A simple bio-economic model of soil natural capital

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A simple bio-economic model of soil natural capital

Robert Lifran, Annie Hofstetter, Mabel Tidball, Oumarou Balarabe

To cite this version:

Robert Lifran, Annie Hofstetter, Mabel Tidball, Oumarou Balarabe. A simple bio-economic model

of soil natural capital. 5. World Congress of Environmental and Resource Economists, Association

of Environmental and Resource Economists (AERE). USA., Jun 2014, Istanbul, Turkey. pp.24. �hal-

02738597�

apital

Robert Lifran

∗

, Oumarou Balarabé

†

,Annie Hofstetter

∗

, Mabel Tidball

∗

Deember2013

Abstrat

Relying on the oneptual framework of natural apital, this paper

buildaoneptofsoilnaturalapitalandimplementitusinganoptimal

ontrol model. Considering soilas aneosystem,webuilda simplebio-

eonomimodelwithtwointerelatedstoks(thesoilorganimatterand

thestok of nutrients diretlyontributing to the plant'sbiomasselab-

oration). Theprodution funtionis of Liebig type, a Linearonewith

plateau. Theeonomi part of themodelrelies onthe longtermprot

maximisation in the ontext of private management. We retained two

ontrols: the mineral fertilization addingto the stok of nutrients,and

the rate of biomassgiven bak to soil to ontributeto the soil organi

matter.

Byombiningboth ontrols,we identiedmanagementregimes and de-

nedthe setof stationaries states. Going beyong that standardstepof

analysis, we simulatedoptimal timepath for dierent initial onditions

anddierentsetofparametervalues. Wespeiallyfousedontherole

oftheprieoffertilizersrelativetotheprieoftheagriulturalproduts.

Resultsshowthatprivatemanagementofsoilnaturalapitaldrivestothe

quasidepletionofsoilorganimatter. Asaonsequene,thereisaneed

forpubliinentivestopromotethoseeosystemsserviesnonsupported

bymarket.

Keywords: Natural Capital, OptimalControl, EosystemsServies,

Environmentalpoliy

Contents

Introdution 2

1 The model 5

1.1 Themodel'sstruture . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Model'sresolution . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Eonomiinterpretation . . . . . . . . . . . . . . . . . . . . . . . 7

∗

INRA,UMR1135Lameta,F-34000Montpellier,Frane

†

UM1,UMR1135Lameta,F-34000Montpellier,Frane

2.1 Parameterssettings. . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Sensitivityanalysisaordingto

φ 1

^or

φ 2

^{. . . . . . . . . . . . . 10}

2.2.1

N 0 < N ¯

^,

φ 1

^{. . . . . . . . . . . . . . . . . . . . . . . . . . 10}

2.2.2

N 0 = ¯ N , ∀ t

^,

φ 1

^{. . . . . . . . . . . . . . . . . . . . . . 10}

2.2.3

N 0 < N ¯

^,

φ 2

^{. . . . . . . . . . . . . . . . . . . . . . . . . 10}

2.2.4

N 0 = ¯ N , ∀ t

^,

φ 2

^{. . . . . . . . . . . . . . . . . . . . . . 10}

3 Optimaltimeproles 11 3.1 Simulationsorganization . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Lowprieoffertilizers(

φ 1 = 0.02)

^{. . . . . . . . . . . . . . . . . 11}

3.2.1

N 0 < N, ¯ 0 < k < 1, n = 0

^then

N 0 = ¯ N , k > 0, n > 0

^{. . . 11}

3.2.2

N 0 > N, k ¯ = 0, n > 0

^then

N 0 = ¯ N, k > 0, n > 0

^{. . . . . 13}

3.3 Highprieoffertilizers(

φ 1 = 0.2)

^{. . . . . . . . . . . . . . . . . . 14}

Conlusion and perspetives 16 Appendies 18 A Steadystates study 18 A.1

N > N ¯

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18}

A.2

N < N ¯

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18}

A.3

N = ¯ N

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19}

B Steadystates summary 21

Introdution

Despite hugeprogress in plantsgeneti and plantsprotetion, the prodution

offood,berorbiofuelsbyagriulturestillreliesonlandasaomplexresoure

made by the mix of an area and a volume of soil. Plants use photosynthesis

to apture solar energytrough areaand soil's volume holds the nutrients the

plantsouldneedtogrowth. Thiseologialmixhasbeenregardedthroughthe

longhistoryofsoilsienesunder variouslensesFelleret al.(2012).

