Theoretical Economics Letters, 2014, 4, 232-234
Published Online April 2014 in SciRes. http://www.scirp.org/journal/tel
http://dx.doi.org/10.4236/tel.2014.43031
An Algebraic Proof of the Existence of a
Competitive Equilibrium in Exchange
Economies
Guang-Zhen Sun
Department of Economics, Faculty of Social Sciences, University of Macau, Macau, China
Email: [email protected]
Received 13 November 2013; revised 13 December 2013; accepted 30 December 2013
Copyright © 2014 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
The standard excess demand argument for existence of competitive equilibria in exchange economies invokes maximizing the market value of the aggregate excess demand function and thereby
adjusting the prices toward equilibrium. By exploiting the Perron-Frobenius theorem on stochastic matrices, we offer an algebraic proof of the existence of a competitive equilibrium without resorting to such a device of excess demand.
Keywords
Exchange Economy, Existence of Competitive Equilibrium, Perron-Frobenius Theorem
1. Introduction
The standard excess demand argument for existence of competitive equilibriain exchange economies invokes
maximizing the value of the excess demand of the economy (the aggregate excess demand) and thereby adjusting the prices toward equilibrium [1]-[3]. At least for pedagogical purpose, one may legitimately interpret the
adjustment procedure, often referred to as tatonnement, as that in which a price-setting agency, the so-called
Walrasian auctioneer, gathers the information as regards to each agent’s excess demand and then fine-tunes the
prices so as to raise the prices of the commodities that are over-demanded (with a positive aggregate excess demand registered) and to lower the prices of the commodities that are under-demanded (with a negative aggregate
excess demand registered). It is true that the excess demand function is a very useful device that renders the
fixed point theory nicely applicable to the proof of the existence of a market-clear price vector. Such a technical
device, powerful though as it is in remarkably simplifying the formulation, nonetheless suggests a centralized
coordination mechanism. Exploiting the Perron-Frobenius theorem on stochastic matrices, a well known result
How to cite this paper: Sun, G.-Z. (2014) An Algebraic Proof of the Existence of a Competitive Equilibrium in Exchange
Economies. Theoretical Economics Letters, 4, 232-234. http://dx.doi.org/10.4236/tel.2014.43031
G.-Z. Sun
in linear algebra, we provide an alternative proof of the existence of a competitive equilibrium without resorting
to such an artificial price-setting mechanism. It is worth pointing out that in so doing our argument does not invoke aggregating the individuals’ excess demand, let alone to maximizing the market value of the aggregate
excess demand. To the extent that the price-setting agency, as is embodied by the Walrasian auctioneer, personifies the coordination of the decentralized price system, “the invisible hand”, and hence rendering it more or less
visible, our approach appears to be conceptually more natural. But a cost of awkwardness in algebraic manipulation has to be paid in our undertaking, compared to the rather neat formulation based on the Walrasian tatonnement. However, such a cost may be well justifiable for an economically appealing argument about the important
idea of the invisible hand.
2. The Proof
We first restate the classical result on the existence of a competitive equilibrium.
Theorem. For any pure exchange economy with n commodities and m agents, in which each agent
τ ∈ {1, , m} has a preference represented by a utility function U τ : R+n → R that is continuous, concave and
strongly monotone and an endowment ωτ ∈ R+n such that the endowment of any commodity for the economy as
m
a whole is positive, i.e., ∑τ =1 ωτ  0, there exists a competitive equilibrium.
m
Proof. First of all, normalize the endowment of each commodity of the community as one, i.e., ∑τ =1 ωiτ = 1 ,
{
}
∀i ∈ {1, , n} 1. Consider the price simplex Sn ≡ p pi ≥ 0, ∀i and ∑ i =1 pi =
1 . For any interior point p , let
n
T
  p xτ ( p )

p xτ ( p )  τ


τ
Y τ ( p ) ≡   1 τ1
, , n τn
1, , m . Notice that for any
 x ( p ) ∈ arg max px ≤ pωτ , x ≥ o U ( x )  ,τ =
p
p
ω
ω
⋅
⋅
 


