A Simple Redued-Form Maroeonomi Model of CO 2 Emissions Dynamis Gerard H. Kuper July 29 Abstrat The present paper explains CO 2 emissions in a non-linear maroeonomi model. We derive a relationship between CO 2 emissions, investment, and eonomi growth, and indiate under what onditions arbon dioxide emissions may be redued. The theoretial model shows that the effet on CO 2 emissions of investment in green tehnologies may redue CO 2 emissions depending on the initial level of CO 2 emissions and on eonomi growth. We test the model with World Bank data for a wide seletion of ountries for the period 196-25, and for the eight rihest ountries (G-8) as well as for the eight biggest polluters (P-8). High eonomi growth leads to more CO 2 emissions unless green investment inreases. Key words: CO 2 emissions, Non-Linear Models, Stability, Investment JEL lassifiation: O13, Q32, Q53. Department of Eonomis and Eonometris, University of Groningen, the Netherlands. Correspondene onerning this artile should be addressed to Gerard H. Kuper, Department of Eonomis and Eonometris, University of Groningen, P.O. Box 8, 97 AV Groningen, The Netherlands. Phone: +31 5 363 3756, E-mail: g.h.kuper@rug.nl. 1
1 Introdution The relationship between eonomi ativity and pollution is a omplex one. In this study we only fous on atmospheri pollution. More speifially,we onsider emissions of one of the greenhouse gases regulated by Kyoto protool: arbon dioxide (CO 2 ) whih is to a large extent responsible for the inrease of CO 2 (Battle et al., 2). 1 We analyze stability of the relationship between eonomi ativity, measured by real prodution, and CO 2 emissions. We also show the effet of investments in tehnologies that redue CO 2 emissions green investment on this relationship. It is shown that these investments are ompatible with positive rates of eonomi growth (f. Azar and Shneider, 22). It should be noted that we are not onerned with optimal CO 2 emissions poliies (f. Wirl, 26, 27) nor with ostbenefit analysis or other (see Azar (1998) for a ritial disussion of these issues). Figure 1 shows how CO 2 emissions per apita inreases in the period 196 27. 2 The mean aross a seletion of 192 ountries steadily inreases, but the standard deviation is very large espeially before the 198s. More reently, the standard deviation of CO 2 emissions per apita inreases again. The ountries with the highest CO 2 emissions per apita inlude Kuwait (> 3), Qatar (> 6) and the United Arab Emirates (> 3). In the United States of Ameria, CO 2 emissions are about 2 metri tons per apita, while in Canada it is over 16 metri tons per apita. In China, CO 2 emissions 1 The differene between CO 2 emission, whether natural or man made, and arbon inrease in the atmosphere is what is absorbed in oeans or by vegetation eah year. 2 The data are downloaded from the online World Development Indiators database. 2
Figure 1: CO 2 emissions in metri tons per apita (mean and 1% standard deviation of 192 ountries); Soure: World Development Indiators by the World Bank 2 15 Mean +/- 1 S.D. 1 5-5 -1 196 1965 197 1975 198 1985 199 1995 2 25 are about 4 metri tons per apita. However, China and India have a large population, so total CO 2 emissions (kt) are high and rapidly inreasing as inome grows (see Auffhammer and Carson, 28). Reently, China has overtaken the USA as the ountry with the highest total CO 2 emissions. This is illustrated in Figure 2 whih shows eight ountries with the highest emissions of arbon dioxide. In this paper we refer to these biggest polluters as the P-8 ountries. Not surprisingly, six of these P-8 ountries belong to the eight rihest industrialized ountries. Ukraine and India are replaed by Frane and Canada to omplete the G-8. This paper is not diretly related to the widely disussed Environmental Kuznets Curve (EKC) that relates per apita arbon dioxide emissions and real Gross Domesti Produt (GDP) per apita (surveyed for instane in 3