Atthe beginning of the XIXthentury, the plantnutrition theory still fo-

usedonthe roleof humus(Humusbeingonorganiompund)Thaer(1809).

Atthesametime,soilfertilityisregardedbyeonomistsasanaturalresoure

inequallyalloatedto dierentsplaes. Observing thegreatrangeofsoilspro-

dutivity,evenunderthesamelimate,earlyeonomistspostulatedthatdier-

enesinagriulturaloutputomesfromthedierenesinthesoil'sfertility. As

aonsequene,theyelaboratedthelandrenttheory. Thatviewpointremained

unhanged, evenafter the demonstration bygerman sientists that plantsuse

only simpleelements likenitrogen, phosphorus and potash fortheir nutrition,

andnotdiretlysoil'sorganimatterSprengel(1838)Liebig(1840). Thatmin-

eralisttheoryofplantnutritionhadlittleimpatonthewayeonomistsregarded

thesoilfertility,untilthegermanhemistsHaberandBoshdisoveredthepro-

ess of extrating nitrogen from atmosphere. From that innovation onwards,

modernagriulturehasbeenthoughtasaorganifree agriultureFelleret al.

(2012). As extrativeindustryis able to supply farmerswith mineral fertiliz-

tutedtothelandrenttheory. Thissubstitutionhasbeenusefullandrobustfor

deades, being reinforedby thesuess of what had beenalled "The Green

Revolution" Borlaug(1970)Borlaug (2007). Beause land, mineralfertilizers,

pestiidesandseedsouldbefoundonmarkets,thereisnoneedto lookatthe

soilsontentsto explaindierenesin agriulturalprodutivity. Farmershave

onlyto alloatefatorsofprodutionaordingtorelativepries,produtivity

under the onstraint of the prodution funtions Heady (1952). As a onse-

quene soilsproperties andsoilsservieswould be oustedfrom theresearhin

agriulturaleonomisforseveraldeades.

Nevertheless,attheendoftheXXthentury,theawarnessofseveralshort-

ommings, ableto balane the benets ofmodern agriulture,wasinreasing.

Firstofall,landresouresdegradationbytheveryeetsofagriulturalinten-

siationpratiesthreatsnononlytheabilityofagriulturetoaordfoodand

ber,but alsoimpairmanyotherseosystemservies. Therstenvironmental

risis,knownasa"dustbowlrisis",hadalreadybeenexperienedintheState

beforetheseondWorldWar. Theeologialandeonomialonsequeneshave

beensodramatithattheFederalGovernmentreatedtherstPubliAgeny

to promotethesoilsonservation. Thepremiss ofthesoilnaturalapitalon-

ept havebeenelaborated at that time and legitimize the publi intervention

Bune(1942)Hiks(1939a)Weitzell(1943). After thewar,thememoryofthe

dust bowl risishas been soondisarded, and we observethat during the fol-

lowingdeadesupto 1980, nopapersontheissue hasbeenpublished. Inthe

years1980,Burt(1981)and MConnell (1983)publishedmodelsof soildeple-

tionbytheeetof produtionintensiation. Theyaretherstpapersusing

anintertemporalframework(dynamisprogrammingoroptimalontrol). After

them,onlyabunhofpapershasbeenpublishedonerosionissues,halfofthem

beingmotivatedbylanddegradationindevelopingountriesShortleandMira-

nowski(1987)Barbier (1997)Miranda (1992)Goetz (1997)Grepperud(1997)

Brekkeetal.(1999)ShiferawandHolden(1999)Hediger(2003)Nakhumwaand

Hassan (2011)Yirgaand Hassan(2010). Inallthat papers,soilis modeledas

asinglestatevariablewithavariousdynamis,undertheontrolofprodution

intensityhoies. Theydonotexpliitlyonsiderthatfarmers,bytheirhoies

andpraties,aremanagingannaturalapital.

