Parameter Identification of a Class of Economical Models

Abstract

In mathematical modeling of economic problems, it is common to assume that economic laws influence the process and analyze the type of solution obtained. Sometimes these laws have unknown parameters and they worth studying if these parameters can be determined. In this paper, a structured system is considered and the identifiability is analyzed. In case the parameters can be uniquely identified, an algorithm for obtaining the model parameters is presented and, finally, the existence of a balanced growth solution is studied.

1. Introduction

Linear ordinary differential or difference equations are frequently used in modeling many engineering, economic, and social processes. These equations describe the system behavior with a relative degree of accuracy and yet are simple enough to be solved. It is well known that any differential or difference equation of order 𝑛 can be written as a system of 𝑛 first-order differential or difference equations. This process works for any higher-order equation, linear or not, and it consists of expressing the top derivative or difference as a function of the lower ones. An 𝑛th order equation gives rise to a first order system in 𝑛 variables whose coefficient matrix is called companion matrix. The characteristic polynomial of an 𝑛th order linear equation is the same as the characteristic polynomial of the companion matrix. Then, a companion matrix𝛼𝒞(𝜶)=010⋯001⋮⋮⋱⋱00⋯01₀𝛼₁⋯𝛼_𝑛−2𝛼_𝑛−1 (1.1) is associated with the monomial characteristic polynomial𝑝(𝜁)=−^𝑛_𝑖=0𝛼_𝑖𝜁^𝑖,with𝛼_𝑛=−1. (1.2)

In the last years, the use of dynamical models in economic theory is quite usual, in stark contrast, with the classical static economic models. Many of these models are mathematically formulated by higher-order difference equations (see [1]), which derive to a structured representation, where the companion matrix can be used.

As already mentioned, economic problems are usually modeled by structured systems, that is, systems where the matrices have a number of fixed zero entries while the rest of the entries are not known. This kind of system is widely considered in the literature, than in control theory (see [2, 3]), as in computing methods to solving it. When the size of the system is large, for solving these structured systems, fast algorithms are used. In [4] an exhaustive analysis on numerical properties of several fast algorithms was shown. In [5] an algorithm for solving dynamic economic models developing first-order and second-order iterations was studied.

In structured systems, the analysis of the unknown entries plays an important role. For example, the economic models shown in [1] have a parameterized structure, where the parameters are given information on the economic process. A first step for the validation of the model is to identify the parameters that appear in the system. Before identifying the parameters, it should be examined whether the parameters can be determined from the data. The identification of parameters allows solving the model. And finally, it is necessary to analyze the obtained solution. In particular, in economic models, the existence of a balanced growth solution is interesting. In the economical sense, balanced growth refers to simultaneous coordinated expansion of several sectors.

The aim of this paper is to study the identifiability problem and the balanced growth solution of an economic problem. The structure of the paper is as follows. In Section 2 we treat the identification problem of a companion structured system. In Section 3, we show a structured Leontief model associated to a determinate economical process and give some results about the identifiability of its parameters. This analysis leads us to construct an algorithm to obtain the parameters of the model. Finally, in the last section we study the balanced growth solution of an economic model.

2. Identification Problem of a Structured System

In the first part of this paper, we discuss a problem of identifiability for a linear differential or difference system𝑥^′(𝑡)=𝒞(𝐩)𝑥(𝑡)+𝑏(𝐩)𝑢(𝑡) (2.1) or𝑥(𝑘+1)=𝒞(𝐩)𝑥(𝑘)+𝑏(𝐩)𝑢(𝑘), (2.2) where the coefficient matrices have a fixed structure, 𝒞(𝐩) is a companion matrix and 𝑏(𝐩) is a monomial vector given by𝑏(𝐩)=00⋯0𝑝^𝑏_𝑛^𝑇, (2.3) with the parameter vector 𝐩=(𝐩_𝑛1⋯𝐩_𝑛𝑛𝐩^𝑏_𝑛)^𝑇 belonging to a subset 𝒫⊆ℝ^𝑟. Note that, the matrix 𝒞(𝐩) is a companion matrix where the last row is the vector (𝐩_𝑛1⋯𝐩_𝑛𝑛)^𝑇.

