doi:10.1111/j.1435-5957.2007.00141.x
Price and quantity competition yield the same
location equilibria in a circular market*
Chia-Ming Yu1
1
Department of Economics, Washington University in St. Louis, Campus Box 1208, St. Louis, MO
63130–4899, USA (e-mail: [email protected])
Received: 13 September 2005 / Accepted: 15 September 2006
Abstract. Gupta et al. (2004) shows that in a circular market under spatial
Cournot discrimination, the location equilibria can be categorised into several
patterns. This paper proves that, given the same number of firms producing
differentiated products, one location pattern constitutes an equilibrium in price
competition if and only if it constitutes an equilibrium in quantity competition.
That is, the location equilibrium is irrelevant to whether the firms compete with
each other on quantities or prices.
JEL classification: D43, L13, R12, R32
Key words: Cournot competition, price competition, spatial competition
agglomeration
1 Introduction
The discussions of location equilibrium initiate with a structure in Bertrand competition, such as Hotelling (1929), d’Aspremont et al. (1979) with a linear market,
and Salop (1979), Martinez-Giralt and Neven (1988), and Kats (1995) with a
circular market. Tabuchi and Thisse (1995), and Tsai and Lai (2005) with a triangle
market, and Dorta-González et al. (2005) with a network market under delivered
pricing. Anderson and Neven (1991) pioneer in analyzing location equilibrium in
a linear market with a spatial Cournot discrimination model. Gupta et al. (2004)
* I would like to thank three anonymous referees for their valuable comments. Also, I would like to
thank Chao-Cheng Mai, Chien-Fu Chou, Kong-Pin Chen, Shin-Kun Peng and Fu-Chuan Lai for their
advice and guidance. However, responsibility for any remaining error rests with the author.
© 2007 the author(s). Journal compilation © 2007 RSAI. Published by Blackwell Publishing, 9600 Garsington Road,
Oxford OX4 2DQ, UK and 350 Main Street, Malden MA 02148, USA.
Papers in Regional Science, Volume 86 Number 4 November 2007.
644
C.-M. Yu
show that the equilibrium location patterns in a circular market can be much more
enriched, so long as the spatial Cournot discrimination is considered. The advantages of a spatial Cournot model lie not only in successfully explaining the overlap
of duopolists’ market shares and the variety of location equilibria, but also in
offering a stable analytical model without giving firms the incentive of undercutting. However, there are several questions left in the literature. First, can this stable
structure be transferred into a model analyzing location equilibrium in price
competition? Second, what’s the difference of equilibrium location patterns, and
in effects of product differentiation on equilibria, in quantity and price competition?
This paper first extends the circular Cournot model to the case of differentiated
products, and then transfers the spatial model to a price competition structure to
examine whether product differentiation yields different effects on location patterns in quantity and price competition. It is proved that the equilibrium location
patterns are unrelated to the degree of product differentiation in a circular market,
and more surprisingly, a location pattern is an equilibrium in price competition if
and only if it is an equilibrium in quantity competition. That is, in a circular
market, the means of competition which firms choose does not play a crucial role
in determining location equilibria. Therefore, the assumption of quantity or price
competition plays no crucial role in explaining the equilibrium location patterns in
a circular market when firms’ products are not completely homogeneous.1
The rest of this paper is organised as follows. Section 2 shows that the
equilibrium location patterns are neutral to product differentiation under spatial
Cournot discrimination. Section 3 shows that in a spatial discrimination model
with price competition, the equilibrium location patterns are the same as the ones
under spatial Cournot competition. The conclusion is provided in section 4.
2 The first-order and the second-order conditions in quantity competition
Suppose there are n firms engaging in spatial Cournot competition, n ⱖ 2, where
consumers are uniformly distributed on a circular market with a perimeter normac
lised to 1. Denote qi and xic as the quantity and the location of firm i in Cournot
competition, respectively, i = 1, . . . , n. Hence, the strategy profiles (qic)in=1 and
( xic)in=1 represent the quantity and the location choices of n firms, respectively.
