Discussion Paper No.

589

GLOBAL STABILITY CONDITIONSON THE PLANE

Anjan Mukherji

July 2003

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Global Stability Conditions on the Plane
Anjan Mukherji ∗Centre for Economic Studies and Planning
Jawaharlal Nehru University, New Delhi
andInstitute of Social and Economic Research
Osaka University
March 2003

Running Title: Global Stability Conditions
Address for Correspondence:
Till July 20, 2003:
ISER, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan.
Telephone: 81-6-6879-8567; Fax: 81-6-6878-2766
After July 20, 2003:
30 New Campus, Dakshinapuram, JNU, New Delhi 110 067, INDIA.
Telephone: 91-11-2618-0982 Fax: 91-11-2616-5886
Email: anjan mukherji@yahoo.com∗

The paper was written during a visit to the Institute of Social and Economic Research of Osaka Uni-

versity. The facilities provided at the Institute are gratefully acknowledged. Discussion with Ken-Ichi
Shimomura and correspondence with Charles Plott have been extremely helpful. I am particularly indebted
to them for making their forthcoming paper available.

Global Stability Conditions

Abstract
The paper considers price adjustment on the plane and derives global stability conditions forsuch dynamics. First, we examine the well-known Scarf Example, to obtain and analyze a
global stability condition for this case. Next, for a general class of excess demand functions,a set of conditions is identified which guarantee not only convergence to some equilibrium
but also robustness of these properties.
Key words: global stability conditions, dynamics on the plane, excess demand functions,
Dulac’s Criterion.
JEL Classification Numbers: C62, C61, D50.

Global Stability Conditions

1

1

Introduction

In economic theory, stability conditions have not been given much importance while inmatters of economic policy, such conditions are sometimes assumed at the outset without
much ado. In matters of theory, it is well established that excess demand functions are
not restricted substantially by routine assumptions such as Walras Law or Homogeneityof degree zero in prices. Consequently since almost “anything goes”, the entire topic ofstability of equilibrium is relegated to texts and forgotten about.
One of the first things in international trade policy, for example, is to assume the wellknown Marshall-Lerner Condition, which is nothing other than a local stability condition.
This is at least a recognition of the fact that unless this condition is met, attainment ofequilibrium cannot be ensured, at least locally. For policy considerations, often, it has
been standard to assume that markets will clear and attain equilibrium. Since these may
entail convergence from arbitrary initial configurations, what one must investigate are global
stability conditions. There is however, a priori, hardly any theoretical reason to assume
that without any such condition, convergence is assured. That there may be seemingly
robust difficulties, for the stability of the tatonnement, has been usually taken for granted,
first, due to the examples in [15] and [5] and then, due to the contributions of [3] and [16].
This dichotomy about the treatment of stability questions, between theory and policy,
indicates that while it is implausible to have a theory of adjustment on disequilibrium
prices which works for every type of excess demand function, it is meaningful to enquire
what conditions would identify a set of excess demand functions which will lead to stableequilibria.
Apart from the intrinsic interest in such an exercise, there is another reason why weshould be interested in such stability conditions. Recent work in experimental economics([1]) has shown that predictions made by tatonnement processes are in fact quite accurate.
So even if tatonnement processes may not converge for every conceivable set of excess
demand functions, it would be of some importance to identify the excess demand functions
for which they do indeed work. This then is the rationale for carrying out the analysis
reported in this paper.
We also confine ourselves to identifying such conditions (stability conditions) in thecontext of adjustment processes on the plane mainly because tools available are best suited
towards that objective. Thus basically we have had to restrict attention to systems ofexcess demand functions involving three goods; one of these goods is identified as numeraireand since prices of other goods are considered relative to this numeraire, the considered

2

Global Stability Conditions

adjustment process will define a motion on the plane. There are some conditions which areavailable for such considerations such as the one in [14] and [11] but, as we hope to show,
it is possible to considerably weaken these conditions.
One of the most well known examples of instability is provided by [15]; we consider thisas a point of departure; by choosing a numeraire, we restrict the dynamics to be on theplane; it is demonstrated that there are closed orbits: this is the counterpart of the resultsin [15]. Next, we analyze a stability condition which ensures that the unique equilibrium is
globally stable. If this condition is violated then the solution is shown to become unbounded.
This exercise allows us to identify the two problems that we will have to encounter in thegeneral case: the first is to guarantee the boundedness of the trajectory or the solution;and the second is to ensure the absence of closed orbits. We next provide, in the light ofthis experience, a set of restrictions which would guarantee that price adjustment will leadto an equilibrium. And we consider price adjustment which could in principle be triggered
off from any initial price configuration. Thus we provide conditions for global stability ofequilibrium.

