Numerical analysis based on STACO model

2.4.1 Background of STACO 3.0

To gain a more explicit and practical insight in our game we apply an integrated assessment model STACO 3.0 (Nagashima et al., 2011; Dellink et al., 2015) for a numerical analysis.

STACO is an integrated model which links a (simple) climate change module with game theory for economic and policy analysis. The climate module of STACO relates GHG emission paths to GHG concentrations and atmospheric temperature change. The model has been calibrated in line with the EPPA model (Paltsev et al., 2005) and the DICE model (Nordhaus, 1994; Nordhaus, 2008). The economic part of STACO specifies regional payoff functions for emissions mitigation, which is composed of abatement costs and benefits.

Regional abatement benefits, which are calculated as the reduced damages, are given as a share of global abatement benefits (Fankhauser, 1995; Tol, 1997; Tol, 2009). STACO formulates a two-stage game of ICA formation among 12 heterogeneous regions: United States (USA), Japan (JPN), European Union-27 & EFTA (EUR), Other High Income countries (OHI), Rest of Europe (ROE), Russia (RUS), High Income Asia countries (HIA), China (CHN), India (IND) and the Middle East countries (MES), Brazil (BRA) and Rest of the World (ROW).

The main equations of STACO 3.0 are presented in Box 2.1. STACO adopts a time horizon of 100 years, ranging from 2011 ( = 1) to 2110 ( = 100). Eqs. (2.20) to (2.22) show the objective functions for coalition members ∈ , market participants ∈ and singletons ∉ , ∉ , which are based on the net present value of payoffs accruing to regions over a period of 100 years. Eq. (2.23) shows a linear functional form of abatement benefits, where represents the regional share of global benefits with ∑ ∈ = 1 , and

gives the aggregated climate change damages in terms of a percentage of gross world product (GWP). Parameter _, reflects two impacts of mitigation adopted in current period : one is the impact on future climate due to the inertia in the climate system; the other is the impact on GDP growth. The calibration of _, is based on the EPPA-5 (Paltsev et al., 2005) model by using a climate module from the DICE (Nordhaus, 1994) model. The regional abatement cost function (Eq. (2.24)) is a cubic function of regional abatement efforts _, , where parameters and are parameters estimated based on the data from EPPA (Morris et al., 2008), which is shown in Table 2.A1. For discount rates , , STACO 3.0 uses the Ramsey

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rule, which implies a changing discount rate over time based on the pure rate of time preference and regional growth rates of GDP. STACO 3.0 uses data from EPPA-5 model to calibrate the regional business as usual (BAU) emission paths (see Table 2.A1 for the BAU emissions in period one).

2.4.2 Modified STACO 3.0: aggregation of regions

The software package MATLAB (R2012a) is used for the numerical analysis. The partition function for the12-region STACO 3.0 model for mitigation coalitions is defined on the power set of the set of players . In our case, since we examine coalition membership and trade participation at the same time, the partition function is defined on the power set of × . For a numerical model with 12 regions the partition function cannot be calculated within reasonable time. We therefore aggregate the 12 regions into 7 regions: United States (USA),

Box 2.1. Main model equations of STACO 3.0 Payoff functions (Objective functions)

max

, ,( ) = ∑∈ ∑ { 1 + , ∙ ( ,( ) − , , )}, ∀ ∈ (2.20) max

, ,( ) = ∑ { 1 + , ∙ ( ,( ) − , , + [ , − ( ̅, − ,)])} , ∀ ∈

(2.21) max

, ,( ) = ∑ { 1 + , ∙ ( ,( ) − , , )}, ∀ ∉ , ∉ (2.22) with global abatement: =∑_{∈ ,}.

Abatement benefits

,( ) = 1 + , ∙ ∙ ∙ , ∙ , ∀ ∈ (2.23) which captures the long term effect of current abatement _, on climate change by including future time periods ⊆ [ , ∞).

Abatement costs

, , = ∙ ∙ _, + ∙ ∙ _, ∙ _,, ∀ ∈ (2.24) where 0 < _, < 1 is declining over time to reflect cost savings due to technological progress.

Ramsey rule for the discount rate

, = + ∙ ( ^,

,

− 1), = 0.015, = 1. ∀ ∈ (2.25)

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European Union-27 & EFTA (EUR), China (CHN), India (IND), the region of High Abatement costs and High Income (HAHI), ¹ the region of Low Abatement costs and Energy Exporting (LAEX) ² and Rest of the World (ROW). This reduces the calculation time by three orders of magnitude and allows for the numerical calculation of all possible coalitions and their payoffs in our model. We have recalibrated the STACO parameters for the abatement cost functions of the seven regions, and the undiscounted marginal cost curves in base year 2011 are shown in Figure 2.2:

Figure 2.2. Marginal abatement cost curves in 2011 for STACO model with 7 regions

All other parameters for STACO 3.0 with 7 regions are shown in Table 2.A1 in 2.7 Appendix II.

2.4.3 Implications from the specified functions

Based on the functions specified in Box 1, the relationship between equilibrium permit price

∗ and the total number of optimal initial permits ^∗ is (the derivation is provided in 2.7 Appendix II):

∗= ̅ − ∑

∙ _, ( ∙ _,) ∙ _,∙ ^∗

∙ _,

∈ . (2.26)

1 The integration of four regions in original STACO model: JPN, OHI, BRA and HIA.

2 The integration of three regions in original STACO model: ROE, RUS and MES.

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Eq. (2.26) cannot be solved analytically in an explicit form of ^∗( ^∗), therefore we resort to a numerical analysis. If we assume, for example, the number of traders is 7 in period = 1, then the numerical value of ^∗ can be obtained for any value ^∗. Figure 2.3 shows a plot of the relationship between ^∗ and ^∗ in the carbon market. As expected, the price of permits falls when the supply of permits increases, and the price decreases to zero when the total initial permits are chosen as the BAU emissions level.

Figure 2.3. Relationship between equilibrium price and total number of initial permits in carbon market

By taking the derivative of both sides of Eq. (2.26) with respect to ^∗, we get a differential equation of permit price ^∗:

∗( ^∗) = − 1

∑ , + 4 , ∗

∈

. (2.27)

Note that the coalition enters the carbon market as a single agent and is included in . From Eq. (2.27), we can see that under higher permit prices, the marginal price due to one more unit change in permits supplying (the absolute value ^∗( ^∗) ) is higher than the one under lower prices. This also can be observed from Figure 2.3.