The renewable resource management nexus: impulse versus continuous harvesting policies

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The renewable resource management nexus: impulse

versus continuous harvesting policies

Alain Jean-Marie, Mabel Tidball, Michel Moreaux, Katrin Erdlenbruch

To cite this version:

Alain Jean-Marie, Mabel Tidball, Michel Moreaux, Katrin Erdlenbruch. The renewable resource

management nexus: impulse versus continuous harvesting policies. 2009. �hal-02821443�

« The Renewable Resource

Management Nexus : Impulse versus

Continuous Harvesting Policies »

Alain JEAN-MARIE, Mabel TIDBALL,

Michel

MOREAUX,

Katrin

ERDLENBRUCH

Impulse versus Continuous Harvesting Poli ies

✩

Alain Jean-Marie a ,MabelTidball

∗

,b ,Mi hel Moreaux ,KatrinErdlenbru h d a

INRIAandUMRLIRMM,161RueAda, 34392MontpellierCedex5,Fran e. b

INRAandUMRLAMETA,2pla e P.Viala,34060Montpellier Cedex1,Fran e.

ToulouseS hoolof E onomi s,IDEIandUMRLERNA,21allée deBrienne,31000Toulouse,Fran e. d

CemagrefandUMRG-EAU,361 rueJFBretonBP5095,34196MontpellierCedex5,Fran e.

Abstra t

We explore the link between y li al and smooth resour e exploitation. We dene an impulse ontrol framework whi h an generate both y li al solutions and steady state solutions. For the

y li al solution, we establish a link with the dis rete time model by Dawid and Kopel [1 ℄. For the steady state solution, we explore the relation to Clark's [2℄ ontinuous ontrol model. Our model an admit onvexand on ave protfun tions and allows theintegration of dierent sto k

dependent ostfun tions. We showthatthe stri t onvexityoftheprotfun tion isonlya spe ial ase of a more general ondition, related to submodularity, that ensures theexisten e of optimal y li alpoli ies.

Keywords: optimal ontrol, impulse ontrol, renewableresour e e onomi s,submodularity JEL lassi ation: C61, Q2.

1. Introdu tion

There aretwoopposingtypesofharvestingpoli ies for renewableresour essu hasasheryor a forest. The rst isa ontinuous harvesting poli y. In a ontinuous time model, at ea h point of

time, some portion of the population is harvested. Thus thesize of thepopulationnever hanges abruptlyalthoughthetimederivativeofthepopulationmaybedis ontinuous. Numerousexamples of su h poli ies have been given by Clark [2 ℄,[3℄ for sheries. The Faustmann harvesting poli y

for a balan ed forest also belongs to this type: only the trees having rea hed the optimal felling age are ut. At the other extreme of the spe trum an impulse poli y onsists inharvesting some signi ant part of the population at dis rete pointsof time while leaving the population to evolve

inits naturalenvironment between anytwo onse utive harvestdates. AnexampleisFaustmann's optimal utting poli yof asingle, even-aged, forest stand.

At an aggregate level, optimal impulse poli ies are quite rare for two main reasons. The rst

beingthatrenewableresour esaregenerallys atteredallovertheworldwithspe i hara teristi s sothatsyn hronizedimpulseharvestingofsomanysour esisunlikely. These ondreasonisthatan

✩

Wearegrateful toOlli TahvonenandFranzWirl forhelpful omments. Wealso thanktheparti ipantsofthe EAERE2007andSURED2008 onferen esforinterestingdis ussions.

∗

Correspondingauthor.

opportunities when sto kpiling osts are high. The arbitrage possibility stems from the very fa t

thatsto kpiling ostsarenil for the resour esleftunexploited. Asaresult, thepri e hikesmaybe arbitragedbymoderately hangingtheharvestdateatalowopportunity ost. Howeveratami ro levelsu himpulse poli ies maybeoptimal, or prot maximizingstrategies.

Termansen [4℄ proposedto onvert Clark's standard ontinuous ontrol model(with one state variable) into an impulse ontrol model by simply hanging the de ision variable: instead of the harvest rate, the ontrol is the harvest amount and the harvest times. Using numeri al solutions with a large but nite time horizon, she showed that this impulse ontrol model may generate

dierent typesofextreme harvestingregimes: those similarto Faustmann-like rotationsand those similar to a steady state solution (see also Touza-Montero and Termansen [5 ℄ and Tahvonen [6 ℄, whoinvestigatesasimilarquestioninanage-stru turedforestrymodel). However, Termansenonly

exploredthe linkbetween theoptimal y li al behaviorand theClark-like solutions inthespe i asewhere harvest ostsareindependent ofsto klevels.

Ourmainobje tiveistodis usstheserelationshipsmoresystemati ally. WerevisitTermansen's

impulse ontrolmodelwithinnitetimehorizonandsolveit ompletelyinaverygeneral ase,with generalpopulationgrowthandsto k-dependent ostfun tions. We hara terizetheoptimalsolution by redu ing it to two oupled optimization problems with two variables ea h. This allows us to

formulate onditions underwhi h optimalharvestingbehavioris y li alor smooth. The literature onrmsthat y les indeterministi models

1

mayo urfor various reasons. In dis rete timemodels, Dawid andKopel [1 ,7℄ showed that stri tly onvex gainfun tions maylead

to optimal y li al solutions, inthe absen e of sto k ee ts. Liski et al. [8 ℄ demonstrated the o - urren eof y les inamodelwithin reasingreturnstos ale andla king adjustment osts, equally inthe absen e of sto k ee ts. Finally, inearly appli ations insheries e onomi s, Hannesson [9℄

suggested the pertinen e of so- alled pulse-shing solutions 2

and explained their existen e by in- reasingreturns to s ale and thepresen eof age lasses inthepopulation. Butalso in ontinuous timemodels,smallmodi ationsofthe standardassumptionsmayleadto y li alsolutions. Lewis

andS hmalensee[10 ,11℄foundthat y les anbeoptimalinpresen eofin reasingreturnstos ale, sto kee ts and modest re-entry osts. Wirl [12 ℄ showed that y les arepossible ina modelwith twostate variablesand sto kee ts(seeClark,Clarke andMunro [13℄.

3

),bysimplyintrodu inga

quadrati (insteadof linear) ost fun tion.

Like Wirl, and in ontrast to most other models withoptimal y li al harvesting poli ies (see for exampleDawid and Kopel [1 ℄,Liskiet al. [8 ℄,Lewis andS hmalensee [10 ,11℄), wesupposethe

harvest ostfun tions tobe sto k-dependent, su h asthe ostsproposed byClark [2℄.

Weshowthat the onditions for theexisten e of y li al solutions involve a lose ombination of the growth fun tion and the ost fun tion, thereby emphasizing that the onvexity of the ost

fun tion,or its dependen e on thesto klevel, arenot theonly issuesworth onsidering. We then dis uss how a Clark-like steady-state solution emerges as a limit of small and frequent harvest operations inour model. We also show that we an reprodu e and generalize Dawid and Kopel's

results, although the latter were obtained with a dis rete-time model (whereas timeis ontinuous inour model)and without sto kee ts.

1

Wedonot onsidersto hasti modelsinthefollowing. 2

Pulseshingand hatteringstrategiesimplyverysmalljumpsinthestatevariable. 3

Inthis model the harvest-gain fun tionis on ave, whi h, a ording to Wirl, renders y li al solutions more di ulttoo ur.

hara terize the type ofsolution inse tion3and the optimal y leinse tion4. Wethenestablish

thelinktoClark's ontinuous ontrol solutionand toDawid andKopel'sdis rete ontrolmodelin se tion5. Thelast se tion isdevotedto the on lusion.

2. The impulse ontrol model

2.1. The Model

The resour e dynami s

We onsidera renewable resour e, for whi h dynami s, inthe absen e of anyharvest, is given by:

˙x(t) = F (x(t)) ,

t ≥ 0,

(1)

where

x(t)

isthesizeofthepopulationat anytime

t

and

F

,stationarythroughtime,isthegrowth ratefun tion. The fun tion

F

isassumedto satisfythefollowing onditions.

