the role of spatial scale in the timing of uncertain environmental policy

Contents lists available at SciVerse ScienceDirect

Journal of Economic Dynamics & Control

Journal of Economic Dynamics & Control 36 (2012) 369–382

0165-18

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jedc

The role of spatial scale in the timing of uncertainenvironmental policy

Charles Sims a,n, David Finnoff b

a Department of Applied Economics at Utah State University, 3530 Old Main Hill, Logan, UT 84322, USAb Department of Economics and Finance at the University of Wyoming, 1000 E. University Avenue, Laramie, WY 82071, USA

a r t i c l e i n f o

Article history:

Received 1 November 2010

Received in revised form

16 May 2011

Accepted 2 September 2011Available online 17 September 2011

JEL classification:

D81

H43

Q58

Keywords:

Reflecting barrier

Brownian motion

Irreversibility

Real options

Spatial boundary

89/$ - see front matter & 2011 Elsevier B.V. A

016/j.jedc.2011.09.001

esponding author. Tel.: þ1 435 797 3863; fa

ail address: [email protected] (C. Sims).

a b s t r a c t

The spatial scale of an environmental problem is dictated by boundaries. Physical

boundaries limit the extent of impacts while the scale of decision making creates

perceived boundaries beyond which impacts are ignored by decision makers. While it is

well understood that uncertainty and irreversibility will alter policy decisions aimed at

alleviating environmental impacts, the effect of spatial scales, both physical and

perceived, is less understood. When spatial scale is included in a real options model

of environmental policy adoption results indicate that the importance and influence of

spatial considerations depends on the level of uncertainty, stringency of the proposed

policy and flexibility of the policy decision. Recognizing spatial scale may force policy

adoption to take place within a window of current damage. When spatial scale is small

or uncertainty high, this window for policy adoption can close precluding policy

adoption entirely. This undermines well-known results demonstrating that changes in

uncertainty will only alter the timing of policy adoption. In other instances, the policy

adoption window remains open but the option value increases faster than the benefits

of the policy creating a scenario where it is always preferable to delay. Here the

inclusion of an option value can prevent adoption of policies that would be adopted

according to traditional cost-benefit analysis. In general policy decisions will be most

affected by spatial considerations when the spatial scale is small, damage is spreading

fast, and the uncertainty in damage spread is high.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

The importance of uncertainty and irreversibility in the design and implementation of policies that mitigateenvironmental damage has been well recognized in the real options literature (Fisher, 2000; Pindyck, 2007). Thisapproach typically assumes a constant sunk cost of implementing a policy that alters a stochastic damage-causing processsuch as global warming (Conrad, 1992, 1997), environmental pollutants (Pindyck, 2000, 2002; Saphores, 2004; Saphoresand Carr, 2000), biodiversity loss (Kassar and Lasserre, 2004), and pest species (Saphores, 2000; Saphores and Shogren,2005). Policy adoption is deemed optimal when damage reaches a critical threshold that incorporates uncertainty andirreversibility, leading to results that differ from traditional cost-benefit analysis. Pollutant concentrations or speciespopulations are typically the only source of uncertainty considered in a spatially independent policy context evaluated ata single scale of decision making.

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dx.doi.org/10.1016/j.jedc.2011.09.001

C. Sims, D. Finnoff / Journal of Economic Dynamics & Control 36 (2012) 369–382370

However, environmental policies are often spatially dependent (Costello and Polasky, 2008; Gaudet et al., 2001;Geoghegan and Gray, 2005; Sanchirico and Wilen, 1999) and encompass multiple scales of decision making (Peltzman andTideman, 1972; Stein, 1971). Spatial dependencies arise when the benefits and costs of environmental policies areinfluenced by the current and potential location of environmental damage. In this case uncertainty is rooted in the abilityto predict where damage will occur in the future. For instance, the dispersal of local, regional, and global pollutants islargely random and involves different zones of influence while invasive species migration is uncertain but bounded by theextent of suitable habitat and possible migration barriers. Since ecological and anthropogenic delineations of space rarelycoincide, physical boundaries may be limited further by the scale of decision making. Water (air) pollution reductionpolicies implemented at a local scale convey uncompensated benefits to those downstream (downwind). Those livingwithin the potential range of an invasive species benefit from the control decisions of currently invaded individuals.Decision makers will naturally ignore these spillover effects creating a perceived boundary dictated by the scale of decisionmaking. Questions of where and to whom damage occurs become key factors in policy decisions, yet are largely neglectedin the real options literature.

The purpose here is to evaluate the timing and stringency of policies that mitigate uncertain environmental damagespreading across a bounded spatial domain. This requires a novel space-based specification of uncertainty that is capableof incorporating physical limits to space and the scale of decision making. In this context the relevant state variable issimply the area impacted by environmental damage. The current area affected is known but its future spread is stochasticfollowing a geometric Brownian motion process subject to an upper barrier. The barrier corresponds to the spatial limit orscale of the problem. Such an approach assumes finite environmental damage yet allows for the possibility of an analyticalsolution. While it has been noted that the omission of upper bounds can bias decision making (Forsyth, 2000; Saphoresand Shogren, 2005; Willassen, 1998), results have been confined to numerical solutions with no derivation of the bias or itsconsequences. Here the bias is analytically derived providing an intuitive explanation of the consequences of boundedenvironmental damage in real options models.

Our results indicate spatial scale influences policy making depending on the stringency of the policy and the flexibilityin the policy decision. First, spatial scale delays adoption in inflexible policy decisions where policy stringency is given.However, if decision makers have some control over the design of policies they are optimally timing, spatial scale alsocreates an incentive for less stringent policies to be adopted more immediately. Second, the inclusion of spatial scalecreates an upper threshold on current damage beyond which the policy becomes obsolete. If environmental damageremains undetected, this upper threshold may be reached in which case the optimal strategy switches from adopting somedamage reducing policy to abandoning the policy and allowing damage to spread to a boundary. This upper threshold hasbeen absent in previous research and places the detection of environmental damage in a key role. Finally, increases in therate and uncertainty of spread as well as decreases in spatial scale may preclude policy adoption entirely. In short a criticalthreshold does not exist for certain combinations of model parameters. This is in contrast to previous literature whichfinds changes in model parameters only influence the timing of policy adoption. However, in the more flexible policydecision where stringency is also chosen, a critical threshold always exists. In general bias resulting from omission ofspatial considerations will be greatest when the spatial scale is small, the spread rate is rapid, and uncertainty in futurespread is high.

