The Supply Side of CO₂ with Country Heterogeneity

Abstract

Several recent articles have analyzed climate policy, paying explicit attention to the fact that carbon resources are non-renewable. In most of this body of literature, the economy is treated as a single unit and, in the context of climate policy, it seems reasonable to interpret this as the whole world. However, carbon taxes and other climate policies differ substantially across countries. With such heterogeneity, the effects on the emission paths of changes in taxes, costs, and subsidies might be very different from what is found for a hypothetical world of identical countries.

I. Introduction

About three-quarters of carbon emissions are caused by the combustion of fossil fuels. Therefore, policies for reducing carbon emissions must, to a large extent, be policies that affect fossil-fuel markets. In much of the policy discussion and some of the academic literature, it is assumed, usually implicitly, that the producer prices of fossil fuels are unaffected by policies directed toward these markets. As shown already by Bohm (1993), in order to study the consequences of climate policies, it might be important to endogenize fuel prices by including the supply side of fossil-fuel markets. While Bohm’s analysis did not explicitly include the dynamic features of the supply side of fossil-fuel markets, an early contribution on such dynamic features was given by Sinclair (1992). Sinclair pointed out that “the key decision of those lucky enough to own oil wells is not so much how much to produce as when to extract it.” Since then, a considerable number of authors have discussed optimal climate policy, with explicit attention given to the fact that carbon resources are non-renewable. These authors either assume a constraint on the amount of carbon in the atmosphere (Chakravorty et al., 2006, 2008, 2011) or explicitly include a climate cost function in the analysis (Ulph and Ulph, 1992; Withagen, 1994; Tahvonen, 1995; Farzin and Tahvonen, 1996; Hoel and Kverndokk, 1996). One of the insights in this body of literature is that the principles for setting an optimal carbon tax (or the price of carbon quotas) are the same as when the limited availability of carbon resources is ignored. At any time, the optimal price of carbon emissions should be equal to the present value of all future climate costs caused by present emissions; this is often called the social cost of carbon.

During the last couple of years, there has been a renewed interest in analyzing climate policy, paying explicit attention to the fact that carbon resources are non-renewable. Much of this later body of literature discusses the so-called “green paradox”, a term introduced by Sinn (2008a,b). Sinn argues that some designs of climate policy, intended to mitigate carbon emissions, might actually increase carbon emissions, at least in the short run. Sinn’s point is that if, for example, a carbon tax rises sufficiently rapidly, profit-maximizing resource owners will bring forward the extraction of their resources. Hence, in the absence of carbon capture and storage (CCS), carbon emissions increase.¹ A thorough analysis of the effects of taxation on resource extraction has been given by Long and Sinn (1985), but without explicitly discussing climate effects. More recently, Hoel (2011, 2012) has studied the relationship between carbon taxes and carbon extraction, emphasizing the fact that governments, in practice, cannot commit to future tax rates.²

A rapidly increasing carbon tax is not the only possible cause of a green paradox. A declining price of a substitute, either because of increasing subsidies or because of technological improvement, can give the same effect (see, for example, Strand, 2007; Grafton et al., 2010; van der Ploeg and Withagen, 2010; Gerlagh, 2011).

As mentioned above, Sinn used the term “green paradox” to describe a situation where policies intending to mitigate climate change actually increase near-term emissions. Gerlagh (2011) uses the term “weak green paradox” for such a phenomenon, whereas he uses the term “strong green paradox” to describe a situation where policies intending to mitigate climate change increase total climate costs. This distinction is important because total climate costs depend not only on near-term emissions, but also on all future emissions. It is therefore possible to imagine policies that increase near-term emissions, but that nevertheless reduce future emissions so much that total climate costs decline.

