EMBODIED ENERGY BASIS FOR
ECONOMICECOLOGIC SYSTEMS
By
ROBERT COSTANZA
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIRE:1EIT7 FOR THE
DEGREE CF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979
Copyright 1979
by
Robert Costanza
AC YrYO w, P r 17'(;
T am greatly indebted to Dr. FT.T. Odum, my com itte?
chairman, for his knowledge, inspiration, insight, and
encouragement. His allencompassing world view gave birth
to many of the concepts which led to this study ani quidel
the wore< to fruition. Many special contributions were made
by the other members of my committee: Drs. S.E. Bayley, B.L.
Capehart, W.C, Huber, and C.D. Kylstra.
B. Hannon and 7. Herendeen at the Center for Advanced
Computation, University of Illinois, contributed experienced
help and encouragement with the inputoutput studies in the
summer o' 1978. 7. Vang and J. Boyles read the manuscript
and provided comments. I would also like to acknowledge tho
valuable interactions with associates and friends,
especially J. Rartholoramel, T. 7ontaine, S. Brown, and D.
Hornbeck.
Work was done at the Center for Wetlands, University of
Florida, and was supported by the Unitad States Department
of Enerqy (Cor tract EY76S054398) project entitled
"Energy Analysis of Models of +he United States," n!.T. Odum
principal' investigator.
TABLE OF CONTFN!TS
page
ACKNOHL EDGE ENTS ............ ............................ iii
LIST OF TABLES ........................................ vi
LIST OF FIG""ER.........................................viii
ABSTRACT................................................ xi
TNTPODrUCTION...................... .. ........ ........... 1
Research Plan.... .. ... ............ . ....... 2
Background of Previous Studies...................... 4
Energy ana Society............... ............. 4
Systems Ecology................................. 5
Energy Analysis..***. *................. ..**** .. 6
Embodied Energy................................. 7
Optimization ..... ........... .... .... ....... ... 13
Economic ,odels... .............................. 13
Spatial economicc Moels......................... 17
Simulation Models............................. 19
Description of the South Florida Area............... 20
METHODS.................................*** ...... 23
Description of the Nodeling Language............... 23
Mo.el Development ...............* ............... 25
Dynamic Op inization...... ..................... 25
Simulation Modeling Methos ..................... 26
Molel Parameter Estimation, Validation and
Testing..................,. ..... .... ......... 27
InputOutput Techniques for Calculating Embodied
Energy. S................... ..........*********** ... 28
Double Counting...........,..................... 41
U.S. Economy Data Assembly and Evaluation........... 43
Government and Households as Endogenous
sectors....................................... 44
Environmental Inputs.................. ..... i.. 52
An Endogenous Environment Sector.................. 55
Capital Plows. ............... ...... .......... 57
South Florida Land Use Data........................ 57
RESULTS.............................. ................. 65
The General Conditions for Maximum Power............ 65
Development of a General, Power Maximizing
Simulation Model... .. ........................... 75
Simulations Using Two Components................ 84
Simulations of Spatial Development Using
25 Cells...................................... 91
page
The U.S. EconomicEcologic System................... 96
Energy Embodied in Goods and Services for 92
U.S. Economy Sectors in 1967................. 96
The Energy to GNP Ratio for the U.S.
From 1920 to 1976................... ..... 114
Total Capital, Investment, and Depreciation
Time Series and a Better Estimate of the
Embodied Energy to Dollar Ratio............... 116
Fourteen Sector Closed System InputOutput
Matrices for 1963 and 1967.................... 126
Five Sector U.S. EconomyEnvironment
Simulation Mocel.............................. 139
The South Florida System............................ 151
Measured Embodied Energy Paps................... 151
Ninty One Cell South Florida Spatial
Simulation Model.................,....,,..... 163
DISCUSSION ..................... ..... .... .. ... . .... 180
The Case for a Constant Embodied Energy to
Dollar Ratio...................................... 180
Conclusions and Predictions from the Simulation
Models.. ............................ .......... 188
rmbodied Energy Analysis and Economics.........,,.. 190
APPENDIX
I SOUTH FLORPDA LAND USE DATA CO"V'F"T TO EMBODIED
T 'EPGY U:.T ... ................................... 193
II ANALOG CCrP"'T? DIAGRAM POPE "H TWO CO"riNFNT
EXCHA r," rODEL.................................... 203
III FORTRAN LISTING FOP THE 25CELL SPATIAL IODEL....... 205
IV ENERGY BODIED INII GOODS AT!D SERVICES FOP 92
U.S. ECO:In"Y SECTORS 'Ir 1967..................... 211
V TIME SERIES DATA 7FO THE U.S.
ECONOHICECOLOGIC SY qln.......................... 219
VI FOrTTAT' LISTIFNG FOP THE 5SECTOR U.S. ECONO1OY
ENVIPONMETIT SIMULATI"!' MODEL....................... 236
VII FOrTF.T: LISTING FOR THE 91CELL SO"[! FLORIDA
SI9MULATION M1ODEL AND DATA ......................... 240
LIST OF Tr ?'?rCES .............................. ...... 247
BIOGRAPHICAL SKETCH..................................... 254
I
LTST OF '"ABLEF
Table 1 p ag
1 Characteristics of the inputoutput and
biosphere embodied energy concepts................ 11
2 Tnputoutput transactions matrix in arbitrary
physical units corresponding to the diagram in
figure 7..................................... .... 32
3 rnputoutput transactions matrix in embodied
energy units corresponding to the diagram in
Figure ........ ............ ... ......... ..... ..... 36
4 Tnputoutput transactions matrix corresponding
to the diagram in Figure 9, using the national
inputoutput accounting conventions............... 39
5 Relationship of inputoutput value added
accounts categories....... ............. .... ...... 49
6 estimatedd lan areas and solar absorption for
major land use types..... ...................... 54
7 Land use subsystem metabolism and structure
estimates in coal equivalents (CE)................ 63
9 Ninty two sector rnmho 'ied energy intensity
statistics............... ,,... ... ............... .103
0 Regression analysis results for total (direct
plus indirect) energy consumption versus total
Dollar output for the four alternative
treatments of labor, government, and solar energy.,113
1I 1197 T.S. business sector capital stock and
investment breakdown (in billions of 1967 dollars).117
11 1967 U.. government sector capital stock and
investments breakdown (in billions of 1967 dollars).118
12 1?67 U.S. household sector capital stock and
investment breakdown (in billions of 1967 dollars) 119
11 1963 aggregate sector net capital stocks, gross
investment, and depreciation (in billions of
1 6" dollars) ..................................... 134
14 1967 aa rare sector net capital stocks, gross
investment, and depreciation (in billions of
1967 dollars)..................................... 136
V1
Table page
15 Sector correspondence............................. 138
16 Initial parameter estimates....................... 149
17 U.S. economyenvironment simulation model
performance statistics for the 19291969 period... 156
18 South Florida simulation model performance
statistics for 1"53 and 1973.................,,... 178
19 South Tlorida land use data converted to embodied
energy units.............................. ...... 193
2V Pmboiedi energy in goods and services for
12 U.S. economy sectors in 1967.................. 211
21 real GrP, total fossil, hydro, and nuclear energy
consumption, and fossil, hydro, and nuclear
energy to real GNP ratio, 19201976............... 219
22 U.7. business sector net capital, investment, and
depreciation time series in constant dollars...... 222
23 TU.. government sector not capital, investment,
and depreciation time series in constant
Dollars ............. .... ........................ 224
24 U.7. household. sector net capital, investment,
and. depreciation time series in constant
Jollars.......... ............................... 226
25 U.S. environment sector, U.S. economy, total
T.S. (Pnvironment plus economy), and rest
of the world net capital stock in
constant dollars.. .................... .. ......... 228
26 Time series or net land stocks in the U.S......... 231
27 mime series of total mineral fuel use and
estimated real dollar value....................,,. 233
i ;
Figure page
1 Solar energy driving the productive processs
of the earth.......................... .........., 8
2 Diagram showing the characteristics of the input
output an" biosphere embodied energy concepts..... 12
3 Diagram showing the standard inputoutput
accounting setup. ........ ......................... 15
4 Location map of south Plorida..................... 22
5 Energy circuit language symbols used in this study 24
6 Single sector energy balance........... 29
7 Hypothetical three sector economy with all flows
in arbitrary physical units.................,,.... 31
S Hypothetical three sector economy with all flows
in embodied energy units.......................... 35
0 Hypothetical three sector economy cast in the
format of the national inputoutput accounting
statistics........................................ 38
10 energy "low diagram of an aggregated 14 sector
U.S. economicecologic system..................... 46
11 Diagram showing definitions of national income
variables.......................................... 47
12 Summary of modifications to the inputoutput
conventions.........,..... ....... ... ............ 53
13 Land use cells for south "'orida.................. 60
it Txample of detailed land use data, showing
cell 45 in Figure 13 for 1973..................... 61
15 Example of detailed land use data, showing
computer coding for cell 45 in Figure 13
for 1973................... ... 62
16 Two production systems and their exchange
pathways. ....................................... 72
17 Energy circuit diagram for a two component
power maximizing model of exchange................ 77
L
v, i i
LITT OF rTGURES
Figure page
1R Differential equations for the model in
Figure 17 ......................... .. .... ....... 79
19 Diagram illustrating the partial production
function relations ............. ................ 81
27 Component i difference equation................... 86
21 Two component nodel analog simulation results..... 88
22 Two component mldel analog simulation results..... 92
21 Digital simulation of the power maximizing
model for a spatial grid of 25 components......... 95
21 Diagram showing the system boundaries and flows
included in 4he four alternatives................. 98
25 "rquncy plots of embodied energy intensities
by sector calculated with and without solar
inputs.... ... ........ ........ .... ..... .. ..... 101
2R 'reguency plots of embodied energy intensities
by sector calculated with and without labor
and government service feedbacks ......,..,..,.. 102
27 Plot of direct plus indirect energy consumption
(calculated excluding solar inputs and labor
and government) versus dollar output for
92 U.S. economy sectors..................,,,..,,, 106
28 Plot of direct plus indirect energy consumption
(calculated including solar inputs but excluding
labor and government) versus dollar output for
92 U.S. economy sectors............................ 108
29 Plot of direct plus indirect energy consumption
(calculated including labor and government but
excluding solar inputs) versus dollar output for
92 U.S. economy sectors........................... 110
3n Plot of direct plus indirect energy consumption
(calculated including solar energy inputs and
labor and government) versus dollar output for
92 U.S. economy sectors....................... 112
31 Fineral, hydro, and nuclear energy consumption
per dollar of real 3NP from 1920 to 1976.......... 115
I
Figure page
32 Time series plot of U.S. business, government,
and household net capital stocks from
1929 to 1969 ...................................... 123
3' Time series plot of 7T.S. environment, U.S.
