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On the stability of the stackelberg oligopoly equilibrium
Economics Letters 3 (1979) 0 North-Holland Publishing
ON THE STABILITY
321-325 Company
OF THE STACKELBERG
OLIGOPOLY EQUILIBRIUM
Koji OKUGUCHI University of New South Wales, Kensington, NSW 2033, Australia Tokyo Metropolitan University, Tokyo, Japan Received
8 November
1979
The stability of the equilibrium in the Stackelberg undifferentiated oligopoly model where each oligopolist strives for leadership is analyzed for general demand and cost functions and for a discrete system with non-instantaneous adjustment.
1. Introduction The stability of the equilibrium of the Stackelberg leader-leader duopoly model has been analyzed by Okuguchi (1971, 1976) for the case of perfect information and by Negishi and Okuguchi (1972) when information is imperfect. Hathaway, Howroyd and Rickard (1979) have derived the stability conditions for the Stackelberg oligopoly model where each firm strives for leadership. In this paper we shall extend the Hathaway-Howroyd-Rickard results for general demand and cost functions. We shall, however, assume away product differentiation and externalities in the cost functions. 2. The stability condition Let the demand function
P=f(Cxi) i
for a homogeneous
product be
3
where p is the price of the homogeneous product and Xi is the output of the ith firm, i = 1,2, . . . . n. The ith firm’s cost function is given as Ci = Ci(Xi),
i = 1 , 2, .... n.
Each firm strives to be a 1 eader and assumes that all other firms will behave as followers. Hence the jth firm’s output in the tth period expected by all other firms, x?(t), is obtained by maximizing
322
K. Okuguchi / The stability of the Stackelberg oligopoly equilibrium
with respect to xi’(t). For the sake of simplicity let us define XMiz Cxj,
i=
The maximizing
condition
j+i
f(X-j(t
1,2, . .. . n.
- 1) +X:(f))
is
+ Xf(t)f’(X-‘(t
- 1) + Xi”(t)) - Ci(Xy(t)) = 0,
(1)
j = 1 , 2, . .., n,
where possibility
of a corner maximum
XF(t)Ehj(Xx-j(t-
is assumed away. From (1) we have
i= 1,2, .. .. n.
l)),
(2)
The ith firm’s expected profit in the tth period n:(t) is therefore given as a;(t) =Xi(f)f(,.
hj(X-j(t
- 1)) + xi(f))
- Ci(Xi(t>)>
i = 1, 2, .. .. n,
(3)
which, maximized with respect to xi(t), yields f$g
hj(X-j(t-
hj(x-j(t - l)) tXi*(t)) +XJ(t)f'(gi
X [lt,z,h;(x-j(t-l))]=O,
i=1,2
where xf 0) denotes the maximizing Xf(t)‘~i(,~,hj(X-j(t-1))))
l>>+XT(t))
,..., n,
(4)
output. We then have i=1,2
,..., n.
(5)
Adjustment of actual output Xi(t) to the profit maximizing instantaneous. Hence Xi(f) - Xi(t - 1) = ki(Xf (t) - Xj(t - l)),
where ki’s are positive constants Xi(t) = (1 - ki)Xi(t - 1) + kigi(]g EZGi(X(t - l)),
i = 1,
(adjustment hj(X-j(t
i = 1,
one is not necessarily
2, . . ., n,
coefficients).
(6)
Define
- 1)))
2, . . . , n,
(7)
where x(l - 1) is the vector of all firms’ outputs in the (t - 1)th period. The stationary value to (7) defines the equilibrium of the Stackelberg oligopoly model where each firm strives for leadership. In the following stability analysis we assume differentiability of relevant functions of any order as desired. Totally differentiate (4) and re-arrange to get [r’ + (1 + ,Gh;(x-j(r
- l)))j-’ + (1 + ,c, h;(x-j(t
- 1))) x’(t)f”
- Ci”] dxf (t)
323
K. Okuguchi / The stability of the Stackelberg oligopoly equilibrium
+ If’
+ (1 + c $(x-j@
- 1))) x&f”1
]Z dxF(t)
j#i
t
Xl?(t)f’
c hl’ d.x-j(l -
j#i
j#i,
1) = 0,
i= 1,2, .. . . IZ.
