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JOURNAL OF ENVIRONMENTAL ECONOMICS AND MAJiAGEhfENT 13 199-211 (1986) Conservation of Mass and Instability in a Dynamic Economy-Environment System’ CHARLES PERRINGS Department of Economics, University of Auckland Private Bag, Auckland 1, New Zealand Received August 10,1984; revised June lo,1985 This paper considers a variant of the Neumann-Leontief-Sraffa general equilibrium models in the context of a jointly determined economy environment system subject to a conversation of mass condition. It shows that the conservation of mass contradicts the free disposal, free gifts, and non-innovation assumptions of such models; that an expanding economy will be associated with continuous disequilibrating change in the material transformations of both economy and environment; and that this change is uncontrollable through the price system. 0 1986 Academic Press, Inc. 1. INTRODUCTION The classical models of general equilibrium-those resting on the foundations laid by Neumann [24], Leontief [18], and Sraffa [29]-are characterized by two extraordinarily powerful assumptions: that technology is fixed, and that the economy functions independently of its environment. The assumption of fixed technology implies a non-innovative system. The assumption that the economy functions independently of its environment implies that resources can be costlessly exacted from nature, and that residuals generated in the economy can be costlessly disposed of in nature. All the classical models of general equilibrium assume, either explicitly or implicitly, non-innovation, free gifts, and free disposals. For some time environmental economists have questioned the usefulness of these highly restrictive assumptions in modelling the links between the economy and the environment. The pioneering work of Boulding [6], Daly [7], and Ayres and Kneese [4] brought to economists the insight that the global system is a closed resource system. The latter were also instrumental in establishing the importance of the conservation of mass condition for the modelling of that system. However, their own analysis was couched in terms of general economic equilibrium, and like many subsequent models missed a crucial implication of the conservation of mass condition for the time behavior of the system.2 The conservation of mass condition in fact contradicts all three basic assumptions of the classical general equilibrium models. The expansion of the economy, or any other subsystem of the global system, implies continuous change in the material ‘I am grateful to Geoffrey Braae, Alan Rogers, Martin O’Connor, and two anonymous referees for comments on an earlier draft of this paper. 2Both d’Arge [9] and Ayres IS], to take only the most eminent examples, construct models on the assumption of the conservation of mass, but ignore the conceptual problems posed by the assumption for the notion of general economic equilibrium. An additional problem with much of the later work building on these foundations is that it has continued to include, explicitly or implicitly, a number of free disposal assumptions. See, for example, Victor [30] and Lipnowski [20]. 199 0095~06%/86 S3.00 Copyright 0 1986 by Academic Press, Inc. All rights of reproduction in any form reserved 200 CHARLES PERRINGS transformations of both economy and environment. This is sufficient to preclude its convergence to an expansion path at which the structure of production and prices of economic goods is stable over time. Moreover, since market prices in an interdependent economy-environment system are inadequate observers of the effects of economic activity on the relative scarcity of environmental resources, this change will be unanticipated. The management or control of economic processes in response to price signals will be insufficient to determine the structure of economic output, since environmental feedbacks will be present even where the economy is technically controllable and observable via the price system. This paper develops a variant of the classical general equilibrium models of Sraffa and Neumann that locates the economy in a materially closed global system, and investigates the implications of the conservation of mass condition on the time behavior of the system. Section 2 describes the axiomatic structure of the model. Section 3 discusses the dynamic implications of the conservation of mass condition. Section 4 considers the place of the economy in the global system and indicates the limits of system controllability through the uses of price signals. Section 5 offers some conclusions. 