Optimal Resource Management with a Safe Minimum
Standard - Conditions for Living on the Edge of Risk
Eric Nævdal1 and Michael Margolis2
Abstract
Safe Minimum Standards (SMSs) has been advocated as a policy rule
for certain environmental problems where uncertainty about risks and
consequences are thought to be profound. The present paper explores
the rationale for such a policies and derives conditions for when a SMS
can be summarily dismissed as a policy choice and for when SMS can
be defended as an optimal policy based on standard economic criteria. It
turns out that these conditions can be checked with quite limited
information about damages and risks.
1. Introduction
Environmental and resource problems that are currently considered critical include many in which
future damages are highly uncertain, possibly severe and likely to be irreversible. In the public debate,
the notion is often expressed that such problems should be dealt with by applying the Precautionary
Principle (PP) and avoid such risks altogether. The Precautionary Principle seems to have great
1
To whom correspondence should be addressed; Dept. of Forestry, Agricultural University of Norway, P.O. Box 5033, N –
1432 Ås, Norway; eric.naevdal@isf.nlh.no.
2
Resources for the Future; margolis@rff.org.
Comments and suggestions from Olvar Bergland, Jason Shogren and Tommy Stamland are gratefully acknowledged.
1
intuitive appeal both to the general public and to non-economist scientists. This may be why the PP has
been adopted in much environmental legislation, such as the Endangered Species Act and the Clean
Water Act in the USA. Ciriacy-Wantrup (1952) suggested that such problems be addressed by
establishing Safe Minimum Standards (SMSs) below which the flow of key ecosystem services should
not be permitted to fall. To prevent an extinction, for example, we might establish a SMS for the size
of the reproducing population, another for the area of its habitat, and a third (actually a safe maximum)
for the extent of human activity within the habitat area. The concept of the SMS has since been refined
by e.g. Bishop (1978), Bishop (1979), Bishop and Ready (1990) and Crowards (1998) and put into
practice in the form of critical habitat designation for endangered species, minimum flow and purity
requirements for water quality, and others. None of these arguments, however, is based on a
conventional social welfare-maximization criterion, which makes SMSs a controversial subject among
economists. The cause of the controversy, as summed up by Farmer and Randall (1998), is that “SMS
processes … cannot be derived from a single direct objective statement that also derives the policy
exception upon which they are superimposed”. The central contribution of the present paper is to
negate this statement and show that it is in fact possible to derive conditions under which an SMS
policy will maximize social welfare and show that surprisingly little information is required to verify
whether these conditions hold.
Economists have suggested alternative justifications for the use of an SMS. Most of these develop the
philosophical position that there are limits beyond which utilitarian calculus ceases to be legitimate.
Bishop’s (1978, 1979) case was built on the assumptions that planners have are unaware of the
probabilities of relevant events and that a planner ignorant of probabilities should follow a “minimax”
strategy, which minimizes the maximum possible loss. This strategy is tantamount to assuming the
worst possible outcome is a certainty, and will not in general maximize welfare. Further investigation
by Smith and Krutilla (1979) and Ready and Bishop (1991) uncovered several plausible circumstances
in which abandoning a species to extinction is clearly preferable to an SMS. Bishop thus suggested the
policy be abandoned when its cost is “intolerably” high, but there is no way to determine exactly how
high cost ought to be tolerated. Norton and Toman (1997) suggested an SMS could play a role “twotiered” decision-making system, in which utilitarian calculus would give way to a more conservative
approach as irreversibility and justice issues increase in importance. This system would be a sort of
compromise, or consensus, among utilitarians and partisans of other philosophical schools. In a similar
vein Farmer and Randall (1998) show that the SMS is a common feature of agreements negotiated
among citizens with varying moral positions.
Ciriacy-Wantrup, (1952) however, based his original argument purely on discontinuities in the physical
systems being managed, or the models being used to predict those systems, rather than the
2
philosophical framework used to value them. He recommended the SMS as a management tool for
what he called “critical zone resources”, a class in which he included animal and plant species, scenic
resources and the storage capacity of ground-water basins. The defining feature of this resource class
was that it the service flow could be smoothly manipulated at certain levels, but that beyond some
critical zone a danger of collapse was introduced. Ecosystems often exhibit such threshold effects, at
least at the level of resolution with which ecologists can understand them. Acid rain and
eutrophication, for example, often leads to destruction or critical alteration of ecosystems if certain
thresholds are violated Mason (1996). In the case of acid rain, the increase in large scale mortality risk
is rapid as ph levels go below 4.5. Biologists and limnologists often treat these thresholds as
deterministic, but since the thresholds are often site specific, the actual location of the threshold in any
given ecosystem would be stochastic until a sufficiently extensive biological study of the relevant
ecosystem had been conducted and a deterministic threshold established. It will however often be a
manageable task to establish bounds where one can say that below3 such a bound, the probability of a
disaster is zero. Such a bound is here referred to as a risk threshold.
In addition to the characterization of natural scientist, risk thresholds are created by the social
organization of resource management. Even if there is no bright line in the underlying risk, an SMS
established by authorities at one level becomes a bright line for those whose compliance is being
monitored. The U.S. Clean Air Act (CAA), for example, places local authorities in danger of losing
control over economic development policy if the annual average concentration of any of several
pollutants crosses a limit set by federal law. Thresholds for private sector decision-makers are similarly
induced by any regulation which places a firm in danger of increased regulatory scrutiny or litigation
exposure conditional on an observable standard – i.e., the Accidental Release Prevention rules of the
CAA (EPA 1997).
