Measuring impact of uncertainty in a stylized macro-economic climate model within a dynamic game perspective

1st IFAC on for 1st Environmental IFAC Workshop WorkshopSystems on Integrated Integrated Assessment Assessment Modelling Modelling 1st IFAC Workshop onBrescia, Integrated Assessment for Environmental Systems University of Brescia, Italy, May 10-11,Modelling 2018 for Environmental Systems Available online at www.sciencedirect.com for Environmental Systems 1st IFAC Workshop onBrescia, Integrated Assessment University of Italy, May 2018 University of Brescia, Brescia, Brescia, Italy, May 10-11, 10-11,Modelling 2018 University of Brescia, Brescia, Italy, May 10-11, 2018 for Environmental Systems University of Brescia, Brescia, Italy, May 10-11, 2018

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IFAC PapersOnLine 51-5 (2018) 138–143 Measuring Measuring impact impact of of uncertainty uncertainty in in a a Measuring impact of uncertainty in a stylized macro-economic climate model stylized macro-economic climate model Measuring impact of uncertainty in a stylized macro-economic climate model within a dynamic game perspective within macro-economic a dynamic game climate perspective stylized model within a dynamic game perspective ∗ Stienen, J.C. Engwerda within aV.F. dynamic game perspective V.F. Stienen, J.C. Engwerda ∗∗