2

The Scarf Contribution

Consider1 an exchange model where there are three individuals h = 1, 2, 3 and three goods
j = 1, 2, 3. The utility functions and endowments are as under:
U 1 (q1 , q2 , q3 ) = min(q1 , q2 ); w1 = (1, 0, 0)
U 2 (q1 , q2 , q3 ) = min(q2 , q3 ); w2 = (0, 1, 0)
U 3 (q1 , q2 , q3 ) = min(q1 , q3 ); w3 = (0, 0, 1)
Routine calculations lead to the following excess demand functions, where good 3 is treatedas numeraire (i.e., p3 = 1):Z1 (p1 , p2 ) =p1 (1 − p2 )(1 + p1 )(p1 + p2 )p2 (p1 − 1)
(1 + p2 )(p1 + p2 )and the tatonnement process, for this example is given by
Z2 (p1 , p2 ) =ṗi = Zi (p1 , p2 ) i = 1, 2
1

(1)

We provide an analysis of the Scarf example which is somewhat different from the one in [15]. This

would set up the groundwork for the later analysis. In particular, it should be pointed out that Scarf did
not use a numeraire.

3

Global Stability Conditions

Notice that equilibrium for this exchange model (and for the process defined above) is givenby p1 = 1, p2 = 1. It would be helpful to transform variables by setting xi = pi − 1 for
i = 1, 2. With this change in variables, our process becomesẋ1 = −

x1 (1 + x2 )x2 (1 + x1 ), ẋ2 =
(x1 + 2)(x1 + x2 + 2)
(x2 + 2)(x1 + x2 + 2)

(2)

Given an arbitrary initial xo = (xo1 , xo2 ), how does the solution x(t, xo ) to ( 2 ) behave
as t → ∞ ? We consider this question, next.

2.1

A Closed Orbit

Defining v : R → R byv(x) =x2+ x − ln(1 + x)2

which is continuously differentiable for all x such that 1 + x > 0, one may note that2
1 For x small, v(x) ≈ x2 .
Next define V (x) = V (x1 , x2 ) = v(x1 ) + v(x2 ). It is straightforward to show that:
2 V (x) is strictly convex function and assumes a global minimum at (0, 0); thus V (x) >V (0, 0) = 0 if x 6= (0, 0).
Further, it is easy to check that
3 Along the solution x(t, xo ) to ( 2 ), V̇ = 0 provided xi (t, xo ) > −1 for i = 1, 2.
These preliminary steps allows to furnish a complete answer to the question framed in
the last section.
First of all, note that since V (t) = V (x1 (t), x2 (t)) = V (xo1 , xo2 ) for all t, it follows that thesolution or trajectory x(t, xo ) = (x1 (t), x2 (t)) is bounded and each xi (t) is bounded awayfrom −1: since if either of these conditions is violated, V (t) would tend to +∞. Hence theω-limit set corresponding to xo , Lω (xo ), is non-empty and compact; also, (0, 0) ∈/ Lω (xo )if x0 6= (0, 0) (remember, (0, 0) is the equilibrium for the system) hence by the PoincaréBendixson theorem3 Lω (xo ) must be a closed orbit. This means that either we have a limit
cycle or the trajectory x(t, xo ) itself is a closed orbit.
If there is a limit cycle L, then by virtue of the Claim 3, it follows that for any y ∈L, V (y) = V (xo ); further, in such circumstances, there would be a neighborhood N of xo
23

For details, [12] and [13]; we mention the steps, for the sake of ease of reference.
See, for instance, [10] p. 248.

4

Global Stability Conditions

such that for any solution x(t, y) originating from any y ∈ N , x(t, y) → L4 . Consequently,
we must have V (y) = V (xo )∀y ∈ N : this of course, is not possible, since the function V
cannot be constant on an open set. Hence no such limit cycle exists. And the solution
x(t, xo ) must be a closed orbit. Thus we have shown the following to be true:
4 For any initial configuration xo , the solution to ( 2 ), x(t, xo ) is a closed orbit aroundthe equilibrium (0, 0). The equation to a typical orbit is V (x) = V (xo ).
We examine, next, a perturbation of this example.

