Interaction in Climate GamesThe Case of Emissions Trading

Jurgen ScheffranPotsdam Institute for Climate Impact Research

Telegraphenberg A31, P.O. Box 60 12 03, 14412 Potsdam, GermanyTel.: (+49) 331 - 288-2528, email: scheffran@pik-potsdam.de

Abstract: Climate games deal with the interaction among multiple actors onglobal, regional and local levels of climate policy, increasing or decreasingemissions of greenhouse gases. Emissions trading is an instrument of the KyotoProtocol to achieve emission reductions in regions and business sectors wherethey are least costly. Emission paths and emission trading flows are analysedwithin a dynamic multi-agent game, with reaction functions depending on in-dividual threshold prices, the level of allowed emission reductions and the pos-sibility to shift to a low-emission production. Data-based computer simulationdepicts the interaction between emission reductions and prices.

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1 Climate games and emissions trading

A major task in international climate negotiations is to find agreed emission li-mits and trajectories that prevent dangerous climate change, in accordance withArt. 2 of the UN Framework Convention on Climate Change (UNFCCC). [Phi-libert/Pershing (2001), Ott etal. (2004)]. One of the key issues is to find a fairallocation of emission permits respecting these limits.[Leimbach (2003)] Eventhough developing countries are less responsible for climate change, they wouldbe affected much stronger and would be less capable to take countermeasures.To deal with these asymmetries, industrialized countries with high per-capitaemissions agreed in the Kyoto Protocol to cut them down.

One of the questions is to define levels and indicators for dangerous climatechange that can be translated into emission trajectories. An established mecha-nism is the tolerable windows approach which defines guardrails for tempera-ture change which in a reverse manner selects emission paths compatible withthe guardrails.[Bruckner etal. (1999), Petschel-Held etal. (1999)]. A more gene-ral mathematical approach is viability theory which applies regulators to keepa dynamic system within viable constraints [Aubin/Saint-Pierre (2004)]. An as-sociated problem is to find mechanisms to allocate emission limits and permitsfrom global levels down to regional, national and local levels, including indivi-dual firms and consumers.

Actors are important in several ways as they can choose targets as well as ac-tions. Target setting is the result of evaluation processes which take into con-

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sideration the selection or exclusion of certain sets of system states or trajec-tories, based on value functions. Actions are chosen according to given rules(rule-based behavior) or in order to achieve given target sets. In many cases theoptimum of a defined value function is sought. Both approaches are combinedin adaptive iteration towards reaction functions which depend on actions takenby other actors.

Climate games deal with the decisionmaking and interaction among multipleactors on global, regional and local levels of climate policy. Game theory pro-vides the terminology and a theoretical framework to analyze interpedendentdecisionmaking, negotiations and coalition formation in climate policy [Svi-rezhev etal. (1999), Finus (2001), Kemfert (2001), Grundig etal. (2001)]. Fora large number of actors and criteria, and complex dynamic interactions othermethods are appropriate, such as multi-criteria decisionmaking, dynamic ga-mes and agent-based modelling.[Brassel etal. (2000), Pickl (2001), Krabs/Pickl(2003), Weber etal. (2003)]. Expert interviews, stakeholder dialogues and ex-perimental gaming provide additional methodologies to link climate model-ling with the socio-economic world, currently a key issue in integrated as-sessment.[Moss (2002)].

Decisionmaking in climate games is complicated by the number of actors andmultiple levels involved which interfer with each other. Multi-actor and multi-level decisionmaking can be analyzed with a top-down approach from globaldecisionmaking bodies, which define global targets for emission reductions, ba-sed on scientific assessment and evaluations of what is tolerable or dangerousclimate change. Decisions are implemented on global levels as well as national

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and sub-levels. In a bottom-up approach, local actors such as citizens, consu-mers and companies pursue their individual interests, having an impact on hig-her levels, e.g. by electing municipal and national governments or by selectingproducts with more or less environmental impact. In reality both approachesinterfer with each other across levels.

Market mechanisms are assumed to provide an efficient and cost-effective allo-cation. Emission trading is designed as a market instrument to achieve emissionreductions in regions and business sectors where they are least costly. Some ofthe models in this field combine general equilibrium models with the selecti-on of policy instruments (see the survey in [Springer (2003)]. The underlyingmicro-macro link is a challenge for the theory of emissions trading as well asfor its implementation.