Given a risk threshold from either natural of social sources, a manager is likely to consider using the
threshold level as an SMS. Our task in this paper is to show examine the circumstances in which this
reaction is consistent with utilitarian calculus, thus developing Ciriacy-Wantrup original argument with
new rigor. We are especially interested in knowing when an SMS is a logical reaction to uncertainty.
We show that an SMS is optimal policy if managers can put lower bounds on two parameters: the
seriousness of the disaster, and how suddenly risk appears.
3
Using the term “below the threshold” is here somewhat unprecise language and depends on the assumption that the
probability of damage is a non-decreasing function. The function being strictly increasing if the argument is larger than the
threshold. If the relevant variable is e.g. ph-levels, where more is “good”, then “above the threshold” is the correct term. For
simplicity it is assumed througout this paper that if a system is “below the threshold” then there is no risk.
3
The remainder of the paper is organised as follows. In Section 2 we present a static model of problem
with a risk threshold, and show that if a necessary condition is satisfied, the attraction of policies that
place the risky variable right at the threshold is substantial. In Section 3 we present the dynamic model
and show that the results from the static model carries over into the mathematically more complicated,
but empirically relevant, dynamic case. Section 4 includes a summary and concluding remarks.
2. A Static Model of Risk Thresholds.
In this section we consider an environmental problem where the variable creating the risk externality is
a flow variable, such as ozone emissions in a specified geographical area.4 In this case, the problem is
may be modelled in a static framework, since it is not truly dynamic, Samuelson (1947). We present a
simple model to illustrate how the optimality of a SMS may be determined.
Consider a two-good world in which society’s welfare function is W = U(x1, x2) - p(x1)A, where U is a
standard utility function, q(x1) is the probability of a disaster, and, A, the severity of. loss if a disater
occurs.5 Note that the probability of disaster depends only on the consumption of x1. Assume this
probability is zero6 up to a threshold x , and continuously increasing in x1 thereafter: q(x1) = 0 ∀ x1∈
[0, x ] and q′( x1) > 0 ∀ x1> x . Note that we have left unspecified the slope of of q(x1) when x1= x ;
whether q′( x1) is zero or positive at the threshold will make a crucial difference in the optimality of an
SMS policy, so both cases will be considered. The production possibility set is given by F(x1, x2) ≤ 0.
Because only interior solutions are of interest, assume that the slope of the production possibilities
frontier, -F1/F2, is zero at the x2 intercept and negative infinity at the x1 intercept. The planner’s
problem is thus
4
The random element with respect to ozone emissions concerns how adverse health effects from a given level of ozone
emissions depend on meteorological variables. Ozone emissions that are perfectly safe under normal conditions may have
severe consequences if e.g. the temperature goes down.
5
This is equivalent to a risk-neutral Von-Neumann-Morgenstern utility function U(x1, x2) – A. The risk neutrality
assumption allows us to put the probability term inside the utility function, which is somewhat more convenient. The same
notational shortcut is used in the dynamic model to follow.
6
An alternative approach would be to let the probability of an accident be constant and/or independent of human activity up
to the threshold and let the probability increase from thereon. This would not alter the results.
4
max W ( x1 , x2 ) = max (U ( x1 , x2 ) − q ( x1 ) A)
x1 , x2
x1 , x2
s.t. F ( x1 , x2 ) ≤ 0
(1)
 0 x<x
q( x1 ) = 
π ( x1 ) x ≥ x
The first order conditions for (1) can be derived by defining W0 as the maximum W possible when the
additional constraint is added that x1 ≤ x , and W1 as the maximum possible if x1 ≥ x . The KuhnTucker conditions for the maximization of W0 are
U1 − µ 0 F1 ≤ 0
(U1 − µ 0 F1 )( x − x1 ) = 0
(2)
and for W1
U1 − µ1 F1 − π '( x1 ) A ≤ 0
(U1 − µ1 F1 − π '( x1 ) A)( x − x1 ) = 0
(3)
Clearly maxW=max(W0,W1). As usual, these conditions are must usefully expressed as ratios which
eliminate the Lagrange multipliers (µ0 and µ1). Consider movement along the productions possibilities
frontier F(x1, x2) = 0 from the x2 intercept towards the x1 intercept. The ratio F1/F2, which is the
opposite of the slope of the frontier, increases smoothly because the production set is strictly convex. If
F U
∃ x10 < x : 1 = 1 , then ( x10 , x00 ) (where F( x10 , x20 ) = 0 ) is the unique optimum: ( x10 , x00 ) satisfies (2) and
F2 U 2
must be preferable to W1, both because W1 must lie on a lower U-indifference curve and because of the
risk of disaster.
If there does not exist such an x10 , then W0= W( x ) . From (3), W1> W( x ) only if ∃ x11 > x such that
1
1
U1 ( x11 , x12 ) − π ′( x11 ) A F1 ( x1 , x2 )
=
U 2 ( x11 , x12 )
F2 ( x11 , x12 )
(4)
It follows from the standard second-order curvature properties of U and F that the left hand side of (4)
decreases with movement towards x2, while the right hand side increases. Thus if the right hand side of
(4) is greater at x , (4) cannot hold for any x > x . Therefore, if
U1 ( x ) − π ′( x ) A F1 ( x ) U1 ( x )
(5)
<
<
U2 (x)
F2 ( x ) U 2 ( x )
the optimal solution is ( x , x2* ) , where F ( x , x2* ) = 0.
