Spatial strategies of firms under Stackelberg competention
The paper explores a spatial duopoly of firms under Stackelberg competition, when one of the firms is a leader in term of product volume. The equilibrium spatial strategies of firms are found. In the process of the analysis of equilibrium stability, it is proved that the transport tariff is a bifurc...
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nasplib_isofts_kiev_ua-123456789-1620582025-02-23T20:16:49Z Spatial strategies of firms under Stackelberg competention Пространственные стратегии фирм в условиях конкуренции Штакельберга Melnikov, S. Аналіз, оцінка та прогнозування в економіці The paper explores a spatial duopoly of firms under Stackelberg competition, when one of the firms is a leader in term of product volume. The equilibrium spatial strategies of firms are found. In the process of the analysis of equilibrium stability, it is proved that the transport tariff is a bifurcation parameter for firms. It was found that the change in the central agglomeration strategy to the differentiation strategy occurs at the point of transcritical bifurcation. The conditions for full coverage of the markets for both strategies are defined. It is obtained that the information asymmetry leads to asymmetry of equilibrium locations of firms. It is established that under differentiation strategy, the follower can due to the space factor overcome information asymmetry in the nearest markets and get a profit more than the leader. В работе исследована пространственная дуополия фирм в условиях конкуренции Штакельберга, когда одна из фирм является лидером по объему продукции. Найдены равновесные пространственные стратегии фирм. В ходе анализа устойчивости равновесия доказано, что транспортный тариф является бифуркационным параметром для фирм. Найдено, что изменение стратегии центральной агломерации на стратегию дифференциации происходит в точке транскритической бифуркации. Определены условия полного покрытия рынка для обеих стратегий. Получено, что информационная асимметрия приводит к асимметрии равновесного расположения фирм. Установлено, что при дифференциации последователь может на прилегающих рынках преодолеть информационную асимметрию за счет фактора пространства и получить прибыль больше, чем у лидера. 2018 Article Spatial strategies of firms under Stackelberg competention / S. Melnikov // Математичне моделювання в економіці. — 2018. — № 4(13). — С. 82-93. — Бібліогр.: 14 назв. — укр. 2409-8876 https://nasplib.isofts.kiev.ua/handle/123456789/162058 519.865:338.518 en Математичне моделювання в економіці application/pdf Інститут телекомунікацій і глобального інформаційного простору НАН України |
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Аналіз, оцінка та прогнозування в економіці Аналіз, оцінка та прогнозування в економіці |
| spellingShingle |
Аналіз, оцінка та прогнозування в економіці Аналіз, оцінка та прогнозування в економіці Melnikov, S. Spatial strategies of firms under Stackelberg competention Математичне моделювання в економіці |
| description |
The paper explores a spatial duopoly of firms under Stackelberg competition, when one of the firms is a leader in term of product volume. The equilibrium spatial strategies of firms are found. In the process of the analysis of equilibrium stability, it is proved that the transport tariff is a bifurcation parameter for firms. It was found that the change in the central agglomeration strategy to the differentiation strategy occurs at the point of transcritical bifurcation. The conditions for full coverage of the markets for both strategies are defined. It is obtained that the information asymmetry leads to asymmetry of equilibrium locations of firms. It is established that under differentiation strategy, the follower can due to the space factor overcome information asymmetry in the nearest markets and get a profit more than the leader. |
| format |
Article |
| author |
Melnikov, S. |
| author_facet |
Melnikov, S. |
| author_sort |
Melnikov, S. |
| title |
Spatial strategies of firms under Stackelberg competention |
| title_short |
Spatial strategies of firms under Stackelberg competention |
| title_full |
Spatial strategies of firms under Stackelberg competention |
| title_fullStr |
Spatial strategies of firms under Stackelberg competention |
| title_full_unstemmed |
Spatial strategies of firms under Stackelberg competention |
| title_sort |
spatial strategies of firms under stackelberg competention |
| publisher |
Інститут телекомунікацій і глобального інформаційного простору НАН України |
| publishDate |
2018 |
| topic_facet |
Аналіз, оцінка та прогнозування в економіці |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/162058 |
| citation_txt |
Spatial strategies of firms under Stackelberg competention / S. Melnikov // Математичне моделювання в економіці. — 2018. — № 4(13). — С. 82-93. — Бібліогр.: 14 назв. — укр. |
| series |
Математичне моделювання в економіці |
| work_keys_str_mv |
AT melnikovs spatialstrategiesoffirmsunderstackelbergcompetention AT melnikovs prostranstvennyestrategiifirmvusloviâhkonkurenciištakelʹberga |
| first_indexed |
2025-11-25T01:58:24Z |
| last_indexed |
2025-11-25T01:58:24Z |
| _version_ |
1849725713017470976 |
| fulltext |
~ 82 ~
Математичне моделювання в економіці, №4, 2018. ISSN 2409-8876
UDC 519.865:338.518
S. MELNIKOV
SPATIAL STRATEGIES OF FIRMS UNDER STACKELBERG
COMPETITION
Abstract. The paper explores a spatial duopoly of firms under Stackelberg
competition, when one of the firms is a leader in term of product volume.