Following Dixit (1979), Anderson and Neven (1991) and Pal (1998), each firm’s
demand function on each point x in a circular market is set to be
n
pic ( x ) = α − β qic ( x ) − γ ∑q cj ( x ),
(1)
j ≠i
where 0 < g < b, x ∈ [0,1]. It is assumed that all firms have the same production
technology and zero production cost. The transportation cost of goods from firm i
1
When firms produce homogeneous good, this conclusion cannot stand, as shown in Matsumura
and Shimizu (2006).
Papers in Regional Science, Volume 86 Number 4 November 2007.
Price and quantity competition yield the same location equilibria in a circular market
645
to one point x is expressed as t ⋅ d ( x , xic ), i = 1, . . . , n, where d ( x , xic) represents
the shortest distance between xic and x in the circular market. To ensure that each
firm serves the whole market, a > t n/2 is assumed; furthermore, for simplicity t is
assumed to be 1. Given the locations and quantities of all firms, the profit function
of firm i in one market point x, where x ∈ [0,1], is expressed as
π ic ( x1c , x2c , . . . , xnc , x ) = ( pic ( x ) − d ( x , xic )) qic ( x ) i = 1, . . . , n.
(2)
Assuming that the arbitrage among consumers is infeasible, the competition
among firms on quantity is strategically independent across different market
points. Therefore, the quantity equilibrium is a composite of the equilibrium
quantities at all market points x ∈ [0,1]. It can be checked that the equilibrium
quantities and profits are
qic ( x1c , x2c , . . . , xnc , x ) =
1
Dc
⎧
c
⎨α (2β − γ ) − (2β + ( n − 2 ) γ ) d ( x , xi ) +
⎩
n
⎫
γ ∑d ( x , x cj )⎬,
⎭
i≠ j
π ic ( x1c , x2c , . . . , xnc , n ) = β qic ( x1c , x2c , . . . , xnc , x ) , i = 1, . . . , n,
2
where Dc = 4b2 - g2 + (n - 2)g(2b - g ) > 0 for all 0 < g < b. Since d ( x , xic) is continuous on x ∈ [0,1], i = 1, . . . , n, it is noticed that both qic and π ic are continuous
on x ∈ [0,1] for all i.
Given the other firms’ locations, each firm chooses its plant location to maximise total profit over the circular market, which is
∏ (x , x , . . . , x
c
i
c
1
c
2
1
c
n
) = ∫ π ic ( x1c , x2c , . . . , xnc , x ) dx,
(3)
0
s.t. xic ∈[0, 1], i = 1, . . . , n.
(4)
For any strategy profile sc ≡ ( xic)in=1 and any i = 1, . . . , n, denote s−c i to be
the vector of all firms’ locations except the location of firm i, i.e.
s−c i ≡ ( x1c , . . . , xic−1, xic+1, . . . , xnc ) . The following proposition presents the necessary
condition for optimisation in a subgame perfect Nash equilibrium (SPNE)
location.
Lemma 1. In a circular market in quantity competition, for all 0 < g < b, given
1
*
s−c i and xic ∈ ⎡0, ⎤ , the necessary condition of an optimal location for firm i is
⎢⎣ 2 ⎥⎦
equivalent to the aggregate-cost-median condition that
Papers in Regional Science, Volume 86 Number 4 November 2007.
646
C.-M. Yu
∫
xic +
1 n
2
xic n
n
∑d ( x, x ) dx = ∫ ∑d ( x, x ) dx + ∫ ∑d ( x, x ) dx.
xic
c
j
j ≠i
0
c
j
j ≠1
1
1
x cj +
2 j ≠1
c
j
(5)
Proof. See Appendix A.
That is, the necessary condition is independent of the degree of product
differentiation. Then, it is natural to examine whether the second-order condition
is also independent of the extent of differentiation.
