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Dynamic Cournot oligopolies with production adjustment costs
ELSEVIER
Journal of Mathematical Economics 24 (1995) 95-101
Dynamic Cournot oligopolies with production adjustment costs Ferenc Szidarovszky
*, Jerome Yen
Department of Systems and Industrial Engineering, University ofArizona, Tucson, AZ 85721, USA
Submitted May 1992; accepted May 1993
Abstract Dynamic oligopolies with discrete time scale are examined with the additional condition that each firm faces a production adjustment cost at each time period. The global asymptotical stability of the resulting discrete system is investigated, and necessary and the sufficient conditions are presented. Keywords: Cournot market; Dynamic games JEL classification: C73
1. Introduction Dynamic oligopolies have been investigated by many authors. A comprehensive summary of earlier works is presented in Okuguchi (1976) and Okuguchi and Szidarovszky (1990) for single-product and multi-product firms, respectively. These results are further extended by Szidarovszky (1991), when market saturation is also considered. Another generalization is presented in Szidarovszky et al. (1992) when it is assumed that at each time period each firm fiit computes its profit optimizing output, and then moves its output in the direction of the profit optimizing output but not necessarily makes the output change up to the profit maximizing one.
* Corresponding author. This research was supported by the National Science Foundation (NSF SES 9023055). 0304-4068/95/$09.50 8 1995 Elsevier Science B.V. All rights reserved SSDI 0304-4068(94)00661-X
In this paper a new extension will be introduced, when a new cost is introduced due to changing the output in any time period. If zero cost is assumed, then our results coincide with the earlier stability results as they should. The paper is developed as follows. In the second section the mathematical model is introduced, and stability conditions are presented in the third section. The last section concludes the paper.
2. The mathematical model Consider an N-fii oligopoly without production differentiation. Let p denote the inverse demand function and let C, be the cost function of firm k. Therefore the profit of firm k is given as N ‘P&q,
X2,.‘.,
%d
=xRP
CXt - Gh), i I= I 1
(24
where xt, x2,. . . , xN denote the output of the firms. Assume that (i) p(s) = b -As, where A and b are positive constants; (ii) for all k, C,(x,) = bkxk + ck, where b, and ck are also positive. Assume that at t = 0, each firm selects an initial output x,(O), and at each later time period t > 0, they first predict the output of the rest of the industry by $01 = c I+ k~r(t - 11, and then they select the profit rn~irn~~g output. Mathematically this selection can be formulated as for each k, 4
0 = argmax{ xkp( %+&Q)
-C,W
-G(W-xk(~-
I))}, (2.2)
where the last term is the cost of changing output. Assume that (iii) for all k, c,( 6 ) = K, 1 6 ‘, where K, > 0. Assume that xk(t) is positive, then simple differentiation shows that x,(t) satisfies equation b-A(x,(t)+s,E(t))-~k(t)-bL-2Kk.(x~(t)-nk(t-1))=0, that is, b-b,+2K,x,(t-l)-As;(f) %(4
=
2Af2K,
I
Substitute the definition of s!(t) into this equation to see that vector .x(t) = (x,(t), f . . , x,(t)jT satisfies the following first-order difference equation: X(t) =H*n(t-
1) i-h,
(2.3)
F. Sziabrovszky, J. Yen/Journal
97
of Mathematical Economics 24 (I 995) 95-101
where h is a constant vector and
K,
A
A+K,
-2(A+K,)
A
K, A+K,
-
2(A +Kd
H=
‘*.
2(A+K,)
- 2(A+K,,,)
- 2(A+K,) A
“.
‘A
‘A
-
A
***
- 2(A +Kd
KN A+&
For all k, introduce the notation a”=-&
(k=l,2
,..., N).
k
Then matrix H can be simplified as /
1-q --
HZ
a2
2
-\
(YN
2
-:
...
-$
l-o,
...
-:
...
1 - ffj
--
(YN
2
(2.4)
In the next part of this paper the stability of system (2.3) will be investigated.
