Carbon Leakage in General and Partial Equilibrium

Transcription

1 Carbon Leakage in General and Partial Equilibrium Larry Karp August 7, 2012 Abstract The general equilibrium effects of stricter environmental policy might reinforce or moderate the policy s partial equilibrium effects. In some cases, the general equilibrium effects can overwhelm the partial equilibrium effects, leading to negative leakage. A partial equilibrium model helps to assess the likely magnitude of leakage, and the magnitude of border tax adjustments (BTAs) needed to offset it. A BTA based on carbon intensity in countries without carbon constraints is an export subsidy and creates negative leakage. Keywords: carbon leakage, border tax adjustment, trade and the environment, environmental policy. JEL classification numbers C72, H4, Q54 I thank Gina Waterfield for research assistance and Meredith Fowlie, Rolf Golembek and Jeff LaFrance for conversations on this topic. The paper benefitted from comments from seminar participants at the September 2010 CESIfo Energy Conference in Munich, the 2011 AERE Summer Conference in Seattle and the December 2011 IATRC Conference in Berkeley. Remaining errors are mine. I thank the Ragnar Frisch Center for financial support. Department of Agricultural and Resource Economics, University of California, Berkeley, and the Ragnar Frisch Center for Economic Research, karp@berkeley.edu

2 1 Introduction Stricter climate policies in one region may cause countries with weaker environmental regulation to increase production of carbon-intensive goods. The possibility of carbon leakage makes it harder to reach sub-global climate agreements. Limited data, due to the dearth of meaningful climate policies, and econometric problems, make it difficult to econometrically estimate the magnitude of leakage. Most of the literature on leakage uses simulation, often with computable general equilibrium (CGE) models. These models yield clear predictions, but their complexity makes them hard for outsiders to evaluate. Partial equilibrium models, which ignore changes in factor prices and income, ignore important leakage channels. I address two questions: (1) Does a partial equilibrium model understate or overstate leakage, relative to a general equilibrium model? (2) What circumstances can cause negative leakage, in a general equilibrium setting? The first question is more involved than it appears, because we may not obtain a partial equilibrium model simply by turning off factor price or income changes. Where a partial equilibrium model is not a special case of a general equilibrium model, we have to decide what it means to compare the leakage estimates. Answering the second question improves our understanding of how partial and general equilibrium models differ. I address these questions using standard Heckscher-Ohlin-Samuelson (HOS) and Ricardian models, but with more general functional forms than in previous studies. Next, I analyze a partial equilibrium model that shows how a few parameters determine the magnitude of both leakage and the border tax adjustment (BTA) needed to offset leakage. Non-regulating countries ability to undermine the regulatory actions of other countries has been a prominent theme in environmental economics for decades (Hoel 1992). Carbon leakage is either a special case of, or closely related to, the pollution haven effect (Copeland and Taylor 2003), (Copeland and Taylor 2004), (Karp 2011). The 1999 Kyoto Protocol special issue of Energy Journal reviews the pre-2000 carbon leakage literature. Five papers in that symposium use CGE models and four use partial equilibrium models. Burniaux and Martins (2012), using a small CGE model, explain the 2% 21% range of leakage estimates produced by earlier models. They conclude that, where fuel markets are integrated and fuel supply quite inelastic, most leakage occurs through the energy markets channel : environmental policies reduce the demand for fossil fuels in abating countries, reducing fuel prices and increasing fuel consumption and non-abating countries emissions. They find that the pollution haven effect (the non-energy markets channel ) is a smaller reason for leakage. 1