In this paper, emission paths and emissions trading are analysed in a game ofmultiple actors which act according to value functions, taking into account netbenefits of economic growth as well as marginal damages of climate change,costs for emission reduction and the selling and buying of emission permits [seeScheffran/Leimbach (2003)]. This expands the authors modelling frameworkdeveloped to analyze dynamic games in climate policy.[Scheffran/Pickl (2000),Ipsen et al. (2001), Scheffran (2002ab)]

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2 The model framework for emissions trading

The task of climate policy is to keep total emissions G(t) =n

i=1Gi(t) ofi = 1, ..., n actors in a time period t below a total allowed limit G(t). andto translate it into admissible targets Gi (t) for each actor. Regulations are de-signed to generate emission reductions Ri(t) = ri(t)Gi(t) from an emissionbaseline Gi(t) for actor i such that actual emissions Gi(t) = Gi(t) Ri(t) =Gi(t)(1ri(t)) Gi (t) stay within the limits. ri(t) is the percentage of reducti-ons from the baseline. Emission reductions are associated with reduction costs,either by loss of production, consumption and thus benefits, or by investing intorestructuring towards low-emission and high-cost technologies.

One option is to impose Gi as legal limits and leave it to actors i to obey theselimits by whatever means, paying a fine if the limit is violated. More gradualis emission tax, i.e. actors pay an amount proportionate to emissions. A marketapproach is emissions trading, based on defined emission permits Gi and allo-cation plans for a group of actors to satisfy the collective limit. Each actor canbuy or sell permits to increase or reduce emissions at a market price that resultsfrom the interplay of supply and demand, depending on the benefits and costs ofemission reductions. While some actors acquire emission permits on the marketif this is beneficial, others sell them.

In the following we develop the modelling framework to analyze the decision-making and interaction processes on emission reductions. Each actor can investinto production and consumption of economic goods, causing emissions and

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damages, and buy or sell emission permits. The value functions of actors areaffected by the following terms:

The benefits generated from production and consumption of economicoutput Q, measured by a utility function U(Q)

Investments (costs and efforts) C(G) associated with emissions G to ge-nerate economic output

The damages and dangers in terms of utility losses D(G) induced byemissions causing climate change

The costs and income for buying or selling emission permits, in excessof allowed emissions G

Expressing all terms in units of utility gains and losses results in the value func-tion V = U C D. In the following, the four value terms are defined asnon-linear functions:

Q = q G, U = uq Q = ugGCG = cg G, CR = cr R, D = d G

Here we distinguish between the production costs CG for generating emissionsand the mitigation costsCR for emission reductions. The factors q, cg, cr, d, uq, ug

represent the respective output on the left-hand side for the first unit of input onthe right hand side. Note that for R < 0 (emission increase) the abatementcosts CR are zero (cr = 0). The parameters ug = uqq and = combineproduction and emissions in the utility function.

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In emissions trading schemes, deviations G G from allowed emissions aretaken into account with a term = pi (G G) where pi is the price peremission unit. Then for i = 1, ..., n actors we have a non-linear value functionof emissions Gi and emission reductions Ri, assuming exponents are equal forall actors:

Vi = UiCiDii = ugi Gi cgi Gi cri Ri di Gpi(GiGi ). (1)

Note that damage is a function of total emissions G =

j Gj. Introducing t asan index for a given time period and using Gi(t) = Gi(t)(1 ri(t)) as redu-ced from baseline emissions in this period, we seek those percentage emissionreductions ri(t) 1 that maximize the value function Vi (negative ri < 0 re-present emission increases). Thus we resolve Vi/ri 0 which leads to

(ugi G1i +cgi G1i cri R1i +diG1+pi)Gipii(GiGi ) 0 (2)

with pii = pi/ri. It is not possible to resolve this general non-linear equationfor ri but in special cases (see section 4).

