Download Conservation of Mass and Instability in a Dynamic

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].
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Copyright 0 1986 by Academic Press, Inc.
All rights of reproduction in any form reserved
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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.
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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).
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