along with Rick Bond, Andy Daughety,Arijit Mukherjee, Martin Osborne, Carolyn Pitchik, Jennifer Reinganum, Oz Shy, and various conferenceparticipants for helpful comments. All errors are our own.
1
0 10.5> >>> >< <<<<
Hotelling (1929)
0 10.25 0.75> > > < < <?? ?
d’Aspremont, Gabszewicz, Thisse (1979)
0 10.26 0.740.5> > >< < < < <<> > >
Osborne, Pitchik (1987)
Figure 1: Previous results in the Hotelling (1929) linear transportation cost setting.
Hotelling apparently solved for the second stage equilibrium in pure strategy prices, and
via backward induction showed that regardless of location, either firm would increase its
profits by moving closer to its competitor. (See Figure 1, first line.) This gave rise to the
“principle of minimum differentiation” – the tendency of competing firms to make similar
choices in geographical or product-characteristic dimensions.
D’Aspremont et al. (1979) critiqued this principle’s validity within the Hotelling model.
They showed that no pure strategy pricing equilibrium exists when the firms are sufficiently
close together (but not at the same location). This is because for firms at close quarters,
the temptation is strong for one firm to cut its price just enough to steal the entire mass of
customers located on the other side of its competitor. In fact, if locations are symmetric,
the Hotelling pricing equilibrium exists only if firms locate in the outer quartiles. Thus,
Hotelling’s backward induction breaks down and there is no guarantee that firms located in
the inner quartiles would want to move closer to each other. (See Figure 1, second line.)
At this point the literature diverges. Much of the literature follows D’Aspremont et al.
(1979), who introduce quadratic transportation costs and show that they provide greater
tractability.1 With quadratic costs, the differentation result is reversed: firms maximally
1For example, Tirole’s textbook (1988, pp. 279-82) focuses on this case.A few other contributions take alternative routes to showing minimum differentiation results in Hotelling-
2
differentiate in equilibrium.2
Osborne and Pitchik (1987) continue with Hotelling’s linear transportation costs, despite
the tractability challenges. They characterize a mixed strategy pricing equilibrium when firms
have located close to each other. In the symmetric subgame perfect equilibrium they compute
where location choices are pure, both firms locate at the edges of the inner quartiles, 0.24 of
the distance from the midpoint. (See Figure 1, third line.) Thus, substantial (near-efficient)
differentiation remains in equilibrium. Evidently, firms’ incentives are to move farther apart
over much of the region where Hotelling’s pricing equilibrium fails to exist.
This paper also retains linear transportation costs, and adds one feature to the model
– negative network externalities. Specifically, consumers are assumed to derive disutility in
proportion to the number of customers of the firm they patronize. This negative externality
can be interpreted as congestion arising from firm capacity constraints – queueing for parking,
customer service, or to pay.3 In many cases, the more crowded a store or restaurant, the
less attractive it is to patronize, all else equal. It can also be interpreted as a specific kind
of snobbery which we term “brand snobbery”: preferring the brand less consumed.4
We find that the greater are congestion costs relative to transportation costs, the more
of Hotelling’s line contains a Hotelling-like pure strategy pricing equilibrium. The centered
interval where the equilibrium does not exist for symmetrically-located firms shrinks as
congestion costs grow in importance, and in the limit vanishes. That is, the Hotelling pricing
equilibrium can be supported at any locations where the firms locate on opposite sides of
the midpoint, given sufficiently high congestion costs relative to transportation costs.
This result has implications for product differentiation. In essence, congestion costs
extend the portion of the line over which Hotelling logic applies, i.e. over which firms can
like models. Among them, de Palma et al. (1985) introduce unobserved, exogenous taste heterogeneity intothe Hotelling model with linear transport costs and show that minimum differentiation holds as long as thetaste heterogeneity is strong enough. Irmen and Thisse (1998) analyze differentiation of multi-characteristicgoods with quadratic transport costs, and find minimum differentiation holds in all but one characteristic.
2In general, sufficiently convex transportation costs restore the existence of a pure strategy price equilib-rium at any location pair (Economides, 1986), and give rise to substantial, if not maximal, differentiation.
3“Congestion costs” are used interchangeably with “negative network externalities”.4See Section 3.3 for a discussion of the snobbery interpretation.
3
increase profits by moving closer to each other. As congestion costs grow in importance,
more and more of the line pushes toward less differentiation, and the centered interval, in
which any (symmetric, pure) equilibrium firm locations must be contained, shrinks. In sum,
we show that the bound on product differentiation in any such equilibrium declines with
congestion costs and approaches zero.5
Without further assumptions, we are unable to prove existence of a subgame perfect
equilibrium with symmetric, pure location choices.6 Our last result adds to the model the
assumption of a minimum distance between firms, justifiable because firms take up non-
negligible space or because of patent or copyright restrictions.7 For any strictly positive
minimum distance, congestion costs that are sufficiently large relative to transportation
costs guarantee that there exists a unique equilibrium in which location choices are pure and
symmetric, and that this equilibrium involves minimum differentiation.
The intuition for these results may seem straightforward: since congestion directly damp-
ens price competition (e.g., de Palma and Leruth, 1989, Grilo et al., 2001), it reduces the
incentive to dampen price competition by moving away from one’s competitor. This intu-
ition is flawed, however. It ignores the fact that congestion also dampens the market-stealing
incentive to move closer to one’s competitor. In fact, wherever Hotelling’s pure strategy pric-
ing equilibrium is played, congestion’s effects on the opposing “competition-dampening” and
“market-stealing” locational incentives exactly cancel out.8
Thus, congestion has its effect not by altering incentives to differentiate in any given
pricing equilibrium, but by changing the kind of equilibrium that gets played. Recall that in
the original model the Hotelling pricing strategies are destabilized when the firms are close
5Though firms may locate very close to each other, profits can still be substantial because equilibriumprices rise with the congestion cost.
6Osborne and Pitchik (1987) also do not provide an analytical proof of existence of a subgame perfectequilibrium, though they argue convincingly for it using a mix of analytics and computation. Vogel (2008)is able to show existence of a subgame perfect equilibrium with linear transport costs using a circle ratherthan a line; his techniques do not apply here because, here, a mixed strategy pricing equilibrium is neededto support any subgame perfect equilibrium in which location choices are pure.
7See Dasgupta and Maskin (1977) and Brekke et al. (2006, 2007) for a similar assumption in Hotellingmodels.
8See Section 3.1 for a discussion in both the linear and quadratic transportation cost setting. This resultis not driven by functional form assumptions on the congestion cost, at least for symmetric locations.
4
together by the temptation toward a significant price cut that steals a mass of consumers.
In these cases, a mixed strategy equilibrium results, with random and significant price cuts
(“sales”) as part of firms’ strategies and with incentives typically tilting toward greater
differentiation so as to relieve price competition (Osborne and Pitchik, 1987). Congestion,
however, dampens the demand swings a firm can capture by price cuts and makes such
deviations less tempting. As a result, it “stabilizes” the Hotelling pure strategy equilibrium
– with its anti-differentiation incentives – for a larger set of firm locations (and eliminates
the “aggressive” mixed strategy equilibrium).