In general, an interpretation of a structured linear system is a family of linear systems, with the same structure, together a set of parameters. In this work, this kind of systems is denoted by𝒮_𝑃(𝐴,𝐵)={𝑆(𝐩)=(𝐴(𝐩),𝐵(𝐩)),𝐩∈𝒫}. (2.4)

A formal definition of structured linear dynamic systems can be found in [6]. Furthermore, 𝑖𝑜(⋅) denotes the input-output behavior of the system 𝑆(⋅)∈𝒮_𝑃(𝐴,𝐵). Note that, 𝑖𝑜(𝐩) of the system 𝑆(𝐩) can completely be characterized by the Markov parameters, defined as 𝑉(𝑗,𝐩)=𝐴^𝑗(𝐩)𝐵(𝐩),𝑗≥0.

Usually, in the analysis of parametric models, studied two problems can be: identifiability and estimability. The difference between estimability and identifiability is that in the identifiability all possible options of the input-output signal are evaluated, while estimability is the ability to estimate accurately the parameters of a given data set.

A model is identifiable if and only if, for every parameter set there is a unique input-output behavior. Several papers deal with identification of models describing behavior of biomedical, chemical, or other systems, [7, 8].

The problem of the structural identifiability of the model consists of the determination of all parameter sets which give the same input-output structure. The structural identifiability of 𝒮_𝑃(𝐴,𝐵) depends on the number of solutions of 𝑖𝑜(𝐩)=𝑖𝑜(𝐪). If the parameters can be determined uniquely from the data, that is, the equation 𝑖𝑜(𝐩)=𝑖𝑜(𝐪) has only the solution 𝐩=𝐪, then the system 𝒮_𝑃(𝐴,𝐵) is structurally globally identifiable. On the other hand, if there exists a finite or contable number of solutions for the parameters, the system 𝒮_𝑃(𝐴,𝐵) is structurally locally identifiable. Otherwise, the system 𝒮_𝑃(𝐴,𝐵) is structurally unidentifiable ( see [9, 10]).

Now, we consider the system 𝒮_𝑃(𝐶,𝑏), with 𝒞(⋅) and 𝑏(⋅) given by (1.1) and (2.3). In order to analyze the identification problem of this system, we use the Markov parameters to know its input-output behavior 𝑖𝑜(𝐩). By the structure of the companion matrix 𝒞(⋅) and by a technical process, we have the following result.

Proposition 2.1. Consider the structured system 𝒮_𝑃(𝒞,𝑏). The Markov parameters 𝑉(𝑘,𝐩)=(𝑣_𝑖^(𝑘)(𝐩))_{𝑖=1,…,𝑛}, 𝑘≥0, are given by 𝑖=1,…,𝑛−1,𝑣_𝑖^(𝑘)𝑣(𝐩)=0,𝑘=0,_{(𝑘−1)𝑖+1}(𝐩),𝑘=1,…,𝑛−1,𝑖=𝑛,𝑣_𝑛^(𝑘)𝑝(𝐩)=^𝑏𝑛,𝑘=0,_{𝑛j=𝑛−𝑘+1}𝑝_𝑛,𝑗𝑣_𝑗^(𝑘−1)(𝐩),𝑘=1,…,𝑛−1. (2.5)

Directly from the above result, we obtain the following proposition which gives a condition to solve the identification problem for the system 𝒮_𝑃(𝒞,𝑏).

Proposition 2.2. The structured system 𝒮_𝑃(𝒞,𝑏) is globally identifiable.

In the next section, we analyze the identification problem for economic models.

3. Economic Dynamic Model

First, we consider the nonlinear discrete-time dynamic model given in [1]𝑦(𝑡)=𝛼₁+(𝛼₂−𝛼₃)𝑦(𝑡−1)+𝛼₃𝑦(𝑡−2)−𝛼₄(𝑦(𝑡−1)−𝑦(𝑡−2))³, (3.1) where 𝑦(𝑡), 𝑡∈ℕ, is the income at time 𝑡. Parameters used in this model have the following economic interpretation: the parameter 𝛼₁ represents the autonomous expenditures, the parameter 𝛼₂∈(0,1), and represents the consumer's reaction against the increase or decrease of his income. Finally, the investment is described using parameters 𝛼₃ and 𝛼₄, where 𝛼₃ represents the difference between the control and the investment function.

This model is interesting because for different values of parameters we obtain well-known linear dynamic models as Hick-Samuelson, Keynes, and the nonlinear dynamic Puu model (see [11]).

