Journal of Mathematical Economics 39 (2003) 27–38
Globally stable price dynamics
Aleksandra Arkit
Institute of Mathematics, University of Zielona Góra, Podgórna 50, 65-246 Zielona Góra, Poland
Received 19 September 2001; received in revised form 26 June 2002; accepted 30 July 2002
Abstract
It is known that the price mechanism whereby the rate of change of a price is proportional to the
excess demand of the corresponding commodity need not converge to a competitive equilibrium for
a pure exchange economy with more than two commodities. In this paper, conditions for the excess
demand, guaranteeing the existence of some price adjustment process, which is globally asymptotically stable and fulfils reasonable compatibility condition with the excess demand, are given.
© 2002 Elsevier Science B.V. All rights reserved.
JEL classification: C62; D51
Keywords: Compatibility condition with the excess demand; Competitive equilibrium; Stable price dynamics
1. Introduction
Let R̄+ = [0, +∞) denote the set of nonnegative real numbers. In a pure exchange
economy with (n) commodities, let p = (p1 , . . . , pn ) ∈ R̄n+ denote the current price
system (pi ≥ 0 for i = 1, . . . , n represents the price of a unit of the ith commodity). By
f , we will denote the aggregate excess demand function from R̄n+ \ {0} into Rn , which
describes the difference between the commodity bundle demanded at price p and the total
supply. When f(p∗ ) = 0, demand is equal to supply and exchange is reached on the
market. Then we call the price system p∗ a competitive equilibrium. Natural hypotheses
on the excess demand function, guaranteeing the existence of the competitive equilibrium
(see Smale (1976)), are:
A1
A2
A3
A4
p → f(p) is continuous,
f(p), p = 0, (Walras’ Law)
f(λp) = f(p) for λ > 0, (positive homogeneity)
if pi = 0 then fi (p) ≥ 0 (boundary condition)
E-mail address: [email protected] (A. Arkit).
0304-4068/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 4 - 4 0 6 8 ( 0 2 ) 0 0 0 8 1 - 2
28
A. Arkit / Journal of Mathematical Economics 39 (2003) 27–38
for all p ∈ Rn+ (·, · denotes the inner product). Let us denote by Pf the set of all equilibrium
points (i.e. all zeros of f ). The essential question is if the price systems, which appear on
the market, will bring nearer to some competitive equilibrium with time going by?
We assume, following Samuelson (1941), that the path of prices, which starts at fixed p0 ,
is a solution of the differential equation
dp
= g(p),
dt
p(0) = p0 .
(1)
The continuous function g : Rn+ \ {0} → Rn , on the RHS of the Eq. (1), which satisfies
Walras’ Law and the condition: g(p∗ ) = 0 if and only if p∗ ∈ Pf , we will call a price
mechanism. We say that a price mechanism is globally asymptotically stable if any price
trajectory, which is a solution of (1) for any initial point p0 , converges to some p∗ ∈ Pf ,
when t tends to infinity.
Samuelson considered the case g = f . It is the simplest rule of the price change since it
describes a natural way of price adjustment, i.e. demand change causes proportional price
change. There are conditions for the excess demand function such as the Weak Axiom of
Revealed Preference, Gross Substituability, some form of Diagonal Dominance, which are
sufficient for the stability of such process, but they are treated as very restrictive. If we assume
about the excess demand nothing more than conditions A1–A4, such price mechanism
have not to be globally asymptotically stable (see Scarf (1960)). We know moreover, from
the Sonnenschein–Mantel–Debreu Theorem, that f could be practically arbitrary and still
could be derived from preference relations of classical type (for details we refer the reader
to Debreu (1974), Mantel (1974), Sonnenschein (1972)). There have been made a lot of
attempts to design price mechanisms which are universal and globally stable. Most of them
are based on algorithms to compute fixed points of arbitrary mapping of some set into itself
and therefore there are useful to compute equilibria of arbitrary economies specified in terms
of excess demand functions. Distinct from Samuelson’s approach such price adjustment
processes need additional information to ensure their efficiency. For example, the universal
but not globally stable Global Newton Method type price mechanism (Smale (1976)) uses
both values of the excess demand function and its derivatives at each point. Saari and
Simon (1978) showed that such informational requirements of this price mechanism cannot
be relaxed by any significant amount. The ingenious attempt of combination of a simple
tatonnément and a Global Newton Method type process (Kamiya (1990)) does not let
avoid informational requirements in order to ensure convergence to equilibrium, since it
depends on values of the excess demand and its derivatives as well as on the location of the
corresponding price system with respect to the initial price system. Similar requirements has
the universal and globally stable price adjustment process of van der Laan and Talman (1987)
(see Herings (1997) also). It depends on values of the excess demand and the location of
the corresponding price system with respect to the initial price system. However, in special
areas, it requires also using of the excess demand’s derivatives. The common feature of
earlier mentioned price mechanisms is manipulative using of the excess demand function in
order to compute equilibria no matter how far is the resulting mechanism from the Walras’
tatonnément.
