Economic Valuation of Aquatic Ecosystems
Anthony Fisher, Michael Hanemann,
John Harte, Alexander Home,
Gregory Ellis, and David Von Hippel
University of California, Berkeley
Final Report to U.S. Environmental
Protection Agency
Cooperative Agreement No. 811847
October 1986
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Table of Contents
Chapter 1. Introduction and Overview
Chapter 2. A Suite of Indicator Variables (SIV) Index for an Aquatic
Ecosystem
Chapter 3. The Hysteresis Effect in the Recovery of Damaged Aquatic
Ecosystems; An Ecological Phenomenon with Policy Implications
Chapter 4: Ecotoxicology and Benefit-Cost Analysis; The Role of Error
Propagation
Chapter 5: Hysteresis, Uncertainty, and Economic Valuation
Chapter 6: The Economic Concept of Benefit
Chapter 7 : Methods of Benefit Measurement
Chapter 8: Further Work
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CHAPTER li INTRODUCTION AND OVERVIEW
In this chapter we indicate the ways in which aquatic ecosystems are
valuable to mankind, and make a first pass at suggesting how these values
might be assessed. Our object is to give an adequate appreciation of the
many and varied kinds of goods and services provided by aquatic ecosystems,
while at the same time beginning the process of organizing the discussion of
methods of measurement of the worth of these benefits. The chapter
concludes with a detailed outline of the plan of the rest of the study.
A. Goods and Services Provided by_ Aquatic Ecosystems
The steps involved in determining the economic value of ecological goods
and services are to identify what benefits ecosystems provide for mankind,
to characterize these benefits in ecological terms, and then to assess their
economic value. Even the first step should not be thought of as completed
for any actual ecosystem. Indeed, it is virtually certain that as our
understanding of ecosystems progresses in the future, we will discover the
existence of presently unrecognized goods and services provided by healthy
ecosystems. The characterization of goods and services by ecologists must
include not only a description of the nature of the good or service, such as
how many trout for sports fishing a particular stream maintains, but also
how the continuing provision of that benefit is linked to the future state
of health of the ecosystem. Generally the ability of ecologists to
characterize the magnitude of the benefit under ambient circumstances far
exceeds their ability to assess how continuing provision is linked to
environmental quality. Finally, valuation must take into account not only
the effect of a change in environmental quality on the ability of an
ecosystem to provide the benefit under discussion, but also its effect on
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the overall health of the ecosystem, which in turn may influence the future
ability of the system to provide benefits not presently identified. This
"insurance" factor is most difficult of all to include in the benefit-cost
calculus because it requires having to guess the value and the ecological
interconnectedness of benefits that we have not even identified as of yet.
In order to guide our thinking about methods of measuring benefits
we have chosen to categorize the goods and services provided by aquatic
ecosystems as being those for which the environment is an input, that is,
the ecosystem provides a factor or means in the production of a good or
service to be consumed, and those for which the environment itself is a
final good. This distinction is,in a sense, artificial, since many goods
and services provided by aquatic ecosystems fall in both categories. It
will, however, be useful because, as explained in section B below and
further in chapter 7, it corresponds in some ways to a distinction between
approaches to economic valuation.
Goods and Services for which Aquatic Ecosystems Provide Inputs to the
Production Process
The most obvious set of goods for which aquatic ecosystems provide basic
inputs are "fisheries" products. These products, as indicated in Table 1,
include harvested fish, shellfish, and crustaceans; aquatic plants such as
kelp, which is used in the manufacture of chemicals and food products; and,
to a small extent, aquatic mammals, now used mostly for garments. The
rivers and reservoirs that allow hydroelectric production and its control
contain aquatic ecosystems. Some types of damage to these ecosystems, e.g.
siltation of reservoirs caused by soil erosion and runoff, can affect the
output of the hydroelectric system. Rivers, lakes, bays, and estuaries are
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TABLE 1: GOODS AND SERVICES PROVIDED BY AQUATIC ECOSYSTEMS
Goods and Services for which the Environment Provides Inputs
Fisheries Products:' Fish, Shellfish, Crustacea, Kelp, Aquatic Mammals
Hydroelectric Power
Transportation
Treatment of Human Wastes
Treatment of Industrial Wastes
Water Purification
Drinking Water Storage
Information Produced via Scientific Research
Goods and Services for which the Ecosystem is a Final Good
Recreational Use of Aquatic Areas (Public Access and Commercial)
Direct Use of Water: Boating, Rafting, Sailing, Canoeing,
Scuba-diving, Swimming, Wading
Recreational Use of Aquatic Organisms: Fishing, Waterfowl
Hunting, Collection of Shellfish and Crustacea
Waterfront Recreational Activities: Strolling, Hiking, Sunbathing,
Team Sports (e.g. Volleyball), Off-Road Vehicle Use,
Horseback Riding, Nature Study (e.g. Birdwatching)
Amenities
Scenic Values
Modulation of Local Climates by Large Bodies of Water
Status and Enjoyment of Owning or Having Access to Aquatic Areas
Informal Education of Children
Psychological Benefit of Availabilty of Pristine Areas
Future Goods and Services
Preservation of Genetic Information: Protection of endangered
Species, Preservation of Gene Pool
Preservation of Wild Areas for Use by Future Generations and for
Future High-Value Development
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also used as transportation arteries, and thus provide an input to the
process of moving people and goods from place to place.
An extremely important and often overlooked set of processes in which
aquatic ecosystems play roles are human and industrial waste-treatment and
water purification. When human wastes are discharged into bodies of water,
biological and physical processes combine to break down organic matter and
release nutrients in the wastes, and to kill pathogenic organisms. In a
similar manner many industrial wastes are broken down when disposed of in
aquatic environments. Coupled with these waste-treatment functions,
wastewaters disposed of in lakes, rivers, marshes, and other aquatic areas
are purified and recycled either by evaporation and subsequent precipitation
or by percolation through benthic (bottom) sediments and soil to groundwater
aquifers. Wastewater added to a lake might undergo biological treatment by
aerobic (oxygen-using) bacteria associated with oxygen-producing algae
growing at the water's surface, chemical treatment by entrapment of metals
and other substances in the anaerobic (oxygen-free) bottom waters and
sediments, and physical treatment by filtering through sediments and soils
before it reaches a subterranean aquifer that supplies fresh water to
consumers. Properly functioning aquatic ecosystems in reservoirs also
provide appropriate conditions for the storage of drinking water. Clean
and/or potable water is an essential input to the production of a vast
number of products and services.
Aquatic environments also provide opportunities for scientific research
and development. In this case knowledge is the product for which the
environment is an input. This knowledge may take the form of information
about the improved cultivation of a valuable organism, for example, or
data that enables prediction of the behavior of other aquatic ecosystems,
and how the goods and services that they provide will vary under changing
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conditions. The study of one small lake, for example, might provide
information valuable in protecting a number of lakes in an area from acid
rairi OP some other pollutant stress.
Uses of Aquatic Ecosystems in which the Environment is the "Final Good"
Perhaps the most obvious set of goods and services in which aquatic
ecosystems are in a sense final goods are the recreational uses of watery
areas. These recreational goods, as listed in Table 1, include direct uses
of water, the recreational pursuit and harvest of aquatic organisms, and
waterfront recreational activities. Examples of activites involving the
direct use of water are boating, rafting, sailing, canoeing, scuba-diving,
swimming, and wading. Fishing, hunting of waterfowl, and collection of
shellfish and Crustacea are examples of the recreational use of aquatic
organisms. Waterfront recreational activities include strolling, hiking,
sunbathing, sports such as volleyball, the use of off-road vehicles,
horseback riding, and nature study (e.g. birdwatching). Many of the
recreational goods mentioned above are available in both public areas and
through commercial interests such as tourist hotels and lodges close to the
water, tour boats, and fishing and other guide services. Virtually all of
these goods and services depend on good water quality for their value.
A much more amorphous class of benefits provided by aquatic ecosystems
can be loosely described as "amenities". These include the pure scenic
value of a waterfront area or lake, the modulation of local climates by
large bodies of water, and the status and enjoyment provided by owning or
having access to areas near the water. While the practical nature of these
amenities is clear to everyone, there is a "spiritual" side to the scenic
value of aquatic ecosystems that may represent the dominant benefit that
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these ecosystems provide. In the informal education of many children,
nature plays an extremely important role. From the autobiographies of
numerous writers, artists, scientists, and others we read often of how early
exposure of pristine wildlands shaped these peoples' minds beneficially.
Such writings reveal the awareness of ecosystem benefits by those that are
most able to express these experiences vividly, but these same benefits
accrue, of course, to a far wider spectrum of people who are not necessarily
as conscious of, or articulate about, their existence.
Beyond the formative years of childhood, amenity values continue to
enrich peoples' lives, but in ways that can be distictly different from the
ways in which children benefit. In particular, a greater awareness of the
amenities occurs as we mature and the experience of nature becomes less
formative than it is restorative. The person in an office in downtown San
Francisco, for example, may take comfort in the fact that pristine areas are
available for him or her to enjoy. This thought, that escape from the "rat
race" is possible, may make it easier to live and work happily in a city.
If such a person were asked what this amenity was worth, he or she might
quote some figure, but it is possible, since the scenic area has always been
available, that the individual would undervalue this amenity relative to
what would be considered his or her "share" of the value of the scenic area
to society as a whole.
A final class of goods and services provided by aquatic ecosystems can be
loosely described as future goods and services, and the preservation
thereof. This includes the preservation of diverse genetic information,
the preservation of ecosystems for future generations of humans to enjoy,
and the preservation of aquatic areas for future development. The
protection of endangered species—for their future commercial use, aesthetic
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value, use as objects for scientific study, and existence value—is one
example in which the preservation of genetic information can provide future
goods and services for society. In preserving a diversity of plants and
animals we are also preserving a library of genes that, with man's growing
ability to manipulate genomes, may someday become tools useful in producing
valuable drugs or chemicals. The preservation of scenic and wild areas for
future generations to use—our National Parks are examples—provides future
goods and services in the form of both recreational opportunities and
aesthetic values, as described above. The knowledge that scenic areas will
be available to their descendants in the future may also provide the benefit
of peace of mind to a person living today. Finally, preservation of some
aquatic areas may allow them to be developed for high-value uses in the
future. Mining in a scenic lake area rich in some ore, for example, might
have to be done today in such a way that the scenic value of the place is
lost indefinitely—through poisoning of the aquatic ecosystem by acids
leached from mine tailings, soil erosion from road constuction, or physical
rearrangement of the area—but it might be possible to mine the same region
at some future time, using an as-yet undeveloped technology, in such a way
that the aesthetic value of the area remains intact. In the latter case
the area continues to provide recreational and aesthetic goods and services
in addition to the valuable ore. As described in chapter 5 the presence of
future-worth considerations can greatly influence regulatory choices
regarding the control of pollution of aquatic ecosystems.
B. Economic Valuation
As the discussion in the first part of this chapter has suggested, the
goods and services that can be provided by aquatic ecosystems are many and
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varied. Yet for the purpose of characterizing evaluation, we must try to
collect them into a manageable number of categories, corresponding to
methods of evaluation. This we attempt in the table, Table 2, below, with
the hope that no major items, at least, are lost in the process. Types of
goods and services are classified into those involving the aquatic
ecosystem, the environment, as input, and those involving it as a final good
(or service). By environment as input, we mean that it enters into a kind
of mixed biological-economic production function, along with conventional
inputs such as labor and capital, to yield some desired final good—as the
table suggests, a supply of fresh water for drinking, perhaps, or a
shellfish harvest. The consumer of the water, or the shellfish, is assumed
to care only about the good he consumes, and not the input mix used to
produce it. By contrast, when the environment is valued as a final good, it
enters directly into the consumer's utility function. Thus improved water
quality can yield benefits both as an input to some production process, and
directly to on-site recreationists, nearby property owners, and so on.
A couple of more exotic, or less tangible, goods are also indicated in
the table. One is the conservation of genetic information. This can be
considered as affecting future commercial harvesting, for example of a plant
or anima'l species for some yet-to-be-discovered medicinal property. The
other intangible good is the existence of an unspoiled environment,
unrelated to any use or consumption of its resources now or in the future.
Some people derive satisfaction simply from the knowledge of existence, and
this has been termed "existence value" in the literature of environmental
economics.
Now, why is it sensible to classify the goods and services provided by
aquatic ecosystems in this fashion? Consider the first column in the table,
headed "method of evaluation." It is our view that a particular method can
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TABLE 2:
METHODS OF VALUATION FOR GOODS AND SERVICES
PROVIDED BY AQUATIC ECOSYSTEMS
Method of Valuation
Type of Benefit
Shifting Supply, Given Demand
Environment as Input
Water Supply and Quality
Commercial Harvesting
(includes genetic conservation
for future harvest)
Travel Cost
Environment as Final Good
Recreation
Comparative Property Values
Amenities
Contingent Valuation
Existence Value
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be identified as best suited to each of the categories. Thus, if the
environment is viewed as an input to a production process, such as the
commercial harvesting of shellfish, an improvement in quality due to reduced
pollution loadings can be expected to lead to a shift (down and to the
right, on a conventional diagram) in the cost or supply of shellfish. Given
an independent estimate of the demand for the particular shellfish product,
the shift in supply generates an increase in combined consumer and producer
surplus, the area bounded by the demand and supply curves. Of course,
establishing the nature of the connection between reduced pollution and the
supply shift is a difficult empirical problem. In section A of chapter 7
below we consider the problem in some detail, and illustrate our method of
solution with some computations based on estimates of relevant demand and
supply parameters in the literature. The use of a change in combiend
surplus to capture the welfare effect of reduced pollution is justified in
chapter 6, a theoretical discussion of the economic concept of benefit.
An aquatic ecosystem can also, as we have noted, be viewed as an input
to the generation of fresh water supplies in a region. Reducing pollution
loadings in the system similarly results in a downward shift in the cost or
supply of providing fresh water. We shall have more to say about this
contribution also in chapter 7.
Turning to the environment as final good, the first item in our table
is recreation. There is a large literature on methods of valuing outdoor
recreation resources, discussed in some detail in section B of chapter 7.
Here we just note that the preferred method, rooted in economic theory and
validated in many empirical applications, is the travel cost method. The
name is derived from the use of travel cost (from the point of visitor
origin to the recreation destination) as the measure of price in an analysis
of the demand for recreation at the site in question. Thus our focus has
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shifted from supply to demand. There is however an interesting parallel to
the analysis of the environment as input. Suppose an improvement in water
quality makes available a site that can be assumed to perfectly substitute
for another (in the provision of recreation). Then recreation at the first
or unimproved site is iri effect available at lower cost, to those who live
nearer the newly available site. Of course, this analytical device requires
the assumption that the newly available site provide the same recreation
services as the other, so that consumers are indifferent as to which is
chosen as "input."
Reducing pollution in an aquatic ecosystem can also lead to enhanced
amenities. Clean water makes nearby residential property more desirable.
An extensive literature has explored the relationship between changes in
environmental amenities and property values—the extent to which it exists,
the circumstances under which it can be estimated, its magnitude in
particular cases, and so on. This literature is reviewed in section B of
chapter 7.
We come, finally, to existence value. This differs in an important way
from all of the other goods, or benefits, discussed thus far in that it is
not associated with use of the resources of an ecosystem. In fact it is
often classified, along with option value, as an "intrinsic", or non-use
benefit of preserving or improving an ecosystem. We shall have more to say
about option value very shortly. With respect to existence value, there is
a double problem for measurement. First, one cannot measure units of
consumption (to which a value might then be imputed). To some extent this
is true also for amenities—as in the case of an improved view. But the
value of the view may be captured by a change in property value, since the
view is associated with a piece of property, and property is valued in
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market transactions.
The second difficulty in measuring existence value is that it is a pure
public good, and one whose consumption is not associated with consumption of
some private good such as residential property. About the only approach
that can be employed here—and has been, in a small number of empirical
studies—is so-called contingent valuation. This is simply asking
individuals what they would be willing to pay for the continued existence of
an area or species. The literature has also addressed the difficulties with
this approach—the hypothetical nature of the question, its unfamiliarity to
respondents, their propensity for strategic behavior, and so on. We provide
a review with special reference to the application to aquatic ecosystems in
section C of chapter 7.
We mentioned option value as the other commonly identified non-use
environmental benefit. Yet it appears nowhere in our table. The reason is
that, in our Judgment, it is not a separate benefit, corresponding to a
separate good or service provided by an aquatic ecosystem. It is instead an
adjustment, or "correction factor," to an estimate of any of the other kinds
of benefits listed in the table, to take account of uncertainty about their
future values. This is a complex issue, however, that has generated
considerable confusion and controversy in the literature. Chapter 5 defines
option value and some of its properties in an analysis of the valuation of
pollution control in a dynamic, uncertain setting. Further discussion,
focusing on different concepts of option value, is provided in chapter 6.
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C. Plan of the Study
In the next chapter we discuss a kind of "quick and dirty" alternate
approach to valuation, the construction of a suite of indicator variables
(SIV) that might be used to characterize the response of an aquatic
ecosystem to reduced pollution or other disruption. This chapter includes a
review of what might be termed ecological scoring methods, such as the HEP
and HES systems. It also introduces concepts which will be useful later on.
Chapter 3 is about one of these: the dynamics of ecosystem recovery.
A model is developed that generates the often-observed and potentially
important hysteresis phenomenon, in which a recovering ecosystem does not
retrace the path of its decline. The point of the model is to enable
prediction of the recovery behavior of ecosystem populations in which we are
primarily interested, higher trophic levels such as fish, from that of the
much more readily observed lower trophic levels such as phytoplankton.
Chapter 4 is an analysis of error propagation in measuring recovery. That
is., suppose we are uncertain about the degree of phytoplankton recovery.
How does this translate into uncertainty about recovery of the fish
population?
Chapters 2, 3 and 4 are primarily about the behavior of aquatic
ecosystems, with no systematic discussion of economic valuation. In chapter
5 we begin this discussion. A model is developed to value the control of
pollution, taking account of key features of the ecosystem behavior
discussed in the earlier chapters: recovery lags, irreversibilities, and
uncertainty. The model does not address the question of how to estimate the
different categories of benefits identified in the preceding section (of the
Introduction). This is the task of chapter 7, divided into three parts,
also noted in the preceding section: the environment as input (water
supply, commercial harvest), the environment as final good (recreation,
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amenities), and non-use benefits (existence value). The discussion of
methods of benefit estimation is preceded, in chapter 6, by a theoretical
analysis of the economic concept of benefit. Specifically, we motivate use
of combined consumer and producer surplus as the preferred measure of a
welfare change following an environmental improvement.
In chapter 8 we consider appropriate directions for further work. Our
present intention is to proceed in two areas: (1) comparative analysis of
models for policy evaluation, and (2) development of a case study. Both
are elaborated in chapter 8.
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Chapter 2. A SUITE OF INDICATOR VARIABLES (SIV) INDEX FOR AN AQUATIC
ECOSYSTEM
I. The Need for a SIV-Index
Assessment of the damage to ecosystems ideally requires
an accurate and precise measurement of the harmful effects. The
results of such measurements are needed to establish a numerical
relationship between pollution and economic damage to the
ecosystem. Although not often used exactly in this way there are
several habitat evaluation procedures available to assess the
"health or state" of the ecosystem. These measures include
several separate procedures (see reviews by U.S. Water Research
Council, 1981; Putnam, Hayes, and Bartless, 1983; Canter, 1984)
and cover most types of aquatic ecosystem but focus on streams
and wetlands rather than large lakes, reservoirs, large rivers,
estuaries or the open ocean. None of these indices is ideal but
they have served well in some circumstances, especially for
evaluation of game habitat used for recreational sport, for
example, deer hunting.
Any of these evaluation systems can be used to give a
numerical value for the ecosystem over a sustained period of
time. The resulting long-term data base is then used to show if
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a decline or improvement has occurred. When compared with an
unaffected or control ecosystem an ecosystem value can be
expressed as a percentage of the optimum even if the evaluation
procedure does not cover all the period of degradation (or
improvement) of the system.
Of considerable practical interest is the need for the
maintenance of a complete habi ta t in the kind of restoration that
occurs when sewage or other wastewaters are cleaned up. For
example, a relatively simple single parameter (e.g., the fish of
concern) or multiple parameters (e.g., the index proposed in this
paper) can be assessed routinely while habitat evaluations are
extensive, expensive, and one-time measurements.
11. The Requirements for an ideal index; Selection of variables
for use in a SIV index
There are three main requirements:
oData must be inexpensive to collect.
oData must already be available for some ecosystems for
use in trial projects.
oThe connection between the variable and its biological
effect must be known from experimental studies.
The purpose of a SIV index is to determine aquatic
ecosystem health over time and/or space. The choice of variables
can change depending on the ecosystem chosen. For example,
dissolved oxygen fluctuations can be deadly in mid-western rivers
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in summer but the same quantity of waste is unlikely to trouble
the temperate open ocean. Since biologically non~functional
variables decrease the precision of any index they should only be
used where important.
III. Review and critique of ecological indexes which could be
used,to estimate ecosystem health.
Critique of existing habitat and other evaluation procedures as
applied to aquatic ecosystems
Existing habitat evaluation methods usually focus on
o the physical structure of the ecosystem -- e.g.,
stream sinuosity, mean depth, percentage of cover,
size of the lake
o indigenous, rare, or sensitive species, diverse
species composition, and
o maintenance of indigenous (native) sport or game
species.
The habitat evaluation procedures are derived from
common sense evaluations once made by wildlife managers. The
purpose was usually to decide what mitigation should be given if
an area was to be physically destroyed -- as for example if a
housing development or a dam were to be built in the area. In
many cases mitigation was the creation, donation, or restoration
of a piece of land which was of comparable ecological worth to
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that being destroyed. An example might be the degradation of a
stream by treated sewage could be compensated for by the creation
of a marshland on the treatment plant property.
The evaluation procedures have a terrestrial bent (e.g.,
deer, partridge) since lakes and streams cover only a small
portion of the landscape. Thus physical features such as trees,
browse, overhanging banks (for fish), are important, even
dominant in existing habitat evaluations -- and rightly so for
terrestrial and some aquatic systems.
However, most lakes, oceans, estuaries, larger streams,
and rivers are structured on the basis of thermal stratification,
the chemical stratification which follows, and an ever-changing
biotic structure. Wetlands are intermediate in this respect
depending on the degree of submersion and the life times of the
plants which constitute the base of the food chain.
Pollution in aquatic systems alters the biotic
structure, sometimes the overall chemical structure, but rarely
the thermal or physical structure of aquatic ecosystems. In this
it differs from terrestrial habital destruction. The rebuilding
of a damaged landscape requires the regrowth of a complex of
physical habitats, while the restoration of an aquatic one may in
principle require only the cessation of pollution. In both cases
it is assumed that the biotic component is readily available to
migrate in from adjacent areas.
Most of the indices, especially the habitat evaluation
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procedure (HEP), the habitat evaluation system (HES), and the
ecosystem scoping method (ESM), also incorporate an implicit
(HEP, HES) or explicit (ESM) belief that diversity = stability =
desirability. That is, the more different types of organisms
there are (or the more links there are in the food web) the
higher the ecosystem will score. Thus the most valuable
ecosystems tend to be the most diverse by this rationale.
The diversity-stability argument has a 20 year history
in ecology. One might sum up the conclusion as the relationship
between diversity and stability depends on the definition of
stability and the time scale of observation. For example, if
stability is equated with constancy over time then, when using
typical northern temperate human time scales of years the simple
non-diverse arctic owl-1emming-grass food chain appears
unstable. When viewed over decades the opposite conclusion can
be drawn (i.e., a perpetually oscillating population). Other
definitions of stability can lead to yet other relationships with
diversity which are not discussed here. It is unfortunate that
the early discoveries of high diversity in tropical forests and
coral reefs were not put in a better perspective for seasonally-
controlled temperate-polar systems.
The intent of this paper is to review in brief existing
habitat evaluation procedures and attempt to derive a
specifically aquatic index which can be used to describe the
"health" of the ecosystem. Such an index will be imperfect but
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is needed if one is to assess change over time, and thus see
effects such as hysteresis•(e.g., Edmondson and Litt, 1982, also
see later in this report), improvement, degradation and ascribe
some economic value to the measured changes.
The choice of variables for an aquatic health index can,
in theory, be made from any or all trophic levels in the
ecosystem. Unfortunately the organisms of most direct economic
interest (recreational or sports fish and shellfish) do not seem
to be either easy or inexpensive to sample or to use for robust
indices. Because of their size, relative rarity and biological
complexity fish and shellfish produce variables which vary widely
from the mean value. These parameters have a high coefficient of
variation and when combined into any index these large errors
propagate to the point of rendering the index useless for
practical purposes.
An example of this is the "scope for growth" (SFG) index
which has been widely proposed for the assessment of the health
of fish and shellfish. In a recent (1983-1984) and costly study
of the effects of the large sewage effluents of Los Angeles, the
California Dept. of Fish and Game (Monterey Office), together
with the local discharger and various other regulatory agencies
use the SFG method. Analysis of this data shows that changes
shown near outfalls using the "scope for growth" (SFG) method are
not statistically significant. Both increases and decreases in
SFG relative to controls occurred at outfalls but similar changes
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occurred between replicated samples in the same place. The SFG
method has a poor and inexplicably variable precision relative to
other methods of growth measurement. SFG can only resolve
changes of 282% (average of all Cal-COMP data) while simple
measurement of length or weight have uniform precision and can
resolve differences of 4% length and 14% weight (Home, 1984).
If we are to detect the biological effects of pollution
near outfalls, a more precise measurement of mussel growth must
be used to replace scope for growth tests. Such a precise method
has been developed for Region #2 (San Francisco Bay Regional
Water Quality Control Board) by scientists at the University of
California, Berkeley. It is clear that SFG is still very much at
the research stage and not a monitoring tool.
Why is the Scope for Growth Test so Imprecise?
The reasons are both physiological and statistical and
both are inevitable. The physiological reason is that it is
common for organisms moved from field to laboratory to experience
long-term stress (see Knight and Foe, report to RWQCB, 1984).
This together with individual genetic variation gives a highly
variable end result.
An implicit assumption in this method is that SFG
represents an obsolute measure of mussel health. For example, it
is assumed that "healthy mussels" are always of approximately 40
joules h~l- Values measured on mussels transplanted to other
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sites are often much lower than this due to transplant effects
alone. In addition, spawning stress will reduce growth. These
stresses and other uncontrolled variables reduce the utility of
SFG as a monitoring tool to almost zero. The statistical reason
for the low precision of scope for growth results from SFG being
a value calculated from a series of ratios and assumed values.
Errors propagate through such an equation. It is usually better
to measure a biologically integrated change directly -- i.e.,
measure growth directly rather than indirectly.
The problem of high variance is apparently inherent in
these higher trophic level indices. Even relatively simple
values, such as th'e percentage survival of animals exposed to an
environmental pollutant, can be variable since animals which
appear identical in size, condition, and amount of pollutant
absorbed may have a very different genetic makeup (Hilvsum, 1983;
Home and Roth, in prep.)
There are two ways to overcome this dilemma. First,
simpler, more abundant organisms can be used to construct a
robust index. Second, functional components of one or more
groups of organisms can be used instead of their abundance.
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IV. Proposed Suite of Indicator Variables (SIV) Index:
strengths, weaknesses.
Lacking any absolute ideal indicator(s) for ecosystem
health an index is an obvious second choice. This has a history
in economics (price index) and in ecology (diversity index,
striped bass index). Again in common with economics (consumer
price index) but not usual in biology, an index with several
components seems desirable. The problem with an index based on
any one variable in ecology is two-fold, lack of robustness and
risk of being misleading. Over the last century several single
indices have been proposed as "master variable". Acidity (pH)
has often been proposed (Schindler et al., 1985) but is
misleading for acid rain studies and alkalinity has been
substituted (Hendriksen, 1979). While alkalinity is an
appropriate guide to the susceptibility of a lake to acid
oligotrophication (acid-induced impoverishment) it is not a good
indicator of the effects of point or non-source wastewater
pollution.
What is required is a suite of independent variables
which would, if taken together, reliably show the current state
of the ecosystem. Only if the majority of variables indicate a
change in the same direction will there be good probability that
the damage is serious (ecologically important) and persistent.
It should be noted that this majority indicator approachimplies
that the "cost" of a false warning is greater or equal to the
9.
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"cost" of a false assumption that all is well. This could be
described in terms of the "crying wolf" paradox. It is not usual
to consider the damage done by false warnings of severe damage.
However, from an ecological viewpoint there is only a certain
amount of public concern for ecosystem preservation. Thus false
warnings can detract from the effort required to respond to true
warnings. An example of this is the hue and cry over DDT and its
environmental effects. The cancers and genetic damage now
ascribed to PCB are not effects of DDT. Although there are
serious effects and a ban on DDT use was appropriate the toxicant
PCB was overlooked for many years since its chromatographic
signature was confused with DDT. A decade was lost when PCB-
filled devices could have been phase out.
The plankton
Large numbers of independent (i.e., physically
unconnected) organisms can be sampled with low statistical
variance. For example, counting 100 single-celled free-floating
phytoplankton gives 95% confidence limits of being within +20? of
the true number (100). It is not always easy to be sure one has
overlooked some algae when examining lots of similar-looking
cells. If a similar number of cells were counted but were
contained in 16 filaments the 95? confidence limit would only be
+50% -- a much larger error (Land, Kipling, and Le Cren, 1959,
pg. 158). In addition to counting errors if the organisms are
10.
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also physically well-mixed then genetic variation between
individuals is muted. These conditions are best met in the
aquatic ecosystem in that group of organisms called the
plankton. The word plankton means wanderer and basically refers
to those small plants and animals which are more or less at the
mercy of water currents. In this paper I will use the term in
its widest extent to cover small unattached organisms in ponds,
lakes, streams, rivers, estuaries, oceans and coastal fringes
including salt marshes. Thus true animal plankton (zoo-
plankton), plant plankton, (phyto-plankton) as -well as the
invertebrate insect drift in rivers and streams is encompassed.
As defined widely plankton includes the young stages of
almost all the commercially valuable fish and shellfish and most
of the sport fish and shellfish. Those which are not included
depend heavily on the plankton for food in the adult stages. For
not fully understood evolutionary reasons the majority of large
valuable fish and almost all shellfish need a planktonic life-
stage and some such as salmon, dungeness crabs, grey mullet or
eels swim or crawl thousands of miles to achieve this planktonic
goal .
The functional components of aquatic ecosystems
The previously mentioned high variance (= high risk of
incorrect predictions) was first recognized in the study of
stream benthos (e.g., Wurtz, 1960). Here extreme patchiness
11.
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(large rock adjacent to gravel, sand etc.) could only be overcome
by very large numbers of replicate samples. Typically 73
replicate collections in a stream riffle might be needed for 95%
confidence in the numbers of invertebrates collected relative to
3-6 replicates which are normally the limit (Needham and Usinger,
1956). This patchiness was later found to be common in most
aquatic ecosystems and remains a partially solved problem
(Richerson, et al., 1970; Riley, 1976; Sandusky and Home, 1978).
