Development with positive externalities: The case of the Russian economy

NORTH- HOLLAND

Development With Positive Externalities: The Case of the Russian Economy Vladimir Matveenko, Russian Academy of Sciences and University of Pittsburgh We consider a model of development of n economic agents with numerous mutual positive externalities. Such a model can serve to describe the economic development of Russia, where the relative underdevelopment of markets led to an important role played by externalities. Extremal algebra is used to study dynamics of the model. The concepts of binding, nonbinding, and actual externalities are introduced. Illegal payments of the agents are explained as policies directed to enlarge binding positive externalities created by other agents.

1. INTRODUCTION The relative underdevelopment of goods and financial markets in the former Soviet Union and the imperfection of the price mechanism have led to an important role played by externalities, in a broad sense of the term. 1 In fact, activities of the majority of economic agents in the country are essentially connected with numerous externalities that form a complex system and became factors and limitations of development. A positive externality created by an agent often consists in diminishing transaction costs for some "We start from a classic definition of externality, namely, any indirect effect that either a production or a consumption activity has on a utility function, a consumption set, or a production set. By "indirect" we mean both that the effect is created by an economic agent other than the one who is affected and that the effect is not transmitted through prices." (Laffont, 1988, p. 6)

Address correspondence to Vladimir Matveenko, St. Petersburg Institute for Economics and Mathematics, Russian Academy of Sciences, 38 Serpukhovskaya Str., 198147St. Petersburg, Russia. I am grateful for useful comments to Irma Adelman, Daniel Berkowitz, Egon Neuberger, and the participants of the 8th S.S.R.C. Summer Workshop on Soviet and East European Economics (Pittsburgh, July 1992). Research was supported in part by a grant from ACTR/ ACCELS, with funds provided by the United States Information Agency. None of these organizations is responsible for the views expressed. Received June 1994; final draft accepted December 1994.

Journal of Policy Modeling 17(3):207-221 0995) © Society for Policy Modeling, 1995

0161-8938/95/$9.50 SSDI 0161-8938(94)00028-E

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other agents, which can be accomplished, for example, by giving them some information inaccessible for others, opening an access to some scarce goods and services, supplying production factors or money credits in favorable terms, and so forth. These external activities, which are not reflected in the price mechanisms, are in a close connection with phenomena that are specific for the Soviet economy, such as blat, sviazi, and nomenklatura, which are still existing after the disintegration of the Soviet Union. An economic agent in Russia often can be more interested in finding a worthy place in the network of externalities than in any otller sort of economizing. The aim of this work is to construct a model of economic development in a non-market environment with externalities. The economy is considered as a complex system with mutual positive externalities. Even in well-developed market economies, such sorts of externalities usually exist between members of one family, neighborhood, or some other bounded community, between people of creative professions (creative influence), or scientists (exchange of ideas). Lucas (1988) called them "externalities of creative professions." The same kind of externalities can be observed between different elements of a city's infrastructure (e.g., housing, public transportation, lighting, emergency services, etc.), and sometimes between departments of a large firm. Our hypothesis is that the mutual positive externalities are much more widespread and are stronger in planned and post-planned economies, where they serve as substitutes and alternatives for usual market relations. Different externalities in cities have been studied, in particular, by Jacobs (1969), Kanemoto (1980), and Abdel-Rahman and Fujita (1990). Externalities play an especial role in development of cities. Robert Lucas comments that "a city, economically, is like the nucleus of an atom: If we postulate only the usual list of economic forces, cities should fly apart. The theory of production contains nothing to hold a city t o g e t h e r . . . " (Lucas, 1988, p. 38). Of course, a city is only one example of a system with mutual creative externalities. "Much Of economic life is creative," Lucas remarks. The continued functioning of the post-Soviet economy even at a lower level was a p h e n o m e n o n no less astonishing than the functioning of a city as a stable (not collapsing) system. Mutual externalities is a natural way to explain such a stability. In our model we consider a system of agents with mutual positive externalities. The development of the system is limited by potential possibilities of development of separate agents and by externalities

