Equilibrium existence in the linear model: Concave
versus convex transportation costs*
Hamid Hamoudi1, María J. Moral2
1
2
Universidad Europea de Madrid, C/ Tajo s/n, Villaviciosa de Odón, Madrid, Spain
(e-mail: [email protected])
Universidad de Vigo, Dpto. Economía Aplicada, Facultad de CC. Empresariales, As Lagoas s/n,
Ourense 32004, Spain (e-mail: [email protected])
Received: 19 November 2003 / Accepted: 3 December 2004
Abstract. We focus on the general linear-quadratic transportation costs in the
linear model. Earlier results have shown that no pure-strategy price equilibrium
exists for whatever firm locations in this context. Since there is no price equilibrium for the whole market, our first objective is to calculate the feasible equilibrium region with concave costs. A crucial change of variables allows us to
explicitly calculate the necessary conditions to obtain the feasible equilibrium
regions. Finally, we compare the equilibrium regions with both the concave and
the convex cases and find that the feasible region with convex costs is bigger than
that with concave costs.
JEL classification: C72, D43
Key words: Hotelling’s model, concave transport costs, equilibrium prices
1 Introduction
After Hotelling’s seminal model (1929) many studies about spatial differentiation
have appeared in economic literature. The model presents a duopoly where firms
compete for a location in the first stage and for prices in the second. The equilibrium concept used in this situation is the sequential price then location equilibrium. Hotelling claimed that competition in differentiated products results in
* Financial support from project OTRI 2004/UEM06 as well as from the Spanish Ministry
of Science and Technology and from FEDER through grant BEC2002-02527 are gratefully
acknowledged.
© RSAI 2005. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street,
Malden MA 02148, USA.
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H. Hamoudi, M.J. Moral
minimal differentiation. D’Aspremont et al. (1979) showed that the minimum
differentiation equilibrium does not exist, which is due to a key calculation in
Hotelling’s model being incorrect. They proposed a slight variation on transportation costs by choosing a quadratic function (instead of a linear function) and
thus obtained a maximum differentiation equilibrium. The article led to a great
number of works that considered alternative variations on the original assumptions of the model (number of firms, demand function, consumer distribution on
the market, transportation costs), and even changed the equilibrium concept by
introducing mixed strategies or Stackelberg’s equilibrium.
Our interest focuses on concave transportation costs in the linear model. There
is a consensus in the economic literature as to convexity on transportation costs
being crucial to determining both the equilibrium existence and the product degree
of differentiation, whatever the type of market, be it linear or circular. In the
circular model, one example with convex transportation costs is the work of
Anderson (1986). In the linear model Economides (1986) analyses the cost function da (with 1 £ a £ 2, and d the distance) and concludes that for a certain
range of a, there exists a price equilibrium that generates maximum differentiation. Neven (1986) explores the spatial competition with n firms. Both Gabszewicz and Thisse (1986) and Anderson (1988) specify the transportation costs
as: c(d) = ad + bd2 with b > 0 and d the distance. Gabszewicz and Thisse were
the first to find that in values of a and b no (pure-strategy) price equilibrium exists
for symmetric fixed locations of firms. Anderson (1988) later extended that result
to any firm location, and showed that a pure-strategy perfect equilibrium in the
two-stage locations-price game exists only under very stringent conditions, which
are reduced to the product function being concave in price for any given location
pair.
Concave transportation costs have nevertheless remained almost unstudied. As
Anderson et al. (1992) affirm,1 Rochet was the first scholar to study such functions. To our knowledge, however, no results have been published in this respect
as yet. We can find real life situations that correspond to concave transportation
costs and hence justify interest in this particular type of transportation cost. Intuitively, concavity of transportation costs captures the idea that the marginal
transportation costs decrease in respect to the distance, and so gives us a more realistic model to work with. We find an example of this kind in the price of a flight:
as the distance increases the price increases but less so. Contrary to the convex
case, which implicitly includes time in the valuation of transportation costs, we
consider that distance is the only relevant variable in the calculation of costs, given
that the evaluation of time is different for each consumer. Consequently, it is interesting to study whether, and how, a firm’s competition is modified by the introduction of this assumption, as compared with the case of convex transportation
costs.
Recent studies have focused on analysing concave transportation costs in the
circular model. The circularity in the space of locations implies that the firms’strategic choices have an equivalent effect on both sides of the spatial market. De Frutos
1
See note number 24 on p. 172.
Equilibrium existence in the linear model
203
et al. (1999) have shown that with the cost function c(d ) = d - d 2, a Nash price
equilibrium exists. Moreover, they have found that with a variable change, the shift
from the concave to the convex case is permitted. Therefore, all previous results in
the literature of convex transportation costs can be applied to the concave case. But
as these authors have demonstrated in a recent article (De Frutos et al. 2002) the
result of equivalence is only useful for the circular model, since it cannot be applied
to the linear model due to the asymmetry in consumer distribution.
