International Capital Markets, Oil Producers and the Green Paradox

International Capital Markets, Oil Producers and the Green Paradox#

Gerard van der Meijden*, Frederick van der Ploeg** and Cees Withagen***

31 January 2014

Abstract In partial equilibrium a rapidly rising carbon tax encourages oil producers to extract fossil fuels more quickly, giving rise to the Green Paradox. General equilibrium analysis for a closed economy shows that a rapidly rising carbon tax negatively affects the interest rate, which tends to weaken the Green Paradox. However, in a two-country world with an oil-importing and an oil-exporting region the Green Paradox may be amplified in general equilibrium if exporters are relatively patient. On the contrary, if oil exporters are relatively impatient, the Green Paradox might be reversed. Furthermore, general equilibrium effects tend to weaken the link between a capital asset tax and the time profile of resource extraction so that the capital asset tax becomes less useful as an instrument to offset the Green Paradox effect associated with the announcement of a future carbon tax. Taking exploration costs into account, we show that the effect of both policy instruments on cumulative extraction is of opposite sign as the effect on current extraction. Moreover, if the change in current extraction is amplified or reversed in general equilibrium, so will be the change in cumulative extraction.

Keywords: Green Paradox, Hotelling rule, oil importers, oil producers, investment, capital markets, carbon tax, asset holding tax JEL codes: D90, H20, Q31, Q38

* Corresponding author: Department of Economics, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. E-mail: [email protected] ** Oxford Centre for the Analysis of Resource Rich Economics, University of Oxford, Manor Road Building, Oxford OX1 3UQ, UK and VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. *** Department of Economics, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands.

# We gratefully acknowledge financial support from FP7-IDEAS-ERC Grant No. 269788. We also appreciate the comments and suggestions of Hassan Benchekroun, Julien Daubanes, Gerard Gaudet, Peter Neary, Erik Verhoef and participants at seminars in Rehovot, St. Petersburg and Montreal. 1

1. Introduction The idea that a rapidly rising carbon tax encourages oil producers to extract oil more quickly1 has gained traction and has become known as the Green Paradox due to the accompanying acceleration of global warming (Sinn, 2008, 2012; Gerlagh, 2011). The underlying mechanism is that a future carbon tax forces oil suppliers to supply less oil in the future and thus to supply more today. This pushes the current oil price down and thus boosts today’s demand for oil. Much of the analysis on the Green Paradox is partial equilibrium, where the interest rate and oil demand function are taken as given.2 However, front-loading of oil extraction influences the global supply of savings and the demand for capital, so that the interest rate must adjust to clear bonds and capital markets. At first blush one might think that a rapidly rising carbon tax increases current output relative to future output, which boosts global savings and thus depresses the global interest rate. This will make it less attractive to extract oil so that acceleration of the burning of oil and the associated acceleration of global warming resulting from the Green Paradox are mitigated. However, in general equilibrium there will be other effects on the income and wealth positions of oil importers and oil exporters, and changes in the interest rate will affect investment and thus will also impinge on future oil demand. In that case, it is possible that the Green Paradox effect is reinforced or even reversed. Our objective is to gain a better understanding of these intricate general equilibrium effects and to investigate under which conditions the Green Paradox is mitigated and under which conditions it is reinforced or reversed in general equilibrium.

For this purpose we construct a two-country, two-period model of oil importers and oil producers, where the interest rate and the current and the future real oil prices are determined from the conditions for clearing the markets for financial assets and oil, and the Hotelling rule governs optimal oil extraction (cf. Dixit, 1981; Marion and Svensson, 1984; van Wijnbergen, 1985; Djajić, 1988). We suppose that all markets operate under perfect competition and that only the oil-importing block of countries produces final goods which can be used for both consumption and investment purposes. We analyze the consequences of changes in future carbon taxes on current oil extraction and on the interest rate. In addition, we are interested in the amount of oil that is left unexploited in the crust of the earth as this affects the ultimate degree of global warming. We focus on changes in future taxes instead of optimal climate policy (typically requiring immediate action instead of a rapidly rising carbon tax), because in reality gradual greening policies are observed and strong current action is lacking. Furthermore, we also analyze the effects of a tax imposed by oil importers on financial asset holdings of oil producers which

1 We refer throughout to ‘oil’ as short-hand for ‘exhaustible fossil fuel resources’ such as oil, natural gas and coal. 2 Green Ramsey models of growth, climate change and the optimal carbon tax (van der Ploeg and Withagen, 2013; Golosov et al., 2013) allow for an endogenous interest rate, but their primary focus is on the optimal carbon tax and not the study of Green Paradox effects.

has been proposed as a second-best alternative for a badly designed carbon tax (cf. Sinn, 2008; Jaakkola, 2012).

Our analysis highlights the crucial role of the interest rate for the Green Paradox. In general equilibrium the interest rate plays a role on the production side as well as on the consumption side of the economy. Demand for oil is determined by oil prices. These prices are intertemporally related by means of the Hotelling rule, which features the interest rate as the opportunity cost of conserving oil. On the demand side for final goods the interest rate depresses the relative price of second-period consumption. Moreover, each region’s wealth and the demand for capital are affected by the interest rate. It is the interplay between these aspects that drives the direction of the change in the interest rate induced by environmental policy and therefore determines the general equilibrium repercussions for the partial Green Paradox.

As already mentioned, one might think that, as a future carbon tax boosts current income relative to future income, it stimulates global savings and thus depresses the rate of interest and mitigates the Green Paradox. In order words, first-period oil use goes up by less in general equilibrium than in partial equilibrium as first-period oil use goes up by less in general than in partial equilibrium. We show that this intuition is correct provided that the world consists of a single jurisdiction, inhabited by a representative agent with homothetic preferences over consumption and there is no investment. However, if we allow for investment in this single jurisdiction world, the Green Paradox may be strengthened (in the sense that first-period oil use goes up by more than in partial equilibrium) despite the fall in the interest rate: a potential decrease in investment (driven by the decrease in future oil extraction, as oil and capital are cooperative in production) depresses future oil demand, which boosts current oil use. If we allow for multiple jurisdictions, namely an oil-producing region and an oil-importing region, the Green Paradox may (even without investment possibilities and with homothetic preferences) be amplified or reversed (i.e., first-period extraction goes down instead of up in general equilibrium compared with the partial equilibrium), if preferences of the representative agents in both jurisdictions are sufficiently different.3

In particular, our results show that the Green Paradox effect associated with a future carbon tax is likely to be weakened in general equilibrium if oil producers are relatively impatient compared to oil importers. Moreover, if in addition global savings depend negatively on the interest rate, the Green Paradox might even be strongly reversed in general equilibrium. On the contrary, if oil exporters are relatively patient, policy makers should be aware of a possible amplification of the Green Paradox. Furthermore, general equilibrium effects tend to weaken the link between an asset tax and the time profile of resource

3 Besides general equilibrium effects, there are other reasons why the Green Paradox might be mitigated or reversed. The emerging abundance of shale gas and other forms of unconventional energy reserves might be enough to eliminate any Green Paradox effects (Michielsen, 2012). The Green Paradox may be also reversed if there are increasing costs in the production of renewables (Grafton et al., 2013).

extraction so that the asset tax becomes less useful as an instrument to offset the Green Paradox effect associated with the announcement of a future carbon tax. If oil producers are relatively patient, using an asset tax to combat climate change might even be counterproductive in general equilibrium. Finally, accounting for partial exhaustion of the stock of oil reserves by imposing exploration costs, we show that in partial equilibrium a future carbon tax ensures that more oil will be locked in the crust of the earth, whereas the opposite happens with a future asset tax. We demonstrate that general equilibrium effects may amplify, mitigate, or even reverse the partial equilibrium effect of climate policy on cumulative oil extraction. The direction of the effect again depends on asymmetries in preferences between oil producers and oil importers and on the possibility to invest in physical capital.

Eichner and Pethig (2011; 2013) also study climate policy in a two-period, multi-country setting, but their focus is on carbon leakage as a result of unilateral action and they abstract from capital accumulation. Moreover, Eichner and Pethig do not deal with effects on cumulative extraction, because they use a model in which the stock of oil reserves is fully depleted. Therefore, they cannot address that a future carbon tax (or renewables subsidy) quickens oil extraction, but also encourages owners of fossil fuel reserves to leave more in the crust of the earth and thus depress cumulative carbon emissions in that way (e.g., Hoel, 2012; van der Ploeg and Withagen, 2012a; van der Ploeg, 2013). Long and Stähler (2013) consider the general equilibrium effects of green technological progress on the speed of oil extraction in a model without capital accumulation, and note the possibility that this may lead to a rise in the interest rate thereby offsetting and possibly reversing the initial increase in the rate of oil extraction.

Our contribution is thus threefold. First, we make explicit the channels through which general equilibrium effects might strengthen or even reverse rather than weaken the partial equilibrium Green Paradox effect of a future carbon tax. We also derive conditions for which each of these possibilities occurs. Second, we show that in general equilibrium the asset tax becomes less useful as an instrument to offset the Green Paradox effect associated with a future carbon tax and show that it might even be counterproductive. Third, we demonstrate that general equilibrium considerations matter for the effect of climate policies on cumulative oil extraction. We show that a naive government that assumes that investment and the interest rate are not affected by its policy might make a serious mistake and we identify the direction of the potential error. Global warming is a major problem and may require non-marginal policies so that the interest rate will be affected and one needs to look at the general equilibrium implications of climate policy.

The remainder of the paper is structured as follows. Section 2 sets up the general model under the assumption that oil producers extract oil costless from the earth. Section 3 solves the model for a given interest rate, derives the key partial equilibrium Green Paradox result, and shows how it can be averted

with a tax on oil producers’ asset holdings. Section 4 employs a general equilibrium analysis with international bond markets to demonstrate that Green Paradox effects may be amplified or reversed rather than weakened. Section 5 introduces capital accumulation and analyzes how investment in physical capital affects the general equilibrium repercussions for the Green Paradox. Section 6 deals with exploration costs and partial exhaustion. Section 7 concludes.

