Mathematical Modeling for Environmental
Economics and Sustainable Management
Outils mathématiques pour la décision en environnement
Master EDDEE
Luc Doyen,Michel De Lara
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Doyen, DeLara
Modeling for the Sustainable Management
Trimester Mathematics of Bio-economics at IHP
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Doyen, DeLara
Modeling for the Sustainable Management
Program
Monday 7, January 10h00-13h00 : IHP : Course Doyen-De Lara Introduction to
decision modelling for sustainable management of natural resources
Monday 14, January 9h00-12h00 : ENGREF : Computer Session De Lara-Doyen
Introduction to the scientific software
Monday 21, January 9h00-12h00 : IHP Course Doyen-De Lara Controlled
dynamics and equilibria
Monday 28, January 9h00-12h00 : ENGREF : Computer Session De Lara-Doyen
Equilibria
Monday 4, February 9h00-12h00 : IHP Course Doyen-De Lara Optimality and
sustainability
Monday 11, February 9h00-12h00 : ENGREF : Computer Session De
Lara-Doyen Optimality
Monday 18, February 9h00-12h00 : IHP Course Doyen-De Lara Viability
Monday 25, February 9h00-12h00 : ENGREF : Computer Session De
Lara-Doyen Viability
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Doyen, DeLara
Modeling for the Sustainable Management
Some References
DeLara M.. & Doyen L., 2008, Sustainable Management of Natural
Resources Mathematical Models and Methods, Springer.
Clark C.W., (1976), Mathematical Bio-economics : The Optimal
Management of Renewable Resource, J. Wiley & Sons, New York
Heal G. (1998) Valuing the Future, Economic Theory and Sustainability.
Columbia University Press, New York.
Conrad J. M. 1999, Resource Economics, Cambridge University Press.
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Doyen, DeLara
Modeling for the Sustainable Management
Outline
General issues
Deterministic models in discrete time
–
–
–
–
General framework ; Examples
Equilibrium and stability.
Viability.
Inter-temporal optimality.
Models under uncertainty
–
–
Robust approach.
Stochastic approach.
Partial information
Strategic interactions
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Doyen, DeLara
Modeling for the Sustainable Management
Environmental issues
Management of exhaustible resources : oil, mining, ...
Pollution management : ex : C02
Management of renewable resources : Fisheries, wildlife,
forest, agroecology
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Doyen, DeLara
Modeling for the Sustainable Management
Sustainable management of biodiversity
Global changes in ecosystems
Alarming trends : MEA, IUCN, ...
Major concerns for ecosystem services
=⇒ bio-economic risk
Sustainable management
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Doyen, DeLara
Modeling for the Sustainable Management
IPBES
"From IPCC"
To
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Doyen, DeLara
Modeling for the Sustainable Management
Biodiversity scenarios
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Doyen, DeLara
Modeling for the Sustainable Management
Bio-economic modeling
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Doyen, DeLara
Modeling for the Sustainable Management
Bio-economic policies
Three major components :
Scientific knowledge :
mechanisms, models, data.
Shared intertemporal objectives :
ex : Johannesburg 2002 : MSY by 2015
Bio-economic instruments
standards : quotas, MPA, ...
monetary : landing taxes, ITQ, ...
informational : MSC,...
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Doyen, DeLara
Modeling for the Sustainable Management
Important goals
Sustainability
–
Reconciliation
Economy–Social-Ecology
–
–
Intergenerational equity
Intragenerational equity
Resilience
Precaution
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Doyen, DeLara
Modeling for the Sustainable Management
Modeling requirements
Systemic approach
Evaluation
Multi-criteria
Dynamics
–
–
Uncertainties
Decision
Ecology
Socio-Economy
Short and long term horizon
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Doyen, DeLara
Modeling for the Sustainable Management
Decision approaches for sustainability
Equilibrium
and stability :
Steady state,
MSY
Schaefer, 1954
Intertemporal
Optimality
Constraints, viability
and risk
Present Value
PVA
CB, CE
Beissinger, 2002
Clark, 1976
Viable control
Maximin criteria
TWA
....