3 Mattoo, Subramanian, van der Mensbrugghe, and He (2009) estimate leakage of less than 4% when high income countries reduce emissions; their second, simpler model, implies 11% leakage. They also examine BTA s effects on leakage and welfare. Fischer and Fox (2009b) and Fischer and Fox (2009a) estimate leakage using both a CGE and a partial equilibrium model. Their CGE estimates of leakage are 28% for energy intensive manufacturing and 14% overall. Like most partial equilibrium models, their s assumes that marginal production costs increase with abatement but are constant with respect to output, implying infinite supply elasticities. Home and foreign firms produce differentiated products. Most CGE models also assume constant returns to scale (CRTS) and differentiated products (the Armington assumption). Their partial equilibrium leakage estimates span 10% 60%. Babiker (2005) builds a model that includes an imperfectly competitive energy intensive tradable goods sectors with increasing returns to scale (IRTS). His leakage estimates range from25% withirtsanddifferentiated products, to 60% with CRTS and homogeneous products, to 130% with IRTS and homogenous products. He concludes that if energy intensive sectors produce homogenous products under IRTS, then unilateral climate policies may increase global emissions. McKibbin and Wilcoxen (2009) examine BTAs between countries with different carbon prices. The EU s effective tariff ranges from 1% 4%, depending on whether they base the BTA on US or China s carbon intensity. They conclude that small estimated leakage and small effective tariffs at moderate carbon prices create modest environmental benefits; these do not justify the BTA s efficiency cost and administrative complexity. Fowlie (2009), Ponssard and Walker (2008) (for the cement industry), and Ritz (2009) (for steel) estimate leakage using partial equilibrium models with Nash-Cournot competition and homogenous products. The technology is CRTS and abatement increases production costs. Estimated leakage ranges from a few percent to over 70%. D ly and Quirion (2008) use a partial equilibrium model to study the effect of EU policy on the EU iron and steel sector. Aichele and Felbermayr (2012) use country and sector level panel data for production and trade to econometrically estimate leakage. Their model includes variables commonly used in gravity models and a dummy to indicate Kyoto ratification. They find that Kyoto membership reduces carbon emissions and increases embodied carbon imports, but does not change the level of emissions embodied in consumption and investment. They conclude that leakage is approximately 100%. Is this interpretation valid? Consider a case where all countries are randomly assigned to accept or reject the Kyoto Protocol (the treatment and control groups, respectively). Suppose that the treatment group s average carbon imports 2

4 are 3 units higher and their carbon emissions (associated with production) 7 units lower than the control group s. What does this information tell about leakage? If Kyoto really reduces a member s emissions by 7 units, then it has no effect on a non-member s emissions, because the 7 units equals the Kyoto-induced emissions difference, between members and non-members. The 3 unit increase in a member s carbon imports must be offset by an equal reduction in non-members consumption, because in this interpretation, the latter s production has not changed. Therefore, leakage is zero. Alternatively, perhaps Kyoto decreased a signatory s emissions by and increased a nonsignatory s emissions by, with + =7. Now, leakage (equal to the number of units non-members increase emissions per unit of a member s abatement) is,whichcantakeany value. In the most favorable case (random assignment), the data tells us the Kyoto-induced difference in emissions across members and nonmembers, but does not tell us the difference in these countries emissions between the observed and the counter-factual (no-kyoto) worlds. A leakage estimate requires an estimate of levels of carbon emissions if no country had signed the Protocol. If we had but-for Kyoto carbon emission levels estimates, we could calculate leakage using those and observed carbon emissions levels, without using trade data. Leakage occurs when members Kyoto-induced actions induce changes in non-members. This causal relation implies that outcomes (levels of carbon emissions) do not satisfy the stable unit treatment value assumption (SUTVA), needed to be able to ascribe, to the treatment, differences in outcomes between the treatment and the control groups. Here, SUTVA states that one country s carbon emissions do not depend on another country s Kyoto status. How can trade connect actions in one country and outcomes in another country (e.g., signing Kyoto and increased emissions) if SUTVA holds? The papers reviewed here shed light on questions about leakage. However, it may still be useful to step back, and reconsider leakage through the lens of simple trade models, in both a general and partial equilibrium setting. 2 General equilibrium and leakage I consider a HOS and then a Ricardian model. A country has a clean sector and a dirty sector, and the relative price of the dirty good is. The clean sector produces the clean good,, and the dirty sector produces the dirty good,, and emissions,, bothusingcapital and labor. Factors are not traded, have fixed supply, and endogenous prices, (capital) and (labor); the emissions tax,, is exogenous, with emissions endogenous. Production 3