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created by different agents as well. To analyze this system, we use techniques of extremal algebra, a modern mathematical discipline that was developed in the context of mathematical economics. (See, for example, Cuningham-Green, 1979, and references in Appendix.) No preliminary acquaintance with these methods is required from the reader. The model is described in Section 2. In Section 3 we contribute to the theory of externalities by introducing the concepts of binding, nonbinding, and actual externalities. The examples o f different patterns of behavior of the system with mutual positive externalities are given in Section 4; we underline the instability of these patterns. The problems of payments directed to improvements of development are discussed in Sections 5 and 6. An attempt to consider contemporary Russian economic development from the point of view of the model is made in Section 7. Appendix contains some results of mathematical studies of the model.

2. A M O D E L OF ECONOMIC D E V E L O P M E N T WITH M U T U A L POSITIVE EXTERNALITIES To simplify understanding the model, the reader can keep in mind a known example of mutual externalities created by bees and flowers (see Meade, 1952; Cheung, 1973; Johnson, 1973; Coase, 1988; Laffont, 1989). A beekeeper (Agent 1) has a positive influence on the production possibilities of an orchard (Agent 2), and, vice versa, the orchard planting flowers creates a food for bees. If each of the agents is interested in its own development, he would, generally speaking, be interested in the development of another agent. Let i = 1 , 2 , . . . , n be n economic agents (firms, regions, authorities). Each of them is characterized at the period t by a single positive number that we shall call a value of the t~h agent. It can be, for example, pro fit, income, welfare, present value of the firm, or some aggregate variable, such as a composite of physical capital and knowledge (see Romer, 1991), or some not directly measurable, latent variable (see Aigner and Goldberger, 1977). Development of the ith agent is being described as a change in his value. We consider discrete time periods t = 0,1, . . . and suppose that the value of an agent depends on his own value and the values of other agents at the previous period: x~ +~ = ~ ( x l , x ~

....

x;),i

= 1.....

n

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The function A will have a special form. We make two assumptions: 1. The development of the l~hagent is limited by his own potential possibilities, described by a fixed coefficient aii> 0: Xi t+t <. aiix], t = O, 1 . . . .

; i = 1 .....

n;

and 2. The development of the ith agent is limited by positive externalities created by other agents: x / + 1 < a o x / ' t = O, 1 . . . .

where au>O;j

= 1.....

i-

1, i +

1.....

n; i=

1.....

n

are fixed coefficients, describing limitations for the tah agent's development caused by restrictiveness of the positive externalities. If an agent j creates no externality for an agent i, we pose au = + oo. So, x I +j = m i n a o x / , t = 0,1 ....

j = I ..... ; i = 1.....

n;

(1)

n.

As an example, consider the case of two agents. The production set of the first agent on the (x~', x( +~)-plane is the intersection of the following regions: >~ O, x'~ + ~ >1 O, X'l + 1 < a~,x~, x f + ' < a ~ .

A small value x~ of the second agent leads to a small externality alzx; and, as a result, to a contracted production set of the first agent. If the value of the second agent x~ increases, the production set expands. These changes in the production set are quite similar to the usual behavior of a production set in presence of an externality (see Laffont, 1989). Now we shall rewrite our model in terms of extremal algebra. The concepts of extremal algebra are, in fact, analogues of the corresponding concepts of usual matrix algebra but with substitution the operation min (or max) for + . For example, an inner product of two vectors a = (al . . . . . a,) and b = (bl . . . . . b,) is defined in usual matrix algebra as

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a b = ~ aibi, i=l

but in extremal algebra as a o b = min[aibi/, i = 1 . . . . .

n

Some other concepts are an extremal product Aob o f a matrix A and a vector b, an extremal product AoB o f two matrices A and B, an extremal t th power A ot o f a matrix A, an extremal eigenvalue, a right and a left extremal eigenvector, and so forth. A n exposition o f these concepts is given in Appendix. The vector xt = (x~, x~. . . . , x t) is said to be a state vector at the period t. Equations (1) are equivalent to xt+ 1 = A o x t,

(2)

where A = (ao) is the n n-matrix o f the coefficients ao. Hence, x t = A°toX °,

(3)

where x ° is an initial state vector.

