Some Notes on the Asymptotical Stability of
Dynamic Economic Systems
Ferenc Szidarovszky∗
University of Arizona

Akio Matsumoto†Chuo University

Abstract
The local and global asymptotical stability of equilibria of dynamic
economic systems is examined. After a survey of the most importantconditions is presented we introduce a new stability condition for discrete
systems, when the iteration map is only piece-wise diﬀerentiable. Some
examples of duopoly illustrate the theoretical findings.

1

Introduction

The long-term behavior of dynamic economic systems is one of the most important problem areas in mathematical economics. In the case of continuous timescales the asymptotic properties of trajectories of ordinary diﬀerential equations
are examined, while in the case of discrete time scales the solutions of diﬀerence equations are studied. There are many diﬀerent methods known from theliterature in investigating the asymptotic properties of dynamic systems. In
the case of time-invariant linear systems the locations of the eigenvalues of thecoeﬃcient matrix determine the stability properties of the system. If the linear
system is time variant, then the system is marginally stable if the fundamental
matrix is bounded, and if in addition the fundamental matrix converges to zeroas t → ∞, then the stability is asymptotical. In the case of linear systems local
and global asymptotic stability are equivalent, however in the case of nonlinear
systems we have to distinguish between local and global asymptotical stability.
The most important results on the stability of linear systems can be found in
all textbooks of linear systems theory (for example, Szidarovszky and Bahill,1998).
The literature on nonlinear systems is less extensive. For continuous systems the stability issues are discussed in many books on ordinary diﬀerential
equations (for example, Brauer and Nohel, 1969), for discrete systems the most
∗ Department of Systems and Industrial Engineering, Tucson, Arizona, 85721-0020, USA.
szidar@sie.arizona.edu† Department of Economics, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, Japan.
akiom@tamacc.chuo-u.ac.jp

1

relevant results are discussed, for example, in Gandolfo (1971), and La Salle(1976).
The asymptotical stability of nonlinear systems can be examined by several
methods. The most common methodology is based on the diﬀerent applicationsof the Lyapunov method. This approach is very useful in many cases however
finding an appropriate Lyapunov function is usually a diﬃcult problem, andthe failure of finding a Lyapunov function does not prove the instability of thesystem. Local asymptotic stability can be shown by locating the eigenvaluesof the Jacobian or by bounding the norm of the Jacobian at the equilibrium.
There was an intensive research on extending the local asymptotical stability
conditions into global stability and relaxing the suﬃcient conditions as much as
possible. Parthasarathy (1983) gives an excellent background of this problem
area in the continuous case, and Cima et al. (1999) discuss its discrete timescales counterpart.
In the classical mathematical literature there are several alternative stability conditions which guarantee global asymptotical stability of dynamic systems.
More recent publications introduce more simple and more general stability conditions which could be very useful in economic studies.
In this paper we will give a brief overview of the current state of this research
field. We will also present an extension of stability conditions to discrete systemwith piece-wise diﬀerentiable maps.

2

Stability conditions using norms

Consider first the one-dimensional discrete systemx(t + 1) = f (x(t))
where f : D → R is a continuously diﬀerentiable function and D is a (finite or
infinite) interval. Let x̄ ∈ D be a fixed point of map f , that is, x̄ = f (x̄). The
following results are well known from numerical analysis:
Fact 1. Assume that |f 0 (x̄)| < 1, then x(t) → x̄ as t → ∞ if x(0) is selected
suﬃciently close to x̄.
Fact 2. Assume that for all x ∈ D, |f 0 (x)| ≤ q < 1 with some constant q, thenx(t) → x as t → ∞ with arbitrary x(0) ∈ D.
Notice that in the case of Fact 1 only local convergence to the equilibrium
is guaranteed based on only local information on the derivative, while Fact 2
guarantees global convergence based on global information on this derivative.
In multidimensional case the derivative f 0 (x) is replaced by the Jacobianmatrix, and naturally the absolute value of the derivative is replaced by thenorm of the Jacobian. Consider therefore the n-dimensional systemx(t + 1) = f (x(t))

