System Dynamics Model of Technology and Economic
Growth: A Preliminary Study
Muhammad Tasrif
Graduate Program in Development Studies
School of Architecture, Planning and Policy Development
Bandung Institute of Technology
Jl. Ganesha 10 Bandung 40132 - Indonesia
email: muhammadtasrif52@ gmail.com
Abstract
The models of technological change and economic growth those have been developed
so far do not provide satisfying directions for policy purposes. In this study, a simple
system dynamics model based on an integration of micro- and macroeconomic theories
is constructed to explore the process of technological change affecting the i
growth. It is hoped that by understanding the process, the developing country may have
some directions more clearly how to design its technology policies. The capital-labor
ratio change is used to represent the technology change and the mathematical
equations of the model are derived from the underlying economic concepts. The main
point of deriving the i is that the production function has a capital intensity
which is not constant. The study resulted in an important finding that the capital
intensity is affected by the average life of capital in a negative direction. The study
shows that the increase in capital intensity is an important source of the economic
growth. This increase will strengthen the accelerator mechanism of the economy and
creates larger multiplier effects. The increase in capital intensity can be obtained
through managing innovation processes base on the development of education and the
R&D capacity of the nation.
Keywords: Capital-labor ratio, Capital intensity, Innovation, System dynamics
1. Introduction
It is an accepted view that technological progress is an extremely important, perhaps
the most important, determinant in the growth in output per man. In the discussions of
the role of technological change in the economy, one of some important questions
naturally arise is how does technological change affect different factors (capital and
labor). Traditionally, some technological changes are thought of as “labor intensive”,
and some as “capital intensive”.
As a milestone in the theory of economic growth literatures, Solow (1956)
modeled the technological change through simply multiplying the production function
by an exogenous increasing scale factor A(t). The term A(t) in the production function
represents all the influences that go into determining output besides capital and labor.
Changes in A over time represent technical progress. Thereafter, there are some studies
those have been trying to replace the term A by some endogenous variables and to
specify the exact real world meaning of those variables. Among others are knowledge
accumulation (education), R&D (Research and Development), and human capital. The
model those have been developed so far do not provide satisfying directions for policy
purposes. For a developing country, the most important question is how to design a
robust strategy of technological changes those can be expected to improve her national
productivity considerably. It is important for policy design that the model has to have an
appropriate policy space to explore the entry points for an evolutionary change. As a
basis for design, the model structure and the behavior of the model and its empirical
relevance has to be fully understood.
A simple system dynamics model based on an integration of micro- and
macroeconomic theories is constucted to explore the process of technological change
(technology) affecting the economic growth. It is hoped that by understanding the
process, the developing country may have some directions more clearly how to design
its technology policies. Firstly, a theoretical framework of technology and economic
growth is described as a basis to construct the model. Secondly, the important features
of the system dynamics methodology is briefly explained. By using the methodology
then, the framework is converted into a system dynamics model of technology and
economic growth; described in the third part of the paper. Furthermore, some
experiments of the sources of economic growth are simulated using the model; and the
long-run growth pattems resulted from the experiments are analyzed. An attempt base
on the process oriented approach is made to build an understanding of the role of
technology in the economy.
The study shows that, for long term strategies, the process of a sustainable
increase in capital intensity of the economy is an important source of the economic
growth. The process, in which the increase in capital intensity of the economy can be
maintained in the long-run, may be the direction of the robust strategies for developing
nations to improve their national productivity considerably. The increase in capital
intensity of the economy can be obtained through managing innovation processes base
on the development of education and the R&D capacity of the nation.
2. Theoretical framework
Hicks and Harrod among others, have proposed the different definitions of a neutral
technical change. If the change of relative shares is used as the measure of bias of
technological change, then the Hicks definition measures the bias along a constant
capital-labor ratio while the Harrod definition measures the bias along a constant
capital-output ratio. In the growth literature, Harrod neutrality has played a more central
role. It has often been alleged that technological change in fact is Harrod neutral.
Stiglitz and Uzawa observed an “almost constant capital-output ratio with an almost
constant rate of interest” (Stiglitz and Uzawa, 1969). Bach (1968) has showed the
validity of their observations. It is based on the patterns of the economic growth in the
US that showed that capital-output ratio and the interest rate were roughly flat in trend
between 1900-1965. On the other hand, the capital-labor ratio was steadily increased in
trend. The fact of the increase in capital-labor ratio can be also observed in the
discussions of the relationship between labor productivity and capital-labor ratio
(Sumanth, 1985).
In this study, using the Harrod definition of technological change, the capital-
labor ratio is used to represent the technology (technology embeds in capital and labor).
Thus, the changes of capital-labor ratio in an economy represent the technological
changes in the economy. An increase in the capital-labor ratio of an economy means the
increase in technology level of the economy. Therefore, the main focus in constructing
the structure (physical and decision making structures) of the model is the mechanism
of changes in capital-labor ratio that may affect the economic growth.