economy, and total [T.S. net capital stock from
1929 to 1969.............................. ........ 124
34 Time series plot of rest of the world net
capital stocks from 1929 to 1969.................. 125
35 1963 14sector transactions matrix with all
values converted to millions of 1967 dollars...... 128
36 1967 14sector transactions matrix with all
values converter to millions of 1967 dollars...... 130
37 Energy flow diagram for a 5sector U.S.
economyenvironment simulation model.............. 141
38 1963 'rsector transactions matrix with all
values converted to millions of 1967 dollars...... 144
39 1967 5sector transactions matrix with all
values converted' to millions of 1967 dollars...... 146
tf Simulation results for the 5sector economy
environment model from 1929 to 2031............... 153
i1 Simulation results for the 5sector economy
environment model from 2031 tO 2131............... 155
42 rbholied energy intensity map for south Florida
for q19C estimated from the 1900 land use map..... 158
43 Embodied energy intensity map for south Fl3rida
for 1913 estimated from the 1953 land use maD..., 160
44 rmhodied energy intensity map for south Florida
for 9"73 estimated from the 1973 land use map..... 162
t5 Simulation results for the 91cell south "lorida
spatial simulation moel.......... ................ 167
46 Analog computer diagram for the two component
exchange model................. ........... ..... 203
Abstract of Tissertation Presented to the CGraduate Council
of the University of Florida in Partial Fulfillment
of the requirements for the Degree of Doctor of Philosophy
'"POrIFD E'pFRGY BASIS FOR
'CONOMTCECOLOGTC SYSTEMS
By
Rober*l Costanza
June, 1979
Chairman: Howard T. Ofum
Major Department: environmentall Fngineering Sciences
The energy basis for economicecologic systems was
investigateO using models of th Urnitel Statos and the south
Florifa area. The energy flow necessary (directly and
indirectly) to produce commodities was termed the embodie;1
energy and was studied as a parameter useful for evaluating
systems and their parts. embodiedd energy was calculated
using inputoutput matrices +o trace the flow of energy
through systems.
The approach was applied to a 92sector U.S. economy
for 1967. Current accounting conventions for government an!
households were modified to make them endogenous sectors.
Solar energy inputs to the economicecologic system were
estimate! and included. These modifications were necessary
to form a closed economicecologic system, with only energy
crossing the boundaries. Te changes were made
incrementally so the effects of each could be studied.
Results indicated a very close correlation between embodied
energy and dollar value of output, with the notable
v
exception of the primary energy sectors (P square = .(? when
the primary energy sectors were omitted). The results
implied a relatively constant embodied energy to dollar
ratio with an estimated value of 47000 kcal fossil/1967S.
Additional supporting data on energy/real GNP ratios for
time series of the U.S. economy and international
comparisons of energy/GDP ratios were collected and
presented.
Embodiecd energy was applied as a common measure to
model dynamic exchanges in combined ecologiceconomic
systems. This approach internalizes all energy
externalities.
Maximization of power (or embodied energy productivity)
was used as an objective function in developing dynamically
optimizing, nonlinear simulation models. The models adjust
their connectivity structure through time in order to
maximize power and the behavior is therefore discontinuous
in a manner analogous to catastrophe theory. The model was
applied to a 5sector U.S. economyenvironment and a 11cell
spatial grid of the souhb "lorida region, generating maps of
predict ed development.
The power maximizing model reproduced the behavior of
the U.S. economicecologic system over the historical period
from 1929 to 19r9 for which data on net capital stocks have
been estimated. Extrapolated into the future, the model
predicted leveling of the U.S. economy at around the year
vii
20n0 vith a subsequent gradual decline in net capital
stocks.
"The south Florida spatial model employed a sequence of
detailed land use naps based on aerial photographs and soils
information for the years 191V, 1953, and 1973. The model
divided the region into 39 cells with three additional cells
to handle Phe embodied energy changee with the rest of th
U.S. and the world. the simulation reproduced the essence
of the historical sequence of development using solar energy
as the only exogenous variable and a power maximizing
decision structure. Development of the east coast, Key Test
and Fort Myers was duplicated by the model based on embodied
energy exchanges between these cells and the U.S. economy
cell. The model predicted leveling of the region as a whole
consistent with the U.S. economyenvironment model.
__i r 1
I ?I T0 'C"'TnN
A fundamental .ssue in ecology is understanding the way
energy and material "lows in ecosystems develop organize'
structures and processes. Mlan's economic systems can be
viewed as subject to many of the same energetic forces as
those shaping ecological systems. Thus, the study of energy
and material flows in combined economicecologic systems can
lead to new insights into the way thes? flows develop
organized patterns.
Several important questions can be identified
concerning the role of energy as it affects organization and
succession in combined ecologiceconomic systems. How does
energy flow through and organize these systems? What is the
relationship between energy flow and money? How is spatial
development related to sources of energy? Vhat are the
general criteria for natural selection? How can selection
be modeled to predict the dynamic behavior of these systems?
These questions were considered using energy systems
analysis (Olum, 1971). Rodels describing the flow of energy
and materials were developed and evaluated. Specific
emphasis was placed on integrating conventional economic
accounting and analysis techniques into the general energy
systems framework. Inputoutput techniques and data were
employed to trace energy flows through combined ecologic
economic systems. ..fforts were made to show relationships
between these different accounting frameworks. Dynamic
simulation models were developed and used to investigate the
temporal and spatial behavior o* complex, selforganizing
systems that can evolve and change their internal structure
and function over time, Lotka's maximum power principle has
been suggested as the fitness criteria for survival of the
system, an.I thus the ultimate goal of evolution (03um 1971).
Can thOse concepts be incorporated in mathematical systems
models? what are the general criteria for survival of
systems? Can optimal control theory be gainfully applied to
this problem? Wha general characteristics do models of
this type exhibit? he simulation models developed in this
study were applied to the growth of the U.S. economy
environment and to the spatially articulated growth of south
Florida in order to predict the general behavior of thes
systems.
Research Plan
his dissertation is a study of the way energy affects,
limits, anrd determines the organized nonequillibrium
phenomenon comprising ecological and economic systems. "o
this end conceptual' and mathematical models were davelope.
to indicate the response of these systems to available
energy inputs. T.near inputoutput models of embodied
energy were developed and evaluated. Ionlinear,
discontinuous optimization models were developed and applied
with Lotka's maximum power principle as the objective
function.
Crosssectional and time series data were collected for
two related examples. The first was the U.S. economy
ecology as a whole. Pata for this example include. input
output transactions at various Ievels of aggregation, time
series of total capital stocks, investment and depreciation,
and time series of energy inputs to the economy and
environment. Most of these data were in dollar terms and a
major part of the study involved integrating them into an
allencompassing energy flow network that included
environmental systems. The second example was the spatial
evolution of the south Florida region. A series of there%
previously compiled, detailed land use maps of the region
for the years 19n^, 1953, and 1973 (along with supporting
data on the characteristics of the mapped units) ware use?
as the primary data base for this application. The models
were run over the historical period and the results compared
with the collected data. Once a reasonable fit was obtained
the models were run into the future and their predictions
interpreted.
background of Previous Studies
This dissertation includes energy analysis, evaluation,
and simulation of economicecologic systems using input
output, optimization, and spatial models. Some background!
of previous work in theso areas is reviewed.
Trnciv andl Society
The thesis that available energy inputs govern and
limit the structure of human societies is not new.
Boltzmann (1986) pointed out that life is primarily a
struggle for available energy. Soady (1933) state: "If w=
have available energy, we may maintain life and produce
every material requisite necessary. That is why the flow of
energy should be the primary concern of economics" (p. 56).
Lotka (1921) also noted the direct relationship between
energy and economics. Cottrell (1955) provided a detailed
analysis of the ways cultures have historically adapted to
their "surplus energy" supplies. Odum (1971) extended
energy concepts to include all systems, thus providing a
conceptual link between man and nature and many new insights
into the workings of man's economic systems.
Daly (1977) discussed the energy limitations which
ultimately lead to steady state economic systems.
Georarscu7'oegen (1071) took a more theoretical approach in
his study of the second law of thermodynamics and its
importance in economic systems. Ophuls (1977) reviewed the
political implications of energy and resource limitations.
Cook (1971, 19"6) and Hannon (1973a) have attempted to
quantify the intricate web of energy flows in industrial
societies.
Systems cology
Systems analysis as practiced in ecology is an
integrative approach used to explain the structure,
function, and interrelationship of all systems as the
product of certain general systems principles which
transcpnd thh boundaries of acalmlc fields. The aim of
general systems theory was formulated by Von Bertalanffy
{1968) as "the formulation and erivation of those
principles, which are valid for 'systems' in general" (p.
32). This paradigm is seen as essential to the
understanding of complex systems with feedback, which has
frustrated the atomisticc" approach of "normal" science.
Odum (1971) produced a unified theory and methodology for
the application oF general systems theory to a broad range
of problems. This is the general approach taken ir this
study.
Zn23rg MI naly!sis
The detailed study of energy flow through systems can
be termed energy analysis. Evaluation of energy flows in
ecosystems has long been an important tool (Juday 1940;
Lineman 1941). Currently in government circles energy
analysis has come to be used more specifically to refer to
the study of energy flows in engineeringeconomic systems
and the resulting policy implications. This application is
new and the concepts and techniques have not as yet
consolidated into a coherent whole. The house committee
print: "Energy analysis as a policy analysis tool" (Gushen
1976) is a goorl survey of the current literature. The
symposium by GilliTanr (1978) highlights points of
controversy, particularly concerning m?!thods of evaluating
embor"ied energy.
