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
(1)
(2)
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
by
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.
Then
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
where
Consequence.
I^{O}. For n=3 the main diagonal of the matrix consists of zeros.
2^{O}. 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
(3)
By similar way we express from the system with number
(4)
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
(5)
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
(6)
(7)
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.
References