Localized competition with heterogeneous …rms
and vertical relations
Marco Alderighia
Claudio A. Pigab
Università della Valle d’Aosta, Italy.
Università Bocconi, Milano, Italy.
Loughborough University, Loughborough, UK.
RCEA, Rimini, Italy.
March 31, 2010
Abstract
The paper analyzes the circular model of localized competition by allowing …rms to
have heterogeneous costs. We provide a closed-form solution for the equilibrium prices
evoking Chamberlin’s “domino e¤ect”, and we discuss some economic implications
and the empirical relevance. We also show that in a simple set-up accounting for
vertically related industries when down-stream markets are chain-linked, market power
of manufacturers is strongly a¤ected by the arrangement of retailers. We …nally present
some competition policy implications.
JEL classi…cation: L11, D61.
Keywords: Localized competition; chain-linked markets; domino e¤ect; cost heterogeneity; vertical relations; mergers.
“Everything is related to everything, but near things are more related to each other”
(known as the ‘…rst law of geography’), Waldo Tobler (1970), p. 236
1
Introduction
Geographic space plays a role in shaping markets and generates preferential interactions
among those agents that are more closely located to each other.1 The recognition of
geographical space as a key determinant of market organization is at the basis of models
of localized competition.
Among the large literature accounting for spatial e¤ects on …rm competition, we distinguish two di¤erent streams of research. A …rst stream is based on the random utility
a
E-mail: [email protected].
Corresponding Author. E-mail: [email protected]
1
This aspect is prominent in retail markets where consumers usually visit shops in the neighbourhood
of their home or working place.
b
1
approach (Anderson, et al 1989). In short, consumers buy their preferred goods comparing
the characteristics of the product (including location). Since far-distance products yield
lower utility, purchase is likely to occur in a localized fashion. This approach has shown
to be useful in structural modelling estimation introduced by Berry (1994), Feenstra and
Levinsohn (1995), and Berry et al. (1995). Some relevant contributions in this ‡ow have
shown that competition in retail markets is localized (Capps et al., 2003; Gaynor and
Vogt, 2003; Thomadsen, 2005; Davis, 2006; Manuszac, 2010).
A second stream of research is the address approach grounding on the seminal works
of Hotelling (1929) and Salop (1979), frequently referred to as the linear city model and
the circular city model, respectively. This approach has been often utilized to derive some
general conclusions on the features of localized competition (pricing, location, welfare
implications) in comparison to standard models of non-localized competition (the Bertrand
and Cournot oligopoly models and the Dixit-Stiglitz model of monopolistic competition).
More recently the Salop model has been employed in mergers analysis (Levy and Reitzes,
1992; Giraud-Héraud et al., 2003), in the study of the e¤ects of cost heterogeneity on
location (Vogel, 2008, 2009) and on the e¢ ciency of dense markets (Syverson, 2004).
This paper builds on this stream of research by making two di¤erent contributions.
The …rst contribution concerns the properties of a circular model when …rms di¤er in
their degree of e¢ ciency. We derive a closed-form solution of equilibrium prices, that turn
out to be a weighted average of all …rms’ costs, plus a spatial di¤erentiation mark-up.
We show that the e¤ects of a change in a …rm’s cost propagate across a series of chainlinked sub-markets evoking the so called “domino e¤ect”(Chamberlin, 1933, pp. 103-104;
Rothchild, 1982). We suggest to rename it the “damped domino e¤ect” to emphasize the
fact that the intensity of the e¤ect decreases rapidly and vanishes quite quickly moving
far away from the location where the change originated. In fact, we …nd that competition
remains highly localized so that only the prices of closely located …rms are linked in an
economically signi…cant way. A direct consequence of the damped domino e¤ect is that
equilibrium prices are a¤ected both by the costs of …rms and by …rms relative to position
to the others. This property thus di¤ers from those in other recent contributions which
introduce heterogeneity in models of localized competition (Syverson, 2004; Vogel, 2008).
From an empirical view-point, evidence for the damped domino e¤ect was recently
found in a some markets (Kalnins, 2003; Mobley, 2003; Thomasen, 2005; Atkinson et al.,
2008; Mobley et al., 2009) but not in others (Pinkse et al., 2002).2
2
A similar propagation e¤ect is documented in other economic …elds. In …nancial markets, international
stock market linkages have been extensively investigated: crisis evolves as a contamination process, starting
from neighbour countries, which usually have more …nancial and economic links, to more distant markets
(Yang et al. 2006). In housing economics, the price propagation is known as “ripple e¤ect”. Empirical
2
Damped propagation requires that two di¤erent conditions are satis…ed. First, competition must be localized, i.e. …rms should care more the behaviour of neighbour with
respect to non-neighbour rivals. Second, markets must be chain-linked, i.e. competing
…rms should not be clustered, but each …rm should compete with a di¤erent subgroup of
rivals. Pinkse et al. (2002) analyze competition among petroleum products terminals by
using a number of closeness measures. Their estimates reveal that direct rivalry decays
abruptly with distance, suggesting that competition is localized. However, they do not
…nd a propagation e¤ect since neighbour competitors are clustered and therefore the price
changes are not transmitted from one market to another.
On the contrary, a large number of studies provides evidence of that a propagation
e¤ect appears in …nal markets. Atkinson et al. (2008) show that there is a spatial price
propagation in the retail gasoline market in Guelph, Ontario. In particular, an initial price
cut generates a response among the closest neighbours in the overnight, the mid-distance
competitors in the next day and the others in the next two or three days. Mobley (2003)
and Mobley et al. (2009) analyze the price transmission among California hospital markets.
Using an econometric spatial lag model of hospital pricing they …nd statistically signi…cant evidence of a spatial lag process in prices using a proximity matrix accounting for
the seventh-closest neighbours. Using a similar technique, Kalnins (2003), analyzing the
pricing behaviour of the four largest fast-food hamburger chains across Texas, concludes
that there is spatial price transmission among neighbour outlets (of separate franchisees)
within the same chain, but not of di¤erent chains. Therefore, price propagation occurs
only when outlets o¤er very close substitutes.3
Thomasen (2005) also analyses the hamburger market in Santa Clara County, California, focusing on the pricing behaviour of the two largest chains. Using the random utility
approach, he estimates that travel costs are about 3 dollars per miles, implying that competition is localized. He also shows that cross-price elasticities are larger for outlets located
close together than for those located further apart. Moreover, he identi…es an asymmetric
result depending on the chain a …rm responsible of the change belongs. In particular,
changes in prices at McDonald’s have much larger e¤ects at either McDonald’s or Burger
King than do changes in prices at Burger King. This result therefore modi…es the …ndings
in Kalnins (2003) where inter-chain competition has no e¤ects. Additional data reported
work documented that change in housing prices in London ripples down around the entire UK (Jones and
Leishman, 2006). In local public economics, estimation results show that a given state’s spending responds
positively to higher spending in neighbour states (Case et al, 1993; Brueckner, 1998).
3
This fact also emerges in Mazzeo (2002) where there is price interdependence among proximate hotels
only when they have the same category but only not between di¤erent categories. Similarly, Manuszak
(2010) …nds that price changes involve substitution to nearby product of the same grade and service level
in the gasoline retail market.
3
in Thomasen (2005) in Table 6 can be used to show that markets are chain-linked and
therefore that there is domino e¤ect.
The second contribution of our paper consists in a simple application of the model
previously presented in order to shed some light on the implications of localized competition on vertically related industries. In particular, we show that when down-stream
markets are chain-linked, market power of manufacturers is strongly a¤ected by the locational arrangement of their retailers; alternatively put, we develop a framework where
as the locational pattern of downstream retailers moves from highly spatially disperses
(neighbouring …rms are associated with di¤erent upstream manufacturers) to highly spatially concentrated arrangements (a cluster of retailers obtain their input from the same
manufacturer), market power enjoyed by manufacturers increases accordingly.
A novel thrust of our paper is to show that when the locational pattern of retailers
is introduced in the analysis, the Her…ndahl-Hirschman Index of concentration provides
poor results in identifying market power; therefore, we derive a measure of geographic
concentration which accounts for up-stream market power in localized contexts, which
depends on the locational pattern of downstream …rms. This measure based on the spatial
a¢ liation of retailers has the required properties to be increasing in the number of retailers
and wholesalers, and decreasing in geographical concentration of retailers patronizing the
same manufacturer.
Moreover, by allowing endogenous a¢ liation of retailers, we …nd that a low-cost manufacturer retains a larger market share and that multiple equilibria involving di¤erent levels
of manufacturer margins can occur. The model predicts more up-stream size heterogeneity,
all other things equal, when the down-stream industry is less spatially concentrated.
Finally, we study the impact of an up-stream merger on the equilibrium prices. We
show that the retail distribution plays an important role. Moreover, contrary to the
…ndings of Levy and Reitzes (1992) and Giraud-Héraud et al. (2003), when the up-stream
market is taken into consideration, we show that a merger involving manufacturers which
have no adjacent retailers, also increases their market power. Some empirical literature
reports similar conclusions on merger outcomes and con…rms the relevance of our set-up in
describing vertical related markets (Sudhir, 2001; Villas-Boas and Zhao, 2005; Thomadsen,
2005; Manuszak, 2010).