In essence, greater congestion allows firms to get closer to their competitors without trig-
gering substantial price cuts. Congestion thus gives firms incentives to locate more closely
together, and arbitrarily closely as congestion costs grow in importance relative to trans-
portation costs. In this sense, greater congestion means Hotelling’s conclusion of minimum
differentiation is valid over more of the unit line – and in the limit, is arbitrarily valid.
Critical to the link between congestion and minimum differentiation turns out to be the
linear transportation cost assumption. The well-known disadvantage of this assumption is
less tractability, and hence, less than ideally sharp results. The advantage is that it brings
out an unnoticed effect of potential empirical importance: network externalities can affect
incentives for product differentiation.
Several papers have examined negative network externalities in a similar setting. Of
these, four are most similar to our location-then-price model with congestion. Kohlberg
(1983) introduces congestion (waiting costs) to the location-only Hotelling model. He focuses
on equilibrium existence as the number of firms varies. De Palma and Leruth (1989) analyze
a capacity-then-price model with congestion. We model congestion exactly as they model
capacity constraints, except that here firm capacity is exogenous and identical across firms
while location is endogenous. Wauthy (1996) adds exogenous capacity constraints – modeled
as a strict limit on number of customers – to the original Hotelling model and shows that
they can restore the Hotelling pricing equilibrium, provided that both they and consumers’
willingness to pay are in an intermediate range. Grilo et al. (2001) analyze both positive
5
and negative network externalities (“conformity” and “vanity”) in a Hotelling setting with
quadratic transportation costs. They focus on how these externalities affect competition.
Our paper differs from all of these in its focus on product differentiation. To our knowledge,
then, this paper is the first to establish a link between negative network externalities and
product differentiation.
While empirical testing is left for future work, some examples can be noted. The results
suggest that since restaurants appear especially prone to congestion, similar restaurants
may locate relatively close to each other. Each stably prices above cost knowing the others
cannot profitably undercut the market because of congestion concerns. These results offer
an explanation for phenomena like China Towns and Little Italys.
In general, the results suggest that any market where congestion concerns are strong rela-
tive to transportation costs or taste heterogeneity can expect to see minimal differentiation of
sellers. In principle, such markets can be identified by evaluating the average customer costs
of waiting to pay, waiting for service, and stockouts, and comparing these with costs from
transportation and mismatch between marketed items and what consumers would prefer.
Several factors would make congestion costs more likely to dominate. One is market den-
sity: increasing density would tend to raise congestion concerns while not affecting trans-
portation costs as much. This would suggest that clustering of similar restaurants (e.g.
China Towns) is more common in cities than it is non-urban areas – a prediction that seems
borne out empirically. A second (and related) factor is land rental prices. Higher rental
prices gives rise to smaller establishments, all else equal, and greater congestion concerns.
If transportation or taste heterogeneity is not as strongly affected, the net result would be
less differentiation. Third, all else equal a higher variance of demand, for example due to
seasonal or time-of-day demand fluctuations, would tend to raise relative congestion costs,
making it more likely that congested conditions are experienced by the average customer.
This would predict that vacation-area establishments, for example, differentiate less. Fi-
nally, taking the product heterogeneity interpretation of transport costs, in markets where
products are more substitutable (so that “transport” costs are lower), differentiation will be
6
0 1
xa
xbδ
a bfirm firm
Figure 2: Firm a locates at distance xa from the left endpoint. Firm b locates at distancexb from the right endpoint. The distance between firms is δ ≡ |1− xa − xb|.
less. Of course, in these circumstances the lack of equilibrium differentiation is less costly
to consumers, implying that market forces have a limiting impact on the overall inefficiency
from differentiation.
Next, Section 2 describes and analyzes the model. Section 3 discusses how and why
results would differ under quadratic transportation costs, positive network externalities, and
a continuous form of snobbery. Section 4 concludes. Most proofs are in the Appendix.
2 Model and Results
There are three stages.
1. In the first stage, firms a and b simultaneously choose locations along a unit line.
2. In the second stage, knowing each other’s locations, the firms simultaneously post
prices for the good, whose production cost is zero.
3. In the third stage, a unit measure of consumers distributed uniformly on the line
observes the firms’ prices and locations and chooses which firm to purchase from.
We restrict attention to subgame perfect equilibria, and analyze the stages in reverse order.
The distances of firms a and b from the left and right endpoints, respectively, are called
xa and xb, and the distance between firms is δ ≡ |1− xa − xb|; see Figure 2. Each consumer
has unit demand for the good. The cost to a consumer who buys from firm i ∈ {a, b}includes the posted price, pi, and a linear transportation cost proportional to τ > 0 and the
consumer’s distance to firm i.
7
To Hotelling’s model we add a third cost due to congestion effects. Specifically, if qi is
the quantity sold by store i, consumers experience an additional cost equal to κqi, κ > 0,
when purchasing from store i. Hence, the disutility to a consumer located at point x on [0, 1]
and buying from firm a is
−Ua(x) = pa + τ |xa − x|+ κqa, (1)
while buying from firm b is
−Ub(x) = pb + τ |1− xb − x|+ κqb. (2)
2.1 Stage 3: Equilibrium Demand
Since consumers observe (pa, pb) and (xa, xb), the choice of where to buy would be mechanical
if not for the network externality; this requires consumers to forecast firms’ sales, qa and
qb = 1 − qa, respectively. Equilibrium in this stage will require that consumers minimize
costs based on their forecasts, and that their forecasts turn out to be correct.
Lemma 1. Given (pa, pb) and (xa, xb), equilibrium quantities sold, qa and qb, are unique.
Uniqueness is due to the congestion costs.9 These do not eliminate the possibility of a
mass of indifferent consumers, but they condition the indifference on a specific division of
consumers across firms;10 any other division of consumers would alter relative congestion
costs and eliminate the indifference of the mass of consumers. It is well known that when
network effects are positive, there can be multiple equilibria as consumers can coordinate
on one or the other firm. Negative network effects, however, guarantee uniqueness because
demand swings away from one firm make that firm more, not less, attractive.
Equilibrium demand as a function of (pa, pb) and (xa, xb) can be found by solving equa-
9In short, if two different quantities sold by firm i were compatible with equilibrium, this would requirestrictly more customers to choose firm i when relative congestion costs there were strictly higher (withtransportation and financial costs unchanged) – a contradiction.
10Division here refers to the numbers of consumers going to each firm, not the specific identities.