5 p S( p; t') S( p; t) p ' D( p) x y Figure 1: The relative supply and demand curves under two levels of tax, 0. The higher taxincreasestheautarkicrelativepriceto 0. is CRTS and preferences homothetic. These assumptions imply that the relative demand and supply ( )arefunctionsof (Figure 1). The tax (possibly zero) induces autarchic equilibrium prices,,. Suppose we hold the factor prices constant and increase the emissions tax to 0. Maintaining zero profits in both sectors, at constant factor prices, requires to rise, say to. If, instead, factor prices adjust after the increase in the emissions tax, a different equilibrium relative price, say 0, maintains zero profits in both sectors. If 0, I say the partial equilibrium leakage estimate overstates the actual level, where actual means when factor prices adjust. With the inequality reversed, I say the partial equilibrium estimate of leakage understates the actual level. The rationale for this definition is that leakage occurs if a higher emissions tax increases marginal production cost, which (with CRTS and zero profits) equals price. The higher the price, the greater the incentive for dirty good production to take place abroad. Comparing the price increase (due to the higher tax) needed to maintain zero profits in both sectors, with and without adjustments of factor prices, determines whether a partial equilibrium estimate understates or overstates leakage. In a partial equilibrium setting, a tax increase raises production costs and reduces the domestic supply curve, without altering the demand curve. The higher tax therefore shifts out the partial equilibrium excess demand curve, increasing production abroad, creating leakage. Leakage is inherent to partial equilibrium models, where the question is the magnitude, not the sign, of leakage. It is harder to determine the sign of leakage in a general equilibrium set- 4

6 y a a' d c b IEP( p) c' b' BOP( p; t) BOP( p, t ') x Figure 2: The balance of payments constraint before, (; ), andafter, (; 0 ), the increase in emissions tax. Corresponding consumption points are and 0. The initial production point is and the post-tax increase production point lies somewhwhere on (; 0 ). ting, which depends on the answer to two questions: Does a higher tax increase the autarkic price (as occurs in Figure 1), and if so, does the higher price create leakage? Copeland and Taylor (2003) and Krishna (2010) find that in the HOS setting, the higher tax does increase the autarchic price. However, a more general and still plausible production function can reverse that conclusion (Section 2.1). The price intercept of a country s import demand function equals the autarchic price. The import demand function shifts in the same direction as the change of the autarchic price, in the neighborhood of the autarchic price. If the country trades at a price in this neighborhood, i.e. if the volume of trade is negligible, then indeed leakage is positive if and only if the higher tax increases the autarchic price. Most previous literature seems to have stopped here. Consider the case of non-negligible trade. General equilibrium models respect the balance of payments constraint. We do not want a change in the balance to drive results, so I assume it is constant, at zero. At constant commodity prices, how does a higher emissions tax effect a county s import demand for the dirty good, allowing domestic factor prices to adjust? The commodity price is arbitrary, i.e. it may not equilibrate world supply and demand. In Figure 2, a country facing relative price and an emissions tax produces at. The line labelled (; ) shows the country s Balance of Payments (BOP) constraint. The 5

7 tax affects affects the production point,, and therefore affects the BOP constraint. The line () shows the country s Income Expansion Path, a straight line due to homothetic preferences. The consumption point is and the trade triangle,, shows initial imports, the length k k. The point (k k) lies on the country s import demand function, at the initial tax. We want to know if a higher tax increases or decreases imports at this price,. The dashed line shows the set of points where relative production of the dirty and clean good equals the ratio at point. Points on this line southwest of experience the same percentage contraction in both sectors. At production points above the dashed line, the dirty sector has contracted more than the clean sector. A higher tax, and resulting lower emissions, decreases dirty sector factor productivity. The higher tax therefore reduces real income, putting aside gains from the cleaner environment. The new consumption point therefore lies on a lower BOP curve, e.g. (; 0 ). Holding constant, consumption occurs at point 0. By construction, triangles and are identical. From the property of congruent triangles, point 0 lies above the dashed line, northwest of. If the actual production point lies northwest of 0 (on ( 0 )) then the dirty sector has contracted more than the clean sector: the relative production has fallen. In this case, the level of imports has increased at the original price : imports exceed k k; the higher tax shifts out the import demand curve for the dirty good, just as in a partial equilibrium model resulting in positive leakage. However, if the actual production point lies southeast of 0 (on ( 0 )) then the higher tax shifts in the import demand function for the dirty good, leading to negative leakage. Point 0 is difficult to identify in a general setting, so I summarize the results using point : Remark 1 Assume preference are homothetic. A higher tax shifts out the import demand for the dirty good (leading to positive leakage) only if the tax causes the dirty sector to contract more than the clean sector (the production ratio falls, so production occurs above point ). If the higher tax causes the ratio of production to rise, i.e. production occurs below point, then it shifts in the import demand for the dirty good, leading to negative leakage. Remark 1 requires homotheticity of demand, but not CRTS. The condition for positive leakage is necessary, whereas the condition for negative leakage is sufficient. If the actual production point lies strictly between points 0 and, the necessary condition for leakage fails, so leakage is negative. 6