The remainder of the paper is organized as follows. Section 2 introduces a spatially dependent policy decision. Section 3outlines the impact of spatial scale on the policy decision and considers the optimal timing of a policy that (i) reverses orstops the accumulation of damage and (ii) slows the accumulation of damage. Section 4 considers the more flexible butcomplex scenario where decision makers can choose the timing and stringency of environmental policy. Section 5discusses the results and concludes.

2. Spatially dependent policy

Consider a permanently established damage-causing process that is uniformly spreading from an origin. Examplesinclude invasive species spreading from an introduction point, pollutants dispersing from a source, habitat loss emanatingfrom a city center, or disease spreading from an infected population. While current affected area A(t) is known withcertainty, future expansion ðdAÞ is unknown and evolves according to a geometric Brownian motion (GBM) process:

dA¼ r0AdtþsAdz ð1Þ

where r0 is the rate of expansion absent any intervention, s is the standard deviation coefficient, and dz is the increment ofa standard Weiner process. Eq. (1) links the temporal and spatial scale in the model. Larger values for r0 imply fasterexpansion corresponding to a more mobile species, expedient fate and transport of pollutants, or contagious disease.Larger values of s correspond to a larger amount of uncertainty in future spread. The generality of GBM is attractive but italso has two particularly desirable properties for many spatially dependent environmental applications. First it rules outnon-positive levels of affected area. This recognizes that complete clean-up of an environmental pollutant, restoration ofdamaged habitat, or eradication of an invasive species or disease is unlikely. Second it allows spread to accelerate which isoften the result of increased human interactions as affected area grows.

Eq. (1) assumes the spread of the damage-causing process is unbounded, which ignores any relevant spatial scale. Ifspread is confined to A by the minimum of either physical boundaries (spatial extent of watershed, airshed, suitable

C. Sims, D. Finnoff / Journal of Economic Dynamics & Control 36 (2012) 369–382 371

habitat, or susceptible population) or perceived boundaries (county, state, or national borders) the stochastic process mustbe bounded. In other applications the logistic Brownian motion and Ornstein–Uhlenbeck processes have been used toincorporate natural limits to an unknown harvestable population (Saphores, 2003) and fund pollutant (Pindyck, 2000).However, for the spatial boundaries we consider these specifications have the undesirable properties that the processslows before reaching a boundary and may exceed the boundary. These specifications also require complex numericalsolution procedures.1 In our context a more appropriate specification treats A as an upper barrier. In this setting spatialboundaries do not affect the stochastic process prior to reaching the boundary.2

In accordance with previous literature (Pindyck, 2000; Saphores and Carr, 2000), the dollar value of environmentaldamage is assumed to have the following convex form:

DðAÞ ¼ gAðtÞy ð2Þ

where y41 and g40. While current damage is known with certainty, future damage is unknown due to uncertaintysurrounding future affected area. Through Ito calculus, a similar GBM process for damage can be found:

dD¼ a0DdtþsDdz ð3Þ

where the drift and variance terms of the damage process are related to those of the affected area process bya0 ¼ yr0þð1=2Þyðy�1Þs2 and s¼ ys. The spatial scale of the problem creates an upper bound D¼ gA

yon damage.

To mitigate damage a decision maker can implement a policy that instantly and permanently reduces the rate ofaffected area expansion from r0 to rp. A policy which results in 0orpor0 will permanently slow spread, a policy that yieldsrpo0 will permanently reverse spread, and a razor’s edge policy (rp

¼0) halts spread. The magnitude of reduction (orpolicy stringency) may be taken as given by decision makers or chosen jointly with the timing of policy adoption. Onceadopted, damage evolves at a lower rate ap ¼ yrpþð1=2Þyðy�1Þs2. lowering expected damage. Due to uncertainty, thepolicy must sufficiently reverse the spread rate ðrpo�ð1=2Þðy�1Þs2Þ in order to reverse the accumation of damage ðapo0Þ.Many environmental problems result in irreversible damage. In this case, the post-policy rate of spread is constrained torp4�ð1=2Þðy�1Þs2.

This policy possesses two common characteristics. First, the policy may be adjusted infrequently once enacted. Ratherthan making arbitrary assumptions about the frequency of adjustment (which may or may not be endogenous) we assumethere is only one opportunity to adopt the policy (Conrad, 1997; Pindyck, 2000; Saphores, 2004). Second, adopting thispolicy would impose an irreversible flow of sunk costs on society. For instance, a policy aimed at controlling pollutionemissions may require emitting firms to alter their production process or invest in end-of-pipe technologies. Invasivespecies policy may prohibit trade in commodities known to harbor invasives. The present value of the expected flow ofsunk costs associated with this policy at time of adoption is proportional to the spatial scale in which it is applied: C ¼ cA

where c40. For instance, policies that address global environmental issues will typically be more costly than policiesaddressing local or regional issues.

There is an opportunity cost of adopting the policy immediately rather than waiting for more information since thepolicy imposes sunk cost on society and it can be delayed. This opportunity cost is known as an option value and ariseswith irreversible investment decisions (Dixit and Pindyck, 1994).3 However, in an environmental context, this option valuemust be balanced against the opportunity cost of delaying action—irreversible or nearly irreversible environmentaldamage to society. At each instant in time a risk-neutral4 decision maker weighs this trade-off by deciding whether apolicy that reduces the spread rate to rp should be adopted immediately or postponed. Immediate policy adoption lowersthe growth in expected damage but incurs C and forfeits the option to adopt the policy in the future. Postponing policyadoption incurs higher expected damage but delays C and retains the option to adopt the policy at the next instant in time.This decision is evaluated at each instant in time given the dependence of rp on A, the dependence of A on D, and thestochastic evolution of D. Immediately adopting the environmental policy becomes the preferred alternative when currentdamage reaches or exceeds a threshold Dn. Adopting the policy at this critical threshold will minimize the present value ofexpected damage and control cost over an infinite time horizon and a finite spatial horizon. When current damage is belowthis threshold it is optimal to delay policy adoption until damage reaches or exceeds Dn creating a policy adoption ‘‘rule’’.