In almost all of the body of literature referred to above, the economy analyzed is a single unit; in the context of climate policy, it seems reasonable to interpret this as the whole world.³ Whether they are optimal or not, policies are thus implicitly assumed to be the same throughout the world. This is in sharp contrast to reality. Carbon taxes and other climate policies differ substantially across countries. In most countries, there are no carbon taxes and not much other climate policy. Many countries actually have quite large explicit or implicit subsidies of fossil fuels.⁴ In contrast, several states in the US and all EU countries have various types of climate policies. In the EU, there is a quota system covering a considerable amount of carbon emissions. From 2013, this will be widened further. The quota price today is about 15 euros per tonne of CO. Several European countries also have carbon taxes for the parts of the economy not covered by the quota system. For instance, Sweden has a carbon tax of about 115 euros per tonne of CO, and Norway has a carbon tax varying from 15 to 50 euros per tonne of CO for a considerable part of the economy not covered by the EU quota system. Moreover, many European countries also have other climate policies that supplement the quota system or the carbon tax, such as subsidies to renewable substitutes for fossil fuels.

With this motivation, in this paper a simple two-country economy is considered, where countries differ with respect to their climate policies.⁵ The differences between countries can be either in carbon taxes (Sections III–VI) or in subsidies (Section VI). There is a given initial stock of a homogeneous carbon resource with a constant unit extraction cost, set to zero for simplicity. There is also a perfect substitute for the resource, supplied competitively at a constant unit cost. The producer price of the carbon resource increases at the rate of interest in accordance with the Hotelling rule. In each country, the carbon resource is the only energy source as long as the consumer price of the resource is lower than the price of the substitute. The substitute is the only energy source once the consumer price has reached the price of the substitute.

Section II gives a brief discussion of climate costs, and shows that for a specific set of assumptions the social cost of carbon will be constant over time. While this feature is used in the formal analysis, most results will also hold under the much weaker assumption that the social cost of carbon increases at a rate less than the rate of interest.

Section III presents the basic model for the carbon resource, and Section IV analyzes the effects of increased carbon taxes. Whatever their level, carbon tax rates are assumed to be constant. In a world of homogeneous countries, an increase in a common tax rate would move resource extraction form the present to the future, and hence reduce climate costs. With heterogeneous countries, the effects of increased taxes are not so simple. In particular, it is found that if the carbon tax is raised in the country that initially has the lowest tax rate, resource extraction might be speeded up, implying increased climate costs. Thus, in this case, there is a strong green paradox.

Section V analyzes the effects of a reduction in the cost of the renewable substitute. In a world of homogeneous countries, such a cost reduction will move resource extraction from the future to the present (see Gerlagh, 2011). Hence, there is a weak and a strong green paradox in this case. With heterogeneous countries the effects of lower costs for the renewable substitute are not so simple. There is also a weak green paradox in this case. However, if the carbon tax rates differ sufficiently between countries and demand is sufficiently price inelastic, the total climate costs might decline as a consequence of the cost reduction. Hence, there is no strong green paradox in this case.

Finally, Section VI analyzes the consequences of subsidizing the renewable substitute. With a common subsidy, the effect on climate costs of an increased subsidy is the same as the effect of a cost reduction, while the effects on social welfare differ between the two cases. If countries initially have different subsidies, the effects of increasing a subsidy in one of the countries are different from a cost reduction affecting both countries. In particular, increasing a subsidy always gives a weak green paradox, and also a strong green paradox if the subsidy is increased in the country that initially has the lowest subsidy.