Lemma 2. In a circular market in quantity competition, for all 0 < g < b, given
1
*
s−c i and suppose xic ∈ ⎡⎢0, ⎤⎥ , the sign of the second-order derivative of
⎣ 2⎦
Π ic( x1c, . . . , xnc) with respect to xic is unrelated to the value of g and b, i.e.,
∂ 2 ∏ i ( x1c , . . . , xnc )
c
∂ 2 xic
2
(
)
n
1 c
c
−
+
d
x
,
x
d ( xic , x cj ) < 0.
∑
∑
i
j
2
j ≠1
j ≠1
n
< 0 iff
(6)
Proof. See Appendix B.
Since the necessary and sufficient conditions of optimisation are both irrelevant to the value of b and g, the following proposition is implied by the previous
two lemmas.
Proposition 1. In a circular market in quantity competition of n firms with
substitutes, the location equilibrium is independent of the degree of product
differentiation.
The above proposition shows that the conclusions in Gupta et al. (2004) are so
robust that they are not altered by the consideration of product differentiation. On
the other hand, the proposition and lemmas in this section offer a benchmark
which can be compared with other variants.
Recall that in a linear market under spatial Cournot discrimination, as shown
in Anderson and Neven (1991) and Pal and Sarkar (2002), the location equilibrium
can be checked by the quantity-median condition: the optimal location for each
firm i is at the point such that the aggregate quantity it supplies to the left
half-circle of that point is equal to the aggregate quantity supplied by it to the right
half-circle of that point. Furthermore, as presented in Gupta et al. (2004), the
location equilibrium in a circular market under the same settings necessarily
satisfies the aggregate-cost-median condition: the optimal strategy for every firm
is to locate at the point such that the aggregate delivered marginal costs of all firms
occurring in the two half-circles divided by a diameter passing through that point
are the same. As noted in Gupta et al. (2004), the aggregate-cost-median condition
is consistent with the quantity-median condition; in fact, the former is a simplified
form of the latter. However, one characteristic of the aggregate-cost-median condition is quite conspicuous: it is independent of firms’ competition variable (quanPapers in Regional Science, Volume 86 Number 4 November 2007.
Price and quantity competition yield the same location equilibria in a circular market
647
tity). Therefore, an examination of whether the equilibria found in Gupta et al.
(2004) are still valid when firms compete with each other in price is naturally of
considerable interest.
3 The first-order and the second-order conditions in price competition
Suppose there are n firms engaging in spatial price competition, n ⱖ 2, where
consumers are uniformly distributed in a circular market with a perimeter normalised to 1.
b
Denote pi and xib as the price and the location of firm i, respectively,
i = 1, . . . , n. The strategy profiles ( pib)in=1 and ( xib)in=1 represent the price and the
location choices of n firms, respectively.
Following Dixit (1979) and Singh and Vives (1984), let Dn = b2 - g2 +
g(n - 2)(b - g ), each firm’s demand function at each point x can be set to
n
qib( x ) = a − bpib( x ) + c ∑ pbj ( x ),
(7)
j ≠1
where a = a(b - g )/Dn, b = (b + (n - 2)g )/Dn, and c = g/Dn. It is noticed that Dn > 0
is implied by 0 < g < b.2 Given the locations and prices of all firms, the profit
function of firm i, at one market point x, x ∈ [0,1], is expressed as
π ib ( x1b , x2b , . . . , xnb , x ) = ( pib ( x ) − d ( x , xib )) qib ( x ), i = 1, . . . , n.