2. ELEMENTS OF THE SYSTEM The following general assumptions underpin the model described below: (i) It is possible to identify discrete physical activities or processes that collectively describe the material transformations of the global system. The global system is materially (and thermodynamically) closed.3 No matter passes into or out of the system. The economy represents a subset of the processes of the global system, and is assumed to be materially open with respect to the environment. Matter passes between the processes of the economy and the processes of the environment. From this it follows that we cannot meaningfully represent the economy as a closed system in the manner of the Neumann or Sraffa models unless we believe that all processes in the global system are “owned” and “controlled” by economic agents. If this is not the case then the complement of the processes of the economy will be the processes of the environment, and the time behavior of the each depends on the links between them.4 (ii) The physical relationship between the economy and the environment is assumed to reflect, on the one hand, the heterotrophic nature of economic agents (as 3There are three types of systems ordinarily recognised in thermodynamics: open systems, closed systems, and isolated systems. An open system is defined to be one which interacts freely with its environment. It exchanges both matter and energy with its environment. A closed system is defined to be one which is materially self-contained but interacts energetically with its environment. There are no transfers of matter between the system and its environment, but there are transfers of energy. An isolated system is defined to be one in which there are no transfers of either matter or energy between the system and its environment. For all practical purposes the terrestial system is a closed system. It exchanges energy in a variety of forms (gravity, radiant heat, etc.) with its environment, but the odd meteorite and space probe not withstanding, it is materially self-contained. “Production” within the system implies the transformation of a set of resources that, to all intents and purposes, are in fixed supply. 4’Ihe environment is accordingly defined in terms of the referent set of processes. The definition is, however, entirely symmetrical. If the universal set of processes in the global system is denoted U, and if the set of processes of the economy, the referent set, is denoted V, then V’, the complement of V in U is the environment. The terms “environment” and “the complement of the referent set” are synonymous. CONSERVATION OF MASS 201 organisms that obtain their nutritional needs by feeding on other organisms), and on the other, the status of the environment as a receptacle for the waste products generated in the economy. In other words, the economy makes both exactions on and insertions into the environment. Of these only the first has traditionally been regarded as a feature of economic activity.5 In addition to these general assumptions, the model rests on the following specific assumptions, each of which is made in one or the other of the classical general equilibrium models: (iii) At any given state of nature there are fixed coefficients of production, implying constant returns to scale.6 (iv) It is possible to identify the same number of linearly independent processes as there are products.’ (v) Consistent with the existence of high and low entropy states of matter, not all resources depreciate/degenerate at the same rate. This makes a basic assumption out of a special case in the Sraffa and Neumann models.’ Assumptions peculiar to the processes of the economy are discussed below. At this point we may formalize assumptions (i) to (v). The technology of the material transformations in an economy-environment system applied in the kth period of its history are described by the pair of non-negative matrices A( k)B(k), related by the equation B(k) = A(k) + G(k). 0) From assumption (iv) all three matrices are n-square. a;(k), the ith row of A(k), is the vector of gross input coefficients for the n products of the system in the i th process in the k th period. laj( k), the jth column of A(k), is the vector of gross input coefficients for the jth product in the n processes of the system in the k th period. bj( k), the i th row of B(k), is the vector of net output coefficients for the n products of the system in the i th process in the k th period. Jbj(k), the jth column of B(k), is the vector of net output coefficients for the jth product in the n processes of the system. c(k) describes the physical change in the mass of the inputs of the system during the kth period of production. gij( k) is unrestricted as to sign. g,Jk) > , = , < 0 ‘The concepts of exaction and insertion each imply actions that are not agreed by all the parties concerned. That is, they imply the impositions of one agent or species on another-a relationship of domination and subordination between agents or species. To the extent that human economies depend on exactions on the environment, they depend on the subordination of the environment. Marx, for an early example, defined the labour process to be one in which man “opposes himself to Nature in order to appropriate Nature’s productions in a form adapted to his own wants” [22, p. 1731. 