The equilibrium spatial strategies of firms are found. In the process of the
analysis of equilibrium stability, it is proved that the transport tariff is a
bifurcation parameter for firms. It was found that the change in the central
agglomeration strategy to the differentiation strategy occurs at the point of
transcritical bifurcation. The conditions for full coverage of the markets for
both strategies are defined. It is obtained that the information asymmetry
leads to asymmetry of equilibrium locations of firms. It is established that
under differentiation strategy, the follower can due to the space factor
overcome information asymmetry in the nearest markets and get a profit
more than the leader.
Key words: linear city, agglomeration, differentiation, Stackelberg
competition, transcritical bifurcation.
Introduction
In search of a solution to the Bertrand paradox, Hotelling proposed to take into
account the factor of space under the price competition of firms. In Hotelling's
linear city model (Hotelling, 1929), two firms compete on a segment with a unit
demand at each point. Firms optimize their prices and location on the segment.
Transportation delivery costs of goods are borne by consumers. Hotelling found
that in an equilibrium state, firms would be minimally spatially differentiated, since
they would be located in the center. This conclusion of the model analysis
subsequently became a famous “principle of minimal differentiation”.
In further research, the Hotelling model has developed in the following areas:
– an increase in the number of firms (Brenner, 2005, Patri and Sacco, 2017);
– increase the dimension of space (Irmen and Thisse, 1998, Mazalov and
Sakaguchi, 2003);
– the complexity of the type of transport costs function (D’Aspremont,
Gabszewicz and Thisse, 1979, Economides, 1986);
– generalization of the consumer’s distribution density (Neven, 1986, Gupta,
Pal and Sarkar, 1997, Tabuchi and Thisse, 1995);
– consideration of the Cournot competition (Hamilton, Klein, Sheshinski and
Slutsky, 1994, Scrimitore, 2011, Hamilton, Thisse and Weskamp, 1989) and
Stackelberg competition (Anderson, 1987).
In the Anderson (1987), a linear city model is investigated under Stackelberg
competition, when firms optimize their locations and prices. It is found that the
outcome is asymmetric in terms of Stackelberg equilibrium locations, prices and
profits. In this paper, we investigate a linear city model in the framework of
Stackelberg competition, when firms optimize their locations and supply volumes.
S. Melnikov, 2018
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Математичне моделювання в економіці, №4, 2018. ISSN 2409-8876
1. The linear city model
Two firms sell homogeneous goods on the unit segment, at each point of which is
the consumer market x , [ ]10,x∈ . The distance of the firms from zero point is
equal 1x and 2x accordingly, and 21 xx ≤ . Each firm faces linear transportation
costs of t to move one good unit per one unit of distance. Consumer arbitrage is
assumed to be prohibitively costly.
The linear demand curve in the market x :
( ) ( ) ( )xqxqxp 211 −−= ,
where ( )xp – the price in the market x , ( )xq1 , ( )xq2 – the quantities supplied of
firms in the market x , a minimum price, at which there is no demand (market
potential), is equal to 1.
Let us assume that firms supply products to all markets, i.e. ( ) 01 >xq ,
( ) 02 >xq .
The profits of firms in the market x :
( ) ( ) ( ) ( )( )
( )
,maxxxtxqxqxqxF
xq,x 11
12111 1 →−⋅−−−⋅=
( ) ( ) ( ) ( )( )
( )
.maxxxtxqxqxqxF
xq,x 22
22122 1 →−⋅−−−⋅=
The competitive game consists of two stages. In the first stage, the firms
simultaneously select their locations. In the second stage, at the given location
decisions, the firms simultaneously choose their supplied quantities. The
equilibrium of the model is solved by backward induction.