2where f : D 7→ Rn is a continuously diﬀerentiable function and D is a convex
subset of Rn . Let x̄ ∈ D be a fixed point of map f , and let J(x) denote theJacobian of f at x. The following results are well known from systems theory:
Fact 3. Assume that with some matrix norm ||J(x̄)|| < 1. Then x(t) → x̄ as
t → ∞ if x(0) is selected suﬃciently close to x̄.
Fact 4. Assume that ||J(x)|| ≤ q < 1 for all x ∈ D, where q is a constant.
Then x(t) → x̄ as t → ∞ with arbitrary x(0) ∈ D.
These results are the most frequently used suﬃcient stability conditions toprove local or global asymptotic stability of the equilibrium. If a given matrix
norm is greater than one, it does not prove instability, since there is the possibility that by selecting another norm stability still can be proved. The mostcommonly used matrix norms are as follows. Let aij denote the (i, j) elementof an n × n matrix A. Thenn
X||A||1= max||A||∞= max||A||2q(k)= max λAT A

j

i

i=1
n
X
j=1

|aij | (column norm)(1)|aij | (row norm)(2)k(Euclidean norm)

(3)

(k)

where the eigenvalues of AT A are λAT A (k = 1, 2, . . . , n).
Example 1. As a simple example consider matrices
µ
¶
µ
¶
µ
¶
0.8 0.8
0.8 00.51 0.51
A1 =, A2 =, and A3 =
.
0
0
0.8 0
0.51
0
Then clearly√||A1 ||1 = 0.8 ||A1 ||∞ = 1.6 ||A1 ||2 = 0.8 · √2 ≈ 1.13||A2 |1 = 1.6 ||A2 ||∞ = 0.8 ||A2 ||2 = 0.8 · 2 ≈ 1.13
and||A3 ||1 = 1.02 ||A3 ||∞ = 1.02 ||A3 ||2 = 0.51 ·

q

√
3+ 5
≈ 0.825.
2

Notice that in each case only one norm is below 1, the other two norms are
greater than one.
O
In many economic models function f is only piece-wise diﬀerentiable on
D because of nonnegativity conditions on production levels and prices as wellas presence of capacity limits. The above mentioned Facts 3 and 4 cannot
be applied in such cases without further consideration. If f is continuously
diﬀerentiable in a neighborhood of x̄, that is, x̄ is in the interior of a subregionof D in which f is continuously diﬀerentiable, then Fact 3 can be used to prove3

local asymptotical stability. However if x̄ is on the boundary between two
subregions, then the Jacobian usually does not exist at x̄, so this result cannot
be applied. Fact 4 assumes the existence of J(x) for all x ∈ D, which is not thecase if f is only piece-wise diﬀerentiable. With some additional considerations
we can however extend these results.
Assume now that D is convex in Rn and it is the union of sets D(1) , D(2) , . . .
with mutually exclusive interiors. Assume furthermore that f is continuous on
D and its restriction f (k) on D(k) is continuously diﬀerentiable in the interior ofD(k) (k = 1, 2, . . . ). Assume in addition that one of the following two conditions
hold:(A) For all k, there is an open set containing D(k) such that f (k) can beextended to it, and f (k) remains continuously diﬀerentiable there. Assume
also that for the linear segment connecting x̄ and any x ∈ D there are
finitely many values 0 = t0 < t1 < · · · < tK(x) = 1 such that all points of
the linear segment connecting x̄ + tl (x − x̄) and x̄ + tl+1 (x − x̄) belongs
to the same D(kl ) (l = 0, 1, . . . , K(x) − 1).
(B) For all linear subsegments connecting x̄ + tl (x − x̄) and x̄ + tl+1 (x − x̄)
defined in the previous assumption there are sequences {ui } → x̄ + tl (x −x̄) and {v i } → x̄ + tl+1 (x − x̄) as i → ∞ such that the entire linear
segment connecting ui and v i is in the interior of D(kl ) for all i.
Assume finally that there is a constant q, such that for all k, and for allx ∈ D(k) (under assumption (A)) or x ∈ int D(k) (under assumption (B)),||J (k) (x)|| ≤ q < 1, where J (k) is the Jacobian of f (k) .
Theorem 1 Under the above conditions x is the only equilibrium and it isglobally asymptotically stable.
Proof. For all x ∈ D,°°°
°K(x)−1
°
° X
°
(f
(x̄
+
t
(x
−
x̄)))
−
f
(x̄
+
t
(x
−
x̄))
||f (x) − x̄|| = ||f (x) − f (x̄)|| = °
l+1
l
°
°
°
° l=0
≤

