American Journal of Applied Mathematics
Volume 3, Issue 3, June 2015, Pages: 146-150

Equilibrium Mechanisms in Models of Reproduction with a Fixed Budget

Sabir Isa Hamidov

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

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To cite this article:

Sabir Isa Hamidov. Equilibrium Mechanisms in Models of Reproduction with a Fixed Budget. American Journal of Applied Mathematics. Vol. 3, No. 3, 2015, pp. 146-150. doi: 10.11648/j.ajam.20150303.21

Abstract: In the study of some models of reproduction of equilibrium mechanisms are used to describe the activity of the economic systems. As the equilibrium mechanisms models with the fixed budgets can be applied. Typically, the control center can not anticipate all situations. Hence, some variations of the initial model parameters are possible. Therefore, in the study of reproduction models a great interest presents the comparative statics, which allows one find out the dependence of the speed of changes of the state trajectories on the changes of the parameters of the trajectories. In this paper, we deal with the Leontief’s  type model with fixed budgets, consisting of n  branches.

Keywords: Reproduction Model, Equilibrium Mechanisms, Lagrange Multipliers

1. Introduction

In studies of the reproduction models describing the activities of economic systems are often used equilibrium mechanisms. In the paper [1] as equilibrium mechanisms a model with fixed budgets is used, depending on the parameters, the prices that determine the utility function, and on the budgets of the elements of the system. Typically, the control center does not have full information about the activities of the system and can not anticipate all situations. Therefore, the study of reproduction models of great interest is a comparative statics, which allows one to find out the dependence of the speed of changes of the state trajectory generated by the mechanism on the changes in the parameters of these states.

First let us give the description of the model [1-6]. Let simulated economy consists of  sectors. Each sector produces one product, and vice versa, each product is made only by one sector. Moreover, under the "product" means the whole total output of the industry. Issue of the i- th sector is described by the function, for which  and in addition, it is assumed this function is twice continuously differentiable and strictly super linear. It means that the function in concave, positively homogeneous and satisfies to the inequality (  if  is not proportional to. The state modeling the sector is given by the -dimensional vector,  –th coordinate of which corresponds to the k –th product at the disposal of this sector. As a result of industrial activity the vector the  partially changes into the vector). The diagonal matrix  having  as a diagonal is called a conservation matrix. Thus, expensing in the beginning of the time interval the vector   at the end time the sector will have the vector ( where

The state of the modeling system is formed by the union of the states of the sectors, and therefore is the vector ..,, where  is a state of -th sector.

Investigating given above model for the reproduction the modelwith fixed budgets is used   [7], that indeed has the form


where is any element of the cones , is an utility function defined as

The functions  indeed are the costs of all assets containing by the corresponding sector with prices

Note that the greatest interest is the case when the prices of the products are the same for all sectors, in other words, depend only on the products themselves. As   in the model  𝔐   we understand the vector  the coordinate of which is a given budget of the –th sector. The model 𝔐 describes the distribution of the produced product between sectors.

The set of the vectors,…, forms a equilibrium state in the model  𝔐, if the vectors    are solutions of the problem

within the conditions , and in addition is valid

2. Main Body

Let () be an equilibrium state for the model. Then as shown in [7], the vector ) is a solution of the following convex programming problem 



Here  is a vector of Lagrange multipliers corresponding to the constraints (2) of this problem.

Set (, ) =

It is clear that

And particularly

(,) =,

where is an equilibrium vector corresponding to the vector of prices .

Let us introduce the denotation  . Usually the quantity  is interpreted as the growth rate of funds in the   – th sector.

Let's give each coordinate  of the price vector   an increment, in other words, we consider the perturbed vector of prices


Consider the problem of convex programming

For some  and . Its solution =,is a component of the equilibrium  state of the model. Thus  and

We choose   such that (if it is possible)

where =.  By this way by prices the budgetsare chosen such that the growth rate of funds in the   – th sector does not change by changing the price .  Consider the function  .

It was shown in [1] that the equilibrium vector = is a solution of the equation , where   is a mapping of  into itself, with coordinate functions


Let us investigate the system   and find out the growth rate change of the trajectory by the constructed by the model.

According to the implicit function theorem [8]

From this we obtain


One can find

The following lemma is true.

Lemma I. The function  is differentiable with respect to  and  coincides with the following block matrix [9-10]


where is a matrix of the second partial derivatives of the function  calculated in the point   .

It is sufficient to restrict the study the case when the increment  of the coordinates of prices, except for one (for example  are zero.

We use the following denotations


where   is some parameter.


The similar changes will have the matrix . In addition we transform to the more convenient form the blocks of this matrix matrix. For the elements   of the matrix   is valid

To express all elements of the matrix  due the element  we use known relations for the first powers of the functions (see[3]):

As a result we get

Then taking  we arrive to the following form of the matrix


Here we give some axillary statements.

Lemma 2.  For the matrix  is invertible and



IO. For n=3 the main diagonal of the matrix consists of zeros. 

2O. For  elements of the diagonals of the matrix  are positive, and all remained ones are negative.

Lemma 3. The matrix is invertible and  where are square matices of order defined by the (3), (4).

Let us sketch the proof of the lemma. First, we note that

If to consider


Then after some transformations we have

From this we see that if

Therefore there exists a matrix that is inverse to .

The matrix we seek in the form  , where  . If to consider that the product of the matrices is a diagonal block matrix with identity block matrices in the main diagonal, one can write out  system of    matrix equations.  As a system we consider the system of equations which is a product all lines of the matrix  by-th column of the matrix   After a sequence of transformations from the system with number   we express


By similar way we express  from the system with number  


Here and later on

Thus the matrix  is now found since all formed blocks are found.  

Let. Then

Let us study the dependence of the sign of coordinates of the vectors     on changes of the price of the first product. Now when the matrix is known we can write out the system of equalities following from the theorem on implicit function


Consider the case   при. This corresponds to the fact that the ratio of the growth is equal to the ratio of the coefficients of safety first product. Namely


One may show that from

Follows the relations


As a result of matrix multiplication we obtain a new matrix at the intersection of k-th line and l-th column of which stands the element


Hence we obtain the formula for expressing the coordinates of the vectors   through the coordinates of the vectors 



Thus we obtain the validity of the following theorems.

Theorem 1. The speed   of change of state of the –th sector  is expressed linearly through the coordinates of the .

Теорема 2. Let . If there exist small enough  such that for all is true , then for arbitrary small enough  is valid the equality

Consequence. Under the conditions of Theorem signs of the coordinates of the vectors  at   coincide with the signs of the coordinates of the vector . Coordinates of the vector have the opposite sign to the corresponding coordinates of other vectors.

In other words, if you change the price of the first product in all sectors, except for the first, occurs simultaneously deceleration or acceleration of the trajectory growth, while in the first branch opposite process takes place.

3. Conclusion

1.   The character of the variation of the speed of the model trajectory growth is defined.

2.   The lemma on the differentiability of the function is proved. The matrix   is constructed.

3.   The invertibility conditions for the matrix  are found.

4.   The invertibility conditions for the matrixare found.

5.   The change of the state speed of the sectors is defined.

Author thanks K.Niftaliyeva for her vary useful assistance in preparation the paper for publication.


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