Optimality of Impulse Harvesting Policies

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

To cite this version:

Katrin Erdlenbruch, Alain Jean-Marie, Michel Moreaux, Mabel Tidball. Optimality of Impulse Har-

vesting Policies. [Research Report] 13031, LIRMM. 2010, pp.28. �hal-00864187�

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Institut d Economie Industrielle (IDEI) Manufacture des Tabacs

Aile Jean-Jacques Laffont 21, allée de Brienne 31000 TOULOUSE FRANCE Tél. + 33(0)5 61 12 85 89 Fax + 33(0)5 61 12 86 37 www.idei.fr contact@cict.fr

April, 2010

n° 603

“Optimality of Impulse Harvesting Policies”

Katrin ERDLENBRUCH, Alain JEAN- MARIE, Michel MOREAUX and

Mabel TIDBALL

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Optimalityof Impulse Harvesting Poliies

KatrinErdlenbruh · ^Alain^Jean-Marie ·

Mihel Moreaux · ^Mabel^Tidball

thedateofreeiptandaeptaneshouldbeinsertedlater

Keywords optimalontrol,impulseontrol,renewableresoureeonomis,

submodularity

JEL lassiation:C61,Q2

Abstrat Weexplorethelinkbetweenylialandsmoothresoureexploita-

tion.Wedeneanimpulseontrolframeworkwhihangeneratebothylial

solutionsand steadystatesolutions. Fortheylial solution,weestablish a

linkwiththedisrete-timemodelbyDawidandKopel(1997).Forthesteady

state solution, we explore the relation to Clark's (1976) ontinuous ontrol

model. Ourmodel anadmit onvexand onaveprotfuntions and allows

theintegrationofdierentstokdependentostfuntions.Weshowthatthe

strit onavity of the prot funtion is only a speial ase of a more gen-

eralondition,relatedtosubmodularity,thatensurestheexisteneofoptimal

ylialpoliies.

1Introdution

Thereexisttwomaintypesofharvestingpoliiesforrenewableresouressuh

asanimalorplantpopulations.Thersttypeofpoliy isthesmoothpoliy.

Inaontinuous timemodel, ateahpointin time,aninnitelysmallpartof

TheauthorsthankLindaNøstbakken,OlliTahvonenandFranzWirlforhelpfulomments

on an early version of this paper. We also thank the partiipants of the EAERE 2007

andSURED2008onferenesforstimulatingdisussions.K.Erdlenbruhaknowledgesthe

nanialsupportfromtheFrenhNationalResearhAgenythroughgrantANR-08-JCJC-

0074-01.M.MoreauxaknowledgesthenanialsupportofINRIA.

Cemagrefand UMRG-EAU,361 rue J.F.Breton, BP5095,34196 Montpellier Cedex5,

Frane.E-mail:Katrin.Erdlenbruhemagref.fr·ÎNRIAândÛMR^LIRMM,¹⁶¹^rueÂda,

34392MontpellierCedex5,Frane.E-mail:ajmlirmm.fr·^Toulouse^Shool^of^Eonomis,

(IDEIandLERNA),21alléedeBrienne,31000Toulouse,Frane.E-mail:mmihelit.fr·

INRAandUMRLAMETA,2 plaeP.Viala,34060 MontpellierCedex1,Frane.E-mail:

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the population is aptured sothat the size of the population never hanges

abruptlyalthoughthetimederivativeofthepopulationsizemaybedisontin-

uous. Numerous examplesofsuh poliieshave been givenin the pioneering

work of Clark and Munro (1975) (see also Clark (1976)) for sheries. The

well-known harvesting poliy of Faustmann (1849) (see also Johansson and

Löfgren(1985))forabalanedforestalsobelongstothis type:onlythetrees

havingreahedtheoptimalfellingageareut.Although foreahtree ohort

thepoliyisanabruptone,fortheforestasawholesuhapoliyisasmooth

one.