2.2

Convergence

Consider a parameter say b, which stands for the amount of second good which individual
2 owns completely. Thus b = 1 would revert back to the example considered above. We
continue to treat good 3 as the numeraire and then compute excess demand functions forthe non-numeraire commodities for the case at hand; it turns out that these are given, using
the same notation as above, by the following expressions:
Z1 (p1 , p2 ) =p1 (1 − p2 )(1 + p1 )(p1 + p2 )p2 (p1 − b) + (1 − b)p1
(1 + p2 )(p1 + p2 )Consequently the system ( 1 ) now takes the form:
Z2 (p1 , p2 ) =p˙1 =p2 (p1 − b) + (1 − b)p1
p1 (1 − p2 )and p˙2 =(1 + p1 )(p1 + p2 )(1 + p2 )(p1 + p2 )(3)

Once more standard computations ensure that the unique equilibrium is given byp∗1 =b= θ say, p∗2 = 1
2−bThus it may be noted that our choice of the parameter places a restriction on its magnitude
0 < b < 2;
and we shall take it that this is met. Notice also that when b = 1, θ = 1 too, and wehave the earlier situation. That there have been some changes to the stability propertyof equilibrium is evident from computing characteristic roots :5 Some tedious calculations
reveal that the characteristic roots of the relevant matrix at equilibrium are given by:√ √1(−b + b2 ± b {−32 + 49b − 26b2 + 5b3 }).
8Consequently, one may claim:
45

See, for instance, [10] p.251.
In fact it was shown in [12] that b = 1 provides a point of Hopf Bifurcation for the process ( 3 ).

5Global Stability Conditions

5 For the process ( 3 ), (θ, 1) is a locally asymptotically stable equilibrium if and only ifb < 1 ; for b > 1, the equilibrium is locally unstable.
A much stronger assertion is possible6 :6 For the system ( 3 ), the unique equilibrium (θ, 1) is globally asymptotically stable whenever b < 1; and any trajectory with (po1 , po2 ) > (0, 0) as initial point remains within thepositive orthant. When b > 1, any solution with an arbitrary non-equilibrium initial point
is unbounded.
Proof. We first note that for the system (3) there can be no closed orbit in <2++ solong as b is different from unity. For this purpose we shall use, Dulac’s Criterion7 . Now
consider the function:f (p1 , p2 ) =(p1 + p2 )(1 + p1 )(1 + p2 )p1 .p2

on <2++ . Notice that:∂f (p1 , p2 )Z1 (p1 , p2 ) ∂f (p1 , p2 )Z2 (p1 , p2 )+= −(1 − b)/p22∂p1∂p2
Thus b 6= 1 implies that Dulac’s Criterion is satisfied by this choice of f (p1 , p2 ) andconsequently there can be no closed orbits when b 6= 1. Applying next, the PoincaréBendixson Theorem, it follows that for any initial po ∈ <2++ , the unique equilibrium
p? = (θ, 1) ∈ Lω (po ) provided the ω-limit set is non-empty.
Recall that for b > 1, the unique equilibrium is unstable; consequently no solution canenter a small enough neighborhood of p? ; consequently, in this situation, Lω (po ) must be
empty, if po 6= p? ; thus the trajectories must be unbounded.
When b < 1, the unique equilibrium p? is locally asymptotically stable; so if Lω (po ) 6= ∅,p? ∈ Lω (po ) ⇒ p? = Lω (po ); since once having entered a small enough neighborhood of theequilibrium, the trajectory cannot leave. Thus all that we need to guarantee convergenceis that trajectories are bounded when b < 1.
6

See [13], p. 89-90. We provide an alternative approach which will indicate what we have to accomplish

in the more general case.
7
See, [2], p. 305. This criterion looks for a function f (p1 , p2 ) which is continuously differentiable onsome region R and for which∂f (p1 , p2 )h1 (p1 , p2 )∂f (p1 , p2 )h2 (p1 , p2 )
+
∂p1
∂p2
is of constant sign on R (not identically zero), then there is no closed orbit for the system p˙i = hi (p1 , p2 ), i =1, 2 on the region R.

