In the paper, the asymptotic distribution of the equilibrium price in markets with the random demand and supply is described. Two special cases - the one with smooth demand and supply curves and the one with jump demand and supply curves - are studied. It is found that in both the cases the fluctuations of the price vanish at the rate O(n^{-1/2}) as the number of the agents n tends to infinity. Finally, a normal approximation of the distribution of the equilibrium price is suggested.equilibrium price, random demand, random supply, asymptotic distribution, normal approximation

Martin Smid

English

Research Papers in Economics

Akademie v· ed · Cesk¶ e republiky
¶ Ustav teorie informace a automatizace
Academy of Sciences of the Czech Republic
Institute of Information Theory and Automation
RESEARCH REPORT
Martin · Sm¶ ³d
e-mail martin@klec.cz
NORMAL APPROXIMATION OF THE
DISTRIBUTION OF THE
EQUILIBRIUM PRICE IN MARKETS
WITH RANDOM DEMAND AND
SUPPLY
No. 2120 November 2004
¶ UTIA AV · CR, P. O. Box 18, 182 08 Prague, Czech Republic
Telex: 122018 atom c, Fax: (+420) 266 053 111
E-mail: utia@utia.cas.czThis report constitutes an unrefereed manuscript which is intended to be
submitted for publication. Any opinions and conclusions expressed in this
report are those of the author(s) and do not necessarily represent the views of
the Institute.Abstract
In the paper, the asymptotic distribution of the equilibrium price in markets
with the random demand and supply is described. Two special cases - the one
with smooth demand and supply curves and the one with jump demand and
supply curves - are studied. It is found that in both the cases the °uctuations
of the price vanish at the rate O(n¡1=2) as the number of the agents n tends to
in¯nity. Finally, a normal approximation of the distribution of the equilibrium
price is suggested.
Keywords: equilibrium price, random demand, random supply, asymptotic
distribution, normal approximation
AMS classi¯cation: 91B26
JEL classi¯cation: C65
Introduction
In many real situations, the demand and/or supply curves have to be taken as
stochastic. Even if there exist models with stochastic demand and/or supply,
I have neot found a work describing the distribution of the equilibrium price.
The reason of this absence could be that the equilibrium price is generally
a non-linear transformation of random parameters, hence its distribution may
be described only in special cases, and even there the resulting formulas are
complex (cf. · Sm¶ ³d [2004a,b]).
One of the possible ways how to overcome these di±culties is studying the
asymptotic properties of the model instead of trying to infer \exact" distribu-
tions.
Until now, no work applying this approach is known to me except for my
own research report (· Sm¶ ³d [2004b]) dealing with the special case of linear de-
mand and supply curves. The present paper generalizes the results of the
report.
The model of a market with stochastic demand nad supply, de¯ned in the
present paper, assumes a ¯nite number number of market participants and
the individual demand and supply curves are taken as identically distributed
independent random elements taking values in function spaces. In accordance
with the standard economic theory, the equilibrium price is assumed to lie in
