The Rio Grande silver minnow (Hybognathus amarus) is exemplary of
many fish species in the arid west. Its stocks have declined markedly
during the past fifty years as Rio Grande water has been increasingly
used to meet agricultural and municipal demands. However, any efforts to
help the species recover will be done in an environment of striking
uncertainty (U.S. Fish and Wildlife Service 2007). How will society
protect the Rio Grande silvery minnow and other endangered species when
so little is known about them, but when the costs of some mitigating
measures are so much more clear? Such costs include the politically
unattractive option of denying water to agriculture and municipalities
to leave more water in the river. Due to extremely limited knowledge,
the management problem society faces does not involve risk, where the
probability distributions of interest are well understood. The problem
we consider in this article is instead one involving pure or Knightian
uncertainty, often deemed ambiguity (Camerer and Weber 1992; Knight
1921; Ellsberg 1961).
Water is a natural resource often taken for granted: it frequently
commands such a low per-unit price that few water users even know what
price they pay. However, the twenty-first century may be the era when
relative water scarcity becomes a fact of life (Brown 2003). Population
growth and economic development place increasing pressure on water
supplies in many arid regions, especially during drought. Idaho's
Snake River below Milner Dam and New Mexico's Gila River and the
Rio Grande below Elephant Butte Reservoir are examples of cases where
water rights holders dry up streams and rivers completely, and it is
legal to do so (Benson 2004).
Societal interest in protecting endangered species creates a
natural conflict between private and public interests. In the United
States, public interests in such species are represented in the
Endangered Species Act. This law has been the focus of many well-known
legal battles over land use rights (Brown and Shogren 1998) and
conflicts over water (Benson 2004). Recent controversies (Barringer
2005) pit the Columbia River salmon runs against agricultural and
municipal interests. Those who want to use the river's water argue
that extraction should be allowed because it cannot be proved that
reductions in current flows would affect species' survival. Because
of the uncertainty in the hydrology and biological situation this claim
cannot be completely ignored. Even some of the biologists who believe
that there is a relationship between instream flows and species survival
admit that there is a fair amount of uncertainty about the exact nature
of that relationship (see Shaw 2005, pp. 264-267).
The instream flow management problem explicitly needs to account
for the depth of the uncertainty. We tackle this building on the
ambiguity and uncertainty literature (Knight 1921; Ellsberg 1961; LeRoy
and Singell 1987). Ambiguity is relevant in many natural resource
problems (Shaw and Woodward 2008) and for endangered species in
particular. As examples, over the period from 1991 to 1999 only about
30% of all fish stocks had known population trends (National Marine
Fisheries Service 2002), and even the well-studied Columbia River Basin
continues to present surprises (Barringer 2005). Even less is known
about the relationship between specific environmental or habitat
conditions and growth.
Our dynamic model of water allocation and fishery management
explicitly introduces ambiguity and the potential for ambiguity aversion
by applying robust control, an approach recently advocated by Hansen and
Sargent (2001; Hansen et al. 2006). Robust control has been used to
examine policies in natural resource problems including water management
(Roseta-Palma and Xepapadeas 2004) and extractive fisheries (Xepapadeas
and Roseta-Palma 2003). In a robust control specification, choices
maximize an objective function relative to the worst-case scenario that
the decision maker admits. Hansen and Sargent have argued that robust
control is an appropriate representation of ambiguity-averse preferences
as defined by Gilboa and Schmeidler (1989).
Some Relevant Economics Literature
In their classic article, Burness and Quirk (1979) demonstrated
that the doctrine of prior appropriation (DPA), which is common
throughout the arid western United States, generally will not allocate
water efficiently. To fully explain allocation of scarce flows under the
doctrine of appropriation several modelers consider the location of the
source of flow and the distance from this by each agent who desires a
diversion (Johnson, Gisser, and Werner 1981). This leads to a
first-order difference equation that can be used to determine water
quality or quantity (Weber 2001), and the spatial dimension allows
game-theoretic equilibrium allocations. As markets for water in the
United States become increasingly prevalent (Howitt and Hansen 2005),
those that value instream flows, typically the residual claimants, are
quite literally left with no flows with which to work (Ward 1987).
The economics literature of fisheries management relates mostly to
commercial harvests. Reed (1979) and Clark and Kirkwood (1986) represent
early contributions to the literature on optimal management under risk,
looking at stock and measurement uncertainty respectively. Reed and
Clarke (1990), Saphores (2003), and Sethi et al. (2005), allow for
multiple sources of uncertainty and Xepapadeas and Roseta-Palma (2003),
on which we will build below, consider the extractive fisheries
management problem under both risk and ambiguity.
