Equilibrium selection in an environmental growth model with a S -shaped production function

ContentslistsavailableatScienceDirect

Chaos,

Solitons

and

Fractals

Nonlinear

Science,

and

Nonequilibrium

and

Complex

Phenomena

journalhomepage:www.elsevier.com/locate/chaos

Equilibrium

selection

in

an

environmental

growth

model

with

a

S

-shaped

production

function

Giovanni

Bella,

Danilo

Liuzzi,

Paolo

Mattana,

Beatrice

Venturi

∗

Department of Economics and Business, University of Cagliari, Italy

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 18 April 2019 Revised 30 July 2019 Accepted 7 September 2019 Keywords:

S -shaped production function Bogdanov–Takens bifurcation Pollution externalities

Policy-induced heteroclinic orbits

a

b

s

t

r

a

c

t

Thispaperpresentsapurelydynamiceconomic-environmentalgrowthmodelwitha S -shapedproduction functionandapollutionexternality.Wefirstoutlinetheroleofthesavingrateininducing/suppressing adualsteady-state,onewithlowercapitalandpollution(the“green” steadystate)thantheother(the “dirty” steadystate).Assumingthesavingrateissuchthattheeconomypossessesadualsteadystate, thesecondpartofthepaperexploitsglobalanalysistecniquestostudyconditionssuchthattheeconomy isabletoescapethedirtysteadystateandapproachthegreensteadystate.

© 2019ElsevierLtd.Allrightsreserved.

1. Introduction

Weconsiderapurelydynamicgrowthmodelá laSolowwhere pollutioncreatesanegativeproductionexternality.Ourmodel dif-fers from existing literature inthat theproduction function isS -shaped (cf.,interal.,[12]), insteadofbeingofthestandard Cobb-Douglastype.1

The model gives rise to a bi-dimensional system of differen-tial equations which, in specific regions of the parameters space ofour interest,admitstwo steadystates,one withhighercapital andpollution(the“dirty” steadystate)thantheother(the“clean” or“green” steady state).Inthiscontext witha dual steadystate, we shedlight onthe characteristicsofpolicy action abletopush the economy out of the dirty steady state andput it on a path convergingtothegreensteadystate.Ofcourse,informationofthis kindcan be achievedonly iflocalanalysis isabandoned infavor of a more global perspective. In this regard, we show that our dynamic systemundergoes, in specific regions of the parameters space,theBogdanov–Takensbifurcation.Whatisinterestingforus isthatagivendynamicsystemthatexhibitstheBTsingularity,can

∗ _{Corresponding author.}

E-mail addresses: bella@unica.it (G. Bella), danilo.liuzzi@unica.it (D. Liuzzi), mattana@unica.it (P. Mattana), venturi@unica.it (B. Venturi).

1 A characteristic S -shaped production function is also considered in [8] _{. In this} contribution, the authors use a system of partial differential equations to explore the diffusion properties of pollution and capital in a similar context to ours. We are surely in debt with [8] for precious background, but we think our work builds on it along a fresh new direction.

beplaced incorrespondencewithasimpleplanar system, whose globalunfoldingisknownineveryaspect.Itisnotthefirsttimein theeconomicsliterature thatthepropertiesofasystem undergo-ingtheBTsingularityare exploited.Wecan cite[1,4]forsystems arising ina R3 _{ambientspace, whereas [2]provides an}

applica-tionto a systemarisingin a R2 _{ambientspace.However, we are}

notawareofcontributionswheretheBTsingularityisusedto de-rivedetailedpolicyprescriptions.

The rest of the paper develops as follows. In Section 2, we present the model and obtain the implied 2-dimensional vector field.InSection3,westudyexistenceandstabilityofsteadystates of the system. We particularly focus on the regions of the pa-rametersspacewherethedynamicsadmitsmultiplesteadystates. Section 4is devoted to theproof that oursystem undergoes the BTbifurcation in specific regions of theparameters space. More-over, we address the main point of the paper, and derive pol-icyprescriptions.Numerical examplesare throughoutprovided. A brief conclusion reassesses the main findings of the paper. The AppendixAprovidesallcalculationsandnecessaryproofs. 2. The model

Considerthepurelydynamicaleconomic-environmentalgrowth model

˙

k t = s f

(

k t

)

d

(

p t

)

(

1−

τ

)

−

δ

kk t

˙

p t =

θ

(

1− u

)

f

(

k t

)

−

δ

pp t

(

S

)

Thefirstequationmodelsthetimeevolutionofphysicalcapital,kt, wherenoexogenoustechnologicalprogressisintroduced.s∈(0,1) https://doi.org/10.1016/j.chaos.2019.109432

2 G. Bella, D. Liuzzi and P. Mattana et al. / Chaos, Solitons and Fractals 130 (2020) 109432 representsthe(fixed)fractionoftheproducedoutputthatgoesto

capitalinvestments,namelythesavingrate.f(kt)isanimplicit pro-duction function withf(kt)>0. d(pt) is an implicit-form damage function,depending solely onthe level of thepollutant, pt, with

d(pt)<0.

τ

istheshareofproductionthatispaidouttothe Gov-ernmentasanenvironmentaltaxaimedtoabateemissions.Finally,

δ

k∈(0,1)measuresthephysicalcapitallineardepreciationrate. Thesecondequationdescribesthetimeevolutionofpollution,2

whereitisassumedthatproductiongeneratesemissionswhich in-creaselinearlythestockofpollution.

