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Full cooperation applied to environmental improvements

Rafa l Marcin Lochowskia joint work with W. Szatzschneider and M. Jeanblanc

Warsaw School of Economics

6th AMAMEF Conference

Rafa l Lochowski (WSE) Environmental improvements 6th AMAMEF Conference 1 / 20

Introduction

We analyse the case of certificates of environmental improvements and fullcooperation of two identical agents and consider three cases of possibleactions:

separate actions - certificates are issued for each agent separately;

collusive actions - there is one certificate that embraces twodomains and each agent has full information about pollution levels inother’s domain at any time 0 ≤ t ≤ 1 but she/he can makeimprovements in her or his domain only;

fusion - there is one certificate for two domains and, in practice, itcould lead to the free transfer of technologies, because two agentsact jointly as a one agent.


Model setting

T = 1 denotes time horizon.

Xi (t) denotes the polution level at time t in the ith domain, i = 1, 2,0 ≤ t ≤ T = 1.

In the first case (separate actions) ith agent minimizes

E(X 2i (1) + cost of improvements

).

In the second and the third case (collusive actions or fusion) agentsminimize

E(

(X1(1) + X2(1))2 + cost of improvements).


Model setting - separate actions

We model pollution levels X1,X2 as independent geometric Brownianmotions with improvements u1, u2 respectively:

dXi (t) = αXi (t)dt + βXi (t)dWi (t)− Aui (t)dt, (1)

where i = 1, 2.

A, α, β > 0 are model parameters,

W1,W2 are Brownian motions,

u1, u2 : [0; 1]→ [0; +∞) are (adapted) improvements,

X1(0) = x1 > 0,X2(0) = x2 > 0 are pollution levels at the moment 0.


Separate actions - costs of improvements

In any case we assume quadratic costs of improvements.

In the first case of separate actions ith agent minimizes

E

(X 2i (1) +

∫ 1

0u2i (s)ds

).

To solve the optimisation problem the simplified version of BSDEapproach will be used.


Separate actions - optimization problem

We will show that there exists a deterministic functionY : [0; +∞)→ [0; +∞) (in the general BSDE approach it is a process)such that

Y (1) = 1

and an adapted process u∗i such that

Zi (t) := Y (t)X 2i (t) +

∫ t

0u∗2i (s)ds

is a martingale

and for any other adapted process ui and t ≥ 0 we have

E

(Y (t)X 2

i (t) +

∫ t

0u∗2i (s)ds

)≤ E

(Y (t)X 2

i (t) +

∫ t

0u2i (s)ds

).

Thus u∗i solves the optimization problem.


Separate actions - optimization problem

We will show that there exists a deterministic functionY : [0; +∞)→ [0; +∞) (in the general BSDE approach it is a process)such that

Y (1) = 1

and an adapted process u∗i such that

Zi (t) := Y (t)X 2i (t) +

∫ t

0u∗2i (s)ds

is a martingale and for any other adapted process ui and t ≥ 0 we have

E

(Y (t)X 2

i (t) +

∫ t

0u∗2i (s)ds

)≤ E

(Y (t)X 2

i (t) +

∫ t

0u2i (s)ds

).

Thus u∗i solves the optimization problem.


Separate actions - optimization problem, cont.

Assuming that dY (t) = y(t)dt we calculate easily the drift part of Zi ,which reads as

Y (t)X 2i (t)(2α + β2) + X 2

i (t)y(t)︸︷︷︸independent of u∗i

+ u∗2i (t)− 2A · Y (t)Xi (t)u∗i (t)︸︷︷︸depends on u∗i

(2)

and is minimized foru∗i (t) = A · Y (t)Xi (t). (3)

Substituting (3) into (2) we get that the drift term equals 0 (i.e. Zi is amartingale) iff

dY (t)

dt= y(t) = A2Y 2(t)− (2α + β2)Y (t). (4)



Equation (4) has unique solution in [0; 1] such that Y (1) = 1, namely

Y (t) =1

A2

B +(

1− A2

B

)eB(t−1)

> 0

where B = 2α + β2.

We also have

u∗i (t) > 0,

for t ∈ [0; 1] since u∗i (t) = A · Y (t)Xi (t) and substituting this into (1), weget that

dXi (t) = (α− A · Y (t))Xi (t)dt + βXi (t)dWi (t),

thus Xi is a gBm and u∗i (t) > 0.



Equation (4) has unique solution in [0; 1] such that Y (1) = 1, namely

Y (t) =1

A2

B +(

1− A2

B

)eB(t−1)

> 0

where B = 2α + β2. We also have

u∗i (t) > 0,

for t ∈ [0; 1] since u∗i (t) = A · Y (t)Xi (t) and substituting this into (1), weget that

dXi (t) = (α− A · Y (t))Xi (t)dt + βXi (t)dWi (t),

thus Xi is a gBm and u∗i (t) > 0.