3. Stability analysis It is well known form the theory of dynamic systems (see, for example, Szidarovszky and Bahill, 1992) that system (2.3) is globally asymptotically stable if and only if all eigenvalues of H are inside the unit circle. Since the eigenvalues of any matrix are bounded by its norm, we have the following theorem. Theorem 3.1. Assume that only two firms are in the market. Then system (2.3) is globally asymptotically stable. Proof Notice first that (YeE (0,l) for all k, therefore the row norm of H is less than one, if 1-a,+$+...
++.
98
F. Szihrovszky, J. Yen/Journal of Mathematical Economics 24 (1995) 95-101
This inequality is equivalent to the following: 1>1-cu,+
(N-
11%
=1+
2 which holds if only if N = 2.
(N-W% 2
’
0
Remark The condition of the theorem is sufficient but not necessary. A neccssary and sufficient condition will be derived next, which will require the determination of the eigenvalues of matrix H. First we prove the following lemma. Lemma 3.1. Let a = (a,> and b = (bi) be real N-dimensional vectors. Then det(l+ab*) = 1 + b*a. Proof Finite induction is used. If N = 1, then I + b*a = 1 + a,b,, assertion holds. Assume now that it is valid for all i < k. Then a,b,
1 + a,b,
a,b, det( I + abT) = det
a,b,
...
so the
l+a,b,
a,b,
...
a,b, a,b,
a,b,
l+a,b,
...
a,b,
a,b,
a,b, ... 1 + hrb, a,b, a,b, Substract the (ak/ak_ ,)-multiple of row k - 1 from row k, then substract the (ak_ ,/a,_,)-multiple of row k - 2 from row k - 1, and so on, and finally, substract the (aJar)-multiple of the first row from the second row to see that +a,b, --
a,b,
a2
a,b,
...
albk-1
a14
1
Ql --
a3
1
a2
det( I + ab*) = de1 --
a4
*
a3
*
--
ak ak-
1 1
Let D, denote this determinant. Expanding this determinant with respect to the last column we see that D,=D,_,.l+(_l)“.$!$
....ak(-l)k+la,bk ak-
=D,_,+a,b,.
1
F. Sziabrovsrky J. Yen /Journal
of Mathematical Economics 24 (1995) 95-101
99
Since D, = 1 + a,b,, the inductive hypothesis implies that k
I** +a,b,=l+
D,=l+a,b,+a,b,+
xaj.bj=l+bTa, j-1
q
which proves the assertion.
CoroiLary. Let G be an N x N rest ma&k, and iet u and b be N-di~e~ioffal real vectors. Then the chara~ter~t~~~oiy~o~ial of G + ab’ is as follows: det( G + ab’ - AZ)=det(G-AZ). =det(G-AZ).(l
det(Z+(G-AZ))‘nbT) +bT(G-AZ)-‘a).
(3.1)
Notice that in our case
1)7
H = diag 1 (
therefore from E@ (3.1) we conclude that the characteristic polynomial of H has the special form ip(A)=(l-~-A)~...+-+
. l-~,l_~~~_A]. )[
(3.2) Assume that the aj’s are ordered so that the distinct values of cy, > cr2 > . . . > a, are repeated with multiplicities M,, m2,. . . , m,, respectively. If mj = 1 with some j, then the factor 1 - cuj/2 - A cancels out, so 1 - aj/2 is not an eigenvalue of ZZ. Otherwise it is an eigenvalue of H with multiplicity brj - 1. The other eigenvalues are the roots of equation
jgl
2 _y
2A
=
(3.3)
l.
I
Let g(A) denote the left-hand side of this equation. Easy to see that g is locally strictly increasing, furthermore A+
lim g(A) = ,!y=g( -02
A)
= 0,
100
F. Szidarouszky, J. Yen /Journal
of Mathematical Economics 24 (1995) 95-101
Notice that by multiplying both sides of Eq. (3.3) by the common denominator an rth-order polynomial equation is obtained, therefore all roots are real, and one real root can be found in each of the intervals 01
(
-cQ,l--,
l-$,1-$ )...) l2 )( ) ( All roots are in (- 1,l) if and only if for all j,
F,I-?. 1
-l
The first condition can be rewritten as 0 < aj < 4, which always holds as a consequence of the definition of aj. The second condition can be simplified as
By using the definition of aj again, this inequality can be rewritten as N A/(A+Kj) c j=14-A/(A+Kj)
<”
that is, jc13A:4K.
