IV. MODELAREA MATEMATICĂ ŞI
OPTIMIZAREA ÎN TRANSPORT
LOCAL COURNOT–NASH EQUILIBRIA FOR DUOPOLY AND
TRIOPOLY WITH LINEAR AND QUADRATIC COSTS
V. GORBACHUK,
B. CHUMAKOV
Cybernetics Institute, National Academy of Sciences
of Ukraine, Kyiv
GorbachukVasyl@netscape.net
Summary. The concept of local Cournot–Nash equilibrium has been introduced. All
local Cournot–Nash equilibria for duopoly with general linear and quadratic costs are
explicitly presented, and all local Cournot–Nash equilibria for triopoly with general linear
costs are explicitly presented.
Key words: Cournot, Nash, homogenous product, market, local equilibria, fixed costs,
duopoly, triopoly.
Let the market price of homogenous product is determined by
(1)
P = a − b Q,
where Q is the quantity of a given product on the market, a , b are some
positive parameters (of demand on the product). If the product is going to
the market from firms 1 and 2 only, such a market is called a duopoly with
Q = q1 + q 2,
(2)
q
where i is the output of product by the firm i = 1, 2 ; if the product is going
to the market from firms 1, 2, 3 only, then such a market is called a triopoly
with
(3)
Q = q1 + q 2 + q3,
where qi is the output of product by the firm i = 1, 2, 3.
Each firm i is maximizing over qi ∈ [0, ∞) its profit
(4)
π i = P qi − C i ( qi ),
where Ci (qi ) is the cost function of firm i . If this function is linear, then
(5)
Ci (qi ) = ci qi + Fi ,
where ci > 0 are average per unit variable costs, Fi > 0 are fixed costs of
the firm i ; if this function is quadratic, then
233
Ci (qi ) = d i (qi ) 2 + ci qi + Fi ,
(6)
where d i qi + ci are average per unit variable costs, Fi > 0 are fixed costs
of the firm i , d i > 0. A duopoly (triopoly) is called heterogeneous if ci ≠ c j
and / or Fi ≠ F j and / or d i ≠ d j for some firms i and j ≠ i of this duopoly
(triopoly) [1].
Taking into account the participation constraint or entry constraint, the
payoff Π i of firm i is equal to π i if π i ≥ 0 ; if π i < 0 and / or qi = 0 , then
Π i = 0 (the firm i does not enter the market):
π i ≥ 0;
⎧π ,
(7)
Πi = ⎨ i
⎩ 0, (π i < 0) ∨ (qi = 0).
A local Cournot–Nash equilibrium for functions Π i , i = 1, 2, 3, is such a
C
combination of q1 , q 2C , q3C ∈ [0, ∞) , that the inequalities
Π1 (q1C , q 2C , q3C ) ≥ Π1 (q1 , q 2C , q3C ) ∀ q1 ∈ [q1C − ε, q1C + ε],
Π 2 (q1C , q 2C , q3C ) ≥ Π 2 (q1C , q 2 , q3C ) ∀ q 2 ∈ [q 2C − ε, q 2C + ε],
C
C
Π 3 (q1C , q 2C , q3C ) ≥ Π 2 (q1C , q 2C , q3 ) ∀ q3 ∈ [q3 − ε, q3 + ε].
(8)
hold true for some ε > 0 . If the relationships (8) hold true ∀ ε > 0, then a
local equilibrium is also the global equilibrium.
C
C
C
C
Denote q = q1 + q 2 + q3 . If the functions π i , i = 1, 2, 3, are of the
form (4), then a Cournot–Nash equilibrium is a solution of the variational
inequality
3 ⎡
⎤
∂ Ci (qiC ) ∂ P (q C ) C
−
qi − P (q C )⎥ (qi − qiC ) ≥ 0 ∀ qi ∈ [0, ∞). (9)
⎢
∂ qi
⎥⎦
i =1 ⎢
⎣ ∂ qi
The relationships (1), (2), (4), (5) imply the value of profit function for
firm i = 1, 2 in the duopoly with linear costs
S i [a b(q1 q2 )] qi ci qi Fi [a b (q1 q2 ) ci ] qi Fi ; (10)
∑
The relationships (1), (3)–(5) imply the value of profit function for firm
i = 1, 2, 3 in the triopoly with linear costs
(11)
π i = [a − b (q1 + q 2 + q3 ) − ci ] qi − Fi ;
the relationships (1), (2), (4), (6) imply the value of profit function for firm
i = 1, 2 in the duopoly with quadratic costs
(12)
π i = [a − b (q1 + q 2 ) − ci ] qi − d i (qi ) 2 − Fi ;
234
the relationships (1), (3), (4), (6) imply the value of profit function for firm
i = 1, 2, 3 in the triooply with quadratic costs
S i [a b(q1 q 2 q3 )] qi d i (qi ) 2 ci qi Fi .
