Local Competition, Entry, and Agglomeration
Supplemental Appendix
January 10, 2011
This supplement contains the technical details and model extensions omitted from the
main text.
S.1 Details of the Before Entry Model
S.2 Extended Model with Strategic Discounter
S.3 Discounter Enters Between Existing Retailers
S.4 Joint Ownership of Incumbent Retailers
S.5 Entry with Single Incumbent
S.1 Details of the Before Entry model
In the main text, we omitted the analysis of the Before Entry model (Section 2.1) because
it follows the basic argument of Lal and Matutes (1989). For the interested reader, we
demonstrate how that argument is applied in our context. Specifically, we provide here
the detailed proof of Proposition 1, which makes us of the following two lemmas.
Lemma S.1 Under Assumption 1, no equilibria exists in which one retailer serves all
value buyers with both products.
Proof
Suppose by contradiction that all value buyers go to A, in equilibrium. Then
p1A  p1B and p2 A  p2 B with pkA  L, k  1, 2 . Then the portion of convenience
shoppers (C) at A is
xC  21t [t  ( p1B  p2 B ]  ( p1A  p2 A )] .
Therefore, profits accruing to B are
 B  [( p1B  p2 B )  2K ] (1  xC ) .
Consider a profitable deviation by retailer B: Set p1B  p1 A   , for   0 sufficiently
small so that p1B  K , and p2 B  p2 B  ( p1B  p1A   ) . Under this pricing strategy,
retailer B enjoys the same number of convenience shoppers since
p1B  p2 B  p1B  p2 B , and all of the value buyers for product 1 since p1B  p1 A .
Profits to B are thus
~B  [ ~
p1B  ~
p2 B  2K ] (1  xC )  ( ~
p1B  K )(1   )   B  ( ~
p1B  K )(1   ) ,
which is strictly greater than  B .
Q.E.D.
Lemma S.2 Under Assumption 1, no equilibrium exists in which a retailer does not
serve value buyers at least one product.
Proof
By contradiction, suppose in equilibrium retailer A does not sell either product to
value buyers. By Lemma S.1, retailer B has a price above L, say pˆ 1B  L . Profits to
each retailer are
 A  xC [ pˆ 1 A  pˆ 2 A  2 K ]
 B  (1   )( pˆ 2 B  K ) I { p  L}   (1  xC )[ pˆ 1B  pˆ 2 B  2 K ],
which are subject to the first order conditions for profit maximization with respect to
p1A and p1B . These conditions imply that pˆ 1 j  pˆ 2 j  2 K  t , for j  A, B , and that
2B
xC  12 . Consider a deviation by retailer A that involves serving value buyers. Set
~
p1A  L and ~
p2 A  2K  t  L . This is profitable since profit from convenience
shopping segment remains the same, but now A earns (1   )( L  k )  0 from value
buyers.
Q.E.D.
With these two results, we provide the final argument to prove Proposition 1.
Proof of Proposition 1
By Lemmas S.1 and S.2 the only possible equilibria has value buyers shopping at
both retailers. Hence, we can restrict attention to outcomes yielding the following
profits:
 A  (1   )( p1 A  K )   xC ( p1 A  p2 A  2 K )
 B  (1   )( p2 B  K )   (1  xC )( p1B  p2 B  2 K ),
where we assume without loss of generality, p1 A , p2 B  L . And xC is defined as
t  p1B  p2 B  p1A  p2 A
. Solving max  A subject to p1A  L , and max  B subject to
p1 A , p2 A
p1 B , p2 B
2t
p2B  L , leads to the system of equations
 A
 t  p1B  p2 B  p1 A  p2 A 
 p1 A  p2 A  2K 
(S.1)
 
  
0
p2 A
2
t
2t




 B
 t  p1B  p2 B  p1 A  p2 A 
 p1B  p2 B  2K 
(S.2)
  1 
  
0
p1 A
2t
2t




2
 A
 t  p1B  p2 B  p1 A  p2 A 
 p1 A  p2 A  2K 
 (1   )   
  
0
p1 A
2t
2t




(S.3)
 B
 t  p1B  p2 B  p1 A  p2 A 
 p1B  p2 B  2K 
 (1   )   1 
  
0.
p2 B
2t
2t




(S.4)
There is no solution such that (S.1)-(S.4) are satisfied with equality. However, under
Assumption 2, (S.1) and (S.2) are satisfied with pˆ1 A  pˆ 2 A  pˆ1B  pˆ 2 B  2K  t . Then
(S.3) and (S.4) hold with strict inequality, suggesting the corner solution in which
pˆ1 A  pˆ 2 B  L . Solving for pˆ 2 A and pˆ1B gives the prices as stated in the proposition.
Assumptions 1 and 2 imply L  pˆ 2 A, pˆ1B  H  s . This solution constitutes an
equilibrium, so long as there is no profitable deviation. We can rule local deviations
by virtue of the maximization. However, we must check whether it is profitable for
one retailer, say A, to undercut B with the price p2 A  pˆ 2 A    L   , for small   0 .
The payoff in this deviation is
 1 2K  t  2L   
(2L    2K )(1   )   (2 L    2 K )  
,
2
2t

