Download equilibrium existence in the linear model: concave versus convex

EQUILIBRIUM EXISTENCE IN THE LINEAR MODEL:
CONCAVE VERSUS CONVEX TRANSPORTATION COSTS.
Hamid Hamoudi
and
Universidad Europea de Madrid
María J. Moral
Universidad de Vigo
November, 2003.
Abstract
In this work we focus on the general quadratic transportation cost function in the linear
model considering both the concave and the convex case. From earlier results in the literature it
is clear that no pure-strategy price equilibrium exists for whatever firm locations with linearquadratic costs in the linear model. Therefore, our first objective is to calculate the feasible
equilibrium region for a general linear-quadratic cost function, since there is no price
equilibrium for the whole market. In order to do this, it is crucial to perform a change of
variables that explicitly allows us to calculate, in an easier way, the necessary conditions to
obtain the equilibrium regions. Finally, we compare the feasible regions equilibrium with both
the concave and the convex cases and we find that the feasible region with convex costs is
bigger than with concave costs.
Keywords: Hotelling’s model, concave transport costs, equilibrium prices.
JEL Classification: C72, D43.
*
Universidad Europea de Madrid, C/ Tajo s/n, Villaviciosa de Odón, Madrid, Spain. Tlf.: +34-912115645, e-mail: hamid.hamoudi@fae.eco.uem.es.
**
Dpto. Economía Aplicada, Facultad de CC. Empresariales, As Lagoas s/n, Ourense 32004. Tlf.: +34988-368760. Fax: +34-988-368923, e-mail: mjmoral@uvigo.es.
1.
Introduction
After Hotelling’s seminal model (1929) many studies about spatial
differentiation have appeared in economic literature. This model presents a duopoly
where two firms compete for a location in the first stage, and for prices in the second.
Hotelling claimed that competition in differentiated products results in minimal
differentiation. D’Aspremont, Gabszewicz and Thisse (1979) showed that the minimum
differentiation equilibrium does not exist, due to a key calculation in Hotelling’s model
being incorrect. They proposed a slight variation on transportation costs choosing a
quadratic function and they obtained maximum differentiation equilibrium. This article
promoted 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 quadratic costs in the linear model. In economic
literature there is consensus as to convexity on transportation costs being crucial for
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 d (with 1    2 , and d the distance),
and concludes that for a certain range of  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. While Gabszewicz and Thisse were
the first to find values of a and b so that 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.
Nevertheless, concave transportation costs have remained almost unstudied. As
Anderson, De Palma and Thisse (1992) affirm1, 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 justifying the interest in this particular type of transportation costs. For example,
there is evidence that flight fares increase in distance but with a decreasing variation
rate.
1
See note number 24 in pp. 172.
1
Recently, some studies have focused on analysing concave transportation costs
in the circular model. De Frutos, Hamoudi and Xarque (1999) have shown that with the
cost function c(d)=d-d2, a Nash price equilibrium exists. Moreover, they have found
that, with a variable change, the pass 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 this 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, as these authors have demonstrated in a recent article (De Frutos, Hamoudi
and Xarque, 2002).
In this work we focus on the quadratic transportation cost function c(d)=ad+bd2
in the linear model considering both the concave case (b<0) and the convex one (b>0).
From the previous results in the literature it is clear that no pure-strategy price
equilibrium exists for any firm locations with linear-quadratic costs in the linear model.
Therefore, our first objective is to calculate the feasible equilibrium region for a general
linear-quadratic cost function c(d)=ad+bd2 with b<0, since there is no price equilibrium
for the whole market. In order to do this, it is crucial to perform a change of variables that
explicitly allows us to calculate, in an easier way, the necessary conditions in order to obtain the
equilibrium regions.
The final aim of the paper consists in comparing the feasible equilibrium regions
with both the concave and the convex cases. Regarding this point, we show that the
feasible region with convex costs is bigger than with concave costs. Indeed, the feasible
region associated to concave costs is totally included in the feasible region
corresponding to convex costs. Consequently, a Nash price equilibrium would fit better
with convex transportation costs than with concave ones in the linear model.
The paper is structured as follows. In Section 2 we describe the model. In
Section 3 we study the feasible regions for firm locations where a pure price
equilibrium exists. In Section 4 we analyse the results obtained from concave
transportation costs versus those from convex costs. Finally, in Section 5 we present our
main conclusions.
2.
The model
There are two firms selling a homogeneous product whose production costs are
equal to zero. Prices p1 and p2 are chosen by firm 1 and firm 2, respectively, given their
locations. In particular, 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 . Consumers are uniformly distributed
along the market, and each of them buys just one unit of the industry good from the
2
firm in which the delivered price (total price that results from the addition of list price
and transportation cost) is the lowest. Let ~
x denote the consumer location along the line.
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’ classic model of horizontal
differentiation consists in considering concave transportation costs as follows:
c (di )  adi  bdi2
 i  1, 2.
(1)
where di is the distance between a buyer and the seller i, and parameters a and b are
non-negative2.
A first consequence of the concavity of transportation costs on the model is that
the length of the market is restricted by the maximum distance at which the cost
function is increasing. Therefore, the length l has to verify the following restriction:
a
a
l
. Without loss of generality, hereafter we will consider l 
for concave costs.
2b
2b
This aspect reveals that there is no equivalence between concave and convex costs,
since with convex transportation costs there are no existing restrictions on space.
A second consequence of the concavity of transportation costs is that the
demand is no longer connected. Consumers bear the transportation cost of the product,
and purchase from the seller with the lowest full price. It makes no difference to a
consumer whether he buys from one firm or another when the full price is the same:
p1 c(d1 )  p2 c(d 2 ) .
With a simple calculation from the equation (1) we find three types of indifferent
consumers whom we shall name m1, m and m2, associated with the intervals (0, x), (x, y)
and (y, l), respectively. In Figure 1 we can observe the three configurations of both
transportations costs. It is clear that for the concave case, three indifferent consumers
can never exist simultaneously. Indeed, there can be two, one or no indifferent
consumers. While in the circular case two or none can exist. Therefore, demand with
concave costs will not be connected, that is, contingent demands are piecewise linear
but not concave.
After checking that these indifferent consumers actually exist in each interval,
we write the demand that firm 1 is facing according to the price dispersion range
( p1  p2 ) as follows:
The assumption a>0 and b=0 corresponds with Hotelling’s classic model with linear transportation
costs.
2
3
l