Fusion - optimization problem

In the fusion case each agent may act in both domains (in fact they act asa one agent) and the dynamics of X1, X2 is given by

dX1(t) = αX1(t)dt + βX1(t)dW1(t)− Au1(t)dt − Au2(t)dt,

dX2(t) = αX2(t)dt + βX2(t)dW2(t)− Au1(t)dt − Au2(t)dt,

and the agents minimize

E

((X1(1) + X2(1))2 +

∫ 1

0(u1(s) + u2(s))2 ds

). (5)

So we have the free transfer of technologies, and the costs ofimprovements correspond rather to thecost of the development of technologies no to the costs of implementingthem on a specific area (domain) - this could be proportional to the area.


Fusion - optimization problem, cont.

To solve this optimization problem we use similar techniques as before.Setting u = u1 + u2 and

Z (t) = X 21 (t)Y (t) + X 2

2 (t)Y (t) + 2X1(t)X2(t)Y (t) +

∫ t

0u2(s)ds (6)

we get that the drift term of Z is minimized for

u∗(t) = A (X1(t) + X2(t))(Y (t) + Y (t)

).



Substituting u∗ into the equation defining Z , (6), and assuming that

dY (t) = f ′(t)dt, dY (t) = g ′(t)dt,

we get that Z is a martingale when f and g satisfy the system of equations{f ′ = A2 (f + g)2 −

(2α + β2

)f ,

g ′ = A2 (f + g)2 − 2αg .(7)

To solve the optimization problem (5), we set the border conditionsf (1) = g(1) = 1.



Now, the problem will be solved if we show that the system (7) (with theborder conditions f (1) = g(1) = 1) has a solution such that

u∗(t) = A (X1(t) + X2(t)) (f (t) + g(t)) ≥ 0.

(We can not make negative improvements...)

But we have

Lemma

The solution of system (7) with the border conditions f (1) = g(1) = 1exists for t ∈ [0; 1] and for such t we have

f (t) + g (t) > 0.

and by the fact that u∗ minimizes (5) we have that X1 + X2 ≥ 0.

Remark

Let (f , g) be a solution of (7) in [0; 1] with the border conditionsf (1) = g(1) = 1 then f is strictly positive but g may be negative.



Now, the problem will be solved if we show that the system (7) (with theborder conditions f (1) = g(1) = 1) has a solution such that

u∗(t) = A (X1(t) + X2(t)) (f (t) + g(t)) ≥ 0.

(We can not make negative improvements...) But we have

Lemma

The solution of system (7) with the border conditions f (1) = g(1) = 1exists for t ∈ [0; 1] and for such t we have

f (t) + g (t) > 0.

and by the fact that u∗ minimizes (5) we have that X1 + X2 ≥ 0.

Remark

Let (f , g) be a solution of (7) in [0; 1] with the border conditionsf (1) = g(1) = 1 then f is strictly positive but g may be negative.


Functions f and g

0.2 0.4 0.6 0.8 1.0t

0.5

1.0

1.5

2.0

2.5

f HtLf for alpha=1, beta=2, A=7

0.2 0.4 0.6 0.8 1.0t

-2.5

-2.0

-1.5

-1.0

-0.5

0.5

1.0

gHtLg for alpha=1, beta=2, A=7


Function f + g

0.2 0.4 0.6 0.8 1.0t

2.05

2.10

2.15

2.20

2.25

f HtL + gHtLf+g for alpha=1, beta=2, A=7


Collusive actions

The dynamics of Xi , i = 1, 2, is given by


The free transfer of technologies does not occur! ;-(

and the agentsminimize

E

((X1(1) + X2(1))2 +

∫ 1

0u2

1(s)ds +

∫ 1

0u2

2(s)ds

). (9)

Again, we proceed as in previous cases but now we work with

Z = X 21 Y + X 2

2 (t)Y + 2X1X2Y +

∫ ·0u2

1(s)ds +

∫ ·0u2

2(s)ds. (10)


Collusive actions

The dynamics of Xi , i = 1, 2, is given by


The free transfer of technologies does not occur! ;-( and the agentsminimize

E

((X1(1) + X2(1))2 +

∫ 1

0u2

1(s)ds +

∫ 1

0u2

2(s)ds

). (9)

Again, we proceed as in previous cases but now we work with

Z = X 21 Y + X 2

2 (t)Y + 2X1X2Y +

∫ ·0u2

1(s)ds +

∫ ·0u2

2(s)ds. (10)


Collusive actions - optimization

The drift is minimimal for u1 = u∗1 = A(X1Y + X2Y

)and

u2 = u∗2 = A(X2Y + X1Y

), and, assuming that

dY (t) = ϕ′(t)dt, dY (t) = ψ′(t)dt,

we get that the drift term vanishes (i.e. Z is a martingale) iff

ϕ and ψsatisfy the system of two equations{

ϕ′ = A2(ϕ2 + ψ2

)−(2α + β2

)ϕ,

ψ′ = 2A2ϕψ − 2αψ.(11)

To solve the optimization problem (9), we set the border conditionsϕ(1) = ψ(1) = 1.