Assumption1. There exist real numbers

x

sup

and

x

s

su h that

0 < x

s

< x

sup

< +∞

. The fun tion

F : (0, x

sup

) → R

is positive over the interval

(0, x

s

)

and negative over the interval

(x

s

, x

sup

)

, with

F (0) = F (x

s

) = 0

, where

lim

x↓0

F (x) = F (0)

. Thefun tion

F

is measurable and bounded above. Itis assumedthatthe dierential equation(1)admits aunique solution for every

initial sto k

x

0

∈ (0, x

sup

)

.

The quantity

x

sup

is the supremum ofthe arrying apa ity oftheenvironment. The long-run maximum sustainablelevel is

x

s

, the level to whi h the population is onverging for any

x

0

su h that

0 < x

0

< x

sup

.

The harvesting pro ess

We are interested in the optimal e onomi exploitation of this resour e by a dis rete harvest pro ess,i.e. within theframework ofimpulse ontrol models.

4

A ordingly,we dene an impulse exploitation poli yIP

:= {(t

i

, I

i

), i = 1, 2, . . .}

as asequen e ofharvestingdates

t

i

and instantaneousharvests

I

i

,onefor ea hdate. Thesequen e ofdatesmay beempty,niteorinnite. Itissu hthat

0 ≤ t

1

,and

t

i

≤ t

i+1

,

i = 1, 2, . . .

and

lim

i→+∞

t

i

= +∞

. By onvention, we shallassume that ifthesequen e is nitewith

n ≥ 0

values, then

t

i

= +∞

for all

i > n

.

The sequen eof harvests mustsatisfy:

I

i

≥ 0

and

x

i

− I

i

≥ 0 ,

(2)

where

x

i

= lim

t↑t

i

x(t) ,

with

x

1

= x

0

given if

t

1

= 0,

(3)

andsu h thatthe following onstraintshold:

˙x(t) = F (x(t))

for

t

i

< t < t

i+1

with

x(t

i

) = x

i

− I

i

,

i = 1, 2, . . .

(4)

4

Impulse ontrolpoli iesininnitehorizon onsistinanunboundedsequen eofde isions. Forthedis ussionof impulse ontrolmodels,seeforexampleLéonardandLong[14 ℄,SeierstaedandSydsaeter[15 ℄.

˙x(t) = F (x(t))

for

0 < t < t

1

with

x(0) = x

0

if

t

1

> 0.

(5)

Inotherwords:

x

i

isthesizeofthepopulationjustbeforetheharvestingdate

t

i

,and

x

i

− I

i

itssize justafter that same date. If

t

1

= 0

,the population

x

1

is supposed to be inherited from the past, anddenoted by

x

0

. Harvests an not be negative nor ex eed totalpopulationsize. The onditions (2)(5)dene the setof feasible IPs,denoted by

F

x

0

.

The harvester's prots

Monetaryprots generatedbyanyharvestdependuponthesizeofthe at handthesizeofthe population atthe at hingtime. Weassumethat the protfun tionis stationarythrough timeso

thatwhatever

t

i

,

I

i

and

x

i

, the urrent prots at time

t

i

amount to

π(x

i

, I

i

)

. The prot fun tion isassumedto have the followingstandard properties.

Assumption2. The fun tion

π(x, I)

is dened on the domain

D := {(x, I), x ∈ (0, x

sup

)

,

I ∈

[0, x]}

. It is of lass

C

1

, positive and bounded, and su h that

π(x, 0) = 0

,

∀x ∈ (0, x

sup

)

. The derivative

π

I

(x, I) := (∂π/∂I)(x, I)

admitsa limit when

I ↓ 0

for all

x ∈ (0, x

sup

)

.

Prots aredis ountedusing a onstant instantaneous interestrate, denotedby

r

,

r > 0

. Themanager'sobje tiveisto hoosesomepoli ymaximizingthesumofthedis ountedprots,

thatis, to solve the problem(P):

(P)

sup

IP

∈F

x0

Π(

IP

) :=

∞

X

i=1

e

−rt

i

_π(x

i

, I

i

) .

Itis assumedherethat thefun tion

Π

iswelldened overthewholeset

F

x

0

. 5

2.2. The Dynami Programming Prin iple

We use the Dynami Programming approa h to solve our problem. The following theorem insurestheexisten eof a uniquevaluefor theproblem.

Theorem 1. The value fun tion

v(x) =

sup

IP

∈F

x

Π(

IP

)

(6)

isthe unique solution of the following variational equation:

v(x) =

sup

y∈[0,xsup)

t≥0

e

−rt

[π(φ(t, x), φ(t, x) − y) + v(y)] ,

(7)

where

φ(t, x)

is the traje tory of the system attime

t

, solutionof the dynami s (1)with

x(0) = x

.

For this standard proofof dynami programming seeDavis[16 , Theorem (54.19),page 236℄.

5

Observethatweformulateourproblemwitha

sup

andnota

max

be auseweareinterestedinthepossibility thatthemaximumisnotrea hedinsidetheset

Fx

₀

.

Inthisse tionweinvestigatethe impulse ontrolmodelandproposeanapproa hfor

hara ter-izing itssolutions. Ourapproa h istodetermine thestru tureof solutions underthequitegeneral assumptionsof the previous se tion. Thepri e to pay for thisgeneralityis thatour results do not guaranteetheuniquenessof solutions,whi h mustbeexamined on a ase-by- ase basis.

Our line of argument will be the following. First of all, the Dynami Programming prin iple implies that, under any optimal poli y for Problem (P), if the sto k rea hes some level already

attained in the past, the a tion hosen in the past (to harvest or not to harvest) should still be optimal. This mere fa t ombined with the positive growth of the sto k's natural dynami s tends to sele t poli ies that are y li al in the sense that they let the sto k grow to some level, harvest it down sosome other level, and repeat. However, it may still be that underthe optimal

poli y, thesto kneverrea hes twi e the same level. We show that when the gain fun tion has a ertainsubmodularity property,su htraje tories annotbeoptimal. Optimal poli iesaretherefore essentially y li al. Moreover,joining theoptimal y lemustbe donewithat most one harvest.

The optimization problem is thenredu ed to nding: a) what isthe optimal y le; b) what is theoptimalwayto rea h the optimal y le fromagiven initial sto k. Finding theoptimal y leis a relatively simple optimization problemwhi h we all the AuxiliaryProblem. But thesolution

to this problemmay orrespond to degenerate y les,whi h we interpretas ontinuous harvesting poli ies à la Clark. We show in thenext se tion thatthe submodularity assumption is againthe keyto determine whethertheoptimal y leis atrue y le or adegenerate one.

Wepro eednowwiththedenitions and thepre isestatementsof theseprin iples.

3.1. Cy li al Poli iesand the Auxiliary Problem

Cy li al poli ies. A y li alpoli yhastwo omponents: a y lewhi his hara terizedbytwovalues

x

and

x

¯

with

x < ¯

x

,orequivalentlybyaninterval

[x, ¯

x]

;andatransitorypartwhi hdes ribeshow thetraje toryevolvesfrom the initial sto kto the y le. The transitory part onsistsin, at most, one harvest, su h thattheremaining populationislessthan

x

¯

. Werst on entrate on the y le.

Hen e, a y le has two main parameters, whi h are su h that

0 ≤ x < ¯

x ≤ x

s

. 6

When in its y li al part, a poli y a ts as follows: a) let the population grow to

x

¯

; b) harvest until

x

; and repeat. Su hapoli yappliesonlytoinitial populations

x

0

≤ ¯

x

. Inotherwords,thetransitorypart an be dispensed withonlyfor su h an initial population.

Gain under a y li al poli y. We will denote by

G(x, ¯

x, x

0

)

the value of dis ounted prots in a poli y without the transitory part, applied to an initial populationof

x

0

. The omplete denition ofthe fun tion

G

involvesseveral ases, orresponding to thelimit asesfor

x

¯

and

x

.