a ( p 2  p1 )

 2b( y  x)(a  b( y  x))

p 2  p1
x y
D1  

2
 2(a  b( y  x))
a ( p 2  p1 )
a

 2b( y  x)(a  b( y  x))  2b

0
p1  p 2  ( x  y )( a  b( y  x))
if
I1 :
if
I 2 : ( x  y )( a  b( y  x))  p1  p 2  ( x  y )( a  b(2l  x  y ))
if
I 3 : ( x  y )( a  b(2l  x  y ))  p1  p 2  ( y  x)( a  b( x  y ))
if
I 4 : ( y  x)( a  b( x  y ))  p1  p 2  ( y  x)( a  b( y  x))
if
I 5 : ( y  x)( a  b( y  x))  p1  p 2
(2)
Obviously, the demand function for firm 2 is calculated as D2 = l – D1. Figure 2
shows the demand for firm 1 given the price of its rival. It is interesting to notice here
that the demand function for each firm is continuous on its own price and depends on
both firms’ location through the market.
On the other hand, when convex transportation costs are considered (that is,
c (d )  ad  bd 2 with a, b>0), one or none indifferent consumer exists in the linear
i
i
i
case (see Panel B in Figure 1). In the circular model, however, there are two or none
indifferent consumers, as with concave costs. Notice that with convex transportation
costs there are no restrictions on the length of the market as the cost function is always
increasing. The demand function associated to convex costs is more familiar and is
written with our notation as:
D1
l
 ( p 2  p1 ) x  y a



