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Proceedings of the 27th Chinese Control Conference July 16-18, 2008, Kunming,Yunnan, China

Stability of Economic Input-output ModelJiang Lei, Fang Jianan *, Zhou Wuneng, Zheng Da, Lu Hongqian

School of Information Science and Technology, Donghua University, Shanghai 201620, P. R. China E-mail: jafang@dhu.edu.cn

Abstract: This paper investigates the condition of stability for the economic discrete-time singular dynamic input-output model. A new mathematic method called as linear matrix inequality is applied to study the singular dynamic input-output systems. With this tool, we need not convert singular systems into general systems. A sufficient stability condition under which the economicdiscrete-time singular input-output model is admissible is proved. This condition is in the form of linear matrix inequality andcan be easily tested by Matlab. Key Words: Economic dynamical input-output model, Linear matrix inequality, Discrete-time singular systems

1 INTRODUCTION

Recently much attention has been paid for the problem of the control of economics. Firstly, economists studied the static economic systems. However, most economic phenomena are dynamic. So the economists want to study the current state of an economic system and research how various policies can be used to move the system from its present status to a future more desirable state when they deal with those dynamic systems. A number of funda-mental notions and methods based on the theory of eco-nomical cybernetics have been extended to the area of economics [1]. Economists and engineers include Holt (1962), Fisher (1962), Zellner (1966), Prescott (1971, 1972), Shupp (1972), Athans (1972), Aoki (1973) have been applying control theory to economic phenomenon in the early period. Their work is mainly characterized by the study of increasingly larger deterministic models and by movements into control theory. These pioneers’ work were followed by many new area of studies by Kendrick (1981), Yang (1985), John (1998), Tang (2000), Cars (2005), Lynda (2006) etc. Robust control is applied to economical cybernetics lately [2]. Many methods and results on robust economic cybernetics have been re-ported by Tang, Cheng and Zhong [3]. Reference [4], [5] investigated the problem of consumption and investment decision with a higher interest rate. A macroeconomic model of one province is established and the problem of finding robust H-infinite control strategy is studied in [6]. When a more detailed description of the production side of an economy is desired with the development of mac-roeconomics, this leads to a so-called input-output analysis (i.e. input-output economics). Harvard Professor Leontief, who opened the door to the input-output eco-nomics, put forward the Leontief “production” (or “in-put-output”) model in 1949. Then in the region of in-put-output economics, many models were established to describe the real economics [7],[8]. Firstly, the general linear system models were well investigated. However, many models are singular linear systems, which are much

*This work is supported by National Nature Science Foundation under Grant 60674088.

*Corresponding author. Email: jafang@dhu.edu.cn (J. Fang).

harder to solve than general linear systems. Singular systems are also referred to as implicit systems, de-scriptor systems, generalized systems, generalized state-space systems, differential-algebraic systems or semi-state systems [9,10]. In the area of cybernetics there was a rapid progress in the control of singular systems because such systems can describe many real systems such as economic systems [10]. Many classic results of regular systems were extended to descriptor systems [11-13]. Recently, an algebraic method via a linear matrix inequality (LMI) has been applied to the singular systems [14-18]. On the other hand, singular systems in economics are generally converted into general linear systems by means of the selection of state vector, control vector and output vector. Little research is about the direct disposal method to singular linear economic models. In this paper we will directly deal with a kind of singular linear sys-tems in input-output economics instead of converting them into general systems. The purpose of this paper is the research of the stability problem of a kind of economic discrete-time singular dynamic input-output model, which belongs to singular linear systems. This paper will directly treat with the singular system via the linear matrix inequality approach. We will consider a sufficient condition under which an economic discrete-time singular dynamic input-output system is admissible. It is hoped that this condition can be easily tested.

2 ECONOMICS MODEL

In this paper, nR is the n-dimensional Euclidean space; m n×

R denotes the m n× real matrices space; I is the identity matrix; AT denotes the matrix transposition of A ; ∗ is used as the term that are introduced by symmetry; when matrices X , Y are symmetric, X Y> means that matrix X Y− is positive-definite.

2.1 Problem Formulation

Consider the economic dynamic input-output model described by

( ) ( ) [ ( 1) ( )] ( )x k Cx k D x k x k Y k= + + − + .

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The vector 1( ) [ (k) ( )]T nnx k x x k= ∈ R is the total output

vector and ( )ix k is the total output from sector i .

1( ) [ (k) ( )]nY k y y k= T is the final net product vector and ( )iy k denotes the final net product of sector i . The ma-

trix [ ]ijC c= is the direct consumption coefficient matrix, [ ]ijD d= is the capital coefficient matrix.

In fact, ( )Y k can be considered as the economic discrete singular dynamic input-output model’s control vector because we can affect the quantity of final net product by controlling the quantity of investment. Then ( )x k can be treated as state vector. Thus the dynamic input-output model can be turned into state space model. In economics, capital coefficient matrix D is not always invertible, be-cause the product of some sectors can not be treat as capital product and applied to invest. So the dynamic input-output model can be rewritten as

( 1) ( ) ( ) ( )Dx k I C D x k Y k+ = − + − (1)

where rankD r n= < . This system is a singular linear system. So model (1) is an economic discrete-time sin-gular dynamic input-output system.

Definition 1: The nominal discrete-time singular dy-namical input-output system:

( 1) ( )Ex k Ax k+ = (2)

1) System (2) is called to be regular if det( )sE A− is not identically zero[9, 23].

2) System (2) is causal if deg(det( ))sE A rankE− = [9, 19].

3) System (2) is stable if any root of det( ) 0sE A− = lies in the interior of the unit disk with center at the origin[9, 19,

24].