≤

K(x)−1 °Z tl+1

X °
°
°
l=0

J

(kl )

tl

K(x)−1 Z tl+1 °

X
l=0

tl

≤ q · ||x − x̄||°°(x̄ + t(x − x̄)) (x − x̄)dt°°°° (kl )°(x̄ + t(x − x̄))° · kx − x̄k dt°JK(x)−1X
l=0(tl+1 − tl ) = q · ||x − x̄||.

Therefore for t ≥ 1,||x(t) − x̄|| ≤ q · ||x(t − 1) − x̄|| ≤ · · · ≤ q t ||x(0) − x̄||,4

which converges to zero as t → ∞. The uniqueness of the equilibrium is a simple
consequence of this convergence.
We will show some applications of this theorem in Section 4.
Local asymptotic stability is guaranteed, if the conditions of the Theorem
are true with D replaced by a neighborhood of the equilibrium that contains theequilibrium in its interior.

3

Stability conditions using eigenvalues

Consider first the discrete system (2).
It is well known that if all eigenvalues of an n × n matrix A are inside theunit circle, then there is a matrix norm such that ||A|| < 1 (see for example,
Ortega and Rheinboldt, 1970). This observation and Fact 3 imply the following
result:Fact 5. Assume f is continuously diﬀerentiable in a neigborhood of x̄ and all
eigenvalues of the Jacobian J(x̄) are inside the unit circle then x̄ is locallyasymptotically stable.
In the case of time invariant linear systems the Jacobian is the constantcoeﬃcient matrix, and in this case the condition of the theorem is suﬃcientand necessary. In the case of nonlinear systems this result gives only suﬃcientconditions, since if some eigenvalues have unit absolute values and all others are
inside the unit circle, then the system can be unstable, or marginally stable, oreven globally asymptotically stable as it is shown in the following examples:
Example 2. Consider systemµ¶1 1x(t + 1) =x(t)0 1with both eigenvalues being equal to one. There are infinitely many equilibria:x̄1 = arbitrary and x̄2 = 0. It is easy to see that
µ
¶t
µ
¶
1 1
1 t
x(t) =
x(0) =x(0)0 1
0 1
which show that if x2 (0) 6= 0, then x1 (t) converges to +∞ or −∞ so the system
is unstable.
Consider next the one-dimensional systemx(t + 1) = −x(t)with eigenvalue -1. Clearly, x(t) = (−1)t x(0), so the zero equilibrium is only
marginally stable.
And finally consider again a one-dimensional system2x(t + 1) = x(t) e−x(t) .
5

Notice that x(0) > 0 implies that for all t, 0 < x(t + 1) < x(t) and if x(0) < 0,then x(t) < x(t + 1) < 0. Hence x(t) is monotonic and bounded, so convergent.
Letting t → ∞ in the systems equation implies that the limit equals zero,
which is the unique equilibrium of the system. Hence the system is globally
asymptotical stable.
O
The extension of Fact 4 in terms of the eigenvalues of the Jacobian is nottrue in general. If the eigenvalues of J(x) for all x ∈ D are inside the unit
circle, then for all x ∈ D there is a matrix norm such that the norm of J(x) is
below one, however this norm is usually diﬀerent for diﬀerent values of x. The
following example gives a general n-dimensional discrete system in which the
eigenvalues of J(x) are inside the unit circle even there is a constant q ∈ [0, 1)
such that the absolute values of the eigenvalues of J(x) for all x ∈ Rn are lessthan q, and the system is not globally asymptotically stable (for more details
see Cima et al., 1999).
Example 3. Consider the n-dimensional system (1) withµf (x) = −