(13)
Note the variational inequality (9) does not take into account the fixed
costs Fi and therefore does not take into account the participation constraints
(7) [2]. Besides, the variational inequality (9) does not take into account the
non-negativity condition for price P, determined by (1).
Theorem 1. Let the duopoly conditions (1), (2), (4), (5) hold true, and
the profit of firm i = 1, 2 is determined by the relationship (10). Assume
a − 2 c2 + c1 > 0.
a − 2 c1 + c2 > 0 ,
Then the firm 1, maximizing over q1 ≥ 0 its profit
S1 [a b(q1 q 2 )] q1 c1q1 F1 [a b (q1 q 2 ) c1 ] q1 F1,
May achieve the only following local maxima of its payoff Π1 :
1a)
2
a − 2 c1 + c2
(a − 2 c1 + c2 ) 2
, if (a − 2 c1 + c2 ) ≥ F1 ,
− F1 under q1 =
3b
9b
9b
(a − 2 c2 + c1 ) 2
≥ F2 ;
9b
2
a − c1
(a − c1 )
(a − c1 ) 2
1b)
, if q 2 = 0 ,
− F1 under q1 =
≥ F1 ;
2b
4b
4b
1c) 0, if q1 = 0, including the case
⎛ a − c1 − 2 b F1 a − c1 + 2 b F1
q2 ∈ ⎜
,
⎜
b
b
⎝
⎞
⎟ ∩ [0, ∞) .
⎟
⎠
The similar takes place for the firm 2.
The condition 1c) shows that fixed costs are barriers to entry.
Corollary 1. The local maximum of Π 1 over q1 ≥ 0 equals to 0:
In the case 1a), if
(a − 2 c1 + c2 ) 2
= F1 ;
9b
in the case 1b), if
(a − c1 ) 2
= F1 ;
4b
in the case q1 = 0 .
Theorem 2. Under the conditions of theorem 1, all the Cournot–Nash
equilibria are given by 4 situations:
235
a − 2 c1 + c2
a − 2 c2 + c1
, q2 =
;
3b
3b
a − c1
(a − c1 ) 2
, q 2 = 0 , if
2L2) in the case 1b1) q1 =
≥ F1 ;
2b
4b 2
a − c2
(a − c2 )
≥ F2 ;
2L3) in the case 1b2) q 2 =
, q1 = 0 , if
2b
4b
2L4) in the case q1 = 0 , q 2 = 0 .
Theorem 3. Assume that under the conditions of theorem 1 we have
a − 2 c1 + c2 ≤ 0 .
Then firm 1 maximizing over q1 ≥ 0 its profit
S1 [a b(q1 q 2 )] q1 c1q1 F1 [a b (q1 q 2 ) c1 ] q1 F1,
May achieve only the local maxima 1b), 1c) of its payoff Π1 .
The similar takes place for firm 2.
Theorem 4. Under the conditions of theorem 3 all the Cournot–Nash
equilibria are given by 3 situations 2L2)–2L4).
Lemma 1. c2 < c1 under the conditions of theorem 3.
Therefore, if a > c1 , then the both situations 2L2), 2L3) are feasible.
If a ≤ c1 , then the situation 2L2) is not feasible. If a ≤ c2 , then the only
Cournot–Nash equilibrium is the situation 2L4).
Theorem 5. Assume that under the conditions of theorem 1 we have
a − 2 c1 + c2 ≤ 0,
a − 2 c2 + c1 ≤ 0 .
Then the only Cournot–Nash equilibrium is the trivial situation 2L4).