Itis onvenient to denethefun tion

τ (x, y)

asthetimene essaryfor thedynami s togofrom value

x

to

y

,

x ≤ y

. Itturns out thatfor all

0 < x ≤ y < x

s

:

τ (x, y) =

Z

y

x

1

F (u)

du.

(8)

Sin e,byAssumption1,

F (x

s

) = 0

,theintegral dening

τ (x, y)

issingularwhen

y = x

s

. Thelimit when

y → x

s

may therefore be nite or innite, depending on the fun tion

F

. Another feature

6

Sin e

x

¯

represents the populationlevel untilwhi hthe resour e grows before harvesting, thereis nopoint in onsidering

x > xs

¯

sin ethepopulation annotgrowtosu halevel.

of Assumption 1 is that

F (0) = 0

. Consequently, if

x(0) = 0

, a solution to the dynami s (1) is

x(t) = 0

for all

t ≥ 0

. Thisimplies the onvention that

τ (0, y) = +∞

if

y > 0

,and

τ (0, 0) = 0

. 7

The value ofthe total protfun tion

G

an beexpressedas: i) If

0 ≤ x < ¯

x ≤ x

s

:

G(x, ¯

x, x

0

) := π(¯

x, ¯

x − x)

e

−rτ(x

0

,¯

x)

1 − e

−rτ(x,¯

x)

.

(9)

The onventionisthat: if

x = 0

,theterm

exp(−rτ (x, ¯

x))

shouldberepla edby0. Likewise,

exp(−rτ (x, ¯

x))

and

exp(−rτ (x

0

, ¯

x))

are0 if

x = x

¯

s

and

lim

y→x

s

τ (x, y) = +∞

. ii) For

x = ¯

x

,Assumption2 allowsto dene

G

by ontinuity as:

G(x, x, x

0

) = π

I

(x, 0)

F (x)

r

e

−rτ

(x

0

,x)

_.

(10)

For the ases

x = ¯

x

, the value

G

is not that of a well-dened impulse ontrol poli y, but that of some ontinuous harvestingpoli y,whi h anbe seenasadegenerate impulse poli y.

Finally, by using the fa t that

τ (x, y)

dened in (8) is also dened for

y ≤ x

, expressions (9) and(10) provide values forthefun tion

G

when

x

0

> ¯

x

aswell. Of ourse,these situationsdo not orrespondto animplementableharvestingpoli y,andthe fun tionlosesitse onomi meaning. In subse tion 3.3 we will study the transitory partof a y li al poli yfor whi hthe ase

x

0

> ¯

x

has ane onomi meaning.

The auxiliaryproblem

Having dened the fun tion

G(x, ¯

x, x

0

)

for all

0 ≤ x ≤ ¯

x ≤ x

s

and all

0 ≤ x

0

≤ x

s

, we now dene theauxiliary problem(AP):

(AP)

:

max

x,

x; 0≤x≤¯

¯

x≤x

s

G(x, ¯

x, x

0

).

Under Assumption2 it turns out that

G

is lower semi- ontinuous asa fun tion of

(x, ¯

x)

. The problem (AP) has therefore always a solution. For the purpose of the dis ussion to ome, it is importanttodistinguishthe asewherethesolutionissu hthat

x = ¯

x

,fromthe asewhere

x 6= ¯

x

. We all therstsituation a diagonalsolution, andthese ond one anon-diagonal solution.

3.2. Submodularity and OptimalTraje tories

In this paragraph,we introdu e a submodularity assumption on theprot fun tion

π

. Appen-di es A.1and A.2 provide results on the onsequen esof this assumption on theshape of optimal traje tories for Problem (P).Consider thefollowing assumption.

Assumption3. Thefun tion

π

issu h that:

π(a, a − c) + π(b, b − d) ≤ π(a, a − d) + π(b, b − c)

(11) for every

d ≤ c ≤ b ≤ a

.

7

Assumption3meansthattheprotgeneratedbyabigharvestinalargepopulation,

π(a, a−d)

, augmented by the prot resulting from a small harvest ina medium sized population,

π(b, b − c)

, is greater than the sum of prots generated by two medium sized harvests, the rst in a large population,

π(a, a − c)

, andthe se ondinamedium sizedpopulation,

π(b, b − d)

. At thelimit, for

c = b

,onebigharvest,

π(a, a − d)

,isbetterthan two mediumharvests,

π(a, a − c)

and

π(c, c − d)

, redu ing the population to the same level after the harvests, i.e.

d

. Assumption 3 implies two essential onsequen es. First, it is more rewarding to harvest a large part of the sto k, rather than two smaller parts. Se ond, at the optimal sto k level, regular harvesting poli ies are more worthwhile than irregular poli ies, withoverlappingsto ks.

In somesituations, we shallrefer to astri t Assumption3,meaning that:

π(a, a − c) + π(b, b − d) < π(a, a − d) + π(b, b − c)

(12) for every

d < c < b < a

.

The following properties arewell-knownor easy to he k.

Lemma 1. Assumethat

π

satises Assumption3. Then:

i) Let

g(x, y) = π(x, x − y)

be dened for

0 ≤ y ≤ x ≤ x

sup

. Then

g

issubmodular. 8

ii) If

π

has se ond-order derivatives, then

π

xI

+ π

II

≥ 0 .

iii) If Assumption2 holds aswell, then the followinginequality holdsfor all

z ≤ y ≤ x

:

π(x, x − y) + π(y, y − z) ≤ π(x, x − z) .

(13)

iv)

If

π(x, I) = R(I)

, then

R

is onvex. Conversely,if

R

is onvex, Assumption3 holds.

These properties are linked to severale onomi assumptions: Initiating theharvesting pro ess is ostly (Lemma 1

iii)

). Hen e, y les are optimal if resour e managers an take advantage of some form of e onomies of s ale. Thisis the ase, for instan e, if therevenue fun tion is onvex, whi his onesub- ondition ofAssumption3 (Lemma1

iv)

) inthe aseof sto k-independent osts. In addition, when

π

is linear in

I

, harvests and resour e sto ks are omplementary (Lemma 1

ii)

) andhen e, any additional harvest, and resulting prots, an only be obtained bywaiting and lettingtheresour ere over,whi h omesat a ost. Notethat ondition(13) , withstri tinequality, is lassi ally required to insure the existen e of optimal impulse ontrol poli ies (see for instan e

Davis[16℄).

In ontrast to usual assumptions on the stri t onvexity of the prot fun tion, Assumption 3

is more general as it overs the ase of obje tive fun tions with multiple variables. It applies to onvex- on ave prot fun tionsandis independent ofanyparti ular form ofthedynami s

F (·)

.

8

Afun tion

g(x, y)

issubmodularifforall

a, b, c, d

su hthat

max(c, d) ≤ min(a, b)

:

Nowwearegoingtoshowtheprin ipalrelationbetween problems(P)and(AP).Theresultsof thisse tionarepartlybasedonthepropertythatsolutionstoProblem(AP)turnoutnottodepend on

x

0

,asstatedinLemma 8,see Appendix A.3. Consequently, we an talkof solutions

(x

∗

_{, ¯}

_x

∗

₎

to

theauxiliary problem(AP)independently of

x

0

. Wethenmake thefollowing assumption: Assumption4. The problem (AP) has a unique solution, denoted with

(x

∗

_{, ¯}

_x

∗

₎

, whi h is su h that

x

∗

_{< ¯}

_x

∗

.

The transitory problem

Under Assumption 4, let us dene thefollowing optimization problem (TP), whi h formalizes

the Transitory Problem. The transitory part of a y li al poli y onsists in a) letting grow the sto k until some value

x

; b) harvesting from

x

down to

y

for

y ≤ ¯

x

∗

; ) applying the y le with theharvesting interval

[x

∗

_{, ¯}

_x

∗

_]

from then on. The question is how to hoose the quantities

x

and

y

. The answeris givenbythesolutions ofthe following optimization problem:

(TP)

:

max

x,y;

0≤y≤x≤xs

x0≤x; y≤¯

x∗

e

−rτ(x

0

,x)

_{[π(x, x − y) + G(x}

∗

_{, ¯}

_x

∗

_{, y)] .}

The following theorem hara terizes thesolutions to theproblem(P).