V.F. Stienen, J.C. Engwerda ∗ V.F. Stienen, J.C. Engwerda Both: Dept. of V.F. Econometrics O.R., P.O. Box∗ 90153, 5000 LE Stienen,and J.C. Engwerda Both: Dept. of Econometrics O.R., P.O. 5000 Both:Tilburg, Dept. ofThe Econometrics and O.R., P.O. Box Box 90153, 90153, 5000 LE LE Netherlandsand (Tel.: +31134662174; e-mail: Both:Tilburg, Dept. of Econometrics and O.R., P.O. Box 90153, 5000 LE The Netherlands (Tel.: +31134662174; e-mail: Tilburg, The Netherlands (Tel.: +31134662174; e-mail: [email protected]). Both:Tilburg, Dept. ofThe Econometrics and O.R., P.O. Box 90153, 5000 LE Netherlands (Tel.: +31134662174; e-mail: [email protected]). [email protected]). Tilburg, The Netherlands (Tel.: +31134662174; e-mail: [email protected]). Abstract: In this paper we quantify [email protected]). the main factors that influence the equilibrium outcome Abstract: Instrategies this paper paper we quantify model the main main factors that influence the equilibrium outcome Abstract: this quantify the influence equilibrium outcome and pursuedIn in we a simplistic for factors the use that of fossil versusthe green energy over time. Abstract: Instrategies this paper quantify model the main factors influence equilibrium outcome and pursued pursued strategies in we simplistic model formacro-economic the use use that of fossil fossil versusthe green energy over time. and in aa the simplistic for the of versus green energy over time. The model is derived using standard Solow growth model in a two-country Abstract: In this paper we quantify the main factors that influence the equilibrium outcome and pursued strategies in a simplistic model for the use of fossil versus green energy over time. The model is derived using the standard Solow macro-economic growth model in a two-country The model is derived usinggame the standard Solow macro-economic model in a two-country setting within a dynamic perspective. After calibrating thegrowth modelgreen for a energy setting of OECD and pursued in game a the simplistic model formacro-economic the use of fossil versus over time. The model is strategies derived using standard Solow model in a two-country setting within dynamic perspective. After calibrating thegrowth model forthe setting of OECD OECD setting within aa dynamic game perspective. After calibrating the model for aa setting of versus non-OECD countries we study what kind of uncertainties affect outcomes of the The model is derived usinggame the standard Solow macro-economic growth model in a two-country setting within a dynamic perspective. After calibrating the model for a setting of OECD versus non-OECD countries we study what kind of uncertainties affect the outcomes of the versus non-OECD countries we study what kinduse of Nash uncertainties affect the with outcomes ofthat the linearized model most, assuming both countries strategies to cope shocks setting within a dynamic game perspective. After calibrating the model forthe a with setting of OECD versus non-OECD countries we study what kind of uncertainties affect outcomes of the linearized model most, assuming both countries use Nash strategies to cope shocks that linearized most, both countries use Nash strategies to cope with shocksinthat impact non-OECD themodel model. Theassuming mainwe outcome of thiskind study is that the parameters occurring the versus countries study what of Nash uncertainties affect the with outcomes of the linearized most, both countries use strategies to cope shocksin impact the themodel model. Theassuming main outcome of used this study study isgrowth that the parameters occurring the impact model. The main outcome of this is that the parameters occurring inthat the objective/welfare function of both players for the model, are the parameters that linearized most, both countries usethe Nash strategies toare cope with shocksinthat impact themodel model. Theassuming main outcome of used this study isgrowth that the parameters occurring the objective/welfare function of both players used for growth model, the parameters that objective/welfare function of both players for the model, are the parameters that influence the outcome of the model (including the strategies) the most. impact the model. Theof main outcome of used this the study isgrowth that the the parameters occurring inthat the objective/welfare function of both players for the model, are the parameters influence the outcome the model (including strategies) most. influence the outcome of the model (including the strategies) the most. objective/welfare function of model both players usedthe forstrategies) the growth model, areLtd. theAll parameters that influence the (International outcome of the (including most. © 2018, IFAC Federation of Automatic Control) Hostingthe by Elsevier rights reserved. Keywords:the differential environmental uncertain dynamic systems; influence outcome games; of the model (includingengineering; the strategies) the most. Keywords: differential differential games; environmental engineering; uncertain dynamic dynamic systems; systems; Keywords: environmental engineering; uncertain linearization; economicgames; systems; open-loop control systems. Keywords: differential environmental engineering; uncertain dynamic systems; linearization; economicgames; systems; open-loop control control systems. linearization; economic systems; open-loop systems. Keywords: differential environmental engineering; uncertain dynamic systems; linearization; economicgames; systems; open-loop control systems. 1. INTRODUCTION of systems. fossil energy and begin to expand their green energy use. linearization; economic systems; open-loop control 1. INTRODUCTION INTRODUCTION of energy to their green energy use. 1. of fossil fossil energy and begin to expand expand theirusing greengreen energy use. This comes withand thebegin fact that currently energy 1. INTRODUCTION of fossil energy and begin to expand theirusing greengreen energy use. This comes with the fact that currently energy This comes with the fact that currently using green energy more expensive than using fossil energy. In particular Climate change is1. aINTRODUCTION key topic on the agenda of most is of fossil energy and begin to expand their green energy use. This comes with the fact that currently using green energy is more expensive than using fossil energy. In particular Climate changeleading is aa key key topic on on the the agenda of ofmost most is more expensive than using fossil energy. In particular which experience period ofusing economic growth Climate change is topic of the world’s governments. Fromagenda reportsof the countries This comes with the fact using thataa currently green energy is more expensive than fossil energy. In particular Climate change is a key topic on the agenda of most countries which experience period of economic growth of the world’s leading governments. From reports of the countries whichsceptic experience achanging period oftheir economic growth be expensive rather climate policy of the world’s leading governments. From reports of the European Environment Agency (2015) andofthe In- could more thanabout using fossil energy. In particular countries whichsceptic experience achanging period oftheir economic growth Climate change is a key topic EEA on the agenda of the world’s leading governments. From reports ofmost the could be about climate policy European Environment Agency EEA (2015) and the the In- is could be rather rather sceptic about changing their climate policy to a more green policy. They have to invest in green energy European Environment Agency EEA (2015) and Intergovernmental Panel on Climate Change (IPCC) the countries whichsceptic experience ahave period oftheir economic growth could be rather about changing climate policy of the world’s leading governments. From reports of the the European Environment Agency EEA (2015) and the Into a more green policy. They to invest in green energy tergovernmental Panel on Climate Change (IPCC) to a more green policy. They have to invest in green energy resources, which costs money and could deteriorate their tergovernmental Panel on Climate Change average global temperature is rising. This can(IPCC) be the seenthe in could be rather sceptic about changing their climate policy to a more green policy. They have to invest in green energy European Environment Agency EEA (2015) and Intergovernmental Panel on Climate Change (IPCC) the resources, which costs money and could deteriorate their average global temperature is rising. This can be seen in resources, which costs money and could deteriorate their economic growth. There are some policies that try to mitiaverage global temperature is rising. This can be seen in Figure 1 of EEA Panel (2015).onHere the global land(IPCC) and ocean to a more growth. green policy. They have to investdeteriorate in green energy resources, which costs money and could their tergovernmental Climate Change the average global temperature is rising. This can be seen in economic There are some policies that try to mitiFigure 1 of EEA (2015). Here the global land and ocean economic growth. There are someintroducing policies that try to mitigate this problem. For instance, a carbon tax, Figure 1 of EEA (2015).are Here the global land and ocean temperature anomalies plotted, with 1940 as a base resources, which costs money and could deteriorate their growth. There are some policies that try to mitiaverage global temperature is rising. Thisland can and be seen in economic Figure 1 of EEA (2015). Here the global ocean gate this problem. For instance, introducing a carbon tax, temperature anomalies are plotted, with 1940 as a base gate this problem. For instance, introducing a carbon tax, the use of green energy and forming coalitions temperature anomalies are plotted, with 1940 as change a base subsidizing year. From this plot we can see that the average economic growth. There are energy someintroducing policies that try to mitigate this problem. For instance, a carbon tax, Figure 1 of this EEA (2015). Here the global land and ocean temperature anomalies are plotted, with 1940 as a base subsidizing the use of green and forming coalitions year. From plot we can see that the average change ◦ subsidizing the use of green energy and forming coalitions of countries to get cooperation gains. Each of these policies year. From this per plot decade we can isseeabout that +0.07 the average change gate in temperature this problem. For instance, introducing a carbon tax, ◦ C.asAccordsubsidizing the use of green energy and forming coalitions temperature anomalies are plotted, with 