(8)
Hence
ax;(t)/ax~(t)=-[f’t
(1 t,$,)Xf(T)f”l
/if’+
(I $$)f’t
(1 t,,.h;)Xl(f)l.“-
3 --(y.I? Totally differentiate h; = dxp(t)/dx-j(t = -(f’
+
~ 1) =
x3)f”)/(2f
From the definition
aXF(t)/dXi(t
+ xF(t)f”
=
% C jti
~
cwiBi,
kG
,..., n.
j#i,
i,j=1,2
,..., n.(lO)
(9)
-
1)
- Ci”)
= % kgi
a+(t)/ax,(t
- 1)
/3j
i = 1 , 2, .... n,
j ad-wxo
= ai(?‘i
i,j=l,2
(5) (9) and (10)
agi/axj(t - 1) = ,z. ag#lxT(t)
=
j#i, (1) to get
E -pi,
agi/axj(t>
Cl,
WXWxj(t
-
(11)
1)
ok
-
i#j,
Pi>,
i,j=1,2
,..., n.(12)
Suppress t - 1 in (7) and consider a mapping G(x) = (Gr(x),
G,(x), . .. . G,(x)),
which is assumed to satisfy the fundamental
5 iaGi/axjt j=1
< 1,
assumption
as given by
i = 1, 2, .., n.
We assume moreover that for any output vector x the following two inequalities
(13) are
K. Okuguchi / The stability of the Stackelberg oligopoly equilibrium
324
satisfied: tXif”(CXi)
f’(Cxj) i
f’ (C
Xi) < C;‘(Xi),
i =
1 , 2, .... n,
(14)
i =
1 , 2, ..., n.
(15)
These assumptions imply that the marginal revenue of any firm is decreasing other firm’s output and that the rate of change of each firm’s marginal cost braically greater than the rate of change of the market price with respect to in the total demand. Under assumptions (14) and (15), /J’s are shown to be From (7) and (1 l), we derive i = 1 , 2, .., n.
aGifaxi = (1 - ki) + ki”iBi,
in any is algechange positive.
(16)
Likewise from (7) and (12), aGi/aXi = kiCYi(Bi - pi),
j#i,
i, j=
1,2, . .. . n.
Taking into account (16) (17) and the positivity 5
l aGi/axjl
= I( 1 ~ ki) + kiaiBi I + lg
(17)
of (Yi’swe get,
Ikiai(Bi - Bi>l
j=l
<
i
1 - kiI + kiBi)CYiI+ kilail ,g, (Bi - Pi) i= 1,2,
= 11 ~ kil + (r~ - l)kiBilail,
n.
(18)
Hence (13) is satisfied provided that 11 -kiit(n-l)kiBiicviI
i=l
>3-3 ...>n.
(19)
Let I/y 11be the maximum norm for any vector y, that is, I/y II = maxi I_Yil,where _Yi is the ith element of y. For any x’ Zx” let IIG(x’) - G(x”)II = IGi, (X’) - Gi,-,(x”)l z [IX’
-
X”ll
=
IXj,,
-
X”ioo
I .
These i. and ioo may not be unique. The assumption value theorem ensures that IIG(x’) - G(x”)ll = IGi, (x’) - Gi, (x”)l
= lgl
aGi,/axi(x;
< ,g
I dGi,/axjIlx;
- xy>l
-
x;I
(13) coupled with the mean
K. Okuguchi / The stability of the Stackelberg oligopoly equilibrium
325
n
G
C I aGjo/axj II-& - Go01 j=l
<
[lx’ - x”Il.
(20)
Hence the mapping G(x) is a contraction and its unique stationary value, the Stackelberg equilibrium, is globally stable under (13) and a fortiori under (19). The inequality is the stability condition we wished to derive. It is easily confirmed that (19) corresponds to the Hathaway-HowroydRickard condition (14) in their paper. From (19) it follows that, other things being equal, increase in the number of firms is destabilizing. In the case of instantaneous adjustment of actual output to the profit maximizing one, ki = 1 for all i, and (19) reduces to i= 1,2, .. .. n.
loilBi < l/(n - l),
As Bi
IaiI
I)*,
(21) (14) and (1.5) (21) is seen to be met if
i = 1, 2, .. .. n.
(22)
In a simple case where the demand function marginal costs are constant and identical, p=a-bCXi,
is linear and all firms’ rate of changes in
i
Ci”=d>O,
i = 1, 2, .. .. n,
we have i= 1,2 ,..., n.
loij =(Yi < 1,
Hence in this case the stability condition Bi = (n - l)b/(2b < ll(n - I),
(21) is satisfied if
+ d) i = 1 ) 2, . . .) n.
(23)
We know further from (22) that in this case, other things being equal, increase in the rate of change in the marginal cost has a stabilizing effect as increase in d is more likely to satisfy (22).
References Hathaway, NJ., T.D. Howroyd and J.A. Rickard, 1979, Dynamic Stackelberg stability, Economics Letters 2, no. 3, 209-213. Negishi, T. and K. Okuguchi, 1972, A model of duopoly with Stackelberg equilibrium, Zeitschrift fir Nationali5konomie 32. Okuguchi, K., 1971, The stability of the Stackelberg duopoly solution, Economic Studies Quarterly 22. Okuguchi, K., 1976, Expectations and stability in oligopoly models (Springer-Verlag, NewYork).