In this study we restrict ourselves to linear quadratic transportation cost function c(d) = ad + bd 2 in the linear model by considering both the concave case (b
< 0) and the convex case (b > 0). We shall then seek a perfect equilibrium in the
two-stage game in which two firms first simultaneously select positions over a
line segment and then, by having observed the positions selected, simultaneously
choose prices. We focus in particular on the study of the second stage. A priori,
one might wonder, how concave costs would modify competition between firms?
We infer that competition in prices should increase in the presence of concave
costs compared with the convex case. Specifically, for the type of linear-quadratic
function we have chosen, for a given distance the transportation costs are lower
in the case of concave costs. This makes location less relevant than prices, therefore increasing competition in the latter.
From the previous results in literature in this research field, it is clear that no
pure-strategy price equilibrium exists for any firm locations with linear-quadratic
costs in the linear model. Since there is no price equilibrium for the whole market,
our first objective is to calculate the feasible equilibrium region for a general
linear-quadratic cost function c(d) = ad + bd 2 with b < 0. In order to accomplish
this, it is crucial to perform a change of variables that explicitly allows us to
calculate, in a simpler way, the necessary conditions needed to obtain the
equilibrium regions.
The final aim of this article consists of comparing the equilibrium regions with
both the concave and convex cases. We show that the equilibrium region with
convex costs is bigger than that with concave costs. Indeed, the equilibrium region
associated with concave costs is entirely included in the equilibrium regions
corresponding to convex costs. Consequently, a Nash price equilibrium would fit
better with convex transportation costs rather than with concave ones in the linear
model; in that case we would have more stability in price with the convex cost
than with the concave cost.
The analysis is structured such that in Sect. 2 we describe the model and
proceed in Sect. 3 to study the feasible regions for firm locations where a pure
price equilibrium exists. In Sect. 4 we analyse the results obtained from concave
transportation costs versus those from convex costs and, finally in Sect. 5, we
present our main conclusions.
2 The model
We consider a different version of the standard spatial paradigm of Hotelling’s
model in which we maintain the assumption of a two-stage game (in the first
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H. Hamoudi, M.J. Moral
stage firms simultaneously decide their locations and next simultaneously decide
prices), but which is then modified in the formulation of transportation costs.
There are two firms selling an homogeneous product whose production costs are
equal to zero. Firms are located in the positions x and y along the linear city with
a length l, and we assume that 0 £ x £ y £ l. Prices p1 and p2 are chosen by firm
1 and firm 2, respectively, given their locations. Consumers are uniformly distributed along the market, and each one of them purchases just one unit of the
industry’s product at the firm at which the delivered price (total price resultant
from the addition of list price and transportation cost) is the lowest. Let x̃ denote
the consumer location in the linear market. The distance between the consumer
and the seller i is defined by: di = x˜ - xi , " xi = x, y.
The variation of our model with respect to Hotelling’s classic model of
horizontal differentiation consists of considering concave transportation costs as
follows:
c(di ) = adi - bdi2
" i = 1, 2.
(1)
where di is the distance between a consumer and the seller i, and parameters a
and b are non-negative.2 The first consequence of the concavity of transportation
costs on the linear model consists of the market length being restricted by the
maximum distance at which cost function is increasing. Therefore, the length l
has to verify the restriction: l £ a/2b. Without loss of generality, hereafter we will
consider l = a / 2b for concave costs.
To derive the demand of each firm we compute the indifferent consumers.
A consumer is indifferent to buying from one firm or another, if and only if the
delivered price (namely also full price) is the same for both cases:
p1 + c(d1 ) = p2 + c(d2 )
(2)
With a simple calculation from Eq. (2), we find three types of indifferent consumers that we denote by m1, m and m2 associated with the interval [0, x], [x, y]
and [y, l]. Depending upon the value of ( p1 - p2) one, two or indeed no indifferent consumers can exist. Appendix I. illustrates in detail how the indifferent consumers are calculated, as are the five regions in which the demand is defined. In
particular, these regions establish the parts in which the demands for both firms
have to be computed. The implication is that the demand is a partitioned function
in the following way:
D1 = l,
D 2 = 0,
if
nd
D1 = m + (1 - m2 ),
D 2 = m2 - m,
if
rd
D1 = m,
D 2 = l - m,
if
th
4 case:
D1 = m - m1 ,
D 2 = m1 + (l - m),
if
5 th case:
D1 = 0,
D 2 = 1,
if
1st case:
2 case:
3 case:
( p1 ( p1 ( p1 ( p1 ( p1 -
p2 ) ΠI1 ,
p2 ) ΠI 2 ,
p2 ) ΠI 3 ,
(3)
p2 ) ΠI4
p2 ) ΠI 5 ,
where each interval Ij " j = 1, . . . , 5 is defined in Appendix I.
