Discussion Papers In Economics
And Business

Large-Scaled Chain Stores versus Small-Scaled Local Stores of
the Local Production for Local Consumption

Hiroaki Sandoh and Risa Suzuki

Discussion Paper 15-16-Rev.3

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANLarge-Scaled Chain Stores versus Small-Scaled Local Stores of
the Local Production for Local Consumption

Hiroaki Sandoh and Risa Suzuki

Discussion Paper 15-16-Rev.3

September 2015

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANLarge-Scaled Chain Stores versus Small-Scaled Local Stores of
the Local Production for Local Consumption∗
Hiroaki Sandoh†

Risa Suzuki‡

Abstract
In some localities, a large-scale chain retailer competes against a small-scale local independent
retailer that specializes in, for instance, vegetables, fruits, and flowers produced locally for
local consumption.
The former usually attracts consumers by emphasizing its width and depth of products
variety, whereas the latter seeks to overcome its limited products assortment by oﬀering
lower prices for them than the chain store. This is possible for the local store partly becauseof lower labor costs and for various other reasons.
This study employs the Hotelling unit interval to examine price competition in a duopoly
featuring one large-scale chain retailer and one local retailer. To express diﬀerences in theirproduct assortments, we assume that the large-scale retailer denoted by A sells two typesof product, G1 and G2 , whereas the local retailer denoted by B sells only G1 . Moreover, we
assume that all consumers purchase G1 at A or B after comparing prices and buy G2 at A .
We examine both Nash and Stackelberg equilibrium to indicate that the local retailer cansurvive competition with the large-scale chain retailer. We also reveal that a monopolistic
market structure, not duopoly, optimize the social welfare if consumers always purchase both
G1 and G2 .

KEY WORDS: Large-scale chain retailer, Small-scale local independent store, Duopoly, Hotelling,
Price competition
JEL Classification: D43, M21∗
This work was partially conducted while the second author was a graduate student in business and management at Osaka University.
†
Graduate School of Economics, Osaka University, E-mail:sandoh@econ.osaka-u.ac.jp‡
Yuki, Co., Ltd., E-mail: rs.k611@gmail.com

1

1.

Introduction

A large-scale chain retailer store customarily oﬀers a wide and deep product variety to attract
more consumers over a wider area. Product width is a term signifying the diﬀerent types of
products a retailer oﬀers; product depth refers to the variety of a product oﬀered. It is a tenet
in business economics that social welfare gains and revenue gains from product width and depth
must be balanced against the prospect of lower consumer prices and lessened product choice.
Lancaster (1990) has surveyed the issue of product variety from an economist’s point of view.
The term product variety is used in his study corresponds broadly to the number of “brands”
as that term appears in the marketing literature.
Kök, Fischer and Vaidyanathan (2006) have extensively reviewed literature on assortment
planning. Kök and Fischer(2007) have proposed a method by which retailers could optimize
product assortment and estimate consumer demand. Cachon and Kök (2007) have examinedthe determination of product assortment among multiple merchandise categories and basket
shopping consumers. Toporowski and Lademann (2014) have reviewed a literature that examines
assortment, price, and location in food retailing.
Some chain retailers set prices according to local markets (Dobson and Waterson 2005) tocompete against local independent retailers. Focusing on grocery retailers, Leszczyc, Shinha,
and Timmermans (2000) have estimated a dynamic hazard model to understand factors influencing timing of consumer purchases, store choices, and the competitive dynamics of retailcompetition. Swoboda, Berg, Schramm-Klein and Foscht (2013) have discussed the relative importance of retail brand equity and store accessibility for determining store loyalty in diﬀerent
local competitive contexts.
In contrast, local independent stores are usually small-scale, for example, discount supermarkets oﬀering everyday-low-pricing and minimal service. They oﬀer fewer product groups andvariants within groups, but they acquire neighborhood consumers who want to reduce theirdependence on automobile (Handy and Cliftono 2001) by oﬀering lower prices than competitors
on targeted groups of products (Kahn and McAlister 1997; Guptil and Wilkins 2002). Lal andRao (1997) have investigated the factors underlying the success of every-day-low-pricing. Belland Lattin (1998) have linked consumer preference for every-day-low-price retailers versus highas well as low-price retailers to the expected dollar cost of a household’s shopping basket.
In some areas of Japan, local independent retailers sell vegetables, fruits, flowers, and fish
produced locally for local consumption. They sell their limited assortments of product groups
at lower prices than that oﬀered by large-scale chain retailer and cater to customers for whomfreshness and prices are essential. They can do so because they buy directly at lower prices
from local growers, farms, and cooperatives and their labor cost are often below those of chain
retailers.
This study uses the Hotelling unit interval model to address price competition in a duopoly
featuring a large-scale chain retailer and a small-scale local independent retailer. We first discuss Nash equilibrium and then investigate Stackelberg equilibrium to indicate that the local
independent retailer can survive competition with the chain retailer. We examine social welfare
at Nash and Stackelberg equilibrium and also social optimality.