SMS
Bishop, 1978
Doyen, DeLara
Modeling for the Sustainable Management
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Control theory
Problem :
Find controls to achieve various goals for states of a system
Three main ingredients :
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–
–
Controlled dynamics
Constraints
Criteria to optimize
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Doyen, DeLara
Modeling for the Sustainable Management
Control theory in discrete time
Control theory
in discrete time
blanco
Economics
decision
under
uncertainty
Ecology
Life-cycle
meta-population
Modeling
Simulations
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Doyen, DeLara
Modeling for the Sustainable Management
Examples for natural resource management
Stylized models :
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Exploitation of an exhaustible resource
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Management of a renewable resource
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Mitigation policies for CO2 emissions
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Doyen, DeLara
Modeling for the Sustainable Management
Exploitation of an exhaustible resource
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Doyen, DeLara
Modeling for the Sustainable Management
Management of a renewable resource
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Doyen, DeLara
Modeling for the Sustainable Management
Mitigation policies for CO2 emissions
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Doyen, DeLara
Modeling for the Sustainable Management
A controlled dynamics in discrete time
Difference equation :
where
–
–
–
n
xi (t + 1) = Fi (t, x(t), u(t)),
t = 0, . . . , T
(1)
the state, x (t) = (x1 (t), . . . , xn (t)) ∈ X = Rn ;
the control or decision, u(t) = (u1 (t), . . . , un (t)) ∈ U = Rp ;
the time Horizon T
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Doyen, DeLara
Modeling for the Sustainable Management
State constraints
The admissible states
x(t) ∈ A(t),
t = 0, . . . , T − 1 ,
(2)
generally specified under inequality form
(
gj (t, x(t)) ≤ 0
gk (t, x(t)) = 0 .
In many examples, a constant set :
x(t) ∈ A
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Doyen, DeLara
Modeling for the Sustainable Management
Final state constraints
Final state reaches some target C ⊂ X :
x(T ) ∈ C
(3)
Generally specified under inequality form
gj (x(T )) ≤ 0
gk (x(T )) = 0.
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Doyen, DeLara
Modeling for the Sustainable Management
Decision or control constraints
The admissible decisions :
u(t) ∈ B(t, x(t)),
t = 0, . . . , T − 1
(4)
generally specified under inequality form
(
gj (t, x(t), u(t)) ≤ 0
gk (t, x(t), u(t)) = 0 .
Frequently, a constant set of feasible control
u(t) ∈ B
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Doyen, DeLara
Modeling for the Sustainable Management
The feasible decisions and sustainability
The feasible strategy : u(.) = (u(0), u(1), . . . , u(T )) such that
–
–
The constraints (3), (2) and (4)
where x (.) with the dynamics (1)
Feasibility set :
Tad =
(x(.), u(.)) (1), (3), (2), (4) hold true
Viability or invariance approach
A particular case : equilibria
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Doyen, DeLara
Modeling for the Sustainable Management
Optimality
An inter-temporal criterion
π x(0), x(1), . . . , x(T ), u(0), u(1), . . . , u(T − 1)
The optimal control problem :
π(x ⋆ (·), u ⋆ (·)) =
max
(x (.),u(.))∈Tad
π(x(·), u(·)).
where
–
–
u ⋆ (·) = (u ⋆ (0), u ⋆ (1), . . .) optimal decision sequence,
x ⋆ (·) = (x ⋆ (0), x ⋆ (1), . . .) optimal state sequence.
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Doyen, DeLara
Modeling for the Sustainable Management
Some criteria
Present value
π(x (·), u(·)) =
T
−1
X
ρt L(x (t), u(t))
t=0
Maximin
π(x (·), u(·)) =
min
t=0,...,T −1
L(t, x (t), u(t)).
Green
π(x (·), u(·)) = M(T , x (T ))
Chichilnisky
π(x , u) = θ
T
−1
X
L(t, x (t), u(t)) + (1 − θ)M(T , x (T ))
t=0
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Doyen, DeLara
Modeling for the Sustainable Management
A useful result
min f (y ) = − max −f (y )
y ∈A
y ∈A
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Doyen, DeLara
Modeling for the Sustainable Management
Simulations can help
Given x0 , compute outputs of specific strategies
u(t)
u(t, x )
Example : scenarios of mitigation policies
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Doyen, DeLara
Modeling for the Sustainable Management
SCILAB Code
t0=1990 ; T=2100 ; M0=354 ;alp=0.471 ; Minf=280 ; d=1/120 ;
EBAU0=7.2 ;g=0.01 ;
//
function y=EBAU(t) ; y=EBAU0*(1+g)^(t-t0); endfunction
//
function y=f(t,M,ab) ; y=M+alp*EBAU(t)*(1-ab)-d*(M-Minf) ; endfunction
//
ab=zeros(1,T) ; // for instance
x(t0)=M0 ;
for (t=t0 :1 :T)
x(t+1)=f(t,x(t),ab(t)) ;
end
//
plot2d(t0 :T+1,x(t0 :T+1),rect=[t0,0,T+1,1000])
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Doyen, DeLara
Modeling for the Sustainable Management
The sensitivity problem
Pb : The outputs g(x(t), u(t)), L(x(t), u(t)), π depend on :
the parameters of the model
the functional forms of the model
the initial conditions
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Doyen, DeLara
Modeling for the Sustainable Management
The sensitivity problem
Example : Different population
dynamics with r = 3, K = 10
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Doyen, DeLara
Modeling for the Sustainable Management
What about complexity ?