In particular, for 𝛼₁=𝛼₄=0, 𝛼₂=1−𝛽, with 𝛽∈(0,1) and 𝛼₃=−𝛾, 𝛾>0, we obtain the Hick-Samuelson model𝑦(𝑡)=(1+𝛾−𝛽)𝑦(𝑡−1)−𝛾𝑦(𝑡−2), (3.2) which matrix expression is given by the companion matrix01−𝛾1+𝛾−𝛽. (3.3) If we do not have investment, that is, 𝛼₃=𝛽∈[0,1], we obtain the Keynes model𝑦(𝑡)=𝛽(𝑦(𝑡−2)−𝑦(𝑡−1))+𝛼₁+𝛼₂𝑦(𝑡−1), (3.4) which has the following coefficient matrix01𝛽𝛼₂−𝛽. (3.5)

Note that these models are structured systems of the kind 𝒮_𝑃(𝒞,𝑏) where the coefficient matrix depends on a vector of parameters which could be identified to analyze its solution. Then, by Proposition 2.2 these models are globally identifiable.

Now, we consider the Leontief model. In the Leontief economic process is assumed that there is a market for different goods in which each industry has only one production process. Assuming that 𝑥(𝑘) is the production level vector and 𝑢(𝑘) is the demand level vector, the control process (see, e.g., [12, 13]), the system has the structured representation given by𝐶𝑥(𝑘+1)=(𝐼−𝑃(𝐩)+𝐶)𝑥(𝑘)−𝐷(𝐩)𝑢(𝑘), (3.6) where 𝐶 is the capital coefficient matrix, 𝑃(𝐩) is the technological coefficient matrix, and 𝐷(𝐩) is the demand coefficient matrix (excluding investment). The (𝑖,𝑗)-entry of the matrices 𝑃(𝐩) and 𝐶 represent the amounts of material input and capital of the 𝑖th good necessary for the production of one unit of the 𝑗th industry, respectively. Since the Leontief model is an economic model, 𝐶≥0,𝑃(𝐩)≥0 and 𝐷(𝐩)≥0. This kind of systems is a dynamic generalized system because the capital coefficient matrix 𝐶 can be or not be singular. The singularity of this matrix arises because no output from one sector is used in the production of some products.

In this paper, the system (3.6) is a structured linear dynamic system where the capital matrix 𝐶 is a nonsingular diagonal matrix, 𝐶=diag(𝑐₁,𝑐₂,…,𝑐_𝑛), and the matrix 𝑃(𝐩) has a companion matrix structure. By the nature of the technological matrix 𝑃(𝐩),0<𝑝^𝑝_𝑛𝑗<1, and ^𝑛_𝑗=1𝑝^𝑝_𝑛𝑗<1, (3.7) where the unknown demand 𝐷(𝐩) with almost one column of type (2.3)exists.

Since capital matrix 𝐶 is invertible, the structured generalized system (𝐶,𝐼−𝑃(𝐩)+𝐶,𝐷(𝐩)) can be transformed into a structured standard system𝑥(𝑘+1)=𝐴(𝐩)𝑥(𝑘)+𝐵(𝐩)𝑢(𝑘), (3.8) where𝐴(𝐩)=𝐼+𝐶⁻¹𝑝(𝐼−𝑃(𝐩))=^𝑎11𝑝^𝑎₁₂0⋯00𝑝^𝑎₂₂𝑝^𝑎₂₃𝑝⋯0⋮⋮⋮⋱⋮_𝑎𝑛1𝑝^𝑎_𝑛2𝑝^𝑎_𝑛3⋯𝑝^𝑎_𝑛𝑛,𝐵(𝐩)=−𝐶⁻¹𝐷(𝐩) (3.9)(3.10) is such that it has almost one monomial column of kind (2.3), with 𝑃={𝐩∈ℝ^3𝑛−1,𝐩≠𝟎)}.

An important point to note here is the structure of matrix 𝐴(𝐩). In the literature, this kind of matrix is known as the modified Leslie (Lefkovitch) matrix, which appears in different areas such as the study of recruitment, survival and population growth rate (see, e.g., [14, 15]). For example, in [16] it is used to describe a discrete age-structured population model where no emigration during harvesting is considered.

Some properties of the spectrum of 𝐴(𝐩) are given in the following result, which are useful to study the stability of the system 𝒮_𝑃(𝐴,𝐵).