Theorem 2. AssumethatAssumptions14hold. Let

(x

∗

_(x

0

), y

∗

(x

0

))

solvethemaximization prob-lem(TP). Then the valuefun tion of (P) is:

v(x

0

) =







G(x

∗

_{, ¯}

_x

∗

_{, x}

0

)

if

x

0

< ¯

x

∗

e

−rτ

(x

0

,x

∗

(x

0

))

_[π(x

∗

_(x

0

), x

∗

(x

0

) − y

∗

(x

0

)) + G(x

∗

, ¯

x

∗

, y

∗

(x

0

))] if

x

s

≥ x

0

≥ ¯

x

∗

.

(14) Moreover there existsa solutionof (P) whi his y li al andgiven by:

t

1

= τ (x

0

, ¯

x

∗

),

t

i

= t

1

+ (i − 1)τ (x

∗

, ¯

x

∗

),

i ≥ 1,

x

i

= ¯

x

∗

,

I

i

= ¯

x

∗

− x

∗

,

i = 1, 2, . . . ,

if

x

0

< ¯

x

∗

, and

t

1

= τ (x

0

, x

∗

(x

0

)),

t

2

= τ (y

∗

(x

0

), ¯

x

∗

),

t

i

= t

2

+ (i − 2)τ (x

∗

, ¯

x

∗

),

i ≥ 2,

x

1

= x

∗

(x

0

),

I

1

= x

∗

(x

0

) − y

∗

(x

0

),

x

i

= ¯

x

∗

,

I

i

= ¯

x

∗

− x

∗

,

i = 2, . . . ,

if

x

0

≥ ¯

x

∗

.

The proof of this result is given in Appendix A.3. The theorem states that any optimal y li al poli y has a y le part with an harvesting interval

[x

∗

_{, ¯}

_x

∗

_]

. It also des ribes the nature of the transitory part of optimal y li al poli ies. In the ase

x

0

< ¯

x

∗

, there is no transitory part, and

the y le isjoined from the start. Inthe ase

x

0

≥ ¯

x

∗

. The transitory part onsistsin lettingthe sto kgrowuntil

x

∗

_(x

0

)

,harvestitdownto

y

∗

_(x

0

)

,thenjointhe y le. Thetypi alformofoptimal traje tories is illustratedin Figure1.

We an nowstate thefollowing relationbetween problems (P)and (AP), theproof of whi his provided inAppendix A.4.

time

(A)

(B)

¯

x

∗

x

(B)

₀

x

∗

x

∗

_(x

0

)

x

(A)

₀

y

∗

(x

0

)

Figure1: Shapeoftheoptimaltraje tory,for

x0

>

x

¯

∗

( ase

(A)

),and

x0

≤

x

¯

∗

( ase

(B)

)

Theorem 3. Let Assumptions 13hold. Then:

i)

If Assumption4 holds aswell, then (P) has a solutionwhi h is y li al.

ii)

If (P) has a solution,then (P) has a solutionwhi h is y li al, and there exists a solutionto problem (AP) when

0 ≤ x < ¯

x ≤ x

s

.

iii)

If the solutionof (AP) is on the boundary

x = ¯

x = x

∗

, then (P) has no solution.

We have therefore shown that there exists a y li al solution to our problem (P) if, and only

if,the solutionto theauxiliary problem(AP)isnon-diagonal. Inotherwords, theexisten eor not of y li al solutions to (P) hinges on the fa t that Assumption 4 holds or not. This question is addressedinthenext se tion.

4. Optimal Cy les

Weinvestigate nowthe problemoflo ating the solutions to Problem(AP). We have seenthat solutions always exist,but they maybe lo atedintheinterior, or on anyoftheboundaries

x = 0

,

¯

x = x

s

or thediagonal

x = ¯

x

.

Itturnsoutthatensuringtheuniquenessofthesolutionisnotaneasytask,evenwithrestri tive yetstandardassumptions,asweargueinse tion4.4. Wethereforelimitourdis ussionto onditions

relatedto the submodularityAssumption 3. We begin inse tion4.1with ne essary onditions for theexisten eofinteriorsolutionsandtheirinterpretation. Westudythe aseofstri tlysubmodular fun tions in se tion 4.2, and the ase of fun tions both submodular and supermodular in se tion

Ne essary onditions for interior solutions to existareprovided bytherst order onditions of theauxiliary problem,whi h we provide as:

Lemma 2. If (

x, ¯

x

) is a solution to the auxiliary problem (AP) with

0 < x < ¯

x < x

s

(interior solution), then the rst order onditions are given by:

π

I

=

r

F (x)

e

−rτ

(x,¯

x)

1 − e

−rτ(x,¯

x)

π(¯

x, ¯

x − x) ,

(15)

π

x

+ π

I

=

r

F (¯

x)

1

1 − e

−rτ(x,¯

x)

π(¯

x, ¯

x − x) .

(16) By rearranging these onditions, weobtain theequivalent:

π

I

F (x)

r

=

e

−rτ

(x,¯

x)

1 − e

−rτ(x,¯

x)

π(¯

x, ¯

x − x) ,

(17)

dπ

dx

= π

x

+ π

I

= π

I

F (x)

F (¯

x)e

−rτ(x,¯

x)

.

(18) The rst ondition states that, at the optimum, the marginal gain from harvesting the resour e, weighted with the growth potential at the new resour e sto k as ompared to the interest rate,

should equal the value of the remaining resour e, 9

out ome of a maximized rotational harvest stream. These ond onditionstatesthatthemarginalgainderivedfromthesto kee tisequalto themarginalgainfromharvestingaugmentedbya orre tingfa tor,whi hdependsonthegrowth

dierential at the lower and upper limit of the rotational y les, thelatter being dis ounted over time. More pre isely,the greater thisgrowth dierential, thegreater the marginalgain due tothe resour esto k.

4.2. Stri t submodularity of the gainfun tion

In this se tion, we show that Assumption 3 in the stri t sense, together with some te hni al assumptions, issu ient to ex lude diagonal solutions to Problem(AP).

Proposition 4. Assume that all maxima

x

m

of the fun tion

x 7→ G(x, x, x

0

)

are su h that

0 <

x

m

< x

s

. If the fun tion

π

has se ond-order derivatives and satises Assumption 3 in the stri t sense (12) , then all solutions to Problem (AP) are non-diagonal.

Theproof isdeferred to AppendixA.5.

4.3. Exa t modularity

We now turn to the ase where Assumption 3 holds with equality in Equation (11) , whi h amountsto requirethatthefun tion

π(x, x − y)

bebothsub-and supermodular. Using Lemma 1, itis not di ult to seethat if

π

admitsse ond-order derivatives, and given that

π(x, 0) = 0

,then itmustbeof the form:

π(¯

x, ¯

x − x) =

Z

¯

x

x

γ(x) dx

(19)

9

for some integrable fun tion

γ(·)

. We shall prove that, under moderate onditions, the problem (AP)doesnot admit non-diagonalsolutions for su h ost fun tions.

Going ba kto the denitions ofSe tion 3.1, we have (see(10) ):

G

d

(x) := G(x, x, x

0

) =

1

r

γ(x)F (x)e

−rτ(x

0

,x)

_,

wherethe hoi eof

x

0

hasnoimpa tonthesolutionoftheoptimizationproblem,aswe have seen. Proposition 5. Assume that the fun tion

G

d

(·)

isof lass

C

1

, and isin reasing, then de reasing for

x ∈ (0, x

s

)

,withan unique maximum at

x

m

. Assumethat the growth fun tion

F (·)

issu h that the integral in (8)diverges when

x ↓ 0

. Assumenally that

G

does not have a maximum at

x = 0

. Thenthe solution of Problem (AP) is unique andgiven by

x = ¯

x = x

m

.