yτ ∈ Y τ ( p ) its i-th component is the percentage share of agent τ ’s wealth spent on commodity i. Monotone
preference implies
τ
= 1 . Convexity of preferences implies that for any agent τ , Y ( p ) is convex at
∑ i =1 yiτ
n
any interior point of Sn . Also note that Y τ ( p ) is upper-semi-continuous by Berge’s maximum theorem. For
any p ∈ ∂Sn , let Y τ ( p ) be the convex hull of the limit points of any possible
k
k
( )}
∈ Y τ pk
where the
{ p } approaches p. That is, Y τ ( p ) ≡ Co { y ∃{ p } and { y } such
( p ) and that p → p, y → y as k → ∞ }. By definition, Y τ ( p ) is
k
sequence of interior points
∀k , p k ∈ Sn  ∂Sn , y k ∈ Y τ
{y , y
k
k
k
k
k
that
up-
per-semi-continuous and convex-valued at any p ∈ ∂Sn .
2
)
(
m
Let F p, y1 ( p ) , , y m ( p ) ≡  ∑τ =1 yiτ ( p ) ωτj 
The sum of each column of F equals

 n× n
yiτ ωτj =
yiτ ) ωτj =
ωτ
∑ ∑=
∑τ 1 (=
∑ i 1=
∑=
τ 1 j
n
m
m
n
m
1 , hence F is a column stochastic matrix. Consequently, the
τ 1
=i 1 =
Perron-Frobenius eigenvalue of F is one and there exists at least one corresponding eigenvector q ∈ Sn , i.e., Fq
1
( w )′ ≡ w ( ∑
One may do so by considering
τ
τ
m
i
i
s =1
wis
)
for any agent τ and commodity 𝑖𝑖and the prices adjusted to the accordingly
changed units used to measure each commodity.
2
A careful reader might be concerned with the closedness of the correspondence graph along ∂S n . But it is not a problem. Consider any
{ p } ⊂ ∂S
k
n
and
{y }
k
suppose that for any k ,
such that ∀k , y k ∈ Y τ ( p k ) and p k → p as k → ∞ . One needs to come to grips with two scenarios. First of all
y k is a limit when some sequence of strictly positive price vectors approaches p . By the definition of Y τ on
{( p ) }
∂S n , for any k there exists
k
s
s ≥1
⊂ Sn  ∂Sn and
One can thus appropriately construct a sequence
{y }
t
t ≥1
, converging to y ∈ Y
τ
{p }
t
t ≥1
( y )
k
s
∈Y τ
(( p ) ) ,
k
s
∀s, such that ( p k ) → p k ,
⊂ S n  ∂S n , with p t ∈
s
{( p ) }
t
s
s ≥1
( y )
k
s
→ y k ∈ Y τ ( p k ) as s → ∞.
, which converges to p and to which a sequence of
( p ) , is associated. Similar argument yet with more awkward notations applies to another scenario in which
k
for some or all k , y is not a limit point but a convex combinations of two vectors each of which is a limit point when one sequence of
strictly positive price vectors approaches p.
233
G.-Z. Sun
= q [4].
Consider any sequences
{ p } and {q }
k
k
(
such that there exist
F k p k = q k for any k where F k ≡ F p k , ( y1 ) , , ( y m )
assume ∀τ ,
( )
(y )
τ k
k
k
)
(y )
τ k
( )
∈ Y τ pk , τ =
1, , m, satisfying
and that p k → p* , q k → q* as k → ∞. WLOG,
→ y*τ . As is established in the above, Y τ ( p ) is upper-semi-continuous. Hence,
y*τ ∈ Y τ p* . Take the limit on both sides of F k p k = q k as k → ∞, F * p* = q* where
(
)
F ≡ F p* , y*1 , , y*m . The correspondence from p to q is closed.
*
We now prove that ∀p ∈ Sn , {q(p)} is convex. Consider any two elements in {q(p)} say q A and q B , i.e.,
there exist y Aτ , y Bτ ∈ Y τ ( p ) , τ ∈ {1, , m} such that  y A1  y Am  Wq A = q A and  y B1  y Bm  Wq B = q B
where W ≡ ωτj  . For any β ∈ [ 0,1] , we claim that there exist ατ ∈ [ 0,1] , τ = 1, , m such that
m× n
(
)
α1 y A1 + (1 − α1 ) y B1 +  + α m y Am + (1 − α m ) y Bm  W β q A + (1 − β ) q B = β q A + (1 − β ) q B
(1)
Note the term in the brackets of LHS of Equation (1) equals
α1  y A1 0n 0n  + (1 − α1 )  y B1 0n 0n  +  + α m 0n 0n y Am  + (1 − α m ) 0n 0n y Bm 
(2)
Also notice that the RHS of Equation (1) equals
{ y
A1
}
{
}
0n  0n  +  + 0n  0n y Am  W β q A +  y B1 0n  0n  +  + 0n  0n y Bm  W (1 − β ) q B
Substituting the above and (2) into (1) yields,
m
n
n
τ A
=
∀s 1, 2,...., n, ∑τ 1  ysBτ =
− ysAτ  (1 − ατ ) β ∑ i 1 ω
− β ) ∑ i 1 ωiτ qiB  0
i qi − ατ (1=
=
=