At dawn oftheXXIst entury, therisisinduedbythe ompeting usesof

landforbiofueldrawnattentiononthefragilityoforganifreeagriulture,and

his dependane onnon renewable(andsometimenonsubstitutable)resoures.

As aonsequene, onerns about the unsustainability of modern agriulture

hasbeenraisingGrion (2006)Conway(1997).

Theonvergenebetweenspeiissuesinagriulture,andtheinternational

onsensusontheroleofarbonemissionsonthelimatehangehasdriventoa

newparadigmshiftinthesientioneptionsofagriulturalprodution,and

morespeially, onthe representationof soil's fertility. Wewill all thenew

paradigm Systemi. Systemi, beausesoil is now onsidered as aneosytem

of his own right. Desertiationand land degradationhasbeen promoted on

theInternationalAgenda,andbeomeamainhaptersintheMEAMillennium

EosystemAssesment(2005).

Mainhangesin therepresentationofsoilfuntions andserviessharedby

soilssientistsare relatedto thesopeof soilserviesandto itsstruture and

to regulatewater yle,and also givesupport to ulturalservies. TheXXIst

Centurybeginning,followingCostanzaetal.(1998)CostanzaandDaly(1992),

somesoilsientistsproposedtoapplytheoneptofnaturalapitaltothesoil

eosystemRobinsonetal.(2009),Dominatietal.(2010),Sanhezetal.(1997).

"Wedenethesoilnaturalapitalasthestokof biotiand abiotimassthat

ontainsenergyandorganization. Furthemore,thestrutureandfuntionality

of soild aross the landsape failitates needed proess for the well-being of

Humanity and the Earth system" Robinson et al. (2009). As soils sientists,

Robinson, Dominati or Sanhez put emphasis on the omponents of the soil

eosystem. Naturally, eventheyhaveseentheveryinterestofthe oneptfor

publi deision making and poliy design, they are not in position to further

developorrespondingmodels suitableformanagement.

Atthat pointof ourinvestigation,wegetaverysurprisingonlusion: to-

day, soilsientistsproposea denition of soil natural apital, and propose to

takeontoaountinthesoil'ssoialmanagementthebulkofgoodsandservies

provided bySES, but donthave the meansto developorrespondinganalyti-

al models. Meanwhile eonomists havedevelopped useful tools to take onto

aountintertemporaltrade-oin theagriulturalprodution,buttheymainly

relyonverysimple,unidimensional,modelsofsoilmainlydened bythedepth

or the volume of topsoil. Naturally, there is no ontradition between both

positions,thereisjust agap tobefullled. Andsomemodellinghurdleahead.

Ouraimsinthatpaperaretotakeadvantageofthereentadvanesbothin

soilsienesandinoptimalontroltheory,inordertoelaborate ansimplebio-

eonomimodelofsoilnaturalapital. Wewilldenesoilnaturalapital(SNC)

asaneonomialonept,aneonomialindiatorusefullto evaluatetheows

ofgoodsandserviesprovidedbythesoilonsideredasaneosystem. SNChelp

in monitoringmanagementations(extrating, renewing,use onversion...) of

privateators,andindesigningpublipoliiesaimedatlongtermonservation

ofsoilsapaities. Thisdenitionrelies ontheeonomialappraisalofowsof

goodsandserviesovertime,andsheisdierentfromthe"naturalistdenition"

elaboratedbysoilsientists. Namely,sheisnotonlyrelatedtotheomponents

of the SES, but to the apitalisation of servies, evaluated either by markets

orby othersevaluation methods. Beause the main serviesprovidedby SES

besidefood and berprodution are non markets servies, and are bynature

publi goods, the valueof SNC is dierent aording to the private or soial

pointofview. Aslongasintertemporalmanagementisinvolved,wewillrelyon

optimal ontroltheory, andbeausemain soil's eosystemservies (SESS) are

relatedtosoil'sorganimatter(SOM)Felleretal.(2012)Vitoria(2012)Miles

etal.(2009),wewillfousonthatstok. Moreover,wewillrelyonthesimplest

representationofSOMdynamisgivenbyHéninandDupuis(1945). Humia-

tion proess inorporates rops residues into SOMstok, while mineralisation

proess nurtur the seond stok of diretly assimilable nutrients. Beause to

theinherentomplexityofSESfuntionning,wewillnotdealatthisstagewith

others SESS besidethe support to the agriulturalprodution. Moreover, we

will fous onprivate management,keepingthesoialvalueofSNC for further

investigations. Assuming that the private management of soils aims at max-

imising the net present value of the soil's asset, we will makeuse of optimal

ontrolmodels,andwillidentifymanagementregimesandstationnariesstates.