Figure 2: CO 2 emissions for the eight biggest polluters (kilotons, log-sale); Soure: World Development Indiators by the World Bank 1,, 7,, 5,, 3,, 2,, 1,, 7, 5, Japan USA China Russia Germany United Kingdom 3, 2, India Ukraine 1, 196 1965 197 1975 198 1985 199 1995 2 25 Dinda, 24). The empirial results of this relationship are ambiguous, whih led some authors to question relevane of the EKC and others to study the robustness of the EKC. Examples are Galeotti, Lanza and Pauli (26) and Vollebergh, Melenberg and Dijkgraaf (29). Reently, Lee and Lee (29) disuss the stationarity properties of per apita arbon dioxide emissions and real GDP per apita whih are important the speifiation and impliations of the EKC. The next setion derives the theoretial model, and disusses various senarios. The impliations for arbon dioxide emissions are analyzed in Setion 3. In Setion 4 we present the data we use in the empirial part. The estimation results are shown in Setion 5. These results are interpreted in Setion 6. 4
2 The Model Denote by C(t) the CO 2 emissions in period t. We assume that the hange of C(t) is affeted by investment in tehnologies I(t) that redue CO 2 emissions (denoted as green investment). These green tehnologies may inlude tehnologies that aptures and stores CO 2 (f. Sulzewski and Juanes, 29). I(t) is the balane of ativities that inrease the emission of arbon dioxide, like fossil fuel ombustion, or redue the ability to onsume these gases. A negative value of I(t) indiates that investment ativities on balane redue the emission of arbon dioxide. Furthermore, it is assumed that the level of CO 2 emissions affets the hange in CO 2 emissions at the rate x(t). x(t) may be negative, but an also be positive. More about this later. In mathematial form, this is expressed as: dc(t) dt Ċ(t) = I(t) + x(t)c(t). (1) Defining (t) C(t)/Y (t) and i(t) I(t)/Y (t), we an rewrite Equation (1) as a fration of real output or prodution Y (t) as follows: Ċ(t) Y (t) The dynamis of the C(t)/Y (t) ratio is desribed by: = i(t) + x(t)(t). (2) where y(t) ln Y (t). ċ(t) = Ċ(t) (t)ẏ(t) = i(t) + [x(t) ẏ(t)] (t), (3) Y (t) We assume that the absorption apaity may inrease of derease. This is 5
indiated by funtion x(t). Green investment may redue CO 2 emissions by inreasing the absorption apaity in whih ase x(t) is negative. We, rather ad ho, introdue a damage funtion d(t), that for sake of onveniene, is assumed to be linear in (t). More realistially the damage funtion may be stohasti to reflet unertainties, and my even inlude jumps to inlude atastrophi events (see Dumas and Ha-Duong, 25). In mathematial form: x(t) = αi(t) + d(t) = αi(t) + β(t). (4) Obviously, the results depend on the sign of feedbak parameter β. For positive β, high values for (t) lead to inreasing arbon dioxide emissions for given values of i(t), while negative values for β redue emissions. Substituting this equation in Equation (5) yields: ċ(t) = Ċ(t) Y (t) (t)ẏ(t) = i(t) + [αi(t) + β(t) ẏ(t)] (t) = i(t) + [αi(t) ẏ(t)] (t) + β 2 (t). (5) This equation is a non-linear first-order differential equation in terms of (t). If the model has real roots then the time profile of state variable (t) depends on the initial value () and the ontrol variables i(t) and ẏ(t). Assuming, for instane, that αi(t) < ẏ(t) and i(t) >, the slope of the ċ(t)-urve is negative for (t) = and rosses the vertial axis for positive values of ċ(t). 3 3 If parameter β =, the model is linear model (see Appendix A). 6