because all value buyers buy both products from A. This payoff is clearly decreasing
in  , so we let   0 and derive the net benefit from this deviation:
2
t
.
   ( L  K )2  (1   )( L  K ) 
t
It can be shown that   0 if
2
1    (1   ) 2  4 2
t.
4
Since f ( ) is increasing in  and lim f ( )  0  f ( )  2t  lim f ( ) , there exists
L  K  f ( ) 
 0
 1
 BE  (0,1) so that 0  L  K  f ( BE ) . Hence, there is no profitable deviation
(i.e.   0 ) for any   ( BE ,1) .
Q.E.D.
S.2 Extended Model with Strategic Discounter and Competition for
Product 3
In the main text, the discounter’s prices were set exogenously. This assumption was made
to simplify the presentation and to illustrate the main effects of the discounter’s entry. In
this section we consider a strategic discounter who sets prices for both its products. Note
also that consumers in the main model had no demand for the alternative product before
entry by the discounter. In order to make the before entry and after entry models
comparable, the extended model includes options for consumers to buy the alternative
product (product 3) before entry.
3
We illustrate here that relaxing these assumptions does not alter the central
notions behind our results. Specifically, in a model in which consumers have access to
product 3 from competing specialty stores before the discounter’s entry, the impact of a
discounter selling an overlapping product with the traditional retailers leads to more (less)
sales for the non-overlapping products the nearby (distant) retailer under conditions
similar to those in the basic model. The logic of the analysis of the extended model and
corresponding economic incentives are similar to that of the basic model. Therefore, we
present here only the crucial distinctions of the extension.
The before entry case of section 2 is augmented by the presence of two competing
retailers carrying product 3. One of these retailers, retailer D is located next to retailer A
and the other, retailer E is similarly located next to B. It may be helpful to imagine a
scenario in which two shopping centers are located in different parts of town. For
instance, both shopping centers contain a traditional grocery store and a specialty home
appliance store. Consumers make periodic trips for groceries and a portion  of these
trips include the need for an appliance. All four retailers compete in price and face a
constant marginal cost of K  0 .
As before, a large presence of value buyers (low  ) can lead to competitive
Edgeworth cycles between retailers D and E, which prohibit the existence of an
equilibrium. Alternatively, if the fraction of convenience shoppers is not too small ( 
sufficiently large), then D and E abandon the value buyer segment and compete in
Hotelling fashion. Proposition S.1 states the result precisely.
Proposition S.2 Under Assumptions 1 & 2, there is a threshold, ˆ BE  (0,1) , which
depends on K, L, and t, such that for all   (ˆ BE ,1) , there exist equilibria
characterized in the before-entry game as follows:
(i)
pˆ 1i  pˆ 2 j  L ; i, j  A, B ; i  j ,
(ii)
pˆ 1 j  pˆ 2i  2 K  t  L  L ,
(iii)
pˆ 3k  K  t  L ; k  D, E .
(iii) Each retailer sells to exactly ½ of convenience shopper segment. All value
buyers visit retailer i for product 1 and retailer j for product 2.
4
The following result will be useful for proving Proposition S.1.
Lemma S.3
There exists ˆ1  (0,1) such that for all   (ˆ1 ,1) , no equilibria exists in the beforeentry game in which either store D or E serve value buyers. That is, no pure Nash
equilibria involves p3k  L , k=D, E.
Proof
Suppose p3D  p3E  L . Then each firm has a profitable deviation by lowering price
slightly in order to capture the entire value buyer segment. Now suppose
p3D  p3E  L . The value buyers will shop at the store with lowest price. Assume,
without loss of generality, that p1 A  p1B , and p2 B  p2 A . This leads to profits
expressed by
 A  ( p1A  K )(1   )  ( p1A  p2 A  2K )[ x3   (1   ) x2 ]
 B  ( p2 B  K )(1   )  ( p1B  p2 B  2K )[ (1  x3 )   (1   )(1  x2 )]
 D   ( p3D  K )[(1   )   x3 ]
(S.5)
 E   ( p3E  K )[ (1  x3 )] ,
where
1 p  p2 B  p1 A  p2 A
x2   1B
,
2
2t
1 p  p2 B  p3 E  p1 A  p2 A  p3 D
x3   1B
.
2
2t
Store D and E’s profits are maximized when p3D  K  853 t and p3E  K  253 t ,
which contradicts the assumption p3D  p3E . Obviously a similar argument shows
p3E  p3D  L leads to a contradiction. Finally, it only leaves to show that the corner
solution p3D  L  p3 E cannot be an equilibrium. Suppose firms D and E price in this
way. Then firm E earns optimal profits  E , with the expression indicated in (S.5)
with
 p3 E  L 
K  t  (1  23 L)
1
p3 E 
and
x