1940 a base year. From this plot we can see that the average change of countries to get cooperation gains. Each of these policies ◦ C. Accord- has in temperature temperature per decade is Centers about +0.07 +0.07 of countries to get cooperation gains. Each of instance, these policies its advantages and disadvantages. For the in is about ing toFrom data this fromper the decade National for average Environmental ◦ C. Accordthe usecooperation of green energy and forming coalitions of countries to get gains. Each of these policies year. plot we can issee that +0.07 the change subsidizing C. Accordin temperature per decade about has its advantages and disadvantages. For instance, the ing to data from the National Centers for Environmental has its advantages and disadvantages. For instance, the disadvantage of a carbon tax is that it will only work well, ing to data from the National Centers for Environmental Information (NCEI), the total CO emission has been ◦ 2 +0.07 C. Accordof countries toofget cooperation gains. Each of these policies has its advantages and disadvantages. For instance, the in temperature per decade is Centers about ing to data from the National for Environmental disadvantage a carbon tax is that it will only work well, Information (NCEI), the total CO emission has been disadvantage of a carbon tax is that it will only work well, 2 it is implemented over the whole world. Next to this Information (NCEI), the total CO has IPCC been if increasing exponentially over time. According to the 2 emission has its advantages and disadvantages. For instance, the disadvantage of a carbon tax is that it will only work well, ing to data from the National Centers for Environmental Information (NCEI), the total CO emission has been if it is implemented over the whole world. Next to this 2 increasing to if it is the implemented over thethis whole Nextemitting to this difficulty to price taxitworld. for increasing exponentially exponentially over over time. time. According According to the the IPCC IPCC comes of a carbon tax is that willlegally only work well, if it is implemented over the whole world. Next to this Information (NCEI), the total CO has IPCC been disadvantage increasing exponentially over time. According to the comes the difficulty to price this tax for legally emitting 2 emission comes the difficulty price this taxtradable for legally emitting CO policy to is to introduce permits if it2 .isAnother implemented over thethis whole world. Next to that this the difficulty to price taxtradable for legally emitting increasing exponentially over time. According to the IPCC comes CO Another policy is introduce permits that CO22 ..companies Another policy is to to introduce tradable permits that2 give the right to emit a certain amount of CO comes the difficulty to price this tax for legally emitting CO . Another policy is to introduce tradable permits that give companies the right emit aa certain amount give companies the should right to tothese emitrights certain amount of of CO CO per 2year. But how be distributed over22 CO .companies Another policy is totointroduce tradable permits that give the should right emitrights a certain amount of CO per year. But these be over per 2world? year. But how how should these rights be distributed distributed over2 the give companies the should right tothese emitrights a certain amount of CO per year. But how be distributed over2 the world? the world? With rapid in computing over the over last per year. Butadvances how should these rightspower be distributed the world? With rapid advances in computing power over the With rapid advancesmodels in computing power over the last last decade, large-scale have become essential to the world? With rapid advancesmodels in computing power over the last decade, large-scale have become essential to decade, large-scale models haveHowever become essential to decision-making in public policy. there are also With rapid advances in computing power there over the last decade, large-scale models haveHowever become essential to decision-making in policy. are also decision-making in public public policy. However there are ecoalso risks in using these models. A central issue in the decade, large-scale models have become essential to decision-making in public policy. However there are also risks in using these models. A central issue in the ecorisks in of using thesechange models. central issue and in the economics climate is A understanding dealing decision-making in public policy. However there are also risks in using these models. A central issue in the economics of climate is and dealing Fig. 1. Global temperature anomalies (EEA) nomics of vast climate change is understanding understanding and dealing with the arraychange of uncertainties. range from risks in of using these models. centralThese issue and in the ecoFig. nomics climate change is A understanding dealing with the vast array of uncertainties. These range from Fig. 1. 1. Global Global temperature temperature anomalies anomalies (EEA) (EEA) with the vast array of uncertainties. These range from those regarding economic and population growth, emisFig. 1. Global temperature anomalies (EEA) nomics of climate change is understanding and dealing with the vast array of uncertainties. These range from those regarding economic and population growth, emisreports, with 90% probability, a doubling (compared to its sion thoseintensities regarding and economic and population growth, new technologies, carbon cycleemisand Fig. 1. Global temperature anomalies (EEA) the vast array of uncertainties. These range from reports, with 90% probability, a concentration doubling (compared to its its those regarding economic and population growth, emission intensities and new technologies, carbon cycle and reports, 90% probability, doubling (compared to value in with the year 2000) of CO2a will lead to with sion intensities and new technologies, carbon cycle and climate response, to the costs and benefits of different reports, 90% 2000) probability, a concentration doubling (compared to those regarding economic and population growth, emisvalue in the year of will lead to ◦ its sion intensities and new technologies, carbon cycle and 2 climate response, to the costs and benefits of different value in with the of year 2000) of CO CO will lead to an increase the average world temperature by 1.5 C. 2 concentration climate response, to the costs and benefits of different policy objectives. Most of the time policy makers must ◦ reports, with 90% probability, a doubling (compared to its value in the year 2000) of CO concentration will lead to ◦ C. sion intensities and new technologies, carbon cycle and 2 an increase of the average world temperature by 1.5 climate response, to the costs and benefits of different policy their objectives. Most of the the time policy makers makers must an increase ofwill theaffect average world temperature byaccepted 1.5◦ C. policy This increase all The broadly objectives. Most of policy must decisions based on time the outcome of adifferent model value in the of year 2000) of countries. CO will lead C. to make an increase theaffect average world temperature byaccepted 1.5 2 concentration climate response, to the costs and benefits of This increase will all countries. The broadly policy objectives. Most of the time policy makers must make their decisions based on the outcome of a model This increase will affect all countries. The broadly accepted consensus is of therefore that actions are needed to by reduce make assumes their decisions on theuncertain outcome parameters. of a model a lot ofbased (possibly) an increase theaffect average world temperature 1.5◦ the C. that This increase will all countries. The broadly accepted policy objectives. Most of the policy makers must consensus is that actions are needed to reduce the make their decisions based on time theuncertain outcome of a model that assumes aa sensitivity lot of (possibly) parameters. consensus is2therefore therefore that actions are needed to reduce the level of CO emission all over the world. For instance, by that assumes lot of (possibly) uncertain parameters. Typically, some analyses on particular paramThis increase will affect all countries. The broadly accepted consensus is therefore that actions are needed to reduce the make their decisions based on the outcome of a model level of CO emission all over the world. For instance, by that assumes a lot of (possibly) uncertain parameters. some analyses on paramlevel CO22green emission allinstead over the world. For instance, by Typically, usingof more energy ofare fossil energy. However, some sensitivity sensitivity analyses on particular particular parameters are executed the policymaker anparameters. indication consensus therefore that actions needed reduce the level COis2green emission allinstead over the Forto instance, by Typically, that assumes lotto ofgive (possibly) uncertain using more energy of fossil However, Typically, somea sensitivity analyses on particular parameters are executed to give the policymaker an indication usingof more green energy instead of world. fossil energy. energy. However, nowadays fossil fuel reserves are abundant. This means eters are executed to give the policymaker an indication of the uncertainty of the model under consideration. Of level COfossil emission allinstead over are the For instance, by Typically, usingof more energy of world. fossil energy. However, 2green some sensitivity analyses on particular paramnowadays fuel reserves abundant. This means eters are executed to give the policymaker an indication of the uncertainty of the model under consideration. Of nowadays fossil fuel reserves are abundant. This means that it is not easy to convince countries to restrict their use of the uncertainty of the model under consideration. Of using more green energy instead fossil energy. However, nowadays fossil reserves areof abundant. This means executed to give model the policymaker an indication that easy to countries to their use of theare uncertainty of the under consideration. Of that it it is is not not easyfuel to convince convince countries to restrict restrict their use eters nowadays fossil fuel reserves are abundant. This means that it is not easy to convince countries to restrict their use of the uncertainty of the model under consideration. Of Copyright © 2018, 2018 IFAC that it is © not easy to convince countries to restrict theirControl) use138Hosting by Elsevier Ltd. All rights reserved. 2405-8963 IFAC (International Federation of Automatic ∗ ∗ ∗ ∗ ∗