2. The model

We consider a two-period model with two blocks of countries (or regions), one block of homogeneous countries being the oil importers and the other block of homogeneous countries representing the oil exporters (denoted by an asterisk). Only the oil importers produce final goods, which are demanded by both oil importers and oil exporters. Oil importers use capital and oil (as well as fixed factors such as land and labour) to produce final goods. Their assets consist of capital and bonds. For the time being, we assume that oil producers extract oil at zero costs but we will discuss exploration costs in Section 6. Oil exporters have a given stock of initial oil reserves, which will thus be fully exhausted. Their other assets consist of capital and bonds. All markets operate under perfect competition and clear in both periods. Oil producers have rational foresight and oil prices obey the Hotelling rule. The governments of the oil- importing block might levy a specific carbon tax on the use of oil by final goods producers and an asset tax on the asset holdings of oil producers.4 Tax revenues are distributed in lump-sum fashion to the households in the oil-importing region. Our aim is to investigate the effects of these two types of taxes on the real price of oil, the world interest rate, investment, and the intertemporal pattern of oil depletion.

Firms

Output of final goods is given by the production function F(,)KRtt, where Kt denotes employed capital and Rt the oil extraction rate in period t (t 1, 2 ). Taking account of other fixed factors such as land and labour, this production function has non-increasing returns to scale and is strictly increasing for positive inputs and concave in each input. With the net rate of return on capital indicated by rt , the world market oil price by qt , the specific carbon tax levied on the producers of final goods by  t , and the constant rate of depreciation by  , profits by firms in the oil-importing region in each of the two periods are given by

ttttttttFK(,) R  r  K  ( q  ), R t  1,2. Profit maximization under perfect competition gives:

(1) FKRK (,)11 r 1 , FKRR (,)11 q 1 1 ,

4 If asset holdings of oil exporters are negative, the asset tax becomes a borrowing subsidy.

(2) FKRK (,)22 r 2 , FKRR (,)22 q 2 2.

This yields the profit-maximizing conditional oil demand Rrq11(, 1 1 ), capital demand Krq11(, 1 1 ), and the profit function 11(,rq 1 1 ). For period two this yields the profit-maximizing factor demands

Krq22(, 2 2 ) and Rrq22(, 2 2 ), which gives the profit function 22(,rq 2 2 ). The factor demand functions are negative functions of the own factor prices. If capital and oil are cooperant factors,

FKR  0 which we assume, future oil demand decreases in the interest rate too.

Profit maximization by oil exporters, facing the real interest rate r22 , with  2 the capital asset tax imposed by the oil-importing region on capital invested in its jurisdiction yields the Hotelling rule:

(3) qrq2221(1 ) .

Hence, the after-tax return on keeping a marginal barrel of oil in the earth, (),rq221 must equal the return on taking a marginal barrel of oil out of the earth, i.e., the expected capital gains, qq21 .

Households

Preferences of the representative household in the oil-importing and oil-exporting regions are given by the

*** * utility functions UC(,12 C ) and UCC(,)12, respectively, where Ct ( Ct ) denotes consumption in period t in the oil importing (oil exporting) region. We assume that these functions are continuously differentiable and concave. The consolidated budget restrictions for both regions read:

* CA12(1 rA 11111 )  R, CrARA2(1 2 ) 2  2 22  22,

** * ** CA12(1 rAqR 1111 )  , CrAqR212222(1 )  ,

* where At and At denote asset holdings at the beginning of period t . The initial asset endowments A1

* 5 and A1 are exogenously given. There is no capital asset tax levied in period one. It follows that the present-value budget constraints for both regions are given by:

* CRA2 2 22  22 (4) CrAR111111(1 )   M, 11rr22

* **CqR222 * (5) CrAqRM11111(1 ) , 11rr22  22

5 Such a tax corresponds to a non-distorting lump-sum tax given the inelastic first-period supply of capital.

where M and M * denote wealth of the oil-exporting and oil-importing region, respectively. Wealth of the oil-importing region is the sum of the net return on assets, the present discounted value of profits and carbon tax refunds, and the present discounted value of taxes on assets held by oil exporters. Wealth of the oil-exporting block consists of the return on assets and the present discounted value of oil revenues.

Equilibrium conditions

Equilibrium on the asset markets requires that capital must be held by one of the two regions:

* (6) KAA111,

* (7) KAA222.

The initial oil stock is S1 , so that oil market equilibrium (OME) requires

(8) SRR112.

The goods market equilibrium conditions (GME) for periods one and two are:

* (9) F(,)(1)KR11 K 1  C 1  C 1  K 2 ,

* (10) F(,)(1)KR22 K 2  C 2  C 2 .

This completes the description of the model with full exhaustion of oil reserves, which we analyze in Sections 3-5. In Section 6 we extend the model to allow for partial exhaustion by introducing exploration costs and an endogenous stock of initial oil reserves.

3. Green Paradox: partial equilibrium analysis

To understand the Green Paradox as it has been discussed so far in a partial equilibrium context, in this

section we study the resource market in isolation thereby taking the interest rate r2 and the capital stocks

K1 and K2 as fixed before moving to a general equilibrium analysis in Sections 3-6. We first consider an

increase in the second-period carbon tax  2 , while keeping both other taxes 1 and  2 constant. To see that oil extraction in period one will increase, substitute oil demand from (1)-(2) and the Hotelling rule (3) in the OME (8):

(11) R11()(1)qRrqS 1 2  2 21  2  1,

where we use that K1 and K2 are given and thus oil demand R1 and R2 are functions of the tax-inclusive oil price only. If the future carbon tax goes up, equilibrium on the oil market does not allow for a higher

current world market oil price q1 , because that would reduce oil demand in both periods so that there would be excess supply. Hence, a future carbon tax increases the future cost of oil and curbs future oil use, but it depresses the current oil price and thus boosts current oil demand and current carbon emissions. This is the key Green Paradox result for the partial equilibrium context with full exhaustion of reserves. Intuitively, a future carbon tax depresses future demand for oil. This pushes down the future producer of oil and thus curbs future supply of oil. But then, given the available stock of oil reserves, oil supply in the present would have to increase. This puts downward pressure on the present oil price and thus boosts current demand for oil. Hence, a future carbon tax forces oil suppliers to supply less oil in the future and thus to supply more today. This is the essence of the Green Paradox.

Next we still keep the capital tax constant but allow both carbon taxes to change in proportion, i.e.,

 21(1 ) . We can rewrite the OME condition (11) as:

()r (12) R ()(1)qRrq  22  S. 11 1 2 2 2 1 1 1 1 1r22

So, if the growth rate of the carbon tax,  , equals the after-tax return on capital for the oil-exporting region, r22 , an increase in the first-period carbon tax does not affect q11 or the intertemporal pattern of oil extraction rates. If the growth rate of the carbon tax is bigger than  , the future cost of oil rises while the current cost of oil falls so that there is a Green Paradox. However, if the growth rate of the carbon tax is lower than  , we deduce immediately from (12) that the future cost of oil falls so that more oil is extracted in the future and less today. Hence, the Green Paradox does not occur.

Finally, we consider the case of given carbon taxes, 1 and  2 , and examine the effect of a higher capital

asset tax  2. Demand for future oil will go up, because equation (11) implies that for a given interest rate

r2 and a higher tax rate  2 , a decrease of R2 would require a higher oil price q1 . But then also R1 decreases, which violates the condition of equilibrium on the oil market. The intuition is that oil producers get a smaller return on their saving, so pump less fast to have less revenue up front. An asset holding tax thus goes in the opposite direction of a future carbon tax, which was the rationale behind Sinn’s original proposal to slow down oil extraction and global warming (cf. Sinn, 2008).

We summarize the above discussion in the following proposition.

Proposition 1: In a partial equilibrium with a given net interest rate r2 the following holds. (i) For given current carbon tax and capital asset tax an increase the future carbon tax accelerates oil extraction and global warming.

(ii) For a given capital asset tax a carbon tax that rises faster (slower) than the net interest rate that the oil exporter faces, accelerates (decelerates) oil extraction and global warming. (iii) For given carbon taxes a tax on oil producers’ asset holdings slows down oil extraction and global warming.

In the sequel we extend the present analysis towards general equilibrium and focus on two policy changes: (i) a higher future carbon tax taking the current carbon tax as given and equal to zero, which captures a carbon tax that rises faster than the net interest rate; (ii) introduction of a capital asset tax.

4. Green Paradox in general equilibrium without capital

Before moving to a full general equilibrium analysis we allow for international bond markets, but abstract from physical capital (cf. Eichner and Pethig, 2011). In this setting aggregate global savings must be equal to zero, since the interest rate adjusts to equilibrate relative aggregate demand for and supply of future and current goods. Section 5 discusses the full general equilibrium model with physical capital.

4.1. Effects of a future carbon tax

For the time being we abstract from the asset tax, so that  2  0 and the future interest rate equals r2 in both regions. We then show under what circumstances general equilibrium effects may lead to strong reversal, weak reversal, attenuation or amplification of the partial equilibrium Green Paradox effects.

Definition 1: Assume the future carbon tax is increased. i. Strong reversal occurs if current oil extraction decreases. ii. Weak reversal occurs if current oil extraction is unaffected. iii. Attenuation occurs if current oil extraction expands by less than in partial equilibrium. iv. Amplification occurs if current oil extraction expands by more than in partial equilibrium. □

Without investment the oil market equilibrium (OME) condition with  21  0 imposed becomes:

(13) R11()qR 2 (1) rq 21 2 S 1.