 .
3
2
2(1  3 )
 2t 
In this case, firms A and B optimally set prices p1 A  L  p2 A and p2 B  L  p1B . It
can be shown that p2 A  p1B  0 , which satisfy the first order conditions:
5
 A
 B

 0 . Suppose firm D deviates by raising its price above L. The most it
 p2 A  p1B
can earn (holding other firms’ behavior fixed) is  D  max p  ( pD  K ) x3 , which is
D
greater than  ( L  K ) x3 . Let    D   ( L  K ) x3 , which is positive. Then, for
firm D, there is a profitable deviation p3D  arg max p ( p  K ) x3 ( p) from p3D  L if
  ( L  K )(1   ) . Since lim 1 ( L  K )  0   , there exists ˆ1  (0,1) such that this
 1
deviation is profitable for all   (ˆ1 ,1) .
Q.E.D.
Lemma S.3 says that the only possible equilibrium of the before entry game involves
p3 D , p3 E  L .
Proof of Proposition S.2
From Lemma S.3, we can conclude that retailers D and E serve only convenience
shoppers. Writing respective profits and optimizing lead to pˆ 3 D  pˆ 3 E  K  t .
Furthermore, the same analysis as in the proof of Proposition 1 leads to the reversed
pricing strategies for retailer A and B. These prices are supported in equilibrium if
there is no profitable deviation for either D or E to attract value buyers with a price of
L. This condition for retailer D (a similar argument can be used for deviation by E)
requires
(S.6)
 ( L  K )(1     x3 )   t / 2 ,
where x3  12 
pˆ1 B  pˆ 2 B  pˆ 3 E  pˆ1 A  pˆ 2 A  L
2t
. It is readily verified that (S.6) holds for all
LK
.
t  (  K 2tt L )( L  K )
Then for   ˆ BE  max{ˆ1 , ˆ 2 } , the only equilibria of the before-entry game are the
ones expressed in the proposition.
Q.E.D.
  ˆ 2 
1
2
1
2
This proposition is the analog of Proposition 1 and establishes that the reversed price
equilibrium of the before entry case occurs in the extended model.
We compare this outcome to the case in which a discounter C locates at (or very
near) the shopping center housing retailers A and D. This discounter, as before, carries
products 2 and 3 and incurs a marginal costs of c, where K  c  0 . Because this
discounter is assumed to have lower costs, the nearby specialty appliance store must drop
out of the market. Hence, we analyze the after-entry case in which there are two shopping
centers, one with a discounter C and traditional retailer A and the other with a specialty
appliance store E and traditional retailer B. (See Figure S.1.) In addition, the presence of
6
competition for product 3 in this after-entry model allows us to meaningfully endogenize
the pricing decisions of the discounter.
Retailer C’s low cost in product 3 makes its price competitive relative to retailer
E. This induces a portion of convenience shopper, big-basket consumers who previously
shopped at E to switch retailers after entry. Not only do they buy product 3 at the
discounter, but while there, they buy product 2. And, as in the original model, have an
incentive to finish their shopping with the purchase of product 1 at A. Because these are
convenience shoppers, it is never optimal for retailer A lower p1A after entry.
Correspondingly, retailer B, will not raise its price p1B . This preserves the basic logic of
Lemma S.2 in the previous analysis and allows us to conclude that if an equilibrium
exists, it must entail a higher price for product 1 and the store nearby the discounter.
For such an equilibrium to exist, two conditions are sufficient. The first is the
familiar condition that the set of value buyers be small enough to prevent retailers A and
B from unilaterally abandoning the high-end of the market and go for the poor segment.
(See the discussion prior to Proposition 2.) The second condition involves the degree of
advantage possessed by the discounter in product 3. Note that in order for some
consumers toward the right-end of the line to travel an extra distance (across town) for
product 3, the discounter must be sufficiently competitive. That is, K  c must be large.
Hence, if an equilibrium exists, it is characterized by an increase in sales of product 1 at
retailer A. Proposition S.2 formalizes this discussion.
Proposition S.3 Suppose that  (1   )  1 and K  c   1   62  t . Then if the
extended model has an equilibrium in the after-entry game, then it has the following
properties.
(i)
Product 1 prices are higher at retailer A than at retailer B: p1*A  p1*B .
(ii)
Sales of product 1 are higher at retailer A than at retailer B.
Proof
In an argument identical to that of Lemma 2, it can be shown that no equilibrium
exists in which p1 A  p1B . Hence, we assume that p1 A  p1B and illustrate conditions
7
that support this as an equilibrium outcome. We start by deriving the demands for
each retailer A, B, C, and D, which we do by consumer segment.
Convenience / Big Basket: Consumers in this segment choose product 1 from either A
or B, product 2 from either C or B, and product 3 from either C or E. There are three
un-dominated shopping plans: (A,C,C), (B,C,C), and (B,B,E), where ( s1 , s2 , s3 ) is
interpreted as buying product i from retailer si . Other combinations can be excluded
because they are strictly dominated by those three plans. Notice that the
transportation cost of plan (B,C,C) does not depend on the location of the consumer
because the this cost is t for all locations. In contrast, the transportation costs for a
consumer located x is xt and (1-x)t for shopping plans (A,C,C) and (B,B,E),
respectively. Define m as the location of the consumer is indifferent between these
two shopping plans. Then
p  p1B
.
p1B  p2C  p3C  mt  p1B  p2C  p3C  t