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course this does not give a good representation of the uncertainties involved into the model. What we often want is to give a measure of uncertainty and to provide a kind of probability distribution for the outcome of the model. This is hardly possible to realize. A more down-to-earth approach is performing an elaborate uncertainty analysis consisting of (see, e.g., Kann et al. (1999)): i) Stochastic parameters, where parameters are assumed to belong to a set of values and corresponding probability distribution; ii) Stochastic relations, where relations are assumed to contain a stochastic element; iii) Deterministic, worstcase scenario, where a new variable is added to the system which can be viewed as nature that is always counteracting the objective(s); iv) Scenario analyses, where model runs are performed for different combinations of assumptions to compare the results; and v) Extending the model: this means that some parts of the model are reconsidered and extended where necessary. The main focus of this paper is to see how (a number of) above uncertainty modeling procedures impact the results in a linear quadratic differential game model that analyzes the ratio between fossil energy an green energy use in a macro-economic growth model. The analysis is limited to a two player setting, where we consider the two players to be the OECD and non-OECD countries. Our main objects of study are the corresponding steady state equilibrium values of the involved economic target variables and instruments. The benchmark model we use is obtained along the lines of a similar model used by Engwerda (2014) to analyze the impact of pollution over time on the fossil fuel/green energy ratio in a dynamic world characterized by four players that have different interests. Already several studies are performed that try to incorporate uncertainty into energy system models. E.g., Pizer (1999) presents a framework for determining optimal climate change policy under uncertainty. They use econometric estimates for some parameters, which are then used to solve the model. They compare the results with those derived from an analysis with best-guess parameter values. Their aim is to show that incorporating uncertainty within a climate model can significantly change the optimal policy recommendations. In particular they suggest that analyses which ignore uncertainty can lead to inefficient policy recommendations. Gillingham et al. (2015) look at model and parametric uncertainties for population, total factor productivity, and climate sensitivity. They estimate the pdfs of key output variables, including CO2 concentrations, temperature, damages, and the social cost of carbon (SCC). They investigate uncertainty of outcomes for climate change using multiple integrated assessment models (IAMs) 1 . This multi-model inter-comparison approach is also preferred by Blanford et al. (2014) and many other research teams. Furthermore, Fragtos (2015) introduce PROMETHEUS, a stochastic model of the world energy system that is designed to produce joint empirical distributions of future outcomes. They represent causal chains for all important variables, with time series analysis for providing patterns of variation over time. Tol (2015) investigates the question whether 1