2. The Nash equilibrium
According to the backward induction method we begin with the second stage. Let
us assume that firms optimize supply volumes under the conditions of the
Stackelberg information asymmetry. The firm 2 (leader) knows the strategy of the
firm 1 regarding the supply volume and has the right of first move. The firm 1
(follower) does not possess such information and makes decisions after leader.
Solving the first-order condition yields the reaction curve of the firm-follower:
( ) ( )
2
1 12
1
xxtxq
xq
−⋅−−
= . (1)
The firm-leader knows the follower's reaction curve (1) and incorporates it
into his profit function:
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Математичне моделювання в економіці, №4, 2018. ISSN 2409-8876
( ) ( ) ( )( )
( )
.max
xxtxxtxqxq
xF
xq,x 222
21 1222
2 →
−⋅+−⋅⋅−−⋅
=
The equilibrium supply volumes of firms to the market x :
( )
4
231 21
1
xxtxxt
xq* −⋅⋅+−⋅⋅−
= ,
( )
2
21 12
2
xxtxxt
xq* −⋅+−⋅⋅−
= .
A coverage conditions for all markets:
( )
21
1 23
10
xxxx
txq*
−⋅−−⋅
<⇔> , (2)
( )
12
2 2
10
xxxx
txq*
−−−⋅
<⇔> . (3)
Substituting into (2) the values: 01 =x , 12 =x , 1=x (maximum value of the
denominator), we find that at 31<t all markets are serviced regardless of the
location of firms. Substituting into (2) the values: 211 =x , 212 =x , 0=x
(minimum value of the denominator), we find that at any location of firms
servicing all markets is possible only at 2<t . Let us note that in the equilibrium
state, the location of firms depends on the transport tariff. Therefore, the analysis of
conditions (2)-(3) will be continued after finding equilibrium.
The equilibrium profits of firms in the market x :
( ) ( )
16
231 2
21
1
xxtxxt
xF* −⋅⋅+−⋅⋅−
= ,
(4)
( ) ( )
8
21 2
12
2
xxtxxt
xF* −⋅+−⋅⋅−
= .
It follows from (4) that the ratio between the profits of the leader and follower
in the market x depends on their location relative to the market x . In the case of
agglomeration or symmetric location relative to the market x , the leader's profit
exceeds the follower's profit exactly 2 times, how in the classical model with one
market. A closer distance to the market x strengthens the market power of the
leader and vice versa:
( ) ( )( ) ( )2112 2 xxxxsignxFxFsign ** −−−=⋅− .
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Математичне моделювання в економіці, №4, 2018. ISSN 2409-8876
In the first stage each firm selects a profit-maximizing location at a given
location of the competitor. In the subsequent analysis, we assume that the
equilibrium location of the firms obeys the condition:
211 ≤x , 212 ≥x . (5)
So, let us start with firm 1. The total profit of firm 1 in all markets:
( ) ( ) ( ) ( )dxxFdxxFdxxFdxxFF
x
*
x
x
*
x
** ∫∫∫∫ ++==
1
11
0
1
1
0
11
2
2
1
1
,
( ) ( )( ) +−⋅⋅+−⋅⋅−=⋅ ∫
1
0
2
211 23116
x
dxxxtxxtF
( ) ( )( ) +−⋅⋅+−⋅⋅−+ ∫
2
1
2
21 231
x
x
dxxxtxxt
( ) ( )( )∫ −⋅⋅+−⋅⋅−+
1
2
21
2
231
x
dxxxtxxt . (6)
After integrating and identical transformations (6), we obtain:
( ) ( )
+
⋅
⋅⋅+⋅⋅−
−
⋅
⋅⋅+⋅⋅−⋅
=⋅
t
xtxt
t
xtxtF
3
231
5
221216
3
21
3
21
1
( ) ( )
t
xtxtt
t
xtxt
⋅
⋅⋅−⋅⋅+−
−
⋅
⋅⋅−⋅⋅+⋅
+
3
231
15
3314 3
21
3
21 .