θ

_>0measuresthedegreeof environmental inefficiency of economic activities. The abatement activitiesreduce a shareu∈(0, 1) ofemissions;thus 1− u repre-sentsunabated emissions. For the sake of simplicity, ina model that already includes nonlinearities and non-convexities, the de-preciationofpollution,thatistheself-cleaningcapacityofthe en-vironment,isassumedtobeproportionaltothestockofpollution. Wethereforehave

δ

ppt,where

δ

p∈(0,1).3

Themodelisinlinewiththeliteraturereviewedin[14]and,on anotherperspective, itcanalsobeconsidered asastripped-down variantof[10,11]DICEmodel.

SystemS canbewritteninamoreconvenientform,wherethe environmental tax rate

τ

is written as a function of the abated shareof pollution u.Consider the following arguments. Environ-mentaltaxation hasbeentakenintoaccount viatheparameter

τ

, theshareofproductionthat ispaidoutto theGovernmentasan environmentaltaxaimed toabateemissions. Thetotal amountof tax revenue is therefore Tt=

τ

s f

(

kt

)

, proportional to saved pro-duction. We assume that the Government maintains a balanced budget at any point in time, so Tt is totally devoted to sustain abatement activities, A(t) which is aimed to abate a share u of pollution. The associated cost is At=C

(

u

)

s f

(

kt

)

, where C

(

u

)

is thecost function.ByequatingTt=At,andassumingthat thecost functiontakesthesuitableform

C

(

u

)

=1−

(

1− u

)



proposedbyBartzandKelly(2008),weconcludethat

τ

=1−

(

1− u

)

 (1)

where



_{≥ 1.}

To the purposes ofthis paper,we now provide explicitforms fortheproductionanddamagefunctionsf(kt)andd(pt).Asforthe productionfunctionf(kt),we noticethat themajorityof the con-tributionsintheeconomic-environmentalliteratureproposes stan-dardCobb-Douglasforms.

This papershallconverselytreatthe caseof aconvex-concave

S-shapedproductionfunction(cf.[12])oftheform

f

(

k t

)

=

α

1 k q_t 1+

α

2k qt

(2)

where

α

1>0,

α

2>0andq>1are parametersdeterminingthe

po-sitionoftheinflection pointandthewidthofthetransitionfrom lowandhighvalueofkt.

Many explicit damage functions have been proposed in the economic-environmentalliterature (see[5],forarecentsurveyon thetopic).Henceforth,we shallassumethat thedamagefunction

d(pt)takestheform

d

(

p t

)

=

(

1+bp t

)

−1; b > 0 (3)

2 In the integrated models literature the word “pollution” generally means green- house gases. We instead interpret it in the broader sense of “undesired by-product” of economic activity.

3 The linear depreciation of pollution, that is the self-cleaning capacity of the en- vironment, is a gross simplification, since, as remarked in some literature (see, for example, [9] ) depreceation is likely to be stock dependant, that is δp = δp(pt) . As

an example, the oceans absortion rate of carbon dioxide, depending strongly on the stock of carbon dioxide itself, can be mentioned.

asin[8].Theformulationimpliesthatthe(negative)pollution ex-ternalityonproductionisnullwhenpollutionisabsent.Italso im-pliesthatproductionfallsnonlinearlywhenpollutionincreases. 3. Long-run equilibria and local stability

In this subsection we study the long-run equilibrium and presentlocal analysisresults.Consider first the followingversion ofsystemS ˙ k t = sk 2 t

(

1− u

)



1+k 2 t



(

1+bp t

)

)

−

δ

kk t ˙ p t =

θ

(

1− u

)

k 2 t 1+k 2 t −

δ

pp t

(

M

)

obtainedbyconsidering(1),theexplicitformsin(2)and(3),and thesimplifyingparametricassumptions

α

1=

α

2=1; q=2;

ε

=1 (4)

Asteadystate ofthesystemisanysolutionofthepair(k,p)that satisfiesthefollowingrelationships

˙

k t=0 (5.1)

˙

p t=0 (5.2)

We firstobservethat (0,0) isa feasiblesolution ofsystemM. However, since therequirementsfor theBogdanov–Takens singu-larityare notsatisfiedatthe origin,weonlyfocusoninterior so-lutions.4

Thefollowingstatementcanbeproved.

Proposition 1 (Conditionsfora fold bifurcation). Lets,thesaving rate,bethebifurcationparameter.Then,thereexistsacriticalvalue

ˆ s

(

δ

k,

δ

p,

θ

, b, u

)

= 2

δ

k



δ

p[b

θ

(

1− u

)

+

δ

p]

δ

p

(

1− u

)

(6) suchthat,if:

1. s>sˆ,systemMpossessestwosteadystates,onewith“low” cap-italandpollution

(

k ∗, p ∗

)

_low≡ P∗ low=



1 2

δ

ps

(

1− u

)

− √



δ

k[b

θ

(

1− u

)

+

δ

p],

θ

(

1− u

)

k ∗2_low



k ∗2 low+1



δ

p



(7.1) andonewith“high” capitalandpollution

(

k ∗, p ∗

)

high≡ Phigh∗ =



1 2

δ

ps

(

1− u

)

+ √



δ

k[b

θ

(

1− u

)

+

δ

p],

θ

(

1− u

)

k ∗2_high



k ∗2_high+1



δ

p



(7.2)

2. s=sˆ, system M possesses two coincident steady states (coales-cenceequilibrium)

(

k ∗, p ∗

)

_ce≡ P∗ ce=



1 2

δ

ps

(

1− u

)

δ

k[b

(

1− u

)

+

δ

p],

θ

_

(

1− u

)

k ∗2ce k ∗2ce+1



δ

p



(7.3)

3. s<sˆ,systemMdoesnotadmitasteadystate.

4 As a matter of facts, the Bogadanov–Takens Theorem requires that, at the fixed point, the Jacobian matrix associated with a specific system presents a double-zero eigenvalue. It is straightforward to see that this cannot happen at the origin for system M .