The paper is organized as follows. Section 2 derives the equilibrium prices, illustrates
the ensuing price propagation and discusses the existence conditions. Section 3 provides
a simple application of the circular model in order to shed some light on the implications
of localized competition on vertically related industries. Section 4 concludes.
4
2
A model of localized competition with a domino e¤ect
In the Section we study the price equilibrium in a market where …rms compete in a localized
fashion. Examples of localized competitions occur especially in the retail market where
consumers sustain transport costs in acquiring goods and therefore there they can visit
a limited number of sellers. Super-markets, petrol stations, theatres, restaurants, co¤ees,
car dealers, hairdressers and barbers are some examples of there markets.
Localized competition is only one of the basic ingredients guaranteeing a price transmission mechanism that resembles the domino e¤ect. In facts, in some cases, activities
are organized in such a way that local markets are self-contained, so that competition is
clustered and does not allow a transmission among di¤erent areas. Empirical evidence
have shown that some markets are chain-linked, although additional evidence is required
to speci…c cases. The set-up we are going to present, follows the approach developed by address models considering one-dimensionally spatial markets. A part from few exceptions,
markets are usually two-dimensional so that, results here derived, should be evaluated
carefully.4
2.1
Baseline model
Consider a circular city of unitary length with uniform density D. There are N
2
equidistant …rms, whose location is denoted by the sequence of integers:
n2L=
l (N ) =
1
N
2
;
1
N
2
+ 1; ; 0; 1; ; l (N ) =
N
1
2
;
where dxe is the approximation of x to its larger integer. As a convention and without
loss of generality, the chosen notation for …rms’ location implies that after attributing
the value l randomly, all the other indexes are assigned in descending order by moving
clockwise (see Figure 1 for an example with N = 12).
Unit variable costs are cn
0 with cn 2 [cL ; cH ]. Thus, we allow …rms to have
di¤erent costs; this may be due to di¤ering managerial abilities across …rms (Bartelsman
and Doms, 2000), to location-speci…c factors (Aiura and Sato, 2008) or to di¤erent costs
in producing varieties (Waterson, 1990). Throughout the paper we assume that …rms
compete by setting their price when costs and locations are common knowledge.
Denote the number of locations separating any two …rms as i 2 L. For …rm n, i < 0
identi…es the …rm which is its i-step clockwise neighbour, while i > 0 refers to …rm n’s
4
Manuszak (2010) analyzes the gasoline market in two islands in Hawaii. Since the vulcanic nature of
the islands, most of the routes are on the coasts and therefore, location of petrol stations and the nature
of geographical location more closely related a one-dimentional geographic dimension.
5
i-step counter-clockwise neighbour. Let hi identify an operator such that hn + ii 2 L and:
8
>
>
n+i
>
<
hn + ii = n + i N
>
>
>
:n + i + N
if l
n+i
l
if n + i > l
if n + i < l
Consumers incur linear unit transport costs t, and have unitary inelastic demand.
Consider a consumer who is located at distance d 2 [0; 1=N ] from …rm n. The utility she
obtains from buying the product from …rm n is Ud;n = v
td
pn , where v is a positive
constant, and pn is the uniform price charged by …rm n. We assume that the reservation
value v is su¢ ciently high so that consumers buy the good in equilibrium.
2.2
Equilibrium prices
Condition 1 (No mill-price undercutting) pn
^ for n 2 L, where k^ = t=N .
phn+1i < k,
Following Eaton and Lipsey (1978), we keep the requirement that competition is localized: the market share and price setting of a …rm located in n is directly a¤ected only
by the behaviour of the two adjacent …rms. Standard computations yield …rm n’s demand
(Tirole, 1988, p. 283):
qn = D
phn
1
+
N
…rm n’s pro…t function is thus:
n
1i
+ phn+1i
2t
= (pn
2pn
; n 2 L;
(1)
cn ) qn . Concavity of the pro…t function in
pn implies that the optimal prices for the N …rms are given by the …rst order conditions:
pn =
2cn + phn
1i
4
+ phn+1i
1^
+ k,
8n 2 L.
2
(2)
Lemma 1 If Condition 1 is satis…ed, then system (2) has a unique solution:
pn =
Xl
i=l
wl
jij chn+ii
+ k, n 2 L.
(3)
Although competition is localized, Lemma 1 highlights the inter-connection of all …rms’
pricing strategies, an aspect that does not feature in a symmetric costs’set-up. Consider
the …rm in n = 0 (Figure 1). From (2), the price charged by …rm 0 depends on its costs
and on the prices charged by its adjacent …rms. The latter’s prices depend on their own
costs and by the price charged by, respectively, …rm 0 and their respective other 1-step
neighbour. Similarly, the price charged by the 2-step neighbours of …rm 0 depends on their
6
own costs and on the price charged by their 1-step neighbours, and so on and so forth. In
other words, despite the localized competition assumption, the total market is the result
of chain-linked sub-markets (Chamberlin, 1933, p. 103-104; Rothchild, 1982), so that each
…rm’s cost in‡uences the pricing of the non-direct competitors through a domino e¤ect.
This property thus di¤ers from those in other recent contributions which introduce
heterogeneity in models of localized competition. In Syverson (2004), the …rms’ price
decisions are taken within a Bayesian set-up (i.e. the rivals’costs are unknown), which is
therefore not conducive to a domino e¤ect. By assuming complete information on costs in
a location-and-pricing game, Vogel (2008) obtains that each …rm’s equilibrium price only
depends on each …rm’s own cost and on the average cost of all the …rms in the market,
i.e., also in this case the price equilibrium does not exhibit a domino e¤ect. The reason is
that …rms’heterogeneity is re‡ected in equilibrium by their location decisions: lower-cost
…rms are more isolated because higher-cost …rms try to locate themselves further away
from their lower-cost competitors to shield themselves from competition.
The degree of interconnection between …rms’costs and prices is captured by the set of
weights wi , while k is an average mark-up re‡ecting the degree of substitutability across
products (as in standard homogeneous cost models).5
Interestingly, the weights are not in‡uenced by the cost di¤erential between …rm n and
its i-step neighbours. That is, each …rm in the market assigns the same weight to the cost
of a …rm regardless of the level of that cost. Furthermore, we also expect that weights
decrease with respect to distance, that is, w0 < w1 < :: < wl .
Proposition 1 (Characterization) Assume the existence of a Nash price equilibrium,
which satis…es Condition 1; then for any N
2, the solution is unique and it is given by
(3) of Lemma 1, where the weights wi and k are:
wi = 4wi
1
w1 = 3
l
4wl = 2 + 2wl
Xl
i=l
wl
jij
wi
2
for i = 1; ::l
1
(4a)
l w0
(4b)
1
(4c)
= 1;
(4d)
k = k^ = t=N:
(5)
(Sketch of the proof.) While a formal proof is shown in Appendix A, here we provide
the intuition behind the derivation of (4) and (5) assuming N is even. From Lemma 1,
5
It is worth noting that the adopted conventional notation assigns the weight with the highest subscript
to a …rm’s own cost, and that, for even N , w0 corresponds to the weight for the cost of the …rm which is
opposite (i.e., furthest) to …rm 0 (see Figure 1); for an odd N , w0 is the weight for both …rms l and l.
7
…rm 0’s equilibrium strategy is of the form:
p0 = wl c0 + wl
1 (c 1
+ c+1 ) + wl
2 (c 2
+ c+2 ) + ::: + w0 cl + k.
(6)
By the same token, using the shift in weights’notation described in Figure 1, the price
charged by the adjacent …rms located at
p
1
= wl c
1
+ wl
p+1 = wl c+1 + wl
1 (c 2
1 (c0
1 and +1, respectively, are:
+ c0 ) + wl
2 (c 3
+ c+1 ) + ::: + w0 cl
+ c+2 ) + wl
2 (c 1
+ c+3 ) + ::: + w0 cl + k.
1
+k
(7)
(8)
Now substituting (6), (7) and (8) in (2), with n = 0, and collecting the similar terms, we
obtain:
c0 4wl
2
2wl
1
+ (c
:: + cl + cl
1
1
+ c1 ) 4wl
(4w1
w0
1
wl
2
wl + (c
w2 ) + cl (4w0
2
+ c2 ) 4wl
2w1 ) + 2k
1
wl
2k^ = 0.
wl
2
+ ::
(9)
In order for this equality to hold for every possible con…guration of cn , n 2 L, all the
expressions in round brackets must be equal to zero, thereby proving (4a)-(4c) and (5).
Finally, (4d) emerges by replacing cn = c for n 2 L in (9).6
Given (4d), when …rms have identical costs cn = c, 8n 2 L, (3) reduces to the standard
^ 8n.
solution in Salop (1979): pn = c + k,
Proposition 1 highlights the relationship among the weights wi , i = 0::l, in Lemma
1; however …nding their numerical values using (4) can be computationally burdensome,
because it would be necessary to solve a system of d(N + 1) =2e equations. We now provide
a simpler way to compute them.