II. Climate Costs and Carbon Resource Extraction

In the subsequent analysis, it is assumed that the total amount of carbon resources is given, and that all of this carbon will eventually be extracted and thus emitted into the atmosphere. Thus, the total emissions over all future years are given. In spite of this, the profile of the carbon extraction is important when looking at the climate. A rapid increase of carbon in the atmosphere will gradually decline over time, as it is transferred to other sinks. However, a significant portion (about 25 percent, according to Archer, 2005, for example) remains in the atmosphere forever (or at least for thousands of years). If a fixed amount of carbon, denoted by , is extracted over any time period, this will therefore give a long-run increase of about  in the atmosphere. With a sufficiently slow rate of carbon extraction, carbon in the atmosphere will grow gradually and monotonically until its long-run level  is reached. This is illustrated by curve A in Figure 1, where  is the amount of carbon in the atmosphere at our initial date 0 (so ).⁶ Clearly, such a development of carbon in the atmosphere will be associated with a gradually changing climate. With a higher rate of extraction, the carbon in the atmosphere will increase more rapidly and will overshoot its long-run value , as shown by curve B in Figure 1. This will give a considerably faster climate change, probably with temperatures above those corresponding to the extraction path for several centuries. It is reasonable to expect that the climate costs associated with the rapid extraction path are much higher than the climate costs associated with the climate development associated with the slow extraction path, even if discounting is ignored. This seems particularly likely if some effects of climate change are irreversible, and if the speed of climate change is also an important consideration.⁷

Figure 1. Carbon in the atmosphere for alternative emission paths

To capture the ideas expressed above, let climate costs at time  be an increasing and convex function of the stock of carbon in the atmosphere above the pre-industrial level, denoted by . Moreover, following Farzin and Tahvonen (1996), let us artificially split  into two components : component 1 remains in the atmosphere forever, and component 2 gradually depreciates at a rate . For each unit emitted, the share that remains in the atmosphere forever is denoted by . The amount of one unit of carbon emissions at time  remaining in the atmosphere at  is thus . If, for example,  and , 45 percent of the original emissions will remain in the atmosphere after 100 years, while 27 percent still remains after 300 years. These numbers are roughly in line with what is suggested by Archer (2005) and others.

Consider next the climate damage caused by one unit of emissions at time . The total additional damage caused by one unit of carbon emissions at time  is the sum of additional damages at all dates from  to infinity caused by the additional stocks from  to infinity. To get from additional stocks at  to additional damages at , it is necessary to multiply the additional stocks at  by the marginal damage at , which is . The marginal damage of one additional unit of emissions at , often denoted the social cost of carbon, is thus given by

• (1)

For , the social cost of carbon will vary over time. While  is increasing as long as emissions are positive,  might be declining for sufficiently low emissions. In any case, , and hence , will change over time.⁸ To simplify the formal analysis, I assume that  (i.e., that damages are linear in the atmospheric stock). When  is constant, equation (1) can be rewritten as

• (2)

which is constant over time. While a constant  simplifies the welfare analysis, most of the subsequent results also remain valid under the much weaker assumption that the present value  is declining over time.

In an optimal world, all countries would have a carbon tax equal to . However, there are many reasons why the actual carbon tax rates will be below . The most obvious reason is that there is little or no international cooperation on climate policy. With  identical countries, the non-cooperative outcome would be for each country to set its carbon tax equal to , provided each country acts individually rational, as is often assumed by economists. In addition, there are also various distributional and other policy reasons why actual taxes differ from their optimal values. These other factors can vary considerably across countries, implying carbon tax rates that differ substantially across countries. It is beyond the scope of this paper to discuss the reasons for such tax differences in more detail. I simply assume that tax rates differ across countries, as they do in the real world.

III. Market for Fossil Fuels

The market for fossil fuels is modeled as a market for an homogeneous non-renewable carbon resource, given in fixed supply and with no extraction costs. The resource is supplied by competitive owners of the resource, and the equilibrium producer price  therefore rises at the interest rate  as long as there are any remaining reserves.

The demand for carbon is given as the sum of demand from two countries. There is a perfect substitute for the carbon resource, supplied competitively at its unit cost . The countries have identical gross utility functions, depending on the sum of the use of carbon and the substitute, , where  and  are the use of carbon and the substitute, respectively. The corresponding demand function is , satisfying , where  is the consumer price of the resource or substitute. As long as , consumers will consume the resource, but will switch to the substitute when . The producer price of the carbon resource develops according to the Hotelling rule, and is thus . The two countries have exogenous and constant carbon taxes  and , respectively, with . The consumer price in country  is hence  until this price reaches .

