This supplementary information describes in more detail the simplified model used to assess options for restricted linking of emissions trading systems (ETSs). We use the model to compute the abatement level, allowance prices, and abatement costs. We apply the model, first for reference, to the cases of no linking and full linking, and then to quotas, exchange rates and discount rates.
Model description
In our model, we use a simple representation of two ETSs in jurisdictions A and B with linear marginal abatement cost curves. We assume that the two ETSs have a long-term target path below business-as-usual emissions (i.e. with no over-allocation of allowances) and allow for banking of allowances, providing entities with a certain long-term emissions budget, and that the two ETSs do not have any price containment mechanisms, such as offsets, reserves, floors, caps and/or triggers. The limitations of these assumptions are discussed in Section 6.5 of the article.
We define each jurisdiction by: (1) the total abatement under no linking, reflected by the difference between its emission cap and business-as-usual emissions, and (2) a simplified, linear marginal abatement cost curve that represents how its cost of reducing emissions increases with the level of abatement:
ππ΄πΆ$(π΄π΅$) = π΄ β π΄π΅$ (Equation 6-1)
ππ΄πΆ*(π΄π΅*) = π΅ β π΄π΅* (Equation 6-2)
where MACA and MACB are the marginal abatement costs in jurisdictions A and B for a given abatement level ABA or ABB. For a given abatement level ABA and ABB in jurisdictions A and B, the corresponding allowance prices and abatement costs in each jurisdiction are computed as:
π$= π΄ β π΄π΅$ (Equation 6-3)
π*= π΅ β π΄π΅* (Equation 6-4)
πΆπππ$= 0.5 β π΄ β π΄π΅$2 (Equation 6-5)
πΆπππ*= 0.5 β π΅ β π΄π΅*2 (Equation 6-6)
where pA and pB are the allowance prices in jurisdictions A and B, and COSTA and COSTB
are the abatement costs in jurisdictions A and B.
We define the cost effectiveness as the total abatement costs in both jurisdictions with no linking divided by the total abatement costs in both jurisdictions with restricted (or full) linking. The cost effectiveness (CE) is thus computed as:
πΆπΈ =456745678,:;4567<,:
8;4567< (Equation 6-7)
where COSTA,0 and COSTB,0 are the abatement costs under no linking.
Where a restricted linking option changes the overall abatement outcome, we use this changed abatement level to calculate COSTA,0 COSTA, COSTB,0 and COSTB. To this end, we assume that the abatement under no linking would change in each jurisdiction proportionally to their emission reduction targets.
For the purpose of illustrating the results in charts in the article, we set the parameters of the model such that, in the absence of any linking, jurisdiction B has an allowance price three times higher than jurisdiction A, similar to the parameterization used by Burtraw et al. (2013):
π΄ = 0.3, π΅ = 0.6, π΄π΅$,@= 20, and π΄π΅*,@= 30
where ABA,0 and ABB,0 are the abatement levels in each jurisdiction under no linking.
No linking
The allowances prices in jurisdictions A and B (pA,0 and pB,0) and the abatement costs in jurisdictions A and B (COSTA,0 and COSTB,0) are computed using Equations 6-3 to 6-6, based on the abatement levels without linking (ABA,0 and ABB,0).
Full linking
Under full linking, both ETSs have equal marginal abatement costs and the overall abatement level is unchanged. The market equilibrium is thus given by two conditions:
π΄ β π΄π΅$,B= π΅ β π΄π΅*,B and (Equation 6-8)
π΄π΅$,B+ π΄π΅*,B= π΄π΅$,@+ π΄π΅*,@ (Equation 6-9)
where ABA,F and ABB,F are the abatement levels in jurisdictions A and B under full linking.
This gives:
π΄π΅$,B=*βD$*8,:;$*<,:E
$;* and (Equation 6-10)
π΄π΅*,B=$βD$*8,:$;*;$*<,:E (Equation 6-11)
The equilibrium allowance price pE = pA,F = pB,F, the abatement costs in jurisdictions A and B (COSTA,F and COSTB,F), and the cost effectiveness under full linking (CEF) are computed with Equations 6-3 to 6-7 respectively, based on the abatement levels under full linking (ABA,F and ABB,F).