3 The price mechanism

Resolving equation 2 for the threshold price leads to:

pi ugi G1i +cgi G1i cri R1i di G1+pii(1GiGi

) pii . (3)

This threshold price increases with productivity ugi and decreases with damageper emission unit di and costs cgi , cri . If marketprice exceeds this threshold (pi >

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pii ) it is profitable for actor i to sell emission permits, below this threshold itis profitable to buy emission permits. The actual permit price adjusts to thisdemand-supply interaction. For each actor the difference pii pi > 0 is anincentive to buy permits while pi pii > 0 is an incentive to sell them. Thus,this difference can be used to define individual demand and supply functionsaround the own threshold price pii for actors i = 1, ..., n:

- Demand G+i = a+i (pii pi) 0 (for pi pii )

- Supply Gi = ai (pi pii ) 0 (for pi pii )

Parameters a+i and ai indicate the reactivity of demand and supply to theprice difference which in the following we assume to be equal for actor i (a+i =ai ai). Summing up all demands and supplies leads to the well-known lineardemand and supply functions. Then market balance (supply=demand) leads to

i

Gi =i

ai(pii pi) = 0.

Thus, for homogenous reactions ai = a for all actors i = 1, ..., n, the marketprice is the average of all marginal value gains

pi =

i aipi

i

i ai=i

piin

(4)

which does not depend on individual reactivity but on productivity, marginalcosts and damages of all actors.

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4 Interaction for linear-quadratic value functions

For analytical purposes we now treat linear-quadratic functions of Gi. In parti-cular, it is assumed that both utility and costs of production are linear functionsin emissions while mitigation costs and damages are quadratic functions. Skip-ping the upper index in cri and using ui = u

gi cgi as utility gain (net growth)

per emission unit, we obtain for constant allowance

Vi = uiGi diG2 ciR2i pi(Gi Gi ) (5)

Further emission reductions increase value for

Viri

= (ui + 2diG 2ciriGi + pi)Gi pii(Gi(1 ri)Gi ) 0. (6)

This leads to the threshold condition for optimal emission reductions

ri (pi ui + 2di[Gi +

j 6=i Gj(1 rj) pii(1Gi/Gi)

2Gi(di + ci) pii= ri . (7)

Below this threshold actor i would continue reduction to increase value, abovethis threshold diminish reduction, thus from both sides would approach the op-timum ri . The reduction threshold increases linearly with emission price pi anddecreases with the amount of own emissions Gi and unit reduction costs ci. Italso decreases with emission reductions rj of other actors j, because marginaldamage is lower in a world with lower emissions.

Resolving the threshold condition leads to a threshold price

pi ui + 2ciriGi 2diG+ pii((1 ri)Gi/Gi) = pii

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If price is above this threshold, further emission reductions increase value. Be-low it, emission reductions decrease value und would be avoided by actor i.

Neglecting pii in a first approximation, according to the analysis in the previouschapter the market price of emissions is the average of the threshold prices ofall actors (with a ai for i = 1, ..., n)

pi =

i pi

i

n=

i ui + 2ciriGi 2diG

n. (8)

which depends on benefits, costs and damages per emission unit of all actors.With the partial derivative of the approximate price function pii = 2Gi(ci+d)/n(with d = j dj) we obtain the price-adjusted optimal reduction for actorsi = 1, ..., n

ri =

j 6=i pi

j + ui 2dGi n(ui 2di(Gi +Gi)) + 2(Gi Gi)(ci + d)

2Gi[(n 1)(di + ci) di] (9)where Gi =

j 6=i Gj(1 rj) and di =

j 6=i dj. The reaction curves ri (ri)

represent targets for emission reduction of actor i as a function of emissionreduction vectors ri = (r1, ..., ri1, ri+1, ..., rn) of all other actors, except ri.The slopes of the reaction curves are given by

rirj

=2Gj(cj + di ndi)

2Gi[(n 1)(di + ci) di] .

The signs of these conjectural variations determine whether emission reductionof actor j induces reduction or increase for actor i. The numerator and deno-minator are both positive for din1 ci di di+cjn . For moderate marginaldamages for actor i and marginal reduction costs ci and cj this condition could

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be satisfied which implies that reductions by one actor j induce reductions byactor i. If however the marginal damages are either excessively high or low, thenthe slope becomes negative and actors behave asymmetrically. For emission in-creases of actor j, the reaction curve changes fundamentally because marginalreduction costs are zero (cj = 0). Then the sign of the numerator is only afunction of the marginal damages.