(8)
It can be checked that the equilibrium prices and profits in price competition
are
pib ( x1b , x2b , . . . , xnb , x ) =
1
Db
⎧
b
⎨a (2b + c ) + b (2b − ( n − 2 ) c ) d ( x , xi ) +
⎩
n
⎫
bc ∑d ( x , x bj )⎬,
⎭
i≠ j
π ib( x1b, x2b, . . . , xnb, x ) = b { pib( x1b, x2b, . . . , xnb, x ) − d ( x , xib )} , i = 1, . . . , n,
2
2
For n > b/g + 3, the own-price effect dominates the cross-price effect. It is also noticed that b and
c are increasing and decreasing with n, respectively, which indicate that a large n and a dominant own
price effect come hand in hand.
Papers in Regional Science, Volume 86 Number 4 November 2007.
648
C.-M. Yu
where Db = 4b2 - c2 - c(n - 2)(2b + c). Furthermore, for all n ⱖ 2 and
2b 2 − c 2
0 < g < b, it can be proved that the inequality 2 +
> n is always valid.3
c (b + c )
2b 2 − c 2
The inequality 2 +
> n implies not only ∂qib ( x ) ∂d ( x, xib ) < 0 in equi
c (b + c )
librium, but also Db > 0.4
Given the other firms’ locations, each firm chooses its location to maximise
total profit over the circular market, which is
∏ (x , x , . . . , x
b
i
b
1
b
2
1
b
n
) = ∫π ic( x1b, x2b, . . . , xnb, x ) dx,
(9)
0
s.t. xib ∈[0,1], i = 1, . . . , n.
(10)
Lemma 3. In a circular market under price discrimination, given s−b*i and
1
xib ∈ ⎡0, ⎤ , the necessary condition of an optimal location for firm i is equivalent
⎢⎣ 2 ⎥⎦
to the aggregate-cost-median condition that
∫
xib +
xib
1 n
2
xib n
n
∑d ( x, x ) dx = ∫ ∑d ( x, x ) dx + ∫ ∑d ( x, x ) dx.
j ≠i
b
j
0
j ≠i
b
j
1
1
xib +
b
j
(11)
2 j ≠i
Proof. See Appendix C.
Proposition 2. In price competition with n firms, a profile of locations
*
*
*
*
sb = ( x1b , x2b , . . . , xnb ) constitutes a subgame perfect Nash equilibrium with n
*
firms if xib is at the aggregate cost median, for all i = 1, . . . , n.
3
First of all, it is noted that b and c are functions of the parameters b, g, and n. Second, it is shown
2b 2 − c 2
⎞
⎛
2
− n⎟ ∂ n = (β + ( n − 2 )γ ) (β + nγ ) / (β + ( n − 1)γ ) > 0 , for all n ⱖ 2, and morethat ∂ ⎜ 2 +
⎠
⎝
c (b + c )
2b 2 − c 2
− n = ( 2β 2 − γ 2 ) [ βγ (1 + γ )] > 0 for all 0 < g < b. Thus, it is proved that
over, at n = 2, 2 +
c (b + c )
2b 2 − c 2
2+
− n > 0 is valid for all n ⱖ 2 and 0 < g < b. Therefore, for all reasonable b and c which
c (b + c )
2b 2 − c 2
− n > 0 is always true. Thanks to
correspond to reasonable b, g and n ⱖ 2, the inequality 2 +
c (b + c )
one of the referees who asked me to clarify this relationship between parameters.
4
It is first noted that in equilibrium qib( x ) = b ( pib( x1b, x2b, . . . , xnb, x ) − d ( x, xib )). Thus, it can be shown
2b 2 − c 2
. On the other hand,
c (b + c )
2
2
2b
2b
bc
⎡ 2b − c
⎤
+ 2⎥ =
> 0, which completes
; moreover, ⎛⎜ 1 + ⎞⎟ − ⎢
Db > 0 if and only if n < 1 +
⎝
c
c ⎠ ⎣ c (b + c )
⎦ c (b + c )
the proof.
b
b
b
that ∂ qi ( x ) ∂ d ( x, xi ) = b [b (2b − ( n − 2 ) c ) − D ] < 0, for all n < 2 +
Papers in Regional Science, Volume 86 Number 4 November 2007.