The optimal location is defined by the necessary condition:
( ) ( ) 021434224
3
16
2221
2
1
1
1 =−+−⋅⋅⋅+⋅−⋅⋅+⋅⋅−⋅⋅=
∂
∂
⋅
⋅
txxttxtxxt
x
F
t
. (7)
The sufficient condition for the existence of profit maximum for the firm 1:
t
txxtxtxt
x
F
t ⋅
−⋅
−<⇔<⋅+⋅⋅−−⋅⋅=
∂
∂
⋅
⋅ 4
2303424
3
8
21212
1
1
2
. (8)
The necessary condition for the existence of the equilibrium location for firm
1 is the nonnegativity of the discriminant of the square equation (7):
( ) ( )( ) 0416208134 22
2
1 ≥+⋅−⋅−⋅−⋅⋅= xtxtD . (9)
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Математичне моделювання в економіці, №4, 2018. ISSN 2409-8876
It is easy to make sure that 01 ≥D at 212 ≥x . Therefore, due to condition
(5), in the equilibrium state the discriminant (9) is always nonnegative.
Let us note that information asymmetry about location of firms can lead to
monopolization of the city. With the right of first move, firm 2 (leader) can will
located in the market 212 <x , create a barrier to entry for firm 1 and monopolize
all markets. Therefore, we assume that firm 2 is the leader only in the second stage,
and in the first stage, when choosing a location, firms compete under Cournot
model.
The roots of the square equation (7) are:
( )
t
D
t
txx*
⋅
−
⋅
−⋅
−=
84
23 1
211 , ( )
t
D
t
txx*
⋅
+
⋅
−⋅
−=
84
23 1
221 .
The root ( )21
*x does not satisfy the sufficient condition (8) and therefore is not
further analyzed. The root ( )11
*x for 212 >x always satisfies the sufficient
condition (8), for 212 =x the condition (8) holds for:
( )
320
4
230
8
2
1 ≠⇔>
⋅
−⋅
⇔>
⋅
t
t
t
t
D
.
The total profit of firm 2 in all markets:
( ) ( ) ( ) ( )dxxFdxxFdxxFdxxFF
x
*
x
x
*
x
** ∫∫∫∫ ++==
1
22
0
2
1
0
22
2
2
1
1
.
( ) ( )( ) +−⋅+−⋅⋅−=⋅ ∫
1
0
2
122 218
x
dxxxtxxtF
( ) ( )( ) +−⋅+−⋅⋅−+ ∫
2
1
2
1221
x
x
dxxxtxxt
( ) ( )( )∫ −⋅+−⋅⋅−+
1
2
12
2
21
x
dxxxtxxt . (10)
After integrating and identical transformations (10), we obtain:
( ) ( )
+
⋅
⋅+⋅⋅−
−
⋅
⋅⋅+⋅⋅−⋅
=⋅
t
xtxt
t
xtxtF
3
21
9
22128
3
12
3
12
2
( ) ( )
t
xtxtt
t
xtxt
⋅
⋅−⋅⋅+−
−
⋅
⋅−⋅+⋅
+
3
21
9
14 3
12
3
12 .
~ 87 ~
Математичне моделювання в економіці, №4, 2018. ISSN 2409-8876
The optimal location is defined by the necessary condition:
( ) ( ) ( ) ( ) 0212222
12
2
1212
2
2 =−−⋅⋅−−−⋅−−⋅=
∂
∂
⋅ xxtxxtxx
x
F
t
. (11)
The sufficient condition for the existence of profit maximum for the firm 2:
( )
t
txxxxt
x
F
t
10111
12212
2
2
2 −
+>⇔<−−+⋅=
∂
∂
⋅ . (12)
The necessary condition for the existence of the equilibrium location for firm
2 is the nonnegativity of the discriminant of the square equation (11):
( ) ( ) ( )( ) 021214 1
2
2 ≥−⋅−⋅+−⋅= xtttD . (13)
It is easy to make sure that 02 ≥D at 211 ≤x . Therefore, due to condition
(5), in the equilibrium state, the discriminant (13) is always nonnegative.
The roots of the square equation (11) are:
( )
t
D
t
txx*
⋅
−
−
+=
2
1 2
112 , ( )
t
D
t
txx*
⋅
+
−
+=
2
1 2
122 .