Actors who seek to adapt to their respective optimal reductions ri can be repre-sented by an iterative adjustment procedure (tatonnement)

ri(t) = ri(t+ 1) ri(t) = ri (ri (t) ri(t)) (i = 1, ..., n).

In this set of dynamic difference equations, the reactivity parameter ri determi-nes the speed of adaptation. For ri = 1, actor i would jump in a single time stepto the optimal reduction which however moves as a result of actions of other ac-tors. Whether a stable Nash equilibrium exists depends on the combination ofmarginal benefits, costs and damages of all actors.

5 Model specifications

After explaining the basic model, we now treat some model specifications todeal with some particular aspects relevant in model applications.

1. So far the threshold prices pii of all actors have the same relevance forthe market price pi, how small an actor may be. This can be modi-

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fied by giving more weight to bigger or more powerful actors (e.g. tho-se with higher emissions or GDP). In particular, setting price reactivi-ty proportionate to emission baselines, ai = aGi, we obtain the mar-ket price pi =

i pi

i Gi/G, weighted with the emission fractions Gi/G

of total emissions. The partial derivative easily modifies to pii/ri =j(pi

i /ri)(Gi/G).

2. We distinguish two different ways of emission reductions where i defi-nes the share for one or the other, allowing to explore different reductionscenarios:

Emission reductions iRi associated with production or consumpti-on losses. Marginal net benefits are replaced by uii.

Emission reductions (1 i)Ri by reducing emissions per produc-tion unit, e.g. by investing into more efficient, low-emission techno-logies. Marginal abatement costs are replaced by ci(1 i)2.

The other two value terms (damage Di, emissions trading i) still dependon complete emission reductions ri = iri + (1 i)ri.

3. Marginal net benefits ui are not treated as constant but vary with the base-line emissions Gi. We assume that for higher emissions marginal benefitsdecline according to ui/Gi with exponents 0 1, which implieshigher marginal benefits at lower emissions.

4. Besides the condition vri = Vi/ri 0 we also take into account thecondition Vi 0 to be satisfied. This defines boundaries for emission

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reductions and emission prices within which net value losses can be avoi-ded. This leads to a threshold price for zero value Vi = 0

pi0i uiGi(1 ri) di(

j Gj(1 rj))2 cir2i G2i

Gi(1 ri)Gi.

Positive value is assured for pi < pi0i and Gi(1 ri) > Gi or for pi >pi0i otherwise. Resolving for ri leads to a quadratic equation of emissionreductions to keep net economic growth Vi > 0. The actions of actorsdepend on both thresholds and their combination, leading to four differenttypes of behavior (Vi > 0 and vri > 0; Vi > 0 and vri < 0; Vi < 0 andvri > 0; Vi < 0 and vri < 0).

5. An essential question is how to allocate global reduction permits G tothe individual actors i = 1, ..., n, depending on global targets G(t) andbaseline emissions Gi(t). Two different mechanisms are taken here intoconsideration:

Allocation of permits is proportionate to population Ni, Gi = GNiN ,where N is the world population.

Allocation of permits is proportionate to the amount currently emit-ted Gi = Gi. where = G/

i Gi. Thus, larger emitters who

demand more emissions have the right for a bigger share but alsowould have to take a greater share of reduction.

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6 Computer simulation of emissions trading

In the following we simulate the dynamic interaction in emissions trading, basedon the outlined model and stylized data used in the ICLIPS model [Leimbach(2003), Leimbach/Toth (2003)], expanding the approach developed in [Schef-fran/Leimbach (2003)] from a single-step optimization to a multi-step dynamicsystem. To compute the threshold and permit prices as well as optimal emissionreductions we apply the Gross Domestic Product (GDP) and emissions data ofthe year 2005, as used in the ICLIPS reference scenario. For the linear-quadraticvalue function defined in equation 5 net economic growth (production minuscosts) is assumed to be 3%. The damage and the mitigation cost functions arecalibrated on the data point assuming that 5% incremental reduction leads to2% loss of damage or economic growth.