1 For more information on numerical solution procedures see Wilmott (1998) or Miranda and Fackler (2002).2 Previous studies have found that the inclusion of lower reflecting barriers (Saphores, 2004) and lower absorbing barriers (Saphores, 2003)

significantly influence the timing of policy adoption in different ways. Saphores (2002) has also shown that the standard real options approach may yield

incorrect decision rules in the presence of lower barriers. This is due to the fact that there is a positive, non-constant probability that the investment may

be rendered worthless if the stochastic variable reaches the lower barrier before reaching the optimal investment threshold. This type of bias is not

present with upper barriers since the probability of reaching the optimal investment threshold is always 1 if the barrier is above the threshold or 0 if the

barrier is below the threshold.3 The Dixit–Pindyck concept of option value is composed of the quasi-option value developed in the environmental economics literature (Arrow and

Fisher, 1974; Hanemann, 1989; Henry, 1974) and the value of postponing a decision to take advantage of more favorable investment circumstances

irrespective of uncertainty (Fisher, 2000; Mensink and Requate, 2005).4 For some agricultural and natural resource applications the decisions of risk-averse firms have been shown to be consistent with risk-neutrality

(Kramer and Pope, 1981; McSweeny and Kramer, 1986; Zacharias and Grube, 1984).


3. Optimally timing policies of fixed stringency

To focus on the timing of policy adoption, policy stringency is initially taken as given by the decision maker.5 This couldarise when policy options are severely limited or when decisions concerning the design and implementation ofenvironmental policy are carried out by different individuals or groups. For instance, the spread of habitat loss maytrigger the listing of certain species as endangered. Alternatively, the spread of an invasive species may cause it to beadded to a ‘‘black list’’ of species whose import and transport is prohibited. These types of decisions imply a given degreeof policy stringency that cannot be adjusted by decision makers. This restriction will be relaxed in Section 4.

Calculating Dn requires defining the lower boundary of the termination (or policy adoption) region and relies onboundary conditions (Dixit, 1991)

WCðDnÞ ¼WT

ðDnÞ ð4Þ

and

WCDðD

nÞ ¼WT

DðDnÞ ð5Þ

where subscripts indicate partial derivatives. The continuation value WCðDÞ represents what is received by postponing

policy adoption: the discounted expected damage from doing nothing and the value of the option to postpone the policydecision (the option value). Because damage is modeled as a positive value, this option value is nonpositive as it has adamage reducing character. The termination value WT

ðDÞ represents the discounted expected damage from immediatepolicy adoption. The boundary condition in Eq. (4) is the value matching condition requiring the payoff from postponingpolicy adoption equal the payoff from immediate policy adoption. The boundary condition in (5) is known as the smoothpasting condition and requires the continuation and termination values meet tangentially at Dn so that payoffs are alsobalanced at the margin.

Following the standard dynamic programming approach, the continuation value arises from the dynamic optimizationproblem in the continuation region where the Bellman equation requires6

rWC¼Dþ

EtðdWCÞ

dt¼Dþa0DWC

Dþ1

2s2D2WC

DD ð6Þ

where time notation is suppressed and subscripts indicate partial derivatives. The obligation to the flow of damage istreated as an asset whose value WC must be optimally managed (i.e., minimized) by a decision maker. The left-hand side isthe return a decision maker would require to delay policy adoption over the time interval dt. The first term on the right-hand side is the immediate damage from delay while the second term is the expected change in the value function. Theexpected change in the value function can be interpreted as capital appreciation of the policy decision. While delaying thepolicy will subject the decision maker to more damage, benefits also arise from being able to control an increasinglydamaging process. Thus the right-hand side is the expected return from delaying policy adoption over the interval dt. TheBellman equation acts as an equilibrium condition ensuring the decision maker’s willingness to delay policy adoption inthe continuation region. Eq. (6) has the well-known solution

WCðDÞ ¼j0ðDÞþZDðtÞe

0

ð7Þ

where j0ðDÞ is expected present value of damage over an infinite time horizon when the policy is never adopted, D(t) isthe current known damage at time t, ZDðtÞe

0

represents the option value associated with the policy with Zo0 an unknownconstant to be determined, and

e0 ¼1

2�a0

s2

� �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0

s2�

1

2

� �2

þ2rs2

s

is the positive root of the fundamental quadratic associated with (6).The inclusion of spatial scale alters the calculation of j0ðDÞ as the problem is framed over an infinite time horizon but a

finite spatial horizon. This cuts off the upside potential for damage making the damage process log-normally distributedover the range ½0,D� only. Following Dixit (1993), j0ðDÞ is comprised of two terms. The first term is discounted expecteddamage over an infinite spatial horizon (ignoring the barrier). This term will be well defined and e041 with the standardassumption r4a0 (Ross, 1983). The second term captures the reduction in expected discounted damage as currentdamage approaches the barrier. At any point in time t, the expected present value of doing nothing is a nonlinear functionof current damage7

j0ðDÞ ¼ Et

Z 1t

DðtÞe�rðtÞdt¼ DðtÞ

ðr�a0Þ�Z0DðtÞe

0

ð8Þ

5 This is also consistent with much of the previous environmental applications of real options.6 See Dixit and Pindyck (1994) p. 106 for the derivation of (6).7 See Appendix 1 for the derivation of (8). This derivation assumes an upper absorbing barrier which implies damage is permanent without policy

adoption. If damage is temporary it may be more realistic to assume a barrier which reflects the process to 0. For a more rigorous treatment of expected

present values with barriers, see Harrison (1985) and Dixit (1993).