Corollary 1 Let:
a (0) = 1, a (1) = 3
a (i) = 4a (i 1)
Xl
AN =
a l
i=l
Then
l
a (i
l, and
(10a)
2) , for i = 2; ::l,
(10b)
jij .
wi = a (i) =AN :
(10c)
(11)
The recursive nature of (10) facilitates the computation of the weights given the initial
conditions (10a). Corollary 1 is used in Table 1 to obtain the numerical values of the
optimal weights for N = 2; 3; ::; 15; 20; 1.
6
Strictly speaking, only two conditions among (4b), (4c) and (4d) are necessary to derive all the weights.
8
[Insert Table 1 approximately here]
From Table 1 one can observe that for N > 7, the weights associated to a …rm’s own
cost and to those of the three closest neighbours are, respectively, about 57:7%, 15:5%,
4:2% and 1:4%, and that such weights remain almost constant as N increases. For N
8,
the costs of the …rms that are farther than 3 steps have a highly negligible impact on the
Pmin(3;l)
price set by some other …rm in the market (R = 1
i=max( 3;l) wl i is less than 1%).
Before concluding we want to draw the following remarks. First, it is possible to
departure from assumption of common knowledge to derive the equilibrium prices. In
fact, …rms’ pricing equilibrium can emerge as a repeated interaction of …rms with their
1-step neighbours just based on their posted prices and therefore without requiring any
additional information on opponents’ costs or on far-distance …rms prices. Intuitively,
assume that …rms have no information on opponent costs and behaves myopically following
this pricing strategy. Initially, each …rm n sets p0 (0) = cn + k^ (as in the Salop framework),
n
and in the next periods, for
1
pn = 2cn + phn 11i + phn+1i
= 1; 2; ::, it adjusts its price in accordance to (2), i.e.
^
=4 + k=2.
Then, since the the algorithm described to obtain
the series fpn g is a contraction mapping, and thanks to the uniqueness of the equilibrium
(see: Propositions 2 and 3, in the next sub-section), the series converge to the equilibrium
prices pn given in (3) and Proposition 1. This simple convergence process is consistent
with the interpretation of geographical propagation of shocks as reported in Atkinson et
al. (2008).
Second, this set-up is closely related to econometric spatial lag models (Anselin, 1988).
Consider this simple equation: p = 1N + c + Wp + ", where p is the price vector, W is
the proximity matrix, where elements are one when two …rms share a common boundary
and zero otherwise,
is the spatial autoregressive coe¢ cient, which captures the inten-
sity of spatial interdependence,
,
, c and " are, respectively, the intercept, the other
parameters, the set of cost variables and the random vector.7 Now, we can express the
system of equations (2) in matrix form and we obtain: AP = 21 c + k2 1N , where symbols
are de…ned in Appendix A. This matricial form hilights that the problem encountered in
the circular model is a special case of the spatial lag model, obtained by setting
=1N =2, " = 0, (1N
W) = A, and
= k=2,
= 1=4. Interesting, Mobley et al. (2009) as-
suming that proximity matrix based on the 7-nearest neighbours …nd that
= 0:231 quite
close to our …ndings. Kalnins (2003) estimates two di¤erent interaction terms using a rowstandardized version of the proximity matrix.8 He …nds that only the row-standardized
7
For instance, Pinkse and Slade (1998), Mobley (2003) and Mobley et al. (2009) assume that weight
matrix is based on the k-nearest neighbours.
8
That is, each element of a row of the proximity matrix is by the value necessary for the sum of all
9
values of same-chain neighbour prices ( c ) are signi…cant and ranges between 0:29 and
0:44. Although the econometric results are not directly comparable to ours since we analyze the spatial price behaviour in an one-dimensional set-up, the magnitude of the
term
are comparable to the empirical ones, providing some support that …rm behaviour does
not substantially di¤er from the Smithian-Hotelling pricing behaviour in contrast to the
competing Löschian predictions, where price changes are matched exactly (Walden, 1990).
2.3
Price equilibrium existence and uniqueness
We now show the existence of the equilibrium in Proposition 1 under two related …rms’
behavioural assumptions; in the …rst …rms commit themselves to Condition 1, in the second
this same condition holds without any commitment on the …rm’s part.9 In both cases we
need to impose some restrictions on the cost di¤erential between two neighbouring …rms
either because otherwise the no-mill price condition would fail to hold, or because the
e¢ ciency gap may be such that a highly e¢ cient …rm can pro…tably undercut a 1-step
ine¢ cient neighbour and drive it from the market.
Proposition 2 (Existence and Uniqueness under …rms’ commitment to Condition 1)
Equations (3)-(5) represent a unique Nash price equilibrium in pure strategies, if the maximum cost di¤ erential is not too large relative to the degree of product di¤ erentiation:
cH
where k^ = t=N and
c (N )
= wl
^
cL <
w0
1
c (N ) k,
(12)
.
Although largely used in the literature, assuming a commitment to the no-mill price
undercutting condition may be restrictive. Without commitment, the e¢ ciency gap may
be such that a highly e¢ cient …rm can pro…tably undercut a 1-step ine¢ cient neighbour
and drive it from the market unless we impose a restriction stronger than (12).
Proposition 3 (Existence and Uniqueness) Equations (3)-(5) represent a unique Nash
price equilibrium in pure strategies, if the maximum cost di¤ erential is not too large relative
to the degree of product di¤ erentiation:
cH
^
cL < (N ) k,
(13)
elements in that row to equal one. In our model the spatial autoregressive coe¢ cient associated to the
row-standardized version of the proximity matrix is 1=2.
9
The study of permissible cost asymmetry is only partially investigated in previous contributions. Syverson (2004) derives a condition satisfying the “no-mill price undercutting” rule, while in Vogel (2008), due
to the complexity of the out of equilibrium strategy, the degree of maximum heterogeneity is not fully
characterized.
10
where,
(1
wl
(N ) = 3 for N = 2;
2
1)
(1
2
wl )
(1
wl
(N ) =
1 ) (wl
wl
1
A
2)
B
p
B2
and B = (1
A
wl
for N
3, with A =
1 ).
First, when N = 2 …rms have no incentive to undercut the rivals, because (2) =
c (2) =
3. That is, the equilibrium in two-…rm Hotelling type game is always undercut-
proof (Hamilton et al., 1991).
Second, the values of
N
3,
c
c
and
as a function of N are reported in Table 1. Clearly, for
is about three times greater than
because an under-cut proof equilibrium is
incompatible with excessive cost heterogeneity. Many authors have noted that, in horizontal product di¤erentiation models, the non-existence of an equilibrium in pure strategies
is due to the small degree of product di¤erentiation (d’Aspremont et. al., 1979).10 Similarly, in our case, as N increases conditions (12) and (13) become more stringent because
k^ = t=N , c and decrease in N .
Finally, both Propositions provide a su¢ cient condition for the existence and uniqueness of the Nash equilibrium in pure strategies which hold under a general costs’structure.
Di¤erent assumptions on the latter would indeed lead to a less stringent speci…cation of
conditions (12) and (13). For instance, if the costs of neighbouring …rms were pair-wise
bounded (e.g., because costs depend on the distance from an input source), then intuitively
a Nash equilibrium would still exist even if the constraint on cH
3
cL were relaxed.
Vertical relations and localized down-stream competition
In this Section we provide a simple application in order to shed some light on the implications of localized competition in vertically related industries. We study a case where retail
prices are under control of down-stream …rms, while up-stream …rms provide the main
input. (The price setting independence of down-stream …rms guarantees that prices propagate among sub-markets). We show that when down-stream markets are chain-linked,
market power of manufacturers is strongly a¤ected by the arrangement of retailers. We
derive a measure of geographic concentration which accounts for up-stream market power
in localized contexts and we use it to discuss the e¤ects of a up-stream merger.
Examples of vertically related markets with down-stream localized competition are
abundant. A …rst group includes those markets characterized by dealer franchise agreements (e.g. petroleum stations, fast foods, restaurants, hotels, and wall papers).11 A
10
An equilibrium in mixed strategies has been proved to exist for the Hotelling linear city case (Dasgupta
and Maskin, 1986).
11
For example, Thomadsen (2005) reports that in the fast food industry during 2003 in North America
92% of Burger Kings and 65% of McDonald’s are franchised and that price dispersion among outlets of
the same chain is conspicuous.
11
second group involves manufacturer-retailer relations especially in the provision of durable
goods (e.g. the automobiles, elevators and copy machines). In both cases, the upstream
market is usually composed of well-established brands operating nationally or internationally, while the down-stream …rms operate at a smaller scale, owning one or few outlets
within a geographic area.12 Moreover, down-stream …rms are usually free to charge independent prices since price …xing agreements are often forbidden by the antitrust law.
3.1
The vertical model
We study two vertically related industries with M up-stream …rms (manufacturers or
wholesalers)13 , A1 ; ::AM , and N down-stream …rms (retailers), D1 ; ::DN . We model vertical relations in a standard way, apart from assuming that competition in the down-stream
market is localized. The competitive environment in the retail market was previously described in Section 2.2. Each retailer n is exclusively supplied by a manufacturer with qn
units of good at the unit cost cn , and she sells the good to the …nal market at a price pn .
