a real option approach to the protection of a habitat ... · context and motivation main model™s...

Context and MotivationMain Model�s features and Results

The ModelEstimation of the stochastic di¤usion process

Application of the RO modelConclusion

A Real Option Approach to the Protection of aHabitat Dependent Endangered Species

Skander Ben Abdallah Pierre LasserreUniversité du Québec à Montréal

September 2009, for presentation at ETH Zurich

Ben Abdallah / Lasserre (UQAM) Endangered Species with Stochastic HabitatSeptember 2009, for presentation at ETH Zurich 1

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The Rangifer tarandusRO in Biodiversity Protection

The Rangifer tarandus caribou (forest caribou)

The Rangifer tarandus caribou is an endangered species inLabrador, Canada;

Main problem: habitat deterioration. The habitat of the forestcaribou is old growth boreal forest: older trees, with lichen onthe ground, on which caribous feed.Habitat evolves stochastically (unpredictable)

Figure: Rangifer tarandus caribou (Source: Animal Diversity Web)


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The Rangifer tarandus caribou is an endangered species inLabrador, Canada;Main problem: habitat deterioration. The habitat of the forestcaribou is old growth boreal forest: older trees, with lichen onthe ground, on which caribous feed.

Habitat evolves stochastically (unpredictable)



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The Rangifer tarandus caribou is an endangered species inLabrador, Canada;Main problem: habitat deterioration. The habitat of the forestcaribou is old growth boreal forest: older trees, with lichen onthe ground, on which caribous feed.Habitat evolves stochastically (unpredictable)



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The Rangifer tarandus caribou (cont�d)

Caribous are a crucial component of the social, economic, andcultural lives of central Labrador�s Innus.

Commercially exploiting a forest which is home to someendangered species increases the later�s probability ofextinction.

A temporary ban on logging may prevent or delay extinctionbut implies foregone timber revenues.

Besides being fraught with uncertain and irreversibleconsequences, forest management decisions are costly.


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RO in Biodiversity Protection

Decision + uncertainty + irreversibility!Real Options Approach,as in:

Pindyck (2000) : Environmental policy adoption.

Conrad (2000) : Land use decisions.

Saphores (2003) : Exploitation of an endangered species.

Kassar and Lasserre (2004) : Biodiversity preservationdecisions.

Saphores and Shogren (2005) : The use of pesticides.


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Model�s featuresMain Results

Model�s features

The state variable is the forest habitat (mature boreal forests)of the Endangered Species.

We use a simulation model (Spatially Explicit LandscapeEvent Simulator: SELES) (Fall et al., 2001) to generatehabitat series.

Habitat series are used to estimate the habitat di¤usionprocess (DP) when "Logging" or "No Logging".

Depending on the value that the community attributes to thespecies existence, some levels of extinction risk are deemedacceptable.

This paper is the �rst one modelling a situation where exercisedecisions a¤ects the DP of that variable.


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Model�s featuresMain Results

Main Results

The Optimum Policy: interrupt or resume logging dependingon the habitat level (observable but not predictable).

The e¤ectiveness is captured by the change in short-run riskof extinction measured at dates when logging is interrupted(decreases by more than 80%).

The expected duration of each forestry regime ("Logging" or"No logging"): short duration of bans (8 y.) relative tologging regimes (80 y.).


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The habitat of the endangered speciesOptimal Policy DescriptionAssumptionsThe extinction riskForest Value FunctionsThe mean durations of the forestry regimes

The habitat of the endangered species

Caribou habitat (=areas covered with mature trees), measureas the log. of the area (in ha) : ht .

Habitat Extinction Threshold he (constant): minimum habitatsize required for survivalAssume ht follows a mean-reversing (MR) di¤usion process(DP):

dht = λa (µa � ht ) dt + σadzat : when logging is alloweddht = λb (µb � ht ) dt + σbdzbt : when logging is banned

E (ht ) = µi + (h0 � µi )e�λi t ) µi is interpreted as long-run

expected habitat level.The stochastic component σidzit = σi εit

pdt accounts for

unpredictable natural variations (wild �res, ...), εit � N(0, 1).