5
Condition (5) makes precise just how “often” the threshold is part of the optimal solution. While every
solution off the threshold holds only for a zero-measure subset of the well-behaved production sets,
each threshold solution holds for a positive-measure subset if π ′( x ) > 0 . Positive measure sets may, of
course, be quite small. The size of the region in production-set space for which threshold solutions hold
depends on π ′( x ) . If the onset of risk is extremely sudden (as with the thresholds created by policy)
then the optimal solution will be found at the threshold quite often. In general, unless π ′( x ) = 0 ,
threshold solutions will be somewhat more common than solutions at other points in the
neighbourhood. The intuition is perhaps best captured with a drawing. Figure 1 shows an indifference
curve, W , for an agent with an utility function given by W(x1, x2). At the risk threshold x there is a
discrete jump in the marginal substitution rate from β to α, so the marginal willingness to pay for x1
measured in units of x2, jumps downward. If the marginal rate of transformation, − F1′ F2′ lies
somewhere in the range [β , α], then implementing a SMS will be optimal.
α
β
x
Figure 1, Risk Thresholds and Kinked Preferences.
Some of the policy implications of these observations can be examined by assuming that the two goods
x1 and x2 are sold in perfectly competitive markets, and that x1 producers disregard the danger of the Adisaster. A policy-maker can in principle fix this externality by setting Pigouvian tax equal to π′(x1)
divided by the marginal utility of money. As is usual with Pigouvian taxes, the optimal tax level will
6
change with every shift in production possibilities, since the equilibrium x1 will shift. If π′(x1) at the
threshold is high, and if A is high, then an alternative policy of prohibiting x1 from crossing the
threshold is also optimal, and need not be recalculated for every little change in technology. If the value
of x1 keeps rising, there will, of course, come a point at which this policy – which is a type of SMS –
must be abandoned, because positive risk of disaster becomes optimal. The marginal rate of
transformation at this point (α in Figure 1) is thus the correct expression for what would constitute “an
intolerably high cost” as defined by Bishop and implemented in e.g. the Endangered Species Act. But
until this boundary is reached, the SMS is an easier way for a policy maker to achieve optimality, since
the Pigouvian tax would have to be recalculated with each change in the marginal utility of income.
The discussion above is summarised by the following proposition.
Proposition 1.
Consider problem (1). Assume that if we solved problem (1) with A = 0, then x1 > x would be optimal.
If π ′ ( x ) > 0 , then there exists a critical value A* such that for all A > A* optimal choice of x1 = x .
Conversely if π ′ ( x ) = 0 , then x1 = x is never optimal for any A < ∞.
„
One way of understanding Proposition 1 is that if we know or can find good estimates for the marginal
rates of substitution − U1′ U 2′ and − F1′ F2′ evaluated at the risk threshold, and we can find lower
bounds on A and π’( x ), then this is sufficient information to establish the optimality of a SMS policy.
3. A Dynamic Model of Risk Thresholds
In this section we consider the problem of a managing a resource over time. Let x(t) represent the state
of the resource the resource at time t, and u(t) actions taken that affect the resource. Risk is represented
by an event we term a “disaster”. If there is a disaster, the state variable x jumps by a quantity
g (τ , x (τ )) . The arrival of this disaster is governed by a Poisson process with intensity Λ( x)
Pr ( τ ∈ (t , t + ∆t ) | τ > t ) ≈ Λ ( x )∆t
 0 if φ ( x (t )) < 0
Λ( x) = 
λ ( x ) if φ ( x (t )) > 0
Otherwise, x(t) evolves according to a the law of motion
7
(6)
x = f ( x, u )
(7)
from a starting point denoted x(0) = x0.
The risk threshold in this model is defined by φ(x(t)), with φ(x(t))<0 being the safe side. The simplest
sort of threshold, in which there is a time-invariant vector x separating the safe side from the
dangerous side, is defined by φ(x(t))= x -x(t) ∀t. The intensity λ(x) is assumed everywhere continuous
and increasing7 except possibly when φ(x(t)) = 0.
The socially optimal path of resource use maximizes
E
(∫
∞
0
f 0 ( x, u ) e − rt dt
)
(8)
where f0 is the instantaneous utility from x and u. The problem defined by (6)-(8) is best envisioned as a
sequence of problems, where one solves a deterministic optimal control problem whenever ϕ(⋅) < 0 and
a piecewise deterministic problem whenever ϕ(⋅) > 0. Figure 2 illustrates this breakdown for a timeinvariant risk threshold. The horizontal line x is the risk threshold, that divides the state-space into
subsets. We will refer to state spaces with risk thresholds as mixed-risk state spaces. The three paths
illustrated show three different cases. x1 (t ) is always below the threshold. If this path is optimal, the
only optimal control method required to derive it is standard deterministic optimal control. If the path
x3 (t ) is optimal, then this path is determined by piecewise deterministic optimal control. If the path
x2 (t ) is optimal, this follows from applying standard deterministic control on the intervals [A, B], [C,
D] and [E, ∞) and piecewise deterministic control in the intervals [B, C] and [D, E].
