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 Title Page
 Copyright
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Appendix
 Reference
 Biographical sketch
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Title: Embodied energy basis for economic-ecologic systems
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Title: Embodied energy basis for economic-ecologic systems
Series Title: Embodied energy basis for economic-ecologic systems
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Language: English
Creator: Costanza, Robert.
Publisher: Robert Costanza
Publication Date: 1979
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Table of Contents
    Title Page
        Page i
    Copyright
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Tables
        Page vi
        Page vii
    List of Figures
        Page viii
        Page ix
        Page x
    Abstract
        Page xi
        Page xii
        Page xiii
    Introduction
        Page 1
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    Methods
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    Results
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    Discussion
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    Appendix
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    Reference
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    Biographical sketch
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    Copyright
        Copyright
Full Text














EMBODIED ENERGY BASIS FOR
ECONOMIC-ECOLOGIC 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 all-encompassing 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 input-output 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 EY-76-S-05-4398) 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
Input-Output 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. Economic-Ecologic 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 Input-Output
Matrices for 1963 and 1967.................... 126
Five Sector U.S. Economy-Environment
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 25-CELL 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.
ECONOHIC-ECOLOGIC SY qln.......................... 219

VI FOrTTAT' LISTIFNG FOP THE 5-SECTOR U.S. ECONO1OY-
ENVIPONMETIT SIMULATI"!' MODEL....................... 236

VII FOrTF.T: LISTING FOR THE 91-CELL 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 input-output and
biosphere embodied energy concepts................ 11

2 Tnput-output transactions matrix in arbitrary
physical units corresponding to the diagram in
figure 7..................................... .... 32

3 rnput-output transactions matrix in embodied
energy units corresponding to the diagram in
Figure ........ ............ ... ......... ..... ..... 36

4 Tnput-output transactions matrix corresponding
to the diagram in Figure 9, using the national
input-output accounting conventions............... 39

5 Relationship of input-output 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 ra-re 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. economy-environment simulation model
performance statistics for the 1929-1969 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, 1920-1976............... 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 process-s
of the earth.......................... .........., 8

2 Diagram showing the characteristics of the input-
output an" biosphere embodied energy concepts..... 12

3 Diagram showing the standard input-output
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 input-output accounting
statistics........................................ 38

10 energy "low diagram of an aggregated 14 sector
U.S. economic-ecologic system..................... 46

11 Diagram showing definitions of national income
variables.......................................... 47

12 Summary of modifications to the input-output
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 rTGUR-ES













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 "r-qu-ncy 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 14-sector transactions matrix with all
values converted to millions of 1967 dollars...... 128

36 1967 14-sector transactions matrix with all
values converter to millions of 1967 dollars...... 130

37 Energy flow diagram for a 5-sector U.S.
economy-environment simulation model.............. 141

38 1963 'r-sector transactions matrix with all
values converted to millions of 1967 dollars...... 144

39 1967 5-sector transactions matrix with all
values converted' to millions of 1967 dollars...... 146

tf Simulation results for the 5-sector economy-
environment model from 1929 to 2031............... 153

i1 Simulation results for the 5-sector 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 91-cell 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
'CONOMTC-ECOLOGTC SYSTEMS

By

Rober*l Costanza

June, 1979

Chairman: Howard T. Ofum
Major Department: environmentall Fngineering Sciences

The energy basis for economic-ecologic 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 input-output matrices +o trace the flow of energy

through systems.

The approach was applied to a 92-sector U.S. economy

for 1967. Current accounting conventions for government an!

households were modified to make them endogenous sectors.

Solar energy inputs to the economic-ecologic system were

estimate! and included. These modifications were necessary

to form a closed economic-ecologic 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 ecologic-economic

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 5-sector U.S. economy-environment and a 11-cell

spatial grid of the souhb "lorida region, generating maps of

predict ed development.

The power maximizing model reproduced the behavior of

the U.S. economic-ecologic 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. economy-environment model.


__i r 1













I ?I T-0 '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 economic-ecologic 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 ecologic-economic 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. Input-output 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, self-organizing

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 input-output 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.

Cross-sectional 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

all-encompassing 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 economic-ecologic 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.

Georarscu-7'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 acal--mlc 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 engineering-economic 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 input-output

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 input-output technique for energy

accounting similar to the one employed by Hannon (1973b) and

Herendeen and Bullard (1974). Isard (1972) suggested the

application of input-output models to scologic-economic

systems but not in terms of energy accounting. The unigu-

feature of -he input-output 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 economic-ecologic

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 input-output 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

input-output 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 input-output an.
biosphere embodied energy concepts,


Characteristic


Tnput-output
embodied energy


Biosphere
embodied energy


Conservation of
embodied energy

All heterogenous
by-products 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
input-output 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. Tnput-output

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. Input-output 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 input-output
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 = (T-A) 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 input-output 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 -cnno-irc 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, cost-minimizing 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

people-trips 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 Kississmee-Fverglades 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. Zucchet-t

(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


--c-r*n-onr of the o'e-li!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)




ON-OFF 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 l-na "'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 no-els

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 mid-sized 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 nern-rv

The application or input-output techniques (Laontief

191) to the study of direct plus indirect energy

consumption was developed and documented by the Fn-ray

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(X-X) (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 three-sector 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 input-output 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. Tnput-output 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 input-output 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(7-X)



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
(X-X) = .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 input-output table exhibits some

of the same characteristics as a dollar value input-output

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 7-0 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. Input-output 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 input-output 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 input-output 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 4-he 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.

\ I-T~AGR.\


Figure 9.


Hypothetical three sector economy cast in
the format of the national input-output
accounting statistics.













Table L.


Tnput-output 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

input-output 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) input-output 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

input-output data. Data from the 1963 input-output study

were also used and reference was made to previous input-

output studies back to 191n.

Leontier's (1941) original exposition of input-output

analysis envisione-d 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, all-inclusive 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 input-output

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

T-O 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 self-employed 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 property-type income category of value

added. Similarly, corporate profits taxes, which are also

embedded in property-type 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 economic-ecologi:

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 property-type 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 r-O 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 input-output value added
components to the national income and product
accounts categories.


Value ad'ied components Value added components in the
in the input-output national income and product
(1-0) 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 property-type 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

input-output 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 input-output 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

= [49-qr6" + 83"nn 389136 816541 / 254060

4 17













Xn = I g- Xh = .4944

Figure 12 summarizes the modifications to the T-0

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 economic-environmental

system. Both of these approaches were utilized in this

study.

An exogenous environment sector was hypothesized for ?2

sector input-output 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 input-output conventions.













"able m. 4stima-c-d .anE areas and solar absorpTion for
major land use types.



Average solar ,Ttal solar
ArPa(a) absorption (b) absorption
( 6 acres) (E Btu/ac-yr) (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 earth-atmospharn 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

input-output 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 input-output 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 7--S 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







I34V|S 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
27-34


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 :: :
I-M+ 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 MM-M+++ 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 /ac-yr) /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 /ac-yr) /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 5-sector U.S.

economy-environment and a 91-cell 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" inpnt-output transactions matrices for the U.S.

economy-rnvironment at the 1-sector 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+ +
a-Q


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

(inclu-ing 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 "cost-benefit"

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
power-maximizing 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
14-b^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 t-At for making decisions

at tinm t. The summation signs indicate that there are

potential exchanges with each of the n-1 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--
Sotet-At

otherwise


if -
j


t-At


otherwise


SiQi,t


n


j=l


t-At


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




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