Information model of economy
A new stochastic model of economy is developed that takes into account the choice of consumers are the dependent random fields. Axioms of such a model are formulated. The existence of random fields of consumer’s choice and decision making by firms are proved. New notions of conditionally independe...
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irk-123456789-1213152017-06-15T03:02:48Z Information model of economy Gonchar, N.S. A new stochastic model of economy is developed that takes into account the choice of consumers are the dependent random fields. Axioms of such a model are formulated. The existence of random fields of consumer’s choice and decision making by firms are proved. New notions of conditionally independent random fields and random fields of evaluation of information by consumers are introduced. Using the above mentioned random fields the random fields of consumer choice and decision making by firms are constructed. The theory of economic equilibrium is developed. 2006 Article Information model of economy / N.S. Gonchar // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. p. 201–210. — Бібліогр.: 7 назв. — англ. 1607-324X PACS: 89.65.G DOI:10.5488/CMP.9.1.201 http://dspace.nbuv.gov.ua/handle/123456789/121315 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A new stochastic model of economy is developed that takes into account the choice of consumers are the dependent
random fields. Axioms of such a model are formulated. The existence of random fields of consumer’s
choice and decision making by firms are proved. New notions of conditionally independent random fields and
random fields of evaluation of information by consumers are introduced. Using the above mentioned random
fields the random fields of consumer choice and decision making by firms are constructed. The theory of
economic equilibrium is developed. |
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Article |
| author |
Gonchar, N.S. |
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Gonchar, N.S. Information model of economy Condensed Matter Physics |
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Gonchar, N.S. |
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Gonchar, N.S. |
| title |
Information model of economy |
| title_short |
Information model of economy |
| title_full |
Information model of economy |
| title_fullStr |
Information model of economy |
| title_full_unstemmed |
Information model of economy |
| title_sort |
information model of economy |
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Інститут фізики конденсованих систем НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/121315 |
| citation_txt |
Information model of economy / N.S. Gonchar // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. p. 201–210. — Бібліогр.: 7 назв. — англ. |
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Condensed Matter Physics |
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AT goncharns informationmodelofeconomy |
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2025-07-08T19:38:22Z |
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2025-07-08T19:38:22Z |
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Condensed Matter Physics 2006, Vol. 9, No 1(45), pp. 201–210
Information model of economy
N.S.Gonchar1
1 Bogoliubov Institute for Theoretical Physics, 14b
Metrolohichna Str., 03143, Kiev, Ukraine
Received July 11, 2005, in final form February 7, 2006
A new stochastic model of economy is developed that takes into account the choice of consumers are the de-
pendent random fields. Axioms of such a model are formulated. The existence of random fields of consumer’s
choice and decision making by firms are proved. New notions of conditionally independent random fields and
random fields of evaluation of information by consumers are introduced. Using the above mentioned random
fields the random fields of consumer choice and decision making by firms are constructed. The theory of
economic equilibrium is developed.
Key words: technological mapping, making a decision by firms, productive process
PACS: 89.65.G
1. Introduction
The proposed stochastic model of economy contains a new approach to the description of
consumer choice and making a decision by firm. This description is based on real observations
of consumer choice that is described by probability measures ensemble given on budget sets of
consumer. This description differs from classical description of consumer choice and making a
decision by firm because consumers and firms make their choice and decision having information
about the state of economy and their choice or decision depend on the available information. Our
approach permits to construct random fields of consumer choice and making a decision by firms
that are dependent random fields. This is very important because as Pareto showed the distribution
of wealth for several nations has a power law. This result it is impossible to obtain if we restrict
ourselves to classical description of consumer choice and take into account that consumers make
their choice independently [6]. We develop the theory of economic equilibrium that is as strict
as in the classical approach [2]. We also develop algorithms of finding equilibrium states that are
constructive and are also applicable to the classical case [4,5].
Such an approach solves the problem that the wealth in several societies is distributed according
to Pareto law [7,6].