At the other extreme of the spetrum an impulse poliy onsists in har-

vestingsome signiant partof thepopulationat somepointsin time while

leavingthepopulationto evolvein itsnaturalenvironmentbetweenanytwo

onseutiveharvestdates. An exampleisagain Faustmann'soptimal utting

poliybutnowforsingle,even-aged,foreststands.

Atanaggregatelevel,optimalimpulsepoliiesarequiterarefortwomain

reasons.Therst isthat renewableresouresaregenerallysattered allover

theworldwithspeiharateristissothatsynhronizedimpulseharvesting

ofsomanysouresisunlikely.Theseondreasonisthatanaggregateimpulse

poliy would indue hikesin the priepath, thus opening thedoor forarbi-

trageopportunitieswhenstokpiling ostsarehigh.Thearbitragepossibility

stems from the veryfat that stokpiling osts are nil for the resouresleft

unexploited. As a result, the prie hikes may be arbitraged by moderately

hangingtheharvestdateatalowopportunityost.Howeveratamirolevel

suhimpulsepoliiesmaybeoptimal,thatis, protmaximizingstrategies.

Weproposeinthispaperamodelofrenewableresouremanagementbased

ontheimpulseontrolframework(f.Vind(1967),LéonardandLong(1998)

orSeierstaedandSydsaeter(1987)).Thismodelgeneralizespreviousdisrete-

time models and ontains, as a limit, the lassial ontinuous-time singular

ontrol model. Weadoptveryweakassumptionsonthe growthfuntion and

ontheprotfuntionwhihisallowedtodependonboththeurrentstokand

thesizeoftheharvest.Inpartiular,wedonotimposeanytypeofonavity.

We haraterize the solution to this problem by reduing it to twooupled

optimization problems with two variable eah. We are then able to disuss

underwhihonditionstheoptimaltrajetoryexhibitsyles ornot.

Cyles in deterministi models may our for various reasons. Thepres-

ene of state variables in addition to the state of the resoure is a well-

doumented reason, both for disrete-time and ontinuous-time models: see

forinstaneBenhabibandNishimura(1985),Wirl(1995)andFeihtingerand

Sorger(1996).The fous ofthe present paperis onone-dimensional models,

where the existene of yles results from other phenomena than hidden

variables.

Indisrete-time, one-dimensionalmodels, yles ourwhen optimaltra-

jetoriesarenotstationary.BenhabibandNishimura(1985)haveshownthat

suhylesourundertheassumptionofonavityandsubmodularityofthe

protfuntion,plusadditionaltehnialassumptions. OlsonandRoy (1996)

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of yles. On the other hand, Dawid and Kopel (1997; 1999)showed that a

stritly onvexgainfuntion dependingonly ontheapture may leadtoop-

timalylialsolutions.Intheliteratureonone-dimensionalontinuous-time

ontrolmodels,ylesmayalsoappear.Indeed,LewisandShmalensee(1977;

1979) found that yles an be optimal in presene of inreasing returns to

sale,stokeetsandmodestre-entryosts.Liskietal.(2001)demonstrated

theourreneofylesinamodelwithinreasingreturnstosaleandmodest

adjustmentosts,intheabseneofstokeets.

Finally,notethatinontinuous-timemodels,therelevaneofimpulseon-

trolhasbeenpointedoutearlyintheliterature,seeClark(1976,p.58)where

it is suggested that optimal poliies mayonsist in one impulse followedby

aontinuous,smoothontrol.Earlyempirial evidenein thesheriessetor

wasprovidedbyHannesson(1975).Ontheother hand,theutting poliy of

Faustmann'sis basedonanimpulseontrolwithyles.

Weshowthat theonditionsfortheexisteneof ylialsolutionsinvolve

a lose ombination of the growth funtion and the ost funtion, thereby

emphasizingthattheonvexityoftheostfuntion,oritsdependene onthe

stok level, are not the only issues worth onsidering.We then disuss how

a Clark-like steady-state solution emerges as a limit of small and frequent

harvest operations in our model. We also show that we an reprodue and

generalizeDawidandKopel'sresults,althoughthelatterwereobtainedwith

adisrete-timemodelandwithoutstokeets.