In addition, particularly in streams, wetlands, and
estuarine-ocean systems the identification of individual
organisms is often impossible. The animals in the above-
mentioned ecosystems are numerically dominated by juvenile stage
of such groups as clams, oysters, polychaetes, insects, fish and
crabs. The taxonomic keys for juveniles in many cases have not
yet been written and even when published require expert
taxonomists. This problem was again first tackled by stream
ecologists who proposed to simplify their ecosystem by using
functional group classification instead of taxonomic
identification. Thus shredders, scrapers, filterers replaced
large crayfish, caddis-flies, and may-flies even though the
functional classification cut across traditional taxonomic lines.
In smaller ecosystems such as ponds and small streams it
has been possible to measure whole-ecosystems variables such as
net photosynthesis or respiration using whole-lake oxygen fluxes,
isotope dispersion, or even carbon depletion. The process has
12
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provisionally been extended to incorporate large lakes (Tailing,
1976).
Functional components have the advantage of built-in
robustness since they incorporate ecosystem homeostasis as
explained below (i.e., the intertia and redundancy in ecosystems
which tend to reduce overall change). A typical example of this
would be the replacement of the attached stream algae Cladophora
by the attached stream algae Tabellaria near the inflow of a
well-treated but nutrient-rich domestic sewage outflow in the
Truckee River, near Lake Tahoe (Home et al . , 1978). Insect and
presumably fish populations did not respond to this food chain
switching presumably because either algae was equally acceptable
(or unacceptable) as food.
Combined plankton-functional component index -- the SIV index
For purposes of monitoring the ecosystem effect of
pollutants a combination of both the plankton and functonal
components will be valuable. Large numbers of individuals (n)
can be measured which will reduce type II errors and concomittant
failure to detect pollution's effects until it is too late. A
large n will also reduce type I errors and risk of overstating an
effect. The use of juvenile stages of commercially and
recreationally important fish and shellfish will assist in the
economic analysis and will also include "sensitive" species
(sensu EPA guidelines on NPOES permits). Both indigenous and
13.
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rare species can be accommodated in such an index. Finally, the
robustness of the index will be ensured by incorporation of
ecosystem homeostasis by the use of functional component
variable s.
The drawbacks to the SIV index in principle are similar
to those of any other environmental scoping or health assessment
namely:
o Require some measurement or knowledge of the
ecosystem.
o Is hard to extrapolate backwards in time to pre- or
low-pollution eras.
o May miss important effects if one component of the
index was capable of indicating serious harm but the
other components lagged behind in their responses.
The proposed SIV index has the advantage for aquatic
ecosystem pollution studies that these drawbacks can be minimized
particularly in the cost of data -collection since the precision
of the index can be very high.
The main purpose of any index is to show changes over
time or space. High precision is vital if change is to be
detected in time for restorative measures to be put into effect.
The literature shows a number of multi-parameter indices
or ranking systems used to measure the "trophic state" of lakes
(i.e., their basic fertility of productivity). These include
14.
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those by Lueschow et al, 1970; Shannon and Brezonik, 1972;
McColl, 1972; Michalski and Conroy, 1972; Sheldon, 1972;
Uttormark and Wall, 1975; Carlson, 1977; and the EPA's own
modified index derived from an extensive study of 757 specially
selected lakes (See Hern, Lambou, Williams, and Taylor, 1981).
The SIV index does not attempt to improve these models especially
those by Carlson 1977 and Hern (EPA) et al., 1981. Our purpose
is to extend their use to cover both toxic and biostimulatory
effects of point and non-point wastewater discharges as well as
extend coverage beyond lakes to all aquatic ecosystems.
For example, one improvement of the model suggested by
EPA (Hern et al., 1981) to use chlorophyll a_ not nutrient levels
as a basis for trophic classification fits directly into the
functional component mechanism of the SIV index.
Multi-parameter indices also exist which attempt to
measure higher trophic level productivity including that of
fish. This is a measure of ecosystem "health". Such attempts
range from pioneering concepts such as those of Thienemann (1927)
and Rawson (1951) to complex but realistic simulation models
(e.g., Steele, 1974; Powell, in press). A "rough indicator of
edaphic (= nutrient) conditions" combined with lake bathymetry
(morphological structure) was the morphoedaphic index (MEI) of
Rawson (1955) and Ryder (1965) and Ryder et al. (1974).
The MEI uses mean depth and fish harvest statistics and
was designed for use in lakes. Since the most productive systems
15.
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(lakes, streams, estuaries and coastal ocean waters) are shallow
and well-stirred this index has limited use when expanded from
typical thermally stratified lakes to all aquatic ecosystems.
Complex simulation models of the plank tonic community
are not yet usable as indices even though multiple parameters are
involved. A primary reason is that such models are not normally
designed to work with the kind of pollution stress normally
imposed by toxic wastewater. Typically, the models will be
perterbed by nutrients or the introduction of a natural change
such as increased predation. Most chemical poisoning or aquatic
habitat structural alteration has few natural analogs and these
are yet little studied. The few potential analogous systems
natural springs with high acidity as toxic metals have been
little studied for metal toxicity dynamics. Almost no examples
of organic biocide accumulation are available in natural aquatic
ecosystems. However, metal or organic toxicants are a prime
cause of aquatic ecosystem degredation, second only to dissolved
oxygen reduction and diversion of water.
The construction of a numerical SIV index with some of
the properties mentioned previously cannot be easily formulated
in the abstract (see e.g., Boesch, 1977). Thus the index must be
built on a case study and then generalized if possible. The task
is formidable but an equal problem is acquiring an adequate data
base which would also be available for other ecosystems. Records
of planktonic and other biological variables are often available
16.
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in the open literature. In contrast, pollution loading values
are hard to find over long periods -- although they are usually
available somewhere in the files of individual dischargers
(Russel and Home, 1977) or in the files of the local regulatory
agency (Home, Fischer and Roth, 1982).
V. Summary
An index which will measure the health of aquatic ecosystems would be
very useful in determining the amount of damage, or recovery from damage, in
aquatic ecosystems. The index should ideally be robust, precise, and multi-
dimensional and reflect changes due to either toxicity or biostimulation.
An index of selected indicator variables—the SIV index—is proposed which
builds on the existing EPA and other indices used to estimate "trophic state".
The SIV index differs from the existing habitat evaluation indices in that the
bias is towards aquatic ecosystems rather than terrestrial ones. This bias
is needed since the damage caused by humans to the two habitats is of a
different kind. The structure of aquatic ecosystems is dynamic and is main-
tained by short-term biological and chemical inputs. Terrestrial ecosystems
depend much more on the physical structures such as trees and hills. Water
pollution usually destroys the chemical and biological structure while ter-
restrial disruption, such as housing developments or dams, destroys the entire
physical structure.
The SIV index follows recent trends to use functional components of the
ecosystem rather than only taxonomic classification. The index is comprehen-
sive in that it uses both types of information. A major difference from other
indices is an emphasis on precision so that small changes in the health of the
ecosystem can be detected with statistical confidence. In this way damage can
17.
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be detected before it is too late and recovery techniques modified during
restoration to maximize benefits. The only way to achieve precision and
avoid both type 1 and type 2 errors is to make a large number of measurements.
This can be done if the variables chosen are inexpensive to measure, and
this concept drives the choice of variables in the SIV index.
Common, numerous, and functionally important variables would be chosen
for the SIV index. In most open-water aquatic ecosystems the plankton provide
a good source of information on the health of the ecosystem. The plankton
include the young stages of most commercially and recreationally important
fishes, their food, and the photosynthetic base of the entire food chain.
The plankton are sufficiently numerous and homogeneous to sample at a reasonable
cost and are most directly exposed to water-born pollutants. For wetlands
and streams the same principles apply but the collection techniques must
;be modified by the use of analogs to achieve the same high precision at a
comparable cost.
Future research should focus on long-term data sets from already damaged
test ecosystems where data are readily available and easily supplemented.
This concept is opposite of the NSF long-term research program which considers
only pristine ecosystems. Thus data from various less accessible "grey"
literature will be the principle source of information.
18.
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References
Cantor, L.W. 1984. A comparison of habitat evaluation
methods. EPA. Environmental Impact Seminar,
Washington, D.C. Feb. 1984. 73 p.
Carlson, R.E. 1977. A trophic state index for lakes. Limno1.
Oceanogr. 22 361-369.
Edmondson, W.T. and A. Litt. 1982. Daphnia in Lake
Washington. Limnol. Oceanogs. 27:272-293.
Hendricksen, A. 1979. A simple approach for identifying and
mea'suring acidification of freshwater. Nature.
278:542-545.
Hern, S.C., V.W. Lambou, L.R. Williams, and W.D. Taylor, 1981.
Modifications of models predicting trophic state of
lakes: adjustment -of models to account for the
biological manifestations of nutrients. (Summary) U.S.
EPA Pb. 81-144 362.
Home, A.J., J.C. Sandusky, J.C. Roth and S.J. McCormick.
1978. Biloogical and chemical conditions in the Truckee
River, California -- Nevada during the low flow
conditions of the 1977-8 severe drought. Rept. to
McLaren Engineering Co., Sacramento, California. 87 p.
Home, A.J., H.B. Fischer, and J.C. Roth. 1982. Proposed
Monitoring Master Plan for the San Francisco Bay-Delta
19.
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Aquatic Habitat Program. Calif. State Water Resources
Board, Sacramento. 180 p.
Lueschow, L.A., J.M. Helm, D.R. Winter, and G.W. Karl. 1970.
Trophic nature of sel«cted Wisconsin Lakes. Wise. Acad,
Sci. Arts Lett. 58:237-264.
Lund, J.W.G., C. Kipling, and E.O. Le Cren. 1959. The inverted
microscope method of estimating algal numbers and the
statistical basis of estimations by counting.
Hydrobiologia. 11:143-170.
Michalski, M.F. and N. Conroy. 1972. Water quality
evaluation. Lake Alert study. Ontario Min. Envir.
Rep. 23 p.
McColl, R.H.S. 1972. Chemistry and trophic state of seven New
Zealand lakes. N.Z. J. Mar. Freshwat. Res. 6:399-447.
Needham, P.R. and R.L. Usinger, 1956. Variability in
the macrofauna of a single riffle in Prosser Creek,
California, as indicated by the Surber sampler.
Hilgar.dia 24, 14:383-409.
Rawson, D.S. 1955. Morphometry as a dominant factor in the
productivity of large lakes. Ve r h. Int. V e r ei n .
Limnol. 12:164-175.
Richerson, P., R. Armstrong, and C.R. Goldman {1970).
Comtemperaneous disequilibrium, a new hypothesis to
explain the "paradox of the plankton." Proc. Na tl .
Acad. Sci. 67:873-880.
20.
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Riley, G.A. 1976. A model of phytoplankton patchiness. L1mno1.
Oceanogr. 16:453-466.
Russell, P.P. and A. J. Home, 1977. The relationship of
wastewater chlorination activity to Dungeness Crab
landings in the San Francisco Bay Area. U.C. Berkeley-
SERHL Rept. 77-1, 37 p.
Ryder, R.A., S.A. Kerr, K.H. Loflus, and H.A. Regier. 1974. The
morphoedaphic index, a fresh yield estimator -- review
and evaluation. J. Fish. Res. Bd. Can. 31:663-688.
Sandusky, J.C. and A. J. Home. 1978. A pattern analysis of
Clear Lake phytoplankton. Limnol. Oceanogr. 23:636-648.
Shannon, E.E. and P.L. Brezonik. 1972. Eutrophication
analysis: a multivariate approach. J . S a n i t. E n g.
Div. ASCE. 98:37-57.
Sheldon, A.L. 1972. A quantitative approach to the
classification of inland waters. pp. 205-261 in Natural
Environments ed. J. V.Kratilla, Johns Hopkins Press.
Baltimore.
Steele, J.H. 1974. The structure of marine ecosystems.
Blackwell, 128 p.
Tailing, O.F. 1976. The depletion of carbon dioxide from Lake
Water by phytoplankton. J. Ecol . 64:79-121.
Wurtz, C.B. 1960. Quantitative sampling. Natil us, 73:131-5.
21.
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CHAPTER 3.
The Hysteresis Effect in the Recovery of Damaged Aquatic Ecosystems:
an Ecological Phenomenon with Policy Implications
Abstract
The individual species or functional components of an ecosystem can be
expected to respond at different rates to the application and/or removal of
pollutant stress. These rates are primarily dependent on the generation
time (a function of body size and complexity) of the organism and its place
in the trophic hierarchy (e.g. producer, grazer or carnivore). Even in the
absence of population extinctions, a non-retraceable behavior (or hysteresis
effect) is expected. Conceptually, the lower trophic levels will follow a
series of nested hysteresis curves, while organisms at higher trophic
levels, such as sports fish, will probably respond more erratically. To
explore these issues, we develop an illustrative hysteresis trophic-link
model (HTLM) that incorporates limited ecological reality but is simple
enough to expand to an arbitrary number of functional groups. This model
is compared to a conceptual model for biotic hysteresis for a system with
three trophic levels. We show how hysteresis might influence population
changes at higher trophic levels (e.g. fish) caused by pollution. These
changes cannot be measured directly because large fish are difficult to
sample with high precision.
Introduction
In most aquatic ecosystems damage occurs by two mechanisms. These are
physical destruction (for example, lake edge filling) or chemical perturbation
(notably, additions of biostimulants and toxicants). With the exception of
1
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sediment loading, most pollutants regulated by the U.S. Environmental
Protection Agency (EPA) cause damage by chemical perturbation of ecosystems.
It is often assumed that above the dose-response threshold, the change in
some component of an aquatic ecosystem is linearly proportional to the amount
of pollution, as for example in the Dillon-Rigler (1974), Vollenweider (1968),
or Vollenweider and Kerekes (1980) phosphorus- (or nitrogen-) chlorophyll
models of lake eutrophication. Studies on lake restoration have shown that
non-linearity and time lags in the recovery of systems perturbed by pollution
occur for at least some lakes (e.g. Shagawa Lake, Maguey et al, 1973; Lake
Washington, Edmondson, 1972). The reasons for non-linearity have not been
well studied, but they appear to be partially due to the varying "turnover-
times" of the physical, abiotic, and simple biotic components of a complete
aquatic ecosystem (Edmondson, 1982; Home, unpublished). Further step-
function-type responses and time lags may be introduced by "higher-order
interactions" that occur far from the site of the pollutant action. Examples
of these interactions are species displacement such as occured for lake trout
in the Great Lakes, or indirect competition from changes in species dominance
(Christie, 1971). Given these complications, it is not surprising that the
recovery of an ecosystem's more complex biotic levels, such as that of a
damaged sports fishery, does not proceed either in a simple linear or
virtually instantaneous manner upon removal of a pollutant load.
It is important to distinguish between the purely phyBiochemical and the
biotic responses to removal of a pollutant from an aquatic ecosystem. All
pollutants will decrease when the source is shut off and the internal
pollutant load is diluted as new clean water flushes out the system. In many
cases the pollutant load will be negligible in months or years—as is the case
following the onset of phosphorus removal by new sewage-treatment plants
(Goldman and Home, 1983, pp. 392-4). In any event the physiochemical
-------
response is generally predictable from a knowledge of the pollutant, the
hydraulic residence time in the system, the mean depth, and the
characteristics of the bottom sediment.
' In contrast, the biotic response may be delayed or may occur in spurts.
In extreme cases the biota may never return to their original states. The
time path of ecosystem recovery is not predictable at present since the
reasons for non-linearity are unclear. The response of the biota to a
decrease in pollution may also fail to mirror the response of the system to
the original increase in pollution, that is, the response may be non-
retraceable. This paper attempts to provide a theoretical basis for a
mathematical description of the biotic restoration of damaged aquatic
ecosystems. In particular the non-linear and non-retracable character of the
process of recovery from pollution—defined here as the hysteresis effect—
will be considered.
In the following sections we present the general methods and theoretical
basis for the hysteresis trophic-link model (HTLM), describe in a theoretical
way our concepts of "ideal" and "non-ideal" biotic hysteresis, show the
specific form used for the HTLM and some initial results from the modelling
effort, and discuss the merits and drawbacks of the HTLM approach in providing
information useful in setting environmental policy.
Methods and Theoretical Basis
Time lag effects may have many sources, but it is most logical (in the
sense of Occam's razor) to examine first the turnover time of the components
of the ecosystem as a possible source. If a population is to recover quickly
when the pollutant load is removed it must grow and breed quickly. Since the
larger organisms depend on the smaller ones as food sources, populations of
larger organisms cannot grow until populations of smaller organisms are in
-------
place. The turnover time for biota is usually the generation time and can
range from a few hours for simple bacteria and algae to decades for very large
fish such as striped bass or sturgeon. Generation time is primarily a
function of two variables: the sexuality of reproduction and the structural
•
complexity of the adults. Asexual reproduction (vegetative or
parthenogenetic reproduction) is typical of simple animals and plants growing
under favorable conditions. Sexual reproduction is typical of more complex
organisms or of simple ones growing under unfavorable conditions. Sexual
reprodiction uses more time than asexual reproduction, and confers few, if
any, short-term benefits. In addition, complex organisms must spend time in
building their large complex body structures. This involves several moults,
a long adolescence, and differing environmental requirements for adult and
young, depending on the species involved. The organisms in the trophic
levels usually present in aquatic ecosystems have the following typical
characteristic sizes (length, 1) and generation times (gt)t
phytoplankton 1 = 0.02 mm, gt = 3 days
zooplankton 1 = 1 mm, gt = 3 weeks
ichthyoplankton 1=1 cm, gt = 1 year
Juvenile piscivorous
and planktivoral adult 1 s 5 cm, gt = 1 year
fish
piscivorous fish 1 = 20 cm, gt = 3 years
large sports fish 1 s 50 cm, gt = 10 years.
The aquatic ecosystem we use in our model is simplified in the sense that
side, across, and multiple-step (omnivory) food-chain links are omitted
(Figure 1). Although this may seem like a major simplification when one
considers the apparently highly cross-linked structure of some aquatic food
webs (e.g. Figure 2), the dynamics of many food webs are in fact much less
cross-linked, in terms of energy or food flow, than they appear to be. This
-------
P2 X2
Figure 1. Schematic diagram of a trophic-link model. Here P describes
the effect of the pollutant on each trophic level, and X is a
measure of the biomass present for each functional class of
organism (e.g. primary producers, filter-feeders, carnivores,
etc.).
Inorgimc
nuuitnts.
t.J.. NO,. CO,
CO,. SO.
Figure 2. A qualitative food web for the Truckee River, California.
Solid lines Indicate measured pathways. Broken lines are
assumed pathways derived from other studies of adjacent
waters. Note that the omnivorous feeders (e.g. dace, trout,
sculpin) use more than one trophic level. Most herbivores
prefer microscopic diatoms to large filamentous green and
blue-green algae. (Reproduced from Goldman and Home, 1983)
-------
is illustrated by one of the few known quantitative examples of an aquatic
food chain, that of the River Thames below Rennet mouth (Goldman and Home,
19&3)> Figure 3a shows the complete food web for the Thames system. As
complicated as this looks, placing of the organisms in this web into
t
functional groups results in the much more simplified structure shown in
Figure 3b. Thus while the assumption of a linear food chain is certainly a
simplification, it may not be a bad starting point for modelling some aquatic
ecosystems.
In the linear food chain depicted in Figure 1, the rate of change of the
phytoplankton population can be described by the equation
dX
(D ~ - rx (x)(1 - (X/KX)) - BxyXY - bxX, where
dt
X = the population density of phytoplankton (e.g. chlorophyll a
per cubic meter of water),
rx = the maximal growth rate of the phytoplankton population,
K = a carrying capacity constant,
Bxy = a rate constant describing predation of zooplankton on
phytoplankton,
Y s the population of zooplankton that feed on the phytoplankton
(X), and
bx = the rate of loss of phytoplankton due to washout and other
linear, donor-controlled mechanisms.
In this system we assume that each organism eaten is killed and that no
significant amount of prey is uneaten.
Analogous equations can be used to describe the rate of change of the
higher trophic-level populations. For example:
-------
Figure 3a.
Light Suspended
729.000 organic matter
C, +H
This figure presents a dynamic food web for a natural system:
an energy flowchart for the River Thames below Rennet mouth.
In general, primary producers are shown at the bottom,
invertebrate animals at the center, and fish at the top of the
chart, but to avoid complex networks of arrows sources of
attached algae, detritus, and al lochthonous materials are
shown in two places. Heavy arrows indicate the largest
channels of energy flow. Note the twin flow of energy to
fish from low-quality attached algae and high-quality animal
food from terrestrial insects and adult chironomids. Energy
input from dissolves organic matter was not measured directly.
(Redrawn from Mann et al, 1972, reproduced from Goldman and
Home, 1983)
Fish (2 species)
Free-floating
Figure 3b.
Top
Carnivore
Saall
Carnivores
Herbivores
Producers
(Plants)
Light +
Nutrients
The major energetic pathways from figure 3a. This diagram
shows that modeling using single-link trophic models is
possible if the organisms in the ecosystem are classified
into functional rather than taxonomic groups.
Attached
-------
dY
(2) — = ExyBxyX* - ByZTZ - byY, where
dt
X, I, and B^ are aa above,
^xy = a factor describing the proportion of biomass consumed from
trophic level x (phytoplankton) that is retained in trophic
level 7 (zooplankton),
By2 = a rate constant describing predation on zooplankton by
icthyoplankton (small fish that feed on zooplankton),
Z s the population of icthoplankton, and
jj_ = the loss rate of zooplankton due to washout, death, or other
donor-related mechanisms.
This pair of equations can be expanded to an arbitrarily large set
describing an arbitrarily long food chain.
Changes in pollution will affect some of the growth rates directly, but all
populations will be affected as a result of trophic interactions. A
straightforward example of such an interaction is the following. Suppose a
pollutant acted so as to decrease the growth rate (r*) of tne phytoplankton in
an aquatic ecosystem. This pollutant could be toxic to the phytoplankton or
could be an inert pollutant, like silt in a lake, that affects rx by
decreasing the light available for photosynthesis. In either case, a
reduction in the phytoplankton growth rate reduces the phytoplankton
population, which reduces the amount of food available to the zooplankton,
which reduces the zooplankton population, which reduces the amount of food
available for small fish, and so on. Alternatively, a pollutant may cause an
overall increase in total phytoplankton (e.g. through eutrophication) but
bring about a decrease in zooplankton levels by allowing undesirable algal
species to dominate at the expense of species that serve as food for the
zooplankton. In this paper we have used mathematical relationships like
those described above to generate a series of "hysteresis relationships"
-------
charting the response of each trophic level in a hypothetical three-level
aquatic food chain to the pollution and subsequent clean-up of the ecosystem.
We have also assumed, in making our calculations, that the onset of
pollution and its clean-up are instantaneous. This is perhaps appropriate
for longer-lived organisms such as fish, but has some inappropriate features
for algae, which turnover rapidly and thus may respond to intermediate as well
as initial and final levels of the pollutant. If it proves important to do
so, a gradual change in pollution may be modeled in future work, but for our
initial analysis the step-function approach is more enlightening and
expedient.
Biotic Hysteresis; Theoretical Concept
The ecological hysteresis response will resemble the physical hysteresis
effect observed in the magnetization of a ferromagnet. When a magnet is
placed next to an unmagnetized bar of iron, the latter becomes magnetized.
When the first magnet is taken away, the iron bar loses its magnetic
properties much more slowly than it gained them. Similarly, as the level of
pollution in an aquatic ecosystem is decreased, the biological response to the
decrease does not trace out in reverse the path it followed in response to the
initial pollution of the system. Nevertheless, ideally, the system, returns
to its starting point. For the purposes of this paper we define "ideal"
biotic hysteresis to occur when a population of organisms perturbed by
pollution returns to its initial population level within a period of time
short enough to be relevant to policy decisions. This time period might be
10 to 20 years. In an ecosystem with several trophic levels (phytoplankton,
large zooplankton and small fish, and large fish, i.e. producers, grazers, and
large carnivores) and a single type of pollutant (such as sewage) a series of
response-and-recovery curves such as those shown in Figure 4 would be
-------
100 P
p
E
R
C
E
N
T
O
F
R
E
S
P
O
N
S
E
80 -
60 -
40 -
20 -
TIME
Figure 4. A theoretical distribution of ideal hysteresis curves for an
aquatic ecosystem with three trophic levels. Curves marked
B1" represent the time-path of the response of a population in
a lower trophic level (e.g. phytoplankton) to a pollutant
stress and the path of recovery once the stress has been
removed. "Response" paths are marked with left-to-right
arrows, while "recovery" paths are indicated by right-to-left
arrows. Curves marked "2" and "3" represent time-paths for
middle (e.g. zooplankton) and higher (e.g. fish) trophic
levels, respectively. Note that populations in higher
trophic levels exibit greater lags in both response and
recovery than those in lower trophic levels.
10
-------
expected. Even the rapidly growning phytoplankton (generation time 1-10
days) can exibit an ideal hysteresis response to the pollutant. For higher
trophic levels (copepod zooplankton and fish), which respond to the altered
phytoplankton population, there will be a delay in the initiation of the
exponential section of the curve in Figure 4 in rough proportion to the
generation time. A delay must occur because complex organisms are incapable
of rapidly increasing their number (that is, they have a slow numerical
response) on a time scale of days. It will thus take at least the adult-to-
birth-to-juvenile period before copepods or small fish can show any numerical
response to the perturbation, and this response period will be slightly
shorter than the complete generation time. This lag in response has the
interesting consequence that the last half of the change will occur more
rapidly for high than for low trophic levels. Such rapid changes would be of
serious concern to resource managers since the response of pollution-control
agencies may be too late to protect the resource before the numbers of
important organisms are seriously depleted. These rapid changes do in fact
seem to happen (see Goldman and Home, 1983). Concern about such changes is
compounded by the fact that it is difficult to measure changes in biomass
stocks at higher trophic levels, such as fish. The statistical resolution
for fish stock estimation is usually so poor that the majority of a fish
population can be lost before biologists can detect the change with any
certainty.
The ideal hysteresis effect is characterized by a cyclic (on a 10-20 year
time scale) non-retracable path when the response of organisms* to pollution
In figures 4 and 5 the response of each trophic level is normalized so that
the "percent response" at each time point is given as a percentage of the
difference between the population of the organism before the system was
perturbed and the population at the point where the pollutant is removed.
Thus these curves show increasingly lagged responses and recoveries from
pollutant stresses, but do not reflect the relative magnitudes of the
responses to pollution that might be shown by the different trophic levels.
11
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100
p
E
R
C
E
N
T
O
F
E
S
P
O
N
S
E
80
60
40
20
TIME
Figure 5. A theoretical distribution of non-ideal hysteresis curves for
an aquatic ecosystem with three trophic levels. Curves
marked "1" represent the time-path of the response of a
population in a lower trophic level (e.g. phytoplankton) to a
pollutant stress and the path of recovery once the stress has
been removed. "Response" paths are marked with left-to-right
arrows, while "recovery" paths are indicated by right-to-left
arrows. Curves marked "2" and "3" represent time-paths for
middle (e.g. zooplankton) and higher (e.g. fish) trophic
levels, respectively. In this case, unlike the ideal
theoretical case presented in figure U, the populations do not
recover completely within a recovery period of the same
duration as the original stress.
12
-------
is plotted against time for a regime in which a pollutant is added (left-to-
right paths in figures U and 5) then removed (right-to-left paths). A damped
hysteresis effect is also possible. This effect, which we have termed "non-
ideal" biotic hysteresis, is characterized by non-retracable and non-cyclic
•
behavior (as shown in figure 5), is also possible. A possible explanation of
such behavior for a specific food chain (rather than a food chain of
generalized trophic levels) is the following. If a species of plant or
animal remains at depressed levels (e.g. as a result of a pollutant-related
stress) for long periods there is in effect a vacant niche that can be
occupied by a pollution-tolerant species or even another species that has no
direct effect on the fish of concern (Christie, 1971). Generally the
replacement species are less highly regarded by sports and/or commercial
fisheries groups and are an economically inferior substitute for the original
species. Thus if the return leg of the hysteresis curve is very flat after
cessation of pollution, organisms at the valuable higher trophic level may be
subject to "species replacement" or "competitive displacement" and never
•4
return to their original dominant position.
Methods and Initial Results from the Hysteresis Trophic-Link Model (HTLM)
Our objective in this modeling effort was to test a simple approach for
describing mathematically the hysteresis phenomenon discussed above. The
purpose of the model described here is solely to illustrate how a generalized
ecological phenomenon of interest (hysteresis) can be demonstrated using
mathematical relationships containing easily identifiable and understandable
parameters. In this approach a food chain with three trophic levels—
phytoplankton, zooplankton, and small fish—was assumed. The rate of change
of the populations in the first two trophic levels were described by
differential equations (1) and (2) above, and the rate of change of the
13
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population in the third trophic level was described by
dZ
(3) — r EyzByZYZ - bzZ, where
dt
•
Y, Z, and Byz are ag previously described,
E s that fraction of biomass in the Yth trophic level that
becomes incorporated in the Zth level, and
bz = a rate constant describing the loss of small fish due to
old-age death and other donor-controlled mechanisms.
The constants in the three equations were obtained by assuming a value of
0.1 for Exy and Eyz> and a value of 2 x X* for Kx. The values for X», Y»,
and Z*, the steady-state biomass populations for the three trophic levels
(that is, the relative amounts of per-unit-area biomass for which dX/dt,
dY/dt. and dZ/dt = 0) were taken to be 50, 10, and 1, respectively.
Generation times for the three trophic levels (Tx> T an(j xz) were taken to
be 3, 20, and 360 days, respectively. The following relationships were used
to derive the values of rx> Bxy, and By2:
rx = Tx"1>
ExyBxyX» = Ty'1,
EyZByzY» = Tsf1-
values for bx> by and bz were derived from the steady-state forms of equations
(1) through (3).
Equations (1) through (3) were incorporated into a fortran computer
program, which was used to approximate the time path of populations X, Y, and
Z in response to a perturbation in rx> tne phytoplankton. The program calls
the NAG (Numerical Algorithms Group, 1984) subroutine D02EBF, which integrates
systems of differential equations using a variable-order, variable-step Gear
method and returns solutions to the system (X(t), Y(t), Z(t)) at specified
-------
time points. Details of the model and a listing of the integration program
are given in the appendix to this paper.
We should note that an analytical approximation to the solution of
equations (1) - (3) can be obtained by adding a fouth equation, namely
•
drx
(U) = 0
dt
to the system, deriving a 4 x 4 "community matrix" using procedures described
by May (1973) and Harte (1985), and using that matrix to explore the effects
of perturbations to the system. A four-level food-chain model was also
developed. This model, which adds a larger piscivorous fish to the three-
tiered food chain, uses equations (1) - (3)» above, with the term - BzfzF
added to equation (3). A fourth equation,
dF
(5) — = E2fBzfZF - bfF-
dt
is added to model the behavior of the population of larger fish (F). In this
system the steady-state biomass ratios in the four trophic levels were taken
to be 500 : 100 : 10 : 1 (X* : I* : Z*: F*), the generation time for the
larger fish (Tf) was taken to be 1080 days, Ezf was taken to be 0.1, and
E2fBzfZ*F* was defined to equal Tj.~l. This four-level system was solved as
above. Details of the model and a listing of the computer program used to
solve it are given in the appendix.