To sum up, the demand for the resource and substitute in the two countries is (for )

• (3)

• (4)

with  determined by

• (5)

Finally, for a given initial resource stock , the extraction paths must satisfy the equilibrium condition

• (6)

Equations (3)–(6) determine the resource extraction paths for any given values of the exogenous variables , , and . In the following sections, it is shown how changes in these variables affect the outcome. The welfare effects of such changes are also discussed.

IV. Increased Carbon Taxes

In the Appendix, it is shown that differentiation of equations (3)–(6) with respect to (w.r.t.)  gives (for  and )

• (7)

• (8)

• (9)

where

• (10)

• (11)

• (12)

It is useful to first consider the case in which both countries initially have the same tax, implying . For this case, it follows that

• 

Hence, an increase in the common tax rate will extend the period of extraction. This result is well known from the theory of non-renewable resources. An increased constant tax rate will make the consumer price path flatter, and therefore the extraction path will also be made flatter. Extraction is therefore postponed in time as a consequence of such a tax increase. This, in turn, reduces climate costs when the present value of the social cost of carbon declines over time.

Consider next the case of increasing the tax in one country, say country , when  initially, implying . It follows from equation (7) that any tax increase will reduce the resource rent. Moreover, it follows from equation (8) that a tax increase in one country will always increase the extraction period in the other country. The reason for this is that the tax increase lowers the time path of the producer price. For country  (which is not increasing its tax), it therefore now takes a longer time for the consumer price to move from  to , when the country switches from the resource to the substitute. Therefore, the total resource use in country  also increases, leaving less total resource use to country , which increases its tax. This tends to make the extraction period in country  decrease. However, the fact that country  has increased its tax has the opposite effect. With a higher tax, the consumer price path is flattened, which tends to move resource use from the present to the future. The net effect on  of an increase in  is hence ambiguous, as confirmed by equation (9).

If , a tax increase in one country has a similar effect as an increase in a common tax rate (i.e., the extraction period increases in both countries). Hence, the total period of extraction increases. Unless the new and old extraction paths intersect more than once, there is an unambiguous reduction in climate costs when the present value of the social cost of carbon declines over time.

The case of  is more interesting. Assume first that the high-tax country increases its tax (i.e.,  is increased). For ,  decreases while  increases. Because  initially, the total period of extraction increases, as it did in the case of an increased common tax rate. If, instead, the low-tax country increases its tax,  will increase and  will decline if . In this case, the total extraction period is shortened, which tends to increase climate costs if the present value of the social cost of carbon declines over time. To illustrate this case further, it is useful to consider the limiting case of completely inelastic demand (i.e., ).

If , then  and

• 

Here,  is the demand for the resource or substitute in each country. Figure 2 illustrates the effect that an increase in  has on resource extraction. Until , the resource is used in both countries. At , country 1 (which has the highest tax) switches to the substitute, while country 2 continues to use the resource until it is exhausted at . If country 2 increases its tax, the date of resource depletion is reduced to , while the period of resource use in country 1 is extended until . Because the total resource extraction is given, squares A and B in Figure 2 are of equal size. Resource extraction of this size is moved from a later period to an earlier period, clearly increasing climate costs if the present value of the social cost of carbon declines over time.

Figure 2. The effect on the extraction path of the reduced cost of the substitute

With a constant social cost of carbon (denoted as ), the total climate costs are (when )

• (13)

The change in emissions described by Figure 2 clearly increases  because . While a completely inelastic demand is unrealistic, continuity implies that  will increase as the carbon tax in the low-tax country increases, for a sufficiently small positive value of . Hence, Proposition 1 follows.

Proposition 1If the demand for the resource plus substitute is sufficiently price-inelastic, the total climate costs will increase if the carbon tax is increased in the country that initially has the lowest tax.