Though there have been important efforts (Ricker 1975; Johnson and
Adams 1988; Jaeger and Mikesell 2002), the connection between instream
flows and fisheries management is poorly developed. Tsur and Zemel
(1994) study a situation in which water has consumptive value, but where
excessive withdrawals can lead to extinction of a species. They find
that risk in the form of a known probability distribution over the
population leads to a cautious strategy that reduces the chance of
extinction.
A Dynamic Model of Instream Flow Allocation
In this section we develop a different model of the problem of
water management in the presence of an endangered species. First, we
incorporate the existence value of a species directly into the benefit
function. Second, the uncertainty we consider relates to the growth
function of the species, not to uncertainty about water flows over time.
Most importantly, we not only introduce risk with a known probability
distribution, but also allow for ambiguity, which we assume arises from
a lack of knowledge about the true dynamics of the relationship between
species' growth and instream flows.
Assume that there is a single fish stock with a population size of
[q.sub.t] living in a river at time t. The available water supply has
two possible uses: it can be used for industry or agriculture,
[a.sub.t], yielding benefits such as profits from farming, D([a.sub.t]),
or it can be left in the river, st, yielding benefits from instream uses
such as recreation or hydroelectricity, W([s.sub.t]). To focus our
attention on uncertainty in the species' dynamics rather than on
randomness in flows due to weather patterns, we assume that the total
flow of water is constant at the rate R, so that [a.sub.1] + [s.sub.t] =
R. For notational simplicity, we delete the time subscripts below except
where necessary for exposition.
[FIGURE 1 OMITTED]
The growth of the fish stock is affected by both the current stock
and the instream quantity of water, s. (1) Our main interest here is in
a protected species, so harvesting is assumed to be illegal or
negligible. We begin with a deterministic model in which the
species' growth depends only on the current stock size and the
current stream flow
(1) [??]/q = f(q, s).
For any value of s, we assume that the dynamics of the species is
characterized by a standard biological growth model with critical
depensation at [q.bar](s) and carrying capacity at [bar.q](s) as in
figure 1. The top two curves in the figure, labeled high and medium
flow, are typical of such dynamics. At flow levels in this range, if q
< [q.bar](s) then growth will be negative and, if the flow does not
change, the stock will decline to extinction. If q starts above
[q.bar](s) it will tend toward the carrying capacity, [bar.q](s). We
assume that an increase in the instream flow improves the species'
rate of growth, shifting the growth curve upward so that [partial
derivative] f(q, s)/[partial derivative]s [greater than or equal to] 0.
(2)
The two lower curves are logical extensions of the idea of a
flow-contingent growth. In the bottom curve in figure 1, the instream
flow is so low that regardless of the species' stock level, its
population will decline over time. The second-lowest curve shows the
growth associated with the lowest possible flow for which it is still
possible to maintain a positive stock. The flow level associated with
this curve, [s.sub.L], is that which would be sought by a planner
seeking to maximize withdrawals for a water course (minimizing instream
flows), while at the same time ensuring that the species survives. The
stock that could be maintained at this flow level is [[q.bar].sub.u].
For constant flows in excess of [s.sub.L], there are three steady
states: stable equilibria at q = 0, and q = [bar.q] and an unstable
equilibrium at q = [q.bar](s), which lies between [[q.bar].sub.l] and
[[q.bar].sub.u]. For flows less than [s.sub.L] the only equilibrium
position is at q = 0. For s = [s.sub.L] there are two equilibria, at
zero and [[q.bar].sub.u]. For some species it will be the case that for
some flow levels [[q.bar](s) = 0; i.e., the growth function is no longer
characterized by critical depensation and recovery can be achieved from
any stock level. In this article we focus on cases in which [q.bar](s)
> 0 for all s so that there is always the possibility of negative
growth.
The fish stock is assumed to have a nonuse, or passive use value,
B(q), sometimes deemed existence or preservation value. This
nonconsumptive value might arise because the current generation wishes
to preserve the species for future generations or simply because it
wishes to know that the species exists (Krutilla 1967). In a model of
pure existence value, the objective function would capture this value
through a penalty paid if q reaches zero, as in Tsur and Zemel (1994) or
Saphores (2003). We assume that B'(q) = 0 for q >
[[q.bar].sub.l]; that is, the marginal value of the fish stock is zero
over the range of stocks where extinction can be avoided. The cost of
extinction is assumed to be finite so that there is a limited
willingness to pay to prevent extinction for the decision maker. This
formulation is consistent with the preferences implicit in the U.S.
Endangered Species Act (ESA), which makes it illegal to place a species
at risk of extinction, but imposes no restrictions once a species is
delisted. Further, since the 1978 ESA amendments, the Endangered Species
Committee (often referred to as the "God squad") can grant
exemptions to the ESA if the Act's costs are excessive.