Results
The time paths traced by the three "populations" (here taken to mean
biomass present in each trophic level per unit area of water) following a -2J
reduction in rx are shown in figure 6. The population of phytoplankton drops
rapidly in response to the reduction in its growth rate, reaching a local
15
-------
p
E
R
C 100
E
N
T
0
p 90
0
R
I
? 80
I
N
A
L
P 70
0
P
60
t — - ,
X ""' —
\
\
\
\
\
\
\
X
X
N.
^^
>.
^
^^
*^
»^. 1>>^
"*•-. ^^
"~~" — «.
II 1 1 1 1
2000
4000 6000
TIME IN DAYS
8000
10000
Figure 6. Calculated time paths for the response of the populations
(measured In blomass per unit area, Initial biomass ratios:
50 phytoplankton: 10 zooplankton : 1 small fish) in a
three-tiered aquatic food chain to a -2% change in the growth
rate of phytoplankton. Solid, dotted, and (partially) dashed
lines give the paths for phytoplankton, zooplankton , and
small fish, respectively. Note that the lower trophic levels
respond more quickly to the stress than higher trophic levels,
but the ultimate effect on higher trophic levels is greater in
magnitude.
16
-------
minimum in 20 days (not visible in figure 6 due to the length of the time
scale). Thereafter the population rises quickly, then falls slowly in
response to the changes in the population of its predator (zooplankton). By
the time 10,000 days (about 30 years) have elapsed, the phytoplankton
population reaches a steady-state value equal to 99J of its original level.
The population of zooplankton drops more slowly, but over a longer period.
For this second trophic level the maximum deviation from the original
population, -3-5J, occurs after 150 days. From there the zooplankton
population rises to a level about 1£ above that originally present. The
population of small fish declines more slowly than those of either of the
lower trophic levels, but in time exibits a greater response, reaching a new
steady-state population 70% as large as the original group. Note that the
deviations in the zooplankton and fish populations are out of phase with each
other. This makes sense ecologically as well as mathematically: as fish
populations decline, grazing pressure on zooplankton is decreased, allowing
that population to expand. Perhaps the most important result shown in figure
6, however, is that a small (-19) perturbation in the phytoplankton growth
rate produces a large (-30t) change in the population at the highest trophic
level.
Figures 7-10 present time paths for the three populations in which a -2%
perturbation in rx is applied at time zero, then removed at 300, 500, 2000,
and 10,000 days, respectively. Paths for which arrows point left-to-right
chart the response of the three populations to the original perturbation,
while paths with right-left arrows chart the return paths for time periods of
the same duration as the original perturbation. Thus in figure 7, for
example, the solid curve labeled with a right-pointing arrow charts the
response of the phytoplankton population to a perturbation applied for 300
days, while the solid path labeled with a left-pointing arrow charts the level
17
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106
P
E
R 104
C
E
N
T
102
0
F
R
1
G
1
N
A.
L
loo
98
P
0 96
P
94
JL
50
100 150 200
TIME IN DAYS
250
300
Figure 7 Calculated time paths for the response and recovery of the
populations (measured in biomass per unit area, initial
biomass ratios: 50 phytoplankton: 10 zooplankton : 1 small
fish) in a three-tiered aquatic food chain when a -2t
perturbation in the phytoplankton growth rate is applied at
time zero, then removed after 300 days. "Response" paths
are indicated by right-pointing arrows, and "recovery" paths
are marked with left-pointing arrows. Solid, dotted, and
(partially) dashed lines give the paths for phytoplankton,
zooplankton , and small fish, respectively. Note that the
population of small fish continues to decline even after the
perturbation is removed, and fails to return to its original
position after 300 days of recovery.
18
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p
E
R 104
C
E
N
T
102
0
F
f\
° 100
Xt
I
G
I
N 98
A
L
P
One
yb
P
94
—
-
— /A
T?. r/ \
r_ -^s \
\\ /?~ ^^
\'-/s "*"' — '•*».
^^v ^""^^"*^^-^
*^^^
"^•^^
^"^^^
— *• "--^
~ ' " >
1 1 1 1 1 1
0 100 200 300 400 50(
TIME IN DAYS
Figure 8 Calculated time paths for the response and recovery of the
populations (measured in biomass per unit area, initial
biomass ratios: 50 phytoplankton: 10 zooplankton : 1 small
fish) in a three-tiered aquatic food chain when a -2J
perturbation in the phytoplankton growth rate is applied at
time zero, then removed after 500 days. "Response" paths
are indicated by right-pointing arrows, and "recovery" paths
are marked with left-pointing arrows. Solid, dotted, and
(partially) dashed lines give the paths for phytoplankton,
zooplankton , and small fish, respectively. Note that the
population of small fish shows a lag of approximately 50 days
before beginning its recovery.
19
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p
E
R
C
E
N
T
0
F
105
100
95
R
I
G
I
N 90
A
L
0 85
P
BO
500
1000
TIME IN DAYS
1500
8000
Figure 9 Calculated time paths for the response and recovery of the
populations (measured in biomass per unit area, initial
biomass ratios: 50 phytoplankton: 10 zooplankton : 1 small
fish) in a three-tiered aquatic food chain when a -2?
perturbation in the phytoplankton growth rate is applied at
time zero, then removed after 2000 days. "Response" paths
are indicated by right-pointing arrows, and "recovery" paths
are marked with left-pointing arrows. Solid, dotted, and
(partially) dashed lines give the paths for phytoplankton,
zooplankton , and small fish, respectively. Note that the
population of small fish fails to return to its Initial
level after 2000 days of recovery.
20
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110 U
p
E
R
C
E
N
T
100
0
F 90
0
R
I
80 -
N
A
L
P 70
0
P
60
2000
4000 6000
TIME IN DAYS
6000
10000
Figure 10 Calculated time paths for the response and recovery of the
populations (measured in biomass per unit area, Initial
biomass ratios: 50 phytoplankton: 10 zooplankton : 1 small
fish) in a three-tiered aquatic food chain when a -2f
perturbation in the phytoplankton growth rate is applied at
time zero,then removed after 10,000 days. "Response" paths
are indicated by right-pointing arrows, and "recovery" paths
are marked with left-pointing arrows. Solid, dotted, and
(partially) dashed lines give the paths for phytoplankton,
zooplankton , and small fish, respectively. Note that the
population of small fish fails to return to its initial
level even after 10,000 days of recovery.
21
-------
of the phytoplankton population after the perturbation is removed. For the
return paths time runs right-to-left, thus the points on the return paths
directly above "SO" on the time axis are actually 250 days from the point
where the perturbation was removed. The presentation of the hysteresis
curves in figures 7-10 are different from those in figures 4 and 5 in that
they are not normalized to the response of each population to the
perturbation, rather they indicate the percentage change in each population.
This allows the relative magnitudes of the population changes in the different
trophic levels as well as the shapes of the hysteresis curves to be compared.
Figures 7-10 present a series of hysteresis curves in which time paths
for the fish populations show a progression from non-ideal- toward ideal-
hysteresis behavior, as those terms are defined above. For each time
interval the phytoplankton population can be seen, after perturbation of the
system, to decline rapidly to Just above 98J of its original level, remaining
near that value for the duration of the perturbation. When the stress is
removed, the phytoplankton population quickly increases to 2% over its pre-
perturbation level, then declines to its original level and remains relatively
stable thereafter. In each of figures 7-10 the zooplankton population
decreases rapidly following perturbation, then drifts slowly higher as fish
populations decline. When the perturbation is removed zooplankton quickly
increase, due to the increased availability of phytoplankton, then decline
slowly to near their original level as fish populations increase. The
population of fish shows a slow and steady decline over a 300-day
perturbation. The decline continues for about 150 days after the
perturbation is removed. In figure 8, the fish population again declines
throughout the perturbation period and into the return period, but starts to
recover approximately 50 days after the perturbation is removed. Figure 9
shows even less lag before the fish population starts to recover. Figures 7-
22
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10 show ideal biotic hysteresis behavior for the populations in the two lower
trophic levels, which return to roughly their original values. Note,
however, that even in this case, where a recovery period of 10,000 days is
allowed, the fish population does not quite return to its original level.
These results suggest the following conclusions. First, organisms at
higher trophic levels show responses to perturbation of the ecosystem that are
less immediate but greater in relative magnitude than the responses of lower
trophic-level organisms. Second, organisms at higher trophic levels exibit
a more pronounced lag in recovery from stress once the perturbation is
removed. This lag has ecological importance beyond what we have been able to
include in our modelling effort, as a period in which the population of an
organism is low may provide an opportunity for another organism, quite often
one that is economically less desirable, to come in and occupy the former's
ecological niche.
Thus far the three-tiered ecosystem has been challenged with only a -2%
reduction in the phytoplankton growth rate. Figure 11 shows the response of
the system over the 3000 days following a "25% perturbation in r jn this
case, the population of small fish declines to less than 10f of its original
level. After the perturbation is removed, the fish population slowly
increases, but remains at less than 10f of its original level even after 2000
days. If the system is allowed 10,000 days of recovery following a 3000-day
-25% perturbation in rx» tne population of small fish gradually rises to 38%
of its original level, still ow enough to constitute an example of non-ideal
biotic hysteresis. It is probable that in a real system a sustained 90%—or
even 60%—reduction in a fish species would result in another, perhaps less
desirable, species occupying its ecological niche. This means that some
component of an aquatic ecosystem may never recover from a stress, even if
23
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p
£
R 100
r
\^
E
N
T BO
0
F
0 80
R
1
G
I
N 40
A
L
p
b 20
P
0
«•
*9
\ — =^ A
\
\
\
\
\
\ ^ :
\
\
N
\
\
\
Xx
\
^^>
^^^
^^^^__
+ \ 1 1 1 1 1
0 500 1000 1500 2000 2500 3000
TIME IN DAYS
Figure 11 Calculated time paths for the response and recovery of the
populations (measured in biomass per unit area, initial
biomass ratios: 50 phytoplankton: 10 zooplankton : 1 small
fish) in a three-tiered aquatic food chain when a -25%
perturbation in the phytoplankton growth rate is applied at
time zero, then removed after 3000 days. "Response" paths
are indicated by right-pointing arrows, and "recovery" paths
are marked with left-pointing arrows. Solid, dotted, and
(partially) dashed lines give the paths for phytoplankton,
zooplankton , and small fish, respectively. Note that the
population of small fish falls to a critical level and falls
to return to its initial level after 3000 days of recovery.
-------
some fraction of the population remains after the stress is removed.
Figure 12 illustrates that assumptions as to the shape of the "biomass
pyramid"—that is, the ratios of biomass-per-unit-area present for each
trophic level--can have a profound effect on the magnitude of the
«
magnification of perturbations down the food chain from producer to carnivore.
Here we show that the effect of a -1t change in the growth rate of
phytoplankton is greater on the fish population in a food chain with biomass
ratios of 100 : 10 : 1 (phytoplankton : zooplankton : fish) than for food
chains in which the trophic level ratios are smaller. It should be
remembered that we know only that this result pertains to the simple predator-
prey model we have been studying: the effect of the shape of biomass pyramids
on responses to stress has yet to be investigated for other types of models.
Figure 13 presents the response of the populations in a four-tiered food-
chain model to a -2% perturbation in the growth rate of the phytoplankton.
Note that, as in the three-tiered case (figure 6) the relative magnitude of
changes in the populations of the various trophic levels increase as the
organisms get larger. Another similarity is that the lag in response to
the perturbation is longer for higher trophic levels. The four-level model
does, however, appear to be more stable: a -2f perturbation in rx results in
only a 10$ decrease in the steady-state value of the larger fish population,
while the highest trophic level in the three-tiered case is decreased 30% in
population. In the four-tiered model all four populations oscillate in a
damped fashion toward a steady state value. This is the sort of behavior
that one might expect from a real ecosystem. It is also gratifying to note
that the oscillations in the populations of each predator-prey pair are out of
phase with each other. This makes ecological as well as mathematical sense.
As the population of larger fish, for example, declines, grazing pressure on
small fish decreases, allowing that population to expand. This increase in
25
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100
P
E
R
c
E 95
N
T
A
0
F
0 90
R
I
G
I
N
A
L 85
P
0
P
80
'• ' V'^x.
J-V \ \
» \ *^»,
\ N ^.
\ N "*— — Icfighic Level Ratios 9:3:1
\ N;
* ^^^^
^ \ Trophic Level Ratios 25:5:1
\ "^ ^^ ^-^.
\ """ ~ — - — .
\
\
\
S
\
\
\
X
\x^
^^^^
*~
Trophic Level Ratios ldb~:TtTfT~" •
rill it
2000
4000 6000
TIME IN DAYS
8000
10000
Figure 12 The response of three different three-tiered aquatic ecosystems to
a -1$ change in the phytoplankton growth rate. The partially
dashed curves give the response of the small fish populations
to the perturbation for food chains in which the initial
biomass ratios (per-unit-area biomass of phytoplankton:
zooplankton: small fish) are as indicated. The solid and
dashed lines give the response of phytoplankton and
zooplankton populations for a food chain with 100:10:1 biomass
ratios.
26
-------
110 -
80
5000 10000
TIME IN DAYS
15000
20000
Figure 13
The response of the populations in a four-tiered aquatic
ecosystem (measured in biomass per unit area, initial blomass
ratios: 500 phytoplankton: 100 zooplankton : 10 small fish: 1
larger fish) to a -2% perturbation in the phytoplankton growth
rate. The paths for the responses of the phytoplankton,
zooplankton, small fish, and larger fish populations are given
by the upper solid curve, the dotted curve, and partially
dashed curve, and the lower solid curve, respectively.
27
-------
small fish creates favorable conditions for the larger fish, which multiply
until the small fish have been overgrazed. At this point the population of
larger fish starts to decline, the small fish start to increase, and the cycle
starts again.
•
Figures 11 and 15 show the response of the four-tiered ecosystem to a -2%
changes in r*» and chart recovery paths for cases in which the perturbation is
removed after 2000 and 1000 days, respectively. These two figures illustrate
how important the timing of the removal of a stress can be. When the stress
is removed after 2000 days there is a pronounced lag in the return path of the
larger fish population. After 2000 days of recovery that population is still
less than its pre-perturbation level. If the stress is removed after 1000
days, the population of larger fish returns to its original level after 2000
days, and is actually 10% above its original level after 1000 days of
recovery. This does not imply, certainly, that it would be prudent to delay
the clean-up of a polluted aquatic ecosystem in the hopes that recovery will
be faster if one waits longer; it merely illustrates that the recovery of a
perturbed ecosystem may not be a simple monotonic function of the length of
time over which it has been polluted.
Our mathematical models tend to validate both the ideal and non-ideal
theoretical hysteresis models. Lower trophic levels tend to return to their
original levels after a relatively short recovery time, and thus show ideal
hysteresis. For higher trophic levels (and especially with more severe
stresses) the non-ideal hysteresis model dominates: larger organisms respond
to a stress more slowly and recover more slowly, and frequently fail to return
to their initial positions within a time-frame relevant to policy decisions.
We should note, however, that by the nature of the mathematics used all of the
populations we have modelled will eventually return to their original levels,
given a sufficiently long recovery period.
28
\1
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120
P
E
R
C
N
T
0
F
R
I
G
I
N
A
L
P
0
P
100
90
60
JL
.L
500
1000
TIME IN DAYS
1500
2000
Figure 14 Calculated time paths for the response and recovery of the
populations in a four-tiered aquatic ecosystem (measured in
biomass per unit area, initial biomass ratios: 500
phy top lank ton: 100 zooplankton : 10 small fish: 1 larger fish)
to a -2> perturbation in the phytoplankton growth rate applied
at time zero and removed after 2000 days. "Response" paths
are indicated by right-pointing arrows, and "recovery" paths
are marked with left-pointing arrows. The paths for the
responses of the phytoplankton, zooplankton, small fish, and
larger fish populations are given by the upper solid curves,
the dotted curves, the partially dashed curves, and the lower
(more highly arched) solid curves, respectively. Note that
the population of larger fish fails to return to its original
position after 2000 days of recovery.
29
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p
E
R
C
E
N
0
F
no -
R
I
G
1
N
A
L
P
0
P
100
90 -
80
1000 2000
TIME IN DAYS
3000
4000
Figure 15 Calculated time paths for the response and recovery of the
populations in a four-tiered aquatic ecosystem (measured in
biomass per unit area, initial biomass ratios: 500
phytoplankton: 100 zooplankton : 10 small fish: 1 larger fish)
to a -2J perturbation in the phytoplankton growth rate applied
at time zero and removed after 4000 days. "Response" paths
are indicated by right-pointing arrows, and "recovery" paths
are marked with left-pointing arrows. The paths for the
responses of the phytoplankton, zooplankton,small fish, and
larger fish populations are given by the upper solid curves,
the dotted curves, the partially dashed curves, and the lower
(more highly arched) solid curves, respectively. Note that
the population of larger fish returns to its original position
after 2000 days of recovery and actually overshoots its
level by 1000 days after the perturbation is removed.
30
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We expect that the addition of higher trophic levels including larger,
longer-lived organisms will show the non-ideal hysteresis model to be more
useful for population changes occuring within a time-frame relevant to policy-
making.
•
Discussion
Mathematical models of ecosystem perturbations are often used in ecology
(Patten, 1975; O'Neill, 1976) and aquatic ecology (O'Melia, 1972; Bierman et
al, 1980; Inoue et al, 1981). The drawbacks of such models are now
sufficiently well understood as to allow for their restricted use.
Our mathematically-derived curves for the pollution and recovery of an
aquatic ecosystem demonstrate a hysteresis effect. These curves agree
closely with the ideal and non-ideal conceptual hysteresis models described
above. We can use the information in our mathematically-derived curves to
choose which of the conceptual models is more realistic.
The non-ideal conceptual model selected by this process is of great
Interest since it forecasts that the most economically valuable species, such
as commercial and sports fish, will not directly and reversibly return to
their original levels. This is due to the time lags that come about in part
because organisms in higher trophic levels are slower to multiply and in part
because increases in these levels must follow recovery of their prey
populations. This type of sustained hysteresis effect is apparently inherent
in ecosystems including linked trophic levels.
Our model differs from many perturbation models (e.g. O'Neill, 1976) in
that we have assumed that the disturbance caused by pollution is small but
continuous. This kind of small change is to be expected from "modern"
pollution, where sophisticated treatment of waste is mandated and disposal of
the end product of the treatment process cannot be postponed or diverted.
31
-------
Sewage and industrial-waste effluents from large cities or companies are
examples of such waste streams. Similarly, it is unlikely that total
restoration of a grossly polluted ecosystem would be considered. Rather, a
small upgrading (e.g. through control of point-sources of toxic metals, a
decrease in suspended solids, or a reduction in chlorine loading) of a
partially restored or partially damaged system is envisaged, as opposed to a
massive ecological change. This sort of approach is typical of pollution-
control strategies currently used in the U.S.
There are, however, two potential drawbacks to our simple mathematical
model. First, pollution-induced changes in real aquatic ecosystems are
unlikely to be quite as steady and continuous as we have modeled them. For
example, many fish scarcely feed over the winter, and are thus unaffected by
decreases in algae or zooplankton populations over that time period. Second,
our model predicts that small fish will rather quickly be forced nearly to
extinction if larger (e.g. 25%) continuous depressions of primary production
are used. This is probably unrealistic due to the patchy nature of the
seasonal and spatial distribution of food for higher-trophic-level organisms.
We expect that some clarification of these drawbacks will result from our
future comparisons of the simple Trophic-Link Model (three trophic and four
levels) with a five-level version, and the comparison of both of these with
real data (yet to be assembled).
Our deterministic TLM may also be insensitive to other likely ecosystem
stresses that are stochastic in nature. A cool spring and summer may, for
example, result in the year's Juvenile fish crop being undersized at the end
of the growth season, leaving them more vulnerable to cannibalism overwinter
(Kipling, 1976). How would such a random event affect the hysteresis loops
we have modeled, especially in the recovery phase? In progressing from a
32
-------
deterministic to a stochastic modelling approach, the major difference we
would anticipate would be that the position of the system would be described
in probabilistic terms. For example, with respect to the -25% perturbation
shown In figure 11, instead of the small fish population becoming critically
*
low after 3000 days with a probablity of one, it might do so with a
probability of 0.9, and have an additional probability of 0.1 of becoming
critical at some other time. Ginzburg et al (1982) present a methodology for
obtaining such extinction probabilities within the framework of a stochastic
single-species population model. We intend to consider whether a similar
approach is feasible for a multi-species model with realistic parameters.
We realize that the results of the HTLM are dependent on the form of the
different differential equations used, the values chosen for the parameters,
the method of solution of the equations, and the functional components of the
ecosystem that the model describes. We intend, in fact to examine how
changes in the form and parameters of HTLM's affect the results of such
models. While no one trophic link model can predict the behavior of a
variety of ecosystems, or even one specific ecosystem, with great certainty,
we hope that advanced forms of the HTLM can be developed that can, when
properly specified and calibrated with field data from a_ specific ecosystem,
yield meaningful insights into the future behavior of that ecosystem in
response to pollutant stresses. This does not mean that we believe any such
model can be used to definitively predict that reducing the annual loading of
compound X by 100 tons per year will result in a 5.5J increase in the number
of game fish. The appropriate use for a properly calibrated model would be
as an aid in making the type of yes/no choices that regulators often face.
Suppose, for example, that a regulator wished to know whether or not to order
the clean-up of a specific lake. If a carefully constructed and calibrated
HTLM indicated that a substantial fraction of the population of an important
33
-------
game fish would be likely to be lost if clean-up were delayed, the regulator
might, after weighing the evidence, decide to proceed with pollution
abatement. In such a case it would not matter if the model predicted a 10{,
60$, 80$, or 100$ reduction in fish: the conclusion drawn by the regulator
would be the same.
We feel that the simplicity of the HTLM framework will make it possible to
easily calibrate models for specific situations. These models could then be
run to yield qualitative information that, because of the simplicity of the
models, can be traced back to allow a better understanding of the ecology
behind the result.
Our initial results suggest that the hysteresis effect may be one reason
why some valuable fisheries resources (e.g. the Great Lakes, where sports
fisheries have failed to re-establish themselves following pollution control
efforts) thave failed to respond to reduction in pollution. An understanding
of hysteresis phenomena may also make it possible to predict (in an
approximate way) how long it will take to see a recovery of a fish resource.
An equally important application of the concept is to use it to gain a
qualitative feeling for why some components of ecosystems and not others fail
to show ideal hysteresis behavior and consequently become locally extinct.
Further calculations using more trophic levels, different values for key
parameters, and generation times derived from data on natural ecosystems, may
show how useful the hysteresis concept can be for economic evaluation of
pollution-control benefits that may be long delayed by ecosystem hysteresis.
-------
References
Bierman, V.J., D.M. Dolan, E.F. Stoermer, J.E. Gannon, and V.E. Smith
(1980). "The Development and Calibration of a Spatially Simplified
Multi-Class Phytoplankton Model for Saginaw Bay, Lake Huron". Great
Lakes Environmental Planning Study Conference No. 3_. U.S. Environmental
Protection Agency, Grosse Pointe, Michigan. 126 pp.
Christie, W.J. (1976). "Change in the Fish Species Composition of the
Great Lakes". J., Fish. Res. Bd. Can.. 31: 827-851*.
Dillon, P.J., and F.H. Rigler (197*0. "The Phosphorus-Chlorophyll
Relationship in Lakes". Limnol. Oceanogr. 19: 767-773.
Edmondson, W.T. (1972). "The Present Condition of Lake Washington". Verh.
Int. Ver. Limnol. 18: 284-291.
Edmondson, W.T. (1982). "Daphnia In Lake Washington". Limnol.
Oceanogr. 27: 272-293.
Ginzburg, L.R., L.B. Slobodkin, K. Johnson, and A.G. Bindman (1982).
"Quasiextinction Probabilities as a Measure of Impact on Population
Growth". Risk Analysis 2: 171-181.
Goldman, C.R., and A.J. Home (1983). Limnology. McGraw-Hill, M62 pp.
Harte J. (1985). Consider a_ Spherical Cow; A Course ijn Environmental
Problem Solving. William Kaufman Inc., Los Altos, CA. 283 pp.
Hedgepath, J.W. (1977). "Models and Muddles: Some Philosophical
Observations". Helgol. Wiss. Meeresunters. 30: 92-10M.
Inoue, Y., S. Iwai, S. Ikeda, and T. Kunimatsu (1981). "Eutrophication of
Lake Biwa—Nutrient Loadings and Ecological Model". Verh. Internat.
Verein. Limnol. 21:2M8-255.
35
-------
Kipling, C. (1976). "Year- Class Strengths of Perch and Pike in
Windermere". Freshwater Biology Association Annual Report No. M, pp.
68-75.
Malueg, K.W., R.M. Brlce, D.W. Schults, and D.P. Larson (1973). The Shagawa
Lake Project. U.S. EPA Report * EPA-R3-73-026, l»9 pp.
May, R.M. (1973). Stability and Complexity iii Model Ecosystems. Princeton
University Press, 235 pp.
Numerical Algorithms Group (1981), "D02EBF - NAG FORTRAN Library Routine
Document". In NAG FORTRAN Library Manual. Mark 11. v. 1. Numerical
Algorithms Group, Downers Grove, Illinois.
O'Melia, C.R. (1972). "An Approach to Modeling of Lakes". Schweiz. Z._
Hydrol. 3^:1-33-
O'Neill, R.V. (1976). "Ecosystem Persistence and Heterotrophic
Regulation". Ecology 57;121U-1253.
Patten, B.C. (Ed.) (1975). Systems Analysis and Simulation in Ecology.
Volume 3. Academic Press, N.Y., N.Y. 601 pp.
Rigler, F.H. (1976). Book Review. Limnol. Oceanogr. 21;H8l-U83.
Vollenweider, R.A. (1969). The Scientific Basis of Lake and Stream
Eutrophication, with Particular Reference to Phosphorus and Nitrogen as_
Eutrophication Factors. Technical Report f DAS/DSI 68.27, OECD, Paris,
France.
Vollenweider, R.A., and J.J. Kerekes (1980). OECD Eutrophication Program,
Synthesis Report. OECD, Paris, France.
Winberg, G.G. (1971). Methods for the Estimation of Production of Aquatic
Animals. Academic Press, N.Y., N.Y., 175 pp.
36
-------
APPENDIX: DETAILS OP MATHEMATICS
-------
DETAILS OP MATHEMATICS
ASSUMPTIONS FOR THREE-LEVEL MODEL;
dX/dt = rxx(1 - X/KX) - B^XY -bxX
dl/dt = EjyB^yXY - By2TZ - byl
dZ/dt = EyZByZYZ - b2Z;
X = Phy top lank ton, I = Zooplankton, Z = Small Fish;
Steady-State Populations: X* = 50, Y* = 10, Z* s 1;
Exy = 0.1, Eyz = 0.1, Kx = 100;
Generation Times: TX = 3 days, Ty = 20 days, Tz = 360 days
Tx = rx~ » Ty = (ExyBxyX *~ » Tz = "
So...
rx =
= 1/TyE^yX* = (20 x 0.1 x 50)-1 = 10~2,
Byz = !/TzEyZY* = (360 x 0.1 x 10)~1 = 1/360
At Steady-State:
rxx*(1 - X*/KX) - BjcyX'Y* -bxx* = 0
- byY* = 0
EyzByZY*Z* - bzZ* = 0.
So...
bx = (1/3)(1 - 1/2) - (10"2 x 10) = 1/6 - 0.1 = .0666667
by = (0.1 x TO'2 x 50) - (1/360) = 0.05 - 1/360 = 0.01722
bz (0.1 x 1/360 x 10) = 1/360.
A-l
-------
ASSUMPTIONS FOR FOUR-LEVEL MODEL;
dX/dt = rxX(1 - X/KX) - B^XY -bxX
dY/dt = E^BxyXY - ByZYZ - byY
dZ/dt = EyzBy2YZ - B2fZF - bzZ
dF/dt = EzfBzfZF - bfF;
X = Phytoplankton, Y = Zooplankton, Z = Small Fish, F = Larger Fish;
Steady-State Populations: X* = 500, Y* = 100, Z* = 10, F* = 1;
Exy =0.1, Eyz =0.1, Ezf = 0.1, Kx = 1000;
Generation Times: Tx = 3 days, Ty = 20 days, Tz = 360 days, Tf = 1080 days.
Tx = rx'1. Ty = (ExyBxyxV1, Tz = (EyzBy2YV1, and Tf = (EzfB2fzV1
So...
rx = (Tx)-1 = (3)~1 = 1/3,
Bxy = 1/TyExyX* = (20 x 0.1 x 500)-1 = 10~3,
Byz = 1/T2EyZY* = (360 x 0.1 x 100)~1 = 1/3600, and
B2f = 1/TfEzfZ* = (1080 x 0.1 x 10)~1 = 1/1080.
At Steady-State:
rxX*(1 - X-/KX) - BxyX*Y* -bxX* = 0
ExyBxyxV - ByzY*Z* - byY* = 0
SyzByz**2* ' Bzfz*F* ' bzz* = °» and
EzfBzfZ*F* - bfF* = 0.
So...
bx = (1/3M1 - 1/2) - (10"3 x 100) = 1/6 - 0.1 = .0666667
by = (0.1 x 10~3 x 500) - (1/3600 x 10) = 0.05 - 1/360 = O.OH722
bz (0.1 x 1/3600 x 100) - (1/1080) = 1/360 - 1/1080 = 1/5^0, and
bf (0.1 x 1/1080 x 10) = 1/1080.