2.1 Positive feedbak For positive values for β the ċ-urve is onvex. Two senarios an emerge whih are shown in Figure 3. Figure 3: Equilibrium and stability in the non-linear model i a 1 2 b The dashed urve shows the situation where there are no real roots. The other urve assumes two real roots, one that is loally stable and a higher one that is unstable. There are no real roots if (αi(t) ẏ(t)) 2 4βi(t) <. (6) in whih ase (t) inreases indefinitely as indiated by the arrows on the dashed urve. Starting in point a and reduing the interept, that is dereasing i(t), we may move to point b on the drawn urve, and ultimately we end in the stable equilibrium in point 1. There are four ases depending on the signs of αi(t) ẏ(t) and i(t). Table 4 summarizes the four ases. 7
Table 1: Possible ases with positive feedbak αi(t) ẏ(t) < αi(t) ẏ(t) > i(t) > ase 1 ase 3 real roots if (αi(t) ẏ(t)) 2 4βi(t) > i(t) < ase 2 ase 4 no imaginary roots 2.1.1 Case 1 Case 1 is the situation desribed above with ẏ > αi(t) > and real roots. In this ase the slope of the ċ(t)-urve is negative for (t) = and intersets the positive part of the vertial axis. In point 1 in Figure 3 CO 2 emissions per unit of prodution are stabilized, but CO 2 emissions still rise as is shown in Figure 4(a), in whih we add the urve that desribes the hange in CO 2 emissions per unit of prodution Ċ(t)/Y (t): Ċ(t) Y (t) = i(t) + αi(t)(t) + β2 (t). (7) The ċ(t)-urve and the Ċ(t)/Y (t)-urve ross at the vertial axis. Sine ẏ(t) > αi(t) >, the slope of Ċ(t)/Y (t)-urve is steeper for (t) =, than the ċ(t)-urve. At the loally stable equilibrium ċ(t) =, but Ċ(t)/Y (t) is positive. Sine ẏ(t) >, emissions inrease faster than the growth rate of prodution. 8
Figure 4: Different CO 2 emissions senarios with positive feedbak (a) Case 1 (b) Case 2,, i 1 1 2 2 () Case 3 (d) Case 4,, 2 1 2 1 2.1.2 Case 2 Case 2 whih is illustrated in Figure 4(b) is haraterized by ẏ(t) > αi(t) i(t) <. Sine i(t) is negative, both urves ross the vertial axis below. At the new loally stable equilibrium 1, Ċ(t)/Y (t) is negative. Like in ase 1, αi(t) < ẏ(t), so that, depending on the initial value of (), positive prodution growth is possible and emissions are redued. 2.1.3 Case 3 Two more ases an be distinguished if αi(t) > ẏ(t). Case 3 is haraterized by i(t) > and real roots. In this ase, there are two negative real roots 9
and Ċ(t)/Y (t) inreases at the loally stable equilibrium. This is shown in Figure 4(). In the loally stable equilibrium ċ(t) =, so that CO 2 emissions grow at the same rate as prodution whih may be negative depending on the initial value of (t). 2.1.4 Case 4 The last ase is haraterized by ẏ(t) < αi(t) <, so that prodution growth is negative. Figure 4(d) illustrates that Ċ(t)/Y (t) inreases. So emissions may inrease or derease depending on the initial value of (t). Figure 5: Different CO 2 emissions senarios with negative feedbak (a) Case 1 (b) Case 2,, 1 2 1 2 () Case 3 (d) Case 4,, 1 2 1 2 1
2.2 Negative feedbak For negative values for β the ċ urve is onave as is shown in Figure 5. Here we assume that there are real roots. Only in ase 2 are CO 2 emissions falling if the model is in its stable equilibrium. The major differene with the model in whih β is positive is that for high initial values of (t) the model moves towards its stable equilibrium. Emissions per unit of output stabilize, but total emissions stabilize in three out of four ases. Only in ase 1 will total emissions inrease for ever. This stands in sharp ontrast with the model in whih β is positive. We will illustrate the differenes in the next setion. 