the intersection of the aggregate demand and supply curves.
2Two special cases of the general model are studied in the paper: the case of
continuous twice di®erentiable demand and supply curves (which may be suit-
able for modeling markets with a divisible good) and the case of jump demand
and supply curves taking integer values (which could be suitable for markets
with an indivisible commodity). In both the cases, the asymptotic behavior
of the equilibrium price, as the number of participants tends to in¯nity, is
described.
In both the \continuous" and \jump" cases, the distribution of the price
Pn converges to a constant, namely to the intersection of the expected demand
curve and the expected supply curve, and the variance of Pn vanishes at the
rate O(n¡1=2) where n is the number of participants.
The asymptotic distribution of
p
n(Pn ¡ p?), where p? is the limit price,
depends directly only on the derivatives of the expected aggregate demand and
supply curves in the point p?, and on the variance of the curves in p?. Rather
surprisingly, the asymptotic distribution is identical in both the \continuous"
and \smooth" cases.
As a main result of the paper, a normal approximation of the equilibrium
price distribution is suggested.
The paper is organized as follows. In Section 1, the general model is de¯ned.
In Section 2, the asymptotic behavior of the price in the \continuous" case is
analyzed. In section 3, the same is done for the \jump" case. In Section 4, the
normal approximation is suggested.
1 De¯nition of the Model
Assume a market peopled by n agents with the individual demand functions
di, i = 1;2;:::;n, and with individual supply functions di, i = 1;2;:::;n,
such that di is a random element taking values in the space of non-negative
non-increasing real functions, and si is a random element taking values in the
space of non-negative non-decreasing real functions for each i = 1;2;:::;n. Let
(di;si), i = 1;2;:::;n, be identically distributed and let (di;si) be independent
of (dj;sj) for each i 6= j. Moreover, let the expectations
±(p)
4
= Ed1(p);
and
¾(p)
4
= Es1(p)
exist and be ¯nite for each p 2 R, and let there exist a unique p? 2 R such
that ±(p?) = ¾(p?).
Further, assume that the equilibrium price Pn \happens" in some point
of the \intersection" of the aggregate demand curve Dn =
Pn
i=1 di with the
3aggregate supply curve Sn =
Pn
i=1 si, i.e. it ful¯ls the condition
P
L
n · Pn · P
H
n (1)
where
P
L
n = inffp : Dn(p) · Sn(p)g;
and
P
H
n = supfp : Dn(p) ¸ Sn(p)g:
2 The Case of Continuous Demand
and Supply Functions
Assume, in the present section, that each realization of di, and each realization
of si are twice di®erentiable for each i = 1;2;:::;n.
Theorem 1 If Ejd1(p?)j < 1, Ejs1(p?)j < 1, 0 < E[s0
1(p?) ¡ d0
1(p?)] < 1,
Ejd
00
1(p?)j < 1, and Ejs
00
1(p?)j < 1 then
p
n(Pn ¡ p
?)
n!1 ¡! N (0;V ) (2)
in distribution where
V =
var(s1(p?) ¡ d1(p?))
[E(s0
1(p?) ¡ d0
1(p?))]
2
independently on the choice of Pn.
If, moreover, there exists an open interval I, such that p? 2 I, and a random
variable g, Eg < 1, such that
s
0
1(p) ¡ d
0
1(p) · g (3)
for each p 2 I almost sure then
V =
var(s1(p?) ¡ d1(p?))
[¾0(p?) ¡ ±0(p?)]
2 : (4)
Proof. The proof is given in the Appendix A. 2