Optimal Water Management under Certainty
First, consider the water allocation problem with no uncertainty of
any kind. The water manager's problem is to choose a level of
instream flow, allocating residual water to extractive uses. This is
tantamount to choosing the growth curve for the species. Assume that the
manager acts as a benevolent social planner seeking to maximize the
present value of net benefits obtained from the water flow over an
infinite horizon subject to the constraint governing the species growth
dynamics, i.e.,
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where p is the social rate of discount. The current-value
Hamiltonian (J) for the problem is
(3) J = D(R - s) + W(s) + B(q) + [mu]qf(q, s).
Letting [f.sub.q] and [f.sub.s] represent partial derivatives with
respect to q and s, the first-order necessary conditions with respect to
[s.sub.t] and [q.sub.t] are:
(4) D' = W'+ [mu][qf.sub.s]
(5) B' + [mu]([qf.sub.q] + f) = [rho] [mu] - [??].
The second term on the right-hand side of (4) captures the value of
water in protecting the species. The costate variable, [mu], is a
measure of the marginal value of the species; it is the stream of future
benefits that arise due to a marginal increase in the stock along an
optimal path. It is multiplied by [qf.sub.s], which is the impact on the
growth rate of the fish stock of a marginal change in s. Hence, the term
[mu][qf.sub.s] is the marginal value of water in terms of future
benefits obtained from the fish stock.
From (4) we see that the manager's optimal allocation of water
between instream and extractive uses sets the marginal value of water to
agriculture, D', equal to the marginal value of the water instream
including its value to instream users, W, plus its value relating to
protecting the species, [mu][qf.sub.s]. Assuming that extractive users
must pay for withdrawals of water from the river, the optimal price of
water, p, would reflect its complete opportunity cost,
(6) p = W' + [mu][qf.sub.s].
Of course, implementation of this rule would require the empirical
analyst to overcome difficult challenges of estimation, but the
conceptual goal above is quite straightforward.
Optimal Water Management under Uncertainty
We now consider the case in which the change in species stock is
stochastic and there is significant uncertainty surrounding the
parameters describing the dynamics of the system. We follow the steps
used by Xepapadeas and Roseta-Palma (2003, hereafter XR) in their
analysis of a commercial fishery, but the current model differs from
XR's in a number of dimensions. First, we focus on a nonextractive
fishery problem in which the species has existence value, adding the
role of instream flow in the fishery's stock dynamics,
characterized by critical depensation.
A stochastic optimization problem seeks a water allocation rule,
s(q, t), that will maximize the present value of expected net surplus,
subject to the constraints of the system. Using standard procedures of
stochastic control, a random variable is added to the state equation,
[d.sub.q/q] = f(q, s) dt + [sigma] dH
where H is a Wiener process, with E(dH) = 0 and var(dH) =dt. This
leads to the stochastic control problem
(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The term [sigma] dH is the stochastic element of the state
equation; although the change in q is centered at qf(q, s), it varies
from that path over time with the variance increasing in a linear manner
with the time horizon considered.
Despite the addition of the stochastic element in (7), the manager
who solves this problem actually has quite a lot of knowledge about the
system's dynamics; the parameters of the system are assumed to be
known with certainty--including the moments of H. We relax this
assumption and follow Hansen et al. (2006) and XR, regarding (7) as a
"benchmark model." That is, we assume below that the manager
does not know the parameters of the model with certainty and, given
finite data available and the manager's limited ability to learn
about the system through experiments, the "true" model cannot
be exactly determined.
To account for this ambiguity, next define an alternative
"perturbed" model that is statistically indistinguishable from
(7) by setting H = z + [[integral].sup.t.sub.0] [Y.sub.[tau]]
[d.sub.[tau]] or dH = dz + [Y.sub.[tau]] dt, where z represents Brownian
motion, and Y is drift distortion in the system's dynamics that
cannot be distinguished from the standard noise. While z is
symmetrically distributed around zero, the model now allows for drift
over time at the rate Y. The variable Y captures the ambiguity the
manager faces because of the uncertainty surrounding the actual model.
The manager does not know that Y actually exists, but neither does she
know that it does not exist. If there is a drift, then the state
equation would be
(8) [d.sub.q/q] = f(q, s) dt + [sigma](dz + Y dt).
Hence, there are two sources of uncertainty above: risk, which has
a known mean and variance; and ambiguity, which reflects what the
decision maker does not know.