Theartileisstruturedasfollows.Wepresenttheimpulseontrolproblem

insetion2,weharaterizethetypeofsolutioninsetion3andtheoptimal

yle in setion 4. We then establish the link to Clark's ontinuous ontrol

solution and to Dawid and Kopel'sdisrete ontrol model in setion 5.The

lastsetionisdevotedtotheonlusion.

2The impulseontrol model

2.1TheModel

The resoure dynamis

Weonsiderarenewableresoure,forwhihdynamis, in theabseneofany

harvest,isgivenby:

˙

x(t) = F(x(t)), t≥0, ⁽¹⁾

where x(t)îs ^the^size ôf ^the^populationât ^time t ândF^, ^stationary ^through

time, is the growth rate funtion. The funtion F ^is ^assumed ^to ^satisfy^the

followingonditions.

Assumption1 There exist numbers xsup ^and xs^, 0 < xs < xsup ≤ +∞^,

suh that the funtion F : (0, xsup) →R îs ^positiveôver^the înterval(0, xs)

and negative over the interval (xs, xsup)^, ^with F(0) = F(xs) = 0^, ^where limx↓0F(x) =F(0)^,^andlimx↑xsupF(x) =−∞^.^The^funtion F ^is^measurable

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andbounded above.It isassumedthat thedierentialequation(1)admits a

uniquesolutionforeveryinitial stokx0∈(0, xsup)^.

The populationlevel xs îs^the ^standard^long-runârrying âpaityôf^the

environmenttowhih,absentanyath,thepopulationisonvergingforany

x0 ^suh ^that 0< x0 < xsup^. ^Note^that ^theassumptions onF ^are^very^weak,

speially the monotoniity assumptions. For instane, F ^is ^not ^neessarily

onave,andmayhaveseveralloal maxima.Continuityover(0, xsup)^is^not

requiredeither,aslongas(1)admitsauniquesolution.

The harvestingproess

We are interested in the optimal exploitation of this resoure by a disrete

harvestproess,i.e.withintheframeworkof impulseontrol models.

1

Aordingly, wedene an impulse exploitation poliy IP := {(ti, Ii), i = 1,2, . . .} âs â ^sequene ôf ^harvesting ^dates ti ând instantaneous harvests Ii^,

onefor eah date. Thesequene of datesmay be empty, nite orinnite. It

is suh that 0 ≤ t1^, ^and ti ≤ ti+1^, i = 1,2, . . . ^and limi→+∞ti = +∞^. ^By

onvention, weshall assumethat ifthe sequeneis nite with n≥0 ^values,

thenti = +∞^for^alli > n^.

Thesequeneofharvestsmustsatisfy:

Ii≥0 ^and xi−Ii ≥0 , ⁽²⁾

where

xi = lim

t↑ti

x(t), ^withx1=x0 ^given^ift1= 0, ⁽³⁾

andsuh thatthefollowingonstraintshold:

˙

x(t) =F(x(t))^forti< t < ti+1 ^withx(ti) =xi−Ii^, i= 1,2, . . . ⁽⁴⁾

˙

x(t) =F(x(t))^for0< t < t1 ^withx(0) =x0 ^ift1>0. ⁽⁵⁾

Inotherwords:xi ^is^the^size^of^the^population^just^before^the^harvesting^date

ti^,ândxi−Ii îts^size^just âfter^that ^same^date.Îft1= 0^,^the^populationx1îs

supposed to beinherited from thepast,anddenoted by x0^.^Harvests^annot

benegativenorexeedthepopulationsize.Theonditions(2)(5)denethe

setoffeasible IPs,denotedbyFx0^.

1

Impulseontrolpoliiesininnitehorizononsistinanunboundedsequeneofdeisions.