For a given  2 equation (13) gives a negative relationship between r2 and q1 and for given r2 , it gives a negative relationship between  2 and q1 . The reason is that a higher interest rate or future carbon tax curbs future oil demand, thus requiring a fall in the current oil price to clear the oil market. Hence, as shown in Figure 1 below, equation (13) corresponds to the downward-sloping OME locus in (,)qr12- space which shifts inwards as the future carbon tax is increased.

To characterize the final goods market equilibrium (GME), we assume without loss of generality that initial bonds endowments are zero. Then the present-value budget constraints (4) and (5) become

CFRrqrqRrq22212212212( ((1 ) )) (1 ) ((1  )  )) CFRqqRq111111(())  ()  (14) 11rr2 2

 Mqr(,),12

* **CqR222 (15) CqRqSMq111111(), 11rr22

where profit maximization, the Hotelling rule (3), and the OME condition (13) have been used. Let the

tax rate be given. For any triplet of prices (,qqr122 ,) such that the Hotelling rule is satisfied, the oil exporting region derives demand for the final good as functions of this triplet. The Hotelling rule makes

sure that (perceived) discounted income of this region equals qS11. For this set of prices (and the given tax rate) the oil importing region determines profit-maximizing demand for oil, and thereby discounted total income M . Then follows demand for final goods for the two periods. So, we can write

** CrMrqt (,2212 (, ; )) and CrMqt (21 , ( )), t 1,2 , where we explicitly indicate that the importing region’s consumption is depending on the carbon tax, through income. The locus of points for which there is equilibrium on the final goods market is defined by

** CrMrq22(, (, 2 12 ; )) CrMq 22 (, ()) 1 FR ( 2 ((1) rq 2 1 2 )) (16) ** . CrMrq12(, (, 2 1 ; 2 )) C 1 (, rM 2 ()) q 1 FR(())11 q

A general equilibrium exists where the GME and OME locus intersect, due to Walras’s Law. The problem is that in general little is known about the location of GME and the shift that occurs in GME following a change in the tax.

One way to get partly around the problem with regard to the effect of a change in the tax rate is to assume

that second-period oil demand equals the remaining supply SRq111 (), irrespective of the second-period price (instead of assuming that second-period oil deployment is determined by the demand function in (2), irrespective of the remaining supply). Using (9)-(10), the GME locus of the initial oil price and interest rate for which this is taken into account is given by the Modified GME or MGME locus:

** CrMrq22(, (, 2 1 )) CrMq 22 (, ()) 1 F SRq111 () (17) ** . CrMrq12(, (, 2 1 )) C 1 (, rM 2 ()) q 1 FR(())11 q

The MGME defined in this way will prove useful, since it does not feature the future carbon tax  2 . To determine the general equilibrium effect of a higher carbon tax, we still need to establish whether the

MGME locus increases or decreases in (,)qr12 space. Note that a higher carbon tax will affect the current oil price which will lead to a reallocation of wealth, as can be seen from (14) and (15), which in turn affects relative total final goods demand.

Figure 1: Four different general equilibrium scenarios

Panel (a) - Strong reversal Panel (b) - Weak reversal

q1 OME q1 OME1 1

OME2 OME2  GE E GE E MGME  MGME  

PE PE 

r2 r2 Panel (c) - Attenuation Panel (d) - Amplification

q 1 OME1 q1 OME1 MGME MGME OME2 OME2

E E   GE 

PE  PE GE

r r 2 2

Notes: The intersection of the solid lines (point E) gives the initial equilibrium. The intersection at point GE gives the equilibrium after the increase in the future carbon tax. The movement from point E to point PE gives the partial equilibrium effect of an increase in the future carbon tax.

Figure 1 shows the MGME and OME loci in (,)qr12- space, in the neighborhood of the equilibrium, for four different cases. Because the OME line shifts down upon an increase in the future carbon tax whereas the MGME line remains unaffected, the comparative statics results depend on the relative slopes of the MGME and OME loci with a crucial role played by the interest rate. In partial equilibrium the interest

rate is exogenous and the equilibrium oil price jumps down from point E to the level corresponding with point PE (partial equilibrium) in each panel, which implies shifting oil extraction from the future to the present. In general equilibrium, however, the interest rate adjusts to the level corresponding with point GE to clear the asset market. The four panels of Figure 1 depict the cases of (a) strong reversal (negative, relatively flat MGME locus), (b) weak reversal (flat MGME locus), (c) attenuation (positively sloped MGME locus), and (d) amplification (negative, relatively steep MGME locus).

The first three outcomes feature a fall in the interest rate. This induces oil exporters to pump more slowly, so that the partial equilibrium Green Paradox is attenuated (panel (c)) or even reversed (panels (a) and (b)). Amplification of the Green Paradox (panel (d)) is associated with a higher interest rate, which further boosts current oil extraction. Proposition 2 summarizes the results we have obtained so far.

Proposition 2: Attenuation, weak reversal and strong reversal of the Green Paradox require a decrease in the interest rate. Amplification occurs if and only if the interest rate increases.

Which of the four cases actually prevails, depends crucially on preferences, and in particular on the wealth effects of a future carbon tax. With homothetic preferences, relative aggregate consumption of future and current final goods in each country is independent of wealth. It follows from the optimality conditions of the households in each country that demand for future relative to current goods depends

** * * solely on the interest rate, i.e. CC21/() r 2 and CC21/() r 2, with '0 and '0. Hence,

** ()/()()CC22 CC 11 r 2 with identical preferences. In that case the MGME locus slopes upwards so that we have attenuation.

Proposition 3: Suppose preferences in the two regions are identical and homothetic, that there is no investment, and that the future carbon tax is increased. Then, in general equilibrium acceleration of oil extraction occurs but is always less than in partial equilibrium. We thus have attenuation of the Green Paradox.

The intuition for oil extraction being brought less forward to the present is as follows. The partial

equilibrium effect of a future carbon tax lowers q1 so that current oil extraction and current production of final goods increase and future oil extraction and final goods production decrease. Because of identical homothetic preferences, wealth effects do not affect the aggregate relative demand for future goods. As a result, an excess demand for future goods materializes at the prevailing interest rate. Hence, the interest rate must decrease to reduce relative demand for future goods, thus eliminating excess demand. The lower interest rate encourages oil-exporters to extract oil less quickly, thereby attenuating the Green Paradox.

A large part of preferences used in dynamic models are of the identical and homothetic class. However, it is interesting to see whether the result of proposition 3 also holds in more general circumstances, in particular if we allow for different preferences across regions. If preferences are homothetic, but not identical, the aggregate relative consumption of future and current goods (at the global level) can be written as

CC ** C (18) 22()rrr  * ()  () 1. **222 CC11 CC 11 The extra term in square brackets in (18) captures a wealth reallocation effect: Even if preferences are homothetic in both countries, a reallocation of wealth between the countries affects the aggregate relative demand for future goods at an unchanged interest rate. For example, if relative current consumption of the country with the highest future-to-current consumption ratio increases, aggregate future-to-current consumption is affected positively and vice versa. In order to clarify the conditions under which each of the three remaining cases (weak reversal, strong reversal, and amplification of the Green Paradox) may occur, let us consider the CES utility functions in assumption 1.6

Assumption 1: The utility functions are given by

CC111  12 if  1 111 UC(,12 C )  ,  1 lnCC12 ln if  1  1 

11**  CC1 * 12if   1  *** *** 111 UCC(,)12  , 1 lnCC** ln if *  12*  1  1 

where   0 and *  0 denote the elasticities of marginal utility (the inverse of the intertemporal

elasticity of substitution), and   0 and *  0 are the rates of pure time preference in both regions.

In first instance, we will restrict ourselves to the case  * 1 , in which the income and substitution effects of changes in the interest rate cancel out against each other, resulting in the fixed expenditure shares

**** * * (19) CMC12, (1 )(1 rMCMC 212 ) ,  ,  (1  )(1 r 22 ) M,

6 Bergson’s theorem asserts that preferences are time separable and homothetic if and only if they are of the CES type in assumption 1.

where  (1 ) / (2  ) and  **(1 ) / (2  * ) . Note that ,[0.5,1)*  due to non-negative discounting. Proposition 4: Suppose that assumption 1 holds with  * 1 , that there is no investment, and that the future carbon tax is increased. Then (i) Strong and weak reversal of the Green Paradox cannot occur. (ii) Attenuation of the Green Paradox occurs if the oil importing region is as impatient as or relatively less impatient than the oil exporting region ()  * . (iii) Amplification of the Green Paradox may occur but requires the oil importing region to be relatively more impatient than the oil exporting region ()  * .

Proof:. Strong reversal implies a higher current world market oil price q1 , in order to have less demand

for current oil, and a lower future world market oil price (1 rq21 ) , in order to have more demand for

future oil (note that the oil price for the final good producer is (1 rq21 )  2, which has to decrease in spite of a higher tax). Hence, given assumption 1, demand for current final goods for the oil exporting region goes up and its demand for future final goods goes down. With lower present production and higher future production, the demand schedule of the oil importer must be the other way around. But this is excluded by (19).

** Equality between first-period consumption and production requires M MFR()1 . Using

* M  qS1 and M FR()12211 FR ()/(1)  r qS, we get

* ( )qS11 (1  ) F ( R 1 ( q 1 ))   F ( R 2 ((1  r 2 ) q 1  2 )) / (1 r 2 ) . Weak reversal occurs if q1 and

(1rq21 )  2 remain unaffected. Hence the income of the oil exporting region is unaffected as well as its first period consumption. Its second period consumption goes down because (1 rq21 ) goes down. So, current consumption in the oil importing region stays unaffected and future consumption increases. This is incompatible with (19), unless   0 , which is outside the admissible range.