The field includes applications of inputoutput
analysis (Herendeen and Bullard 1974) and an evaluation of
energy systems diagrams (Gilliland 1975, 1978; Kylstra 1974;
Odum 1968, 1973). "ho~ applications involve different ways
of accounting for energy flows through systems. This
dissertation develops an inputoutput technique for energy
accounting similar to the one employed by Hannon (1973b) and
Herendeen and Bullard (1974). Isard (1972) suggested the
application of inputoutput models to scologiceconomic
systems but not in terms of energy accounting. The unigu
feature of he inputoutput energy flow models in this
dissertation is the application to combined economic
ecologic systems. The relationship between energy analysis
and economic analysis has been the subject of recent debate
and is a major topic or this dissertation. Leach (1975) 3nd
Webb and ?earce (1n75) have challenged the ability of energy
analysis to provide useful information beyond that available
to standard economics. By treating the economicecologic
system as a unit and by following energy flows through it,
many of these questions may be clarified.
Fmbodied Enerav
embodied energy is .defined as the total amount of
energy of a single type required directly and indirectly to
produce the substance of interest. For example, Pigure 1
shows solar energy as the primary energy input to the earth.
rMost flows and storage of free energy on the earth can be
thought of as embodied past and present sunlight, since
sunlight has been the most significant source of free energy
for the earth, Tides, nuclear energy, and residual heat
contribute much smaller amounts. "hus the sunlight of past
cons is embodied in the current storage of fossil fuel, raw
materials, soil, etc. that are employed by industrial
society. Tt is convenient to divide the continuum of energy
sources into renewable sources of free energy (embodied
present sunlight) and nonrenewable storage (embodied past
sunlight) on the basis of their relative rates of production
Figure 1.
Solar energy driving the productive processes
of the earth.
and consumption. renewablee sources are those whose rate of
production roughly equals their rate of consumption.
Obvious examples are sunlight itself, rain, wine!, and the
shorter time scale products of the interaction of these
inputs, such as forestry, fishery, and agricultural
products. Fonrenewable sources of free energy (embodied
past sunlight) are hose whose ratl of consumption far
exceeds their ra+e of production. Nonrenewable sources ar?
mine substances, such as the fossil fuels, soil storage,
and mineral eposits, which are the results of slow
biogeologic production cycles over long time periods.
embodied energy is linked to ability to Fdo work (or
available energy) by the theory that the energy used
(directly and indirectly) to produce a substance shows up as
a proportional increase in the stored order or departure
from equillibrium of the substance, and thus its ability to
do work. 7or example, oil mined and delivered to the point
of use would have more embodied energy (that consumed by the
recovery and transportation process) than oil in the ground.
The delivered oil would also have a larger ability to do
work than the untapped oil, due to its increased proximity
to an oil consuming economic system. A ceremonial mask
would have more embodied energy than the piece of wood from
which it was carved and would also presumably do more work
by performing an important symbolic function in the
ceremonies of the culture of its manufacture.
Several concepts of embodied energy have thus far been
proposed. One employs inputoutput techniques (Leontief 
1941) to trace input energy flows through the complex webs
of interactions in economic and ecological systems (Hannon
1973b; Tferendeen and ullard 1971) This can be termed the
inputoutput embodied energy. Tt assumes that embodied
energy is a conservative quantity, or that the sum of the
embodied energy inputs to each sector are emdodied in the
output. When pathways diverge, the total embodied energy is
partitioned among them so that the system of interconnected
flows maintains the conservation constraint. Figure 2a
shows an example of this approach. Another approach reasons
that since all processes are interconnected and in that
sense required for each other's production (either directly
or indirectly), the amount or input necessary to produce any
one product must be taken as the total input (Odum 1978).
This approach assigns equal embodied energy to all by
products of a process but partitions the embodied energy if
the same commodity is simply divided. The embodied energy
inputs and outputs for each sector do not necessarily
balance. This approach has been applied (Odum 1978) to
estimating the embodied energy in natural energy flows in
the biosphere and can thus be termed the biosphere embodied
energy. Figure 2b shows an example of the concept. Table 1
and figure 2 compare the characteristics of these two
approaches.
I
Table 1.
Characteristics of the inputoutput an.
biosphere embodied energy concepts,
Characteristic
Tnputoutput
embodied energy
Biosphere
embodied energy
Conservation of
embodied energy
All heterogenous
byproducts of a
production process
assigned equal.
embodied energy
yes
no
yes
(except for
degraded heat)
(a) INPUT OUTPUT
Figure 2. Diagram showing the characteristics of the
inputoutput and biosphere embodied energy
concepts.
The first concept was used in this study, with some
modification and extensions. A complete description of the
technique with examples is given in the methods section.
_Oti mization
Optimization is the search for maxima or minima usually
subject to some constraints. Wilde ani Beightler (1967)
provide a good introduction to the method. Cody (1974)
reviews some of th'e applications of optimization models to
ecological systems. Rapport and Turner (1977) discuss the
underlying similarities of economic and ecologic processes
as an explanation for the success of optimization models in
both fields.
'he maximization of useful energy flow {or maximum
power) was suggest, as an objective function by Lotka
(1922). olum (1971) has elaborated and generalized on this
theme. Oster and 7ilson (1978) employ what they trm
ergonomic (or work) efficiency as an objective function in
the study of colonial insects. Wang, Odum, and Costanza
(1978) showed an example application of the maximum power
principle to a land and water management problem.
Economic Mofels
current microeconomic theory can all fit under the
umbrella of optimization. In the general optimization
problem there is some objective function to be maximized or
I
minimized usuallyy profit, utility or cost) subject to
constraints dealingg with resource availability, income or
levels of production). he partial equillibrium theorists
deal with small pieces of the system taken in isolation with
the ubiquitous "all else boing equal" frequently invoked.
Most of the analysis focuses on graphical solutions. Becker
(1971) is a good text along these lines. Tnputoutput
analysis and linear programming are important approaches for
determining optimum, equillibrium flows of commodities and
money in an economy. Oorfman, Samuelson, and Solow (1958)
summarize these approaches. Inputoutput is a general
equillibrium technique developed by Leontief (1911). It is
a tabular accounting system with balance constraints. Tn
the typical application the economy is disaggregated into n
sectors and ;th production of each sector is expressed as:
Xi = 1ij + Yi (i=1,2,...,n) (1)
j=1
where
7i = total production o' sector i
Xij = production of sector i to be used as input
to sector j
Yi = output of sector i to consumers (final demand)
Figure 3 illustrates this setup.
A set of direct requirements coefficients can be defined as:
A ij = 7ij/ j (2)
or:
Xi = EXij Yi
Figure 3.
Diagram showing the standard inputoutput
accounting setup.
ij = A jXj (3)
substituting (3) in (1) yields:
n
i = ij j + Yi (4
j=1
or in matrix notation for all n sectors:
7 = AX + Y (5)
or solving for the sector outputs in terms of the final
demand (Y) and the direct requirements matrix (A)
1
X = (TA) Y (6)
The set of simultaneous linear equations represented by
(6) is useful for tracing interdependencies at a point in
time or for making predictions of the effects of small
departures from equillibrium. Programming models are
similar to inputoutput models except that more than one
solution to the equations is possible. "he approach
originated as a strategic planning mooei for directing Air
Force activities (Tantzig 1951) The mathematical problem
is finding the best (or optimal) solution from among the set
of feasible solutions. The approach initially was applied
to linear systems (linear programming), and this is still
the major practical application. Gradually, more
mathematically difficult problems were attempted with
current applications to dynamic, nonlinear systems with
stochastic elements. Baumol (1977) reviews these methods,
5Bpatial cnnoirc models
A good review of models of the spatial distribution of
economic activity can be found in Chorley and Haggett
(196) most of these models can be divided into three main
groups. Central, pace. thlorv is based on the fact that some
goods and services must be produced at "centers" and
transported to users (or the users must be transported to
the center). For a specific mix of goods and services there
is a limited range of distance from the center defining a
market area inside which the marginal revenue exceeds the
marginal cost. In a homogeneous plane with only one type of
qood or service produced, hexagonal market areas are
theorized, since these would represent the closest packing
of the market areas with no overlap. Production of
different types of goods and services leads to hierarchical
arrangements of hexagons, with different levels of central
places producing goods and services with different market
areas. Original works in this area are by Christaller
(1331) and Losch (1941). Berry and ?red (1961) provide a
review. Location theory oostulates that economic activity
will occur at the point of minimum total transport cost.
Transport cost surfaces for each of several inputs and
outputs of a specific economic activity are estimated and
overlaid to find the point of minimum total transport cost
(Weber 1909). The approach is highly amenable to computer
applications as well as to the inclusion of costs other than
those normally taken into consideration. See Smith and Lee
(19'7) for an example. The overlay system used by McHarg
(1960) is essentially a location theory model in which
environmental degradation costs are to be minimizes,
Objective procedures for estimating environmental costs have
limited the application o' this technique. Rent theory is
another extension which attempts to minimize the sum of rent
and transportation costs (vujnovsky 1972).
Previous studies of economic location have concentrated
on static, costminimizing models. The spatial simulation
models in this dissertation combine cost and benefit
concepts (in energy terms) in a dynamic framework. The
aE2vitZ morli is so called because its mathematical form is
analogous to the gravity equation in physics. It predicts
that the amount of exchange between two centers will be
proportional to the product of center sizes divil1nl by somn
power of distance between centers.
Or:
a
7ij = KSiSj / ij (7)
where
Yij is some measure of exchange between
centers
Sir,j are some measure of the sizes of centers
,ij is distance between centers
K,a are parameters of the model
In empirical studies, Yij is often the number of
peopletrips between centers and Si and Sj might bo the
populations of the centers. Tsard (1975) reviews these
concepts and applications. The generalized gravity relation
was incorporated in the spatial simulation models developed
in this study.