8 cy w, r 1 c wr,, t x p c w, r, t' p x Figure 3: Graphs of zero profit conditions when the clean sector is relatively capital intensive. At constant commodity price,, the higher tax shifts factor prices from to. 2.1 A Heckscher-Ohlin-Samuelson model I invert the dirty sector joint production function, writing dirty good output as a function of capital, labor and emissions. The clean and dirty sector unit cost functions are ( ) and ( ). The envelope theorem implies: =, =, =, =, and = where istheamountoffactor {capital, labor, emissions} used to produce one unit of sector output, { }. The zero profit conditionsare ( ) =1 and ( ) = Figure 3 graphs the zero profit conditions for a given emissions tax,, andcommodity price. For this tax, point shows the combination consistent with zero profits in both sectors. The relative slopes of the tangent of the and isocost curves imply that the clean sector is relatively capital intensive; the following results do not depend on that assumption. Holding constant, a higher tax, 0, increases dirty sector production costs, causing the 0-profit isocost curve to shift down (the light dashed curve). Point shows the new equilibrium factor price at the original commodity price and the higher tax. The higher tax raises the rental rate and lowers the wage. This figure shows that the general equilibrium 7

9 effect, operating through changed factor prices, moderates the partial equilibrium effect of ahighertax. Atpoint, the dirty sector maintains zero profits without any rise in the commodity price. Here, the change in factor prices offsets the change in the tax, causing production costs to remain constant. With constant factor prices, in contrast, the higher tax causes dirty-sector profits at the original commodity price to be negative. At constant factor prices, the higher tax would require higher to sustain zero profits in the dirty sector. If both commodity prices and factor prices adjust, the factor prices change in the direction shown, partially offsetting the higher production costs, and requiring a smaller increase in (relative to the partial equilibrium setting where factor prices are constant). Remark 2 Factor price changes moderate the higher dirty good production cost resulting from the higher emissions tax. The general equilibrium effects moderate the partial equilibrium effect of the higher tax, and a partial equilibrium model overstates the magnitude of leakage. I need additional structure to determine if a higher tax creates positive leakage. The difficulty involves identifying point 0 in Figure 2. Therefore, I consider the simpler question: How does a tax increase shift the relative supply,forafixed relative commodity price? Answering this question requires finding if production occurs above or below point in the figure. From Remark 1, a necessary condition for positive leakage at the initial commodity priceisfor to fall with the higher tax, and a sufficient condition for negative leakage is for to rise. The economy s capital/labor ratio is. Using the full employment conditions, the relative supply of the dirty good, ( ; ( ) ( )) =, is ( ; ( ) ( )) = = µ 0 where and are the capital/labor ratios in the two sectors. Prices and determine the relative supply and factor prices,. The sign of,andremark1tellushowa higher tax affects the import demand function. Holding the commodity price fixed and differentiating with respect to the tax, I obtain ( ) 2 = + with (1) µ 8 0 and