3. BINDING A N D NONBINDING EXTERNALITIES Harvey Rosen notes, " M a n y people who have never heard the term 'positive externality' nevertheless have a good intuitive grasp o f the concept and its policy implications. T h e y understand that if they can convince the government their activities create benefit spillovers, they m a y be able to dip into the treasury for a subsidy" (Rosen, 1988, p. 143). This observation can serve as some confirmation o f a hidden and complex character o f m a n y externalities. First, it can be not easy to find that some agent at all has some external influence to other agents in the economy. Secondly, even if an externality is found, it can appear to be u n i m p o r t a n t or insignificant for other economic agents. Generally, an economic agent perceives an externality o f some other agents only if his worth and the size o f the externality are in some sense comparable. Thirdly, in an economic system with complex mutual interrelations, externalities can act indirectly through a chain o f agents. Even if an agent realizes a presence o f an externality, it can be difficult for him to find an origin o f the externality, because externalities can be created not only by separate agents but by groups o f agents as well. (We shall see it further.)

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Let us go back to the case of two agents. Depending on the values of the agents, the externality can matter or not for the first agent. Suppose that the value of the first agent is equal to xf = x'. Two cases are possible. If a12x~ ~ alex', then the value xf +1 at the next period is defined by the restriction of the externality. In this case we call the externality binding. If a~2x~> aHx', then the next period value x[ +~ does not depend on the externality but is defined only by the value and the possibilities of the first agent himself. In this case we call the externality nonbinding. The difference between binding and nonbinding externalities has important consequences for the first agent's policy. If the externality is binding, his development is limited by the size of the externality but not by his own possibilities to develop, and he will be interested in enlarging the externality or in development of another agent who creates this externality. It is possible that the first agent will be ready to make some payment (maybe even in an illegal form), directed to increasing either the coefficient of the externality a12 or the second agent's value x~. If the externality is nonbinding, the first agent probably will be primarily interested in his own development but neither in enlarging the externality nor in the development of the second agent. (Of course, in dynamics, the agent can begin to enlarge a potentially binding externality in advance, before it becomes binding.) The relative easiness of receiving a subsidy in Rosen's example is based on the incapacity of the government to distinguish between binding and nonbinding externalities. Another interesting analogy is connected with the question why the American aid to Russia in the 1990s is not so considerable as the American aid to Germany at the end of the 1940s. A possible answer is that although in both cases the externality was considered as binding, for Germany the policy of enlarging the second agent's value x~ was chosen, and for Russia, the policy of enlarging the coefficient a12. In the model with n agents, the externality of the agent j is binding for the agent i if aux~ < aj~v~. The externality is nonbinding if ai~v~< a;~x~. The nonbindingness of the externality can be interpreted as "too low" value x~ of the fh agent. The positive externality presents, but it is sufficiently high, that is, it does not limit the development of a "small" agent. It becomes visible when this agent "grows up" and his further development is limited by the externality. Several externalities can simultaneously be binding for the ith agent. For example, if n = 4, and for some period t alzxt < allx[, a13x~< azlx[, al4x[ > allx~,

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then the externalities, created by agents 2 and 3, are binding for agent 1, but the externality, created by agent 4, is nonbinding. It can be difficult for an agent to find which of the binding externalities actually defines his production set. Such a binding externality can be called actual. If our example with four agents, if al2x~ < a l 3 x J ,

then X~ +j = al2x~,

and the externality, created by agent 2, is actual. The externality, created by agent 3, is binding but not actual. Generally, the externality of the agent j for the agent i is actual if aux~

= min aikx~,

k = 1, 2 . . . . .

n,

or, in terms of extremal algebra, aijx~ = ai o x t ,

where al is the fh row of the matrix A = (au), x ~is the state vector of the system at the period t. 4. PATHS OF MUTUAL DEVELOPMENT The dynamics of the system is described by any of Equations 1, 2, or 3. The following examples show some possible patterns of the behavior of the paths.