8
0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Price of firm i, pi
Quan
tity de
mand
ed fro
m firm
i, q i
Demand −− qi
xa = x
b = 2/5
pj = 1
κ + τ = 1
κ/τ = 0
κ/τ = 0.5
κ/τ = 1
IIIi
Vi
Ii
IIi
IVi
0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Profits −− pi*q
i
Price of firm i, pi
Profi
ts of
firm i
κ/τ = 0
κ/τ = 0.5
κ/τ = 1
xa = x
b = 2/5
pj = 1
κ + τ = 1
IIIi
Vi
Ii
IIi
IVi
Figure 3: Firm i’s demand and profits as a function of its price pi, for various κ/τ ratios,given κ+ τ = 1, pj = 1, and xa = xb = 2/5. Where not visible, the dashed and dash-dotteddemand curves and profit functions coincide with the solid lines.
tions 1 and 2 for the prices at which one firm captures the entire market and, for other prices,
the location of an indifference point, to the left (right) of which consumers patronize firm a
(b). Assume without loss of generality that xa + xb ≤ 1, as in Figure 2.11 Firm-i demand is
qi =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
1 if pi ≤ pIi ≡ pj − τδ − κ | region Ii
pj−pi−τδ+κ
2κif pIi ≤ pi ≤ pIIi ≡ pj − τδ − κ(1− 2xj) | region IIi
pj−pi+τ(1+xi−xj)+κ
2(κ+τ)if pIIi ≤ pi ≤ pIIIi ≡ pj + τδ + κ(1− 2xi) | region IIIi
pj−pi+τδ+κ
2κif pIIIi ≤ pi ≤ pIVi
≡ pj + τδ + κ | region IVi
0 if pIVi≤ pi | region Vi
, (3)
for (i, j) = (a, b) and (b, a). See Figure 3. For brevity, throughout the paper statements
involving i are taken to apply for i equal to a and b, and statements involving (i, j) are taken
to apply for (i, j) equal to (a, b) and (b, a).
11This is without loss of generality since competition at firm locations (xa, xb) is isomorphic to competitionat firm locations (1− xa, 1− xb).
9
In regions Ii/Vj,12 the price discount at firm i is sufficient to overcome the transportation
cost of traveling from one firm to the other (τδ) and the disutility from consuming the same
product as all other consumers (κ). Firm i captures the whole market. In regions IIIi/IIIj,
there is a point between the two firms such that all customers to the left (right) of this point
strictly prefer, and patronize, firm a (b). The novel cases are regions IIi and IVi. In regions
IIa/IVb, for example, all consumers to the left of firm b strictly prefer firm a. Consumers
to the right of firm b are indifferent between firms a and b, conditional on a specific number
of them patronizing each firm; any other number would raise congestion at one firm and
make the other strictly preferable. Within these regions, a lower pa or higher pb pushes
more customers to firm a, to the point that the resulting higher congestion at firm a restores
indifference of this mass of consumers.
2.2 Stage 2: Pricing Equilibrium at Fixed Locations
With congestion costs, the demand curve is continuous, unlike in the original Hotelling model.
This can be seen in Figure 3, which graphs firm i’s demand and profit functions against its
price for several values of κ/τ , with (κ+ τ) fixed. The original model, κ/τ = 0, corresponds
to the solid lines. Demand and profits are discontinuous, as is well-known. For example, firm
i can lower its price below firm j’s just enough to cover the cost of transportation between the
two firms and win the entire mass of consumers on the other side of firm j. With congestion
effects, however, demand responses to price cuts are continuous and dampened. This can be
seen in the dashed and dash-dotted demand curves and profit functions of Figure 3.
Analysis of this pricing subgame reveals that a Nash equilibrium in pure strategies exists
for any strictly own-sided locations, i.e. for any xa, xb < 1/2 – as long as congestion effects
are large enough relative to transportation costs.
Lemma 2. Assume xa, xb ∈ [0, 1/2). If and only if congestion costs are high enough rel-
ative to transportation costs, (that is, there exists a function C(xa, xb) such that iff κ/τ ≥12It is easily verified that firm i pricing in region Ii, IIi, and IIIi respectively, is equivalent to firm j
pricing in region Vj , IVj , and IIIj , respectively.
10
C(xa, xb),) the following Nash equilibrium exists:
p∗a = κ+ τ + τ(xa − xb)
3p∗b = κ+ τ + τ
(xb − xa)
3. (4)
No other pure strategy Nash equilibrium exists.
As in Hotelling’s model, there is at most one pure strategy pricing equilibrium. In
contrast to Hotelling’s model, a pure strategy equilibrium exists regardless of how (own-
sidedly) close to the midpoint the firms locate, as long as relative congestion costs are
high enough.13 Intuitively, congestion costs support the equilibrium by muting the demand
response to a price cut. This is illustrated in the profit function graph of Figure 3; the
candidate equilibrium is at pi = 1, but only when κ/τ is high enough is it robust to a price
cut deviation.
The equilibrium, when it exists, involves a point of indifference between the two firms
and strict preference for firm a (b) to the left (right) of this point. The equilibrium prices are
identical to Hotelling’s, except for the addition of κ. Higher congestion costs thus dampen
competition and allow firms to charge more and earn more in equilibrium, as others have
found.14
Note that the key conditions involve κ/τ , not κ by itself, so the results do not require
congestion costs to be high in absolute terms. Indeed, relative congestion costs κ/τ can
achieve any positive value with κ and τ bounded, and hence also equilibrium prices.
The exact bound function guaranteeing existence, C(xa, xb) (defined by equation 10 in
the Appendix), comes from ensuring that deviation profits are no greater than equilibrium
profits (equation 9 in the Appendix). The bound increases the closer either firm gets to
the other (i.e. C(xa, xb) increases in xa and xb), and approaches infinity as either firm
approaches the midpoint. Thus, greater relative congestion costs are needed to “stabilize”
the pure strategy pricing equilibrium when firms are closer to each other. The (own-sided)
13One can also show that when both firms locate on the same half of the line, but neither is at the midpoint,high enough κ/τ guarantees a pure strategy pricing equilibrium. If exactly one firm is at the midpoint, nopure strategy pricing equilibrium exists regardless of the value of κ/τ .
14Even if firms share the same location, congestion gives rise to positive prices and profits in equilibrium.
11
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Firm a location, xa
Firm
b lo
catio
n, x
b
Pure strategy pricing equilibrium existence
κ/τ = 0
κ/τ = 1/3
κ/τ = 1
κ/τ = 2
κ/τ = 5
Figure 4: The pure strategy pricing equilibrium of Lemma 2 exists when (xa, xb) falls belowthe solid lines, for various values of κ/τ .
locations for which this equilibrium exists are graphed for various values of κ/τ in Figure 4.
The main importance of Lemma 2 is that it facilitates backward induction by helping
map out the payoffs of stage-1 locational choices. As Figure 4 makes clear, the extent of
our understanding of second-stage pricing behavior grows the greater are relative congestion
costs. There are two unresolved issues, however.
First, what type of pricing behavior results where the pure strategy pricing equilibrium
does not exist? It is straightforward to show existence of a pricing equilibrium – in mixed
strategies – at these locations.15 Unfortunately, characterizing the payoffs in these mixed
strategy equilibria enough to guide backward induction and guarantee a subgame perfect
equilibrium is hard; it has not been done purely analytically even in the original Hotelling
model.16 One approach we will take is to introduce a simplifying assumption that, combined
with conditions on congestion costs, can guarantee existence.
15Theorem 1.3 of Fudenberg and Tirole (1991), due to Glicksberg (1952), can be used since the payofffunctions are continuous.
16Osborne and Pitchik (1987) nearly do so, but must partly rely on computation.
