A quantitative approach to carbon price risk modeling - CiteSeerX

Given the fuel switching policy (ξt) i T t=0 −1 ∈ U i of the agent i, replace the de facto
allowances trading (θt) i T t=0 ∈ L 1 × L 1 by the virtual trading (ϑ i t) T t=0 ∈ L 1 × L 1
given by
t∑
ϑ i t = θt i + ξs, i t = 0, . . . , T. (24)
Next, for (ϑ i t) T t=0 ∈ L 1 × L 1 , (ξ i t) T −1
t=0 ∈ U i introduce the objective
s=0
I A,i
T (ϑi , ξ i ) := V ϑi ,A
T
− ϑ i T A T − π(Γ i − ϑ i T )+ + ∑ T −1
t=0 ξi t(A t − Et). i (25)
which, as shown in the proof of the following proposition, expresses (7) in terms
of the virtual trading in the sense that
if (ξ i t) T −1
t=0 , (ϑi t) T t=0 , (θi t) T t=0
fulfill (24), then IA,i
T
(θi , ξ i ) = I A,i
T (ϑi , ξ i ). (26)
Consequently, we have the following equilibrium characterization for the new parameterization
(compare with Definition 1)
Proposition 2. A ∗ = (A ∗ t ) T t=0 is an equilibrium carbon price process, if for i =
1, . . . , N there exist (ϑ i∗ , ξ i∗ ) ∈ L 1 × L 1 × U i satisfying
E(I A∗ ,i
T
(ϑ i∗ , ξ i∗ )) ≥ E(I A∗ ,i
T
(ϑ i , ξ i )) for all
(ϑ i , ξ i ) ∈ L 1 × L 1 × U i , i = 1, . . . , N
(27)
and
N∑
i=1
ϑ ∗i
t =
N∑
t∑
i=1 s=0
ξ ∗i
s holds at each time t = 0, . . . , T . (28)
Proof. Since the re-parameterization mappings
L 1 × L 1 × U i → L 1 × L 1 × U i , (θ i , ξ i ) → (ϑ i , ξ i )
from (24) are bijections for all i = 1, . . . , N and (θ i , ξ i ) N i=1 fulfills (11) if and only
if (ϑ i , ξ i ) N i=1 satisfies (28), it suffices to prove (26). This assertion is derived as
follows:
V θi ,A
T
− θ i T A T =
T∑
θ s (A s+1 − A s ) − θT i A T
s=0
= −θ 0 A 0 +
= −ϑ 0 A 0 +
T∑
(θ s−1 − θ s )A s
s=1
T∑
T∑
−1
(ϑ s−1 − ϑ s )A s + ξ s A s
s=1
s=0
T∑
−1
= V ϑi ,A
T
− ϑ i T A T + ξ s A s , (29)
17
s=0