where Z0¼ a0D=½rðr�a0ÞD

e0

� is the constant associated with the correction term that arises due to the upper absorbingbarrier D. When current damage is small, the probability of reaching the upper barrier in any reasonable future time issmall and the second term in Eq. (8) is small. As a result, the current expected present value of damage assuming no barrier(the first term in Eq. 8) is a good approximation. As current damage increases, the second term in Eq. (8) increases as theprobability of hitting the upper barrier increases. Conversely a larger barrier decreases the probability of hitting the upperbarrier and results in expected damages increasing at a decreasing rate:

@j0

@D¼ðe0�1Þa0

rðr�a0ÞDe0 40 and

@2j0

@D2¼�

e0ðe0�1Þa0

rðr�a0ÞDe0�1

o0

The termination value equals the expected present value of damage if the policy is adopted immediately plus the cost ofthe policy: WT

ðDÞ ¼jpðDÞ. Due to spatial scale, the calculation of jpðDÞ depends critically on whether post-policy damageis reversed or slowed. For exposition, consider two polices that vary by degree of stringency. Policy 1 reverses or stops theaccumulation of damage ðapr0Þ at cost C ¼ c1A. In this case, a decision maker recognizes that damage is approaching anupper barrier prior to policy adoption but also recognizes that adopting such a stringent policy will prevent furthermovement toward the upper barrier. However, a positive probability of reaching the upper barrier still remains since asufficiently large positive stochastic shock may move D to D. Since the policy has already been adopted, D will not remainat D. If apo0, D will move down at the next instant in time and if ap ¼ 0 a negative shock may move D below D. To ensurethe effect of the policy is preserved, the absorbing barrier D is transformed into a reflecting barrier following the adoptionof policy 1.8 The expected present value of damage and policy cost after adopting policy 1 becomes

jpðDÞ ¼DðtÞ

r�ap�Zp

RDðtÞep

þc1A: ð9Þ

where

ep ¼1

2�ap

s2

� �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiap

s2�

1

2

� �2

þ2rs2

s41

and the reflecting barrier term is

ZpR ¼

D

epðr�apÞDep :

In contrast, policy 2 only slows the accumulation of damage ðap40Þ at cost C ¼ c2A where c2oc1. With this lessstringent policy, the decision maker is certain that damages will reach the upper barrier even under policy adoption. Sincethe less stringent policy is incapable of moving D below D, the upper barrier remains absorbing following the adoption ofpolicy 2. The calculation jpðDÞ proceeds in the same fashion as Eq. (8):

jpðDÞ ¼DðtÞ

ðr�apÞ�Zp

BDðtÞep

þc2A ð10Þ

where the absorbing barrier term is

ZpB ¼

apD

rðr�apÞDep :

If costs C are low enough for policies to be economically feasible, the payoff from policy adoption will be positive whendamages are above DL, which can be interpreted as the policy adoption threshold arising from traditional cost-benefitanalysis. However, the curvature of j0 and jp implies the payoff from policy adoption may be positive only within aninterval ½DL,DH

�. The upper bound on the interval, DH, signals the highest level of damage the policy maker should beprepared to withstand and still implement the policy. When DðtÞ4DH the benefits of policy adoption do not justify thecost of the policy since damage will soon stop at D at no cost. When a critical damage threshold exists, the intervalboundaries can be found by setting j0 ¼jp. While it is possible for there to be any number of roots which satisfy thiscondition (it can be of any order in D depending on e0 and ep) for all cases considered here we have found at most two inthe interval ½0,D� where the smaller root corresponds to DL and the larger root to DH. Since policy adoption should not takeplace if the expected payoff is negative: DLrDnrDH . The inclusion of spatial scale forces policy adoption to take placewithin a window or interval of current damage. If damage remains undetected and is allowed to inadvertently exceed DH

a previously desirable policy becomes economically undesirable.

8 If the effect of a policy is nullified when the damage-causing process has spread to all potentially affected areas, D would continue to be treated as

an absorbing barrier following policy adoption; the solutions for type 1 and type 2 policies would then be equivalent.


Dermining where Dn falls within this interval requires the boundary conditions in (4) and (5). Policy 1 ðapr0Þ shouldonly be adopted when current damage satisfies9:

Dn¼

e0

e0�1

C

ða0�apÞ=ððr�a0Þðr�apÞÞþðep�e0Þ

ðe0�1Þ

ZpRDnep

ða0�apÞ=ððr�a0Þðr�apÞÞð11Þ

which is subject to

Z¼� C

ðe0�1ÞDne0 �ðep�1Þ

ðe0�1ÞZp

RDnep�e0

þZ0o0: ð12Þ

The solution to Eq. (11) can be found numerically and provides the critical threshold at which one would optimallyimplement policy 1 when incorporating spatial scale. Results for policy 2 will be identical to (11) and (12) but with Zp

R

replaced by ZpB. By comparison, the critical threshold and the option value constant will equal the first terms on the right-

hand side of (11) and (12) when spatial scale is ignored (damage follows an unbounded GBM process).There are two cases where the critical threshold Dn will not exist. The first is when (11) yields a positive real root but

that root causes the option value constant in (12) to be positive. As shown in Fig. 1a, holding the option increases costsfaster than not holding the option. As jp is always higher (more costly) than jo it is preferable to do nothing. In this case,the boundary conditions in (4) and (5) lead to an un-economic solution. The second case where a critical threshold will notexist is when (11) does not yield a positive real root. In this instance, there is no value for D in the interval ½0,D� that allowsthe boundary conditions in (4) and (5) to hold. Since boundary condition (5) is an optimality condition for Dn (Dumas,1991) and WC

D4WTD for D 2 ½0,D�, value is always gained by delaying policy adoption. However, there are two reasons this

can occur, with very different economic implications. Fig. 1b considers a small spatial scale where the decision windowbetween DL and DH does not exist. Damage accrues over such a small area that the cost of the policy cannot be justified atany possible level of damage. This provides an incentive to delay policy adoption indefinitely. Fig. 1c considers a largerspatial scale coupled with a higher level of uncertainty. The decision window between DL and DH exists, but the optionvalue is increasing faster than the value of adopting the policy. This makes it optimal for the decision maker to delayindefinitely even though the payoff from policy adoption is positive. The inclusion of an option value in this case preventsthe adoption of policies that would be adopted based on traditional cost-benefit analyses.