For such purpose, the mathematical equation of the model is derived from the
underlying economic concepts (see Appendix A). The first main point of deriving the
equations is that the production function, for society’s output as a whole, has a capital
intensity which is not constant. The second is that the study uses the standard
neoclassical assumption of profit-maximizing behavior to model the decision making
structures of the acquisition of production factors i.e. capital and labor.
The important equations used to develop the model are as follows (taken from
Appendix A).
Equation (1) the production function:
Kia
q = 4o* G cP
4, Yo = production, initial production [unit/year]
K, Ko = capital, initial capital [unit]
L, Lo = labor, initial labor [person]
a = capital intensity [dimensionless], not constant
B = labor intensity [dimensionless], not constant.
Equation (5) the optimum capital K:
Peace
(em
alk = average life of capital [year], not constant
R_ =real interest rate [1/year], constant.
Equation (8) the optimum labor L:
—_ Bea
rr
rw =real wage rate [unit/year/person].
Equation (9) the capital intensity a:
L
a= KoR+(—+ R)
KOR =capital-output ratio, constant [year].
And Equation (24) the production (economic) growth rate:
Gy= @ Gx + —@) G, + KOR « In (2) « [Gre— Gel,
where G;, is the growth rate of production gq, Gx is the growth rate of capital K, G; is
the growth rate of labor L, Gxzp is the growth rate of capital-labor ratio KLR, and G,,
is the growth rate of real wage rw.
3. A brief description of system dynamics methodology
Some references in system dynamics literatures, concerning the structure (physical and
decision making structures) of system dynamics model, are considered in constructing
the model of this study as follows.
.
“System dynamics is a methodology for studying and managing complex
feedback systems, such as one finds in business and other social systems.”
[System Dynamics Home Page.htm]
Forrester (Forrester, 1990, pp. 4-2 - 4-5):
“In concept a feedback system is a closed system. Its dynamic behavior arises
within its internal structure. Any interaction which is essential to the behavior
mode being investigated must be included inside the system boundary. Within
the system boundary, the basic building block is the feedback loop. The
feedback loop is a path coupling decision, action, level (or condition) of the
system, and information, with the path returning to the decision point. Every
decision is made within a feedback loop. The decision controls action which
alters the system levels which influence the decision. There are two fundamental
types of variable elements within each loop--the levels, and the rates. The level
variables accumulate the results of action within the system. As flows
influencing the levels, the rates are the results of action that cause the level to
change.”
[Therefore, in constructing a model for policy analysis using the system
dynamics methodology, the model has to reflect the way decision is actually
made in the system.]
Sterman (Sterman, 1981):
“1. Desired states and actual states must be distinguished.” [p. 50]
“The variables and relationships should have real world meanings; equations
should balance dimensionally without the addition of scaling factors or
parameters.” [p. 52]
Richardson & Pugh III (Richardson & Pugh III, 1981):
“The system dynamics approach to complex problems focuses on feedback
processes. It takes the philosophical position that feedback structures are
responsible for the changes we experience over time. The premise is that
is of system structure and will become
meaningful oad powerful. At this point, it may be treated as a postulate, or
perhaps as a conjecture yet to be demonstrated.” [p. 15]
[There are two structures: physical and decision-making structure.]
Saeed (Saeed, 1994):
“Empirical evidence is the driving force both for delineating micro-structure
of the model and verifying its behaviour, although the information conceming
the behavior may reside in the historical data and that concerning the micro-
structure in the experience of the people [Forrester 1979].” [p. 22]
“The dynamic hypothesis must incorporate causal relations based on
information about the decision mules used by the actors of the system, and not
on correlations between variables observed in the historical data.” [p. 22]
“The model structure must be “robust” to extreme conditions and be
“jdentifiable” in the “real world” for it to have credibility, where real world
consists both of theoretical expositions and experiential information.” [p. 22]
“When a close correspondence is simultaneously achieved between the
structure of the model and the theoretical and experiential information about
the system, and also between the behavior of the model and the empirical
evidence about the behavior of the system, the model is accepted as a valid
representation of the system. [Bell & Senge 1980, Forrester & Senge 1980,
Richardson & Pugh III 1981].” [p. 23]
Based on the mentioned references above, the main (important) features of the
system dynamics methodology in constructing the structure of the model are
summarized as follows.
(a) Is the model structure consistent with relevant descriptive knowledge of the
system?
(b) Does the model conform to basic physical laws?
(c) Do the decision rules capture the behavior of the actors in the system?
(d) Is each equation dimensionally consistent without the use of parameters
having no real world meaning?
(e) Do all parameters have real world counterparts?