For the disussion of impulseontrolmodels,see forexampleLéonard and Long(1998),

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The harvester'sprots

Monetaryprotsgeneratedbyanyharvestdependuponthesizeoftheath

andthesizeofthepopulationattheathingtime.Weassumethattheprot

funtionisstationarythroughtimesothatwhateverti^,Ii ^andxi^,^the^urrent

protsattimetiâmount^toπ(xi, Ii)^.² ^The^prot^funtionîsâssumed^to^have

thefollowingstandardproperties.

Assumption2 The funtion π(x, I)îs ^dened ôn^the ^domain D:={(x, I)^, x∈(0, xsup)^,I∈[0, x]}^.Îtîsôf^lassC¹^,^positiveând^bounded,ând^suh^that π(x,0) = 0^,∀x∈(0, xsup)^.^The^derivativeπI(x, I) := (∂π/∂I)(x, I)âdmits â

nitelimitwhenI↓0 ^for^allx∈(0, xsup)^.

Prots are disounted using a onstant instantaneous disount rate, de-

notedbyr^,r >0^.

The manager's problem is to hoose somepoliy maximizing thesum of

thedisountedprots,thatisto solvetheproblem(P):

(P) sup

IP∈F_x0

Π(^IP) :=

∞

X

i=1

e^−rtⁱ π(xi, Ii).

ThefuntionΠ îsâssumed^to^be^well^denedôver^the^whole^set Fx0^.

3

Approximation ofaontinuous ontrol

Thelassialmodelingofaontrolledrenewableresoureinvolvesthemodied

dynamis

˙

x(t) = F(x(t)) − h(t),

where h(t)îs^the^rate ôf^harvestât ^timet^.^The harvester'sprotis somein- stantaneousprotfuntionp(x, h)^dependingôn^theûrrent^stokând^the^rate

ofextration. It ispossibleto approximate thetrajetoriesofaontinuously

ontrolledsystembyanimpulse-ontrolledone.Forinstane,byhoosingthe

impulsessothatthetwotrajetoriesareperiodiallysynhronized,say,every

δt^units^of^time.^When^the^periodδt^tends^to^0,^the^distane^between^trajeto-

riesshouldgoto0.Thegainofsuhamiro-impulsepoliyanbeestimated

asfollows:

4

during theinterval [t0, t0+δt]^,^the ^resoure^under ^the^dynamis

(1) goesfrom x^to x+δtF(x) +o(δt)^. ^The ^ontrolled ^resoure^goes^from x

tox+δt(F(x)−h(t0)) +o(δt)^.^The^disrepanyîsôrreted^withânîmpulse

of I =δt×h(t0)^. Âording^to Âssumption^2, ^we^haveπ(x,0) = 0 ^forâllx^,

2

Thusweassumethattheresourestokperseisnotgeneratinganysurplusowasin

Hartman(1976),Smith(1977)andBerk(1981)toquoteafewpioneeringworksalongthis

way.Thiseetanbenegletedforawidespetrumofrenewableresoures.Forexample,

mostsheriesdonotgeneratesuhsurplus.

3

Observethatweformulateourproblemwithasup ând^notâmax ^beause^weâre

interestedinthepossibilitythatthemaximumisnotreahedinsidethesetFx0^.

4

Wedonotpursueherethetaskofformallyprovingtheselaims,sinethisisnotessential

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whihimpliesthatπx(x,0) = 0âlso^forâllx^.^Therefore,^theîmpulse^generates

againof:

π(x+δt×F(x), δt×h(t0)) = (δt×h(t0))πI(x,0) + o(δt).

In the limit, the gain obtained by the series of impulses is the same asthe

ontinuouslyaumulatedgainwithprot funtion p(x, h) =hπI(x,0)^. ^This

funtionisofthespeiformusedinthesingularontrolmodelofClark.We

omebaktothispropertyin Setion5.1.

2.2TheDynamiProgrammingPriniple

We use theDynami Programming approah to solvethe problem. The fol-

lowingtheoreminsurestheexisteneofauniquevaluefortheproblem.