The proofis deferredto Appendix A.6.

4.4. An example of multiple interiorsolutions to Problem (AP)

A ase where Problem (AP) has two distin t interior solutions is onstru ted as follows. The standard logisti fun tion

F (x) = x(1 − x)

is hosen as the growth fun tion. It is on ave. The gainfun tion is hosenas

π(x, I) = a(¯

x) × I

,with,for some onstant

A > 0

,

a(x) = 1 + min



x

100

, A × (x −

2

3

)



.

It anbeeasilyveriedthat

π

satisesAssumption3,sin ethefun tion

a

isstri tly in reasing. Fi-nally,set

r = 0.01

. Numeri alinvestigationthenrevealsthatthefun tion

G(x, ¯

x, x

0

)

orresponding tothisdatahastwolo almaxima: onewith

x < 2/3

¯

andonewith

x > 2/3

¯

. Thelo aloptimalityof therstoneresultsfromthe ombinationofalargegrowthratewithasmallgainper y le. Cy les

are short for this solution. The se ond lo al optimum results from the ombination of a smaller growthratewithalargergainat ea hharvest. Thetwo lo almaxima anbegiventhesame value bysetting the onstant

A

to approximately 1.23.

5. Links between ImpulseControl Models and Other Control Models

5.1. Comparison with Clark'sModel

Wemaynowestablisharstlinkbetweenourgeneralimpulse ontrolmodelandthe ontinuous ontrolmodel,asproposed byClark [2℄.

Consider a solution of problem (AP) on the boundary

x = ¯

x

. The maximization problem be omes:

max

0≤x≤x

s

G(x, x, x

0

),

where

G

isgiven by (10). The rstorder onditionfor this problemis:

π

Ix

(x, 0)F (x) + π

I

(x, 0)[F

′

(x) − r] = 0 .

(20)

This ondition oin ides with the well-known marginal produ tivity rule of resour e exploitation when

π

I

(x, 0)

is the instantaneous prot fun tion (see for example Clark [2 ℄ or Clark and Munro

optimal ontrol problem: (CP)

max

h(·)

Z

∞

0

e

−rt

π

I

(x(t), 0) h(t) dt ,

˙x = F (x) − h,

for

x

0

given and

0 ≤ h(t) ≤ h

max

for all

t

. This means that the onditions of a Clark-like steady state solution an also be triggered bytheimpulse ontrol modelthatwe propose.

5.2. Comparison with DawidandKopel's model

Inthisse tion,weshowthatDawidandKopel'smodel[1℄ anbeembeddedwithinours,through a judi ious hoi e of the dynami s, the ost fun tion and thedis ount rate. Then, we explainthe

orresponden ebetween theresults of[1 ℄ and ours.

5.2.1. Growth fun tion and timespan asso iated tothe growth

The modelofDawid and Kopelisindis rete time. The populationdynami s hastheform:

x

t+1

= f (x

t

) − u

t

= min[1, (1 + λ)x

t

] − u

t

with

x

t

, u

t

≥ 0 ∀t ≥ 0

. Wepro eedbyreprodu ingthisbehaviorforourmodel. Whennoharvesting takespla e,wemust have:

˙x(t) = F (x(t))

. Suppose:

F (x) = Ax

if

x < x

s

= 1

and

F (x) = 1 − x

if

x ≥ 1.

It an beveried thatthis fun tion satisesAssumption1. 10

Integratingthedierential equation, we ndthatthe sto kevolvesa ordingto thefollowing fun tion:

x(t) = φ(t, x

0

) = min(x

0

e

At

, 1) .

Inordertoreprodu ethedynami sofDawidandKopel'sdis rete-timemodel,wexatimeduration

∆

,and set:

x

t+1

= φ(∆, x

t

)

. The dynami s areequivalent when

f (x

t

) = φ(∆, x

t

)

for all

x

t

,whi h isthe ase when:

(1 + λ)x

t

= x

t

e

A∆

.

Wededu ehowthe marginalgrowthfa tor

A

mustbedenedintermsofDawidandKopel'sfa tor

1 + λ

:

A =

log(1 + λ)

∆

.

Letus ompute the time span ne essary for the dynami s to get from

x

to

y

in terms of thenew notation: for every

x ≤ y ≤ 1

,

τ (x, y) =

Z

y

x

1

F (u)

du =

Z

y

x

1

Au

du =

1

A

log

y

x

.

10

For the undis ounted gains

π

, the orresponden e withDawid and Kopel's model is made by setting

π(x, I) = R(I)

. Note that for this parti ular form of the gain fun tion, Condition (11) is equivalent to the onvexityof

R

,a ording toLemma 1

iv)

.

Next, the orresponden e for dis ounting rates in both models is established as follows. The dis rete-timedis ountfa torbeing

δ

andthe ontinuous-timedis ountratebeing

r

,weshouldhave:

δ

t

_{= e}

−rt∆

, that is:

log δ = −r∆

. Finally, Dawid and Kopel's introdu e a threshold quantity

a

dened as:

a = −

log δ

log(1 + λ)

=

r∆

A∆

=

r

A

.

5.2.3. The maximization problem

We pro eed with the denition of the fun tion

G

whi h is the basis of the auxiliary problem (AP). Two ases mustbe onsidered: diagonal or non-diagonal.

Non-Diagonal where

x < ¯

x

. Inthis ase,

G(x, ¯

x, x

0

) =

R(¯

x − x)(

x

0

¯

x

)

r

A

1 − (

x

_x

_¯

)

A

r

=

R(¯

x − x)(

x

0

¯

x

)

a

1 − (

x

_¯

_x

)

a

.

(21)

Thisexpressionholds even when

x = x

¯

s

= 1

and

x = 0

.

Diagonal where

x = ¯

x

. Given that

π(x, I) = R(I)

,we have

lim

I→0

π

I

(x, I) = R

′

₍₀₎

,when e:

G(x, x, x

0

) = R

′

(0)

Ax

r



x

₀

x



_A

r

= R

′

(0)

x

a

0

a

x

1−a

_.

(22)

Dawid andKopeldene the elasti ityof gainsasthefun tion:

ε(x) =

R

′

_(x)x

R(x)

.

(23)

We have:

Lemma 3. The following results holdfor all

0 ≤ x ≤ ¯

x ≤ 1

:

i)

If

a < 1

, then:

0 < ε(¯

x − x) − 1 <

∂G

∂ ¯

x

(x, ¯

x, x

0

) < ε(¯

x − x) − a .

(24)

ii)

If

a > 1

, then:

ε(¯

x − x) − a <

∂G

∂ ¯

x

(x, ¯

x, x

0

) < ε(¯

x − x) − 1 .

(25)

iii)

If

a = 1

, then:

1

G(x, ¯

x, x

0

)

∂G

∂ ¯

x

(x, ¯

x, x

0

) =

1

¯

x − x

(ε(¯

x − x) − 1) > 0 .

(26)

Lemma 4. If

a > 1

, then

∂G

∂x

< 0 .

Theproofoftheselemmas followsfromstandard al ulations. Thefa tthat

ε > 1

followsfrom the onvexityof

R

andthefa tthat

R(0) = 0

.

Asa onsequen e ofLemmas 3and 4,we havethe following optimization results:

Lemma 5. The fun tion

G

given by (21) and (22) has the followingproperties:

i)

If

a < 1

, then there existsa unique

(x, ¯

x)

, with

0 < x ≤ ¯

x = 1

, solutionto:

max

0≤x≤¯

x≤1

G(x, ¯

x; x

0

) .

(27)

ii)

If

a > 1

, then for all

x

¯

,

arg

max

0≤x≤1

G(x, ¯

x; x

0

) = 0.

(28)

iii)

If

a = 1

, then arg

max

0≤x≤¯

x≤1

G(x, ¯

x; x

0

) ∈ [(0, 1)]

2

_.