Example 1. A = (125 11),Initial statex °= (116) Time periods

0

1

2

3

State vectors

(11.6)

(:156)

(111~)

(111~)

No No

Yes No

Yes No

Presence of binding externalities: for agent 1 for agent 2

The path stabilizes at the point (1.6, 1.6)'.

Yes No

V. Matveenko

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Example 2.

A:(: Time periods

0

2

sttevecto,s

(:)

1

Presence of binding externalities: for agent 1 for agent 2

Yes Yes

Yes No

Yes No

3

Yes No

The path enters a ray and then grows stationary with a growth-rate factor equal to 1.1.

Example 3.

Time periods

Presence of binding externalities: for agent 1 for agent 2

0

1

2

Yes Yes

Yes Yes

Yes Yes

3

Yes Yes

The path enters a limit cycle.

These examples show both the variability of patterns and their instability. A little change in coefficient matrices in Examples 2 and 3 is enough for a sharp change both in character of the paths and in bindingness of externalities. The latter leads to changes in the policies of the agents. It can be shown (see Appendix) that a path x t of the dynamic system (2) will always reach either a limit cycle or a limit ray (or a steady state, in particular) as it was in these examples. After that the rate of mutual development o f the system will be defined by

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the extremal eigenvalue B of the matrix A. The type o f the behavior (a limit ray or a limit cycle) depends on the structure of the matrix A, which can be described in terms of the graph theory. Some details of this analysis can be found in Appendix. A limit case of the model is the case of competition, when the agents do not "help" (and do not prevent) the development of each other. In this case external limitations are absent, so aij = + o o for all i = j , and the development of each agent is defined solely by his own individual coefficient ai~. So, actual rates of development for different economic agents are, generally, different. It is important that the development of any agent i can be defined, in fact, by indirect externalities of agents or groups of agents that can be connected with agent i not directly but through a chain of other agents. For example, in the case of three agents, agent 3 can create a binding externality for agent 2, and the latter for agent 1. Agent 1 can feel a single externality, created by agent 2, but in fact his development can be limited by agent 3. Another possible situation is a cycle o f binding externalities created by agent 1 for agent 2, by agent 2 for agent 3, and by agent 3 for agent 1. These externalities are not related to any single agent.

5. D I R E C T I O N OF P A Y M E N T S As we have seen, an agent facing a binding externality can be interested in some payments allowing to improve his development. In the Russian economy such payments have diverse forms: income redistribution through the mechanism of taxes and transfers, direct supply of scarce resources and products (e.g., in a form of barter or credit), providing with information, and, simply, bribes. Not excusing illegal forms of payments, we should recognize that if they are correctly oriented in the system with a dense network of externalities, they can lead to improvements in the mutual development. However, as the model shows, the choice of a proper direction of payments is a serious problem for an agent that can have important political consequences. For detailed study of this problem, a model with changeable coefficients could be studied. In a simple case we can assume that an agent i is able to decrease his own coefficient of development aii and, instead, to increase a coefficient of development ajj for any other agent, or a coefficient of external limitation akt for any pair of agents k,l. In the simplest case we can suppose that the coefficients are being exchanged on the "one-to-one" base.

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To find an effective direction of such payments, an agent has to find a source o f an actual externality. It is shown in Appendix that it corresponds to, so called, leading contour o f the matrix A.