6The state of nature describes the technologies of both the economy and the environment obtaining at the commencement of the reference period. ‘1 make the very strong assumption that the number of products remains constant over time. The system is n-dimensional in all periods. This implies that only the input and output mix of different processes changes. In reality, the number of distinct products produced by the system will change over time. If we define the dimensions of the system to be time-variant, however, the results of this paper are only strengthened. *Simple production-the case where all but one of the means of production advanced in each process is completely “ used up” in each period of production-is the special case. 202 CHARLES PFRRINGS implies that the jth input is augmented, unchanged, or diminished in the i th process in the k th period. The period of production, indexed k, is of uniform duration. Since, from assumption (iii), there are constant returns to scale, it is entirely arbitrary. From assumption (iv) the rank of A(k), B(k) is n. From assumption (v) the system is one of joint production, implying that B(k) is not diagonal except as a special case. The elements of A(k), B(k) are coefficients on the resources (products of past periods) available to the system at the commencement of the reference period. To see their construction, let us first define an n-dimensional, time indexed, row vector q(k), in which q,(k) denotes the quantity (mass) of the ith product available to the system at the commencement of the k th period. Let us further define a non-negative n-square gross input matrix X(k), in which xii(k) denotes the quantity of the jth resource employed in the ith process in the k th period. We then have aji( k) = qi(k)-‘xij(k). aij(k) denotes the gross input of the jth resource in the ith process per unit of the ith resource available to the system at the commencement of the kth period. bij(k) is similarly obtained and denotes the net output of the jth resource from the ith process per unit of the ith resource available to the system in the kth period. The tune path for the physical system is thus given by the first order difference equation q(k) = q(k - l)B(k - 1). (2) Hence the outputs of the k - lth period comprise the stock of resources available to the system at the commencement of the kth period. Notice that if there is no technical change, B(k) = B(0) for all k > 0, and the physical system has the very simple general solution q(k) = q@)B(0)k. 3. THE CONSERVATION OF MASS Consider now the dynamic effects of the conservation of mass condition. Notice, first, that the condition implies that for all k 2 0, (9 q(k)e = qGWk)e; (ii) qi(k)ai(k)e = qi(k)bi(k)e for all i E {1,2,. . . , n}; (4) W q(k) = q(k)A(k); where e is the unit or summing vector. (4i) means that a closed physical system has a zero growth rate. Although any subsystem within a closed physical system may be able to expand, i.e., qi(k) < q(k)h(k) for some i and some k, it will not be able to expand without limit. Sooner or later it will be bound by the conservation of mass condition. This is, of course, what is implied by the so-called “doomsday model.“’ (4ii) means that the mass of inputs in every process will be exactly equal to the mass ‘Meadows et al. [23] and Forrester [12]. CONSERVATION 203 OF MASS of outputs. This is the precise meaning of the Neumann dictum that nothing can be produced out of nothing.1o (4iii) means that the gross input matrix, A(k), will fully account for all resources in the system in the kth period. This follows from the fact that in a closed system there is no free disposal of resources. Waste material cannot be ejected from the system. Every residual must go somewhere. It follows that the dominant effect of the conservation of mass is that the system will be time variant. To see this notice that the quantity of resources available to the system at the beginning of the kth period is given by q(k), but the quantity of resources which are required by the system in terms of the technology inherited from the previous period is given by q(k)A(k - 1). If there is full employment of all resources under this technology, that is if q(k) = q(k)A(k - l), then the inherited technology will obviously satisfy the conservation of mass condition. But if there is less than full employment of ail resources or if there is unfulfilled excess demand for any resource, that is if q(k) f q(k)A(k - l), implying that q(k) # q(k - l), the inherited technology will not satisfy the conservation of mass condition. It follows from this that a jointly determined economy-environment system satisfying the conservation of mass condition may be technologically stationary only if