The manufacturer m produces the good at the constant unit cost rm , and sell it at the
wholesale price of Cm . Manufactures and retailers are vertically separated, and collusive
behaviours in the up-stream as well as in the down-stream industries are excluded. Let
Nm be the number of retailers supplied by the manufacturer m and let nm = Nm =N be the
share over the total. With a little abuse of notation we call Am the set of retailers which
patronize the wholesaler Am . Market density is set equal to 1 to simplify the notation,
i.e. D = 1.
We analyze the following two-stage game. First, manufacturers simultaneously choose
the wholesale prices C1 ; ::; CM and, afterwards, retailers (which are exogenously assigned to
wholesalers) compete in prices in accordance to the pricing game previously illustrated.14
Since there is perfect information, the relevant equilibrium concept is the subgame
12
The peculiarity of some durable goods is that the reseller is not only responsable for the initial sale
of the product but also for the after-sale services (i.e. repair, maintenance and upgrades). In most of
cases, the retailer, who sells the product, also provides after-sale services, either because of contractual
agreements (e.g. a leasing contract) or of preferential relationship with the buyer (Cohen and Whaing,
1997). Therefore, in these industries, down-stream competition emerges to be localized more due to aftersale services needs than from the purchase reasons.
13
Throughout the paper, these terms are used interchangebly.
14
We consider a very simple contractual structure. In general, vertical agreements involve more complex
contracting schemes. However, in many vertical agreements, franchisors require the payment of a royalty
(fR ) on the revenue generated by the retailer and intermediate inputs are provided to the retailer at the
wholesale unit cost (r). As noted by Thomadsen (2005) this pricing scheme is equivalent to applying a
mark-up on the wholesale cost. In fact, by dividing the retailer pro…t by 1 fR , the implicit wholesale
unit costs becomes Cm = r= (1 fR ). Moreover, Lafontaine (1992) found that franchisors generally o¤er
the same contract terms to their franchisees at a given point in time, so that non-discriminatory price
behaviour of upstream …rms is empirically relevant.
12
perfect Nash equilibrium (SPNE). We therefore solve the model by backward induction,
starting, as usual, from the second stage.
After replacing cn with the corresponding wholesale prices Cm in (3), we obtain the
equilibrium prices in the …nal market:
pn =
where Wnm =
Pl
i=l
wl
m
jij Ihn+ii
XM
m=1
Cm Wnm + k; n 2 L,
(14)
and Inm = 1 if n 2 Am , and 0 otherwise. From (1) it follows
that the quantity demanded by the retailer n is:
qn =
where Ynm = 2Wnm
m
Whn
the up-stream …rm m is:
where
u
m
The term
=
P
u
m
n2Am
(15)
m
Whn+1i
, and the total quantity demanded by retailers to
1i
Qm =
1 XM
Cm Ynm ,
m=1
2t
1
N
X
n2Am
q n = nm
1 XM
Cu
u=1
2t
u
m,
Ynu , and the pro…t of wholesaler Am is simply:
(16)
m
= Qm (Cm
rm ).
is a mixture of the weights coming from the down-stream price competition.
These weights are responsible for the change in the wholesale market shares caused by the
pricing behaviour of down-stream a¢ liates in response to a variation in their unit costs
(i.e. the wholesale prices).
Note that, by construction,
m
m
> 0,
u
m
=
m,
u
m
u
< 0 when m 6= u, and
PM
m=1
u
m
=
0 (See: Appendix B for a proof). That is, market shares of wholesaler m decreases in its
own price (Cm ) and increase in the price of the opponents (Cu ), and an identical shift
of prices all wholesalers has no impact on shares. These properties will be extensively
employed in the next sub-sections, where we provide additional intuition on the role played
by
u.
m
A change in wholesale prices induces a modi…cation in the down-stream equilibrium
prices, which varies in response to the geographic arrangement of retailers, and therefore
produces di¤erent e¤ects on the wholesale market shares. If down-stream a¢ liates cover
contiguous locations, an increase in the wholesale prices generates a small reduction in
the wholesale market share, while if the down-stream a¢ liates are spatially dispersed, a
similar strategy induces a larger negative e¤ect.15 In other terms, the geographic location
15
In fact, only down-stream demand coming from exposed retailers (i.e those competing with a¢ liates
of rival wholesalers) is negatively a¤ected, while down-stream demand coming from the other (potential)
a¢ liates, as a whole, is shielded from competition and therefore does not vary.
13
of retailers a¤ects the up-stream elasticity of demand. To conclude,
u
m
provides an
indication on the strength of the rivalry between wholesalers u and m. The larger in
absolute value and the stronger is the competition between wholesalers. Moreover, since
m
m
u,
m
corresponds to the negative sum of the other
it summarizes the exposure of …rms
to competition.
Previous relations can be succinctly reported in matrix form:
1
2t
Q=n
and
C
= Q (C
r) ,
(17)
where the symbol ‘’stands for the one-by-one product between elements of two vectors,
Q = [Qm ],
=[
m ],
n = [nm ], C = [Cm ], r = [rm ],
=[
First order conditions imply that the optimal wholesale prices
C = 2t •
where • =
+
1
n+ •
1
(
u ], with
m
are:16
u; m = 1; ::; M .
(18)
I) r,
I, and I is the identity matrix. In the next sub-sections we investigate
speci…c cases to better characterize and discuss the properties of the equilibrium.
3.2
Duopolistic up-stream market
We start considering the case with only two manufactures, A = A1 and B = A2 . De…ne
P
m
A + B as a measure of overall geographic interdependencies, which
= M
m=1 m =
A
B
capture the general exposure of the whole up-stream …rms to competition. From previous
A
A
considerations, we know that
=
B
B
A
B
=
=
B
A
> 0; and from (18), the optimal
wholesale prices are:
Cm =
1
4
(2rm + ru ) + (1 + nm ) t
3
3
1
,
(19)
with m; u = A; B and u 6= m. The equilibrium outcome inherits the characteristics both
of the up-stream and of down-stream markets. Indeed, geographic and non-geographic
components are simultaneously at work. Equation (19) shows that wholesale equilibrium
prices are positively a¤ected by: wholesale costs (rm ), wholesale share of retailers (nm ),
the intensity of product di¤erentiation (t) and a measure of geographic arrangement of
retailers (
1 ).
Using (17) and (19), we obtain the equilibrium quantities and pro…ts:
Qm =
m
16
=
First order conditions are: n
1
1
(1 + nm ) +
(ru
3
12
4
1
(1 + nm ) + (ru
9
4
1
2t
• c+ 1 (
2t
rm ) t
1
,
(20)
2
rm ) t
1
t
1
.
IM ) r = 0 therefore: • c = 2tn+ (
14
(21)
IM ) r = 0.
Note that if wholesale costs are identical, i.e. rA = rB = r, then (19) and (21) imply
that wholesale margins and pro…ts are proportional to t
1.
This suggests that
may
play a role in the analysis of vertically related markets and deserves more investigation. In
the following analysis, we will show that
1
(that capture the geographic arrangement of
retailers) is indeed a measure of spatial market concentration, e.g. it have some remarkable
properties such as the fact that it is decreasing in the number of retailers and wholesalers,
and decreasing with respect to those geographic arrangements which can be commonly
classi…ed as more concentrated. Therefore, market power of up-stream …rms may be
directly related to the geographic concentration in the down-stream market. The reason
is that because down-stream concentration reduces the demand elasticity of up-stream
…rms, and therefore the up-stream …rms can excise more market power.
1
In the following analysis, we refers to
1
=
1N
as the spatial concentration index, and to
as the average spatial concetration index. Similarly we call
(and
) the
(average) spatial dispersion index.
We now analyze how the location of retailers a¤ects this measure of geographical
arrangement. We impose some symmetry to descriptional reasons and we represent con…gurations which can be described by the triple (a; b; h), where a and b are, respectively,
the number of consecutive retailers (CR) patronizing …rm A and B, s = a+b is the pattern
size, h is number of the repetitions, and N = hs.17 Interesting, we …nd that the average
spatial concentration index
1
is independent from h. This has important implications
since for any geographic pattern, wholesale margins and pro…ts which we …nd proportional to t
1
are also proportional to k
1
=t
1.
Thus, wholesale market power can
be expressed as a function of the average spatial concentration of retailers (
1)
and the
retailers’mark-up rising from spatial product di¤erentiation (k). In addition, it emerges
that, for any geographic pattern,
is proportional to the number of retailers.
Table 2 provides some evidence that
tration.18
In fact, the value of
1
1
is a consistent measure of geographic concen-
increases: a) moving from a more spatially dispersed
con…guration to a more concentrated one, and b) moving from balanced con…gurations
to more unbalanced ones (with the same pattern size).19 The latter property show that
1
also accounts for non-geographic aspects of concentration in a similar way of e.g. the
17
For example, (2; 1; 4) is given by the circular sequence A A B A A B A A B A A B.
In Table 2, we limit our analysis to those cases where Condition 1 is satis…ed, i.e. where in equilibrium
qn > 0.