6

Global Stability Conditions
This last step may be accomplished by considering the function8 :W (p1 , p2 ) = 2(1 − b)p1 + (2 − b)p21 /2 − b log p1 + p22 /2 − log p2
and noting that its time derivative, along any solution to the system ( 3 ):Ẇ = {((2 − b)p1 − b)(1 + p1 )}= −(1 − p2 )2

p˙1
p˙2+ (p22 − 1)
p1
p2

p1 (1 − b)≤0p2 (p1 + p2 )whenever b < 1. Thus for b < 1, W (p1 (t), p2 (t)) ≤ W (po1 , po2 )∀t, where we write (p1 (t), p2 (t))
as the solution to (3). Note that if pi (t) → +∞, for some i, W (p1 (t), p2 (t)) → +∞ and theboundedness and positivity of the solution are established. This establishes the claim. •There are thus two things to be noted from the above result: first that choosing a valueof b different from unity negates the existence of a closed orbit; and a value of b less thanunity is required to ensure that trajectories remain bounded. In a sense to be made precise
below, these are the two aspects we need to account for if we are interested in identifying
global stability conditions.

3

General Global Stability Conditions

If there are three goods and one of them is the numeraire, then the price adjustment
equations of the type used for the Scarf example introduces dynamics on the plane. For
motion on the plane, along with Poincaré-Bendixson Theorem, there is a result reported
in [14] and its refinement [11]. We show next that it is possible to substantially weaken
the conditions under which a global stability result may be deduced. This would allow
us to conclude global stability for a competitive equilibrium as well as providing a generalstability result which would be of some general interest, as well.
Consider the following systems of equations:ẋ = f (x, y) and ẏ = g(x, y)(4)

where the functions f, g are assumed to be of class C 1 on the plane <2 . For any pair offunctions f (x, y), g(x, y) let J(f, g) or simply J, if the context makes it clear, stand for theJacobian9 :8

One may show that this function is, in addition, a Liapunov function for the system (3); see, for instance[13], p. 89-90.
9fx for any function f will refer to the partial derivative of f with respect to the variable x.

7

Global Stability Conditions
fx

fy

gx

gy

!

Consider, next, the following:• O1: There is an unique equilibrium (x̄, ȳ) to (4).
• O2: Trace of J(f, g) = fx + gy < 0 for all (x, y) ∈ <2 .
• O3: Determinant of J(f, g) = fx .gy − fy .gx > 0 for all (x, y) ∈ <2
• O4: Either fx .gy 6= 0 for all (x, y) ∈ <2 or fy .gx 6= 0 for all (x, y) ∈ <2 .
Under the conditions O1 - O4, Olech’s Theorem, [14], shows that the unique equilibrium(x̄, ȳ) is globally asymptotically stable. The contribution in [11] provides conditions which,in addition, guarantee that the solution remains positive, a requirement which has great
importance to economic theory.
We shall use the setting of the tatonnement to investigate motion on the plane and forthis purpose we introduce the notion of the excess demand functions Zi (p1 , p2 , p3 ) : <3++ →<, i = 1, 2, 3 which are required to satisfy the following:
A. Each Zi (.) is continuously differentiable with continuous partial derivatives and is boundedfrom below on <3++ ; further for any (p1 , p2 , p3 ) ∈ <3++ , p1 .Z1 (.) + p2 .Z2 (.) + p3 .Z3 (.) = 0(Walras Law); and further for any (p1 , p2 , p3 ) ∈ <3++ , ∀i, Zi (λp1 , λp2 , λp3 ) = Zi (p1 , p2 , p3 )for any λ > 0 (Homogeneity of degree zero in the prices); finally, for any sequence,
P s = (ps1 , ps2 , ps3 ) ∈ <3++ , psi = 1, ∀s for some index i, say i = io and ||P s || → +∞ as
s → +∞ ⇒ Zio (P s ) → +∞10 (Boundary Condition).
The conditions listed under A are all routine; however they do imply some consequencesof interest. First of all under these conditions, the set of equilibria for the economy E ={p ∈ <3++ : Zi (p) = 0∀i} =6 ∅; an independent demonstration of this assertion would follow
as a by product of the analysis of the dynamics.
To study the dynamics on the plane, we shall investigate the solutions to a system of
equations of the following type:p˙i = hi (p), i = 1, 2 with p3 ≡ 1(5)where the functions hi (p) are assumed to satisfy the following: (we write p = (p1 , p2 ) ∈ <2++ )B hi (p) = Zi (p1 , p2 , 1), i = 1, 2.
10||x|| stands for√ 2(x1 + x22 + x23 ), when x = (x1 , x2 , x3 ).