Collusive actions - optimization

The drift is minimimal for u1 = u∗1 = A(X1Y + X2Y

)and

u2 = u∗2 = A(X2Y + X1Y

), and, assuming that

dY (t) = ϕ′(t)dt, dY (t) = ψ′(t)dt,

we get that the drift term vanishes (i.e. Z is a martingale) iff ϕ and ψsatisfy the system of two equations{

ϕ′ = A2(ϕ2 + ψ2

)−(2α + β2

)ϕ,

ψ′ = 2A2ϕψ − 2αψ.(11)

To solve the optimization problem (9), we set the border conditionsϕ(1) = ψ(1) = 1.


Collusive actions - optimization problem, cont.

Now, the problem will be solved(recall that we can not make negative improvements) if we show that thesystem (11) (with the border conditions ϕ(1) = ψ(1) = 1) has a solutionsuch that ϕ ≥ 0 and ψ ≥ 0 for t ∈ [0; 1] and then

u∗1 = A (X1ϕ+ X2ψ) ≥ 0,

u∗2 = A (X2ϕ+ X1ψ) ≥ 0.

(By (8) X1 and X2 are gBm.)

But we have

Lemma

The solution of system (11) with the border conditions ϕ(1) = ψ(1) = 1exists for t ∈ [0; 1] and for such t we have ϕ > 0 and ψ > 0.


Collusive actions - optimization problem, cont.

Now, the problem will be solved(recall that we can not make negative improvements) if we show that thesystem (11) (with the border conditions ϕ(1) = ψ(1) = 1) has a solutionsuch that ϕ ≥ 0 and ψ ≥ 0 for t ∈ [0; 1] and then

u∗1 = A (X1ϕ+ X2ψ) ≥ 0,

u∗2 = A (X2ϕ+ X1ψ) ≥ 0.

(By (8) X1 and X2 are gBm.)But we have

Lemma

The solution of system (11) with the border conditions ϕ(1) = ψ(1) = 1exists for t ∈ [0; 1] and for such t we have ϕ > 0 and ψ > 0.


Optimizaton results

Separate actions. In this case, due to the martingale property of Zi ,

E(X 2i (1) +

∫ 10 u2

i (s) ds)

= y (0)X 2i (0) = y (0) x2

i

and

E

(X 2

1 (1) +

∫ 1

0u2

1 (s)ds + X 21 (1) +

∫ 1

0u2

1 (s) ds

)= y (0)

(x2

1 + x22

).

Fusion. In this case

E

((X1 (1) + X2 (1))2 +

∫ 1

0(u1 (s) + u2 (s))2 ds

)= f (0) (x1 + x2)2 + 2g (0) x1x2.

Collusive actions. In this case

E

((X1 (1) + X2 (1))2 +

∫ 1

0u2

1 (s)ds +

∫ 1

0u2

1 (s)ds

)= ϕ (0) (x1 + x2)2 + 2ψ (0) x1x2.


Optimizaton results


E(X 2i (1) +

∫ 10 u2

i (s) ds)

= y (0)X 2i (0) = y (0) x2

i and

E

(X 2

1 (1) +

∫ 1

0u2

1 (s)ds + X 21 (1) +

∫ 1

0u2

1 (s) ds

)= y (0)

(x2

1 + x22

).


E

((X1 (1) + X2 (1))2 +

∫ 1

0(u1 (s) + u2 (s))2 ds

)= f (0) (x1 + x2)2 + 2g (0) x1x2.


E

((X1 (1) + X2 (1))2 +

∫ 1

0u2

1 (s)ds +

∫ 1

0u2

1 (s)ds

)= ϕ (0) (x1 + x2)2 + 2ψ (0) x1x2.


Optimizaton results


E(X 2i (1) +

∫ 10 u2

i (s) ds)

= y (0)X 2i (0) = y (0) x2

i and

E

(X 2

1 (1) +

∫ 1

0u2

1 (s)ds + X 21 (1) +

∫ 1

0u2

1 (s) ds

)= y (0)

(x2

1 + x22

).


E

((X1 (1) + X2 (1))2 +

∫ 1

0(u1 (s) + u2 (s))2 ds

)= f (0) (x1 + x2)2 + 2g (0) x1x2.


E

((X1 (1) + X2 (1))2 +

∫ 1

0u2

1 (s) ds +

∫ 1

0u2

1 (s) ds

)= ϕ (0) (x1 + x2)2 + 2ψ (0) x1x2.


Optimizaton results - numerical examples

Recall the optimal values.

Separate actions.y (0) x2

i .

Fusion.f (0) (x1 + x2)2 + 2g (0) x1x2.

Collusive actions.

ϕ (0) (x1 + x2)2 + 2ψ (0) x1x2.

Numerical examples

Model parameters Quantityα = 1, β = 0.7

A = 0.7α = 1, β = 2

A = 2

Separate actions y (0) 3.80 1.5

Fussion f (0) , g (0) 2.16, 0.19 16.7, −13.81

Collusive optima ϕ (0) , ψ (0) 3.05, 1.10 1.49, 0.00


Thank you!


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