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

                               Table of Contents
Chapter 1.    Introduction and Overview
Chapter 2.    A Suite of Indicator Variables (SIV) Index for an Aquatic
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
Chapter 5:    Hysteresis, Uncertainty, and Economic Valuation
Chapter 6:    The Economic Concept of Benefit
Chapter 7 :     Methods of Benefit Measurement
Chapter 8:    Further Work


   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


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


Goods and Services for which the  Environment Provides Inputs

     Fisheries Products:'  Fish, Shellfish, Crustacea, Kelp, Aquatic Mammals

     Hydroelectric Power


     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)


          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

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

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

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

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

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


     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


                            TABLE 2:


     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
  Comparative Property Values
    Contingent Valuation
        Existence Value

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


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

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.

                         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


     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,


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.

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

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

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
         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

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

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

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

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

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.

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
         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

"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

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

(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

provisionally been extended  to  incorporate large lakes (Tailing,
         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

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
         o   Require some measurement or knowledge of the
         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

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

(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

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

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.


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.
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

         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.

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.
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.
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                               CHAPTER 3.

     The Hysteresis Effect in the Recovery of Damaged Aquatic Ecosystems:

            an Ecological Phenomenon with Policy Implications


   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.


   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


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


     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,
                        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

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


   In the linear food chain  depicted in  Figure 1, the rate of change of the

phytoplankton population can  be described  by  the equation

  (D         ~ - rx (x)(1  -  (X/KX)) - BxyXY   - bxX, where

          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

          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)
Figure 3b.
                                                              Light +
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.


  (2)         — =  ExyBxyX* - ByZTZ - byY, where

         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


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


80  -
60  -
40  -
20  -
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.

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


   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.



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.

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

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

population in the third trophic  level was described by


  (3)                   — r EyzByZYZ - bzZ,   where


          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

  (U)                           	= 0


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,


  (5)                      — = E2fBzfZF - bfF-

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.


   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

C 100

p 90

? 80
P 70

t 	 	 — - 	 , 	 	
X 	 ""' 	 	 — 	
»^. 1>>^
"*•-. ^^
"~~" — «.

II 1 1 1 1
4000       6000

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

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


   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



R 104
 0  96
                        100      150      200

                             TIME IN DAYS
  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.

R 104
° 100
N 98



— /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.


N 90

0  85

                            TIME IN DAYS
 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.

  110  U
F  90

   80  -

P  70
                        4000       6000

                         TIME IN DAYS
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.

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-


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

R 100
0 80
N 40
b 20

\ — =^ 	 A

\ 	 	 ^ 	 :
+ \ 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



E 95

0 90
L 85

'• ' V'^x.
J-V \ \
» \ *^»,
\ N ^.
\ N "*— — Icfighic Level Ratios 9:3:1
\ N;
* ^^^^
^ \ Trophic Level Ratios 25:5:1
\ "^ ^^ ^-^.
\ """ ~ — - — 	 .

Trophic Level Ratios ldb~:TtTfT~" 	 • 	
rill it
4000       6000

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

 110  -
                   5000         10000

                             TIME IN DAYS
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.

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.







                          TIME IN DAYS
 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.

no  -

 90  -
                    1000          2000

                              TIME IN DAYS
 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.

   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-



   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.


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

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


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.


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.


Kipling, C.  (1976).    "Year- Class Strengths  of Perch and  Pike in

   Windermere".   Freshwater  Biology Association  Annual  Report  No.  M,  pp.


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,


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.



                           DETAILS OP MATHEMATICS


     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  =           "
     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.
     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.


     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

     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.
     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.



 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        ..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
         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
         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


       subroutine pederv(x. y, PW) •
 c      ..scalar arguments..
       double precision z
 c      ..array arguments..
       double precision PW(3.3). y(3)
       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)
       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       ..scalars in common..
       double precision H, xend
       integer I
c       ..local scalars..
       integer J, nout
       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
99999  format (1H , FB.2. 3E13.5)

            If (toUt.o) write (nout.99995)
            If ( 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 ( write (nout.99995)
            write (8.99997) Ifail
            If (toLlt.o) write (8.99995)
         40 continue
      99999 format (22hOCALCULATION WITH TOL=.  e8.1)
      99997 format (8h Ifail= II)
      99996 format (4(lx/). 31h D02EBF EXAMPLE PROGRAM RESULT
      99995 format (24h RANGE TOO SHORT FOR TOL)
            subroutine fcn(T. y. F)
     %       ..scalar arguments-
           double precision T
r             ..array arguments..
           double precision F(3). y(3)
             .  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)



 c      ..scalars in common
       implicit double precision (a-h.o-z)
       double precision H, xend
       integer I
 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
       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 ( write (nout.99995)
       If ( 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 ( write (nout.99995)
      write (8.99997) Ifail
      If (toUt.o) write (8.99995)
   40 continue
99999 format (22hCALCULATION WITH TOL=. eB.l)
99997 format (8h Ifail= II)
99996 format (4(lx/), 31h D02EBF EXAMPLE PROGRAM RESULTS/lx)
99995 format (24h RANGE TOO SHORT FOR TOL)
      subroutine fcn(T. y. F)
c      ..scalar arguments..
      double precision T
c      ..array arguments..
      double precision F(4). y(4)
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))

       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)
       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       ..scalars in common..
       double precision H.  xend
       integer I
c       ..local scalars..
      integer J. nout
      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
99999 format (1H . F8.2. 4E13.5)


                                CHAPTER ^

                  Ecotoxicology and Benefit-Cost Analysis:

                       The Role of Error Propagation
   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


 Change in a polluting activity
 (e.g., placement of scrubbers
 in power plants)
(combustion science)
I  Change in emission levelsI
  (e.g. reduction in SOg output)    |
(atmospheric  sciences)
 Change in primary stress on
 ecosystem (e.g., increase in pH
 of precipitation at a particular
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
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
 Indirect ecological  changes
 stimulated by the direct  biological
 effects (e.g.,  improvement in
 fish productivity)
                                         (economics  and  the
                                          political  process)
             (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)
                Table 1.  The stages of ecosystem  impact  assessment

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

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

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

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

                           Figure  1

Illustration of a linear (a), a threshold (b), and a saturation (c) process
relating variables describing successive  stages in the assessment chain.