Optimization under Ambiguity--Robust Control
Before discussing the solution of the optimization problem
presented above, additional background is provided on ambiguity. (3)
Since Savage (1954), most economists have assumed that decision makers
act as if they have personal subjective probabilities when making
choices. These probabilities are known by the decision maker, but we
perhaps cannot directly observe them. The assumptions of Savage's
model are more restrictive than they may first appear. Since
Ellsberg's (1961) work there has been a large body of work (mostly
laboratory experiments) showing that individuals often violate the
Savage axioms when there are conditions of ambiguity.
In response to these violations numerous alternatives to
Savage's axioms of rationality have been proposed. One approach is
provided by Gilboa and Schmeidler (1989), who axiomatically show that a
decision maker who faces ambiguity can rationally choose to maximize
relative to the worst possible probability distribution. (4) If we
accept this broader definition of economic rationality, it leads to
important changes in policy rules.
In a series of articles (Hansen and Sargent 2001; Hansen et al.
2006) it has been argued that the "maxmin" optimization
criterion of Gilboa and Schmeidler (1989) can be operationalized by
using robust control. Robust control explicitly recognizes that there
may be uncertainty in the underlying model that cannot be captured
through the use of probabilities alone. It is a general form of dynamic
optimization that encompasses a continuum of optimization criteria, from
expected net-present value at one extreme, to a maximin criterion at the
other. Hansen et al. (2006) provide two possible specifications of the
robust control problem, which they call the constraint and penalty
problems and show that the two are mathematically equivalent. Although
the constraint problem specification is most closely tied to Gilboa and
Schmeidler (1989), as in XR we use the penalty problem specification in
which the degree to which ambiguity affects decisions is captured
through a parameter [theta] in the optimization problem.
Optimal Water Management under Ambiguity
The robust control specification of our instream flow problem is
(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(10) [d.sub.q/q] = f(q, s)dt + [sigma] dz + [sigma] Y dt.
The differences between this problem and the benchmark problem (7)
are related to the robust control variable, Y. Values of Y should be
"close enough to the approximating model that they are
statistically difficult to distinguish from it after having observed a
continuous data record of only finite length" (Hansen et al. 2006,
p. 57).
There are two ways that the variable Y enters the optimization
problem. First, Y enters in the state equation. Hence, by minimizing
over Y, the manager behaves as if in a game against nature, in which
nature is choosing Y so as to make the manager as bad off as possible.
This is the maximin element of the Gilboa and Schmeidler (1989)
criterion.
The second way that Y enters the robust control problem is in the
objective function, through the term [theta] [Y.sup.2]/2. This is
essentially a penalty for pessimism. As we will discuss below, the
parameter [theta] can be thought of as the inverse of the weight placed
on ambiguity aversion, which we will treat as exogenously determined.
The way that ambiguity aversion enters into the solution can be
understood by looking at two extremes. First, consider the case of there
being no ambiguity aversion, [theta] [right arrow] + [infinity]. In this
case the penalty for choosing a high value of Y dominates so that the
objective function is minimized at Y = 0. Without ambiguity aversion,
therefore, the manager accepts the baseline model and ignores the
possibility of drift; s is chosen to maximize the expected present value
of net benefits, the standard stochastic control problem. At the other
extreme, ambiguity aversion increases as [theta] gets smaller. If
ambiguity aversion is severe, s will be chosen to maximize welfare in
the worst-case setting. Intermediate values of [theta], therefore,
capture a continuum of degrees of aversion to ambiguity.
Following the standard approach as in XR, we next rewrite the
Hamilton-Jacobi-Bellman equation for the optimization problem,
(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [J.sub.q] and [J.sub.qq] are the first and second derivatives
of J respectively and P is the discount factor as above. (5) The term
[J.sub.q] in equation (11) is analogous to [mu] in the deterministic
formulation (3). Robust water extraction rules can be found by solving
this optimization problem. Setting the first derivatives of equation
(11) with respect to s and Y equal to zero, we obtain
(12) D'(R - [s.sup.*]) - W'([s.sup.*]) =
[J.sub.q][qf.sub.s](q, [s.sup.*])
(13) [Y.sup.*] = [sigma]q[J.sub.q]/[theta].
These two equations can be used to explore the relative roles of
risk and ambiguity.
The parameter [theta] captures, in the words of XR (p. 12),
"the maximum specification error ... that the social planner is
willing to accept." We see from (13) that as [theta] decreases,
[Y.sup.*] increases in absolute value and, in the words of Whittle
(2002, p. 9), a breakdown point can be reached where the decision maker
is "so pessimistic that his apprehension of uncertainties
completely overrides the assurance given by known statistical
behavior." XR discuss [theta] in much more detail. We treat it as a
primitive parameter reflecting the decision maker's preferences and
information, and assume that it is strictly positive and sufficiently
large to avoid the breakdown point.