2
The assumption a > 0 and b = 0 corresponds with Hotelling’s classic model with linear
transportation costs.
Equilibrium existence in the linear model
205
Given that one of our objectives here is to compare the results which come
from concave and convex transportation costs in the linear model, in Fig. 1 we
present the different cases in which demand has to be partitioned both for concave
costs (Panel a) and convex costs (Panel b). In respect to the concave case, it is
noteworthy that in the second case the demand of firm 1 is not connected, that is,
firm 1 has separated markets. Intuitively, that expresses the idea that although the
two extreme consumers buy at firm 1, firm 2 continues having a positive demand
(similarly, this is found in the case 4 for firm 2). However, it is important to realise
that, in the convex case, this situation never happens.
From expression (3) we calculate the demands for both firms merely by
substituting m1, m and m2 by their values (see Eq. (A1) in Appendix I) and
l = a/2b. Thus the demand function for firm 1 can be written as:
Ïa
Ô 2b
Ô
a( p2 - p1 )
Ô
(
Ô 2 b y - x )(a - b(y - x ))
Ô
Ô
Ô
p2 - p1
x+y
+
D1 = Ì
2(a - b(y - x ))
2
Ô
Ô
Ô
a( p2 - p1 )
a
Ô
+
Ô 2 b(y - x )(a - b(y - x )) 2 b
Ô
Ô
Ó0
if
p1 £ p2 - (y - x )(a - b(y - x ))
if
p2 - (y - x )(a - b(y - x ))
£ p1 £ p2 - b(y 2 - x 2 )
if
p2 - b(y 2 - x 2 ) £ p1
(4)
£ p2 + (y - x )(a - b(x + y))
if
p2 + (y - x )(a - b(x + y))
£ p1 £ p2 + (y - x )(a - b(y - x ))
if
p2 + (y - x )(a - b(y - x )) £ p1
Obviously the demand function for firm 2 is calculated as D2 = l - D1 =
(a/2b) - D1. We see that the demand functions are piecewise linear with five
different price domains. Figure 2 shows the demand for firm 1, given the price
of its rival ( p2 ) , the firms’ locations (x, y) and parameters (a, b). Moreover,
the demand function for each firm is continuous on its own price, and that it
depends on both firms’ location within the market.
Conversely, when convex transportation costs are considered (that is,
c(di ) = adi + bdi2 with a, b > 0) just one or no indifferent consumers exist in the
linear case3 as we can see in Fig. 1. With convex transportation costs there are no
restrictions on the length of the market because the cost function is always increasing. For the sake of comparison we also use l = a/2b in the convex case. The
demand function associated with convex costs is more familiar in the literature
and with our notation is written as:
3
In the circular model, however, there are two or no indifferent consumers both with concave costs
and convex costs (De Frutos et al. 1999).
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H. Hamoudi, M.J. Moral
Panel a: Concave costs
Panel b: Convex costs
Case 1
Case 1
x
y
y
x
Case 2
Case 2
my
x
m2
y
x
Case 3
x
Case 3
m
y
x
m
y
Case 4
Case 4
x
m1
m2
m
Case 5
y
x
m1
y
Case 5
x
y
x
FIRM 1
y
FIRM 2
FIRM 1
FIRM 2
Fig. 1. Price and transportation costs with respect to each firm with concave and convex costs
Note: m1, m, m2, are indifferent consumers associated with the intervals (0, x), (x, y) and (y, l),
respectively.
Equilibrium existence in the linear model
207
D(p1, p2 )
l
I1
I2
I3
0
u1
I4
u2
I5
u3 u4
Fig. 2. Demand for firm 1, given the price of firm 2*
*Note: Intervals Ij for j = 1, . . . , 5 are defined in Appendix I. The slopes in each interval are equal
to:
Ï0
Ô
-a
Ô
Ô 2b( y - x )( a - b( y - x ))
1
∂ D1 ( p1 , p2 ) ÔÔ
=Ì
∂ p1
Ô 2( a - b( y - x ))
Ô
-a
Ô
Ô 2b( y - x )( a - b( y - x ))
ÔÓ0
Ïa
Ô 2b
Ô
Ô ( p2 - p1 ) + x + y - a
2
2b
Ô 2 b( y - x )
Ô
Ô
Ô
p2 - p1
x+y
+
D1 = Ì
2(a + b(y - x ))
2
Ô
Ô
Ô p -p
x+y a
1
Ô 2
+
+
2
2b
Ô 2 b( y - x )
Ô
Ô
Ó0
if
p1 ΠI 1
if
p1 ΠI 2
if
p1 ΠI 3
if
p1 ΠI 4
if
p1 ΠI 5
if
p1 £ p2 + (x - y)(2 a - b(x + y))
if
p2 + (x - y)(2 a - b(x + y)) £ p1 £ p2
+ (x - y)(a + b(y - x ))
if
p2 + (x - y)(a + b(y - x )) £ p1 £ p2
+ (y - x )(a + b(y - x ))
if
p2 + (y - x )(a + b(y - x )) £ p1 £ p2
+ (y - x )(a + b(x + y))
if
p2 + (y - x )(a + b(x + y)) £ p1
(5)
From this demand function the profit function is piecewise concave for given
firm locations (Anderson 1988). Notice that the five critical intervals are different
208
H. Hamoudi, M.J. Moral
to the defined ones in Eq. (4) for the demand associated with concave costs.