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Caribou habitat (=areas covered with mature trees), measureas the log. of the area (in ha) : ht .Habitat Extinction Threshold he (constant): minimum habitatsize required for survival

Assume ht follows a mean-reversing (MR) di¤usion process(DP):




pdt accounts for



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Caribou habitat (=areas covered with mature trees), measureas the log. of the area (in ha) : ht .Habitat Extinction Threshold he (constant): minimum habitatsize required for survivalAssume ht follows a mean-reversing (MR) di¤usion process(DP):




pdt accounts for



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expected habitat level.

The stochastic component σidzit = σi εitpdt accounts for



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pdt accounts for

unpredictable natural variations (wild �res, ...), εit � N(0, 1).Ben Abdallah / Lasserre (UQAM) Endangered Species with Stochastic Habitat

September 2009, for presentation at ETH Zurich 7/ 32





The habitat of the endangered species (cont�d)

SELES provided Monte Carlo data on habitat under thelogging regime and the ban regime:

µa � µb : The LR e¤ect of logging versus not logging.


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The habitat of the endangered species (cont�d)

SELES provided Monte Carlo data on habitat under thelogging regime and the ban regime:

µa � µb : The LR e¤ect of logging versus not logging.Ben Abdallah / Lasserre (UQAM) Endangered Species with Stochastic Habitat






Decisions, bene�ts, and costs

Logging provides a constant �ow of bene�ts ω except attimes when it is banned.

Caribous provide a constant instantaneous utility �ow s aslong as they are in existence.

The decision maker has control over whether or not logging isallowed to take place at any given time.

When logging is resumed after some interruption, a cost ofIa � 0 is incurred.When logging is interrupted, a cost of Ib � 0 is incurred.


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When logging is resumed after some interruption, a cost ofIa � 0 is incurred.

When logging is interrupted, a cost of Ib � 0 is incurred.


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When logging is resumed after some interruption, a cost ofIa � 0 is incurred.When logging is interrupted, a cost of Ib � 0 is incurred.


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Interrupting and resuming logging optimally

Logging may be interrupted if habitat decreases to some lowthreshold level: ha

If habitat recovers correctly, then logging may be allowedagain when it increases to some high threshold level: hbThis alternance will go on for as long as the species is notextinct.

The problem is time autonomous: The decision depends onlyon the current habitat level (observable).


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Logging may be interrupted if habitat decreases to some lowthreshold level: haIf habitat recovers correctly, then logging may be allowedagain when it increases to some high threshold level: hb

This alternance will go on for as long as the species is notextinct.



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Logging may be interrupted if habitat decreases to some lowthreshold level: haIf habitat recovers correctly, then logging may be allowedagain when it increases to some high threshold level: hbThis alternance will go on for as long as the species is notextinct.



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Assumptions

Costs Ia and Ib do not depend on the length of any previouslogging or ban periods.

µb > he ; µb > µa (In the empirical application, in log, he , µaand µb respectively represent 18%, 29%, and 62% of the totalforest area of 1, 134, 000 ha).

Ia � ω/r (N and S for logging to be resumed if the speciesgoes extinct).

Ib � s/r (N but not S for logging ever to be banned).


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The extinction risk

Habitat will reach any �nite level with probability one (morethan regular); extinction is certain under both regimes.

The probability of extinction over any �nite period is lowerunder a ban (i = b) than under logging (i = a): How tomeasure it?Phe ,µii (h): probability for habitat to reach µi before he , givenh0 = h and assuming forestry regime (i) forever:Phe ,µii (x) = Pr[T µi

i (x) < Thei (x)].

T yi (x) is the date at which habitat level reaches y for the �rsttime under forestry regime (i) given habitat level x at timezero.The risk that habitat hits he before µi is

Ri (h) = 1� Phe ,µii (h).


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The extinction risk

Habitat will reach any �nite level with probability one (morethan regular); extinction is certain under both regimes.The probability of extinction over any �nite period is lowerunder a ban (i = b) than under logging (i = a): How tomeasure it?

