We suggest a model of (a thin) market at which the number of participants is random with Poisson distribution. We provide a formula for joint distribution of the market price and the traded volume. We derive an asymptotic distribution of the quantities. We find that, according to our model, with increasing intensity of the participants' number, the fluctuations of the market price vanish while the variance of the traded volume increases.thin market, market price, traded volume, asymptotic distribution

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
STOCHASTIC MODEL OF THIN
MARKET WITH DIVISIBLE
COMMODITY
No. 2106 September 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
We suggest a model of (a thin) market at which the number of participants
is random with Poisson distribution. We provide a formula for joint distri-
bution of the market price and the traded volume. We derive an asymptotic
distribution of the quantities. We ¯nd that, according to our model, with in-
creasing intensity of the participants' number, the °uctuations of the market
price vanish while the variance of the traded volume increases.
Keywords thin market, market price, traded volume, asymptotic distribution
AMS classi¯cation: 91B26
JEL classi¯cation: C65
1 Introduction
The standard economic theory teaches us that changes of the market price are
caused by movements of the aggregate demand and of the aggregate supply
curves (the movements itself are caused by changes of agents' preferences) or
by a delay of producers' reaction to the demand (various cobweb models are
trying to describe the situation, see [1]).
We point at another possible cause of the price (and of the traded volume)
°uctuation - the varying number of market participants.
We suggest a model in which the individual demand function of each buyer
and the individual supply function of each seller remains constant but the
number of buyers coming to trade and the number of sellers coming to trade
are random. We assume Poisson distribution of both quantities.
We suppose our model to be quite realistic (especially on the side of the
demand). For instance, a consumer may have constant demand for milk but
he/she may not come to the same milkman every day. Another exemplary
situation is the following: The consumers have constant (weekly) demand for
carrot but they buy it only once a week. Similarly, the farmers may provide
constant supply of carrot but they come to sell it only once a week. In this
situation, both the number of buyers and the number of sellers may be modelled
using random variables so that our model may be suitable.
2 De¯nitions
Consider a market with perfectly divisible commodity being peopled by two
types of agents - the buyers and the sellers. Suppose all buyers to have the
2same (linear) demand function
d(p) = max(d1(d2 ¡ p);0)
where d1 > 0, d2 > 0 are some constants. Further, suppose that all sellers have
the same (linear) supply function
s(p) = max(s1(p ¡ s2);0);
where s1 > 0, s2 > 0 and s2 < d2.
Denote by X the number of buyers coming to trade and by Y the number
of sellers coming to trade and assume that X and Y are random both having
Poisson distribution with parameters ±, ¾ respectively. We recall that the
Poisson distribution of agents' arrival is usually used in mathematical models
of situations with a large number of agents, each coming with the same small
probability.
Under our assumptions, the (random) aggregate demand function and the
(random) aggregate supply function are
D(p) = Xd(p); S(p) = Y s(p)
with expected values ED(p) = ±d(p);ES(p) = ¾s(p): Assume further, that
the market price is determined by the intersection of the aggregate demand
function with the aggregate supply function, provided that at least one buyer
and at least one seller came to trade. If no seller came or no buyer came then
the price remains unde¯ned. Then, the price equals to
P =
8
> > <
> > :
d1d2X+s1s2Y
d1X+s1Y = d2
³
1 ¡
1¡s2d¡1
2
1+d1s¡1
1 XY ¡1
´
= s2
³
1 +
d2s¡1
2 ¡1
s1d¡1
1 Y X¡1+1
´ if X > 0 ^ Y > 0
unde¯ned otherwise
(1)
(symbol ^ denotes logical and) while the traded volume is
Q =
(
D(P) = S(P) =
d2¡s2
s¡1
1 Y ¡1+d¡1
1 X¡1 if X > 0 ^ Y > 0
0 otherwise
:
Since X > 0 ^ Y > 0 ) Q > 0, we have
[P = unde¯ned] , [Q = 0] , [Y = 0 _ X = 0] (2)
(symbol _ means logical or).