there is full employment of all resources in all periods. Since, by the conservation of mass condition, q(k) = q(k)A(k) for all k 2 0, if q(k) # q( k)A( k - l), it follows that A(k) # A(k - 1). Hence, A(k) = A(k - 1) only if the vector of residuals q(k)[I - A( k - l)] is equal to zero. Moreover, from (4ii) a change in ai( k) implies a change in bi( k). The first order difference equation which defines the time path of the physical system (2) may accordingly be written in the form q(k) = q(k - l)[B(k - 2) + B&k - 2)] (5) where B(k - 2) represents the technology inherited from the previous period, and B,(k - 2) represents the changes brought about in the elements of B(k - 2). Wherever q(0) # q(O)B(O), the conservation of mass condition implies that the general solution of the physical system will be defined by the expression k-l q(k) = q(O) (6) ,FoBB(i)- A corollary of considerable importance is that the equilibrium associated with any given technology will be stable if and only if there is free disposal of residuals. Physical equilibrium is defined to be the state in which qi(k)/qi(k - 1) = b*, for all i E {1,2,..., that n } and for all k 2 0. The stability of this equilibrium lim k-tocqi(k)/qi(k for all i E {1,2,..., q(k)B(O)= bti(k) - 1) = b*, bk,,q(k)B(O) (7) implies = lim,,,b*q(k) (8) n ). Hence a system operating a technology given by B(0) is “The Neumann version requires only that ai, (k) > 0 for at least one j, and that b,,(k) least one i. > 0 for at 204 CHARLES PERRINGS defined to be stable if, in the limit, the vector q(k) converges to a left eigenvector corresponding to the dominant eigenvalue of B(O), b*, for any initial vector q(0). At equilibrium the structure of production will be constant over time, and all products in the system will be expanding at the rate given by b*. Free disposal is defined to mean that the spectrum of the net output matrix will be constant in the face of the existence of a non-negative vector of residuals. More precisely, free disposal is defined to mean that q( k)[I - A(k)] > 0 implies that h(k) = h(0) for all k 2 0, where h(0) and h(k) denote the set of eigenvalues of B(0) and B(k), respectively. In other words, free disposal means that the existence of residuals in the system has no effect on the technology applied. It can be appreciated that this definition carries over very easily to cover the case of technological externalities in the economy-environment system. If the global system is partitioned to distinguish the economy from its environment, so that B(0) = [1 2 (0) where B,(O), describing the output coefficients of the economy, is m x n, and B,(O), describing the output coefficients of the environment, is (n - m) X n, and if q(k) and A(k) are partitioned conformably, then free disposal of economic goods implies that B,(k) = B,(O) for all q,(k)[I - A(k)] > 0. A proof of the proposition that the global system will be convergent by these definitions if and only if there is free disposal of residuals is offered in the Appendix. It is sufficient to note here that a system applying a given technology will be convergent if and only if residuals generated in the process of convergence have no feedback effects. Since the conservation of mass condition ensures that any change in the structure of production will be associated with feedback effects, the existence of residuals and the instability of the global system under the conservation of mass condition are synonymous. The conservation of mass condition implies that there will be such technical change as is necessary to dispose of all residuals in all periods. But notice that it implies nothing about the nature of this change. It is of interest, therefore, to consider whether the result holds in the presence of controlled technical change in a dynamic economic system. 4. ECONOMY AND ENVIRONMENT In order to distinguish between the processes of the economy and those of the environment, I now identify a price system involving the construction of two additional vectors. The first of these, p(k), is a semi-positive time-indexed n-dimensional column vector of prices, in which p,(k) is the price of the ith resource in the kth period, and p,(k) > 0 for i E {1,2,. . . , m}, and pi(k) = 0 for j E {m + 1, m + 2,..., n }. The first m components of p(k) are positive, indicating that the first m resources “produced” in the general system are positively valued. The last n-m components are all zero, indicating that the last n-m resources “produced” in the system are zero valued. The first m resources are thus scarce economic resources, the last n-m resources are non-scarce resources: either the waste products of economic processes or unvalued environmental products. Since p(k - 1) is positive in its first CONSERVATION 205 OF MASS m components only, it follows that if aij(k) > 0, i E {1,2,. . . , m}, j E {m + 1, m + 2,..., n }, then the agents operating