[FIGURE 2 OMITTED]
Optimal Policies under Certainty
The optimal policy choices under certainty, risk, and ambiguity can
now be compared. Much of the intuition for this comparison can be
developed by first considering the optimal policy under certainty in
more detail. This policy is identified by (4), which is equivalent to
(12) when [sigma] = Y = 0. Table 1 summarizes the important variables at
various parts of the state space, coinciding with the regions with
different state dynamics as presented in figure 2.
Consider first the policy at q = [[q.bar.].sub.l], the lowest
possible value for the point of critical depensation in figure 2. (6) At
this point the stock sits at the precipice of extinction. If the stock
falls below [[q.bar].sub.l] then it will unavoidably decline to
extinction. Assuming extinction is not optimal as in Clark (1973), at
this point the shadow price, [mu], will be high enough to induce
sufficient stream flow to at least maintain the stock level. The level
of s chosen at this point will be at least as high as that chosen at any
other level of q. Hence, we indicate the optimal level of s at
[[q.bar].sub.l] as [bar.s].
By definition, for q < [[q.bar].sub.l], [??] is strictly
negative and the stock will decline to extinction regardless of the
level of instream flow. The marginal value of q in this range is
positive only because the time of extinction is deferred somewhat. If
the decline is sufficiently fast, over this range [mu] will tend to be
small and the manager will choose an s that is quite close to [s.sub.M],
the level of instream flow that would be chosen if the manager were
myopically focused only on use-oriented benefit, i.e., where
D'([s.sub.M])--W'([s.sub.M]) = 0.
Next consider the upper end of the stock levels. Recall from figure
1 that [s.sub.L] is the lowest flow level for which nonnegative growth
is possible. If [s.sub.M] > [s.sub.L] (figure 2A) then even without
regard for the species, a flow level will be chosen that allows for
positive growth. In this case any increment to the stock above
[q.bar]([s.sub.M]) will not cause a change in s, nor will it have any
impact on future existence values experienced. Hence, if [s.sub.M] >
[s.sub.L] the marginal value of the stock will be zero for all q
[greater than or equal to] [q.bar]([s.sub.M]). If q starts at
[q.bar]([s.sub.M]), then growth is zero so the stock will remain at that
unstable equilibrium indefinitely. If q starts above [q.bar]([s.sub.M]),
the species population will converge to [bar.q]/([s.sub.M]).
A more interesting situation occurs if [s.sub.M] < [s.sub.L]
(figure 2B). In this case the myopic choice would lead to extinction.
The optimal policy will, therefore, balance the desire to increase
extractions with the need to avoid collapse of the species to
extinction. In this case the marginal value of species will be positive
over the entire domain since an increase in the stock allows the manager
to decrease s closer to [s.sub.M], at least for a time, before
inevitably having to sacrifice extractive value for the sake of the
species. Assuming that extinction is not optimal, we know that the
long-run equilibrium will be reached at some flow level, say s*, and a
stock level of either [q.bar]([s.sup.*]) or [bar.q]([s.sup.*]). It turns
out, however, that [bar.q](s*) is not optimal since the sustainable path
with s = [s.sup.*] and q = [bar.q](s*) is dominated by a path in which s
< [s.sup.*] for a finite period, followed by a sustainable path in
which s = [s.sup.*] and q = [q.bar]([s.sup.*]). This alternative path
leads to a higher present value of net benefits and an equilibrium at
the biologically unstable point of critical depensation. In the
Appendix, we show that such a point can be locally economically stable
because it is possible to instantaneously adjust the flow levels to push
the stock back to some point [q.bar](s).
[FIGURE 3 OMITTED]
Deviations of s from [s.sub.M] arise due to the marginal value of
the stock, [mu] in equation (4) or [J.sub.q] in equation (12). A
stylized representation of how this parameter changes over the state
space is presented in figure 3. The solid line there, which we refer to
as [J.sup.C.sub.q], represents the marginal value of the species under
certainty.
Optimal Policies under Risk and Ambiguity
Equation (12) presents the first-order condition with respect to s,
showing that optimal instream flow is set where the immediate marginal
net benefit of extraction, D'(x)--W'(x), is equal to the
marginal cost in terms of foregone future benefits obtained from the
species, [[J.sub.q][qf.sub.s]. The term [J.sub.q][qf.sub.s] is the
marginal user cost. The effect of risk and ambiguity on optimal instream
flows is captured in [J.sub.q]; as long as the functions D and W are
monotonic and concave, an increase in [J.sub.q] will lead to increased
instream flows.
Recall that here the species is exclusively valued for its
existence. In this model, therefore, its direct marginal value is
positive only as q approaches zero. For q > [[q.bar].sub.l], an
increase in q is valuable because it lowers the probability that the
stock size will move toward zero, where the optimal policy is to make
costly increases in s.