12
Second, where the pure strategy pricing equilibrium does exist, is it unique? Lemma 2
guarantees uniqueness among pure strategy equilibria, but does not rule out existence of
additional mixed strategy equilibria. Of course, the possibility of multiple equilibria can
complicate backward induction. We use two alternative strategies to address this. One
strategy is to follow much of the literature in ignoring the potential existence of mixed
strategy equilibria where a pure strategy equilibrium exists. Specifically, it can be assumed
that a pure strategy pricing equilibrium is selected whenever possible. The second strategy
is to derive a bound on relative congestion costs that rules out mixed strategy equilibria
when the pure strategy pricing equilibrium exists.
Lemma 3. Assume xa, xb ∈ [0, 1/2). If congestion costs are high enough relative to trans-
portation costs, (that is, there exists a function D(xa, xb) such that iff κ/τ ≥ D(xa, xb),) the
Nash equilibrium of Lemma 2 exists and is unique.
The proof uses dominance arguments (drawing on Osborne and Pitchik, 1987) to limit
a firm’s pricing support in any equilibrium to a single point, thus ruling out mixed strategy
equilibria. These arguments require quasi-concavity of firm i’s profit function when pj ≤ p∗j ;
necessary and sufficient for this is κ/τ ≥ D(xa, xb).17
Thus, any strictly own-sided location pair has as its unique pricing equilibrium the
Hotelling one, as long as relative congestion costs are large enough. Though the bound
function guaranteeing existence and uniqueness, D(xa, xb) (equation 11 in the Appendix), is
greater than the bound guaranteeing only existence, C(xa, xb), it has the same qualitative
features. In particular, it increases the closer either firm gets to the other (i.e. D(xa, xb)
increases in xa and xb), and approaches infinity as either firm approaches the midpoint.
Looking ahead, there will be two parallel sets of location equilibrium results: one with
quantitatively tighter bounds on differentiation aided by an assumption on equilibrium se-
lection, the other with looser bounds but no additional assumption. Qualitatively, both sets
17Quasi-concavity, however, may not be necessary for uniqueness; it clearly is not necessary for existenceof a pure strategy pricing equilibrium (see, for example, the dash-dotted profit function of Figure 3). Inparticular, it may be that no mixed strategy equilibrium exists even under the weaker bound, C(xa, xb).Supporting this conjecture, Osborne and Pitchik find that the pure strategy pricing equilibrium is the uniquepricing equilibrium whenever it exists (in the model without congestion).
13
of results will be the same – greater relative congestion costs reduce differentiation, in the
limit to zero.
2.3 Stage 1: Location Equilibrium
The pricing equilibrium results have implications for subgame perfect equilibrium firm loca-
tions. Specifically, congestion costs extend the portion of the line over which the Hotelling
pricing equilibrium exists and minimum differentiation logic applies.
Define a pure strategy, symmetric location equilibrium (PSSLE) as a subgame
perfect equilibrium of the entire game in which location strategies (but not necessarily pricing
strategies) are pure and symmetric. Our main result is that
Proposition 1. The distance between firms in a PSSLE is bounded by a quantity that strictly
decreases and approaches zero in relative congestion costs, κ/τ . A tighter bound for this
distance, which assumes that a pure strategy equilibrium is selected in the pricing game if
one exists, is
1−κτ
2if κ/τ ≤ 1/3
1− 2(√
κτ(1 + κ
τ)− κ
τ
)if κ/τ ≥ 1/3
.
A looser bound with no assumption on pricing equilibrium selection is
1
1 + 2κτ
.
Proof. At any symmetric firm locations xa = xb = x ≤ 1/218 with inter-firm distance δ
strictly greater than the second expression, κ/τ > D(xa, xb) (where D(xa, xb) is defined in
equation 11). By Lemma 3, there is a unique pricing equilibrium for any firm locations in a
neighborhood of (x, x). By inspection of firm-i equilibrium profits (equation 9), firm i would
raise profits by locating closer to firm j. Thus (x, x) cannot be a location equilibrium.
18Given symmetry, own-sidedness is without loss of generality since competition at firm locations (x, x) isisomorphic to competition at firm locations (1 − x, 1− x).
14
0 10.33 0.670.25 0.75
κ/τ = 1/3
> > > > < < <<?
0 10.41 0.590.25 0.75
κ/τ = 1
> > > > > < < <<<?
0 10.48 0.520.25 0.75
κ/τ = 5
> > > > > > < < <<<<?
Figure 5: Bounds on differentiation in any PSSLE for several values of κ/τ , under theassumption that pure strategy pricing equilibria are selected if they exist. Where the lineis solid, a pure strategy pricing equilibrium exists; where the line is dashed, only a mixedstrategy pricing equilibrium exists. Wherever the pure strategy pricing equilibrium exists,locational incentives are for lessened differentiation. Higher relative congestion costs κ/τreduce possible differentiation, in the limit to zero.
If the distance between firms with xa = xb = x ≤ 1/2 is strictly greater than the first
expression, κ/τ > C(xa, xb) (where C(xa, xb) is defined in equation 10). By Lemma 2, the
pure strategy pricing equilibrium exists and is unique among pure strategy equilibria for any
firm locations in a neighborhood of (x, x). The above logic, with the assumption that pure
strategy equilibria are selected first, rules out this outcome as a location equilibrium.
Thus, the greater are congestion or snob effects relative to transportation costs, the closer
firms must be in any PSSLE. Asymptotically, the distance between them must go to zero.19
Proposition 1’s tighter bound on firm differentiation is graphed for several values of κ/τ
in Figure 5. The case of κ/τ = 0 is known: symmetric locations in the outer quartiles are
ruled out (d’Aspremont et al., 1979). As κ/τ increases, a smaller centered interval remains
in which any PSSLE must be located. In this sense, maximum firm differentiation declines
toward zero.
Intuitively, greater congestion “stabilizes” the pure strategy equilibrium with respect to
19Note that limx→∞(√
x(1 + x)− x)= 1/2.
15
price cuts, for more locations; it also rules out equilibria involving random price cuts for
more of the line. As a result, it paves the way for the market-stealing incentive to dominate
the competition-dampening incentive for all but a smaller centered interval. With the added
stability that congestion brings to competition, Hotelling’s (1929) original argument that
competition pushes firms toward minimum differentiation has greater validity.
It would be ideal, in addition to bounding differentiation in any PSSLE as Proposition 1
does, to prove existence of a PSSLE. As noted, showing existence of equilibrium in the pricing
stage is straightforward; the difficulty lies in gaining a sufficient handle on firm payoffs when
only mixed pricing equilibria exist to prove existence at the stage of location choice.
Instead, we prove existence of a minimum differentiation PSSLE with the addition of one
assumption to the model: that there is some exogenous minimum distance between firms,
δ ∈ (0, 1/2].20 If Hotelling’s line is interpreted as physical distance, this positive minimum
differentiation can represent the impossibility of firms co-locating since firms take up space.
If the line is taken to represent characteristic space, this minimum differentiation can be
thought of as resulting from copyright or patent restrictions. Specifically, assume that
Profits of both firms are identically zero if δ ∈ [0, δ), for some δ ∈ (0, 1/2]. (A1)
Proposition 2. Under assumption A1 and for sufficiently high relative congestion costs,
κ/τ , a unique PSSLE exists and involves minimum differentiation, that is the inter-firm
distance equal to δ.