Global Stability Conditions

8

Thus the equation (5) defines motion on the positive quadrant of the plane. A typicaltrajectory or solution to (5) from an initial po ∈ <2++ will be denoted by φt (po ); the priceconfiguration will be (φt (po ), 1) for each instant t; this is just to signify that the numeraire(the third good) price is always kept fixed at unity. Also we note that any equilibrium
for the dynamical system (5), say p̄ where hi (p̄) = 0, i = 1, 2, implies that (p̄, 1) is anequilibrium for the economy, in the sense that (p̄, 1) ∈ E and conversely. We shall denotethe equilibrium for (5) by ER .
We are interested in the structure of the ω-limit set Lω (po ) i.e., the limit points of the
trajectory φt (po ) as t → +∞. On the plane, the structure of non-empty ω-limit sets is
known to be one of the following11 :i. Consists of a single equilibrium orii. Consists of one closed orbit oriii. an union of equilibria and paths tending to them.
It is because this classification offers some hope of obtaining general results that weshall investigate this situation more closely. We need to guarantee that the solution remains within the positive quadrant, which was the main item of concern in [11], as we
mentioned above; then we need to guarantee that the ω-limit sets are non-empty; this will
be accomplished by ensuring that the solution or trajectories are bounded; if a meaningful
set of conditions allow us to rule out possibilities listed at (ii) and (iii), we have then astability result. The conditions [14] mentioned above contain one such set of conditions;these need to be refined a bit if we want to ensure positivity as has been indicated in [11].
As should be apparent, even for motion on the plane, the requirements are fairly stringent.
First, we note:
7 For each i = 1, 2, there exists εi > 0 such that Zi (pi , pj , 1) > 0 if pi ≤ εi for any pj ,j 6= i, j = 1, 2.
Proof: Suppose to the contrary that there is no such ε1 i.e., for any sequence ps1 > 0∀s, ps1 →0 as s → +∞, it is possible to find some ps2 > 0 such that Z1 (ps1 , ps2 , 1) ≤ 0 for all s large
enough, say s > S1 . Consider the sequence q s = (1, ps2 /ps1 , 1/ps1 ) notice that ||q s || →+∞ as s → +∞; hence by the boundary condition, Z1 (q s ) → +∞ or by homogeneity,
Z1 (ps1 , ps2 , 1) → +∞; thus for all s large enough, say for s > S2 , Z1 (ps1 , ps2 , 1) > 0; we thus
arrive at a contradiction for s > M ax(S1 , S2 ). This establishes the claim. •Given the above claim, note that any trajectory of (5), φt (po ) = (p1 (t), p2 (t)), say, where
11

See, for instance, [2], p. 362.

Global Stability Conditions

9

poi > εi , i = 1, 2 satisfies pi (t) > εi for all t > 0. Thus the trajectory remains within thepositive orthant. This is important enough to be noted separately.
8 Given A and B, the solution φt (po ) from any po > (0, 0) remains within the positive
orthant for all t and remains bounded away from the axes.
The next step is to note that the solution has to remain within some bounded region.
We claim:
9 Under A and B, any solution φt (po ) to (5) for po > (0, 0) remains bounded.
Proof: Suppose ||φt (po )|| → +∞ as t → +∞; then by the boundary condition, Z3 (φt (po ), 1) →+∞ as t → +∞. Writing the solution as p1 (t), p2 (t)), we have by virtue of Walras Law:
p1 (t).Z1 (φt (po ), 1) + p2 (t).Z2 (φt (po ), 1) = −Z3 (φt (po ), 1) → −∞as t → +∞. Recall that the excess demands Zi (.) are bounded below; hence, it follows thatfor some i = 1, 2, pi (t).Zi (φt (po ), 1) → −∞; this is possible only when pi (t) → +∞ andZi (.) < 0. Thus for all t ≥ T , say, pi (t).Zi (.) < 0 which means that for all t ≥ T , p˙i < 0 orhence pi (t) ≤ pi (T ) for all t ≥ T : a contradiction. This establishes the claim. •On the basis of the above claims, we know then that there is a rectangular region R ={(p1 , p2 ) : εi ≤ pi ≤ Mi } in the positive quadrant within which the solution gets trapped.
Incidentally, this fact together with Poincaré’s theory of indices for singular points12 , implies
that R contains equilibria; i.e., ER 6= ∅; recall that, by virtue of our assumptions on excess
demands, (p1 , p2 , 1) ∈ E ⇔ (p1 , p2 ) ∈ ER .
We shall assume now the following:
C i. Trace of the Jacobian J(h1 , h2 ) is not identically zero on R nor does it change sign on
R.
C ii. On the set ER , the Jacobian J(h1 , h2 ) has a non-zero trace and a non-zero determinant.
Notice that while the contributions in [14] and [11] demand an unique equilibrium, we
do not. They demand a lot of other restrictions as well13 . We have of course the propertiesof the excess demand function in A which have helped us to isolate a region such as R; C
i and C ii appear weaker than the requirements demanded [11] and [14]. C ii ensures thatthe equilibria in ER have characteristic roots with real parts non-zero: this ensures that all
equilibria for the dynamic system (5) are hyperbolic or nondegenerate or simple14 . Thus
12