Theorem 6. Let the triopoly conditions (1), (3)–(5) hold true, and the
profit of firm i = 1, 2, 3 is determined by the relationship (11). Suppose
a − 3 c2 + c1 + c3 > 0 ,
a − 3 c1 + c2 + c3 > 0 ,
2L1) in the case 1a) q1 =
a − 2 c1 + c2 > 0 ,
a − 3 c3 + c1 + c2 > 0 ,
a − 2 c2 + c1 > 0,
a − 2 c 2 + c3 > 0 ,
a − 2 c3 + c1 > 0 ,
Then the firm 1, maximizing over q1 ≥ 0 its profit
S1
[a b(q1 q 2 q3 )] q1 c1q1 F1
a − 2 c1 + c3 > 0 ,
a − 2 c3 + c 2 > 0 .
[a b (q1 q 2 q3 ) c1 ] q1 F1 ,
May achieve only the following local maxima of its payoff Π1 :
2
6a) (a 3 c1 c2 c3 ) F1 under q1
16 b
236
a 3 c1 c2 c3
, if
4b
(a 3 c2 c1 c3 ) 2
(a 3 c1 c 2 c3 ) 2
t F2 ,
t F1 ,
16 b
16 b
(a 3 c3 c1 c 2 ) 2
t F3 ;
16 b
6b)
a − c1
(a − c1 ) 2
(a − c1 ) 2
, if q 2 = 0 = q3 ,
− F1 under q1 =
≥ F1;
2b
4b
4b
2
2
a − 2 c1 + c2
, if (a − 2 c1 + c2 ) ≥ F1 ,
6c) (a − 2 c1 + c2 ) − F1 under q1 =
3b
9b
9b
(a − 2 c2 + c1 ) 2
≥ F2 , q3 = 0;
9b
2
a − 2 c1 + c3
(a − 2 c1 + c3 ) 2
6d)
, if (a − 2 c1 + c3 ) ≥ F1,
− F1 under q1 =
3b
9b
9b
2
(a − 2 c3 + c1 )
≥ F3, q 2 = 0 ;
9b
6e) 0, if q1 = 0 , including
⎛ a − c1 − 2 b F1 a − c1 + 2 b F1 ⎞
⎟ ∩ [0, ∞).
q 2 + q3 ∈ ⎜
,
⎜
⎟
b
b
⎝
⎠
The similar takes place for firms 2 and 3.
Remark 1. Note that case 6e) presumes a duopoly of firms 2 and 3 and
therefore applies the theorems 1–5.
Therefore, in the case similar to 1a) we have
a − 2 c 2 + c3 a − 2 c3 + c 2
q 2 + q3 =
+
=
3b
3b
=
2 a − c2 − c3 ⎛⎜ a − c1 − 2 b F1 a − c1 + 2 b F1
∈
,
⎜
b
b
3b
⎝
⎞
⎟ ∩ [0, ∞) .
⎟
⎠
wherefrom
3 a − 3 c1 − 6 b F1 < 2 a − c2 − c3 < 3 a − 3 c1 + 6 b F1 under
a − c1 − 2 b F1 > 0 ,
0 ≤ 2 a − c2 − c3 < 3a − 3c1 + 6 b F1 under a − c1 − 2 b F1 ≤ 0 .
It means that
237
− 6 b F1 < a − 3 c1 + c2 + c3 < 6 b F1 under a − c1 > 2 b F1 ,
− 6 b F1 ≤ a − 3 c1 + c2 + c3 under a − c1 ≤ 2 b F1.
As the conditions of theorem 6 imply the latter automatically, then under
6e) in the case similar to 1a) we obtain
a − 3 c1 + c2 + c3 < 6 b F1 .
Under 6c) in the case similar to 1b1) we have q 2 = 0,
a − c3 ⎛ a − c1 − 2 b F1 a − c1 + 2 b F1 ⎞
⎟ ∩ [0, ∞),
q3 =
∈⎜
,
2b
⎜
⎟
b
b
wherefrom
⎝
⎠
2 a − 2c1 − 4 b F1 < a − c3 < 2 a − 2 c1 + 4 b F1 under
a − c1 − 2 b F1 > 0,
0 ≤ a − c3 < 2 a − 2 c1 + 4 b F1 under a − c1 − 2 b F1 ≤ 0 .
It means that
− 4 b F1 < a − 2 c1 + c3 < 4 b F1 under a − c1 > 2 b F1 ,
− 4 b F1 ≤ a − 2 c1 + c3 under a − c1 ≤ 2 b F1 .
As the conditions of theorem 6 imply the latter automatically, then under
6e) in the case similar to 1b1) we have
a − 2 c1 + c3 < 4 b F1 .