ON THE STABILITY
321-325 Company
OF THE STACKELBERG
OLIGOPOLY EQUILIBRIUM
Koji OKUGUCHI University of New South Wales, Kensington, NSW 2033, Australia Tokyo Metropolitan University, Tokyo, Japan Received
8 November
1979
The stability of the equilibrium in the Stackelberg undifferentiated oligopoly model where each oligopolist strives for leadership is analyzed for general demand and cost functions and for a discrete system with non-instantaneous adjustment.
1. Introduction The stability of the equilibrium of the Stackelberg leader-leader duopoly model has been analyzed by Okuguchi (1971, 1976) for the case of perfect information and by Negishi and Okuguchi (1972) when information is imperfect. Hathaway, Howroyd and Rickard (1979) have derived the stability conditions for the Stackelberg oligopoly model where each firm strives for leadership. In this paper we shall extend the Hathaway-Howroyd-Rickard results for general demand and cost functions. We shall, however, assume away product differentiation and externalities in the cost functions. 2. The stability condition Let the demand function
P=f(Cxi) i
for a homogeneous
product be
3
where p is the price of the homogeneous product and Xi is the output of the ith firm, i = 1,2, . . . . n. The ith firm’s cost function is given as Ci = Ci(Xi),
i = 1 , 2, .... n.
Each firm strives to be a 1 eader and assumes that all other firms will behave as followers. Hence the jth firm’s output in the tth period expected by all other firms, x?(t), is obtained by maximizing
322
K. Okuguchi / The stability of the Stackelberg oligopoly equilibrium
with respect to xi’(t). For the sake of simplicity let us define XMiz Cxj,
i=
The maximizing
condition
j+i
f(X-j(t
1,2, . .. . n.
- 1) +X:(f))
is
+ Xf(t)f’(X-‘(t
- 1) + Xi”(t)) - Ci(Xy(t)) = 0,
(1)
j = 1 , 2, . .., n,
where possibility
of a corner maximum
XF(t)Ehj(Xx-j(t-
is assumed away. From (1) we have
i= 1,2, .. .. n.
l)),
(2)
The ith firm’s expected profit in the tth period n:(t) is therefore given as a;(t) =Xi(f)f(,.
hj(X-j(t
- 1)) + xi(f))
- Ci(Xi(t>)>
i = 1, 2, .. .. n,
(3)
which, maximized with respect to xi(t), yields f$g
hj(X-j(t-
hj(x-j(t - l)) tXi*(t)) +XJ(t)f'(gi
X [lt,z,h;(x-j(t-l))]=O,
i=1,2
where xf 0) denotes the maximizing Xf(t)‘~i(,~,hj(X-j(t-1))))
l>>+XT(t))
,..., n,
(4)
output. We then have i=1,2
,..., n.
(5)
Adjustment of actual output Xi(t) to the profit maximizing instantaneous. Hence Xi(f) - Xi(t - 1) = ki(Xf (t) - Xj(t - l)),
where ki’s are positive constants Xi(t) = (1 - ki)Xi(t - 1) + kigi(]g EZGi(X(t - l)),
i = 1,
(adjustment hj(X-j(t
i = 1,
one is not necessarily
2, . . ., n,
coefficients).
(6)
Define
- 1)))
2, . . . , n,
(7)
where x(l - 1) is the vector of all firms’ outputs in the (t - 1)th period. The stationary value to (7) defines the equilibrium of the Stackelberg oligopoly model where each firm strives for leadership. In the following stability analysis we assume differentiability of relevant functions of any order as desired. Totally differentiate (4) and re-arrange to get [r’ + (1 + ,Gh;(x-j(r
- l)))j-’ + (1 + ,c, h;(x-j(t
- 1))) x’(t)f”
- Ci”] dxf (t)
323
K. Okuguchi / The stability of the Stackelberg oligopoly equilibrium
+ If’
+ (1 + c $(x-j@
- 1))) x&f”1
]Z dxF(t)
j#i
t
Xl?(t)f’
c hl’ d.x-j(l -
j#i
j#i,
1) = 0,
i= 1,2, .. . . IZ.