3 Impliations 3.1 Positive feedbak In the previous setion we illustrate that it is possible to redue CO 2 emissions, even at positive rates of prodution growth, as long as investment is aimed at ativities that redue the emission of arbon dioxide, so that i(t) <. There are two unambiguous ases, these are ases 1 and 2. Only the seond ase leads to a redution in the emission of arbon dioxide provide the initial value of the state variable is lower than the unstable equilibrium. In ases 3 and 4 the effet on CO 2 emissions is ambiguous. Case 4 obviously is the more favorable one sine investment is green and aimed at reduing CO 2. Obviously the effet depends on the initial onditions. Espeially if CO 2 emission per unit of prodution in the initial period is high, there an 11
only be lower CO 2 emissions if there are large investments aimed at ativities that redue the emission of arbon dioxide. Figure 6: Time paths of CO 2 emissions senarios with positive feedbak and with favorable initial onditions () = 1 (a) Case 1 (b) Case 2 ê ê.35.7.15.1.3.25.2.15.1.5 1 1 19 28 37 46 55 64 73 82 91 1 period.6.5.4.3.2.1.1 -.1.5 -.2 1 1 19 28 37 46 55 64 73 82 91 1-.3 -.5 -.4 -.1 -.5 period () Case 3 (d) Case 4.7.6.5.4.3.2.1 1 1 19 28 37 46 55 64 73 82 91 1 period ê.35.3.25.2.15.1.5.1.9.8.7.6.5.4.3.2.1 1 1 19 28 37 46 55 64 73 82 91 1 period ê.18.16.14.12.1.8.6.4.2 The effets are dependent on the initial value of the state variable and the values of the ontrol variables. Assuming positive roots, Figure 6 simulates the time profiles of Ċ(t)/Y (t) and total emissions Ċ(t)/Y (t) starting at a very favorable initial values () = 1, in whih ase total arbon in the atmosphere is redued. Figure 7 simulates similar paths starting at less favorable initial values () = 1, and assuming real roots. In two of these ases emissions inrease without bounds. 12
Figure 7: Different time paths of CO 2 emissions with positive feedbak and with unfavorable initial onditions () = 1 (a) Case 1 (b) Case 2 ê ê.35.7.1.1.3.25.2.15.1.6.5.4.3.2.5 -.1 1 1 19 28 37 46 55 64 73 82 91 1-.2 -.5 -.3.5 1 1 19 28 37 46 55 64 73 82 91 1 period.1 -.1 -.15 period -.4 -.5 () Case 3 (d) Case 4 ê ê 2,25 8 2,5 18 16 14 12 1 8 6 4 2 1 1 19 28 period,2,15,1,5 7 6 5 4 3 2 1 1 1 19 28 period 2 1,5 1,5 13
3.2 Negative feedbak The effets in ase β is negative are shown in Figure 8. In ases 3 and 4 emissions are ultimately stabilized. Only ase 2 shows a redution in CO 2 emissions. Case 1 leads to ever inreasing arbon dioxide emissions. Figure 8: Time paths of CO 2 emissions senarios with negative feedbak and initial onditions () = 1 (a) Case 1 (b) Case 2 ê ê.15.1.5.4 1 1 19 28 37 46 55 64 73 82 91 1 -.5 -.1.5.3.2 1 1 19 28 37 46 55 64 73 82 91 1 -.5.1 -.1 -.15 -.2 -.25 -.2 -.3 -.4 -.1 -.3 -.5 -.15 period -.1 -.35 period -.6 () Case 3 (d) Case 4 ê -.1 1 1 19 28 37 46 55 64 73 82 91 1 -.2 -.2 -.4 -.3 -.6 -.4 -.8 -.5 -.1 -.6 -.7 -.12 -.8 -.14 -.9 -.16 -.1 -.18 period ê 1 1 19 28 37 46 55 64 73 82 91 1 -.1 -.5 -.2 -.1 -.3 -.15 -.4 -.2 -.5 -.25 -.6 -.3 -.7 -.35 period 14
4 Data The model above suggest a relationship between the hange in arbon dioxide per unit of output on the one hand and investment per unit of output, the square of CO 2 emissions per unit of output, and the interation between investment per unit of output and output growth with CO 2 emissions per unit of output on the other hand. The panel data we use to estimate the model are from the World Development Indiators database ompiled by the World Bank. These series are: [ 1 ] CO2 emissions (metri tons per apita); [ 2 ] Total population; [ 3 ] GDP (urrent US$); [ 4 ] GDP growth (annual %); [ 5 ] Inflation, GDP deflator (annual %); [ 6 ] GDP deflator is alulated from [5] with 25=1; [ 7 ] Gross apital formation (% of GDP); [ 8 ] Adjusted net savings (% of Gross National Inome); [ 9 ] CO 2 per unit of GDP is alulated as [1]*[2]/([3]/[6]). Adjusted net savings are equal to net national savings plus eduation expenditure and minus energy depletion, mineral depletion, net forest depletion, and arbon dioxide. This series exludes partiulate emissions damage. Adjusted net saving also known as genuine saving (Hamilton and Clemens, 1999 is a proxy for sustainability. In the empirial model we use gross fixed apital formation (% of GDP) and the adjusted net savings rate (% of GNI) for investment, denoted below 15
as s t and k t, respetively. The latter is more relevant for analyzing sustainable growth, although it is ritized. For example Ferreira and Vinent (25) point at measurement errors and show that adjustments for depreiation of produed apital, depletion of natural apital, and investment in human apital do not add muh preditive power. Critique by Pillarisetti (25) and Valente (28) is more fundamental. Pillarisetti (25) shows that the measure is oneptually and empirially imperfet sine it is based on weak sustainability assuming perfet substitutability between different types of apital inluding physial, natural and human apital. Valente (28) argues that the sign of urrent genuine savings might deliver a false message: Positive genuine savings in the present does not guarantee sustainability in the future. The series are shown in Figure 9. The panel is unbalaned with data for CO 2 emissions per unit of output ( t ) for 177 ountries and 6,213 observations, Annual % GDP growth (ẏ t ) data for 198 ountries and 7,69 observations, data for adjusted net savings (% of GNI) for 15 ountries and 3,87 observations, and data for gross apital formation (% of GDP) for 182 ountries and 6,491 observations. The graph shows a strong inrease in CO 2 emissions per unit of GDP sine 199. The standard deviation also inreases whih indiates that aross the full sample of ountries the variation aross ountries inreases. Adjusted net savings seems to drop towards the end of the sample, whereas gross apital formation remains at a high level. Applying a range of panel unit root tests reveals that for ċ t, ẏ t, s t and k t the null of a unit root is rejeted. 16
Figure 9: Variables used in the model (means and 1% standard deviation bounds); Soure: World Development Indiators by the World Bank.2 Carbon dioxide emissions per unit of GDP 2 Annual GDP growth (%).15.1.5. 15 1 5-5 -.5 196 197 198 199 2-1 196 197 198 199 2 3 2 1-1 Adjusted net savings (% GDP) 36 32 28 24 2 16 12 Gross apital formation (% GDP) -2 196 197 198 199 2 8 196 197 198 199 2 5 Estimation results The empirial ounter part of Equation (5) in disrete time, where the state variable is lagged, is: ċ j,t = γ j + γ t + γ 1 i j,t + γ 2 (i j,t j,t 1 ) + γ 3 (ẏ j,t j,t 1 ) + γ 4 2 j,t 1, (8) where j = 1,..., N with N the number of ountries, and i t is either k t or s t. Parameter γ 4 is the feedbak parameter β from the theoretial model. The 17