Proposition 3.1. Consider the matrix 𝐴(𝐩) given in (3.9). (i)If 𝑝^𝑎_{𝑖0,𝑖0+1}=0 or 𝑝^𝑎_𝑛1=⋯=𝑝^𝑎_𝑛𝑖0=0, with 𝑖₀>𝑛−1, then {𝑝^𝑎₁₁,𝑝^𝑎₂₂,…,𝑝^𝑎_𝑖0,𝑖0}⊆𝜎(𝐴(𝐩)). (3.11)(ii)If 𝑝^𝑎_{𝑛−1,𝑛}=0 or 𝑝^𝑎_𝑛1=⋯=𝑝^𝑎_{𝑛,𝑛−1}=0, then {𝑝^𝑎₁₁,𝑝^𝑎₂₂,…,𝑝^𝑎_𝑛,𝑛}⊆𝜎(𝐴). (3.12)

Proof. If we structure 𝐴(𝐩) as (^{𝐴1(𝐩)𝐴2}_𝐴^(𝐩)_{3(𝐩)𝐴4(𝐩)}), with 𝐴₁(𝐩)∈ℝ^{𝑛−1×𝑛−1}, 𝐴₂(𝐩)∈ℝ^𝑛−1×1, 𝐴₃(𝐩)∈ℝ^1×𝑛−1, and 𝐴₄(𝐩)∈ℝ, then |𝜆𝐼−𝐴(𝐩)|=|𝐴₁(𝐩)||𝐴₄(𝐩)−𝐴₃(𝐩)𝐴₁⁻¹(𝐩)𝐴₂(𝐩)|=|𝐴₁(𝐩)||𝜆−𝑝^𝑎_𝑛𝑛−(𝑝^𝑎_𝑛1𝑝^𝑎₁₂…𝑝^𝑎_{𝑛−1,𝑛}|𝐴₁|+⋯+𝑝^𝑎_{𝑛,𝑛−2}𝑝^𝑎_{𝑛−2,𝑛−1}𝑝^𝑎_{𝑛−1,𝑛}(𝜆−𝑝^𝑎𝑥_{𝑛−2,𝑛−2})(𝜆−𝑝^𝑎_{𝑛−1,𝑛−1})+𝑝^𝑎_{𝑛,𝑛−1}𝑝^𝑎_{𝑛−1,𝑛}𝜆−𝑝^𝑎_{𝑛−1,𝑛−1})|. (3.13)(i)If 𝑝^𝑎_{𝑖0,𝑖0+1}=0 or 𝑝^𝑎_𝑛1=⋯=𝑝^𝑎_𝑛𝑖0=0, with 𝑖₀>𝑛−1, then we simplify |𝐴(𝐩)|=(𝜆−𝑝^𝑎₁₁)⋯(𝜆−𝑝^𝑎_𝑖0,𝑖0)𝑞_𝑛−𝑖0(𝜆), (3.14) where 𝑞_𝑛−𝑖0(𝜆) is a polynomial of degree 𝑛−𝑖₀.Hence, {𝑝^𝑎₁₁,𝑝^𝑎₂₂,…,𝑝^𝑎_𝑖0,𝑖0}⊆𝜎(𝐴(𝐩)).(ii)By (i) it is straightforward.

In order to solve the identifiability problem of a family of structured systems of kind (3.6), we show some results on structured standard system 𝒮_𝑃(𝐴,𝐵).

The system (3.9)-(3.10) belongs to a kind of structured systems. In particular, the identifiability of the parameters of a standard system 𝒮_𝑃(𝐴,𝐵) is concerned with the determination of them from the external behavior of the system. That is, to determine the input-output behavior (𝑖𝑜) of a model 𝒮_𝑃(𝐴,𝐵), we can use the Markov parameters associated to the system 𝒮_𝑃(𝐴,𝐵).

The parameter identification process for the structured system is followed from the structure of the vectors obtained in the following proposition.

