Pure and Applied Mathematics Journal
Volume 5, Issue 6, December 2016, Pages: 205-210

On a Consumer Problem

Sabir Isa Hamidov

Mathematical-Cybernetics Department, Baku State University, Baku, Azerbaijan

Sabir Isa Hamidov. On a Consumer Problem. Pure and Applied Mathematics Journal. Vol. 5, No. 6, 2016, pp. 205-210. 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

Contents

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 [5-7]

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 [8-10]

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 k-th 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)

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

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