m  1  1A
t
Similarly define n as the location of the consumer indifferent between (B,C,C) and
(B,B,E).
p1B  p2 B  p3E  (1  n)t  p1B  p2C  p3C  t

p2 B  p3 E  ( p2C  p3C )
.
t
If m, n  (0,1) and m  n then plans (A,C,C), (B,C,C), and (B,B,E) will be
implemented by consumers depending on the locations. We take up the case when
m  n later. For a convenience/Big Basket consumer located at x, her choice
corresponds to the shopping plan as follows:
x  [0, m)  ( A, C , C )
x  [m, n)  ( B, C , C )
x  [n,1]  ( B, B, E ).
Convenience / Small Basket: Consumers in this segment buy products 1 and 2 only.
A consumer will choose A if x  x2  12  ( p1B  p2 B )2t( p1 A  p1B ) .
Value Buyers: These consumers will buy product 1 at B and products 2 and 3 (if Big
Basket) at C.
Hence, retailers’ profits can be specified as follows when m  n :
 A  ( p1A  K ) m  ( p1A  p2 A  2K ) (1   ) x2
n
 B  ( p1B  K )[(1   )   (m  n)]  ( p1B  p2 B  2K )[ (1   )(1  x2 )   (1   )(1  n)]
 C  ( p2C  c)(1   )  ( p3C  c)  (1   )  ( p2C  p3C  2K ) n
 E  ( p3E  K ) (1  n) .
First-order conditions with respect to the relevant prices lead to a solution described
as follows:
8K  (1   ) y 3  5   (  2  10  3)
p1*A 

t
7
3 (7   )
6 K  (1   ) y (      1)
p2* A 

t
7
 (7   )
8
(9   ) K  2(1   ) y 6  10   (  2    6)

t
7
3 (7   )
(5   ) K  2(1   ) y 2(      1)


t,
7
 (7   )
p1*B 
p2* B
where
3K (1   )  2c(7   ) (  2  11  4)  2 (  2  5  2)

t.
17  
 (17   )
The total quantity of good 1 sold at store A at these prices is
 (1   ) x2   m  (2     ) / 6 . The assumption  (1   )  1 implies that the
sales of product 1 are higher at A than at B.
Finally, we examine the case when m  n . In this case, shopping plan (B,C,C) is
dominated by (A,C,C). Define x1  12  ( p1B  p2 B  p3 E )2t( p1 A  p2 C  p3C ) . Then retailers’ profits
are
 A  ( p1A  K ) x1  ( p1A  p2 A  2K ) (1   )
 B  ( p1B  K )(1   )  ( p1B  p2 B  2K )[ (1   )(1  x2 )   (1  x1 )]
 C  ( p2C  c)(1   )  ( p3C  c)  (1   )  ( p2C  p3C  2c) x1
 E  ( p3E  K ) (1  x1 ) .
First-order conditions lead to a solution described as follows:
K (9  2 )  c(3   )  (9  2 )  (3   )
p1*A 