Phe ,µii (h): probability for habitat to reach µi before he , givenh0 = h and assuming forestry regime (i) forever:Phe ,µii (x) = Pr[T µi

i (x) < Thei (x)].




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The extinction risk

Habitat will reach any �nite level with probability one (morethan regular); extinction is certain under both regimes.The probability of extinction over any �nite period is lowerunder a ban (i = b) than under logging (i = a): How tomeasure it?Phe ,µii (h): probability for habitat to reach µi before he , givenh0 = h and assuming forestry regime (i) forever:Phe ,µii (x) = Pr[T µi

i (x) < Thei (x)].




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The extinction risk


i (x) < Thei (x)].

T yi (x) is the date at which habitat level reaches y for the �rsttime under forestry regime (i) given habitat level x at timezero.

The risk that habitat hits he before µi is



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The extinction risk


i (x) < Thei (x)].




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The extinction risk (comments)

The short run ends the �rst time the process reaches itslong-run level (no arbitrariness).

The short-run risk is appropriately zero when habitat is higherthan its long-run level and is higher the closer the extinctionthreshold.Lemma: When µi > he , P

he ,µii (h) = [Si (h)] [Si (µi )]

�1 anincreasing function of h 2 [he , µi ] with

Si (h) =R hhee

λiσ2i(x�µi )

2

dx : the scale function of the DP.xxh 2 [he , µi ]?Means S-R risk decreases as h increases: Good!However the gap Rb(h)� Ra(h) is negative when h isbetween µa and µb : Bad bec. may be interpreted to meanthat risk increases if a ban is imposed!


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The short run ends the �rst time the process reaches itslong-run level (no arbitrariness).The short-run risk is appropriately zero when habitat is higherthan its long-run level and is higher the closer the extinctionthreshold.

Lemma: When µi > he , Phe ,µii (h) = [Si (h)] [Si (µi )]


Si (h) =R hhee

λiσ2i(x�µi )

2



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The short run ends the �rst time the process reaches itslong-run level (no arbitrariness).The short-run risk is appropriately zero when habitat is higherthan its long-run level and is higher the closer the extinctionthreshold.Lemma: When µi > he , P

he ,µii (h) = [Si (h)] [Si (µi )]


Si (h) =R hhee

λiσ2i(x�µi )

2

dx : the scale function of the DP.xxh 2 [he , µi ]?Means S-R risk decreases as h increases: Good!

However the gap Rb(h)� Ra(h) is negative when h isbetween µa and µb : Bad bec. may be interpreted to meanthat risk increases if a ban is imposed!


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The short run ends the �rst time the process reaches itslong-run level (no arbitrariness).The short-run risk is appropriately zero when habitat is higherthan its long-run level and is higher the closer the extinctionthreshold.Lemma: When µi > he , P

he ,µii (h) = [Si (h)] [Si (µi )]


Si (h) =R hhee

λiσ2i(x�µi )

2



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Notation:Vb (h): Forest Value Function under forestry regime b (loggingban);Va (h): Forest Value Function under forestry regime a when thethreshold ha at which it is optimal to interrupt logging exists.Va(h): Forest Value Function under forestry regime a when hadoes not exist (too costly to ban; the species is not valuableenough; logging is too valuable)


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Forest Value Function under logging ban:

There exists a �nite habitat level hb > he above which loggingshould be allowed.