We would like to refer to Samuelson (1941) approach. The aim of this paper is to formulate sufficient conditions for the excess demand, guaranteeing the existence of some price
A. Arkit / Journal of Mathematical Economics 39 (2003) 27–38
29
adjustment process which is globally asymptotically stable and fulfils reasonable compatibility condition with the excess demand. These conditions are satisfied by the broad
class of the excess demand functions, for which the Weak Axiom of Revealed Preference,
Gross Substituability or some form of Diagonal Dominance are violated (in particular
Scarf’s example (1960)). Our proof of the existence of globally asymptotically stable price
adjustment process is based on the construction of the adequate price mechanism. This
construction gives particular, but not necessarily unique rule of price change M, for which
g(p) = M(f(p), p, p∗ ) (it depends on values of the excess demand and the location of the
corresponding price system with respect to the equilibrium price system). Thus different
economies can have different rules of price adjustment respecting their own peculiarity
(compare Saari (1995)). Of course, this approach is not relevant for computing equilibria,
but gives an answer to the question, which was formulated at the begining of this section.
It is easy to show that every price trajectory of the price mechanism g = f is located on
n−1
the nonnegative part of the sphere S+
(|p0 |) = {p ∈ Rn+ : |p| = |p0 |}. Since f satisfies
A3, we can treat such price adjustment process as a continuous tangent vector field on
n−1
S+
= {p ∈ Rn+ : |p| = 1}. This is the reason why we assume the price mechanism also
satisfies Walras’ Law. Whenever g will fullfil reasonable compatibility condition with the
excess demand, we can analyse the character of the price dynamics in the context of the
n−1
excess demand function as relations between two tangent vector fields on S+
. From now
on let Pf denote the set of norm one equilibrium points.
2. Problem
The case when g = f was investigated by Uzawa (1961). He considered the stability
of a price adjustment process, where the price mechanism was defined by some function
having the same signs as the excess demand. Such price mechanism represents a general
rule of price change on the market and it imposes on both vectors: excess demand and price
change to be in the same orthant of Rn . In special cases, it permits the situation when these
directions are relatively divergent (the angle between these vectors could be nearly π/2).
This motivates a little more general question. What can we say about the stability of the
price adjustment process if we assume that the price change vector forms with the excess
demand vector an acute angle and not necessarily the price mechanism has to have the same
signs as the excess demand?
n−1
For any p ∈ S+
and s ∈ R let us denote Hs (p) = {v ∈ Rn : p, v = s}, Hs+ (p) = {v ∈
n
−
R : p, v > s}, Hs (p) = {v ∈ Rn : p, v < s}. For α ∈ [0, π/2] and z ∈ Rn let Cα (z) =
{v ∈ Rn : z, v ≥ |z||v| cos α} if z = 0 and Cα (0) = {0}. Let Q be any subset of Rn .
Definition 1. We say that a function g : Q → Rn satisfies α-compatibility condition with
given function f : Q → Rn if
g(p) ∈ Cα (f(p)) for all p ∈ Q.