See, for instance, [2] p. 305.
See, for example conditions listed as O1-O4, above.
14
See, for instance, [6] p. 13; this helps in determining the nature of the fixed points locally by considering
13

the linearized version.

Global Stability Conditions

10

the only fixed points are either focii or nodes (Poincaré index +1 for both) or saddle points(Poincaré index - 1)15 . It follows that ER contains a finite odd number of equilibria since
the sum of the indices of all must add up to +116 .
Proposition 1 Under A, B and C, for any po ∈ R, Lω (po ) = p? ∈ ER . Thus all solutions
converge to an equilibrium.
Proof: Consider any po ∈ R and the trajectory φt (po ): the solution to (5); by virtue of
the Claim 8, the ω-limit set Lω (po ) is not empty. Again by the criterion of Bendixson
17 , C i implies that there can be no closed orbits in the region R.

Thus there can be nolimit cycle and hence the Poincaré-Bendixson Theorem implies that Lω (po ) ∩ ER 6= ∅. It
follows therefore that p? ∈ Lω (po ) for some p? ∈ ER ; consequently there is a subsequence{ts }, ts → +∞ as s → +∞ such that φts (po ) → p? as s → +∞.
Since we know that the only types of equilibria are focii, nodes and saddle-points, thecharacteristic roots of the Jacobian J(h1 , h2 ) at p? have real parts either both positive or
negative, or they are real and of opposite signs, given C ii.
In the first case, there would be an open neighborhood N (p? ) which no trajectory or
solution could enter; consequently since our trajectory φt (po ) does enter every neighborhoodof p? , it follows that at p? , the characteristic roots of the Jacobian, if complex, have real parts
negative; and if real, then at least one must be negative. Thus the fixed point p? is eithera sink or at worst, saddle-point. If it is a sink, then any trajectory once having entered asmall neighborhood of the equilibrium, can never leave. Consequently, the trajectory φt (po )has no other limit point. Thus Lω (po ) = p? . In the case of a saddle-point, there is only asingle trajectory which converges to the equilibrium; if po happens to be on this trajectory,Lω (po ) = p? but otherwise it is not possible for a trajectory to have a saddle-point as a limit
point. In any case therefore, the trajectory must converge to an equilibrium, as claimed.
•
15

See, for instance [2] p.301 or [6] p. 51. Consider the characteristic roots of the Jacobian evaluated at

equilibrium. A focus is an equilibrium or fixed point with the characteristic roots are complex conjugates;
the equilibrium is a stable focus when the real parts of these roots are negative; it is an unstable focus when
the real parts are both positive; the equilibrium is called a node when these characteristic roots are both
real and of the same sign; again it is a stable node if the real roots are both negative and an unstable node
if the real roots are positive. Sometimes stable focii and nodes are called sinks; unstable nodes and focii arecalled sources. A saddle-point is an equilibrium when the characteristic roots are both real but of opposite
sign.
16
See, for instance, [2] p. 305.
17
See, for example, [6] p. 44.