marginal effets are: ċ j,t j,t 1 = γ 2 i j,t + γ 3 ẏ j,t + 2γ 4 j,t 1, (9) ċ j,t i j,t = γ 1 + γ 2 j,t 1, (1) ċ j,t ẏ j,t = γ 3 j,t 1. (11) We estimate the model using World Bank data and apply panel least squares inluding ross setion fixed effets and period fixed effets. 4 The results are shown in Table 2. In all estimations we use White ross setion robust standard errors and ovarianes, making the estimator robust to rossequation (ontemporaneous) orrelation. The estimation results lead to some onlusions: 1. Parameter γ 4 is negative, and even signifiant for the G-8 and the P-8. This implies a negative feedbak: high emission levels lowers emissions per unit of GDP (everything else being equal). 2. For the full set of ountries γ 4 does not differ signifiantly from zero, implying a linear ċ-urve. 3. Adjusted net savings have a negative impat on the hange in CO 2 per unit GDP, as expeted. However, the level of signifiane is only 1%. 4. Gross fixed apital formation does not affet the hange in CO 2 per unit GDP diretly. 4 The hoie of ross setion fixed effets and period fixed effets is onfirmed by likelihood ratio tests. 18
Table 2: Estimation results for different seletions of ountries, with t-values between brakets Dependent variable: Change in CO 2 emissions per unit of GDP All ountries G-8 P-8 Variable (a) (b) () (d) (e) (f) onstant 6.3E-5 4.1E-5 1.6E-4 1.4E-4 2.7E-4 2.9E-4 (8.697) (4.942) (1.991) (.888) (2.718) (1.34) k t -4.5E-5-9.E-4 -.1 (-.889) (-1.87) (-1.268) s t -1.2E-4 -.1 -.2 (-1.737) (-1.758) (-1.891) ẏ t t 1-1.37 -.689 1.63 -.12.251 -.1 (-2.557) (-3.48) (2.18) (-.69) (.425) (-.6) k t t 1.63.877.755 (.647) (2.339) (1.769) s t t 1 -.111.628.688 (-.892) (1.561) (1.455) 2 t 1-9.944-1.544-76.442-12.124-43.94-64.396 (-1.51) (-1.587) (-2.711) (-3.79) (-2.178) (-2.46) Countries 151 17 8 8 8 8 Obs. 376 559 233 36 218 289 R 2.19.137.221.192.271.266 These are the diret effets. There are also indiret effets: 1. For the full sample of ountries a high level of CO 2 emissions leads to a drop in emissions for high rates of GDP growth. 2. For the G-8 and the P-8 high adjusted net savings redue CO 2 emissions, but the effet is moderated if the level of emissions initially is high. 19
3. For the G-8 and the P-8 gross apital formation inreases CO 2 emissions if the level of emissions is high. 6 Disussion In this setion we interpret the estimated equation in terms of the theoretial model and fous on the interept and slope of the ċ-urve. The interept is alulated as γ 1 s, where s is the mean of adjusted net savings. Higher adjusted net savings is interpreted as green investment (negative i(t) in the theoretial model). The slope is alulated as γ 2 s + γ 3 ẏ, where ẏ is mean GDP growth. For all ountries, the interept and the slope is negative, whih resembles ase 2 with negative feedbak, whih may lead to falling CO 2 emissions. For the G-8 and also for the P-8 the interept is negative, but the slope is positive. This is ase 4 in terms of the senarios desribed in the theoretial model, whih leads to lower CO 2 emissions. It should be noted that, beause of various issues with the onept of genuine savings, positive adjusted net savings is no guarantee for sustainability and that in terms of our model the interept is positive and we may end up in ase 1 with inreasing CO 2 emissions. High growth requires positive genuine savings (measured in the orret way) in order to redue CO 2 emissions permanently. The sample ends before the start of the urrent worldwide eonomi reession. Taking into aount that GDP growth is estimated to be -4% in 29, this turns the estimated slope positive for the full sample. So, we move to ases 3 or 4 where CO 2 emissions are lower, but stabilize. The preferred 2
ase is ase 2 where CO 2 emissions ontinue to drop whih requires higher net adjusted savings and investments, whih is probably a more diffiult task in a reession. 21
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A Linear model Again, there are four ases whih are shown in Figure A.1. Cases 1 and 2 where αi(t) ẏ(t) < are stable. Figure A.1: Different CO 2 emissions senarios for the linear model (a) Case 1 (b) Case 2,, 1 1 () Case 3 (d) Case 4,, 1 1 24