7
Assuming λ to be increasing implies that more of x is bad. If x was ph value in a lake threatened by large scale fish
mortality due to high acidity we would have to turn the definition around in an obvious way.
8
x(t )
x 3(t )
A
B
E
D
C
x
x 2 (t )
x 1 (t )
t
Figure 2, State-Variable Paths through a Mixed-Risk State Space
Necessary conditions for optimality in the deterministic intervals are well known. Necessary conditions
for the piecewise deterministic intervals are less well known, and are given in the appendix. The
remaining question is the behaviour of the solution at the times the threshold is crossed. The state
variable x, of course, simply follows the differential equation unless the disaster occurs. The behaviour
of the co-state variable is potentially more subtle. The problem at hand is an isoperimetric problem,
since we require x(s) to be on a certain surface. Such problems often have jumps in the co-state
variables when t = T. In the appendix we show that whenever the threshold is crossed from a state
where ϕ(⋅) < 0 to a state where ϕ(⋅) > 0, the co-state jump according to the following equation.
1
p (T − ) − p (T + ) = − ( f 0+ − f 0− ) e − rT + p (T + )( f + − f − )
f
(
(
+ λ x (T
+
)) ( J (T
+
, x + g ( x ) | τ ) − J (T , x )
+
))
(9)
where T is the time when the threshold is crossed; p(⋅) is the co-state; and J denotes the criterion
function evaluated at T. That is, J(T, x) = maxu E
(∫
∞
T
f 0 ( x (t ) , u ) e − rt dt
) s.t. x(T) = x . Superscript +
indicates that the function is evaluated at the limit from above and superscript − indicates that the
9
function is evaluated at the limit from below. Thus J (T + , x + g ( x ) | τ ) − J (T + , x ) is the loss from the
disaster if it happens at time T.
The expression for the case where the state goes from ϕ(⋅) > 0 to a state ϕ(⋅) < 0 is given by
(( f
+
0
−λ x (T
−
p (T − ) − p (T + ) =
1
f−
(
− f 0− ) e − rT + p (T + )( f + − f − )
)) ( J (T
−
, x + g ( x ) | τ ) − J (T , x )
−
(10)
))
The central insight from (9) and (10) is that there may be a jump in the co-state variable when x(t)
crosses from one side of ϕ = 0 to the other. As always, the co-state variable may be thought of as the
shadow cost in terms of current and future welfare incurred when an additional unit of the state
variable is made freely available. The discontinuous shift in this shadow cost at the threshold is an
indication, although not a proof, that the intuition of the static example carries over to the dynamic:
sometimes one should reach the threshold then stop, since the price of going further jumps, but this is
never the case if the onset of risk is slow in the sense that if λ ′ ( x ) = 0 at the threshold .
If an SMS is applied then x = 0 for all t after the threshold has been reached. For this case (9) and (10)
are not well defined, because their derivation involves division by x (T + ) = 0. When an SMS is
applied, the co-state will jump according to
p (T
−
) − p (T ) =
+
(
(
)) (
(
f (T , x (T ) , u (T , x (T ) , p (T )))
f 0 T + , x (T + ) , u T + , x (T + ) , p (T + ) − f 0 T − , x (T − ) , u T − , x (T − ) , p (T − )
−
−
−
−
))
−
(11)
That is, the jump in the shadow cost of resource damage is the jump in instantaneous social welfare
divided by the speed with which the state variable is changing as the threshold is approached.
We now turn to the question of when an SMS is optimal. To dispose of the trivial problems, assume x0
is on the safe side of the threshold and that without considering the disaster the optimal path crosses the
threshold. One possible response to the threshold is simply to require that φ(x(t)) ≤ 0 ∀t. This is
essentially what we shall mean by an SMS. 8 A policy obeying a SMS would have to look something
8
Strictly speaking, a slightly more general definition is possible and, as argued below, in fact required if necessary
conditions are to hold. Defining Ψ(t) = λ(x(t)) if φ(x(t)) > 0 and Ψ(t) = 0 if φ(x(t)) < 0. A path will be considered to satisfy a
SMS from the time s if and only if the Lebesque integral
∞
∫s Ψ (t ) dt . This means is that we may allow the threshold to be
violated as long as the violations are not large enough to give positive probability to the possibility that the disaster occurs.
Thus an SMS can include a policy of forcing the system back below the threshold as soon as it is seen to have crossed,
10
like Figure 3. In the interval [0, s) the system is approaching the risk threshold. In the interval [s, ∞) the
SMS is binding.
x(t )
x
t
T
Figure 3, State - path satisfying SMS.
For clarity we derive our results within a parameterised model. Let the equation of motion
be x = f ( x) = u − δ x ; let the constant A be the cost of the disaster (in utility terms) if it occurs; and let
the intensity of the disaster process be the quadratic function λ(x) = λ 0 ( x − x ) + λ1 ( x − x ) , where λ1
2
is strictly positive. The instantaneous cost of reducing u is given by
c
2
(u0 – u) 2 . We can apply the
principle of optimality. If a path similar to the one drawn in Figure 3 is optimal, we will get the same
path if solve the problem over the time interval [T, ∞) with x(T) = x . We therefore first solve the
problem when the state variable is exactly at the threshold:
provided the observation and reaction is instantaneous. An infinite number of such jumps may be allowed, and will in
ceratin circumstances be required if a path is to observe necessary conditions. This point is further discussed below.
