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Modeling of Sustainability
Kaoru Yamaguchi*
Osaka Sangyo University, Japan
E-mail: kaoruGdi:
osaka-sandai.ac.jp
Abstract
The purpose of this paper is twofold. (1) A step-by-step procedure
of system dynamics (SD) modeling is developed from a viewpoint of a
mathematical system of difference equations. Through this procedure,
essential concepts for building a SD model are developed such as the
difference between a moment and a period of time, a unit check, a com-
putational procedure for feedback loops, an expansion of boundaries, and
a limit to an analytical mathematical model. (2) To exemplify the above
procedure, a macroeconomic growth model is employed. Then a meaning
of sustainability is clarified by expanding a model step by step from a
simple macroeconomic growth model to a complicated ecological model.
To be specific, sustainability is represented in terms of physical, social
and ecological reproducibilities by a system of difference equations. As
an implementation of the analysis, it is shown that a sustainable eco-
nomic development is unsustainable in the long run with non-renewable
resources being taken into the model.
1 A Macroeconomic Growth Model
An step-by-step construction of a system dynamics (hereafter called SD) model
in this paper starts with a simple macroeconomic growth model which can
his paper is presented at the 19th International Conference of the System Dynamics
y, Atlanta, Georgia, USA, July 23 — 27, 2001. The paper is mostly written while I'm
visiting the Hawaii Research Center for Futures Studies, University of Hawaii at Manoa, in
March, 2001. I truly thank Dr. James Dator, its Director, and his colleagues who kindly
hosted my visit and provided a wonderful working environment for multidisciplinary studie
Hawaii turned out to be a good place for me to deeply consider sustainability. Since it became
the 50th US state in 1959, only less than a half century has passed. Yet, its economy with 1.2
million islanders needs more than five million tourists and their food annually for its survival
by dumping as much garbage. Can it be sustainable for this 21st century? Why did the
Easter Island in the southern Pacific Ocean suddenly collapse: overpopulation, lack of food,
water and natural resources, wars, epidemics ?
Table 1: Unknown Variables and Constants (1)
Category _| Symbol | Notation Unit
Ki _| Capital Stock machine
Unknown Y, | Output (or Income) food [year
Variables % | Consumption food /year
S| Saving food /year
L ment machine/year
v Capital-Output Ratio (= 4) __| machine/(food/year)
Constants ¢ Marginal Propensity to Con- dimensionless
sume (= 0.8)
Tnitial value | _K, _| Initial Capital Stock (=400) machine
be found in many macroeconomic textbooks. It consis
equations.
s of the following five
Kin =Kith (Capital Accumulation) ()
1
Y= -Ki (Production Function) (2)
0
G=e% (Consumption Function) (3)
S%=%-C (Saving Function) (4)
Lh=S (Equilibrium Condition) (5)
Equation (1) represents a capital accumulation process in which capital stock
is increased by the amount of investment. Output is assumed to be produced
only by capital stock in a macroeconomic production function (2). The amount
of consumption is assumed to be a portion of output - a well-known macroeco-
nomic consumption function (3). Saving is defined as the amount of output le
consumption in (4). At the equilibrium investment has to be equal to saving as
shown in (5), otherwise output would not be sold out completely or in a state
of shortage.
These five equations become simple enough to d
growth process. And most of the symbols used in the above equations should be
familiar for economics and business students. Precise meaning of these variables,
however, are usually left unexplained in the textbooks. SD modeling, on the
other hand, requires precise specification of these variables, as defined in Table 1,
without which it is impossible to construct a model. It is thus worth considering
these specifications in detail.
vibe a macroeconomic
Time
As emphasized in [8], it is fundamental in SD modeling to make a distinction
between two different concepts of time. One concept is to represent time as a
moment of time or a point in time, denoted here by r; that is, time is depicted
as a real number such that 7= 1, 2, 3, .... It is used to define the amount of
stock at a specific moment in time. The other concept is to represent time as
a period of time or an interval of time, denoted here by t, such that t = Ist,
2nd, 3rd, ... , or more loosely t = 1, 2, 3, ... . It is used to denote the amount
of flow during a specific period of time. Units of the period could be a second,
a minute, an hour, a week, a month, a quarter, a year, a decade, a century, a
millennium, ctc., depending on the nature of the dynamics in question. In a
macroeconomic analysis, a year is usually taken as a unit period of time.
With these distinctions in mind, the equation of capital accumulation (1),
consisting of a stock of capital and a flow of investment, has to be precisely
described as
Koy HK +h 7 and t = 2001, 2002, 2003, - - - (6)
A confusion, however, might arise from these dual notations of time, 7 and ¢, no
matter how precise they are. It would be better if we could describe stock-flow
relation uniformly in terms of either one of these two concepts of time. Which
one should, then, be adopted? A point in time 7 could be interpreted as a limit
point of the interval of time ¢. Hence, t can portray both concepts adequately,
and usually be chosen.
When ¢ is used to represent a unit interval between 7 and 7 +1, the amount
of stock K, thus defined at the t-th interval could be interpreted as an amount
at the beginning point 7 of a period t or the ending point 7 +1 of the period #;
that is,
Beginning amount of stock (7)
or
Ky = Kyy1 Ending amount of stock (8)
When the beginning amount of the stock equation (7) is applied, a stock-flow
equation of capital accumulation (6) is rewritten as
Kin = Kieth t = 2001, 2002, 2003, - (9)
In this formula, capital stock Ky41 is evaluated at the beginning of the period
t+ 1; that is, a flow of investment J; is to be added to the present stock value
of K;, for the evaluation of the capital stock at the next period.
When the ending amount of the capital stock equation (8) is applied, the
stock-flow equation (6) is rewritten as
K=Khith t = 2001, 2002, 2003, -- - (10)
Two different concepts of time - a point in time and a period of time - are
in this way successfully unified. It is very important for the beginners of SD
modeling to understand that time in system dynamics usually implies a period
of time which has a unit interval. Periods need not be discrete and could be
continuous. In this paper, the beginning amount of capital accumulation (9) is
employed as many macroeconomic textbooks do!.
Unit
In SD modeling, units of all variables, whether unknowns or constants, have
to be explicitly declared. In equation (5) of equilibrium condition, inv
is defined as an amount of machine per year, while saving is measured by an
amount of food per year. Therefore, in order to make the equation (5) con-
gruous, a unit conversion factor £ of a unitary value has to be multiplied such
that
sstment
Lh= Sr, (13)
food unit of
in which € conver aving to a machine unit of capital investment;
that is to say, it has a unit of machine/food dimension. This tedious procedure
of unit conversion could be circumvented by replacing machine and food units
with a dollar unit as many macroeconomic textbooks implicitly presume so.