2. General notions
The paper presents some results expounded in detail in [1]. Let S ⊆ Rn
+ be a set of possible
goods that are ordered and B(S) is Borell σ-algebra of subsets of S. For example,
S =
{
x = {xi}
n
i=1, x ∈ Rn
+, 0 6 xi 6 ci, i = 1, n
}
, ci > 0, i = 1, n.
We assume that the set of possible prices Kn
+ is also ordered and it is a certain subcone of the
cone Rn
+, B(Kn
+) is Borell σ-algebra of subsets of the set Kn
+. Suppose that in an economic system
there are m firms that are described by technological mappings Fi(x), Xi, i = 1,m, where Xi is
an expenditure set of the i-th firm, Fi(x) is the set of plans at the expenditure vector x ∈ Xi. It is
convenient to assume that the firms are ordered and for further consideration a set of productive
processes Γi = {(x, y), x ∈ Xi ⊆ S, y ∈ Fi(x)} of the i-th firm is only important. By Γm =
m
∏
i=1
Γi
c© N.S.Gonchar 201
N.S.Gonchar
we denote the direct product of the sets Γi, i = 1,m. By [Γm]k we denote the k-multiple direct
product of the set Γm, B([Γm]k) is Borell σ-algebra of subsets of the set [Γm]k, k = 1, 2, . . . . Budget
set of an i-th insatiable consumer is given by the formula
X i
(p,z) = {x, x ∈ S, 〈p, x〉 = Ki(p, z)}, p ∈ Kn
+, z ∈ Γm, i = 1, l,
where Ki(p, z) is a profit function of the i-consumer. Further we use the following notation:
{Xk,s
(p,z)k
, B(Xk,s
(p,z)k
)} is the direct product of budget spaces
{Xs
(pi,zi)
, B(Xs
(pi,zi)
)}, i = 1, k, k = 1,∞
for every s = 1, l.
Definition 1 A technological mapping F (x), x ∈ X, belongs to a class of CTM (compact techno-
logical mappings), if the domain X ⊆ S is a closed bounded convex set such that 0 ∈ X, 0 ∈ F (0),
and F (x) is a Kakutani continuous technological mapping that takes the value in the set of the
closed convex bounded subsets of the set S. Moreover, there exists a compact set Y ⊆ S such that
F (x) ⊆ Y, for all x ∈ X.
Definition 2 A set of functions K0
i (p, u), i = 1, l, defined on the set Kn
+ × Γm, that take the
values in the set R1, we call profit prefunctions of consumers if it satisfies conditions:
1. K0
i (p, u) is a measurable mapping of the measurable space {Γm,B(Γm)} into the measurable
space {R1,B(R1)} for every p ∈ Kn
+, i = 1, l;
2. For every p ∈ Kn
+ the set D(p) =
l
⋂
i=1
Di(p) is non empty, where
Di(p) =
{
u ∈ Γm, K0
i (p, u) > 0
}
, i = 1, l;
3. K0
i (tp, u) = tK0
i (p, u), t > 0, (p, u) ∈ Kn
+ × Γm, i = 1, l.
Definition 3 Let m firms be described by technological mappings Fi(x), x ∈ Xi, i = 1,m,
and an i-th consumer have the vector of property bi(p, z) > 0, i = 1, l. Assume that for every
(p, z) ∈ Kn
+ × Γm there exist productive processes
(Xi(p, z), Yi(p, z)), Xi(p, z) ∈ Xi, Yi(p, z) ∈ Fi(Xi(p, z)), i = 1,m,
that satisfy the conditions:
1. (Xi(p, z), Yi(p, z)) is a measurable mapping of the measurable space {Γm,B(Γm)} into the
measurable space {Γi,B(Γi)} for every p ∈ Kn
+, i = 1,m, where B(Γi) is Borel σ-algebra of
subsets of the set Γi.
2. (Xi(tp, z), Yi(tp, z)) = (Xi(p, z), Yi(p, z)), t > 0, (p, z) ∈ Kn
+ × Γm.