A-2
-------
LISTING OF COMPUTER PROGRAM USED TO CALCULATE TIME PATHS FOR
THREE-LEVEL AQUATIC BCOSTSTB4 MODEL
c This program, which incorporates the NAG subroutine do2ebf, can
c be used to solve three coupled differential equations.
c ..scalars in common
implicit double precision (a-h.o-z)
double precision H. xend
integer I
c
c ..local scalars..
double precision tol, x
integer Ifail. JR. 1W. mped, nout
c ..local arrays..
double precision W(3.2l). Y(3)
c ..subroutine references..
c d02ebf
c
external fen. out. pederv
common xend, H, ]
open(8. file='output')
c opens file, named "output", in which results are to be placed
data nout /6/
write (nout.99996)
write (8.99996)
write (nout.99994)
write (8.99994)
N = 3
IW=21
MPED = 0
1R = 2
tol = 10.0dO«»(-5)
write (nout.99999) tol
write (nout.99998)
write (8.99999) tol
write (8.99998)
x = 0
zend = 1.0d4
c Program is now set to calculate a "response" path. To calculate a
c "return" path one would substitute post-perturbation values for
c y(l-3) below
y(l) = SO.OdO
y(2) = lO.OdO
y(3) = l.OdO
H = (xend-x)/50
c Prints out solution at 49 evenly spaced points between x(0) and xend
1 = 49
Ifail = 1
call D02EBF(x. xend. N. y. tol. IR. fen. mped, pederv,
• out. W. IW. Ifail)
write (nout.99997) Ifail
write (8.99997) Ifail
A-3
-------
subroutine pederv(x. y, PW) •
c ..scalar arguments..
double precision z
c ..array arguments..
double precision PW(3.3). y(3)
c
PW(l.l) = -1.00dO'2.0dO"(1.0dO/(3.0dO«100.0dO))«y(l) +
+ + 1.00dO'(1.0dO/3.0dO)-0.0666666666666667dO
+ - (1.0dO/100.0dO)"y(2)
PW(1.2) = (I.0d0/100.0d0)«y(l)
PW(1.3) = O.OdO
PW(2.1) = (1.0dO/100.0dl)»y(2)
PW(2.2) = -(I.0d0/36.0dl)*y(3)- (4.72222222222d-2) +
+ (1.0dO/100.0dl)»y(l)
PW(2.3) = -(1.0dO/36.0dl)*y(2)
PW(3.1) = O.OdO
PW(3.2) = (I.0d0/36.0d2)»y(3)
PW(3.3) = (I.0d0/36.0d2)*y(2) - 1.0000dO»(l.OdO/3.8d2)
return
end
subroutine out(x. y)
c ..scalar arguments..
double precision x. u
c ..array arguments..
double precision y(3)
double precision z(3)
c u allows time to be counted "backwards" (for return paths), while z(3)
c is a set of variables that allow the populations, y(t). to be normalized
c with respect to one another. The equations for z(l-3) below express
c each y(t) as a percentage of the initial population in that trophic level
c
c ..scalars in common..
double precision H, xend
integer I
c
c ..local scalars..
integer J, nout
c
common xend. H. I
data nout /6/
z(l) = y(l)/0.5dO
z(2) = y(2)*10.0dO
z(3) = y(3)M.Od2
u = 1.0d4 - x
write (nout.99999) x. (z(J).J=l,3)
write (8.99999) x. (z(J).J=1.3)
x = xend - dble(I)*H
1 = 1-1
return
99999 format (1H , FB.2. 3E13.5)
end
A-4
-------
If (toUt.o) write (nout.99995)
If (tol.lt.o) write (8,99995)
20 continue
mped = 1
c mped = 1 indicates that routine is using supplied J accsi^ni^r- {in PEDERV)
c rather than calculating it internally (which happens wrrssrrn mped = 0)
write (nout,99993)
write (8.99993)
tol = 10.0dO**(-5)
write (nout.99999) tol
write (8,99999) tol
write (8,99998)
write (nout.99998)
x = 0
xend = 1.0d4
y(l) = SO.OdO
y(2) = lO.OOdO
y(3) s l.OdO
H = (xend-x)/50
1 = 49
Ifail = 1
call D02EBF(x. xend. N. y. tol, IR, fen, mped. pederv.
• out. W. IW, Ifail)
write (nout.99997) Ifail
If (tol.lt.o) write (nout.99995)
write (8.99997) Ifail
If (toLlt.o) write (8.99995)
40 continue
stop
99999 format (22hOCALCULATION WITH TOL=. e8.1)
99998 format (40b T AND SOLUTION AT EQUALLY SPACED POINTST
99997 format (8h Ifail= II)
99996 format (4(lx/). 31h D02EBF EXAMPLE PROGRAM RESULT
99995 format (24h RANGE TOO SHORT FOR TOL)
99994 format (32hOCALCULATING JACOBIAN INTERNALLY)
99993 format OlhoCALCULATING JACOBIAN BY PEDERV)
end
subroutine fcn(T. y. F)
% ..scalar arguments-
double precision T
r ..array arguments..
double precision F(3). y(3)
I
. j = 1.00dO*(1.0dO/3.0dO)*y(l)*(l.OdO-(y(l)/100.0dC -
+ (I.0d0/100.0d0)*y(l)'y(2)
+ - 0.066666666666667dO*y(l)
F(2) = (l.OdO/100.0dl)*y(D*y(2) - ((l.OdO/36.0dl)«y(2)T
+ (4.72222222222d-2)»y(2)
F(3) = (I.0d0/36.0d2)»y(2)*y(3) - 1.00dO«(l.OdO/3.6d2)J
Program is now set at steady state. To model a
in the phytoplakton growth rate, replace "l.OOdO" in the ?~
for F(l) (and also in the expression for PW(l.l). below)
c with, for example, "0.98dO" (for a 2% decrease)
(return
end
I
A-5
-------
LISTING OP COMPUTER PROGRAM OSKD TO CALCULATE TIMB PATHS FOR
POUR-LEVEL AQUATIC ECOSYSTEM MODEL
c ..scalars in common
implicit double precision (a-h.o-z)
double precision H, xend
integer I
c
c ..local scalars..
double precision tol. x
integer Ifail. JR, IW, mped, nout
c ..local arrays..
double precision W(4.22). Y(4)
c ..subroutine references..
c d02ebf
c
external fen, out. pederv
common xend. H. I
open(8. flle='output')
c Places the output of this program into a file named "output"
data nout /6/
write (nout.99996)
write (8.99996)
write (nout.99994)
write (8.99994)
N = 4
IW = 22
MPED = 0
IR = 2
tol = 10.0dO*'(-5)
write (nout.99999) tol
write (nout.99998)
write (8.99999) tol
write (8.99998)
x = 0
c Program is now set to calculate time paths starting with steady-state
c conditions. To calculate "return" paths, replace the values of
c y(l-3) below with post-perturbation values
xend = 2.0d4
y(l) = 500.0dO
y(2) = lOO.OdO
y(3) = lO.OdO
y(4) = l.OdO
H = (xend-x)/50
1 = 49
Ifail = 1
call D02EBF(x. xend. N, y. tol. IR. fen. mped. pederv,
• out. W. IW. Ifail)
write (nout.99997) Ifail
write (8,99997) Ifail
If (tol.lt.o) write (nout.99995)
If (tol.lt.o) write (8,99995)
20 continue
c This section, which is optional, calculates time points based on values
c of the Jacobian matrix of the system supplied in "PEDERV". below
mped = 1
write (nout.99993)
write (8.99993) A_6
-------
tol = 10.0dO»*(-5)
write (nout.99999) tol
write (8.99999) tol
write (8.99998)
write (nout.99998)
x = 0
xend = 2.0d4
y(l) = SOO.OdO
y(2) = lOO.OdO
y(3) = lO.OdO
y(4) = l.OdO
H = (xend-x)/50
1 = 49
Ifail = 1
call D02EBF(x, xend. N. y. tol. IR. fen. mped. pederv.
• out. W. IW. Ifail)
write (nout.99997) Ifail
If (tol.lt.o) write (nout.99995)
write (8.99997) Ifail
If (toUt.o) write (8.99995)
40 continue
stop
99999 format (22hCALCULATION WITH TOL=. eB.l)
99998 format (40h T AND SOLUTION AT EQUALLY SPACED POINTS)
99997 format (8h Ifail= II)
99996 format (4(lx/), 31h D02EBF EXAMPLE PROGRAM RESULTS/lx)
99995 format (24h RANGE TOO SHORT FOR TOL)
99994 format (32hCALCULATING JACOBIAN INTERNALLY)
99993 format (31hCALCULATING JACOBIAN BY PEDERV)
end
subroutine fcn(T. y. F)
c ..scalar arguments..
double precision T
c ..array arguments..
double precision F(4). y(4)
c
c To calculate response to a perturbation in the phytoplankton growth rate,
c replace "l.OOdO" in F(l). and PW(1,1) below with, for example "0.98dO"
c (for a -2% perturbation
F(l) = 1.00dO*(1.0dO/3.0dO)»y(l)*(1.0dO-(y(l)/100.0dl)) -
+ (1.0dO/100.0dl)*y(l)*y(2)
-«• -0.066666866666667dO»y(l)
F(2) = (1.0dO/100.0d2)*y(l)*y(2) - ((I.0d0/36.0d2)»y(2)'y(3)) -
+ (4.72222222222d-2)*y(2)
F(3) = (1.0dO/36.0d3)*y(2)-y(3) - ((1.0dO/1080.0dO)»y(3)'y(4)) -
+ 1.00dO«(1.0dO/5.4d2)-y(3)
F(4) = (1.0dO/1080.0dl)»y(3)»y(4) - ((l.OdO/1080.0dO)*y(4))
return
end
A-7
-------
subroutine pederv(x, y. PW)
c ..scalar arguments..
double precision z
c ..array arguments..
double precision PW(4,4), y(4)
c '
PW(l.l) = -1.00dO«2.0dO*(1.0dO/(3.0dO*100.1dO))*y(l) +
+ + 1.00dO«(l.OdO/3.0dO)-0.0866686666668887dO
+ - (I.0d0/100.0dl)»y(2)
PW(1,2) = (1.0dO/100.0dl)»y(0
PW(1,3) = O.OdO
PW(1,4) = O.OdO
PW(2,1) = (1.0dO/100.0d2)»y(2)
PW(2.2) = -(1.0dO/36.0d2)«y(3) - (4.72222222222d-2) +
+ (1.0dO/100.0d2)»y(l)
PW(2,3) = -(1.0dO/36.0d2)"y(2)
PW(2,4) = O.OdO
PW(3,1) = O.OdO
PW(3,2) = (1.0dO/36.0d3)»y(3)
PW(3,3) = (I.0d0/36.0d3)*y(2) - 1.00dO*(1.0dO/5.4d2) -
+ (1.0dO/1080.0dO)»y(4)
PW(3,4) = (1.0dO/1080.0dO)»y(3)
PW(4.1) = O.OdO
PW(4.2) = O.OdO
PW(4.3) = (I.0d0/1080.0dl)*y(4)
PW(4.4) = (1.0dO/1080.0dl)'y(3) - (I.0d0/1080.0d0)
return
end
subroutine out(x, y)
c ..scalar arguments..
double precision x, u
c ..array arguments..
double precision y(4)
double precision z(4)
c "u" allows time to be counted "backwards" for return time paths; z(l-4)
c is a set of variables that allow the time points for y(l-4) to be
c expressed as percentages of the initial populations in each trophic
c level
c
c ..scalars in common..
double precision H. xend
integer I
c
c ..local scalars..
integer J. nout
c
common xend. H. I
data nout /8/
z(l) = y(l)/0.5dl
z(2) = y(2)
z(3) = y(3)*1.0dl
z(4) = y(4)»1.0d2
u = 2.0d3 - x
write (nout,99999) u, (z(J).J=1.4)
write (8.99999) u. (z(J).J=1.4)
x = xend - dble(I)*H
1 = 1-1
return
99999 format (1H . F8.2. 4E13.5)
end
A-3
-------
r-,
CHAPTER ^
Ecotoxicology and Benefit-Cost Analysis:
The Role of Error Propagation
Introduction
An understandable desire exists on the part of policy makers to devise a
set of procedures, an analytical approach, that can be used to guide policy.
Such an approach would obviate the need for trusting to historical practice,
or to the intuition of wise but inevitably fallible and probably biased
individuals, or to the awkward and time-consuming process of making every
decision by plebiscite. It would "rationalize" policy making and, if the
procedure were appropriately chosen, optimize the well-being of the affected
sector of the public. Pollution abatement policy is a prime example, for
it is here that a vigorous effort is underway to promote benefit-cost analysis
as the appropriate analytical approach for determining proper emission levels
(see U.S. Executive Order 12291).
Despite the advantages in efficiency of decision making, and possibly in
enhancement of societal welfare, that may accrue to a society that employs the
benefit-cost approach to set pollution emission levels, there are major
pitfalls lurking that need to be identified and discussed. These pitfalls
fall into two categories: limitations in the ability of ecologists to describe
precisely the ecological consequences of pollutant emission rates, and
limitations in the ability of economists to describe precisely the economic
consequences of ecological changes.
Quite generally, the economic and ecological analyses that are required to
characterize and quantify costs and benefits of particular pollutant abatement
strategies consist of a sequence of steps. Table 1 shows what a typical
sequence of steps would have to look like for a believable benefit-cost
1
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Change in a polluting activity
(e.g., placement of scrubbers
in power plants)
i.
(combustion science)
I Change in emission levelsI
(e.g. reduction in SOg output) |
2.
(atmospheric sciences)
Change in primary stress on
ecosystem (e.g., increase in pH
of precipitation at a particular
watershed)
3
(biogeochemistry)
Change in secondary stresses (which I
act directly on biological populations
and processes) (e.g., increase in pHl
of surface waters and soils) I
4 I
(biological toxicology)
Direct biological effects of changes
in secondary stresses (e.g.,increase
in populations of acid-sensitive
plankton)
5.1
Direct market value of
changed use patterns and
of indirect benefits (e.g.,
value of user-day fees and
additional water supply);
value of other benefits
(e.g., feelings of civic
accomplishment, spiritual
satisfaction)
(ecology)
Indirect ecological changes
stimulated by the direct biological
effects (e.g., improvement in
fish productivity)
(economics and the
political process)
6.
(environmental sciences,
sociology, ...)
Change in pattern of direct use of ecosystem
(e.g., fishermen flock to site)
Change in indirect ecological benefits to
society (e.g., hydrologic integrity of
watershed is enhanced, leading to reduction
in fluctuations of water supplies to people)
7.
Table 1. The stages of ecosystem impact assessment
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analysis, with the example of acid rain used to provide specificity. The
information that must be used to quantify any given step in the sequence must
come from analysis at the preceding stage. Thus the possibility exists that
error may propagate through the sequence to the point where the final output—
for example, the economic benefit of a particular level of pollution
abatement—is so uncertain as to be of little or no use in a benefit-cost
analysis or related procedure.
Whether or not this occurs will depend in part on the degree to which
ecologists and other environmental scientists can characterize the uncertainty
in a manner that can be used by economists. To take a simple example,
consider the statement that the decrease in fish mortality following pollution
abatement in a particular lake is uncertain. This statement may mean that
the decrease in mortality cannot be predicted accurately but that the odds of
any specified degree of decrease in mortality are known (from some combination
of measurement and modeling). Or it may mean that only the range of
uncertainty is known but that the probabilities of any particular value of
mortality within that range are not known. In the former case, economists
may be able to estimate an expected value of benefit of any particular degree
of abatement (using methods such as those described elsewhere in this report),
whereas in the latter case the opportunity to characterize the benefit of any
particular degree of abatement is considerably more limited.
In the remainder of this chapter we discuss in a systematic and general
manner the subject of error propagation in environmental impact assessment,
with an emphasis on impacts involving ecosystems. We deduce some general
results about error propagation that are independant of the method of
analysis. One key result is that error tends to "biomagnify" in ecological
food chains, so that a small degree of uncertainty about the effect of a
pollutant on the lowest trophic level is likely to translate into much more
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substantial uncertainty about the effects on higher trophic levels, in which
we are often more interested. We also explore the origin of some of the
most refractory types of error in impact assessment. To relate the analysis
to the specific concerns of practitioners of economic evaluation we also show
how the relevant issue is not merely one of the magnitude of the range of
uncertainty but also of the type of uncertainty; this is because economic
analysis, which must begin where ecological analysis leaves off, can cope with
some kinds of uncertainties better than others. Of particular concern in the
context of benefit-cost analysis is the degree to which sources of ecological
uncertainties can be characterized in ways that will be of use to economists.
The overall dimensions and a few critical elements of this problem are
discussed here, but it will be shown that considerable work on the part of
ecologists will be necessary to bridge the gap between what is now known and
what needs to be known to provide a plausible underpinning for the successful
application of benefit-cost methods of decision-making.
Uncertainty iii Impact Assessment; an Example
Examples of error propagation in environmental science abound. Consider
the acid rain example from Table 1. Analysts have attempted to establish the
existence and valuse of a threshold level of precipitation pH, below which
lakes would become acidic and above which the natural restorative capacity of
lakes and surrounding soils would afford protection. The existence of such a
threshold would make the task of setting standards easier because such a
threshold would provide a natural level to aim for—tightening the standard
beyond the threshold would lead to diminishing returns.
However, uncertainties in impact assessment render the threshold notion a
highly dubious one in this context. It is likely, in fact, that one's
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perception of the location of the threshold for a particular class of lakes
depends on how long one has been observing those lakes under various levels of
exposire; whereas precipitation with a pH of, say, 1.5 might acidify the lakes
in 10 years, precipitation with a higher pH of, say, 4.9 might acidify the
lakes in 30 years, a. period longer than anyone has had the opportunity to
observe. Thus the threshold concept is time-dependent and intrinsic
uncertainty characterizes its evaluation
The threshold value for one class of lakes might not be of much use for
others. For example, in eastern North America it has been pointed out that
over several decades, the period over which observations have been made, lakes
receiving precipitation with a pH of less than about U.7 have had their
chemistry altered by the precipitation. Even if we accept this relatively
short time-frame for that particular group of lakes, there is still
uncertainty as to the value of this "threshold" in other areas. In the
mountains of the western United States, for example, the susceptibility of
lakes to acidification appears to be greater than in watersheds of the
northeastern U.S. (Roth et al, 1985). A more complete discussion of
uncertainties plaguing the use of the threshold concept in ecotoxicology is
found in Cairns and Harte (1985).
Even if we had confidence in the location of a pH threshold, we would still
not know exactly what the effect on precipitation pH would be for any
specified emissions reduction plan. Here the uncertainty stems from the
complexity of the source-receptor relation.
The uncertainty in deducing the effect of a particular level of emissions
reduction on precipitation pH must be combined with the further uncertainty in
deducing the effect of a reduction in precipitation pH on surface water
acidity. By combining these two uncertainties, the overall uncertainty in
steps 2 to U of Table 1 can be determined. At the other stages in the impact
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assessment further opportunity for error arises. The combined error is
almost invariably sufficiently large to make it difficult to obtain a precise
characterization of the ecological benefits from a particular emissions-
reduction plan.
The fact that one cannot precisely characterize the benefits of a
pollution-abatement policy should not he taken to mean that the policy is
unwarranted. Even though an economic analysis might not produce a reliable
cost-benefit ratio, it can lead to a range of uncertatinty in that ratio,
which can then be evaluated through the political process to determine what
policy action is warranted. The first step, however, must be to have a
systematic approach to the analysis of uncertainty; this is discussed in the
following section.
^ Framework for Analysis
The sequence of steps, in an environmental impact assessment as shown on the
left hand side of Table 1 provides a convenient framework for analysing the
propagation of error in such assessments. Generally, the relation between
the ith and the i+1st stage in the sequence is likely to look like one of the
three graphs shown in figure 1. In each of the graphs, the horizontal axis
represents the variable describing the i^h stage and the vertical axis
represents the subsequent one down the chain. The first of these three
graphs illustrates a linear relation, in which the response, or output, at the
subsequent stage is proportional to the input from the one before, as, for
example, if the loss of organisms is proportional to the concentration of a
pollutant. The second one illustrates a threshold process, in which an
output is only weakly dependent on an input for small values of the input, but
when the input exceeds a critical value, then the output rises sharply. The
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a.
b.
Xi
c.
Xi
Figure 1
Illustration of a linear (a), a threshold (b), and a saturation (c) process
relating variables describing successive stages in the assessment chain.
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third graph in Figure 1 illustrates a saturation process, in which an output
ceases 'to be strongly dependent on input once the input exceeds a critical
value.
These three basic types of relations between sequential stages in the
impact chain can be modified or combined to describe, generically, most
real processes. For example, the graphs can be turned upside down to
describe processes in which an output is a decreasing function of input. Or
graphs 1-b and 1-c can be combined to describe a process with a threshold at a
relatively low value of the input and a saturation effect at a higher one.
If knowledge of the functional relation between two sequential stages in
the chain were complete, and the input data were known with perfect precision
and accuracy* then a graph of the function describing the relation might,
indeed, look something like one of the plots in Figure 1, But, in reality,
there is always uncertainty in both knowledge of functional relations and in
the data needed to substitute into those functions. These uncertainties will
propagate down the impact chain, sometimes leading to a surprisingly high
level of uncertainty at the end.
Two types of uncertainty were alluded to above. one results from poor
knowledge of the dynamics of the processes--!.e. uncertainty in our
understanding of the form of the relation between variables—and one results
from incertain numerical values for data. For example, suppose that we are
interested in estimating the uncertainty in our knowledge of the lessening of
damage to plankton populations due to an expected decline in the rate of input
of a pollutant to a lake. Because it is difficult to predict with high
"Precision" refers to the detail with which a number is expressed—the number
of significant figures. "Accuracy" refers to how close the number is to the
true, or real, value. Thus if I state my height is 3-47258 meters, I am
being precise but inaccurate. Oftentimes authors will substitute precision
for accuracy, providing more significant figures than the data deserve and
giving the illusion that they are highly accurate.
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accuracy how the concentration of a pollutant in a lake will respond to a
change in the input rate, there will be uncertainty in our knowledge of what
the concentration of pollutant in the lakewater will be. On top of that we
will have, at best, only partial knowledge of how the plankton population will
respond to any precisely stated change in the pollutant concentration. In
other words, even with perfectly accurate data describing the pollutant, our
knowledge of the functional form of the relation between pollutant
concentration and plankton survivability is uncertain.
Because of the uncertainty in our knowledge of functional relations, the
graphs shown in Figure 1 must be modified as in Figure 2. Furthermore,
because the input data (the horizontal axis variable) are likely to be
uncertain, the output (the vertical axis variable) is also going to have an
uncertainty that reflects the fuzziness of the input data. At each stage in
the chain, the uncertainty may be amplified or damped as uncertainty in the
output from one stage becomes uncertainty in the input to the next. Figure 3
provides a generic illustration of how the error will propagate down the
chain. The range of uncertainty is shown to broaden in the figure, a result
of the width and steepness of the functional forms assumed. If probability
distributions characterizing the likelihood of the parameters taking on
particular values within the range of uncertainty are known, then a more
sophisticated analysis can be carried out; shown here is the simpler case in
which only the propagation of the range of uncertainty is described.
A useful analysis of the consequences for policy makers of this sort of
error propagation is given in Reckhow (1984). In the following section, we
discuss some general results about uncertainty that can be deduced from the
above considerations.
-------
xi
xi
a.
b.
c.
Figure 2
Examples of error bands in the curves shown in Figure 1,
10
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Ax,
Figure 3
Illustration of the propagation of error along the assessment chain,
in Xj is "passed along" to Xi+1 in the manner shown.
In each graph, the uncertainty
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General Results; The "Fallacy of the Mean" and "Error Biomagnification"
Quantities such as fish productivity or water clarity, indeed any parameter
to which a numerical range can be ascribed, can be characterized by a mean
value and a range of uncertainty about that mean. Because it is much simpler
to focus on a mean value, which is a single number, rather than on the range
of uncertainty, which is at the very least a range of numbers (often with a
complicated interpretation attached explaining what that range really refers
to) it is not uncommon for analysts to be asked questions such as "if I take
the mean value of the pollutant concentration and substitute that into the
formula relating concentration to plankton survivability, then what mean value
will I obtain for plankton survivability?" This question reflects a
fundamental confusion: a function evaluated at the mean value of its
independent variable is generally not equal to the mean value of the function.
Indeed, as shown below, considerable error can result if mean values are
estimated by commiting this "fallacy of the mean".
How will the general shape of the graph (as is Figure 1) of the relation
between two successive stages in impact assessment influence the error
committed by assuming that a function of the mean equals the mean of the
function? Figure *l illustrates the answer to this question. In this
figure, the parameter, a, has an equal probability of lying anywhere in the
range from B to C and its mean is midway between at E. At the upper end of
this range, x(a) takes on the value D while at the lower end it takes on the
value A. As the figure shows, if the relation between an independent
variable, a, and a dependent variable, x, is linear, then despite uncertainty
in our knowledge of a, the mean value of x, denoted by x, is equal to x(a)
evaluated at g, the mean value of a. In equation form, X = x(a). For the
case of a threshold-type relation, this figure shows why X > x(a), while for a
saturation process, X < x(a).
12
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X(a)
X(a)
B E
a. IF BE=EC, THEN AB=CD
b. IF BE=EC, THEN DOBA
c. IF BE=EC, THEN AB>CD
Figure 4
The relation between the mean value of X and the value of X evaluated at the
mean value of the parameter, a, upon which it depends, is shown for the
three cases of a linear (a), upward curving (b), and downward curving (c)
relation between X and a.
13
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This can be very important in practice; for relations characterized by very
steeply curved functions, the use of the mean value of the independent
variable for evaluating the mean value of the dependent one can lead to a
gross under- or over-estimation, depending on the type of curvature in the
functional relation. To illustrate this, we present the following example.
The attenuation of light with depth in a relatively transparent lake obeys
a simple formula: I(d) = IQ exp(-vd), where I(d) is the intensity at depth d,
Io is the intensity of light at the surface, and v is a constant
characterizing the transparency of the water. The more opaque the water, the
larger the value of v . Primary productivity of aquatic plants at any
particular depth will be roughly proportional to the value of I at that depth,
although it also depends, of course, on concentrations of essential nutrients
such as nitrate and phosphate. Suppose siltation results in a large value of
v . We will assume that the mean value of v is 0.3/meter and that the range
of uncertainty is +_ 0.02/meter. We will interpret this range to mean (for
the sake of simplicity) that the actual value of v is equally likely to lie
anywhere in the range from 0.28 to 0.32/meter. Suppose erosion control is
expected to reduce the value of v to 0.17 +. 0.09, with the range of
uncertainty increased because it is not known how effective the control
program will be. At a depth of, say, 20 meters, the mean value of I prior to
the erosion control that would be calculated (incorrectly) by substituting the
mean value of v into the formula for I(d) is Io exp(-6.0) or 0.0025 IO'
After the control is implemented, the similarly incorrect value is
Io exp(-3.M) = 0.033IO» an increase of I by a factor of about 12. However,
if the actual mean value of I is calculated properly, not by substituting into
exp(- vd) the mean value of v but rather averaging over the range of
uncertainty in v , then we find that erosion control results, on the average,
in twice as great an increase in mean light intensity at 20 meters. Leaving
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aside subtleties such as whether plants respond to the average light intensity
they receive or to some more complicated value that depends on the
fluctuations, there is clearly a large potential for error in naively
estimating mean values by being oblivious to the uncertainties.
We emphasize that the propagation of error by this means can result either
from a situation where one knows what the uncertainties are but uses the
incorrect formula relating mean values, or from a situation where one simply
under- or overestimates the magnitudes of the uncertainties but uses a correct
averaging procedure for estimating mean values.
In the modular approach to error propagation discussed in the previous
section, there is an opportunity for errors of this type to either be
reinforced or to cancel. If a sequence of relations between the variables
describing the successive stages in the impact chain are all of, say, the
threshold type, or more generally, of any similar curvature, then the error
propagation that results from ignorance of the true range of uncertainty will
be reinforcing, leading to greater and greater error as one moves along the
chain. In contrast, if curves of types 1.b and 1.c from Figure 1 are equally
represented in the chain, then the tendency will be for the errors of that
type to cancel.
Next, we turn to the topic of "error biomagnification". Error, like many
a toxic substance, will frequently increase as one probes higher up the food
chain (not to be confused with the impact assessment chain in Fig. 1),
although the mechanism that accounts for error biomagnification is quite
different from that for toxic substance biomagnification. To see how error
biomagnification arises, consider the following relatively simple model for a
food chain. Figure 5 illustrates the model, showing the inflows and outflows
of biomass from each link in the chain. The links can be thought of as
15
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aN XN
6N-1,N XN-1 XN
] 6N-1,N XN-1 XN
al Xl
e34 x3 x4
23
12
a2 X2 + Y2 X2
Figure 5
A trophic chain and the rates of biomass input and output from each link in
the chain as described by a simple Lotka-Volterra model.
16
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species (for example, grass, which is eaten by rabbits, which are eaten by
lynx, etc.) or as functional groupings of species (for example, primary
producers, herbivores, first carnivores, . . . and on up to top carnivores).
In equation form, the model reads as follows:
dX1
--- = 01X1 - Y1X12 - 612X1X2
dt
dX2
--- = E12&12X1X2 . a2X2 _ Y2X22 - ^23X2X3
dt
- 3^X3X4
dt
"*
dt
In these equations, the X^ are the biomasses of the components; the
coefficients gji are rate constants describing the predation of species j
upon species i; the coefficients E^j describe the efficiency of incorporation
of prey biomass by the predator; and the coefficients at and yi are srowth and
death rates for the individual species. The presence of the y^ terms
represents a negative feedback mechanism induced by the finite carrying
capacity of any realistic environment. They result in steady-state solutions
that are stable against perturbations such as the removal of some percentage
of the biomass of the system. Indeed, the only solution to these equations
is one in which all the Xi approach time-independent values. Although real
populations are not found in steady-state (that is, the numbers of
individuals in real populations generally exibit both cyclic and random time
dependence), models with steady-state solutions are often used to study the
17
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time-averaged behavior of such populations. Although simple models of this
sort are generally unreliable for making detailed predictions of the values of
the variables, Xi(t)> they are useful for exploring the qualitative features
of ecosystems.
Suppose that the growth rate of the primary producers is affected by a
pollutant, but that there is some uncertainty about the magnitude of the
effect. In other words, suppose that the value of a1 ig known only to be in
the range between $1 + a and a-) -a where $1 *s fche mean value and o is a
measure of the uncertainty in the mean. How will the uncertainty in affect
the uncertainty in the steady-state values of the individual variables, Xj.?
A simple two-level model illustrates the general idea:
--- = aiX-| - Y1X-T - 612X1X2
dt
dX2
--- = E12$12X1X2 - a2X2 -
dt
For this case the steady-state solutions for the X^ are:
--------------- and
E-I2&122 + Y1Y2
A measure of the relative uncertainty in the Xi caused by the uncertainty in
»i is (a/XiXSXi/Sotj). Thus the ratio of the relative uncertainty in X-| to
that in X2» which we denote by Ri2» *3
(a/X2)(3X2/3a1)
This can be shown to equal (Y2X2)/(a2 + Y2X25' wnich is less than unity-
18
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other words, the relative error in X2 induced by the uncertainty in
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110 -
50
2000
4000 6000
TIME IN DAYS
8000
10000
Figure 6a
The response of the populations in a three-tiered aquatic
ecosystem (measured in biomass per unit area, Initial blomass
ratios: 50 phytoplankton: 10 zooplankton : 1 small fish) to -1f,
-2%, and -3f changes in the phytoplankton growth rate. Solid,
dotted, and (partially) dashed lines give the paths for
phytoplankton, zooplankton , and small fish, respectively. This
figure corresponds to a situation in which the degree of
perturbation in the growth rate, caused, for example by pollution,
is uncertain, but is known to lie within some range. The effect
of this uncertainty on the relative magnitudes of population
changes in the three trophic levels is shown.
20
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+5X
50
2000
4000 6000
TIME IN DAYS
8000
10000
Figure 6b
The response of the populations in a three-tiered aquatic
ecosystem (measured in biomass per unit area, initial biomass
ratios: 50 phytoplankton: 10 zooplankton : 1 small fish) to +2%,
+3>5f, and +5% changes in the rate at which fish die off. Solid,
dotted, and (partially) dashed lines give the paths for
phytoplankton, zooplankton , and small fish, respectively. This
figure corresponds to a situation in which the degree of
perturbation in the die-off rate is uncertain, but is known to lie
within some range. The effect of this uncertainty on the
relative magnitudes of population changes in the three trophic
levels is shown.