Thus,21 as long as there is some inability of firms to co-locate and provided congestion
costs are strong enough relative to transportation costs, the unique PSSLE involves minimum
differentiation. In this sense, Hotelling was right – the principle of minimum differentiation
20Dasgupta and Maskin (1977) also introduce a minimum distance to the original Hotelling model, andBrekke et al. (2006, 2007) make the same kind of assumption in a location-then-quality Hotelling model.
21The proof mainly follows previous logic, with additional assumption A1 devaluing locational deviationsinto the centered interval where only a mixed strategy pricing equilibrium exists. The one additional com-plication comes in ruling out leapfrog deviations from minimum differentiation, where both firms end up onthe same side of the line. This would not seem especially profitable; however, we have not ruled out mixedstrategy pricing equilibria when both firms are on the same side of the line. We rule out these deviations bybounding the deviation payoff using dominance arguments.
16
holds, at least for goods exhibiting sufficient congestion.
3 Discussion
3.1 Linear vs. quadratic transportation costs
It turns out that under quadratic transportation costs, congestion does not affect differenti-
ation, which remains maximal. Here we discuss the difference in results.
Interestingly, even under the original linear transportation cost assumption, congestion
does not alter differentiation incentives over much of the locational space. To see this,
note that the change in profits from changing location has two components:22 one from the
change in demand directly due to the locational change, ∂qi/∂xi, and the other from the
change in demand due to the competitor’s equilibrium price response, (∂qi/∂pj)(dpj/dxi).
(Own-price changes can be ignored by the Envelope Theorem.) The first is the market-
stealing incentive to differentiate less; the second corresponds to the competition-dampening
incentive to differentiate more. In the linear-cost model where a pure strategy equilibrium
exists and is played,
∂qi∂xi
=τ
2(κ+ τ)=
1
2(1 + κτ), (5)
using the region-IIIi expression for demand from equation 3; and
∂qi∂pj
dpjdxi
=1
2(κ+ τ)
−τ
3=
−1
6(1 + κτ), (6)
using the region-IIIi expression for demand and the equilibrium prices of equation 4.
Clearly, the balance of incentives for differentiation where the pure strategy pricing equi-
librium exists is unaffected by relative congestion, κ/τ . Congestion reduces the demand effect
of the competitor’s price response, thus dampening the incentive to differentiate (equation 6);
but it also reduces the demand effect of encroaching on one’s competitor (equation 5). Net
incentives are scaled down but continue to tilt toward less differentiation (the sum of the
22This follows the discussion in Tirole (1988, pp. 281-2).
17
two effects is positive), as in Hotelling (1929).23
The same basic result can be shown in the quadratic transportation cost case. With
congestion,
∂qi∂xi
=3− 5xi − xj
6(δ + κτ)
−(xi − xj)
κτ
δ+κτ
6(δ + κτ)
, (7)
and
∂qi∂pj
dpjdxi
=−2 + xi
3(δ + κτ). (8)
As24 in the linear-cost case, both types of differentiation incentives are dampened by conges-
tion, and for the same reasons. Net incentives are merely scaled toward zero by congestion,
at least for symmetric locations. For asymmetric locations, it is easy to show that incentives
continue to tilt toward maximal differentiation for any parameter values.
In both setups, then, congestion costs do not alter differentiation incentives where the
pure strategy pricing equilibrium exists. This is the end of the story in the quadratic-cost
case, since the pure strategy pricing equilibrium exists everywhere. Maximal differentiation
continues to result.
In the linear-cost case, the pure strategy pricing equilibrium does not exist everywhere.
Congestion reduces the temptation toward price cuts and, thus, in more locations allows
existence of the pure strategy pricing equilibrium and eliminates mixed strategy pricing
equilibria with their aggressive price cuts. In short, congestion operates in this context
not by changing incentives in the domain of the pure strategy pricing equilibrium, but by
extending its domain while maintaining its anti-differentiation incentives.
23Interestingly, this result is not an artefact of the linear congestion cost specification we have used.Consider a strictly increasing, twice continuously differentiable congestion function g(q), say, and assumefirms are symmetrically located and that the Hotelling pricing equilibrium (i.e. the pure strategy pricingequilibrium with an indifferent consumer between the two firms) exists and is played. Then it can be shownthat the market-stealing incentive is always three times as big in absolute value as the competition-dampeningincentive, exactly as in the case above where g(q) = κq.
24Compare to equations (7.10) and (7.11) in Tirole (1988, p. 281).
18
3.2 Positive network externalities
From the preceding discussion, it is clear that we can only conjecture about the effect of
positive network externalities (κ < 0) in this framework. This is because positive externalities
tend to make price cuts more attractive. They would thus decrease the portion of the line
containing a pure strategy pricing equilibrium and widen the centered interval where only
mixed strategy pricing equilibria exist. Our lack of understanding of the mixed strategy
equilibria would then limit what could be said. It would seem reasonable to conjecture,
however, that a PSSLE would be located near the edge of the centered interval where only
mixed pricing equilibria exist, as in the original Hotelling model (Osborne and Pitchik,
1987); and if so, that differentiation would increase with the extent of the positive network
externalities (at least until multiple equilibria at the consumer stage became an issue).
3.3 A snobbery interpretation
One can interpret the negative externalities here as snobbery, since consumers experience
disutility the more others consume of the same good. However, it is snobbery of a specific
form, which can be called “brand snobbery” – disutility is experienced only from others’
consumption from the exact same firm/brand, but not nearby brands.25 The paper predicts
that the greater is brand snobbery, the less products are differentiated in equilibrium.
A potential application of this counterintuitive prediction is fashion. If polo shirt con-
sumers have strong preference for brand uniqueness, the model predicts producers will dif-
ferentiate little – e.g. via a horse or alligator, rather than substantial color or pattern
differences. This could also help explain fashion trends, in which designers market similar
fashions (e.g. 70’s-era) simultaneously with relatively moderate twists, instead of a case in
which very different decades and styles were on offer from one designer or another.
This result, though, depends critically on the form of snobbery. Consider a more contin-
uous snobbery, under which consumers experience disutility from others consuming similar
25This is also the approach to modeling “vanity” in Grilo et al. (2001).
19
products also. Specifically, let snob disutility be proportional to the number of customers
of a product and to a function of the distance of that product from a consumer’s chosen
product, h(δ). Let h(δ) be continuously differentiable and strictly decreasing, with h(1) = 0
and h(0) = 1. Then snobbery disutility for a customer of store i would be
κqi + κ h(δ)qj .
Here there is disutility from customers of store j also, to the extent that firm j markets a
product similar to firm i’s (low δ).
Since qj = 1− qi, snobbery disutility for a customer of store i equals
κ h(δ) + κ[1 − h(δ)]qi .
Only the utility differential across firms matters, so the κ h(δ) term can be ignored and this
model is the same as the one analyzed with a modified congestion cost κ′ = κ [1 − h(δ)].