the i th process are able to obtain quantities of the jth resource without advancing positively valued resources in order to do so. The non-scarcity of resources means that they can be obtained without surrendering positively valued products in the process. Conversely, the scarcity of resources means that their utilization by economic agents implies the commitment of positively priced products to gain their possession. The second vector, v(k), is a semi-positive time-indexed n-dimensional row vector of resource values, in which ui( k) indicates the value of the i th resource produced in the system in the kth period. The two vectors are related by v(k) = q(k - l)B(k where Q(k) tion as = diagonal [ pr, p2,. , . , p,](k). v(k) = q(k - l)[I - l)Dp(k) (9) v(k) is related to the costs of produc- + Dr(k - l)]A(k - l)Dp(k - 1) 00) where b(k) = diagonal [rr, r,, . . . , r,,](k) denotes the rates of profit earned in all processes. As with the price vector, Dr(k) is positive in the first m elements on the principal diagonal only. ri( k), i E { 1,2, . . . , m }, is an increasing function of the level of excess demand for the outputs of the i th process, where the level of excess demand for the ith product in the kth period is given by q(k))aJk - 1) - q,(k). The time path of the price vector may be described by B(k - l)p(k) = [I + Dr(k - l)]A(k - l)p(k - 1). (11) The only property of this system that we need to note here is that prices may be stable over time only if there is zero excess demand for all resources, and if there is no technical change. The problem of this paper is to determine the role of prices as observers and instruments of control in a time-varying system. To see the capacity of the price system to regulate change in a time-varying system, let us redefine the model discussed in Sections 2 and 3 as a control system.” We have already seen that wherever a system generates a set of residuals there exists a set of resources, the disposal of which has the effect of changing the technology of the system. When residuals are disposed of with a particular impact on output in view (as a purposeful act of investment), we have a controlled feedback process; the application of a linear combination of the state variables (the available resources) in order to transform the system from one state to another. The time path for the physical system may be described in terms of the state-space representation: (i) q(k) = q(k - l)B(k - 2) + j(k - l)M(k (ii) v(k - 1) = q(k - l)Dp(k - 1). - 1) (12) j( k - 1) in (12i) denotes an n-dimensional row vector of control variables applied in “Although control theory has been applied to economy-environment problems by Smith [28], it is uncommon to find technological change conceptualized as a control process. It is, however, well established in other disciplines. The process of evolution by natural selection, for example, has been convincingly conceptual&d by biologists as a control process, initially only implicitly, as by Lotka [21], but later explicitly, as by Rendel [27]. 206 CHARLES PERRINGS the k - lth period. It is a linear combination of the state variables, q(k - 1). M(k - 1) is an n-square feedback matrix describing the change brought about in the elements of B(k - 1) as a result of the controlled application of the residuals to the system. More particularly, the vector of control variables is a linear combination of the vector of residual resources generated by the system in the k - lth period under the technology of the k - 2th period: j(k - 1) = q(k - l)[I - A(k - 2)]K(k). (13) K(k) is discussed below. v(k) in (12ii) denotes the control system “outputs.” A non-stationary system of this type is said to be controllable if it is possible to transform it into a system in which none of the state variables, the qi(k), are independent of the control vector [13]. More particularly, the controllability of such a system implies that the kn x n controllability matrix constructed for an n-dimensional system controlled over k periods, J(k), is of rank n.12 The controllability matrix is formed from the sequence of state and feedback matrices as follows: J(k) M(O) Nww B@)Btl)M@) . = k-l (14) . ,G Btt)Mtk - 1) This matrix describes the effects of the controls applied to the system over the k periods of the control sequence. Its importance in the determination of the final state may be seen from the equation giving the general solution of the controlled non-stationary system-the system transition equation: k-l k-l q(k) = q(O) tnoB(t) + c t-o i(t) M(t). 