18
Although these results can be derived analytically, since the computation is quite cumbersome, we
prefer to follow a more informal way and to show the numerical result. In Appendix B, we provide the
basis intuition for the analytical calculus.
19
To complete the description of the properties of this geographic concentration measure, in Table 3, we
will show that when the number of wholesalers increases, the spatial concentration reduces.
15
Her…ndahl-Hirschman Index (HHI). In fact the HHI (which is the sum of the square
market shares of each …rm in a industry) increases by moving from symmetric to asymmetric situations. Remember that the HHI is widely utilized in order to determine the
market concentration of industries. The di¢ sion of this concentration index is because in
the Cournot set-up, it directly relates the industry market power (measured by the Lerner
index, i.e. the price-cost margin of an industry) with market concentration.
3.3
Endogenous a¢ liation of retailers
In this sub-section we endogenize the retailer a¢ liation, assuming that retailers are free
to choose which wholesaler to patronize. Following previous literature on monopolistically
competitive environments, we assume that each retailer’s decision has a negligible impact
on wholesalers pricing behaviour (it is reasonable when the number of retailers is large),
and therefore in its a¢ liation strategy she considers wholesale prices, cA and cB , as given.
This fact implies that a dominant strategy for retailers is to patronize the cheaper wholesaler, since retail pro…tability is monotonically decreasing in own costs. Therefore the
market equilibrium is given by the equalization of wholesale prices: cA = cB . This result
can be obtained by the following three-stage game. First, retailers choose which wholesaler to patronize. Second, knowing the a¢ liations of the retailers, wholesalers choose
their wholesale prices. Finally, for given wholesale prices, retailers compete in a localized
fashion.
Disregarding the integer problem, this implies that:
nA =
1 1
+ (rB
2 8
rA ) t
1
.
(22)
The wholesaler with lower costs retains a larger market share. Since
depends on the dis-
tribution of retailers, it also emerges that multipla equilibria may occur. In particular, in
those cases in which market con…gurations are such that end up to be more spatially concentrated (retailers patronizing the same wholesaler are more likely to be close each other)
and therefore
is smaller, all other things equal, wholesale market shares are more similar.
This implies that we observe more up-stream size heterogeneity, for a given wholesale cost
di¤erential, when the down-stream industry is less spatially concentrated. Moreover, it
clearly emerges that being geographic concentrated is a dominant strategy for each wholesaler. Using (21) and (22), in fact, we obtain that
Taking …rst order conditions: d
m =d
= (rB
is a reciprocal advantage in dividing the markets.
16
m
rA ) t
= 1 + 41 (ru
1 2
rm ) t
1 2
1 t.
1 < 0 and therefore there
3.4
Oligopolistic up-stream market
We now complete the analysis of the properties of the spatial concentration, by investigating the e¤ects of an increase in the number of wholesalers. The spatial concentration
1
PM
m
1 =
. In Table 3, we present
index in the oligopoly case is de…ned as
m=1 m
the value of the average spatial concentration index
1
=
1N
for symmetric con…g-
urations characterized by sequences of an identical number of consecutive retailers (CR)
patronizing the same up-stream …rms.20 From the table, it clearly emerges that an increase in the number of retailers patronizing the same producer, as well as a reduction in
the wholesale number, causes an increase of concentration. However, the e¤ects are not
linear. In fact, the number of wholesalers in the market seems to have small impact on
spatial concentration for M > 4, and in any case when N C > 4. Results concerning the
properties of
1
are here summarized.
Proposition 4 The spatial concentration index
1,
increases when:
1. moving from a more spatially dispersed con…guration to a more concentrated one
2. moving from balanced con…gurations to more unbalanced ones
3. the number of retailers decreases
4. the number of wholesalers decreases
In sub-section 3.1 we have shown that
1
is proportional to wholesale industry margin
when there are two wholesalers. In this paragraph we now show that
is a good approx-
imation of up-stream margins also when there are more than two wholesalers. Moreover,
we show that a-spatially based measures of concentration (e.g. the Her…ndahl-Hirschman
Index, HHI) provides pour results in identifying market power in the up-stream market
when down-stream competition is localized. The importance of this result rests on the fact
that HHI is also utilized, prima facie, to identify the market concentration and therefore
market power, in the DOJ analysis of the impact of mergers.21
In Figure 5, we investigate the relationship between market power (average wholesalers’margins) and market concentration. We have simulated 700 drawn from a random
20
For instance, if M = 4 and CR = 3 the geographical arrangement is A A A B B B C
C D D D.
21
Incidentally, it is worth noting that the goal of this exercise is not that to provide a directly operazionalizable geographic measure
for concentration which accounts for localized competition, but to show
the importance to account geographic aspects. The introduction of geographical measures of localized
competition has important consequences for the antitrust discipline as well as for the industrial organization literature as it calls for additional research for …nding suitable market de…nitions in spatial contexts
(f.e. contemplating the existence of chain-linked markets). The advantage of structural econometric models for policy evaluations is that they can encompass this de…nitional problem by directly de…ning …rm’s
competitive environment through the estimation of …rm demand using consumer data.
C
17
con…guration of retailers with 2 to 10 wholesalers and 12 to 24 retailers. Half of the
drawn refers to situations where there is a potential dominance of one wholesaler, i.e. the
probability of …nding the dominant …rm is one half.22 In the left and central panel, we
plot the average wholesale margins and, respectively, the HHI computed on the wholesale
(HHIQ = QQT ) and retail quantities (HHIq = qqT ). The right panel consider a measure
P
m
of spatial concentration ( = M
m=1 m ). Non acceptable con…gurations are market with
a grey dot.
The …gure shows a positive relation between the three measures of concentration and
market power, although the level of correlation in the last case is superior with respect to
the previous case. Note that Gans (2005) shows that in case of vertical markets and in
absence of vertical integration, a general version of HHI to measure concentration in that
markets simply coincides with the HHI computed for the retail market. Indeed, HHIq
seems to better perform with respect of HHIQ .
Since
has shown to be quite e¤ective in identifying market power in vertically related
markets, we draw some qualitative considerations concerning the simulated values of
1.
In …gure 6, left panel we have simulated 700 drowns for a random con…guration of retailers
with two wholesalers and a random number of retailers ranging uniformly from 4 to 100.
Dashed lines correspond to the case of symmetric con…gurations with a = b = 1; ::; 4
(see Table 2). Non acceptable con…gurations are marked with a grey dot. Most of the
observations are between 2 and 2:70 indicating that random con…gurations have a spatial
symmetric analogue in con…gurations with 2 or 3 consecutive retailers patronizing the
same wholesaler. For smaller values of N we observe more variability in
as well as the
number of non acceptable con…guration is higher. In the other two panels we consider the
case of 3 and 4 wholesalers. Dashed lines correspond to symmetric con…gurations with
sequence of 1 to 4 retailers patronizing the same wholesaler in a similar way of the previous
case. Clearly, as the number of wholesaler increases, the inverse of the average measure of
geographical arrangement reduces.
3.5
Mergers
Figure 6 also provides some information on the impact of mergers and acquisitions. Reducing the number of wholesalers, on average, increases the market power of remaining
…rms and their pro…ts.23 However, the high variability of margins for a given number of
22
This choice is driven by the need to account for di¤erent market situations. Other simulations considering di¤erent market structures ends up with similar results.
23
Positive e¤ects of mergers are due to e¢ ciency gains (e.g. the elimination of some duplicate costs), due
to economies of scale. Negative e¤ects of mergers are the rationalization of the o¤er due to the reduction
of the total number of retailers. Our arguments do not consider these aspects. They can however be easily
18
wholesalers implies that to provide sound indications on the e¤ect of mergers, information
on the retailer locations patronizing the merging …rms is very important.
fully employed to obtain this information. In particular for a given
between a set of …rms can be obtained by pre-multiplying
it by
TT ,
can be use-
of size M a merger
by T and post-moltiplying
where T is a manipulation of the identity matrix where rows corresponding to
…rms involved in the merger are summed up. Moreover, simple computations imply that
the change in
m,
u
due to a merger is given by 2
wholesalers m and u in terms of
retailers.24
follows that every merger reduces
that is a measure of proximity between
Note that since
m
u
< 0, when m 6= u, it
and therefore increases the margins of the …rm. This
result holds even if merged manufactures do not have adjacent retailers as we show in the
following example.
Example 1 In a context of vertical relations with localized competition, consider the case
where there are four manufacturers and …ve retailers located in the following way: A
B
A
C
D. The matrix of geographical con…guration is:
0
B
B
=B
B
@
1:4737
0:6316
0:6316
0:4211
0:8421
0:4211
0:1053
0:4211
0:1053
0:1053
0:8421
0:3158
0:4211
1
C
0:1053 C
C.
0:3158 C
A
0:8421
(23)
B
A
Note that …rm B is more connected with A than with C or D since
>
B
C
C
B
Firm C has as direct competitors A and D and with indirect competitor B. Indeed
smaller to
C
A
or
C
D
B
C
=
.
is
. Consider, …rst a merger between …rms C and D. The resulting
matrix of geographical con…guration becomes:
CD =
0
B
T TT = @
1:4737
0:6316
0:8421
0:6316
0:8421
0:2105
0:8421
1
the post-merger situation, it becomes
1 0 0 0
1
C
B
C
0:2105 A , with T = @ 0 1 0 0 A .