4, Model description
By using the system dynamics methodology mentioned above, the theoretical
framework of the model is converted into a simple system dynamics model of
technology and economic growth. The model is formulated from a familiar theoretical
model, the multiplier-accelerator model of Samuelson (Samuelson, 1939), with several
minor modifications. The model also considers the inventory adjustment model of
Metzler (Metzler, 1941). The model consists of a single-sector two-factor production
system that incorporates national income accounting at an aggregate level and an
important mechanism to determine capital-labor ratio. There are 4 sub-models namely
Income Sub-model, Labor & Unemployment Sub-model, Wage Sub-model, and
Innovation Sub-model as shown in Figure 1. In this preliminary study, the model uses a
constant price measures (real terms) so that the price is not included in the model;
therefore there is no need to consider the money balance in the model. It means that
there is no influence of money availability to economic decisions. The model does not
also consider imports and exports.
The Innovation Sub-model has not yet been developed. In developing a model of
technology and economic growth one has to include the innovation activities because
innovations (basic innovations) create a new type of human activity as stated by Mensch
(Mensch, 1979, p. 47):
“innovations which produce new markets and industrial branches... or open new
realms of activity in the cultural sphere, in public administration, and in social
services. Basic innovations create a new type of human activity.”
In the real world the above statement can be interpreted that the innovations will
produce new products (goods) in turn, in macro (global) term, making the average life
of goods (including capital) becoming more shorter (empirical evidences support this
interpretation). In the developed model, due to the innovation sub-model has not been
yet constructed; the life of capital is treated as an exogenous variable and becoming one
of the growth sources. Besides this, there are 2 more exogenous variables i.e. population
and goverment spending fraction as shown in Figure 1.
Income a——— Government
| Submodel spending fraction
Population (GDP,C,G,Capital,
inventory) \ eset
Submodel
Desired (Education,
labor R&D)
Cs .
Labor &
Submodel
increase
Wage
Submodel
Desired
capital
Life of capital
Labor productivity
Figure 1: The global structure of the model
In the Innovation Sub-model there are 2 important sectors those have to be
included into the technology and economic growth model, i.e. education sector and
R&D (Research and Development) sector. The nation education level produced by the
education sector represents the “repetition capability” of the nation, meanwhile the
effective R&D activities of a nation represents the “generating capability” of the nation.
There is a positive feedback relationship between those two sectors. In system dynamics
methodology those two sectors create a growth behavior of the system and becoming a
powerful structure that has to be considered seriously by a developing country.
Figure 2 is the causal loop diagram of Income Sub-model. As shown in Figure 2,
the multiplier-accelerator principle is represented through 2 positive feedback loops
namely M/+ loop and A/+ loop respectively. These two positive loops are the growth
engine of the economy. When there is an increase in aggregate demand through the
increase in investment, or consumption, or government spending; these increases will be
multiplied and accelerated by those two loops. Due to the assumption in the model that
capital intensity Alpha («) can vary (not constant) caused by the change in life of capital
(as illustrated in Figure 2), an increase in Alpha will augment the positive accelerator
loop through the increase in desired capital (illustrated in Figure 2). The change in
Alpha is determined by the change in life of capital alk which can be explained through
Equation (9)
a= KOR«(=+ R).
In this equation capital-output ratio KOR and real interest rate R are constant. Based on
this equation a decrease in life of capital, caused by the effective innovations, will
increase the a. Meanwhile Desired capital is determined through Equation (5)
= aoe
re
Production q is replaced by long-run expected demand accommodating the main
important features of system dynamics model (described before) in constructing the
structure of the system dynamics model. The change of Alpha that is considered in the
model of technology and economic growth can be thought as a mechanism how the
technology which is produced by the effective innovations affecting the economic
growth.
"
. . Permanentincome_+ Government
yo. ae spending fraction
Potential output Production Desired
<Capital> ‘i Het outpal + ++ consumption:
Desired Desired govt +
M/+
+
Short-run.
expected
demand
inventory spending
+ Consumption
Govt spending
Desired inventory
investment
Tong-run
expected
demand
st Desired capital (a/y
Desired investmel
Capital-labor ratio
Figure 2: Causal loop diagram of Income Sub-model
The following figure of Figure 3 is the causal loop diagram of Labor &
Unemployment Sub-model and Wage Sub-model. As shown in Figure 3, an increase in
Alpha may reduce desired labor when at the same time there is also a decrease in short-
tun expected demand and an increase in wage. In turn, the reduced desired labor will
increase the unemployment rate and then will decrease the wage. At the end this will
increase the desired labor (balancing behavior of the negative feedback loop between
desired labor through the pool of unemployment). This behavior can be explained
through Equation (8) which is used to determine the desired labor as follows. Equation
(8) is
| hg i)
rw
where beta is 1-Alpha represents the labor intensity of the economy. The output q in
Equation (8) in the model is replaced by short-run expected demand. The increased
unemployment rate due to the increase in Alpha can be prevented if the augmented
economic growth caused by this increased Alpha (technology) is able to maintain the
growth of the aggregate demand.
ze Life of capital
Alpha
ta Employment time
Figure 3: The causal loop diagram of Labor & Unemployment Sub-model and Wage
Sub-model
5. Simulation results and analysis
As an attempt to explore the process of technological change affecting the economic
growth that the developing country may have some directions more clearly how to
design its technology policies; there are three sources of growth (growth scenarios) are
considered in this study i.e.: population, government spending, and innovation (through
life of capital). The model is initialized in the full equilibrium. In this equilibrium the
population growth rate is equal to zero, the government spending fraction is 0.15, and
life of capital is 14 years providing the capital intensity Alpha is 0.25. The model is
simulated for 350 years, and the changes of the source of growths are introduced into
the model in year 50.