33 Distribution of the market price and the
traded volume
If p > s2 then
[P 6= unde¯ned] ^ [P · p]
, [X > 0 ^ Y > 0] ^ [S(p) ¸ D(p)]
, [X > 0 ^ Y > 0] ^
·
s
¡1
1 Y
¡1d2 ¡ p
p ¡ s2
· d
¡1
1 X
¡1
¸
: (3)
Further, if q > 0 then
[P 6= unde¯ned] ^ [Q · q]
, [X > 0 ^ Y > 0] ^
·
d2 ¡ s2
q
· s
¡1
1 Y
¡1 + d
¡1
1 X
¡1
¸
: (4)
Using the equivalences above and the fact that P 6= unde¯ned ) P ¸ s2 we
may describe the distribution of the random vector1 (P;Q) as follows:
Lemma 1 If p > s2 and q > 0 then
P f0 < Q · q;P · pg
= Pf[X > 0;Y > 0]
^[max
µ
d2 ¡ s2
q
¡ s
¡1
1 Y
¡1;s
¡1
1 Y
¡1
µ
d2 ¡ s2
p ¡ s2
¡ 1
¶¶
· d
¡1
1 X
¡1]g;
if p · s2 then
P f0 < Q · q;P < pg = 0 (5)
and it holds that
P fQ = 0g = P fP = unde¯nedg = P f[Q = 0] ^ [P = unde¯ned]g
= P fX = 0 _ Y = 0g:
1Since P may take not only real values but also the value unde¯ned, it is not a real
random variable but the random element de¯ned on space R[funde¯nedg with ¾-algebra
¾(B(R);funde¯nedg) where the symbol B denotes Borel ¾-algebra (see [3] for information
how to handle random elements on general spaces).
4Lemma 2 If X and Y are independent, then
P f0 < Q · q;P · pg
= e
¡¾¡±
1 X
i=1
X
½
j2N:1·j·
³
d1 max
n
d2¡s2
q ¡s¡1
1 i¡1;s¡1
1 i¡1
³
d2¡s2
p¡s2
¡1
´o´¡1¾ ±i
i!
¾j
j!
(6)
for each p > s2 and q > 0 and it holds that
P fQ = 0g = e
¡± + e
¡¾ ¡ e
¡±¡¾: (7)
Proof. We may get (6) by summing the probabilities of all combinations of X
and Y values ful¯lling conditions (3) and (4). The relation (7) is straightfor-
ward. 2
4 Asymptotic distribution of market price and
traded volume
Fix d1;d2;s1;s2 and denote Pn and Qn the market price and the traded volume
reached at the market de¯ned by d1;d2;s1;s2, X = Xn » Po(n·) and Y =
Yn » Po(n¸) where n 2 N, · 2 R
+, ¸ 2 R
+ are some constants (the symbol
Po(z) denotes Poisson distribution with parameter z). Assume that X and
Y are independent. The subject of our interest is the limit behavior of vector
(Pn;Qn) as n ! 1.
Lemma 3 Denote
p
¤ =
·d1d2 + ¸s1s2
·d1 + ¸s1
; q
¤ =
d2 ¡ s2
d
¡1
1 ·¡1 + s
¡1
1 ¸¡1:
It holds that µ
p
n(Pn ¡ p
¤);
1
p
n
(Qn ¡ nq
¤)
¶
n!1 ¡! Z
(we mean the convergence in distribution) where Z = (Z1;Z2) is a random
vector de¯ned by
P fZ1 < c1;Z2 < c2g
= P
©
»1 < minf·
1=2Ac1 + ·
1=2¸
¡1=2»2;s
¡1
1 ·
1=2¸
¡1Ac2 ¡ s
¡1
1 d1·
3=2¸
¡3=2»2g
ª
=
Z C
¡1
Á
¡
·
1=2Ac1 + ·
1=2¸
¡1=2x
¢
dÁ(x)
+
Z 1
C
Á
¡
s
¡1
1 ·
1=2¸
¡1Ac2 ¡ s
¡1
1 d1·
3=2¸
¡3=2x
¢
dÁ(x)
5A =
(·d1 + ¸s1)2
·¸d1s1(d2 ¡ s2)
; C = A
s
¡1
1 ¸¡1c2 ¡ c1
¸¡1=2 + s
¡1
1 d1·¸¡3=2
for each c1 2 R and c2 2 R, where »1 » N (0;1), »2 » N (0;1) are independent
random variables and Á is standard normal distribution function.