kx32
kx31
1
1
,
, x3 , . . . , xn
1 + x21 + x22 1 + x21 + x22 2
2

¶T

³
´
where k ∈ 1, √23 . This function is continuously diﬀerentiable in Rn , x̄ = 0 is
a fixed point and it has the following properties:(a) The eigenvalues of the Jacobian are λ1 = λ2 = 0, λ3 = · · · = λn = 12 ifx√1 x2 = 0, otherwise λ1 and λ2 are complex with absolute values less than3
1
2 k < 1 and λ3 = · · · = λn = 2 ;
´ ³
´
³
1
1
(b) f 4 √k−1
, 0, . . . , 0 = √k−1
, 0, . . . , 0 .
OIn an important special case however such counter example cannot be found,
as the following results state (see Cima et al., 1999).

Fact 6. Assume that f : Rn 7→ Rn is a continuously diﬀerentiable triangularmap, that is,
f (x) = (f1 (x1 ), f2 (x1 , x2 ), . . . , fn (x1 , x2 , . . . , xn )) .
Let x̄ be an equilibrium of system (2) and assume that the absolute valuesof the eigenvalues of J(x) are less than one for all x ∈ Rn . Then x̄ is
globally asymptotically stable.
Fact 7. Let f : R2 7→ R2 be a polynomial map with all eigenvalues of J(x)having absolute values less than one for all x ∈ R2 . Then system (2) has
a unique equlibrium that is globally asymptotically stable.
The following example (see Cima et al., 1997) shows that Fact 7 does nothold for all higher dimensional systems.

6

Example4. Assume now thatµf (x) =

1
1
1
1
x1 + x3 d(x)2 , x2 − d(x)2 , x3 , . . . , xn
2
2
2
2

¶T

with d(x) = x1 + x2 x3 . This is clearly a polynomial map with zero fixed point
and it satisfies the following properties:(a) For all x ∈ Rn , the eigenvalues of J(x) are equal to 12 ;(b) If x(0) =

¡ 147

¢

T
63
, then for t ≥ 1,
32 , − 32 , 0, . . . , 0x(t) =

Ã147 t 63 t
· 2 ,− · 2 ,
32
32

!T
µ ¶t
1
, 0, . . . , 0,
2O

so the system is unstable.

The following statement gives a suﬃcient condition for the instability of theequilibrium
Fact 8. Assume that at least one eigenvalue of J(x̄) has absolute value larger
than one. Then x̄ is unstable.
A simple elementary proof of this result is given in Li and Szidarovszky(1999).
We now turn our attention to the continuous systemẋ = f (x),where f : D 7→ Rn with D ⊂ Rn . Assume that x̄ is an equilibrium of thissystem, which is in the interior of D.
The continuous counterpart of Fact 5 can be given as follows.
Fact 9. Assume f is continuously diﬀerentiable in a neighborhood of x̄ and all
eigenvalues of J(x̄) have negative real parts. Then x̄ is locally asymptocally stable.
Similarly to the discrete case, this condition is suﬃcient and necessary fortime invariant, linear systems. However in the case of nonlinear systems theabove condition is only suﬃcient, since if some eigenvalues have zero real parts
and all other eigenvalues have negative real parts, then the system can be unstable, or marginally stable, or even globally asymptotically stable, as it isillustrated in the following examples.
Example 5. Consider first the two-dimensional system
µ
¶
0 1
ẋ =
x
0 0
7

where the equilibrium is x̄1 = arbitrary and x̄2 = 0. Both eigenvalues of theJacobian are equal to zero, and it is easy to see thatµ¶1 tx(t) =x(0),0 1
so the system is unstable.
Consider next the one-dimensional systemẋ = 0with zero eigenvalue. Then all solutions are constant, the equilibrium is any
real number and all are marginally stable.
Consider finally the systemẋ = −x3
where x = 0 is the only equilibrium, and the eigenvalue is zero at x = 0. Thisequation is separable, so it can be easily solved:x(0)x(t) = p.
1 + 2tx(0)2