Fig. 1. Parameters combinations implying s = ˆ s with δk = 0 . 05 .

Proof. Solving (5.1)and(5.2) givesthe coordinatesof the steady states in(7.1), (7.2) and(7.3) depending on the value of the dis-criminant



=

δ

p



δ

ps 2

(

1− u

)

2− 4b

δ

k2

θ

(

1− u

)

− 4

δ

2 k

δ

p



which vanishes for the critical value of the saving rate in (6). Since d_ds=2

δ

2

p

(

1− u

)

2s>0the statements in proposition are implied. _

Foramoreimmediatepolicyidentification,weshallnameP_low∗

as the “green” or “clean” steadystate, whereas P_high∗ shall be re-ferredtoasthe“dirty” steadystate.

Itisinterestingheretoobservethatinourmodelthefollowing occurs.

Remark 1. Considerthe casethe economyhastwo steadystates. Then,inthelong-run,capitalandpollutionarecomplements.

Starting from baseline parameter values, we provide now a parametricexamplegivingrisetotwosteadystates.

Example 1. Set

δ

k=0.05. In Fig. 1, dots depict combinations of the(

δ

p,b, u) parameterssuch thatsˆ, namelythecriticalvalue of thesavingrategivingrisetothecoalescenceequilibria,isbetween zeroandone.Thebiggerthedots,thehighersˆ.Noticethatthe re-gionwithplausiblevaluesofsˆismainlylocatedincorrespondence ofsmallvaluesofb.

Setnow

(

δ

p,b,u

)

=

(

0.02,0.01,0.5

)

asin[8].Thus,sˆ∼=0.2236. At sˆ∼₌0.2236, the coalescence equilibrium has coordinates Pce∗ ∼=

(

0.8944,11.1111

)

. In Fig.2,we depictthe nullclines in thephase space(k,p) fors=sˆinpanel (b),andforsmalldeviationsof the saving rate around sˆ∼₌0.2236. Specifically, in panel (a) we use

s₌0_.2232_,whereasinpanel(c)wehaves₌0_.2240.

Noticethat theveryhighsensitivityofthebifurcationto devi-ationsinthesavingrate.Namely,verysmalldeviationsofthis pa-rameterwithregardtoitscriticalvalueimplylargedistancesofthe twoequilibria,bothintermsofk∗andp∗.Weleavethecomments regardingthedirectionoftheflowinFig.2tothenextSubsection.

3.1. Stabilityanalysis

Thelocalstabilitypropertiesofagivensteadystatecanbe de-scribedintermsofthesignsoftheinvariantsoftheJacobian ma-trix, J ,evaluatedatahyperbolicequilibriumpoint.

Fig. 2. Nullclines in the phase space ( k , p ).

Let Tr(J) and Det(J) ,betheTraceandDeterminantof J , respec-tively.ExplicitformulascanbefoundinAppendix1.Thefollowing statementscanbeproved.

Proposition 2. RecallProposition 1. Assume s>sˆ. Thensystem M hasa dualsteadystate.P_low∗ _,thegreensteadystate,isalwaysa sad-dleequilibrium,whereasP_high∗ ,thedirtysteadystate,isalwaysanon saddleequilibrium.

Proof. Bystandardarguments,theresultsonstabilityareobtained byevaluating theinvariantsof J atthetwo equilibriumpoints. In

4 G. Bella, D. Liuzzi and P. Mattana et al. / Chaos, Solitons and Fractals 130 (2020) 109432 A.1,we show thatatP_low∗ , Det(J) <0 .In thecaseofP_high∗ ,we

con-verselyhave Det(J) >0. ThestatementsinPropositionaretherefore implied. 

4. Global analysis

In this section, we discuss the application of the Bogdanov– Takens(BT) bifurcation to system _M. The theorem allows usto detectaparticulartypeofglobalphenomenon,namely,the homo-clinicbifurcation,by whichorbitsgrowingaround thenon-saddle steadystate collidewith the saddle steadystate. The interesting point is that a given dynamicsystem, undergoing a BT singular-ity,can beplacedincorrespondencewitha simpleplanarsystem whoseglobalunfoldingisknownineveryaspect.Implicationsare veryrelevantforus:asshownhereafter,whenthe economyisin proximity of a BT singularity, the details of appropriate policies abletopushtheeconomyawayfromthedirtysteadystatetowards thegreensteadystatecanbefullydevised.

Considerthefollowingpreliminaryresult.

Proposition 3 (SystemMmayundergotheBTbifurcation). Recall

Proposition1andassumes=sˆ.ThensystemMpossessestwo coinci-dentsteadystates(coalescenceequilibrium)atwhich Det

(

J

)

=0.Let furthermore u =u ¯≡ 2

δ

2 p b

θ

(

δ

k−

δ

p

)

(8.1) bethevalueoftheabatedemissionswhichsatisfies Tr

(

J

)

=0. Substi-tutingu=u¯ intosˆ,weobtain

s = ¯s≡ b

δ

k

θ

(

δ

k−

δ

p

)

δ

2 p

1+ 2

δ

p

δ

k−

δ

p (8.2) Then,atu¯=u¯ands= ¯s_,Det (J )and Tr (J )simultaneouslyvanish.Since technicalnon-degeneracy conditionsare alsomet,system _M under-goes a BT bifurcation. The set of the parameters for which

(

¯s,u¯

)

∈

(

0,1

)

2_isnotempty.