Thus the pricing equilibrium Lemmas 2 and 3 go through with the bounds for κ/τ divided
by [1 − h(δ)]. However, the first stage analysis, location choice, differs since firms can raise
the relative snobbery cost κ′ = κ [1 − h(δ)] by raising the inter-firm distance. Specifically,
firm-i profits in the pure strategy pricing equilibrium are (see equation 9 in the appendix):
[κ′ + τ
(1 +
xi−xj
3
)]22(κ′ + τ)
=
{κ[1− h(δ)] + τ
(1 +
xi−xj
3
)}2
2{κ[1− h(δ)] + τ} .
While a formal analysis is beyond the scope of this paper, two suggestive points can be
made. First, it can be shown that if κ/τ is large enough (all else fixed), firm-i profits are
decreasing in xi, meaning firm i would prefer to move away from firm j, raising (relative)
snobbery costs by differentiating. Thus with a more continuous form of snobbery, greater
snobbery costs seem likely to push toward greater differentiation, not less.
Second, and on the other hand, if snobbery costs fall off quickly enough with distance,
then minimum differentiation can result from high levels of snobbery – even with continuous
20
snobbery. To illustrate this, assume h(δ) = (1− δ)n. Here as n gets large, snobbery costs at
nearby locations go to zero, and a result in the spirit of Proposition 2 holds. The argument
is as follows. Fix κ/τ high enough so that the conditions of Proposition 2 (which do not
assume a continuous snobbery function) hold with strict inequality. Then, one can show
that for n high enough, a) the conditions of Proposition 2 are also satisfied in the continuous
snobbery model at all symmetric locations not violating the lower bound on differentiation;
and b) equilibrium profits are increasing in xi meaning firm i can gain from moving closer
to its competitor. Thus, for this fixed κ/τ and n high enough, the only PSSLE is minimum
differentiation.
In sum, differentiation implications of snobbery depend on the form of snobbery. Loosely,
holding fixed the shape of a continuous snobbery function, a high enough magnitude of peak
snobbery costs relative to transportation costs appears to lead toward maximum differen-
tiation. On the other hand, holding fixed peak snobbery costs at a high enough level, and
concentrating snobbery more and more locally results in minimum differentiation – exactly
as in the baseline model of this paper. Thus, greater snobbery can weaken differentiation –
the preference for uniqueness stabilizing competition at close quarters.
4 Conclusion
We have shown that the existence of negative network externalities is a force for decreased
firm differentiation. Intuitively, these additional costs stabilize firm competition at close
quarters by dampening demand responses to price cuts. The more important are these
costs relative to transportation costs, the less differentation can be. In the limit, minimum
differentiation comes arbitrarily close to holding. Thus the existence of congestion costs gives
Hotelling’s Principle of Minimum Differentiation greater validity.
The model provides interesting testable implications. It suggests that differentiation be-
tween firms can depend on the type of good being sold, in particular its network externalities.
Variation across goods in the degree of congestion or brand snobbery would thus be related
21
to variation in product differentiation. Exploring this relationship empirically is left for fu-
ture work.
A Proofs
Proof of Lemma 1. Suppose not. Then for some (pa, pb) and (xa, xb), there exist two
equilibrium demand values for firm a, qa and q′a > qa. Let A, I, B denote the measures of
consumers that strictly prefer firm a, that are indifferent between firms, and that strictly
prefer firm b, respectively, when firm a sells qa; and similarly for A′, I ′, B′. Since q′a > qa,
relative congestion costs are strictly lower at firm a under qa than under q′a, and hence
A′ + I ′ ≤ A: all consumers who weakly prefer firm a at some relative congestion costs must
strictly prefer firm a at strictly lower relative congestion costs. Since A′ + I ′ + B′ = 1, this
implies that 1 − B′ ≤ A. Also, consumer optimization implies that qa ∈ [A, 1 − B] and
q′a ∈ [A′, 1 − B′]. Combining, q′a ≤ 1 − B′ ≤ A ≤ qa, which contradicts qa < q′a. Hence, the
values for qa and qb = 1− qa must be unique.
Proof of Lemma 2. We first find C(xa, xb) such that κ/τ ≥ C(xa, xb) is equivalent to
(p∗a, p∗b) – defined in equation 4 – constituting a Nash equilibrium. Fix pj = p∗j . It can be
verified that p∗i is in the interior of region IIIi (of equation 3) and is the best response to pj
in this region. It can also be shown that firm i profits are decreasing in price for pi ≥ pIIIi.
Thus the only potential profitable deviations are downward, in regions Ii and/or IIi.
Some algebra verifies that iff κ/τ ≥ Mhi≡ (xi + 2xj)/[3(1 − 2xj)], firm i’s profits are
monotonically increasing up to pIIi; this guarantees no profitable downward deviation. Iff
κ/τ ≤ Mli ≡ (xi + 2xj)/3, profits are monotonically decreasing in region IIi, so the best
response below pIIi is pIi. Deviation profits also equal pIi, since demand is one. Using the
expressions for p∗a and p∗b , and demand from equation 3, firm-i equilibrium profits are
[κ+ τ
(1 +
xi−xj
3
)]22(κ+ τ)
. (9)
Some algebra shows that deviation to pIi does not strictly raise profits iff
κ/τ ≥ Kli ≡xi + 2xj
3− (1− xj) + 2
√xj
(xi + 2xj
3
).
Iff Mli ≤ κ/τ ≤ Mhi, there is a price in region IIi with a zero (region-IIi) derivative, which
22
must then be the best response below pIIi: (pj − τδ+κ)/2, with associated deviation profits
[κ+ τ
(xi+2xj
3
)]22κ
.
This is not strictly higher than equilibrium profits iff
κ/τ ≥ Khi≡
xj
(2xi+xj
3
)− (1− 2xj) + (1− xj)
√(2xi+xj
3
)2
+ 1− 2xj
2(1− 2xj).
Summarizing, firm i has no profitable deviation iff one of three sets of conditions holds:
Kli ≤ κ/τ ≤ Mli , max{Mli , Khi} ≤ κ/τ ≤ Mhi
, or κ/τ ≥ Mhi. One can show that
Mli , Khi≤ Mhi
; thus the latter two conditions are equivalent to max{Mli , Khi} ≤ κ/τ . Also,
if 4xixj ≤ (3− 6xj − 5x2j ), then both Kli and Khi
are less than Mli , and the conditions are
equivalent to Kli ≤ κ/τ . If 4xixj ≥ (3− 6xj − 5x2j ), then both Kli and Khi
are greater than
Mli , and the conditions are equivalent to Khi≤ κ/τ . The condition for existence thus boils
down to
κ/τ ≥
⎧⎪⎪⎨⎪⎪⎩
xi+2xj
3− (1− xj) + 2
√xj
(xi+2xj
3
)if 4xixj ≤ 3− 6xj − 5x2
j
xj
(2xi+xj
3
)−(1−2xj)+(1−xj )
√(2xi+xj
3
)2+1−2xj
2(1−2xj)if 4xixj ≥ 3− 6xj − 5x2
j
(10)
for (i, j) = (a, b) and (i, j) = (b, a). The implied bound, C(xa, xb), is finite for xa, xb ∈[0, 1/2).26
Conversely, the above arguments make clear that condition 10 exhausts the parameter
space for which neither firm a or b has a strictly profitable deviation. Thus the existence of
this equilibrium implies the stated conditions.