(29)

5.2.4. Relations withthe Results by Dawid andKopel

As Dawid and Kopel, we an now showthefollowing results. Aslong theelasti ityof gains

ε

, averagedoverthe harvestofoneperiod,islargerthanthethreshold

a

,theoptimalpoli yistowait anddeferharvesting. Inversely,whenthe average elasti ityofgains

ε

issmallerthan

a

,immediate harvestingisoptimal(see[1,Lemma 5℄). When

a = 1

,the de isionmakerisindierentin hoosing between harvestingimmediately or harvesting inthenextperiod.

Other results of Dawid and Kopel address the question of whether immediate extin tion is optimalor not. These resultsarereprodu ed byour analysisaswell.

6. Con lusion

We have proposed an impulse ontrol framework for the management of renewable resour es whi h isgeneral enough to in lude on ave and onvex gainfun tions, as well assto k dependent

ostfun tions. Theoptimal management oftheresour eisexpressedasoptimizationproblem(P), thesolution of whi h is shownto satisfythedynami programming prin iple. By introdu ingthe lassof  y li al poli ies, we have redu ed the solution of Problem (P) to thesequential solution

of two stati optimization problems with two variables ea h, whi h we an solve. With the help of theAuxiliary Problem, we an dene the optimal y le. Withthe Transitory Problem, we an des ribe theevolutionfrom the initial sto kto the y le.

Central to our solution framework is the submodularity ondition, whi h is ne essary for the existen eof y les. This onditionis moregeneral than thestri t onvexity of theprotfun tion, as it also overs the ase of obje tive fun tions with multiple variables. Thus, the existen e of e onomies of s ale is only one possible ondition for the o urren e of y les, whi h depends on

themore omplex intera tion between dis ounted gains,(sto kdependent) ost fun tions andthe population growthdynami s.

y li alsolutionswhi h orrespondtoasmoothsteadystatesolution. Thee onomi andbiologi al

onsequen esof thesetwo typesof equilibria might be verydierent,espe ially ifthreshold values exist. For example, the y li al solution may temporarily deplete the population underneath the level that would be desirable for the maintenan e of the food- hain. These onsequen es are not

taken into a ount inour model.

Our impulse ontrol model an generate thesteady state solution thatClark des ribed for his onestatevariablemodelwitha on avegrowthfun tion. We analso repli atethe y li alpoli ies des ribed by Dawid and Kopel ina dis rete time framework witha quasi-linear growth fun tion.

Thisallows us to laim that our model is a meta-model. The linkbetween these models an be expressedthroughtheir responsiveness tothe submodularity ondition.

Re entbioe onomi modelshavestrengthenedtheimportan eofun ertainty,forexamplelinked

towhether onditionsortotheavailabilityofsto ks. Furtherresear h ouldin ludesu hun ertainty andalso onsiderthemanager's riskaversioninasimilarimpulse ontrol framework. E onometri appli ations ould help to he k whether ontinuous or dis rete representations of the harvest

de isionsaremoreappropriateinpra ti e,andhowtospe ifygrowthand ostfun tions. Depending onthefun tionalforms hosen,theoptimalharvestingpoli ies anthenbedenedwithintheabove framework.

Referen es

[1℄ H. Dawid,M. Kopel,Onthe e onomi allyoptimalexploitationof arenewable resour e: The aseof a onvex environmentanda onvexreturnfun tion,J.E on.Theory76(1997)272297.

[2℄ C. Clark, Mathemati al Bioe onomi s, The Optimal Management of Renewable Resour es, John Wiley and Sons, 1990.

[3℄ C.Clark, G.Munro,Thee onomi sofshingand modern apitaltheory: A simpliedapproa h,J.Environ. E on.Manage.2(1975)92106.

[4℄ M. Termansen, E onomies of s ale and the optimality of rotational dynami sinforestry, Environ. Resour e E on.(2007)643659.

[5℄ J. Touza-Montero, M. Termansen,The Faustmann modelas a spe ial ase, Te h.rep., unpublishedworking paper,EnvironmentDepartment,UniversityofYork,UK(2002).

[6℄ O. Tahvonen, Optimal hoi e between even-and uneven-agedforestry, Te h.Rep. 60, Working Paperof the FinnishForestResear hInstitute(2007).

[7℄ H. Dawid, M. Kopel,Onoptimal y les indynami programmingmodelswith onvexreturnfun tion, E on. Theory13(1999)309327.

[8℄ M. Liski, P. M. Kort, A. Novak, In reasing returns and y les inshing, Resour e Energy E on. 23 (2001) 241258.

[9℄ R.Hannesson,Fisherydynami s: Anorthatlanti odshery,Can.J.E on.8(2)(1975)151173.

[10℄ T. R. Lewis, R. S hmalensee, Non onvexity and optimal exhaustion of renewable resour es, Int. E on. Rev. 18(3)(1977)535552.

[11℄ T. Lewis, R. S hmalensee, Non- onvexity and optimal harvesting strategiesfor renewable resour es, Can. J. E on.12(4)(1979)677691.

[12℄ F.Wirl,The y li alexploitationofrenewableresour esto ksmaybeoptimal,J.Environ.E on.Manage.29 (1995)252261.

[13℄ C.Clark,F.Clarke,G.Munro,Theoptimalexploitationofrenewableresour esto ks: Problemsofirreversible investment,E onometri a47(1979)2547.

[14℄ D.Léonard,N.V.Long,OptimalControlTheoryandStati OptimizationinE onomi s,CambridgeUniversity Press, 1998.

[15℄ A.Seierstad,K.Sydsaeter,OptimalControlTheorywithE onomi Appli ations,North-Holland,Amsterdam, 1987.

A.1. Submodularity andTraje tories

We prove here traje tory omparison results whi h are a onsequen e of the submodularity Assumption3. Before statingthe results,we need some preliminaryexplanations.

Consider animpulse ontrolpoli y

ICP

whi h issu hthatthereexists

i

and

j

with

i < j

and:

x

j

− I

j

≤ x

i

− I

i

≤ x

j

≤ x

i

,that is, overlapping harvests. Denotewith

a = x

i

,

b = x

j

,

c = x

i

− I

i

and

d = x

j

− I

j

. Let

ℓ = j − i

and

δt = t

j

− t

i

. Consider the following two modi ations of the referen epoli yICP:

Poli y A ( opyapie e oftraje toryfrom

c

to

b

): for

k < j

,

t

A

k

= t

k

,

I

A

k

= I

k

; for

k = j

,

t

A

j

= t

j

,

I

A

j

= b − c

; for

k > j

,

t

A

k

= t

k−ℓ

+ δt

,

I

A

k

= I

k−ℓ

.

Poli y B (remove thepie eof traje tory from

c

to

b

): for

k < i

,

t

B

k

= t

k

,

I

B

k

= I

k

; for

k = i

,

t

B

k

= t

k

,

I

B

k

= a − d

; for

k > i

,

t

B

k

= t

k+ℓ

− δt

,

I

B

k

= I

k+ℓ

.

Thesepoli ies anbevisualizedinFigure2,whi hrepresentstheevolutionofthepopulationunder

ea hofthethreepoli ies. Thetrianglerepresentstherestofthetraje tory,whi histhesameforall three poli ies, ex eptfor a shift intime. There tangle represents an arbitrarypie e of traje tory, whi h an possiblyexitthe range

[b, c]

.

11

The resultis:

Lemma 6. Consider an impulse ontrol poli y ICP whi h is su h that there exists

i

and

j

with

i < j

and:

x

j

− I

j

≤ x

i

− I

i

≤ x

j

≤ x

i

. Then:

i)

If Assumption3 holds in the stri tsense (12),then one of poli iesA or B onstru ted above yields stri tlylargerprotsthan ICP.

ii)

If Assumption3 holds withequality in (11) andif ICP isoptimal, then poli ies Aand B are optimalas well.