11
(
)
 ∞

2
max  E ∫ − 2c (u 0 − u ) − a e − rt dt 
u
 T

s.t.: x = u − δ x, a = 0, x (T ) = x , a (T ) = 0, a (τ + ) − a (τ − ) = A
(
(12)
)
Pr (τ ∈ (t , t + ∆t ) | x ≥ x ) ≈ λ ( x ) ∆t = λ 0 ( x − x ) + λ 1 ( x − x ) ∆t
2
All constants are assumed strictly positive except λ0 the sign of which does not matter.9 It is assumed
that the shape of λ is such that λ′ ≥ 0 for all feasible values of x. The Maximum principle for piecewise
deterministic problems is given in the appendix. Using (31) to derive the first order conditions gives
p
(13)
u = u 0 + x e − rt
c
(14)
p x = δ px + λ ( x ) ( px − px (t | t > τ )) + λ ′ ( x ) ( J (t , x | a = 0 ) − J (t , x | a = A ))
x = u 0 +
p − rt
e −δ x
c
(15)
px (t ) is the present value co-state variable for x.10 J (T , x | a = A ) is the criterion evaluated from time t
given that the disaster has occurred and J (T , x | a = 0 ) is the expected value of the criterion given that
the disaster has not occurred. If a path satisfying (14) and (15) is to stay exactly at the risk threshold for
all t > s, then we can characterise the solution as follows. Let µ x (t ) = px (t ) e rt be the current value costate. If the optimal solution is going to be exactly at the risk threshold, then there is some instant T,
such that x(t) = x ∀ t ∈ [T, ∞). If x is equal to x for all t > s, then x = 0 and x = 0. This implies that
µ x = 0 . Also, the criterion function value without the disaster, J (T , x | a = 0 ) , and with the disaster,
J (T , x, A ) , are deterministic integrals. J (T , x | a = 0 ) is deterministic since the solution is assumed to
stay at the threshold and λ( x ) = 0. J (T , x | a = A ) is deterministic since this is the pay-off after the
disaster has occurred and if the disaster occurs, there is no more randomness in the problem. These
expressions are given by:
2
c
(16)
J (T , x | a = 0 ) = − (u 0 − δ x ) e − rT
2r
and:
A
(17)
J (T , x | a = A ) = − e − rT
r
9
The parameterisation of λ(x) may be considered resulting an appropriate Taylor expansion of the underlying risk structure.
10A
co-state variable for a(t) may also be computed, but does not affect the solution and has no important economic
interpretation. Discussion of this variable is therefore omitted.
12
For all t ∈ [T, ∞) we get the quantities J (t , x | a = 0 ) and J (t , x | a = A ) by substituting T with t in (16)
and (17). Using these results allow us to rewrite (14), the differential equation for p x , as
Solving for µ x gives
2
A
 −c
0 = ( r + δ ) µ x + λ 1  (u 0 − δ x ) + 
r
 2r
µx =
2
λ1  c 0
A
 (u − δx ) − 
r
( r + δ )  2r
(18)
(19)
µx is the current value shadow price on x consistent with a SMS being optimal. Inserting from (19) into
the expression for x gives
x = u 0 +
2
A
λ1  c 0
 (u − δx ) −  − δx = 0
c ( r + δ )  2r
r
(20)
Solving (20) with respect to A yields a critical value of A which we denote A* .
A* =
cr (u 0 − δx ) ( r + δ )
λ
1
+
c 0
(u − δx )
2
(21)
If A is known to be equal A*, then an SMS at the threshold is optimal policy. Note that if λ1 → 0 then
A*→ ∞. Interestingly, λ0 does not even enter the expression for A*. No matter how large λ0 is, if λ1 is
zero, the derivative intensity function λ(x) is exactly zero at the threshold and it is optimal accept some
risk. This finding mirrors the static case where SMS would only be optimal if the derivative of the
probability of disaster was strictly positive at the threshold.
Suppose, then, that a regulator knows all parameters in the model with certainty except A and λ1, and is
able to put lower bounds on A and λ1. Denote those lower bounds AL and λL1 . It is clear that if
AL ≥
c (u 0 − δ x ) ( r + δ )
λ
1
L
+
c 0
(u − δ x )
2r
(22)
then it will always be optimal to let the SMS be binding from some point in time. We thus get a result
that is qualitatively similar to the static case. If lower bounds can be established for the relevant
parameters A and λ1 and these obey (22), then a SMS policy will be optimal. Note that (22) is a
sufficiency condition, but not a necessary condition for a SMS to be binding. If a regulator is able to
form a probability distribution over possible values of A, then a SMS is optimal if E(A) ≥ A* even if A
> A* with some probability.
13
It is interesting to note that calculating the optimality of a SMS is requires less information than what is
needed to calculate the optimal path if we know that SMS is not optimal, e.g. if λ1 = 0. For that, we
either need to know A (and all the other parameters) exactly or we must at least be able to form
meaningful probability distributions over the values that they may be able to take. One context in
which this insight may be usefully applied is one in which a private agent has superior information, but
the cost of the disaster is borne externally – i.e., a land-owner with endangered species habitat. This
observation is really what links the utilitarian theory in this paper to the SMS as practiced and
defended, as a response to uncertainty and irreversibility. What we have derived is one structure of
uncertainty – perhaps not only one – which can justify the application of an SMS on purely utilitarian
grounds. This is true Frankian uncertainty: the probability of disaster is not known, although a lower
bound for it is.