In trying to understand the paths of the economic growth due to technology
development in the economy, the first experiment of 3 scenarios are simulated using the
model to show whether those three sources of the growth considering in this study can
produce an increase in production. First, in the absence of changes in life of capital and
government spending fraction, the model is simulated with introducing 1% per year
growth in the population namely “Population”. Second, only the change in government
spending fraction is introduced into the model (becoming 20% from 15%) namely
“Goverment”. And the third, only the gradually change in life of capital is introduced
into the model from 14 years to 10 years in the period of 100 years, namely
“Technology”. The behavior comparisons of those 3 scenarios are shown in Figure 4,
Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, and Figure 10.
Life of capital scenarios
0 50 100 150 200 250 300 350
Life of capital : Populati
Life of capital : Government
Life of capital : Technology
Figure 4: Life of capital scenarios
Kessler Syndrome: System Dynamics A pproach
Jakub Drmola!, Tomas Hubik?
'Division of Security and Strategic Studies at the Faculty of Social Sciences of Masaryk
University in Bmo, Czech Republic
2Department of Theoretical Computer Science and Mathematical Logic at the Faculty of
Mathematics and Physics of Charles University in Prague
Yostova 10, 602 00 Bmo, Czech Republic
?Malostranskw namesti 25, 118 00 Praha 1, Czech Republic
4drmola@mail.muni.cz
?hubik.tomas@qmail.com
The present paper explores the Kessler Syndrome (the potentially catastrophic
accumulation of debris in the Low-Earth Orhit) through System Dynamics methodology. It
models satellites and three classes of debris, their fragmentation, interactions and gradual
decay over 50 years. It presents three scenarios: a) a “business as usual” approach, which
leads to exponential accumulation and growing rate of satellite losses, but no catastrophic
chain reaction; b) a conflict with a large-scale deployment of Anti-Satellite Weapons,
leading to accelerated accumulation and losses, but still no chain reaction; and c) cessation.
of all LEO satellite launches, illustrating high inertia of the system, which continues to
produce more debris. Both b) and c) take place in 2040. The paper demonstrates the gravity
of the situation and the necessity for a sustainable long-term solution, as orbital debris
poses a threat to our future space operation even without triggering a catastrophic chain
reaction.
Keywords: System Dynamics, Kessler Syndrome, LEO, Satellites, Orbital Debris, Anti-
Satellite Weapons, Debris Evolutionary Model
Introduction
There are over 29,000 man-made objects greater than 10cm in size in the orbit.
These objects include defunct satellites, pieces of spacecraft, mission related debris and
other pieces of space junk. The number of these objects is continuously growing due to
the continuing launch activity and spontaneous space collisions and breakups.
Atmospheric drag force is not sufficient to stop this trend. Methods and techniques on
how to stop this growth are becoming a more and more relevant topic when talking
about space programs. Precise tracking of all these objects is critical for any space
mission to succeed.
As the number of objects in the orbit increases, the likelihood of collisions
increases as well. A typical space object experiences several close flybys per day. By
close we mean a couple of kilometers in distance. Each flyby has a certain probability of
Dmnl
Alpha scenarios
0.5
0.375
0.25
0.125
0
0 50 100 150 200 250 300 350
Year
Alpha : Population Alpha : Technology —————_
Alpha : Govermment—————
Figure 5: Alpha scenarios
Population growth scenarios
2
15
1
0.5
0
0 50 100 150 200 250 300 350
Population growth : Populati
Population growth : Goverment
Population growth : Technology
Figure 6: Population growth scenarios
Production
4e+013
3e+013
g 2e+013
=)
0
0 50 100 150 200 250 300 350
Time (Y ear)
Production : Technology
Production : Government
Production : Populatic
Figure 7: Production
Production per capita
60,000
i 45,000
& 30,000
ts)
2
& 15,000
0
0 50 100 150 200 250 300 350
Time (Y ear)
Production per capita : Technology
Production per capita : Goverment:
Production per capita : Populatic
Figure 8: Production per capita
Unemployment rate
10
75
Pol A
2:5
0
0 50 100 150 200 250 300 350
Time (Y ear)
Unemployment rate : Technology
Unemployment rate : Government
Unemployment rate : Populatic
Figure 9: Unemployment rate
Capital labor ratio
600,000
450,000
g
i 300,000
eB
v
150,000
0
0 50 100 150 200 250 300 350
Time (Y ear)
Capital labor ratio : Technology
Capital labor ratio : Goverment:
Capital labor ratio : Populatic
Figure 10: Capital labor ratio
Figure 4, Figure 5, and Figure 6 illustrate the changes of input scenarios of those
three scenarios: the life of capital followed by its effect on the Alpha (the capital
intensity), and the population growth respectively. As sources of economic growth,
those three scenarios show that they can produce an increase in production with
different growth paths (Figure 7). However, the increase in production per capita
(income per capita) can be only obtained by “Government” scenario and “Technology”
scenario (Figure 8). These mean that in “Population” scenario the growth rate of
production is equal to the population growth rate. Besides this constant production per
capita, the long-term unemployment rate of this scenario is higher compared with its
initial and the other two scenarios (Figure 9). The level of technology (the capital-labor
ratio) is also constant in the “Population” scenario, meanwhile in the “Government” and
the “Technology” scenarios the technology levels are increasing (Figure 10).