2
2b
 2b ( y  x )
x y
 p 2  p1


2
 2( a  b( y  x ))
 p 2  p1 x  y a
 2b ( y  x )  2  2 b
0

if
I1 :
p1  p 2  ( x  y )( a  b ( 2l  x  y ))
if
I2 :
( x  y )( a  b ( 2l  x  y ))  p1  p 2  ( x  y )( a  b ( y  x ))
if
I3 :
( x  y )( a  b ( y  x ))  p1  p 2  ( y  x )( a  b ( y  x ))
if
I4 :
( y  x )( a  b ( y  x ))  p1  p 2  ( y  x )( a  b ( x  y ))
if
I5 :
( y  x )( a  b ( x  y ))  p1  p 2
(3)
From this demand function the profit function is piecewise concave for given
firm locations (see, for example Anderson, 1988). Notice that the five critical intervals
are different to the defined ones in equation (2) for the demand associated with concave
costs. Moreover, as we have commented before in the linear market, the number of
4
indifferent consumers is not the same with concave and convex costs, again justifying
the non-equivalence between both cases, as occurs in the circular model.
3.
The feasible region for the existence of a Nash-equilibrium price with
concave costs.
From the previous results it is clear that no pure-strategy price equilibrium exists
for any firm locations with concave costs in the linear model. In this section our
objective is to calculate the feasible equilibrium region for a general concave cost
function, since there is no price equilibrium for the whole market (see De Frutos,
Hamoudi and Xarque, 2002). As it is well-known, “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, pp. 30). Precisely, with concave transportation costs the
profit function is a non-quasi-concave function, which is the reason that justifies the
non-existence of a Nash equilibrium price for any firm locations.
We shall now study the feasible regions for the existence of a Nash equilibrium
price with concave costs assuming that firm locations are fixed. The Nash equilibrium
prices are defined as a pair ( p1N , p2N ) , where p1N maximises B1 ( p1 , p2N ) and p2N
maximises B2 ( p1N , p2 ) , respectively. It is convenient to mention that with these
equilibrium prices, in addition to the fact that no firm can increase its profits by varying
its price unilaterally, we must check that both firms are operating in the market (e.g.,
there are no incentives to be out of business).
Assuming without loss of generality that production costs are zero, the profit
function for firm i-th is equal to Bi ( p1 , p2 )  Di ( p1 , p2 ) pi  i=1, 2. A simple calculation
leads to the profit function defined by intervals in the same way as the demand in
equation (2). Figure 3 shows the profit function of firm 1 given values of firm locations
(x=0.1, and y=0.3) and the price of firm 2 (p2=0.2). It is clear that the profit function is
continuous and non-quasi-concave.
In order to obtain the feasible regions associated to a given value for firm
locations (x and y) in which a Nash price equilibrium exists, we first derive the
necessary and sufficient conditions on those variables in which an equilibrium exists,
and secondly, we calculate all equilibria in prices for each firm and each critical
interval. Finally, we check the prices are actually a Nash equilibrium.
Due to the complexity of the demand function for obtaining a representation of
the feasible region, we need to reduce the number of parameters. Therefore, we assume
5
that the linear term of the transportation cost function is equal to the quadratic term, that
is, a=b. Then, concave transportation cost function is3:
c ( d i )  a ( d i  d i2 )
 i  1, 2.
(4)
In spite of restricting a=b, to be able to define the equilibrium in prices we need
to introduce a variable change in the following way: z  y  x and q  x  y . This
change of variables is crucial in calculus for identifying the equilibrium prices because
these new variables allow us to draw the feasible regions in an easier way4. The
transportation cost function defined in equation (4) implies that the maximum length of
the city is 0.5. These new variables are defined on the following intervals: z  (0, 0.5], q
 (0, 1].
In order to facilitate the exposition hereafter, we will use these new variables (z,
q) rather than the firm locations (x, y).
Proposition 1:
Under the concave cost function defined in expression (4), for z and q such as z
 (0, 0.5], q  (0, 1] and z < q there is a Nash-price equilibrium if and only if,
2
 2q 
z
 when q  0.5 , and
 2(1  q ) 
2
 1 q 
z
 when q  0.5 ,
 2( 2  q ) 
and whenever it exists, a pure price equilibrium is uniquely determined by:
a
p1N  (1 z)(1q)
3
and
a
p2N  (1 z)( 2q)
3
(4)
The demonstration of this proposition is in Appendix I. The feasible region for
firm locations in which a price Nash equilibrium exists is shown in Figure 4.
Given that Nash equilibrium in pure strategies exists, profits for each firm are
a
a
B1 ( p1N , p2N ) (1  z)(1q) 2 and B2 ( p1N , p2N ) (1 z)(2 q) 2 .
18
18
3
Notice that with this transportation cost function a price equilibrium exists in the circular model.
This variable change permits us to reduce the dimension of the problem to solve second order equations
rather that fourth or fifth order equations.
4
6
Several comments arise from this proposition:
Firstly, the Nash equilibrium prices all differ among themselves except when
q=0.5, contrary to the circular model where prices are always the same. This
characteristic is specific to the linear market with quadratic transportation costs because
there is no symmetry.
Secondly, the Nash equilibrium prices are included in the interval I3 defined in
the demand equation. Carefully analysing the feasible region in terms of firm locations,
we can verify that it is possible to obtain a maximum differentiation (z=q=0.5  x=0
and y=0.5) like a Nash equilibrium (point A in Figure 4). On the other hand, the feasible
minimum differentiation will correspond with the smallest feasible value for z that is
equal to 0.25 (point B in Figure 4). Indeed, it is possible to obtain the exact firm
locations in this situation: x=1/8 and y=3/8. Moreover, this location corresponds with
the optimum in terms of profits for both firms.
Thirdly, we see that p1N  p 2N only if q  0.5 , and this only occurs when firm 1
is located closer to the inferior market boundaries ( x1  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 therefore, firm 2
will sell at a higher price (the opposite situation exists when q  0.5 ).
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. First of all, we present the
results associated to a convex transportation cost function in order to facilitate the
comparison with the concave case.
Interestingly it turns out to be the results of Gabszewicz and Thisse (1986) and
Anderson (1989) who analysed convex transportation costs. While Gabszewicz and
Thisse were the first to find such values of a and b that 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 only exists under very stringent
conditions that are reduced to the product function being concave in price for any given
location pair.
7
Proposition 2:
c (di )  a(di  di2 )
Under the convex cost function defined as
, for z and q such
as z  (0, 0.5], q  (0, 1] and z < q there is a Nash-price equilibrium if and only if,
 4q 2  q  4  2(1  q) 7  7q  4q 2 
q  0.5