SimulatI on 1od els
Simulation of dynamic, nonlinear systems of equations
can be accomplished by solving differential or difference
equations using a computer. Examples of simulations of
economic and ecologic systems are those by orrester (1961,
1969, 1171) and Odum (1971). The approach has been
expanding rapidly in recent years with the decreasing cost
and increasing availability of computers. Hall and Day
(197") provide a compendium of recent ecological simulation
studies. Aifeld anI Graham (1976) is a recent example of
simulation applied to urban systems. In outline, the
techniquee involves deciding on "state variables" or storage
for the system of interest and then writing a differential
or difference equation for the time rate of change of each
of these storage in terms of the other storage and any
external inputs. qiven initial conditions for the storage
and a set of parameter values the computer simulates the
time course of each variable in the system. The solution
can then be compared with reality and adjustments made to
the parameter values.
fi Sout.h Florida Jrea
Figure ft is a location map of the south Florila area,
The region boundaries were taken as the drainage basin
boundaries of the KississmeeFverglades basin. Extensive
background information was developed on the detailed spatial
evolution of land uses in this region as part of a study by
the Center for Wetlands, University of Florida, funded
through the Department of t'e. Interior and the State
Department of Administration. This dissertation developer
as an attempt to answer some of the questions raised by
these previous studies. The reader is referred to Costanza
(1975) for characteristics of the mapped categories, land
use, energy data, and a detailed description of the mapping
procedure. Prowder, Litttlejohn, and Young (1975) provide
full color land use maps and a general overview of the
application of energy analysis to a region. The technical
report of the south Florida study (Odum and Brown 1975)
provides detailed data encompassing the full range of energy
and environmental problems facing south Florida. Zucchett
(1975) provides a retailed systems analysis of the Miami
urban area.
ORLANDO I
82 %OR A E 0t o
$, \ \ 0 to 10 20 0 40 Miles
27
261..
2 l
    
ST THODS
cr*nonr of the o'eli!n LanuaSSess
The energy circuit language developed by H. ". Odum
(1971) was us.l for illustrating the structure of the
models used in this dissertation. The symbols of the
language have associated mathematical functions which allow
the energy circuit molel to be translated directly to
differential or difference equations for computer
simulation. he symbols used in this thesis are summarized
in Figure r. A complete description of the symbols and
their mathematical connotations can be found in Oun (1971)
and Odue an5 (VIum (1906).
Symbolic modeling languages, such as Otum's (1971)
energy circuit language, Porrester's (1961) industrial
dynamics language, analog computer diagrams, and others are
useful for concise conceptualization and presentation of
complex networks of flows and storage. Diagrammatic
languages allow immediate comprehension (once the language
is understood) of the connectivity structure of the model.
while conveying the same mathematical content as
differential equations which require much closer inspection
before the overall structure is apparent.
0
GROUP SYMBOLS (1) AUTOCATALYTIC SELF
MAINTENANCE UNITS, (2) PRODUCTION UNITS,
AND (3) GENERAL PURPOSE BOX FOR
MISCELLANEOUS SUBSYSTEMS.
Figure 5.
Energy circuit language symbols used in
this study.
ENERGY SOURCE (FORCING FUNCTION),
HEAT SINK, OUTFLOW OF USED ENERGY.
ENERGY INTERACTION, ONE TYPE OF ENERGY
AMPLIFIES ENERGY OF A DIFFERENT
QUALITY.
ECONOMIC TRANSACTION AND PRICE
FUNCTION.
STORAGE (STATE VARIABLE)
ONOFF CONTROL WORK (DIGITAL ACTIONS)
 ( 2) (3)
MoI1l D development
A model is an abstract representation of a structure or
process which is constructed to aid understanding.
Development of an appropriate model is guided by the
questions being as!:ed, the general principles which th
model employs in answering these questions, and the amount
of time and effort which can be devoted to the modeling
effort, including data collection and evaluation. There are
several classes of models but this dissertation focuses on
mathematical models of the dynamic nonlinear type. Several
authors, notably Forrester (161) Nicolis and Prigogine
(1977) and Odum (1971) have argued that questions of time
behavior in living systems can be adequately addressed only
with models of this type.
The topological structure of each model was developed
by deciding on the internal components and external forcing
functions to be considered and then making assumptions about
the interconnections. The models were evaluated with data
from real situations. The simulation results were compared
with observed historical trends to determine the accuracy of
the original assumptions. These were modified as necessary
to improve the 4it.
The general meBhod of Lagrange multipliers was employed
in the development o the power maximizing simulation
models. Baumol (1977) contains a readable description of
this technique. Tr. essence it allows a static, constrained
optimization problem to be translated into an equivalent
unconstrained problem using the device of the Lagrange
multipliers. This method yielded the general conditions
necessary for optimum (maximum power) behavior of the system
at each point in time. An algorithim, which employed these
conditions in a dynamic simulation framework, was then
developed and teste.
Simulation r'Q lna "'nho:'
Both analog and digital simulation procedures were
utilized in this study. The main advantage of the analog is
the "hanis on" interaction with the molel that its small
size and continuous operation facilitate. For these reasons
an 717T inic analog computer was used to simulate a
simplified, two component, unscaled version of the model.
This allowed investigation of some theoretical aspects o1
the model and the range of behavior which the model could
produce. An analog diagram of the model is given in
Appendix T7.
Digital simulation requires integration by discrete
approximation and is therefore theoretically less accurate
than the continuous integration possible on an analog
machine. discrete integration quickly approaches the
accuracy of continuous integration as the size of the
integration "nterval is renirc~ or the order of the
numerical method is increased, however. The main advantage
of the digital machine is its large capacity, allowing th
simulation of much more complex models than possible on
available analog machines.
An Amrdahal $471 digital. computer was utilized, for
running the large models of the U.S. economy and s'ith
Florida for which detailed data were available. The noels
were written in 70??TAN using a rectangular integration
scheme. Listings of the POPTIP76 programs are given in
Appendices T77, VI, and VI7. An Tntecolor microcomputer was
also utilized for testing some midsized versions of the
models in BASTC.
flodel Parameter Estimati..on, validation and Testing
Dynamic simulation models of the type used in this
study require a large number of parameters. Frequently,
there are not enough data available to calculate statistical
"best fit" estimates of the parameter values. Therefore,
initial estimates of the parameter values were generated
from the available data and these initial estimates were
adjusted, iteratively until a reasonable fit was obtained
between the model and reality. The adjustment process was
limited to the least well known parameters and required
additional information about the historical behavior of the
system being modeled to compare with the model's output.
This calibration or validation of the model was performed by
manually adjusting the mod,1's parameters. The
discontinuous nature of the simulation models made the use
of nonlinear parameter optimization computer programs for
fitting the model to the historical data impractical. This
was because all of these algorithims (short of brute force)
require a continuous error surface +o operate efficiently.
I rplptIltu Techniqanes for cal.culatijn Tmbod3ied nernrv
The application or inputoutput techniques (Laontief
191) to the study of direct plus indirect energy
consumption was developed and documented by the Fnray
Research nroup at the Center for Alvanced Computation,
University of Illinois (alrendeen and Bullard 197(t). 7he
technique consists of defining a set of energy balance
equations (one for each sector) and solving the resulting
set of simultaneous linear equations for the energy
intensity coefficients vector e, which is the energy
required directly and indirectly to develop a unit commodity
flow. The underlying assumption of this technique is that
embodied energy is a conservative quantity. Figure 6 shows
the basic "energy balance" for sector j.
wher
Xij is the transaction from sector i to sector j,
X is the total output of sector j, part of which
may be net change in storage.
ej is the embodied energy intensity per unit of X ,
I
Ej j e xj
(a)
EXTERNAL
ENERGY
SOURCES .ejxj
(b)
Single sector energy balance.
Figure 6.
for this concept of embodied energy.
7. is the external direct energy input to sector j.
Thus the ernrqy balance for the jth component is:
n
7j = ejxJ nixij (8)
i=1
In matrix notation 'or all components:
= n(XX) (9)
Here 7 is a vector of direct external energy inputs, X
is a diagonalized matrix of output flows, X is a matrix of
input flows and e is the vector of total (direct plus
indirect) energy embodied in a unit of outflow.
We can solve for e as:
A simple example will clarify the procedure. Consider
a threesector economy consisting of an agriculture sector,
a manufacturing sector, and a consumers sector as shown in
Figure 7 and Table 2.
The economy is represented both in energy flow diagrams
(Odum '9"1) and corresponding inputoutput tables with all
the steps Irom physical flow units to embodied energy units
detailed. For simplicity the economy is at steady state
Figure 7.
Hypothetical three sector economy with all
flows in arbitrary physical units.
Table 2. Tnputoutput transactions table in arbitrary
physical units, corresponding to the diagram in
Figure "

'"o ?.gri Mlanufac Con Net m tal
cultr o touring sumers Output Output
From 1 2 3 c

Agriculture 1 10 5 5 10 30
Manufacturing 2 1n 0 30 10 "00
Consumers 3 .25 .25 1 .5 2
Energy input F 30n 70 
I
implying no net change in storage over the accounting
period. For systems not in steady state, any change in
storage can be accounted for in the net output column.
In reading the inputoutput table, the output from a
sector to other sectors is road as a row. In this example
agriculture sectorr 1) delivers 1n units of output to
itself, 5 units to manufacturing (sector 2) 5 units to
consumers (sector 3) and 11 units to depreciation (net
output). Inputs to a sector are read as a column. In this
example consumers (sector 3) receive 5 units of agricultural
products (from sector 1), 31 units of manufactured products
(from sector 2), an, 1 unit from themselves.
To convert to embodied energy units, first calculate
the energy intensity vector e, by applying the equation:
1
e = E(7X)
Tn this example:
S 0 2 .25 .25 1
S 5 5
X) = 5 3 E = [30 70
.25 . 25 1
.0618187 ,? 9009 .581P1R2
I1
(XX) = .0254055 .0272?27 .9454545
.0218182 .no9en090 1.3818182
S= (X) [31 6.364 21.818 836.364]
To convert the original physical units into embodie!
energy units multiply the energy intensities (e's) by the
appropriate flows. This yields the values shown in Fiqure 8
and Table %.