The second experiment of 5 scenarios is done to understand the power of
technology (innovation) in producing a higher sustainability of economic growth. In this
experiment, for those 5 scenarios, the population growth rate is set at 1% per year.
Those 5 scenarios are namely: (1) “Population” (population growth rate of 1% per year
only), (2) “Population+Government” (“Population” scenario plus an increase in
government spending fraction from 15% to 20%), (3) “Population+Technology 10”
(“Population” scenario plus a gradually change in life of capital from 14 years to 10
years), (4) “Population+T echnology 10+Government”, and (5)
“Population+Technology 7” (“Population” scenario plus a gradually change in life of
capital from 14 years to 7 years). The simulation results are shown from Figure 11 until
Figure 17.
Life of capital scenarios
0 50 100 150 200 250 300 350
Life of capital : Population
Life of capital Gi
Life of capital : Population#Technology 10
Life of capital : PopulationTechnology
Life of capital : Population Technology 7
Figure 11: Life of capital scenarios
Alpha scenarios
0.5
0.375
E025
0.125
0
0 50 100 150 200 250 300 350
Year
Alpha : Population
Alpha : Populati mment
Alpha : Population+Technology 10
Alpha : Population+Technology 10+Government
Alpha : Population+Technology 7
Figure 12: Alpha scenarios
Population growth scenarios
2
15
g
i 1
0.5
0
0 50 100 150 200 250 300 350
Population
Population growth : Population+Technology 10
Population growth : Population+Technology
Population growth : Population+Technology 7
Figure 13: Population growth scenarios
‘UniY ear*Person)
Production
4e+014
3e+014
8
= 2e+014
5
1e+014
0
0 50 100 150 200 250 300 350
Time (Y ear)
Production : Popul: Technology 7
Production : Popul: Technology 1
Production : Population+Technology 10
Production : Populati
p
Production : Population
Figure 14: Production
Production per capita
60,000
45,000
30,000
15,000
0
Production per capita
Production per capita
Production per capita
Production per capita
Production per capita
50 100 150 200 250 300
Time (Y ear)
350
Population+Technology 7
Population+Technology
Population+Technology 10
Population
Figure 15: Production per capita
Unemployment rate
20
15
10 f\
. Ya
0
0 50 100 150 200 250 300 350
Time (Y ear)
t Plain Teck
U te: Popul nology 10
U te : Popul
l te : Population
Figure 16: Unemployment rate
Capital labor ratio
600,000
450,000
5
£ 300,000
5
150,000
0
0 50 100 150 200 250 300 350
Time (Y ear)
Capital labor ratio : Population+Technology 7
Capital labor ratio : Population+Technology
Capital labor ratio : Population+Technology 10
Capital labor ratio
Capital labor ratio : Population
Figure 17: Capital labor ratio
Figure 11, Figure 12, and Figure 13 illustrate the changes of input scenarios of
those five scenarios: the life of capital followed by its effect on the Alpha (the capital
intensity), and the population growth respectively. The results show differences in
production patterns (Figure 14), production per capita (Figure 15), and capital-labor
ratio representing the technology level (Figure 17). The differences in unemployment
rate patterns (Figure 16) emerge in the transition periods from its initial value to the
slightly higher of new equilibrium value. An analysis of the five scenarios reveals that
the economic growth (production per capita) resulted by the model in the scenarios
those augmented with technology (innovation) surpass those in the other scenarios
without technology augmentation. The innovation in the long term will decrease the
average life of capital of the economy; in turn this will increase the capital intensity of
the economy. The increase in capital intensity will strengthen the accelerator
mechanism of the economy and creates larger multiplier effects. Apparently, in this
scenario the behavior of the unemployment rate fluctuates in the transition periods from
its initial value to the slightly higher of new equilibrium value. This indicates that
developing countries have to develop policies for establishing more efficient labor
market. The simulation results show the power of the technology (innovation),
represented by an increase in the capital-labor ratio of the economy (Figure 17), in
producing the higher sustainability growth of the economy.
6. Concluding remarks
This study is an attempt to investigate and to understand the dynamic effects of
technological changes (technology development through innovation activities) on the
growth of the economy. Using the changes in capital-labor ratio as a representation of
technological changes, three sources of economic growth (and its combination) are
simulated, i.e.: population growth, increasing in government spending, and innovation.