1:0526
0 0 1 1
In the pre-merger situation and the spatial concentration index is
1
CD
0
1
= 0:250 while in
= 0:297, meaning that there is an increase in
geographical concentration. We now consider a merger between …rms B and C. In this
case the two wholesalers have no adjacent retailers. The resulting matrix of geographical
included in the set-up.
24
See: Appendix B for a proof.
19
con…guration is now:
BC =
0
1:4737
B
T TT = @
1:0526
0:4211
1:0526
0:4211
1
0
1 0 0 0
1
C
B
C
0:4211 A , with T = @ 0 1 1 0 A .
0:8421
0 0 0 1
1:4737
0:4211
Clearly, in this case we register an increase in concentration, but it is less relevant than
the previous case:
1
BC
= 0:264. Note that using (23), we can easily obtain the impact of
merger of the other cases. For example
therefore
1
AD
AD
=
2
14
=4
2 0:4211 = 3: 1578, and
= 0:317.
Theoretical literature on mergers in markets with localized competition has shown
that if there is no proximity between retailers, mergers have no e¤ects (Levy and Reitzes,
1992 and Giraud-Héraud et al., 2003). In the previous example we have shown that a
merger involving manufacturers, who have no adjacent retailers, produce an increase in
the spatial concentration index, and therefore a positive e¤ect on market power. This result
is general and does not depend on the particular retailer arrangement (see: Appendix B
for more details). The di¤erence of our …ndings with the previous literature depends on
the fact that in that models, mergers directly occur between retailers, while in our set-up
merger only occurs among up-stream …rms. Manuszac (2010), in a recent work analyzing
vertical related industries in a similar way to ours, has empirically shown that a merger
involving manufacturers who have no adjacent retailers, produce an increase in market
power, although the e¤ect is less relevant with respect to involving manufacturers who
have adjacent retailers.25
4
Conclusions
In the …rst part of the paper, we have introduced …rms’ cost heterogeneity in the traditional model of the circular city, showing that equilibrium prices are a weighted average
of all …rms’costs (plus the usual di¤erentiation premium). We found that the weights decrease quite sharply with distance and are largely independent of the number of …rms. In
the second part, we have extended the basic set-up in order to analyze the impact vertical
relations in localized markets on market power. Our …ndings have important implications
for competition policy. First, we show that when the locational pattern of retailers is introduced in the analysis, the Her…ndahl-Hirschman Index of concentration provides poor
25
On the contrary, Davis (2005) …nd no signi…cant evidence that geographical concentration impacts
on prices in the U.S. motion picture exibition market using a panel data approach considering the period
1993-1997.
20
results in identifying market power. Moreover, we derive a measure of geographic concentration which accounts for up-stream market power in localized contexts, which depends
on the locational pattern of downstream …rms. This measure based on the spatial a¢ liation of retailers has the required properties to be increasing in the number of retailers and
wholesalers, and decreasing in geographical concentration of retailers patronizing the same
manufacturer. In addition, by allowing endogenous a¢ liation of retailers, we …nd that a
low-cost manufacturer retains a larger market share and that multiple equilibria involving
di¤erent levels of manufacturer margins can occur. The model predicts more up-stream
size heterogeneity, all other things equal, when the down-stream industry is less spatially
concentrated. Finally, we study the impact of an up-stream merger on the equilibrium
prices. We show that the retail distribution plays an important role. Moreover, contrary
to the …ndings of Levy and Reitzes (1992) and Giraud-Héraud et al. (2003), when the
up-stream market is taken into consideration, a merger involving manufacturers which
have no adjacent retailers, also increases their market power.
References
[1] Aiura, H. and Y. Sato. 2008. “Welfare properties of spatial competition with locationdependent costs”. Regional Science and Urban Economics, 38(1): 32–48.
[2] Ammer, J., Mei J., 1996. “Measuring international economic linkages with stock
market datas”. Journal of Finance, 51(5): 1343–63.
[3] Anselin, L.,1988. Spatial Econometrics: Methods and Models. Kluwer Academic, Dordrecht.
[4] d’Aspremont C., J. J. Gabszewicz, and J.-F. Thisse, 1979, “On Hotelling’s ‘Stability
in Competition’”. Econometrica, 47(5): 1145-1150.
[5] Atkinson, B., A. Eckert and D. S., West, 2008, “Price Matching and the Domino E¤ect
in a Retail Gasoline Market”. Economic Enquiry, forthcoming, DOI 10.1111/j.14657295.2008.00149.x
[6] Bartelsman, E. J. and M. Doms, 2000, “Understanding Productivity: Lessons from
Longitudinal Microdata”. Journal of Economic Literature, 38(3): 569-594.
[7] Berry, S., 1994. “Estimating discrete choice models of product di¤erentiation”. Rand
Journal of Economics, 25(3), 242–262.
21
[8] Berry, S., Levinsohn, J., Pakes, A.,1995. ”Automobile prices in market equilibrium”.
Econometrica, 60 (4), 889–917.
[9] Bhatia, R., 1997. Matrix Analysis. Springer-Verlag: New York.
[10] Brueckner, J.K., 1998. “Testing for strategic interaction among local governments:
The case of growth controls”. Journal of Urban Economics, 44: 438-467.
[11] Capps, C., Dranove, D.D., Satterthwaite, M.A., 2003, “Competition and market
power in option demand markets”. Rand Journal of Economics, 34: 737-763.
[12] Chamberlin, E. H., 1933, The Theory of Monopolistic Competition. Harvard University Press: Cambridge.
[13] Capps, C., Dranove, D., Satterthwaite, M., 2003. “Competition and Market Power in
Option Demand Markets”. Rand Journal of Economics, 34(4): 737-763.
[14] Case, A.C., Rosen, H.S., Hines, J.R., 1993. “Budget spillovers and …scal policy interdependence: Evidence from the states”. Journal of Public Economics, 52: 285-307.
[15] Cohen, M.A., Whang, S., 1997. “Competing in Product and Service: A Product
Life-Cycle Model”. Management Science, 43: 535-545.
[16] Dasgupta, P. and E. Maskin, 1986. “The Existence of Equilibrium in Discontinuous
Economic Games, II: Applications”. Review of Economic Studies, 53(1): 27-41.
[17] Davis. P., 2005. “The e¤ect of local competition on admission prices in the US. motion
picture exhibition market”. Journal of Law and Economics, 58: 677-708.
[18] Draganska M., M. Mazzeo and K. Seim, 2009, “Beyond plain vanilla: Modeling joint
product assortment and pricing decisions”. QME-Quantitative and Marketing Economics, 7:105-146.
[19] Davis, P., 2006. “Spatial competition in retail markets: movie theatres”. Rand Journal
of Economics, 37 (4), 964–982.
[20] Eaton, C. B. and R. G. Lipsey, 1978, “Freedom of Entry and the Existence of Pure
Pro…t”. Economic Journal, 88(351): 455–69.
[21] Feenstra, R., Levinsohn, L., 1995. “Estimating Markups and Market Conduct with
Multidimensional Product Attributes”, Review of Economic Studies, 62: 19-52.
22
[22] Gaynor, M., Vogt, B., 2003. “Competition among hospitals”. Rand Journal of Economics, 24(4):764–85.
[23] Giraud-Héraud, E., Hammoudi, H., Mokrane, M., 2003. “Multiproduct Firm Behaviour in a Di¤erentiated Market”. Canadian Journal of Economics, 36(1): 41–61.
[24] Hamilton, J. H., W. B. MacLeod, J.-F. Thisse, 1991, “Spatial Competition and the
Core”. Quarterly Journal of Economics, 106(3): 925–937.
[25] Hotelling, H., 1929. “Stability in Competition”. Economic Journal, 39(153): 41–57.
[26] Jones, C., Leishman, C., 2006. “Spatial dynamics and the housing market: an interurban perspective”. Urban Studies, 43(7): 1041–1059.
[27] Kaldor, N., 1935.
[28] Lafontaine, F., 1992. “Agency theory and franchising: some empirical results”, Rand
Journal of Economics, 23: 263–283.
[29] Levy, D.T., Reitzes, J.D., 1992. “Anticompetitive e¤ects of mergers in markets with
localized competition”, Journal of Law, Economics, & Organization, 8: 427–440.
[30] Manski, C., 1993. “Identi…cation of endogenous social e¤ects: the re‡exion problem”,
Review of Economic Studies, 60: 531–542.
[31] Manuszak, C., 2010. “Predicting the impact of upstream mergers on downstream
markets with an application to the retail gasoline industry”, International Journal of
Industrial Organization, 28: 99–111.
[32] Mobley, L.R.., 2003. “E stimating hospital market pricing: an equilibrium approach
using spatial econometrics”, Regional Science and Urban Economics, 33: 489–516.
[33] Mobley, L.R.., Frech III, H.E., Anselin, L., 2009. “Spatial Interaction, Spatial Multipliers and Hospital Competition”, International Journal of the Economics of Business, 16: 1–17.