Proof. TheapplicationoftheBogdanov-Takenstheoremtosystem

M follows the discussion in [7]. First we are required to show thatthereexistregionsintheparameterspaceatwhichthe Jaco-bian matrix associated with system M has Det

(

J

)

= Tr

(

J

)

=0 at a coalescence equilibrium. Parameters combinations in (8.1) and (8.2) ensure that this can happen. Further non-degeneracy con-ditionsneed tobe satisfied.In A.2we discussthepositive appli-cationof thesenon-degeneracy conditionsto systemM.The ex-amplebelow showsthat theset ofparameters forwhich

(

¯s_,u¯

)

∈

(

0,1

)

2_{isnotempty. }

Interestingly,weobservethat

Remark 2. From (8.1)and(8.2),since

(

u¯,¯s

)

∈

(

0,1

)

2_,theBT

sin-gularitycanonlyoccurfor

δ

k−

δ

p>0.

In Fig. 3 we depict combinations of the (b,

δ

p,

θ

) parameters implying ¯s_∈

(

0,1

)

2 _{- panels(a), (b), (c) -and u¯∈}

₍

_0,1

₎

_-panels

(d),(e),(f)-forgivenvaluesof

δ

_k>

δ

p.Inallpanels,thebiggerthe dots,thehighertheimpliedvaluesof ¯soru¯.

Furthermore, is worth pointingout some features ofthe sub-regionoftheparametersspaceatwhichtheBTbifurcationoccurs. Wefirstnoticethat high

δ

ktypicallymeansrelative highervalues of ¯s withregard to u¯;the converseappliesforlow valuesof

δ

k. Moreover,therelevantregionflattensthehigher

δ

_kwhen

θ

is suf-ficientlyhigh.Thismeansthat,irrespective of

δ

p,when

θ

is suffi-cientlyhigh,

(

u¯,¯s

)

∈

(

0,1

)

2 _{onlyforsmallvaluesofb.When}

_δ

kis lower,therelevant regionissteeper.Thisimpliesthat,when

θ

is sufficientlyhigh,

(

u¯,¯s

)

∈

(

0,1

)

2 _{alsoforhighvaluesofb.}

ConsidernowthefollowingCorollaryofProposition3.

Corollary 1 (Existence of a simple planar system topologically equivalenttosystemM). RecallProposition3.Assumetheeconomy issufficiently closeto theBTsingularity.ThensystemMis topologi-callyequivalenttothesystem

˙ w 1 = w 2

˙

w 2 =

η

1+

η

2w 2+w 12+

σ

w 1w 2

σ

=±1

(

Z

)

Theunfoldingparameters,

η

1 and

η

2,satisfy

η

1= D 1

μ

− D2

ν

(9.1)

η

2= D 3

μ

− D4

ν

(9.2)

where

μ

₌s_{− ¯s}and

ν

₌u_{− ¯u}areadditionalvariablesexpressing de-viationsofthebifurcationparametersfromtheircriticalvaluesandDi,

withi=1...4,areintricatecombinationsoftheoriginalparameters ofthemodel.

Proof. InA.3 we have detailedall necessarysteps. See also [13], p.321–330. 

TheresultsinCorollary1areofgreathelpforouranalysis:the globalanalysisofsystemMcanbebasedonasimpleplanar sys-temwhose unfolding (depending on thesign of

σ

)is completely known.

Considerthefollowingdetails ofthedynamicsimplied by sys-temZ.

Proposition 4 (Unfolding of system Z). Recall Proposition 3 and

Corollary 1. Assume

σ

=+1. Then, for

μ

and

ν

sufficiently small,

thereexistsasmoothcurveN thatoriginatesat

(

η

1,

η

2

)

=

(

0,0

)

and

hasalocalrepresentationgivenby

N ≡

{

(

η

1,

η

2

)

:

η

1=−

η

22,

η

2 > 0

}

(10)

suchthat:

• abovethecurve_N,thenon-saddlesteadystateisasource.There also exists a family of heteroclinic connections leadingfrom the non-saddlesteadystatetothesaddlesteadystate;

• belowthecurveN,thenon-saddlesteadystateisasink.

Proof. See[13],p.321–330,fordetails. _

Tousethe knowledgeofsystem Z forourpurposes,we need firsttoestablishthesignofthecoefficients.Considerthefollowing propedeuticalresults.

Lemma 1 (Signs of thecoefficients ofthe planar systemZ). Our computations show that

σ

=+1. Furthermore, for all combina-tions of thetriplet (

δ

p,

θ

,b)(forgiven

δ

k)implying

(

¯s,u¯

)

∈

(

0,1

)

2,

sign(D1)<0 whereassign

(

D2

)

=sign

(

D3

)

=sign

(

D4

)

>0.

Proof. Thesignofthecoefficient

σ

canbeeasilyobtainedby ap-plyingthealgorithmof[3]toourcase.AsshowninA.2,weobtain

sign

(

σ

)

=+1.ToestablishthesignsofthevariousDi,wehave re-peated the numerical simulations made inFig. 3 andfound that

D1<0andthatsign

(

D2

)

=sign

(

D3

)

=sign

(

D4

)

>0. 

4.1. Policy-inducedheteroclinicorbits

Ourresultsprovideaninnovativeframeworkinwhichitis pos-sibletodiscussseveralcrucialpolicyissues.Inthisregardwefind very appropriatethe useof

ν

=u− ¯u ascontrol elements. There-fore,assume

μ

=s− ¯sand

ν

=u− ¯uaresufficientlysmall. Proposition 5 (Escapingthedirtysteadystatewhentheeconomy is above the N curve). Assume theeconomy belongs to the region abovetheN curve. Then, anappropriatechoice oftheinitial condi-tionsputstheeconomyontheunidimensionalstablemanifoldofthe

Fig. 3. Regions of the parameter space giving rise to ( ¯s , ¯u ) ∈ (0 , 1) 2 .

greensteadystatesuchthataneconomywanderingin the neighbor-hood of the dirty steady state is able to approach thegreen steady state.Thesetofparameters atwhichthis phenomenonoccursis not empty.