Combination of the scientific and economic aspects of climate change used to assess policy options for climate change. Details can be found in Kelly (1998).

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uncertainty about climate change is too large for running an expected cost benefit analysis. The approach taken is to test whether the uncertainties about climate change are infinite. This is done by calculating the expectation and variance of the marginal costs of CO2 emissions. In short, the conclusion is that climate change is an area that tests decision analytic tools to the extreme. In this paper we differ from the above mentioned papers by several aspects. All models above are trying to quantify uncertainty within an IAM that does not incorporate interrelations between players. The models are developed to optimize a policy for a country, without incorporating the interrelations between countries. One of the first papers that treat global warming as a multi-agent problem is Nordhaus (1996). They develop a dynamic, multiagent, general-equilibrium model incorporating climate and economy. They use Nash equilibria to predict the economic/environmental variables used in the model. They compare a cooperative (and a noncooperative) approach in which all countries choose climate policies to maximize global (respectively own) income. We want to investigate which parts of such models (parameters, relations, scenarios etc.) carry the most uncertainty in the model outcomes. That is, we want to get a broad overview of the uncertainty involved, by applying and evaluating multiple approaches as described above. The results can be used to conclude which parts need special attention when calibrating. The outline of the rest of the paper is as follows. In Section 2, we create our simple dynamic linear two country growth model along the lines of Engwerda (2014) based on the standard Solow growth model introduced by Solow (1956). We integrate the impact of CO2 emission on economic growth in this model to get a world energy model. Using an extensive model calibration, we arrive at our benchmark model. In Section 3 we perform some experiments with our benchmark model. This to illustrate the basic operation of the model and explain the outcome of the model by investigating the use of the different forms of energy for both players under different scenarios. Next, in Section 4, we perform an extensive uncertainty analysis of this model. Approaches i), ii) and iv) are used to analyze this impact. Section 5 concludes. Due to space limitations many details in all sections are skipped. They can be found in an extended report of this paper, Stienen (2018). 2. THE MODEL In this section we formulate our benchmark endogenous growth model. The model is based on the standard Solow exogenous growth model introduced by Solow (1956). The model can be obtained along the lines of Engwerda (2014). Therefore, details are omitted. Also a more extensive discussion on the ins and outs of some equations used can be found there. We consider two countries, identified as the OECD and non-OECD countries from now on. With Yi denoting the output, Fi the production/use of fossil energy, Gi the production/use of green energy; Ki the amount of capital, Li the total population, Ti the state of technology, Ei the total CO2 emission, and Ai measuring the total factor productivity, all in country i, i = 1, 2, the basic model equations are as follows. Yi = Ai Kiαi Lβi i Eiγi Tiκi ,

αi , βi , κi ≥ 0,

(1)

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K˙ i = si Yi + sij Yj − δi Ki + τi Ti , T˙i = gi Ti + gij Tj + i Ki , E˙ i = ζi Fi + ζij Fj − ξi Ei ,

L˙ i = ηi Li , with j = 2 if i = 1, and j = 1 if i = 2.

(4)

linear differential equations. Based on our initial variable calibration we determine the steady-state values of the variables satisfying these conditions. These equilibrium values are tabulated in Table 1.

(5)

Table 1. Equilibrium variables

(2) (3)

That is, we assume that production is provided by a CobbDouglas function (1); the change in capital is endogenous and depends on domestic and foreign investment, depreciation of the current capital and domestic technology (2); and technological progress depends on both domestic and foreign technology and the amount of domestic capital (3); the change in CO2 emission is endogenous too and increases due to domestic and foreign use of fossil fuels and depreciation of the current stock of CO2 emission (4); and labor supply grows at a constant rate ηi (5). Under the assumption that the Cobb-Douglas production functions satisfy constant returns to scale (i.e. αY (K, L) = Y (αK, αL), or, in this specific case the production function parameters satisfy αi + βi + κi + γi = 1), above equations can be rescaled in terms of effective labor. Taking Yi i natural logarithms, with yi := log( L ), ki := log( K Li ), ti := i Ti Ei Fi i log( Li ), ei := log( Li ), fi := log( Li ) and gi := log( G Li ), equations (1-5) can be rewritten as yi (t) = log(Ai ) + κi ti (t) + αi ki (t) + γi ei (t) k˙ i (t) = −(ηi + δi ) + e−ki (t) ·   si eyi (t) + sij eyj (t)+t(ηj −ηi ) + τi eti (t) t˙i (t) = −ηi + gi + e−ti (t) ·   gij etj (t)+t(ηj −ηi ) + i eki (t)

e˙ i (t) = −(ξi + ηi ) + e−ei (t) ·   ζi efi (t) + ζij efj (t)+t(ηj −ηi ) .

(6)

Furthermore, with U := µi Yi −(Fi +Gi ) and E := E1 +E2 , we assume both countries like to maximize next total discounted welfare  ∞   Wi = e−θi t −U 2 (t) − πi E 2 (t) − ρi G2i (t) dt. (7) 0

Here, µi is the proportion of output in country i that can only be produced with the use of energy. In this welfare function the weight of meeting the energy requirements is set equal to 1 in order to emphasize the need for realizing this objective. Factor ρi represents the actual disadvantages of using green energy for country i. For instance, when the price of using green energy is higher than the price of using fossil energy. Furthermore, each country has its own availability of resources. It might be difficult to use green energy, because there are no resources in the neighborhood. Factor πi expresses that the higher the CO2 emission, the more it is disliked. For instance, emitting lots of CO2 will entail costs. For the calibration of the parameters in the above model (6,7), we refer to Stienen (2018). We choose to concentrate on the OECD countries and the non-OECD countries as our two parties involved. Assuming that both countries play a non-cooperative open-loop strategy, we next determine the corresponding necessary conditions. This gives rise to a set of non140