Price and quantity competition yield the same location equilibria in a circular market
649
1
*
Lemma 4. In price competition with n firms, given s−b i and suppose xib ∈ ⎡0, ⎤,
⎣⎢ 2 ⎦⎥
the sign of the second-order derivative of Π ib( x1b, . . . , xnb ) with respect to xib is as
follows.
∂ 2 ∏ i ( x1b , . . . , xnb )
b
b2
∂ 2 xi
n
< 0 iff
∑d
j ≠i
(
)
n
1
xib + , x bj − ∑d ( xib , x bj ) < 0.
2
j ≠i
(12)
Proof. See Appendix D.
Similar to Proposition 1, since the aggregate-cost-median condition and the
second-order condition of optimisation are irrelative to the value of b and c, the
following proposition can be summarised.
Proposition 3. In price competition with n firms all producing substitutes, the
location equilibrium is independent of the degree of product differentiation.
Since the first-order and the second-order conditions of optimisation in price
competition are exactly the same as the ones in Cournot competition, though
unexpected, the following proposition is true in a circular market.
Proposition 4. (Competition measures neutral to location equilibrium) In a circular market with discrimination, the location equilibria in price competition are
the same as those in quantity competition. In other words, no matter whether firms
compete with each other on quantity or price, the same location equilibria are
yielded.
The above proposition says that all the location equilibria in price competition,
given the same number of firms, are the same as the equilibria in quantity competition; they are comprehensively presented and categorised in Gupta et al. (2004).
Numerical examples are also shown in previous lectures. In both quantity and price
competition, similar to Kats (1995), who pioneers in analyzing location equilibrium
in a circular market with homogeneous products, ‘equal distance’location pattern is
one of the multiple equilibria. In fact, it is the unique location pattern which can
constitute an SPNE for all number of firms, which presents an evidence for
supporting assumption when a general case of n firms is considered. It can also be
seen that some location equilibrium shows unusual characteristics: for example,
when the number of firms equals 5, one of the equilibrium location patterns is
exactly the same as a combination of a location equilibrium with 2 firms and a
location equilibrium with 3 firms.5
5
This characteristic is called ‘Nash combination’ in Yu and Lai (2003a).
Papers in Regional Science, Volume 86 Number 4 November 2007.
650
C.-M. Yu
4 Conclusion
This paper proves that, in a circular market with discrimination, the first-order and
the second-order conditions for firms’ optimal locations are the same in both
quantity and price competition. That is, though the competition in each point is
tighter in price competition than that in quantity competition, given the same
number of firms, the location equilibria in price competition are the same as the
ones in quantity competition. Therefore, a circular market can be a neutral platform where the patterns of location equilibria do not depend on the assumption of
price or quantity competition.
The elegant results of this paper come under the assumptions of linear and
sufficiently large demand, identical cross price effect, unit transportation cost, and
zero production cost. One natural extension of this paper is to examine whether the
equivalence claimed in this paper is still valid when these assumptions are relaxed.
Another extension of this paper is to consider the feasibility of multiple planes.
Chamorro-Rivas (2000) and Yu and Lai (2003b) show that when each duopolist has
two plants, four plants disperse equally when firms produce substitutes. In recent
literature, Pal and Sarkar (2006) show that when there are n multi-plant firms with
homogeneous products, each firm maximises its profit only when every plant is at
the plant’s quantity median. When an equal number of plants are owned by each of
an even number of firms, there is a continuum of subgame perfect Nash equilibrium
(SPNE) locations; when there are more than two firms with different numbers of
plants, the equidistant location pattern may or may not be sustained as an SPNE
outcome. Therefore, it is interesting to further examine whether the equivalence
result of this paper still holds if each firm has multiple plants.