Model Consistency
A model consistency has to be examined as a next step in SD modeling, following
consistent if it has the same
time and a unit check. A model is said to be at le
number of equations and unknown variables. This is a minimum requirement for
any model to be consistent. The above macroeconomic growth model consists
of five equations with five unknowns and two constants. Thus, it becomes
consistent.
Let us now consider how these equations are computationally solved. Star'
ing with the initial condition of the capital stock K,, numerical values are as
signed from the right-hand variable to the left-hand variables. We can easily
trace these value assignments as follows:
Kio ¥, > C1 8 he Kean (14)
This is how a computer solves equations of dynamic systems. .
To show the difference between stock and flow explicitly, it would be informative to
decompose the capital accumulation equation (1) as follows:
Key = Ki t+ AKe (Idenatity of Capital Stock Accumulation) (11)
AK ah (Investment as a Flow of Capital) (12)
Feedback Loop
In our macroeconomic growth model, there are two types of equations. One
type is the equation of stock-flow relation which specifies a dynamic movement.
Capital accumulation equation (1) is of this type. The other type is the equation
of causal relation in which a left-hand variable is caused by right-hand variables
(and constants). The remaining four equations in the macroeconomic growth
model are of this type.
These two types are clearly distinguished in SD modeling. A. stock-flow
relation is illustrated by a box that is connected by a double-lined arrow with a
flow-regulating faucet, while a causal relation is drawn by a single-lined arrow.
Then, we can easily trace a loop of arrows starting from a box and coming back
to the same box. Such a loop is called a feedback loop in . Tt is
called positive feedback if an increase in stock results in an increase in a coming
back stock, while negative feedback if an decreased amount of coming back stock
results in.
A feedback loop corresponds with a computational trace of the equation
(14). By drawing a SD model, we could easily find two such feedba
that start from a capital stock box. A feedback loop has to include at least
one stock-flow equation. Simultaneous equation system, on the other hand, has
only equations of causal relation and, accordingly, cannot have feedback loops.
Without a feedback loop,
stem dynami
ystem cannot be dynamic.
Initial Capital
Capital
| |
/
la
Output (Income)
Marginal Propensity /
io Consume Capital-Output Ratio
Figure 1: A Simple Macroeconomic Growth Model
A Steady State Equilibrium
Since SD modeling is by its nature dynamic, it is very important to find out a
: ate equilibrium as a structural consistency of the model. Steady state
s that all st top changing, which in turn means that values of flows
net flows) become zero. In other words, it ate of no growth.
In our mode a steady state equilibrium of capital accumulation is attained for
Ki41 = Ky. To attain the steady state analytically, equations of the model are
first reduced to be a single equation of capital accumulation:
fe
Ria = (1+ *) Ki. (as)
Then, a steady state is easily shown to exit for ¢ = 1; that is, output is all
consumed and no saving is made available for investment. In our numerical
example, a steady state equilibrium is attained at the values of K* = 400,Y* =
C* = 100, and S* = I* =0.
Simulation for an Economic Growth
Let us try to drive the economy out of this steady state equilibrium. A growth
path can be easily found by setting “a marginal propensity to consume” to be
less than unitary; say, c = 0.8. Then 20% of output (or income) is saved for
investment, which in turn inc: es the capital stock by the amount of 20, which
then contributes to the increase in output by 5 next period, driving the economy
toward an indefinite growth. Table 2 shows how capital, output, consumption
and investment grow at the rate of 5% for c = 0.8.
Table 2: Macroeconomic Growth Model
Year ] Capital Output Consump- Invest-
tion. ment
2001 | 400.00 100.00 80.00 20.00
2002 | 420.00 105.00 84.00 21.00
2003 | 441.00 110.25 88.20 22.04
2004 | 463.04 115.76 92.61 23.15
2005 | 468.20 121.55 97.24 24.31
2006 | 510.51 127.62 102.10 25.52
2007 | 536.03 134.00 107.20 26.80
2008 | 562.84 140.71 112.56 28.14
2009 | 590.98 147.74 118.19 29.54
2010 | 620.53 155.13 124.10 31.02
Let us consider another growth path in which maximum amount of saving
is made first at the cost of consumption, then, by accumulating capital stock
as fast as possible, a higher level of consumption is enjoyed later. This type of
growth path can be built by making “a marginal propensity to consume” as a
function of a normalized output level such that
e(¥%1/Ynorm) (16)
c
where Ynorm is a normalized reference level of output with which a current
level of output is compared. Usually an initial value of output is selected as a
reference level: Ynorm = Yinitiat = 100. This function is called a table function,
or a graphic function, or a lookup function in SD modeling.
09
08
07
06
Figure 2: A Table Function of A Marginal Propensity to Consume
A simple example is the following linear function as illustrated in Figure 2:
Yi
+0.2 (17)
initial
‘a marginal propensity to consume” is set to be a lowest (or
c(1) = 0.6, to allow for a maximum growth rate, then
, enabling more consumption.
and no further saving and in-
vestment are made; that is, a maximum consumption level is enjoyed. Figure 3
y to consume”
At the beginning
a subsistence) level,
it gradually becomes higher as income incree
When income level doubles, we have ¢(2) =
illustrates a gradual increase in the value of “a marginal prope
and a gradual decrease in the growth rate.
Building up a table function is to connect the variable Output by a single-
lined arrow to the constant Marginal Propensity to Consume in Figure 1. And
a constant of “marginal propensity to consume” which has been residing out-
side the model now becomes an inside unknown variable whose value is to be
determined by the behavior of the model itself. Better modeling is to reduce the
number of constants and make a model self-determined by itself without relying
on the outside values of constants. In this sense, a capability of introducing
table functions is one of the most powerful features in SD modeling. In fact,
1 Dmnt
10 Dut
0.8 Dmnt
5 Dmnl
0.6 Dmnt
Dmnt
2001 2006 2011 2016 2021 20262031 2036
Time (Year)
Marginal Propensity to Consume : macroeconomy mnt
Growth Rate : macroeconomy ——— Dn
Figure 3: Growth Rate and Marginal Propensity to Consume
an introduction of nonlinear and/or numerical table functions can make many
diversified dynamic behaviors possible for analytical simulation:
2 Physical Reproducibility
Sustainability
In the above macroeconomic growth model, depreciation of capital stock is
not considered, or J; is regarded as net investment. In reality, capital stocks
depreciate, and for maintaining the current level of output, some portion of the
income has to be saved to replace the depreciation. When a depreciation rate
high, a higher portion of income has to be saved at the cost of the consumption.