A measurable mapping Q(p, z) of the measurable space {Γm,B(Γm)} into itself for every p ∈ Kn
+,
given by the formula
Q(p, z) = {(Xi(p, z), Yi(p, z))}
m
i=1 , (1)
we call a productive economic process if for every p ∈ Kn
+ the set of values Q(p,Γm) of the mapping
Q(p, z) belongs to the set
G(p) = {z ∈ Γm, R(p, z) ∈ S} , R(p, z) =
m
∑
i=1
[yi − xi] +
l
∑
k=1
bk(p, z),
where bi(p, z) > 0, i = 1, l, is an initial vector stock of goods of the i-th consumer at the initial
moment of economy functioning.
202
Information model of economy
Definition 4 A set of functions Ki(p, z), i = 1, l, given on the set Kn
+ ×Γm, that are measurable
mappings of the measurable space {Γm,B(Γm)} into the measurable space {R1
+,B(R1
+)} for every
p ∈ Kn
+, we call profit functions of consumers, if there exist a set of profit prefunctions of consumers
K0
i (p, z), i = 1, l, a productive economic process Q(p, z), given on the set Kn
+ × Γm, such that
Q(p,Γm) belongs to the set D(p) from the definition 2 and for every p ∈ Kn
+ there hold equalities:
1. Ki(p, z) = K0
i (p,Q(p, z)), (p, z) ∈ Kn
+ × Γm, i = 1, l;
2.
l
∑
i=1
Ki(p, z) =
=
〈
p,
m
∑
i=1
[Yi(p, z) −Xi(p, z)] +
l
∑
k=1
bk(p,Q(p, z))
〉
, (p, z) ∈ Kn
+ × Γm. (2)
Definition 5 Let l consumers be described by profit functions Ki(p, z), i = 1, l, given on the
set Kn
+ × Γm, and m firms be described by technological mapping Fi(x), x ∈ Xi, i = 1,m. The
description of consumers is completely given if for every s = 1, l and on every direct product
{Xk,s
(p,z)k
, B(Xk,s
(p,z)k
)} of budget spaces
{
Xs
(pi,zi)
, B(Xs
(pi,zi)
)
}
, i = 1, k, k = 1,∞
a probability measure
F s
p1,...,pk
(As|z1, . . . , zk), As ∈ B(Xk,s
(p,z)k
), s = 1, l,
is given for all
{p1, . . . , pk} ∈ [Kn
+]k, {z1, . . . , zk} ∈ [Γm]k, k = 1, 2, . . . .
The measure F s
p1,...,pk
(As|z1, . . . , zk) is the probability that the s-th consumer chooses the collection
of goods from the set As ∈ B(Xk,s
(p,z)k
) on the assumption that in the economic system the productive
processes {z1, . . . , zk} ∈ [Γm]k were realized on conditions that the prices vector {p1, . . . , pk} ∈
[Kn
+]k was carried out correspondingly.
Definition 6 Let l consumers be described by profit functions Ki(p, z), i = 1, l, defined on the
set Kn
+ × Γm, and m firms be described by technological mapping Fi(x), x ∈ Xi, i = 1,m.
If there exists a probability space {Ω,F , P}, l random fields ξi(p), p ∈ Kn
+, i = 1, l, de-
fined on it, that take the values in the set of possible goods S and m random fields ζ(p) =
{η0
1(p), . . . , η
0
m(p)}, p ∈ Kn
+, on the same probability space that take the values in the set of possible
productive processes Γm such that
P
(
{ξi(p1), . . . , ξi(pk)} ∈ Ai|ζ(p1) = z1, . . . , ζ(pk) = zk
)
= F i
p1,...,pk
(Ai|z1, . . . , zk),
Ai ∈ B(Xk,i
(p,z)k
), i = 1, l,
then the random field ξi(p) is called the random field of choice of the i-th consumer that is described
by the probability measures ensemble
F i
p1,...,pk
(Ai|z1, . . . , zk), Ai ∈ B(Xk,i
(p,z)k
), i = 1, l,
the random field η0
s(p), s = 1,m, is called the random field of decision making by s-th firm relative
to productive processes.
203
N.S.Gonchar
Theorem 1 Let X be a bounded closed convex set every point of which is internal for a set X1
and F (x) is down convex technological mapping from the CTM class, given on the convex compact
set X1. For every sufficiently small ε > 0 there exists a continuous strategy of firm behaviour
(x0(p), y0(p)), y0(p) ∈ F (x0(p))
such that
sup
p∈P
|ϕ(p) − 〈y0(p) − x0(p), p〉| < ε,
where
ϕ(p) = sup
x∈X
sup
y∈F (x)
〈y − x, p〉.
3. Axioms of random consumer’s choice and decision making by firms
Under uncertainty conditions the description of economy is given if for every fixed s, s = 1, l,
a family of finite dimensional conditional distributions
F s
p1,...,pk
(Ak|z1, . . . , zk)
satisfies the conditions:
1) F s
p1,...,pk
(Ak|z1, . . . , zk) is a probability measure on the σ-algebra of Borell subset Ak ∈
B(Xk,s
(p,z)k
) for every fixed values of variables
{p1, . . . , pk} ∈ Knk
+ , {z1, . . . , zk} ∈ [Γm]k, k = 1, 2, . . . ,
and for every fixed As ∈ B(Sk) F s
p1,...,pk
(As ∩X
k,s
(p,z)k
|z1, . . . , zk) is a measurable mapping of
the measurable space {[Γm]k,B([Γm]k)} into measurable space {[0, 1],B([0, 1])};
2) for every permutation π of indexes {1, . . . , k} there holds the equality
F s
pπ(1),...,pπ(k)
(
Π0
kA
k|zπ(1), . . . , zπ(k)
)
= F s
p1,...,pk
(Ak|z1, . . . , zk), k = 1, 2, . . . ,
where Π0
kA
k is the image of the set Ak ∈ B(Xk,s
(p,z)k
) under transformation Π0
k of the set Sk
into itself: Π0
kx = {xπ(1), . . . , xπ(k)}, x = {x1, . . . , xk} ∈ Sk, where π is a permutation of
indexes {1, . . . , k};
3) F s
p1,...,pk
(
Aj ×
k
∏
i=j+1
X̂s
(pi,zi)|z1, . . . , zk
)
= F s
p1,...,pj
(
Aj |z1, . . . , zj
)
, Aj ∈ B
(
X
j,s
(p,z)j
)
;
4) F s
tp1,...,tpk
(
Ak|z1, . . . , zk
)
= F s
p1,...,pk
(
Ak|z1, . . . , zk
)
, ∀t > 0
and a family of unconditional finite dimensional distributions
ψp1,...,pk
(Bk), Bk ∈ B([Γm]k), pi ∈ Kn
+, i = 1, k, k = 1, 2, . . . ,
satisfies the conditions:
1) ψp1,...,pj
(Bj) = ψp1,...,pk
(
Bj × [Γm]k−j
)
, Bj ∈ B
(
[Γm]j
)
,
k = j + 1, j + 2, . . . , ψp1(Γ
m) = 1;
204
Information model of economy
2) for every permutation π of indexes {1, . . . , k}
ψpπ(1),...,pπ(k)
(Π1
kB
k) = ψp1,...,pk
(Bk), k = 1, 2, . . . ,
where Π1
kB
k is the image of the set Bk under transformation Π1
k of the set [Γm]k into itself:
Π1
kz = {zπ(1), . . . , zπ(k)}, z = {z1, . . . , zk} ∈ [Γm]k, where π is a permutation of indexes
{1, . . . , k};
3) ψtp1,...,tpk
(Bk) = ψp1,...,pk
(Bk), ∀t > 0, k = 1, 2, . . . ;
4) If bi(p, z) is a stock vector of goods of the i-th consumer, i = 1, l, at the initial moment of
the economy functioning that is a measurable mapping of the measurable space {Γm,B(Γm)}
into the measurable space {S,B(S)} for every p ∈ Kn
+, and
G(p) = {z ∈ Γm, R(p, z) ∈ S} ∈ B(Γm),
where
R(p, z) =
m
∑
i=1
[yi − xi] +
l
∑
k=1
bk(p, z), zi = (xi, yi) ∈ Γi ,
then for all p ∈ Kn
+
∫
G(p)
ψp(dz) = 1.
Definition 7 A family of functions of sets
Φp1,...,pk
(D ×A1 × . . .×Al) =
∫
D
l
∏
i=1
F i
p1,...,pk
(
Ai ∩X
k,i
(p,z)k
|z1, . . . , zk
)
dψp1,...,pk
(z1, . . . , zk),
where D ∈ [B(Γm)]k, Ai ∈ B(Sk), i = 1, l, we call the finite dimensional distributions of l
consumers choice and decision making by m firms, where
Ai =
k
∏
s=1
Ai
s, D =
k
∏
i=1
Di, Ai
s ∈ B(S), i = 1, l, s = 1, k.
The economic sense of Φp1,...,pk
(D×A1× . . .×Al) is the probability that the i-th consumer chooses
a set of goods from the set Ai
s ∈ B(S) on the assumption that the firms have made a decision as to
the productive processes that belong to the set Ds ∈ B(Γm) and the price vector in the economic
system is ps ∈ Kn
+, i = 1, l, s = 1, k.
Theorem 2 Let a set of conditional finite dimensional distributions
F i
p1,...,pk
(
Ai|z1, . . . , zk
)
, Ai ∈ B
(
X
k,i
(p,z)k
)
, i = 1, l, s = 1, k,
and a set of unconditional finite dimensional distributions ψp1,...,pk
(D), D ∈ B([Γm]k), k = 1,∞,
satisfy the above formulated axioms. The function of sets given by the formula
Φp1,...,pk
(D ×A1 × . . .×Al) =
∫
D
l
∏
i=1
F i
p1,...,pk
(
Ai ∩X
k,i
(p,z)k
|z1, . . . , zk
)
dψp1,...,pk
(z1, . . . , zk)
on the sets of the kind
D1 × . . .×Dk × A1
1 × . . .×Ak
1 × . . .×A1
l × . . .×Ak
l , As
i ∈ B(S), Di ∈ B(Γm),
205
N.S.Gonchar
where the set Ai has the form Ai = A1
i × . . . × Ak
i , i = 1, l, and the set D is of the kind D =
D1 × . . .Dk, admits an extension on the measurable space
V1 =
{
[Γm × Sl]k,B(Γm) × B(Sl)]k
}
,
that is, there exists a family of measures µ̄p1,...,pk
z1,...,zk
(E), given on the measurable space V1, such
that for every fixed {p1, . . . , pk} ∈ Knk
+ and E ∈ [B(Γm) × B(Sl)]k every measure of the family
is a measurable mapping of the space L = {[Γm]k, [B(Γm)]k into the space {R1,B(R1)} and the
extension is given by the formula
Φ̄p1,...,pk
(E) =
∫
[Γm]k
µ̄p1,...,pk
z1,...,zk
(E)dψp1,...,pk
(z1, . . . , zk).
The extension satisfies the conditions:
Φ̄pπ(1),...,pπ(k)
(
Π2
kE
)
= Φ̄p1,...,pk
(E), E ∈
[
B(Γm) × B(Sl)
]k
, (3)
Φ̄p1,...,pk
(
A× (Γm × Sl)k−r
)
= Φ̄p1,...,pr
(A), A ∈
[
B(Γm) × B(Sl)
]r
, (4)
where Π2
kE is the image of the set E under transformation Π2
k of the set [Γm × Sl]k into itself:
Π2
k{w1, . . . , wk} = {wπ(1), . . . , wπ(k)}, wi = {zi, x
i
1, . . . , x
i
l} ∈ Γm × Sl, and π is a permutation of
indexes {1, . . . , k}.
Theorem 3 The family of finite dimensional distributions Φ̄p1,...,pk
(E), where {p1, . . . , pk}∈ [Kn
+]k,
and E ∈ [B(Γm) × B(Sl)]k, that was constructed in the theorem 2, satisfies the conditions of the
Kolmogorov theorem with the full separable metric space of state X = Γm × Sl and the σ-algebra
subsets Σ = B(Γm)×B(Sl) and, thus, the family generates a unique measure P on the measurable
space {XT ,ΣT } such that the family of finite dimensional distributions of a random field
νp(ω) = {ζ0(p), ξ1(p), . . . , ξl(p)} = ω(p), ω(p) ∈ XT , p ∈ Kn
+,
coincides with the family Φ̄p1,...,pk
(E), that is,
P
(
ω ∈ XT , {νp1(ω), . . . , νpk
(ω)} ∈ E
)
= Φ̄p1,...,pk(E), E ∈
[
B(Γm) × B(Sl)
]k
.
By XT we denote the set of all functions, given on the set T = Kn
+, with the values in the set
X = Γm × Sl, ΣT is the minimal σ-algebra, generated by cylindrical sets of the kind
{
ω(p) ∈ XT , {νp1(ω), . . . , νp1(ω)} ∈ E
}
, E ∈
[
B(Γm) × B(Sl)
]k
.
4. Conditionally independent random fields
Definition 8 Let {Ω,F , P} be a measurable space. A family of sub σ-algebras {Bi, i ∈ I} of the
σ-algebra F is called conditionally independent with respect to a sub σ-algebra B ⊆ F , if
M
∏
j∈Is
Xj |B
=
∏
j∈Is
M{Xj|B}
for every finite subset Is ⊆ I and a family of random values {Xj , j ∈ Is}, where Xj is Bj-measurable
positive random value.
Definition 9 Let l random fields of consumers choice ξ1(p), . . . , ξl(p), p ∈ Kn
+, be given on the
probability space {Ω,F , P} and take the values in the set of possible goods S, that is, they are
measurable mappings of the measurable space {Ω,F} into the measurable space {S,B(S)} for every
fixed p ∈ Kn
+, and let random fields of decision making by m firms ζ(p) = {η0
1(p), . . . , η
0
m(p)} be
206
Information model of economy
measurable mappings of {Ω,F} into {Γm,B(Γm)} for every fixed p ∈ Kn
+. The random fields of
consumers choice are conditionally independent relative to random fields of decision making if
σ-algebras
Fi = F{ξi(p), p ∈ Kn
+}, i = 1, l,
conditionally independent relative to σ-algebra F0 = F{ζ(p), p ∈ Kn
+}, where σ-algebras Fi, i =
0, l, are minimal σ-algebras generated by the family of random values {ζ(p), p ∈ Kn
+} for i = 0
and the family of random values {ξi(p), p ∈ Kn
+} for i = 1, l.
In the next theorems, we assume that the productive economic process Q(p, z) and vectors of
initial stock of goods bk(p, z), k = 1, l, are continuous functions of variables (p, z) ∈ Kn
+ ×Γm with
the values correspondingly in the sets Γm, S. The next theorem is very important for a construction
theory of economic equilibrium.
Theorem 4 Let a random field η0
i (p, z, ωi), (p, z) ∈ Kn
+ × Γm, given on the probability space
{Ωi,Fi, Pi}, be a continuous function of (p, z) ∈ Kn
+ × Γm for every ωi ∈ Ωi, take values in
S, i = 1, l, and a random field ζ0(p, ω0), p ∈ Kn
+, given on the probability space {Ω0,F0, P0},
take values in the set Γm and every realization of the field be continuous function of p ∈ Kn
+.
Moreover, let η0
i (tp, z, ω) = η0
i (p, z, ω), i = 1, l, t > 0, ζ0(tp, ω0) = ζ0(p, ω0), p ∈ Kn
+, t > 0, and
Ki(p, z), i = 1, l, be profit functions of consumers that satisfy all the conditions of definition 4 and
be continuous functions of variables (p, z) ∈ Kn
+ × Γm. If
〈
η0
i (p, z, ωi), p
〉
> 0, ηi(p, z, ωi) = η0
i (p,Q(p, z), ωi),
(p, z, ωi) ∈ Kn
+ × Γm × Ωi, i = 1, l,
Pi
(〈
η0
i (p, z, ωi), p
〉
<∞
)
= 1, (p, z) ∈ Kn
+ × Γm, i = 1, l,
then the random fields
ξi(p, ω) =
Ki(p, ζ0(p, ω0))ηi(p, ζ0(p, ω0), ωi)
〈ηi(p, ζ0(p, ω0), ωi), p〉
, i = 1, l, (5)
are continuous on the probability space {Ω,F , P} for each realization, where
Ω =
l
∏
i=0
Ωi, F =
l
∏
i=0
Fi, P =
l
∏
i=0
Pi,
that can be identified with random fields of choice of insatiable consumers on the same probability
space under the condition that ζ(p, ω0) = Q(p, ζ0(p, ω0)) are identified with random fields of decision
making by firms as to the productive processes.
The random field η0
i (p, z, ωi) is called the random field of evaluation of information by an i-th
consumer, i = 1, l.
Definition 10 Let
ηi(p, ζ0(p, ω0), ωi) = η0
i (p, ζ(p, ω0), ωi) = {η0
ik(p, ζ(p, ω0), ωi)}
n
k=1, i = 1, l,
and ζ(p, ω0) = Q(p, ζ0(p, ω0)) be the random field as in the Theorem 4. By demand vector of an
i-th insatiable consumer we denote a random field
γi(p) = γi(p, ω0, ωi) = {γik(p)}n
k=1, i = 1, l,
where
γik(p) = γik(p, ω0, ωi) =
pkη
0
ik(p, ζ(p, ω0), ωi)
n
∑
s=1
η0
is(p, ζ(p, ω0), ωi)ps
, k = 1, n, i = 1, l.
207
N.S.Gonchar
A random field of choice of the i-th insatiable consumer is connected with the demand vector of
the i-th insatiable consumer by the formula
ξi(p) =
{
Di(p)γik(p)
pk
}n
k=1
, Di(p) = Ki(p, ζ0(p, ω0)), i = 1, l. (6)
γik(p) has the following economic sense: the part of the profit of the i-th consumer he spends to
buy the k-th good.
A demand of society is described by a demand matrix ||γij(p)||
l,n
i,j=1. All the statements proved
for the case of insatiable consumers are valid if not all consumers are insatiable.
Definition 11 An economic system is in the Walras equilibrium state if there exists a price vector
p∗, m productive processes
(x∗i (p
∗), y∗i (p∗)), x∗i (p
∗) ∈ Xi, y∗i (p∗) ∈ Fi(x
∗
i (p
∗)), i = 1,m,
such that the following inequalities
φ(p∗) 6 ψ(p∗), (7)
〈φ(p∗), p∗〉 = 〈ψ(p∗), p∗〉, (8)
hold, where p∗ = (p∗1, . . . , p
∗
n) is an equilibrium price vector, Fi(x) is a technological mapping of
the i-th firm, i = 1,m.
Inequality (7) means that in the equilibrium state there exists a price vector p∗ while the demand
of society does not exceed proposition and the equality (8) means that the value of goods that
society wants to buy is equal to the value of goods offered for consumption. The price vector p∗,
that guarantees the fulfilment of (7) and (8), is called equilibrium price vector.
Definition 12 Walras equilibrium state of economy is called optimal if the following conditions
〈y∗i (p∗) − x∗i (p
∗), p∗〉 = sup
x∈Xi
sup
y∈Fi(x)
〈y − x, p∗〉, i = 1,m,
are valid where Xi is the expenditure set of an i-th firm, Fi(x) is its technological mapping.
5. Theory of economic equilibrium
In the next theorem we assume that the matrix ||γik(p)||l,ni=1,k=1 is not necessarily generated
by random fields of evaluation of information by consumers and it is arbitrary which satisfies the
conditions of this theorem.