Proof. RecallProposition4.Considerthecasesanduarechosenso that(

η

1,

η

2)areabovethecriticalNcurve.Then,byProposition4,

we know that there is a heteroclinic connection going from the non-saddleto thesaddlesteadystate.The restoftheProposition follows. Example 2 below will show that this phenomenon can happenforparametervaluesinsidetheadmissibleregion. 

Thesestatementsarecomplementedbythefollowing.

Proposition 6 (Escapingthedirtysteadystatewhentheeconomy isbelow theN curve). Assumethe economy belongsto the region below theN curve. Then, theabatement share ucan besufficiently increasedtopushtheeconomyabovethe_N curve.Additionally,asin

Proposition5,an appropriatechoiceoftheinitialconditionsputsthe economyon theunidimensional stable manifoldsuchthatthe econ-omyescapesthedirty steadystate andapproachesthegreen steady state.Thesetofparametersatwhichthisphenomenonoccursisnot empty.

Proof. Considersanduare chosensothat (

η

1,

η

2) arebelowthe

criticalN curve.Then,byProposition4,therearenopathsleading tothegreensteadystate.Consider however,substituting(9.1)and (9.2) into Eq. (10). Solving for

μ

and substituting back into the

6 G. Bella, D. Liuzzi and P. Mattana et al. / Chaos, Solitons and Fractals 130 (2020) 109432

Fig. 4. The heteroclinic connection.

formulafor

η

2,weobtain5

¯

η

2= 1 2



D 2 1+4D 3

(

D 2D 3− D1D 4

)

ν

− D1 D 3

whosederivativewithregardto

ν

gives

∂

η

¯2

∂ν

= D 2D 3− D1D 4



D 2 1+4D 3

(

D 2D 3− D1D 4

)

ν

which,recalling fromLemma2 thesigns ofthe various Di, gives ∂ η2 ¯

∂ν >0.Anexample belowwill show thatthisphenomenon can happenforparametervaluesinsidetheadmissibleregion. 

Theseresultsareinnovativeinthefield:publicinterventioncan appropriatelycalibrateitsinstrumentstoinducemarketresponses suchthatthedesiredobjectiveisreached.

WeclosethisSectionbydiscussinganumericalexamplegiving risetoaheteroclinicorbit.

Example 2. Set

(

δ

_k,

δ

p,

θ

,b

)

=

(

0.05,0.02,0.93,0.03

)

. Then, we easily derive

(

¯s,u¯

)

∈

(

0.1598,0.0442

)

and D1=−12.4865, D2=

848.5521, D3=0.4380, D4=0.0732. By Proposition 1, we know

that for a dual steady state,

μ

=s− ¯s>0. Consider the case of

μ

=s− ¯s=0.0011.Usingtheformulaofthelocalrepresentationof thecurveN inProposition4,weknowthatif

ν

=u− ¯u=0.0002,

theeconomy is exactlyon the curve N.Letnow set

ν

=0.0094. Then,since ∂ _∂νη2 ¯ >0,weknowthattheeconomybelongstothe re-gionabove thecurve N.InFig.4,wedepicttheheteroclinicorbit goingfromthedirtysteadystatetothegreensteadystate.

We complement the analysisby showing that a wrongpolicy maypushthe economybelow thecurve N.Ifthis happens,P_high∗

becomesasinkandtheeconomyremainstrappedaroundthedirty steadystate,asshowninFig.5forthesameparametervaluesused intheExampleabove.

5 Notice that the solutions for _{μare two, one positive and one negative. Since, in} our case, the dual steady states requires s > ¯s , and therefore a positive μ, we have neglected the negative solution.

Fig. 5. P ∗

high when the economy is below the curve N .

5. Concluding remarks

This paper sheds light on the role of fiscal policy in models wheremultiplelong-runequilibriaemergebecauseofthepresence of pollution externalities. Results are based on a purely dynami-calgrowthmodel `alaSolowwithaS-shapedproductionfunction, wherepollutionismodeledthroughanonlineardamagefunction. The saving ratio and the ratio of abated emissions are naturally usedaspolicyvariable.

We first outline the role of the saving rate in induc-ing/suppressingthemultiplicityofequilibria.Inparticular,wefind that there exists a specific level of thesaving rate such that the systemof dynamiclaws implied by our model eitherhas: (i) no steadystates,(ii)adualsteadystate(coalescenceequilibrium),and (iii)two steadystates,one withlower levelsofcapitaland pollu-tion(the“green” or“clean” steadystate)withregard totheother (the “dirty” steadystate).Bymeans ofthelocal analysis,we also findthat,inthecaseofadualequilibrium,thegreensteadystate

isalwaysasaddle,whereasthedirtysteadystateisalwaysa non-saddlerestpoint.

Assumingthesavingrateissuchthattheeconomypossessesa dualsteadystate,thesecondpartofthepaperexplorestheroleof adequatepoliciesindrivingtheeconomytowardsthecleansteady state. To do so, we need to switch to global analysis.In this re-gard, we first prove that thesystem ofdynamiclaws implied by ourmodelmayundergotheBogdanov–Takens(BT)singularity.The usefulnessofthisphenomenonlaysonthepossibilityofputtinga generic non linear system in correspondence with a simple pla-nar systemwhoseglobalunfoldingis knowninevery aspect.We find that, at least in proximity of the bifurcation, there are two qualitativelyseparatedregionsintheparameterspace.Inthefirst region,theeconomyisabletoreachthegreensteadystateby sim-plystartingoninitialconditionslocatedontheunidimensional sta-blemanifoldofthegreensteadystate.Inthe secondregion,only appropriately-dosed incrementsof theabatement share(apolicy, in our definition) put the economy in the condition of escaping thedirtysteadystateandfinallyapproachthegreensteadystate. Declaration of Competing Interest