Here arises a sustainability issue of the economy: how to maintain a level of
income for sustainable development. In this sense a sustainability is
human history.
ne has
been as old a
Let us now consider what is meant by sustainability from an economic point
of view. After the UN Conference on Environment and Development (UNCED),
widely known as the Earth Summit, in Rio de Janciro, Brazil, 1992, sustain-
able development becomes a fashionable word in our daily conversations. This
might be an indication that our awareness on environmental crises such as global
warming, acid rain, depletion of the ozone layer, tropical deforestation, deser-
tification, and endangered species has deepened. How should, then, a state of
sustainable development be defined? Some proposed definitions are the follow-
ing:
Sustainable development is development that meets the needs of the
present without compromising the ability of future generations to
meet their own needs. [4, p.43].
The simplest definition i iety is one that can per-
sist over generations, one that is far-seeing enough, flexible enough,
and wise enough not to undermine either its physical or its social sys-
tems of supports. (Italic emphases made by the author) (2, p.209].
These definitions are articulated so as to be understood even by children.
However, from an economi:
's point of view, these definitions lack an interrelated
view of production, consumption, society and environment.
A sustainability is comprehensively defined when all activities in economy,
society and nature are interpreted as reproduction processes; that is, in terms
of physical, social and ecological reproducibility [7]. A merit to this approach is
that an economic structure such as in the gencral equilibrium framework [6]
be applied, since the most basic activity in any society is a reproduction proc
in which inputs are repeatedly transformed into outputs for consumption and
investment cach year. The same approach is followed in this paper. In this way
the interrelationship between economic activities and environment is integrated
wholistically.
Capital Depreciation
Let us introduce depreciation in the macroeconomic growth model. The equa-
tion of capital accumulation (1) is expanded as follows:
Ki =Ki+h-D (Capital Accumulation) (18)
Dy = 6K; (Capital Depreciation) (19)
As Figure 4 shows, th sily done in SD modeling by adding an out-
flow arrow of depreciation from the capital stock. J, in equation (18) is now
Table 3: Unknown Variable and Constant Added (2)
New Variable | D; | Depreciation machine/year |
New Constant [5 | Depreciation Rate (0.02) T/year_|
reinterpreted as gross investment.
Physical reproducibility implies that gross investment is greater than or
equal to the depreciation.
-D,>0 [Physical Reproducibility] (20)
The macroeconomic growth model with depreciation, which is here called phy
cal reproducibility model, now consists of 6 equations with 6 unknown variable:
Kisi.Ye,Ct, St, 1, Di and three constants: v,¢,6 «
Initial Capital
SS vial:_ |- —
Investment Depreciation
\ —_ \
Saving J
a Depreciation Rate
Consumption Output (Income) Capital-Output Ratio
7
\ Raw Material Initial Non-Renewable
\ Input Rate Resources
Marginal Propensity /
to Consume
Non-Renewable
- Resources
Raw Material
Figure 4: Physical Reproducibility Model
A Steady State Equilibrium
A steady state equilibrium is attained at Ky. = Ky or I,
from the equation (18). To obtain the steady state analytically, all equations in
the model have to be reduced to a single capital accumulation equation:
Kin = ( + 1% - ‘) Ki. (21)
A steady state condition is then easily obtained as follows (asterisks are added
to the constants that meet this condition):
(22)
At the steady state, “a marginal propensity to consume” becomes less than
unitary; c* = 1 — 6*v" < 1, which implies that a portion of output has to be
saved to replace the capital depreciation. One possible combination of numerical
values for the steady state is (v*,c*,6*) = (4, 0.8, 0.05).
Simulation for an Economic Growth
For the economy to grow out of the steady state; that is, Kj41 > Ky, at |
one of the following three actions has to be taken.
(1) Increase productivity (+ > 4) or vu <u".
(2) Decrease consumption (or increase saving and investment) ¢ < c* .
(3) Improve capital maintenance 5 < d* .
1
8,000 machine
2,300 food/Year
4,000 machine
1,150 food/Year
0 machine
0 food/Year = :
2001 2026 2052 2077 2102
Time (Year)
Capital : macro growth machine
"Output (Income)" : macro growth——__— food/Year
Consumption : macro growth ——— food/Year
Investment : macro growth ~ food/Year
Figure 5: A Simulation for Economic Growth
As one such numerical example, let us take the case (3) and set a rate
of depreciation at 6 = 0.02. In this cas conomic growth becomes 3%.
As Figure 5 illustrates, during the 21st century capital stock keeps increasing
from 901 = 400 to K’2191 = 7,687 and so does output from Y2o91 = 100 to
Yo191 = 1,921, more than 19 folds! Can such a growth be sustainable?
an
Non-Renewable Resource Availability
The physical reproducibility condition (20) presupposes an availability of non-
which is represented by the following equation:
renewable natural resource
Ruy = Re - AR (Non-Renewable Resource Depletion) (23)
AR, = AY; (Non-Renewable Raw Material Input) (24)
ame that non-renewable resources are represented
and oil whose units are uniformly measured by
ary for
For simplicity, let us here ass
by fossil fuels such as coal, gas
a ton. Then, ) is interpreted as an input amount of fossil fuels ne
producing a unit of output.
Table 4: Unknown Variable and Constant Added (3)
New Variable_| Ri | Non-Renewable Resource ton
New Constant | A | Raw Material Input Rate | ton/ food
(=0.05)
Tnitial value Ry | Initial Non-Renewable Re [ton
source (=1,00)
Assuming that equations (23) and (24) are reduced to one equation, we have
now 7 equations for 7 unknown variables and four constants. Hence the model
is shown to be consistent.
Let us next consider the existence of a steady state equilibrium. There are
two state variables Ky41 and Ri41 in the model. A steady state of capital
accumulation is not affected by the introduction of non-renewable resources,
while a steady state of non-renewable resources implies Ry41 = Rr, which in tum
means AR, = XY; = 0 or ¥; = 0. However, a steady state equilibrium of capital
stock implies a positive amount of output; that is, Y; > 0. A contradiction arises!
Hence it is concluded that a macroeconomic growth model with non-renewable
cannot have a steady state equilibrium by its nature. To make the
sible, the existence of a steady state equilibrium of non-renewable
resources has to be conceptually given up. Or, non-renewable natural resources
have to be assumed to be available at any time in the economy so that the
resources
model fe:
earth’s limited source of non-renewable resources is not depleted; that is,
SS AR: < Reo01 [Non-Renewable Resource Availability] (25)
t=2001
Simulation for Sustainability
Non-renewable resources are continuously deleted even at a steady state equi-
librium of capital accumulation, contrary to a general belief that they are not
in a non-growing economy.