Under 6e) in the case similar to 1b2) we have
a − 2 c1 + c 2 < 4 b F1 .
Theorem 7. Under the conditions of theorem 6 all the Cournot–Nash
equilibria are given by 8 situations:
a − 3 c1 + c2 + c3
a − 3 c2 + c1 + c3
7L1) in the case 6a) q1 =
, q2 =
,
4
b
4
b
a − 3 c3 + c1 + c2
q3 =
;
4b
a − c1
(a − c1 ) 2
, q 2 = 0 = q3 , if
7L2) in the case 6b1) q1 =
≥ F1;
2b
4b
7L3) in the case 6b2) q 2 =
(a − c2 ) 2
a − c2
, q1 = 0 = q3 , if
≥ F2 ;
4b
2b
7L4) in the case 6b3) q3 =
a − c3
( a − c3 ) 2
, q1 = 0 = q 2 , if
≥ F3;
2b
4b
238
7L5) in the case 6c) q1 =
a − 2 c1 + c2
a − 2 c2 + c1
, q2 =
, q3 = 0 , if
3b
3b
(a − 2 c2 + c1 ) 2
(a − 2 c1 + c2 ) 2
≥ F2;
≥ F1 ,
9b
9b
7L6) in the case 6d) q1 =
a − 2 c1 + c3
a − 2 c3 + c1
, q3 =
, q 2 = 0 , if
3b
3b
(a − 2 c3 + c1 ) 2
(a − 2 c1 + c3 ) 2
≥ F3 ;
,
≥ F1
9b
9b
7L7) in the case 6e) q 2 =
a − 2 c 2 + c3
a − 2 c3 + c 2
, q3 =
, q1 = 0 , if
3b
3b
( a − 2 c3 + c 2 ) 2
( a − 2 c 2 + c3 ) 2
≥ F3 ;
≥ F2 ,
9b
9b
7L8) in the case q1 = 0 , q 2 = 0 , q3 = 0 .
Theorem 8. Assume that under the conditions of theorem 6 we have
a − 3 c1 + c2 + c3 ≤ 0 .
Then the firm 1 maximizing over q1 ≥ 0 its profit
π1 = [a − b (q1 + q 2 + q3 ) − c1 ] q1 − F1 ,
may achieve only the local maxima 6b)–6e) of its payoff Π1.
The similar takes place for firms 2 and 3.
Theorem 9. Under the conditions of theorem 8 all the Cournot–Nash
equilibria are given by 7 situations 7L2)–7L8).
Lemma 2. c3 < c1 , c2 < c1 under the conditions of theorem 8.
Therefore, if a > c1, then the situations 7L2)–7L4) are feasible.
If a ≤ c1 , then the situation 7L2) is not feasible.
If a ≤ c3 , then the situations 7L2), 7L4) are not feasible.
Lemma 3. Under the conditions of theorem 6, the inequalities
a − 3 c1 + c2 + c3 ≤ 0 ,
a − 3 c 2 + c1 + c3 ≤ 0
are inconsistent.
Proof. The first inequality gives
0 < a − 2 c1 + c3 ≤ c1 − c2 ,
and the second inequality –
239
0 < a − 2 c2 + c3 ≤ c2 − c1 .
Remark 2. For n firms, 7L1) has the form
a − n qi +
n
∑ q j , i = 1,..., n .
j =1, j ≠i
qi =
(n + 1) b
Theorem 11. Let the duopoly conditions (1), (2), (4), (6) hold true, and
the profit of firm i = 1, 2 is determined by the relationship (12). Suppose
that
A12 { 2 (b d 2 )(a c1 ) b (a c2 ) ! 0 ,
A21 { 2 (b d1 )(a c2 ) b ( a c1 ) ! 0 ,
B12 { 4 (b d1 )(b d 2 ) b 2.