2

2.

Model

2.1

Notations and assumptions

Our notations and assumptions are as follows:(1) Homogeneous consumers are uniformly distributed along the Hotelling unit interval [0, 1](Hotelling 1929; Basicaia and Mota 2013).
(2) Large-scale retailer A is located at 0 and a small-scale local independent retailer B is at
1 on the horizontal unit interval [0, 1].
(3) To express the diﬀerences in the product breadth, we assume that A sells two types of
products(G1 and G2 ), whereas B sells only G1 .
(4) Consumers purchase only G1 with probability α and purchase G1 and G2 simultaneouslywith probability 1 − α.
(A)

(B)

(5) A sells G1 and G2 at prices p1 and p2 , respectively, and B sells G1 at price p1 . Since
we concentrate on price competition between A and B, p2 is assumed to be determinedby some suitable criterion
(A)

(B)

(6) The raw prices of G1 and G2 are a and b, respectively, where p1 , p1 ≥ a and p2 ≥ b.
Since we concentrate upon the price competition between A and B, p2 is assumed to be
determined by some suitable criterion.
(7) Consumer willingness to pay for G1 and G2 are u1 and u2 , respectively.
(8) Traveling cost for each consumer is c per unit of distance.
()(A) (B)(9) u1 − max p1 , p1
− 2c ≥ 0 and u2 − p2 − 2c ≥ 0.
(10) u1 − a ≥ 5c.
(A)

Note that assumption (9) provides an upper bound for p1()(A) (B)pmax:= u1 − 2c ≥ max p1 , p1
,
1

(B)

and p1

given byand assumption (10) is introduced to avoid mathematical intricacies.

2.2
2.2.1

Indexes
Boundaries

When consumers purchase only G1 , the boundary by which consumers are categorized delineates,
(A)

x̃1 =

1 p1
−
2

(B)

− p1
,
4c

(1)

where consumers with x ∈ [0, x̃1 ] travel to retailer A , and consumers having x ∈ (x̃1 , 1] visit B.
Note that Eq. (1) considers consumers’ round-trip travel costs from their home to the store.
3

If consumers purchase G1 and G2 , the boundary is given by(A)x̃2 = 1 −

p1

(B)

− p1
,
2c

(2)

where consumers with x ∈ [0, x̃2 ] travel to retailer A to bur both G1 and G2 , and consumers
with x ∈ (x̃2 , 1] visit B to obtain G1 and then A to purchase G2 .
Equations (1) and (2) engender Proposition 1;
Proposition 1. x̃2 = 2x̃1 .
Proof. Trivial. ■2.2.2

Expected profits

When consumers behave as observed above, expected profit to A is given by
(
)
(
)
[ (
)
]
(A) (B)(A)(A)
ΠA p1 , p1
= αx̃1 p1 − a + (1 − α) x̃2 p1 − a + (p2 − b)()(A)
= [αx̃1 + (1 − α)x̃2 ] p1 − a + (1 − α)(p2 − b),and B earns his expected profit as()()(A) (B)(B)
ΠB p1 , p1
= [α (1 − x̃1 ) + (1 − α) (1 − x̃2 )] p1 − a .