At the steady state equilibrium set by the condition (22), as Figure 6 illus-
trates, the initial non-renewable resources R901 = 1,000 constantly diminishes
to one half a century later R191 = 500. This can be easily examined by a simple
calculation. Since the economy is at the steady state, the output level becomes
constant at Y; = 100. Hence, AR, = AY; = 0.05 - 100 = 5 and non-renewable
resources are depleted by 5 tons every year. Over a century they are depleted
by 500 tons. It is very important, therefore, to understand that a steady state
equilibrium is not sustainable in the long run. In fact, a simple calculation s
that non-renewable resources will be totally exhausted over two centuries; that
is, by the year 2201 we have Ro901 = 0.
To show how fast non-renewable resour
a depreciation rate is
ows
deplete under a growing
set to 6 = 0.02 and the economy starts growing at the
onomy,
1,000
750
500
250
o
2001 2026 2052 2077 2102
Time (Year)
‘Non-Renewable Resources” : dep.rate=5% on
"Non-Renewable Resources" : dep.rate=2% ton
Figure 6: Depletion of Non-Renewable Resources
rate of 3%. In this case, non-renewable resoure
year 2066; that is, at the beginning of the next
as Figure 6 illustrates.
How can we circumvent such a faster depletion of non-renewable resources
and stay within a limit to resource availability and physical reproducibility?
First, an efficient use of non-renewable natural resources has to be invented.
For this, an introduction of long-term management: of resources will be nec-
stitutes for non-renewable resources have to be discovered
or newly invented through technological breakthroughs. For this, re
development of new technology have to be oriented toward this direction. The
issue of substitutes for non-renewable resources will be more fully analyzed in
will be totally depleted in the
ar we have Roogr = —5.813,
arch and
the next section.
Feedback Loop for Non-Renewable Resource Availability
What will happen if the development of substitutes
To overcome a diminishing non-renewable resources, two self-regulating forces
might appear in the economy. The first and more direct force is to curb down
a raw material input rate \. In a market economy, this might emerge as an
increase in prices of non-renewable resources so that their use will be regulated.
In SD modeling, this self-regulating force can be easily implemented by draw-
ing an arrow from a stock of non-renewable resources to a constant of the raw
material input rate \ and defining a table function as follows:
rA=X (a) (26)
The second and more indirect force might appear as a reduction of pro-
ductivity as non-renewable resources begin to be exhausted. In other words,
productivity which is defined as 4 might begin to slide down. In SD modeling,
are delayed or failed?
a
thi ogulating force can be casily implemented by drawing an arrow from
non-renewable resources to a capital-output ratio and defining a table function
v=0( 8)
ond force of self-regulation is considered as an exam-
as follows:
In this paper, the
ple of the effect of diminishing non-renewable resources on the economy. Let
us assume that a productivity is not affected until non-renewable resow
depleted up to 40%. Then it begins
continue to be depleted. Table 5 indicates one such numerical example of di-
are
to decrease as non-renewable resources
Table 5: A Table Function of Capital-Output Ratio
Ri/Rintiat | 001 02 03 04 05 06-1
v 0 16 2 8 6 5 q
minishing productivity (or an increasing capital-output ratio).
2,000 machine
450 food/Year
1,000 ton
1,000 machine
225 food/Year
500 ton
0 machine
0 food/Year
0 ton
2001 2026 2052 2077 2102
Time (Year)
Capital : non-renewable machine
"Output (Income)" : non-renewable food/Year
Consumption : non-renewable food/Year
Investment : non-renewable food/Year
"Non-Renewable Resources" : non-renewable ton
Figure 7: Physical Reproducibility with Non-Renewable Resources
Figure 7 illustrates the effect of such self-regulating forces. Output level
attains its highest peak in the year 2043; Yoo43 = 337.53, then begins to decrease.
In the year 2093, the output level becomes less than its initial output level;
Yo093 = 98.15 < Y2o01 = 100. Apparently at this lower level of output the initial
number of population would not be sustained. In other words, non-renewable
resource availability and population growth become a serious trade-off, and
cither the preservation of non-renewable resources or population growth has to
be sacrificed in the long run. To sce this trade-off relation of sustainability more
explicitly, the equation of population growth has to be brought into the model,
which will be done in the next section.
3 Social Reproducibility
Population growth is embodied in the model as follows:
(Population Growth) (28)
(Net Birth = Birth - Death) (29)
Notations of new variables and constants are shown in Table 6.
Table 6: Unknown Variables and Constants Added (4)
New Variables Population pe
Workers (Labor Force) person
Birth Rate (= 0.03) T/year
B__| Death Rate (= 0.01) 1/
New Constants [~ 9 | Participation Ratio to Labor | dimensionl
Force(= 0.6)
T_| Output-Labor Ratio (= 0.4) _| (food/year)/person
© | Subsistence Consumption | (food/ycar) /person
Level (= 0.16)
Tnitial value | N; _| Initial Population (=500) person
For a survival of any socicty a minimum amount of consumption has to be
at least produced cach period to reproduce its population. This amount needs
not be a subsistence amount, but has to be enough “to maintain the minimum
standards of wholesome and cultured living (Article 25, The Constitution of
Japan).” Let ¢ be such a minimum amount of consumption per capita. Then,
a total amount of consumption defined in the consumption function (3) has to
be replaced with the following:
oP
Ni, (Minimum Consumption) (30)
With the introduction of this minimum amount of consumption demanded
irrespective of the output level, the amount of saving defined in the saving
function (4) might become negative as population increases. To warrant a non-
negative amount of saving, the saving function also has to be technically revised
as follows:
S; = Max{¥; — C;,0} (Non-Negative Saving) (31)
Social reproducibility is now defined as a reproduction process in which a
minimum amount of consumption is always secured out of the net output?; that
is,
¥; -D,-cN, >0 [Social Reproducibility] (33)
Note that whenever this social reproducibility condition is met, ph
ducibility (20) also holds; that is,
ical repro-
(34)
With the introduction of population, the number of workers or labor force
is easily defined as a portion of the population:
(Work
(35)
Production function (2) is then replaced with the following revised production
function which allows an inclusion of labor force explicitly as a new factor of
production®.
al
Y, =Min{=K,, Li} (Production Function) (37)
v
A Steady State Equilibrium
Our macroeconomic growth model is now getting a little bit complex. From the
tables of unknown variables and constants (1) through (4), 9 unknown variables
and 9 constants are enumerated for 9 equations. Therefore, the model is shown
to be consistent.
Let us now consider a steady state equilibrium. There are three variables of
Kea, New, Rip.