None. Appendix A

A1. Localstabilityanalysis

The JacobianmatrixofthesystemM,evaluatedatthe steady stateis J=

⎡

⎣

2s( 1−u) k∗

(

1+k∗2

)

2₍ bp∗₊₁₎ −

δ

k − sbk∗2( 1−u)

(

1+k∗2

)

_{( bp}∗₊₁₎ 2 2θ ( 1−u) k∗

(

1+k∗2

)

2 −

δ

p

⎤

⎦

(A.1) Therefore Tr

(

J

)

=

_

2s

(

1− u

)

k ∗ 1+k ∗2



2

₍

bp ∗₊1

)

−

δ

k−

δ

p (A.2) Det

(

J

)

=

_

2s

(

1− u

)

k ∗ 1+k ∗2



2

₍

1+bp ∗

₎



θ

b

(

1− u

)

k ∗2



1+k ∗2



₍

_1+bp ∗

₎

−

δ

p



+

δ

k

δ

p (A.3)

Wecannowproceedtoevaluatethesignsof Det (J) atthegreen and dirtysteady states.We report hereonly the final stage of a numberof algebraicmanipulation performedwithMaple 18.The completesequenceofcommentedMapleisavailable fromthe au-thorsuponrequest.

Westartbyconsideringthegreensteadystate.Wefindthat Det

(

J

)

P∗

low=C low[s

(

1−u

)

δ

p



δ

ps 2

(

1−u

)

2−4

δ

k2[b

θ

(

1− u

)

+

δ

p]



+ −√





δ

ps 2

(

1− u

)

2− 2

δ

k2[b

θ

(

1− u

)

+

δ

p



] (A.4)

wherethepositiveconstantC_lowreadsas

C low=

2

δ

k

δ

p

s

(

1− u

)

[√



−

δ

ps

(

1− u

)

]2 > 0

Ifweimpose Det

(

J

)

_P∗

low<0in(A.4),thisimpliesthat





δ

ps 2

(

1− u

)

2− 2

δ

k2[b

θ

(

1− u

)

+

δ

p



> s

(

1− u

)

δ

p[

δ

ps 2

(

1− u

)

2− 4

δ

k2[b

θ

(

1− u

)

+

δ

p]

ByrecallingfromtheproofinProposition 1inthemaintext that



=

δ

p



δ

ps 2

(

1− u

)

2− 4b

δ

_k2

θ

(

1− u

)

− 4

δ

_k2

δ

p



(A.5) (A.4)reducesto

δ

ps 2

(

1− u

)

2− 2

δ

2k[b

θ

(

1− u

)

+

δ

p] > s

(

1− u

)

√



(A.6)

Multiplyingbothmemberof(A.6)by

δ

p,weget:

δ

p



δ

ps 2

(

1− u

)

2− 2

δ

k2[b

θ

(

1− u

)

+

δ

p]



>

δ

ps

(

1− u

)

√



(A.7)

Further, by adding and subtracting to the first member in (A.7)thequantity 2

δ

p

δ

2_k[b

θ

(

1− u

)

+

δ

p], andrecalling the defini-tionof



in(A.5),aftersomesimplealgebra,weget



+2

δ

p

δ

2k[b

θ

(

1− u

)

+

δ

p] >

δ

ps

(

1− u

)

√



whichis always satisfied,since we are assuming b>0 and

θ

>0, (

δ

p,s,u)∈(0,1)3,and



>0.

Nowitistheturnof Det

(

J

)

_P∗

high.Repeatingthestepsabove,we findnowthat Det

(

J

)

_P∗ high =C high[s

(

1− u

)

δ

p



s 2

₍

_{1− u}

₎

2

_δ

p− 4

δ

2_k[b

θ

(

1− u

)

+

δ

p]



+√





s 2

₍

_{1− u}

₎

2

_δ

p− 2

δ

2k[b

θ

(

1− u

)

+

δ

p]



]

wherethepositiveconstantChigh readsas

C high₌ 2

δ

k

δ

p

s

(

1− u

)

[√



+

δ

ps

(

1− u

)

]2 > 0

Contrarilyto what happensin P_low∗ , Det

(

J

)

_P∗

high ispositive. Indeed, theterm s

(

1− u

)

δ

p



s 2

₍

_{1− u}

₎

2

_δ

p− 4

δ

2_k[b

θ

(

1− u

)

+

δ

p]



= s

(

1− u

)

> 0

whiletheterm s 2

(

1−u

)

2

_δ

p−2

δ

k2[b

θ

(

1−u

)

+

δ

p]=



+2

δ

k2[b

θ

(

1−u

)

+

δ

p]> 0 when



>0,namelywhenthereexistsadualsteadystate.

A2.ExistenceoftheBTsingularity

InordertodetecttheBTbifurcationinplanarsystems,[3] out-lineaneasy-to-checkalgorithmbasedon[7].Weneedfirstto per-formthecoordinatechange

˜ k = k − k∗ ce ˜ p = p − p∗ ce

μ

= s − ˜s

ν

= u − ˜u System_Mbecomes ˙ ˜ k =

(

s ˜+

μ

)

(

1− ˜u −

ν

)



_˜ k t+k ∗ce



2 [1+b

(

p ˜t+ p ∗ce

)

]



1+



˜_{k t+k} ∗ ce



2



−

δ

k



_˜ k t+k ∗ce



˙ ˜ p =

θ

(

1− ˜u −

ν

)



_˜ k t+k ∗ce



2 1+



˜_{k t+k} ∗ ce



2 −

δ

p

(

p ˜t+ p ∗ce

)

(A.8) Nowthesystemisintheform

˙

x = f

(

x,

α

)

, x ∈R2_,

_α

_∈R2

where x=



˜_k,p˜



_and

_α

₌

₍

_μ

_,

_ν

₎

_{. In primis the algorithm checks} if there are regions in the parameter space such that the point

(

_k˜_t,p˜t

₎

₌

₍

_0,0

₎

_{is a bifurcation point with a double zero} eigen-value at

(

μ

,

ν

)

=

(

0,0

)

.As shown inProposition 3, thishappens at u =u ¯= 2

δ

2 p b

θ

(

δ

k−

δ

p

)

; s= ¯s=b

δ

k

θ

(

δ

k−

δ

p

)

δ

2 p

1+ 2

δ

p

δ

k−

δ

p which,providedthat



δ

_k−

δ

p



>0,mayoccur insidethe admissi-bleparameterspace.