* * From ( )qS11 (1  ) F ( R 1 ( q 1 ))   F ( R 2 ((1  r 2 ) q 1  2 )) / (1 r 2 ) we also see that    is a

necessary condition for amplification. Occurrence of the Green Paradox implies an increase in R1 and a

decrease in R2 . Hence, q1 must go down. Furthermore, amplification of the Green Paradox requires an increase in r2 (proposition 2). Therefore, the right hand side of the expression increases. Given that q1

goes down, the left hand side can only increase if    * . Finally, given existence, the conditions given in (iii) are sufficient. □

In fact, as long as the oil-importing region is more patient or not too much more impatient than the oil- exporting region, Green Paradox effects are attenuated in general equilibrium. Figure 2 shows that if the oil-importing region is much more impatient than the oil-exporting region, the Green Paradox may be amplified. The production function underlying the results in the figure is given by

(20) FR() [ R(1)/ (1) L (1)//(1)  ] , with   0.1 and L  1.

Figure 2: Attenuation versus amplification of the Green Paradox

Panel (a): S1  2 Panel (b): S1 1

1 1

0.9 0.9

0.8 0.8

0.7 attenuation 0.7 attenuation

0.6 0.6   0.5 0.5

0.4 0.4

0.3 0.3 amplification amplification 0.2 0.2

0.1 0.1 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -* -*

Notes: Parameters are set to   0.1 and L  1 . The carbon tax  2 is increased from 0 to 0.01. The horizontal axis features a mean-preserving spread with * 0 ,   1/2 x , *  1 x , x  (0,1/ 2) ; the shaded area indicates amplification of the Green Paradox.

The intuition for obtaining amplification of the Green Paradox is as follows. At a given interest rate, the future carbon tax induces excess supply of current goods (the partial equilibrium Green Paradox effect summarized in proposition 1), but also increases the relative wealth of the oil-importing region due to the decline in the oil price. If the oil-importing region has a relatively high current-to-future consumption rate (which is the case if   * ), the wealth reallocation effect positively affects aggregate current-to-future consumption, which diminishes the excess supply of current final goods that is caused by the partial equilibrium Green Paradox effect. If strong enough, this wealth reallocation effect even leads to excess demand for current goods. As a result, the interest rate then needs to rise instead of fall in order to restore the general equilibrium so that the Green Paradox is amplified. The strength of the wealth reallocation effect depends on the difference between the current expenditures shares,   * (or, equivalently on how much more impatient the oil-importing region is relative to the oil-exporting region, as measured by

  * ). The strength of the supply effect (the change in current output relative to future output due to the lower current world price of oil) depends crucially on the elasticity of factor substitution,  , through its effect on the price elasticity of oil demand 1 :

 1 R qR   R  dF[()]' R q F R () q1 dq R dq with  1110 1 . 11 11 1 111 1  q1 Rq111   L 

The grey areas in panels (a) and (b) of Figure 2 give combinations of   * and  for which the Green Paradox is amplified. In panel (a), the initial oil reserves amount to two, whereas in panel (b) they are equal to unity. The non-convexity of the attenuation area in panel (b) arises because although 1 is a convex, monotonically increasing function of  if RL1 /1 , it is a convex, non-monotonic function of

 with lim11 lim  if R1 /1L  . Hence, if R1 /1L  the supply effect dominates the wealth 0 reallocation effect not only for large, but also for small values of  , which explains that attenuation occurs in the upper right as well as in the lower right corner of panel (b). The intuition behind this result becomes clear when considering the extreme cases of   0 and    . If SL1  , the case with a

Leontief production function YLR11 min{ , } (i.e.,   0 ) is essentially similar to the case with linear production YRL11 (i.e.,  ): in equilibrium q1 1 and qr22211 , so that dr22/1 d   , implying that the Green Paradox is attenuated.

We have shown that reversal of the Green Paradox in the model without capital cannot be obtained if the elasticities of marginal utility equal unity (see proposition 4), which implies constant current expenditure shares as income and substitution effects of changes in the interest rate cancel out against each other. However, if the income effect dominates the substitution effect, so that current expenditure shares depend positively on the interest rate, reversal of the Green Paradox may occur. The intuition behind this result is as follows. Recall that if the oil-importing region has a relatively low current-to-future consumption ratio, the wealth reallocation effect amplifies the excess supply of current goods resulting from the increase in the future carbon tax. Hence, the interest rate must fall to equilibrate relative demand for and relative supply of current goods. However, if current expenditure shares fall together with the interest rate (i.e., if the income effect dominates the substitution effect of a change in the interest rate), this constitutes a counteracting effect on the excess supply of current goods. Hence, the required decrease in the interest rate to clear the intertemporal goods market is larger than in the case of constant expenditure shares. If the negative effect on current expenditure shares is strong enough, the Green Paradox will be reversed in general equilibrium.

As an example, consider the case of assumption 1 with  * 1 . The current expenditure shares resulting from utility maximization are given by

1 * 1 11 11  *    CC11 **  1(1)rr22 (1)  , 1(1) (1)  , M M *     which depend positively on the interest rate if the income effect dominates the intertemporal substitution effect, i.e., if   1 and * 1 . Figure 3 shows simulation results for different combinations of  ,

  * , and  * . The underlying CES production function is again given by (20). In panel (a)  and

* are fixed at 10, and in panel (b) we impose   0.5 and  *  0.99 . Attenuation occurs in the shaded areas of the two panels, whereas strong reversal of the Green Paradox upon an increase in the future carbon tax occurs in the white areas.7 The figure illustrates that a combination of relatively patient oil importers, a sufficiently high elasticity of marginal utility, together with a sufficiently low elasticity of factor substitution can give rise to reversal of the Green Paradox. The last result of this subsection is summarized in proposition 5.

Proposition 5: Suppose that assumption 1 holds with  * 1 , that there is no investment, and that the future carbon tax is increased. The Green Paradox may then be strongly reversed in general equilibrium.

Figure 3: Attenuation versus reversal of the Green Paradox with a future carbon tax

Panel (a): * 10 Panel (b):   0.5 and  *  0.99

1 1

0.9 0.9

0.8 0.8

0.7 0.7 attenuation 0.6 0.6

 attenuation  0.5 0.5

0.4 0.4

0.3 0.3

0.2 strong reversal 0.2 strong reversal 0.1 0.1 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 2 4 6 8 10 12 14 16 18 20 -* =*

Notes:   0.1 , L  1, and S1  1 . The carbon tax  2 is increased from zero to 0.01. The white area gives the region in which the Green Paradox is strongly reversed. In the shaded area the Green Paradox is attenuated. Along the solid line between the white and the shaded region, the Green Paradox is weakly reversed.

7 It has been checked that all equilibria (including those with strong reversal) are locally stable under a standard Walrasian tâtonnement mechanism as described in appendix A2.

As we have shown in this section that even for the special class of CES utility functions the Green Paradox can be amplified or reversed in general equilibrium, this result will also be true for more general utility functions.

4.2. Effects of a tax on asset holdings of oil exporters

The partial equilibrium analysis of Section 3 has shown that a capital asset tax works in the opposite direction as a carbon tax: oil owners pump more slowly because the real rate of return on their financial assets is curbed by the asset tax. Here, we show that in general equilibrium without investment, this outcome is likely to be attenuated, and cannot be reversed.

With an asset tax  2 in the model and no physical capital, the OME condition (8) becomes

(21) R11()qR 2 (1 r 2 21 ) q 2 S 1.

The consolidated budget conditions (4) and (5) in the absence of initial debt positions become

* CFRqRA222222() (22) CFRqR1111()   M, 11rr22

* **CqR222 (23) CqRM111. 11rr22  22

* The asset tax revenues ( 22A ) are refunded to the consumers in the oil-importing region.. Combining (21) -(23), using (9)-(10), and taking profit and utility maximization into account, the MGME condition becomes

** CrMrq22(, (, 2 1 , 2 )) CrMq 22 (, ()) 1 FS ( 1 Rq 11 ()) (24) ** , CrMrq12(, (, 2 1 , 2 )) C 1 (, rM 2 ()) q 1 FR(())11 q

** where we have used the Hotelling rule and the OME condition to obtain M Mq()111 qS. The OME and MGME conditions (21) and (24) can again be used to determine the general equilibrium. A comparative static analysis with respect to the asset tax is more cumbersome than with respect to the carbon tax, because the MGME line may shift upon a change in  2 . The reason is that M depends on the asset tax. Note that in the case at hand strong reversal would mean an increase in the first-period extraction rate. The definitions of the other possible outcomes should be modified in a similar way.

Proposition 6: Suppose assumption 1 holds with  * 1 , that there is no investment , and that the capital asset tax is increased. Then the partial equilibrium effect is attenuated.

Proof: See Appendix A2. □

The intuition underlying this proposition is as follows. The partial equilibrium effect of a future asset tax is a relative increase in future oil extraction and production, because the lower after-tax interest rate makes conservation of oil relatively more attractive. Abstracting from wealth reallocation effects, in general equilibrium this would imply excess supply of future goods, so that the interest rate must increase to clear the asset market. This increase in the interest rate partly offsets the partial equilibrium effect because it makes conservation of oil relatively less attractive. If we allow for wealth reallocation effects, amplification is still impossible, because this would require reallocation of wealth towards the region with a relatively high future-to-current consumption rate and the resulting demand effect must be strong enough to dominate the supply effect. In other words, oil exporters must both have a relatively high current expenditure share and positive savings, which is impossible in the extreme cases in which a strong perverse demand effect can materialize. Therefore, the interest rate always goes up. A reallocation of wealth towards the region with the lowest relative future consumption would magnify the excess supply resulting from the partial equilibrium effect of an asset tax and therefore would call for an even larger increase in the interest rate. However, the increase in the interest rate cannot be so strong to reverse the partial equilibrium effect. This latter result also survives under general preferences.

Proposition 7: Suppose there is no investment, then general equilibrium effects cannot strongly reverse the partial equilibrium effect of an asset tax.