However, no steady state equilibrium is possible for non-renewable resources as
stocks such as capital, population and non-renewable resource
already mentioned above. A steady state of population growth Niz1 = N; is
2To be precise, a unit of depreciation (machine/year) has to be converted to a unit of food
(food/year) as follow
Yt — Dt/€ (32)
as in the equation (13)
3.Alternatively, a neoclassical production function such as a Cobb-Douglas production func-
tion can be used without any difficulty in SD modeling as follows:
Yom AKeri (36)
Capital
Investment -
Initial Capital |
|
Lr saint |. -
/ \ ‘a Capital-Output Ratio
Output
Output-Labor Ratio
Workers
Raw Material
Input Rate
Initial Non-Renewable
\
\ Resources
AS sate oe
Participation Rate }
Subsistence \ ”
Consumption Level |
st ollie & | < Zt ‘Non-Renewable
Resources
Death gq
AON
= \
Birth Rate Death Rate
Figure 8: Social Reproducibility Model
ay, a* = 9" = 0.01. A
steady state of capital stock can be attained as before for the values of constan
(v*,c*,5*) = (4, 0.8, 0.05). Due to the introduction of the new production func-
tion (37), two cases of steady state equilibria may emerge.
(1) A case in which output is constrained by capital stock: ¥; = 1K;. In
this case, from a simple calculation we have
attained when a birth rate is equal to a death rate
=0.8 (38)
For N;, = 500, capital stock has to be K, = 400 at the steady state. Hence, a
steady state equilibrium is summarized as (K*, N*,Y*,C*, S*, I*) = (400,500,
100, 80, 20, 20), except that non-renewable resource depleting by the amount
of 5 tons every year as analyzed under the previous section of physical repro-
ducibility.
(2) A case in which output is constrained by workers: Y; = ¢L;. In this case,
we have
Ky
N=
(39)
For N; = 500, capital stock this time has to be K; = 800 at the steady state.
Hence, a steady state equilibrium is summarized as (K*, N*,Y*,C*, S*, 1°) =
(800, 500, 120, 80, 40, 40).
Neoclassical Golden Rule of Capital Accumulation
How are these two steady state equilibria related? To figure out these relations
analytically, neo-classical economists invented a very neat concept of per capita
capital stock, which is defined as ky = Ki/N;. Let n(= a — 8) be a net growth
rate of population. Then the above 8 equations, except for the non-renewable re-
sources, are very compactly reduced to a single equation. To do so, the equation
(18) is first rewritten as
Key
(40)
~R(%_
=a ™
tt Ne
Then, a simple calculation results in the following capital growth equation.
1 pls, aay :
het = het (stint, 00} —e-(n +5)hy) (41)
A steady state equilibrium is obtained at ki41 = ke, which in tum yields two
equilibrium levels of per capita capital.
For a smaller level of equilibrium we have
(42)
This corresponds to the previous case in which output is constrained by capital
stock (38). For a larger level of equilibrium we have
_ ol
~ ntd
ke
(43)
This corresponds to the previous case in which output is constrained by the
number of workers (39). Note that even at a steady state of per capita equilib-
rium, population is allowed to grow at a net growth rate of n > 0.
It is easily shown that & is an unstable state of equilibrium, since kiy1 < kt
for ky <k, and kiy1 > ky for k < ky < k*. Thus, per capita capital ky is, once
displaced with the equilibrium, shown to decree
k*. Meanwhile, k* is a stable state of equilibrium, s
¢ toward zero or converge to
ce kyr < ky for ky > k*.
—_ 0, ithe <k (44)
kt, ifky >k
Let us examine the stability of per
economy to grow out of the initial s
birth rat
population are allowed to start growing. It is then calculated that
and k* = 2. Since the initial population is 500, an unstable equilibrium level of
initial capital stock is obtained as K2991 = kN2901 = 380.95. This means that if
the initial capital stock is less than this amount, per capita capital stock tends
pita capital numerically by allowing the
cady state equilibrium. Depreciation and
are now reset to (6,a) = (0.02, 0.03) so that both capital stock and
0.7619
to diminish toward zero, and the economy will get stuck eventually. Figure 9
numerically illustrates that when A291 = 380 (and k2001 = 0.76) per capita
capital decreases to k2190 = 0.0296, and eventually to zero.
#
1s
1
as
0
2001 2026 2052, 2077 2102
Time (Year)
Per Capita Capital : for K=380 ——————machine/person
Per Capita Capital : for K=381] ———_machine/person
Figure 9: Golden Rule of Capital Accumulation
On the other hand, if the initial capital stock is greater than this amount,
per capita capital tends to converge towards a s
capital in neoclassical growth theory [1]. Figure 9 a that when
Ko01 = 381 (and kgo01 = 0.762) per capita capital increases to k2199 = 0.1898,
and eventually converges to a golden rule level of capital: k* = 2.
It is interesting to know that only one unit difference of capital stock in our
numerical example will result in a big difference in the growth paths. When the
initial capital stock is K2991 = 380, the economy will be destined to be trapped
forever to a stagnant state, while an additional unit of capital stock will drive
the economy to its prosperity. The importance of an initial level of capital
a well recognized feature in development
. To circumvent the situation of this economic trap, the initial capital
ot so far to K’291 = 400 in our numerical example.
stock for an economic development i
economi
From now on let us reset the initial value of capital stock, without losing
generality, at its critical value of Kg991 = 381. Figure 10 illustrates how capital
stock, population, output, consumption and investment grow simultancous
Population grows at 2%, so does a minimum amount of consumption regardless
of the growth of capital stock and output level. Output is first constrained by
the availability of capital stock, then from the year 2042, it is constrained by the
availability of workers, which is in turn constrained by the population growth.
Thus, the economy continues to grow at an increasing rate as the capital stock
grows up to the year 2042 (from 2% to 5%), then it grows at a constant rate
of population growth of 2% . This is why there are some bumps on the output
and investment growth paths around 2042.
8,000 machine
4,000 person
1,400 food/Year
4,000 machine
2,000 person
700 food/Year
0 machine
0 person
0 food/Year
2001 2026 2052 2077 2102
Time (Year)
Capital : for K=381 machine
person
Output : for food/Year
Consumption : for K=381 ‘food/Year
Investment : for K=381 ‘food/Year
Figure 10: Golden Rule of Economic Growth
Even at this growth rate of population, output level is still maintained at a
higher level than a minimum amount of consumption so that social reproducibil-
ity is constantly sustained. Eventually, per capital capital growth will converge
to a steady state of k*, showing a long-run stability of capital accumulation.
This is what is meant by a neoclassical economic growth of golden rule: a very
elegant and optimistic theory of economic growth!
Can such a growth be sustainable in the long run, indeed? The answer
would be yes as long as non-renewable resources are disregarded and left out of
the model. Remembering, however, that neoclassical concept of a steady state
allows a constant growth rate of 2% , and the economy still keeps growing, de-
s, the answer would be absolutely no. In fact,
non-renewable resources will be totally depleted in the year 2077 in our numer-
ical example and become negative for the next year; that is, Roors = —13.31.
