A Cost-Effectiveness Differential Game Model for Climate Agreements

Abstract

In this paper, we propose a differential game model with a coupled constraint to represent the possible effects of climate agreements between industrialized, emerging and developing countries. Each group of countries is represented by an economic growth model where two different types of economies, called, respectively, ‘low-carbon’ and ‘carbon,’ can co-exist, each of which having different productivities of capital and of emissions due to energy use. We assume that each group of countries participating in the negotiations has identified a damage function, which determines a loss of GDP due to warming and has also a possibility to invest in a capital permitting adaptation to climate changes. The climate agreements we consider have two main components: (1) They define a global emission budget for a commitment period and impose it as a limit on cumulative emissions during that period; (2) they distribute this global budget among the different coalitions of countries taking part in the agreement. This implies that the game has now a coupled constraint for all participants in the negotiations. The outcome of the agreement is therefore obtained as a generalized or ‘Rosen’ equilibrium which can be selected among a whole manifold of such solutions. We show that the family of Nash equilibria in the games obtained through a distribution of the total budget among the different parties corresponds to the manifold of normalized equilibria. We then propose an equity criterion to determine a fair division of this total emission budget or equivalently to select a proper weighting for a normalized equilibrium.

Appendices

Appendices

1.1 Appendix 1: Model Calibration

The calibration of the three-player Ada-BaHaMa model follows the approach detailed in [3] for a two-player model. It is done for the Pareto scenario.

In short, the different economic and climate parameters [Eqs. (1)–(10)] are mostly from the DICE model (version 2007,⁵ thereafter referred to as DICE2007). Compared to the carbon economy, production in the low-carbon economy has higher energy costs but a better energy efficiency. As a result, the overall production of the three-player Ada-BaHaMa reproduces the economic output of DICE2007.

In addition, some regional parameter values have been adapted in the spirit of the RICE model. In particular, the three players have different population levels and initial values for capital accumulation in the carbon economy:

$L(j,0)$::

initial value for population level of player $j$, in millions of persons; $L(1,0)=1{,}043.9$; $L(2,0)=2{,}731.5$; $L(3,0)=2{,}635.5$;

$K_1(j,0)$::

initial value for carbon-intensive capital of player $j$, in trillions USD; $K_1(1,0)=60.2$; $K_1(2,0)=20.6$; $K_1(3,0)=16.6$.

Damages and adaptation parameters [Eqs. (11)–(13)] are from the AD-DICE model [7] and the World Bank [15]. Note that the maximal adaptation effectiveness is assumed to be 0.33 in all three regions. As a result, Ada-BaHaMa reproduces the overall magnitude of climate change damages estimated by DICE2007 and AD-DICE.

1.2 Appendix 2: GAMS Code

The different GAMS codes used to perform our numerical experiments are available from http://www.ordecsys.com. We provide in particular:

Ada_Bahama-3pBAU.gms::

the code to run our baseline (BAU) scenario;

Ada_Bahama-3pPareto.gms::

the code to run our Pareto scenario;

Ada_Bahama-3pNash.gms::

the code to run our Nash scenario;

Ada_Bahama-3pRosen.gms::

the code to run our Rosen scenarios.