O n-O

ye 15.07 16.94

ke 29.98 31.64

te 31.47 32.74

ee 13.94 14.15

fe 9.11 11.79

ge 11.89 12.17

Assuming that both countries operate within the neighborhood of above steady-state values, we can approximate the dynamics around this nonlinear model by a linear model (see Stienen (2018)[Appendix C]). We will use this linearized model in our subsequent analyses. Note that the parameters and the objective are therefore also reformulated. For details see Stienen (2018). 3. BENCHMARK MODEL SIMULATIONS In this section we illustrate, by considering a couple of scenarios, how the model (6,7) will approximately respond if it is out of equilibrium. To that end we will perform two different kind of shocks to the equilibrium, symmetric shocks and asymmetric shocks. Symmetric shocks are shocks that hit all countries at the same time. Asymmetric shocks are shocks that occur to just one of both countries. Furthermore, we distinguish between two forms of cooperation. We have a cooperative situation and a non-cooperative situation. In the cooperative situation we discuss a regime where both countries form a coalition. In the non-cooperative situation we discuss the regime where both countries play actions in the Nash sense. We only analyze the impact of an emission shock, as such a shock is mostly related to our control variables. In this paper, we only consider the asymmetric emission shock. For the elaboration of a symmetric emission shock see Stienen (2018). To perform the simulations, we used the numerical toolbox developed by Michalak (2011) to solve N -player affine linear-quadratic open-loop differential games. Clearly, the use of open-loop strategies is made to simplify the analysis. A discussion of pros and cons using this setting can be found in, e.g., Engwerda (2014). In particular we recall some observations from literature suggesting that the difference between open-loop and feedback policies in practice might not be that large (see, e.g., Maler et al. (1998), Basar et al. (2011)). 3.1 Asymmetric emission shock We start with an asymmetric positive CO2 emission shock which hits the non-OECD countries. Figure 2 shows the response of both countries in a noncooperative setting in terms of energy consumption, f and g. We see that the non-OECD countries immediately start to use more fossil energy and less green energy. The reason for this is that fossil energy is less expensive than using green energy for them. On the other hand, the OECD countries start to increase their use of green energy and reduce their fossil energy consumption due to the high cost involved on CO2 emissions for them. Figure 3 shows the corresponding evolution of capital,

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OECD countries

0.1

Fossil energy Green energy

0.4

0.05

Table 2. Losses under asymmetric shock

non−OECD countries

0.6

Fossil energy Green energy

0.2

O n-O

0

0

−0.2

−0.6 −0.1

0

20

40

60

80

−0.8

100

0

20

40

60

80

NC 4.1684 1.9403

C 0.72411 2.5121

Loss reduction (%) 82.6 -22.8

this can be explained by the fact that in a cooperative mode of play more green energy is used to compensate the shock. As this instrument is more costly than using fossil fuels for the non-OECD countries, their losses increase.

−0.4

−0.05

141

100

Fig. 2. Control variables OECD countries

0.3

Capital Technology Emission

0.25 0.2

0.8 0.6

0.1

0.4

0.05

0.2

0 −0.05

Capital Technology Emission

1

0.15

4. UNCERTAINTY ANALYSIS

non−OECD countries

1.2

0 0

20

40

60

80

−0.2

100

0

20

40

60

80

100

Fig. 3. State variables technology and stock of emissions for both countries. We see that as well as in the non-OECD and OECD capital and technology are unaffected, both the technology and the capital curve (behind the technology curve) do not deviate from zero. Furthermore, note that the emission stock lags behind in the OECD countries with reaching a peak after approximately 10 years. Output, y

0

4.1 The production function

−0.01 −0.02 −0.03 OECD countries non−OECD countries

−0.04 −0.05

Clearly in arriving at our linear adjustment model several approximations were made and the question is how sensitive results obtained in this linear model are to inaccuracies in the original model specification (1-5,7). In subsections 4.1-4.3 below we consider some potential inaccuracies and analyze how they impact the results presented in the previous section. To that end we distinguish two kinds of impact. The impact on the equilibrium values and the impact on the optimal strategies. In Section 4.1, we address the consequences of our assumption that our production function satisfies constant returns to scale. Section 4.2 considers the effects if we assume that some parameters are stochastically determined. Finally, in Section 4.3 we consider a scenario where both the initial use of green energy and parameter ρ, which represents the disadvantage of using green energy, are correlated. Details on the presented results can be found in Stienen (2018).

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Fig. 4. Output variables From Figure 4 we see that output drops significantly initially for the non-OECD countries, but that the shock has less impact for output in the OECD countries. Also clear again is the lagged reaction by the OECD countries. The output drop in the OECD countries is caused by the increased use of green energy, needed to compensate the large CO2 emission in the non-OECD countries. Since this impact on capital and technology growth is rather small, output is affected not too much in the OECD countries. Next we consider the case how both countries respond if they decide to fight the shock collectively. This is modeled by assuming that control instruments by both countries are determined such that the weighted sum of both welfare functions is collectively minimized. We assumed weights to be equal, i.e. 12 . The simulation results are similar to the figures presented above for the non-cooperative case. The only major difference is that all curves reach their equilibrium earlier than in the non-cooperative setting. So we omit the figures and focus on the losses of both countries incurred under the two cooperation scenarios. Table 2 reports them. Here we use the acronym NC (C) for the non-cooperative (cooperative) setting. We see that cooperation for OECD countries would be profitable. However non-OECD countries do not profit from it. Likely, 141