Example 4. In the matrix A = (14 2l) there are three contours: the loop of the first agent (its length is equal to 4), the loop of the second agent (the length is equal to 1), and the contour f o r m e d by two mutual externalities (its average length is equal t o (a12a21) 1/2 = (2.1) 1/2 = x/2)" The leading contour (which possesses the m i n i m u m average length) is the loop of the second agent. The actual m u t u a l growth factor is equal to 1, the average length of the leading contour. (It is equal to the extremal eigenvalue ~t of the matrix.) Some payments, m a d e by the first agent, would diminish his own development coefficient 4 but, if they are directed to increase the coefficient a22, they would rise the actual rate of mutual development.

Example 5. For the matrix A = (24 22) the average lengths of the three contours are equal to 4, 2, and x/2. The leading c o n t o u r is formed by externalities al2, a21 o f both the agents. A n y changes in the o w n development factors a11, a22 are not able to improve the mutual development, but it can be achieved by increasing the effect of any (or both) o f externalities.

Example 6. Let us consider a path of m u t u a l development for two agents with the matrix A = (21) 2 2 and the initial state x ° = ( 1 , 1 ) '. At the initial state no one agent faces binding externalities. We can say that the values of the agents are small. But further x t = (2,1)', t >i 1, the first agent has grown, and the second agent creates now a significant externality for him. The first agent is interested in some payments. The leading contour is the loop of the second agent. If the first agent can "share" his development coefficient, so that a l l + a22 = 3, then (in this example) his optimal policy is to equalize these coefficients. As a result, the matrix A turns into (12s 125), and the path, starting f r o m x r = (2,1)' increases: x r+l =

(2,1.5)',

x r+2 =

(3,2.25)',

x r+3 =

(4.5,3.38)', . . .

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6. TAXES AND TRANSFERS More realistic is a situation when the agents can exchange not the coefficients of development but the values. Such a case takes place when the value of an agent is his income. Suppose that the government has total control over the values (using taxes and transfers) and is interested in increasing the summary value. Again assuming that the payments are costless, we come to the following problem of the optimal income redistribution: for a state x t find a redistributed state yt to m a x i m i z e ~, ( a i o y ' ) , i = 1 . . . . .

n

i=l n

subject to~yt= i=1

~ x t, i = 1 . . . . .

n.

i=1

Here ai is the ith row of the matrix A. This problem is an extremalalgebraic analogue for the linear programming problem. Example 7. Let us return to the previous example. For the matrix (~ ~) the problem takes the form: M a x [(2,2) o (YJ,y2) + (2,1) o (Yl,Y2)] s u b j e c t t o y , + Y2 = c o n s t .

Independently of the value of the constant, the optimal solution is achieved for y~ = Y2. This solution guarantees development of the system with the growth factor equal to 1.5. To ensure the realization of this solution, the government can impose a tax on the first agent and transfer it to the second. The path x t in Example 6 will be continued in the following way, starting from the state x r = (2,1)': x r~

(1.5,1.5)'

x r+ ~ = A o (1.5,1.5)' = (3,1.5)' ~ (2.25,2.25)', x r+2 = A o (2.25,2.25)' = ( 4 . 5 , 2 . 2 5 ) ' ~ ( 3 . 3 8 , 3 . 3 8 ) ', x r+3 = A o (3.38,3.38)' = ( 6 . 7 5 , 3 . 3 8 ) ' ~ ( 5 . 0 7 , 5 . 0 7 ) ' , ....................................................................

The arrows show the results of the government intervention. The value of the optimal tax imposed on the first agent is here 25 percent. The first agent would be agree to this payment so far as it let him move up from the stationary state x r -- (2,1)'.

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In the general case with n agents, the model can be used for choosing the agents being subject to taxes, the values of the taxes, and the distribution of the transfers.