10 ( ) 2 ( ) µ The definition of uses = and = the sector tax elasticities of unit inputs (holding commodity price fixed, but allowing factor prices to adjust). As Figure 3 shows, a higher tax induces a higher rental rate to wage ratio,,causingthe clean sector to use a more labor intensive process: 0. These inequalities are strict except in the limiting case of Leontieff production, where the factor mix is independent of factor prices. These inequalities and imply 0. The term can be positive or negative. The term in square brackets is positive because the clean sector is relatively capital intensive ( ). However, the sign of the second term, in parenthesis, is ambiguous in general. The higher emissions tax reduces emissions, making production of one unit of the dirty good infeasible at the initial level of capital and labor. One or both factors increase per unit of output of the dirty good. Possibly, both 0 and 0. A production function that is separable in emissions and a composite of capital and labor (hereafter, separable ) can be written as = (( )), withthefunction positive and increasing in both arguments. Previous papers implicitly assume separability. With constant returns to scale, also has constant returns to scale, and therefore is homothetic. Denote as the optimal level of emissions per unit of output under the the tax. The assumption 2 1 implies 2 1. Denote as the level necessary to produce one unit of the dirty good when = : the solution to ( )=1. Thus, 2 1. Given the role of the separability assumption, it is important to understand its economic meaning. With separable production, we can think of the dirty firm as using capital and labor to produce the joint products, potential output and emissions, and then using a fraction of potential output to reduce emissions (Copeland and Taylor, 2004). Here, production and abatement use the same capital-labor ratio. With non-separable production in the dirty sector, this kind of two-stage process does not occur. A reduction in emissions does not merely use some potential output to abate, but possibly requires a different capital labor ratio even without changes in factor prices. To assess the plausibility of the separability assumption, we can ask whether, in the absence of factor price changes, we would expect firms to change their capital/labor ratio, following a policy-induced emissions reduction. If 9

11 capital h 1 a b w w r r 2 1 h 2 labor Figure 4: At constant commodity price, a higher tax and resulting changes in factor prices shifts unit capital and labor requirements form to. The percentage increase in labor per unit of the dirty good is greater than the percentage increase in capital per unit of the dirty good. firms change their input mix in this circumstance, then their production function is not separable. Figure4shows: (i)two isoquants, 2 1 ; (ii) the equilibrium ratios, and 1 2 corresponding to taxes 1 2 ; and (iii) the capital and labor per unit production, and, in the two cases. The higher tax and lower wage/rental ratio decreases the labor per unit of output,, and might increase or decrease. However, the proportional increase in is less than the proportional increase in, implying 0. 1 This inequality and 1 imply 0; 0 unless is Leontieff. Thus, in the case where the dirty sector production function is separable in emissions and in the capital labor composite, a higher emissions tax reduces the relative supply of the dirty good. With homothetic preferences, the higher tax therefore increases relative excess demand for the dirty good. I restate this conclusion as: Remark 3 With homothetic preferences and separable production, a higher emissions tax lowers the country s comparative advantage in the dirty good, satisfying the necessary condition for positive leakage. 1 A straight line through the origin and point gives the set of capital and labor requirements at different levels of for the same factor price ratio. This line passes through the curve 2 northwest of ; denotethat point as (not shown in Figure 4). At points and the capital/labor ratios are constant, so in moving from to capital and labor increase in equal proportions. Because lies southeast of, themovement from to capital s proportional increase exceeds labor s. 10

12 If production is not separable, then the general equilibrium effect may oppose and overwhelm the partial equilibrium effect. For example, suppose the elasticity of substitution between capital and labor in the clean sector is small, so 0. 2 Ignoring that term, equation (1) implies 0 0 (2) A higher emissions tax and resulting fall in emissions can create a large increase in dirty sector capital intensity and a fall in labor per unit of production. In that case ( 0) inequality (2) holds. If both unit labor and capital requirements in the dirty sector increase with the fall in emissions, inequality (2) requires 1 When the fall in emissions increases both capital and labor unit requirements, this inequality puts a lower bound on the ratio of changes, required to obtain the counter-intuitive result. In summary: Remark 4 If the production function in the dirty sector is not separable, and if in addition the elasticity of substitution between inputs in the clean sector is low, then an increase in the emissions tax can promote a country s comparative advantage in the dirty sector. In this case, the higher tax creates negative leakage. 2.2 A Ricardian model Chau (2003) uses Cobb Douglas functions to study a three-sector model with dirty and clean tradable goods, and non-tradable abatement services. General functions make the results more transparent and more easily compared with the results above Each sector has constant returns to scale in capital and labor. With three sectors and two factors of production, the model generalizes the textbook Ricardian model of two sectors and one factor. The imbalance between sectors and factors means that technology determines the equilibrium relative commodity price under incomplete specialization. Here, the exogenously chosen emissions tax determines the autarchic relative commodity price between the two tradeable goods,, independent of preferences. 2 This example helps illuminate the general equilibrium effects. However, it is well known that a large difference between the two sector s elasticities of substitution can create factor intensity reversals. This complication does not appear central to the issue at hand. 11