Moreover, as we have commented before, in the linear model the number of indifferent consumers is not the same in concave as with convex costs. Indeed, all of
this justifies the non-equivalence between both cases, such as occurs in the circular model.
Next, given that the demand function is piecewise linear with five different
price domains following the usual analysis of Hotelling’s model, we shall confine
our study to the price equilibrium in the second stage.
3 The region for the existence of a Nash-equilibrium price with
concave costs
In this section we investigate the price equilibrium in the last-stage of the game.
Locations are supposedly already given, so firms decide only upon their prices.
Assuming without loss of generality that production costs are zero, we denote the
profit function by Bi (p1, p2) = pi · Di(p1, p2) " i = 1, 2.
From this profit function, a Nash price equilibrium is defined as a pair:
( p1N (x, y), p2N (x, y)) , such that:
B1 (x , y, p1N (x , y), p2N (x , y)) > B1 (x , y, p1 , p2N (x , y));
for all p1 other than p1N
B2 (x , y, p (x , y), p (x , y)) > B2 (x , y, p (x , y), p2 ); for all p2 other than p2N
N
1
N
2
N
1
(6)
The pure-strategy price equilibrium can be interpreted as a pair of expectations.
Neither one of the two companies wants to change its price strategy, even when
one reveals its decision on prices. Consequently, a pair ( p1N , p2N ) is a Nash price
equilibrium for a given location (x, y) when p1N maximises B1 ( p1 , p2N ) on R+
and p2N maximises B2 ( p1N , p2 ) on R+.
However, the previous results in literature have shown that with concave costs
in the linear model, a pure strategy price equilibrium does not exist for all potential pairs of firm locations (De Frutos et al. 2002). As Gabszewicz and Thisse have
shown, “the non-existence of a price equilibrium is not necessarily related to the
existence of discontinuities in demand; rather it is the non-quasi-concavity of the
profit functions which may pose problems” (Gabszewicz and Thisse 1986, p. 30).
To better understand this aspect, let us examine the profit function. A simple
calculation where the demand function is substituted into the profit function leads
to a profit function defined by pieces, with exactly the same intervals as the
demand in Eq. (4). Logically, we would expect that the profit function of firm 1
can have at most three local maxima ( p1* , p1**, p1***) : The first lying on I2, the
second on I3 and the third one on I4. Consequently, depending on the values of
the parameters a, b and the locations (x, y), three cases may arise if firm 2 chooses
p2. The global profit maximum of firm1 can therefore be reached in I2, I3 or I4.
More precisely, we will show that this is the reason why a Nash equilibrium price
does not exist for all possible pairs of firm locations. Figure 3 shows some configurations of firm1’s profits B1(p1, p2). In Fig. 3.a, the global profit maximum of
Equilibrium existence in the linear model
209
Panel a
Bi
P1*
P1**
P1***
P1
Panel b
Bi
P 1*
P1**
P1***
P1
Panel c
Bi
P 1*
P1**
P1***
Fig. 3. Some configurations of profit functions
P1
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H. Hamoudi, M.J. Moral
0.04
0.03
B1
0.02
I3
I2
0.01
I4
I1
I5
0
0.04
0.12
0.32
0.36
Fig. 4. Profit function of firm 1*
*Note: The values for firm locations and for price of firm 2 are x = 0.1, y = 0.3 and p2 = 0.2,
respectively.
firm1 occurs in I2; in Fig. 3.b, the global profit maximum of firm1 occurs in I3; in
Fig. 3.c, the global profit maximum of firm1 occurs in I4.
Since there is no price equilibrium for the whole market, our objective is to
calculate the feasible equilibrium region. This is the set of pairs of firm locations
for which a Nash price equilibrium does exist in the linear model and for a general
concave cost function.
Due to the complexity of the demand function, in order to obtain a representation of the feasible region, we need to reduce the number of parameters. The
first step consists of assuming that the linear term of the transportation cost function is equal to the quadratic term, that is, a = b. The concave transportation cost
function becomes:4
c(di ) = a(di - di2 )
" i = 1, 2.
(7)
Despite this assumption and in order to define the equilibrium in prices, we
need a second step that requires the introduction of a variable change in the following way: z = y - x and q = x + y. The change of variables is crucial in calculus for identifying the Nash equilibrium prices, as the new variables allow us to
draw the feasible regions in an easier way and reduce the dimension of the numerical problem. The transportation cost function defined above implies that the
maximum length of the city is 0.5. Therefore, these new variables are defined on
the intervals: z Π(0, 0.5] and q Π(0, 1]. Hereafter, to facilitate the exposition we
will use these new variables (z, q) rather than the firm locations (x, y).