If the ban is not to be lifted immediately, then 8h 2]he , hb ]

Vb (h) = maxx�he

E�Z T xb (h)

0se�rτdτ + e�rT

xb (h) (Va (x)� Ia) jh0 = h

�If the ban is to be lifted immediately, then

Vb (h) = Va (h)� Ia


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Vb (h) = maxx�he

E�Z T xb (h)



�

If the ban is to be lifted immediately, then



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Vb (h) = maxx�he

E�Z T xb (h)



�If the ban is to be lifted immediately, then



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Forest Value Function under logging ban (cont�d):

Proposition 1:

1 Vb(h) satis�es on ]he , hb [ BEσ2b2V 00b (h) + λb (µb � h)V 0b(h)� rVb(h) + s = 0

2 At hb , Vb satis�es8<:Vb(hb) = Va(hb)� Ia: the value-matching condition (VM)V 0b(hb) = V

0a(hb): the smooth-pasting condition (SP)

Vb(he ) = ωr � Ia:a boundary condition (BC)

9=;


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Forest Value Function with logging (ban possible):

Proposition 2(#1,3,4): If 9 habitat level ha where it isoptimal to ban logging, then

Va(h) = maxx�he

E�Z T xa (h)

0(s +ω) e�rτdτ + e�rT

xa (h) (Vb (x)� Ib) jh0 = h

�, 8h 2 [ha,+∞[

1 Va(h) satis�es on]ha,+∞[ BEσ2a2V 00a (h) +

λa (µa � h)V 0a(h)� rVa(h) + s +ω = 0

2 At ha, Va satis�es

8><>:Va(ha) = Vb(ha)� Ib : VMV 0a(ha) = V

0b(ha): SP

limh!+∞

Va(h) = s+ωr : BC

9>=>;3 9 s of s / ha exists for s � s and does not exist for s < s.4 If ha exists then ha = hb () Ia = Ib = 0.


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Va(h) = maxx�he

E�Z T xa (h)



�, 8h 2 [ha,+∞[


λa (µa � h)V 0a(h)� rVa(h) + s +ω = 0



0b(ha): SP

limh!+∞

Va(h) = s+ωr : BC

9>=>;

3 9 s of s / ha exists for s � s and does not exist for s < s.4 If ha exists then ha = hb () Ia = Ib = 0.


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Va(h) = maxx�he

E�Z T xa (h)



�, 8h 2 [ha,+∞[


λa (µa � h)V 0a(h)� rVa(h) + s +ω = 0



0b(ha): SP

limh!+∞

Va(h) = s+ωr : BC

9>=>;3 9 s of s / ha exists for s � s and does not exist for s < s.

4 If ha exists then ha = hb () Ia = Ib = 0.


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Va(h) = maxx�he

E�Z T xa (h)



�, 8h 2 [ha,+∞[


λa (µa � h)V 0a(h)� rVa(h) + s +ω = 0



0b(ha): SP

limh!+∞

Va(h) = s+ωr : BC

9>=>;3 9 s of s / ha exists for s � s and does not exist for s < s.4 If ha exists then ha = hb () Ia = Ib = 0.


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Forest Value Function with logging (No ban ever):

Proposition 2(#2) If ha > he does not exist then8h 2 [he ,+∞[

Va(h) = E

"Z T hea (h)

0(s +ω) e�rτdτ +

Z +∞

T hea (h)ωe�rτdτ jh0 = h

#

1 Va(h) satis�es on ]he ,+∞[ the DEσ2a2V 00a (h) + λa (µa � h) V 0a(h)� r Va(h) + s +ω = 0

2 Va(h) satis�es�Va(he ) = ω

r : BClimh!+∞ Va(h) = s+ω

r : BC

�


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Forest Value Function with logging (No ban ever):

Proposition 2(#2) If ha > he does not exist then8h 2 [he ,+∞[

Va(h) = E

"Z T hea (h)

0(s +ω) e�rτdτ +

Z +∞

T hea (h)ωe�rτdτ jh0 = h

#

1 Va(h) satis�es on ]he ,+∞[ the DEσ2a2V 00a (h) + λa (µa � h) V 0a(h)� r Va(h) + s +ω = 0

2 Va(h) satis�es�Va(he ) = ω

r : BClimh!+∞ Va(h) = s+ω

r : BC

�Ben Abdallah / Lasserre (UQAM) Endangered Species with Stochastic Habitat






Va(h) vs. Vb(h)

ha increases with the existence value s and decreases with Ib .

For s not too high, a ban reduces the short-run risk fromRa(ha) = 1� Phe ,µaa (ha) to Rb(ha) = 1� P

he ,µbb (ha).