Corollary
p
n
A
p
·¡1 + ¸¡1 (Pn ¡ p
¤)
n!1 ¡! N (0;1) (8)
1
p
n
A
p
·2¸¡1d2
1 + ·¡1¸2s2
1
(Qn ¡ nq
¤)
n!1 ¡! N (0;1) (9)
Proof. See Appendix A. 2
5 Interpretation
Not surprisingly, randomness of market participants' number causes °uctua-
tions both of the price and of the traded volume. Another not surprising fact is
that if the demand or the supply function shifts to the right (i.e. the parameter
d2 or s2 increases) then the price also increases (for all possible positive values
of X and Y ) and vice versa (those facts follow from (1)).
It may be seen from (1) that the distribution of P does not depend directly
on d1 and s1 but on their ratio s1=d1. Hence, the distribution of the price
(not of the volume) remains unchanged if the slope of the individual demand
function and the slope of the individual supply function are multiplied by the
came constant.
As the market tends to be more liquid (i.e. the intensity of buyers arrival
and the intensity of sellers arrival increase at the same rate) then it follows
from Corollary of Lemma 1 that the market price converges to the intersection
of expected demand and supply curves (which is not changing if the intensities
grow accordingly). Hence, at liquid markets, the °uctuations of the price
caused by random arrival of the agents are small, hence may be neglected.
The situation is di®erent in case of the traded volume: even if the expecta-
tion of traded volume is roughly equal to the intersection of expected demand
and supply curves, the variability of the traded volume is increasing with in-
creasing intensity. Hance the °uctuations of the traded volume may not be
neglected at the liquid market if the number of agents is random.
66 Concluding remarks
We have assumed Poisson distribution of sellers and buyers. However, Lemma
1 holds for each positive discrete distribution of X and for each positive discrete
distribution of Y including deterministic X and/or Y .
Moreover, the model may be easily generalized to the situation that the
slopes of the individual demand and supply functions are random: If we denote
by Di the (possibly zero) slope of the i-th demand function and by Sj the
(possibly zero) slope of the j-th supply function then the aggregate demand
function is D(p) = R(d2 ¡ p), R =
P
i Di whilde the aggregate supply one is
S(p) = T(p ¡ s2), T =
P
j Sj and it cam be easily shown that Lemma 1 holds
with R instead of d1X and with T instead of s1Y .
A Appendix - proof of Lemma 3 and its
Corollary
Ad. the Lemma. Fix c1 and c2 and denote pn
4
= p¤+c1=
p
n and qn
4
= nq¤+
p
nc2.
It holds that
´n
4
= P
½
£p
n(Pn ¡ p
¤) 2 (¡1;c1)
¤
^
·
1
p
n
(Qn ¡ nq
¤) 2 (¡1;c2)
¸¾
= P f[Pn 2 (¡1;pn)] ^ [Qn 2 (¡1;qn)]g
= P f[Pn 6= unde¯ned] ^ [Yns1(pn ¡ s2) > Xnd1(d2 ¡ pn)]
^([Qn = 0] _ [(Qn 6= 0) ^ (d2 ¡ s2)d1Xns1Yn < qn (s1Yn + d1Xn)])g
(2)
= P f[Pn 6= unde¯ned] ^ [Yns1(pn ¡ s2) > Xnd1(d2 ¡ pn)]
^[(d2 ¡ s2)d1Xns1Yn < qn (s1Yn + d1Xn)]g
for each c1 2 R and c2 > ¡
p
nq¤. The subject of our interest is limn!1 ´n.