Proposition 3.2. Consider the structured system 𝒮_𝑃(𝐴,𝐵). The Markov parameters 𝑉(𝑘,𝐩)=(𝑣_𝑖^(𝑘)(𝐩))_{𝑖=1,…,𝑛}, 𝑘≥0 are given by 𝑘=0,𝑣_𝑛⁽⁰⁾(𝐩)=𝑝^𝑏_𝑛,𝑘=1,…,𝑛−1,𝑣_𝑖^(𝑘)(𝐩)=0,𝑖=1,…,𝑛−𝑘−1,_1𝑗=0𝑝^𝑎_{𝑖,𝑖+𝑗}𝑣^(𝑘−1)_𝑖+𝑗(𝐩),𝑖=𝑛−𝑘,…,𝑛−1,_𝑛𝑗=1𝑝^𝑎_{𝑖,𝑖−𝑘+𝑗}𝑣^(𝑘−1)_{𝑖−𝑘+𝑗}(𝐩),𝑖=𝑛. (3.15)

We solve the identification problem for the system 𝒮_𝑃(𝐴,𝐵) in the next result.

Proposition 3.3. The structured system 𝒮_𝑃(𝐴,𝐵) where 𝐴(𝐩) and 𝐵(𝐩) are defined by (3.9) and (3.10) with 𝐩∈𝒫 is globally identifiable.

Proof. We consider two structured systems 𝑆(𝐩) and 𝑆(𝐪) defined by (3.9)-(3.10) with 𝐩input-output behavior(𝑖𝑜)𝑉(𝑘,𝑝)=𝑉(𝑘,𝐪),𝑘≥0and we shall prove that 𝐩=𝐪. By the structure of the Markov parameters obtained in Proposition 3.2, for 𝑖=1,…,𝑛, it is show that 𝐴(𝐩)^𝑘𝐵(𝐩)=𝐴(𝐪)^𝑘𝐵(𝐪)⇒𝑝^𝑎_𝑘,𝑘=𝑞^𝑎_𝑘,𝑘,𝑝^𝑎_𝑘,𝑘+1=𝑞^𝑎_𝑘,𝑘+1. (3.16) Moreover, the rest of unknown entries of 𝐵(𝐩) are identified from the first equality 𝐵(𝐩)=𝐵(𝐪). Hence, 𝐩=𝐪.

Consider the associated standard Leontief model (3.8), in this case given by𝐴(𝐩)=𝐼+𝐶⁻¹=(𝐼−𝑃(𝐩))1+1−𝑝^𝑝₁₁𝑐₁−𝑝^𝑝₁₂𝑐₁0⋯001+1−𝑝^𝑝₂₂𝑐₂−𝑝^𝑝₂₃𝑐₂⋯0⋮⋮⋮⋱⋮−𝑝^𝑝_𝑛1𝑐_𝑛−𝑝^𝑝_𝑛2𝑐_𝑛−𝑝^𝑝_𝑛3𝑐_𝑛⋯1+1−𝑝^𝑝_𝑛𝑛𝑐_𝑛, (3.17) and 𝐵(𝐩)=−𝐶⁻¹𝐷(𝐩) has the column(00⋯0−𝑝^𝑑_𝑛𝑐_𝑛)^𝑇. (3.18) By Proposition 2.2 the entries of these matrices are identified and hence we can also identify the initial parameters of the vector 𝐩. Thus, the initial Leontief model is globally identifiable.

Proposition 3.4. The Leontief economic model given by (3.6) where 𝐶 is a non-singular diagonal matrix, 𝐶=diag(𝑐₁,𝑐₂,…,𝑐_𝑛) and 𝑃(𝐩) is a companion matrix, is globally identifiable.

The following algorithm allow us determine the parameters 𝐩 of the Leontief model.