(2)
Let f denote the excess demand function of some pure exchange model. If some price
mechanism g satisfies α-compatibility condition with given excess demand function f the
30
A. Arkit / Journal of Mathematical Economics 39 (2003) 27–38
price change vector forms with the excess demand an angle less than or equal to α. Let us
observe that if α = 0 then α-compatibility condition implies Uzawa’s condition. On the other
hand, if α = π/2, this relation changes radically. Uzawa’s condition yields α-compatibility
condition. In other cases, there are no direct connections between them. In case of three
commodities, we can interpret this condition in the following way: let p = (p1 , p2 , p3 )
(p1 > 0, p2 > 0, p3 > 0) denote a price vector and p = (p1 , p2 , p3 ) denote a vector
of the price change. Consider the situation f1 (p) > 0, f2 (p) < 0, f3 (p) = 0. Suppose
moreover that p1 > 0. If now g(p) characterizes the price change following from the
excess demand f(p), we have p = g(p). Then we can rewrite the condition (2) as:
f(p), g(p) = f(p), p = f1 (p)p1 + f2 (p)p2 ≥ |f(p)||p| cos α,
and using Walras’ Law for f we get:
p1
p2
|f(p)| |p|
p2
|f1 (p)| p1
≥
+
cos α ≥
+
cos α,
p1
p2
f1 (p) p1
p2
f1 (p) p1
and hence
p1
p2
(1 − cos α) ≥
.
p1
p2
(3)
This inequality implies that a relative change of the price of the second commodity does
not exceed a relative change of the price of the first commodity. In particular, we allow for a
possibility of a growth of the price of the second commodity although its supply outnumbers
demand, but relatively not more than the change of the price of the first commodity times
a factor 1 − cos α (which is less or equal to 1).
The problem reads as follows. Let α ∈ [0, π/2]. Can one find such price mechanism g,
which is globally stable and fulfils the α-compatibility condition with the excess demand f ?
In other words, we ask about the existence of the continuous selection g of the multivalued
map p → Cα (f(p)) ∩ H0 (p), such that any trajectory of an autonomous equation ṗ =
n−1
g(p) with initial condition p(0) = p0 , (p0 ∈ S+
) is asymptotically convergent to some
equilibrium point p∗ ∈ Pf .
3. Stability
For any z, x ∈ Rn and α ∈ [0, π/2] let Kα (z, x) = Cα (z) ∩ Lin{z, x} denote a cone
consisting of all vectors from Lin {z, x} = {v ∈ Rn : v = az + bx, a, b ∈ R} which form
with the vector z = 0 an angle less than or equal to α (if x is parallel to z or x = 0 then
Kα (z, x) = {λz : λ ≥ 0}) or of the vector 0 if z = 0.
Let us denote by C(Rn ) the class of continuous functions f : Rn → Rn with the property:
f(x) = 0 if and only if x = 0.
For β ∈ [0, π/2] and g ∈ C(Rn ) which is (π/2 − β)-compatible with f we consider
an autonomous equation ẋ = g(x) with initial condition x(0) = x0 analogous to (1).
In this section, we give sufficient conditions under which solutions of this equation are
asymptotically convergent (to x ≡ 0).
A. Arkit / Journal of Mathematical Economics 39 (2003) 27–38
31
For every β ∈ [0, π/2] let fβ : (Rn \ {0}) × (Rn \ {0}) → Rn be defined by: fβ (z, x) =
argmax{v, x : v ∈ Kβ (z, x) ∩ S n−1 } if β > 0 and f0 (z, x) = cz/|z| for some fixed c ∈
(0, 1). For β ∈ (0, π/2], this map assigns to every couple (z, x) such that z, x = −|z||x|
(it means that the angle between x and z is less than π) a unique norm one vector from
Kβ (z, x), which forms with the vector x the smallest possible angle. It is easy to verify
that fβ is continuous at all such couples (z, x). If z, x = −|z||x| then fβ assigns to (z, x)
two different vectors fβ1 (z, x) and fβ2 (z, x) and at such points this maps is not singlevalued.
However, (z, x) → {fβ1 (z, x), fβ2 (z, x)} (fβ1 (z, x) = fβ2 (z, x) whenever z, x = −|z||x|),
is an upper semicontinuous multivalued map.