Now we can obtain the regions associated with a given value for firm locations (x and y) in which a Nash price equilibrium exists. Firstly, we calculate the
necessary and sufficient conditions for the maximisation of profits in respect to
4
With this transportation cost function a price equilibrium exists in the circular model.
Equilibrium existence in the linear model
211
z
A
0.5
È 2-q ˘
z≥Í
˙
Î 2(1 + q ) ˚
2
B
È 1+ q ˘
z≥Í
˙
Î 2( 2 - q ) ˚
2
0.5
0
1
q
Fig. 5. Feasible region for a Nash equilibrium with concave transportation costs
prices. Next, from the solutions of these conditions, we deduce the Nash-price
equilibria regions.
Proposition 1. Under the concave cost function defined in expression (6), for z
and q such as z Π(0, 0.5], q Π(0, 1] and z < q there is a Nash-price equilibrium
if and only if,
z ≥ [(2 - q) (2 + 2 q)]
2
z ≥ [(1 + q) (4 - 2 q)]
2
when q £ 0.5, and
(8)
when q ≥ 0.5,
(9)
and whenever it exists, a pure price equilibrium is uniquely determined by:
p1N = a(1 - z)(1 + q) 3
and
p2N = a(1 - z)(2 - q) 3
(10)
The demonstration of this proposition is in Appendix II.
Given the Nash equilibrium prices, the profits for each firm with concave costs
are:
2
2
B1 ( p1N , p2N ) = a(1 - z)(1 + q) 18 and B2 ( p1N , p2N ) = a(1 - z)(2 - q) 18 (11)
Several comments arise from this proposition. Firstly, the Nash equilibrium
prices are included in the interval I3 (see the demand function). Indeed, we
should not expect price equilibrium elsewhere since that would imply that one of
the two firms has an incentive to cut prices in order to capture the entire market.
Figure 5 shows the feasible region for firm locations in which a price Nash
equilibrium exists with concave transportation costs. A careful analysis of this
feasible region reveals that it is possible to obtain a maximum differentiation like
a Nash equilibrium. This situation corresponds with point A where z = q = 0.5,
that is, x = 0 and y = 0.5. Conversely, the feasible minimum differentiation will
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H. Hamoudi, M.J. Moral
correspond with the smallest feasible value for z. This is achieved with z = 0.25
(point B) that matched with the firm locations: x = 1/8 and y = 3/8.
Secondly, we observe that the Nash equilibrium prices ( p1N , p2N ) are different each other ( p1N π p2N ) except when q = 0.5. The result is contrary to the circular model where prices are always equal to each other (De Frutos et al. 1999).
In fact, this characteristic is specific to the linear market with quadratic transportation costs because there is no symmetry in the model.
Thirdly, we see that p1N < p2N if and only if q < 0.5, and this just occurs when
firm 1 is located closer to the inferior market boundaries x1 Æ 0 because z > 0.25.
It is precisely this proximity of firm 1 to the inferior boundary of the market that
generates an asymmetry in costs which gives greater market power to firm 2, and
so firm 2 will sell at a higher price (the opposite situation exists when q > 0.5).
Finally, recall that a perfect price-location equilibrium is defined as a par
( p1N , x N ), (P2N , y N ) such that:
(i) P1N = P1N (x N , y N ) and P2N = P2N (x N , y N ),
(ii) B1 (x N , y N , p1N (x N , y N ), p2N (x N , y N )) ≥ B1 (x , y N , p1N (x N , y N ), p2N (x N , y N ))
for "x Œ[0,1/2],
(iii) B2 (x N , y N , p1N (x N , y N ), p2N (x N , y N )) ≥ B2 (x N , y, p1N (x N , y N ), p2N (x N , y N ))
for "y Œ[0,1/2].
This concept is meaningful only if, for any location choice by firms, there is
one and only one corresponding price equilibrium, otherwise, either payoff would
be undefined or multivalued (Gabszewiccz and Thisse 1986, pp. 42). However,
in our case if conditions (8) and (9) are violated, we know that no price equilibrium can exist for any pair of location (x, y). Thus, it is clear that no location
equilibrium can exist in the sequential game. Consequently, the only thing that
we can be said is that in the equilibrium region the firms tend to a minimum
differentiation. Indeed, define (P1N (x , y), P2N (x , y)) as a price equilibrium (if it
exists), corresponding to a pair of location (x, y) profits for each firm are in Eq.
(11). We can easily check that ∂ B1 ( p1N (x , y), p2N (x , y)) ∂ x is strictly positive, and
∂ B2 ( p1N (x , y), p2N (x , y)) ∂ y is strictly negative, which implies a tendency of both
firms towards the pair location (x = 0.125, y = 0.375), which represents their
optimum.