21
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75
-1%
-2%
-3%
5000 10000
TIME IN DAYS
15000
20000
Figure 6c
The response of the populations in a four-tiered aquatic ecosystem
(measured in biomass per unit area, initial biomass ratios: 500
phytoplankton: 100 zooplankton : 10 small fish: 1 larger fish) to
-1%, -2%, and -3% changes in the phytoplankton growth rate. The
paths for the responses of the phytoplankton, zooplankton, small
fish, and larger fish populations are given by the upper solid
curve, the dotted curve, and partially dashed curve, and the lower
solid curve, respectively. This figure corresponds to a situation
in which the degree of perturbation in the growth rate, caused,
for example by pollution, is uncertain, but is known to lie within
some range. The effect of this uncertainty on the relative
magnitudes of population changes in the four trophic levels is
shown.
22
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nature study (or the public itself, which occupies the top carnivore spot in
the global ecosystem!). The increase in error as it propagates up the chain
will tend to render difficult the prediction of the magnitude of precisely
those effects that the public is most concerned about. While an enormous
effort is sometimes expended trying to determine precisely the environmental
concentration of a pollutant, the effort may be misplaced if error propagation
leads to large uncertainties higher up in the food chain where the public
welfare is more directly and obviously involved.
Like toxic substance biomagnification, this magnification of error is
unavoidable. It is a consequence of the fundamental ecological dynamics of a
food chain and can not be circumvented. Like toxic substance
biomagnification, whose effects at the higher trophic levels can be minimized
by keeping the level of the toxicant in the environment to a minimum, the
effect of error propagation up a food chain can be minimized by keeping to a
minimum the initial error in our knowledge of the effect of the toxicant on
the growth of the primary producers.
We have not discussed here the question raised in the Introduction
concerning the probability distribution of the quantity of interest within its
range of uncertainty. As mentioned previously, when a parameter such as a
fish population is uncertain, but a probability distribution for it is
calculable, then economic valuation is easier than when such a probability
distribution is unknown. Consider an uncertainty in the effect of a toxicant
on the growth rate of a species of phytoplankton, as in our simple food chain
model, that has the characteristic that the error in our knowledge of it is
gaussian-distributed. What will the distribution of biomagnified error be in
the fish population? Unfortunately, no general statement that is model-
independent can be made about this at present. The particular, unabashedly
unrealistic, model used to motivate the existence of the phenomenon of error
23
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biomagnification provides a precise answer to this question, but other models
will generally provide other answers. Because we lack confidence in any
particular model or class of models for the analysis of complex ecosystems,
further work is clearly needed here.
Since our ability to characterize ecological uncertainty with probability
distributions is presently limited, it might seem like a sensible strategy for
ecologists to place more emphasis on reducing the range of uncertainty. As
we show in the following Section, that approach, too, has its limits and,
indeed, they are even more stubborn than are the problems discussed
heretofore.
Refractory Error in Ecology
Some types of uncertainty in impact assessment are easily remedied. If a
few more observers spend a little more time gathering data or improving their
models, a noticeable improvement will result and these remediable types of
errors will be eliminated or at least greatly reduced in magnitude. A more
interesting class of errors can not be pushed to zero, however, or even
significantly reduced in magnitude regardless of how much effort is expended
to do so. These are the refractory or intrinsic uncertainties whose origin
we now discuss. In a general sense, they stem from two sources: uniqueness
and sensitivity to Initial conditions. We explain these in turn.
The uniqueness of individual ecosystems and of the planetary environment in
its entirety renders it impossible to achieve the sina qua non of the
classical scientific experimental approach—replication of the system under
Investigation. Without the benefit of replicable systems, a statistically
meaningful analysis of the effect of a toxin on an ecosystem is unattainable.
The reason is that in any dose-response study, be it at the level of an
individual organism or at the ecosystem level, one's interest is always in the
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difference between a treatment and a control system. Inherently, this
requires at least two initially identical systems. If replication of the
treatment and control systems is also desired so that a measure of the
statistical significance of the dose-response relarion can be derived, then
even more identical systems are required. Ecosystems, unfortunately, are not
so obliging. Two nearby lakes, two forests in the same region, and even two
patches of meadow close by one another differ in myraid ways; ecologists
will never be aware of all of them, let alone be able to quantify them.
To attempt a resolution of this dilemma, interest in ecological microcosms
has recently accelerated. Microcosms are segments of natural ecosystems of a.
size convenient for laboratory replication and analysis. Lake microcosms,
for example, consist of containers filled with lake water and possibly lake
sediments taken from a real lake. If appropriate precautions are taken
in the design, initiation, and operation of these systems, they can be
replicated adequately for periods of up to several months and used for
toxicological testing. Because they can be put together in such a way that a
large fraction of the natural ecological diversity in the parent system is
present in the microcosms, they offer a partial solution to the problem of
uniqueness. Valuable as the microcosm approach is for ecotoxicological
testing, problems of size or scale inherently limit its usefulness. Most
importantly, it is not feasible to place large plants an animals in them; to
do so would result in wildly unrealistic behavior, both with respect to
chemical concentrations and population densities in the microcosms.
Therefore, the very types of organisms of greatest interest to the public can
not be studied in such systems. In addition, long-term microcosm
investigations (usually of more than a few months duration) are not possible
without jeopardizing the ecological realism (that is, the degree of similarity
between the control microcosms and the parent ecosystem from which the
25
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microcosms were derived) of the microcosms.
Which brings us to the second refractory source of uncertainty—
sensitivity to initial conditions. Ecosystems, like the global climate
system, are complex at many spatial and temporal dimensions. That is, within
such systems microscopic behavior and macroscopic behavior are present and are
strongly coupled. For example, the population dynamics of microbes can
affect the health of fish in a lake, and at a molecular level, the diffusion
of nutrients and the turbulence of the water can affect the microbe
populations. In the global climate system, atmospheric turbulence influences
climate on a macroscopic scale. In systems where such different dimensions
are coupled and chaotic or turbulent behavior is important, the ability to
predict the future consequences of the system is severly limited. In a
profound analysis of the effect of turbulence on climate prediction, Lorenz
(1969) showed that microscopic turbulence introduces an intrinsic source of
error in the prediction process. In particular, it renders the future
behavior of the climate incredibly sensitive to initial conditions. The
amount of detailed initial conditions one needs to measure in order to predict
future climate with any specified degree of accuracy increases faster than
exponentially with the period of time into the future one wants to predict the
climate. Long term prediction with the same detail and accuracy as we now
can achieve for one or two day predictions thus becomes intrinsically
impossible for a practical reason: we can not gather sufficiently detailed
measurements on today's climate.
The deep reason for this phenomenon is the extreme sensitivity of complex
systems possessing many scales of motion, such as systems with turbulence, to
small changes in initial conditions. Platt et al. (1977) investigated marine
ecosystems and found a similar sensitivity to initial conditions. It is
likely, in fact, that ecosystems, generally, are characterized by such a
26
-------
sensitivity, although this has not been investigated yet.
Conclusion
The major advances in environmentally relevant ecological research in the
past decade have not been in the direction of developing models that can
*
predict with greater accuracy the future state of a disturbed ecosystem or the
distribution of values of some uncertain parameter within its range of
uncertainty. Rather the direction of progress has been in characterizing the
features of ecosystems that render them either vulnerable or susceptible to
change when subjected to stress and in identifying the major sources of
uncertainty. Rather than making substantial progress in the development of
one "correct" mathematical model for predicting the future behavior of an
ecosystem, the effort has been to search for relatively model-independent
truths. Valuable as this information is, it does not necessarily provide the
type of information economists need if they are to apply valuation procedures
to realistic situations. Error propagation and the existence of refractory
sources of uncertainty in ecology must be taken into account if realistic
goals for benefit-cost analysis in environmental policy are to be set. Perhaps
most importantly, uncertainty about uncertainty—that is, uncertainty about
the probability distribution of ecological variables within their range of
uncertainty—limits progress toward more rational decision making. Perhaps
error distributions can be better characterized and refractory uncertainties
can be reduced by more intensive analysis of ensembles of models in
conjunction with properly designed laboratory and field studies. In any
event, progress toward the goal of more rational decision making will
require that economists and ecologists working at the interface of these two
discriplines are aware of the internal constraints of each others' field,
while at the same time they sharpen their tools within their own.
27
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References
Cairns, J., and J. Harte, 1985. "The Myth of the Threshold in
Ecotoxicology", in preparation.
Lorenz, E., 1969. "The Predictability of a Flow Which Possesses Many Scales
of Motion". Tellus 21. (3), pp. 289-307.
Platt, T., K. Denham, and A. Jasby, 1977. "Modelling the Productivity of
Phytoplankton", in The Seas; Ideas and Observations on Progress in the
Study of the Seas. E. Goldberg, ed. Vol. 6. Wiley, N.Y., N.Y..
Reckhow, K., 198M. "Decision Theory Applied to Lake Management", Duke
University School of Forestry and Environmental Studies, Durham, N.C..
Roth, P., C. Blanchard, J. Harte, H. Michaels, and M. El-Ashry, 1985. "The
American West's Acid Rain", World Resources Institute, Research Report #1.
Washington, D.C..
U. S. Executive Order 12291, 1981.
28
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Chapter 5
Hysteresis, Uncertainty, and Economic Valuation
I. INTRODUCTION
The purpose of this chapter is to investigate some issues that arise
when one attempts to conduct a benefit evaluation for the control of pol-
lution in an aquatic ecosystem. Obviously, the extent of the benefits de-
pends on the nature of the ecosystem's response to control. We are concerned
with two aspects of ecosystem behavior in particular. The first is the
phenomenon known as "hysteresis", as discussed in chapter 3. Recall that
this is the notion that a damaged ecosystem may not respond immediately to
a cessation in pollution discharges and, when it does respond, may not
exactly retrace the trajectory of its decline. Indeed, because of some
irrecoverable losses from the system, it may never return to its original
state. The second aspect of ecosystem behavior we focus on is the stochas-
ticity of natural phenomena which, as emphasized in chapter 4, implies that
the ecosystem response is inherently uncertain.
Both the uncertainty and the dynamic constraints on ecosystem behavior
need to be taken into account in evaluating the benefits of control and in
the related decision on whether, or when, to control. Recovery dynamics, for
example, may favor doing nothing, as in the case where the system is so far
gone that recovery is impossible, or they may favor early action precisely
to forestall more damaging, long-lasting consequences.
When uncertainty is factored into the analysis, an additional considera-
tion arises which is sometimes overlooked. The temporal resolution of uncertainty-
-1-
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the possibility of acquiring better information about the future consequences
of controlling or continuing pollution—adds an extra element to the decision
calculus. Regardless of whether the decisionmaker exhibits risk aversion or
risk neutrality, if further information is forthcoming, there is a premium on
those initial actions which preserve future flexibility and a discount on those
which reduce flexibility and preclude the exploitation of the additional infor-
mation at a later date. In the present context, this could be information
about either the dynamics of ecosystem behavior or the social valuation of eco-
system products. If we control pollution now and, subsequently, learn that the
ecosystem was not at a threshold of irreversible damage, we can always resume
pollution later; but if we do not control now and then observe irreversible
changes in the ecosystem, we cannot undo them by controlling later. Similarly,
if we control now and then learn that future generations place a low value on
ecosystem services, we can resume pollution; but if we do not control now and
the ecosystem is irreversibly damaged, it is too late to act if we subsequently
discover that future generations place a high value on the ecosystem. In each
case there is an asymmetry in our ability to exploit future information and a
premium associated with the action that preserves flexibility.
This flexibility premium has been recognized in the environmental valuation
literature under the name of "quasi option value" (Arrow and Fisher [1974]) or
"option value" (Henry [1974]). Within the context of an irreversible land
development decision where the future benefits of preservation in an unde-
veloped state are uncertain, these authors show that, when a decisionmaker
ignores the possibility of acquiring further information about the future
value of undeveloped land, he inevitably understates the net benefit of preser-
vation over development and prejudices the decision somewhat in favor of im-
mediate development.
-2-
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The present wo^k extends these results in several ways. First, we con-
sider a decision framework where the irreversibility is associated with not
taking action now (i.e., not controlling): In effect, we are dealing with the
sin of omission rather than commission. More importantly, we consider a multi-
period decision problem, rather than the two-period problem of previous work.
This change is important not merely because it is a step in the direction of
greater realism--most practical policy issues involve a sequence of decision
points—but also because it enables us to investigate some questions that are
obscured within a two-period framework.
Suppose continued stress on a system is certain to trigger irreversible
changes, beyond some critical point or period, but we do not know the period.
Is there an analog to the two-period option value? Or suppose the critical
period is known, but the damaging consequences are delayed as with certain
kinds of health impacts. How does this affect the control decision? Still
another issue we can consider in a multiperiod setting is the distinction
between ordinary lags and irreversibility. Irreversible environmental de-
gradation may be regarded as an extreme form of .a lagged recovery in which the
lag period is infinite (or, at any rate, longer than the effective planning
horizon). What about less extreme lags where, if pollution continues beyond a
certain point, the ecosystem is disabled for a certain (finite) period of time
but then recovers: Do the option value arguments still apply?
Uncertainty, or more precisely the nature of learning, is necessarily
treated differently in a multiperiod setting. In the two-period models, un-
certainty is assumed completely resolved by the start of the second period.
By contrast, we assume that the decisionmaker acquires some, but not all of
the information over the first period, more over the second, more still over
-3-
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the third, and so on. Partial, not perfect, information at any time is
accordingly part of the structure of our model.
The chapter is organized as follows: In the next section we develop a
model to evaluate pollution control, taking account of both the relevant
physical constraints and the uncertainties. The model is used in sections
III and IV to study the implications of various interesting combinations of
recovery dynamics and uncertainties, of the sort just noted. Conclusions
are offered in section V.
II. A FRAMEWORK FOR BENEFIT EVALUATION AND DECISION
We model the decision on whether or not to control pollution from the
point of view of an environmental authority concerned with the net present
value (benefits minus costs) of control. The optimal control is defined as
the choice that maximizes this value. The important contraints are those
that emerge from the discussion of the preceding section: (1) Beyond some
-4-
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point in time, failure to control is not readily reversible; and (2) the
benefits of control are uncertain due to a lack of knowledge about the timing
and nature of ecosystem recovery and the willingness of individuals to pay for
the goods and services it can produce.
Though recovery is a continuous process, evaluation and control take place
in a discrete setting. Thus, we assume that a decision to control pollution
can be made in each period t = 1, 2, 3, The outcome of the decision can
be represented by a sequence X,, X2, X3, ..., where X. = 1 corresponds to
building a treatment plant, say, and X = 0 corresponds to not building. Note
that we are considering a binary choice, neglecting intermediate levels of
control. The results we obtain can be extended to the case of continuous con-
trol, but this is somewhat beside the point and comes at a substantial cost in
complexity.
Associated with the choice of Xt is a set of benefits and costs. The
capital and operating costs of the control facility in period t are denoted by
Ct, and the benefits are denoted Bt; the net benefits are NB = B - C . In
the most general model, the benefits and costs accruing during any time period
depend not only on the current pollution control decision, X , but also on
all previous decisions, X,, ..., Xt_p
An essential feature mentioned above is that the benefits and costs of
ecosystem recovery are uncertain. Thus, we write the overall net benefit
function as
NB(X1,X2,X3, ...; 9) = NB^Xp- 6) + 6 NB2(XpX2; 0) + &2 NB3(XpX2,X3; 0) + ..
-5-
-------
where
NBt(X1, ..., Xt; 6) = Bt(X1, ..., Xt; 6) - Ct(Xp ..
Here 8 is a one-period discount factor, and 0 is a random variable (or vector
of random variables) representing the present uncertainty concerning the fu-
ture consequences of pollution control.
With regard to the cost functions, it seems reasonable to assume that,
with probability 1,
Ct(0, ..., 0; 9) = 0
and
Li. ^ A-i , . . . , ^* _ i > 1 > 9 J ^ f^ 1 ' •••> t_1> ' « J •
That is to say, pollution control is costly. Finally, in order to keep the
decision problem simple while still making it interesting, we focus on a three-
period model. This is significantly more general than the two-period models
which have been used in irreversibility literature so far (for example, Arrow
and Fisher [1974], Henry [1974], Epstein [1980]). With minimal notational
clutter, it permits us to consider scenarios involving a variety of types of
irreversibility, which is our primary objective in this paper.
Given this structure, the social decision problem is to maximize the dis-
counted present value of expected net benefits:
(1) max E{NB(X,, X7, X,; 9)}.
1. £. j
-6-
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Two aspects of this problem need to be addressed, both pertaining to the treat-
ment of uncertainty. First, what about attitudes toward risk? Should one
assume risk aversion on the part of the social decisionmaker and, therefore,
include a risk-premium term when taking the expectation in (1), or should one
assume risk neutrality following the arguments, for example, of Samuelson
[1964] or Arrow and Lind [1970]? Although it clearly makes a difference in
practice, the question of risk aversion is not fundamental to the results that
we will obtain: They are qualitatively independent of any assumption about
risk preferences. The second aspect of modeling uncertainty in a dynamic set-
ting is its behavior over time. Uncertainty means a lack of information; yet,
it is likely that this situation changes—that information is acquired over
time. Our analysis is largely concerned with the consequences of a failure on
the part of the decisionmaker to take this prospect into account. We will
show how this affects the social decision and how conventional benefit-cost
analysis must be adjusted to incorporate this consideration.
Suppose, first, that the decisionmaker does not have to commit himself in
the first period to an entire intertemporal control strategy; he can postpone
the choice of X2 to t = 2 and the choice of X, to t = 3. Suppose, moreover,
that in each time period (except t = 3), he recognizes that further informa-
tion about the future consequences of control will become available which he
can exploit in making these future decisions. Define
(2a) V3(X3|X1, X2) E E3{NB3(Xlf X2, X3; 9)}
(2b) V2(X2|X1) = E2(NB2(X1, X£; 0) + max 6 V^X^, X2)>
X3
-7-
-------
(2c) V^X^ E E^NB^X^ 6) + max 3
X
where E {•} denotes an expectation with respect to the information set avail-
able at time t--i.e., E, is the expectation with respect to the decision-
maker's prior distribution for 6, £2 is the expectation with respect to his
posterior distribution in t = 2 which is updated in a Bayesian manner on the
basis of the information obtained by the beginning of the second period, etc.
One point must be emphasized: We assume that the acquisition of information
does not depend on the choice of X ; it emerges either with the passage of
time (e.g., as period 2 approaches, one can make a more accurate assessment
about the social value of environmental quality in the second period) or as
the result of a separate research program on ecosystem dynamics.
•
Following the Backwards Induction Principle of dynamic programming, in the
third period the decisionmaker selects
(3a) X3 = arg max V^^, X£),
in the second he selects
(3b) X2 E arg max V^X^X^,
and in the first he selects
(3c) X: = arg max V^).
In each case we are assuming that, however X,, ..., X , are chosen, Xf is
chosen optimally in the light of these previous decisions. Where it is neces-
S*
sary to emphasize this dependence, we shall write X as an explicit function of
-8-
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the previous choice variables--e.g., X = X-(X ). In the terminology of sto-
chastic control theory, (X.., X?, X,) represents a closed-loop policy: At each
decision point, both current information and all future anticipated informa-
tion are considered in choosing a control.
We wish to contrast this with a policy in which the prospect of future
information is disregarded. There are two ways to model this. One is to as-
sume that, although the decisionmaker is still free to postpone his choice of
X? and X, until the second and third periods, respectively, in each period
Lf J
he ignores the possibility of future learning and deals with uncertainty about
future consequences by replacing random variables with his current estimate of
their mean. Define
(4a) V3(X3lXi> V = E3{NB3(X1' X2» X3;
(4b) V2(X2|X1) = max E^NB^X^ X2; 9) + 6 NB^Xj, X2, X3; e)}
X
(4c) v(X1) = max E^NB^X^ 9) + B NB2(Xp X£; 0) + 6 NB3(Xp X2> X3; e)}
X2'X3
In the third period, the decisionmaker selects
(5a) X* = arg max V*(X3|X1, X2):
in the second he selects
(5b) X* = arg max V^X^X^,
and in the first he selects
(5c) X* = arg max V*(X1).
-9-
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In the terminology of stochastic control theory, this is an open-loop feedback
policy: As new information becomes available, the decisionmaker incorporates
it in his choice of a control; but he assumes that no further information will
become available.
The other approach to modeling the disregard of future information is to
assume that the decisionmaker does not wait (or cannot wait) until the second
and third periods to choose X2 anc* X, but, instead, chooses them in the first
period along with X,. This decision, denoted (X, , X_ , X, ), is the solution to
(6) max E1{NB1(X1; 6) + B NB2(Xp X2; 9) + 62 NB3(\1, X2, X3; 8)}.
X1'X2'X3
This is known as an open-loop control where all decisions are made simul-
taneously on the basis of the information available at the beginning of the
A &A
initial period. Comparing (5) and (6), it is clear that X, = X. , but in
general, X~ / X- and X, / X,--there is no difference between the open-loop
and open-loop feedack controls in the first period but in subsequent periods
/\ j.
they differ. Thus our discussion below of the relation between X, and X-, also
applies to X, , but it does not apply to relations in t = 2 and t = 3.
Since, in a three-period model, unlike a two-period model, the choice of X- is
of substantive interest, the sharp distinction between open-loop and open-loop
feedback policies is one of the benefits that we gain by switching to a multi-
period setting. It will become clear below that, for our purposes, useful re-
sults can be obtained by comparing the closed-loop policy with the open-loop
feedback policy.
We can pursue this comparison in two ways. We can ask a policy question:
How do X and X differ? In particular, under what circumstances is it
t U
S\ £
true that Xt >_ X (i.e., the case for intervening to control pollution is
-10-
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strengthened when the prospect of further information is recognized)? Or we
^ A
can ask a benefit evaluation question: How do V (•) and V (•) differ? What
correction is required when expected benefits are estimated by replacing un-
certain future quantities with a current estimate of their expected value?
Given the constraint that X = 0 or 1, these questions can be answered by
observing that, from (2)-(4),
(7a) Xx >_ (<) X* as OV1 >_ (•<) 0
and, for any given X,,
(7b) LCX,) > (<) X*(Xn) as OV?(X,) > (<) 0
£ J. ~" ~" " _L L* ±. "~" ""^
where
(8a) OV1 = [VjU) - V^O)] - [V*(l) - V*(0)j
(8b) = [^(1) - V*(l)] - [V^O) - V*(0)];
and, given X-,,
(9a) OV
(9b)
The quantities 0V, and OV_(X.) are the correction factors required when the pros-
pect of future information is disregarded and benefits are measured in terms of
* ~
VtO) instead of Vt(»); they are multiperiod generalizations of the Arrow-
Fisher-Henry concept of option value.
-11-
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^
To interpret them, consider (8b) and (9b) and observe that the term [V (X ) -
V.(X )] can be cast in the form of
(10) Vt(») - V*(») = Et{ max Ft(-;6)}- max Et{Ft(.;8)}.
This is a measure of the value of information acquired after the beginning of
period t that can be exploited in the subsequent choice of X ,, X 2> •••>
•"• *
conditional on the choice of X in period t. Thus, in (8b), [V,(l) - V,(l)]
is the expected value of the information that might be acquired in time to in-
fluence the second- and third-period choices conditional on controlling pollution
s*. £
in the first period, while [V,(0) - V,(0)] is the expected value of sub-
sequent information conditional on not controlling pollution in the first period.
The correction factor 0V, is simply the difference between these two condi-
tional values of information; similarly, for OV_. Thus, if 0V >^ 0, the value
of information associated with setting X = 1 exceeds that associated with a
decision to set X = 0 and the case for controlling pollution in period t is
strengthened when the prospect of future information is considered. Conversely,
if 0V. <_Q, the case for pollution control is weakened.
However, without placing further structure on the model, it is impossible to
determine which outcome is the more likely. From the convexity of the maximum
operator and Jensen's Inequality applied to (10), it follows that Vt(») -
V (•) >^ 0. Thus, each component of 0V is nonnegative; but this tells us
nothing about the sign of their difference. In the following sections we con-
sider some alternative model structures embodying features of ecosystem dynamics
discussed in section II and explore their effect on 0V and their implications
for pollution control policy.
-12-
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III. CRITICAL PERIOD IRREVERSIBILITY
Suppose that, at some point in the evolution of the ecosystem, if the
policymaker does not intervene and control pollution at that time, it could
never be optimal for him to control pollution subsequently. We shall call a
time period with this property a "critical" period. Whether such a phenomenon
exists and what factors bring it about depends on the specifics of the eco-
system structure. In the context of the three-period model, suppose that,
while it might pay to introduce controls after pollution has continued un-
checked for one more period, it could never pay to introduce controls after
pollution has continued unchecked for two more periods in a row. More for-
mally, we assume that, with probability 1,
(11) Et(NB3(0, 0, 1; 0)} <_ Et{NB3(0, 0, 0; 9)} t = 2, 3.
Thus, if pollution is not controlled in the first period (X, = 0), the second
period becomes critical.
From (2a,b) and (4b), when X, = 0, we have
(12a) V2(0|0) = E2{NB2(0, 0; 6) + 0 max [E3 NB3(0, 0, 1; 0), E3 NB3(0, 0, 0; 9)]},
(12b) V*(0|0) = E2 NB2(0, 0; 6) + 0 max [E£ NB3(0, 0, 1; 6), EZ NB3(0, 0, 0; 9)].
Applying (11) yields
(13a) V2(0|0) = E2 NB2(0, 0; 9) + 0 E2(E3 NB3(0, 0, 0; 9)}.
(13b) V*(0|0) = E2 NB2(0, 0; 6) + 6 E£ NB3(0, 0, 0; 0).
-13-
-------
However, by the Total Probability Theorem, Et{h(e)} = Et(E h(e)} for any
function of a random variable, h(e). Therefore, we obtain the key result that
(14) V2(0|0) - V*(0|0) = 0.
Because the second period is critical when X, =0, it follows that, if the
decisionmaker does not control pollution in that period, he anticipates that
he will never choose to control it subsequently. Since the anticipated future
decisions are exactly the same under both the closed-loop and open-loop feed-
back policies, the expected future benefits are identical under both policies.
In effect, any subsequent information is expected to have no economic value
because it is not anticipated to have any effect on future decisions; hence,
(14). Substituting this into (9) yields
(15) OV2(0) = V2(l|0) - V*(l|0) >_ 0.
From (7b), this implies that X2(0) ^X2(0). That is, if pollution is not con-
trolled in the first period, we have a situation where, once the potential for
the acquisition of future information is recognized, the case for controlling
pollution in the second period is strengthened, and there is a positive flexi-
bility premium associated with setting X2 = 1.
The key to this analysis is equation (11) which embodies our particular as-
sumption that the second period is critical when X, = 0. Without imposing any
additional restrictions, it is impossible to determine the signs of OV-^ or OV7(1)
For example, from (11), one cannot infer that V2(0|l) = V2(0|l). Therefore,
the indeterminacy concerning the relation between X, and X,, or X2(l) and
X,(1), remains.
-14-
-------
Generalizing from this particular example, a period is critical whenever an
equation analogous to (11) holds, i.e., whenever the situation is such that, if
the decisionmaker does not control in that period, with probability 1 he antici-
pates that it would never pay to control in future periods regardless of the
information subsequently acquired. By construction, when a period t is critical,
* *
we have V (0|») = V (OH which implies that 0V (•) > 0 and X (•) > X.(-).
U C L ^" L. *"" L
It may be useful to compare our notion of a critical period with the concept
of irreversibility employed by Arrow and Fisher [1974] and by Henry [1974] which,
in the present context, would be represented by a constraint of the form
(16) X, = 0 -»• X7 > X,.
J_ £ — ^
Our assumption (11) implies (16) but is somewhat broader and illuminates the
two crucial ingredients required to extend their results to more general
settings. First, what is irreversible is the policy, not the fate of any
particular biotic components. The ecosystem dynamics may be such that, if
X2 = 0, the lake trout become extinct without this necessarily implying (11)
as long as the trout are sufficiently unimportant relative to the decision-
maker's other objectives. The truth or falsity of (11) depends on values as
well as biology. Second, what is at issue is economic rather than technical
irreversibility. The technology may be such that the decision on X2 is
physically reversible in later periods (e.g., setting X^ = 0 corresponds to
permitting the construction of a steel mill on the edge of a lake which could
subsequently be converted to a nonpolluting bowling alley); the question is
whether it could ever pay to reverse the, current decision. Moreover, what
matters is the present anticipation of whether it could ever pay to reverse
-15-
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that decision. Our assumption (11) does not preclude the possibility that,
ex post, at the end of period 3, it might actually turn out that it would have
been optimal to choose X3 = 1 even with X2 = 0. What is required is that,
ex ante, this choice is always deemed implausible. Thus, we can admit the
possibility that
NB3(0, 0, l; 9) > NB3(0, 0, 0; 9)
for some realizations of 9 as long as the prior density on 9 and the subse-
quent updated posterior densities are sufficiently bounded to ensure that the
expected benefits satisfy the inequality in (11).
IV . DELAYED AND TEMPORARY IRREVERSIBILITY
In this section we consider two forms of irreversibility which are weaker
than the critical-period concept introduced above and yield somewhat different
results. First, we consider what might be called "delayed" irreversibility:
If pollution is not controlled, the consequences are (economically) irrevers-
ible, but the irreversibility sets in only after a lag. Thus, if pollution is
permitted to continue now, there is an intermediate period during which it may
or may not be optimal to impose controls; but, after this intermediate period,
it can never pay to control. Within the framework of our three-period model,
we identify "now" with period 1, the intermediate period during which it may
or may not be optimal to control with period 2, and the subsequent future with
period 3. The assumption of delayed irreversibility is captured by combining
(11) together with the assumption that
(17) Et(NB3(0, 1, 1; 6)} <_ Et(NB3(0, 1, 0; 9)} t = 2, 3
-16-
-------
with probability 1. The question to be addressed is how this type of irre-
versibility affects the pollution-control decision in period 1.
Substituting (11) and (17) into (2c) and (4c) yields the following expres-
sions for V^O) and V*(0):
V1(0) = El NB1(0; 9) + 3 EX {max [E2 NB2(0, 0; 0) + 8 EZ NB3(0, 0, 0; 9),
(18a)
E2 NB2(0, 1; 9) + 3 E2 NB3(0, 1, 0; 9)]}
V*(0) = EL NB^O; 9) * 3 max [EL NB2(0, 0; 9) + 3 E: NB,(0, 0, 0; e),
(18b)
E: NB2(0, 1; 9) + 3 ^ NB3(0, 1, 0; 9)].
* f.
By inspection, it can be seen that, while V,(0) - Vl(0) ^ 0, it is not true
~ * ~
in general that V,(0) = V,(0). Since it can also be shown that V,(l) -
V1(l) >_ 0, from (8a,b), this is a situation where the sign of 0V, and the re-
~ *
lation between X, and X, are indeterminate.