Clearly with x(0) 6= 0, x(t) → 0 as t → ∞ showing the global asymptoticalstability of the equilibrium.
O
The extension if Fact 9 to global asymptotical stability is true only in thetwo-dimensional case (see Gutierrez, 1995; Fernandes et al., 2004).
Fact 10. Assume that f : R2 7→ R2 is diﬀerentiable and x̄ is an equilibriumof system (7). Assume furthermore that for all x ∈ R, the eigenvalues if
J(x) have negative real parts. Then x̄ is globally asymptotically stable.
This result however is not true in higher dimensions as the following example
shows.
Example 6. Let¡¢Tf (x) = −x1 + x3 d(x)2 , −x2 − d(x)2 , −x3 , . . . , −xnwith d(x) = x1 + x2 x3 as in Example 4. It is easy to see that the eigenvalues ofJ(x) are equal to −1 for all x ∈ Rn , andx1 (t) = 18 et , x2 (t) = −12 e2t , x3 (t) = · · · = xn (t) = 0
is a solutions of system (7) which shows that the zero equilibrium cannot be
globally asymptotically stable.
O
The continuous counterpart of Fact 6 remains valid for continuous systems(see Markus and Yamabe, 1960).

8

Fact 11. Assume that f : Rn 7→ Rn is a continuously diﬀerentiable triangular∂fi
map such that for all i and x ∈ Rn , ∂x(x) < 0. Then the equilibrium ofisystem (7) is globally asymptotically stable.
Another useful suﬃcient condition is the following (see Markus and Yamabe,1960).
Fact 12. Assume f : Rn 7→ Rn is continuously diﬀerentiable, and for all x ∈Rn , M (x) = J(x) + J(x)T is negative definite. Assume furthermore that
there are positive constrants α and β such that
| Trace M (x)| < α and | Det M (x)| > β.
Then x̄ is globally asymptotically stable.
Similarly to the discrete case, a simple suﬃcient instability condition is
given in the following result.
Fact 13. Assume f is continuously diﬀerentiable in a neighborhood of x̄, andat least one eigenvalue of J(x̄) has positive real part. Then x̄ is unstable.

4

Stability of special duopolies

It was earlier mentioned that if the eigenvalues of a matrix are inside the unit
circle, then there is a matrix norm such that the norm of the matrix is below
unity. In the 2-dimensional case for a large group of such matrices a very special
type of norm can be selected. Assume that all eigenvalues of matrix
µ
¶
a b
A=
c d
are inside the unit circle. The characteristic polynomial of A is given as ϕ(λ) =(a − λ)(d − λ) − bc = λ2 − λ(a + d) + (ad − bc), so the matrix elements satisfy
relations (see for example Bischi et al., 2009)±(a + d) + (ad − bc) + 1 > 0
ad − bc < 1.
Select a diagonal matrix
T =

µ

¶
x 0
0 1

with x > 0, and consider the row norm of A generated by matrix T :
°µ
½
¶°
¾
° a bx °
|c|
°
=
max
|a|
+
|b|x,
||T AT −1 ||∞ = °+|d|.
° c d °
xx∞
This is below unity, if |a| < 1, |d| < 1, furthermore|a| + |b|x < 1 and9|c|+ |d| < 1,x

(4)which occurs if and only if|c|
1 − |a|<x<.
1 − |d||b|(5)