13 The relative supply of tradables is discontinuous in the commodity price. Under trade, the economy is completely specialized in one of the two tradable goods unless the world price equals the country s autarchic price, a function of the tax. This model shows that the general equilibrium effects do not always moderate the partial equilibrium effects of a higher tax, in the sense used in Remark 2. It also provides another example of the possibility that a higher emissions tax promotes a country s comparative advantage in the dirty sector (leakage is negative). At constant factor prices, a large tax causes the dirty firm to abate. If the firm continues to emit some pollution, it also pays taxes. These two effects decrease profits. At constant factor costs, the relative price of the dirty good must rise to maintain 0 profits in the dirty sector. The partial equilibrium response to stricter environmental policies increases the relative price of the dirty good. In the general equilibrium setting, the tax changes factor prices. For example, suppose the dirty sector is the most labor intensive, and the abatement sector is the most capital intensive, of the three sectors. The tax creates the demand for abatement, which requires relatively large amounts of capital. This increased demand for capital tends to increase the price of capital relative the price of labor. Because (by assumption) the clean sector is capital intensive relative to the dirty sector, this change in factor prices raises costs in the clean sector by more than in the dirty sector. In order to maintain 0 profits in all sectors, this change in costs causes pressure for the price of the clean good to rise relative to the price of the dirty good. In this example, the general equilibrium effect moderates and possibly overwhelms the partial equilibrium effect. The emissions tax may promote the country s comparative advantage in the dirty good, leading to negative leakage. In other cases, the general equilibrium effect possibly reinforces the partial equilibrium effect,unlike inthehosmodelabove. In the absence of abatement, the dirty firm produces one unit of emissions per unit of output. The clean good is the numeraire and the price of the dirty good is. The dirty firm faces a unit emissions tax of and can remove a unit of emissions by buying a unit of abatement services at price. A firm that produces and buys units of abatement services has profits = ( ) = ( ) + (3) Assume the firm abates at a positive level but does not eliminate emissions: 0. 12

14 w c w, r A a c w, r p x b cy w, r 1 r Figure 5: Zero profits for three sectors with incomplete specialization. The abatement sector is most capital intensive and the dirty sector is least capital intensive. At constant factor price,, a higher tax moves the factor price equilibrium from, along the clean sector isocost curve, in the direction of. Profit maximization then implies = (4) If this equality did not hold, the firm would abate all or none of its emissions. The cost of producing one unit of output in sector = (the dirty, clean, and abatement sectors, respectively) is ( ). The assumption that all sectors operate, implies 0profits in each sector: = ( ) 1 = ( ) = ( ) (5) System (5) contains three equations in three unknowns, and ; under incomplete specialization, the unique solution determines the relative price of tradables, as in the Ricardian model. Given fixed and the assumption that the abatement sector is more capital intensive than the clean sector, the solid curves in Figure 5 show the zero-profit isocost curves for the clean sector ( =1) and abatement services ( = ). Point shows the equilibrium factor price. The isocost curve for the dirty good sector (the dashed curve, = ), shows the price that is consistent with incomplete specialization at the initial tax. This price causes 13