4) System (2) is called to be admissible if it is regular, causal and stable[9, 19].

Remark 1: Definition1 is widely used in the area of the control problem of singular systems[9], [19], [23].

In this paper, we try to find conditions of the stability for the economic discrete-time singular dynamic in-put-output systems. In other words, we need to prove conditions under which system (1) is admissible.

3 MAIN RESULT

In this section, we will prove a kind of condition of the stability of the economic discrete-time singular system (1) by using linear matrix inequality approach. Firstly, we study the case of which means that all products of each sector are moved to other sectors. Then the system (1) turns into:

( 1) ( ) ( )Dx k I C D x k+ = − + . (3)

Using the similar method to [19], we have the following results.

Theorem1: The discrete-time singular dynamic in-put-output system (3) is admissible if there exist matrices P with proper dimensions such that

0P

0T TP C PCβ β+ + + < (4)

whereTPD PC C PDβ = − − .

In order to get our result, we need the following lemmas:

Lemma1: Any singular matrix is congruent with the ma-

trix0

0 0I

.

Lemma2: Any nonsingular matrix is congruent with the unit matrix I .

Proof of Theorem1: We define rankd D= . For system (3), according to Lemma1, there exist two nonsingular matrices M and N such that

00 0dI

=D M N (5)

whered d

dI×

∈ R is identity matrix.

Then, premultiplying ( )I − +C D by 1−M and postmul-tiplying ( )− +I C D by 1−N , one can get a new matrix:

1 21 1

3 4

( )T TT T

− −− + =M I C D N .

It can be written as:

1 2

3 4

dT I TT T−

= −C I M N . (6)

We can define:

1 2 1

2 3

TT

P PP P

− −=P M M . (7)

Together with (4)~(6), (3) is equivalent to:

11 12

21 22

0

T T

T T T T T

T

P C PCP PC C P C PC PD C PD D P D PC

N N

β β+ + + =

− − + + − + −Ξ Ξ

= <Ξ Ξ

where

11 1 1 1 3 2 1 1 2 3 3 3 3 1

12 1 1 2 3 2 2 1 2 4 3 3 4

21 2 1 1 4 2 1 2 2 3 4 3 3

22 2 1 2 4 2 2 2 2 4 4 3 4

T T T T T

T T T T T

T T T T T

T T T T T

T PT T P T T P T T PT P

T PT T P T T P T T PT

T PT T P T T P T T PT

T PT T P T T P T T PT

Ξ = + + + −

Ξ = + + +

Ξ = + + +

Ξ = + + +

.

From this, we can get 22 0Ξ < . From formula (6), we can get 1 0P . Then we can assert

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4 2 2 2 2 4 4 3 4 0T T T TT P T T P T T PT+ + < . (8)

If this conclusion is not true, there exists a vector 0 0Χ ≠such that

0 4 2 2 2 2 4 4 3 4 0( ) 0T T T T TT P T T P T T PTΧ + + Χ .

Obviously, it is easy to show 0 1 0 0T PΧ Χ since 1 0P .So we have 0 22 0 0TΧ Ξ Χ . This is impossible because

22 0Ξ < . Thus formula (7) is true. Then utilizing the re-duction to absurdity, it can be easily proved that the ma-trix 4T is nonsingular. Suppose, 4T is singular. Then we can find a vector 1 0Χ ≠ such that 4 1 0T Χ = . So we have:

1 4 2 2 2 2 4 4 3 4 1( ) 0T T T T TT P T T P T T PTΧ + + Χ = .

This is also impossible for 4 2 2 2 2 4 4 3 4 0T T T TT P T T P T T PT+ + < .So the matrix 4T is nonsingular. Now, define:

T I C D= − + .

Then, for system (1-1) it is easy to see:

1 2

3 4

1 2

3 4

det( ) det( )

det( )det( )

d

d

sI T TsD T M N

T T

sI T TMN

T T

− −− = =

− −

− −− −

.

By Lemma2, there exist two nonsingular matrices L and R such that

'1 2 1 2

'3 4 3 2

d dsI T T sI T TT T T I− − − −

=− − − −

L R .

So, it can be easily see that '

1 2'

3 2

det( )dsI T TT I− −

− − is not

identically zero and the degree is rankD . For M , N , Land R are nonsingular, det( )sD T− is not identically zero and deg(det( ))sD T rankD− = . So this economic discrete singular input-output system (3) is regular and causal.

Next, we will prove that system (3) is stable. Since sys-tem (3) is regular and causal proved above, according to [9, 19] we can find two nonsingular matrices M and Nsuch that

00 0dI

D M N= ,2

00T

I C D M NI

− + = (9)

where ( ) ( )2

n d n dI − × −∈ is the identity matrix. Then define

11 2

2 3

TT

P PP M M

P P− −= . (10)

Substituting (8)-(9) into (3), we can acquire

1 1 2

2 3

0TT

TPT P TPN N

P T P−

< . (11)

Using the Schur complement formula, we have 1 1 0TPT P− < . From [9, 19, 25], we know that every root

of det( ) 0sB T− = lies in the interior of the unit disk with center at the origin. Therefore, we know system (3) is stable. It follows immediately that system (3) is admis-sible. This is the end of proof.

Remark2: Theorem 1 offers a sufficient condition for the discrete-time singular dynamic input-output system to be admissible. Based on this, we can assert whether an economic discrete input-output model is admissible.

4 CONCLUSION

In this paper, the stability of the economic discrete sin-gular dynamic input-output model has been studied. This economic singular linear model is investigated without transformed into the general linear system. A sufficient condition has been proved. The analysis of the problem was solved via linear matrix inequality approach.

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