Proof: See Appendix A2. □

4.3 Summing up

The results so far have shown that the Green Paradox effect associated with the announcement of a future carbon tax is likely to be weakened in general equilibrium if the oil-importing region is relatively more patient than the oil exporting region. Moreover, if in addition current expenditure shares depend positively on the interest rate (due to strong income effects), the Green Paradox might even be strongly reversed in general equilibrium. On the contrary, if oil exporters are relatively more patient, policy makers should be aware of a possible amplification of the Green Paradox. Furthermore, general equilibrium effects tend to weaken the link between asset taxes and the time profile of resource extraction so that the asset tax becomes less useful as an instrument for combating climate change.

5. Investment and the role of capital markets

This section investigates the importance of capital markets for the Green Paradox in general equilibrium. The existence of physical capital allows for nonzero aggregate investment and may therefore affect the results of Section 4 through two channels.8 First, a change in aggregate investment affects the interest rate, and therefore the extraction path through the Hotelling rule, as in the model of Section 4. Second, a change in aggregate investment affects the second-period capital stock and therefore shifts the resource demand function in the second period. We show that amplification of the Green Paradox no longer requires an increase in the interest rate. Moreover, we argue that reversal of the Green Paradox is less likely in the model with physical capital. We again focus on a future rather than a current carbon tax, so that 1  0 .

5.1. The effects of a future carbon tax when there is investment

In this section, we again abstract from the asset tax, so that  2  0 and the second-period net interest rate equals r2 in both countries. The OME condition (8), with  21  0 imposed, becomes:

(25) R11()qRrrq 22 (,(1) 21 2 ) S 1.

It should be noted that second period oil demand decreases if both input prices r2 and (1rq21 )  2 increase. With the use of (9)-(10), the GME locus is now given by

** CrMrq22(, (, 2 1 )) CrMq 22 (, ())(1) 1 Kr 22 (,(1) rq 2 1 2 )) FKr ( 22 (,(1) rq 2 1 2 ),(,(1) Rr 22 rq 2 1 2 )) ** . CrMrq12(, (, 2 1 )) CrM 1 (, 2 ()) q 1 (1 )KFRqKr11122212 ( ( ))  ( , (1  rq ) )) As in the previous section it can be turned into an MGME locus without the tax rate appearing. This is slightly more complicated now. The procedure is as follows. First, we replace Rr22(,(1) rq 2 1 2 ) by

SRq111 (). Second, for every q1 we know SRq111 (), so that, with a little abuse of notation, second- period demand for capital can be written as KrS22(, 1 Rq 11 ()). Third, national wealth levels given by (4) and (5) also contain the first-period interest rate. This interest rate can be written as a function of q1 only,

* because it is determined by FKRqK (,())111 r 1  , with KAA111  given. So, we arrive at

** CrMrq22(,(,;))(,())(1)(,())((,()),()) 2 12 CrM 22 q 1 K 22 rS 1 Rq 11 FKrS 22 1 Rq 11 S 1 Rq 11 (26) ** . CrMrq12(, (, 2 1 ; 2 )) C 1 (, rM 2 ()) q 1 (1 )KFRqKrSRq11122111 ( ( ))  ( ,  ( ))

The changes in the equilibrium prices again depend crucially upon the relative slopes of the OME and MGME curves. In terms of the direction of changes in the equilibrium prices, the different possibilities

8 To have an interesting problem, we assume that capital demand reacts to a change in the rental rate for any given future oil price.

are still fully described by the four panels of Figure 1. Nevertheless, there is an important difference with the analysis in Section 4. An endogenous change in the second-period capital stock shifts the second- period oil demand function. As a result, the movement from E to PE does no longer correspond with the partial equilibrium effect. Inspection of the OME equation leads to the following modification of proposition 2, entailing that amplification does not necessarily imply an increase in the interest rate.

Proposition 8: Strong reversal, weak reversal, and attenuation require a decrease in the interest rate. An increase in the interest rate is a sufficient condition for amplification.

Proof: Strong and weak reversal can only occur if dq1  0 for d 2  0. If dr2  0 then dR2  0 so that dS1120( d R  R )0 , a contradiction. Let (,rq21 ) be the equilibrium prices corresponding to a given tax rate  2 . Assume that an increase in the tax rate to  2 , while keeping r2 fixed, leads to a new equilibrium price qqˆ11 , so that first-period oil demand increases. Suppose we have attenuation. Then the new equilibrium prices, allowing for a change in r2 , are (,rq21 ) with qqqˆ111  . But then it must be the case that rr22 because otherwise we would have excess supply of oil.

Rq11()ˆˆ Rr 22 (,(1) rq 21 2 ) Rq 11 () Rr 22 (,(1) rq 21 2 ). To prove the final part of the proposition we suppose that upon an increase of the carbon tax, the interest rate increases. Then it must be the case that demand for first period oil increases, by more than in the case of a constant interest rate. Hence, we have amplification. □

As we did in the previous section, we continue by discussing several specific examples to develop the intuition between the different possible outcomes. In first instance, we will again restrict ourselves to the case in which the income and substitution effect of changes in the interest rate offset each other, resulting in constant expenditure shares  and  * .

Proposition 9: Suppose assumption 1 holds with  * 1 and that the future carbon tax is increased. Then reversal of the Green Paradox cannot occur.

Proof: See Appendix A1. □

An extension of proposition 9 so as to include statements regarding amplification and attenuation as in

* proposition 4 is not possible, for two reasons. First, if A1  0 in the model with physical capital, the

* condition    is no longer required for obtaining a rise in the interest rate r2 upon an increase of the future carbon tax rate, because the marginal product of capital in the first period, and hence the interest rate r1 , depend on first-period oil use. This establishes an additional channel for the wealth reallocation effect.

Corollary 1: In the model with physical capital,    * is no longer a necessary condition for obtaining an increase in the interest rate.

Proof: See Appendix A1. □

Second, in the model with physical capital, an increase in the interest rate is no longer necessary to obtain an amplification of the Green Paradox. The reason is that current investment shifts the future resource demand function, because capital and the resource are cooperative in production. Hence, if investment goes down upon an increase in the future carbon tax, the Green Paradox may be amplified even without an increase in the interest rate. Consider, for example, the situation in which expenditures shares are

** * constant and identical, i.e. 0.5CM11 / C / M  1. Although in this case the interest rate r2 always goes down upon an increase in  2 , the Green Paradox can still be amplified. Figure 4 shows the case for a CES production function,

 11   1 1 (27) FR() K R (1  ) L , with   1/ 3,   0.1, and L  1. 

* * We have set   0.1, 1 , A1 1, A1  0 , S1  2 (panel (a)) and S1 1 (panel (b)). The carbon tax  2 is increased from zero to 0.01. The shaded area in the figure indicates the combinations of the elasticity of factor substitution  , and the common current expenditure share  , for which the Green Paradox is amplified, despite the decrease in the interest rate.

Figure 4: Attenuation versus amplification under identical homothetic preferences

Panel (a): S1  2 Panel (b): S1 1

0.9 0.9

0.8 0.8

0.7 0.7 attenuation attenuation 0.6 0.6   0.5 0.5

0.4 0.4

0.3 0.3

0.2 amplification 0.2 amplification

0.1 0.1 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 =* =*

* * Notes:   1/3, 1 ,   0.1 ,   0.1 , A1  1 , A1  0 , and L  1. The carbon tax  2 is raised from zero to 0.01. In the shaded area the Green Paradox is amplified. In the white area the Green Paradox is attenuated.

Like in the model without physical capital, a strong reversal of the Green Paradox may occur if current expenditure shares depend positively on the interest rate, which is the case if the elasticities of marginal utility exceed unity. However, once we take investment into account, it is less likely that a strong reversal of the Green Paradox occurs. To see this, consider a strong reversal in the model without physical capital, implying that R2 goes up and r2 goes down. In the model with physical capital, this would induce an increase in investment. The resulting increase in the second-period capital stock has two opposing effects on the Green Paradox. First, relative supply of future consumption goods goes up, which positively affects the equilibrium rate of interest and therefore current extraction through the Hotelling rule, enhancing the Green Paradox. Second, the increase in the second-period capital stock induces an upward shift in the future resource-demand function, working against the Green Paradox. The first effect dominates the second one, so that on balance, a strong reversal of the Green Paradox is less likely if the possibility of investment in physical capital is taken into account. When allowing for investment, the area in Figure 3 for which strong reversal occurs, disappears. However, it is still possible to obtain strong

* reversal in our example with CES utility and CES production. By choosing A1  0 and A1 1 to increase the share of the oil exporting region in global consumption, Figure 5 shows that strong reversal occurs if  * is sufficiently high and the elasticity of factor substitution sufficiently small.

Figure 5: Attenuation versus reversal of the Green Paradox   0.5 and  *  0.999

0.05

0.045

0.04

0.035 attenuation

0.03  0.025

0.02

0.015

0.01 strong reversal

0.005 1 2 3 4 5 6 7 8 9 10 * =

* Notes:   1/3,   1,   0.1 ,   0.1 , A1  0 , A1  1 , L  1, and S1  1. The carbon tax  2 is increased from zero to 0.01. In the white area the Green Paradox is strongly reversed. In the shaded area the Green Paradox is attenuated.

The last result of this subsection is summarized in proposition 10.

Proposition 10: Suppose that assumption 1 holds with  * 1 and that the future carbon tax is increased. The Green Paradox may then be strongly reversed in general equilibrium.

Comparing the results of the models with and without physical capital, we find that the parameter space for which a reversal occurs is smaller in the model with physical capital. Taking physical capital into account also affects the conditions required for amplification of the Green Paradox: in the model with investment amplification no longer requires an increase in the interest rate. A downward shift in the future resource demand function puts upward pressure on the extraction increase in the first period so that the Green Paradox may be amplified even if the interest rate does not increase. This does not imply, however, that the parameter space for which amplification occurs is necessarily larger in the model with physical capital. The reason is that the parameter space for which an increase in the interest rate occurs is smaller in the model with physical capital, because the situation of a decrease in future extraction combined with an increase in the interest rate induces a decrease in investment in the model with physical capital, which lowers relative supply of future goods, thereby putting downward pressure on the interest rate.