Even so, neoclassical growth theory keeps silent about this point, giving the
impression that our macroeconomy can continue to grow and be stable in the
long run.
pleting non-renewable resources
Capital
Investment Depreciation.
/ MR
vw
|
Depreciation Rate
Output. we Net Output
Substitutes
Ratio —
Productivity
ayy
Capital-Output Ratio
@ x Consumption
‘onsumption
. fA
[ | Workers |
" Ae Raw Material
~ Output-Labor Ratio
\ Input Rate
\ \ \ Initial Non-Renewable
\ ‘ \ Resources
\. Labor Force
\
Subsistence
Consumption
Level
Participation
Rate
Non-Renewable
Resources
Level of
Substitutes
_® Birth
fi
__ Death Rate
we Table
Birth Rate Death Rate
Figure 11: Social Reproducibility Feedback Model
Feedback Loop for Non-Renewable Resource Availability
is now taken into consideration. To
Availability of non-renewable resour
make non-renewable resources available for future generation:
a similar feedback mechanism as implemented in the previous section of physica
reproducibility; that is, as the non-renewable resources continue to be depleted.
productivity becomes worsened, and accordingly output is curbed, resulting in
preserving non-renewable resources. Two constants in the production function
(37) could influence productivity separately; that is, capital-output ratio v and
output-labor ratio ¢. Instead of affecting these separately, we introduce a table
function which affects output level directly such that
, let us introduc
Ry
1
Y= Productivity ( ) Min{=K;, (L:} (Production Feedback) (45)
v
tinitial
A table function of the productivity is defined in Table 7. It is assumed that
the productivity is not affected until non-renewable resources are depleted up
to 40% , beyond which, then, it begins to decrease gradually. Figure 11 shows
| feedback loop for non-renewable resour
arev
Table 7: A Table Function of Productivity
RifRinitia | 0 0.1 0.2 03 04 0.5 06-1
Productivity [0 0.1 02 04 06 08 1
2,000. machine
4,000 person
1,000 food/Year
1,000 ‘ton
0 machine
0 person |
0 food/Year
0 ton
2001 2026 2052 2077 2102
Time (Year)
Capital : feedback machine
Population +f person
‘Net Output ‘food/Year
Consumption : feedback ‘food/Year
"Non-Renewable Resources” : feedback ton
Figure 12: Golden Rule with Non-Renewable Resource Feedback
As expected by the introduction of a productivity feedback loop, Figure
12 illustrates how growth paths of capital stock and net output (output less
tion) are curved as non-renewable resources continue to be depleted. In
rved.
y non-renewable resources are to be pres
Feedback Loop for Social Reproducibility
Figure 12 also illustrates that population increases exponentially at a net growth
rate of 2% , so does a minimum amount of consumption for maintaining a per-
capita wholesome and cultured living standard: C, = cN;. Since net output
is curved by a negative feedback loop of non-renewable resources, social re-
producibility condition (33) will be eventually violated, and a portion of the
population might be forced to be starved to death.
The violation of social reproducibility implies
Y%—D,-eN, <0. (46)
In our numerical example, this occurs in the year 2057 when C2957 = 242.49 and
Yo057 — Daos7 = 234.33, so that consumption exceeds net output by the amount
of 8.16 as roughly illustrated in the Figure 12. The violation of social repro-
ducibility implies that a smaller amount of net output has to be shared among
people, forcing their level of living standards to be reduced. How far can such
a per capita consumption be lowered? For maintaining physical reproducibility,
it is desirable to keep its level at which per capita consumption is equal to per
capita net output. It would be imaginable, however, that starving people would
eat up everything available out of the output, including the reserved amount of
capital stock for depreciation.
Reflecting the situation of food shortage, per capita consumption is recalcu-
lated as follows!:
Per Capita Consumption = Min ( (48)
This formula enables per capita consumption to be lowered from the level of
¢ = 0.16. A decrease in per capita consumption may increase a death rate
due to a food shortage. This could happen equally among weaker people and
children, or among the countries whose economy is not wealthy enough to buy
food, or among the countries that are politically weaker and neglected. A table
function of death rate in Table 8 is created to reflect such situations. For
Table 8: A Table Function of Death Rate
Per Capita Consumption =0.16 0.14 0.12 01 0.08 0.06 0.04
Death Rate 3 0.01 0.015 0.02 0.03 0.05 0.07 0.1
instance, whenever a per capita consumption is reduced by half from the original
minimum amount, a death rate is assumed to jump to 5% from 1% . In this
way, a negative feedback loop of social reproducibility is completed. Figure 11
a revised feedback vei ial reproducibility model.
ed growth paths that reflect the feedback loop rela-
tion to the death rate. The amount of consumption exceeds net output during
the year 2056, and accordingly capital stock begins to decay. The difference
between consumption and net output is the amount of capital depreciation that
is allowed to be consumed by hungry people.
show sion of so
Figure 13 illustrates re
4On the other hand, for maintaining the physi
consumption has to be changed to the following:
val reproducibility, the equation of per capita
ee)
Per Capita Consumption = Min (« ) (47)
2,000 machine
2,000 person
1,000 food/Year
1,000 ton
0 machine
0 person i
0 food/Year
0 ton
2001 2026 2052 2077 2102
Time (Year)
Capital : feedback? machine
person
food/Year
‘food/Year
ton
Figure 13: Growth Paths with Social Reproducibility Feedback
A century later, output level becomes only one quarter of its initial level; that
is, Yo101 = 25.71 from Y291 = 95.25. Population is almost pulled back to its
original level of Nooo: = 500; that is, it increases to its peak at Noose = 1,793,
then begins to decline to Noi. = 541.51. Per capita income level has been
maintained at ¢ = 0.16 until the year 2059, then begins to decline to the level
of 0.0474 in the year 2101 (a 70% de !), and the death rate jumps up to
almost 10 %. In this way, all economic activities will be trapped. Is there a way
to escape from this economic trap?