There is a technical issues concerned with how to characterise an optimal solution if we know that A >
A*. That is, the cost of a disaster is strictly larger than the minimum cost required for the optimality of
SMS. If this is the case, then a path with x = 0 does not, strictly speaking, satisfy necessary conditions.
The optimal x (T + ) < 0 and will therefore be bounded towards the no-risk region. But as soon as we
enter the no risk region any path that satisfies necessary conditions is bounded towards the risk region.
This is resolved by letting the control be a chattering control, i.e. a control that is discontinuous at
every point in time after the threshold has been reached. See Zelikin and V.F. Borisov (1994) for an
extensive discussion of chattering controls. The control will jump according to (9) and (10) in such a
way that whenever x > x , then x < 0 and vice versa. The SMS will then be observed in the sense
discussed in footnote 7. This is obviously an impractical control, but in the present context there is
little loss from ignoring this issue when the threshold is reached and simply setting u = δ x at the
threshold and thus keeping x at the threshold forever. However, see Nævdal (2001) for a deterministic
model where chattering is a fundamental and unavoidable property of the control.
Optimal Policies Prior to Reaching the risk Threshold
So far we have examined the path the optimality of a path satisfying a SMS from the time the system
reached the risk threshold. If A and λ1 are known with certainty to have values such that (21) holds
with equality, this would be a straightforward procedure. A jump in the adjoint variable could be
calculated according to (11) and p(T-) be solved from:
.
14
2
− p (T − ) +
2
A
λ1  c 0
 (u − δx ) −  =
r
( r + δ )  2r
2
A  − rT  1
1  λ1  c 0
−
− rT
− 
 (u − δx ) −  e  + P (T ) e
c  ( r + δ )  2r
r
 c
u −
0
P (T − )
c
e
− rT
(
)
2
(23)
− δx
Calculating the u(t), x(t) prior to T and T itself would be a straight forward, if algebraically tedious,
problem. However if we base the optimality of a SMS on lower bounds derived from (22) and do not
have any more information about A and λ(x), then we cannot calculate the optimal path prior to
reaching the SMS. In this case we must form expectations about relevant parameters, no matter how
vaguely founded, and approach the risk threshold as best we can. It should however be noted that in
most cases where a SMS is considered as a policy option, optimal policies prior to reaching the risk
threshold is generally considered a problem of secondary importance if discussed at all.
Uncertainty About the Location of Risk Threshold
So far the paper has assumed that the location of the risk threshold, x , is known to the regulator. In
many cases this is not the case. It could be that the definition of Λ(x) would have to be modified to take
account for a stochastic x with support [ xL , xH ] ⊂ [0, ∞). It turns out that the results derived above
can be modified if we replace Λ(x) (and therefore also λ(x)) with a modified risk process given by
( x )∆t
Pr ( τ ∈ (t , t + ∆t ) | τ > t ) ≈ Λ
 0 if φ ( x (t )) < 0
( x) = 
Λ
λ ( x ) if φ ( x (t )) > 0
(24)
( x ) = λ 0 ( x − x )2 + λ1 ( x − x )
Here φ(x) = x(t) – xL and λ
L
L
Are Necessary Conditions for Safe Minimum Standards ever
fulfilled?
The empirical relevance of the model must yet be verified. The most important necessary condition for
the optimality of SMS is that q′ ( x ) > 0 in the static model or λ′ ( x ) >0 in the dynamic model. As we
have shown, if these conditions are not satified, then it is optimal to assume at least some risk. If these
conditions do not hold for any environmental problem, then SMSs are never optimal. We present an
example where this condition does in fact hold to show that SMSs may be optimal in real world
15
problems. Consider the case where a regulator is concerned that the arrival of ships to a particular
represents an environmental hazard. It may be that the arrival of boats brings the risk of introducing
invasive species, as boats departing from the Caspian sea did when they introduced the Zebra mussel
into the great lakes or there may be a risk of oil spills from decrepit oil tankers. Let 1 - θ be the risk of
an accident accociated with the arrival of a ship. Assuming that that ships are homogenous and that
accident risk is independent of the number of ships, the probability of an accident if there are x ships
entering is Θ = 1 - θx. Figure 4 shows a graph of Θ as a function of x. The figure is drawn for θ = 0.99.
Θ
x
Figure 4, Aggregate accident risk as a function of the number of arriving ships.
Figure 4 shows that Θ is a concave function of x. If we allow ourselves to treat x as a continuous
variable and calculate a second degree Taylor series of Θ we find that Θ ≈ (-lnθ)x + (-½ln2θ)x2. Note
that the approximation is a special case of λ(x) in the dynamic model above with x = 0, λ0 = (-½ln2θ)
and λ1 = ( d Θ dx )x = 0 = (-lnθ). The important fact to note here is that λ1 = (-lnθ) > 0 for all θ > 0. Thus
the necessary condition for the optimality of SMS is satisfied in this case. Of course, λ1 > 0 is only a
necessary condition, but the fact that it holds for this risk structure implies that SMS should be
16
considered potentially an optimal policy whenever we deal with environmental problems where the
problem is invasive species or oil spills due to shipping.