In this section we reconsider the assumption that the production function (1) satisfies constant returns to scale. After calibration it turns out that the sum of the involved parameters, αi + βi + γi + κi , equals 0.905 for OECD countries and 1.03 for non-OECD countries. The used numbers for the analyses are obtained by normalizing these parameters for both countries. In this section we consider how equilibrium values of (6) change if we fix all but one of these parameters to their calibrated value, and estimate the remaining parameter as the difference between one and the sum of the calibrated parameters. That is, if e.g. we calibrated αi = α ¯ i , βi = β¯i , γi = γ¯i , ¯ we fix κi at 1 − α ¯ i − βi − γ¯i . We calculated for all four possible combinations corresponding equilibrium values of (6). Table 3 reports the average of all equilibrium variables for all these four possibilities. For the non-OECD countries Table 3. Weighted equilibrium variables O n-O

y 16.29 16.82

k 30.18 31.85

t 31.68 32.95

e 14.00 14.22

f 9.38 11.84

g 13.31 11.96

we see that the differences are marginal. For the OECD we see some more substantial differences in output (8%) and energy variables (f ≈ 3%, g ≈ 12%). The average of all differences for non-OECD variables turns out to be 0.79% and for the OECD countries 4.14%. All details on this calculation can be found in Stienen (2018) again. So, the equilibrium values lie, on average, for the OECD countries within a range of 5% of the variables we used and within a range of 1% of the variables we used for the non-OECD countries.

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4.2 Stochastic parameters Next, we consider the case that one of the key parameters in the model, ξ, is in fact only approximately known. More precisely, we add a distribution function to this parameter and use 100 simulated realizations (based on this distribution), for analysis. We initially assumed, based on Inman (2008), a CO2 lifetime of 30 years for 50% of the CO2 emission today. The IPCC, on the other hand, estimates a CO2 lifetime of 50 years for 50% of the CO2 emission today. This results in a 20 year difference between the two studies. Therefore, we add a distribution to the lifetime of CO2 emission as visualized in Figure 5.

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Fig. 7. Control variables

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the effect on the optimal control variables if the lifetime of CO2 is increased from 30 years (Old ) to 50 years (New ) in the non-cooperative asymmetric shock benchmark case.

We see that both trajectories do not change significantly.

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probability

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4.3 Scenario analysis

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Fig. 5. Distribution added to ξ After some lengthy simulations it turns out that apparently this assumption does not have a large impact on the resulting value of the objective function. Complementary to this approach we also calculated the equilibrium values for the complete, specified range of CO2 lifetimes. It turns out that the CO2 lifetime is not affecting the equilibrium values much. Figure 6, shows the corresponding plot of the equilibrium values for OECD and non-OECD countries.

Fig. 6. Left: OECD. Right: non-OECD, CO2 lifetimes As we can see in this figure, the equilibrium values are rather constant for the specified range of CO2 lifetimes. By comparing equilibrium outcomes just for extremal choices of this parameter we get the percentage changes tabulated in Table 4. We see that the average percentage difference y 3.04% 3.77%

k 6.02% 5.69%

t 5.74% 5.52%

e 2.21% 2.13%

f 3.61% 2.97%

g 3.26% 4.78%

Average 3.98% 4.14%

In this section we want to investigate the impact of considering a larger value for initial use of green energy by both countries. However, using more green energy, in percentage, will typically be an outcome of good availability of resources and a smaller price for it. This can be modeled by assuming that the parameter that represents the disadvantages of using green energy, ρi , becomes smaller. Based on some calibrations, we estimated that to investigate a 5% increase in the use of green energy this should be accompanied by a decrease of parameter ρi by 5%. We calculated the new equilibrium variables under this scenario and found that this adjustment has no large impact on them. What attracts the attention is that all equilibrium values changed in a negative direction. The interpretation of this is that if we change our initial green energy use towards its original equilibrium value and decrease, for instance, the difficulty of accessing the green energy market, all variables are converging to an equilibrium value that is lower than the one we had in the beginning. This also implies that, as expected, the total welfare corresponding to this equilibrium is higher. Furthermore, this scenario has no impact on the optimal strategies. Also for this scenario analysis we calculated, for all possible combinations of ratio’s between 0 and 5 % for both countries, corresponding equilibrium outcomes. This means that we look at the equilibrium results where the initial f and g are changed. For the worst-case scenario the results are tabulated in Table 5. y 3.08% 3.77%

k 6.19% 5.83%

t 5.88% 5.62%

e 4.70% 4.58%

f 7.06% 5.63%

g 2.26% 3.48%

Average 4.86% 4.82%

Table 5. %-differences from orig. equilibrium So the maximal percentage difference of both countries is, on average, just below 5%. 5. CONCLUDING REMARKS

Table 4. %-differences from orig. equilibrium of both countries is around 4%. Finally, we also determined the effect of this parameter on the optimal strategies. As an example, Figure 7 shows 142

In this paper we consider a simplistic model that analyzes the ratio between fossil energy use and green energy use within a context of OECD and non-OECD countries. For that purpose we developed from some basic economic relationships a growth model including both these factors.