A critical threshold will exist at sufficiently large spatial scales. Fig. 2a considers an expanded spatial scale where DL andDH exist. At this larger spatial scale, the option value is negative as desired but the payoff from policy adoption is positiveonly when DLrDðtÞrDH . In this case, policy adoption should take place when current damage reaches Dn given byEq. (11). However, if damage exceeds DH the decision marker has missed the window for policy adoption. Unless astochastic shock brings D below DH, the decision maker chooses to forgo the policy since damage will soon be stopped at D

at no cost. Further increases in spatial scale cause results for policy 1 and policy 2 to diverge. Fig. 2b considers policy 1 atan even larger spatial scale where DH

ZD. This eliminates the upper bound on the interval ½DL,DH� since DðtÞ is incapable of

exceeding it. In other words potential damage is so extensive at this large spatial scale in relation to costs of the policy thatthe net payoff from reversing damage is positive even when the damage-causing process has spread to all potentiallyaffected areas. When a critical threshold exists, DH oD when j0ðDÞojpðDÞ, which only occurs when spatial scale isless than

eprðr�apÞc1

ðr�apepÞg

� �1=ðy�1Þ

However, policy 2 implies jpðDÞ4j0ðDÞ ensuring DH oD at all spatial scales (Fig. 2b is not possible). Adoption of policy 2must always take place within a policy adoption window: DnrDrDH .

The impact of spatial scale on Dn depends on the net of four effects, two on the benefit of policy adoption and two onthe opportunity cost of policy adoption. The barrier D lowers j0, which reduces the benefit of adopting the policy by Z0De0

.This is reflected as a larger value of DL creating an upward pressure on Dn. Spatial scale also reduces the opportunity cost ofadopting since the option value is lower in absolute value by Z0De0

, which places a downward pressure on Dn. Being able todelay the adoption of the policy is less valuable since any delay brings one closer to the barrier, which will stop the processwith certainty at no cost. These two effects exactly offset. The remaining two effects do not offset because they arise fromthe constants Zp

R and ZpB, which unlike Z0 do not have a one for one influence on the option value constant Z. For example,

for policy 1 the benefit of adopting is increased as jp is lower by ZpRDep

. This induces a downward influence on Dn. Theopportunity cost of adopting policy 1 is also increased since the option value is increasingly negative by

ZpRDnep D

Dn

� �e0

ep�1

e0�1

� �

The result is a larger upward influence on Dn. This is also true for policy 2. In net the inclusion of spatial scale results ina delay in policy adoption for both policies.

9 See Appendix 2 for the derivation of (11) and (12).

WC

WC

WC

∗

Fig. 1. Cases where a critical threshold Dn will not exist. (a) The expected payoff from policy adoption is always negative but the value matching and

smooth pasting conditions can be met with a positive option value. (b) The expected payoff from policy adoption is always negative and the value

matching and smooth pasting conditions cannot hold. (c) The expected payoff from policy adoption is positive over a limited range of current damage but

the option value is growing faster than the benefit of policy adoption.


3.1. Numerical example

To illustrate these results, suppose a process that causes damage such as a pollutant or invasive species has recentlybeen discovered and is currently affecting a 1 km2 area. Affected area is increasing at a rate of 2% annually (r0

¼0.02),damage is quadratic in affected area (y¼2), g¼0.001 (in millions of dollars/km2), c1¼0.125 ($125,000/km2 to reversedamage), c2¼0.005 ($5,000/km2 to slow damage), and r¼0.08. Fig. 3 partitions the solution space for both policy 1 and 2across intervals of uncertainty and spatial scales. The figure illustrates each case considered above. For either policy thefigure shows that with high uncertainty and a small spatial scale, a critical threshold is less likely to exist. Even with nouncertainty, policy 1 will never be adopted at spatial scales smaller than 19 km2 and policy 2 will never be adopted atspatial scales less than 31

2km2. For larger spatial scales a critical threshold exists for both policies as long as the level ofuncertainty is low enough. If a threshold exists, the termination region may be constrained to an interval (DH exists)depending upon the spatial scale and type of policy.

Fig. 4 contrasts optimal policy adoption strategies for policies 1 and 2 at various spatial scales with a low enough levelof uncertainty (s¼0.01) that a critical threshold exists. The upper barrier on damages D is given by the solid gray line. Thedashed black line represents the threshold damage must reach to trigger immediate policy adoption when spatial scale isignored. Solid black lines represent the upper (DH) and lower (Dn) boundary of the termination region when spatial scale isincorporated. For purposes of comparison we assume policy cost (C) is the same when spatial scale is ignored andincorporated.

Termination region = immediate adoption

Continuation region = delay

WC

*

Terminationregion =

immediateadoption

Continuation region

= delay adoption

Abandonmentregion = never

adopt

WC

*

Fig. 2. Optimal policy timing with spatial boundaries. For policy 1, smaller spatial scales produce a policy adoption window between Dn and DH (a) but at

larger spatial scales DHZD eliminating the policy adoption window (b). For policy 2 a policy adoption window is always present.


First consider policy 1 and assume this policy would cause affected area to shrink by 3% per year: rp1¼�0.03. At spatial

scales greater than approximately 23 km2 the optimal strategy is to delay policy adoption until current damage reaches orexceeds Dn. From 19–23 km2 there is a termination interval such that DH exists. Shrinking the spatial scale below 19 km2

completely eliminates the termination region as DH and Dn converge. In this example, policy 1 should never be adopted atspatial scales less than 19 km2. The range of spatial scales where policy adoption is possible will decrease under morestringent and thus costly policies. For instance, when rp

1 is reduced from –0.03 to –0.12 (a more stringent policy) and c1 isincreased from 0.125 to 0.98, policy adoption should never take place at spatial scales less than 100 km2. The implication isthat more stringent policies may be optimal at a national level and not optimal at a state level.