Substitutes for Non-Renewable Resources
The economic trap mentioned above is basically caused by a diminishing avail-
ability of non-renewable r . To see the effect, let us modify the equation
of non-renewable resource depletion (23) so that it allows an inflow of substitutes
source
for non-renewable resources. Let SU; be an inflow amount of non-renewable
Table 9: Unknown Variable and Constant Added (5)
[New Variable [SU; | Non-Renewable Substitutes
New Constant |v | Level of Substitutes (0 <v <
1)
p | Output-Substitutes Ratio | food/ton
measured by a unit of ton/year, that can be added to the stock of
and v be a level of the substitutes such that 0 <v <1.
substitut
non-renewable resourc
Then the equation (23) is replaced with the following:
Risi = R,+SU,—AR, —(Non-Renewable Resource Depletion) (49)
SU, =vAR, (Substitutes for Non-Renewable Input) (50)
Or, combining these two, we have
Rua =R,—(1—v)AR, — (Non-Renewable Resource Depletion) (51)
Where do the substitutes come from? For simplicity it is
are converted from the output by a factor of output-substitutes ratio. Saving
function (4) then has to be revised as follows:
sumed that they
5. =Y¥, — Cy — pSU; (52)
Figure 11 illustrates the SD modeling implementation of the substitutes for
non-renewable resout
Several simulations are done, under such circumstances, to attain the growth
paths of the golden rule of capital accumulation illustrated in Figure 10. It
turned out that at least 400 unit machines of initial capital stock are needed to
drive an economic growth initially. So the initial capital stock is reset again to
K2991 = 400. Even so, if a level of substitutes is set high above 80%, the economy
again turns out to be trapped. T.
higher rate of substitut: pposed to preserve the non-renewable resoure
moment of thought clarifies the reason. A higher level of substitutes subtr
more portion of output, and capital accumulation begins to decline with less
saving and investment. On the other hand, a lower level of substitutes depletes
non-renewable resources faster, reducing productivity and output. Again, the
economy is trapped.
Tf a level of substitutes is 80%, the economy can recover from the economic
trap that is caused by a negative feedback loop of non-renewable resources as
illustrated in Figure 13, and once again attain the growth paths of golden rule
for the entire 21st century. Figure 14 illustrates such golden rule growth paths
up around the turn of the 21st century.
However, this is nothing but postponing a problem of economic trap to the
22nd century, and there will be no way to escape from the economic trap in the
long run. Figure 14 shows exactly the same structure as in the Growth Paths
with Social Reproducibility Feedback in Figure 13, except that a time scale is
clongated over two centuries in the present case. Substitutes of non-renewable
annot be an economic savior in the long run.
s a little bit surprising result, because a
A
resourt
8,000 machine
8,000 person
2,500 food/Year
1,000 ton
machine 5
person x
food/Year ae
ton :
2001 2035-2068 = 20221352169 202
Time (Year)
Capital : substitutes machine
Population : substitutes person
Net Output : substitutes foot/Year
Consumption : substitutes ‘food/Year
"Non-Renewable Resources” : ton
Figure 14: Social Reproducibility Economic Trap in the Long Run
4 Ecological Reproducibility
Production and consumption activities as well as capital accumulation formal-
ized above produce as by-products consumer garbage GC;, industrial wastes GY;
and capital depreciation dumping GK;. These by-products are in turn dumped
into the earth or s an artificial
attered around atmosphere and accumulated as
environmental stock called sink $K141. Some portion of the sink will be natu-
rally regenerated (or recycled) and made available as renewable resource stock
that is called source SRr41. As al example, we can refer to photosynth
is processes in which tropical forests and trees grow by taking carbon dioxide
(industrial wastes) as inputs and producing oxygen as by-product output.
These three dumping processes together with an extracting process of non-
renewable resources now form an entire global environment Env, consisting of
the earth’s sink and source. Hence, the formation of the entire global environ-
ment might be appropriately considered as an ecological reproduction proc
which is symbolically illustrated a
(CAR: © GC, © GY, @ GK) => Env(SKiy1 > SRi41). (53)
we need to add the
To describe such an ecological reproduction proces
following seven equations.
SK = SK, + ASK; (Accumulation of Sink) (54)
ASK, = GC, + GY; + Gk; — (e+ )SK, (Net Change in Sink) (55)
GCL = eC (Consumer Garbage) (56)
OY, = wh (Industrial Wastes) (57)
GK, = 12: (Depreciation Dumping) (58)
SRiz1 = SR, + ASR, (Accumulation of Source) (59)
ASR, = (+ WSK, — Yi (Net Change in Source) (60)
Table 10: Unknown Variables and Constants Added (6)
SK | Sink source
New SRizi | Source
GC;_| Consumer Garbage
GY, Industrial Was
Gk, | Capital Depreciation Dump-
ing
€ Natural Rate of Regeneration
(= 0.15)
mn Recycling Rate (= 0.05)
New M1 Renewable Raw Material In-
put Rate (= 0.6)
Constants Ye Garbage Rate(= 0.5)
yy | Industrial Wa Rate (=
0.1)
7x | Depreciation Dumping Rate | source/machine
(= 0.5)
Initial values | SK, Initial Sink (=300) source
SR, Initial Source (=3,000) source
In order for an ecological reproduction process to continue, total amount of
consumer garbage, indu
be less than the earth’s ecological ca
those newly regencrated source have to add enough amount to renewable source
for continued production activities
accumulate, and the accumulated
and capital depreciation dumping have to
pacity to absorb and dissolve the sink, and
Otherwise, the amount of sink begins to
ink will eventually cause the environment
to collapse, or renewable source will be completely depleted. Therefore, for a
sustainable ecological reproducibility, the following two conditions have to be
met.
pe] Capital x —_
Invcximent Depreciation
ry aro
Initial Capitat | \ Depreciation Rate
| &
| Net Ouput
— "Productivity
x ay
Per Copia it Output Raio |
> Consumption
Consumption g \
OuputLaborRaio |
4 PT wae a | prada
Works aie
\ { \ Rave Material \
\ ‘Input Rare | \
Subsistence ‘ \ Initial Non-Renewable
Consumprion Labo Foci Rewources
Level ~ Participation
Consumer <é ‘Rate \
Garbage |
| / __ Wasting
| si Rare
| : \
Mi r — Level of
| Inatustiat Substitutes
| Wastes
\ | i Death Rate
- Table
Recycling —
Rate Renewable Rew
Material Input Rate
Recycling
a
sink 2 | Source
Regeneration
Raw Material
AA Natura Rate of
Regeneration Initial Source
Figure 15: Ecological Reproducibility Model
Ye (GC. + GY +E) <¢ S2 SKiy1 [Ecological Reproducibility]
122001 22001
(61)
SRiy1 > 0, t= 2001, ++ [Renewable Resource Availability]
(62)
ling sink into source and
a self-
Fortunately, the ecological reproducibility of rec
restoring the original ecological shape has been built in the earth ¢
regulatory mechanism of Gaia [3]. Consumer garbage, industrial wastes and
capital depreciation dumping have been taken care of and disintegrated by a
natural reproduction process, and the environment so far seems to have contin-
ued to restore itself to a certain degree. Therefore, a sustainable development
might be possible for the time being so long as the accumulated sink which the
ecological reproduction process fails to disintegrate does not reach the environ-
mental capacity of regeneration.