Then firm 1, maximizing over q1 ≥ 0 its profit
S1
[a b(q1 q 2 )] q1 d1 (q1 ) 2 c1q1 F1
= [a − b (q1 + q 2 ) − c1 ] q1 − d1 (q1 ) 2 − F1 ,
may achieve only the following local maxima of its payoff Π1 :
11a)
(b d1 )( A12 ) 2
( B12 ) 2
(b d 2 )( A21 ) 2
( B12 ) 2
F1
under
q1
A12
,
B12
if
(b d1 )( A12 ) 2
( B12 ) 2
t F1 ,
t F2 ;
2
a − c1
(a − c1 ) 2 b
11b) (a − c1 ) b − F1 under q1 =
, if q 2 = 0 ,
≥ F1;
2 (b + d1 )
4 (b + d1 ) 2
4 (b + d1 ) 2
11c) 0, if q1 = 0 , including the case
⎛ a − c1 − 2 (b + d1 ) F1 a − c1 + 2 (b + d1 ) F1
q2 ∈ ⎜
,
⎜
b
b
⎝
In the case 11a) we obtain
S1
F1 ⎞
⎟ ∩ [0, ∞) .
⎟
⎠
F1 ( a c1 ) q1 b q1q 2 (b d1 )(q1 ) 2
(a c1 ) A12 B12 b A12 A21 (b d1 )( A12 ) 2
( B12 ) 2
240
A12 {( a c1 ) B12 b A21} (b d1 )( A12 ) 2 .
[4 (b d1 )(b d 2 ) b 2 ]2
Here the nominator is
(a c1 ) B12 b A21
F1 4 (a c1 )(b d1 )(b d 2 ) b 2 (a c1 ) 2 b (b d1 )(a c2 ) b 2 ( a c1 )
wherefrom
2 (b d1 )[2 (b d 2 )(a c1 ) b (a c2 )] 2 (b d1 ) A12 ,
(b d1 )( A12 ) 2 .
( B12 ) 2
The theorem 1 is a particular case of the theorem 11 under d1 = 0 = d 2 .
Lemma 3. (a c1 ) B12 b A21 2 (b d1 ) A12 under the conditions of
theorem 11.
S1
F1 In particular, under d1 = 0 = d 2 we have: B12
A12
2 b (a c1 ) b (a c2 ) ,
(a c1 ) B12 b A21
A21
3b 2 ,
2 b (a c2 ) b (a c1 ),
3 b 2 (a c1 ) 2 b 2 (a c 2 ) b 2 (a c1 )
4 b 2 (a c1 ) 2 b 2 (a c2 )
2 b [2 b (a c1 ) b (a c2 )] 2 b A12.
The statement, similar to the theorem 11, takes place for firm 2.
Theorem 12. Under the conditions of theorem 11, all the Cournot–Nash
equilibria are given by 4 situations:
A21
A12
, q2
;
12Q1) in the case 11a) q1
B12
B12
a − c1
(a − c1 ) 2 b
12Q2) in the case 11b1) q1 =
, q 2 = 0 , if
≥ F1 ;
2 (b + d1 )
4 (b + d ) 2
1
a − c2
(a − c2 ) 2 b
12Q3) in the case 11b2) q 2 =
, q1 = 0 , if
≥ F2 ;
2 (b + d 2 )
4 (b + d 2 ) 2
12Q4) in the case q1 = 0 , q 2 = 0 .
Theorem 13. Assume that under the conditions of theorem 11 we have
A12 d 0 . Then the firm 1, maximizing over q1 ≥ 0 its profit
2
π = [a b (q1 q 2 ) c1 ] q1 d1 (q1 ) F1 ,
1
241
may achieve only local maxima 11b), 11c) of its payoff Π1 .
The similar takes place for the firm 2.
Theorem 14. Under the conditions of theorem 13, all the Cournot–Nash
equilibria are given by 3 situations 12Q2)–12Q4).
Lemma 4. 2 d 2 (a − c1 ) − 2 d1 (a − c2 ) < b (c1 − c2 ) under the conditions
of theorem 13.
Therefore, if d1 = d 2 = d , then 0 (b 2 d )(c1 c2 ) , under a > c1 the
both situations 12Q2), 12Q3) are feasible.
If a ≤ c1 , then the situationт 12Q2) is not feasible. If a ≤ c2 , then the
only Cournot–Nash equilibrium is the situation 12Q4).
References
1. Gorbachuk V. Cournot–Nash equilibria and Bertrand–Nash
equilibria for a heterogeneous duopoly of differentiated products
// Cybernetics and systems analysis. – 2010. – Vol. 46. – № 1. –
P. 25–33.
2. Gorbachuk V. Monotone operators for Cournot–Nash equilibria
search / Generalized convexity and monotonicity. – Cluj-Napoca,
Romania: Babes-Bolyai University, 2011. – P. 54–55.
242