3 The Case of Jump Demand and Supply
Functions
Assume, in this section, that each realization of di, i = 1;2;:::;n, is left-
continuous with integer values, and that each realization of si, i = 1;2;:::;n,
is right-continuous with integer values. Further, assume that there exist twice
di®erentiable functions ^ ± and ^ ¾ such that
4(a) d1(p) » Po(^ ±(p)) for each p 2 R,
(b) d1(p) ¡ d1(p0) » Po(^ ±(p) ¡ ^ ±(p0)) for each p < p0,
(c) s1(p) » Po(^ ¾(p)) for each p 2 R,
(d) s1(p) ¡ s1(p0) » Po(^ ¾(p) ¡ ^ ¾(p0)) for each p0 < p.
Evidently, ±(p) = ^ ±(p) and ¾(p) = ^ ¾(p) for each p 2 R under these assumptions.
Theorem 2 If ¾0(p?) ¡ ±0(p?) > 0 then
p
n(Pn ¡ p
?)
n!1 ¡! N (0;W) (5)
in distribution where
W =
var(s1(p?) ¡ d1(p?))
[¾0(p?) ¡ ±0(p?)]
2 :
If, moreover d1 is independent of s1, it holds that
W =
¾(p?) ¡ ±(p?)
[¾0(p?) ¡ ±0(p?)]
2: (6)
Proof. The proof in given in the Appendix B. 2
4 Normal Approximation
Our asymptotic results indicate that, if the assumptions of Theorem 1 (together
with (3)) or the assumptions of Theorem 2 are ful¯lled, and if n is su±ciently
large, we may write
Pn _ »N
µ
p
?;
1
n
¢
var(s1(p?) ¡ d1(p?))
[¾0(p?) ¡ ±0(p?)]
2
¶
:
Rewritten \in the language of aggregate functions", the formula sounds
Pn _ »N
µ
p
?;
var(Sn(p?) ¡ Dn(p?))
[§0
n(p?) ¡ ¢0
n(p?)]
2
¶
where ¢n = EDn, §n = ESn.1
1Indeed,
var(Sn(p?) ¡ Dn(p?)) = nvar(s1(p?) ¡ d1(p?));
and
E(S0
n(p?) ¡ D0
n(p?)) = n(¾0(p?) ¡ ±0(p?)):
5Conclusion
In the paper, the asymptotic behavior of the equilibrium price in markets
with random demand and supply, and with the independent behavior of its
participants was studied. In particular, markets with smooth demand and
supply curves, and markets with jump demand and supply curves were taken
into account. As a main result, a normal approximation of the distribution
of the equilibrium price, depending only on the aggregate demand and supply
curves, was suggested.
Acknowledgement: This paper was partially supported by the grant no
402/04/1294 of the Grant Agency of the Czech Republic and by the grant no.
402/03/H057 of the same institution.
A Proof of Theorem 1
The core of the proof consists in the determination of the asymptotic distribu-
tion of the vector
¥n
4
=
p
n(P
H
n ¡ p
?;P
L
n ¡ p
?):
It could be shown analogously to e.g. Araujo and Gin¶ e [1980], p. 8, that
Sc1;c2 = (¡1;c1] £ (¡1;c2); c1 2 R;c2 2 R; (7)
is a convergence determining system of sets, i.e. it su±ces to study the con-
vergence on the sets (7).
Let us do it: From the continuity of Dn and Sn, it follows that
P
L
n · p () Sn(p) ¸ Dn(p)
for each p 2 R. Using it and the Taylor expansions
di(p
? + ¢) = di(p
?) + d
0
i(p
?)¢ + d
00
i (p
?)
¡
»¢
i
¢2
2
;
and
si(p
? + ¢) = si(p
?) + s
0
i(p
?)¢ + s
00
i (p
?)
¡
´¢
i
¢2
2
6where »¢
i and ´¢
i are constants lying between 0 and ¢, we get
p
n(P
L
n ¡ p
?) · c1 () P
L
n · p
? +
c1 p
n
() Dn
µ
p
? +
c1 p
n
¶
¡ Sn
µ
p
? +
c1 p
n
¶
· 0
()
1
p
n
·
Dn
µ
p
? +
c1 p
n
¶
¡ Sn
µ
p
? +
c1 p
n
¶¸
· 0
() An ¡ c1Bn + R
c1
n · 0
(8)
where
An =
1
p
n
n X
i=1
[di(p
?) ¡ si(p
?)]; Bn =
1
n
n X
i=1
[s
0
i(p
?) ¡ d
0
i(p
?)];
and
R
c
n =
1
2
p
n
"
n X
i=1
d
00
i (p
?)(»
c=
p
n
i )
2 ¡
n X
i=1
s
00
i (p
?)(´
c=
p
n
i )
2
#
:
Since
P
H
n < p () Sn(p) > Dn(p)
we could prove, similarly to the case of P L
n , that