Algorithm
Step 1. Introduce the size of the state vector: 𝑛. Introduce the capital matrix 𝐶. Introduce the matrices {𝑉(𝑘,𝐩),𝑘=0,…,𝑛} that determine the known external behavior of the system 𝒮_𝑃(𝐴,𝐵) with the structure (3.9)-(3.10). And introduce the position of the monomial column of matrix 𝑉(0,𝐩): 𝑗, for 𝑘=0,…,𝑛−1. Step 2. Choose the 𝑗th column of 𝑉(𝑘,𝐩) and denote it as 𝑣^(𝑘).Step 3. Introduce 𝑣^(𝑘−1)_𝑛+1=0.Step 4. Construct the following system: 𝑣_𝑛⁽⁰⁾=𝑝^𝑏_𝑛,𝑣_𝑖^(𝑘)(𝐩)=^1𝑗=0𝑝^𝑎_{𝑖,𝑖+𝑗}𝑣^(𝑘−1)_𝑖+𝑗(𝐩),𝑖=𝑛−𝑘,…,𝑛−1,_𝑛𝑗=1𝑝^𝑎_{𝑖,𝑖−𝑘+𝑗}𝑣^(𝑘−1)_{𝑖−𝑘+𝑗}(𝐩),𝑖=𝑛. (3.19)Step 5. Solve the above system and obtain the parameters 𝑝^𝑎_{𝑛,n−𝑘+𝑗} for 𝑗=1,…,𝑛 and 𝑝^𝑏_𝑛 and 𝑝^𝑎_{𝑖,𝑖+𝑗}, for 𝑗=0,1, and 𝑖=𝑛−𝑘,…,𝑛.Step 6. Use the parameters obtained in Step 5 and construct the matrices (3.17)-(3.18) and from them obtain 𝑝^𝑑_𝑛 and 𝑝^𝑝_{𝑖,𝑖+𝑗}, 𝑗=0,1, and 𝑖=1,…,𝑛.
With this process, we have determinate all parameters of the system.Step 7. Finally, construct the matrices of the system (3.6).Note that, this algorithm can also be applied when the Hick-Samuelson and the Keynes models are considered, that is, when 𝑝^𝑎_𝑖𝑖=0 and 𝑝^𝑎_𝑖,𝑖+1=1, 𝑖=1,…,𝑛−1.
Next we present an academic example to clarify the above algorithm. The model considers a number of consumers such that at any given time 𝑡 each consumer holds quantities of different services or activities. We assume that this kind of market can be modeled by a structured Leontief model. Using the data, the problem is to assure the uniqueness in the parameter of the dynamical model. In this example to determine these parameters we use the above algorithm.
Example 3.5. Tourism demand in a country involves different activities. In this example the external behavior of the foreign demand for tourist services during a period of years is known and we want to apply the algorithm in order to identify the structured model.
The external behavior of the foreign demand for tourist services during a period of four years is given by 000000𝑉(0)=,0.4000,𝑉(1)=−0.04000.4214,𝑉(2)=0.0049−0.08610.4445𝑉(3)=−0.00130.0160−0.13920.4695,𝑉(4)=−0.00580.0349−0.20010.4964 (3.20) and consider that the capital coefficient matrix (given in thousands) 𝐶=diag(3.8,8.2,10,12.5). (3.21)
Then we apply the above algorithm.Step 1.  Introduce 𝑛=4, the matrices 𝐶, {𝑉(𝑘),𝑘=0,…,4} and 𝑗=1, for 𝑘=0,…,3.Steps 2-3
Denote 𝑉(𝑘) as 𝑣^(𝑘) and introduce 𝑣₅^(𝑘−1)=0.Steps 4-5
Solving the system 𝑣₄⁽⁰⁾=𝑝^𝑏₄,𝑣_𝑖^(𝑘)(𝐩)=^1𝑗=0𝑝^𝑎_{𝑖,𝑖+𝑗}𝑣^(𝑘−1)_𝑖+𝑗(𝐩),𝑖=𝑛−𝑘,…,𝑛−1,_𝑛𝑗=1𝑝^𝑎_{𝑖,𝑖−𝑘+𝑗}𝑣^(𝑘−1)_{𝑖−𝑘+𝑗}(𝐩),𝑖=𝑛, (3.22) obtain 𝑝^𝑏₄=0.4000,𝑝^𝑎₃₄=−0.1000,𝑝^𝑎₄₄=1.1000,𝑝^𝑎₂₃𝑝=−0.1220,_𝑎33=1.1000,𝑝^𝑎₄₃=−0.0139,𝑝^𝑎₁₂=−0.2632,𝑝^𝑎₂₂𝑝=1.1220,_𝑎42=−0.0114,𝑝^𝑎₁₁=1.2632,𝑝^𝑎₄₂=0.1430,𝑝^𝑎₄₃𝑝=0.1740,_𝑎41=−0.0249,𝑝^𝑎₁₄=0,𝑝^𝑎₂₄=0,𝑝^𝑎₁₃𝑝=0,_𝑎32=0,𝑝^𝑎₂₁=0,𝑝^𝑎₃₁=0. (3.23)Steps 6-7
From (3.17)-(3.18) the system is identifiable and its coefficient matrices are 0005.𝐶=diag(3.8,8.2,10,12.5),𝑃=0100001000010.31110.14300.17400.3310,𝐷= (3.24)
An interpretation of these results can be, for example the following. It seems reasonable that the fact that capital matrix is diagonal implies that to obtain one unit of production in each industry, an investor is only necessary. On the other hand, the technological matrix is determined as each service depends on one material input except the last service, which needs a part of all material input involved in the process. Finally, the structure of the demand matrix indicates that only the last service has consumers, given in thousands. This service includes all others.