The figures show results from simulations of the linear-quadratic approach (equa-tion 5), where the final prices and optimal reductions are based on the iterativeadjustment procedure outlined in section 4. For each time step t = 1, ..., 25,starting with a given emission baseline Gi(t) and allowed emissions Gi (t), welet the interactive iteration run with an iteration index l = 1, ..., L to determinereductions rli(t) gradually moving towards the optimal reductions ri (t) of allactors. The results ri(t) = rLi (t) from the iteration are used as input into thenext time period t+ 1, to determine the increased or reduced emission baselineGi(t + 1) = Gi(t)(1 ri(t)) = Gi(t) to calculate emission price and optimalreductions for the next period. Allowed emissions Gi (t) are based on the equi-ty principle to gradually achieve equal emissions per capita in 25 years in all

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regions (using population of 2005 as fixed), certainly a tough requirement.

The results are depicted in the following figures where the following acronymsare used for the 11 regions: AFR Sub-Saharan Africa; CPA China, Mongolia,Vietnam, Cambodia, Laos; EEU Eastern Europe; FSU Former Soviet Union;LAM Latin America and the Caribbean; MEA Middle East and North Africa;NAM North America; PAO Pacific OECD (Japan, Australia, New Zealand);PAS Other Pacific Asia; SAS South Asia (mainly India); WEU Western Europe.

Depicted are emissions per capita and values for each region as well as totalemissions and emission price. By using the parameter we distinguish twodifferent mixes of emission reductions. In the first case (Figure 1), representedby = 0.75, one quarter of emission reductions is realized through restructuringand modernization of production while the rest results from production losses.The second case is the opposite (Figure 2), three quarters of emission reductionsis realized through modernization ( = 0.25).

The results are quite different. In case 1, only two regions significantly buy per-mits and thus increase emissions per capita (WEU and Pacific OECD) whichare also the regions with the highest value gains. The other regions either keepemissions almost constant or sell emission permits (first of all Russia) and thusreduce emissions per capita. This includies North-America which starts withhighest per capita emissions and reaches value losses during the process. Final-ly for all regions values go up, due to avoided damage and diminishing miti-gation costs at lower emission levels. Total emissions slightly go down (fromabout 8 to 7 Gigaton Carbon) while the permit price goes up from about 60 to

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80 Dollars per ton of carbon. In case 2, basically no region significantly incre-ases emissions, leading to an overall reduction of about 50% in 25 years, whilepermit price increases from about 25 to 40 Dollars/tonC. However, due to highmitigation costs the values of all regions start in the negative range and moveup to zero values.

The analysis shows the significance of finding the proper mix between the twooptions of reducing emissions. Shutting down production facilites is a way ofreducing emissions if they are replaced by lower-emission technologies at mo-derately higher costs. The results should not be overstated in this reduced modelsince both the equal emissions per capita assumption and the square functionsin damage and mitigation cost overdraw the effect. Further analysis can take thisinto consideration by modifying our approach with more appropriate assumpti-ons.

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5 10 15 20 25t

1

2

3

4

5

6

Emission scapita

WEUSASPASPAONAMMEALAMFSUEEUCPAAFR

0 5 10 15 20 25l

-0.1

-0.075

-0.05

-0.025

0.025

0.05

0.075

0.1

Values

WEUSASPASPAONAMMEALAMFSUEEUCPAAFR

0 5 10 15 20 25t

2

4

6

8

10

Emission , Price

price

emis

Emissions/capita [Tons Carbon]

Emissions

[Gigaton C]

Dollar x 10 / tonC

t

Net value growth [1012 $]

Total emissions / Price

Figure 1: Computer simulation of emission tradings among 11 world regions for = 0.75.Depicted are emissions per capita and net value growth for each region as well as total emissionsand emission price.

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0 5 10 15 20 25l

-0.1

-0.075

-0.05

-0.025

0.025

0.05

0.075

0.1

Values

WEUSASPASPAONAMMEALAMFSUEEUCPAAFR

5 10 15 20 25t

1

2

3

4

5

6

Emission scapita

WEUSASPASPAONAMMEALAMFSUEEUCPAAFR

0 5 10 15 20 25t

2

4

6

8

10

Emission , Price

price

emis

[Tons Carbon]

Dollar x 10 / tonC

t

Emissions/capita [Tons Carbon]

Net value growth [1012 $]

Total emissions / Price

Emissions

[Gigaton C]

Figure 2: Computer simlation of emission tradings among 11 world regions for = 0.25.

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