Proposition 2. If a function g ∈ C(Rn ) is (π/2 − β)-compatible with f ∈ C(Rn ) and
satisfies the condition
x
(4)
+ fβ (f(x), x), g(x) < 0 for all x ∈ Rn \ {0}
|x|
then every solution to an initial value problem ẋ = g(x), x(0) = x0 , x0 ∈ Rn , is globally
asymptotically convergent to 0.
Proof. We use the Asymptotic Stability Theorem, see e.g. Lefschetz (1965). To prove this
proposition, it suffices to show that there exists differentiable Lyapunov function, i.e. a
function V : Rn → R, which satisfies the following conditions for every x ∈ Rn :
(1) V(x) ≥ 0,
(2) V(x) = 0 if and only if x = 0,
(3) if x = 0 then ∇V(x), g(x) < 0.
We show that V(x) = 1/2|x|2 (∇V(x) = x) is the Lyapunov function for our problem.
Under our assumptions, we obtain:
x
∇V(x), g(x) = x, g(x) = |x|
, g(x) < −|x|fβ (f(x), x), g(x) ≤ 0,
|x|
for x ∈ Rn \ {0}, since fβ (f(x), x), g(x) ≥ 0 (the angle between two vectors u, v from
䊐
Rn such that u ∈ Cβ (z) and v ∈ Cπ/2−β (z) cannot be greater than π/2).
Remark 3. If we consider some function g ∈ C(Rn ) which is (π/2 − β)-compatible with
f ∈ C(Rn ) and satisfies condition (4) we know that at every x ∈ Rn the vector g(x) is
directed inward the ball B(|x|) = {y ∈ Rn : |y| ≤ |x|} (it means g(x) ∈ TB(|x|) (x) =
{λ(b − x) : b ∈ B(|x|), λ ≥ 0}). It implies that if assumptions of Proposition 2 are
satisfied for f, g ∈ C(Q), where Q is some ball of constant radius centred at 0 then solutions of the equation ẋ = g(x) with initial condition x(0) = x0 ∈ Q never leave the
set Q.
We will show now that the condition (4) allows us to characterize these functions f ∈
C(Rn ) for which there exists a function g ∈ C(Rn ), which is (π/2 − β)-compatible with
32
A. Arkit / Journal of Mathematical Economics 39 (2003) 27–38
f and such that an equation ẋ = g(x) with initial condition x(0) = x0 has asymptotically
convergent solutions.
Lemma 4. Let β ∈ [0, π/2] and f ∈ C(Rn ). If
x∈
/ Cβ (f(x)) for all x ∈ Rn \ {0}
(5)
then there exists a continuous function g ∈ C(Rn ) which is (π/2 − β)-compatible with f
and satisfies (4).
(The condition (5) implies that at every x = 0 an angle among vectors f(x) and x cannot
be less than or equal to β.)
n
Proof. Let β ∈ [0, π/2] and consider the multivalued map Gβ : Rn → 2R defined by
Gβ (0) = 0 and
x
−
Gβ (x) = Kπ/2−β (f(x), x) ∩ H0
+ fβ (f(x), x) ,
|x|
if x = 0.
It is easy to see that x ∈
/ Cβ (f(x)) is equivalent to x ∈
/ Kβ (f(x), x). Moreover, the set
/ Kβ (f(x), x). Otherwise
Gβ (x) is nonempty for all x such that x ∈
Kπ/2−β (f(x), x) ⊂ H0+
x
+ fβ (f(x), x) .
|x|
Then for α = ∠(x; f(x)) ≤ π1 the inequality: α + (π/2 − β) ≤ 1/2(α − β) + π/2 yields
α ≤ β and we receive a contradiction with (5). Now we will construct a continuous selection
g : Rn → Rn of Gβ .