Notice that this proposition will hold even if the social cost of carbon is not constant, as assumed in equation (13). As long as the present value of the social cost of carbon declines over time, early emissions are worse for the climate than later emissions. Therefore, the total climate costs will also increase with this less restrictive assumption if the conditions in Proposition 1 hold. The same is true for the subsequent propositions.

How relevant is Proposition 1 in the more realistic setting of many countries instead of only two? To answer this, it is again useful to consider the limiting case of completely inelastic demand. For this case, each country’s resource use is its demand over the time period it takes for the country’s consumer price to go from  to . As one country  reduces its carbon tax,  declines, which implies that  increases for all other countries. The reduced resource use by the country that increases its tax is thus moved to all other countries. Some of these countries might have higher taxes than the country increasing its tax, and the move in resource use to them will increase climate costs defined by . However, there might also be countries that have lower carbon taxes than the country that is increasing its tax. The move in resource use to these countries will reduce climate costs defined by . The net affect on  is ambiguous. The possibility of climate costs increasing seems more likely the lower the initial tax is in the country increasing its tax, because this means that more of the resource use is moved to higher-tax countries and hence is used earlier.

Climate costs are only one component of a country’s total welfare. To study the effects of a tax increase on a country’s total welfare, let us return to the simple two-country case. The welfare in country  is

• 

The first two terms in the curly brackets give utility from resource and substitute use minus the costs of substitute production. The third term is the value of country ’s net export of the resource, if country  owns a share  of the resource. Each country is assumed to bear half of the total climate costs, giving the green welfare term  for both countries.

Differentiating  w.r.t.  and using the fact that  for  and  for  gives

• (14)

The first term is a pure terms-of-trade effect; if , the sum of these terms for the two countries is zero. The second term reflects the fact that a carbon tax gives a distortion in the economy if climate effects are ignored. As the time path of resource use is changed, because of a change in the carbon tax, a negative or positive welfare effect is obtained, depending on how this time path is changed. Finally, the last term is the change in the country’s climate costs.

The total welfare change for the two countries is (assuming )

• (15)

Consider first the case of increasing the common tax rate . Using equation (13), we find

• (16)

It has been shown above that an increase in a common tax delays extraction, implying  for small  and  for large . Because , the integral is therefore negative, which implies that the whole expression is positive for . The welfare-maximizing carbon tax is, of course, the Pigovian rate . This (rather obvious) result can be summarized as follows.

Proposition 2Increasing a common carbon tax will reduce climate costs, and will also increase social welfare in both countries if the common tax rate is initially below the Pigovian tax rate.

When tax rates differ, there is not much to be said, in general, about the welfare effects of changing one tax rate. However, if , the last terms in equations (14) and (15) can dominate all other terms if  is sufficiently large, because  is proportional to . Therefore, Proposition 3 follows from Proposition 1.

Proposition 3If the social cost of carbon is sufficiently high, and the demand for the resource plus substitute is sufficiently price-inelastic, the total welfare for both countries will decline if the carbon tax is increased in the country that initially has the lowest tax.

From the discussion after Proposition 1, it is clear that welfare can decline for all countries in a multicountry setting if the carbon tax is raised in a country that initially has a relatively low tax rate.

V. Lower Costs of Producing the Substitute

In the Appendix, it is shown that differentiation of equations (3)–(6) w.r.t.  gives

• (17)

• (18)

• (19)

Notice that the initial consumer price in both countries decreases as  is reduced. Hence, the near-term emissions increase and there is a weak green paradox. As can be seen below, the total climate costs can nevertheless decline, in which case there is no strong green paradox.

It is useful to consider first the case in which both countries initially have the same tax, which implies . For this case, it follows that

• 

Hence, a reduction in  will shorten the period of extraction. This result is well known from the theory of non-renewable resources. If a substitute has a lower cost, this will reduce the price path of the resource and hence speed up resource use. It is straightforward to see that this result remains valid for positive but small tax differences, implying a small value for . The change in the extraction path implied by the reduction in  will increase climate costs when the present value of the social cost of carbon declines over time. As can be seen below, the total welfare can nevertheless increase.