Appendix A. Proof of Proposition 1
1
Without loss of generality, suppose xic ∈ ⎡0, ⎤ , the aggregate profit of firm i can
⎢⎣ 2 ⎥⎦
be expressed as
∏
xc +
xc
1
i
i
( x1c , . . . , xnc ) = ∫0 π ic ( x1c , . . . , xnc , x ) dx + ∫xc 2π ic ( x1c , . . . , xnc , x ) dx +
i
c
i
∫
1
xic +
π ( x , . . . , x , x ) dx .
1
2
c
i
c
1
c
n
Since d ( x , xic) equals xic − x, x − xic and 1 − x + xic, for all x ∈[0, xic] ,
1
1
xic, xic + ⎤ , and xic + , 1⎤ , respectively, it can be shown that
2 ⎦⎥
2 ⎥⎦
(
(
∂π ic( x1c, . . . , xnc, x )
∂q c ( x c , . . . , x c , x )
= 2β qic( x1c, . . . , xnc, n ) i 1 c n ,
c
∂xi
∂xi
Papers in Regional Science, Volume 86 Number 4 November 2007.
(13)
Price and quantity competition yield the same location equilibria in a circular market
651
where
∂qic ( x1c , . . . , xnc , x )
2β + ( n − 2 ) γ ∂d ( x , xic )
=
−
Dc
∂xic
∂xic
(
⎧− 2β + ( n − 2 ) γ , ∀x ∈[0, x c ] ∪ x c + 1 , 1⎤,
i
i
⎪⎪
Dc
2 ⎦⎥
=⎨
1
⎪ 2β + ( n − 2 ) γ ,
∀x ∈ xic , xic + ⎤.
⎪⎩
Dc
2 ⎦⎥
(
(14)
The first-order derivative of Π ic( x1c, . . . , xnc) with respect to xi is
∂∏ i ( x1c , . . . , xnc )
c
∂xic
=∫
xic
0
+∫
xic + ∂π c ( x c , . . . , x c , x )
∂π ic ( x1c , . . . , xnc , x )
dx
+
∫xic 2 i 1 ∂xic n dx
∂xic
1
1
1
xic +
2
∂π ic ( x1c , . . . , xnc , x )
dx
∂xic
2β + ( n − 2 ) γ
= − 2β
Dc
−∫
xic +
xic
{∫
xic
0
(15)
q ( x , . . . , x , x ) dx
c
i
1
2 c
i
c
1
q ( x1c , . . . , xnc , x ) dx + ∫
c
n
1
xic +
⎫
q ( x1c , . . . , xnc , x ) dx ⎬.
⎭
c
1 i
2
Hence, the first-order condition of optimisation for firm i is equivalent to
∫
xic +
xic
1
2 c
i
xic
q ( x1c , . . . , xnc , x ) dx = ∫ qic ( x1c , . . . , xnc , x ) dx + ∫
0
1
xic +
q ( x1c , . . . , xnc , x ) dx .
c
1 i
2
(16)
Equation (16) shows that the optimal location for firm i coincides with its
1
quantity-median. Furthermore, it is always true for all xic ∈ ⎡0, ⎤ in a circular
⎢⎣ 2 ⎥⎦
market that
∫
xic +
1
2
xic
xic
α (2β − γ ) − (2β + ( n − 2 )γ ) d ( x , xic ) dx = ∫ α (2β − γ ) −
0
1
(2β + ( n − 2 )γ ) d ( x , xic ) dx + ∫ c 1α (2β − γ ) −
xi +
2
(2β + ( n − 2 )γ ) d ( x , x ) dx .
c
i
Therefore, the quantity-median condition can be further simplified as
∫
xic +
xic
1 n
2
∑d ( x, xic ) dx = ∫
j ≠1
xic n
0
∑d ( x, xic ) dx + ∫
j ≠i
1
xic +
n
∑d ( x, x
1
2 j ≠i
c
i
) dx,
(17)
which is the same as the aggregate-cost-median condition in Gupta et al. (2004).