p
n(P
H
n ¡ p
?) < c2 () An ¡ c2Bn + R
c2
n < 0: (9)
Further, it follows from the Central Limit Theorem for the sums of i.i.d. vari-
ables that
An
n!1 ¡! U; (10)
where U » N (0;var(d1(p?) ¡ s1(p?))), in distribution. Further, due to the
Law of Large Numbers,
Bn
n!1 ¡! E(s
0
1(p
?) ¡ d
0
1(p
?)) (11)
in probability, hence in distribution. Finally, since
jR
c
nj ·
1
p
n
Ã
1
n
n X
i=1
·
c2jd
00
i (p?)j
2
¡
c2js
00
i (p?)j
2
¸!
;
we may use the Law of Large Numbers to get that
R
c
n
n! ¡! 0 (12)
7in distribution.2 By a combination of (8), (9), (10), (11) and (12) and by using
the Continuous Mapping Theorem (cf. Pollard [2002], p.175), we get that
lim
n!1
P f¥n 2 (¡1;c1] £ (¡1;c2)g
=P fU ¡ c1E[s
0
1(p
?) ¡ d
0
1(p
?)] · 0;U ¡ c2E[s
0
1(p
?) ¡ d
0
1(p
?)] < 0g
=P
½
U
E[s0
1(p?) ¡ d0
1(p?)]
· c1;
U
E[s0
1(p?) ¡ d0
1(p?)]
< c2
¾
=P
½
U
E[s0
1(p?) ¡ d0
1(p?)]
· c1;
U
E[s0
1(p?) ¡ d0
1(p?)]
· c2
¾
;
(13)
i.e.
¥n
n!1 ¡!
1
E[s0
1(p?) ¡ d0
1(p?)]
(U;U)
in distribution.
Now that we have the asymptotic distribution of ¥n, it is easy to prove the
rest. It follows from (13) that
p
n(P
L
n ¡ p
?)
n!1 ¡! N (0;V ): (14)
in distribution, and, by the Continuous Mapping Theorem, that
¯
¯p
n(Pn ¡ P
L
n )
¯
¯
(1)
·
¯
¯p
n(P
H
n ¡ P
L
n )
¯
¯ =
¯
¯p
n(P
H
n ¡ p
?) ¡
p
n(P
L
n ¡ p
?)
¯
¯
n!1 ¡!
1
E[s0
1(p?) ¡ d0
1(p?)]
jU ¡ Uj = 0
in distribution, which, together with (14), gives
p
n(Pn ¡ p
?) =
p
n(Pn ¡ P
L
n ) +
p
n(P
L
n ¡ p
?)
n!1 ¡! N (0;V )
by the Continuous Mapping Theorem.
Ad (4): The formula follows from the well known theorem on the exchange-
ability of an integral with a derivative (see e.g. Luke· s and Mal¶ y [1995], par.
9.2.).
2Indeed, if X1;X2 are i.i.d. such that EjX1j < 1 then
P
n
n¡1=2j ¹ Xnj ¸ ²
o
· P
n
n¡1=2(j ¹ Xn ¡ EX1j + jEX1j) ¸ ²
o
· P
©
j ¹ Xn ¡ EX1j ¸ ²=2
ª
+ P
n
n¡1=2jEX1j ¸ ²=2
o
n!1 ¡! 0
from the Law of Large Numbers.
8B Proof of Theorem 2
It follows from the de¯nition of P L
n that
P
L
n · p () Sn(p
+) ¸ Dn(p
+):
which is equivalent to
P
L
n · p () Sn(p) ¸ Dn(p
+):
due to the right-continuity of Sn. Moreover, since
P
©
Dn(p) ¡ Dn(p
+) > 0
ª (b) = 0; (15)
it holds that
P
©
P
L
n · p
ª
= P
©
Sn(p) ¸ Dn(p
+)
ª
=P fSn(p) ¸ Dn(p)g ¡ P
©
[Sn(p) ¸ Dn(p
+)] ^ [Dn(p) ¡ Dn(p
+) > 0]
ª
(15)
= P fSn(p) ¸ Dn(p)g
i.e.
P
L
n · p () Sn(p) ¸ Dn(p) almost sure: (16)
Similarly we get that
P
H
n ¸ p () Sn(p) · Dn(p) almost sure
which is equivalent to
P
H
n < p () Sn(p) > Dn(p) almost sure: (17)
Using it, we are getting
P
©p
n(P
L
n ¡ p
?) · c1;
p
n(P
H
n ¡ p
?) < c2
ª
=P
½
P
L
n · p
? +
c1 p
n
;P
H
n < p
? +
c2 p
n
¾
(16)(17)
= P
½
1
p
n
·
Dn
µ
p
? +
c1 p
n
¶
¡ Sn
µ
p
? +
c1 p
n
¶¸
· 0;
1
p
n
·
Dn
µ
p
? +
c2 p
n
¶
¡ Sn
µ
p
? +
c1 p
n
¶¸
< 0
¾
:
(18)
Let us expres the aggregate curves similarly to the proof of Theorem 1: Thanks
to the the di®erentiability of ±(²), we may expand
±(p
? + ¢) = ±(p
?) + ±
0(p
?)¢ + ±
00
(p
?)
(»¢)2
2
9where »¢ lies somewhere between 0 and ¢. Hence, and thanks to the divisibility
of the Poisson distribution, we may write
di
µ
p
? +
c
p
n
¶
= ai + b
c
i + r
c
i (19)
where
ai » Po(±(p
?)); b
c
i » Po
µ
±
0(p
?)
c
p
n
¶
; r
c
i » Po
µ
±
00
(p
?)
(»c=
p
n)2
2
¶
:
Similarly, we can write
si
µ
p
? +
c
p
n
¶
= ®i + ¯
c
i + ½
c
i (20)
where
®i » Po(¾(p
?)); ¯
c
i » Po
µ
¾
0(p
?)
c
p
n
¶
; ½
c
i » Po
