Meta-Modeling to Assess the Possible Future of Paris Agreement

Abstract

In the meta-modeling approach, one builds a numerically tractable dynamic optimization or game model in which the parameters are identified through statistical emulation of a detailed large scale numerical simulation model. In this paper, we show how this approach can be used to assess the economic impacts of possible climate policies compatible with the Paris Agreement. One indicates why it is appropriate to assume that an international carbon market, with emission rights given to different groups of countries will exist. One discusses the approach to evaluate correctly abatement costs and welfare losses incurred by different groups of countries when implementing climate policies. Finally, using a recently proposed meta-model of game with a coupled constraint on a cumulative CO₂ emissions budget, we assess several new scenarios for possible fair burden sharing in climate policies compatible with the Paris Agreement.

Appendix: Mathematical Description of the Meta-models

1.1 Appendix 1: The Game Design Problem.

Design variables

θ_j, share of the safety emission budget given to group of country j. These variables define the key element of the negotiations, namely the sharing of the safety emission budget.

Strategic variables

ω_j(t), supply of quotas by coalition j during period t. We assume that once a player (group of countries) has been given a share of the emission budget, it can supply this amount of quotas (emission rights) on the emissions trading markets organized at each period of the planning horizon. These supplies are strategic variables. They influence the market structure, determining price of carbon, then emission levels by each player, and, finally the transfers (buying and selling of permits) and the net surplus variations.

Secondary (passive) variables

These are variables that will be computed from the values given to the strategic variables. They will be used to describe the permits market functioning. Using statistical emulation an abatement cost is estimated as a function of the abatement realized w.r.t. the BaU scenario.

e_j(t):

emission level for group of countries j in period t;

q_j(t):

abatement level for player j in period t;

p(t):

carbon price in period t;

AC^t_j(q_j(t)):

abatement cost for player j in period t;

MAC^t_j(q_j(t)):

marginal abatement cost for player j in period t;

GTT_j(t):

gains from the terms of trade for player j in period t;

ν _j :

multiplier associated with the share of budget given to group of countries j.

Parameters

Bud:

safety budget: global safety emission budget;

bce_j(t):

BaU emissions for group of countries j in period t;

α_0j(t), α_2j(t), α_2j(t), α_3j(t), α_4j j(t)):

coefficients in the abatement cost function;

μ_0j(t), μ_1j(t):

coefficients in the gain from the terms of trade function;

β :

periodic discount factor;

hc_j:

discounted household consumption in BaU scenario over the planning horizon.

Payoffs for the game of quotas supply

The players (groups of countries) try to minimize (resp. maximize) the discounted sum of net surplus losses (resp. gains). The payoff is therefore defined as the discounted sum of the gains from the terms of trade plus the gains from the permit trading (can be negative) minus the abatement cost, given the actions taken by the other players.

$$ {W}_j=-\sum \limits_t{\beta}^t\left\{{AC}_j^t\left({q}_j(t)\right)-p(t)\left[{w}_j(t)-{e}_j(t)\right]-{\mathrm{GGT}}_j(t)\right\}, $$

(1)

where q_j(t) = bce_j(t) − e_j(t)

Functions estimated by statistical emulation of GEMINI-E3: They are the abatement cost

$$ {AC}_j^t\left({q}_j(t)\right)={\alpha}_{0j}(t)+{\alpha}_{1j}(t){q}_j(t)+{\alpha}_{2j}(t){q}_j{(t)}^2+{\alpha}_{3j}(t){q}_j{(t)}^3+{\alpha}_{4j}(t){q}_j{(t)}^4 $$

(2)

and the gains in the terms of trade

$$ {\mathrm{GTT}}_j(t)={\mu}_{0j}(t)+{\mu}_{1j}(t)\sum \limits_i{q}_i(t). $$

(3)

The statistical emulation of GEMINI-E3 is based on a sample of 200 scenarios that simulate different possible world climate change policies.

Objective of the game design problem

One applies a Rawlsian criterion of fairness [39].

$$ z={\max}_{\theta }{\min}_j\frac{W_j^{\ast }}{{\mathrm{hc}}_j}, $$

(4)

where ${\mathrm{W}}_j^{\ast }$ is the Nash equilibrium payoff for the game designed by the choice of the θ’s. So we select the sharing which, in the Nash equilibrium solution of the game of quotas supply, maximizes the worst surplus gain among the players.