t
6
 (6   )
(3   ) K  3c 3(1   )
p2* A 

t
6
 (6   )
4(3   ) K  2 c 3 (2   )  2
p1*B  p2* B 

t
6
 (6   )
(3   ) K  (9   )c 9  3  
p2*C  p3*C 

t
6
 (6   )
3K  (3   )c 3  3  
p3*E 

t.
6
 (6   )
These prices imply a level of sales at store A of
6 t  2 ( K  c)  (  2 )t
,
 (1   ) x1   (1  x2 ) 
12t  2t 
y  p2*C  p3*C  p3*E 
which is greater than 1/2 under the assumption that K  c   1   62  t .
Q.E.D.
Proposition S.3 confirms that entry by the discounter can lead to a shift in consumer
shopping patterns indicated in Proposition S.2 (ii) and a pricing reaction by incumbent
retailers consistent with the basic model. That is, even when product 3 is available before
entry and the discounter’s prices are endogenous, the key results of section 2 are retained.
9
S.3 Alternative Discounter Location
In the main text, we model the case when the discounter locates at one endpoint of the
“line”, atop of one of the incumbent retailers. Alternatively, in this section, we consider
an after entry scenario in which the discounter situates directly between the incumbent
retailers. We show that in equilibrium, incumbent retailers abandon low-income
consumers altogether.
Formally, consider an after-entry scenario with the discounter, retailer C, located
at x  12 . Consider the situation in which A serves value buyers in product 1. That is,
assume p1 A  p1B and p1A  L . We consider demand from each of the four segments.
The Convenience Big-basket consumers buy products 2 and 3 from retailer C,
independent of their location. Alternatively, the choice between A and B for product 1 is
determined by the consumer’s location. Specifically, a consumer located at x visits
retailer A if
p1A  max  12 , x t  p1B  max  12 ,1  x t ,
(S.7)
and otherwise visits retailer B. This implies that a shopping trip to A and C yields a
transportation cost of
t
2
for consumers with x  12 and of xt for consumers with x  12 .
Under the assumption p1 A  p1B , (S.7) is always satisfied for x  12 . For x  12 , (S.7)
holds with equality at x1 
p1 B  p1 A
2t
 12 , which is greater than
1
2
. Thus, x1 and 1  x1
represent the demand of rich / Big-basket consumers at A and at B, respectively.
Convenience Small-basket consumers buy products 1 and 2 only. For simplicity,
we focus on the possibility that consumers in this segment visit one store. This is the case
if shopping costs s is large enough. These consumers choose between retailers A and B
based only on their location and the sum of prices of products 1 and 2. It is readily shown
that consumers x  x2 
p1 B  p2 B  p1 A  p2 A
2t
 12 shop at A and the remaining 1  x2 shop at B.
The remaining consumer segments, big and small basket consumers value
shoppers, buy product 1 at A and product 2 at C. This analysis yields the profits at the
incumbent retailers expressed by
 A  ( p1A  K )[ x1  (1   )]  ( p1A  p2 A  2K ) (1   ) x2
(S.8)
 B  ( p1B  K )[ (1  x1 )]  ( p1B  p2 B  2K ) (1   )(1  x2 ) .
(S.9)
10
First-order conditions imply that optimal prices must satisfy
p1 A  K 
4  4  3
t
6
p1B  K 
and
4  4  3
t,
6
which contradicts the initial assumption p1 A  p1B . This argument rules out any
asymmetric equilibrium with poor consumers being served in product 1 after entry. All
symmetric strategies with K  p1 A  p1B  L
lead to Edgeworth cycle deviations and
thus also cannot be sustained in equilibrium. We have thus proved the following.
Lemma S.4 No equilibria exists in which either retailer serves poor consumers with
product 1.
Using this lemma, we now assume p1A and p1B exceed L. This implies profits to
the incumbent retailers expressed as follows:
 A  ( p1A  K )[ x1 ]  ( p1A  p2 A  2K ) (1   ) x2
 B  ( p1B  K )[ (1  x1 )]  ( p1B  p2 B  2K ) (1   )(1  x2 ) .
First-order conditions lead to the symmetric pricing strategies with pik  K  2t , i = 1, 2,
and k  A, B . Furthermore, these prices exceed value shoppers’ reservation price, L by
Assumption 1. As the following proposition confirms, these price are sustained in
equilibrium if  is sufficiently large.
Proposition S.4 Under Assumption 1, if  
4( L  K ) t
4( L  K ) t   (2 K  2 L  t ) 2
pricing strategy pik*  K  2t , i = 1, 2, and k  A, B is an
, then the symmetric
equilibrium. In this
equilibrium, retailers A and B have equal market share among convenience shoppers,
x1  x2  12 , serve no value buyers, and earn  k*  14  (2   )t .
Proof: It has already been argued that these prices maximize the profits expressed in
(S.8) and (S.9). It remains to show that there is no unilateral deviation for either retailer
to profitably serve poor consumers. Suppose A deviates in this way. Then A can earn no
more than
 A   ( L  K )[1  ( L  K ) / t ]   (1   ) 2t  (1   )( L  K ) ,