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Va(h) vs. Vb(h)

ha increases with the existence value s and decreases with Ib .For s not too high, a ban reduces the short-run risk fromRa(ha) = 1� Phe ,µaa (ha) to Rb(ha) = 1� P

he ,µbb (ha).


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The mean durations of the forestry regimes

Proposition 3 If the species is not extinct and ha exists:1 The mean duration of logging eTa, de�ned as the expectedtime for the DP(a) to reach ha for the �rst time when itsh0 = hb :

eTa = 2 Z hb

ha(Sa(ξ)� Sa(ha))ma(ξ)dξ

2 The mean duration of a ban eTb , de�ned as the expected timefor for the DP(b) to hit either hb or he when h0 = ha:eTb = 2nPhe ,hbb (ha)

R hbha(Sb(hb)� Sb(ξ))mb(ξ)dξ

+�1� Phe ,hbb (ha)

� R hahe(Sb(ξ)� Sb(he ))mb(ξ)dξ

o


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The mean durations of the forestry regimes

Proposition 3 If the species is not extinct and ha exists:1 The mean duration of logging eTa, de�ned as the expectedtime for the DP(a) to reach ha for the �rst time when itsh0 = hb :

eTa = 2 Z hb

ha(Sa(ξ)� Sa(ha))ma(ξ)dξ

2 The mean duration of a ban eTb , de�ned as the expected timefor for the DP(b) to hit either hb or he when h0 = ha:eTb = 2nPhe ,hbb (ha)

R hbha(Sb(hb)� Sb(ξ))mb(ξ)dξ

+�1� Phe ,hbb (ha)

� R hahe(Sb(ξ)� Sb(he ))mb(ξ)dξ

oBen Abdallah / Lasserre (UQAM) Endangered Species with Stochastic Habitat





Estimation of the DP

The Spatially Explicit Landscape Event Simulator (SELES)generates caribou habitat series simulating forest managementwith logging and without logging alternatively.

The logging rate assumed constant at 62 000 m3/y , ie MSY(Forsyth et al., 2003).

The main natural perturbation a¤ecting caribou habitat: wild�res (�re rotation of 343 years).

Autoregressive correlation coe¢ cients are decreasing; partialautoregressive correlation coe¢ cients negligible except for the�rst-order coe¢ cient.


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Estimation of the DP (cond�t)

Hence both habitat series with logging (i = a) and withoutlogging (i = b) assumed to be AR1.

Discrete version of DP: hit = µi (1� ρi ) + ρiht�1 + ηi εit with

i = a, b, λi > 0, ρi = e�λi < 1, and η2i = σ2i

1�e�2λi

2λi

Box-Pierce and Breusch-Godfrey Lagrange multiplier testsvalidate this representation (uncorrelated error terms).


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Estimation of the DP (cond�t)

The Maximum Likelihood estimators are:F. Reg. logging allowed: i = a logging banned: i = b

mean st. dev. mean st. dev.

bµi 12.7034 0.00497 13.4504 0.00513bλi 0.0575 0.011 0.0532 0.0106

bσi 0.0028 0.0028


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Application of the RO model:

VM and SP conditions linking Va (h) and Vb (h) at ha and hb :


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Application of the RO model (cont�d):

Sensitivity to s of habitat thresholds ha and hb and short-runextinction risks at the time logging is interrupted:

ha does not exist for values of s below s; s = (1+ 10%) �ω.Ben Abdallah / Lasserre (UQAM) Endangered Species with Stochastic Habitat






Mean durations of bans and logging regimes according to s:


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Bans involve values of h in [he , hb ] ; µb /2 [he , hb ]

Logging regimes involve h 2 [ha,∞]; µa 2 [ha,∞]. At µa thedeterministic component of the motion of h is zero: onaverage h does not move as much in a logging regime thanduring a ban.