Ourpaperisorganizedasfollow:

In the rst setion, we will explain the model's struture, the resolution

method andtheresults,andwillgiveeonomialinterpretation.

Intheseondsetion,wewilllookatthestationariesstateorrespondingto

dierentmanagementregimes.

Wewilldevotethethird setiontothe simulationsof optimaltimeproles

for relevant initial onditions and sets of parameters values. We will give a

speiattentionto theimpatofthepriesoffertilizers.

Finallywewilldrawonlusionsandtraesomeresearhperspetivesonsoil

naturalapital.

1 The model

1.1 The model's struture

The SNC model usesthe optimal ontrol theory to maximize overan innite

horizontheprotsfromtheagriulturalativity. Themanagerisentitledwith

twoontrols: oneisthefertilizerappliationrate,andtheotheroneistherate

of biomass restitution to soil. The soil eosystem struture and dynamis is

representedbytwointerelatedstoks: therstonerepresentstheSOMontents,

the seond is the soil's nutrients ontents. Fertilizers just add to the seond

stok,whilerestitutionsofropsresiduestosoilontributes,troughhumiation

proess,tothebuildingoftherst. TroughthemineralizationproessofSOM,

therststokontributestothebuildingofthestokofnutrients

N

^.

Aording to the mineralist theory of plants nutrition, the rop'sbiomass

depends only on the stok of nutrients. SOM plays yet no diret role in the

prodution. As a onsequene, the manager fae an intertemporal trade-o

betweenharvestingtodayallthemarketablebiomass,orleaveasharetothesoil,

in ordertospareinthefuturesomefertilizeraddition. Inordertoaomodate

thattrade-o,wediretlymodelthebiomassprodution,notonlythegrainor

beryield. Thebiomassprodutionfuntionispieewiselinear,withaplateau,

aordingtothelimitingfatortheoryParis(1992):

f (N ) =

βN

^if

N < N ¯ β N ¯

^if

N ≥ N . ¯

Therststatementholds when thenutrients availableare notsuientto

providefullgrowthtotherop;inthatase,

N

^{isalimitingfator. Theseond}

oneholdsassoonas

N

^{issuient,whileothersfatorslikewateror}temperature remainlimiting. Inthatonditions,thebiomassprodutionisonstant,andthe

prodution's plateauours. Thethresshold valueforN will benoted

N

^{. It's}

notneessarytoaddfertilizersassoonastheSOMstokisabletoprovides

N

ormoretotheplants.

β

^{isthetehnialoeientof}transformationofnutrients

N

^intobiomass.

ν

^{oeientexpress the} orrespondingonsumption of nutrients. As

N ≥ N ˇ

^,

thenutrientsonsumptionremainsonstantandisvalueis

N

^:

ǫ(N ) =

νN

^if

N < N ¯

N ¯

^if

N ≥ N . ¯

humiationof therop'sresiduerestitutions, ononeside, andhis owndegra-

dationtrough bateria,known asthemineralization. Thelater mineralization

ontributes to the building of thenutrientsstok,

N

^. Fertilizersappliation,

n

^{, diretly ontribute to}

N

^{, with some losses, so as the eient appliation}

beomes

χ(n)

^{. Whiletherateofbiomass}restitutions,

k

^{,ontributesto}

N

^indi-

retly,trough

M

^dynamis:

M ˙ = kf(N) − γM, M (0) = M 0 ,

⁽¹⁾

N ˙ = γM + χ(n) − ǫ(N ), N (0) = N 0 ,

⁽²⁾

The timepoint prot funtion is quadrati in

k

^and

n

^{, due to operating}

variable ostsfor harvesting and inorporatingrops remainsto thesoil. Fer-

tilizersappliationissubmittedtothesameonstraints,beauseappliationof

oneinreasingquantityoffertilizerrequiresmoretimeandenergy.

a(1 − k)f (N) − b [(1 − k)f (N)] ² − Φ(n)