which is achieved by setting p1A  L and p2 A  2K  t  L . The condition on  in the
statement of the proposition guarantees that  *A   A .
11
Q.E.D.
When the discounter locates in the middle of the line, the Convenience big-basket
segment is no longer competitive ground for the incumbent retailers, contrary to the case
in the main text, when C locates next to A. Specifically, all Convenience big-basket
consumers shop at C for product 2. It is only the Convenience small-basket consumers
who have interest for this product at the incumbent retailers. However, their interest in
product 2 is strictly related to the bundle product 1 plus product 2. In this sense, C’s
center location alleviates one dimension of competition between incumbent retailers. This
price increasing incentive encourages incumbents to abandon poor consumers.
S.4 Joint Ownership
In the section 2 of the main text, it is assumed that retailers A and B are independently
owned, and therefore maximize individual profits. The Dominick’s Finer Foods (DFF)
data used in the empirical verification of section 3 has the two retailers belonging to the
same grocery store chain. In what follows, we consider the theory when the two retailers
A and B have a single owner. The intention of this supplement, therefore, is to establish
that the same basic result holds when the stores are jointly owned. Specifically, we
illustrate that if there is a single owner of the two retailers, the impact of the dominant
retailer’s entry on pricing strategies and market share distribution is consistent with
independent ownership case.
Formally, maintain the model structure developed in section 2 of the main text
except that retailers A and B are owned by a single decision maker who can coordinate
prices. As is done in the main text, we first consider the before-entry model and compare
it with the after entry.
S.4.1 Before Entry
When retailers A and B are jointly owned, then it is optimal for the retailers to set prices
such that all surplus from value buyers is extracted. Under the condition that all
convenience consumers are served, it optimal to split the market equally and raise prices
until the convenience consumer located x  1/ 2 obtains zero utility. These conditions
imply the equilibrium pricing strategy:
12
pˆ 1i  pˆ 2 j  L ;
pˆ 1 j  pˆ 2i  p ,
where p solves H  2 p  2t  0 . We summarize in the following proposition.
Proposition S.5
Suppose retailers A and B have a single owner. Then the equilibrium pricing strategy
before entry by the discounter is
pˆ 1i  pˆ 2 j  L ;
pˆ 1 j  pˆ 2i 
1
2
H  2t  .
In equilibrium, each retailer sells to exactly ½ of convenience buyers. All value
buyers visit retailer i for product 1 and retailer j for product 2.
Hence, the ownership structure considered here yields the reversed pricing strategy in
equilibrium. In addition, the distribution of consumers is identical to that described in
Proposition 1. The only difference from independent ownership is that the common
owner can coordinate prices to extract the maximum possible surplus from consumers.
Note that in the competitive case, retailers competed over convenience consumers only.
Thus, joint ownership improves price coordination in one each of the retailer’s product.
S.4.2 After Entry
Entry by the discounter forces the owner to revise the optimal coordinated pricing
strategy. If retailers can coordinate prices, then the optimal pricing strategy must extract
all surplus from value buyers on product 1 since there is no competing seller. Hence, the
price for product 1 must be equal to L at one of the retailers. Because the convenience big
basket consumers are traveling to C, it is optimal for the owner to use A to serve these
consumers in product 1. This implies an equilibrium condition that retailer B has the
lowest price in this product:
p1B  L .
(S.10)
Also note that we assume that retailer A carries product 2 in equilibrium. In that case, it
must be that p2 A is no more than p 2C  s . Thus, in any equilibrium in which A serves in
product 2 must satisfy
13
p2 A  K  s .
(S.11)
To determine the remainder of the joint owner’s optimal pricing strategy, we derive the
owner’s profit:
 AB   (1   )( p1 A  p 2 A  2 K ) x S  ( p1B  p 2 B  2 K )(1  x S )
  ( p1 A  K ) x B  ( p1B  K )(1  x B )
where x B and x S are the locations of the marginal Big Basket and Small Basket
consumers, respectively, as given in (3) of the main text. The equilibrium pricing strategy
is therefore determined by solving
max  AB
p1 A , p2 A
p1 B , p 2 B
subject to (S.10) and (S.11).
The first order necessary conditions with respect to p1A and p2 B lead to
 1 ( p  p 2 B )  ( p1 A  p 2 A ) 
 2( p1B  p1 A ) 
(1   )  1B
   1 
0,
t
t
2