µ
¾
00
(p
?)
(´c=
p
n)2
2
¶
for some ´c=
p
n lying between 0 and c=
p
n.
By summing the individual curves, we obtain
1
p
n
µ
Dn
µ
p
? +
c
p
n
¶
¡ Sn
µ
p
? +
c1 p
n
¶¶
= An ¡ B
c
n + R
c
n (21)
where
An =
1
p
n
n X
i=1
[ai ¡ ®i]; B
c
n =
1
p
n
n X
i=1
[¯
c
i ¡ b
c
i]; R
c
n =
1
p
n
n X
i=1
[r
c
i ¡ ½
c
i]:
Therefore and due to (18),
P
©p
n(P
L
n ¡ p
?) · c1;
p
n(P
H
n ¡ p
?) < c2
ª
= P fAn ¡ B
c1
n + R
c1
n · 0;An ¡ B
c2
n + R
c2
n < 0g:
Finally, since
An
n!1 ¡! N (0;var(d1(p
?) ¡ s1(p
?)));
in distribution,
Bn
n!1 ¡! c(¾
0(p
?) ¡ ±
0(p
?))
10in distribution3 for each c 2 R, and since
R
c
n
n!1 ¡! 0
in distribution4 for each c 2 R, we may use the Continuous Mapping Theorem
to get that
An ¡ B
c
n + R
c
n
n!1 ¡! N (0;var(d1(p
?) ¡ s1(p
?))) ¡ c(¾
0(p
?) ¡ ±
0(p
?))
in distribution for each c 2 R. The rest of the proof of (5) is identical to the
proof of Theorem 1.
Ad. (6). The formula follows from basic rules for computing variances.
References
A. Araujo and E. Gin¶ e. The Central Limit Theorem for Real and Banach
Valued Random Variables. J. Wiley & Sons, New York, 1980.
J. Luke· s and J. Mal¶ y. Measure and Integral. Matfyzpress, Prague, 1995.
D. Pollard. A User's Guide to Measure Theoretic Probability. Cambridge Univ.
Press, Cambridge, 2002.
3Indeed, if X1;X2;:::;Xn are i.i.d. such that Xi » Po(d=
p
n) for each i = 1;2;:::;n
then there exist independent random variables Y1;Y2;:::;Yn;Zn such that Yi » Po(d) and
Zn » Po(°n), °n < d, and that
n X
i=1
Xi =
b
p
nc X
i=1
Yi + Zn:
Consequently,
1
p
n
n X
i=1
Xi =
1
b
p
nc
b
p
nc X
i=1
Yi +
µ
1
p
n
¡
1
b
p
nc
¶ b
p
nc X
i=1
Yi +
Zn p
n
:
Since the limit of the ¯rst summand is d due to the Law of Large Numbers, and the variance
of the second and the third summand goes to zero, the limit of the whole sum is d.
4The convergence follows from the fact that
var(Rc
n) = var(rc
i ¡ ½c
i)
Schwarz
·
µq
var(rc
i) +
q
var(½c
i)
¶2
=
Ã
»c=
p
n
r
±
00(p?)
2
+ ´c=
p
n
r
¾
00(p?)
2
!2
·
c2
n
Ãr
±
00(p?)
2
+
r
¾
00(p?)
2
!2
! 0
for each c 2 R.
11M. · Sm¶ ³d. Stochastic model of thin market of nondivisible commodity, 2004a.
Research report no. 2100, Institute of Informaton Theory and Automa-
tion, April 2004, http://econwpa.wustl.edu:80/eps/ge/papers/0406/
0406003.pdf.
M. · Sm¶ ³d. Stochastic model of thin market with divisible commodity, 2004b.
Research report no. 2106, Institute of Information Theory and Automa-
tion, September 2004, http://econwpa.wustl.edu:80/eps/ge/papers/
0409/0409006.pdf.
12

3Indeed, if X1;X2;:::;Xn are i.i.d. such that Xi » Po(d= p n) for each i = 1;2;:::;n then there exist independent random variables Y1;Y2;:::;Yn;Zn such that Yi » Po(d) and Zn » Po(°n), °n < d, and that n X

A User's Guide to Measure Theoretic Probability. Cambridge Univ.

Stochastic model of thin market of nondivisible commodity,

Stochastic model of thin market with divisible commodity,

The Central Limit Theorem for Real and Banach Valued Random Variables.

Normal Approximation of the Disrtibution of the Equilibrium Price in Markets with Random Demand and Supply

http://128.118.178.162/eps/ge/papers/0411/0411001.pdf

Normal Approximation of the Disrtibution of the Equilibrium Price in Markets with Random Demand and Supply

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics