On a Consumer Problem
Sabir Isa Hamidov
MathematicalCybernetics Department, Baku State University, Baku, Azerbaijan
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To cite this article:
Sabir Isa Hamidov. On a Consumer Problem. Pure and Applied Mathematics Journal. Vol. 5, No. 6, 2016, pp. 205210. doi: 10.11648/j.pamj.20160506.15
Received: February 10, 2015; Accepted: October 20, 2016; Published: November 15, 2016
Abstract: The model of economic dynamics of the Leontief type is considered. The problem of determining the equilibrium state of the model with a fixed budget is investigated. It is proved that the state of equilibrium exists, if the trajectory model is a solution to the consumer problem.
Keywords: Production Mapping, Utility Function, Equilibrium
1. Introduction
Let at the moment the production mapping [1, 4] be given
(1)
where is a diagonal matrix the main diagonal of which has a form
are production functions of the branches:
(2)
Production mapping [4] of the branch has a form:
(3)
Note. The mapping is completely defined by the set
(4)
where
Let
(5)
If consider (2) then the utility function of the branch takes the form [2]
(6)
where is a cost vector, the set is defined by the formula (5).
By the definition the set is an equilibrium state if and is a solution of the consumer problem [57]
subject to (7)
where is in the form (6), is a component of the budget vector
Let the vector be a solution of the consumer problem
(8)
Then the equilibrium vector has a form [4]
(9)
Note that from (4) and (9) follows that
This means that
In the future, we will be interested in the following problem. Given a mapping i.e. and are a set of the vectors such that Determine whether there is model M with given in which the set is an equilibrium, and, if so, to find it, that is, specify and such that this set is a state of equilibrium in the model . From this problem, it follows that it is a problem with unknowns
2. Materials and Methods
Let . Throughout the following notation will be used below
(10)
Before talking about equilibrium, we examine the consumer problem. Along with set in the consumer problem (8) we can consider the set
(11)
Due the homogeneity of the functions their maximums on the sets and coincide.
Let be a maximum point in the consumer problem (8). Then this point satisfies the necessary and sufficient conditions for an extremum differentiable on the direction function
(12)
where [9]
And the set is defined by the formula (11).
Introduce the set
Since then from follows that From the condition or for follows that
a) If then consequently
b) If then for small enough
From the foregoing, we find that the set can be written as
(13)
where the set is defined by the formula (10).
Let us solve the properties of the solution of the consumer problem. Particular attention is paid to how these properties are associated with the structure of the set .
Lemma 1. Let be a solution of the consumer problem. Then if _then
Proof. Let be a seeking solution of the consumer problem and i.e. there exists an index such that Then suppose that But since we get that is impossible. If then for all that is also impossible. Consequently, .
The lemma is proved.
Consequence. If then
Let us study in detail the th consumer problem. Let be a solution of this problem and the vector be given where for any .
The utility function of the th branch in the point has a form [2]
(14)
where is a given cost vector.
Introduce the vector
Then (14) takes the form
To investigate the th consumer problem, we apply the necessary and sufficient conditions for the extremum, according which the maximum is reached in the point if and only if
where the cone is defined by the formula (13).
It is well known that, where
(15)
Introduce the denotations
(16)
Thus if in the point the maximum is reached then
(17)
where the set is defined by (10).
Consider some particular cases.
1. Let
(18)
where is a maximum pint in the th consumer problem and the set is defined by the formula (10).
In this case from (13) follows that , where
(19)
Then the necessary and sufficient condition for the optimality in the branch takes the form
(20)
where the function is defined by the formula (16).
Lemma 2. The following conditions are equivalent:
1)
2)
where –is a superdifferential of the function
Proof. The function is concave. Let the inequality take place.
Since then in the point the function reaches its maximum in the set It is well known that necessary and sufficient conditions for the maximum of the concave function in the point on the set
Consist in the existing of the element such that
But it means that in the same place where It implies that for some takes place the following equality
Since and are positive we get
Citing the same arguments, but in reverse order, it is easy to show that from condition 2) of the lemma follows condition 1).
The proof is complete.
Lemma 3.Superdifferential of the function defined by the formula (15) has a form [810]
where
moreover
(21)
Proof. From (15), (16) we have
Define the vector where is coordinate ort. Then
and
From the definition of the superdifferntial we obtain (21).
The lemma is proved.
Lemma 4. The number defined in the Lemma 2. Is equal to
(22)
where
Proof. Let be such that i.e. condition 2) of the Lemma 2 is satisfied. Using (21) one can obtain from this that for some the following equality holds true
(23)
where
It follows from the last that
(24)
Let’s fix the index and express all through
Due the conditions
From this we obtain
(25)
Substituting the obtained values of into the first equalities of (24) we get (22).
Lemma is proved.
Theorem 1. Let strictly positive vector index и and a number defined by the formula (22) be given. The vector is a solution of the th consumer problem (8), satisfying the relation
If and only if when
(26)
(27)
when are defined in the lemma 3.
Proof. Necessity. Let is a solution of the problem (8), satisfying the relation (18). From lemmas 2 and 3 we get that there exists such that i.e. (22) is satisfied. Then using the proof of lemma 4 (namely formula (25))
the condition for all and formula (22) we get the system of inequalities in the system (26). The first system of equalities of (26) follows from (23).
Sufficiency. Let the conditions (26) and (27) take place. Let us choose by the formula (25) which due (26) satisfies to the relations moreover Then as follows from the lemmas 2, 4 and (27) the number has a form (22) and which indeed is a necessary and sufficient condition for optimality of in the th branch.
Theorem is proved.
Remark. If then the number defined by the formula (22) is the maximum growth rate of the total wealth of the th branch.
2. Consider the case when

Then according to Lemma 1
Introduce the projection operator taking for
Consider the vectors and functions defined on as below
(28)
Note that the functions , are superlinear.
Take
(29)
The cone adjoint to the cone has a form
(30)
We’ll consider the functional only on the cone
Proposition 1. For any the conditions
1. for all ;
2. for all ;
are equivalent.
Proof. Let i.e. (see (15))
(31)
Due to (13) e have fixing the index , expressing and substituting into the left hand side on the last inequality, after some eliminations we get (31).
Having the same argument in reverse order, we get the contradiction.
Proposition is proved.
In the case when the subdifferential has a form [8, 10]
(32)
where are defined in the formulas (28), (30) [2, 3].
Theorem 2. Let the strictly positive vector be given. If the vector for which
(33)
is a maximum point in the th consumer problem (8), then for the following relation is satisfied
(34)
for is satisfied:
(35)
2) Let and for some is valid (34) (if ) or (35) (if ). Then the vector is a solution of the th consumer problem (8).
Proof. Necessity. Let strictly positive vector be given and is the point at which the utility function of the kth branch of (8) satisfying (33) takes its maximum.
Then due the proposition 1 necessary and sufficient optimality conditions for the vector in the th branch take the form
where the cone is defined by the formula (29).
By the definition of the superdifferential we have
where superdiffeential has a form (32).
Consequently
(36)
Depending on the choice of the index the function may have various forms (see (28)).
1) Let Then from (28) we have
Superdifferential of this function has form (21).
Substituting (21), (30) and (28) into the relation (36) we get that there exist the numbers such that
(37)
From this immediately follows the inequality (34).
We claim the opposite. Suppose that the conditions of the theorem hold true in the case when i.e. we can choose the numbers and by such way that the inequality (37) would be satisfied and It means that the relation (36) is satisfied or Consequently, due to the definition of the superdifferential the following condition takes place , that indeed is necessary and sufficient condition for the optimality of the vector in the th branch.
2) Consider the case when
Then from (28) we get
Superdifferential of which has a form [8, 9]
(38)
The according to (28), (30) and (38) from (36) follows that there exists numbers such that
(39)
These lead us to
Since the first inequality of the last system holds true for it turns to the equality. As a result, we obtain the desired result.
Let’s claim the opposite. Suppose that the conditions of the theorem in the case when i.e. we can choose the numbers and such that the inequality (39) would be satisfied or This is equivalent to the condition , that is indeed a necessary and sufficient condition for the optimality of the vector in the th branch. Theorem is proved.
3. Conclusion
(1). The form of the superdifferential of the utility function is defined.
(2). Necessary and sufficient condition for the existence of the maximum of this function is derived.
(3). The maximum rate of the growth of the industries total wealth determined.
(4). A necessary and sufficient condition is obtained or optimality of the state vector of the branches.
References