Policy 2 will only slow the spread of the damage-causing process from 2% to 1% per year: rp2¼0.01. This less stringent

and less costly policy will never be adopted at spatial scales smaller than approximately 3.5 km2. At larger spatial scalespolicy adoption should be delayed until damage reaches Dn. However, an abandonment region is always present.

The inclusion of spatial scale not only partitions the termination regions; it can also induce a significant difference inthe critical threshold. Fig. 5 contrasts optimal adoption strategies for policy 1 at a 30 km2 spatial scale and policy 2 at a5 km2 spatial scale with various degrees of uncertainty and rates of initial spread. As in previous research we find that theeffect of uncertainty on Dn is ambiguous and may delay or expedite policy adoption (Hanemann, 1989; Saphores, 2003,2004; Saphores and Carr, 2000; Shackleton and Sødal, 2010). In regions when a critical threshold exists with spatial scale(low levels of uncertainty in Fig. 5) there is only a slight difference in the critical threshold with and without spatial scale.However, considering the difference across a range of spread uncertainty sees significant differences in the thresholds. Atlow levels of uncertainty the optimal strategy for policy 1 is to delay adoption until damage reaches or exceeds Dn and forpolicy 2 delay until damage reaches Dn but is less than DH. As uncertainty increases, the critical threshold incorporatingspatial scale diverges from the threshold where spatial scale is ignored. When uncertainty is greater than 0.1, Eq. (11) does

Spatial scale (A) Spatial scale (A)

Spre

ad u

ncer

tain

ty (

s)

Spre

ad u

ncer

tain

ty (

s)

Policy 2: = 0.01 and c2 = 0.005

Adopt when D* ≤ D

Adopt when D* ≤ D ≤ DH

Never adopt: expected payoff

from policy adoption is

negative

Never adopt: option value is positive

Never adopt: option value is positive

Never adopt: expected payoff

from policy adoption is

negative

Policy 1: = -0.03 and c1 = 0.125

Never adopt: option value

growing faster than policy

benefits

*

Adopt when

0.20

0.15

0.10

0.05

0.00

0 10 20 30 40 50

0.20

0.15

0.10

0.05

0.00

1 2 3 4 5

Fig. 3. Combinations of uncertainty and spatial scale that preclude or permit policy adoption.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 5 10 15 20 25 30

Cur

rent

dam

age

(mill

ions

)

delay adoption

0

0.002

0.004

0.006

0.008

0.01

0 1 2 3 4 5

Cur

rent

dam

age

(mill

ions

)

delay adoption

never adopt

D*

DH

adopt immediately

D*

DH

adopt immediately

never adopt

never adopt

Policy 1: = -0.03 and c1 = 0.125

Policy 2: = 0.01 and c2 = 0.005

–

–

Spatial scale (A)–

Spatial scale (A)–

Fig. 4. Policy adoption strategies at various spatial scales. Dashed line represents policy adoption threshold ignoring spatial scale. Solid lines represent

policy adoption thresholds incorporating spatial scale. Results generated with r0¼0.02, s¼0.01, g¼0.001, y¼2, and r¼8%.


not yield a real root for either policy 1 or policy 2. For policy 1 the option value is growing faster than the benefit ofadoption over this small spatial scale. Even though the payoff is positive, policy 1 should never be adopted at a spatial scaleof 30 km2 when uncertainty reaches 10%. For policy 2, the payoff from policy adoption is always negative once uncertaintyreaches 10%. Much like uncertainty, sufficiently high initial rates of spread will prevent policy adoption for both policies.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.05 0.1 0.15 0.2

Cur

rent

dam

age

(mill

ions

)

Spread uncertainty (s)

adopt immediately

never adopt

never adopt

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.01 0.02 0.03 0.04

Cur

rent

dam

age

(mill

ions

)

Initial rate of spread (r0)

adopt immediately

delay adoption

neveradopt

never adopt

D*

DH

0

0.005

0.01

0.015

0.02

0 0.05 0.1 0.15 0.2

Cur

rent

dam

age

(mill

ions

)


DH

D*

never adopt

adoptimmediately

0

0.005

0.01

0.015

0.02

0.01 0.02 0.03 0.04

Cur

rent

dam

age

(mill

ions

)

Initial rate of spread (r0)

adopt immediately

never adopt

delay adoption

never adopt

DH

D*

delay adoption

D*

Policy 1: = -0.03, c1 = 0.125, and = 30

–

delay adoption

never adopt

–

–

–

Policy 2: = 0.01, c2 = 0.005, and = 5A–

A–

Fig. 5. Policy adoption strategies with various degrees of spread uncertainty and initial rates of spread. Dashed lines represent policy adoption thresholds

ignoring spatial scale. Solid lines represent policy adoption thresholds incorporating spatial scale. Benchmark results generated with r0¼0.02, s¼0.01,

g¼0.001, y¼2, r¼8%.


4. Convex costs and optimal policy stringency

In contrast to the analysis up to this point, decisions makers may have some degree of flexibility over the stringency ofpolicies they evaluate. For example, decision makers may choose to limit habitat loss by preserving habitat or limitingdevelopment. Instead of prohibiting the import and transport of a potentially invasive species, decision makers maychoose to limit spread through inspections and public information campaigns. Expanding the model to allow decisionmakers to choose policy timing (Dn) and stringency (measured as a reduction in spread rate r0�rp*), two decisions aremade at each instant in time: (1) the optimal stringency of the policy to be adopted based on current known damage and(2) whether a policy with that level of stringency should be adopted immediately. At each instant in time optimalstringency minimizes expected damage and cost from that point forward by lowering the spread rate to rp* at a cost C.Immediate policy adoption lowers the growth in expected damage but incurs C and forfeits the option value. Otherwisepolicy adoption is postponed until the next instant in time where the decision maker determines a new rp* based oncurrent damage and is faced with the same binary choice concerning policy adoption. Now timing and stringency of thepolicy decision are linked.10 The solution in this case requires finding all local solutions when apr0 and ap40 andcomparing resulting values of expected damage and control costs to find the global solution.