This embodied energy inputoutput table exhibits some
of the same characteristics as a dollar value inputoutput
table. "he total output from any sector equals the total
input to that sector and the total net output, or "final
demand,' in the economic terminology (1000 in this case), is
equal to the total net input, or "value added" (th E
vector, also 1 in this case). Final demand refers to thp
dollar value of the reat output of the system, while value
addel refers to the dollar payments for the net inputs to
the system. The total final demand or the total value added
is defined in the national income accounts as the gross
National Product (GNP) This would imply a GNP for the
hypothetical economy of 100". However, the conventions used
in the national income accounts are not the same as those
followed! here, 'o demonstrate the relationships, our
example economy's 70 fable can be converted into one
consistent with the national accounting conventions.
Figure 8.
Hypothetical three sector economy with all
flows in embodied energy units.
Table 3. Inputoutput transactions matrix in emiboied
energy units, corresponding to the diagram in
Figure 8.

"o Agri ,anufac Con Nrt :)tal
culture during sumers Output Output
From 1 2 3

Agriculture 1 313.6 181.q 181.8 363.6 13q0.8
Manufacturing 2 218. 1'0 0.9 654.5 218.2 2181.8
Consumers 3 ?9. 1 2n9.1 836.9 418.2 1672.7
Fnprgy input 3I0 7f) 100"
Total input 1"9r.8 ?181.8 1672.7

The major differences concern the treatment of
depreciation, the exogenous energy inputs, and the consumers
sector. In the national inputoutput accounts, the
depreciation is credited to the value of the output to
consumers. The consumers plus the net output are the final
demand sector. finally, the feedbacks from consumers are
considered to be exogenous and are added to the other
exocenous *nputs. These modifications lead to the flow
diagram anr inputoutput table given in Figure 9 and Table
4, respectively.
"he "interindustry" flows are not affected, and neither
are the total inputs and outputs from the remaining
endogenous sectors, "he modifications have affected only
the "final demand" and "value added" categories and their
common sum, the 5 72. The GNP is now 1418.1, which is
greater than 4he previous total of 1000 by 418. 1, the
depreciation o* consumers. The economic accounts aggregate
the consumers sector with final demand and value added.
Tt is interesting to note how the results for the
energy intensities (e's) would differ if the standard, input
output conventions were followed. Returning to the original
physical flow ratrix (Table 2) and ignoring the input from
consumers yields:
[ [o 13f5
 4< _
/
/
\\ 218.2 AG.
\ IT~AGR.\
Figure 9.
Hypothetical three sector economy cast in
the format of the national inputoutput
accounting statistics.
Table L.
Tnputoutput transaction matrix corresponding to
thp diagram in Fig. 9 using the national input
outpnt conventi ons.
,o ?gri Manufac Consumers + net Total
culture uring output or output
From 1 2 "final demand"
Agriculture 1 353.6 181.P 545.5 1n9.,8
Manufacturing 2 21q8.2 1fnn.9 872.7 21p1.8
energy input
+ Consumers or
"value added" 509.1 909.1 1481.1
Total input 1^q0. 21q1.8
;7 = ; = (300 700)
 i.0526316 .005632
.^105263 .0210526
9 = (X)71 = (23.158 16.316)
This is substantially different from the result with
consumers endogenous.
'he lower energy intensities that result from an
exogenous consumer sector are due to the fact that the
consumer services (labor) contain embodied energy that is
ignore when this approach is used. An alternative to
having an endogenous consumer sector would be to calculate
independently the consumer services energy intensity, and
use the coefficient to include the energy embodied in
consumer services as an input.
In this example, the intensity of consumer services
(03) was calculated as 836.364 when consumers were
considered to be endogenous, This number can be used to
include the energy contained in consumer services in the
direct energy input vector (the r vector) while leaving
consumers exogenous. "he new r vector is:
S= [300 + .25(836.364) 700 + .25(836.364)]
= rnn,. nnl 90.q9. 91]
Recalculating the energy intensities using this vector
yields:
.0526316 .0052632
1
.0105263 .0210526
e = [36.3=4 21.818]
These are the original energy intensity figures.
This approach requires an independent calculation of the
energy intensity of consumer services, however.
The approach considering consumers endogenous is more
in keeping with Leontief's original conception of a closed
(except to energy) economic system. It also does not
require the indepen'ent calculation of the energy intensity
of consumer services. It does, however, require some
manipulations to extract the relevant data from the current
accounting scheme.
Double. Countin
An often raised question concerning any accounting
scheme involves double counting. This is especially true of
inputoutput schemes tha display all intermediate flows.
The question becomes clear with reference to the preceding
diagrams and discussion. It is strictly a question of
accurately defining boundaries and making note of those
flows crossing the boundaries (net flows) and those flows
remaining within the boundaries (which when added to net
flows yield gross flows). Double counting problems are
encounter" when the boundary is shifted, but the
redecinitions of gross and net flows (which are defined only
with reference to the boundary) are not made. For example,
consider Figure 9. Here a boundary has been drawn around
the "industrial" sectors of the economy with the consumers
outside the boundary in the manner of conventional
macroeconomics. '"he net output of the industrial sectors
(that which crosses the boundary to consumers) is iefined as
the gross national product (CGNP). The confusion starts with
this misnomer, since the S'P is really a net flow. The
total output or "total transactions" would be a measure of
the true gross product. If the boundary is expanded to
include the consumers, then the GNP is no longer a net
outflow but an internal transaction. The net output with
the expanded bounlarr would be depreciation plus net exports
plus any change in internal storage. Conceptual problems
with double counting arise when this is not realized and th?
now internal transaction from producers to consumers is
still considered to be a ne4 outflow. rAdding the flow from
consumers to producers to the flow from producers to
consumers would obviously be double counting the GP as
previously defined. With the expanded boundary, however,
the GCP is no longer the net output from the system and
shouId be treated like any other internal transaction.
U.7. rconomv Data AssmblZ atn 7valua,ion
The major data sources for the U.S. economy model wers
the Bureau of economic Analysis' {B'A) inputoutput tables
(along with their associated amplifying articles) and
Kendrick's (1976) estimates of capital stock and investment
time series. Other statistical sources were consulted as
needed.
'he year 1967 was used as the base year for data
collection since this was the most recent year with measurer
inputoutput data. Data from the 1963 inputoutput study
were also used and reference was made to previous input
output studies back to 191n.
Leontier's (1941) original exposition of inputoutput
analysis envisioned a completely closed economic system.
Since then the convention has been to view households and
government as part of "final demand," and to treat them as
exogenously determined. The original, allinclusive view of
the economy was deemed necessary in light of the objectives
of this study. Thus, certain modifications to current
accounting conventions were required. The inputoutput
statistics were modified to achieve a completely "closed"
(in the thermodynamic sense) system. This means that only
energy crosses thm system boundaries. To achieve this goal,
households an' government were brought within the system
boundary (made enrogneous) as were a "U.S. environment"
I
sector and a "res* of the world" sector. The conventional
TO sectors were aggregated to 10 major groups, making a
total of 1i sectors. Figure 1n is an energy circuit diagram
summarizing the accounting scheme employed in this study.
All flows and storaqges of energy and matter in the world are
included (at least in an aggregated form) in this accounting
framework.
ov2ernment ant Ho7 sl. as ndoSaenous Sectors
Tn orger to make households and government internal
endogenouss) components in the accounting framework, certain
modifications to current accounting conventions and
approximations were necessary. Figure 11 illustrates the
problem. The household sector's inputs from the other
sectors were measured as personal. consumption expenditures
(PC"), which are the dollar payments of individuals for
goods and services from the "producer" sectors. The
exception is the input to households of government services,
which are paid for with federal income taxes, along with
state and local government taxes on households. The outputs
of the household sector to the other sectors are labor
services, which show up in the accounts as the employee
compensation category. Complications arise since the input
output accounts separate value added (or VA, defined as the
payments to the factors of production) into only three
categories: t() employee compensation, (2) indirect
L / V, I GOVERNMENT \ GOV.
TOTAL VALUE EXPENDITURES SALARIES
ADDED (VA)
= GNP
PART OF PROPERTY TYPE INCOME (PTI)
Figure 11. Diagram showing definitions of national income variables.
business taxes, and (?) property type income. Table 5
shows the relationship of these categories to the national
income and pronduc accounts categories. The implied wages
of selfemployed people or unpaid family workers are not
directly included in the employee compensation category but
show up as a portion of proprietor's income, which is
em',dded in the propertytype income category of value
added. Similarly, corporate profits taxes, which are also
embedded in propertytype income, should be added to
indirect business taxes to determine the total taxes paid by
business to government.
A detailed examination of questions relevant to the
proper distribution of value added to the economicecologi:
sectors was not possible during this study. Certain
approximations were therefore made to derive the estimates.
Value added was distributed by crediting all employee
compensation (rC) plus a fraction of propertytype income
(PTT) to householAs and all indirect business taxes (IBT)
plus a fraction of PLm to government. The fractions were
calculated using balance considerations, and the fraction of
PTI remaining after government and household's shares were
remover was considered a net profit attributable to inputs
from the environment (see the following section).
The rO accounting frameworkk requires that the sum of
the total dollar value of the outputs from a sector equal
the sum of the dollar value of the inputs. This requirement
"able 5.
Relationship of inputoutput value added
components to the national income and product
accounts categories.
Value ad'ied components Value added components in the
in the inputoutput national income and product
(10) accounts (NIP) accounts

Employee compensation mployee compensation
Indirect business taxes Indirect business taxes
Property type income Proprietor's income
rental income of persons
Corporate profits (before taxes)
Inventory valuation adjustment
Net interest
Business transfer payements
Surplus of government enterprises
Capital consumption allowances
was used to set up accounting identities for the new
household and government sectors, which could be solved for
the percentages of propertytype income to be credited to
each sector in order to balance the accounts. This is
admittedly only an approximation which was necessary due to
the lack of data on the allocation of value added in the
inputoutput accounts. Gross investment and net exports
were assumed to exactly balance against net profits. This
left the following identities. For the government sector:
TB" + Xq*(PT) + PT = GP + GS (11)
where
Xg = reactionn of PTI to Government
IBT = directt business taxes
P"T = Property type income
Im = Personal taxes
GP = Government purchases
G! = Government salaries
So the fraction of PTT to government necessary to balance the
sector's accounts is:
Eq = [GP + GS TB" P]l / PTT (12)
For the household sector:
'C + Xh*( TI) +GS = PCE + PT (13)
where
Th = Traction of PTI to households
C = Fmployee compensation
PTm = Property type income
G7 = Government salaries
PC7 = Personal consumption expenditures
P7 = Personal taxes
So, the percentage of PTT to households necessary
to balance the sector's account is:
Ch = [p + P eC GS! / PTT (14)
The remaining fraction (call it Xe) was considered a net
profit:
XP = 1 xg h 15)
Using data from the statistical abstract of the U.S,
(Unite5 States Department of Commerce 1971) and the bureau
of economic analysis inputoutput tables (United States
Department of Commerce 1969b, 1974a, 1975) the following
values for 7g and 71 for 1963 and 1967 were estimated.