As production and consumption activitie
expand exponentially, however,
And natu-
rally built-in ecological reproducibility of Gaia eventually begins to fail to regen-
crate the sink so that a portion of the sink will be left unprocessed Eventually, an
environmental catastrophe occurs, and the earth might become uninhabitable
for many living species, including human beings. In fact, many environment
scientists warn us that such a catastrophe has already begun. For instance, see
5].
Accordingly, to be able to stay within a limit to ecological reproducibility,
such environmental sink also continues to accumulate exponentially.
fi
within an environmental regenerating capacity. Second, new development of
st of all, the total amount of environmental sink has to be directly regulated
recycling-oriented products has to be encouraged so that the amount of en-
vironmental sink is reduced at every cycle of reproduction and consumption
which are not naturally disposed
cycled safely at all costs. Then, the
ity (61) is expanded as follow
process. Third, hazardous and toxic was
of have to be chemically processed and re
equation of ecological reproduc’
Ye (GCr+ GY + Gh) <(e+n) S2 $Kiy1 [Recycling of Sink] (63)
t=2001 t=2001
A Steady State Equilibrium
A steady state equilibrium of of the ecological reproducibility is attained at
SK = SK, and SRis1 = SR;; that is, ASK, = ASR, = 0. From the above
equations of ecological reproducibility this implies
GC. + GY, + Gk; = (+p) SK = MY
cumulation is already obtained under the secti
ng the same numerical values of that steady state,
and constant values assigned in Table 10, we have
GC; + GY; + Gky = 0.5-80 + 0.1- 100 +0.5- 20 = 60. (65)
(€ + p)8Ki = (0.15 + 0.05)300 = 60. (66)
MY; = 0.6 - 100 = 60. (67)
A steady state of population growth is attained when rates of birth and death
are equal as shown under the cial reproducibility. Hence, a steady
sate of ecological reproducibility is shown to exist and our model of the ec
logical reproducibility
non-renewable resources are considered explicitly. Figure 16 illustrates that an
ection of si
becomes consistent. However, this is no longer true if
400 source
4,000 source |
100 food/Year
600 person
1,000 ton
0 source
source
food/Year
person
ton
2001-2035 «2068 = 2102, 213521692202
Time (Year)
Sink : steady state source
Source
‘food/Year
person
ton
Figure 16: A Steady State of Ecological Reproducibility
ecological steady state equilibrium is sustained almost throughout the 21st cen-
tury until net output starts d ing in the year 2082. This decrease in the
net output is caused by a diminishing productivity, which is in turn caused by
the depletion of non-renewable resources. Accordingly, per capita consumption
deer s, resulting in a decline of population growth
that begins to start in the year 2091, a decade later. Hence, an ecological steady
state equilibrium becomes impossible in the long run if non-renewable resources
are taken into consideration.
and a death rate increas
Simulations for Sustainable Growth
that is
When a depreciation rate and a birth rate are reset at the original values;
6 = 0.05 and a = 0.03, respectively, the economy begins to grow. How
growth paths are eventually curbed by the depleting non-renewable resources
as illustrated in Figure 17.
To avoid such restrictions of the growth paths, a level of substitutes might
be set to be 80% as in the previous section. Then the net output and popu-
lation once again continue to grow for the entire 21st century. However, this
sustained growth paths begin to cause a problem of ecological unsustainability.
The amount of sink continues to accumulate and source is completely depleted
in the year 2077 as illustrated in Figure 18.
Eventually some negative feedback loops might emerge to prevent such envi-
ronmental catastrophe umulated amount of the sink
such as chemical wastes will surely affect human health and a birth rate will be
reduced as a result: a feedback loop from the sink to the birth rate. Meanwhile,
as renewable source continue to be depleted, output will be curbed as in the case
; this
For instan
1,100 source
4,000 source
500 food/Year
1,800 person
1,000 ton
0 source
0 source \
0 food/Year
0 person
0 ton
2001 2026 2052 2077 2102
Time (Year)
‘Sink : growth source
Source : growth source
Net Output : food/Year
Populatio person
"Non-Renewable Resources” : growth ton
Figure 17: Growth Paths of Ecological Reproducibility
of the depletion of non-renewable re a feedback loop from the source to
output.
Such loops can be built by introducing appropriate table functions. In this
ity might be restored by avoiding the problems
nk and the depletion of the source. Due to a
, we only show one such
growth paths in Figure 19 without specifying here numerical values of table
functions for the negative feedback loops. With the introduction of 80% level of
s together with ecological feedback loop:
ainable paths over the next two centurie:
It is worth warning, however, that the economic trap illustrated in Figure 14
will eventually emerge in the 23rd century so long as non-renewable resources
continue to be depleted!
ourc
way an ecological reproducib
of the over-accumulation of the
tainable
References
{1] Olivier Blanchard. Macoeconomics. Prentice Hall, Inc., New York, 1997.
{2] Donella H. Meadows, Dennis L. Meadows and Jorgen Randers. Beyond the
Limits. Chelsea Green Publishing, Post Mills, Vt., 1992.
[3] J.B. Lovelock. The Ages of Gaia: A Biography of Our Living Earth. Bantam
Books, New York, 1988.
[4] World Commission on Environment and Development. Our Common Future.
Oxford University Press, Oxford, 1987.
[5] Theo Colborn, Dianne Dumanoski and John Peterson Myers. Our Stolen
Future. A Plume/Penguin Book, New York, 1997.
[6] Kaoru Yamaguchi. Beyond Walras, Keynes and Marx ~ Synthesis in Eco-
nomic Theory Toward a New Social Design. Peter Lang Publishing, Inc.,
New York, 1988.
[7] Kaoru Yamaguchi, editor. Sustainable Global Communities in the Informa-
tion Age ~ Visions from Futures Studies, chapter 5, pages 47 — 65. Adaman-
tine Press Limited, 1997.
ca
Kaoru Yamaguchi. Stock-flow fundamentals, dealt time (dt) and feedback
loop — from dynamics to system dynamics. OSU Journal of Business Ad-
ministration, 1(2):57 — 76, 2000.
4,000 source
4,000 source
900 food/Year
4,000 person
1,000 ton
0 source
0 source
0 food/Year
0 person
600 ton
2001 2026
2052 2077 2102
Time (Year)
"Non-Renewable Resources" : substitutes
food/Year
person
ton
Figure 18: Growth Paths with Non-Renewable Substitutes
1,200 source
4,000 source
900 food/Year
1,800 person
1,000 ton
0 source
source
food/Year
‘person
ton
2001-2035 2068
2102-2135 21692202
Time (Year)
Sink : feedbach3 source
Source : feedback3 source
Net Outpi food/Year
Population : person
"Non-Renewable Resources” : feedback
Figure 19: Growth Paths
ton
with Ecological Feedback
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