A theoretical framework of the model is derived using the standard assumption of
profit-maximizing behavior to model the decision making structures of the acquisition
of production factors (capital and labor). The framework is then converted to a system
dynamics model which is used to explore the power of the technology in maintaining
the higher sustainability growth of the economy.
The study shows that the economy, in which a sustainable increase in capital
intensity of the economy can be maintained through innovation activities in the long
tun, can be expected to improve the economic growth considerably and continuously.
Therefore, developing countries have to manage the innovation processes based on the
development of education and the R&D capacity of the nation. Besides, to reduce the
increase in unemployment rate in the transition periods of the improved of economic
growth due to the increase in technology level; developing countries have to develop
policies for establishing more efficient labor market.
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APPENDIX A
The Mathematics of Technology-Economic Growth Model
Given a production function for society’s ouput as a whole
4= 40° Get a)
4,40 = production, initial production [unit/year]
K, Ko = capital, initial capital [unit]
L, Lo = labor, initial labor [person]
a = capital intensity [dimensionless], not constant
B = labor intensity [dimensionless], not constant
And profit in the model is total revenues (output times price of output) less the holding
cost of capital (capital times the price of capital times the depreciation rate plus the
interest rate) less the cost of labor (labor [employment] times the wage rate), as follows.
Profit = q * P,—K * Py * (=) +R] —L *(rw * P,) (2)
Profit = [$/year]
P, == price of output [$/unit]
Py = =price of capital [$/unit]
alk — =average life of capital [year]
R = real interest rate [/year]
rw =real wage [unit/year/person]
The standard neoclassical assumption of profit-maximizing behavior in the acquisition
of production factors are obtained by setting the partial derivative of profit with respect
to capital equal to zero and setting the partial derivative of profit with respect to labor
(employment) also equal to zero, and solving for capital and labor:
Profit _ anid aProfit _ g
aL
age OProfit
The condition of orroftt
aq 1 Ho
ts p,— Px(t+R)=0. (3)
Based on Equation (1), the partial derivative of output q with respect to capital K is
Ke
aq _ x) 1 Lye
24 = ergy (p)-) * (+) + (4)
K
OK @ Lo
=0 gives
where go» (oe * cy =q [Equation (1)], hence the derivative can be simplified as
aq _ aq 4)
a,
Putting Equation (4) into Equation (3) [(4) > (3)], the Equation (3) can be written as
aq ue _
“1 p,- Px (4+ R)=0,
and assuming that P, = P; (one goods), hence the equation for capital K is obtained as
K=—+. 5
Gm) (5)
The condition of orrefit =0 gives
19
becoming a collision. Every collision creates more debris making the probability even
higher in the future. When the number of objects in the space is sufficiently high, there
is a chance of forming a self-sustaining collisional cascading process, the so called
Kessler Syndrome, named after D. J. Kessler (1978).
In order to better understand this problem and facilitate possible future policy
discussions, we have decided to build a model based on aggregate values of existing
models and datasets to predict the numbers of space debris grouped into four groups -
inactive satellites, large debris (larger than 10 cm), medium debris (between 1 and 10
cm) and small debris (smaller than 1 cm). This categorization is consistent with the one
used by NASA, ESA, and other space agencies.
Model
There are basically two dominant groups of feedback loops in our model. The
first group is reinforcing, meaning that the more debris there is in the space, the more
collisions will occur, creating even more debris (see Figure 1). The secondary group is
balancing the reinforcing loops for bigger objects and strengthening the loops for
smaller objects meaning that the collision will usually destroy some bigger object
creating a cloud of smaller ones.
Active satellites
Gas
ras
=.
aC in LL &
a
Fig. 1 - Simplified Causal Loop Diagram
a
a +P,-(rw* P,)=0 . (6)
Based on Equation (1), the partial derivative of output q with respect to labor L is
a ny", (y+
2 = pxq(Z) * ey? GD
where qo« oy * ~ =q [Equation (1)], hence the derivative can be simplified as
4 ee (7)
aL it
Putting Equation (7) into Equation (6) [(7) > (6)], the Equation (6) can be written as
pea * P, — (rw * P,) = 0 ; hence the equation for labor L is obtained as
L= xa (8)
Assuming that capital-output ratio (KOR) [KOR = K/q] is constant, hence Equation (5)
= —"1__ can be written as
(ant 2)
K__a@ _ 2
a= (en) a 7) (where K/q = KOR), and then gives
a= KOR+(4++R) . (9)
Dividing capital (K) by labor (L) [Equation (5)/Equation (8)], mentioned as capital-
labor ratio, KLR; gives
aerw
KLR=7,—_ : (10)
(at R) 6
Putting Equation (9) into Equation (10) [(9)=> (10)], the Equation (10) can be written as
KoR~ (z+ R) «rw .