 when
, and
z
1
3 (1  4q)
z
1
  4 q 2  7 q  7  2( q  2 ) 4 q 2  q  4 
 when q≥0.5, and
3 (5  4q) 
whenever it exists, a pure price equilibrium is uniquely determined by:
a
~
p1N  (1 z) (1q)
3
and
p2N 
a
(1 z) ( 2q)
3
(6)
The demonstration of this proposition is in Appendix II.
Given the Nash equilibrium prices, the profits for each firm with convex costs
a
a
B1 ( p1N , p 2N ) (1  z)(1q) 2
B2 ( p1N , p 2N ) (1 z)(2 q) 2
18
18
are:
and
.
Several comments arise from this proposition 2:
Firstly, as with concave costs, the optimal prices are different between firms.
Consequently, 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 Figure 5), in contrast with the concave case that is 0.25. In particular, 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 to
fix higher prices, the behaviour is similar to that obtained for concave costs.
Figure 5 presents the feasible regions associated to convex and concave costs.
Clearly, we can check that the feasible region is more restrictive for concave costs
(ACBD) than for convex costs (ALMN). Indeed, the feasible region for convex costs
8
completely contains the region for concave costs. The variation range for q is limited by
[0.292, 0.7081] in convex case versus [0.3618, 0.6382] in concave case.
Therefore we can conclude that it is more probable to find a Nash equilibrium in
prices with convex than with concave transportation costs. The reason for this is that the
equilibrium region in the convex case is larger than in the concave costs for both firms
and for any locations.
5.
Conclusions.
One of the first conclusions that we can make from this study is the difficulty of
the search for the feasible regions in the linear model when linear-quadratic
transportation costs are considered (concave or convex). This difficulty arises due to the
very high degree of polynomials that need to be solved. For this reason, our contribution
consists in the use of a change of variables that is crucial in order to reduce the
dimension of the problem. Until we know the calculus of the feasible regions has been
avoided. Precisely, the use of that 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 can 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 with concave costs.
9
Appendix 1: Demonstration of Proposition 1.
( p1N , p2N )
First of all, we must evaluate what the pair of prices
generates a
global maximum in profit functions of both firms. To do that it is necessary consider all
local maxima obtained for every interval in equation (2) due to profit function is defined
( p1N , p2N )
at the same intervals. Prices
are in the interval I3 therefore we have to
compare with intervals I1, I2 and I4, respectively.
p1*  p2  z(1  z )
In the first interval the local maximum occurs when
. In the
2q
2q
z
5q  2 , and it is given by
interval I2 a local maximum exists if and only if 6
1
p1**  (1  z )( 2  q)
6
. Finally, in the interval I4 the local maximum is
p1***  p2  z(1  q)
expressions, given
. In sum, for firm 1 we have to prove that
p2N
p1N
verify the following
, z and q.
B1( p1N , p2N )  B1( p1* , p2N )
B1( p1N , p2N )  B1( p1** , p2N )
B1( p1N , p2N )  B1( p1*** , p2N )
(A.1.a)
(A.1.b)
(A.1.c)
The last condition always occurs because profits continuously decrease on all
values in the corresponding interval (see Figure 3). In relation to the other two
conditions, they are true if and only if firm locations comply, respectively, with:
z
5  5q  q 2
9
 2q 
z