(3)

(4)

2.2.3

Best responses()(A) (B)(A)
The derivative of ΠA p1 , p1
with respect to p1 is()(A) (B)∂ΠA p1 , p1
(A)

=

(
)
(A)
(B)
(2 − α) a + 2c − 2p1 + p1
4c

∂p1

,

(
)
(A) (B)(B)
while that of ΠB p1 , p1
in reference to p1 is()(A) (B)∂ΠB p1 , p1
(B)
∂p1

By solving=)((A) (B)∂ΠA p1 ,p1
(A)
∂p1

α
+
2

(
)
(A)
(B)
(2 − α) a + p1 − 2p1
4c

.

(A)

= 0 with respect to p1 , we obtain the best response of A against

B which is written as
(B)

(A)

p1

=

p1

+a
+ c.
2

(5)

4

The best response of B against A , which is a solution to()(A) (B)∂ΠA p1 ,p1
(B)

∂p1

is
(A)
p1 + a
(B)
p1 =
+

2

(B)

= 0 with regard to p1 ,

cα
.
2−α

(6)

Equations (5) and (6) prompt Proposition 2.
(A)

Proposition 2. The best responses of A and B satisfy p1
when α = 1.

(B)

≥ p1 , where the equality holds

Proof. See Appendix A.
(B)
Note in Proposition 2 that the price p1 oﬀered by B as its best response against A does(A)not exceed p1 . Further, if α = 1, all consumers purchase only G1 , where the pricing problem
examined here agrees with the simple pricing problem confronting a single product in a duopoly.
2.2.4

Consumer surplus

Consumer surplus is an essential index for evaluating the equilibrium. Its intricate structure isgiven by()(A) (B)CS p1 , p1{∫ x̃1= α0(A)(u1 − p1 − 2cx)dx ++(1 − α)[∫ x̃2
0

∫ 1
x̃1

}
(B)
[u1 − p1 − 2c(1 − x)]dx(A)(u1 + u2 − p1 − p2 − 2cx)dx +

[
(
)]
(A)
(B)
= [αx̃1 + (1 − α)x̃2 ] 2c − p1 − p1

∫ 1
x̃2

]
(B)
(u1 + u2 − p1 − p2 − 2c)dx()(B)+ u1 − p1 − 2c + (1 − α) (u2 − p2 ) + cα(1 − 2x̃21 ) − c(1 − α)x̃22 .

3.

Equilibrium

3.1

Nash equilibrium(A)

Solving Eqs. (5) and (6) with respect to p1
equilibrium as follows:(B)

and p1(7)

simultaneously, we obtain the Nash

p1

(A)∗

= a+

2c
4c
+
(< pmax),1
3
3(2 − α)

(8)

(B)∗

= a−

2c
8c
+
(< pmax).
1
3
3(2 − α)

(9)

p1

5

Hence, the boundaries x̃1 and x̃2 in Eqs. (1) and (2) becomex̃∗1 =11+,
6 3(2 − α)x̃∗2 =

12+.
3 3(2 − α)Profits of A and B at the Nash equilibrium are
(

Π∗A

:= ΠA

(A)∗ (B)∗
p1 , p 1)(= (1 − α)(p2 − b) + c

2 α
−
3
9

)
+

4c
,
9(2 − α)() c(2 + α)2(A)∗ (B)∗Π∗B := ΠB p1 , p1=,9(2 − α)and the consumer surplus is
CS

∗

(
:= CS

(A)∗ (B)∗
p1 , p 1

)

]
22
17α
−
.
= u1 − a + (1 − α)(u2 − p2 ) − c 1 −189(2 − α)[The indexes derived above initiate Proposition 3.
Proposition 3.
(A)∗

(B)∗

(i) Both p1
and p1
with increasing α.