For 1963: (in millions of dollars)
Xg = [GP + GS TBT PT] / PTT
= [68167 + %553' 5627 61000] / 194248
= ,39n
Xh = rPCE + PT 7C GS] / PTI
= [375540 + 61'r0 341514 55030] / 184248
Xe = 1 Xg 'h = .7551
For 1967: (in millions of dollars )
Vg = [GPr + GS TBT PT] / PCT
= r[9465 + P1659 70239 8300"] / 254060
sh = [PC7 + PT rC GS] / PTT
= [49qr6" + 83"nn 389136 816541 / 254060
4 17
Xn = I g Xh = .4944
Figure 12 summarizes the modifications to the T0
conventions made for this study.
environmental ~ n unDts
Ts with household and government services, there are two
ways of including environmental services. One is to treat
the environment as an exogenous entity and quantify its
inputs to the economy, The second is to treat the
environment as an endogenous sector with flows to and from
the other sectors in an integrated economicenvironmental
system. Both of these approaches were utilized in this
study.
An exogenous environment sector was hypothesized for ?2
sector inputoutput studies of energy flow through the U.S,
economy. thesee studies were carried out in collaboration
with the Energy research Group, University of Illinois at
Champaign. For this analysis the solar energy absorbed by
the p.3, was partitioned to the economic sectors according
to land and water area. Table 6 shows the land and water
use distribution for the U.s and estimates the total solar
absorption (including atmospheric) for the various uses,
The agriculture sector was credited with the solar
absorption over all agricultural land plus 341 of the
absorption over the wetlands, desert, and tundra category,
as agriculture represents 7t, of the remaining land and
Figure 12.
Summary of modifications to the inputoutput conventions.
"able m. 4stimacd .anE areas and solar absorpTion for
major land use types.
Average solar ,Ttal solar
ArPa(a) absorption (b) absorption
( 6 acres) (E Btu/acyr) (E18 Btu/yr)
Total land 2254 28 33.9
Agriculture 1212
Cropland 38
nrassland pasture 54"
Grazing lan? 288
Forestry 587 '8 16.4
roofland~ pasture 62
Voo dand (not pastured) 50
Forest land 475
Wotlanrs, desertt r, tunrra 272 20 5.
urban F mining 193 20 3.9
Total water 1550 28 43.4
Inland G estuarine 50
Offshore (2?n mile limit) 15F0
Total land plus wa4er 3P14 103.0
a. rrom United states Department of Commerce (1976a).
b. .stimater solar absorption of the earthatmospharn system
Fru'Iyko o 807, Haar and Suomi loq)).
water use,. his amounted to 35.74 F18 Btu solar/yr. The
forestry an7 fisheries sector was credited with the
absorption over all forested areas plus estuaries and
coastal water to the 200 mile limit plus 60n of the
wetlands, desert, and tundra absorption. This amounted to
63.06 E"8 ntu solar/yr. The remaining 4.20 E18 Btu solar/yr
represents direct utilization by the remaining industrial,
commercial, residential, and governmental sectors of the
economy. "his should be distributed to the remaining
sectors according to their total land areas. Accurate land
use data are not available at this level of disaggregation,
however. As an approximation, the entire 4.20 E18 Btu
solar/yr was credited to the household category since this
category represents about "" of the remaining land area,
Pn nogonv "nv'nrnmnrt s sector
7 more conceptually satisfying method of including
environmental services is to treat the environment as an
endogenous sector, mhis sector contains all the land, air
and water in the U!S. and performs the essential task of
capturing solar energy and converting it into other forms
more usable by the economic sectors. Since the environment
sector is not completely "owned" by economic agents and
competitive markets do not exist for many of its products,
economists have difficulty evaluating many of the flows and
storage in this sector. A broader perpsective based on
energy flows has proved useful (Odum, 1971; Bayley et al.
1975) in conceptualizing this problem.
For the purposes of this study it was assumed that,
where competitive markets exist, market values wer3
proportional to embodied energy content and that both of
these could he considered to be conservative quantities.
Evidence for the validity of this assumption is presented in
the results section. Conservation of dollars and embodied
energy allows many of the flows to and front the environment
sector to be estimate from balance considerations, The
inputoutput accounts are arranged such that the total
dollar value of all inputs to a sector equals the total
dollar value of all outputs from a sector.
Once the payments to households and government, and
capital flows have been accounted for internally, there
still remains an imbalance between the dollar values of th"
sum oc the inflows to each sector and the dollar value of
the sum of the outflows. There is still a "net input" to
the sector or "profit." This net input was attributed to
services provided by the environment sector. This is
essentially a "pure economic rent" conception of the origin
of profits. Under this view entrepreneurial capacity is a
component of the labor services input necessary to
effectively capture environmental inputs. At steady state
these environmental inputs would just cover the depreciation
of the economic system, "h1 approach can also be viewed as
a form of' "shadow pricing" (Dorfman, Samuelson, and Solow
1958) of environmental services.
Capital Flows
Capital flows are normally not included explicitly in
the inputoutput tables. Data recently available from the
Bureau of economicc Analysis (SCB Sept. 1975) on
interindustry transactions in new structures and equipment
combined with data from !endrick (1976) on investment and
depreciation of human and government capital allowed the
inclusion oc capital flows in parts of this study.
"Dr the purposes of this study the capital floors were
simply aided to the existing interindustry flows. This
increase' the total input to each sector by the amount of
capital purchased hy that sector during the year and
embedrled th, year's nross Private Fixed Capital Pormation
column in the current transaction's matrix. The capital
purchases by each sector were then added as a "capital
maintenance and growth" column in final demand to balance
the additional input.
South Florida Land Use Data
A time series of three full color land use mans for the
years 19'0, 1953, and 19"3 for the south Florida region wren
produced as part of the study, "Carrying capacity for man
and nature in south Florida", edited by H.T. Odum and M.
Brown (197) The maps are also included in Browder,
Littlejohn and Young (1975) and Costanza (1975) with
supporting data. "he maps were manually digitized using a
cell size of 128 acres for computer manipulation. For this
study, the land use maps were aggregated to 88 larger square
cells, 1( miles on a side, as shown in Figure 13.
For example, Figure 1i is a full size copy of cell 45
in Figure 13 from the 1913 land use map. Figure 15 is a
computer printout of the same data to show how it was
digitized. The correspondence between the symbols on the
printout and the numerical codes for the land use subsystems
listed in Table is given below the printout. Since not
all cells had the same land area, the area include? in each
cell was calculated and recorded. The data in Table 7 (from
Costanza 10"") were employed to perform the aggregation.
The subsystem structure intensities listed in Table 7 were
multiplied by the number of acres of that land use type in
each o* the 8R south Florida cells, and these values wer
accumulated for each cell to yield estimates of th total
embodied energy in each of the cells. This was done for
each of the three land use maps. These data are listed in
Appendix I, along with the 'and area, and latitude and
longitude of the centroi' of each cell,
I
U
60
1 I
0 10 2 0 4
s i 4 I I
4 16 17
I .27
. 27 
25 26 Z7 287 I9 0 31 32
45 46 47 4? 49 SC 5! 52
71 7S I
53 5 4 55 56 57 58 593
263 63 5 66 67 G2'
S0 7! 72 7 74
>I 26 7 I 28! i 23032I^31
I34VS 736 37 8 3 I 1 771 4 43
1 i i
[1 5 j i
^[ 54 &5J56! 57 5R j5 JSCi
2S 6! 62 63 64 S
25"
84 LS.JI
e?82 8I 80: 1
! ___ _ J ... ______ J __
Figure 14.
Example of detailed land use data, showing
cell 45 in Figure 13 for 1973.
19
20
21
22
23
24
25
26
2734
Figure 15.
Example of detailed land use data, showing
computer coding for cell 45 in Figure 13
for 1973.
++.1 C+ +
+. : : : ++++ : ++: CC +
::+ : : : +hM 'M M 4 ++ =+  ='I K 4+++ : :C : :C+ U
:++++: ++1,~~M + ++ 1++::+i+t+++tCC:+: ::
+++ +:+ ,+MM.M +++3 1 1 : : :++1++ C: :C :: :
IM+ hM++MMMMM: +++53 11i 1++++ C. ::C
M M++MM+M R++5=1 I +++ : : : :
*MM M.M+I^MM ,MMM:3355=; : 1 1 *^*f^ !<(C +: : : :+;
S hM'A+ M+ MM : : M+ ++ C*+ + c C *+ + x 
55,' MM M mexiSB s ::::: ++:: *:::+ i+ C+
5S ,e5 v. 4 +ig + + + 4c++*CC++*CC+ +
5 ^'^^M + t ++L4AA 4AA ::+++s*+++*1CCC+
I M MMM++++ :ex++ +++++C//+
^ ^MNMA++++: : :++++CC/++
1 ,MM a+++++++++++++CC.CCCC///
A ^M+ ++++ +++ =* 't *++C*CC C/
+ MM+++++ ++ ++ + ** /C/ /UU
M MMM+++ 1111 1*U=+'*CUUUUU
M I.M I U++ I 1111 :: C+ cCUUUU
M M4++++11A : x:: :k*CUUUUU
M IMM4M+: : +11 Ca ::.: : CCIJUUUU
MC+CCC+ +CC ::: : :=CCCCUUUJ
,CCC+C5C'a**: ::CCCC///U
.MMM&CC+CC l.a : : CCCC/ /
M''CCCC I I s55A4A CCCCC'
Mi1M5 4+5555554++. + CC
554++A +++i5 s ++++ C
e 55+++++ati *xxxC *ZC
PxM+++CC *a**CCa
S3+:5 5*xxx :::++x yCC
Table 7. Land use subsystem metabolism ann structure
estimates in coal equivalents (CE).