We finally argue that there are no pure strategy equilibria beside the proposition’s.
Clearly there is no equilibrium with the firms pricing in regions Ii/Vj; firm j could lower
its price to earn strictly positive, instead of zero, profits. Writing out best responses, it is
clear that at least one firm is not best-responding when the firms are pricing in the interior
of regions IIi/IVj. One can also show that firm i would not price on the boundary between
regions IIi and IIIi, i.e. at pi = pIIi, since the slope of the profit function jumps strictly
up there (if pIIi > 0). It is also straightforward to show that there are no mutual local best
responses in the interior of regions IIIi/IIIj besides (p∗a, p∗b).
26Equation 10 implies two bounds, for (i, j) equaling (a, b) and (b, a), respectively. C(xa, xb) is the maxof these two bounds.
23
Proof of Lemma 3. We argue that under a condition stronger than 10, namely
κ/τ ≥ xi + 2xj
3(1− 2xj)(11)
for (i, j) = (a, b) and (i, j) = (b, a), the pure strategy pricing equilibrium of Lemma 2 exists
and is unique. The implied bound, D(xa, xb), is finite for xa, xb ∈ [0, 1/2).27
Existence is clear from the proof of Lemma 2 (since this bound equals max{Mhi,Mhj
}).We also have established that there are no other pure strategy equilibria. Consider a candi-
date mixed strategy equilibrium with pricing supports [pa, pa] and [p
b, pb].
We first show that pi ≤ p∗i , where again the p∗i ’s are defined in equation 4. Note that
pi > 0, since a firm can guarantee strictly positive profits regardless of its opponent’s pricing
strategy.
Define p∗i (pj) as the best response(s) of firm i to firm-j price pj . Define p∗IIIi(pj) as the
firm-i price pi that optimizes against firm-j price pj based on the region-IIIi expression
for demand: p∗IIIi(pj) = [pj + τ(1 + xi − xj) + κ]/2. Define p∗IIi(pj) and p∗IVi(pj) similarly:
p∗IIi(pj) = (pj − τδ + κ)/2 and p∗IVi(pj) = (pj + τδ + κ)/2. One can show that firm i’s
best response function follows perhaps p∗IVi(pj), then perhaps pIIIi(pj), then p∗IIIi(pj); then
it either jumps down to p∗IIi(pj) and follows it until joining pIi(pj) forever, or jumps down
to pIi(pj) and follows it forever. It is continuous and increasing everywhere except at the
downward jump.
Let Zi ≡ 3κ + τδ + 2τ(1 − xi), and assume pj ≤ Zj. Examination of the best response
functions and the profit functions makes clear that for any pj ∈ [0, Zj], firm i’s best response
is no greater than p∗IIIi(pj) and firm i’s profits are declining in pi beyond p∗IIIi(pj), strictly
so up to pIVi(pj). (Condition 11 guarantees that p∗IIIi(pj) < pIVi
(pj) for any pj ∈ [0, Zj].)
Hence, since p∗IIIi(pj) is increasing, p∗IIIi
(pj) dominates all prices pi > p∗IIIi(pj) for any firm-j
price pj ∈ [pj, pj], and strictly so for pj in a left-neighborhood of pj. Thus, pi ≤ p∗IIIi(pj).
Algebra verifies that if pa ≤ p∗IIIa(pb) and pb ≤ p∗IIIb(pa), then pi ≤ p∗i . Thus, pi ≤ Zi,
i = a, b, implies that pa ≤ p∗IIIa(pb) and pb ≤ p∗IIIb(pa), which implies that pi ≤ p∗i , i = a, b.
Assume next that pi > Zi, i = a, b. This implies that p∗i (pj) = pIi(pj) > p∗IIIi(pj),
(i, j) = (a, b), (b, a), which guarantees that p∗a(pb) < pa or p∗b(pa) < pb. But, we argue that
p∗i (pj) < pi is not possible in equilibrium. For pj ≤ Zj, the previous paragraph argues
that firm i’s best response is no greater than p∗IIIi(pj) and firm i’s profits are declining in
pi beyond p∗IIIi(pj). For pj > Zj, firm i’s best response is pIi(pj), and one can show that
firm-i profits are declining in pi beyond pIi(pj), strictly so up to pIVi(pj). Note that for
pj ∈ [0, pj ], p∗IIIi
(pj) and pIi(pj) are less than pIi(pj). Thus, pIi(pj) = p∗i (pj) dominates all
prices pi > p∗i (pj) for any firm-j price pj ∈ [pj, pj], and strictly so for pj in a left-neighborhood
27Clearly, D(xa, xb) = max{ xa+2xb
3(1−2xb), xb+2xa
3(1−2xa)}.
24
of pj . Hence, pi > p∗i (pj) and, thus, pi > Zi for i = a and b, cannot occur in equilibrium.
Consider finally pi ≤ Zi and pj > Zj. By above reasoning, pi ≤ Zi implies that pj ≤p∗IIIj(pi), and condition 11 and pi ≤ Zi ensure that p∗IIIj(pi) ≤ Zj ; together, these imply
pj ≤ Zj , a contradiction.
We have ruled out every case but pi ≤ Zi, i = a, b, which implies that pi ≤ p∗i , i = a, b.
Given condition 11 and that pi ≤ p∗i , i = a, b, one can show that for any pj ∈ [pj, pj ],
firm-i profits are quasi-concave in pi, strictly so up to pIVi(pj); and have a unique maximum
p∗i (pj), which is continuous and strictly increasing in pj. This allows us to establish that
pi≥ p∗i (pj) and pi ≤ p∗i (pj). This follows since p∗i (pj) dominates all pi > p∗i (pj) for all
pj ∈ [pj, pj], strictly so for pj in a left-neighborhood of pj ; and p∗i (pj) strictly dominates all
pi ∈ [0, p∗i (pj)) for all pj ∈ [pj, pj].
Note that p∗i (pj), for pj ∈ [0, p∗j ], is continuous and piecewise linear with slopes (where
they exist) equal to 1/2 or 1; this is because for pj ∈ [0, p∗j ], p∗i (pj) follows perhaps p
∗IVi
(pj),
then perhaps pIIIi(pj), then p∗IIIi(pj). Thus, for some λi, λj ∈ [1/2, 1], this fact gives rise to
the equalities in the following set of claims:
p∗i (pj)− p∗i (pj) = λi(pj − pj) ≤ λi[p
∗j(pi)− p∗j (pi)] = λiλj(pi − p
i) ≤ λiλj [p
∗i (pj)− p∗i (pj)].
The inequalities come from the previous paragraph’s result. This set of claims implies that
if pi< pi, then p
j< pj and λi = λj = 1. Further, p
a= p∗a(pb) and p
b= p∗b(pa) (also using
the previous paragraph’s result). Since pi< pi ≤ p∗i , this contradicts the fact that (p
∗a, p
∗b) is
the unique pure strategy equilibrium. Hence, it must be that pi= pi, i = a, b; that is, any
equilibrium is in pure strategies.