I <”
(3.4)
Hence we proved the following theorem. Theorem 3.2. holds. Remark.
System (2.3) is globally asymptotically stable if and only if (3.4)
If N = 2, then
Therefore this theorem implies our earlier Theorem 3.1. Assume next that N = 3. If all Kj = 0, then (3.4) does not hold. However, if Kj > 0 for at least one j, then
Therefore the system is globally asymptotically stable.
F. Szidarovsrky, J. Yen/Journal of Mathematical Economics 24 (I 995) 95-101
101
Consider next the special case, when Kj = K, that is, the firms have identical costs of changing outputs. In this special case inequality (3.4) can be rewritten as
NA 3A+4K
< 1,
that is, (N - 3)A < 4K. If N = 2, then this relation always holds. If N = 3 and
K = 0,then this inequality is false, however if N = 3 and K > 0,then this relation holds again. If N > 3, then it holds if and only if K>
w-v 4
*
This condition can be interpreted as sufficiently large value of K implies the global asymptotical stability of the system.
4.Conclusions Dynamic single product oligopolies with Coumot expectations were examined under the additional condition that each firm faces an additional quadratic cost for changing the output. The necessary and sufficient condition for the global asymptotical stability of the resulting discrete system was derived, and special cases examined. Multi-product oligopolies, and other types of expectations (such as, adaptive, extrapolative, etc.) can be discussed in a similar way.
Okuguchi, K., 1976, Expectations and stability in oligopoly models (Springer-Verlag, Berlin). Okuguchi, K. and F. Szidarovszky, 1990, The theory of oligopoly with multi-product firms (SpringerVerlag, Berlin). Szidarovszky, F., 1991, Multiproduct Coumot oligopolies with market saturations. Pure Math. and Appl., Ser. B, 1, no. 1, 3-15. Szidarovszky, F., S. Rassenti and J. Yen, 1992, Modified Coumot expectations in dynamic oligopolies, Paper presented at the IFAC/IFORS/IJASA Conference on Dynamic National Economics, Aug. 18-22, 1992, Beijing, China. Szidarovszky, F. and A.T. Bahill, 1992, Linear systems theory (CRC Press, Boca Raton, FL).
Journal of Mathematical Economics 24 (1995) 95-101
Dynamic Cournot oligopolies with production adjustment costs Ferenc Szidarovszky
*, Jerome Yen
Department of Systems and Industrial Engineering, University ofArizona, Tucson, AZ 85721, USA
Submitted May 1992; accepted May 1993
Abstract Dynamic oligopolies with discrete time scale are examined with the additional condition that each firm faces a production adjustment cost at each time period. The global asymptotical stability of the resulting discrete system is investigated, and necessary and the sufficient conditions are presented. Keywords: Cournot market; Dynamic games JEL classification: C73
1. Introduction Dynamic oligopolies have been investigated by many authors. A comprehensive summary of earlier works is presented in Okuguchi (1976) and Okuguchi and Szidarovszky (1990) for single-product and multi-product firms, respectively. These results are further extended by Szidarovszky (1991), when market saturation is also considered. Another generalization is presented in Szidarovszky et al. (1992) when it is assumed that at each time period each firm fiit computes its profit optimizing output, and then moves its output in the direction of the profit optimizing output but not necessarily makes the output change up to the profit maximizing one.
* Corresponding author. This research was supported by the National Science Foundation (NSF SES 9023055). 0304-4068/95/$09.50 8 1995 Elsevier Science B.V. All rights reserved SSDI 0304-4068(94)00661-X
In this paper a new extension will be introduced, when a new cost is introduced due to changing the output in any time period. If zero cost is assumed, then our results coincide with the earlier stability results as they should. The paper is developed as follows. In the second section the mathematical model is introduced, and stability conditions are presented in the third section. The last section concludes the paper.