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Mean durations (duration of logging increases with s)

1 Since hb " with s, bans last longer, the higher s; hence ittakes longer for h < hb to reach hb ;

2 Since ha " with s it takes less time from any h > ha for h toreach ha; this factor reduces the expected duration oflogging; However,

since hb " with s, bans are also introduced at higher levels ofh; this means that it takes longer for h to reach any givenlower level from hb ; this factor increases the expected duration.Furthermore hb being closer to µb , logging starts in a zonewhere h is not moving fast, and more so at higher s: this isanother factor which increases the expected duration oflogging.

In this application, the combination of these three e¤ects isthat the expected duration of logging regimes increases with s.


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since hb " with s, bans are also introduced at higher levels ofh; this means that it takes longer for h to reach any givenlower level from hb ; this factor increases the expected duration.

Furthermore hb being closer to µb , logging starts in a zonewhere h is not moving fast, and more so at higher s: this isanother factor which increases the expected duration oflogging.



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since hb " with s, bans are also introduced at higher levels ofh; this means that it takes longer for h to reach any givenlower level from hb ; this factor increases the expected duration.Furthermore hb being closer to µb , logging starts in a zonewhere h is not moving fast, and more so at higher s: this isanother factor which increases the expected duration oflogging.



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The impact of the discount rate:


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Sensitivity of habitat thresholds and extinction risks hb and hato costs Ia and Ib :


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Optimal Policy versus (1) logging forever (11% at ha) or (2)banning forever until extinction (7% at hb and 9% at +∞):


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Conclusion:

This RO endangered species protection model applies whenhuman activity may a¤ect a species or a natural site adversely.

It optimizes the trade-o¤ between bene�ts from humanactivity and risks to the natural environment.

Partial or total irreversibility is present not only as extinctionis �nal but also as policy changes are costly to introduce andto undo.

Uncertainty a¤ects the evolution of the species or site.

In the empirical application presented, the habitat of thespecies is a stochastic variable which is currently observablebut whose future level is unknown.


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Conclusion (cont�d):

The sole instrument to achieve the forest value is toban/resume logging.

As usual in RO models, the optimal rule takes advantage ofuncertainty in such a way as to increase exposure to favorableoutcomes (when the habitat grows more than expected), whileseeking protection from unfavorable outcomes (low habitatlevels).Interrupting logging when habitat is dangerously low doesnot guarantee that extinction will not occur but reduces itsprobability, thus providing (partial) protection against thatunfavorable outcome.In that respect the model just presented is a rigorousapplication of the precautionary principle.


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The sole instrument to achieve the forest value is toban/resume logging.As usual in RO models, the optimal rule takes advantage ofuncertainty in such a way as to increase exposure to favorableoutcomes (when the habitat grows more than expected), whileseeking protection from unfavorable outcomes (low habitatlevels).

Interrupting logging when habitat is dangerously low doesnot guarantee that extinction will not occur but reduces itsprobability, thus providing (partial) protection against thatunfavorable outcome.In that respect the model just presented is a rigorousapplication of the precautionary principle.


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The sole instrument to achieve the forest value is toban/resume logging.As usual in RO models, the optimal rule takes advantage ofuncertainty in such a way as to increase exposure to favorableoutcomes (when the habitat grows more than expected), whileseeking protection from unfavorable outcomes (low habitatlevels).Interrupting logging when habitat is dangerously low doesnot guarantee that extinction will not occur but reduces itsprobability, thus providing (partial) protection against thatunfavorable outcome.

In that respect the model just presented is a rigorousapplication of the precautionary principle.


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The sole instrument to achieve the forest value is toban/resume logging.As usual in RO models, the optimal rule takes advantage ofuncertainty in such a way as to increase exposure to favorableoutcomes (when the habitat grows more than expected), whileseeking protection from unfavorable outcomes (low habitatlevels).Interrupting logging when habitat is dangerously low doesnot guarantee that extinction will not occur but reduces itsprobability, thus providing (partial) protection against thatunfavorable outcome.In that respect the model just presented is a rigorousapplication of the precautionary principle.


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a real option approach to the protection of a habitat ... · context and motivation main model™s...

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