The root ( )12
*x does not satisfy the sufficient condition (12) and therefore is
not further analyzed. The root ( )22
*x for 211 <x always satisfies the sufficient
condition (12), for 211 =x the condition (12) holds for:
( ) 1010
2
2
2 ≠⇔>
−
⇔>
⋅
t
t
t
t
D
.
Thus, we received the reaction curves of firms:
( ) ( )
t
xtxtt
xx
⋅
+⋅−⋅−⋅−⋅−⋅−
+=
4
4162081332 22
2
21 , (14)
( ) ( ) ( )
t
xtttt
xx 1
2
12
21211 −⋅−⋅+−+−
+= . (15)
To solve the system of equations (14)-(15) we introduce a new variable:
( ) ( ) 41620813 22
2 +⋅−⋅−⋅−⋅= xtxtw , 0≥w .
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Математичне моделювання в економіці, №4, 2018. ISSN 2409-8876
Then we obtain from (14):
t
wtxx
⋅
−⋅−
+=
4
32
21 , ( )tt
ttwx
−⋅⋅
−⋅+⋅−
=
28
42013 22
2 . (16)
Substituting (16) into (15), and solving with respect to w , we obtain:
tw ⋅−= 321 ,
3
27
2
−⋅
=
tw . (17)
Substituting (17) into (16), we find solutions of the system (14)-(15):
2121 == aggagg xx , (18)
t
txdis
⋅
⋅−
=
18
78
1 ,
t
txdis
⋅
−⋅
=
18
417
2 . (19)
So, we obtained two equilibrium strategies for the location of firms: central
agglomeration and dispersion.
It follows from condition 0≥w that the solution (18) is defined for 32≤t ,
the solution (19) is defined for 72≥t . For 21=t , the solutions (18) and (19)
coincide. From the location condition, 21 xx ≤ , it follows that firms can apply the
dispersion strategy only when 21≥t .
Let us finish off an analysis of the market coverage conditions (2)-(3). Under
central agglomeration strategy, firms will be able to serve all markets on
condition:
2
23
1
21
<⇔
−⋅−−⋅
< t
xxxx
t
aggagg
.
Under dispersion strategy, firms will be able to serve all markets on condition:
( ) ( ) 7350
1213
1
23
1
1121
<⇔
−⋅−−⋅
<⇔
−⋅−−⋅
< t
xx
t
xxxx
t disdisdisdis
.
Under central agglomeration strategy the firms minimize a total distance of
traffic, therefore full market coverage may be possible with a higher transport
tariff.
In previous studies (Gupta, Pal and Sarkar, 1997, Hamilton, Klein, Sheshinski
and Slutsky, 1994, Hamilton, Thisse and Weskamp, 1989) it is proved that firms in
the Cournot equilibrium are always located symmetrically with respect to the
center. We have obtained that the information asymmetry about supply volumes
will lead to the asymmetry of equilibrium location of firms. It follows from (19)
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Математичне моделювання в економіці, №4, 2018. ISSN 2409-8876
that in the equilibrium state the firm-leader is 2 times closer to the center than the
firm follower:
( )21221 21 −⋅=− disdis xx .
3. The analysis of the stability of equilibrium
Let us analyze a stability of the solutions (18)-(19). For this we consider a two-
dimensional map:
( ) ( ) ( )( ) ( )( )
t
nxtnxtt
nxnx
⋅
+⋅−⋅−⋅−⋅−⋅−
+=+
4
4162081332
1 22
2
21 ,
(20)
( ) ( ) ( ) ( ) ( )( )
t
nxtttt
nxnx 1
2
12
21211
1
−⋅−⋅+−+−
+=+ ,
where n is a time moment, ...,,,,n 210= ( ) 001 =x , ( ) 102 =x .
As is known, the nature of the stability of fixed points is determined by their
multipliers. The multipliers are eigenvalues of the Jacobian matrix in a fixed point,
and their number is equal to the dimension of map.
The Jacobian matrix of the map (20) in the fixed point (18):
( )
−⋅
−
⋅−
⋅
−
=
0
12
32
20
t
t
t
t
J . (21)
From (21) we obtain two real multipliers:
( ) ( ) 32
321
2
21 <
⋅−⋅−
±= t,
tt
t
,µ . (22)
For 121 <,µ the fixed point is stable, for 121 >,µ the fixed point is unstable,
for 121 =,µ the bifurcation occurs.