7. THE CASE OF THE RUSSIAN ECONOMY

Following the model, we can see how the transition process influenced the development of the economic system of the former Soviet Union. On the level of the national economy, where the agents are republics, regions, and industries, the last years of the existence of the Soviet Union were characterized by sharp decreases in the values of some agents (in particular, as a result of strikes and separate events, such as the Chernobyl catastrophe, the Armenian earthquake, and the Karabakh conflict). According to the model, these decreases immediately spread throughout the economy, which led to a decrease in the total level of development. Simultaneously, a strengthening of external limitations took place as a result of numerous conflicts between the agents. The result was a change in patterns and a decline in rates of development. Naturally, many agents tried to leave the c o m m o n economic system. More complex processes took place on the level of individual firms. Part of the external bounds remained, and some new ones appeared, but the values of external limitations considerably changed, and many traditional directions of payments became ineffective. As was said, in a non-market environment with a network of externalities, an agent is often interested in some payments, allowing an improvement of his development pattern. Such sort of payments is an objective necessity for such an economy. However, the relaxation of the central redistribution and the absence of other legal bases for payments led in Russia to a situation where such payments are insufficient and often have some illegal forms. The governments of Russia and other former Soviet republics still were not successful in creating effective public finance systems that could substitute for the traditional system of horizontal payments. REFERENCES Abdel-Rahman, H., and Fujita, M. (1990) Product Variety, Marshallian Externalities, and City Sizes, Journal of Regional Science 30: 165-183.

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Aigner, D.J., and Goldberger, A.S., Eds. (1977) Latent Variables in Socio-Economic Models. Amsterdam: North-Holland. Cheung, S.N.S. (1973) The Fable of the Bees: An Economic Investigation, The Journal o f Law and Economics 16: 11-33. Coase, R.H. (1988) The Firm, the Market, and the Law. Chicago: University of Chicago Press. Cuninghame-Green, R.A. (1979) Minimax Algebra, Lecture Notes in Economics and Mathematical Systems 166 Berlin: Springer Verlag. Jacobs, J. (1969) The Economy o f Cities. New York: Random House. Johnson, D.B. (1973) Meade, Bees and Externalities, The Journal of L a w and Economics 16: 35-52. Kanemoto, Y. (1980) Theories o f Urban Externalities. Amsterdam: New-Holland. Laffont, J.-J. (1989) Fundamentals of Public Economics. Cambridge, MA: The MIT Press. Lucas, R.E, Jr. (1988) On the Mechanics of Economic Development, Journal o f Monetary Economics 22: 3-42. Matveenko, V.D. (1990) Optimal Paths of Dynamic Programming Procedure and Extremal Degrees of Non-negative Matrices, Diskretnaya Matematika 2:59-71 (in Russian). Meade, J.E. (1952) External Economies and Diseconomies in a Competitive Situation, The Economic Journal 62: 54-67. Romer, P. (1991) Increasing Returns and New Developments in the Theory of Growth. In Equilibrium Theory and Applications (W.A. Barnett et al., Eds). Cambridge: Cambridge University Press, 83-110. Rosen, H. (1989)Public Finance. Homewood: Irwin. Takayama, A. (1985) Mathematical Economics. New York: Cambridge University Press. Vorob'ev, N.N. (1963) Extremal Algebra of Positive Matrices, Doklady Akademii Nauk SSSR 152:24-27 (in Russian). Vorob'ev, N.N. (1967) Extremal Algebra of Positive Matrices, Electronische Informationsverazbeitung and Kybernetik 3:39-71 (in Russian). Zimmermann, K. (1984) On Max-Separable Optimization Problems, Annals o f Discrete Mathematics 19: 357-362.