KLR = ——“*,——___ ; and then gives
oe (att 8) 8
+Tw
b= aa (1)
From the Equation (11) B = a the change rate of f (labor intencity) df/dt, can
be derived as follows.
rw rw
B _ cons na (ger) dow) . (Rez) d(KLR),
dt a(rw) dt Q(KLR) dt °*
OR 1 d(rw) rw d(KLR)
= :—— ee es Se
(TR dt (KLR)? dt
d(rw) d(KLR)
_ Tw dt TW dt
=KOR* Rw” IR KR!
gives “eb = KOR * a * [Gav — Geral. (12)
d(rw)
where Gv =—— , growth rate of real wage [/year] and
dur)
Garr = a , growth rate of capital-labor ratio [/year].
For a Cobb-Douglas production fuction (i.e. a+ B = 1 or a= 1 —f), the Equation (12)
gives
the change rate of a (capital intensity) da/dt, as
20
d
= KOR + 2 + [Gare - Grek
The growth rate of the output (production) or the economic growth rate (i.e.
be derived as follows.
Given a Cobb-Douglas production fuction of the economy as
a) =4qo« AQ, Oyo:
dq/dt
q
(13)
) can
(14)
and assuming that f(K,a) = G" and g(L,B) = oN, hence the Equation (14) can be
Ko
written as
q= hs
and then gives
=f
=r,
[Note: go (initial production) is a constant]
Differentiating Equation (15) with respect to time gives
dq_, af dg.
asta 7. wl 8 *y 4 where
@) _ Of dK | of da
iin eee
g(Lp)_ Og dl | gdp
dt ‘AL dt + Op dt 3; hence
dq_ q of dK | af da. ag dL , ag dp
ae ja ana * aaa 9 * a ae + ap ae fh
Dividing by q gives
dg/dt _ af/oK dK , df/da da , dg/dt dt , ag/ap ap
qf at fd g at gat
_(af/aK ),) dK/dt af/aa) da | (ag/aL ,\ dL/dt | ,dg/ap, ap
=} ee eel ee + ee
but
af K\@D 4
aK Ge) Ko
K/Ko)%
(K/Ko)Ko then
Of/AK yy _ (K/Ko)® 4 1
f K= (K/Ko) Ko Koen ** oF
of/aK
K=a:
f 5
and the same procedure gives
20/0 =,
In addition, some terms of Equation (18) are derived as follows:
of (K\* K
aa~ (ao) (zs)
where f= (2), and then in f= a in x)
anf _
f da Ko. Ko.
of (K\* K
da Go in G) ’
hence
Of/aa _ (K/Ko)*In(K/Ko)
fo (K/Ko)*
(15)
(16)
(17)
(18)
(19)
(20)
21
af/aa _
or MRE In (< “) (21)
and the same procedure gives
49/08 _ ib
2 = in(¥), 2
Putting Equation (19), Equation (20), Equation (21), and Equation (22) into Equation
(18); the Equation (18) can be written as
dq/dt _ oy Atle dL/dt L)\ dp
“a +n eae Bae ein) es
and using B = 1 — a (Cobb- Douglas production function), hence
dq/dt _ oy Slat K tae L)\ da
AU = +n(2)\S+a-g)S"-m(Z)4 — @3)
Some terms of the above equation, Equation (23), are simplified as follows:
in(£) fin (E ae auela)- afar sted) Sele)
and replacing da/dt in the above equation by Equation (13) & — = KOR* ae * [Geir —
Gy], hence the Equation (23) can be written as
dq/dt _ sia tla
its #(1— +KOR+™ In ce *[Grir— Gr] or
G,= a Ga + ‘fh — a) G_+ KOR * ae In () * [Grr — Gry] - (24)
where Gx is the growth rate of capital K, G, is the growth rate of labor L, Gxzr is the
growth rate of capital-labor ratio KLR, and G,,, is the growth rate of real wage.
22
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m ethiopian hi
We also assume that in order to fragment an object the collision must consist of
objects of similar size. It practice, this means that a satellite can be fragmented only by
another satellite or large object, a large object can be fragmented only by another large
or medium object and a medium object can be fragmented by a medium or small object.
Collision of an active satellite and medium object will disable the satellite making it an
inactive satellite that is not operating anymore and loses its ability to maneuver. The
collision of small debris and a satellite usually does not cause any harm as satellites are
equipped with a shielding protecting them against these small fragments. We are also
not modeling collisions of two active satellites as their positions are known precisely,
their trajectories plotted ahead of time, both can maneuver, and the evasion success rate
is so high that such a collision has never occurred so far.
Besides these spontaneous collisions we are also modeling a certain satellite
malfunction rate and an effect of anti-satellite weapons. Both effects are simulated as
exogenous factors with parameters estimated based on real-world data. The model also
includes variable solar activity (11-year cycle) and its impact on decay times, as it heats
up the atmosphere and increases drag on orbiting objects.