Global Stability Conditions

11

We provide, next a set of remarks which highlight the implications of the above result.
Remark 1 The above result provides a set of conditions under which an adjustment on
prices on disequilibrium, in the direction of excess demand, will always lead to an equilibrium. Notice also that these conditions guarantee that there will always be at least one sinki.e., an equilibrium at which the Jacobian has characteristic roots with real parts negative.
To see this note that if no such equilibrium existed, then the only equilibria are saddle-points
and sources. Also in aggregate they are finite in number and moreover, as argued above, no
trajectory can come close to sources; so the only possibility for a limit is a saddle-point; buteach saddle-point has only one trajectory leading to it and there are an infinite number ofpossible trajectories. Thus there must be a sink.
More importantly:
Proposition 2 Under A, B and C, if there is a unique equilibrium, it must be globally
asymptotically stable.
Remark 2 As mentioned above, there must be at least one equilibrium where the characteristic roots have real parts negative. Hence the trace of the Jacobian at that equilibrium
must be negative; further, since the trace at that equilibrium will be negative and the tracecannot change sign nor can it be zero at equilibria, it follows that the trace of the Jacobianat every equilibrium must be negative.
Consequently, we have:
Proposition 3 Under A, B and C, at every equilibrium, the sum of the characteristicroots of the Jacobian will be negative18 .
Remark 3 If we consider hi (p1 , p2 ) to have the same sign as Zi (p1 , p2 , 1), i = 1, 2 then theassumptions in C are restrictions placed on the functions hi . Of course these become difficult
to interpret. One may show that the Jacobian of (h1 , h2 ) at equilibria is related to theJacobian of (Z1 , Z2 ), where the partial derivatives are with respect to (p1 , p2 ), again atequilibria by means of the following:
h11

h12

h21

h22

!

=

Z11

Z12

Z21

Z22

!

.

d1

0

0

d2

!

where all partial derivatives are evaluated at an equilibrium and di > 0, i = 1, 2 are somepositive numbers. This would provide some link between the equilibria for the dynamic
process and equilibria for the economy.
18

Thus, if roots are complex, the real parts must be negative.

Global Stability Conditions

12

Let us reconsider the system (4); assume that the set of equilibria for this system
E = {(x, y) : f (x, y) = 0, g(x, y) = 0} is non-empty. The following general result follows
from our analysis:
Proposition 4 If
i. There is a rectangular region R = {(x, y) : 0 ≤ x ≤ M, 0 ≤ y ≤ N } such that any trajectory of (4) on the boundary of R is either inward pointing or coincides with the boundary;ii. Trace of J(f, g) is not identically zero and does not change sign in the positive quadrant;iii. On the set E, the trace and determinant of the Jacobian J(f, g) do not vanish;then any trajectory φt (xo , y o ) where (xo , y o ) > (0, 0) converges to a point of E.
A final remark considers the weakening of the assumption C i.
Remark 4 If we can find a function θ(p1 , p2 ) which is continuously differentiable on theregion R and for which∂θ(p1 , p2 )h1 (p1 , p2 ) ∂θ(p1 , p2 )h2 (p1 , p2 )
+
∂p1
∂p2
is of constant sign on R, then there is no closed orbit for the system (5) on the region R19 .

In some situations, the above may provide a weakening of the condition C i. It may berecalled that the sole purpose of C i was to rule out closed orbits in R. If, for example,hi (p1 , p2 ) = Zi (p1 , p2 , 1) = pi .gi (p1 , p2 ), i = 1, 2, then we may replace C i. by requiringthat p1 g11 (p1 , p2 ) + p2 g22 (p1 , p2 ) be of constant sign on R; note that we do not require thetrace of J(p1 g1 , p2 g2 ) being constant on R. This follows by virtue of the fact that we may−1consider θ(p1 , p2 ) = p−1
1 p2 and then the condition in Remark 4 is satisfied for this choiceof θ(p1 , p2 )20 .

4

Conclusion

The above analysis shows, first of all, that cyclical behavior around equilibrium, notedby Scarf, is not robust particularly with reference to perturbation of the endowments. In
an identical set-up, results exist which show that a redistribution of the goods among
19

This is Dulac’s criterion; the Bendixson’s Criterion is a special case when θ(p1 , p2 ) = 1. Recall that the

perturbation for the Scarf example, we used the Dulac’s criterion to rule out the existence of closed orbits.
20
Of course, the application of Dulac’s criterion raises problems similar to the search for Liapunov functions.