We describe now how to characterize the Nash equilibrium in the game of quotas supply. There are m players (groups of countries) indexed j = 1, …, m, that generate emissions e_j (t) on periods t ∈ {0, 1, …, T − 1}. Let Ω (t) = ${\sum}_{j=1}^m{\omega}_j(t)$ denote the total supply of permits on the market at period t and p(t, Ω(t)) the clearing permit price at period t.

We assume a competitive market for emissions permits, which clears at each period. Given a price p(t), each player chooses emissions so as to (5)

$$ {\max}_{e_j(t)}\left\{{\pi}_j^t\left({e}_j(t)\right)+p(t)\left({w}_j(t)-{e}_j(t)\right)\right\}. $$

where ${\pi}_j^t\left({e}_j(t)\right)=-{AC}_j^t\left({\mathrm{bce}}_j(t)-{e}_j(t)\right)$ is the economic benefit associated with emission level e_j (t) at period t. The equilibrium conditions of profit maximization and market clearing are then

$$ {\displaystyle \begin{array}{c}p(t)=M{AC}_j^t\left({\mathrm{bce}}_j(t)\hbox{-} {e}_j(t)\right)=\frac{\partial }{\partial {e}_j}{\pi}_j^t\left({e}_j(t)\right)\\ {}t=0,\dots, T-1;\kern0.5em j=1,\dots, m,\end{array}} $$

(5)

$$ \Omega (t)=\sum \limits_{j=1}^m{e}_j(t),\kern1.5em t=0,\dots, T-1. $$

(6)

These conditions define after-trade equilibrium emissions, e_j (t, Ω(t)), and the permit price p(t, Ω(t, t). Differentiating (5) and (6), we can compute the derivatives

$$ \frac{\partial }{\mathrm{\partial \Omega }}\mathrm{p}\left(t,\Omega (t)\right)=\frac{1}{\sum_{j=1}^m\frac{1}{\pi_j^{t\hbox{'}\hbox{'}}\left({\mathrm{e}}_j^t\left({\Omega}^t\right)\right)}} $$

(7)

$$ \frac{\partial }{\mathrm{\partial \Omega }}{\mathrm{e}}_j\left(t,\Omega (t)\right)=\frac{1}{\sum_{i=1}^m\frac{\pi_j^{t\hbox{'}\hbox{'}}\left({\mathrm{e}}_j^t\left({\Omega}^t\right)\right)}{\pi_i^{t\hbox{'}\hbox{'}}\left({\mathrm{e}}_j^t\left({\Omega}^t\right)\right)}}. $$

(8)

Applying standard Kuhn-Tucker multiplier method, with multipliers ν_j, and exploiting the equality (5), we obtain the following first order necessary conditions for a Nash equilibrium

$$ {\displaystyle \begin{array}{l}{v}_j={\beta}_j^tM{AC}_j^t\Big({\mathrm{bce}}_j(t)-{\mathrm{e}}_j\left(t,\Omega (t)\right)+\frac{\partial }{\mathrm{\partial \Omega }}\mathrm{p}\left(t,\Omega (t)\right){\upomega}_j(t)-{\mathrm{e}}_j\left(t,\Omega (t)\right)\\ {}t=0,\dots, T-1;\kern0.5em j=1,\dots m.\end{array}} $$

(9)

$$ 0={v}_j\left({\theta}_j\mathrm{Bud}-\sum \limits_{t=0}^{T-1}{\upomega}_j(t)\right) $$

(10)

$$ 0\le {\theta}_j\mathrm{Bud}-\sum \limits_{t=0}^{T-1}{\upomega}_j(t) $$

(11)

$$ 0\le {v}_j. $$

(12)