Papers in Regional Science, Volume 86 Number 4 November 2007.
652
C.-M. Yu
Appendix B. Proof of Proposition 2
From equation (15), the sign of the second-order derivative of firm i’s profit
function with respect to xic is the same as the sign of
Φc ≡ −∫
xic
∂qic ( x1c , . . . , xnc , x )
∂xic
0
−∫
∂qic ( x1c , . . . , xnc , x )
1
1
xic +
{
dx + ∫
∂xic
2
xic +
1
2
∂qic ( x1c , . . . , xnc , x )
∂xic
xic
dx
dx −
(18)
(
2 qic ( x1c , . . . , xnc , xic ) − qic x1c , . . . , xnc , xic +
)}
1
,
2
where
∂qic
∂d ( x , xic )
1
= − c (2β + ( n − 2 ) γ )
.
c
∂xi
D
∂xic
(19)
Therefore, it can be proved that
((
Φ c ≡ 2 qic x1c , . . . , xnc , xic +
)
)
1
2β − ( n − 2 ) γ
− qic ( x1c , . . . , xnc , xic ) +
.
Dc
2
(20)
Finally, substituting equilibrium quantities into Fc yields
(
)
n
n
1
Φ c = ∑d xic + , x cj − ∑d ( xic, x cj ).
2
j ≠i
j ≠i
This finishes the proof.
Appendix C. Proof of Lemma 3
The first-order derivative of aggregate profit with respect to xib is
∂∏ i ( x1b , . . . , xnb )
b
∂x
b
i
=∫
xib
0
∫
∫
∂π ib ( x1b , . . . , xnb , x )
dx +
∂xib
xib +
1
2
xib
1
1
xib +
2
It can first be checked that
Papers in Regional Science, Volume 86 Number 4 November 2007.
∂π ib ( x1b , . . . , xnb , x )
dx +
∂xib
∂π ib ( x1b , . . . , xnb , x )
dx .
∂xib
(21)
Price and quantity competition yield the same location equilibria in a circular market
653
∂π ib( x1b, . . . , xnb, x )
= 2b ( pib( x1b, . . . , xnb, x ) − d ( x , xib ))⋅
∂xib
⎧ ( pib ( x1b, . . . , xnb, x ) − d ( x , xib )) ⎫
⎨∂
⎬,
∂xib
⎩
⎭
where
∂
( pib( x1b, . . . , xnb, x ) − d ( x, xib )) 2b2 − c 2 − c (b + c ) (n − 2 ) ∂d ( x, xib )
=
∂xib
∂xib
Db
.
(22)
(
1
In detail, ∂d ( x , xib ) ∂xib = 1 for x ∈[0, xi ] ∪ xi + ,1⎤ , and ∂d ( x , xib ) ∂xib = −1
2 ⎦⎥
1⎤
2b 2 − c 2 − c (b + c ) ( n − 2 )
for
; moreover,
for all
x ∈ xi, xi +
>0
b
2 ⎦⎥
D
2b 2 − c 2
n<
+ 2 . Substituting ∂ d ( x, xib ) ∂ xib into the first-order derivative yields
2 (b + c )
(
∂∏ i ( x1b , . . . , xnb )
b
∂xib
=∫
xib + ∂π b ( x b , . . . , x b , x )
∂π i ( x1b , . . . , xnb , x )
dx
+
∫xib 2 i 1 ∂xib n dx +
∂xib
1
xib
0
∫
∂π ib ( x1b , . . . , xnb , x )
dx
∂xib
1
1
xib +
2
= 2b
2b 2 − c 2 − c ( n − 2 ) (b + c )
⋅
Db
{∫
−
xib
0
xib +
∫
xib
∫
xib +
1
1
2
pib ( x1b , . . . , xnb , x ) − d ( x , xib ) dx +
pib ( x1b , . . . , xnb , x ) − d ( x , xib ) dx −
⎫
p ( x1b , . . . , xnb , x ) − d ( x , xib ) dx ⎬ .