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


   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.

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.

                               Figure  2
Examples of error bands in the curves shown in Figure 1,

                                            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

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).


                                                               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.


   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

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

                                              aN XN
                                                            6N-1,N XN-1 XN
                             ]   6N-1,N XN-1 XN
        al  Xl
                                e34 x3 x4
                                              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.

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:


                 ---  = 01X1 - Y1X12 - 612X1X2


                 ---  = E12&12X1X2 . a2X2 _  Y2X22 - ^23X2X3
                                               -  3^X3X4
   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

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

                      --- = E12$12X1X2 - a2X2  -
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

This can be shown to equal (Y2X2)/(a2 + Y2X25' wnich is  less than unity-

 other words, the relative  error in X2  induced by the uncertainty in 
  110  -
4000       6000

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.

4000       6000

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.


                5000          10000

                           TIME IN DAYS
 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

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


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


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

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


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


sensitivity, although this has not  been investigated yet.


   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.

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.

                                  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-

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.


    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

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.


     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

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


    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) + ..


     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


               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

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)>

(2c)              V^X^  E E^NB^X^ 6) + max 3

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-
sary to emphasize this dependence,  we shall write X   as an explicit function of

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)}
(4c)  v(X1) =  max  E^NB^X^  9)  +  B NB2(Xp  X£;  0)  +  6  NB3(Xp X2> X3;  e)}
 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).


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)}.

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

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* ±. "~"  ""^


(8a)                 OV1 = [VjU)  - V^O)] -  [V*(l) -  V*(0)j
(8b)                     = [^(1) - V*(l)]  -  [V^O)  - V*(0)];
and, given X-,,

(9a)        OV

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.


    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.


    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).

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.

    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

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).


    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

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),


            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),


              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.
    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

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),
         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),
             Ex NB3(1, 1; 0)], Ej_ NB2(0, 0; 0) + 6 EI  NB3(0, 0; 0)]}.

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)


                              =  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)


                     '  + B2 Ej_ {max  [E3 NB3(0; e),  Ej NB3(1;  e)]}

                V*(0) = El NB1(0; 9)  + 6 Ej_ NB2(0,  0; 9)


                        + 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):

      V^l) = El  NB1(1;  6) + 6 EI  {max  [E£ NB2(1, 0; e), EZ NB2(1, 1; 9)]}


              + 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)]


                + 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]}


                    - 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.

    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

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,

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.

     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].

     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



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


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).

                                       Page 1



        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

                                       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

                                       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


                              page  5

where "the p.  "s are the prices of the marketed goods,  and  y  is  the
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.  =
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

                              page 6

welfare change are the quantities C and E defined, respectively,

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

                              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

                              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

                             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. ^


                    A V
PROPOSITION 2:   If V\ --O for all the q's which  change,  C = E.

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


                             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

                             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.


     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

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

                             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
                             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
       =  \    Lf.,oO-fvO]d. t  ^(,^0-™.0)      00

                             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  
                             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,

                             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
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).
     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

                             page  17
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
                             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*
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
                        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.

                             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.
If we apply WC to u(x,q), this requires that
But, by itself, this is not enough to ensure that
                        5  O,                               (25)
which is what one requires in order to rule out the

                             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


     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

                             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.


     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 ,

although,  in general,  there is no determinate relation between C

                             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):

                              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
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


 PROPOSITION 5:  If u(x,q) has the form given in (30),

                             page 25
It follows from (31b) that
while the change in expenditure on x is
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?

                             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

                             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
where the cut-off price is
In this case,  therefore,  all of the benefit from a change in q is

                             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.


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.)



    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


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

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

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


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[

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
    , 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
u(x, q) = u
mm ( q, — ] , ..., mm [ q.
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

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
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


price, TT.  It must be emphasized that this  market is entirely hypothetical
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.

                                                       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.


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

(3")                           E = / Q n(p,  q,  u1)  dq .


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

FIGURE J.   WTP and WTA for a  Change  in  q


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


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

£  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 -
(iii)  If £U <  1, or if £u > 1 and 1 + (1 -
1°' f 1
1 + (1 - 51
J, A
                                                                              - 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
    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

of the income elasticity,  £--could it happen, for example, that £L = °°?  To

answer that question, differentiate (11) implicity
                                   ,  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

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

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.



     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).

     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.

     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)].


    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).



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,


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.

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


    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


    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


    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.


            FIGURE 1
         THE ECOSYSTEM   .

      FIGURE 2

     FIGURE 3

    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

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


    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

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),

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.

     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
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

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
        5^= W -  X A XT a Xf~  = 0
        dA,            £     1
           = 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,

(5)     Xj_ =
(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):


(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

                         TABLE 1

Welfare Gain Associated with an Increase in Wetland Acreage
          (From an Initial Base of 25,000 Acres)
Number of
Change in

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
    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


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

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


    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)}


(13)    X(0) = XQ


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,

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,


 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


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

(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


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.


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.,


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,


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.

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.


               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.)

                              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


     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


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

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