5.2. The effects of an asset holding tax when there is investment

Although the imposition of a future asset tax always leads to a decrease in current oil extraction in the model without physical capital (though to a smaller extent than in a partial equilibrium analysis, as shown in Section 4.2), in the model with physical capital the endogenous response of investment opens up the possibility of a reversal of the partial equilibrium effect. Intuitively, an increase in the interest rate does not only speed up extraction through the Hotelling effect, the increase in the interest rate also reduces the demand for capital in the second period. As a result, the second-period resource demand function shifts down, which provides an additional channel for more oil extraction in the first period. As a result, in the full model with physical capital, an asset tax may actually speed up oil extraction instead of slowing it down.

Consider again our example with constant expenditure shares and production function (27). The shaded area in Figure 6 shows the combinations of  and   * for which a reversal of the partial equilibrium effect is obtained, with the asset tax  2 increased from 0 to 0.01. In the white area, the partial equilibrium effect is attenuated.

The reason for reversal occurring in the region where  *  0 is that the imposition of an asset tax implies a reallocation of wealth from the exporter towards the importer. Because future relative to current consumption is smaller for the importer than for the exporter, this reallocation reduces aggregate relative future consumption at a given interest rate. Hence, the excess supply of future goods as a result of the partial equilibrium effect of a future asset tax is increased. Therefore, a relatively large increase in the

interest rate is required to clear the goods market. The results show that the increase may even be so strong that the partial equilibrium effect is reversed.

Figure 6: Attenuation versus reversal of the partial equilibrium effect

0.9

0.8

0.7

0.6 attenuation  0.5

0.4

0.3

0.2 strong reversal 0.1 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -*

* Notes:   1/3,   0.1,   0.1 , A1  1 , A1  0 , L  1, and S1  1. The asset tax  2 is increased from zero to 0.01. The shaded area gives the region in which the partial equilibrium effect is strongly reversed. In the white area the partial equilibrium effect is attenuated.

Proposition 11 summarizes the results of this section.

Proposition 11: In the model with physical capital, general equilibrium effects may strongly reverse the partial equilibrium effect of an asset tax.

5.3 Summing up

Comparing the models with and without investment, our results stress that it is crucial to take the possibility of investment in physical capital into account. Changes in investment shift the future resource demand function; whereas changes in future oil use affect the return to capital. We have shown that strong reversal of the Green Paradox upon an increase in the future carbon tax is less likely than in the model without investment possibilities. Moreover, in the model with investment, the effect of an asset tax can be strongly reversed in general equilibrium, whereas this is not possible without taking investment into account.

6. Exploration costs and partial exhaustion

So far, the only margin for oil producing countries is how fast to extract a given stock of oil reserves. But for climate policy it is crucial to also consider how much of reserves to lock in the crust of the earth forever (e.g., Hoel, 2012; van der Ploeg and Withagen, 2012a; van der Ploeg, 2013). To introduce this

additional margin, we assume that the total recoverable stock of oil S1 depends on exploration investment at the beginning of the first period:

(28) SGI1  (), with G ' 0 , G '' 0 , where I denotes the amount of the final good devoted to exploration.9 Hence, the return on oil exploration falls as less accessible fields have to be explored. Profit maximization gives, in addition to the Hotelling rule (3), a condition for optimal exploration investment giving the initial stock of oil reserves as an increasing function of the initial oil price:

1 ' (29) qG111'( I ) 1 I  G (1 / q )  I ( q ) , I '0 and SSq111 () with Sq11() 0.

The OME condition (8) becomes

(30) R11()qRrr 22 (,(1 2 21 ) q 2 ) Sq 11 (),

where we have imposed 1  0 . In a partial equilibrium, where the interest rate r2 and the capital stocks

K1 and K2 are fixed, an increase in the future carbon tax  2 enhances current oil extraction, because the return to conserving oil becomes lower, just like in the case without exploration costs. In addition, however, cumulative extraction goes down as the current oil price falls so that the return to exploration becomes lower. Increasing the asset tax rate has the opposite effect. On the one hand, current oil extraction falls, because the return to leaving oil in situ increases relative to investing oil revenues on the bonds or capital market. On the other hand, cumulative extraction goes up, as the current oil price goes up so that the return to exploration becomes higher. We have thus established the following proposition.

Proposition 12: In partial equilibrium with a given net interest rate r2 the following holds.

(i) An increase in the future carbon tax  2 increases current oil extraction and decreases cumulative oil extraction.

(ii) An increase in the asset tax  2 decreases current extraction and increases cumulative extraction.

Proof:

(i) Given d 2  0 , suppose that dq1  0 . This would imply that the left-hand-side of (30) goes down,

whereas the right-hand-side goes up. Hence, we need dq1  0 so that dR1  0 and dS1  0 .

(ii) Given d 2  0 , suppose that dq1  0 . This would imply that the left-hand-side of (30) goes up,

whereas the right-hand-side goes down. Hence, we need dq1  0 so that dR1  0 and dS1  0 . □

9 Assuming that total oil reserves in the crust of the earth are given by SS01 , it follows that SS01 units of oil remain untapped in the market equilibrium, because they are not worthwhile to exploit.

Hence, for both instruments the effects on current oil extraction and cumulative oil extraction work in opposite direction. As a result, moving our focus to global warming, there is a trade-off between the speed of emissions and cumulative emissions (cf. van der Ploeg and Withagen, 2012a; van der Ploeg, 2013) when the announcement of a future tax is used for climate policy.

Using the Hotelling rule (3), the OME condition (30), and taking investment in exploration and the dependence of the initial oil stock on the oil price into account, the MGME condition is given by

** CrMrqCrMq12(, (,)) 21 12 (, ()) 1 (1) KFRqKrSqRqIq 1 (()) 11 2211 (,() 11 ())() 1 (31) ** , CrMrq22(, (, 2 1 )) CrMq 22 (, ()) 1 (1)(,()())((,()()),()()) K2211rSq Rq 11 FKrSq 2211 Rq 11 Sq 11 Rq 11 where the wealth levels for the oil-importing and exporting-region are, respectively,

* FK(22 (,),() r q 1 S 11 q R 12 (,)) r q 1 22 A M (1rA11 )  FKRq ( 111 , ( ))  ( r 1  ) K 1   qSq 111 ( ) , 1 r2

** M (1rA11 )  qSq 111 ( )  Iq ( 1 ) .

Depending on the relative slopes of the OME and MGME line, we again can get reversal, attenuation, or amplification of the Green Paradox in general equilibrium. In terms of the direction of changes in the equilibrium prices, the different possibilities are once more described by the three panels of Figure 1. Nevertheless, there is an important difference with the analysis in Section 4 and 5: the general equilibrium consequences for the equilibrium price of oil also affect cumulative resource extraction, so that we get the following result.

Proposition 13: If the partial equilibrium effect on current oil extraction of an increase in one of the two tax rates,  2 or  2 is (i) strongly reversed, so will be the effect on cumulative extraction. (ii) weakly reversed, so will be the effect on cumulative extraction. (iii) attenuated, so will be the effect on cumulative extraction. (iv) amplified, so will be the effect on cumulative extraction.

Proof: In case of strong reversal, q1 changes in the opposite direction, compared to the partial equilibrium outcome. In case of weak reversal, q1 remains constant. In case of attenuation, q1 changes in the same direction, but by less than in partial equilibrium. In case of amplification, q1 changes in the same direction, but by more than in partial equilibrium. Because S1 depends positively on q1 , the same qualifications hold for cumulative extraction. □

As an example, consider the case of attenuation of the Green Paradox upon an increase in the future carbon tax, corresponding with panel (c) of Figure 1. The first-period oil price drops by less than in partial equilibrium. As a result, cumulative extraction goes down by less than in partial equilibrium, so that general equilibrium effects weaken both the increase in current oil extraction and the decrease in cumulative oil extraction. Similarly, if the Green Paradox is strongly reversed, as in panel (a) of Figure 1, cumulative oil extraction goes up upon the announcement of a future carbon tax.

Given that the effect on cumulative extraction works in opposite direction as the change in current extraction, proposition 13 implies that that the general equilibrium consequences of future taxes on global warming are (partly) offset by the general equilibrium consequences of future taxes on cumulative extraction.

7. Conclusion

The Green Paradox implies that a climate policy that becomes more ambitious during the next decades can be regarded as an announced expropriation of oil reserves. Decreased future demand for oil forces oil producers to accelerate oil extraction and carbon emissions and thus to exacerbate the problem of global warming. Since a gradual tightening of climate policy exerts a stronger downward pressure on future prices than on current ones, it curbs the expected rate of capital gains on oil reserves. Oil producers will try to avert this by extracting more quickly and putting the sales revenue into investments in the capital markets, which offer higher yields. Hence, a climate policy which becomes more aggressive over time accelerates carbon emissions. Most of the discussion of the Green Paradox has been cast in a partial equilibrium framework and has taken the interest rate and the resource demand function as given. Since the interest rate is the key intertemporal price driving saving and investment decisions as well as oil depletion decisions and future oil demand shifts upon changes in the future capital stock this seems a serious shortcoming.

Our main point is thus that the effects on bonds and capital markets and the fact that carbon taxes and oil prices are set in different jurisdictions require a thorough re-thinking of the Green Paradox. Reversal of the Green Paradox is impossible if preferences of oil importers and oil producers are homothetic and identical or, equivalently, if the world consists of a single jurisdiction with homothetic preferences. The reason is that in this case wealth (reallocation) effects do not play a role, so that global current consumption demand relative to global future consumption demand only depends negatively on the interest rate. Hence, in such a setting the partial equilibrium Green Paradox effect is attenuated.