Full linking involves a net transfer of allowances from jurisdiction A to jurisdiction B (TF), which is computed as:
πB= π΄π΅$,Bβ π΄π΅$,@. (Equation 6-12)
Quotas
Quotas restrict the amount of units from other jurisdictions that can be used for compliance. Quotas can be formulated and implemented in different ways. Here we define the quota level Q as the fraction of allowances that can be imported by jurisdiction A divided by the amount of allowances that are imported under full linking. The allowance transfer under the quota (TQ) is thus given by
πG= π β πB (Equation 6-13)
The abatement levels under the quota in each jurisdiction (ABA,Q and ABB,Q) are then computed as:
π΄π΅$,G= π΄π΅$,@+ πG and (Equation 6-14)
π΄π΅*,G= π΄π΅*,@β πG. (Equation 6-15)
The allowance prices in jurisdictions A and B (pA,Q and pB,Q), the abatement costs in jurisdictions A and B (COSTA,Q and COSTB,Q), and the cost effectiveness under the quota (CEQ) are computed with Equations 6-3 to 6-7 respectively, based on the abatement levels under the quota (ABA,Q and ABB,Q).
Exchange and discount rates
Exchange rates imply that units from one jurisdiction can be used for compliance in another, but their value is adjusted by a conversion factor. Exchange rates operate in a symmetrical fashion. If an exchange rate is set such that two units from jurisdiction A could be used in place of one unit in jurisdiction B, then in jurisdiction A, one jurisdiction B unit would be worth two jurisdiction A units.
Discount rates could be regarded as a variation on exchange rates. They also involve a conversion factor, but such that more than one unit from another jurisdiction is required to meet a compliance obligation in the own jurisdiction, thereby placing a greater value on units of the own jurisdiction. While exchange rates inherently require a symmetrical
relationship in the value of jurisdictionsβ allowances, discount rates do not. Jurisdictions could apply one discount rate in one direction of allowance flow (e.g., 3:1 from system A to system B) and parity (1:1) or a rate of different magnitude in the other direction (e.g., 2:1 from system B to system A).
We define the exchange or discount rate R as the number of allowances from jurisdiction A that are needed for compliance in jurisdiction B.
In contrast to full linking and quotas, exchange and discount rates can alter the overall abatement from both jurisdictions. The market equilibrium is given by three conditions:
First, the marginal abatement costs in jurisdiction B are R times higher than in jurisdiction A (Equation 6-16). Second, the abatement in jurisdiction A under the exchange or discount rate (ABA,R) corresponds to the abatement without linking, plus the allowances transferred to jurisdiction B (TR) (Equation 6-17). Third, the abatement in jurisdiction B under the exchange or discount rate (ABB,R) corresponds to the abatement without linking, minus the allowances imported, however, adjusted for the exchange or discount rate R (Equation 6-18):
π β π΄ β π΄π΅$,J= π΅ β π΄π΅*,J (Equation 6-16)
π΄π΅$,J= π΄π΅$,@+ πJ (Equation 6-17)
π΄π΅*,J= π΄π΅*,@β7JK (Equation 6-18)
These three conditions give the abatement levels under the exchange rate in each jurisdiction:
π΄π΅$,J=$*8,:;Jβ$*<,:
L;8βKM< (Equation 6-19)
π΄π΅*,J=$*8,:;Jβ$*<,:
J;8βK< (Equation 6-20)
The allowance prices (pA,R and pB,R) and the abatement costs (COSTA,R and COSTB,R) in each jurisdiction under the exchange or discount rate are computed with Equations 6-3 to 6-6 respectively.
The allowance prices in jurisdictions A and B (pA,R and pB,R), the abatement costs in jurisdictions A and B (COSTA,R and COSTB,R), and the cost effectiveness under the exchange or discount rate (CER) are computed with Equations 6-3 to 6-7 respectively, based on the abatement levels under the exchange or discount rate (ABA,R and ABB,R).