Since q A , q B ∈ Sn and ωτ ∈ [ 0,1] , ∀τ , there apparently exists ατ ∈ [ 0,1] , τ = 1, , m, such that
n
τ A
τ B
(1 − ατ ) β ∑ in 1ω
i qi − ατ (1 − β ) ∑ i 1 ωi qi , i.e., (1) holds. That is, for any ∈ S n , {q(p)} is a convex set.
=
=
n
(
( )
( ))
By Kakutani’s theorem, there exists p* ∈ Sn , such that F p* , y1 p* , , y m p* p* = p* where
then by the definition of Y τ ( p )
1, , m . We now show that p* ∉ ∂Sn . For otherwise,
yτ ( p* ) ∈ Y τ ( p* ) , τ =
k
such that
on ∂Sn , one possibility is that there exist sequences { p k } and ( yτ )
( )
τ k
∀k , p ∈ Sn  ∂Sn , y
k
( ) ( )
lim  y1  y m

∈Y
τ
(p )
k
and that p → p ∈ ∂Sn ,
k
*
{ }
(y )
τ k
( )
→ yτ p*
as k → ∞, that is,
 Wp* = p* . WLOG, assume p* > 0 but some other components of p* are zeros. Then
1

k
k
k
m
we obtain from lim  y1  y m  Wp* = p* that lim ∑τ =1 yτs ω1τ = 0 for any s such that ps* = 0 .


k →∞
k →∞ 
Thus for an agent 𝜏𝜏 endowed with a positive amount of commodity 1 (existence of such agent is guaranteed
k
k →∞
k
( )
( ) ( )
by the assumption that
∑τ =1ω1τ
m
( )
τ
> 0 ) lim k →∞ ys
k
= 0 for anys such that ps* = 0 , an impossibility in light of
the strongly monotone preferences. In the case that Y τ ( p ) itself is not a limit point for any sequence of interior price vectors that approaches p* , but a convex combination of two such limit points, the above application
applies to each of these two limit points. Thus, p*  0. By simple algebraic manipulation one obtains from
( )
( )
xiτ p* =
1=
F p* p* = =
p* that ∀i, ∑τ 1 =
∑τ 1ωiτ , i.e., all markets clear.
m
m
References
[1]
Debreu, G. (1983) Four Aspects of the Mathematical Theory of Economic Equilibrium. In: Mathematical Economics:
Twenty Papers of Gerard Debreu, Cambridge University Press, New York, 217-231.
[2]
Mas-Colell, A., Whinston, M.D. and Green, J. (1995) Microeconomic Theory. Oxford University Press, Oxford.
[3]
Jehle, G.A. and Reny, P.J. (2000) Advanced Microeconomic Theory. 2nd Edition, Addison Wesley, Boston.
[4]
Aldrovandi, R. (2001) Special Matrices of Mathematical Physics. World Scientific, Singapore.
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