~
Subsystem Su bsyst m
Metabolism Structure
(E1 CE kcal (T'6 CE kcal
Subsystem /acyr) /ac)
1. Cleared land 0.5 5.0
2. Lakes an~ reservoirs 0.7 5.5
3. rPcreational space 7.7 24.7
4. Residential (light) 250.0 750.0
5. residential (meF. aens.e) 520.0 2,250.
6. Commercial/Industrial 1,600. 11,125.0
7. Transportation 500.0 2,000.0
8. Pownr plants 4,n0o.0 126,e00.0
9. Improved pasture 5.1 24.7
10. Vegetable crops 21.3 294.8
11. "reo crops 9.6 74.9
12. Sugar cane 22.2 313.1
13. Grassy scrub systems 4.0 16.5
14. Pinoeland systems 6.4 80.1
15. Hardwood systems 7.7 235.9
16. Lakes and ponds 1.4 7.4
17. Cypress domes and strands 7,3 214.5
18. ret prairie 5.4 51.6
19. Scrub cypress 5.8 61.3
2M. Freshwater marsh 7.4 228.7
Table 7. fCortinued).
Subsystem Subsystem
Metabolism Structure
(E6 CE kcal (' 6 CE kcal
Subsystem /acyr) /ac)
~
21. Sawgrass marsh 8.1 273.7
22. Beach and anne system n.3 u,0
23. Salt flats 0.3 4.
24. Scrub mangroves 1.0 7.2
25. Salt water marsh 5.0 29.5
26. Mangroves 7.3 218.,
Source: Cotanza 
Source: Costanza (1T'5)
PResults include a derivation of the general conditions
for maximum power, developmentt of an algorithim for
approximating the maximum power conditions in a dynamic
simulation model, and applications of the model. Some
features of the model were demonstrated using hypothetical
examples and the model was applied to a 5sector U.S.
economyenvironment and a 91cell spatial array for south
Florida. The embodied energy intensity of goods and
services was calculated for 92 U.S. economic sectors for
four different alternatives concerning the treatment of
labor and government services and solar energy inputs. Data
ver assembled on total capital stocks and flows and were
used to determine a better estimate of the mean energy
intensity for goods and services and to create "closed
system" inpntoutput transactions matrices for the U.S.
economyrnvironment at the 1sector level.
e .en1ral Conditionls for Maximum Powser
A major hypothesis of the simulation models in this
study is that complex living systems evolve so as to
maximize their productivity (or power as defined earlier),
There is a large literature on the various aspects of
optimization and specifically dynamic, nonlinear
optimization but these methods are generally not integrated
with simulation studies. Wagner (1975) views simulation as
a last resort to be used only if all else fails. "he
potential benefits of the integration of simulation with
optimal control theories are great, however. In this study
a simple algorithm for achieving this goal is developed.
The objective function is taken as the maximization of total
system power (Lotka's power principle as discussed earlier)
and thz constraints deal with limits on the total amount of
free energy in the system and the exchange of energy between
components of the system. Power is maximized when an
optimal exchange network is used, and this network changes
through time,
The problem can be stated in the mathematical framework
of nonlinear programming for each point in time as follows.
maximize PT = P1 l(QQ2'***,QnT1) +
P2 (Q1 Q2 a* 'n,rE2) + ***
Pn (QI Q2, ***Qn Fn)
subject to Q1 + Q2 +n'+ Qn = Ct
1 = Kit (16)
P2 2t
7n = nt
where
PT = Total power of the system, equal to the sum
of the n individual components
Pl*,P2Fr,** n = Power of the individual components as
functions of the embodied energy storage
in the system (QI1Q2,.. ,Qn) and the dira:t
energy inputs (EI 2,.. ,'n)
The constraints indicate that at any point in time the total
embodied energy in the system is equal to some constant
value Ct, and the direct energy inputs are equal to
constants (Knt ). "he optimization of the system involves
moving the scarce (limited) embodied energy around to elicit
the maximum total system power.
The problem can theoretically be solved using the
method of T.agrange multipliers. Certain additional
conditions must be satisfied to assure the existence of a
solution. Wagner (1975, p. 604) outlines these conditions,
They are divided into two groups, one for the constraints
and one for the objective function. Since the constraints
are all linear in 4the above system, only the objective
function nee" be addressed. The conditions for th?
objective function are:
(i) ny is single valued and finite for each Q and F
satisfying the constraints
(ii) Every partial derivative (DPT/aQi' i) is single
valued, finite, and continuous at each Q and
satisfying the constraints
(iii) PT possesses a finite maximum PT ) over all
values of Q and satisfying the constraints
(iv) PT is concave over all values of Q and I
satisfying the constraints
These conditions guarantee that
(A) There exists at least one feasible solution
(B) Tf PT is strictly concave, then there is a unique
optimal solution
(C) Tf Q, is a constrained stationary point, then
Q, E is a global optimum
Tt will be shown in a later section that the specific
objective function chosen nets the above conditions.
The Lagrange multiplier technique involves creating a
substitute problem tha~ incorporates the constraint
equations into the objective function. This new equation,
called the Lagrangian, can then be maximized (or minimize3)
using standard calculus techniques. The Lagranglan
expression for the above system is:
L = PI(QIQ2,...,Qn' 1) + P2(Q01Q2,*" Qn, 2) + ***
Pn(Q 1Q2'***'Qn,1 n) + 71(Ct Qi Q2 ** Qn) +
"2ilt 1) + V3(K2t 2) +' Vn+l(Knt 7n) (17)
where
V1' 2'" ,* n+1 are unknown Lagrange multipliers
To maximize the original constrained system, one then
maximizes the unconstrained Lagrange expression (L) by
writing the partial derivitives of I. with respect to all the
69
variables including the V's) and setting them all equal to
zero:
aL a P
= ?1
aQ2 a02
a%
aQP
2
2 r
a?1
Pn
aP
n
a2
 v
1
1
(18)
aT.
a5n
San
+
n
aP
n
n

1
aL aPl
i" ai1
a2,
?2
aP2
= 2
2
 2
3
(19)
* S
Sn
9 9 nn+l
n n
 I1
t
aP1
a"n
c
c
S

a
L
= C Q Q ... Q = 0
S t 1 2 n
1
>L
=7 1 =
2
TL
S =
nt n
'n+1
Thus there are 3n + 1 equations in 3n + 1 unknowns. Tn
this example the equations in groups (19) and (21) can be
iqnortd since they are simply restatements of the
constraints which specified that for a single small time
intFrval, the direct energy inputs can be considered as
constants. "hus to maximize or minimize the system the
following relations must hold:
P2
Q1
2
+ +,,,+
3aQ2
2 2
+ n+ +
aQ
an
n = 1
1
3P
n
n =
Q1
2
aP
n
n
3Q1
aPT
a Q
02
1P1
aoQ
aP1
3 2
aPT
n
(21)
P1
3Qn
^n
71
Or,
DPT PT aPT
= =...= r22)
l Q2 an
which says that the marginal total power of all the storage
should be equal in order to optimize the system. 'he
problem is then, how lo living systems go about adjusting
these marginal total power conditions in a fluctuating
dynamic environment? One possibility involves adjusting the
interconnection network of the system by switching selected
pathways on and off, as shown in Figure 16, This does not
violate the condition (ii) that the partial derivatives be
single valued and continuous since at each point in time the
functions are continuous. Mhis approach simply modifies the
problem from one time step to the next.
Consider the potential exchange from component 2 to
component in Figure 16. The diagram indicates that the
pathway is open if:
aPT PT
3 > @ ( 2 3 )
Q1 a802
Tf this condition does not hold then the pathway is switched
off. This would eliminate the term aP1/aQ2 from the
equation for aPT/a2 since Q2 would no longer be a variable
in the equation for Pl" This would lower aPT/aQ2 so that
the condition (23) would hold. By applying this decision
O PT f PT
on if aQ aQ2
aPT d PT
on if Qz aQ0
EXCHANGE
EXCHANGE
PT = P + P2
Figure 16.
Two production systems and their exchange
pathways.
73
structure to all pathways in a system over time one coulV
prevent it from diverging too far from the optimum. This
may be considered a form of the "feasible directions" method
outlined in vagner (1975).
possibly more accurate but operationally more
difficult approach involves adjusting the model parameters
to achieve the desired partial derivative relations at each
point in time. "'e approach using the switches has an
effect similar to continuous adjustment of the parameters
but is operationally easier and acknowledges the physical
limits to parameter changes.
An alternative derivation of the conditions in (23) can
be formulate, as follows. hhe change in total power caused
by the exchange terms (Y12 and Y21) are composed of direct
and indirect effects. Tn the two component model shown in
Figure 16, the total power is a function of the storage,
direct inputs, and exchange flows.
PT = f(01' O2 9' 2 Y12 Y21) (24)
The rules for total differentiation can be used to determine
dPT/dY12, or the change in total power caused by a change
in the exchange flow Y12"
Using (24) one can write:
PT DPT aT T aPT
P = 3Q + dQ2 + d 1+ RE + dY +
T  1  2  j 2 2I2
QI Q2 ~12 12
aT
 21 f)
3Y21
or:
74
e0 9Pp aQ DP (I ap a1L 3p dE aE p
T T TQ1 + T 2 T 1 + T 1 T
= + + + +
Yii2 QI 11]I2 3Q2 dY12 EI1 ~Y12 3'2 1(Y2 ~Y12
aP dY
+ T 21 (26)
Y21 (l12
The third and fourth terms on the right hand side can he
dropped since E1 and "2 are exogenous and.Y12 has no effect
on them, thus:
ds SE
1 2 = (27)
O12 y 12
Since embodied energy units are used throughout, some
additional simplifying relations can be made for this model.