1.2 Appendix 2: Imposing the Benchmark Abatement Scenario

To obtain in the game the same global abatement schedule as in the benchmark scenario, it suffices to add, for the periods 2020, 2030, and 2040, new coupled constraints imposing the global supply of permits on the world market to be greater than or equal to the corresponding emissions level in the benchmark scenario. More formally, we introduce the new parameters

$$ \mathrm{BMGE}(t):\kern0.5em \mathrm{global}\ {\mathrm{CO}}_2\kern0.5em \mathrm{emissions}\ \mathrm{at}\ \mathrm{period}\ t\ \mathrm{in}\ \mathrm{benchmark}\ \mathrm{scenario},t=1,\dots, T-1; $$

and the new constraints

$$ \sum \limits_{j=1}^m{\omega}_j(t)\ge \mathrm{BMGE}(t),\kern1.5em t=1,\dots, T-1. $$

(13)

The NOCs will be modified in an obvious way.

1.3 Appendix 3: Full Optimization vs. Equilibrium

To find the fully optimal solution, one solves the following problem:

$$ {\tilde{V}}^{\star }{=}_{\mathrm{q}\left(\cdot \right)}^{\mathrm{min}}\sum \limits_t\sum \limits_j{\beta}^t\left\{{AC}_j^t\left({q}_j(t)\right)\right\}, $$

(14)

$$ {}_{\mathrm{Bud}\kern0.5em \ge \kern0.5em \sum \limits_t\sum \limits_j\left\{{\mathrm{bce}}_j(t)-{q}_j(t)\right\}}^{s.t.}, $$

(15)

where, as defined above, bce_j (t) denotes the BaU emissions and q_j (t) the abatement level for group of countries j in period t; Let q^∗(·) denotes the optimal abatement path for all countries. The optimal emissions levels are determined by

$$ {e}_j^{\ast }(t)={\mathrm{bce}}_j(t)-{q}_j^{\ast }(t). $$

We form the Lagrangian

$$ \mathrm{\mathcal{L}}=\sum \limits_t\sum \limits_j{\beta}^t{AC}_j^t\left( qj(t)\right)-\pi \left(\mathrm{Bud}-\sum \limits_t\sum \limits_j\left[{\mathrm{bce}}_j(t)- qj(t)\right]\right), $$

(16)

where π is the multiplier (dual variable) associated with the global emissions budget constraint.

The optimality conditions are

$$ {\pi}^{\ast }={\beta}^t{MAC}_j^{\ast}\left({q}_j^t(t)\right),\kern0.5em \forall j,t, $$

(17)

$$ {\pi}^{\ast}\ge 0, $$

(18)

$$ \mathrm{Bud}\ge \sum \limits_t\sum \limits_j\left\{{\mathrm{bce}}_j(t)-{q}_j(t)\right\}, $$

(19)

$$ 0={\pi}^{\ast}\left(\mathrm{Bud}-\sum \limits_t\sum \limits_j\left[{\mathrm{bce}}_j(t)-{q}_j^{\ast }(t)\right]\right) $$

(20)

These are also the market clearing conditions for an international emissions trading scheme. The carbon price at t will thus be p^⋆(t) = π^⋆β^−t.

1.3.1 Fair Allocation of Emission Permits

We consider that fairness is obtained when the net costs relative to the respective consumption levels in the BaU scenarios are equalized. The balance in net cost per unit of consumption is obtained at period t if the following relation holds true

$$ {K}^{\ast }(t)=\frac{W_j^{\ast }(t)}{{\mathrm{hc}}_j(t)}=\frac{W^{\ast }(t)}{\mathrm{Hc}(t)}, $$

(21)

where

$$ {W}_j^{\ast }(t)={AC}_j^t\left({q}_j^{\ast }(t)\right)-{p}^{\ast }(t)\left[{\omega}_j(t)-{e}_j^{\ast }(t)\right]-{\mathrm{GTT}}_j^{\ast }, $$

(22)

and W^⋆(t) with Hc (t) are world level values. The fair allocation of permits at period t would then be

$$ {\upomega}_j(t)=\frac{1}{p^{\ast }(t)}\left\{{AC}_j^t\left({q}_j^{\ast }(t)\right)+{p}^{\ast }(t){e}_j^{\ast }(t)-{\mathrm{GTT}}_j^{\ast }-{K}^{\ast }(t){\mathrm{hc}}_j(t)\right\} $$