Let W = {(z, x) ∈ R2n : x ∈
/ Kβ (z, x), z = 0}. By the definition of Kβ (z, x), we
have that (z, 0) ∈
/ W for all z, and certainly (0, x) ∈
/ W for all x. For (z, x) ∈ W let
u1 (z, x), u2 (z, x) ∈ Lin{z, x} be the norm one vectors defined in the following way. First
we set u1 (z, x) ∈ Kπ/2−β (z, x) ∩ H0 (x/|x| + fβ (z, x)) ∩ S n−1 if z ∈ H0+ (x/|x| + fβ (z, x))
and u1 (z, x) = z/|z| if z ∈ H0− (x/|x| + fβ (z, x)). Recall that fβ (z, x) is not unique for
β > 0 and (z, x) such that z, x = −|z||x| (we have then fβ (z, x) ∈ {fβ1 (z, x), fβ2 (z, x)}),
however, it is easy to check that z ∈ H0− (x/|x| + fβ1 (z, x)) ∩ H0− (x/|x| + fβ2 (z, x)) in this
case. Thus u1 is uniquely defined function, given by formulas:
u1 (z, x) = if z ∈ H0+
1
M(x/|x|)z/|z| − M(z/|z|)x/|x|
M(x/|x|)2
− 2M(x/|x|)M(z/|z|)x/|x|, z/|z| + M(z/|z|)2
x
+ fβ (z, x)
|x|
The (u; v) denotes the angle between u and v.
A. Arkit / Journal of Mathematical Economics 39 (2003) 27–38
33
where M(y) = y, x/|x| + fβ (z, x) and
u1 (z, x) =
z
|z|
if z ∈ H0−
x
+ fβ (z, x) .
|x|
Let us observe that u1 is continuous at (z, x) such that z ∈ H0− (x/|x| + fβ (z, x)) or z ∈
H0+ (x/|x|+fβ (z, x)), since then u1 is defined by the same formula on some neighbourhood
of (z, x). If z0 ∈ H0 (x0 /|x0 | + fβ (z0 , x0 )), then (z, x) → (z0 , x0 ) implies z/|z|, x/|x| +
fβ (z, x) → 0, i.e. M(z/|z|) → 0 and z/|z| → z0 /|z0 |. Thus u1 (z, x) → u1 (z0 , x0 ) in
each case. Now we set
u2 (z, x) = argmin {v, x : v ∈ Kπ/2−β (z, x) ∩ S n−1 }.
The function u2 is also uniquely defined:
 x

−


|x|




u2 (z, x) = N(x/|x|)z/|z| − N(z/|z|)x/|x|


 N(x/|x|)2 − 2N(x/|x|)N(z/|z|)




×x/|x|, z/|z| + N(z/|z|)2
if x ∈ −Kπ/2−β (z, x)
if x ∈
/ −Kπ/2−β (z, x)
where N(y) = y, fβ (z, x). Analogously, we can show that u2 is continuous at every
(z, x) ∈ W. First let us observe that
x z
x
, fβ (z, x) = ∠
,
− β.
(6)
∠
|x|
|x| |z|
If x0 ∈
/ −Kπ/2−β (z0 , x0 ) then x ∈
/ −Kπ/2−β (z, x) for (z, x) from some neighbourhood of
(z0 , x0 ). For x0 ∈ −Kπ/2−β (z0 , x0 ), we consider two cases. The first one if −x0 , z0 >
|x0 ||z0 | cos (π/2 − β) then x ∈ −Kπ/2−β (z, x) for (z, x) from some neighbourhood of
(z0 , x0 ) as well. The next one if −x0 , z0 = |x0 ||z0 | cos (π/2 − β) (it means that
x0 /|x0 |, z0 /|z0 | = cos (π/2 + β)) then (z, x) → (z0 , x0 ) implies x/|x| → x0 /|x0 | and
N(x/|x|) = x/|x|, fβ (z, x) → x0 /|x0 |, fβ (z0 , x0 ) = 0 since by (6)
x0
π
π
x0 z0
∠
, fβ (z0 , x0 ) = ∠
,
−β = +β−β = .
|x0 |
|x0 | |z0 |
2
2
Thus u2 (z, x) → −x0 /|x0 | = u2 (z0 , x0 ). Therefore, we conclude that u2 is continuous.