[34] Pinkse, J., Slade, M.E., Brett, C., 2002, “Spatial Price Competition: a Semiparametric Approach.” Econometrica, 70(3): 1111–1153.
[35] Pinkse, J., Slade, M.E., 1998. “Contracting in space: an application of spatial statistics to discrete-choice models. Journal of Econometrics, 85: 125–54.
23
[36] Rothschild, R., 1982, “Competitive Behaviour in Chain-Linked Markets”. Journal of
Industrial Economics, 33(1–2): 57–67.
[37] Salop, S. C., 1979. “Monopolistic Competition with Outside Goods.”Bell Journal of
Economics, 10(1): 141–156.
[38] Syverson, C., 2004. “Market Structure and Productivity: A Concrete Example”.
Journal of Political Economy, 112(6):1181-1222.
[39] Tirole, J., 1988. “The Theory of Industrial Organization". MIT Press: Cambridge,
Mass., USA.
[40] Tobler, W., 1970. “A Computer Movie Simulating Urban Growth in the Detroit
Region”, Economic Geography, 46(2):234-240.
[41] Thomadsen, R., 2005. “The E¤ect of Ownership Structure on Prices in Geographically
Di¤erentiated Industries”, Rand Journal of Economics, 36(4):908-929.
[42] Thomadsen, R., 2007. “Product positioning and competition: the role of location in
the fast food industry”, Marketing Science, 26(6):792-804.
[43] Vogel, J., 2008, “Spatial Competition with Heterogeneous Firms”. Journal of Political
Economy,116(3):423-466.
[44] Vogel, J., 2009, “Spatial Price Discrimination with Heterogeneous Firms”. NBER
Working Paper, No. w14978.
[45] Walden, M., 1990. “Testing implications of spatial economics models: Some evidence
from food retailing”. Journal of Consumer A¤ airs, 24(1): 24-43
[46] Waterson, M., 1990, “Product Di¤erentiation and Pro…tability: An Asymmetric
Model”. Journal of Industrial Economics, 39: 113-130.
[47] Yang, J., Hsiao, C., Qi, L., Wand, Z., 2006. “The emerging market crisis and stock
market linkages: further evidence”. Journal of Applied Econometrics, 21: 727–744.
24
Appendix A
Proof of Lemma 1. Express the system of equations (2) in matrix form:
1
k
c + 1N ; where
2
2
0
0
1
1
1
1
1
p
c
1
0
::
::
::
::
0
l
l
4
4 C
B
B
C
C
B
B
C
C
C
1
1
B
B
C
C
C
1
0
::
::
::
::
0
p
c
4
4
B l+1 C
B l+1 C
C
B
B
C
C
C
1
1
B :: C
B :: C
0
1
0 :: ::
::
0 C
4
4
B
B
C
C
C
B
B
C
C
C
::
::
::
:: :: ::
:: C ; P = B p0 C ; c = B c0 C .
B
B
C
C
C
B
B
C
C
C
B
B
C
C
::
:: :: ::
::
0
::
:: C
B
B
C
C
C
B
B
C
C
1
1 C
B
B
C
C
C
0
::
::
:: :: 0
1
p
c
l+1
l
1
4
4 A
@
@
A
A
1
1
0
::
:: :: :: 0
1
cl 1
pl
4
4
P
is strictly diagonal dominant by rows (i.e. jaii j > j6=i jaij j), it is non singular and
AP =
0
B
B
B
B
B
B
B
B
A=B
B
B
B
B
B
B
@
Since A
hence the solution of the system is unique (Bhatia, 1997, p. 251). Linearity of (2) also implies
Pl
that the solution is linear in cn , i.e. pn = i=l wi;n chn+ii + kn . In addition, the structure of A
also implies that: a) wi;n = wi and kn = k, for n 2 L, as every i-step shift of the position of all
…rms on the circle must not a¤ect the result; b) wi = w i , as mirroring the position of the …rms
on the circle, the result must remain unchanged.
Proof of Proposition 1.
Under Condition 1, the best reply function of …rm n is given by (2), for n 2 L. From Lemma
(1), the solution of the system is unique, and must be of the form given in (3). We show that if
…rm n’s adjacent rivals located at hn
1i and hn + 1i choose phn
1i
and phn+1i , where the weights
wi and k are given by (4) and (5), respectively, then for …rm n it is optimal to set:
pn =
Xl
i=l
wl
with weights given by (4) and (5). Replacing phn
pn =
Xl
1
2cn + chn+ii +
w
i=l l
4
jij
chn
1+ii
jij chn+ii
1i
(24)
and phn+1i in (2), we obtain:
+k +
We derive the proofs for N even and odd, separately.
25
+ k,
Xl
i=l
wl
jij
chn+1+ii + k
1
+ k^
2
Even case: N = 2 is trivial. Consider N
pn
=
4. Collect the similar terms for chn+ii ; i 2 L:
1 X 1
wl jij 1 + wl jij+1 chn+ii + 2 1 + wl 1 cn +
i=l
4
Xl 1
1 ^
wl jij 1 + wl jij+1 chn+ii + 2w1 chn+li +
k+k
+
i=+1
2
(25)
Requiring all coe¢ cients of chn+ii to be the same in (24) and (25), we obtain (4a)-(4c). A
similar reasoning can be used to derive (5). Finally, it is easy to show that (4d) is satis…ed, by
replacing cn = c and k = k^ in (24) and (25), and equating the two expressions.
Odd case: The proof is very similar to the even case. Collecting the similar terms for cn :
pn
=
X 1
1
wl jij 1 + wl jij+1 cn+i + 2 1 + wl 1 cn +
(w0 + w1 ) cn+l +
i=l+1
4
Xl 1
1 ^
wl jij 1 + wl jij+1 cn+i + 2 (w0 + w1 ) chn+li +
k+k .
+
i=+1
2
The only di¤erence with the even case is the second equality of system (4).
Proof of Corollary 1. We show that wi described in (4) can be generated from system
(10) and (11). Equations (4a), (4b) and (4d) are su¢ cient to uniquely determine wi (thus 4c
is unnecessary). Let w
^i = a (i) =AN . Rewriting (10b), (10a) and (10c) in terms of w
^i yields,
respectively, (4a), (4b) and (4d), woth wi replaced by w
^i . Thus: w
^i = wi .
Proof of Proposition 2.
(Existence) We start by computing the maximal costs’di¤erentials cH
(1) is satis…ed, i.e.:
k^ < pn
phn
=
w0
1i
cL for which Condition
^ Without loss of generality, assume that p > p
< k.
n
hn
1i .
Using (3)-(5) yields:
pn
phn
1i
+
Xl
wl+l chn+li +
1
i=0
wl
jij
wl
X
1
i= l
ji+1j
wl
jij
wl
ji+1j
chn+ii
chn+ii .
Because all the terms multiplying the ch:i in the …rst square bracket are negative, and all those in
the second bracket are positive, the maximal price di¤erence for a given cost di¤erential is given
by choosing chn+ii = cL , for i = l; ::; 1; l, and chn+ii = cH for i = 0; ::; l
h
i 1 hP
P 1
l 1
w0 wl+l
wl ji+1j
=
c =
i=l wl jij
i=0 wl jij
dition (1) is satis…ed by (12).
(Uniqueness) It is given by Lemma 1.
26
1. Let:
wl
ji+1j
i
1
> 1. Con-
Proof of Proposition 3.
^
cL < k.
(Existence) We focus on the situation in which costs di¤erentials are such that: cH
From (12), p = pl ; ::pl
given by (3)-(5) satis…es Condition (1). In order to be an equilibrium,
p must be such that, a generic …rm n has no unilateral incentive to deviate from pn . Assume
that p^n is a unilateral deviation potentially pro…table for …rm n. Since Condition 1 ensures the
uniqueness of the solution, we require that p^n is such that it does not satisfy this Condition.
o
n
If p^n > min phn 1i ; phn+1i + k^ then …rm n is undercut by at least one of the opponents, so
that its pro…t is null: therefore this strategy is not pro…table.
o
n
^ it can grab the market of at least one adjaBut if …rm n chooses p^n < max phn 1i ; phn+1i k,
cent opponent. Clearly, the maximum advantage from deviation emerges when phn
1i
and phn+1i
are as large as possible, so that after undercutting a …rm retains the largest margin. Equations
k^ < cH , n 2 L. Hence, for a …rm with costs su¢ ciently low relative to
(3)-(5) imply that cL < pn
phn+1i k^ and p^n
the 1-step rivals, it is possible to choose p^n > cn such that p^n
^ it is not possible that p^n > cn and simultaneously p^n
cL < k,
if we assume that cH
or p^n
phn
^ But
k.
1i
phn
2k^
2i
^ So …rm n can pro…table undercut one or both its 1-step opponents but not
2k.
phn+2i
any i-step rival, with i > 1.
Assume the best possible situation for …rm n; i.e., cn = cL and chn+ii = cH for i 2 L0
For symmetry phn+ii = phn
of:
n
=
D
t
cn ) phn
(pn
ii
for i 2 L0 . Further, by choosing pn = pn , …rm n obtains the pro…t
pn + k .