More complex models :
Exploitation of an exhaustible resource with capital (DHS)
A trophic web and sustainable use values
An exploited metapopulation and PA management
SSVPA Model
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Doyen, DeLara
Modeling for the Sustainable Management
Exploitation of an exhaustible resource with capital (DHS)
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Doyen, DeLara
Modeling for the Sustainable Management
A trophic web and sustainable use values
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Doyen, DeLara
Modeling for the Sustainable Management
Simulations can help
Cheung et al, Fish and Fisheries, 2009
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Doyen, DeLara
Modeling for the Sustainable Management
Simulations can help : Fisheries in French Guiana
Cissé et al., Environment and Development Economics, 2013
Biodiversity
Seafood production
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Doyen, DeLara
Modeling for the Sustainable Management
Simulations can help ! ! Farming land-use and birds
Mouysset et al., Ecological Economics, 2011
2008
ւ ց
HQE2 2050
ENERGY 2050
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Doyen, DeLara
Modeling for the Sustainable Management
Simulations can help ! ! Fisheries in Bay of Biscay
Doyen et al., Ecological Economics, 2012
Co-viability
Status Quo 2008
25
20
15
10
50
250
200
150
100
30
25
20
15
10
50
2020
time(years)
3.5e+007
3.0e+007
2.5e+007
2.0e+007
1.5e+007
1.0e+007
2010
2015
2020
time(years)
0
2025
1.2e+007
2010
2015
2020
time(years)
0
2025
4.5e+007
4.0e+007
1.0e+007
3.5e+007
8.0e+006
3.0e+007
2.5e+007
6.0e+006
2.0e+007
4.0e+006
1.5e+007
1.0e+007
2012
2016
2020
2024
0.0e+000
2008
2028
2012
2016
2020
2024
2028
200
150
100
2015
2020
time(years)
0
2025
8e+006
6e+006
4e+006
2e+006
0e+000
-2e+006
-4e+006
-6e+006
2010
6e+007
5e+007
4e+007
3e+007
2e+007
1e+007
0e+000
-1e+007
2008
-1e+007
2008
2024
2028
30
25
20
15
10
0
2025
7e+007
-8e+006
2020
2020
time(years)
8e+007
0.0e+000
2008
2016
2015
9e+007
5.0e+006
2012
35
5
2010
2.0e+006
5.0e+006
0.0e+000
2008
0
2025
profit fleet 2 (Keuro)
2015
profit fleet 2 (Keuro)
2010
250
50
5
profit fleet 1 (Keuro)
5
0
40
35
SSB species 2 (Tons)
100
30
profit fleet 2 (Keuro)
150
35
SSB species 1 (Tons)
200
SSB species 2 (Tons)
250
NPV scenario
40
SSB species 1 (Tons)
SSB species 2 (Tons)
SSB species 1 (Tons)
40
2012
2016
2020
2024
2028
2010
2015
2020
time(years)
9e+006
8e+006
7e+006
6e+006
5e+006
4e+006
3e+006
2e+006
1e+006
2012
2016
2020
2024
2028
0e+000
2008
2012
2016
2020
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Doyen, DeLara
2025
Modeling for the Sustainable Management
2024
2028
But which synthetic indicators ? ? ?
Mouysset et al., Ecological indicators, 2011
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Doyen, DeLara
Modeling for the Sustainable Management
Open loop or closed loop ?
Open-loop : time-dependent sequences :
u : t → u(t) .
Closed-loop ≈ feedback
u : (t, x) → u(t, x) .
–
–
More robust
Dynamic programming methods.
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Doyen, DeLara
Modeling for the Sustainable Management
Decision tree
Finite countable decisions
(ex u ∈ U = {0, 1})
Given x0
Tree :
–
–
nodes = states x (t)
edges = decisions u(t)
Evaluate the performance π on
every path
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Doyen, DeLara
Modeling for the Sustainable Management
The curse of the dimensionality
A binary decision u ∈ {0, 1} on horizon T
=⇒ 2T possible sequences u(.)
On a computer :
–
–
ram : 8 GBytes = 8(1 024)3 = 233 bytes
a double-precision real : 8 bytes = 23 bytes.
=⇒ 230 double-precision reals
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Doyen, DeLara
Modeling for the Sustainable Management