05) Notice first that this differs from (6) in the second term describing the contribution of the controls over the interval [0, k - 11. This term is the product of the kn x n controllability matrix J(k) and the 1 x kn vector j( k, 0), formed by combining the control vectors j(t) over the same interval. It follows that if the vector k-l q(k) - do) tgo B(t) = j(k O)J(k) (16) has any zero valued components, that is if J(k) has any columns comprising only zeros, or is less than full rank, the general system will not be controllable [l]. The rank of the controllability matrix is limited by the rank of each matrix in the pair B(k), M(k). B(k) is of full rank by assumption. Hence if the feedback matrices describing the technological changes associated with the controls are of less than full rank, the controls will not reach all the processes in the system. The system will not be controllable. ‘*See, for example, Atham and Falb [2]. CONSERVATION OF MASS 207 What is interesting here is that the controls in an economic system are triggered by changes in the control system outputs, the price signals. In other words, the system is one of linear output feedback control. K( k - 1) in (13) depends on Dp(k - 1) in (11). More particularly, K(k - 1) = U(k - l)Dp(k - l), where the columns of U(k - 1) indicate the effect of a particular resource price on the demand for the resource in each of the m processes of the economy. A necessary condition for the complete controllability of the system is therefore that it be completely observable, where the conditions for the observability of the system parallel those for its controllability. That is, the complete observability of the system requires that the rank of an observability matrix of similar construction to (14) be n. Now in all economic systems the control instruments are the residuals or available resources in the system, but the observers differ between economic systems. The most basic form of control is that in which the physical system is observed directly through the level of residuals it generates. This type of control has been called by Komai and Martos [17] “vegetative control,” and its chief characteristic is that each agent has access to a very limited set of observations: “It is a characteristic of vegetative control that it always takes place at the lowest level between producers and consumers, without the intervention of higher administrative organizations. It is autonomous i.e. not directly connected to any social process.. . the firm or household only watch their own stock levels” [17, pp. 60-611. The rank of the observability matrix confronting each agent in the system is not much greater than zero. In the market economies the price system provides each agent with a more complete, though less direct, measure of the residuals of the system. Consequently, the rank of the observability matrix confronting each agent is much greater, implying that the controllability of the system is similarly greater. However, since p,(k) = 0 for i E {m + 1,m + 2,..., n }, the control vector j( k, O)J(k) may be positive in its first m components only, implying that the observability and hence the controllability matrices are of rank m at the most. The last n-m resources in the system are not touched by the controls. It follows that technical change described by feedback control informed by the signals of the economy, the price system, will not have determinate effects in respect of the environment. More importantly, wherever the economy and its environment are mutually dependent and are bound by the conservation of mass condition, such technical change will not have determinate effects even in respect of the economy. If the controlled allocation of resources does not satisfy the conservation of mass condition (rtiii), then there will be uncontrolled disposal of residuals, and there will exist unanticipated feedback effects. It is these unanticipated feedback effects that are the basis of all of the so-called external effects.13 It is, indeed, only if the economy and the environment are completely disjoint, implying that the Sraffa-Neumann models or the environmental models in the spirit of Coase accurately reflect reality, and if all residuals are allocated as control variables implying that there are no uncontrolled residuals, that technical change will not produce unanticipated effects. Moreover, while it is more realistic to postulate a process of “parameter adaptive control” in which economic agents gradually uncover the parameters of the system, it is misleading to substitute 13The notion that “technological externalities” underlie all of the external effects reported in the literature, including those associated with the common property problem, is well established. See Bator [S], Dasgupta and Heal [8], Fisher and Peterson [lo], and Fisher [ll]. 208 CHARLES PERRINGS the perfect information assumption normally made in control processes by the assumption of stochastic variation of the system parameters. These variations are not random, merely unobserved and unobservable given the structure of property rights prevailing in the system. 