1i
If …rm n undercuts its rivals, it must choose a price p^n
become hn
phn
consumers; consequently, the pro…t it gains is:
D
t
(^
pn
deviation pro…t
n
d
n
k^
1i
d
n
(satifying p^n
cn
phn
2i
phn
phn
1i
1i
2i
+1
wl
wl
1
2i
p^n + 2k^
^ the maximum
k,
1i
1i
k^ with
d
n
=
k^ + 2k^ . In order for deviation to be unpro…table,
+1
1
= (cH
wl
1
After some algebra, it emerges that this is satis…ed if:
(cH
phn
1i
p^n + 2k . Since the previous
^ is reached when p^n = p
k)
hn
analitical expressions provided by (3)-(5), and set
wl
2i
+ 12 cn + k^ > phn
> 0. Divide the latter by D=kt and replace pn , phn
1
D
t
1i and sells to
cn ) phn
expression, which is concave in p^n , is maximized in 12 phn
phn
k, so that its next competitors
1i
2i and hn + 2i. For analytical convenience, we assume that by choosing p^n = phn
^ …rm n is able to “steal” all the customers of …rm hn
k,
D
t
Ln f0g.
^
cL ) < (N ) k,
27
1i
and phn
2i ,
with the respective
cL ) =k^ to obtain:
wl
1
wl
2
+3
> 0.
where
and B =
(2) =
c
(N ) =
1
A
1
1
wl
p
B
B2
A and A = 1
2
wl
1
1
wl
2
1
wl
wl
wl
1
2
. When N = 2 …rms have no incentive to undercut the rivals, so that
(2) = 3.
(Uniqueness) We show that every pro…le strategy p^ 6= p cannot be an equilibrium. Assume
that another price strategy equilibrium vector p^ 6= p exists. It means that for some n, Condition
1 is not satis…ed, i.e. there exists at least one p^n such that p^n
^ Let Lnp be the set
p^hn+1i > k.
of …rms that in equilibrium do not sell to any consumer. Note that for any p^, at least one but
not all …rms must sell a positive quantity, i.e. 1
#Lnp
N
1. In equilibrium, a …rm cannot
have negative pro…ts, otherwise it would prefer to charge a su¢ ciently high price and sell nothing.
Let n 2 Lnp . Because n sells nothing, then at least one of the following conditions must hold:
^ for 0 6= i 2 L. Now, choose " > 0 su¢ ciently small. If n charges pn = cn + ", it
p^n > p^hn+ii + jij k,
can pro…tably deviate and reach a positive pro…t, (otherwise at least one active opponent is gaining
negative pro…ts) Hence, assuming the existence of another equilibrium p^ 6= p is a contradiction.
5
Appendix B
In this Section we provide a more formal treatment of the properties of the
=[
ij ]
matrix.
We start considering a case where for every retailer there is a corresponding manufacturer.
To simplify the notation we assume that retails are indexed by n with n ranging from 1 to N ,
and manufacturers are indexed by m, ranging from 1 to M . Without loss of generality, retailer n
patronizes manufacturer m = n. In this case, Wnm = wl
Ynm
between retail …rms m and n. Therefore,
= 2wl
d
where d is the d-step (shortest) distance
wl
d
d
wl
d+
is simply the double of the
weight associated to the …rms at the d-step distance less the weight associated to a d -step and to
a d+ -step (shortest) distances between retail …rm m and retail …rms, respectively on the left and
on the right of n. Finally,
0
2wl
ij
2wl
=
j
i
1
B
B
B 2wl 1 wl wl 2
B
B
B 2wl 2 wl 1 wl 3
=B
B
B
B
B
B
@
2wl 1 wl wl 2
= Yij . Therefore:
2wl
1
2wl
wl
2wl
wl
1
2
2wl
2wl
1
2
2wl
28
wl
wl
wl
1
2wl
wl
1
2
3
1
C
C
C
C
C
C
C
C
C
C
C
C
A
This matrix has the following properties:
1.
ij
= 2wl
2.
ij
= 2wl
3.
ij
=
4.
2wl
d
1
wl
> 0.
d
wl
d+
< 0 when i 6= j.
ji .
PM
ij
i=1
= 0.
The …rst property follows from wl > wl
i 6= j max l
+
d ;l
d
> l
1.
The second one derives from the fact that when
d and from the fact that wl
h
> 2wl
h 1
(equation 4c). The
third property comes from the symmetry of …rms. The last property derives from the fact that for
every row (or column), each coe¢ cient is counted twice with the positive sign and twice with the
negative sign.
Starting from this matrix, it is possible to obtain every possible geographical retail con…guration
by merging up-stream …rms. In particular, a merger between up-stream …rms m and n implies
that the
ij
with i or/and j equal to m and n should be added because Wnm , and all the following
variables derived from it, is additive with respect the action of merging. In sub-section 3.5 we have
shown that this result can be obtained by pre-multiplying
by T and post-moltiplying it by TT ,
where T is a manipulation of the identity matrix where rows corresponding to …rms involved in the
merger are summed up. Therefore, it is possible to obtain any con…guration by simply reiterate
the merging procedure. Thanks to this fact, all the proprieties of the initial matrix are inheritated
by the following matrixes.
increasing in N and M and decreasing in geographical concentration.
29
30
N
2
3
4
5
6
7
8
9
10
11
12
13
14
15
20
1
l
0
1
1
2
2
3
3
4
4
5
5
6
6
7
9
1
l
1
1
2
2
3
3
4
4
5
5
6
6
7
7
10
1
wl
66:667
60:000
58:333
57:895
57:778
57:747
57:738
57:736
57:735
57:735
57:735
57:735
57:735
57:735
57:735
57:735
wl 1
33:333
20:000
16:667
15:789
15:556
15:493
15:476
15:471
15:470
15:470
15:470
15:470
15:470
15:470
15:470
15:470
2
8:333
5:263
4:444
4:225
4:167
4:151
4:147
4:146
4:145
4:145
4:145
4:145
4:145
4:145
wl
3
2:222
1:408
1:190
1:132
1:116
1:112
1:111
1:111
1:111
1:111
1:111
1:111
wl
4
0:595
0:377
0:319
0:304
0:299
0:298
0:298
0:298
0:298
0:298
wl
5
0:159
0:101
0:085
0:081
0:080
0:080
0:080
0:080
wl
6
0:043
0:027
0:023
0:022
0:021
0:021
wl
7
0:011
0:007
0:006
0:006
wl
R
0:000
0:000
0:000
0:000
0:000
0:000
0:595
0:755
0:798
0:809
0:814
0:813
0:813
0:813
0:813
0:813
c
300:00
250:00
200:00
190:00
180:00
177:50
175:00
174:34
173:68
173:51
173:33
173:29
173:24
173:23
173:21
173:21
Table 1: The sequence of equilibrium prices’weights (percentage values).
300:00
69:99
62:44
61:36
61:08
61:00
60:88
60:98
60:97
60:97
60:97
60:97
60:97
60:97
60:97
60:97
Table 2: Spatial concentration index
1 to 10.
1
2
3
4
5
6
7
8
9
10
1
1:500
2
1:875
2:000
3
2:400
2:375
2:700
4
2:969
2:813
3:106
3:500
1
for a sequence (a; b) where a and b range from
5
3:553
3:270
3:537
3:923
4:342
6
3:733
3:975
4:354
4:771
5:200
7
4:415
4:788
5:204
5:632
6:063
8
5:233
5:637
6:065
6:496
6:929
9
6:071
6:498
6:929
7:361
7:794
10
6:931
7:373
7:794
8:227
8:660
1 when M ranges from 2 to 8 and the number of
Table 3: Spatial concentration index
consecutive retailers ranges from 1 to 10.
2
3
4
5
6
7
8
1
1:500
1:260
1:200
1:187
1:184
1:183
1:183
2
2:000
1:875
1:867
1:866
1:866
1:866
1:866
3
2:700
2:650
2:650
2:649
2:649
2:649
2:649
4
3:500
3:482
3:482
3:482
3:482
3:482
3:482
5
4:342
4:336
4:336
4:336
4:336
4:336
4:336
31
6
5:200
5:198
5:198
5:198
5:198
5:198
5:198
7
6:063
6:063
6:063
6:063
6:063
6:063
6:063
8
6:929
6:928
6:928
6:928
6:928
6:928
6:928
9
7:794
7:794
7:794
7:794
7:794
7:794
7:794
10
8:660
8:660
8:660
8:660
8:660
8:660
8:660
Figure 1: The bold line describes the location of the …rms. The external dashed circumference denotes the weights given by …rm 0 to the costs of the …rms located at any other
point. The internal …rst (second) dashed circumference denotes the weights given by …rm
+1 ( 1) to the costs of any other …rm.
32
33
Figure 2: Average wholesale prices and concentration indexes: HHIQ (left panel), HHIQ (central panel)
1
(right panel).
34
Figure 3: Geographical concentration measures with di¤erent retail con…gurations with two (left panel), three (central panel)
and four (right panel) manufacturers.