Proof. The dis ounted prots

G

asso iatedwith theoriginal poli yICP an bewritten as:

G = V

0

+ R

i

π(a, a − c) + R

i

V

1

+ R

j

π(b, b − d) + R

j

V

d

where

R

i

and

R

j

arethe dis ounts:

R

i

= e

−rt

i

R

j

= e

−rt

j

,

11

Thesituationwhere

b

= c

is allowed,inwhi h ase thepie eoftraje torymaybeempty. Inthat ase,thereis adoubleharvestatthesameinstantintime.

andwhere

V

0

,

V

1

and

V

d

arethe urrent-valuegainsasso iatedwiththerstpartofthetraje tory, andthe pie es ofthe traje tory,respe tively,intheintervals

(t

i

, t

j

)

and

(t

j

, +∞)

:

V

0

=

i−1

X

k=1

e

−rt

k

_π(x

k

, I

k

)

V

1

=

j−1

X

k=i+1

e

−r(t

k

−t

i

)

_π(x

k

, I

k

)

V

d

=

∞

X

k=j+1

e

−r(t

k

−t

j

)

_π(x

k

, I

k

) .

Thetotal dis ountedgains asso iated withpoli ies Aand Bare:

G

A

= V

0

+ R

i

π(a, a − c) + R

i

V

1

+ R

j

π(b, b − c) + R

j

V

1

+ R

j

ρ π(b, b − d) + R

j

ρ V

d

G

B

= V

0

+ R

i

π(a, a − d) + R

i

V

d

,

with

ρ = R

j

/R

i

= exp(−r(t

j

− t

i

))

. A ordingly,modi ationsinprotsimpliedbyswit hingfrom theoriginal poli yto either Aor Bare:

G − G

A

= R

j

(π(b, b − d) − π(b, b − c) + V

d

− ρπ(b, b − d) − V

1

− ρV

d

)

G − G

B

= R

i

(π(a, a − c) − π(a, a − d) + V

1

+ ρπ(b, b − d) + ρV

d

− V

d

) .

Asa onsequen e, we have the following identity:

π(a, a − c) + π(b, b − d) − π(a, a − d) − π(b, b − c) =

1

R

j

(G − G

A

) +

1

R

i

(G − G

B

) .

Under Assumption 3, the left-hand side is negative. If the inequality in (11) is stri t, it is even stri tlynegative. Thisimpliesthat one atleastof

G

A

or

G

B

isstri tly larger than

G

.

If equality holds (11) and the poli y ICP is assumed to be optimal, then

G

A

= G

B

= G

and poli ies A and Bare optimalaswell.

Consequen es of Lemma6 on theoptimalityof poli ies an bestated as:

Corollary 6. Consider an impulse ontrol poli y ICP whi h is su h that there exists

i

and

j

with

i < j

and:

x

j

− I

j

≤ x

i

− I

i

≤ x

j

≤ x

i

. If Assumption3 holds in the stri t sense (12), then ICP annot be optimal.

A.2. Dynami Programming andTraje tories

In this appendix, we propose a te hni al result whi h isuseful in a varietyof situations. This

omparisonoftraje toriesissimilartoLemma6butitisprovidedbytheappli ationoftheDynami Programming prin iple of Theorem1.

Before stating the result, we need some preliminary explanations. Assume that a poli y

P

is su h that

x

i+1

≥ x

i

. Let

δt = τ (x

i

− I

i

, x

i

)

. Consider the following modi ations of thereferen e poli y

P

:

Poli y A (remove the harvestingat

t

i

) for

k < i

,

t

A

k

= t

k

,

I

A

k

= I

k

; for

k ≥ i

,

t

A

k

= t

k−1

− δt

,

I

A

k

= I

k−1

.

t

A

j

ti

t

A

i

t

B

i

tj

t

A

j+ℓ

(A)

d

c

b

(B) (A)

a

Figure2:Theoriginalpoli yanditsmodi ationsAandB

Poli y B ( opyon ethe harvestingo urring at

t

i

) for

k ≤ i

,

t

B

k

= t

k

,

I

B

k

= I

k

; for

k > i

,

t

B

k

= t

k+1

+ δt

,

I

B

k

= I

k+1

.

Poli y C (reprodu e innitelytheharvesting o urringat

t

i

) for

k < i

,

t

C

k

= t

k

,

I

C

k

= I

k

; for

k ≥ i

,

t

C

k

= t

i

+ (k − i)δt

,

I

C

k

= I

i

.

Assume now that the poli y

P

is su h that

x

i+1

∈ (x(t

+

i−1

), x

i

]

,where by onvention,

t

−1

= 0

in the ase

i = 1

. In that ase, there exists a time

T = t

i

− τ (x

i+1

, x

i

)

su h that

x(T ) = x

i+1

. As above,let

δt = τ (x

i

− I

i

, x

i

)

and dene thepoli ies A,Band Cexa tlyasabove.

Thesepoli ies areillustrated inFigure3

a)

inthe rst ase,and

b)

inthese ond one. We an nowstate the result:

Lemma 7. Consider an impulse ontrol poli y

P

whi h is su h that either

x

i

∈ (x(t

+

i

), x

i+1

]

or

x

i+1

∈ (x(t

+

i−1

), x

i

]

for some

i

. Then, the gain of poli y

P

is smaller than that of poli ies

A

or

C

onstru ted above.

Proof. Assume rst that poli y

P

is su h that

x

i

∈ (x(t

+

i

), x

i+1

]

, whi h implies

x

i+1

≥ x

i

. Let

G

P

,

G

A

,

G

B

and

G

C

bethetotalprotsforpoli iesP,A,BandC.Denotewith

V

0

the urrent-value gains asso iated with the part of the traje tory before

t

i

(whi h is ommon to all these poli ies) andlet

G

π

= V

0

+ e

−rt

i

_G

˜

π

for poli ies

π ∈ {

P, A,B,C

}

. It iseasy to seethat

˜

G

P

= π(x

i

, x

i

− I

i

) + e

−rδt

G

˜

A

˜

G

B

= π(x

i

, x

i

− I

i

) + e

−rδt

G

˜

P

˜

G

C

= π(x

i

, x

i

− I

i

) + e

−rδt

G

˜

C

.

(C) (P) (A) (C) (B) (P) (B) a) b) (B) (C) (C) (A) (B) (B) (B) (P) (C) (P)

xi

xi+1

xi+1

xi

Figure3: Theoriginal poli yICP anditsmodi ations A,Band C.Thetrianglerepresentsthe remainderof the traje tory,whi his ommontoICP,AandB,uptoashiftintime.

Consequently, we have the identity:

G

˜

P

− ˜

G

B

= e

−rδt

_{( ˜}

_G

A

− ˜

G

P

)

. This implies that

G

˜

P

≤

max( ˜

G

A

, ˜

G

B

)

. Next, if we have

G

˜

P

≤ ˜

G

B

, then we have

G

˜

B

≤ π(x

i

, x

i

− I

i

) + e

−rδt

_G

_˜

B

so that:

˜

G

B

≤

π(x

i

, x

i

− I

i

)

1 − e

−rδt

= ˜

G

C

.

Thisprovesthestatement.

Consider now the ase

x

i+1

∈ (x(t

+

i−1

), x

i

]

. As argued above, thetime

T = t

i

− τ (x

i+1

, x

i

)

is su hthat

x(T ) = x

i+1

. Let

˜

G

i

be the urrent-value gainsof thedierent poli ies at time

t = T

. It is lear that:

˜

G

P

= e

−r(t

i

−T

)

π(x

i

, x

i

− I

i

) + e

−rδt

G

˜

A

˜

G

B

= e

−r(t

i

−T

)

π(x

i

, x

i

− I

i

) + e

−rδt

G

˜

P

˜

G

C

= e

−r(t

i

−T

)

π(x

i

, x

i

− I

i

) + e

−rδt

G

˜

C

.

Asa result, we have thesame identity:

G

˜

P

− ˜

G

B

= e

−rδt

_{( ˜}

_G

A

− ˜

G

P

)

,and therest of theprevious reasoningapplies.