From (22) we find that the fixed point (18) is stable at 21<t and is unstable
at 21>t . The loss of stability occurs at the bifurcation point: 21=t .
The Jacobian matrix of the map (20) in a fixed point (19):
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Математичне моделювання в економіці, №4, 2018. ISSN 2409-8876
( )
+⋅
−⋅
−⋅
−⋅
=
0
12
45
27
8100
t
t
t
t
J . (23)
From (23) we obtain two real multipliers:
( )
( ) ( ) 72
271
45 2
21 >
−⋅⋅+
−⋅
±= t,
tt
t
,µ . (24)
From (24) we find that the fixed point (19) is unstable at 21<t and is stable
at 21>t . The acquisition of stability occurs at the bifurcation point: 21=t .
Thus, at the value of the transport tariff 21=t , occurs a transcritical
bifurcation, in which fixed points change a nature of stability (Fig. 1).
The equilibrium profit dynamics of the firm-leader, depending on the transport
tariff, is presented in Fig. 2.
Fig. 1 – Equilibrium spatial strategies of firms depending on transportation tariffs
21, xx
*
2x
*
1x
stable unstable
t
~ 91 ~
Математичне моделювання в економіці, №4, 2018. ISSN 2409-8876
Fig. 2 – Dynamics of the equilibrium profit of the firm-leader, depending on the
transport tariff
The Fig. 2 illustrates the effects that affect spatial strategies of firms. Before
bifurcation point the effect of minimizing transport costs is dominate (Scrimitore,
2011). Firms choose the central agglomeration strategy to minimize a total distance
of transportation. The growth of the transport tariff leads to a decrease in the total
profit. Due to information asymmetry, the leader's profit in all markets is twice as
high as the profit of the follower (Fig. 3).
In the bifurcation point begins to dominate the effect of reducing competition.
Firms choose a differentiation strategy to monopolize adjacent markets. The
growth of the transport tariff leads to an increase in total profits. The growth of
total profit with growth of the transport tariff is due to the fact that when
differentiation strategy, the firms supply more to adjoining markets and less to
distant markets. Due to information asymmetry, the leader almost monopolizes the
markets to the right of himself and at the same time is present in the markets to the
left of the follower. This is clearly seen in Fig. 4. In the equilibrium state, for
32=t , the leader in the follower "territory", [ ]disx,x 10∈ , receives 19,2% of the
total profit both firms, and the follower in the leader "territory", [ ]12 ,xx dis∈ ,
receives only 1,9% of the total profit both firms.
2F
t
( )aggagg x,xF 212
( )disdis x,xF 212
~ 92 ~
Математичне моделювання в економіці, №4, 2018. ISSN 2409-8876
Fig. 3 – Equilibrium profits of firms under central agglomeration
Fig. 4 – Equilibrium profits of firms under differentiation
( )aggagg* x,x,xF 211
( )aggagg* x,x,xF 212
31=t
x
( ) ( )xF,xF 21
aggagg x,x 21
x
32=t
( )disdis* x,x,xF 211
( )disdis* x,x,xF 212
( ) ( )xF,xF 21
disx1
disx2
~ 93 ~
Математичне моделювання в економіці, №4, 2018. ISSN 2409-8876
Conclusions
The paper explores the spatial duopoly of firms under Stackelberg competition, when
one of the firms is the leader in term of supply volume. The equilibrium spatial
strategies of firms are found. In the process of the analysis of equilibrium stability, it
is proved that the transport tariff is a bifurcation parameter for firms. It was found
that the change in the central agglomeration strategy to the differentiation strategy
occurs at the point of transcritical bifurcation. The conditions for full coverage of the
markets for both strategies are defined. It is obtained that the information asymmetry
leads to asymmetry of equilibrium locations of firms. It is established that under
differentiation strategy, the follower can due to the space factor overcome
information asymmetry in the nearest markets and get a profit more than the leader.
It is proved that spatial differentiation enhances a market power of firms and allows
monopolization of neighboring markets.
The purpose of further research is to analyze the competitive interaction of firms
in the Hotelling's linear city model under the conditions of other asymmetry types.
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Стаття надійшла до редакції 29.10.2018.
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