(
)
(A)∗
(B)∗
are increasing in α, whereas their diﬀerence p1 − p1
decreases(ii) Boundary x̃∗1 increases from 13 to 12 as α increases, whereas boundary x̃∗2 is increasing in
α from 64 to 1.
(iii) Π∗A is decreasing in α, but Π∗B is increasing in α.
Proof. Trivial. ■In Proposition 3–(ii), when all consumers purchase both G1 and G2 simultaneously, they visit
not B but A because of x̃2 = 1.

3.2

Stackelberg equilibrium

In the real circumstances, the large-scaled chain retailer can be considered a price leader andthe small-scaled local retailer a price follower in a Stackelberg game. Moreover, the relation(A)(B)
p1 ≥ p1 in Proposition 2 agrees with the underlying idea of a Stackelberg game, suggestingthat the Stackelberg equilibrium is more realistic than the Nash equilibrium. We derive theStackelberg equilibrium as follows.
Substituting Eq. (6) into Eqs. (1), (2) and (3) gives
(
)[
(
)
]
(A)
(A)
(
)
p1 − a (2 − α) p1 − 2c − a − 4c
(A)
.
(10)
ΠA p1
= (1 − α)(p2 − b) −
8c
(
)
(A)
(A)
By diﬀerentiating ΠA p1
with respect to p1 , we have
(
)
(
)
(A)
(A)
p
−
a
(2 − α)∂ΠA p1
1
α
−
.
=
1
−
(A)
4
4c
∂p
1

6

The solution to
(A)∗∗

p1

(
)
(A)
∂ΠA p1
(A)

∂p1

=a+c+

= 0 is2c(< pmax
),
1
2−α

(11)

which is the optimal price of G1 for A against the best response of B.
When A adopts the selling price in Eq. (11), the optimal price of G1 for B becomes
(B)∗∗

p1

=a−

c
3c
+
(< pmax
).
1
2 2−α

(12)

Hence, the boundaries x̃1 and x̃2 in Eqs. (1) and (2) become
x̃∗∗
1 =

1
1
+
,
8 4(2 − α)

x̃∗∗
2 =

1
1
+
.
4 2(2 − α)Profits to A and B are
(
)
(
)
3 α
c
(A)∗∗ (B)∗∗
Π∗∗
:=
Π
p
,
p
=
(1
−
α)(p
−
b)
+
c
−
+
,
2
A
A
1
1
4
8
2(2 − α)()
c(4 + α)2(A)∗∗ (B)∗∗
Π∗∗
:=
Π
p
,
p
=
.
B
B
1
1
16(2 − α)Finally, the consumer surplus at the Stackelberg equilibrium is
CS

∗∗

(
:= CS

(A)∗∗ (B)∗∗
p1
, p1

)

[

]
(42 − 31α)23
= u1 − a + (1 − α)(u2 − p2 ) − c−.
328(2 − α)The indexes in this subsection suggest Proposition 4.
Proposition 4.

(
)
(A)∗∗
(B)∗∗
(A)∗∗
(B)∗∗
(i) Both p1
and p1are increasing in α, whereas their diﬀerence p1− p1
decreases with increasing α.

1
3
1
3
∗∗
(ii) When α increases from 0 to 1, x̃∗∗1 increases from 4 to 8 , and x̃2 increases from 2 to 4 .
∗∗(iii) Π∗∗A decreases and ΠB increases with increasing α.