A.3. Proof of Theorem2

Theproofisseparatedintotwo ases. If

x

0

issmallenough,theproofisprovidedbytraje tory omparisonarguments. For the aseoflarge

x

0

,theproof onsistsinembeddingtheoptimization problems(AP)and(TP) into amoregeneral optimizationproblem, thensolving thismore general problem. The solution turns out to be provided by (AP) and (TP), and satisfy the dynami

programmingequation.

Throughout the rest of this se tion, Assumptions 1 and 2 hold, sothat the fun tion

G

is well dened,and Assumption4is assumedtoholdaswell, sothattheoptimalvalues for(AP),

x

∗

and

¯

x

∗

Let

w(·)

bedened, asin(14) , as:

w(x) =







G(x

∗

_{, ¯}

_x

∗

_{, x)}

if

x < ¯

x

∗

e

−rτ(x,x

∗

_(x))

[π(x

∗

_{(x), x}

∗

_{(x) − y}

∗

_{(x)) + G(x}

∗

_{, ¯}

_x

∗

_{, y}

∗

_(x))]

if

x

sup

≥ x ≥ ¯

x

∗

(30) where

(x

∗

_{(x), y}

∗

_(x))

isany solutionof theproblem(TP) withinitial population

x

0

= x

.

The following resultwill be useful for the proof. Consider problem(AP). Itssolution does not

dependon theinitial sto kvalue

x

0

:

Lemma 8. Assumethat

(x

∗

_{, ¯}

_x

∗

₎

solves (AP) for somevalue of

x

s

> x

0

> 0

. Then it solves (AP) for every value of

x

0

.

Proof. The resultfollows fromthe fa tthatfor all

x

0

,

x

1

:

G(x, ¯

x, x

0

) = e

−rτ(x

0

,x

1

)

G(x, ¯

x, x

1

) .

Thereforethetwofun tionsareproportional,withaproportionalityfa torwhi hisstri tlypositive

if

0 < x

0

< x

s

and

0 < x

1

< x

s

. The problems (AP) for

x

0

and (AP) for

x

1

have therefore the same solutions. If

x

1

= 0

, or if

x

1

= x

s

and

lim

y↑x

s

τ (x, y) = +∞

, then

G = 0

and any

(x

∗

_{, ¯}

_x

∗

₎

maximizesit.

A.3.1. Proof for

x

0

< ¯

x

∗

Lemma 9. IfAssumptions3and 4hold,thenthefun tion

w(x

0

)

solvesthedynami programming equation (7)for all

x

0

< ¯

x

∗

.

A ording to Theorem 1,thevalue fun tion ofproblem(P)veries:

v(x) =

max

t≥0

0≤y≤φ(t,x)

e

−rt

[π(φ(t, x), φ(t, x) − y) + v(y)]

(31)

= max



max

0≤y≤x

[π(x, x − y) + v(y)] ,

(32)

max

¯

x,y;

x<¯

x≤xs

0≤y≤¯

x

e

−rτ(x,¯

x)

[π(¯

x, ¯

x − y) + v(y)]



.

(33)

This breakdown is obtained by separating the ase

t = 0

(expression (32) ) from the ase

t > 0

, and performing the hange of variable

t = τ (x, ¯

x)

in(33) . This hange of variable maps thetime interval

t ∈ (0, +∞)

totheintervalonpopulations

x ∈ (x, x

¯

s

)

or

x ∈ (x, x

¯

s

]

,dependingonwhether

τ (x, y)

diverges ornot when

x ↓ 0

.

Wemustshow thatthefun tion

w(x)

,dened in(30) , isa solutionof Equation(31) . By assumption,

x < ¯

x

∗

M

= max



max

0≤y≤x

[π(x, x − y) + G(x

∗

_{, ¯}

_x

∗

_{, y)] ,}

(34)

max

x<¯

x≤xs

0≤y<¯

x∗

e

−rτ(x,¯

x)

[π(¯

x, ¯

x − y) + G(x

∗

, ¯

x

∗

, y)] ,

(35)

max

x<¯

x≤xs

¯

x∗≤y≤¯

x

e

−rτ(x,¯

x)



π(¯

x, ¯

x − y)

(36)

+ e

−rτ(y,x

∗

(y))

[π(x

∗

_{(y), x}

∗

_{(y) − y}

∗

_{(y)) + G(x}

∗

_{, ¯}

_x

∗

_{, y}

∗

_(y))]



.

Were ognizeinthe term(35)theproblem(TP).Weproverstthatthisisthelargestofthethree.

Consider, for some

y = y

0

, the value in bra kets in (36). It orresponds to a poli y P with two harvests

x → y

¯

0

and

x

∗

_(y

0

) → y

∗

(y

0

)

. Two asesmayhappen, a ordingto whi h of

x

¯

and

x

∗

_(y

0

)

isthelargest.

Case

x ≥ x

¯

∗

_(y

0

)

: in this ase, these two harvests are overlapping (sin e

y

∗

_(y

0

) < ¯

x

∗

≤ y

0

), in whi h aseLemma6applies. Thepoli yPisdominatedbyatleastoneoftwomodi ations. Ifthe dominatingpoli yistheoneex ludingthese ondharvest,thenitsvalueispresentin(35)when

y

has thevalue

y

∗

_(y

0

)

. Ifthe dominatingpoli yis the one withanadditional harvest, thenitis obvious (see for instan e the proof of Lemma 7) that the poli y with a y li al harvesting with interval

[¯

x

∗

_{, y]}

is even better. But this poli y provides a gain equal to

π(¯

x, ¯

x − y) + e

−τ(y,¯

x

∗

₎

G(y, ¯

x

∗

_{, y) ≤}

π(¯

x, ¯

x − y) + e

−τ(y,¯

x

∗

₎

G(x

∗

_{, ¯}

_x

∗

_{, y)}

. Poli yPisthereforeagaindominatedbysomepoli yrepresented in(35) .

Case

x < x

¯

∗

_(y

0

)

: in this ase, Lemma 7 applies, and poli y P is dominated by at least one of twomodi ations. Either the dominating poli yis themodi ation A without ase ond harvest: itsgainisone ofthe values in(35). Orthedominatingpoli yistheonewitha y li alharvesting. The reasoning above thenapplies and there is a value in(35) whi h dominates thevalue in (36) .

We have shown that(36)is smallerthan (35) .

Next,weshowthat(34)isdominatedby(35) . Ea h

y

in(34) orrespondstosomepoli y

P

y

for whi h thetwo rst harvests are

x → y

and

x

¯

∗

_{→ x}

. Sin e

x

issmaller than

x

¯

∗

, we areon e more

inthe situation of Lemma 7. Thepoli y

P

y

istherefore dominated: eitherbythe poli y A whi h onsistsindire tly applying the y le withinterval

[x

∗

_{, ¯}

_x

∗

_]

, or bythe y li al poli ywith interval

[y, x]

. This one is inturndominated bythe y li alpoli yA a ording to Assumption4. Inboth ases,

P

y

isdominated byC. Sin e thegainasso iated with

C

is present in(35)(with

¯

x = ¯

x

∗

and

y = x

∗

),theterm in(34)isdominated by theterm in(35) .

At thisstage, wehave proved that(35) dominatesthe two other terms,sothat:

M =

max

x<¯

x≤xs

0≤y<¯

x∗

e

−rτ

(x,¯

x)

[π(¯

x, ¯

x − y) + G(x

∗

_{, ¯}

_x

∗

_{, y)] .}

It now remains to be proved that the maximum in the right-hand side is rea hed at

x = ¯

¯

x

∗

and

y = x

∗

. Ea h value of the right-hand side is the gain of some poli y P for whi h the two rst harvests are

x

1

= ¯

x

and

x

2

= ¯

x

∗

. Whether

x < ¯

¯

x

∗

or

x > ¯

¯

x

∗

, theappli ation of Lemma 7implies thatPis dominated: either bypoli yA whi hhasthevalue

G(x

∗

_{, ¯}

_x

∗

_{, x)}

,or bypoli y C whi h

hasthe value

G(y, ¯

x, x) < G(x

∗

_{, ¯}

_x

∗

_{, x)}