4 Comparing results from concave versus convex costs in the linear model
In this section we compare the results obtained in a linear market regarding two
types of transportation cost functions: concave and convex. We first present the
results associated with a convex transportation cost function in order to facilitate
the comparison with the concave case.
As far as convex transportation costs are concerned, Gabszewicz and Thisse
(1986) and Anderson (1989) have provided the most important results in the
analysis of convex transportation costs. Gabszewicz and Thisse first discovered
Equilibrium existence in the linear model
213
that in values of a and b no (pure-strategy) price equilibrium exists for symmetric
fixed locations of firms. Anderson (1988) later extended their result to any
firm location and showed that a pure-strategy perfect equilibrium in the two-stage
locations-price game only exists under very stringent conditions that are reduced
to the demand being concave in price for any given location pair.
Proposition 2. Under the convex cost function defined as c(di ) = a(di + di2 ) , for
z and q such as z Π(0, 0.5], q Π(0, 1] and z < q there is a Nash-price equilibrium if and only if,
z ≥ [ -4 q 2 + q - 4 + (2 + 2 q) 7 - 7q + 4 q 2 ] (3 + 12 q) when q £ 0.5, and (12)
z ≥ [ -4 q 2 + 7q - 7 - 2(q - 2) 4 q 2 - q + 4 ] (15 - 12 q) when
q ≥ 0.5,
(13)
and whenever it exists, a pure price equilibrium is uniquely determined by:
p˜ 1N = a(1 + z)(1 + q) 3 and
p˜ 2N = a(1 + z)(2 - q) 3
(14)
The demonstration of this proposition is in Appendix III.
Given the Nash equilibrium prices, the profits for each firm with convex costs
are:
2
B1 ( p˜ 1N , p˜ 2N ) = a(1 + z)(1 + q) 18 and
2
B2 ( p˜ 1N , p˜ 2N ) = a(1 + z)(2 - q) 18
(15)
Comments arising from proposition 2 are as follows. Firstly, as with concave
costs, the optimal prices are different between firms. Consequently, we can conclude that this is a characteristic for the linear model, since optimal prices are
identical for both firms in the circular model. Secondly, with convex costs it is
possible to obtain a lesser degree of differentiation (distance between firms) since
the minimum value for z is 0.2071 (see point M in Fig. 6), in contrast with the
concave case that is 0.25. In fact, the firm locations in this feasible minimum
differentiation are: x = 0.14645 and y = 0.35355. Finally, with respect to the
existence of market power for a firm that allows the fixing of higher prices, the
behaviour is similar to that obtained for concave costs.
Figure 6 presents the feasible regions associated with convex and concave
transportation costs. We can clearly see that the feasible region is more restrictive
for concave costs (ACBD) than for convex costs (ALMN). Indeed, the feasible
region for convex costs completely contains the region for concave costs. The
variation range for q is limited by [0.292, 0.7081] in the convex case versus
[0.3618, 0.6382] in the concave case.
We can conclude that there is a higher probability of finding a Nash equilibrium in prices with convex rather than concave transportation costs. Indeed, the
equilibrium region in the convex case is larger than in the concave costs for both
firms and for any location. In the concave case the incremental transportation cost
is decreasing in distance, which means that for consumers located far from the
two firms, price is the only relevant variable. The implication is that competition
in prices increases in the presence of concave costs and firms therefore have to
differentiate sufficiently to avoid the Bertrand solution.
214
H. Hamoudi, M.J. Moral
z
A
0.5
C
D
L
N
B
M
0
0.5
1
q
Fig. 6. Feasible regions for the Nash equilibrium with concave and convex transportation costs
5 Conclusion
We first conclude from this study the difficulty in the search for the feasible regions
in the linear model when linear-quadratic transportation costs are considered
(concave or convex). Difficulty arises due to the very high degree of polynomials
needing to be solved. According to our knowledge, the calculus of the feasible
regions had not been examined until now. For this reason our contribution to this
literature consists of a change of variables that is crucial in order to reduce the
dimension of the problem. In fact, the use of this change of variable allows us to
calculate the feasible regions of both concave and convex transportation costs. In
particular, inside each equilibrium region we obtain the best locations for firms.
Finally, we conclude that the price behaviour is rather similar in both transportation costs (concave and convex), although it is clear that the feasible region
of equilibrium with convex costs is larger than that of equilibrium with concave
costs.