Discussion and Conclusions
The paper has analysed the likelihood of a Safe Minimum Standard (SMS) being optimal policy in
environmental regulation. Our most significant result is that the use of SMS can lead to optimal
resource management only if the first derivative of the risk function (probability in the static case,
intensity in the dynamic) is strictly positive at the threshold. This result makes intuitive sense - a
threshold at which risk begins to rise at rate zero is almost no threshold at all. These are the first set of
results to show that there are any circumstances in which an SMS can be dictated by a conventional
social-welfare maximization criterion, rather than the minimax or non-utilitarian arguments resorted to
by earlier authors. An important feature is that little knowledge about the nature of a disaster is
required. If lower bounds for the relevant unkown parameters can be established, one can calculate
whether sufficiency conditions for the optimality of a SMS hold. Also, as mentioned in the
introduction, such stark thresholds are created often by federal systems. Thus, it may well be optimal
for a state regulator to impose an SMS on local polluters given that the federal government has created
an air-quality threshold, even if it is not the case that the threshold approach was the best for the federal
authorities to have taken in the first place.
It is somewhat paradoxical that the informational requirements to calculate that a SMS is optimal are
less than the requirements to calculate the optimal path given that the SMS is not optimal. It may be the
case that the true optimal path is very close to a SMS, even if it can be established that the SMS is not
optimal. In this case, one may argue that there are excessive research costs associated with deriving the
optimal policy and that, when these costs are considered, it is optimal to implement a SMS rather than
implementing a risky policy based on flawed data. Also, one should not discount the alternative
arguments for SMSs, such as those presented by Farmer and Randall (1998) or Rolfe (1995).
Appendix: Optimal Control in Mixed-Risk State Spaces.
This section briefly reviews some mathematical results needed for the problem at hand.
Deterministic Control Problems
The problem Ad(t0, x(0), φ, S) is defined by:
17
J (t0 , x (0 )) = max ∫ f 0 (t , x, u ) dt + S (t1 , x (t1 ))
t1
u∈U ,t1
t0
(25)
s.t.: U ⊆ , x ( 0 ) ⊆ n , x = f (t , x, u ) , φ ( x (t1 )) = 0
m
Necessary conditions (see Seierstad and Sydsæter (1987) for details) are usually given in terms of the
Hamiltonian, H(t, x, u, p) = f0(t, x, u) + f(t, x, u). They are:
∂
u ( x, p ) = arg max H (t , x, y, p ) p = − ( H (t , x, u, p )) x = f (t , x, u ( x, p ))
(26)
∂x
y
The transversality condition and the equation determining optimal t1 are given by:
pi (t1 ) = S x′ (t1 , x (t1 )) + γφi′ ( x (t1 )) H (t1 , x, u ( x, p ) , p ) = − St (t1 , x (t1 ))
γ is a scalar. The following properties are well known and are often used in economics:
∂J (t0 , x (t0 ))
= p ( t0 )
∂x (t0 )
∂J (t0 , x (t0 ))
∂t0
(
= − H t 0 , x ( t0 ) , u ( x ( t 0 ) , p ( t 0 ) ) , p ( t 0 )
(27)
(28)
)
Piecewise Deterministic Problems
Here we consider a problem Qs(t0, x(0), φ, S) given by
J (t0 , x (t0 )) = max E
u∈U ,t1
(∫
t1
t0
f 0 (t , x, u ) dt + S (t1 , x (t1 ))
)
s.t.: U ⊆ m , x ( 0 ) ⊆ n , x = f (t , x, u ) , φ ( x (t1 )) = 0
(
−λ x t
τ ~ λ ( x (t )) e ( ( )) on (t0 , ∞ ) , x (τ + ) − x (τ − ) = g τ , x (τ − )
(29)
)
Necessary conditions for problems like Qs(t0, x(t0), φ, S) can be found in various forms. Note that the
random process determining τ obeys (6). We will present conditions due to Seierstad (2001). Nævdal
(2000) has shown that the conditions derived by Seierstad can be stated in terms of a Risk-Augmented
Hamiltonian, defined by:
(30)
H r = f 0 (t , x, u ) + pf (t , x, u ) + λ ( x ) J (t , x + g ( x ) | τ ) − J (t , x )
(
)
Here J(t, x + g(x)|τ) is the criterion to a problem of type Qd(t, x + g(x) , 0, 0). Jr(t, x) is the objective
function to a problem Qr(t, x, φ, S). Applying the Maximum principle to Hr gives the following
conditions.
18
u ( x, p ) = arg max H r (t , x, y, p )
p = −
y
∂
( H r (t , x, u, p )) x = f (t , x, u ( x, p ))
∂x
(31)
The transversality condition and the equation determining t1 are given by:
pi (t1 ) = S x′i (t1 , x (t1 )) + γφi′ ( x (t1 )) H r (t1 , x, u ( x, p ) , p ) = − St (t1 , x (t1 ))
(32)
Here γ is some real number roughly corresponding to a Lagrange multiplier on the constraint ϕ(x(t1)) =
0. By using the results ∂J d (t , x + g ( x ) | τ ) ∂x = p (t; t , x + g ( x )) (1 + g ( x )) and that ∂J r (t , x ) ∂x
= p (t )
p = −
∂f 0
∂f
− p + λ ( x ) p − p (t , x + g ( x ) | τ ) ( I n − g ′ ( x ))
∂x
∂x
(
(
+ λ ′ ( x ) J ( t , x ) − J (t , x + g ( x ) | τ )
)
(33)
)
Here In is the n-dimensional identity matrix. The following results will be used later:
∂J r (t0 , x (t0 ))
= p ( t0 )
∂x (t0 )
∂J r (t0 , x (t0 ))
∂t0
(
= − H r t0 , x ( t0 ) , u ( x ( t0 ) , p ( t0 ) ) , p ( t0 )
(34)
)
There is a conceptual problem with (33). J (t , x + g ( x ) | τ ) is easy to calculate but J(t, x) is still
undefined. Seierstad has shown that J(t, x) is given by the solution to the differential equation:
z ′ ( s; t , x (t )) = − f 0 ( s , x , u )
(
)(
(
+ λ x ( s; t , x ( t ) ) z ( s; t , x ( t ) ) − z s , t , x + g ( t , x ( t ) ) | τ = t
))
(35)
Equation (35) must be coupled with an end point condition z(T) = S(T, x(T)). Then J(t, x) = z(t; t, x).