When tax rates differ, a lower renewable cost will speed up extraction in the high-tax country. However, resource extraction lasts until , and the direction in which  moves as  is reduced is ambiguous. If , the total period of extraction will increase as  is reduced. In this case, the reduction in  will give increased early emissions (because  decreases), increased late emissions (because  increases), and thus lower medium-term emissions (from the resource constraint). Therefore, it is not obvious how climate costs are affected by the reduction in .

To illustrate the possibility of , it is useful to consider again the limiting case of completely inelastic demand (i.e. ), which implies . This gives

• 

where  is the demand for the resource or substitute in each country. A reduction in  has a similar effect on resource extraction as was seen in Figure 2, except now the initial switch dates are given by  and , while the switch dates after the reduction in  are given by  and . Some of the resource extraction is thus moved from A to B, reducing climate costs if the present value of the social cost of carbon declines over time.

From this analysis, Proposition 4 immediately follows.

Proposition 4If the differences in carbon tax rates are sufficiently small, the total climate costs increase if the cost of the substitute declines. For larger differences in carbon tax rates, this need not be true. In particular, if the demand for the resource plus substitute is sufficiently price-inelastic, the total climate costs will decline if the cost of the substitute declines.

The effect of reduced  on total welfare is found by proceeding as for the case of a tax change. The following equation is now found (ignoring the terms-of-trade term):

• (20)

The first term, which is negative, is the direct effect of a change in . This term tends to make welfare increase as  is reduced. The second and third terms have exactly the same interpretations as in equation (14).

For the case of a common tax rate , using equation (13), the total welfare change for the two countries is

• (21)

From the reasoning above, it is clear that both integrals are negative. If , it follows that welfare increases as  declines. However, if , it cannot be ruled out that the increased climate costs, implied by a reduction in , dominate the positive direct effect.

Proposition 5If carbon taxes in both countries are at the Pigovian rate, a lower cost of the substitute is welfare-enhancing for both countries. With a lower common tax rate, the welfare for both countries will decline if the cost of the substitute is reduced, provided the social cost of carbon is sufficiently high.

When considering different tax rates, there are not many general conclusions regarding overall social welfare. Perhaps the most interesting case is that described in Proposition 4, which implies that lower  might reduce climate costs. In particular, if , and if  is sufficiently low, it is clear from equation (20) that , while the sign of  will depend on . This gives the following proposition.

Proposition 6If the tax rate in the low-tax country is sufficiently low and demand for the resource plus substitute is sufficiently price-inelastic, a reduced cost of the substitute will increase welfare in the low-tax country. The sign of the welfare change in the high-tax country will depend on the tax rate in this country and on the social cost of carbon.

VI. Subsidizing the Renewable Substitute

In Section V, the cost reduction of the renewable substitute was a real cost reduction. Alternatively, a cost reduction to the users of the substitute could be considered, which is the result of a subsidy  reducing the private cost of the substitute from  to . Notice that such a subsidy in this model is a promise or commitment from the government to hold the future price of the substitute at . There could be good reason to believe that such a promise or commitment is not credible, as it might be in the interests of the government to terminate the subsidy once the carbon resource is depleted. It is nevertheless useful to consider how such a subsidy would work if it were possible to convince resource owners that the subsidy would continue “forever”, or at least sufficiently beyond the date of resource depletion.