Global Stability Conditions

13

individuals may also help to restore stability to the Scarf example: [8] and [9] both contain
illuminating results in this connection21 .
More importantly, in the realm of micro-economic theory, it is well known that thesubstitution effects are all in the proper direction and it is income effects which may ruinstability. The Scarf example is an effort at ruling out all substitution effects; the instability
noted by Scarf might then have been assumed to be due to this fact; that this is not the casemay be seen by our result, since without introducing any substitution effects, the economy
has a globally stable equilibrium when b < 1.
A more recent paper [1], points out the existence of an endowment distribution whichleads to global stability. The identified endowment distribution is the one where eachindividual has an unit of the good that he is not interested in: that is individual one has aunit of good 3; individual 2 has an unit of good 1 and individual 3 has an unit of good 222 .
The important and significant part of the contribution made in [1] lies in their discovery thatexperiments conducted with agents with similar preferences and endowments, but engaging
in double auctions would lead to price movements which are predicted by the tatonnement
model. Thus the results provided by the tatonnement process, they argue, should be lookedat with greater care because they seem to predict what price adjustments might actuallyoccur.
As we showed in our analysis of the Scarf Example, the perturbation allowed us to get
rid of closed orbits; for convergence, we needed to show that the solution was bounded. Oneof the reasons for our being able to obtain such a different result was due to the fact that at
the original equilibrium, the relevant matrix had purely complex characteristic roots, withzero real parts. It is not surprising that in such a situation, a perturbation changed the realparts of the characteristic root from zero to positive or negative.
Notice that C ii rules out the Jacobian of the excess demand functions from having
characteristic roots with zero real part or from being singular; both serve to ensure that theproperties we observe are robust; non-singularity of the Jacobian, some times called regularity preserves static properties of the equilibria of the economic system for small changes inparameters; the trace being non-zero at equilibrium, preserves the dynamic properties fromsmall changes in parameters. While C i rules out the trace of the Jacobian from changing
21

The interesting contribution in [4] looks at a slightly different question. It is shown that a different pricemechanism is able to attain equilibrium for the Scarf example; the price mechanism is discrete and considersa weighted average of past prices, together with the current level of excess demand in determining revised
prices.
22
See in this connection, the example in [5] with two goods and two individuals with similar tastes.

Global Stability Conditions

14

signs on the positive quadrant which eliminates cycles. Thus C i rules out periodic behavior
and C ii ensures robustness. These two together imply that the process will always leadto an equilibrium, provided trajectories are bounded; the particular equilibrium approachedwill depend on the initial configuration of prices, of course. It is also important to note that
if there is a unique equilibrium, then that has to be globally asymptotically stable. Thus
the feature of the original Scarf example, of a unique equilibrium which cannot be attained,
is removed. However, these conclusions are for motion on the plane. Their interest lie in
the fact that in many applications in economics, only such motions are considered.
In [7], there is an enquiry relating to the following questions: if a market is stable byitself, can it be rendered unstable from the price adjustment in other markets ? Alternatively, if a market is unstable when taken by itself, can it be rendered stable by the priceadjustment in the other markets ? To both an answer was provided in the negative. Noticethat C i essentially ensured (together with C ii) that the trace of the relevant Jacobianremained negative; notice that this would be implied by assuming that Zii < 0 for each i,that is when each market when taken in isolation, was stable. This in turn has been seen to
imply that the markets together must also be globally stable. Under certain conditions, C
i may be weakened further; this involves the existence of a function θ() satisfying Dulac’s
criterion, as in the case of the perturbation of the Scarf example. But this is a matter ofserendipity rather than design.
General results in this area are difficult to obtain due to two reasons: first of all, theexcess demand functions are not expected, a priori to satisfy any other property apartfrom Homogeneity of degree zero in the prices and Walras Law; secondly, dynamics indimensions greater than 2 may be quite difficult to pin down. Even on the plane, a variety
of dynamic motions are possible. It is not surprising that in higher dimensions matters
become a lot more complicated and Walras Law and Homogeneity do not help too much.
And consequently, we must impose additional restrictions which may be called global
stability conditions ; we have shown that an easy such condition for the Scarf example
is b < 1. For the general case, on the plane, the conditions in C serve the same purpose.

Global Stability Conditions

15

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