(8)
Hence
ax;(t)/ax~(t)=-[f’t
(1 t,$,)Xf(T)f”l
/if’+
(I $$)f’t
(1 t,,.h;)Xl(f)l.“-
3 --(y.I? Totally differentiate h; = dxp(t)/dx-j(t = -(f’
+
~ 1) =
x3)f”)/(2f
From the definition
aXF(t)/dXi(t
+ xF(t)f”
=
% C jti
~
cwiBi,
kG
,..., n.
j#i,
i,j=1,2
,..., n.(lO)
(9)
-
1)
- Ci”)
= % kgi
a+(t)/ax,(t
- 1)
/3j
i = 1 , 2, .... n,
j ad-wxo
= ai(?‘i
i,j=l,2
(5) (9) and (10)
agi/axj(t - 1) = ,z. ag#lxT(t)
=
j#i, (1) to get
E -pi,
agi/axj(t>
Cl,
WXWxj(t
-
(11)
1)
ok
-
i#j,
Pi>,
i,j=1,2
,..., n.(12)
Suppress t - 1 in (7) and consider a mapping G(x) = (Gr(x),
G,(x), . .. . G,(x)),
which is assumed to satisfy the fundamental
5 iaGi/axjt j=1
< 1,
assumption
as given by
i = 1, 2, .., n.
We assume moreover that for any output vector x the following two inequalities
(13) are
K. Okuguchi / The stability of the Stackelberg oligopoly equilibrium
324
satisfied: tXif”(CXi)
f’(Cxj) i
f’ (C
Xi) < C;‘(Xi),
i =
1 , 2, .... n,
(14)
i =
1 , 2, ..., n.
(15)
These assumptions imply that the marginal revenue of any firm is decreasing other firm’s output and that the rate of change of each firm’s marginal cost braically greater than the rate of change of the market price with respect to in the total demand. Under assumptions (14) and (15), /J’s are shown to be From (7) and (1 l), we derive i = 1 , 2, .., n.
aGifaxi = (1 - ki) + ki”iBi,
in any is algechange positive.
(16)
Likewise from (7) and (12), aGi/aXi = kiCYi(Bi - pi),
j#i,
i, j=
1,2, . .. . n.
Taking into account (16) (17) and the positivity 5
l aGi/axjl
= I( 1 ~ ki) + kiaiBi I + lg
(17)
of (Yi’swe get,
Ikiai(Bi - Bi>l
j=l
<
i
1 - kiI + kiBi)CYiI+ kilail ,g, (Bi - Pi) i= 1,2,
= 11 ~ kil + (r~ - l)kiBilail,
n.
(18)
Hence (13) is satisfied provided that 11 -kiit(n-l)kiBiicviI
i=l
>3-3 ...>n.
(19)
Let I/y 11be the maximum norm for any vector y, that is, I/y II = maxi I_Yil,where _Yi is the ith element of y. For any x’ Zx” let IIG(x’) - G(x”)II = IGi, (X’) - Gi,-,(x”)l z [IX’
-
X”ll
=
IXj,,
-
X”ioo
I .
These i. and ioo may not be unique. The assumption value theorem ensures that IIG(x’) - G(x”)ll = IGi, (x’) - Gi, (x”)l
= lgl
aGi,/axi(x;
< ,g
I dGi,/axjIlx;
- xy>l
-
x;I
(13) coupled with the mean
K. Okuguchi / The stability of the Stackelberg oligopoly equilibrium
325
n
G
C I aGjo/axj II-& - Go01 j=l
<
[lx’ - x”Il.
(20)
Hence the mapping G(x) is a contraction and its unique stationary value, the Stackelberg equilibrium, is globally stable under (13) and a fortiori under (19). The inequality is the stability condition we wished to derive. It is easily confirmed that (19) corresponds to the Hathaway-HowroydRickard condition (14) in their paper. From (19) it follows that, other things being equal, increase in the number of firms is destabilizing. In the case of instantaneous adjustment of actual output to the profit maximizing one, ki = 1 for all i, and (19) reduces to i= 1,2, .. .. n.
loilBi < l/(n - l),
As Bi
IaiI
I)*,
(21) (14) and (1.5) (21) is seen to be met if
i = 1, 2, .. .. n.
(22)
In a simple case where the demand function marginal costs are constant and identical, p=a-bCXi,
is linear and all firms’ rate of changes in
i
Ci”=d>O,
i = 1, 2, .. .. n,
we have i= 1,2 ,..., n.
loij =(Yi < 1,
Hence in this case the stability condition Bi = (n - l)b/(2b < ll(n - I),
(21) is satisfied if
+ d) i = 1 ) 2, . . .) n.
(23)
We know further from (22) that in this case, other things being equal, increase in the rate of change in the marginal cost has a stabilizing effect as increase in d is more likely to satisfy (22).
References Hathaway, NJ., T.D. Howroyd and J.A. Rickard, 1979, Dynamic Stackelberg stability, Economics Letters 2, no. 3, 209-213. Negishi, T. and K. Okuguchi, 1972, A model of duopoly with Stackelberg equilibrium, Zeitschrift fir Nationali5konomie 32. Okuguchi, K., 1971, The stability of the Stackelberg duopoly solution, Economic Studies Quarterly 22. Okuguchi, K., 1976, Expectations and stability in oligopoly models (Springer-Verlag, NewYork).