Consider first a common subsidy . An increase in  has an identical effect on resource extraction as a reduction in . Therefore, the results in Section V (up to and including Proposition 4) remain valid. However, the welfare effects of increasing  differ from the welfare effects of a reduction in . The difference is in the first term in equation (20). Clearly, this term vanishes when the reduction in  is not caused by a reduction in  but instead by an increase in . If  is initially positive, a term similar to the second term in equation (20) is obtained. The exact expression is (ignoring, as before, the terms-of-trade term)

• (22)

For the case of a common tax rate , using equation (13), the total welfare change for the two countries is

• (23)

The term in square brackets in the first integral is positive: An increased subsidy implies an earlier start of the use of the subsidy, and once it is used a larger subsidy implies larger use. If , it follows that welfare decreases as the subsidy is increased. The second integral is positive, because an increase in the subsidy will speed up carbon extraction. For any , the whole expression is hence negative.

Proposition 7If carbon taxes in the two countries are equal and do not exceed the Pigovian tax rate, subsidizing the renewable substitute will reduce social welfare in both countries.

Notice that this proposition implies that introducing a small positive tax on the substitute in this case would increase social welfare, as is also pointed out by van der Ploeg and Withagen (2010).

When considering different tax rates, there are not many general conclusions regarding overall social welfare. Perhaps the most interesting case is that described in Proposition 4, which implies that higher  might reduce climate costs. In particular, if , and if  and  are sufficiently small, it is clear from equation (20) that , while the sign of  will depend on . This gives the following proposition.

Proposition 8If the tax rate in the low-tax country is sufficiently low and the demand for the resource plus substitute is sufficiently price-inelastic, the introduction of a small subsidy for the renewable substitute will increase welfare in the low-tax country. The sign of the welfare change in the high-tax country will depend on the tax rate in this country and on the social cost of carbon.

Finally, it is useful to consider the case where carbon taxes are equal in the two countries, but subsidies can differ. For this case, equations (3) and (4) are changed to (for )

• (24)

• (25)

with  determined by

• (26)

It is assumed that , implying that . In the Appendix, it is shown that

• (27)

• (28)

• (29)

An increased subsidy in any country will reduce the initial consumer price in both countries. Just as for a reduction in , the near-term emissions therefore increase, giving a weak green paradox. If the subsidy in country 1 is increased, the total period of extraction is increased (because ). In this case, the increased subsidy will give increased early emissions (because  decreases), increased late emissions (because  increases), and hence lower medium-term emissions (from the resource constraint). It is thus not obvious how climate costs are affected by the increased subsidy.

If, instead, the subsidy in country 2 is increased, the total period of extraction is shortened. Moreover, the total extraction at any point in time up to the exhaustion date must increase, because the consumer price path becomes lower in both countries. This immediately gives the following proposition.

Proposition 9If the subsidy is increased in the country that initially has the lowest subsidy, the total climate costs increase.

For the same reason as for a carbon tax increase in a single county, this result is also relevant in the more realistic setting of many countries instead of only two countries. Each country’s resource use is its demand over the time period it takes for the country’s consumer price to go from  to . As one country  increases its subsidy,  declines, implying that  increases for all other countries. The reduced resource use by the country that increases its subsidy is thus moved to all other countries. Some of these countries might have higher subsidies than the country increasing its subsidy, and the move in resource use to them will increase climate costs defined by . However, there could also be countries that have lower subsidies than the country that is increasing its subsidy. The move in resource use to these countries will reduce climate costs defined by . The net affect on  is ambiguous. The possibility that climate costs will increase seems more likely the lower the initial subsidy is in the country that is increasing its subsidy, because this means that more of the resource use is moved to countries with higher subsidies and is thus used earlier.

To simplify the discussion of total welfare, assume that the common carbon tax rate is zero. Then, the following expression is obtained, which is similar to equation (22).

• (30)

The integral in equation (30) is positive for the country that increases its subsidy, because this country switches to the substitute earlier and uses more of the substitute as a consequence of the increased subsidy. However, it is negative for the other county, because this country delays its switch to the substitute as a consequence of the increased subsidy. These results are summarized in the following proposition.