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As this model is highly non-linear, we determined for this model its equilibrium points, under the assumption that both players want to maximize their welfare. To see how both players will react to distortions, we derived the corresponding linear dynamics around the equilibrium. Some shock simulations with this benchmark model turn out to provide results that are not too unrealistic. We also considered the question if a coalition of OECD countries and non-OECD countries could be profitable for both countries. It turns out that this is not the case. The nonOECD countries will in general not profit from this, where the OECD countries will. Given the large number of uncertainties involved in modeling this kind of problems, the next step was to perform an extensive uncertainty analysis. We found that small changes to the parameters used in the dynamics of the model do not affect the outcome of the model much. Adding, for instance, stochastics to such a particular parameter carries in the worst-case, on average, 5% uncertainty in the equilibrium values. If we add stochastics to a complete state equation, we get on average an extra 6.5% uncertainty in the equilibrium values. This means that changing the set-up of one of the state equations in our model with a small amount, has a larger impact on the outcome of the model than changing the parameters within this state equation with a small amount. So far, the uncertainty involved seems to have no direct effect on the optimal strategies of both players after an emission shock. However, though not reported in this paper, we also investigated the uncertainty involved in the parameters that occur in the objective function of both players. In particular, we investigated the effect on the outcome of the model by changing the preference rate for emitting CO2 . This parameter seems to have a larger effect on the optimal strategies we just discussed. However, this effect consists mainly of getting faster/slower back to the equilibrium. So the structure of the path of the variables stays the same, only the time it takes to get back to the equilibrium decreases/increases. In Table 6, a short overview is given where the approximate uncertainty is tabulated for each analysis. This uncertainty is divided in uncertainty on the equilibrium value and uncertainty on the optimal strategies of both players. Shares of income Parameter (dynamics) Parameter (objective) Relation Scenario

Equilibrium ≈ 5% ≈ 5%

Strategies ≈ 0% ≈ 0%

≈ 6.5% ≈ 5%

≈ 0% ≈ 0%

≈ 50% 2

≈ 75% 3

Table 6. Overview of the uncertainties What we can conclude is that the calibration of the parameters that occur in the objective of the players needs special attention. These parameters carry the most uncertainty for the outcome of the model. Both in the equilibrium and in the optimal strategies. Secondly, we see that the structure of the optimal strategies after an emission shock occurred, does not variate much if we perform some uncertainty analyses. Changing the parameters of the objective does 2 3

Based upon the average difference of the most extreme value of π. Based upon the maximal absolute change between strategies.

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not affect the path of the variables much. It only changes the size of the reaction of both players. The direction seems to be very stable against the uncertainty analyses performed. REFERENCES Ba¸sar, T., & Zhu, Q. (2011). Prices of anarchy, information, and cooperation in differential games. Dynamic Games and Applications, 1, 50-73. Blanford, G. J., Kriegler, E., & Tavoni, M. (2014). Harmonization vs. fragmentation: overview of climate policy scenarios in EMF27. Climatic Change, 123 (3), 383-396. Engwerda, J.C., & Michalak, T. (2014). Economic growth and choice of energy: a simplistic strategic approach. Environmental Modeling & Assessment, 20 (4), 321-342. European Environment Agency, http://www.eea.europa.eu/. Fragkos, P., Kouvaritakis, N. & Capros, P. (2015). Incorporating uncertainty into world energy modelling: the PROMETHEUS Model. Environmental Modeling & Assessment, 20 (5), 549-569. Gillingham, K., Nordhaus, W., Anthoff, D., Blanford, G., Bosetti, V., Christensen, P., McJeon, H., Reilly J. & Sztorc, P. (2015). Modeling Uncertainty in Climate Change: A Multi-Model Comparison (NBER Working Paper No. 21637). Retrieved from National Bureau of Economic Research website: http://www.nber.org/papers/w21637 Impact Data Source, http://www.impactdatasource.com/ choosing-a-discount-rate. Inman, M. (2008). Carbon is forever. Nature Reports Climate Change, 2, 156-158. Kann, A., & Weyant, J. P. (1999). Approaches for performing uncertainty analysis in large-scale energy/economic policy models. Environmental Modeling & Assessment, 5 (1), 29-46. Kelly, D. L., & Kolstad, C. D. (1998). Integrated assessment models for climate change control. International Yearbook of Environmental and Resource Economics 1999/2000: A Survey of Current Issues, 171-197. M¨aler K.-G. and Zeeuw A.J. de, 1998, The acid rain differential game, Environmental and Resource Economics, Vol.12, no.2, pp.167-184. Michalak, T., Engwerda, J., & Plasmans, J. (2011). A numerical toolbox to solve N-player affine LQ open-loop differential games. Computational Economics, 37 (4), 375-410. Nordhaus, W., & Yang, Z. (1996). A regional dynamic general-equilibrium model of alternative climate-change strategies. The American Economic Review, 741-765. Pizer, W. A. (1999). The optimal choice of climate change policy in the presence of uncertainty. Resource and Energy Economics, 21 (3-4), 255-287. Solow, R. (1956). A contribution to the theory of economic growth. Quarterly Journal of Economics, 70 (1), 65-94. Stienen, V.F., & Engwerda J.C. (2018). Measuring impact of uncertainty in a stylized macro-economic climate model within a dynamic game perspective. Tilburg University, CentER DP Series No. 2018-007. Tol, R. (2015). Is the uncertainty about climate change too large for expected cost-benefit analysis?. Climatic Change, 56 (3), 265-289.