Since
P fPn = unde¯nedg = P fXn = 0 _ Yn = 0g · P fXn = 0g + P fYn = 0g
= e
¡n· + e
¡n¸ n!1 ¡! 0;
we have
lim
n!1
´n = lim
n!1
Pf[Yns1(pn ¡ s2) > Xnd1(d2 ¡ pn)]
^[(d2 ¡ s2)d1Xns1Yn < qn (s1Yn + d1Xn)]g (10)
(indeed, if limn P fAng = 0, then it holds that limn P f:An ^ Bng = limn[P fBng
¡ P fAn ^ Bng] = limn P fBng). Further we have
7Yns1(pn ¡ s2) > Xnd1(d2 ¡ pn)
, Xnd1(d2 ¡ pn) ¡ Yns1(pn ¡ s2) < 0
, (Xn ¡ n·)d1(d2 ¡ pn) ¡ (Yn ¡ n¸)s1(pn ¡ s2)
< n¸s1(pn ¡ s2) ¡ n·d1(d2 ¡ pn)
, (Xn ¡ n·)d1(d2 ¡ pn) ¡ (Yn ¡ n¸)s1(pn ¡ s2)
< n[pn(¸s1 + ·d1) ¡ (¸s1s2 + ·d1d2)]
, (Xn ¡ n·)d1(d2 ¡ pn) ¡ (Yn ¡ n¸)s1(pn ¡ s2)
<
p
nc1(¸s1 + ·d1)
,
Xn ¡ n·
p
n
d1(d2 ¡ pn)
¸s1 + ·d1
¡
Yn ¡ n¸
p
n
s1(pn ¡ s2)
¸s1 + ·d1
< c1
, Un·
1=2d1(d2 ¡ pn)
¸s1 + ·d1
¡ Vn¸
1=2s1(pn ¡ s2)
¸s1 + ·d1
< c1 (11)
where
Un =
Xn ¡ n·
p
·n
; Vn =
Yn ¡ n¸
p
¸n
:
and, since
·
d1(d2 ¡ p¤)
·d1 + ¸s1
= ¸
s1(p¤ ¡ s2)
·d1 + ¸s1
=
·¸d1s1(d2 ¡ s2)
(·d1 + ¸s1)2 =
1
A
(12)
we have
Yns1(pn ¡ s2) > Xnd1(d2 ¡ pn)
(11);(12)
, Un·
¡1=2 ¡ Vn¸
¡1=2 < Ac1: (13)
Further, since
n
¡3=2d1Xns1Yn = n
¡3=2(Xn ¡ ·n + ·n)(Yn ¡ ¸n + ¸n)
= n
¡1=2p
·Un + ·
p
n)(
p
¸Vn + ¸
p
n)
= n
¡1=2p
·¸UnVn + ·
p
¸Vn + ¸
p
·Un + ·¸
p
n
= n
¡1=2p
·¸UnVn + ·
p
¸Vn + ¸
p
·Un + ·¸
p
n
= O(n
¡1=2) + ·
p
¸Vn + ¸
p
·Un + ·¸
p
n (14)
and
n¡3=2qn (d1Xn + s1Yn)
= n
¡3=2qn [d1(Xn ¡ ·n + ·n) + s1(Yn ¡ ¸n + ¸n)]
= n
¡1qn(d1
p
·Un + s1
p
¸Vn) + n
¡1=2qn(d1· + s1¸)
= (q
¤ + O(n
¡1=2))(d1
p
·Un + s1
p
¸Vn) + n
¡1=2qn(d1· + s1¸)
= (q
¤ + O(n
¡1=2))(d1
p
·Un + s1
p
¸Vn) + n
1=2q
¤(d1· + s1¸) + c2(d1· + s1¸)
= (q
¤ + O(n
¡1=2))(d1
p
·Un + s1
p
¸Vn) + d1s1(d2 ¡ s2)·¸
p
n
+c2(d1· + s1¸) (15)
8we may write, for a = d1s1(d2 ¡ s2),
aXns1Yn < qn (s1Yn + d1Xn)
(14);(15)
, a(¸
p
·Un + ·
p
¸Vn) + a·¸
p
n
< q
¤(d1
p
·Un + s1
p
¸Vn) + c2(d1· + s1¸) + a·¸
p
n + O(n
¡1=2)
,
p
·(a¸ ¡ q
¤d1)Un +
p
¸(a· ¡ q
¤s1)Vn < c2(d1· + s1¸) + O(n
¡1=2)
,
p
·
a¸ ¡ q¤d1
d1· + s1¸
Un +
p
¸
a· ¡ q¤s1
d1· + s1¸
Vn < c2 + O(n