Strictly speaking, the stock will never reach zero within a finite
horizon because of the assumption of geometric growth. To simplify our
discussion, we will refer to a species as being extinct if it falls
below some small stock size in the neighborhood of, but is not exactly
at, zero (e.g., a single individual). Because the stochastic process is
a Weiner process, there is a nonzero probability that in any finite
period the species will drift into the range of critical depensation
from which it will, on average, tend toward zero. Over an infinite
horizon, therefore, extinction is unavoidable. Over any finite horizon,
however, even a very long one, the chance of actually going extinct can
be near zero or near one. For any stock greater than [[q.bar].sub.l] it
is possible to allocate enough water to instream flow that the
probability of an immediate increase in the stock is greater than 50%.
Hence, if the stock starts near [[q.bar].sub.u] and if from
[[q.bar].sub.u] to [[q.bar].sub.l] the optimal policy at every point
stock leads to a better than 50% chance of the stock increasing, then
the probability of reaching [[q.bar].sub.l] in the foreseeable future
would be very small.
The effects of risk and ambiguity on [J.sub.q] are presented in
figure 3. The deterministic case discussed above, [J.sup.C.sub.q], is
indicated by the solid line. The case of risk without ambiguity
aversion, [sigma] > 0 and [theta] [right arrow] [infinity], is
represented by the dotted line in figure 3 and will be referred to as
[J.sup.R.sub.q]. The introduction of risk means that for some values of
q < [[q.bar].sub.l] the marginal value of s is positive as it
increases the probability that the stock will grow in the immediate
future. As a result, the [J.sup.R.sub.q] curve slopes upward from zero
in figure 3 before [[q.bar].sub.l]. To the right of [[q.bar].sub.l], it
is possible for the average rate of growth to be positive. Risk means
that there is never a range in which the stock is completely
"safe" from extinction, but the greater the stock, the less
likely it is that extinction will occur in any finite time period.
Because risk of extinction in the near future declines gradually as q
increases, [J.sup.R.sub.q] will descend more slowly than
[J.sup.C.sub.q]. From (12), if follows that at any stock level where
[J.sup.R.sub.q] > [J.sup.C.sub.q], the flow s will be greater under
risk than under certainty.
Next, consider the introduction of ambiguity aversion by reducing
the penalty parameter, [theta] (which again, is the inverse of the
weight placed on it). Ambiguity aversion changes the nature and solution
to the problem in two important ways. Under ambiguity aversion the
manager will behave as though nature is playing against her by choosing
[Y.sub.t] following (13). Because [J.sub.q] [greater than or equal to] 0
across the entire domain, the minimizing value of Y will be negative,
meaning that the manager will optimize relative to a growth function
that is being pulled down compared to the baseline model. This leads to
the [J.sup.A.sub.q] curve in figure 3, the dashed line, which is a
multiplicative transformation Of [J.sup.R.sub.q], so that
[J.sup.A.sub.q] is "stretched" to the right.
Because of the assumption of critical depensation with
[[q.bar].sub.l] > 0, all the [J.sub.q] curves are nonmonotonic. This
means that there is a point (indicated [q.sup.c] in the figure) at which
[J.sup.A.sub.q] crosses [J.sup.R.sub.q]. To the right of [q.sup.c],
[J.sup.A.sub.q] lies above [J.sup.R.sub.q], meaning that ambiguity
aversion leads managers to place more weight on the value of the species
and allocate a greater share of the available water to instream flows.
In this range, ambiguity aversion makes it optimal to be extra cautious;
even when the population appears (on average) to be secure, ambiguity
causes the manager to allocate more water to species protection. Hence,
to the right of [q.sup.c] (and assuming that the baseline model is true)
the probability of the stock falling to extinction in any finite horizon
is less under ambiguity aversion than it is under risk.
Figure 3 also shows, however, that ambiguity aversion has a very
different effect when the species population is below [q.sup.c]. In this
range ambiguity aversion actually reduces the marginal value given to
the species in (12). The intuition behind this result is that for low
stocks, the pessimistic manager will view the prospects for the species
as increasingly bleak and will tend to reallocate water to the
extractive use that generates tangible benefits. As a result, for the
lowest and most vulnerable stocks, ambiguity aversion in our model
actually leads to a reduction in instream flows, decreasing the mean
rate of growth. For a given initial stock below [q.sup.c], it is
possible that the policy under risk might have a better than 50% chance
of avoiding extinction in the near future, while the ambiguity averse
policy might tend to push the species toward a quick extinction.