Suitable x exists, if|bc| < (1 − |a|)(1 − |d|)or−(|a| + |d|) + |ad| − |bc| + 1 > 0
which is consequence of the first condition of (8) if ad ≥ 0 and bc ≥ 0. Hence
we have the following result.
Fact 14. Assume |a| < 1, |d| < 1, ad ≥ 0 and bc ≥ 0. Assume furthermore thatthe eigenvalues of matrix A are inside the unit circle. Then there is a row
norm generated by a diagonal matrix with positive diagonal, such thatthe norm of A and that of all other matrices which can be obtained fromA by decreasing the absolute value of at least one of its elements, are allless than unity.
Consider first a duopoly with linear cost and price functions. Then the profit
of firm k has the form
ϕk (x1 , x2 ) = xk (A − Bx1 − Bx2 ) − (ck xk + dk )where all coeﬃcients are positive. Assume that both firms have finite capacity
limits, so 0 ≤ xk ≤ Lk for k = 1, 2. The best response of firm k is⎧kif xl ≥ A−c⎨ 0
B
A−ck −2BLkRk (xl ) =Lk
if xl ≤B
⎩ A−ck −Bxlotherwise
2Bwith l 6= k. By assuming partial adjustment to best responses, the dynamic
system can be written as follows:xk (t + 1) = xk (t) + αk · (Rk (xl (t)) − xk (t))(k = 1, 2).

Notice that the value of Rk0 is either − 12 or 0 in the diﬀerent segments, so
the four possible Jacobians are
¶ µ
¶ µ
¶
µ
¶
µ
1 − α1
1 − α1 − α21
1 − α100
1 − α1 − α21
,
,
and
.
− α22
0
1 − α2− α22
1 − α2
1 − α20
1 − α2
It is always assumed that 0 < αk ≤ 1, so the first matrix satisfies thecondition of Fact 14 on the matrix elements. The eigenvalues are inside the unit
circle if and only if
α1 α2(1 − α1 )(1 − α2 ) −
< 1,
4
α1 α2
+ 1 > 0.
±(2 − α1 − α2 ) + (1 − α1 )(1 − α2 ) −4
10

The first inequality is clearly satisfied, the second is true with both positive
and negative signs if
3−(2 − α1 − α2 ) + 1 − α1 − α2 + α1 α2 + 1 > 0
4or3
α1 α2 > 04which always holds. Hence all conditions of Fact 14 are satisfied, consequently
system (11) is globally asymptotically stable.
Assume again linear price function but quadratic costs. The payoﬀ of firm
k is now as follows:ϕk (x1 , x2 ) = xk (A − Bx1 − Bx2 ) − (ck xk + dk x2k ).
If B + dk > 0 for k = 1, 2, then ϕk is strictly concave in xk , so the best responseof firm k is given as⎧k⎪
if xl ≥ A−c⎨ 0
B
A−ck −2(B+dk )Lk
Lk
if xl ≤Rk (xl ) =B⎪⎩ A−ck −Bxl otherwise.
2(B+dk )By assuming again partial adjustment to best responses (dynamic equations(11)), the four possible Jacobians are
Ã
! µ
¶ µ
µ
α1 B
α1 B ¶
1 − α1− 2(B+d1 − α10
1 − α1 − 2(B+d1 − α1
1)
)
1
,
,
and
α2 B
α2 B
−
1
−
α
0
− 2(B+d2 )1 − α220
1 − α22(B+d2 )The first matrix satisfies the conditions of Fact 14 concerning the matrix
elements. The eigenvalues are inside the unit circle if and only if(1 − α1 )(1 − α2 ) −α1 α2 B 2
< 1,
4(B + d1 )(B + d2 )±(2 − α1 − α2 ) + (1 − α1 )(1 − α2 ) −α1 α2 B 2
+ 1 > 0.
4(B + d1 )(B + d2 )The first inequality is clearly satisfied, and the second holds if
µ
¶
B2
+1>0
−2 + α1 + α2 + 1 − α1 − α2 + α1 α2 1 −
4(B + d1 )(B + d2 )which is valid if
B 2 < 4(B + d1 )(B + d2 ).
Hence under this condition the equilibrium is globally asymptotically stable.
Let’s turn our attention to general concave duopolies. Let p be the pricefunction and Ck the cost function of firm k. It is assumed that these functions
are twice continuously diﬀerentiable and
11