Theorem1 The valuefuntion

v(x) = sup

IP∈F_x

Π(^IP) ⁽⁶⁾

isthe uniquesolutionof the following variational equation:

v(x) = sup

t≥0 0≤y≤φ(t,x)

e^−rt[π(φ(t, x), φ(t, x)−y) +v(y)] , ⁽⁷⁾

whereφ(t, x)îs^the^trajetoryôf^the^systemât^timet^,^solutionôf^the^dynamis

(1)with x(0) =x^.

Forastandardproofofthisdynamiprogrammingresult,see(Davis,1993,

Theorem(54.19),page236).

3Redution to Cylial Poliies

Inthis setion weinvestigatethe impulseontrol model and propose anap-

proahforharaterizingitssolutions.Ourapproahistodeterminethestru-

tureofsolutionsunder thequitegeneralassumptionsoftheprevioussetion.

The prieto payfor this generality is that our resultsdo notguaranteethe

uniquenessofsolutions,whih mustbeexaminedonaase-by-asebasis.

Ourline ofargumentwillbethefollowing.First ofall,theDynami Pro-

grammingpriniple impliesthat,under anyoptimal poliy forProblem (P),

ifthestokreahessomelevelalreadyattainedin thepast,theationhosen

in the past(to harvest ornotto harvest) should still beoptimal. This mere

fatombinedwiththepositivegrowthofthestok'snaturaldynamistends

to selet poliies that are ylial in the sense that they let the stok grow

to somelevel, harvest it down so someother level, and repeat. However, it

maystillbethat under theoptimalpoliy,the stok neverreahestwiethe

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property, suh trajetoriesannot beoptimal. Optimalpoliies aretherefore

essentiallyylial.Moreover,joining theoptimalylemustbedonewithat

mostoneharvest.

Theoptimizationproblemisthenreduedtonding:a)whatistheoptimal

yle;b)whatistheoptimalwaytoreahtheoptimalylefromagiveninitial

stok. Finding the optimal yle is arelativelysimple optimization problem

whihwealltheAuxiliaryProblem.Butthesolutiontothisproblem may

orrespondto degenerateyles,whih weinterpretasontinuousharvesting

poliies à la Clark.We show in the nextsetion that the submodularity as-

sumption is again the key to determine whether the optimal yle is atrue

yleoradegenerateone.

Weproeed nowwith thedenitions and the preise statementsof these

priniples.

3.1CylialPoliiesandtheAuxiliaryProblem

Cylial poliies Aylialpoliy hastwoomponents:aylewhihishar-

aterized by two values x^and x¯ ^with x < x¯^, ^or equivalently by an interval

[x,x]¯^; ândâ^transitory^part^whih ^desribes^how^the^trajetory êvolves^from

theinitialstoktotheyle.Thetransitorypartonsistsinanite(possibly

empty)sequeneof harvests, suh that,after thelast harvest, theremaining

populationislessthanx¯^.^We^rst^onentrate^on^the^yle.

Hene,aylehastwomainparameters,whiharesuhthat0≤x <x¯≤ xs^.

5

When in itsylialpart, apoliy ats asfollows: a)let thepopulation

growtox¯^;^b) ^harvest x¯−x^; ând ^repeat.^Suh â^poliy âppliesônly^to înitial

populationsx0≤x¯^.Înôther^words,^the^transitory^partân^be^dispensed^with

onlyforsuhaninitialpopulation.

Gain under a ylial poliy We will denote by G(x,x, x¯ 0) ^the ^value ^of ^dis-

ounted prots in apoliy without the transitorypart, applied to an initial

populationof x0^. ^The ômplete ^denition ôf ^the ^funtion G învolves^several

ases,orrespondingtothelimitasesforx¯^andx^.