Proof of Proposition 2. We argue that the following condition guarantees existence
of a unique PSSLE, which involves minimum differentiation:
δ ∈ (1/3, 1/2] and κ/τ ≥ 1−δ2δ
, or
δ ∈ (0, 1/3] and κ/τ ≥ (3−δ)2
12δ.
When κ/τ ≥ (1− δ)/2δ and firm locations are symmetric and own-sided, condition 11 is
satisfied with strict inequality for any δ > δ (i.e. κ/τ > D(xa, xb)). Thus the Hotelling-like
pricing equilibrium exists and is unique in a neighborhood of every symmetric location pair
outside the central interval of width δ; so by the Hotelling logic of the proof of Proposition 1,
there can be no PSSLE with δ > δ.
There can also be no PSSLE where δ < δ, when δ ≤ 1/2, since a firm could deviate to
locate at a distance δ from its competitor and earn strictly positive, instead of zero, profits.28
28When δ > 1/2, it may not be possible for a firm to unilaterally achieve the minimum distance. Hence,
25
It remains to show that own-sided, symmetric firm locations at distance δ apart constitute
an equilibrium. When δ ∈ (1/3, 1/2] the condition κ/τ ≥ (1−δ)/2δ guarantees this. Moving
away from the other firm would lower profits, due to Hotelling logic, since the pure strategy
pricing equilibrium exists and is unique at equilibrium locations and if either firm deviates
away from the other; while deviation toward or beyond the other firm lowers profits to zero
by violating minimum differentiation, δ.
When δ ≤ 1/3, the condition κ/τ ≥ (1 − δ)/2δ and previous arguments apply to rule
out deviations away from or toward the other firm, but not “leapfrog” deviations beyond
the other firm that do not violate minimum differentiation. After a leapfrog deviation, both
firms are located on the same side of the line’s midpoint, a scenario for which we have no
proof of a unique pricing equilibrium. Our strategy here is to bound deviation payoffs in any
pricing equilibrium at these locations. The condition we find to guarantee that a leapfrog
deviation is not profitable,κ
τ≥ (3− δ)2
12δ, (12)
will also guarantee κ/τ ≥ (1− δ)/2δ.29
Consider the candidate minimum differentiation equilibrium, where xa = xb = (1− δ)/2.
Consider a deviation by firm i to some location xi ∈ [1 − xj + δ, 1], i.e. xi ∈ [(1 + 3δ)/2, 1].
Competition at these locations is isomorphic to competition when xi ∈ [0, (1 − 3δ)/2] and
xj = (1 + δ)/2, so we consider the latter.
First, one can show that if δ ≤ 1/3 and firms at these deviation locations are randomizing
over (possibly degenerate) pricing supports [pi, pi], i = a, b, then pi ≤ p∗i , i = a, b, where
the p∗i are calculated using the deviation locations. This uses the exact reasoning of the
part of the proof of Lemma 3 that establishes the upper bound of the pricing support in
any equilibrium; the only difference is that whenever condition 11 is used there, here we
substitute condition 12.
Next we let pj = p∗j . Since pj ≤ p∗j , and since firm i’s profits are increasing in firm j’s
price pj , for any price pi, this allows for an upper bound on firm i’s profits. One can show
that given condition 12 and the fact that xj = (1 + δ)/2 and pj = p∗j , firm i’s profits are
maximized the closer it is to firm j, i.e. at xi = (1−3δ)/2, and at a price pi in the interior of
region IIi (defined in equation 3). Firm i’s profits in this scenario equal (2κ+τ−τδ/3)2/8κ,
and are less than equilibrium profits (κ + τ)/2 under condition 12.
in this case there are PSSLE with δ < δ, but by the previous paragraph’s logic, there are no PSSLE withδ > δ as long as κ/τ ≥ (1− δ)/2δ.
29Less restrictive bounds on κ/τ can be derived by assuming that a) pure strategy pricing equilibria areselected if they exist and/or that b) firms must locate on their own half of the line. However, the qualitativeflavor of the bounds does not change: they increase in δ and approach infinity as δ approaches zero.
26
References
[1] Kurt R. Brekke, Robert Nuscheler, and Odd Rune Straume. Quality and location choicesunder price regulation. Journal of Economics and Management Strategy, 15(1):207–227,2006.
[2] Kurt R. Brekke, Robert Nuscheler, and Odd Rune Straume. Gatekeeping in health care.Journal of Health Economics, 26(1):149–170, January 2007.
[3] Partha Dasgupta and Eric Maskin. The existence of economic equilibria: Continuityand mixed strategies. Technical report no. 252, Institute for Mathematical Studies inthe Social Sciences, Stanford University, 1977.
[4] Claude d’Aspremont, J. Jaskold Gabszewicz, and Jacques-Francois Thisse. OnHotelling’s “Stability in Competition”. Econometrica, 47(5):1145–1150, 1979.
[5] A. de Palma, V. Ginsburgh, Y. Y. Papageorgiou, and J.-F. Thisse. The principle ofminimum differentiation holds under sufficient heterogeneity. Econometrica, 53(4):767–781, July 1985.
[6] Andre de Palma and Luc Leruth. Congestion and game in capacity: A duopoly analysisin the presence of network externalities. Annales d’Economie et de Statistique, (15-16):389–407, July-December 1989.
[7] Nicholas Economides. Minimal and maximal differentiation in Hotelling’s duopoly. Eco-nomics Letters, 21(1):67–71, 1986.
[8] Drew Fudenberg and Jean Tirole. Game Theory. MIT Press, Cambridge MA, 1991.
[9] I. L. Glicksberg. A further generalization of the Kakutani fixed point theorem, withapplication to Nash equilibrium points. Proceedings of the American MathematicalSociety, 3(1):170–174, February 1952.
[10] Isabel Grilo, Oz Shy, and Jacques-Francois Thisse. Price competition when consumer be-havior is characterized by conformity or vanity. Journal of Public Economics, 80(3):385–408, June 2001.
[11] Harold Hotelling. Stability in competition. Economic Journal, 39:41–57, 1929.
[12] Andreas Irmen and Jacques-Francois Thisse. Competition in multi-characteristic spaces:Hotelling was almost right. Journal of Economic Theory, 78(1):76–102, January 1998.
[13] Elon Kohlberg. Equilibrium store locations when consumers minimize travel time pluswaiting time. Economics Letters, 11(3):211–216, 1983.
[14] Martin J. Osborne and Carolyn Pitchik. Equilibrium in Hotelling’s model of spatialcompetition. Econometrica, 55(4):911–922, July 1987.
[15] Jean Tirole. The Theory of Industrial Organization. MIT Press, Cambridge MA, 1988.
27
[16] Jonathan Vogel. Spatial competition with heterogeneous firms. Journal of PoliticalEconomy, 116(3):423–466, June 2008.
[17] Xavier Wauthy. Capacity constraints may restore the existence of an equilibrium in theHotelling model. Journal of Economics, 64(3):315–324, 1996.
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