Discussion of Results
Our models of optimal instream flow management under certainty,
risk, and ambiguity provide a number of insights into the kinds of
trade-offs that are made in making choices associated with species
habitat. Several of these results are intuitively obvious: optimal
management will balance net marginal benefits to extractive use with the
marginal cost in terms of species loss; species management becomes
smoother when risk is introduced, and ambiguity can lead to even more
cautious policies. However, one key result is more difficult to
understand. As shown in figure 3, there is a range of stocks where the
ambiguity averse strategy is less cautious in terms of protecting the
species and seem to give up on the species. What is driving this
counterintuitive result that our formulation offers?
First, it may be that ambiguity as defined here is still too
restrictive a notion. Quiggin (2005) argues that the precautionary
principle, which has become an important heuristic for environmental
planning, can be explained by reference to the incompleteness
hypothesis, which allows for uncertainty that is even more broadly
defined than it is here. Second, in the model specification there is a
lower bound on the point of critical depensation, [[q.bar].sub.l]; if
the stock falls below this point there is nothing that can be done to
avoid extinction. Although this seems biologically reasonable, we know
of no real species management problem in which managers have consciously
given up all hope of species recovery. If [[q.bar].sub.l] = 0, then
[J.sup.A.sub.q] would dominate [J.sup.R.sub.q] across the entire domain
so that ambiguity would always lead to higher levels of instream flow.
Third, recall that the robust control solution follows from an
assumption of ambiguity aversion. Admittedly, the degree of ambiguity
aversion for any decision maker, just like risk aversion, is ultimately
an empirical question. Nevertheless, to the right of [q.sup.c], the
robust-control policy reduces ambiguity by pushing the stock upward, to
the region where ambiguity becomes irrelevant to the manager. Near
[[q.bar].sub.l], on the other hand, the ambiguity surrounding the
species' survival becomes most salient to the manager--a point is
reached where a reduction in s will reduce ambiguity, which would be
perceived as a benefit to the ambiguity averse decision maker. For the
manager who is ambiguity averse a point is reached where focus is
shifted back to foregone benefits of water's extractive use. In
other words, ambiguity aversion does not mean "preserve at any
cost." Instead, the maximin strategy can move a manager from making
aggressive efforts to protect the species to giving up on the species as
the chance of survival falls.
A final interpretation of these results might be found in a
critique of robust control itself. We have proposed that policy
makers' preferences might be consistent with the axioms of Gilboa
and Schmeidler (1989) and that they might exhibit aversion to ambiguity.
However, for very low species stocks, when extinction is most imminent,
these assumptions lead to choices that seem inconsistent with what we
see in the real world. Real-world manager or policy-maker actions may be
"wrong," or they do not adhere to these axioms, or they are
not averse to ambiguity.
Nevertheless, we believe our analysis actually highlight the merits
of the use of robust control. Robust control is derived from strong
normative foundations based on optimal decision making under ambiguity.
Robust control is not simply a rationalization of policies that are
preordained as "conservative." This echoes the findings of
Giannoni (2002) who, in the context of monetary policy, identified
policies that were inconsistent with "conventional wisdom."
Finally, we emphasize that the robust optimal policies do not make
extinction more likely. On the contrary, to the right of [q.sup.c]
ambiguity increases the marginal value placed on the species, resulting
in optimal polices that allocate more water to instream flows. The
robust policy will, therefore, be proactive to the threats to the
species and reduce the possibility that species will become endangered.
Summary and Conclusions
The connection between endangered species, water allocation, and
laws such as the ESA are evident today (Benson 2004). Controversy runs
high, with some arguing for compensation of damages from forgoing water
use, and concerns that this will ultimately weaken the ESA (Boxall
2004), and that the burden of proof will fall to those arguing that
instream flows are necessary for species protection. As an example,
consider again what we know about the silvery minnow. First, Berrens,
Ganderton, and Silva (1996) estimate that New Mexico households would be
willing to pay between $28 and $90 per year to restore instream flows
for the silvery minnow, a fairly wide range in values. Second, Ward and
Booker (2006) find that augmenting Rio Grande instream flows
sufficiently to protect the silvery minnow would actually increase over
all net benefits by moving water from lower- to higher-valued users.
Third, however, the U.S. Fish & Wildlife Service (2007) estimates
that the undiscounted cost to restore the silvery minnow's
population would exceed $114 billion over twenty-five years. (7) Because
of the enormous uncertainty surrounding the population dynamics of many
endangered species such as this, we may never be certain whether a
particular plan of action is either necessary or sufficient to avoid the
minnow's extinction. It remains to be seen whether society will
allocate enough resources in time to actually save several species.
Recognition of the importance of ambiguity and the development of
appropriate frameworks for decision making in light of it is an
important challenge to economists today. Here we have considered the
problem of water management where an endangered species exists, but
where there is significant ambiguity about the species' dynamics.