4. Balanced Growth Solution

In economic problems it is important to assure the existence of a solution which output of each sector increases by a constant percentage per unit of time. This kind of solution is called balanced growth solution, and it satisfies𝑥(𝑘)=(1+𝛿)^𝑘𝑥₀≫0 (4.1) with the balanced growth rate 𝛿>0.

The balanced growth problem is solved in [17] for autonomous systems. The interest in obtaining balanced growth solutions for some dynamic model has to do the following question: is it possible to obtain a feedback such that the closed-loop system has a balanced growth solution? A first approximation of the solution to this problem has been given in [18]. In this work a characterization to the existence of a balanced growth solution has been obtained. This characterization is based on the existence of a feedback such that the closed-loop system has a positive eigenvalue with a positive eigenvector.

To analyze the balanced growth solution of the system (3.6), we consider the associated system 𝒮_𝑃(𝐴,𝐵) given by 𝑥(𝑘+1)=𝐴(𝐩)𝑥(𝑘)+𝐵(𝐩)𝑢(𝑘). (4.2)

The balanced growth problem for Leontief structured model is treated in the following result.

Proposition 4.1. The balanced growth problem for Leontief structured model (3.6) is solvable.

Proof. Note that technological matrix 𝑃(𝐩) is an irreducible matrix, that is, its associated directed graph is strongly connected, which follows by the companion matrix structure of 𝑃(𝐩). Moreover, as 0<𝑝^𝑝_𝑛𝑗<1, and ∑^𝑛_𝑗=1𝑝^𝑝_𝑛𝑗<1, we can assure that 𝜌(𝑃(𝐩))<1. These conditions on 𝑃(𝐩) lead us to the fact that matrix (𝐼−𝑃(𝐩))⁻¹ is positive. If we construct a feedback 𝑢(𝑘)=𝛿𝐹𝑥(𝑘) with 𝐹>𝑂, then (𝐶+𝐷(𝐩)𝐹)≥0. Therefore, the coefficient matrix (𝐼−𝑃(𝐩))⁻¹(𝐶−𝐷(𝐩)𝐹) of the closed-loop system is positive. Hence, there exists a positive eigenvalue 𝜆 with a positive eigenvector 𝑥₀, that is, (𝜆𝐼−(𝐼−𝑃(𝐩))⁻¹(𝐶+𝐷(𝐩)𝐹))𝑥₀=0, (4.3) and taking 𝛿=1/𝜆 we have ((𝐼−𝑃(𝐩))−𝛿(𝐶+𝐷(𝐩)𝐹))𝑥₀=0,(−𝛿𝐼+𝐶⁻¹(𝐼−𝑃(𝐩))−𝛿𝐶⁻¹(𝐷(𝐩)𝐹))𝑥₀=0,(1+𝛿)𝑥₀=(𝐼+𝐶⁻¹(𝐼−𝑃(𝐩))−𝛿𝐶⁻¹(𝐷(𝐩)𝐹))𝑥₀,(1+𝛿)𝑥₀=(𝐴(𝐩)−𝛿𝐵(𝐩)𝐹)𝑥₀,(1+𝛿)^𝑘+1𝑥₀=(𝐴(𝐩)−𝛿𝐵(𝐩)𝐹)(1+𝛿)^𝑘𝑥₀. (4.4)
Considering the state-feedback 𝑢(𝑘)=−𝛿𝐹𝑥(𝑘), we show that 𝑥(𝑘)=(1+𝛿)^𝑘𝑥₀ is a solution of the system (3.6). Thus, the balanced growth problem for Leontief structured model (3.6) is solvable.

The economic meaning of the existence of a positive eigenvector can be given, for instance, when there is one economy where each sector of this economy depends on all others directly or indirectly for either its intermediate product or its capital. We refer to [17] for more details for it and in the case when the economy can be divided into several sub-conomies.

Acknowledgments

The authors are grateful to the anonymous referees for their constructive remarks. This paper is supported in part by Grant MMT2007-64477.