Jump Conditions for the Adjoint Function at the Risk Threshold
In this section we show how the Maximum Principle can be used to “glue” together problems like Qr(t,
x, φ, S) and Qs(t0, x, φ, S) in order to solve problems like (25). This is less straightforward than it would
appear, since, depending on the shape of λ(x), p may have discrete jumps whenever the risk threshold
is crossed. We first examine the case where the state variables cross from a no-risk regime to a risk
19
regime. To avoid lengthy algebra the jump condition is proved for x ∈ . Let T be a point in time
when the threshold is crossed. Define T − = limt →T − t and T + = limt →T + t . Also define the following
(
)
(
)
(
)
= f (u (T ) , x (T )) . By combining equations (27), (34) it is clear that the conditions in (27) and
abbreviations: f 0− = f 0 u (T − ) , x (T − ) e − rT , f 0+ = f 0 u (T + ) , x (T + ) e− rT , f + = f u (T + ) , x (T + ) and
−
f−
−
+
−
(34) may be rewritten
(
p (T − ) = p (T + ) + γφ ′ ( x (t ))
)
(36)
(
H d T − , x (T − ) , u (T − ) , p (T − ) = H r T − , x (T − ) , u (T − ) , p (T − )
)
(37)
Treating (36) and (37) as two equations with two unknown variables, p (T − ) and γ,
enables us to form the following expression.
1
− p (T + ) + p (T − ) = − f 0+ − f 0− + p (T + )( f + − f − )
f
(
(
+ λ ( x ) J (T , x + g ( x ) | τ ) − J (T , x )
+
+
))
(38)
It is straight forward to verify that the in the case where x ∈ n , the jump in the co-state variables are
given by:
(
− p (T + ) + p (T − ) = µ f 0+ − f 0− + p (T + )( f + − f − )
(
+ λ ( x ) J (T , x + g ( x ) | τ ) − J (T , x )
+
+
Here µ is a n×1 vector with each element µj determined by:
∂φ
∂x j
µj =
, j = 1, 2,
∂φ
−
−
−
f t , x (T ) , u (T ) , T
∂x
(
)
))
", n
(39)
(40)
The denominator is a dot-product. Applying the same method on the case where one moves from a
regime with risk to a regime without risk, it can be shown that the appropriate equation determining the
jump in the adjoint function is given by:
(
− p (T + ) + p (T − ) = µ f 0+ − f 0− + p (T + )( f + − f − )
(
−λ ( x ) J (T + , x + g ( x ) | τ ) − J (T + , x )
20
))
(41)
with µ still determined by (40). Note that the conditions here presented are only necessary conditions.
They can be used only to determine how and when jump in the adjoint function should look like.
Sufficiency conditions turns out to be not very useful, since in addition to require that J() satisfy certain
curvature properties, require that we check all possible combinations of the state variables crossing the
threshold. Note from (39) and (41) that if λ(x) = 0 at the threshold (when φ(x) = 0) then setting
p (T + ) = p (T − ) will always be a solution. This may however not be the correct solution. If solving
(39) or (41) gives rise to several solutions for p (T − ) , these solutions must be compared analytically or
numerically for optimality. This is particularly important in the context of SMSs, since a path satifying
a SMS in such a way that the SMS is binding from some time T is not compatible with a continuous
adjoint variable. To see this, assume that we go from a no-risk zone to the risk threshold and then stay
there forever as one would if a SMS policy is optimal. If we insert the solution p (T + ) = p (T − ) into
(38), we see that it fits, and therefore the control is continuous at the threshold. But if the control is
continuous at the threshold, then f − = f + = 0 and from (38) we see that this implies that we have
divided by zero which is a contradiction. Thus we must look at the equations in (36) and (37) again and
assume that f + = 0. Assuming that λ( x ) is continuous at the threshold means that we can write the
jump in the co-state as:
p (T
−
) − p (T ) =
+
(
(
)) (
(
f (T , x (T ) , u (T , x (T ) , p (T )))
f 0 T + , x (T + ) , u T + , x (T + ) , p (T + ) − f 0 T − , x (T − ) , u T − , x (T − ) , p (T − )
−
−
−
−
−
)) (42)
Equation (42) is the equation used in the text.
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Boyle, K. J. and R. C. Bishop (1987). Valuing Wildlife in Benefit-Cost Analyses: A Case Study
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21
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Farmer, M.C. and A. Randall, (1998), “The Rationality of a Safe Minimum Standard,” Land
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