5. CONCLUSIONS The assumptions of non-innovation and the independence of the economy stand or fall together. Once the free gifts and free disposals assumptions are abandoned, then non-innovation fails too. Symmetrically, innovation cannot be conceived except in the presence of disposable real resources. If we agree to excise these assumptions from the axiomatic foundations of the classical general equilibrium models, however, we lose the determinacy (and much of the formal elegance) of the closed time-invariant system. The global system takes on the character of an imperfectly observed, imperfectly controlled set of competing subsytems, lurching from one disequilibrium state to another. Even if it is possible to draw the boundaries that demarcate the processes of the economy and its environment, there exist a multiplicity of material flows between them that are unsignalled by the prices informing the decisions of the agents of the economy. The feedback effects of these flows, with varying time delays, are the basis of the externalities surrounding every economic activity. The source of the difficulty is the conservation of mass condition. From this flow all the results that preclude the convergence of the system, or, if it is indecomposable, any constituent part of the system. In a materially closed system the conservation of mass condition ensures that any equilibrium path is one in which the absolute values of the components of the quantity vector will be constant over time. That is, if a Sraffa-Neumann system is indeed closed, then the only rate of growth compatible with the conservation of mass condition is the zero rate. The dominant eigenvalue of the net output matrix, B(k), will have an absolute value of unity. It follows immediately that any arbitrary set of physical processes to which corresponds a (notional) equilibrium growth rate greater than zero is not a materially closed system. If it is not a materially closed system then there will be material flows into and out of the system, and it will be jointly determined with its environment. Whether or not residuals generated in the system are allocated in a controlled or purposeful manner, the system will be subject to change resulting from the disposal of residuals in its environment. There is no reason why a particular subset of processes within a materially closed system should not have a positive growth rate over some finite period, but it will necessarily be at the expense of some other set of processes in its environment. An expansion in the mass of resources at the command of a particular group of agents implies a contraction in the mass of resources at the disposal of some other group of agents. It also implies an expansion in the mass of wastes generated by the former. High rates of growth in one subset of processes imply high rates of exaction on other processes, and high rates of residuals disposals in both sets of processes. Consequently, high rates of growth in one subset of processes imply high rates of change in the system as a whole. Not only is the growth-oriented economy itself an unstable CONSERVATION 209 OF MASS system, it is directly responsible for destabilizing the global system of which it is a constituent part. It is worth pointing out the parallels between this conclusion and the more general results of Prigogine and Stengers’ [25,26] analysis of the time behaviour of farfrom-equilibrium thermodynamically closed systems. The effect of energy flows between a referent system and its environment in such cases is a seemingly chaotic sequence of unstable dissipative structures. l4 The effect of material flows between the economy and its environment in a materially closed system are remarkably similar. We do not, however, need the force of the second law of thermodynamics to show that investment and waste disposal in an expanding economy leads to (irreversible) change in the material transformations of the global system. APPENDIX The proposition that a time-invariant system will converge to an equilibrium growth path if and only if there is free disposal of residuals implies that in a physical system satisfying assumptions (i) to (v), if cu(k) = max qi(k)/qi(k - l), P(k) = minq,(k)/q,(k - l),fori E {1,2,..., n},thenlim,,,a(k)andlim,,,B(k) = b* for any initial vector q(0) if and only if q(k)[I - A(k)] > 0 implies that B(k) = B(0) for all k 2 0. To prove sufficiency, let B(k) = B(0) = B for all k 2 0. By assumption, B has a dominant eigenvalue which is real and positive. Let the set of all eigenvalues in B be ordered in such a way that b,, = 6,. There exists a non-singular matrix S such that B = SDbS-’ (Al) where Db = diagonal {b,, b,, . . . , b,,}, and where the first row of S’, sl, and the first column of S, br, are the left and right eigenvectors of B corresponding to b,,. By the Frobenius theorem the components of sr, and Is, are strictlv oositive. From (3) the ith component of q(k) may-be defined by ’I dI qi(k) = q(0)Bkei 642) where e, is the ith