2. The mathematical model Consider an N-fii oligopoly without production differentiation. Let p denote the inverse demand function and let C, be the cost function of firm k. Therefore the profit of firm k is given as N ‘P&q,
X2,.‘.,
%d
=xRP
CXt - Gh), i I= I 1
(24
where xt, x2,. . . , xN denote the output of the firms. Assume that (i) p(s) = b -As, where A and b are positive constants; (ii) for all k, C,(x,) = bkxk + ck, where b, and ck are also positive. Assume that at t = 0, each firm selects an initial output x,(O), and at each later time period t > 0, they first predict the output of the rest of the industry by $01 = c I+ k~r(t - 11, and then they select the profit rn~irn~~g output. Mathematically this selection can be formulated as for each k, 4
0 = argmax{ xkp( %+&Q)
-C,W
-G(W-xk(~-
I))}, (2.2)
where the last term is the cost of changing output. Assume that (iii) for all k, c,( 6 ) = K, 1 6 ‘, where K, > 0. Assume that xk(t) is positive, then simple differentiation shows that x,(t) satisfies equation b-A(x,(t)+s,E(t))-~k(t)-bL-2Kk.(x~(t)-nk(t-1))=0, that is, b-b,+2K,x,(t-l)-As;(f) %(4
=
2Af2K,
I
Substitute the definition of s!(t) into this equation to see that vector .x(t) = (x,(t), f . . , x,(t)jT satisfies the following first-order difference equation: X(t) =H*n(t-
1) i-h,
(2.3)
F. Sziabrovszky, J. Yen/Journal
97
of Mathematical Economics 24 (I 995) 95-101
where h is a constant vector and
K,
A
A+K,
-2(A+K,)
A
K, A+K,
-
2(A +Kd
H=
‘*.
2(A+K,)
- 2(A+K,,,)
- 2(A+K,) A
“.
‘A
‘A
-
A
***
- 2(A +Kd
KN A+&
For all k, introduce the notation a”=-&
(k=l,2
,..., N).
k
Then matrix H can be simplified as /
1-q --
HZ
a2
2
-\
(YN
2
-:
...
-$
l-o,
...
-:
...
1 - ffj
--
(YN
2
(2.4)
In the next part of this paper the stability of system (2.3) will be investigated.
3. Stability analysis It is well known form the theory of dynamic systems (see, for example, Szidarovszky and Bahill, 1992) that system (2.3) is globally asymptotically stable if and only if all eigenvalues of H are inside the unit circle. Since the eigenvalues of any matrix are bounded by its norm, we have the following theorem. Theorem 3.1. Assume that only two firms are in the market. Then system (2.3) is globally asymptotically stable. Proof Notice first that (YeE (0,l) for all k, therefore the row norm of H is less than one, if 1-a,+$+...
++.
98
F. Szihrovszky, J. Yen/Journal of Mathematical Economics 24 (1995) 95-101
This inequality is equivalent to the following: 1>1-cu,+
(N-
11%
=1+
2 which holds if only if N = 2.
(N-W% 2
’
0
Remark The condition of the theorem is sufficient but not necessary. A neccssary and sufficient condition will be derived next, which will require the determination of the eigenvalues of matrix H. First we prove the following lemma. Lemma 3.1. Let a = (a,> and b = (bi) be real N-dimensional vectors. Then det(l+ab*) = 1 + b*a. Proof Finite induction is used. If N = 1, then I + b*a = 1 + a,b,, assertion holds. Assume now that it is valid for all i < k. Then a,b,
1 + a,b,
a,b, det( I + abT) = det
a,b,
...
so the
l+a,b,
a,b,
...
a,b, a,b,
a,b,
l+a,b,
...
a,b,
a,b,
a,b, ... 1 + hrb, a,b, a,b, Substract the (ak/ak_ ,)-multiple of row k - 1 from row k, then substract the (ak_ ,/a,_,)-multiple of row k - 2 from row k - 1, and so on, and finally, substract the (aJar)-multiple of the first row from the second row to see that +a,b, --
a,b,
a2
a,b,
...
albk-1
a14
1
Ql --
a3
1
a2
det( I + ab*) = de1 --
a4
*
a3
*
--
ak ak-
1 1
Let D, denote this determinant. Expanding this determinant with respect to the last column we see that D,=D,_,.l+(_l)“.$!$
....ak(-l)k+la,bk ak-
=D,_,+a,b,.
1
F. Sziabrovsrky J. Yen /Journal
of Mathematical Economics 24 (1995) 95-101
99
Since D, = 1 + a,b,, the inductive hypothesis implies that k
I** +a,b,=l+
D,=l+a,b,+a,b,+
xaj.bj=l+bTa, j-1
q
which proves the assertion.