¡1=2)
,
p
·a
¸ ¡
·¸d1
d1·+s1¸
d1· + s1¸
Un +
p
¸a
· ¡
·¸s1
d1·+s1¸
d1· + s1¸
Vn < c2 + O(n
¡1=2)
,
p
·as1¸2
(d1· + s1¸)2Un +
p
¸ad1·2
(d1· + s1¸)2Vn < c2 + O(n
¡1=2)
, (s1·
¡1=2¸Un + d1·¸
¡1=2Vn) < Ac2 + O(n
¡1=2) (16)
From the computations above it follows
´n
(10);(13);(16)
= Pf[Un·
¡1=2 ¡ Vn¸
¡1=2 < Ac1]
^[s1·
¡1=2¸Un + d1·¸
¡1=2Vn < Ac2 + O(n
¡1=2)]g
and, since (Un;Vn)
n!1 ¡! (»1;»2) by the Multivariate Central Limit Theorem, we
have, using the Continuous Mapping Theorem (see [2] for both the Theorems),
lim
n!1
´n = P
©
[»1·
¡1=2 ¡ »2¸
¡1=2 < Ac1] ^ [s1·
¡1=2¸»1 + d1·¸
¡1=2»2 < Ac2]
ª
= Pf»1 < minf·
1=2Ac1 + ·
1=2¸
¡1=2»2;
s
¡1
1 ·
1=2¸
¡1Ac2 ¡ s
¡1
1 d1·
3=2¸
¡3=2»2gg
=
Z
Á(minf·
1=2Ac1 + ·
1=2¸
¡1=2x
;s
¡1
1 ·
1=2¸
¡1Ac2 ¡ s
¡1
1 d1·
3=2¸
¡3=2xg)dÁ(x)
=
Z C
¡1
Á
¡
·
1=2Ac1 + ·
1=2¸
¡1=2x
¢
dÁ(x)
+
Z 1
C
Á
¡
s
¡1
1 ·
1=2¸
¡1Ac2 ¡ s
¡1
1 d1·
3=2¸
¡3=2x
¢
dÁ(x): (17)
Ad. the Corollary. From the Lemma it follows that
P
©p
n(Pn ¡ p
¤) < c1
ª n!1 ¡! P
©
Z
1 < c1
ª
= lim
c2!1
P
©
Z
1 < c1;Z
2 < c2
ª
(17)
= P
©
·
¡1=2»1 ¡ ¸
¡1=2»2 < Ac1
ª
= P
©
N
¡
0;·
¡1 + ¸
¡1¢
< Ac1
ª
= P
np
·¡1 + ¸¡1A
¡1N (0;1) < c1
o
9which proves (8). Similarly,
P
©
n
¡1=2(Qn ¡ nq
¤)
ª n!1 ¡! P
©
[s1·
¡1=2¸»1 + d1·¸
¡1=2»2 < Ac2]
ª
= P
©
N
¡
0;s
2
1·
¡1¸
2 + d
2
1¸
¡1·
2¢
< Ac2
ª
= P
½
A
¡1
q
s2
1·¡1¸2 + d2
1¸¡1·2N (0;1) < c2
¾
:
2
Acknowledgement: This work was partially supported by grant no 402/04/1294
of the Grant Agency of the Czech Republic and by grant no. 402/03/H057 of
the same institution.
References
[1] R. G. D. Allen. Mathematical Economics. Mcmillan & Co. Ltd., Prague,
1963.
[2] D. Pollard. A User's Guide to Measure Theoretic Probability. Cambridge
Univ. Press, Cambridge, 2002.
[3] J. · St· ep¶ an. Probability Theory (in Czech). Academia, Praha, 1987.
10

A User's Guide to Measure Theoretic Probability.

St· ep¶ an. Probability Theory (in Czech).

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