Let the sunk cost of policy adoption be a quadratic function of stringency: C ¼ cAðr0�rpÞ2. Optimal stringency when

apr0 solves:

minrpfWTðDn,rpÞg ¼

Dn

r�ap�

D

epðr�apÞDep Dnep

þcAðr0�rpÞ2: ð13Þ

10 While it is well understood that the degree and timing of control actions are linked (Clark, 1990), surprisingly few studies have considered the

coupled nature of timing and magnitude of uncertain environmental policies in the real options literature. The exception is Pindyck (2000) which

considers the effect of uncertainty on optimal timing of a policy that reduces pollution emissions to an optimal level.


The first-order condition for this problem determines rp*ðDnÞ, which is the optimal rate of spread at the critical threshold:

yDn

ðr�ap*Þ2�

yD

epðr�ap*Þ2

Dn

D

� �ep

1� lnDn

D�

1

ep

� �rap*

O� �

¼2

r0�rp*Cn

ð14Þ

where Cn¼ cAðr0�rp*Þ

2, ap* ¼ yrp*þð1=2Þyðy�1Þs2, and

O¼ap*ðr�ap*Þ

s2r1�

ðap*=s2Þ�ð1=2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiððap*=s2Þ�ð1=2ÞÞ2þð2r=s2Þ

q264

375o0

The left hand side of (14) is the marginal cost of choosing a less stringent policy (increase in expected damage resultingfrom a less stringent policy or higher rp) and the right hand side is the marginal benefit of a less stringent policy (reductionin policy costs from a less stringent policy or higher rp). Spatial scale impacts the policy stringency decision through thesecond term on the left-hand side of (14). The imposition of spatial scale decreases the marginal cost of higher post-policyspread rates causing decision makers to choose a less stringent policy compared to the case where spatial scale is ignored.

Substituting the termination and continuation values and the first-order condition into the boundary conditions (4)and (5) provides the critical threshold that triggers adoption of a policy that stops or reverses damage11:

Dn¼

C

ða0�ap*Þ=ððr�a0Þðr�ap*ÞÞ2þ

e0

e0�1

a0�ap*

ðr�ap*Þ

� �þ

Dðr�a0Þ

epðr�ap*Þ

Dn

D

� �ep

1� lnDn

D�

1

ep

� �rap*

Oþep�e0

e0�1

� �ð15Þ

subject to Zo0. The optimal policy decision can be found by simultaneously solving Eqs. (14) and (15) to yield Dn andrp*ðDn

Þ. Only values of Dn corresponding to a negative option value are permissible in an economic sense, limiting thenumber of feasible roots.

Optimal stringency when ap40 solves

minrpfWTðDn,rpÞg ¼

Dn

ðr�apÞ�

apD

rðr�apÞDep Dnep

þcAðr0�rpÞ2: ð16Þ

This is equivalent to (13) except that the correction term is due to an absorbing barrier rather than a reflecting barrier. Thefirst-order condition for this problem implicitly defines rp*ðDn

Þ:

yDn

ðr�ap*Þ2�

yD

ðr�ap*Þ2

Dn

D

� �ep

1� lnDn

D

� �O

� �¼

2

r0�rp*Cn

ð17Þ

where O40 since ap40. In the same fashion the left (right) hand side is the marginal cost (benefit) of choosing a lessstringent policy or higher rp. From the second term on the left-hand side of (17), the imposition of spatial scale decreasesthe marginal cost of higher post-policy spread rates causing decision makers to choose a less stringent policy compared tothe case where spatial scale is ignored.

A less stringent policy that slows the accumulation of damage should be adopted if current damage satisfies12:

Dn¼

Cn

ða0�ap*Þ=ððr�a0Þðr�ap*ÞÞ2þ

e0

e0�1

a0�ap*

ðr�ap*Þ

� �þ

Dðr�a0Þ

ðr�ap*Þ

Dn

D

� �ep

1�lnDn

D

� �Oþ

ep�e0

e0�1

ap*

r

� �ð18Þ

subject to Zo0. The combined policy decision can be found by simultaneously solving Eqs. (17) and (18) to yield Dn and rp*ðDnÞ.

The first term on the right-hand side of (15) and (18) is the critical threshold when spatial scale is ignored. Comparingthis term to (11), we find that the ability to choose policy stringency leads to a delay in policy adoption all else constant.Intuitively, the additional flexibility allows the decision maker to absorb more damage before exercising the policy option.In keeping with previous results the inclusion of spatial scale creates an upward influence on Dn. However, incorporatingspatial scale also triggers a less stringent policy. This reduces the cost of policy adoption creating an opposing incentive formore immediate adoption but of a less stringent policy.

4.1. Numerical example

To contrast these results with those of exogenous stringency, return to the numeric example with c¼50 to ensure thecost of the policy remains consistent. For the parameter values we consider, no solution exists when apr0 so that it is onlyoptimal to adopt a policy that slows the accumulation of damage. Fig. 6 contrasts optimal policy adoption strategies atvarious spatial scales when ap40. When policy stringency was fixed at a 50% reduction (rp

¼0.01 in Fig. 4), policy adoptionoptimally occurred when current damage reached $4000 to $6000 and never occurred at spatial scales less than 3.5 km2.Spatial scale in that case had no effect on policy stringency but resulted in delayed policy adoption. When policystringency is a choice variable, the optimal policy decision involves a less stringent policy adopted at a lower level of

11 See Appendix 3 for the derivation of (15).12 See Appendix 4 for the derivation of (18).

0

0.00002

0.00004

0.00006

0.00008

0.0001

Cur

rent

dam

age

(mill

ions

)

delay adoption

adopt immediately

never adopt

0

0.005

0.01

0.015

0.02

0.025

Cur

rent

dam

age

(mill

ions

)

adopt immediately

delay adoption

DH

D*

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

0 1 2 3 4 5

% r

edcu

tion

in s

prea

d ra

te

Spatial scale (A)

0%

20%

40%

60%

80%

100%

0 0.05 0.1 0.15 0.2

% r

educ

tion

in s

prea

d ra

te


never adopt

DH

D*

–

Fig. 6. Policy timing (a) and stringency (b) at various spatial scales and various degrees of uncertainty. Dashed line represents the policy decision

ignoring spatial scale. Solid lines represent the policy decision incorporating spatial scale. Benchmark results generated with r0¼0.02, s¼0.01, A¼5,

g¼0.001, y¼2, c¼50, and r¼8%.


current damage. This highlights the tradeoff between policy stringency and policy timing. Even though the policy is lessstringent, its more immediate adoption reduces discounted damages and policy cost.