 2(1  q ) 
(A.2.a)
2
(A.2.b)
From these conditions and together with the feasible regions defined by each
local maximum, we check that the expression (A.2.b) is the most restrictive when
q  0.5
, while the expression (A.2.a) is the most restrictive when q>0.5.
On the other hand, carrying out a similar analysis for the firm 2 we find that the
p2N
p1N
maximization problem is symmetrical respect to q=0.5. Let be
such as, given
,
10
p2* p2**
z, and q, maximises
comparing with all local maxima defined as
,
***
N
p2
p2
and
for each interval. In a similar way for firm 1,
is a global maximum if and
B2 ( p1N , p2 )
2
 1 q 
 q 2  7q  1
q  0.5
z
z

9
 2( 2  q)  when q>0.5. In
only if
when
, and
( p1N , p2N )
p1N
consequence, given that
is the vector of Nash equilibrium prices,
must
N
p2
be the best reply against
given locations, and viceversa. Finally, we select the
2
 2q 
z
q  0.5

common region for both firms and it is represented by
,
 2(1  q )  when
2
 1 q 
z
q  0.5

and
.
 2( 2  q )  when
Q.E.D.
11
Appendix II: Demonstration of Proposition 2.
Let us follow the same structure used in Appendix I, taking into account that the
critical intervals are now defined according to equation (3). The Nash equilibrium prices
in expression (6) belong to the third interval, then we calculate that its feasible region is
1  2q
2q  1
q  0.5
q  0.5
z
z
3 when
3 when
given by
, and
. Afterwards, we have
~
p2N
to calculate local maxima in the other intervals for firm 1 given values
, z and q.
1
~
p1*  ( 2  q)(1  2 z )
3
In the first interval the local maximum is equal to
. In the
2  q  z ( 2q  1)
~
p1** 
6
second interval, there exists a local maximum equal to
, if and
 1  4q  16q 2  16q  49
2q
z
12
only if firm locations verify 7  2q
. Finally, the
2