8 G. Bella, D. Liuzzi and P. Mattana et al. / Chaos, Solitons and Fractals 130 (2020) 109432 Secondly, the algorithmchecks ifa(0)b(0) =0, wherea(0)and

b(0)arecertainquadraticcoefficients(generatedbythealgorithm itself).Thealgorithmshowsthat

a

(

0

)

=−

δ

k

(

δ

p+

δ

k

)

δ

2 p

δ

k−

δ

p 1



δ2 p( δp+δk) δk−δp b

(

0

)

=−

δ

2 k−

δ

2 p

δ

k

δ

p



δ

2 p

(

δ

p+

δ

k

)

δ

k−

δ

p Asaconsequence,a(0)b(0) =0.

Finally,themap(x,

α

)→(f(x,

α

),Tr(fx(x,

α

)),Det(fx(x,

α

)))must beregularat

(

x_,

α

)

₌

(

0_,0

)

.Sincethedeterminantofthemapis

−1 2

δ

2 p

(

δ

k−

δ

p

)

2

δ

p+

δ

k



δ

2 p

(

δ

p+

δ

k

)

δ

k−

δ

p theconditionsissatisfied.

As aconsequenceofthesteps above,systemM isacandidate fortheonsetofaBTsingularity.

Moreoverthealgorithmgives sign

(

σ

)

= sign



δ

p

(

δ

p+

δ

k

)

2



=+1

A3.TransformationofsystemMintonormalform

Theprocedurerequires5steps.

Step 1. ConsidersystemA.8andperformthesecondorder Tay-lorexpansionon



k˜,p˜,

μ

,

ν



.Weobtain



_˙ ˜ k ˙ ˜ p



=J



˜ k ˜ p



+B

(

μ

,

ν

)



˜ k ˜ p



+



F 1



_˜ k , p ˜,

μ

,

ν



F 2



_˜ k , p ˜,

μ

,

ν





(A.9)

where J containsthefirstorderterms,whilethematrix B andthe vector F =[F1

F2

] groupthesecondordertermsamongwhich B col-lects theones linearlydependent on



k˜,p˜



. In ourcase we have

J=

⎡

⎢

⎢

⎢

⎣

2s ˜

(

1− ˜u

)

k ∗ce



1+k ∗2ce



2

(

1+bp ∗ce

)

−

δ

k − ˜ s

(

1− ˜u

)

k ∗2ceb

(

1+bp ∗ce

)

2



1+k ∗2ce



2

θ

(

1− ˜u

)

k ∗_ce



1+k ∗2_ce



2 −

δ

p

⎤

⎥

⎥

⎥

⎦

B=

⎡

⎢

⎢

⎣

2

(

μ

u ˜+

ν

s ˜

)

k ∗ce

(

1+bp ∗ce

)



1+k ∗2ce



2 −

(

μ

u ˜+

ν

s ˜

)

bk ∗2ce

(

1+bp ∗ce

)

2



1+k ∗2ce



2

θν

k ∗_ce



1+k ∗2_ce



2 0

⎤

⎥

⎥

⎦

F1= 1 1+k ∗2ce



˜ s

(

1− ˜u

)

1+bp ∗_ce − ˜ s

(

1− ˜u

)

k ∗2_ce

(

1+bp ∗_ce

₎



_1+k ∗2_ce



−4k ∗2ce

(

1− ˜u

)

s ˜

(

bp ∗ce +s ˜

)

(

1+bp ∗ce

)

2



1+k ∗2ce



2



˜ k 2₊ 1 1+k ∗2ce ×



−k∗ ce

(

2bp ∗ces ˜

(

1− ˜u

)

+2s ˜

(

1− ˜u

)

)

b

(

1+bp ∗_ce

₎

3 +2k ∗3ce

(

1− ˜u

)

s ˜

(

b +k ∗ce

)

(

1+bp ∗ce

)

2



1+k ∗2ce



2



˜ k p ˜ + s ˜

(

1− ˜u

)

k ∗2ceb

(

1+bp ∗ce

)

3



1+k ∗2ce



p ˜2₋ k ∗2ce

(

1+bp ∗ce

)



1+k ∗2ce

μν

F2= 1 1+k ∗2ce



θ

(

1− ˜u

)

−

θ

(

1− ˜u

)

k ∗2ce 1+k ∗2ce −4

θ

(

1− ˜u

)

k ∗2ce



1+k ∗2_ce



2



˜ k 2

Step 2. TogetthenormalformdiscussedinCorollary 1,weneed toperformthesimilartransformationofvariables



w 1 w 2



=T



˜ k ˜ p



where T=



0 0 J

(

1, 1

)

J

(

1, 2

)



isthetransformationmatrixwhichtransforms(A.7)into



˙ w 1 ˙ w 2



=



0 1 0 0



w 1 w 2



+M

(

μ

,

ν

)



w 1 w 2



+



_˜ F 1

(

w 1, w 2

)

˜ F 2

(

w 1, w 2

)