The probabilities and rate of collisions of objects from different groups were
calculated using a conversion coefficient converting the rate of collisions between
objects from one group to the rate of collisions between objects from another group.
The initial rate was estimated using repetitive simulations and comparison of the
resulting runs with real data.
Modeled scenarios and preliminary results
Our baseline scenario is best described as “business as usual”, where we simply
extrapolate ongoing trends into the future. Running it for full 50 years (2016-2066)
yields the expected result of perpetually growing amount of debris in the LEO. We can
observe a nearly than 3-fold increase in the large debris (over 10 cm) and a 10-fold
increase in medium debris (1-10 cm) quantities (Figure 2). Perhaps surprisingly, even
such a dramatic increase in numbers still does not result in full realization of the Kessler
Syndrome as most of the satellites being launched remain intact for their full lifetime.
However, it comes with a significant increase in risk to satellites which is manifested by
their higher yearly losses, making satellites operations riskier and more expensive for
govemments and private companies alike.
2016 2026 2036 2046 2056 72066
year
Fig. 2 - Satellites and debris during “business as usual” run
In our second scenario, we imagine some major military conflict erupting in the
year 2040, during which roughly half of all military satellites is destroyed by intentional
kinetic impacts using Anti-Satellite weapons (ASATs). With military and dual-use
satellites generally representing a little over one third of all satellites (depending on the
operating country), this equals to some 200 satellites destroyed by ASATs in 2040
(Figure 3). But even this event is not enough to trigger a chain reaction of satellites
disintegrating in LEO, at least according to our model. Nevertheless, the number of
collisions with active satellites ends up nearly twice as high at the end of the simulation
(i.e. 25 years after the conflict and ASAT strikes) when compared to the previous run.
This shows that the damage would be long-term and would negatively impact satellite
operations (including commercial and scientific ones) for many years after any conflict
nvolving ASATs.
2046 2056 72066
2016 2026 2036
year
Fig. 3 - Satellites and debris during run with assumed conflict
Next, we ran the model without any launches after year 2040 (Figure 4). This serves to
demonstrate the high inertia of the system. Counterintuitively, even with no satellites
being launched, the amount of debris in LEO keeps growing for at least another 5 years.
Only another 10 years later the orbital decay removes enough debris from LEO to retum.
the total amount back to what it was in 2040 (but still much higher than what it was in
the very beginning). This is caused mostly by the ongoing disintegration of already
launched and inactive satellites, which essentially serve as reservoirs of future debris,
ready to be scattered.
2016 2026 2036 2046 2056 72066
year
Fig. 4- Satellites and debris with no launches after year 2040
Our fourth scenario represents an attempt to mitigate the situation and somehow “fix”
the debris accumulation problem. We model this by removing some inactive satellites
from the LEO (again, starting from 2040) before they get fragmented. We do not detail
or discuss any specific method of possible debris mitigation, we simply deorbit the
satellites “somwhow”. We find that removing 8 inactive satellites per month would
stabilize the debris populations (Figure 5).
2016 2026 2036 2046 2056 72066
year
Fig. 5 - Debris mitigation policy
Our fifth and final scenario (Figure 6) is one modeling an EMP gping off,
leading to loss of control over satellites en masse. To model this, we tum one third of
active satellites into inactive ones over a period of one year. This is notably less
catastrophic (from the debris point of view) than using ASAT’.
Saas 2
00k ogee
0.0 -
oo} 7
oof
0.0
2016 2026 2036 2086 2056 2066
year
Fig. 6 - Satellite loss due to EMP
Conclusion
The results show that we are not as close to a catastrophic chain-reaction in LEO
as it might seem. At the same time, the trend is quite clear. And we are not only
approaching the catastrophic cascade at an accelerating rate (despite not being quite
there yet), we are also making our existing and future space operations less safe and
more expensive. Simply put, larye amounts of debris make losses of equipment more
likely. It is also conceivable that orbital collisions with debris could endanger the life of
people working in, or passing through, LEO. Orbital debris can damage critical systems
on space stations, weaken heat shields on spacecrafts, and even small pieces can hit
astronauts during spacewalk.
Today, losses of satellites due to collisions are rather rare, but they might
become quite common within a couple of decades, unless we change how we operate in
LEO. Possibly even more alarming is the long-term effect of debris accumulation. Its
great inertia means we need some sort of solution sooner rather than later, lest we harm
our own future, when we will most likely be even more dependent on satellites and
space technology than we are now.
Despite its high level of abstraction, this work demonstrates that System
Dynamics is a viable approach to modeling the Kessler Syndrome and the associated
issues of accumulation of orbital debris and the threat this poses to our satellites.
Compared to other models, using different methods, it is probably more accessible and
comprehensible to academics and practitioners from other, less technical fields, as well
as to involved policymakers. Furthermore, it can be readily modified to include specific
critical events (such as wars), changes in solar activity, rate of space launches, new
industries and possible future debris mitigation attempts designed to either slow down
its accumulation or even to remove certain fraction of the debris by somehow deorhiting
it
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