15 the three curves to intersect at point. At any other value of there is no solution to system (5) and one tradable sector shuts down. The flatter slope of the curve = indicates that the clean sector is more capital intensive than the dirty sector. A higher emissions tax shifts out the isocost curve for abatement services, causing the equilibrium factor prices to move along the =1curve toward point. To maintain incomplete specialization, must also change so that the three curves intersect at the new equilibrium factor price. The direction of change of following an increase in is ambiguous. The difference falls in order for the dashed curve to pass through the new equilibrium factor price, but the higher might increase or decrease. To determine the comparative statics, I totally differentiate the system (5) and manipulate the result to obtain the expression for the change in the price needed to retain incomplete specialization: µ =1 (6) At constant factor prices, a unit increase in the tax requires a unit increase in the commodity price to maintain 0 profits in the dirty sector (under incomplete specialization). The first term (1) on the right side of equation (6), reflects this partial equilibrium effect. The second term can be positive or negative, depending on the sign of. If this expression is negative, the general equilibrium effects reinforce the partial equilibrium effect. If this expression is positive, the general equilibrium effect tends to offset, and might overwhelm, the partial equilibrium effect. In summary Remark 5 The general equilibrium effect moderates the partial equilibrium effectifandonly if sign ( )=sign ( ). For, the general equilibrium effects oppose and overwhelm the partial equilibrium effect, leading to negative leakage. In the limiting case where the dirty sector and the abatement sector use the same capital labor ratio, the general equilibrium effect reinforces the partial equilibrium effect: = A comparison In the HOS setting, the general equilibrium effect moderates the partial equilibrium effect of stricter environmental policies. The higher tax increases dirty sector costs. Factor prices adjust to decrease those costs, moderating the partial equilibrium effect. In the Ricardian setting, a higher tax increases the price of abatement services, increasing the relative price of the factor used intensively in that sector. If, for example, the abatement sector is relatively capital intensive, the higher tax leads to a higher rental/wage ratio. If the dirty sector is 14

16 more capital intensive than the clean sector, the change in factor prices increases labor and capital costs in the dirty sector more than in the clean sector, thus reinforcing rather than moderating the partial equilibrium effect. When the clean sector is more capital intensive than the dirty sector, the general equilibrium effect moderates the partial equilibrium effect. In both the HOS and the Ricardian setting, the general equilibrium effect may overwhelm the partial equilibrium effect. When that occurs, stricter environmental policies increase a country s comparative advantage in the dirty good, and decreasing trading partners emissions (negative leakage). Section 2.1 notes that a necessary condition for negative leakage in the HOS setting is non-separability of the dirty good production function, in emissions and a composite of capital and labor. Separability is analytically convenient, but not especially plausible. In the Ricardian model, production of the dirty good and abatement services are different activities, and therefore naturally have different capital labor ratios. There, the general equilibrium effect moderates the partial equilibrium effect if and only if the clean sector capital labor ratio lies between the ratios in the dirty sector and the abatement sector. If (for example) the dirty sector is much less capital intensive then the clean sector, which is only slightly less capital intensive than the abatement sector, then the general equilibrium effect overwhelms the partial equilibrium effect. In this example, the increased demand for capital caused by higher abatement takes resources chiefly from the clean sector, causing a relative expansion of the dirty sector. If the dirty sector and the abatement sector in the Ricardian model have the same capital labor ratios as is implicitly the case in the HOS model with separable production then the general equilibrium effect reinforces the partial equilibrium effect. 3 A partial equilibrium model A general equilibrium model is probably appropriate for studying the effects of broad climate policy. However, these models do not tell us much about the magnitude of leakage, unless we move to a CGE framework. The partial equilibrium assumption that factor prices do not adjust to environmental policies provides a good approximation in many cases, although probably not for climate policy. However, the partial equilibrium model helps reveal the relation between parameters and results, and sometimes permits back-of-the-envelope calculations of the magnitude of leakage. This information is useful when confronting CGE models, where the relation between assumptions and outcomes is not transparent. 15