Moreover, we have x/|x| + fβ (z, x), u2 (z, x) < 0. Indeed if x ∈ −Kπ/2−β (z, x) then
x
x
x
x
+ fβ (z, x), u2 (z, x) =
+ fβ (z, x), −
= −1 −
, fβ (z, x)
|x|
|x|
|x|
|x|
= −1 + c cos β < 0,
34
A. Arkit / Journal of Mathematical Economics 39 (2003) 27–38
where c = 1 for β > 0 and c ∈ (0, 1) for β = 0 (c = |fβ (z, x)|). On the other hand, if
x∈
/ −Kπ/2−β (z, x), then
x
+ fβ (z, x), u2 (z, x)
|x|
x/|x| + fβ (z, x), N(x/|x|)z/|z| − N(z/|z|)x/|x|
=
N(x/|x|)2 − 2N(x/|x|)N(z/|z|)x/|x|, z/|z| + N(z/|z|)2
x/|x|, z/|z|N(x/|x|) − N(z/|z|)
=
N(x/|x|)2 − 2N(x/|x|)N(z/|z|)x/|x|, z/|z| + N(z/|z|)2
N(z/|z|)(1/cN(x/|x|) − 1)
<
≤ 0.
2
N(x/|x|) − 2N(x/|x|)N(z/|z|)z/|z|, x/|x| + N(z/|z|)2
We have used the fact that ∠(z/|z|, x/|x|) > ∠(z/|z|, fβ (z, x)) and therefore
z x
1 z
z
1
,
<
, fβ (z, x) = N
= cos β ≤ 1.
|z| |x|
c |z|
c
|z|
Let g̃ : W ∪ {(0, 0)} → Rn be given by g̃(z, x) = |z|(u1 (z, x) + u2 (z, x)) for (z, x) = (0, 0)
and g̃(0, 0) = 0. It is clear that g̃(z, x) ∈ Kπ/2−β (z, x) for all (z, x). Moreover, g̃(z, x) ∈
H0− (x/|x| + fβ (z, x)), because we have
x
+ fβ (z, x), g̃(z, x)
|x|
x
x
= |z|
+ fβ (z, x), u1 (z, x) +
+ fβ (z, x), u2 (z, x)
|x|
|x|
x
≤ |z|
+ fβ (z, x), u2 (z, x) < 0
|x|
since x/|x| + fβ (z, x), u1 (z, x) ≤ 0 by the definition of u1 .
Let Φ : Rn → Rn × Rn be the continuous map defined by Φ(x) = (f(x), x). Clearly
Φ(x) ∈ W ∪ {(0, 0)} for all x, because of (5). Then let g = g̃ ◦ Φ. Observe that x → g(x)
is continuous at every x = 0. Since |g(x)| ≤ 2|f(x)| and f(x) → 0 as x → 0, we have
also g(x) → 0 as x → 0. Thus, the constructed function g is a continuous selection
of Gβ .
䊐
The important property of the continuous selection constructed in the proof of Lemma 4
is established by the next Remark. This property will be used in the proof of Theorem 8.
Remark 5. Under the assumption (5), the continuous selection g of Gβ , which was constructed in the proof of Lemma 4, satisfies condition
g(x) ∈ Cone {f(x), −x} = {η1 f(x) − η2 x : η1 , η2 ≥ 0}.
(7)
Proof. Let us first observe that u1 (z, x) ∈ Cone {z, −x} for all (z, x) ∈ W. It is obvious if z ∈ H0− (x/|x| + fβ (z, x)). If z ∈ H0+ (x/|x| + fβ (z, x)) then M(z/|z|) > 0 and
A. Arkit / Journal of Mathematical Economics 39 (2003) 27–38
35
M(x/|x|) = 1 + x/|x|, fβ (z, x) ≥ 1 − c ≥ 0. Moreover, u2 (z, x) ∈ Cone {z, −x} for
/ −Kπ/2−β (z, x) then N(z/|z|) =
all (z, x) ∈ W. It is clear if x ∈ −Kπ/2−β (z, x). If x ∈
z/|z|, fβ (z, x) = c cos β ≥ 0 and N(x/|x|) > 0. Indeed, x ∈
/ −Kπ/2−β (z, x) implies
π
x z
π
x z
,
> cos π −
−β
thus ∠
,
< + β.