Proposition 10Assume that both countries have zero carbon taxes but positive subsidies. An increased subsidy will reduce social welfare, excluding climate costs, in the county that increases its subsidy, but it will increase social welfare, excluding climate costs, in the other county. Because climate costs increase if a country that has a low subsidy increases its subsidy, the total social welfare for this country will decline as it increases its subsidy.

VII. Conclusions

The analysis in this paper is performed using an extremely simple model. Perhaps the most drastic simplification is that carbon resources are homogeneous, and that they have constant unit costs up to a physical upper limit on total extraction. A much more realistic assumption would be to let extraction costs rise in cumulative extraction. Van der Ploeg and Withagen (2010), Gerlagh (2011), and Hoel (2011, 2012) have shown that the effects on the emission paths of changes in carbon taxes and the costs of a renewable substitute might depend significantly on the properties of the extraction cost function. Moreover, the extraction cost function can differ between different types of fossil fuels.

A second drastic simplification is that the substitute for the carbon resource is assumed to be a perfect substitute, and that it has a constant unit cost of production. Relaxing these assumptions could change the conclusion that a lower cost of the substitute will speed up extraction in a world of homogeneous countries (see, for example, Grafton et al., 2010; van der Ploeg and Withagen, 2010; Gerlagh, 2011). Even if the substitute is perfect, the properties of the extraction path could change if there is a time lag between investment in the substitute and its availability (see Gerlagh and Liski, 2011).

The focus of this paper has been to show that the degree of country heterogeneity can significantly affect the relationship between carbon taxes, costs, and subsidies, on the one hand, and emission paths, on the other hand. To focus on this issue, it has been useful to keep the model as simple as possible in all other dimensions. The analysis has shown that the effects on the emission paths of changes in taxes, costs, and subsidies could be very different in a world of heterogeneous countries than in a hypothetical world of identical countries. Although details will differ, it seems reasonable to also expect similar differences in more general models of carbon resources with a substitute.

Appendix: Changes in Taxes, Costs, and Subsidies

Inserting equation (3) into equation (6) gives

• 

For each county,  is determined by equation (5) or, for the more general case with the possibility of subsidies, by equation (26). Together with the above equation, there are three equations determining  , and  as functions of , , , , and . Differentiation of this equation system gives

• 

where

• 

and  and  are as defined by equations (10) and (11), but with the subsidy rates included in the demand function.

These equations give equations (7)–(9) and (27)–(29), where , defined by equation (12) with the subsidy rates included, is the determinant of the matrix .

For , the above equations imply

• 

• 

• 

These equations are identical to equations (17)–(19).

Footnotes

• 1

  Throughout this paper, CCS is ignored. Amigues et al. (2010), Le Kama et al. (2009), and Hoel and Jensen (2010) have discussed climate policy when there is a possibility of CCS and when the carbon resource scarcity is taken into consideration.

• 2

  Hörner and Kamien (2004) and Liski and Montero (2011) have shown that the assumptions made about commitment might strongly affect the equilibrium extraction path under non-competitive conditions.

• 3

  Papers that consider two or more countries in the context of the green paradox include Eichner and Pethig (2009) and Grafton et al. (2010). None of the papers discusses the effects of exogenous tax changes or cost reductions, as the present paper does.

• 4

  According to a recent International Energy Agency report, the total world direct subsidies of fossil fuels amounted to US$312 billion in 2009. Eliminating these subsidies would cut global carbon emissions by about 7 percent (see http://www.worldenergyoutlook.org/docs/second_joint_report.pdf).

• 5

  In a previous version of the paper, I assumed several countries. However, going from two to many countries adds to the complexity without giving much new insight.

• 6

  Strictly speaking,  is the long-run level of carbon in the atmosphere if emissions were zero for .

• 7

  Tahvonen (1995), Hoel and Isaksen (1995), and Hoel and Kverndokk (1996) explicitly consider the speed of climate change in their analyses.

• 8

  Farzin and Tahvonen (1996) give a detailed analysis of how  might develop over time when .