CoroiLary. Let G be an N x N rest ma&k, and iet u and b be N-di~e~ioffal real vectors. Then the chara~ter~t~~~oiy~o~ial of G + ab’ is as follows: det( G + ab’ - AZ)=det(G-AZ). =det(G-AZ).(l
det(Z+(G-AZ))‘nbT) +bT(G-AZ)-‘a).
(3.1)
Notice that in our case
1)7
H = diag 1 (
therefore from E@ (3.1) we conclude that the characteristic polynomial of H has the special form ip(A)=(l-~-A)~...+-+
. l-~,l_~~~_A]. )[
(3.2) Assume that the aj’s are ordered so that the distinct values of cy, > cr2 > . . . > a, are repeated with multiplicities M,, m2,. . . , m,, respectively. If mj = 1 with some j, then the factor 1 - cuj/2 - A cancels out, so 1 - aj/2 is not an eigenvalue of ZZ. Otherwise it is an eigenvalue of H with multiplicity brj - 1. The other eigenvalues are the roots of equation
jgl
2 _y
2A
=
(3.3)
l.
I
Let g(A) denote the left-hand side of this equation. Easy to see that g is locally strictly increasing, furthermore A+
lim g(A) = ,!y=g( -02
A)
= 0,
100
F. Szidarouszky, J. Yen /Journal
of Mathematical Economics 24 (1995) 95-101
Notice that by multiplying both sides of Eq. (3.3) by the common denominator an rth-order polynomial equation is obtained, therefore all roots are real, and one real root can be found in each of the intervals 01
(
-cQ,l--,
l-$,1-$ )...) l2 )( ) ( All roots are in (- 1,l) if and only if for all j,
F,I-?. 1
-l
The first condition can be rewritten as 0 < aj < 4, which always holds as a consequence of the definition of aj. The second condition can be simplified as
By using the definition of aj again, this inequality can be rewritten as N A/(A+Kj) c j=14-A/(A+Kj)
<”
that is, jc13A:4K.
I <”
(3.4)
Hence we proved the following theorem. Theorem 3.2. holds. Remark.
System (2.3) is globally asymptotically stable if and only if (3.4)
If N = 2, then
Therefore this theorem implies our earlier Theorem 3.1. Assume next that N = 3. If all Kj = 0, then (3.4) does not hold. However, if Kj > 0 for at least one j, then
Therefore the system is globally asymptotically stable.
F. Szidarovsrky, J. Yen/Journal of Mathematical Economics 24 (I 995) 95-101
101
Consider next the special case, when Kj = K, that is, the firms have identical costs of changing outputs. In this special case inequality (3.4) can be rewritten as
NA 3A+4K
< 1,
that is, (N - 3)A < 4K. If N = 2, then this relation always holds. If N = 3 and
K = 0,then this inequality is false, however if N = 3 and K > 0,then this relation holds again. If N > 3, then it holds if and only if K>
w-v 4
*
This condition can be interpreted as sufficiently large value of K implies the global asymptotical stability of the system.
4.Conclusions Dynamic single product oligopolies with Coumot expectations were examined under the additional condition that each firm faces an additional quadratic cost for changing the output. The necessary and sufficient condition for the global asymptotical stability of the resulting discrete system was derived, and special cases examined. Multi-product oligopolies, and other types of expectations (such as, adaptive, extrapolative, etc.) can be discussed in a similar way.
Okuguchi, K., 1976, Expectations and stability in oligopoly models (Springer-Verlag, Berlin). Okuguchi, K. and F. Szidarovszky, 1990, The theory of oligopoly with multi-product firms (SpringerVerlag, Berlin). Szidarovszky, F., 1991, Multiproduct Coumot oligopolies with market saturations. Pure Math. and Appl., Ser. B, 1, no. 1, 3-15. Szidarovszky, F., S. Rassenti and J. Yen, 1992, Modified Coumot expectations in dynamic oligopolies, Paper presented at the IFAC/IFORS/IJASA Conference on Dynamic National Economics, Aug. 18-22, 1992, Beijing, China. Szidarovszky, F. and A.T. Bahill, 1992, Linear systems theory (CRC Press, Boca Raton, FL).