⎭
b
1 i
2
Therefore, the first-order condition of optimisation for firm i is equivalent to
∫
xib +
xib
1
2
xib
pib( x1b, . . . , xnb, x ) − d ( x , xib ) dx = ∫ pib( x1b, . . . , xnb, x ) −
0
1
d ( x , xib ) dx + ∫ b 1 pib( x1b, . . . , xnb, x ) − d ( x , xib ) dx ,
xi +
(23)
2
which can be further simplified to
Papers in Regional Science, Volume 86 Number 4 November 2007.
654
C.-M. Yu
∫
xib +
xib
1
2
xib
pib ( x1b , . . . , xnb , x ) dx = ∫ pib ( x1b , . . . , xnb , x ) dx +
0
∫
1
xib +
b
b
b
1 pi ( x1 , . . . , xn , x ) dx ,
(24)
2
The above equation shows that the optimal location for firm i coincides with its
‘delivered-price-median’. Furthermore, the delivered-price-median condition can
be further simplified to
∫
xib +
1 n
2
xib
xib n
n
∑d ( x, x ) dx = ∫ ∑ ( x, x ) dx + ∫ ∑d ( x, x ) dx.
b
j
0
j ≠i
1
b
j
b
j
1
xib +
j ≠i
(25)
2 j ≠i
The above equation is the same as the aggregate-cost-median (of other firms)
condition presented in the previous section (Lemma 1).
Appendix D. Proof of Lemma 4
From (23), the first-order derivative of firm i’s profit function with respect to xib
is equivalent to
xib +
−∫ b
1
2
xib
pib ( x1b, . . . , xnb, x ) − ( xib − x ) dx + ∫ pib( x1b, . . . , xnb, x ) −
0
xi
1
( x − xib ) dx − ∫xb + 1 pib( x1b, . . . , xnb, x ) − (1 − x + xib ) dx.
i
(26)
2
Hence, the sign of the second-order derivative of firm i’s profit function with
respect to xib is the same as the sign of
Φb ≡ −∫
xib
0
∫
1
1
xib +
2
xib + ∂p b ( x b , . . . , x b , x )
∂pib ( x1b , . . . , xnb , x )
dx
+
∫xib 2 i 1 ∂xib n dx −
∂xib
1
{
∂pib ( x1b , . . . , xnb , x )
dx − 2 pib ( x1b , . . . , xnb , xib ) −
∂xib
(
{(
pib x1b , . . . , xnb , xib +
1
2
)}
(27)
)
}
1
b (2b − ( n − 2 ) c )
− pib ( x1b , . . . , xnb , xib ) −
2
Db
n
⎞
1
2bc ⎛ n
= b ⎜ ∑d xib + , x bj − ∑d ( xib , x bj )⎟ .
2
D ⎝ j ≠i
⎠
j ≠i
= 2 pib x1b , . . . , xnb , xib +
(
)
Therefore, it is proved that,
Papers in Regional Science, Volume 86 Number 4 November 2007.
Price and quantity competition yield the same location equilibria in a circular market
∂ 2 ∏ i ( x1b, . . . , xnb )
b
b2
∂ 2 xi
∑d ( x
n
< 0 iff
b
i
j ≠i
)
n
1
+ , x bj − ∑d ( xib, x bj ) < 0.
2
j ≠i
655
(28)
It is interesting to note that, the second-order condition indicates that firms in this
model tend to on average avoid competition with other firms.6
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1
First of all, it is noted that xib is at the aggregate cost median if and only if xib + is at the
2
1
aggregate cost median. Given two candidate points ( xib and xib + ) which both satisfy the first-order
2
condition, firm i’s optimal choice is to locate at a point on average farther away from other firms’
locations.
6
Papers in Regional Science, Volume 86 Number 4 November 2007.