q

z
(
2
q

5
)
~
~
p1*** 
p1N
6
local maximum in interval I4 is defined by
. Then
will be
the global equilibrium for firm 1 if and only if verifies the following expressions:
B1( ~
p1N , ~
p2N )  B1( ~
p1* , ~
p2N )
B1( ~
p1N , ~
p2N )  B1( ~
p1** , ~
p2N )
B1( ~
p1N , ~
p2N )  B1( ~
p1*** , ~
p2N )
(A.3.a)
(A.3.b)
(A.3.c)
The last condition always occurs because the profit function is continuously
decreasing in that interval. The other conditions are true if and only if locations comply
respectively with:
z
z
5  5q  q 2
q 2  4q  13
(A.4.a)
 4q 2  q  4  2(1  q) 4q 2  7q  7
3(1  4q)
(A.4.b)
From these conditions and together with the feasible regions for each local
maximum, we have checked that the expression (A.4.b) is the most restrictive but only
q  0.5
when
.
12
References
Anderson, S., (1987), “Spatial competition and price leadership”, International Journal
of Industrial Organization, Vol. 5 (2), pp. 369-398.
Anderson, S., (1988), “Equilibrium existence in the linear model of spatial
competition”, Economica, Vol. 55, pp. 479-491.
D’Aspremont, C., Gabszewicz, JJ. And Thisse JF., (1979), “On Hotelling’s stability in
competition”, Econometrica, Vol. 47, pp. 1145-1150.
De Frutos, A., Xarque, X. and Hamoudi, H., (1999), “Equilibrium existence in the circle
model with linear quadratic transport costs”, Regional Science and Urban
Economics, Vol. 29 (5), pp. 605-615.
De Frutos, A., Xarque, X. and Hamoudi, H., (2002), “Spatial competition with concave
transport costs”, Regional Science and Urban Economics, Vol. 32, pp. 531-540.
Economides, N., (1986), “Minimal and maximal differentiation in Hotelling’ duopoly”,
Economics Letters, Vol. 21, pp. 67-71.
Gabszewicz, JJ., and Thisse, JF., (1986), “On the nature of competition with
differentiation products”, The Economic Journal, Vol. 96, pp. 160-172.
Hotelling, H. (1929), “Stability in competition”, Bell Journal, Vol. 39, pp. 41-57.
Neven, D., (1985), “In Hotelling’s competition with non-uniform customer
distributions”, Economic Letters, Vol. 231, pp. 121-126.
13
Figure 1: Price and transportation costs respect to each firm with concave
and convex costs.
Panel a: Concave transportation costs.
x
y
m
x
m1
x
m
x
y
y
m
y
m2
Panel b: Convex transportation costs.
x
m
y
m1
x
y
x
y
x
m2
y
Note: m1, m and m2 are indifferent consumers associated with the intervals (0,x), (x,y) and (y,l),
respectively.
14
Figure 2: Demand for firm 1, given the price of firm 2.
D( p1 , p2 )
I1
I2
I3
I5
I4
P1
15
Figure 3: Profit function of firm 1*.
I2
I1
I3
I4
I5
* Note: The values for firm locations and for price of firm 2 are x=0.1, y=0.3 and p2= 0.2,
respectively.
16
Figure 4: Feasible region for a Nash equilibrium with concave
transportation costs.
0,6
A
0,5
0,4
z
0,3
 1 q 
 2( 2  q ) 


2
B
0,2
 2q 
z
 2(1  q ) 


0,1
2
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
17
Figure 5: Feasible regions for the Nash equilibrium with concave and convex
transportation costs*.
0,6
Concave costs: ACBD
A
0,5
0,4
C
D
Convex costs: ALMN
0,3
B
L
N
0,2
M
0,1
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
18