Appendix I. Indifferent consumers and critical intervals for demand
From Eq. (2) we compute the indifferent consumers. We find three types of
indifferent consumers denoted by m1, m and m2 and which are associated with
intervals [0, x], [x, y] and [y, l], respectively. The indifferent consumers are characterised by the following expressions:
Equilibrium existence in the linear model
p1 - p2
x+y a
+
+
2 b( y - x )
2
2b
m1 =
m=
215
p1 - p2
x+y
+
2(a + b(y - x ))
2
m2 =
[A.1]
p1 - p2
x+y a
+
2 b( y - x )
2
2b
Each one has to verify, respectively, the following conditions:
m1 Œ[0, x] ¤
m Œ[ x , y]
¤
m2 Œ[ y, 1]
¤
(y - x )(a - b(x + y)) £ p1 - p2 £ (y - x )(a - b(y - x ))
-(y - x )(a - b(y - x )) £ p1 - p2 £ (y - x )(a - b(y - x ))
-(y - x )(a - b(y - x )) £ p1 - p2 £ - b(y - x )(y + x )
[A.2]
Notice that we can order these critical values as:
-• £ - (y - x )(a - b(y - x )) £ - b(y 2 - x 2 ) £ (y - x )(a - b(y + x ))
£ (y - x )(a - b(y - x )) £ •
As consequence, it is possible to define five intervals that will be critical in the
calculation of the demand and are denoted by:
I1 = [ -•, -(y - x )(a - b(y - x ))]
I2 = [ -(y - x )(a - b(y - x )), - b(y 2 - x 2 )]
I3 = [ - b(y 2 - x 2 ), (y - x )(a - b(x + y))]
[A.3]
I4 = [(y - x )(a - b(x + y)), (y - x )(a - b(y - x ))]
I5 = [(y - x )(a - b(y - x )), •]
Taking into account both these critical intervals and depending on the
value of (p1 - p2) one, two or no indifferent consumers could exist, we can
compute the demand of each firm by distinguishing five cases that correspond
with:
D1 = l,
D2 = 0,
if
nd
D1 = m + (1 - m2 ),
D2 = m2 - m,
if
rd
D1 = m,
D2 = l - m,
if
th
D1 = m - m1 ,
D2 = m1 + (l - m),
if
th
D1 = 0,
D2 = 1,
if
1st case:
2 case:
3 case:
4 case:
5 case:
( p1 ( p1 ( p1 ( p1 ( p1 -
p2 ) ΠI1 ,
p2 ) ΠI 2 ,
p2 ) ΠI 3 ,
p2 ) ΠI4
p2 ) ΠI 5 ,
[A.4]
Therefore the demand of each firm is partitioned.
Appendix II. Demonstration of Proposition 1
As expressed in Eq. (6), the pair of equilibrium prices ( p1N , p2N ) must be a
global maximum for every one of the firms: p1N = Arg Max B1 ( p1 , p2N ) , and
p1
216
H. Hamoudi, M.J. Moral
p2N = Arg Max B2 ( p1N , p2 ) . Given location (x, y) and price p2 of firm 2, the profit
p2
function of firm 1 is as follows:
Ï p1
Ô2
Ô
Ô p1 ( p2 - p1 )
Ô 2 z(1 - z)
ÔÔ
p2 - p1 q ˆ
B1 = Ì p1 Ê
+
Ë 2(1 - z) 2 ¯
Ô
Ô Ê p2 - p1 1 ˆ
Ô p1 Ë 2 z(1 - z) + 2 ¯
Ô
Ô0
ÔÓ
p1 - p2 £ - z(1 - z)
- z(1 - z) £ p1 - p2 £ - zq
- zq £ p1 - p2 £ z(1 - q)
z(1 - q) £ p1 - p2 £ z(1 - z)
z(1 - z) £ p1 - p2
Recall that z = y - x and q = x + y. We see that B1(p1, p2) can
have three local maxima ( p1* , p1**, p1***). Therefore:
p*1 = Arg Max B1 ( p1 , p2 ),
p1 ŒI2
p**
1 = Arg Max B1 ( p1 , p2 ),
p1 ŒI3
p***
= Arg Max B1 ( p1 , p2 ).
1
p1 ŒI4
In this way the global maximum can be reached in the region I2, I3, or I4, so it is
precise to evaluate the three possibilities.
Case 1. Price equilibrium in region I2
Given p*2 as the local maximum of firm 2 in the region I2, p1* can not be
the global maximum of the firm 1 since some simple calculation shows that
B1 ( p1* , p2* ) £ B1 ( p1** , p2* ) . Consequently, no price equilibrium belongs to I2.
Case 2. Price equilibrium in region I4
An argument similar to the above applies here, thus it does not exist a price
equilibrium in region I4.
Case 3. Price equilibrium in region I3
We consider that
p1N = p**
= Arg Max B1 ( p1 , p**
1
2 )
p1 ŒI3
and
p2N = p2** = Arg Max B1 ( p1** , p2 )
p1 ŒI3
are candidates for the price equilibrium. Solving simultaneously the first-order
conditions, we obtain (11). Then ( p1N , p2N ) must satisfy the following:
Equilibrium existence in the linear model
217
(i) p1N - p2N ΠI3
(ii) B1 ( p1N , p2N ) ≥ B1 ( p1 , p2N ), for all p1 ≥ 0
(iii) B2 ( p1N , p2N ) ≥ B2 ( p1N , p2 ), for all p2 ≥ 0
Condition (i).