 1 ( p  p 2 A )  ( p1B  p 2 B ) 
(1   )  1 A
  0.
t
2

Solving the above pair of equations gives the optimal pricing strategy for the owner of
the two retailers.
Proposition S.6
Suppose retailers A and B have a single owner. Then the equilibrium pricing strategy
after entry by the discounter is
 1 1  

p1*A  L  t  
 2 8 
p2* A  K  s ;
p1*B  L
 1  
.
p 2* B  L  s  t 1 
8 

Market shares are characterized by
xS 
3
4
and
xB 

1
2

 18 .
Note that the optimal pricing strategy does not depend on the portion  of the
convenience consumers. The joint owner has a monopoly on the sale of product 1 and is
14
thus able to extract all surplus from the 1   value buyers on product 1 with p1*B  L .
The relevant maximization with regard to the remaining pricing decisions concerns only
the convenience consumers and is thus invariant to the constant  .
Comparing the two propositions, we see that as a consequence of the discounter’s
entry, the retail chain has similar incentives as independent retailers. Specifically, it
wants to raise A’s price on product 1 to exploit the gain in convenience consumers
generated by the demand externality of product 3 at the discounter. That is, consistent
with Proposition 2, we have p1*A  p1*B .
We now verify that the shift in consumer distribution due to the dominant
retailer’s entry is consistent with the sales change data (and the theory of the original
model). In particular, recall that under Proposition S.1, both retailers A and B split the
market in two, each receiving half of the total amount of consumers. Post entry, retailer A
sees sales in product 1 according to
After Entry Sales of
product 1 at Retailer A

x B   (1   ) x S   (5   ) / 8 ,
which is greater than ½ under the conditions of Proposition 2.1 Note that these sales are
all to the  -portion of convenience consumers. This means, as in the original model, that
the retail channel nearest to the discounter is enjoying more convenience consumers than
before entry. It is straightforward to show that under the conditions of Proposition 2, the
after-entry sales of product 2 at retailer A as well as both products at retailer B are all less
than ½. Hence, entry by the dominant retailer causes sales of these products to decrease.
Summarizing, when the traditional retailers have the same owner, the impact of
the dominant retailer’s entry is directionally consistent with the case of independently
owned retailers. The main distinction between the two cases is that with joint ownership,
the retail chain can coordinate prices and extract more consumer surplus.
1
Recall that    AE implies that   1 /(1   / 4) . Therefore,  (5   ) / 8  (5   ) / 8(1   / 4)  1 / 2 if
  ½ . On the other hand, if   ½ then 5    9 / 2  4 /  since    AE (  1 / 2)  0.921311 
 AE (  1 / 2) .
15
S.5 Entry with Single Incumbent
In the main text we examine how entry by the discount retailer alters the competitive
incentives between the two incumbent retailers. This is done in order to examine the
impact of entry as a function of its location relative to the incumbent competitors. As
shown there, being closer to the entering discounter has benefits relative to being further
away. In particular, the main text demonstrates that the entering discounter, when
carrying a partially overlapping assortment, can provide an agglomeration effect –
increased store traffic at the nearby store – and a segmentation effect – less price sensitive
demand, for the non-overlapping product. The question arises as to whether the presence
of the distant store is necessary for these effects. In this section we show that the answer
to this question is no. Specifically, we consider a single retailer A and compare its pricing
decisions before and after entry by a discount retailer, C, who co-locates with A. As we
demonstrate below, the same benefits arise even without the second incumbent.
All aspects of the model set up are as in the main text, except where noted. Note
that in the main model, retailer A benefit from gain in store traffic at the cost of B. Here,
in order to have a gain in traffic for A after entry, we assume that the market is not fully
covered in before entry. This requires the following assumption.
Assumption S.1 : 2H  s  p1  p2  t
Under Assumption S.1,the retailer before entry profits are:
2 ( p1  p2  2K )(2H  2K ) / t  (1   )( p1  p2  2K ) . Any pair of prices ( pˆ1, pˆ 2 ) that
satisfies pˆ 1  pˆ 2  12 (2H  s  2K )  2t (1   ) is an interior maximize of profits. This
implies incumbent profits, before entry, of

t 2 (1   ) 2 
ˆ A  (2 H  s  2 K )  2t
(2 H  s  2 K ) 
4t 

 2 
1 
t (1   ) 

(2 H  s  2 K ) 
.