We find that when ambiguity aversion is present, the ambiguity-averse
strategy will change policy relative to the policy under risk. For some
levels of the fish stock, the potential for extinction will be given
more weight in the water allocation decision, even if current
populations appear to be fairly safe from extinction. On the other hand,
we find that if the stock reaches a point where extinction becomes
likely, then ambiguity-averse decision makers might prefer to shift
resources to alternative economic activities. At least in this context,
the "optimal" policy is not necessarily to try to save the
species regardless of the cost.
We have left unanswered the critical question of how ambiguity
should be treated by a benevolent social agent or manager. Individual
decision makers regularly demonstrate ambiguity aversion. Should policy
decisions also be based on ambiguity aversion, reflecting an assumption
that the public exhibits ambiguity aversion? Or should policy instead be
based on cold probabilities and the best available science? These are
good questions to ask. There is already evidence of ambiguity aversion
in public policy; the ESA requires that protection programs be robust to
the uncertainty surrounding the species' survival and one could
argue, therefore, that the U.S. Congress revealed a strong aversion to
ambiguity when overwhelmingly approving the ESA at its inception.
Ambiguity aversion provides a justification for placing weight on
species preservation in water allocation problems, even in the absence
of conclusive evidence of a relationship between the species and
instream flow. It does not necessarily warrant extreme behavior, but it
does suggest caution can be "optimal."
Appendix
Stability of Equilibrium in the Deterministic Model
The equations of motion of the deterministic system are in
equations (1) and (5). Noting that s is implicitly a function of q and
[mu], we obtain a linear approximation of these equations around
equilibrium, [q.sub.0], [[mu].sub.0] from the first-order condition,
equation (4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
The qualitative properties of this system are equivalent to the
properties of the associated homogeneous system, which can be written in
matrix notation,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The Eigen values for this linear system of equations (8) are
(A.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
A necessary and sufficient condition for stability of the
equilibrium is that the Eigen values have negative real parts. This can
only occur if the sum of the first five terms in parentheses in (A.1) is
less than zero. That is, stability will be achieved if
(A.2) [f.sub.q] + [rho] + [q.sub.0][f.sub.s][s.sub.q] +
[mu][f.sub.qs][s.sub.[mu]] + [q.sub.0][f.sub.q] <0.
Economic intuition for the conditions in which this is satisfied
can be obtained by first considering the case where [f.sub.qs] = 0. At a
point of critical depensation [f.sub.q] > 0, so we can sign the
various parts of this inequality as follows:
(A.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
If [q.sub.0] is large, then (1 + [q.sub.0]) [approximately equal
to] [q.sub.0] and 1/[q.sub.0] [approximately equal to] 0. So (A.3) will
hold approximately if [f.sub.q] < - [f.sub.s][s.sub.q]. The left-hand
side of this inequality is the rate at which growth increases as the
stock increases. The right-hand side is the rate at which growth
declines due to the decrease in the instream flows as a result of the
marginal increase in q. If, at the margin, the effect on the growth rate
caused by the change in s outweighs the effect on the growth rate caused
by the increase in the stock, then a point of critical depensation can
be economically stable.
Stability is much more easily satisfied at a point of
flow-contingent carrying capacity, for in this case [f.sub.q] < 0 and
the first term in (A.3) will also be negative. Using the approximating
assumptions, the inequality will always hold.
Note that if [s.sub.M] > [s.sub.L] , then at the point of
critical depensation, [q.bar]([s.sub.M]) a marginal increase in the
stock will not induce a change in s. In this case, therefore, [s.sub.q]
= 0 so that equation (A.3) cannot be satisfied so that it is quite
unlikely that [q.bar]([s.sub.M]) would be a stable equilibrium. In this
case the carrying capacity associated with [s.sub.M],
[bar.q]([s.sub.M]), would be the only stable equilibrium in addition to
the extinction.
When [f.sub.qs], [not equal to] 0, the relationship becomes more
complicated, but the essential economic intuition remains. From equation
(4), it follows that an increase in [mu] would cause an optimal increase
in the water allocated to instream purposes if D" < 0. Hence
both [s.sub.[mu]] and [mu] are greater than zero, so that if [f.sub.qs]
< 0 it is easier to satisfy (A.2) (or harder if [f.sub.qs] > 0).
The authors gratefully acknowledge Michele Zinn's editorial
assistance. They have received very helpful comments on earlier drafts
of this article from Robert Berrens, Trudy Cameron, Catarina
Roseta-Palma, and especially from Stephen Swallow, the editor who
handled the manuscript, and three anonymous reviewers for the journal.
Financial support for this work was provided by the Texas Agricultural
Experiment Station.
[Received August 2006; accepted December 2007.]
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