(A.10) where M=T−1BT and



˜ F 1

(

w 1, w 2

)

˜ F 2

(

w 1, w 2

)



=˜F=T−1F



T



˜ k ˜ p



In particular, ifwe rename the entries of theJacobian matrixas J

(

i_,j

)

₌J_{i j,}fori_,j₌1_.2_,thenwehave

M=

⎡

⎢

⎢

⎢

⎣

(

2

μ

(

1− ˜u

)

+2s ˜

(

1−

ν

)

)

k ∗ce

(

1+bp ∗ce

)



1+k ∗2ce



2 −

(

μ

(

1− ˜u

)

+s ˜

(

1−

ν

)

)

bk ∗2ceJ 11

(

1+bp ∗ce

)

2



1+k ∗2ce



−

(

μ

(

1− ˜u

)

+s ˜

(

1−

ν

)

)

bk ∗2ceJ 12

(

1+bp ∗ce

)

2



1+k ∗2ce



−J 11

(

2

μ

(

1− ˜u

)

+2s ˜

(

1−

ν

)

)

k ∗ce J 12

(

1+bp ∗ce

)



1+k ∗2ce



2 + 2

θ

(

1−

ν

)

k ∗ ce J 12



1+k ∗2ce



2 + J 2 11

(

μ

(

1− ˜u

)

+s ˜

(

1−

ν

)

)

bk ∗2ce J 2 12

(

1+bp ∗ce

)



1+k ∗2ce



(

μ

(

1− ˜u

)

+s ˜

(

1−

ν

)

)

bk ∗2ceJ 11

(

1+bp ∗_ce

₎

2



₁ +k ∗2_ce



⎤

⎥

⎥

⎥

⎦

˜ F1 = 1



1+k ∗2_ce





˜ s

(

1− ˜u

)

α

(

1+bp ∗ce

)

− s ˜

(

1− ˜u

)

k ∗2ce

(

1+bp ∗_ce

₎



_1+k ∗2_ce



2 +2 k ∗2_ce

(

−2bp ∗_ces ˜

(

1− ˜u

)

− 2s ˜

(

1− ˜u

)

)

(

1+bp ∗_ce

₎

2



_{1+k ∗2} ce



2



w 2 1+



−k ∗ce

(

2bp ∗ces ˜

(

1− ˜u

)

+2s ˜

(

1− ˜u

)

)

b

(

1+bp ∗_ce

₎

3 +2 k ∗2_ce



bk ∗3_ces ˜

(

1− ˜u

)

+bk ∗_ces ˜

(

1− ˜u

)



(

1+bp ∗ce

)

2



1+ k∗2_ce



2





J 11w 21+J 12w 1w 2





1+k ∗2_ce



+ ˜ s

(

1− ˜u

)

k ∗2_ceb 2

₍

_J 11w 1+J 12w 2

)

2

(

1+bp ∗ce

)

3



1+k ∗2ce



− k ∗2ce

μν

(

1+bp ∗_ce

₎



_1+k ∗2_ce



˜ F2 = 1 1+k ∗2ce



θ

(

1− ˜u

)

−

θ

(

1− ˜u

)

k ∗2ce 1+k ∗2ce −4

θ

(

1− ˜u

)

k ∗2ce



1+k ∗2_ce



2



w 2 1

Step 3 . To simplify the nonlinear parts in (A.10), we follow [6]. Hence,(A.10)canberewrittenas



˙ w 1 ˙ w 2



=



0 1 0 0



w 1 w 2



+M

(

μ

,

ν

)



w 1 w 2



+



0 a 2w 21+b 2w 1w 2



(A.11) where a 2=− 1 2

∂

2_F˜ 2

(

w 1, w 2

)

∂

w 2 1 b 2=

∂

2_{F ˜} 2

(

w 1, w 2

)

∂

w 1

∂

w 2 +

∂

2_{F ˜} 1

(

w 1, w 2

)

∂

w 2 1

Step 4 .Atthisstage,[13]showsthatsystem(A.11)issimilarto



˙ w 1 ˙ w 2



=



0 1

η

1

η

2



w 1 w 2



+



0 a 2w 21+b 2w 1w 2



(A.12) where



η

1

η

2



=



D 1 D 2 D 3 D 4



μ

ν



given D 1= 2J 11

(

1− ˜u

)

k ∗ce J 12

(

1+bp ∗ce

)

(

1+k ∗ce

)

2 − J 112

(

1− ˜u

)

bk ∗2ce J 12

(

1+bp ∗ce

)

2

(

1+k ∗ce

)

D 2= 2J 11s ˜k ∗ce J 12

(

1+bp ∗ce

)



1+k ∗2ce



2 − 2

θ

k ∗2ce J 2 12



1+k ∗2ce



− J 112s ˜bk ∗2ce J 12

(

1+bp ∗ce

)

2



1+k ∗2_ce



D 3= 2

(

1− ˜u

)

k ∗ce

(

1+bp ∗ce

)



1+k ∗2ce



2 D 4= 2s ˜k ∗_ce

(

1+bp ∗ce

)



1+k ∗2ce



2 Since





D 1 D 2 D 3 D 4





= 4

θ

(

1− ˜u

)

k ∗2ce J 12



1+k ∗2_ce



4

₍

_1+bp ∗_ce

₎

doesnotvanish,theversaldeformationsatisfiesthetransversality condition.

Step 5 . Finally, [13] shows that (A.10) can be further trans-formedinto ˙ w 1= w 2 ˙ w 2=

η

1+

η

2w 2+a 2w 21+b 2w 1w 2+O



|

·

|

3



andrescaledtoobtainthesystem

˙ w 1= w 2

˙

w 2=

η

1+

η

2w 2+w 21±

σ

w 1w 2

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