unit vector. From (Al) this may be written qi( k) = q(O)SDb%‘e, for any k and all i E {1,2,..., q,(k) (A3) n}. (A3) may also be written in the form = bfq(O)SDc+S-‘e, (A4) where DC-~ = diag[l, b/b,, Accordingly, b,/b,, . . . , b,/b,] k. (A5) for all i E { 1,2,. . . , n }, we have in the limit: q,(k) lirnk+rn4i(k - 1) b: = lirnk+c-a bf-’ q(0)SDc-kS-‘ei q(O)SDc- W’)s-lei l4 This confirms the highly suggestive work of Georgescu-Rcegen [14,15]. ’ (fw 210 CHARLES PERRINGS Since q(0) is positive by assumption, since s1 and Isi are positive by the Frobenius theorem, and since lirn,,, DC-~ = diadl, 0, * * . ,O], q(0)SDc-kS-lei = q(0)SDc-(k-l)S-k,. Hence, defining b* = b,, lim k+coqi(k)/qi(k foralliE{1,2 ,..., - 1) = b* (A71 n}.Moreover, limk&dk) = zS1 (A@ for z > 0. If B(k) = B(0) = B for all k 2 0, then the rate of growth of all resources converges to the dominant eigenvalue of B, and the quantity vector converges to a left eigenvector of B corresponding to b,,. Necessity follows directly from (4). A semi-positive vector of residuals q(k)[I A( k - l)] implies that, in order to satisfy (43, there will exist a matrix A,( k - 1) with at least one positive element. From (43 there will exist a matrix B,(k - 1) with at least one positive element, implying that B(k) # B(k - l), and, if B(k - 1) is indecomposable, that SDb(k - l)S-’ # SDb(k)S-‘. The eigenvectors and so the equilibrium structure of production corresponding to B(k) and B(k - 1) will be different. So if q(k)[I - A(k)] > 0 does not imply that B(k) = B(0) for all k 2 0, lim k - ,q( k) will not be an eigenvector of B(0). REFERENCES 1. M. Aoki, “Optimal Control and System Theory in Dynamic Economic Analysis,” North-Holland, Amsterdam (1976). 2. A. Athans and P.L. Falb, “Optimal Control,” McGraw-Hill, New York (1966). 3. R.U. Ayres, A materials-process-product model, in “Environmental Quality Analysis” (Kneese and Bower, Eds.), Johns Hopkins Press, Baltimore (1972). 4. R.U. Ayres and A.V. Kneese, Production consumption and externalities, Amer. Econ. Reu. 59, 282-297 (1969). 5. F.M. Bator, The anatomy of market failure, Quart. J. Econ. 72, 351-379 (1958). 6. K.E. Bounding, The economics of the coming spaceship earth, in “Environmental Quality in a Growing Economy” (Jarrett, Ed.), Johns Hopkins Press, Baltimore (1966). 7. H. Daly, On economics as a life science, J. F’olit. Econ. 76, 3, 392-406 (1968). 8. P.S. Dasgupta and G.M. Heal, “Economic Theory and Exhaustible Resources,” Cambridge Univ. Press, Cambridge (1979). 9. R.C. d’Arge, Economic growth and the natural environment, in “Environmental Quality Analysis” (Kneese and Bower, Eds.), Johns Hopkins Press, Baltimore (1972). 10. A.C. Fisher and F.M. Peterson, The environment in economics: A survey, J. Econ. Lit. 14, l-33 (1976). 11. A.C. Fisher, “Resource and Environmental Economics,” Cambridge Univ. Press, Cambridge (1981). 12. J.W. Forrester, “World Dynamics,” Wright-Allen Press, Cambridge, Mass. (1971). 13. H. Freeman, “Discrete Time Systems,” Wiley, New York (1965). 14. N. Georgescu-Roegen, “The Entropy Law and Economic Process,” Harvard Univ. Press, Cambridge, Mass. (1971). 15. N. Georgescu-Roegen, Energy analysis and economic valuation, Southern Econ. J. 45,4, 1023-1058 (1979). 16. A.V. Kneese, R.U. Ayres, and R.C. d’Arge, Economics and the environment: A materials balance approach, in “The Economics of Pollution” (Wolozin, Ed.), General Learning Press, Morristown, N.J. (1974). 17. J. Komai and B. Martos, Vegetative control, in “Non-Price Control” (Komai and Martos, Eds.), North-Holland, Amsterdam (1981). CONSERVATION OF MASS 211 18. W. Leontief, “Studies in the Structure of the American Economy,” Oxford Univ. Press, Oxford (1953). 19. W. Leontief, Environmental repercussions and the economic structure: An input-output approach, Rev. Econ. Statist. 52, 262-271 (1970). 20. I.F. Lipnowski, An input-output analysis of environmental preservation, J. Environ. Econ. Manage. 3, 205-214 (1976). 21. A.J. Lotka, “Elements of Mathematical Biology,” Dover, New York (1956). 22. K. Marx, “Capital I,” Lawrence and Wishart, London (1954). 23. D.H. Meadows et al., “The Limits to Growth,” Universe Books, New York (1972). 24. J. von Neumann, A model of general equilibrium, Rev. Econ. Stud. 13, l-7 (1945-1946). 25. I. Prigogine and I. Stengers, The new alliance, Scientia 112, Part I, 319-332, Part II, 643-653 (1977). 26. I. Prigogine and I. Stengers, “Order out of Chaos,” Heinemann, London (1984). 27. J.M. Rendel, The control of development processes, in “Evolution and environment” (Drake, Ed.), Yale Univ. Press, New Haven, Corm. (1968). 28. V.L. Smith, Control theory applied to natural and environmental resources, J. Environ. Econ. Manage. 4, l-24 (1977). 29. P. Sraffa, “Production of Commodities by Means of Commodities,” Cambridge Univ. Press, Cambridge (1960). 30. P. Victor, “Pollution, Economy and Environment,” Allen & Unwin, London (1972).