|x| |z|
2
|x| |z|
2
Combining this with (6) we conclude that
π
π
x
, fβ (z, x) < + β − β =
∠
|x|
2
2
x
x
hence
, fβ (z, x) = N
> 0.
|x|
|x|
Let us remind that g̃ : W ∪ {(0, 0)} → Rn was given by g̃(z, x) = |z|(u1 (z, x) + u2 (z, x))
for (z, x) ∈ W. Thus we have g̃(z, x) ∈ Cone {z, −x} for all (z, x) from W. It implies that
g(x) = g̃(Φ(x)) ∈ Cone {f(x), −x} where Φ : Rn → Rn × Rn was the continuous map
䊐
defined by Φ(x) = (f(x), x).
Theorem 6. Let β ∈ [0, π/2] and f ∈ C(Rn ). If (5) holds, then there exists a continuous
function g ∈ C(Rn ) which is (π/2 − β)-compatible with f and such that every solution to
the equation ẋ = g(x), x(0) = x0 , x0 ∈ Rn is asymptotically convergent to 0.
Proof. It follows immediately from Proposition 2 and Lemma 4.
䊐
4. The stability of the price dynamics
4.1. The case of unique equilibrium
In present subsection, we will assume that the excess demand function of the considered
pure exchange economy satisfies hypothesis A1–A4 and it has the only zero, i.e. Pf = {p∗ }.
In Section 3, we have considered an autonomous differential equation ẋ = g(x), x(0) =
x0 , with some function g, which was (π/2 − β)-compatible with given function f and
we have stated conditions under which the asymptotic stability of solutions is guaranteed
for such differential equation. Since we are going to use Theorem 6 in order to prove the
n−1
existence of the price mechanism, i.e. the continuous function g : S+
→ Rn which is
(π/2 − β)-compatible with the excess demand f and such that g(p) ∈ H0 (p) for all p ∈
n−1
S+
, we have to project conformally trajectories of price dynamics characterized by the
n−1
excess demand f on hyperplane tangent to the sphere S+
at the point p∗ . The advantage
of this projection is at each point the angle between two trajectories on the sphere is the
same as the angle between their projections on the hyperplane tangent to this sphere. Let
us define the conformal projection formally.
Definitaion. A one-to-one smooth mapping Ψ of S n−1 \ {−p∗ } onto H1 (p∗ ) defined by
2
(8)
Ψ(p) = Φ(p)(p + p∗ ) − p∗ , where Φ(p) =
1 + p, p∗ for p ∈ S n−1 \ {−p∗ } we call the conformal projection (see Fig. 1).
36
A. Arkit / Journal of Mathematical Economics 39 (2003) 27–38
Fig. 1.
Because the conformal projection preserves angles we can use α-compatibility condition
for analysis of the properties of the price dynamics projected on H1 (p∗ ).
Theorem 8. Let β ∈ [0, π/2]. If f is the excess demand function of some pure exchange
economy, fulfilling A1–A4, with the only equilibrium point p∗ , and for all p = p∗ we have
∠(f(p), p, p∗ p − p∗ ) > β, i.e.
/ Cβ (f(p)) for all p = p∗
p, p∗ p − p∗ ∈
(9)
then there exists the globally stable price adjustment mechanism g which is (π/2 − β)-compatible with f .
n−1
Proof. Our proof starts with the conformal projection of price trajectories from S+
on
n−1
at p∗ . Let Q be defined by
the hyperplane tangent to S+
n−1
}
Q = {q ∈ H0 (p∗ ) : q = Ψ(p) − p∗ for all p ∈ S+
(where Ψ(p) is given by (8)). Then each q ∈ Q describes location of corresponding Ψ(p)
with respect to the point of tangency of H1 (p∗ ) to S n−1 .
n−1
For any price trajectory p(t) on S+
, there exists the trajectory q(t) on Q, such that for
all t, we have q(t) = Φ(p(t))(p(t) + p∗ ) − 2p∗ . It is easy to check that
dp
dq(t) dΦ(p(t))
=
(p(t) + p∗ ) + Φ(p(t))
dt
dt
dt
1
dp
2
∗ dp
= − Φ(p(t)) p ,
(p(t) + p∗ ) + Φ(p(t)) .