= = 1 (28)
xY 12 Y 21
1 = 1 (29)
12
dQ
S2 = 1 (30)
d 12
Using (29) andf (3C):
dY21 dQ2 2 ()
= / = 1 31)
12 d 12 (21
then:
dP
T P,T 8P
 = 1 + 1 (32)
dY 12 Q 1 2Q 2
or:
T T T (33)
Y 12 1 aQ2
whore the firstt term on tho right hand side of (33)
represents the "benefit" and the second term the "cost" of
the transaction Y12. rn a dynamic simulation framework, a
transfer from component ? to 1 (Y12) is seen as beneficial
(leading to increased total power) if:
dP
T
d 1 .1
or (using 33) if:
T T
T > T ( 3 5 )
aQl 0Q2
which is equivalent to (23). Thus, allowing the pathway
switches in figure 16 to remain open as long as conditions
(35) an (23) ho'li will tend to maximize the total power of
the system.
n'n l _n4. of a Power MaximiZino Simuowateion oi33el
A specific motel structure and an algorithm for
approximating the maximum power conditions in a dynamic
framework must now be developed for application to real
systems, "he model equations will always represent a
compromise between simplicity (and therefore manageability)
and accuracy. Here the mathematical form of the madel
(incluing the power maximizing algorithm) is laid out,
first for a simple two component case and then for the
general case of n components. It should be noted that this
specific mosel is not the only conceivable way to achieve
the maxi.mum power conditions derived eariler in a dvnamic
system, It is only one of a number of possible approaches,
Figure 17 is an energy circuit diagram of a simple two
component version of the model in Figure 16, showing the
specific production function chosen. The differential
equations for the m'iel are given in Figure 18. The choice
of a production function was difficult, since it involved a
compromise between accuracy and simplicity. The production
funclion chosen was built up from simple, slightly nonlinear
"partial production functions," which exhibit the important
characteristic of diminishing returns.
The algorithm involving the switches on the exchange
pathways in the model is a simplified method for
approximating the maximum power conditions in a complex
dynamic system. The switches are intended to maximize the
total power (PI+P2) in the system by allowing only those
exchanges that lead to a net increase in total power during
a particular small time interval. This function may be
handled in nature by the mechanism of natural selection.
The system of equations is allowed to "evolve" by changing
its connectivity structure as it progresses through time,
The technique is analogous to a "costbenefit"
calculation for each potential exchange pathway for every
point in time. The "cost" is the loss of productivity due
aP PT aPT
on if 8Q' Q2
aQi a Q2~
a P. a P
on if I> a
Q2 a QI
PT = P + P2
Figure 17.
Energy circuit diagram for a two component
powermaximizing model of exchange.
Differential equations for the model in
Figure 17.
where
Ql,' Q2
E1, E2
a1, a2
b12 b21
bl b22
cl, c2
P=P1 +P2
T 1 '2
= embodied energy storage in
components 1 and 2
= direct energy inputs to
components 1 and 2
= direct energy input co
efficients for components
1 and 2
= transfer coefficients
for exchanges from com
ponent 2 to component
1 and from 1 to 2
respectively
= internal transfer
coefficients
= depreciation rates for
components 1 and 2
= total embodied energy
productivity (power)
of the system, given
by the first three terms
in the equations
Y12' Y21 = exchange flows from com
ponent 2 to 1 and 1 to 2
respectively, given by the
third term in the equations
aPT
aQ2
change in total power
_ with respect to the system
storage (Q1 and Q2) all
else being equal
aPT
aQl
I
Figure 18.
2
bll+
+ +
+ a Q1 1 + blaQ
1b2 1 Q2
1 + bl2Q1
aT 3T
if >
aQI 3Q2
otherwise
9PT
if 
3Q2
QPT
>Q
otherwise
2
b22Q2
+i +~
1 + b22Q2
?T PT
if > 
'Ql "Q2
o ther.wse I
b21Q2Q1
S+ b21
14b^O,
aPT aPT
if > 
9Q2 aQ1
otherwise
 c2Q2
(37)
alEl
Q =
1
b21Q2Q1
1 + b21Q2
 clQ
(36)
2E2 2
1 + a2Q2
b 12QlQ2
1 + b+l2Q
to a decrease in the contributing component's storage, while
the "benefit" is the gain in productivity due to an increase
in the receiving component's storage. Since the model's
production functions are differentiable, single valued,
finite and continuous at each point in time, an optimum
distribution of the storage exists for any point in time,
and the exchange network is adjusted to move the system in
that direction. As already noted, it is necessary to have a
common currency in which P1 and P2 are expressed in order to
perform this calculation, mhis study employs embodied
energy as the common currency.
The equations require some explanation. Sach of the
individual partial production functions (indicated by the
work gate symbols) are given a relatively simple "limiting
factor" form. This is equivalent to saying that there are
infinitesimal storage in the flow (shown by the small tanks
in the diagram), that limit the amount of source material
which can he used. A derivation of the partial production
equation follows. Consider a system given by the energy
circuit diagram in figure 1 and the equations below (Odum
and Odum 1)76).
Q1 lQQT k2Q1 (38)
Q = k3QT k4Q1QT (39)
Now assume that QT is an infinitesimally small storage with:
Q = 0 and k3 = 1 (turnover = 10n%). This yields:
Q = 0 = Q k4QT )
Solving for OT
Figure 19.
Diagram illustrating the partial production
function relations.
r
QT = 7/(1 + '4Q1) (41)
Substituting (41) in (38) yields:
Q1 = klQ1/(1 + k4Q1) k2Q1 (42)
A further simplification was that kI = k4 since in
embodied energy terms all of the inflow is embodied in the
input to the tank. The total production function for each
component was created by adding together partial production
functions of the form given above. The power maximizing
logic built into the model was us,"' to decide which partial
production functions were included in the total at any point
in time. Tn differential form the logic is hard to follow,
since it simultaneously uses one decision to make a second!
decision, which is used in making the first decision. The
equations can also be expressed in difference form (which is
necessary for digital computer simulation and is done in a
following section) to clarify the logic. For now one can
imagine? a tiny delay between one recision and the next.
The equation (36) for the rate of change of storage in
component 1 has five terms, mhe first term determines the
rate of capture of direct external energy (7i) as a function
of the amount of stored assets (QI) and the capture
coefficient al., 'he second term determines the amount of
internal interactions within component 1 as a function of Q1
and the coefficient bll. "Te third term determines the
amount of transfer from component 2 to component 1 with a
maximum power constraint. Tf the transfer is deemed to be a
net increase in total power at a particular time, then th2
rate or transfer is the given function of the stored assets
of the two components (Q1 and 02) and the transfer
coefficient b12. "rom ('6) and (37) the following
expressions can be derived for the above partial
derivitives:
3pT ai 2(b111) + (bl111) b 121
= + 4
SQj (^+ l 1 2 11 h~ 1 5 (1b+ 12
  i~~le;Z r~h207
Q b2ll ( 2Q2
+ (43)
1 + +21Q2
2
T = a2R2 2(b22Q2) + (h22 2) b21Q2
1+ b 1 l 2
1 + bl2Q1
The fourth term in equation (36) is the (potential)
outflow to component 2. It is subject to decisions
analogous to thosp discussed above for the inflow from
component 2 to component 1. The last term in equation (36)
is the depreciation term, which was assumed to be a linear
function of the quantity stored. Thus, cl is the
depreciation rate for component 1's storage.
The morel can he easily expanded to n components.
Figure 21 is a difference gqation representation for one
component in an n component version of the model, In
applications of the model the components can be sectors in
an economy, areas of land, trophic levels in an ecosystem,
or any other suitable subdivision of the system under stuly.
The difference equation representation makes statement of
the logic sequence easier. In Figure 20 the partial
derivatives are calculated at time tAt for making decisions
at tinm t. The summation signs indicate that there are
potential exchanges with each of the n1 other components in
the system, at each time step.
Simulations Usinq wo Comoonents
To investigate the range of behavior that a two
component version of the power maximizing simulation moael
can exhibit, some hypothetical situations were set up and
simulated on an EA7 ?iniAc analog computer. An analog
computer diagram of the model is given in Appendix T. The
simulations also served to test the power maximization
algorithm. This was done by constraining the system to
operate vith the switches either always on (always
exchanging) or always off (never exchanging) and comparing
this with the "unconstrained" situation where the switches
were allowed to perform their normal role. The algorithm
was deemed successful if, for the same coefficient settings
the unconstrained mode consistently led to a higher total
power in the model than a constrained mode. The model was
also run on digital computers as a check.
Four hypothetical cases were investigated with the
model, Tn the figuress the plots labeled "with exchange"
Figure 20. Component i difference equation
where
Q, = embodied energy storage in
t component i at time t
E. = direct energy input to
L,t component i at time t
a. = direct energy input coefficient
1 for component i
b.. = transfer coefficient for
13 exchange from component j
to component i
c. = depreciation rate for
1 component i
3P T
j= rate if change of total system
\Q i power (P ) with respect to
embodied energy storage in
component i at time t
a T
= rate of change of total system
\3Q ) power (P ) with respect to
embodied energy storage in
component j at time t
i, t+At i, t
aiEi,tQi,t
1 + aiQi,t
1 1, t
b jtQj ,t
1 + b. .Q
1 +b..Q.
3bjij,tQi,t
1 + bjiQjt
0
if
SotetAt
otherwise
if 
j
tAt
otherwise
SiQi,t
n
j=l
tAt
J
indicate the model's behavior with the switches in the
unconstrained mode, while those labeled "without exchange"
indicate the behavior with the switches constrained to the
off position. The time and quantity scales are in arbitrary
units. Tn general this two component version produced
almost identical behavior when the switches were Isft on as
when they were allowed to function normally. This indicated
that with only two possible exchange pathways, it almost
always "pays" in a maximum power sense to exchange, 'his
was not the case for a larger number of components where
more intricate exchange networks were possible.
Case 1: resource consumption. One relevant application
of the mo?!l is to 'he question of resource consumption.
Here one of the components represents a resource pool. A
resource pool can be operationally defined in the context of
this model as a situation where the external energy input
and the depreciation rate are both very low. For example,
the oil deposits of the ?rabian deserts occur in low
productivity areas protected from deterioration by overlying
rock. Tn the language of the model this is a large storage
of structure whose incremental effect on productivity in its
local environment is very small. If another component
capable of utilizing the resource exists in close enough
proximity, the logic of the model would decide to export to
this component in order to maximize power. Figure 21a shows
some simulation results for this situation. Both the
1