However, when oil producers and oil importers have different preferences, the Green Paradox may be reversed, even with homothetic preferences in both jurisdictions. The reason is that environmental policy induces a wealth reallocation effect between the oil producers and the oil importers. If the degree of impatience, in terms of the pure rate of time preference, differs between the two jurisdictions, the reallocation of wealth affects current expenditure shares at a given interest rate. If oil exporters are relatively impatient, the wealth reallocation effect amplifies the excess supply of savings resulting from the increase in the future carbon tax. Hence, the interest rate must decrease further than under identical preferences to clear the bonds and capital market. If, in addition, current expenditure shares fall together with the interest rate (i.e., if the income effect dominates the substitution effect of changes in the interest rate), the required decrease in the interest rate to clear the intertemporal goods market is even larger, so that the Green Paradox may be reversed in general equilibrium. We should stress here that, although we have demonstrated that reversal can occur in our example with CES production and utility, in order to do so we had to impose rather stringent conditions, especially in the model with capital accumulation: an elasticity of factor substitution close to zero and an extreme difference in the pure rate of time preference between the two regions.

Amplification of the Green Paradox either requires an increase in the interest rate (which induces oil producers to extract faster, according to the Hotelling rule), or a downward shift in the future resource demand function. An increase in the interest rate may occur if oil exporters are relatively patient, so that, at the prevailing interest rate, the wealth reallocation effect of a carbon tax increases current expenditure shares. A downward shift in the future resource demand function materializes if investment drops because of the carbon tax, as oil and capital are cooperative factors in production.

A withholding tax on the capital gains on the financial investments of oil producers or a tax on asset holdings of oil producers may be a more fruitful way to slow down oil extraction and carbon emissions. This certainly holds true in our trade model with a fixed interest rate. In the general equilibrium setting without man-made capital and constant expenditure shares we still get this result, but it is less pronounced than in the partial equilibrium setting, because the interest rate will increase. However, when taking capital accumulation into account, introducing an asset tax to combat climate change might work counterproductive in general equilibrium, because the induced increase in the interest rate negatively affects investment so that the future resource demand function shifts downwards, encouraging oil producers to supply more oil initially.

Finally, we have studied the case of partial exhaustion in the presence of exploration costs. In practice, exploration costs rise as less accessible fields are explored and then it is seldom optimal to fully exhaust all oil reserves. An effective climate policy must therefore focus on the supply side of the carbon market

as well as on the demand side because it is crucial that not all oil that is in the crust of the earth is burned. We show that in partial equilibrium, a carbon tax will reduce cumulative emissions, whereas an asset tax has the opposite result. In general equilibrium, however, this effect may again be amplified, mitigated, or even reversed.

Our analysis has demonstrated that general equilibrium effects may amplify or reverse the effect of carbon and asset taxes on current oil use, but also the effect on cumulative carbon emissions. Therefore, our results call for a careful general equilibrium approach to climate change policy.10 A first complicating factor, which is subject to research, is that the model discussed in this paper abstracts from strategic behavior in the design of climate policies. Another qualifying comment is that Green Paradox effects require that oil is scarce and generates substantial Hotelling rents. The recent discovery of huge shale gas and other unconventional energy reserves makes fossil fuel a lot less scarce and thus Green Paradox effects will become less important.

References Dixit, A. (1981). A model of trade in oil and capital, Discussion Papers in Economics No. 16, Princeton University. Djajić, S. (1988). A Model of Trade in Exhaustible Resources, International Economic Review. 29(1): 87- 103. Eichner, Th. and R. Pehtig (2011). Carbon leakage, the Green Paradox, and perfect future markets. International Economic Review. 52(3): 767-804. Eichner, Th. and R. Pethig (2013). Flattening the carbon extraction path in unilateral cost-effective action, Journal of Environmental Economics and Management, 66(2): 185-201. Gerlagh, R. (2011). Too much oil. CESifo Economic Studies. 57(1): 79-102. Golosov, M., J. Hassler, P. Krusell and A. Tsyvinski (2013). Optimal taxes on fossil fuel in general equilibrium, Econometrica, forthcoming. Grafton, R.Q., T. Kompas and N.V. Long (2013). Substitution between biofuels and fossil fuels: Is there a green paradox?. Journal of Environmental Economics and Management, 64(3): 328-341. Hoel, M. (2012). Carbon taxes and the Green Paradox, chapter 11 in R.W. Hahn and A. Ulph (eds.), Climate Change and Common Sense: Essays in Honor of Tom Schelling. Oxford: Oxford University Press.

10 Our results have been derived for non-marginal changes in carbon taxes and the tax on asset holdings. A more conventional, but less general, approach would have been to derive the marginal effects around an outcome with zero carbon and asset holding taxes. Appendix A2 shows that this confirms the more general results of our paper.

Jaakkola, N. (2012). Can we save the planet by taxing OPEC capital wealth?, chapter 3, PhD thesis, University of Oxford. Long, N. van and Stähler, F. (2013). Resource Extraction and Backstop Technologies in General Equilibrium. In K. Pittel, F. van der Ploeg and C. Withagen (Eds.), Climate Policy and Exhaustible Resources – The Green Paradox and Beyond, Ch. 5. MIT Press, Cambridge, MA Marion, N.P. and L.E.O. Svensson (1984). World equilibrium with oil price increases: an intertemporal analysis, Oxford Economic Papers, 36(1): 86-102. Michielsen, T.O. (2012). Brown backstops versus the Green Paradox, mimeo, Tilburg University. Ploeg, F. van der (2013). Cumulative carbon emissions and the Green Paradox, Annual Review of Resource Economics, 5: 281-300. Ploeg, F. van der and C. Withagen (2012a). Is there really a Green Paradox? Journal of Environmental Economics and Management. 64(3): 342-363. Ploeg, F. van der and C. Withagen (2012b). Growth, renewables and the optimal carbon tax. International Economic Review. 55(1): 283-311. Sinn, H.W. (2008). Public policies against global warming. International Tax and Public Finance. 15(4): 360-394. Sinn, H.W. (2012). The Green Paradox. Cambridge, Mass.: MIT Press. Wijnbergen, S. van (1985). Taxation of international capital flows, the intertemporal terms of trade and the real price of oil, Oxford Economic Papers, 37(3): 382-390.

Appendix

A1. Proofs of Propositions

Proof of Proposition 6: We first prove that strong reversal cannot occur. Strong reversal requires

** dr2  0, since dq1  0 and dr((12212 ) q  )  0 . Hence, dC111 d q S  0 and

** * * dC22211 d(1 )(1  r ) q S  0. But dC()011 C  because dF()0 R1  , whereas dC()022 C .

Hence, dC1  0 and dC2  0. But dC(/)21 C d (1)(1)/0 r 2    because dr2  0 . This gives a contradiction.

To prove that weak reversal cannot occur, we show that weak reversal occurs if and only if 00*  or 11* , whereas both   0 and   1 are outside the admissible range.

* Suppose   0 so that C1  0 . Then CqSFRq11111(()). This uniquely determines q1 independent of

r2 . An increase of the asset tax rate only leads to a pro tanto increase in the interest rate r2 . It has no impact on extraction in any period. So, the partial equilibrium effect is weakly reversed. Similarly, if

 1, we get C2  0 . Hence,

* CFRrq2( 2 ((1  2212 )  ))  (1 r 22111222212  ) qRqqRrq ( )  ((1   )  ))  (1 r 2211  ) qS. This uniquely determines (1rq221 ) for any given carbon tax. Moreover, in equilibrium, R1 and, consequently, R2 are uniquely determined, irrespective of the capital asset tax. Hence, we have weak reversal again.

* Necessity. A weak Green Paradox requires dR1  0 and hence dq1  0 and dM  0 . Moreover,

** dr((12212 ) q  )  0 , implying dr22  d 0 . Hence, both dC111 d q S  0 and

** ** dC22211 d(1 )(1  r ) q S  0. Because dC()()011 C dC 2 C 2 , we also have dC12 dC 0 .

Suppose 0 1. Because CM1   , dC1  0 requires dM  0 . But, CrM22 (1 )(1 ) , so that dC2  0 . Hence, we have a contradiction.

(iii) Given d 2  0 and d 2  0 , it is immediate from (21) that amplification requires dq1  0 and

** * dr((12212 ) q  )  0 . Hence dC111 S dq 0 and dC22211 d(1 r ) q S  0 . But

* * dC()()011 C dFR 1 , whereas dC()022 C  . Therefore we need dC1   dM  0 and dC22(1 ) d (1  r ) M  0 . It follows that dr2 >0 , implying that R1 goes down by less than in partial equilibrium, so we obtain a contradiction. □

Proof of Proposition 7: First, note that the relationship between  2 and q1 is monotonic. This can be seen as follows. Suppose two capital tax rates yield the same first period oil price q1 : dq1  0 . Then also dr(122 )  0 , because R11()qRrr 22 (,(1 2 21 ) q 2 ) S 1. Hence, since wealth and prices are

** unaffected, dC12 dC 0 . Consequently, since dF() R12 dF ()0 R  by assumption, also dC12 dC 0 However, dr(12 ) 0 and therefore, generically, also optimal consumption. A contradiction.