Proof. Trivial. ■3.3

Comparison

This subsection compares the Nash and Stackelberg equilibrium. Table 1 summarizes the indexes
derived above. Comparing the Nash equilibrium with the Stackelberg equilibrium presents
Proposition 5:
Proposition 5.
(A)∗∗(i) p1

(A)∗

> p1

(B)∗∗

and p1

(B)∗

> p1

,
7

∗
∗∗
∗
(ii) x̃∗∗1 < x̃1 and x̃2 < x̃2 ,∗(iii) Π∗∗A > ΠA > 0,∗(iv) Π∗∗B > ΠB > 0,(v) CS ∗∗ < CS ∗ .
(vi) (CS ∗ − CS ∗∗ ) decreases with increasing α.
Proof. Trivial. ■Proposition 5 reveals that the local independent retailer B can survive competition against
the large-scaled chain retailer A because it earns a profit both at the Nash and Stackelberg
equilibrium. In addition, A and B earn more at the Stackelberg equilibrium than at the Nashequilibrium by oﬀering higher prices, thereby reducing consumer surplus. However, the diﬀerencein consumer surplus (CS ∗ − CS ∗∗ ) between the Nash and Stackelberg equilibrium is decreasing
in α.

4.

Social WelfareThis section confines itself to social welfare to explore socially optimal prices of the product G1 .
Social welfare is given by
(
)
(
)
(
)
(
)
(A)
(B)
(A)
(B)
(A)
(B)
(A)
(B)
W p1 , p1= Π A p1 , p 1+ Π B p1 , p 1+ CS p1 , p1

= u1 − a + (1 − α)(u2 − b) −(2 − α)c
−
2

(
)
(A)
(B) 2
(2 − α) p1 − p1
8c

. (13)The social welfare at the Nash equilibrium is given by
W

∗

(
:= W

(A)∗
(B)∗
p1 , p1

)

]
[
2
13α
−
, (14)= (u1 − a) + (1 − α)(u2 − b) − c 1 −189(2 − α)and at the Stackelberg equilibrium, it becomes
(
)
(A)∗∗
(B)∗∗
W ∗∗ := W p1
, p1
[

=

]
19 35α1−−.
(u1 − a) + (1 − α)(u2 − b) − c16328(2 − α)Hence, we reach Proposition 6:√

Proposition 6. If (0 ≤)α ≤ 74−471716 = 0.69309, then we haveW ∗∗ ≤ W ∗ ,√while if 74−471716 < α(≤ 1), we haveW ∗∗ > W ∗ .
Proof. Trivial. ■
8

(15)

9

Consumer surplus

Profits

PricesBoundary

IndexesCS ∗ = u1 − a + (1 − α)(u2 − p2 )][22−
−c 1 − 17α189(2−α)

Π∗B = c(2+α)
9(2−α)

28c= a − 2c
3 + 3(2−α))(
4cΠ∗A = (1 − α)(p2 − b) + c 23 − α9 + 9(2−α)

(B)∗

p1

2c
= a + c + 2−αCS ∗∗ = u1 − a + (1 − α)(u2 − p2 )][23−
−c (42−31α)328(2−α)2c(4+α)
Π∗∗B = 16(2−α)

3c= a − 2c + 2−α
(3 α)
c
Π∗∗
A = (1 − α)(p2 − b) + c 4 − 8 + 2(2−α)

(B)∗∗

p1

(A)∗∗

p1

1
1
x̃∗∗
2 = 4 + 2(2−α)

2x̃∗2 = 13 + 3(2−α)
4c= a + 2c
3 + 3(2−α)

1
1
x̃∗∗
1 = 8 + 4(2−α)

1x̃∗1 = 16 + 3(2−α)

(A)∗

p1

Stackelberg

Nash

Table 1: Comparison

Proposition 6 signifies that the Nash equilibrium provides greater social welfare than the
Stackelberg equilibrium when the probability 1 − α of purchasing both G1 and G2 tends toward
1.
(A)

Now,
let us)maximize the social welfare with respect to p1
(
(A)
(B)
(A)
(B)
W p1 , p 1
with respect to p1 and p1 , we have()(A) (B)∂W p1 , p1

= −

(A)

∂p1
(
)
(A) (B)∂W p1 , p14c(2 − α)

=

(B)

(
)
(A)
(B)
(2 − α) p1 − p1
(

(A)
(B)
p1 − p1

By letting()(A) (B)∂W p1 ,p1
(A)

∂p
(1

maximizing W(A)∗∗∗

p1

=

(A)
(B)
p1 , p 1(B)∗∗∗= p1()(A) (B)∂W p1 ,p1
(B)

)

∂p1

,

(16)

)

4c

∂p1

(B)

and p1 . By diﬀerentiating

.