XV
Observe that the formula for V^O) in (18a) involves information acquired
between the first and second periods but not that acquired between the second
and third periods—the expectation £,{•} does not appear. The latter informa-
tion has no economic value when X, = 0 because the irreversibility has set in by
then, but the former does have some value because it can be exploited during the
intermediate period (t = 2) where there is still some flexibility. Of course,
if X, = 1, there is sufficient flexibility to exploit both sets of information.
But this fact, by itself, does not guarantee that the overall value of informa-
tion associated with setting X, = 1 necessarily exceeds that associated with
setting X = 0. The point is that, with delayed irreversibility, the first
-17-
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period is not critical because, if one does not control, then it is not^ true
that it can never be optimal to control subsequently; it may still be optimal
to control during the intervening period before the irreversibility sets in.
Thus, with delayed irreversibility, the introduction of future learning into
the decision calculus need not tilt the balance in favor of immediate control.
We now examine what might be called "temporary" irreversibility as opposed
to the "permanent" irreversibility considered so far. We consider two
scenarios. In the first we suppose that, if pollution is not controlled in
any period, the consequences are temporarily irreversible and are felt in the
following period but not necessarily thereafter. In effect, the system has a
one-period memory with
(19) Et{NB2(0, 1; 0)} < Et{NB2(0, 0; 0)}
(20a) NB3(X2> X3' Q) E NB3(°» X2> X3' e) = ^(l, X2, X3; 8)
(20b) Et(NB3(0, 1; 6)} <_ Et {NB3(0, 0; 9)}.
In this case V.(0) and V,(0) are given by
VjCO) * El NB1(0; 6) + 6 E^max^ NB2(0, 1; 6) + & EZ max [E3 NB3(1, 0; 0),
(21a)
E3 NB3(1, 1; 0)], E2 NB2(0, 0; 0) + 6 EZ NB3(0, 0; 0)J \ ,
V*(0) = EJ_ NB1(0; 0) + 6 max {El NB2(0, 1; 0) + 6 max [EX NBjCl, 0; 0),
(21b)
Ex NB3(1, 1; 0)], Ej_ NB2(0, 0; 0) + 6 EI NB3(0, 0; 0)]}.
-18-
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It follows that, while V^O) - V*(0) >_ 0, it is not true in general that V,(0) =
V.(0). Thus, with this type of temporary irreversibility, the sign of 0V and
* A
the relation between X, and X, are indeterminate.
We now change the scenario by assuming that, if pollution is not con-
trolled in the first period, the consequences are temporarily irreversible in
the second period but the third period is entirely independent of what has hap-
pened previously, i.e., the system makes a fresh start and has no memory in the
third period. Thus, we retain (19) while assuming that the third-period bene-
fit functions satisfy the restrictions
NB3(X3; 6) = NB3(1, 1, X3; 8) = NB3(1, 0, X3; e)
(22)
= NB3(0, 1, X3; 6) = NB3(0, 0, X3; e).
~ *
The new formulas for V,(0) and V,(0) are
VL(0) = Ej_ NBjCO; 9) + 6 El NB2(0, 0; e)
(23a)
' + B2 Ej_ {max [E3 NB3(0; e), Ej NB3(1; e)]}
V*(0) = El NB1(0; 9) + 6 Ej_ NB2(0, 0; 9)
(23b)
+ 62 max [E1 NB3(0; 9), EI NB3(1; 9)].
Similarly, substitution of (19) and (22) into (2c) and (4c) yields the following
formulas for V^l) and V*(l):
-19-
-------
V^l) = El NB1(1; 6) + 6 EI {max [E£ NB2(1, 0; e), EZ NB2(1, 1; 9)]}
(24a)
+ B2 Ex (max [E3 NB3(0; 9), ES NBjCl; 9)]}
V1(l) = Ex NB1(1; 9) + B max [EX NB2(1, 0; 9), EX NB2(1, 1; 9)]
(24b)
+ B2 max [Ex NB3(0; 9), EI NBj(l; 9)].
In this case, although it is still true that [V^l) - V*(l)] ^0 and [V
V,(0)] ^ 0, we can determine the sign of 0V. since application of (8) yields
OV1 = 0 Ex {max [E2 NB2(1, 0; 9), EZ NB2(1, 1; 9]}
(25)
- B max [Ej_ NB2(1, 0; 9), EX NB2(1, 1; 9)] >_ 0.
^ *
It follows, therefore, that X, >_X,.
In the first scenario, based on (19) and (20a,b), if one fails to control
in the first period, it may nevertheless be optimal to control in the second,
despite the irreversibility embodied in (19), because second-period decisions
influence third-period outcomes. Thus, when X, = 0, information acquired
between the first and second periods still has some economic value because it
may shed light on third-period outcomes and can, therefore, affect the second-
period decision. When X, = 1, information acquired between the first and
second periods also has an economic value. Consequently, the net effect of
incorporating future learning into benefit estimation is ambiguous: it may
strengthen or weaken the case for initial control.
-20-
-------
By contrast, in the second scenario, based on (19) and (22), the second-
period decision cannot affect third-period outcomes at all because of the
total lack of memory between these two periods. Therefore, the temporary
irreversibility in (19) ensures that it is never optimal to control in the
second period when one has not also controlled in the first. As a result, the
information acquired between the first and second periods has some value when
X, = 1 but none when X, = 0. Moreover, because the system makes a fresh start
in the third period, the information acquired between the second and third
periods is equally valuable regardless of whether X = 0 or 1, t=l, 2.
Hence, the case for initial control is unambiguously strengthened when one
recognizes the possibility of future learning.
While it is clear that the first scenario of temporary irreversibility is
incompatible with the concept of a critical period, the second scenario can
still be related to that concept, albeit in a somewhat unusual manner. Under
the second scenario, if the decisionmaker decides not to control in the first
period, he anticipates that it could never be optimal for him to reverse this
decision during the subsequent interval lasting until the system's memory is
"reset." Once that has occurred, all future decisions are entirely
independent of prior events. Thus, there is a sense in which the first period
is "locally" critical.
V. CONCLUSIONS
It has long been recognized that the selection of an optimal pollution
control or other environmental policy is highly dependent on the treatment of
time and uncertainty in the benefit cost calculus. A delay in ecosystem
recovery, for example, may reduce the present value of the benefits from
-21-
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control; but if the recovery lags caused by continuing pollution are growing
faster than the discount rate, this would tilt the balance in favor of early
control, as shown in a somewhat different context by Krutilla and Fisher
(1975). Similarly, depending upon one's view of the degree of risk aversion
appropriate for public policy decisions, the presence of uncertainty may
require an adjustment to the expected monetary benefits and costs of control.
Since there may be uncertainty about the consequences of both control and no
control, this could cut either way.
While not denying the importance of these issues for empirical policy
analysis, in this chapter we have focused on a different aspect of benefit
evaluation involving flexibility, the temporal resolution of uncertainty, and
the value of information. In a dynamic system, information about the conse-
quences of previous actions may arrive over time, and this prospect must be
taken into consideration when one makes policy decisions. Future observations
have no economic value, however, if (1) they are entirely uninformative in the
sense that the prior and posterior distributions coincide or (2) they are
informative but they cannot affect subsequent decisions because the policy-
maker lacks freedom of action. Thus, flexibility is a necessary ingredient
for information to have economic value. This must be borne in mind when one
contemplates an action with irreversible consequences, because the resulting
lack of flexibility nullifies the value of any subsequent information.
In many pollution control issues this may be a relevant consideration be-
cause the ecological consequences of a failure to control may be irreversible.
Actually, we have shown that what is crucial is economic irreversibility.
That is to say, if in some time period the decisionmaker anticipates that,
unless he controls then, it would never pay to control in the future,
-22-
-------
regardless of the subsequent information, a decision not to control then would
effectively eliminate future flexibility. In that case, there is a positive
flexibility premium associated with a decision to control: When future learn-
ing is taken into account, the balance is tilted in favor of control. We have
termed this a critical-period irreversibility. In other cases, however, the
issue is less clear cut. For example, it may happen that the irreversible
consequences are delayed in their onset or are only temporary in their effects.
In such cases, we show that the conditional value of future information when
one fails to control now is not necessarily zero; conceivably it may exceed
the value of information associated with a decision to control. The prospect
of future learning then has an ambiguous effect--it may strengthen or weaken
the case for control. Our intuition is that the value of information condi-
tional on control will ordinarily exceed the value of information conditional
on no control but this is an empirical issue to be resolved through specific
case studies. Such an application is the focus of our current research and
will be reported separately.
-23-
-------
FOOTNOTES
The term "option value" has also been used in connection with a differ-
ent concept related to risk version in an atemporal setting. Major references
include Schmalensee [1972], Bohm [1975], Graham [1981], Bishop [1982], Smith
[1983], and Freeman [1984].
2
Obviously, if the control decision itself generates information, this
may alter the balance of the argument. If, by not controlling now, one gener-
ates potentially useful information which can be exploited in future decisions
(for example, because the major uncertainty concerns the consequences of not
controlling), this would weaken the case for control. If, on the other hand,
one generates useful information by controlling now (because the major uncer-
tainty concerns the consequences of control), this would strengthen the case
for control. In the absence of a specific case study, it is difficult to say
a priori whether or not there is dependent learning and, if there is, which
form it takes. For this reason we have focused on the case of independent
learning. For a further discussion of this issue see Fisher and Hanemann
[1985].
-24-
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REFERENCES
Kenneth J. Arrow and Anthony C. Fisher, "Environmental preservation, uncer-
tainty, and irreversibility," Quart. J. Econ. 88, 312-319 (May 1974).
Kenneth J. Arrow and R. C. Lind, "Uncertainty and the evaluation of public
investment decisions," Amer. Econ. Rev. 60, 364-378 (1970).
R. C. Bishop, "Option value: An exposition and extension," Land Econ. 58,
1-15 (February 1982).
Peter Bohm, "Option demand and consumer's surplus: Comment," Amer. Econ.
Rev. 65_, 733-736 (September 1975).
Larry G. Epstein, "Decision making and the temporal resolution of uncer-
tainty," Inter_._Jcon_._^ev. 21, 269-283 (1980).
Anthony C. Fisher and W. Michael Hanemann, "Quasi-option value: Some
misconceptions dispelled," J. Environ. Econ. Manag. (forthcoming).
A. Myrick Freeman, "The size and sign of option value," Land Econ. 60, 1-13
(February 1984).
D. A. Graham, "Cost-benefit analysis under uncertainty," Amer. Econ. Rev. 71,
715-725 (September 1981).
Claude Henry, "Investment decisions under uncertainty: The irreversibility
effect," Amer. Econ. Rev. 64, 1006-1012 (December 1974).
A. J. Home, "A Suite of Indicator Variables (SIV) Index for an Aquatic
Ecosystem," Energy and Resources Group, University of California, Berkeley
(August 1985).
A. J. Home, J. Harte, and D. F. Von Hippel, "Predicting the Recovery of
Damaged Aquatic Ecosystems: A Hysteresis Trophic Link Model (HTLM),"
Energy and Resources Group, University of California, Berkeley (August
1985).
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J. V. Krutilla and A. C. Fisher, The Economics of National Environments:
Studies in the Valuation of Commodity and Amenity Resources, Johns Hopkins
Press, Baltimore (1975).
P. A. Samuelson, "Discussion," Amer. Econ. Rev. M., 93-96 (1964).
Richard Schmalensee, "Option demand and consumer's surplus: Valuing price
changes under uncertainty," Amer. Econ. Rev. 62, 813-824 (December 1972).
V. Kerry Smith, "Option value: A conceptual overview," Southern Econ. J.
654-668 (January 1983).
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Page 1
CHAPTER 6 THE ECONOMIC CONCEPT OF BENEFIT
INTRODUCTION
Consider the question: "v/hat is the value of the wetlands of San
Francisco Bay?". V/hy might such a question be addressed to an economist rather
than a philosopher or 3 poet? To explain this it is vital to distinguish
between tv/o different meanings that might be attached to the original question:
(i) Hew much value do people place on the wet lands (assuming an adequate base of
information)? (ii) How much value ought they to place on them? The latter
question is certainly the province of the philosopher or the poet; the economist
too may have some thoughts about the question, but these arise from his private
sentiments, not from his professional discipline. The former question - the
positive question - is the one that the discipline of economics addresses. When
we talk of benefits and benefit measurement in this report, we have this
- interpretation in mind - the values that people actually place on ecosystems.
This itself raises a host of questions: Which people? In what units
should values be measured? Why do people have these values? Just how do we
ascertain them? We will comment briefly on each of the first three questions.
The answers to the fourth question will take up the remainder of this chapter,
as well as Chapter 7. Which people? This is specified, in principle, by the
agency commissioning the benefit assessment. A related, and more complex
question, is: How do we add up different people's values? Again, this is
specified, in principle, by the agency commissioning the study; however, here
there is a body of economic theory which can guide the answer - see, for
example, Sen (1973), Blackorby and Donaldson (1973), and Bcadway and Sruce
(1984, Chapter 9). To save space, we will duck this issue here. V/hat units?
Values can be measured in monetary units or in units of any commodity that
people happen to value. For example, we could measure the value to an
-------
Pago 2
individual of aquatic ecosystems in unite- of chocolate truffles - Lake Tahoo is
worth 100 truffles, say, while Mono Lake is worth only 82 truffles. Different
systems cf units will generate the ssnie ordinal ranking of ecosystems, but not
necessarily the same cardinal index of value. We choose to adopt noney -
purchasing power - as our unit of measurement because this is the predominant
convention. It is possible to develop an analogous theory of benefit
measurement based on chocolate truffle units, but we shall not explore this
here (aggregation across individuals would presumably be more difficult).
How do wo ascertain values? In principle there are two ways to proceed:
(i) Ask people directly, and (ii) Rely on revealed preference - observe their
behavior when they make choices on which the aquatic ecosystem somehow impinges,
and infer their values from this behavior. In this chapter we focus on the
latter approach exclusively. An immediate implication is an answer to the
question: Why do people have these values? The answer is that it doesn't
matter. V/e rely on preferences as revealed by actual behavior, without needing
to know how these prefences might be decomposed into alternative motives. Or
rather, there are two circumstances in which motives might matter. The first
is when a knowledge of motives gives us reason to believe that preferences (and
behavior) will be different in the future. Stability of preferences is
essential to extrapolation from observed behavior. If preferences are not
stable, this poses both a philosophical and a practical problem. The
philosophical problem is: Which set of preferences do we rely on? The practical
problem is: How can we predict what the new preferences will be if it is
decided to rely on them? The other circumstance in which we might care about
motives has to do with aggregation across individuals: specifically, a knowledge
of motives may help us to identify groups of individuals who have different
preferences. For empirical purposes, it might bo more appropriate to analyze
the behavior of each group separately, rather than to aggregate them into a
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Page 3
single group.
Given the focus oft revealed preference, why is the presence of markets
required for the success of our endeavor? One answer common among non-
econor.tists, but erroneous, is that values are embodied in market prices and
expenditures. Markets are needc-d because market prices establish values: if a
commodity sells for 510, that is the value of the commodity, i-icv/over, this is
not exactly true. If I buy the commodity at a price of S10, then it certainly
must be worth 510 to me - but it may be worth even more; i.e., the price is a
lower bound en value. If I do not buy the commodity at this price, it is not
worth $10 to rna; i.e., the price is an upper bound. Let us switch from prices
to expenditures and focus on the first case. Suppose I buy 5 units of the
commodity at the going price of $10, so that my total expenditure is 550. This
expenditure is clearly a lower bound on the value of the commodity to me. The
problem, however, is that this lower bound may be inadequate for our purposes.
Ultimately we are interested in net benefits - i.e. benefits minus costs. If
the cost of supplying the commodity is also 310 a unit, the cost amounts to
S50 and the difference between that and our lower bound estimate of benefits is
zero - because we underestimate benefits when we use expenditures, we
underestimate net benefits, possibly to the point of absurdity. Moreover,
consider some change in the supply of the commodity (for example an improvement
in its quality) which leads me to spend $70 on it. For the same reason as
before, this $70 is a lower bound on the value of the improved commodity to me.
But the change In expenditure conveys absolutely no information about the
channo m vaIue; the difference between two lower bounds is not necessarily a
lower bound on the difference in the quantities being bounded.
In short, we do not care about markets because market expenditures
directly indicate values. At best they provide bounds on values, but these
bounds are frequently so imprecise as to be useless, and the channos in market
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Page 4
expenditures provide no information about changes in values. Instead, wo cara
about markets because they provide a forurn for choice behavior - perform inn
tradeoffs between goods and money - from which v/e can indirectly infer
preferences. That is the essence of the revealed preference approach.
iloracvar, as will be shov;n in the next section, these market transactions, or
tradeoffs, can convey information about preferences for other items of value
which are not themselves traded in a market, as long as the preferences for the
latter items interact (in a sense to be made specific below) with preferences
for the traded items. V/e turn, now, to an elaboration of this argument.
THE 3ASIC FRAMEWORK
The revealed preference approach to benefit assessment can be explained
in terms of two basic consumer choice models. Both models pertain to an
individual consumer - we want to avoid the complications associated with
estimation and interpretation of aggregate demand functions. In the first
model, the individual has preferences for various marketed commodities, whose
consumption is denoted by the vector x, and for various environmental resources,
which are denoted by q: this could be a vector but, for simplicity of notation,
we treat it as a scalar. These preferences are represented by a utility
function u(x,q) which is continuous and non-decreasing in al! arguments (we
assume that the x's and q are all "goods"), and strictly quasiconcave in x
(we assume strict quasiconcavity rather than quasiconcavity in order to rule
out demand correspondences). At this point, we do not .assume that u( ) is
(strictly) quasiconcave in q. The individual chooses his consumption of the
marketed goods - the x's - by maximizing his utility subject to a budget
constraint
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page 5
where "the p. "s are the prices of the marketed goods, and y is the
i
individual's income. Note that he does not determine the level
of the q variables. These are in the nature of public goods for
him, and he takes them as given.
The utility maximization generates a pattern of consumption
behavior represented by the ordinary demand functions x.=hv(p,
q,y) i=l,...,N. For convenience we assume that these represent
an interior solution, so that problems associated with corner
solutions (discussed in Bockstael, Hanemann, and Strand [1984,
Chapter 9] can be ignored. Substitution of these demand
functions into the direct utility function yields the indirect
utility function v(p, q,y )su[h(p,q,y ) , q] . Alternatively, as a
dual to (1) there is an expenditure minimization problem
C.K VA(X')SU. X>0 (2)
which yields a set of compensated demand functions, x. =
i*
gMp^q/u), and the expenditure function m(p,q,u) = £"p. gi (p, q, u) .
These constructs can be employed to define what we mean by
the benefits to the individual from a change in q. Suppose that
q changes from q° to q1 , while prices and income remain constant
a"t (p,y). Accordingly, the individual's welfare changes from u°2
v(p,qe>,y) to u'= v(p,q' ,y). Two alternative measures of this
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page 6
welfare change are the quantities C and E defined, respectively,
by
Each of these represents an adjustment to the individual's income
calculated to offset the effects of the change in q. C, the
compensating variation, is the amount of money by which the
individual's income must be adjusted after the change in order to
render him as well off as he was before the change. If u1 -C u° ,
so that C <. 0, this is the minimum compensation that he would
require in order to acquiesce in the change. Similarly, E, the
equivalent variation, is the amount of money by which the
individual's income must be adjusted before the change in order
to render him as well off as he would be after it. If u1 > u° ,
so that E > 0, this is the minimum compensation that he would
require in order to forego the change while, if u1 < u° so that E
< 0, this is the most he would be willing to pay to avoid the
change .
The second model is based on the household production
approach, in which the individual gains utility from "composite
commodities" which he produces himself from private goods. One
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page 7
version of this model is
uJxZ-} s.K
where z denotes the vector of composite goods, f ( . ) is the
production function for these goods written in implicit form, and
w( . ) is a utility function defined over the z's and, perhaps,
some of the x's. In this formulation we are assuming that the
individual derives utility from q not directly, but indirectly,
in so far as they contribute to the production of z's. The
utility maximization in (5) can be solved in two stages. In the
first stage one obtains
=: "^ ^(^^ s.K Hx^^ =0, (6)
while in the second stage one solves (1) using the function
u(x,q) derived from (6). That is to say, a household production
model can always be "collapsed" into a model in the form given in
(1). Moreover, welfare measures for changes in q can be defined
as in (3) and (4) using the indirect utility function v(p,q,y)
associated with u(x,q) in (6). One consequence of the household
production approach, however, is that it generates demand (and
supply) curves for the z's - as well as demand curves for the x's
- which are of some empirical as well as theoretical interest.
Given this framework, our analysis will be concerned with
three sets of issues that have arisen in the literature on
environmental benefit evaluation; (i) What is the relation
between C and E - we know they must have the same sign, but how
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page 8
much can they differ in magnitude? (ii) How can we measure C and
E from observed demand behavior - after all, since we do not
observe utility directly we cannot estimate the indirect utility
function v(p,q,y) directly? (iii) Is there any relation between
C or E and expenditures on some of the private goods - the x's -
which might be specially related to the q's in terms of either
consumer preferences or household production technology? Can we
use expenditure on some goods as proxies for C or E?
To answer these questions, it is convenient to consider
three possible markets. One is the market for x's, in which
there are observable demand curves. The second is the market for
z's, which may arise in connection with the household production
model (5). The third market is entirely hypothetical. Suppose
that the individual could actually buy q in a market at some
given price,TT . instead of (1) he would now solve
mcv* xxlx fl^ s.V. ^OX.JT rto r u (7)
X a " «•• V .J
(at this point we assume strict quasiconcavity of u(.) with
respect to q in order to ensure an interior solution). Denote the
resulting ordinary demand functions for the x's by hu (p,TT,y),
A q
and the ordinary demand function for q by h ^ (p,tT, y). The
>
associated indirect utility function is denoted by v (p,fT,y)s
A A 6.
u[h(p,rt,y), h/(p,IT,y)]. Similarly, we could define a dual
expenditure minimization problem analogous to (2), in which both
the x's and q are the choice variables. The resulting
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page 9
/* __ A _
compensated demand functions are denoted by g(p,n,u) and gq(p,"
,u), and the expenditure function is >* (p, fT,u.^ = £. P^fciT.wV fg^fp.iT,
These utility maximization and expenditure minimization
problems are hypothetical because, in fact, environmental
quality, q, is not a marketed commodity. Nevertheless, they are
of theoretical interest because they shed light on the solutions
to (1), (2), (5), and (6). For example, it is convenient to
introduce the following:
7 v f \
DEFINITION: q is normal (inferior) if hu >O ( G. ^
Moreover,
A V
PROPOSITION 2: If V\ --O for all the q's which change, C = E.
J
Suppose, however, that there are income effects in the demand
functions for q; the question remains: just how much can C and E
differ? To answer this, we must investigate the q-market in more
detail.
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page 10
HOW MUCH CAN C AND E DIFFER?
Willig (1976) established that, unless the income elasticity
of demand for a commodity is very high, the compensating and
equivalent variations for a price change will not differ
considerably. Some environmental economists do not believe that
the same holds true of compensating and equivalent variations for
change in q - see, for example, Maler (1985, p.39) or Knetsch and
Sinden (1984), who ...-(present empirical evidence of a considerable
disparity between C and E. However, Randall and Stoll (1980) have
shown that Willig's analysis carries over to changes in fixed
parameters such as the q's, and Brookshire, Randall and Stoll
(1980) have interpreted this result as implying that C and E
should not be very different in value. How can these divergent
views be explained or reconciled?
In the paper reproduced in the Appendix to this chapter I
reexamine randall and Stoll's analysis and show that, while it is
indeed accurate, its implications have been misunderstood. There
is no presumption that C and E must be close in value and, unlike
price changes, the difference between them depends not only on an
income effect but also on a substitution effect. Specifically,
the magnitude of the difference depends on (i) the magnitude of
the change in q, (ii) the size of the income effects, and (iii)
the degree of substitutability between private consumption
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page 11
activities (the x's) and the level of environmental quality q in
the individual's preferences, all of which are empirical issues.
Moreover, I suggest that the substitution effects are likely to
exert far greater leverage, in practice, on the relation between
C and E than the income effects. Thus, large empirical
divergences between C and E may be indicative not of some failure
in the survey methodology but of a general perception on the part
of the individuals surveyed that the private market goods
available in their choice set are, collectively, a rather
imperfect substitute for the public good under consideration.
MEASURING C AND E FROM DEMAND CURVES
Analysis of the market for q is useful in that it gives us
an idea of the factors that affect the relation between C and E,
but it is of no value when it comes to measuring C or E in
practice because, by definition, no such market exists - the
demand curve for q can never be observed. What can be observed
is behavior in the x market - the market for private goods. This
raises the question, therefore, of whether the values of C and E
can be inferred from knowledge of the demand curves from the x's.
There are two ways in which this can be accomplished. The first
in
is to uncover the /direct utility function from the fitted demand
curves for the x's, and then employ the formulas in (3) and (4).
The second is based upon results developed by Maler (1971, 1974)
which establish a relation between areas under demand curves for
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page 12
the x's and the quantities C and E.
In the first approach one postulates a specific functional
form for either the direct utility function u(x,q) or the
indirect utility function v(p,g,y), and derives the appropriate
formula for the corresponding ordinary demand functions - by
analytically solving the direct utility maximization problem or
by differentiating the indirect utility function and applying
Roy's Identity. Alternatively, one can start out with a given
system of ordinary demand functions hf* (p,q,y) i.*\}-.,tJt and then
attempt to recover the corresponding indirect utility function by
applying the integrability techniques developed by Hurwicz and
Uzawa. As a simple example, suppose that N=2 and the demand
function for the first good takes the semi-log form
«M-X( , of- $(>,/£) + Y(«j/Pj * £^ ; (8)
in Hanemann (1980a, 1981) it is shown that the indirect utility
function is
"
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page 13
E one first fits the demand function (8) and then substitutes the
estimated values of the coefficients ^ (?, f and o into the
formulas in (10a,b).
The alternative approach to computing C and E- developed by
Maler, is based on the following decomposition of the formula for
C (a similar analysis applies to E)
c - ^- ^fp-V,^
<£t,V) - Airp.^.oO J
where p is an arbitrary price vector. Assuming that q1 > q° , we
know that C > 0. Since mq£. 0, we also know that the second term
in (11) is non-negative. The first term is the sum of areas
between compensated demand curves corresponding to q1 and q° ,
tPi *••*
between the actual price p± and the i element of p (this line
integral is path-independent). It should be emphasized that the
first item is not necessarily positive; it can be shown that the
increase in q raises the compensated demand for the i* h private
good (^gi/3q>0) if this good is a complement to q in the Hicks-
Allen sense, and lowers the compensated demand (^gi/Sq<0) if the
good is a substitute. Moreover, if q is a scalar, at least one
of the private goods must be a Hicks-Allen substitute for q.
Nevertheless, we know that the sum of the two terms in (8) must
P
= \ Lf.,oO-fvO]d. t ^(,^0-™.0) 00
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page 14
be positive.
Maler's trick is to select p in such a way that the second
term in (| [) vanishes. For this purpose, he introduces two
assumptions. The novel assumption is that there exists a set of
commodities with the property that, if these commodities are not
being consumed, the marginal utility of q is zero. Let I be the
index set of these commodities, and I its complement. Partition
the vector x accordingly: x = (x 'X7 )• Maler's assumption,
which he calls weak complementarity, is:
(WC) There exists a non-empty set I such that <3_^( Q*?'^' — &
H (12)
His second assumption is:
(NE) The commodities in I are non-essential: there exists some
price vector such that g1 ( . ) = 0 and hi(.) = 0 all i e. I.
We can now apply these assumptions to (11) by choosing the price
s^ *V *"" ***
vector p so that p± = p.^ for i E. I while, for it. I, p± is simply
the cut-off price of the i* h compensated demand function - i.e.
}]* ^' SillCe S±9n (mq)= - S±9n
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page 15
This proposition establishes a relationship between C and
the areas between two sets of compensated demand functions. It
is useful here to make a distinction between two sets of
circumstances: (i) there is a set of goods with the property
that q has no value only when none of these goods is being
consumed, and (ii) there is a set of goods with the property that
q has no value when any one of them is not being consumed. In
the first case, C is measured by the area between compensated
demand curves summed over all of the goods in I; in the second
case it is measured by the area between compensated demand curves
for any one of the goods in I, and we obtain the same answer
regardless of the particular good selected. Note that, in order
to make use of the proposition, one still needs to know something
more than ordinary demand functions unless there are no income
effects in the demand for the goods in I, in which case the
compensated and ordinary demand functions coincide. If there are
income effects and one attempts to calculate the area in ( 13 )
using ordinary instead of compensated demand functions, i.e. one
calculates the area
this is likely to be of limited value. The issue is examined in
Hanemann (1980b), where it is shown that under some circumstances
S may not even have the correct sign. The requirement that one
employ the compensated demand function in (13) implies that,
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page 16
wherever there are income effects, Maler's method for calculating
C and E has the same information requirements as the method based
on direct application of (3) and (4). Finally, as an
illustration, it turns out that semi-log demand function (8)
satisfies the WC condition since, on differentiating the indirect
utility function (9), one finds that
u
which is equivalent to (12). The compensated demand function
corresponding to (13) is
*^YP,*?,^ . Mi- ^- e.^Pi/p^~ ^ 1
1 J ' * J y L vA J
and it is straightforward to verify that (lOa) and (lOb) combine
to satisfy (13).
THE LIMITS TO REVEALED PREFERENCE
Both of the methods for measuring C and E from observed
demand functions rely on the assumption that all the relevant
components of the indirect utility function can be recovered from
demand functions. However, that assumption is not always true:
it holds when the underlying direct utility function has the form
u = u(x, ^} (17)
as has implicitly assumed up to now, but not when the utility
function can be cast into the form
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page 17
<18>
where T( . ) is increasing in its first argument and u(x,q) is a
conventional direct utility function. It can be shown that both
utility models imply exactly the same ordinary demand functions
for x's
C.TO mox U (x( ^) r arg. mw T L ^ ( *, ^\ <^ J
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page 18
where C satisfies v(p,q1,y-C) = v(p,q°,y), v( . ) being the
indirect utility function corresponding to~u(x,q), and C*
satisfies
(20)
Assuming that q1 > q° and T(.,q) is increasing in q, it can be
shown that C* > 0, so that
C > C? > 0. (21)
A similar result can be shown to hold for equivalent variation
measures:
E = E? + E*> E > 0, (22)
where E is the true equivalent variation associated with the full
utility function u(x,q) in (18), E is the equivalent variation
associated with the sub-function u(x,q), and E* is calculated
from the transformation function T(.,q), along the lines of (20).
Since C and E are derived from the sub-function containing the
interactions between the x's and q, we can regard them as the
"consumption - or use - related" components of benefits.
Similarly, we can regard C* and E* as the "non-consumption
related" or "non-use related" components of benefits - they arise
from that part of the individual's preferences which do not
affect his choice of x.