⁽³⁾

with:

a

^{: prieofbiomasssoldonmarkets}

b :

^{harvestingosts}

Φ 1 :

^{prieoffertilizer}

Φ 2 :

appliationsosts

Φ(n) = Φ 1 n + Φ 2

2 n ²

Themanager'sproblem is oneof maximizing thepresentvalueof the ow

ofprotsoveroneinnitehorizon:

n≥0,k∈[0,1] max Z ∞

0

e ^−ρt h

a(1 − k)f (N ) − b [(1 − k)f (N )] ² − Φ(n) i

dt

⁽⁴⁾

where

f (N ) =

βN

^if

N < N ¯ β N ¯

^if

N ≥ N . ¯

suhthat

M ˙ = kf(N) − γM, M (0) = M 0 ,

⁽⁵⁾

N ˙ = γM + χn − ǫ(N ), N (0) = N 0 ,

⁽⁶⁾

ǫ(N ) =

νN

^if

N < N ¯ N ¯

^if

N ≥ N . ¯

In order to solves that intertemporal optimization problem, the manager

ouldombinetheontrolsinseveralways.Asaonsequene,besidetheinterior

solution (

n > 0; 0 < k < 1)

^{, theyare abunh ofothers possiblemanagement}

regimes. Naturally,someofpotentialregimesdonothaveanypratialhane

tobeimplemented,beausetheyindue hargesandnoprots.

In order to solve the manager's problem, we now form the Lagrangean in

his moregeneralexpression. Fromthat Lagrangean,wewill omputetherst

order onditionsandwill giveeonomialinterpretation. Inafurtherstep, we

Controls

n = 0 n > 0

k = 0

^Mining CompensatedMining

0 < k < 1

^{AttenuatedMining} Complementarity

k = 1

^{Fallow ImprovedFallow}

Table1: Managementregimes

1.2 Model's resolution

TheLagrangean:

L = a(1 − k)f (N ) − b [(1 − k)f (N )] ² − (Φ 1 n + ^Φ ₂

²

n ² ) + λ [kf (N ) − γM ] + µ [γM + χn − ǫ(N )] + +λ k=0 k + λ k=1 (1 − k) + λ n=0 n,

where

λ

^and

µ

^{aretheadjointvariables}orrespondingto

M

^and

N

respetively.

λ k=0

^,

λ k=1

^,

λ n=0

^{aretheLagrange}multipliersorrespondingtotheonstraints

k ≥ 0

^,

k ≤ 1

^and

n ≥ 0

respetively:

λ k=0 k = 0 λ k=0 > 0 λ k=1 (1 − k) = 0 λ k=1 > 0 λ n=0 n = 0 λ n=0 > 0

Firstorderonditions,whenderivativesexists, are:

∂L

∂k = f (N ) [ − a + 2b(1 − k)f (N ) + λ] + λ k=0 − λ k=1 = 0

⁽⁷⁾

∂L

∂n = − Φ 1 − Φ 2 n + µχ + λ n=0 = 0

⁽⁸⁾

λ ˙ = ρλ − ∂H

∂M = (ρ + γ)λ − γµ

⁽⁹⁾

˙

µ = ρµ − ∂H

∂N = (ρ + ǫ ^′ (N ))µ − f ^′ (N )

a(1 − k) − 2b(1 − k) ² f (N ) + kλ

(10)

Remarque 1 Notethatwhen

0 < k < 1

^as

f (N ) > 0

^{,equation(13) beomes}

− a + 2b(1 − k)f (N ) + λ = 0.

Replaing in (10) wehave,

˙

µ = (ρ+ǫ ^′ (N ))µ − f ^′ (N ) [(1 − k)(a − 2b(1 − k)f (N )) + kλ] = (ρ+ǫ ^′ (N ))µ+f ^′ (N)λ.

1.3 Eonomi interpretation

As rops do not make a dierene among nutrients oming from dierents

soures, the question arises to dene the optimal ombining of bothontrols.

Its worthremarking that rop residues restitutions ontributes with aertain

timelagto providenutrients. Naturally,thenutrientsprovidedbythemineral-

ization ofSOMare notfree,theyhaveanopportunityostrepresentedbythe