This condition is equivalent to (1 - z)/(z + 2) £ q £ (2z + 1)/(z + 2).
Condition (ii).
Given p2N , we now study if p1N is the best reaction of firm 1. To assess
this we must consider all the local maxima obtained for every interval in Eq.
[A.3] due to the fact that profit function is defined at the same intervals.
In the first interval I1 the local maximum occurs when p1 = p2N - z(1 - z).
In the interval I2 a local maximum exists if and only if [(2 - q)/6] £ z £
[(2 - q)/(5q + 2)], and it is given by p1* = a (1 - z)(2 - q) 6 . Finally, in the
interval I4 the local maximum is p1*** = p2N + z(1 - q). In summary, for firm
1 we have to check that p1N verifies the following expressions, given p2N , z
and q.
B1 ( p1N , p2N ) ≥ B1 ( p, p2N )
(A.5.a)
B1 ( p1N , p2N ) ≥ B1 ( p1* , p2N )
(A.5.b)
B1 ( p1N , p2N ) ≥ B1 ( p1***, p2N )
(A.5.c)
The expression (A.5.c) always occurs because profits continuously
decrease on all values in the corresponding interval (Fig. 4). In relation to the
other two conditions, they are true if and only if firm locations comply respectively, with:
z ≥ (5 - 5q - 5q 2 ) 9
z ≥ [(2 - q) 2(1 + q)]
2
(A.6.a)
(A.6.b)
From expression (A.5) and together with the feasible regions defined
by each local maximum, we see that the expression (A.6.b) is the most
restrictive when q £ 0.5, while the expression (A.6.a) is the most restrictive
when q > 0.5.
Condition (iii).
Carrying out a similar analysis for firm 2 we find that the maximisation
problem is symmetrical with respect to q = 0.5. Let p2N , given that p1N , z and
q, maximises B2 ( p1N , p2 ) . These are compared with all local maxima defined
as p2 , p2* and p2*** for each interval. In a similar way for firm 1, p2N
is a global maximum if and only if z ≥ (-q2 + 7q - 1)/9 when q £ 0.5, and
z ≥ [(1 + q)/(4 - 2q)]2 when q > 0.5. In consequence, given that ( p1N , p2N )
is the vector of Nash equilibrium prices, p1N must be the best reply to p2N
given locations, and vice-versa.
218
H. Hamoudi, M.J. Moral
Finally, we select the common region for both firms; it is represented as:
z ≥ [(2 - q)/(2 + 2q)]2 when q £ 0.5, and z ≥ [(1 + q)/(4 - 2q)]2 when q > 0.5.
Q.E.D.
Appendix III. Demonstration of Proposition 2
Let us follow the same structure used in Appendix II, taking into account that the
critical intervals are now defined according to the demand function represented
in Eq. (4). The Nash equilibrium prices in expression (14) belong to the third
interval, then we calculate that its feasible region is given by z ≥ (1 - 2q)/3 when
q £ 0.5, and z ≥ (2q - 1)/3 when q > 0.5. Afterwards, we have to calculate local
maxima in the other intervals for firm 1 given values p̃2N , z and q.
In the first interval the local maximum is equal to p1 = (2 - q)(1 - 2 z) 3.
In the second interval, there exists a local maximum equal to
p˜ 1* = (2 - q + z(2 q - 1)) 6, if and only if firm locations verify (2 - q)
(7 - 2 q) £ z £ [ -1 + 4q + 16q 2 - 16q + 49 ] 12 . Finally, the local maximum in
interval I4 is defined by p˜ 1*** = (2 - q + z(2 q + 5)) 6. Then p̃1N will be the global
equilibrium for firm 1 if and only if it verifies the following expressions:
B1 ( p˜ 1N , p˜ 2N ) ≥ B1 ( p1 , p˜ 2N )
(A.7.a)
B1 ( p˜ 1N , p˜ 2N ) ≥ B1 ( p˜1*, p˜ 2N )
(A.7.b)
B1 ( p˜ , p˜ ) ≥ B1 ( p˜ 1*** , p˜ )
(A.7.c)
N
1
N
2
N
2
The last condition always occurs because the profit function is continually
decreasing in that interval. The other conditions are true if and only if locations
comply respectively with:
5 - 5q - q 2
q 2 - 4 q + 13
(A.8.a)
-4 q 2 + q - 4 + 2(1 + q) 4 q 2 - 7q + 7
3(1 + 4 q)
(A.8.b)
z≥
z≥
From these conditions and together with the feasible regions for each local
maximum, we have proved that expression (A.8.b) is the most restrictive but only
when q £ 0.5.
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