Incorporating spatial scale reduces optimal policy stringency further as decision makers recognize that damage is nowbounded by the spatial domain. Decision makers that ignore spatial scale assume damages are unbounded and chooseexcessively stringent policies delayed further into the future. While more stringent, these policies require a longer delay tojustify their larger cost causing society to incur greater damages and costs. Policy makers also find it optimal to choose lessstringent policies at smaller spatial scales rather than forgo management altogether. This ensures a critical thresholdalways exists in this more flexible policy scenario.

The consequences of ignoring spatial scale are small with uncertainty at 1%. A decision maker ignoring a 5 km2 spatial scalewould choose to slow spread by 0.75% as opposed to 0.2% and would adopt this policy when current damage reaches $60 asopposed to $17. As shown in Fig. 6, the consequences of ignoring spatial scale increase as more uncertainty is introduced to theproblem. When uncertainty reaches 20% at a 5 km2 spatial scale, a decision maker ignoring spatial scale would choose to nearlystop spread but would only choose to do so when current damage reaches $64,000. In contrast, a decision maker incorporatingthe 5 km2 spatial scale would chose to merely slow damage by 4% and does so when current damage reaches $1250.

5. Conclusions

A wealth of research has addressed the optimal timing of environmental policy under uncertainty and irreversibility.However, by assuming the policy decision is spatially independent and occurs at a single level of decision making, theseapproaches may not accurately reflect environmental problems that unfold over space or decision-making that occurs atmultiple scales. When environmental damage is spatially dependent, physical limits to space and the scale of decisionmaking become influential factors in the timing and stringency of policies to mitigate this damage. This paper representsthe first attempt to include these spatial considerations in a real options model. The results have important implicationsfor environmental decision making in general and real options models in particular.


Depending on the stringency of the policy being considered and the flexibility of the policy decision, ignoring spatialscale may result in untimely and excessive environmental policy. When policy stringency is given, ignoring spatial scaleresults in premature policy adoption and fails to recognize that policy adoption is forced to take place within a window ofcurrent damage, which will contract and eventually close as spatial scale shrinks or uncertainty and spread rate increase.The inclusion of spatial scale also allows for the possibility the option value always grows faster than the benefit of policyadoption creating an incentive to delay policy adoption indefinitely. If decision makers are able to select the stringency ofthe policy they are considering, the policy adoption window will always remain open. In this more flexible policy scenario,ignoring spatial scale results in delayed and excessive policies.

The inclusion of spatial considerations shines new light on the role of uncertainty in the timing of environmental policy. Whileprevious literature has found that increases in uncertainty only alter the timing of policy adoption, our findings indicate two waysthat increased uncertainty may preclude policy adoption entirely. First, for sufficiently high levels of uncertainty the policyadoption window will close eliminating the optimal stopping threshold. In this case the policy should not be adopted since thepayoff from adoption is negative over the relevant spatial scale. Second, high levels of uncertainty may cause the option value togrow faster than the benefit of policy adoption over a fixed spatial scale. In this case the payoff from policy adoption is positive,but the policy will never be adopted since the benefit of delaying policy adoption is always greater than the benefit of immediateadoption. This implies that high levels of uncertainty may render certain policies obsolete as opposed to simply affecting whenthey are adopted. These results also show that the effect of uncertainty is dependent on the spatial scale in which the policydecision is framed. A given amount of uncertainty on a large spatial scale may result in policy adoption while the same amount ofuncertainty on a small spatial scale may render the policy economically undesirable. Thus, a given amount of uncertainty will notinfluence all decision makers in the same way.

One of the attractive aspects of real options models are their ability to produce a closed-form ‘‘rule’’ for policy adoptionin the form of the optimal stopping threshold. However, spatial scale precludes a monotonic increase in policy benefitsallowing the payoff from policy adoption to move from negative to positive to negative as current damage increases. Thusbasing policy decisions solely on the optimal stopping threshold may be misleading since a level of current damage thatexceeds this threshold does not ensure the policy should be adopted. With the inclusion of spatial scale decision makersmust identify an additional abandonment threshold. This abandonment threshold places detection in a key role and mayrepresent the more relevant factor when decision making has been inadvertently delayed.

In short, environmental policy decisions differ when evaluated at different spatial scales just as decisions differ whenevaluated at different temporal scales. If spatial scale corresponds to a decision maker’s jurisdiction, a given environmentalpolicy should be adopted at a local level before it should be adopted at a regional or federal level. If local decision makerscan choose policy stringency they will choose less stringent policies than those chosen at a federal level. If spatial scalecorresponds to a physical barrier such as the suitable habitat of an invasive species, a given invasive species policy shouldbe applied to species with small potential ranges before species with large ranges. If post-policy spread is a continuouschoice, species with smaller ranges will also see less stringent policies. Such a result cautions against developing optimalpolicy at one spatial scale and then applying that policy at a different spatial scale.

Acknowledgments

We would like to thank Jim Sanchirico and two anonymous referees for helpful comments. This research was supportedby the Program of Research on the Economics of Invasive Species Management (PREISM), Economic Research Service, U.S.Department of Agriculture, grant #58-7000-6-0080.

Appendix A. Supplementary material

Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jedc.2011.09.001.

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