Itisonvenienttodenethefuntionτ(x, y)^as^the^time^neessary^for^the

dynamistogofromvaluex^toy^,x≤y^.Ît^turnsôut^that ^forâll0< x≤y <

xs^:

τ(x, y) = Z y

x

1

F(u) du. ⁽⁸⁾

Sine, by Assumption 1, F(xs) = 0^, ^the ^integral ^dening τ(x, y) ^is ^singular

when y = xs^. ^The ^limit ^when y → xs ^may ^therefore ^be ^nite ^or ^innite,

dependingonthefuntionF^.Ânother^featureôfÂssumption¹îs^thatF(0) = 0^.Consequently,ifx(0) = 0^,â^solution^to^the^dynamis⁽¹⁾îsx(t) = 0^forâll t≥0^.^Thisîmplies^theônvention^thatτ(0, y) = +∞îfy >0^,ândτ(0,0) = 0^.⁶

5

Sinex¯^represents^the^population^levelûntil^whih^the^resoure^grows^beforeharvesting, thereisnopointinonsideringx > x¯ s^sine^the^populationânnot^grow^to^suhâ^level.

6

Thisonventiondoesnotmeanthatlim_x↓0τ(x, y) = +∞ⁱⁿ^every^situation.

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ThevalueofthetotalprotfuntionGân^beêxpressedâs:

i) If0≤x <x¯≤xs^:

G(x,x, x¯ 0) :=π(¯x,x¯−x) e^−rτ(x⁰^,¯^x)

1−e^−rτ(x,¯^x) . ⁽⁹⁾

Theonventionisthat:ifx= 0^,^the^termexp(−rτ(x,x))¯ ^should^be^replaed

by 0. Likewise, exp(−rτ(x,x))¯ ând exp(−rτ(x0,x))¯ âre ⁰ îf x¯ = xs ând

limy→xsτ(x, y) = +∞^.

ii) Forx= ¯x^,Âssumption²âllows^to^deneG^byôntinuityâs:

G(x, x, x0) =πI(x,0) F(x)

r e^−rτ^(x⁰^,x). ⁽¹⁰⁾

Fortheasesx= ¯x^,^the^valueGîs^not^thatôfâ^well-dened împulseôntrol

poliy.AswehaveseeninSetion 2.1,thisvalueisthat ofaontinuous har-

vestingpoliy,whihanbeseenasadegenerateimpulsepoliy.Theharvest

rateofthisontinuouspoliy isonstantandequalto F(x)^.

Finally, by using the fat that τ(x, y) ^dened ⁱⁿ ⁽⁸⁾ ^is ^also ^dened ^for y≤x^,expressions(9)and(10)providevaluesforthefuntionG^whenx0>x¯

as well. Of ourse, these situations do not orrespond to an implementable

harvesting poliy, andthefuntionloses itseonomimeaning.Insubsetion

3.3 we will study the transitorypart of a ylial poliy for whih the ase

x0>x¯ ^has^an^eonomi^meaning.

The auxiliaryproblem

Having dened thefuntion G(x,x, x¯ 0) ^for âll 0 ≤ x≤ x¯ ≤xs ând âll 0 ≤ x0≤xs^,^we^now^dene ^theâuxiliary^problem^(AP):

(AP): max

x,x; 0≤x≤¯¯ x≤xs

G(x,x, x¯ 0).

Under Assumption 2 it turns out that G îs ^lower semi-ontinuous as a funtionof (x,x)¯ ^.^The^problem^(AP)^has^thereforeâlwaysâ^solution.^Fôr^the

purposeofthedisussiontoome,itisimportanttodistinguishtheasewhere

thesolutionis suh thatx= ¯x^,^from ^the^ase^where x6= ¯x^.^We^all^the^rst

situationadegenerateylesolution,andtheseondoneanon-degenerate

solution.

3.2SubmodularityandOptimalTrajetories

In this paragraph, we introdue a submodularity assumption on the prot

funtion π^.^Consider^the^followingassumption.

Assumption3 Thefuntion π^is^suh^that:

π(a, a−c) +π(b, b−d) ≤ π(a, a−d) +π(b, b−c) ⁽¹¹⁾

foreveryd≤c≤b≤a^.