Theorem 5 Let technological mappings Fi(x), x ∈ X1
i , i = 1,m, be down convex, belong to CTM
class, a productive economic process Q(p, z) and a family of profit prefunction K0
i (p, z), i = 1, l,
be continuous mappings of variables (p, z) ∈ Rn
+ × Γm and random fields of decision making by
firms satisfy the conditions of the theorem 4. Moreover, if the productive economic process Q(p, z)
satisfies the condition
R(p,Q(p, z)) > 0, p ∈ Rn
+, p 6= 0 z ∈ Γm, (9)
R(p, z) =
m
∑
i=1
[yi − xi] +
l
∑
j=1
bj(p, z),
then for every continuous matrix ||γik(p)||l, n
i=1,k=1, given on Rn
+, the rows of that satisfy the condi-
tions
n
∑
k=1
γik(p) = 1, i = 1, l, (10)
208
Information model of economy
and continuous realization z(p) = {zi(p) = (xi(p), yi(p))}
m
i=1 of random fields of decision making
by firms η0
i (p), i = 1,m, the set of equations
l
∑
i=1
γik(p)Di(p) = pk
[
m
∑
i=1
[yik(p) − xik(p)] +
l
∑
i=1
bik(p, z(p))
]
, k = 1, n (11)
is solvable in Rn
+, where Di(p) = K0
i (p, z(p)), i = 1, l.
Theorem 6 Let technological mappings Fi(x), x ∈ X1
i , i = 1,m, be down convex, belong to CTM
class, a productive economic process Q(p, z), a family of profit prefunctions K0
i (p, z), i = 1, l, be
continuous mappings of variables (p, z) ∈ Rn
+ ×Γm, and random fields that describe the consumers
and firms satisfy the conditions of the Theorem 4. Then with probability 1 there exists the Walras
equilibrium state, that is, for every realization of random fields that describe consumers and firms
there exists a corresponding price vector p∗ ∈ Rn
+ such that the economic system is in Walras
equilibrium state. Moreover, if realization of random fields that describe the consumers and firms
is such that between them there exist realizations that are arbitrary close to optimal behaviour
strategies of firms in the sense of the Theorem 1, then with probability 1 there exists an optimal
Walras equilibrium state.
Theorem 7 Let technological mappings Fi(x), x ∈ X1
i , i = 1,m, a productive economic process
Q(p, z), a family of profit prefunctions of consumers K0
i (p, z), i = 1, l, satisfy the conditions of
the Theorem 6. Random fields that describe the consumer’s choice and decision making by firms
are continuous with probability 1. Moreover, if a productive economic process Q(p, z) satisfies the
condition
R(p,Q(p, z)) > 0, p ∈ Rn
+, p 6= 0 z ∈ Γm, (12)
R(p, z) =
m
∑
i=1
[yi − xi] +
l
∑
j=1
bj(p, z),
then for every continuous on Rn
+ demand matrix ||γik(p)||l, n
i=1,k=1, that satisfies conditions of the
Theorem 6, and a realization of random fields of decision making by firms z(p), that is, with
probability 1 every Walras equilibrium price vector p̄ satisfies the set of equations
l
∑
i=1
γik(p)Di(p) = pk
[
m
∑
i=1
[yik(p) − xik(p)] +
l
∑
i=1
bik(p, z(p))
]
, k = 1, n. (13)
References
1. Gonchar N.S., Mathematical foundations for information model of economy (in preparation).
2. Arrow K.J., Debreu G., Econometrica, 1954, 22, No. 2, 265.
3. Debreu G. Handbook of Mathematical Economics, North-Holland Publishing Company, II, 1982, 698.
4. Scarf H.E., Handbook of Mathematical Economics, North-Holland Publishing Company, II, 1982, 1007.
5. Kehoe T.J., Handbook of Mathematical Economics, North-Holland Publishing Company, IV, 1991,
2049.
6. Feigenbaum J., Rep. Prog. Phys., 2003, 1611.
7. Cont R., Bouchaud J-P., Microeconom. Dyn., 2000, 170.
209
N.S.Gonchar
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PACS: 89.65.G
210
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