(17)

(
) (
)
(A)
(B)
(A)∗∗∗(B)∗∗∗= 0, the optimal prices p1 , p1
= p1
, p1

satisfies,(18)

along with()
(2 − α)c(A)∗∗∗(B)∗∗∗W ∗∗∗ := W p1
, p1
= (u1 − a) + (1 − α)(u2 − b) −.
2
Consequently, we have Proposition 7:
(A)

(B)

Proposition 7. If we have p1 = p1 to maximize social welfare, consumers buying both G1and G2 simultaneously visit only A . Particularly when α = 0, social welfare is maximized not(A)in a duopoly but in a monopoly dominated by A with p1 = a.
Proof. See Appendix B.
Proposition 7 indicates that when maximizing the social welfare, it becomes more diﬃcult
for B to enter the market as α increases. However, monopoly is generally considered undesirable for other economic reasons and consequently the maximization of social welfare would bemeaningless.
Table 2: Comparison of social welfare
Social welfare

Nash equilibrium
Stackelberg equilibrium
Socially optimum[]2
W ∗ = (u1 − a) + (1 − α)(u2 − b) − c 1 − 13α−189(2−α)
[
]
1
19
− 35α
−
W ∗∗ = (u1 − a) + (1 − α)(u2 − b) − c 16328(2−α)W ∗∗∗ = (u1 − a) + (1 − α)(u2 − b) − (2−α)c
2

10

5.

Conclusion

This study employed the Hotelling unit interval model to examine price competition betweena large-scale chain retailer A and a small-scale local independent retailer B. To represent thediﬀerence in product assortment between A and B, we assumed that A sells product G1 andG2 and that B deals in only product G1 . We also assumed homogeneous consumers purchaseG1 from A or B and buy G2 at A . Then we focused on price competition over G1 . Moreover,
we assumed that each individual consumer purchased G1 only with probability α and purchased
both G1 and G2 with probability 1 − α.
Nash and Stackelberg equilibrium were examined, and the main results in this study are as
follows:(1) The local retailer can earn profits both at the Nash and Stackelberg equilibrium andsurvive competition with a large-scale chain retailer.
(2) Both A and B earn more at the Stackelberg equilibrium than at the Nash equilibrium.
However, consumer surplus diminishes more at the Stackelberg equilibrium than at the
Nash equilibrium.
(A)

(B)

(3) Maximization of social welfare suggests p1 = p1 , where consumers buying both G1 andG2 visit only A . Particularly when α = 0, social welfare is maximized not in a duopoly
but in a monopoly dominated by A . Given the general disrepute in which monopoly isheld, however, the maximization of social welfare would be meaningless.

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AppendixA.

Proof of Proposition 3

From Eqs. (5) and (6), we have
(
(B)
(A)
p1 − p1
(A)
(B)
+c 1−
p1 − p1 =
2

α
2−α

)
,

which agrees with())
3 ( (A)α(B)p − p1
=c 1−
.
2 1
2−α
(A)

α
Since 1 − 2−α≥ 0, we have p1
(A)

p1

(B)

− p1

α
≥ 0. In addition, if α = 1, we have 1 − 2−α= 0, i.e.,(B)= p1 . ■12

B.

Proof of Proposition 7(A)∗∗∗(B)∗∗∗

When p1= p1, Eq. (2) reveals that x̃2 = 1. Accordingly all consumers would visit only
A to purchase both G1 and G2 simultaneously. This signifies social welfare is maximized in amonopoly dominated by A , especially when α = 0. ■13