APPENDIX Extremal Algebra Operators In this Appendix we quote some basic results of extremal algebra that have been used in our analysis. The concepts of extremal algebra were developed by numerous authors, among them Vorob'ev (1963, 1967), Cuninghame-Green (1979), Zimmermann (1984), and Matveenko (1990). Let a, b be nonnegative n-vectors. Their extremal product (in the sense of minimum) is defined as a number a o b = min (ajbi),

i =

1 .....

n

An extremal product of nonnegative matrices A,B (where the number of columns in A is equal to the number of rows in B) is a matrix D = A oB, of which the element du is the extremal product

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o f the ith row of the matrix A and the jth column o f the matrix B. The extremal k th power o f a square matrix A is the extremal product o f k copies o f matrix A: A °k= A o A o . . . o A .

A positive number ~t is called an extremal eigenvalue o f a square matrix A, and a vector v is called its right eigenvector if A°v

= ~t v.

The extremal eigenvalue for a nonnegative matrix is always single. Our task is to study the behavior o f the paths (2), generated by an extremal matrix operator A o. A n operator A o is said to be stable if for every nonnegative vector x there exists such a vector y ( x ) and such a n u m b e r T that i.t-t ( A °t o X) = y (x) for allt >I T.

A n operator Ao is said to be global stable if it is stable, and the right eigenvector v is single. (It follows that y ( x ) is proportional to v.) We will show, how the extremal eigenvalue can be calculated, and how the structure o f the matrix A defines the character of behavior o f the sequence (2). First, we shall remind some concepts o f usual linear algebra (see T a k a y a m a , 1985). A n n x n-matrix A = (ao) with positive (maybe, infinite) elements is called irreducible if there is no such subset of indices s C [1 . . . . . n] that a,j = + co for all i ~ S, j e S. To a square matrix A an oriented graph < M , N > corresponds with a set of nodes M = I1, • • .,nl and a set of arcs N = [(i,j): aij < + col. The matrix A is irreducible if and only if the graph < M , N > is strongly complete, that is, it is possible to pass f r o m each node i to each node j. A matrix A is called imprimitive with a period h if the set [1 . . . . . . n] can be separated into h disjoint subsets $1 . . . . , Sh in such a way that if a0 =~ + co, i ~ Sk, t h e n j ~ Sk+l, where Sh+~ =S~. Otherwise, the matrix A is called primitive. For each contour il,i2 . . . . . ie,il in the graph its "average length" 1 can be calculated: I = (aili2ai2i3...

aieil )l/e.

A contour (maybe not single) with the m i n i m u m average length will be called a leading contour, its average length Ix is equal to the extremal eigenvalue of the matrix A.

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m

Now we can introduce a new matrix A, corresponding to all (if t h e y a r e several) leading contours. It can be called a contour matrix. Let N be the set o f all arcs (pairs o f indices) which enter some leading c__ontours, and M t__hecorresponding set o f the indices. The matrix A is defined as [ M [ x [M[-matrix with elements aij

= I a~i, if(i,J) e ~r, L+ oo, if (i,j.~ ~ ~ .

A contour matrix A exists for every positive matrix A . A matrix A is called extremal irreducible if its contour matrix A is irreducible. A matrix A is called extremal primitive if its contour matrix A is primitive. The following statements are reformulations o f theorems proved in Matveenko (1990). T H E O R E M 1: Let a nonnegative matrix A be irreducible and extremal primitive. Then the operator Ao is stable. I f a matrix A is irreducible, extremal irreducible, and extremal primitive, then the operator Ao is global stable. T H E O R E M 2: Let a nonnegative matrix A be irreducible. Then there exists such a natural n u m b e r s that for every nonnegative vector x there are s vectors v~,v2 . . . . . v2 and a n u m b e r T, such that ~t- (t+k)A*(t+k)*x = Vk, k :

1, 2 , . . . ,

s

for all t >i T. Moreover, vk ~ V, k = 1,2 . . . . ,s, where Vis the set o f all extremal eigenvectors o f the matrix A os. (This set V coincides with the set o f all extremal eigenvectors for all powers A °t, t = 1,2 . . . . ).