2 
 
2
1
16
After entry, retailer A loses some sales of product 2. If C sets price p3C  K then A
cannot compete for any other buyers on product 2. However, the Small Basket
convenience buyers still may buy product 2 from A. In addition, Big Basket convenience
buyers buy product 1 from A if obtaining a positive utility from shopping. Note that they
pay K for the product 2 at C. Denote by v3 , the net surplus from the purchase of product
3 from C. In order to ensure an interior maximization of the A’s profit, we require that the
demand from this segment is downward sloping (partial market coverage). Thus, we
make the following assumption.
Assumption S.2 : v3  2H  2s  p1  K  t
Under these conditions, retailer A’s profit after entry can be expressed as follows:
A 
 (1   )

t

t
2 H  s  p1  p2  p1  p2  2 K 
v3  2 H  2s  p1  K ( p1  K )  (1   )( p1  K ).
Retailer A’s optimal prices are
v3  2 H  2 s t (1   )

2
2
2H  s  2K
p2* 
 p1* ,
2
p1* 
both of which are positive as long as 2K  v3  s . Furthermore, note that p1*  p2* under
the condition 2K  2(v3  s)  (2H  s)  2t (1   ) /  . (This condition holds simply as
long as the consumption value of products 1 and 2 exceed their costs.) These prices imply
an optimal after entry profit of
 *A 

 (1   )
4t
 
2VH
 s  2K  
2
1
2
(V3  2VH  2 s  2 K ) 2 
4t 

t (1   ) 
V3  2VH  2 s  2 K   


2t (1   )

17
(V3  2VH  2 s  2 K ) 
t 2 (1   ) 2 
.
 2  2 
Comparing prices p1* and p̂1 is impossible since there are multiple maximizers to the BE
profits. However, if we assume that A sets equal prices for products 1 and 2 before entry,
then
p1*  pˆ 1  V3  s 
t (1   )(1  1 )

,
which is positive for V3  s and   1 . This demonstrates that as long as it there is
benefit from purchasing product 3, net of the additional shopping costs, then A can raise
its price for its non-overlapping product.
Finally, we compare sales of product 1, before and after entry. The sales level of
product 1 before entry and after-entry is computed as follows:
qˆ1 A 
1
2t
(2H  s  2K )  (1   )1  21  ,
and the sales level of product 1 after entry is
q1*A 
1
2t


1
(v3  s  2H  s  2 K )  (1   ) 1  2
  (1   ) 21t (2 H  s  2K ) .
Thus, entry by C implies a sales boost in product 1 for v3  s and  close to 1.
By comparing before entry and after entry profit, we can show that under certain
conditions, retailer A will benefit from C’s entry. We simplify the comparison by
examining profits for  and  arbitrarily close to 1:
lim 
 
( , )  (1,1)
*
A

 ˆ A 
1
2 H  s  2 K 2  1 v3  2 H  2s  2 K 2 .
4t
4t
The expression above is positive so long as v3  s . Finally, note that under the condition
that v3  s , Assumption S.2 implies Assumption S.1. Therefore, the conditions for A to
benefit from C’s entry are consistent with the assumptions of partial market coverage.
The above argument establishes the following result.
Proposition S.7 Consider the model of the main text with retailer A, but not retailer
B. Under Assumption S.2, there exist   (0,1) and   (0,1) such that if
   ,    , and v3  s then
(i) entry increases the sale of product 1 at retailer A: q1*A  qˆ1 A
(ii) entry increases the profit of retailer A:  *A  ˆ A .
18
(iii) Furthermore, if A sets prices for the symmetric products 1 & 2 before entry, then
after entry, the p1*  p̂1 .
The proposition establishes that the basic insight of the model in the main text does not
require competition before entry. In particular, retailer A, a monopoly before entry, can
experience a boost in sales of its non-overlapping product and increase its price because
of entry by the discount retailer C.
This proposition also suggests that entry can increase profits of retailer A if the
agglomeration benefits are sufficiently high. This stands in contrast to the model of the
main text simply for the reasons that we have assumed here that the market must expand
because of entry and there is no competitive presence from B. Note that entry by C
creates added value to consumers in the sale of the additional product, 3. Part of this
value accrues to A in the case considered here. However, with a rival B, as considered in
the main text, this extra value is competed away, leaving both retailers worse off after
entry.
Figure S1: Location of Stores in Extended Model
19