Second, we show that dq1  0 as d 2  0 . In a general equilibrium qq11 defined by F '(Sq11 )  , because otherwise there is excess demand on the oil market. We have q1  0 due to the Inada condition

F '( ) 0 . Also (1r22 ) is finite in equilibrium. If it would be unbounded from above, demand for second period oil and, hence, second period final production can be made arbitrarily small (from

F '(0) ). But R2 would be without bound generically (from the Inada condition on the utility function),

resulting in excess demand. Moreover, second-period oil demand is only defined for a positive oil price, so that (1rq2212 )  cannot become negative. Finiteness of (1 r22 ) implies that r2 goes to

infinity as  2 goes to infinity. For large  2 first and second period demand for final goods by the oil exporters is finite, and so must be first and second period production. But with the relative price of second

period consumption for the oil importer going to zero ( (1 r2 ) ), second period demand for final goods goes to infinity. Hence we have excess demand, which contradicts the existence of general equilibrium. □

Proof of Corollary 1: The analogue of the argument used in the proof of proposition 4 would run as follows. We have

FK(,)(22 R r 2  ) K 2 M {(1rA11 )  FKR ( 11 , )  ( r 1 ) K 1   qS 11 } 1 r2

** M {(1rA11 )  qS 11 }

and

*** CC11(1 ) KFKSK 1  ( 112 , )  M  M

This yields the following expression

(1 ){FK (11 , R ) (1 ) K 1 }

(32) ** F(,)(KR22 r 2 ) K 2 ( )((1rA11 ) qS 11 ) K 2  1 r2

An increase of the tax rate yields a Green Paradox (proposition 9), so that the left hand side of (32) increases. And if the second-period interest rate would go up (as was the case with amplification in the

* model without capital), the second and third term of the right hand side go down. Then, with A1  0 , it

* ** would be necessary to have    . However, M {(1rA11 )  qS 11 } could go up now because of the increase of the interest rate of the first period, which is necessary to meet capital supply, in view of the lower first-period oil price. So, even if the second-period interest rate increases, amplification does not require    * . □

The intuition behind Corollary 1 is that it is not a priori clear what happens to the exporter’s national wealth upon an increase in the second-period tax. Without man-made capital a higher tax leads to a higher first-period price of oil, and thereby to a larger national wealth of the oil exporter. However, now also the rate of return on first-period capital plays a role.

Proof of Proposition 9: We first show that weak reversal cannot occur. Weak reversal of the Green

Paradox requires dR1  0 and dR2  0 (from (25)), therefore dr1  0 and dq1  0 (from (1)), and

**** dC111 dM dq S 0 . Furthermore, note that Rrq22(, 1 (1) r 2 2 ) decreases if r2 and

qr122(1 )  increase. Hence, the constancy of R2 and q1 imply that dr2  0 , so that dK2  0 (from

* * (2)). Consequently, (9) implies that dC()011 C  . Given that dC1  0 , this requires dC1  0 . However, we also have CM1111111222221(1 rAFKRr )  ( , )  (  ) KFKRr [ ( , )  (  ) K ] / (1  rqS )  .

Note that dr1  0 , due to dq1  0. The change in the bracketed term is given by

[(,)(FK KR22 r 2 )] dKKdr 2  22  Kdr 22, where the equality uses (2). Therefore, dC1  0 , so that we get a contradiction.

A consequence of the absence of weak reversal is that the use of first-period oil is monotonic in the second-period tax rate. In order to exclude strong reversal we therefore only need to show that for the tax large enough there exists an equilibrium with second-period oil use close to zero, implying first-period oil use close to the total available stock of oil. Hence, first-period oil use is increasing in the tax rate, and no

* strong reversal occurs. Define q1 by FR (,)AASq1111  . Hence, it is the first-period world market oil price such that the entire stock of oil is demanded in the first period, with full employment of all capital.

* * Define r1 by FAK (,)1111 AS r  Suppose r2    0 . Then, with KAA111  , the equilibrium in this economy can be characterized as follows:

* CC11(1 ) KFKSK 1  ( 112 , ) ,

* CC22(1 ) K 2.

Moreover

M {(1rA11 )  FKS ( 11 , )  ( r 1 ) K 1  qS 11 } ,

** M {(1rA11 )  qS 11 }.

We also have

CC***(1 ) K  (1 ) M  (1 ) M  (*) 22 2 * (1 ){(1rA11 ) FKS (,)( 11  r 1 ) K 1 qS 11 }(1 ){(1 rA 11 ) qS 11 }.

From this final condition we can solve for K2 . We now have a set of prices and allocations. It is claimed that this set constitutes the limit of an equilibrium of an economy for the second-period carbon tax going

to infinity. Given these prices consumers maximize their utility, subject to their budget constraints. First- period profits are maximized. Demand for final goods as well as for oil equal supply in both periods. The difficulty lies in second-period profit maximization and demand for second-period capital in production.

Profits can be written as F(,)(KR22 r 2 ) K 2  ((1) rq 21  22 ) R. Given the solution K2 of (*) the optimal oil input goes to zero as  2 goes to infinity. Moreover, given zero oil input, the capital stock K2 maximizes profits. This establishes the absence of a strong reversal. □

A2. Tâtonnement and comparative statics around an equilibrium with zero taxes

This appendix discusses the local Walrasian stability of the different equilibria and derives the comparative statics around an equilibrium outcome with zero taxes. The tâtonnement versions of the OME and GME conditions, respectively,

 qRqRrrq111122() ,(1 2212 ) S 1 ,

rCrMrqCrMqKrRrrq  (, (, ))** (, ()) (, ,(1 ) ) FKRqK ( , ())(1) , 22122212 1 22222212 111 1 where dots above variables denote changes and the wealth levels for both regions are given by

M1 rRq111 [()] A 1  FRq [()] 11  rRq 111 [()]   K 1  qS 11

FKrRrrq2222[ , ,(1 2 21 )  2 ], Rrrq 22 ,(1  2  21 )  2  1 r2 * (rKrRrr22222221222211222212 ) ( , ,(1 ) q ) ArqqRrr ( , )  ,(1 ) q  , 1 r2

**q2 * M 1[()]rRq111 A 1  qR 11  R 2  1[()] rRq 111 A 1  qS 11 . 1r22

Standard Walrasian tâtonnement implies that the auctioneer would raise oil prices if demand for oil exceeds the supply of oil, hence 1 > 0. It also implies that the auctioneer would raise the current final goods price which corresponds to a fall in the future final goods price and a rise in r2 if current demand for final goods exceeds current supply of final goods, hence 2 > 0.

Linearizing this system around a steady state with  22  0 we get the following system:

qdqd11211qr   11     ,        rdrd22222qr  22    with the stationary state of this system giving the following steady-state solution

  dqqq d  d , drrr d  d , 122|| || 222|| ||

where | |qr  rq , and we have defined the following coefficients and elasticities:

*  Rr Rr  CY12() r 2 K 2 K 2 222 qR  C 1 C 1 C 12  qR 22 22 K KR 2 2   r 22 22K , MrrrrMrrrrr1(1) 1 1  1 r2 22 22 222 22 *  *  CC11  C 1   CC 11* rR r 1  KR K 1  r 2 q SR111  1122 RR   A 111  2 22 R, * * MM  M   MM q12R r  

2 KR KR CR122 22 K 2 1 CK1222*    ,  22qR 2 C 2  , Mr1 22 q 11rM22   r 

Rr  R qR  RR 0  22qR 0,   22 0   12 2 0, q 11 2 2 , r  12  ,  rr22 1 1 r2 1 r2

RCYr() K K  C C*   K  22 1 2 2 2 2 1 1 2 2 qr   r 2      , 11rM1 r  r  r  r r  222222   

**   22RCC 1 1   CC 1 1* rR r 1  KR K 1  r 2 rqq     SR111      A 111     211  R  , **  1rMM2     MM q12R r 

()rK  qR    ,    , ||    ,  22(0,1),  22 (0,1), qrvrrqq  qr rq K Y R Y

Rq11() q 1 Rrq22(, 2 ) q 2 Rr Rrq22(, 2 ) r 2  rR rKR111(,) R 1 1  0,  2  0,  2  0, 1   0, qR11 qR22 rR22 Rr11

KrRr22(, 2 ) 2  KR KrR22(, 2 ) R 2 2  0, 2 0. rK22 RK22

The terms in curly brackets drop out if there is no investment, the terms in square brackets drop out if expenditure shares are constant (i.e., if the elasticity of intertemporal substitution (EIS) equals unity in both regions), and the terms in double square brackets drop out if preferences are identical and homothetic. We discuss comparative statics induced by a future carbon tax in four different cases.11 Before we do this it is helpful to use correspondence principle and make an assumption which ensures stability of the tâtonnement mechanism. Stability requires that the Jacobian matrix of the dynamic system for the tâtonnement process, i.e.,

11qr J  , 22qr  has two eigenvalues with negative real parts. This requires det(J ) 12 rq   qr  0 or | | 0 and trace(J )12qr   0 .

11 Results for the future asset tax are available from the authors upon request.

Assumption A1: ||0 12qr   0.

Identical homothetic preferences, EIS=1 and no investment: Terms in curly, square and double square

RCY() r K K  RR brackets drop out, so 22 1 2 2 2  2 0 ,  112 2 0 . Therefore, a q 11rM2  r r 1 r 221 r2 2 future carbon tax pushes down the interest rate and the Green Paradox is attenuated.

Heterogeneous homothetic preferences, IES=1 and no investment: Terms in curly and square brackets

* 22RCY 1 2() r 2 K 2 K 2  22RCC 1 1 drop out, so   0 ,  SR  . We q 11rM2  r r 1rMM * 111 221 r2 2 

12 thus see that r can become negative if oil exporters are much more patient than oil importers. In that case, a future carbon tax pushes up the interest rate and the Green Paradox is amplified.

Heterogeneous homothetic preferences and no investment: Terms in curly brackets drop out, so

RCYr() K K  C C*   RCC* 22 1 2 2 2 2 1 1 22 1 1 q      , r  SR111  . 11rM2  r  r  r 1rMM * 22221 r2   2 

Hence, q can become negative if the IES < 1 and importers are not too impatient. In that case, the

13 Green Paradox is strongly reversed. Under these conditions,  r might become negative. However, as long as 0 | | we can always find a pair of 1  0 and 2  0 such that the tâtonnement process remains stable (see assumption A1).

Investment: The terms in curly brackets appear. Investment makes it more likely for q to be positive. Hence, strong reversal is less likely with endogenous investment.

12 As in the main text, we use the first period expenditure share, CM1 /  , as a measure of impatience. 13 We interpret IES here as a weighted average of the intertemporal elasticities of substitution in both regions, such * that IES<1 if Cr12//0 C 1 r 2 .