2
dt
dt
A. Arkit / Journal of Mathematical Economics 39 (2003) 27–38
37
Thus in particular, if p(t) is a solution to ṗ = f(p), then q(t) solves q̇ = fˆ(q), where fˆ(q)
is defined implicitly by the equality
1
2 ∗
ˆ
f (q) = − Φ(p) p , f(p) (p + p∗ ) + Φ(p)f(p).
2
Certainly fˆ(q), p∗ = 0, so fˆ(q) ∈ H0 (p∗ ) for all p. Calculations give
ˆ
|f (q)| = Φ(p)|f(p)|, |q| = Φ(p) 1 − p, p∗ 2 .
We conclude from (9) that for all p
−p∗ , f(p) < 1 − p, p∗ 2 |f(p)| cos β
hence
q, fˆ(q) = −Φ(p)2 p∗ , f(p) < Φ(p)2 1−p, p∗ 2 |f(p)| cos β = |q||fˆ(q)| cos β
and finally q ∈
/ Cβ (fˆ(q)) for all q = 0. According to Theorem 6, there exists the continuous
function ĝ : Rn → Rn which is (π/2 − β)-compatible with fˆ and such that every solution
of the equation q̇ = ĝ(q), q(0) = q0 , q0 ∈ Q is asymptotically convergent to 0. According
to Remark 5, we have ĝ(q) ∈ Cone {fˆ(q), −q} at every q ∈ Rn . Vectors of the projected
excess demand function fˆ(q) at q from the boundary of Q are directed inward the set Q
because of A4 and properties of the conformal projection. Thus we have guaranteed that
solutions of the equation q̇ = ĝ(q), q(0) = q0 , q0 ∈ Q do not leave the set Q. After inverse
n−1
, we receive a globally stable price
translation and projection of these solutions on S+
mechanism.
䊐
Remark 9. Let us remind that under usual assumptions guaranteeing global stability of the
price mechanism g = f (if the Weak Axiom of Revealed Preference for the excess demand
holds between the equilibrium point and any disequilibrium point), the excess demand at
any disequilibrium price situation weighted by equilibrium prices is always positive (see
e.g. Uzawa (1961)). The same conclusion we receive from (9) for g = f (β = π/2).
Moreover, we have for β ∈ [0, π/2]: if p∗ ∈ Pf is an equilibrium and if the value of the
excess demand f at p ∈
/ Pf weighted by p∗ satisfies
p∗ , f(p) > −|f(p)| 1 − p, p∗ cos β,
then there exist the globally stable price adjustment mechanism g, which is (π/2 − β)-compatible with f .
4.2. The case of multiple equilibria
When we assume that the set Pf has more than one equilibrium point, the following
result may be proved in much the same way as Theorem 8:
38
A. Arkit / Journal of Mathematical Economics 39 (2003) 27–38
Theorem 10. Let β ∈ [0, π/2]. If f is the excess demand function of some pure exchange
economy, fulfilling A1–A4, with arbitrarily fixed equilibrium point p∗ ∈ Pf and for all
p∈
/ Pf we have ∠(f(p), p, p∗ p − p∗ ) > β, i.e.
p, p∗ p − p∗ ∈
/ Cβ (f(p)),
for all p ∈
/ Pf
(10)
then there exists the globally stable price adjustment mechanism g which is (π/2 − β)-compatible with f .
The main idea of the proof remains the same. Let us observe that the construction of
continuous selection g in the proof of Theorem 6 ensured f(x) = 0 ⇔ g(x) = 0. Since the
constructed price adjustment process has this property, we can expect two cases:
(i) its trajectory never reach any equilibrium point in finite time and by Theorem 8 it
converges to p∗ when time tends to infinity;
(ii) its trajectory reach an equilibrium point in finite time and then the price mechanism
stops.
Thus we receive the globally stable price adjustment process, because for any initial price
system its price trajectory converges to one of equilibrium points of the excess demand f .
Acknowledgements
I would like to thank L.E. Rybiński for suggesting the problem and the concept of a
price mechanism α-compatible with an excess demand as well as for subsequent helpful
discussions during preparation of this paper.
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