¶
0
.
1 − α2a) p0 < 0;b) p0 + xk p00 ≤ 0;c) p0 − Ck00 < 0for all feasible values of the relevant variables. It is known from oligopoly
theory that in the case of finite capacity limits the best response of firm k is asfollows:⎧
if p(xl ) − Ck0 (0) ≤ 0⎨ 0
Lk if p(xl + Lk ) + Lk p0 (xl + Lk ) − Ck0 (Lk ) ≥ 0
Rk (xl ) =⎩ ∗xk otherwise

where x∗k is the unique solution of the monotonic equationp(xl + xk ) + xk p0 (xl + xk ) − Ck0 (xk ) = 0
inside interval (0, Lk ). By implicite diﬀerentiation it is easy to see that in thecase of interior best response−1 < Rk0 (xl ) ≤ 0
and in the first two cases Rk0 (xl ) ≡ 0. By assuming dynamic equations (11), theJacobian has again four possibilities
µ
¶ µ
¶ µ
¶
µ
¶
1 − α1 α1 R10
1 − α1
1 − α1 α1 R10
1 − α1
0
0
,
,
and
.
α2 R20 1 − α201 − α2
α2 R20 1 − α201 − α2
We will prove by using Fact 14, that there is a row-norm generated by a diagonal
matrix with positive diagonal such that the norms of these Jacobians are below a
constant q < 0 on the entire region. Since the feasible region and all subregions
are compact, there are constants r1 , r2 such that −1 < −r1 ≤ R10 (x2 ) ≤ 0
and −1 < −r2 ≤ R20 (x1 ) ≤ 0 everywhere in the feasible region (except on theboundaries between the subregions). Consider a diagonal matrix T = diag(x, 1),
then in the first caseµ¶1 − α1 α1 R10 x−10
T J(x)T = α2 R2
1 − α2x
The row norm of the first Jacobian generated by matrix T isnn
α2 0 oα2 r2 omax 1 − α1 + α1 x|R10 |; 1 − α2 +|R2 | ≤ max 1 − α1 + α1 xr1 ; 1 − α2 +
=q<1
x
x
if

r2 < x <1.
r1

Feasible x can be found if r2 < r11 or r1 r2 < 1, which is obviously true. Hencethe equilibrium of a concave duopoly is always globally asymptotically stable,
since if any one or both Rk0 is replaced with zero, the norm can only decrease,so it will always remain under q.
12

References[1] Bischi, G.I., C. Chiarella, M. Kopel and F. Szidarovszky (2009) Nonlinear Oligopolies: Stability and Bifurcations. Springer-Verlag, Berlin/New
York (to appear)[2] Brauer, F. and J. A. Nohel (1969) Qualitative Theory of Ordinary Diﬀerential Equations. Benjamin, New York.
[3] Cima, A., A. Gasull and F. Mañonas (1999) The Discrete Markus-Yamabe
Problem. Nonlinear Analysis 35, 343-354.
[4] Fernandes, A., C. Gutierrez and R. Rabanal (2004) Global Asymptotic
Stability for Diﬀerentiable Vector Fields at R2 . Journal of Diﬀerential
Equations 206, 470-482.
[5] Gandolfo, G. (1971) Economic Dynamics: Methods and Models. North
Holland, Amsterdam.
[6] Gutierrez, C. (1995) A Solution to the Bidimensional Global Asmyptotic
Stability Conjecture. Annales de 2’ I.H.P., section C 12(6), 627-671.
[7] La Salle, J. P. (1976) The Stability of Dynamical Systems. SIAM Publ.,
Philadephia.
[8] Li, W. and F. Szidarovszky (1999) An Elementary Result in the Stability
Theory of Time Invariant Nonlinear Discrete Dynamical Systems. Appl.
Math. and Comp. 102, 35-49.
[9] Markus, L. and H. Yamabe (1960) Global Stability Criteria for Diﬀerential
Systems. Osaka Math. J. 12, 305-317[10] Ortega, J. and W. Rheinboldt (1970) Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.
[11] Parthasarathy, T. (1983) On Global Univalence Theorems. Lecture Notes
in Mathematics, Vol. 977. Springer-Verlag, Berlin/New York.
[12] Szidarovszky, F. and T. Bahill (1998) The Theory of Linear Systems. CRC
Press, Boca Raton/London (2nd edition).

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