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page 19
The practical implications of (18) for the revealed
preference approach - the measurement of C and E on the basis of
observed demands for the x's - are highly important. If we only
have data on ordinary demand functions for the x's, we can only
recover u(x,q), but never T(.,q) nor the full utility function
u(x,q) in (18). That is, we can only measure C and E - not C* or
E* and, therefore, not the full value of C or E. This is a
significant limitation to the revealed preference approach.
It is sometimes thought that Maler's Weak Complementarity
(WC) assumption eliminates this problem, but I would dispute
this. Differentiate (18) to obtain the marginal utility of q.
(23)
act
If we apply WC to u(x,q), this requires that
(24)
But, by itself, this is not enough to ensure that
5 O, (25)
which is what one requires in order to rule out the
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page 20
representation in (18). Suppose, for example, that
fi xr -°
This satisfies (24) but not (25), and therefore C* > 0 and
E* > 0. In this case WC does not eliminate the problem.
To summarize, the only circumstance in which the revealed
preference approach to the measurement of C and E is fully
satisfactory is when (25) holds - i.e. the utility function is
represented by (17) rather than (18). But there is no way to
verify this from data on ordinary demand functions for x's. It
could be verified if there were a market for q and one could
observe demand functions for q as well as the x's. Indeed, in
that case, T(.,q) could be recovered along with u(x,q) so that,
if (25) were violated, C and E could still be calculated because
one would obtain the full indirect utility function associated
with (18). But, in the absence of a market for q, the problem
remains.
In practice, there are two possible solutions. The first is
simply to assume that the utility function takes the form of (17)
and not (18) - which is what is generally done. The second is to
collect additional behavioral data besides ordinary demand
functions for the x's. For example, after measuring C by the
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page 21
revealed preference approach one could conduct interviews to
elicit the willingness to pay for an improvement in q directly;
if the interviews yielded an estimate close to C in value one
would conclude that C* = 0 and hence, the utility model
corresponds to (17) rather than (18). If they yielded an estimate
much greater than C one would take the difference to be a measure
of C* . Alternatively, instead of contingent valuation exercises,
one could conduct what has been called [Hanemann (1985)]
"contingent behavior" exercises in which one attempts to elicit a
hypothetical demand function for q. Both of these approaches
remain subjects for future research.
THE SIGNIFICANCE OF EXPENDITURE DATA
In the theory of the welfare measurement of price changes it
is well known that calculation of expenditure changes provide
bounds on the compensating and equivalent variations, even if
they are not exactly equal to these welfare measures. If prices
change from p° to p1 and the quantities demand change
correspondingly from x° to x1 , then the compensating variation
? a
for the price change, C , and the equivalent variation, E ,
satisfy
and
although, in general, there is no determinate relation between C
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page 22
or E and the overall change in expenditure 5p.x° -
When dealing with changes in q, as opposed to price changes,
some authors have wondered whether one can obtain a relation
between the welfare measures C and E and the change in
expenditures on some or all of the private market goods, £. p \.W(p.G
- V\"(PI«J/, jj} J • In general, I do not believe that this is a
useful approach; with one exception described below, there does
not appear to be any determinate relation between changes in
expenditure on x's and either C and E. Indeed, the effect of an
increase in q on the demand function for any of the x's is by no
means obvious. Given that (Su/^q) > 0, it is sometimes assumed
that ^h1" /^q i. 0 all i - an increase in quality can never lower
the demand for any of the x's. In fact, this is not true; in
general, an increase in q will affect the demand for the x's, but
note that the effect could be in either direction, depending on
the specifics of the utility function. Even if q is a Hicks-
Allen complement with some private good -say, x - it is not
necessarily true that an increase in q will raise the demand for
that good.
This pessimistic conclusion is based on the following
proposition which links the demand functions x. = h>"(p,q,y) to
A L
the hypothetical demand functions x = hl(p,(T,y) associated with
the utility maximization problem (7):
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page 23
PROPOSITION 4: Let fi~ =n (p,q,y) be defined implicitly by
Appendix equation (11). Then,
• , ^ (27)
It follows as a corollary that
(28)
U^/dr?
Given that uq > 0, /T > 0. If u(x,q) is quasiconcave in q, the
denominator of the second term on the RHS is negative. Thus, the
sign of 3hc /^q depends upon a complex set of factors. The
numerator of the term in braces on the RHS will be recognized as
the cross-price derivative of the compensated demand curve from q
and this is positive or negative according as x . and q are
substitutes or complements. Moreover,
rtr^-l .r U) <-| - | ^ O CLS nf?^"^"
where _ qn. % Thus, if d^/jti >O and
>o (29)
this is a sufficient condition for ^(rx^/c^s >0 . Even if ^W/ita^O, it
-------
page 24
can still happen that ^ W^ "> 0 if (29) holds and that term is
sufficiently large.
Without belaboring it further, the point is that an increase
in q could either lower or raise the expenditure on x. This
should make us cautious about expected any simple relation
between the change in expenditure on some of the x ' s and C or E
since it is quite possible that C and E are positive while the
change in expenditure is negative. One case in which more
definitive results can be obtained is where q is a perfect
substitute for some of the x's - say x.^ . In that case the direct
utility function takes the form
where i|^ (.) is some increasing function of q. Let h1(p,y) and
v(p,y) be the ordinary demand function for good 1 and the indirect
utility function associated with u( . ) . The following may be
shown:
PROPOSITION 5: If u(x,q) has the form given in (30),
Ola)
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page 25
It follows from (31b) that
while the change in expenditure on x is
(32)
as
Thus, if X-L is a normal good and a perfect substitute for q, the
change in the expenditure on x1 understates the true benefit from
an increase in q. In this case, moreover, there are no income
effects in the demand curve for q, so that the compensating and
equivalent variations coincide. Apart from this special case,
however, there does not appear to be any determinate relation
between A and C or E.
NON-USE VALUES
This above framework can be used to shed some light on the
concept of existence value due originally to Krutilla (1967).
This is based on the notion that, even if he did not consume any
of the x's that are associated with q, an individual might still
feel some improvement in q and be willing to pay something to
secure it. How can this be explained in terms of the utility
model discussed above?
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page 26
Smith and Desvousges (1986) have made an important
distinction between existence values under conditions of
certainty and uncertainty. The phenomenon of consumer choices
under uncertainty -e.g. the individual does not know whether or
not he will want in the future to consume certain x's that are
associated with q - raises many important issues that transcend
the theory developed above, which is firmly rooted in the context
of decisions under certainty. Accordingly, I focus here on the
concept of existence values under the conditions of certainty -
an individual places some value on an improvement in q even
though he does not himself consume any of the x's that might be
associated with q, and has no doubt that he will never consume
these goods in the future. Under these circumstances, how can we
use the theoretical framework developed above to give some
operational meaning to this concept?
Two quantities identified above may have some bearing on
this question. The first is based on the decomposition in (11).
Suppose that Weak Complementarity does not apply so that<)u/^q > 0
even when there is zero consumption of x's that are
conventionally associated with q. In that case one could regard
the quantity
(33)
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page 27
as a measure of the non-use benefits associated with the
improvement in q - these are the benefits that would accrue to
the individual even if he were consuming none of the x ' s .
Operationally, one would measure them by computing C from the
indirect utility function using (3), and then subtracting the
area between the compensated demand curves represented by the
integral on the RHS of (33). of course, if Weak Complementarity
holds, this quantity is zero. As already noted, that would apply
to the semi-log demand function (8). Interestingly, it does not
apply to another common functional form, the linear ordinary
demand function
It can be shown that the corresponding compensated demand
function Q(p,q,u) is independent of q so that the integral in
(11) and (33) is zero and
(35)
where the cut-off price is
In this case, therefore, all of the benefit from a change in q is
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page 28
associated with term jWp',
-------
page 29
extreme form in (37), a similar conclusion would apply: "the only
way to measure the non-use benefits C* and E* is by contingent
valuation and/or contingent behavior procedures.
-------
REFERENCES
W. Michael Hanemann (1980a) "Measuring the Worth of Natural
Resource facilities: Comment" Land Economics Vol. 56 November
1980 pp. 482-486
W. Michael Hanemann (1980b) "Quality Changes, Consumer's Surplus
and Hedonic Price Indices," University of California, Department
of Agricultural and Resource Economics, Working paper No. 116,
Berkeley, November 1980.
W. Michael Hanemann (1981) "Some Further Results on Exact
Consumer's Surplus" University of California, Department of
Agricultural and Resource Economics, Working paper No. 190,
Berkeley, December, 1981.
W. Michael Hanemann (1985) "Some Issues in Discrete - and
Continuous - Response Contingent Valuation Studies,"
Northeastern Journal of Agricultural Economics, April 1985
John V. Krutilla (1967) "Conservation Reconsidered," American
Economic Review Vol. 57, September 1967, pp 777-786
Karl-Goran Maler (1971), "A Method of Estimating Social Benefits
From Pollution Control," Swedish Journal of Economics Vol. 73,
No. 1, pp 121-133
Karl-Goran Maler (1974), Environmental Economics: A Theoretical
Inquiry (Baltimore: The Johns Hopkins University Press)
V. Kerry Smith and William H. Desvousges (1986), Measuring Water
Quality Benefits (Boston: Kluwer-Nijhoff Publishing Co.)
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APPENDIX
WILLINGNESS TO PAY AND WILLINGNESS TO ACCEPT: HOW MUCH CAN THEY DIFFER?
Consider an improvement in the exogenous variables comprising an indi-
vidual's choice set. Two possible monetary measures of the gain in her wel-
fare are the compensating variation (C) and the equivalent variation (E). In
the present context, these correspond, respectively, to the maximum amount the
individual would be willing to pay (WTP) to secure the change and the minimum
compensation that she would be willing to accept (WTA) to forego the change.
How much can the two differ, and what are the factors that determine the dif-
ference? These Questions were addressed by Robert Willig (1976) in his path-
breaking paper on the welfare measurement of price changes. Willig argued
that C and E are likely in practice to be fairly close in value, and he showed
that the difference depends directly on the size of the income elasticity of
demand for the commodity whose price changes.
In many empirical studies, however, analysts seek to obtain money measures
of welfare changes due not to price changes but to changes in the availability
of public goods or amenities, changes in the qualities of commodities, or
changes in the fixed quantities of rationed goods. Karl-Goran Maler (1974)
was perhaps the first to show that the concepts of C and E can readily be ex-
tended from conventional price changes to quantity changes such as these.
Subsequently, Alan Randall and John Stoll (1980) examined the duality theory
associated with fixed quantities in the utility function and showed that, with
appropriate modifications, Willig's formulas for bounds on C and E do, indeed,
carry over to this setting. Within the environmental literature and else-
where, Randall and Stoll's results have been widely interpreted as implying
that WTP and WTA for changes in environmental amenities should not differ
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-2-
greatly unless there are unusual income effects. However, recent empirical
work using various types of interview procedures has produced some evidence of
large disparities between WTP and OTA measures--for example, Richard C. Bishop
and Thomas A. Hebertein (1979) and several studies described by Irene M.
Gordon and Jack L. Knetsch (1979), and by Knetsch and Sinden (1984). This has
led to something of an impasse: How can the empirical evidence of significant
differences between WTP and OTA be reconciled with the theoretical analysis
suggesting that such differences are unlikely? Can they be explained entirely
by unusual income effects or by peculiarities of the interview process?
In this note I reexamine Randall and Stoll's analysis and show that, while
it is indeed accurate, its implications have been misunderstood. For quantity
changes there is no presumption that OTP and OTA must be close in value and,
unlike price changes, the difference between OTP and OTA depends not only on
an income effect but also on a substitution effect. By the latter, I mean the
ease with which other privately marketed commodities can be substituted for
the given public good or fixed commodity, while maintaining the individual at
a constant level of utility. I show that, holding income effects constant,
the smaller the substitution effect (i.e., the fewer substitutes available for
the public good) the greater the disparity between OTP and OTA. This surely
coincides with common intuition. If there are private goods which are readily
substitutable for the public good, there ought to be little difference between
an individual's OTP and OTA for a change in the public good. But, if the pub-
lic good has almost no substitutes (e.g., Yosemite National Park or, in a dif-
ferent context, your own life), there is no reason why OTP and OTA could not
differ vastly—in the limit, OTP could equal the individual's entire (finite)
income while OTA could be infinite. My argument is developed in the following
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-3-
two sections. Section I deals specifically with the two polar cases of per-
fect substitution and zero substitution between the public good and available
private goods. Section II deals with Randall and Stoll's extension of
Willig's formulas and shows that their bounds are, in fact, consistent with
substantial divergences between WTP and WTA.
I. Two Polar Cases
The theoretical setup is as follows. An individual has preferences for
various conventional market commodities whose consumption is denoted by the
vector x as well as for another commodity whose consumption is denoted by
q. This could represent the supply of a public good or amenity; it could
be an index of the quality of one of the private goods; or it could be a
private commodity whose consumption is fixed by a public agency. The key
point is that the individual's consumption of q is fixed exogenously, while
she can freely vary her consumption of the x's. These preferences are repre-
sented by a utility function, u(x, q), which is continuous and nondecreasing
in its arguments (I assume that the x's and q are all "goods") and strictly
quasiconcave in x. The individual chooses her consumption by solving
(1) max u(x, q) subject to £pjX- = y
x
taking the level of q as given. This yields a set of ordinary demand func-
tions, x- = h1(p, q, y), i = 1, ..., N, and an indirect utility function,
v(p, q, y) E u[h(p, q, y), q], which has the conventional properties with
respect to the price and income arguments and also is increasing in q. Now
suppose that q rises from a to q > q while prices and income remain constant
-------
-4-
at (p, y) . Accordingly, the individual's utility changes from u = v(p, q , y)
to u = v(p, q , y) > u . Following Nfaler, the compensating and equivalent
variation measures of this change are defined, respectively, by
(2) v(p, q1, y - C) = v(p, q°, y)
(3) v(p, q1, y) = v(p, q°, y + E) .
Dual to the utility maximization in (1) is an expenditure minimization: Mini-
mize £p-x- with respect to x subject to u = u(x, q), which yields a set of
compensated demand functions, x. = g1(p, q, u), i = 1, — , N, and an expendi-
ture function, m(p, q, u) = Zp-g1(p, q, u), which has the conventional proper-
ties with respect to (p, u) and is decreasing in q. In terms of this function,
C and E are given by
(2') C = m(p, q°, u°) - m(p, q1, u°)
(31) E = m(p, q°, u1) - m(p, q1, u1) -
It is evident from (2) and (3) that 0 <_ C _< y while E >_0. The questions
at issue are: (1) Is it true that E/C « 1? (2) What factors affect this
ratio? As a first cut at an answer, I compare two polar cases. In the first
case at least one private good--say, the first—is a perfect substitute for
some transformation of q. Thus, the direct utility function assumes the
special form
(4) u(x, q) = U[
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-5-
where \jj(-) is an increasing function and u(«) is a continuous, increasing,
strictly quasiconcave function of N variables. As W. M. Gorman (1976) has
shown, the resulting indirect utility function is
(5)
, q, y) =
where v(«) is the indirect utility function corresponding to u('). Substi-
tution of (5) into (2) and (3) yields the following:
PROPOSITION 1: If at least one private market good is a perfect substitute
for q, then C = E.
At the opposite extreme, I assume that there is a zero elasticity of sub-
stitution not just between q and x-, but between q and ajJ. the x's. Thus,
the direct utility function becomes
(6)
u(x, q) = u
mm ( q, — ] , ..., mm [ q.
al
where a-,, ..., a^ are positive constants and u(«) is a conventional direct
utility function. In this case the indirect utility function v(p, q, y) has a
rather complex structure and changes its form in different segments of (p, q, y)
space. It will be sufficient for my purposes to focus on just one of these seg-
ments. Suppose that q <_y/Zp. a-; then the maximization of (6), subject to the
budget constraint, yields ordinary demand functions and an indirect utility func-
tion of the form x^^ = h1(p, q, y) = c^ q, and u = v(p, q, y) = u~(q, ..., q) =
w(q). In this region of (p, q, y) space, the individual does not exhaust her
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-6-
budget, and her marginal utility of income is therefore zero. Now suppose that
q £y/Zp-a- and q > q . Since v(p, q , y) > w(q ), it is evident from (2)
that the individual would be willing to pay some positive but limited amount C
to secure this change. However, for any positive quantity E, no matter how
large, v(p, q , y + E) = v(p, q , y) = w(q ). This implies the following:
PROPOSITION 2: If there is zero substitutability between q and each of the
private market goods, it can happen that, while the individual would only be
willing to pay a finite amount for an increase in q, there is no finite com-
pensation that she would accept to forego this increase.
It should be emphasized that this result obtains only in a portion of
Q
Cp> Q> y) space; in other regions, even with (6), E would be finite. How-
ever, the result in Proposition 2 can also be established for other utility
functions that permit some substitutability between q and the x's as long as
the indifference curves between q and each of the x's become parallel to the
q axis at some point. The lesson to be learned from these two propositions is
that the degree of substitutability between q and private market goods signifi-
cantly affects the relation between C and E. In the next section, I show how
this observation can be reconciled with the bounds on C and E derived by
Randall and Stoll.
II. Randall and Stoll's Bounds
In order to extend Willig's bounds from price to commodity space, Randall
and Stoll focus on a set of demand functions different from those considered
above. Suppose that the individual could purchase q in a market at some given
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-7-
price, TT. It must be emphasized that this market is entirely hypothetical
g
since q is actually a public good. Instead of (1), she would now solve
(7) max u(x, q) subject to Zp-x^ + irq = y.
x,q
s\ -I
Denote the resulting ordinary demand functions by x- = h (p, TT, y), i = 1,
..., N and q = hq(p, IT, y). The corresponding indirect utility function is
v(p, TT, y) = u[h(p, TT, y), hq(p, TT, y)]. The dual to (7) is: Minimize
Zp.x. + Trq with respect to x and q subject to u = u(x, q). This generates
a set of compensated demand functions, x- = gHp, TT, u), i = 1, ..., N and
q = gq(p, TT, u), and an expenditure function, m(p, TT, u) = Zp-g^p, TT, u) +
Trg^tp, TT, u). These functions are hypothetical since q is really exogenous to
the individual, but they are of theoretical interest because they shed light
on the relation between C and E.
For any given values of q, p, and u, the equation,
(8) q = gq(p, TT, u^
may be solved to obtain TT = nCp, q, u), the inverse compensated demand (i.e.,
willingness to pay) function for q: 9(0 is the price that would induce the
individual to purchase q units of the public good in order to attain a utility
level of u, given that she could buy private goods at prices p. Let TT =
Ti(p, q , u ) and TT = n(p, q , u ) denote the prices that would have supported
q and q , respectively. The two expenditure functions dual to (1) and (7) are
related by:
(9) m(p, a, u) E m[p, Ti(p, q, u), u] - Ti(p, q, u) • q.
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-8-
This implies that*0
(10) mq(p, q, u) = -ir(p, q, u) .
Combining (10) with (2') and (31) yields these alternative formulas for C and
E expressed in terms of the willingness -to -pay function:
(2") C = / 0 irCp, q, uU) dq
q
(3") E = / Q n(p, q, u1) dq .
q
It can be shown that sign (TT ) = sign (h^) . Therefore, for given (TT, q), the
" 1 "^ 0
graph of Ti(p, q, u ) lies above (below) that of TT(P, q, u ), and E > (<) C,
accordingly as q is a normal (inferior) good. Figure 1 shows E and C for the
case where q is normal: E corresponds to the area q a y q while C corre-
sponds to the area q 3 6 q .
Using the technique pioneered by Willig, Randall and Stoll establish
bounds on the difference between each of C and E and the area under an inverse
ordinary demand function for q. From this, they derive bounds on the differ-
ence between C and E. However, the requisite inverse ordinary demand function
is obtained in a rather special manner. Given any level of q, we can ask what
market price TT would induce the individual to purchase that amount of public
good if it were available in a market, while still allowing her to purchase
the quantity of the x's that she actually did buy at market prices p with in-
come y. In conducting this thought experiment, one needs to supplement her
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-9-
FIGURE J. WTP and WTA for a Change in q
-------
-10-
income so that she can afford q as well as the x's. Thus, for given (p, q, y) ,
we seek the price IT that satisfies
(11) q = hq(p, TT, y + TO) .
/\
The solution will be denoted by IT = irCp, q, y). This inverse function is
related to the inverse compensated demand function by the identities11
(12a) 7i(p, q, y) = ir[p, q, v(p, q, y) ]
(12b) TT(P, q, u) = Tr[p, q, m(p, q, u)].
0 A 0 0 "^ 0 1
It follows from (12a) that TI = ir(p, q , u ) = ir(p, q , y) and TT =
~ 1 1 A 1 •*
rr(p, q , u ) = Tr(p, q , y)--i.e., the graph of irCp, q, y) as a function of q
intersects the graph of TT(P, q, u ) at q = q , and the graph of TT(P, q, u ) at
1 12
q = q . This is depicted in Figure 1.
/\
Using the inverse demand function irCp, q, y), define the quantity
(13) A = / Q TT(P, q, y) dq
q
which corresponds to the area q $ y <$ q in Figure 1. This is a sort of
Marshallian consumer's surplus, which is to be compared with C and E. Let
31n rcC
be the income elasticity of Tr(p, q, y); Randall and Stoll call this the "price
flexibility of income." Assume that, over the range from (p, q , y) to
(p, q , y), this elasticity is bounded from below by £ and from above by
-------
-11-
£ with neither bound equal to 1. Using the mean-value theorem, as in
Willig's equation (18), and the above equations (21), (3'), (10), (12b), (13),
and (14), yields Randall and Stoll's result—namely,
PROPOSITION 3: Assume £L <_ £ <_ £U where £L £ 1 and £U £ 1. Then,
(i) 0 <
(i - r) 9
(ii) 0 < 1 -
1 - (1 -
-u
(iii) If £U < 1, or if £u > 1 and 1 + (1 -
1°' f 1
1 + (1 - 51
J, A
y
- i
(iv) If C > 1, or if C < 1 and 1 - (1 -
y ^ 0, ^ £ 1 -
1 - (1 -
-Ll A
' y
Applying a Taylor approximation, as in Willig, and assuming that the condi-
tions in (iii) and (iv) are satisfied, one obtains
(15)
This is commonly interpreted as implying that C and E are close in value,
but whether or not that is correct clearly depends on the magnitudes of (A/y)
and the bounds f; and E, . The magnitude of (A/y) depends in part on the size
of the change from q to q . But what can be said about the likely magnitude
-------
-12-
of the income elasticity, £--could it happen, for example, that £L = °°? To
answer that question, differentiate (11) implicity
(16)
37
, TT, y + irq) + qhtp, TT, y + Trq)
By the Hicks-Slutsky decomposition, the denominator is equal to the own-price
derivative of the compensated demand function for q and is nonpositive
= h°(p, TT, y + Trq) + q h£(p, TT, y + Trq) <_ 0.
Converted to elasticity form, (16) becomes
(16') g = - n" a)
where n = (y + rrq) h^(p, rr, y + qir)/q is the income elasticity of the direct
ordinary demand function for q, a = qrr/(y + qir) is the budget share of q in re-
lation to "adjusted" income, and e = Trgq[p, TT, v(p, q, y)]/q is the own- price
elasticity of the compensated demand function for q. The last term can be re-
lated to the overall elasticity of substitution between q and the private mar-
ket goods x^, ..., x^. By adapting W. E. Diewert's (1974) analysis, it can be
shown that, if the prices p,, ___ , p^ vary in strict proportion (i.e., p. = e~p.
for some fixed vector p), the aggregate Allen-Uzawa elasticity of substitution
between q and the Hicksian composite commodity XQ E Zp-x., denoted OQ, is
related to the compensated own-price elasticity for q by the formula: e =
-ov,(l - a). Hence, (16') may be written
-------
-13-
(16")
C0
where OQ >_ 0 .
This provides an explanation of the results in the previous section. For
changes in q, unlike changes in p, the extent of the difference between C and
E depends not only on income effects (i.e., n) but also on substitution ef-
fects (i.e., an). If, over the relevant range, either n = 0 (no income ef-
fects) or an = co (perfect substitution between q and one or more of the x's),
then E, = £ = 0 and, from Proposition 3, C = A = E. On the other hand,
if the demand function for q is highly income elastic, or there are very few
substitutes for q among the x's so that aQ is close to zero, this could
generate very large values of £ and substantial divergences between C and E.
Suppose, for example, that, over the relevant range, a lower bound on the income
~ i
elasticity of TT(') is E, = 20 (e.g., r\ = 2 and aQ = 0.1) and A/y = 0.05.
Then, from Proposition 3 (i and iv), C/y <_ 0.0345 while 0.1708 <_ E/y, so that E
is at least five times larger than C. Higher values of £ would imply even
greater differences between C and E.
III. Conclusion
A recent assessment of the state of the art of public good valuation con-
cludes "Received theory establishes that . . . WTP . . . should approximately
equal . . . OTA. ... In contrast with theoretical axioms which predict
small differences between WTP and WTA, results from contingent valuation
method applications wherein such measures are derived almost always demon-
strate large differences between average WTP and WTA. To date, researchers
-------
-14-
have been unable to explain in any definitive way the persistently observed
differences between WTP and WTA measures" (Cummings, Brookshire, and Schulze,
p. 41). This paper offers an explanation by showing that the theoretical
presumption of approximate equality between WTP and WTA is misconceived. This
is because, for public goods, the relation between the two welfare measures
depends on a substitution effect as well as an income effect. Given that the
substitution elasticity appears in the denominator of (16") and the Engel
aggregation condition places some limit on the plausible magnitude of the
numerator, this suggests that the substitution effects are likely to exert far
greater leverage, in practice, on the relation between WTP and WTA than the
income effects. Thus, large empirical divergences between WTP and WTA may be
indicative not of some failure in the survey methodology but of a general
perception on the part of the individuals surveyed that the private market
goods available in their choice set are, collectively, a rather imperfect
substitute for the public good under consideration.
-------
-15-
FOOTNOTES
This view is expressed by, for example, Myrick Freeman (1979, p. 3);
Mark A. Thayer (1981, p. 30); Jack L. Knetsch and J. A. Sinden (1984, p. 508);
and Don L. Coursey, William D. Schulze, and John J. Hovis (1984, p. 2).
2
I am treating q as a scalar here, but it could be a vector without
seriously affecting the analysis in this section. In the next section, how-
ever, the analysis would become significantly more complex if q were a vector
and more than one element of q changed.
These alternative interpretations are offered, respectively, by Maler,
W. Michael Hanemann (1982), and Randall and Stoll.
4
These properties are established in my earlier paper.
I have taken the liberty of defining C and E as the negative of quan-
tities appearing in Willig and in Randall and Stoll, so that sign (C) =
sign (E) = sign (u - u ).
I assume throughout that q > q and u ^ u . The analysis could be
repeated for a case in which quality decreases and u £ u . In that case, C
and E are both nonpositive and correspond, respectively, to the compensation
that the individual would be willing to accept to consent to the change and
the amount that she would be willing to pay to avoid the change. This would
reverse the inequalities presented below, but it would not affect the sub-
stance of my argument.
This result carries over, of course, if more than one private good is a
perfect substitute for q. In the most general case, u(x, q) = u[x, + ^-.(q),
..., XN + v|»N(q)] and C = E = Ep^Cq1) - ^(q0)].
-------
-16-
Q —: f\
Indeed, if h (cup,, ..., a.^ p^, y) <_ q , i = 1, ..., N, it can be shown
that v(p, q , y) = v(p, q , y) = vta^, ..., «NPN, v) and C = E = 0, where
h1(«) and v(») are the ordinary demand functions and indirect utility function
associated with u(0.
It is now necessary to assume that u(*) is strictly quasiconcave in both
x and q.
Using subscripts to denote derivatives, differentiate (9) and note that
q = g Cp> IT, u) = m (p, IT, u) by Shephard's Lemma. Equations similar to (9)
through (12) are presented by J. P. Neary and K. W. S. Roberts (1980).
11 A
Note that TT(P, q, y) is not an inverse ordinary demand function in the
sense of Ronald W. Anderson (1980) because it involves an income adjustment as
well as a price effect.
It is commonly supposed that TT > TT when q < q --see, for example,
Figure 7.12 in Richard E. Just, Darrell L. Hueth, and Andrew Schmitz (1982)--but
this is not correct. It can be shown that TT ^ TT according as n ^ (I/a).
Since Za^rh + an = 1 by the Engel aggregation condition, where a^ = Pj*./(y +
Trq) and r^ = (y + irq) hVx^ TT° < ir1 if and only if Xa^^ n^^ _< °-
•"•^This is actually the order of magnitude by which WTA measures exceed
WTP measures in the empirical studies summarized in Table 3.2 of Ronald G.
Cummings, David S. Brookshire, and William D. Schulze (forthcoming).
-------
-17-
REFERENCES
Anderson, Ronald W., "Some Theory of Inverse Demand for Applied Demand
Analysis," European Economic Review, 1980, 14, 281-90.
Bishop, Richard C. and Hebertein, Thomas A., "Measuring Values of
Extra-Market Goods: Are Indirect Measures Biased," American Journal of
Agricultural Economics, December 1979, 61, 926-30.
Coursey, Don L., Schulze, William D. and Hovis, John J., "On the Supposed
Disparity Between Willingness to Accept and Willingness to Pay Measures of
Value: A Comment," mimeo., University of Wyoming, Department of
Economics, Laramie, Wyoming, January 1984.
Cummings, Ronald G., Brookshire, David S. and Schulze, William D., Valuing
Public Goods: An Assessment of the Contingent Valuation Method, Totowa,
N. J.: Rowman and Allanheld, forthcoming.
Diewert, W. E., "A Note on Aggregation and Elasticities of Substitution,"
Canadian Journal of Economics, February 1974, 7, 12-20.
Freeman, A. Myrick, The Benefits of Environmental Improvement: Theory and
Practice, Baltimore: Johns Hopkins University Press, 1979.
Gordon, Irene M. and Knetsch, Jack L., "Consumer's Surplus Measures and the
Evaluation of Resources," Land Economics, February 1979, 55, 1-10.
Gorman, W. M., "Tricks With Utility Functions," in M. Artis and R. Nobay,
eds., Essays in Economic Analysis, New York: Cambridge University Press,
1976.
Hanemann, W. Michael, "Quality and Demand Analysis," in Gordon C. Rausser,
ed., New Directions in Econometric Modeling and Forecasting in U. S.
Agriculture, Amsterdam: North Holland Publishing Co., 1982.
-------
-18-
Just, Richard E., Hueth, Darrell L. and Schmitz, Andrew, Applied Welfare
Economics and Public Policy, Englewood Cliffs, New Jersey: Prentice-Hall,
Inc., 1982.
Knetsch, Jack L. and Sinden, J. A., "Willingness to Pay and Compensation
Demanded: Experimental Disparity in Measures of Value," Quarterly Journal
of Economics, August 1984, 507-21.
Maler, Karl-Goran, Environmental Economics: A Theoretical Inquiry, Balti-
more: Johns Hopkins University Press, 1974.
Neary, J. P. and Roberts, K. W. S., "The Theory of Household Behavior Under
Rationing," European Economic Review, 1980, 13, 25-42.
Randall, Alan and Stoll, John R., "Consumer's Surplus in Commodity Space,"
American Economic Review, June 1980, 71, 449-57.
Thayer, Mark A., "Contingent Valuation Techniques for Assessing Environmental
Impacts: Further Evidence," Journal of Environmental Economics and
Management, 1981, 8, 27-44.
Willig, Robert, "Consumer's Surplus Without Apology," American Economic
Review, September 1976, 66, 589-97.
-------
CHAPTER 7
METHODS OF BENEFIT MEASUREMENT
In the two preceding chapters, we have spoken of benefits in a rather
general sense not specifying where they come from or how they might be meas-
ured in practice. In Chapter 5, for example, we assume the existence of a
benefit function for ecosystem recovery and examine how a decision on pollu-
tion control is affected by the dynamics of recovery and the uncertainties
surrounding it. In this chapter we look behind the benefit function. What
kinds of benefits are provided by aquatic ecosystems, and how might they be
measured? Here we take up the discussion begun in Chapter 1 drawing upon the
classification of benefits and measurement approaches suggested there.
I. Aquatic Ecosystems as an Input to Production
Aquatic ecosystems function as an input to production whenever changes in
an ecosystem's characteristics affect the costs of providing a good or serv-
ice. For example, the number of wetland acres available as a habitat for fish
may influence the cost of harvesting commercially valuable species. The
quality of water withdrawn from rivers and lakes for municipal water supplies
and irrigation determines the cost of subsequent water treatment and level of
agricultural productivity. Finally, just as air pollution may lead to the
chemical deterioration of materials, diminished water quality can lead to the
corrosion of household appliances and industrial equipment. Valuing the bene-
fits from improved environmental quality when the environment acts as an input
to production is the focus of this section. We critically review a number
of earlier studies in the area and go on to suggest (and illustrate) some
improvements.
-------
We focus on the examples identified in Chapter 1: supply of clean water
and harvest of commercial species. Consider the former. Wetlands reduce the
cost of water treatment by removing or settling pollutants. This can be
represented as a shift in a marginal cost or supply curve along a given demand
curve. An environmental improvement, such as provision of additional wet-
lands, would then involve a supply shift down and to the right, as from S to
S' in Figure 1, where the shaded area between old (S) and new (S1) supply
curves indicates the net welfare gain, the change in consumer and producer
surplus.
This is probably a typical case, but others are possible—and, it turns
out, relevant to some of the existing literature. One, in particular, is
worth noting. Suppose the new cost or supply curve is simply the horizontal
axis. In other words, creation of the wetlands completely eliminates the need
for human inputs, at least up to a point (represented by Q" on Figure 2).
Then the welfare gain, illustrated in the figure, is the shaded area between
old and new supply curves up to the point (Q1 on the figure) where demand
equals the old supply and between demand and new supply thereafter (up to
Q"). Note that this is less than the area between the two supply curves.
Beyond Q', consumer willingness-to-pay for water is less than the old cost of
treatment so that the latter is no longer relevant.
This same point is made more dramatically in Figure 3. There the old cost
of treatment or supply curve lies everywhere above the demand curve. The
benefit of the environmental improvement, represented as a shift in the supply
curve to coincide with the horizontal axis, is then simply the area under the
demand curve (up to Q"). The area between the two supply curves, which is
just the area under the old curve, or the cost of providing treatment in the
absence of the wetlands, would overstate the benefit of having the wetlands
for this purpose.
-2-
-------
p
FIGURE 1
WASTE ASSIMILATION
BENEFIT PROVIDED BY
THE ECOSYSTEM .
S=MC
S'=MC
Q
-3-
-------
FIGURE 2
WASTE ASSIMILATION
BENEFIT PROVIDED BY
THE ECOSYSTEM
-4-
-------
FIGURE 3
WASTE ASSIMILATION
BENEFIT PROVIDED BY
THE ECOSYSTEM
-5-
-------
This is essentially the difficulty with the pioneering and influential
study of the value of estuarine wetlands by Gosselink, Odum, and Pope (GOP,
1974). They claim that an acre of estuarine wetland provides benefits which
would cost $2,500 per year if produced by man-made treatment plants. Shabman
and Batie (1977) are justifiably critical of this figure:
"... the use of alternative estimates should be governed by three
considerations: (1) the alternative considered should provide the
same services; (2) the alternative selected for the cost comparison
should be the least-cost alternative; and (3) there should be sub-
stantial evidence that the service would be demanded by society if it
were provided by the least-cost alternative. GOP failed to subject
their estimate to any of these important tests."
Park and Batie (1979) contend that GOP not only failed to test whether the
least-cost alternative would be demanded, but that their identification of
waste treatment plants as the least expensive alternative may be incorrect.
They argue that recent evidence suggests that adjustments in agricultural
practices (e.g., restriction on the application of fertilizers which "run off"
into estuarine waters) may be a less costly alternative to the construction of
treatment plants. The criticism of the work of GOP is not to suggest that
waste assimilation is not an important service provided by wetlands; however,
care must be taken when determining just how society values that service.
Problems have also plagued efforts to value benefits which might be pro-
vided by aquatic ecosystems sometime in the future but which are not currently
provided. Instead of valuing the option to use a resource as an input to pro-
duction in the future in the way suggested in Chapter 5, some studies have
calculated benefits as if the resource were already being used. What is miss-
ing here is an estimation of the likelihood that the resource will ever be
-6-
-------
used and the timing of its use. Gupta and Foster (GF, 1976) attempt to value
wetlands as a potential source of water supply for the state of Massachusetts
and find that the state's wetlands could provide an annual benefit of $2,800
per acre. Unfortunately, GF's estimated benefit of wetlands1 preservation in
this regard is calculated as though the cost savings of using wetlands instead
of current sources were already realized. Their finding, that wetlands would
provide a cheaper supply of water for Massachusetts, can be questioned in two
respects. First, if wetlands are a cheaper alternative to current sources,
why are they not used? Second, if it is the existence of institutional bar-
riers which block their use, why won't those barriers continue to preclude the
tapping of wetlands as a supply of water in the future? Although it is cer-
tainly true that the preservation of wetlands may be valuable because the
option to use them as a water source would be retained, this is not the bene-
fit GF estimate. As a final point, their estimate of the total value of
undeveloped wetlands may be plagued by double counting problems. If water
were taken from Massachusetts' wetlands, would the same wetlands continue to
generate the recreational and amenity benefits they add to the water supply
benefits?
We now turn to the commercial harvest example. A substantial amount of
previous empirical \vork has sought to value the environment as input for this
purpose in ways not fully consistent with the deceptively simple approach dis-
cussed thus far and summarized in Figure 1. The estimated benefits variously
fail to analyze changes in the relevant cost structure, ignore price effects
of a change in production, and rely on ad hoc measures like total or net reve-
nue. As a measure of change in social welfare, revenue figures exhibit at
-7-
-------
least two problems. First, they do not reflect the opportunity cost of pro-
ducing goods and services. Second, demand for many fish and shellfish species
is relatively price inelastic (Bell, 1970), so an increase in production due
to an environmental improvement results in a decrease in total revenue, incor-
rectly implying that the improvement does not lead to a welfare gain. About
the best that can be said for the revenue calculations (with or without price
effects) is that they are not relevant to the determination of a change in
combined consumer and producer surplus--our preferred welfare measure.
A Council on Environmental Quality (CEQ, 1970) study illustrates the same
difficulties in a somewhat different way. The study reports that, due to the
practice of ocean dumping, one-fifth of the nation's shellfish beds are con-
taminated and closed. Assuming the closed shellfish beds would be as produc-
tive as their open counterparts, the study concludes that an improvement in
water quality would result in a 25 percent increase in quantity produced and a
subsequent 25 percent increase in total revenues. The increase in total reve-
nues are claimed as the gain to society of cleaning up the shellfish beds.
However, as long as demand is not perfectly elastic, an additional 25 percent
in the amount of shellfish supplied to the market could only be sold if the
price of shellfish fell. The estimate of CEQ of an additional $63 million in
shellfish revenues (the additional 25 percent) is clearly an overstatement.
But in any case the revenue figures do not reflect costs or willingness to pay
for nonmarginal units and, hence, are not adequate measures of welfare.
An important question to address, in valuing commercial fishing benefits,
is this: What is the contribution of the ecosystem to the production proc-
ess? It is a question some studies have failed to address. Thus, GOP (1974),
-8-
-------
in assessing the value of wetlands as a fish nursery, divide annual dockside
values of fish products landed by the total number of wetland acres to arrive
at a value per acre in production of fish. Imputing all of the revenue from
commercial fishing to wetland acreage, however, ignores the contribution of
other fishing inputs like labor and capital.
The more recent study by Lynne, Conroy, and Prochaska (LCP, 1981) suggests
that it may be possible to isolate the contribution of environmental inputs to
production. They develop a bioeconomic model in which human effort and marsh-
land are distinct inputs in the production of blue crab off Florida's Gulf
Coast. The population of blue crabs is assumed to be a function of the quan-
tity of local marshland acres. Since the successful .harvesting of the crabs
is modeled to be dependent on their population level, marshlands, which act to
define the carrying capacity for blue crabs, appear as an input in the produc-
tion function. The reduced form production function is estimated according to
the ordinary least-squares criterion; and, using the appropriate estimated
coefficients, a marginal product for an acre of wetlands is calculated.
Finally, the value of the marginal product for an acre is computed using cur-
rent dockside prices. The study is laudable for valuing both marshland acre-
age and human input in the production of blue crabs. However, the authors'
contention that the value of the marginal product is the relevant measure of
benefits provided by wetlands is incorrect. Let us take up the analysis at
this point and develop an example in which notions of consumer and producer
surplus are correctly employed, as in Figure 1, to evaluate the commercial
fishing benefits produced by the marshland.
-9-
-------
In keeping with the spirit of LCP, consider the optimization problem faced
by a price-taking firm or industry where price is P and the unit cost of the
human effort input is, W:
(1) max P FCX.^ Y2) - W
X
The production process is posited to be a function, F(»), of two inputs:
one (X. ) which captures the efforts of man to harvest shellfish and another
.(X?) which represents the contribution of an ecosystem variable like marsh-
land acreage. The bar over X_ indicates that, for the time being, the acre-
age is fixed. Although we, like LCP, model human effort as a single input,
the number of traps set, one many prefer to explicitly model the use of sev-
eral inputs so that substitution among them can be studied.
We assume that the production of blue crabs can be represented as a Cobb-
Douglas process. Although the Cobb-Douglas form is no doubt a simplification
of the true production process (and is probably a poor approximation to
reality for extreme values of either input), we use it here because our main
purpose is to demonstrate the procedure for calculating changes in combined
consumer and producer surplus. Therefore, substituting for the production
function in equation (1) the Cobb-Douglas form and noting that cost minimiza-
tion is the dual problem to profit maximization, the optimization problem can
be rewritten as
(2) min <£ = W X, + X(Q - A X? X)
i l L
-10-
-------
where X is the Lagrange multiplier; Q is output; and A, a, and b are parame-
ters. Differentiating the Lagrangian with respect to the effort variable and
the Lagrange multiplier yields
(3)
5^= W - X A XT a Xf~ = 0
dA, £ 1
(4)
= Q - A X X = 0.
Since the production function is characterized by only one decision variable,
X,, equation (4) is the only one needed to solve for the cost function,
C(O.
(5) Xj_ =
xr
I/a
(6) C(W, Q, XJ = W A"1/a X:b/a Q1/a.
Differentiating the cost function with respect to output generates the mar
ginal cost expression
MC _ 9C _ W -I/a rb/a n(l-a)/a
ML - - - -A X2 Q
The blue crab industry also presumably faces a demand curve for its product.
A simple constant elasticity demand function is given in (8), and the corres-
ponding inverse demand function in (9):
-11-
-------
(8) Q = KP"m
(9) P = K1/m Q"1/m
where K is a parameter and m is the (constant) elasticity. The profit-
maximizing firms will equate price and marginal cost so that the equilibrium
level of blue crabs sold is given by
, , Ta 1/m I/a K/O! ma/[m+(l-m)a]
(10) Q = - Ki/m Ai/a
I W
The result in (10) holds for all relevant values of marsh acreage, X~,
available for the biological promotion of the blue crab population. There-
fore, we first calculate the equilibrium output associated with various levels
of wetland acreage, then we compute the equilibrium price corresponding to the
output by use of equation (9).
We proceed to calibrate the parameters of the model in order to construct
an example which is reasonably compatible with the price, input, and output
data used by LCP. We also incorporate their econometric finding that the mar-
ginal product of an acre of marsh is roughly 2-1/2 pounds of blue crab (annu-
ally). Although the demand for shellfish has been found to be relatively
price-inelastic, as we noted earlier, we assume in this case a high elasticity
since the Gulf Coast fishery is presumably not the sole source of blue crab in
the market. Welfare gains associated with an increase in marshland habitat
(remember we are considering only gains in the blue crab industry for purposes
of this example) are calculated as the change in consumer and producer sur-
plus. These measures are presented in Table 1. For example, for a demand
elasticity of -2.05, the net gain associated with an increase from 25,000
-12-
-------
TABLE 1
Welfare Gain Associated with an Increase in Wetland Acreage
(From an Initial Base of 25,000 Acres)
Elasticity
(m)
2.05
2.05
2.05
2.05
2.05
Wetland
acreage
OCj)
100.000
200,000
300,000
400,000
500,000
Number of
traps
(xL)
33.610
33,332
33,170
33,056
33,000
Change in
combined
surplus
191,389
294,290
356,843
402,316
435,829
-13-
-------
acres to 100,000 is $191,389. Successive increments in acreage add less to
estimated benefits due to diminishing returns to the marshland input.
The results of a sensitivity analysis, in which different price elastici-
ties of demand [ranging from (-.25) to (-3.45)] are used to calibrate the
model, indicate that, in this particular model, the estimates of welfare gain
are reasonably robust to the choice of an assumed price elasticity.
The purpose of this exercise has been to demonstrate that a theoretically
correct measure of welfare can be constructed and calculated on the basis of
empirical information about the impact on product supply (given demand) of a
change in ecosystem characteristics (here the number of wetland acres) which,
in turn, might be related to pollution control.
Of course, this has been a hypothetical exercise; and, in an actual case
study, one would econometrically estimate the demand and production functions
necessary to conduct the welfare analysis. Moreover, if the estimated demand
function includes an income variable, simple Marshallian consumer surplus is
no longer the appropriate welfare measure. Fortunately, for a variety of
functional forms for the demand function, exact surplus measures are known and
available.
A still more recent study, by Kahn and Kemp (KK, 1985), appears to follow
the procedure we have outlined, though they use it to calculate a welfare
loss. Specifically, they are concerned with the effect the decline in sub-
merged aquatic vegetation (SAV) is having on the various fisheries supported
by Chesapeake Bay. SAV serves as an important link in the estuarine food
chain, and KK attempt to quantify the welfare loss primarily to the striped
bass commercial fishery and, also, to other commercial and sport fisheries
stemming from the reduction in SAV caused by agricultural runoff, discharges
-14-
-------
from sewage treatment plants and soil erosion, and the consequent reduction in
the carrying capacity of the Bay. Unlike LCP, KK are fortunate to have popu-
lation data on the striped bass. With this, they can estimate a supply func-
tion which includes a population variable for the fish and an equation which
relates SAV to fish. After estimating a demand function for striped bass, KK
calculate the losses in consumer and producer surplus following incremental
reductions in SAV. One criticism that can be made of their procedure is that,
since demand is estimated as a function of per capita income, a more exact
welfare measure than Marshallian consumer surplus could have been calculated.
Just for purposes of comparison with the welfare gains that we calculated for
the Florida Gulf Coast blue crab fishery, we observe that a 50 percent reduc-
tion in SAV is associated with an annual loss of approximately $4 million.
This is substantially larger than the numbers in our example. It is important
to note that KK are casting a wider net, so to speak: both commercial and
sport fishing, for several species, are considered.
The studies just described are limited by their static nature. Both exam-
ine the contribution of an environmental input to production assuming the
fishery is in bioeconomic equilibrium (i.e., the harvest rate of the marketed
species equals its growth rate). To the extent that their data are comprised
of observations for years in which the fisheries were not in a steady state,
the regression coefficients they obtain will be biased as parameters of
steady-state models. In addition, static approaches to fisheries economics
fail to evaluate the stream of benefits generated by fisheries as they move
from one equilibrium to the next. As demonstrated in Chapter 3, the higher
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trophic levels of damaged ecosystems may respond slowly to pollution control
measures, and attempts to value control need to take this into account.
The need for dynamic analysis arises from the recognition that fishery
resources constitute capital assets which yield a stream of benefits over
time, and it is in this framework that we can view proposed environmental
cleanup policies as potential investments. Although much of the literature
now recognizes the dynamic nature of fishery resources, with a few articles
even explicitly recognizing the dynamic links between predator and prey
species (see Clark, 1976, and Ragozin and Brown, 1985), the literature has not
considered the management of fisheries' environmental problems in a dynamic
context.
A framework for finding an optimal management strategy when a fishery is
confronted with pollution and open-access problems might look something like
the following. The management problem is one of simultaneously determining
harvesting and pollution control policies to maximize the present discounted
value of net benefits generated by the fishery. In the most general notation,
i.e., making no assumptions about the forms of economic or biologic functions,
the management problem is
00 t
(11) Max Z (1 + r)~ NB[E(t), Z(t), X(t)]
E(t),Z(t) t=0
subject to
(12) X(t+l) - X(t) = f{E(t), Q[Z(t)], X(t)}
and
(13) X(0) = XQ
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where r is a discount rate, NB(») is a net benefit function (e.g., combined
consumer and producer surplus), E is fishing effort, Z is pollution control,
X is the stock of the, harvested species, and Q is the level of environmental
quality. Further realism may be given to the model by including additional
equations of motion [like equation (12)] which represent the growth rates of
other species in the ecosystem and establish links between distinct levels of
the food chain. Modeling species interaction may be of particular importance
if pollution directly affects growth rates at the lower trophic levels, as
demonstrated in Chapter 3. However, the introduction of biological inter-
action among species also poses the problem of selecting an appropriate model
from the available alternatives (see May, 1973, for a description of the vari-
ous ways in which species interaction may be modeled). Interactions can be
complex and models like the Lotka-Volterra used in Chapters 3 and 4 and also
in the studies reviewed in this section which imply simple feeding hierarchies
rather than complex food webs may be misleading (see Harte, 1985).
A key feature of the solution of the optimization problem stated in equa-
tions (11) through (13) may be the interdependence of the two control vari-
ables, allowable fishing effort, and pollution control. For example, if the
level of the fish stock is below the optimum, the derived solution to the
management problem may include the enactment of stringent pollution controls
to enable the fish population to recover. The solution may also include con-
current restrictions on fishing effort (possibly even prohibition) so that the
eventual benefits of costly pollution control may be realized.
The fisheries management problem is further complicated by the fact that
decisions must be made in the face of uncertainty. As discussed in Chapter 4,
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uncertainty pervades the modeling of species interaction; and this is com-
pounded by uncertainty about ecosystem responsiveness to pollution control.
When uncertainty about .the values of economic variables is introduced, the
optimization problem becomes a very difficult stochastic control problem in-
deed. If it is the case that uncertainty about the parameters of the model
can be reduced by research or the acquisition of information through experi-
ence, management strategies should ideally be evaluated with the aid of
closed-loop models in which policy decisions are subject to revison as new
information becomes available, as discussed in Chapter 5 (see also Rausser,
1978).
II. Aquatic Ecosystems as a Final Good
When an aquatic ecosystem is conceived of as ;a final good the benefits
of enhancing the ecosystem typically take the form of improved opportunities
for water-related recreation. These benefits can be estimated using the
methodologies discussed in Chapter 6—either contingent valuation/behavior
experiments or the revealed preference approach based on fitting demand
functions for visiting alternative recreation sites (also called the "travel-cost"
approach). Some of the methodological issues involved in contingent valuation
experiments are discussed in Cummings, Brookshire and Schulze (1986),
Hanemann (1985), and Carson and Mitchell (forthcoming). Issues involved
in the travel-cost approach are discussed in Bockstael, Hanemann and Strand
(1984) and Smith and Desvousges (1986).
The main challenge confronting practitioners of travel-cost studies is
the need to handle the allocation of water-based recreation activities among
multiple sites differing in their environmental quality attributes in a manner
consistent with the utility maximization hypothesis. Two particular aspects
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stand out — the selection of appropriate functional forms for the ordinary
demand functions, and the need to deal with corner solutions. Taking the
question of functional forms first, the problem is to select a set of functions
for the ordinary demands, x. = h (p.q.y), i = 1,..,N, defined at the be-
ginning of Chapter 6. In this context x. is the number of visits to re-
creation site i by a household over some period of time (e.g., the fishing
season), p = (p,,.-,pN) where p. is some measure of the cost of visiting
the i site, q = (q1,..,qN) where q is some vector of attributes of the
i site (including water quality, etc.) and y is either the household's
total income or its total expenditure on recreation activities. The problem
is that, if these demand functions are to be consistent with some utility
maximization hypothesis, they must satisfy certain economic integrability
conditions, including (i) the adding up condition and (ii) the symmetry and
(iii) negative semidefiniteness of the matrix of Slutsky terms, S =iS,, 3 , where
These requirements are by no means trivial and impose significant
restrictions on the eligible functional forms. For example a demand system
of the form
***.•-- *;-£;?;, * JfjJ ia',-->M (15a)
where *t* «• + $ *>^* (15b)
• + . (15c)
.
which is employed in Smith and Desvousges (1986), would appear to violate
the symmetry of the s., terms. Other generalizations of the semi-log form
to systems of multiple demand equations are examined by Hanemann and Lafrance
(1983), where it is shown that the symmetry conditions place very stringent
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(and empirically implausible) restrictions on the underlying direct utility
function. This does not mean that there are no suitable functional forms:
systems such as the Linear Expenditure System — Binkley and Hanemann (1975)
and other members of the Generalized Gorman Polar Form family of indirect
utility functions
can certainly be employed.
The second issue — the phenomenon of corner solutions — is more trouble-
some. This refers to a situation where some of the x.'s are zero — a household
visits some of the available sites, but not all of them. The conventional theory
of consumer behavior is developed under the assumption of an interior solution
to the utility maximization problem (1) in Chapter 6 — i.e., a solution where
all the x.'s are positive. Modifying this theory to deal with non-consumption
of certain goods (non- visitation of certain sites) — a phenomenon that is
overwhelmingly apparent in micro-data sets — is a rather complex task. The
problems involved, and some possible solutions, are examined in Chapters
8-10 of Bockstael, Hanemann and Strand.
A common approach to modelling corner solutions is to decompose consumer
choices into two elements: the selection of a total level of recreation activity,
x =Zx-. and then the allocation of this total among the alternative possible
sites based on some type of shares model
^ 0=1,.., A) (17)
where ft., the share of total visits assigned to the i site, satisfies
... (18)
Statistical models such as logit and probit can be used to estimate the share
equations, and these models can be related to a utility maximization hypothesis.
But, at the present time, it is often difficult to obtain a utility-theoretic
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justification for the "macro visitation equation" determining x, and to
integrate it with the share equations in a theoretically consistent manner.
That is to say, one would like the determination of "x and iTl , .. , 'T^j to
originate in a single, simultaneous utility maximization procedure. Some
models which permit this have recently been developed, but they are relatively
difficult to estimate. The resolution of these issues represents one of the
frontiers of research for the travel cost approach.
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BIBLIOGRAPHY
Bell, F. W. The Future of the World's Fishery Resources: Forecasts of De-
mand, Supply and Prices to the Year 2000 with a Discussion of Implications
for Public Policy, U. S. National Marine Fisheries Service, No. 65-1, 1970.
Clark, C. W. Mathematical Bioeconomics. New York: John Wiley and Sons, Inc.,
1976.
Council on Environmental Quality. Ocean Dumping: A National Policy. Washing-
ton, D. C.: Government Printing Office, 1970.
Gosselink, J. G., Odum, E. P., and Pope, R. M. The Value of the Tidal Marsh,
Center for Wetland Resources, Louisiana State University, No. LSU-SG-74-03,
1974.
Gupta, T. R., and Foster, J. H. "Economics of Freshwater Wetland Preservation
in Massachusetts," in J. S. Larson, ed., Models for Assessment of Fresh-
water Wetlands, Water Resources Center, University of Massachusetts,
No. 32, 1976.
Harte, J. Consider a Spherical Cow. Los Altos: William Kaufmann, Inc., 1985.
Kahn, J. R., and Kemp, W. M. "Economic Losses Associated with the Degradation
of an Ecosystem: The Case of Submerged Aquatic Vegetation in Chesapeake
Bay," Journal of Environmental Economics and Management, Vol. 12 (1985),
pp. 246-63.
Lynne, G. D., Conroy, P., and Prochaska, F. J. "Economic Valuation of Marsh
Areas for Marine Production Processes," Journal of Environmental Economics
and Management, Vol. 8 (1981), pp. 175-86.
May, R. M. Stability and Complexity in Model Ecosystems. Princeton, New
Jersey: Princeton University Press, 1973.
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Park, W. M., and Batie, S. S. "Methodological Issues Associated with
Estimation of Economic Value of Coastal Wetlands in Improving Water
Quality," Virginia Polytechnic Institute and State University,
No. VPI-SG-79-09, 1979.
Ragozin, D. L., and Brown, G., Jr. "Harvest Policies and Nonmarket Valuation
in a Predator-Prey System," Journal of Environmental Economics and
Management, Vol. 12 (1985), pp. 155-68.
Rausser, G. C. "Active Learning, Control Theory, and Agricultural Policy,"
American Journal of Agricultural Economics, Vol. 60 (1978), pp. 476-90.
Shabman, L. A., and Batie, S. S. "Estimating the Economic Value of Natural
Coastal Wetlands: A Cautionary Note," Coastal Zone Management, Vol. 4
(1977), pp. 231-47.
Binkley, Clark S. and Hanemann, W. Michael, (1975) The Recreation Benefits
of Water Quality Improvement. Cambridge, Mass. :Urban Systems
Research & Engineering, Inc.
Bockstael, Nancy E., Hanemann, W. Michael and Strand, Jr., Ivar E., (1984)
Measuring the Benefits of Water Quality Improvements Using Recreation
Demand Models. University of Maryland, Department of Agricultural
and Resource Economics, College Park, MD.
Carson, Richard T. and Mitchell, Robert C., Using Surveys To Value
Public Goods: The Contingent Valuation Method, Washington, D.C.:
Resources for the Future, Inc. (forthcoming).
Cummings, R.G., Brookshire, D.S. and Schulze, W.D., (1986) Valuing
Environmental Goods: An Assessment of the Contingent Valuation Method
Tolowa, NJ: Rowman and Allanheld.
Hanemann, W. Michael (1985), "Some Issues in Discrete - and Continuous -
Response Contingent Valuation Studies," Northeastern Journal of Agri-
cultural Economics , April 1985.
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and Lafrance, Jeffery T., (1983) "On the Integration
of Some Common Demand Systems," Staff Paper No. 83-10, Dept. of
Agricultural Economics and Economics, Montana State University, Bozeman.
Smith, V. Kerry and Desvousges, William L. (1986), Measuring Water Quality
Benefits, Boston: Kluwer-Nijhoff Publishing Co.)
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Chapter 8.
Further Work
Our present intention is to proceed in two areas: (1) comparative
analysis of models for policy evaluation; and (2) development of a case
study.
The first task, the comparative analysis, is intended to further
integrate the ecologic and economic models developed in earlier chapters,
and to compare the results obtained with those of variant versions of the
models. Both aspects of this task are important. The first involves a
tighter linking (than any in the present report) of a model of ecosystem
recovery with a model of dynamic optimization under uncertainty. The idea
is to develop the capability to evaluate control policies leading to
ecosystem recovery, taking account of the (probabilistic) state of the
system over time and at any point in time.
The second aspect of this task, comparative analysis of different
models, is dictated by our lack of knowledge about population dynamics in a
recovering aquatic ecosystem. In chapters 3 and 4 these dynamics were
described by perhaps the simplest model for the purpose, the Lotka-Volterra.
This was sufficient to obtain interesting results about qualitative features
of recovery dynamics and the propagation of uncertainty. But as we move
toward application (as in the case study described below) it becomes
important to determine whether the results are robust, i.e., whether they
continue to hold for equally plausible, though more complex, specifications
of ecosystem population dynamics. Further, we need to explore the notion of
robustness itself. Two models may yield seemingly quite different
predictions about the nature and timing of recovery, yet imply the same
1
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ranking of policy alternatives. For example, one model may predict recovery
of a fish population to 50% of its pre-pol lution level (ignoring
uncertainty) within five years of the imposition of some control measure,
whereas another may predict recovery to just 10%. But the net present value
of control may be positive in both cases. In any event, considerable
further work is needed, in our judgment, on model development, integration,
and comparative analysis, before we are ready to tackle a case study.
Turning now to the case study, we wish to pose a basic question: What
do we want to get out of a case study? Two things, it seems to us. First,
of course, we want quantitative results. What are the benefits of a
particular control option? Second, however, we want to know what the
results depend on. Partly, this is traditional sensitivity analysis. How
are results affected by changes in assumptions about the discount rate,
about a parameter describing interaction between the first and second
trophic levels, and so on. But more importantly, we want to try to
establish links between results and the types of models used to generate
them. This task clearly links back to our proposed work in the first area,
comparative analysis of models for policy evaluation. The difference is
that now we are proposing to go through the exercise in a real case, with
real numbers.
With these objectives in mind, we wish to propose a "double-barreled"
study. First, we would look at a relatively simple lake ecosystem, and one
for which there also exists fairly good data on pollution control and
subsequent recovery. A leading candidate here is Lake Washington, in the
state of Washington. The idea would be to "field-test" our modeling
approach in a relatively favorable setting.
Second, we would like to tackle San Francisco Bay. The Bay is of
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course a much larger and more complex aquatic ecosystem, a marine estuary
with substantial wetlands. Further, existing data are less reliable than
for Lake Washington. Yet even with these difficulties, we feel the Bay is
an appropriate subject for study by this project, for several reasons.
First, it is economically important, a major influence on the natural
resource base (including climate) of a metropolitan area of more than five
million people. Second, the Bay is the subject of considerable current
research and policy interest, at both the state and national levels. Third,
a related point, the Bay ecosystem includes the major remaining wetlands in
Northern California, and wetlands are themselves the subject of much current
interest. Fourth, a study of San Francisco Bay would nicely complement
existing work on the major east coast marine estuarine system, the
Chesapeake Bay. Fifth, clearly travel costs would be minimized by choice of
the Bay. Sixth, and finally, despite, or perhaps because of, the
difficulties, we regard the proposed study as an exciting challenge.
We should note that, again because of the magnitude of the task and the
potential difficulties, we do not propose to complete a study of the Bay
within 12 to 18 months following submission of the final report on the
current study. But we certainly would anticipate completion of parts of the
task, which might stand on their own as interesting and useful research
results.
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