17 The previous section shows general equilibrium effects always moderate the partial equilibrium effects in the HOS setting, and often in the Ricardian setting. A partial equilibrium model builds in the assumption of positive leakage, whereas leakage can be negative in the general equilibrium setting. These results suggest a partial equilibrium model is likely to provide an upwardly biased estimate of leakage. The results are no more than suggestive, because the partial equilibrium model need not be a special case of the general equilibrium model. The partial equilibrium models discussed in Section 1 assume CRTS (infinite supply elasticities), and rely on either imperfect competition or product differentiation to close the model. I consider an alternative, in which markets are competitive and producers have increasing marginal production costs, so their supply curves have positive slopes (finite supply elasticities). In this setting, stricter environmental policies in one country elicit a supply response elsewhere, only to the extent that the output price changes. This model makes it possible to relate leakage to supply elasticity. A group of insiders reduce their emissions, and the remaining countries, the outsiders, follow business as usual. I study two versions the model; in both, all countries have the same the demand function for the dirty good. In the first version, with general functional forms, the production costs and thus the carbon intensity and the supply functions can be different between the insiders and the outsiders even before the insiders reduce their emissions. Here I use comparative statics to approximate leakage. Then I use a linear model, for which I calculate the effects of a non-marginal change in insiders emissions. With this model it is easy to calculate two kinds of border tax adjustments. For the linear model I assume at the outset that countries are ex ante identical, i.e. they all have the same supply functions prior to reducing their emissions. 3.1 The approximation There are countries; insiders adopt a carbon constraint, increasing their cost of producing the carbon-intensive commodity, and outsiders face no carbon constraint. All countries have the same demand function, and there is free trade. Leakage equals the number of units outsiders increase emissions per unit of insiders decreased emissions. If each insider reduces emissions by and each outsider increases emissions by, leakage equals = ( ) 16

18 Evaluating this derivative at BAU gives an approximation of leakage. In country { } (for insiders and outsiders) the industry cost function is ( ), where equals s output (supply) of the carbon-intensive commodity and equals its emissions: 0, 0, and 0. The index allows insiders and outsiders to have different cost functions, and thus different supply functions and carbon intensity, even before the insiders reduce emissions. Outsiders choose to minimize costs, so their level of emissions satisfies ( )=0 (7) where equals the outsider s supply and their emissions; insiders emit at the constrained level, =. Country s inverse supply function equals = where is the common price. Insiders and outside have different levels of, sotheyhave different supply functions even if they have the same cost function. Each country has the demand function (), implying the market clearing condition () = +( ) Differentiating the market clearing condition with respect to, the insiders emissions level, yields. I then differentiate outsiders equilibrium condition (7) to find expressions, the definition of elasticity of supply with respect to price, elasticity of BAU emissions with respect to output, = the approximation of leakage: 3 = (1 ) +(1 ) + = (1 ) ³ + +(1 ) 1. = These,the, and notation in Table 1, yield If demand is more elastic than supply, then 1. The ability to choose emissions freely 3 Both and depend on the cost function, but one is not the inverse of the other. For example, it might be the case that under BAU one unit of output creates one unit of emissions, in which case =1. Unless production happens to be Leontieff, a one unit reduction in the emissions constraint reduces output by less than one unit. 17

19 increases the elasticity of supply, so it is reasonable to expect that 1. 4 The second line shows the effect on leakage of countries relative carbon intensity,. Leakage depends on the fraction of insiders,, multiplied by the average production share of an insider,,not on the two share parameters independently. The elasticity of the estimate of leakage, with respect to, equals1. Ifoutsidersaremore carbon intensive, 1. The example below sets =1=, i.e. I evaluate the estimate of leakage at a point where the insiders and outsiders are identical before the former reduce their emissions. If, for example, the outsiders are 30%, more carbon intensive then the insiders, then the estimates below increases by 30%. The elasticity of the estimate of leakage with respect to equals = 1 µ (( 1+ )(1 )) + whichislessthan1 if 0. Thus, if the typical outsider produces a smaller fraction of the carbon intensive good than the typical insider ( 1), then the estimate below exaggerates leakage. parameter name, =, =, = and = meaning elasticity of BAU emissions wrt output absolute value of elasticity of demand elasticity of supply wrt price in country = elasticity of output wrt constrained emissions (constant price) output in country relative to average output per country output in country and average output per country fraction of countries that constrain emissions outsider s emission intensity = insider s emission intensity Table 1: Notation By evaluating the approximation at a symmetric equilibrium, where = =1,the expression for leakage simplifies to SYM = Ã 1 +(1 )+! (8) Leakage decreases in the membership ratio,, and in the elasticity ratios and,and 4 If insiders and outsiders cost functions are different, this inequality need not hold. 18

20 0.10 z x y 2 3 Figure 6: The approximation of leakage using equation (8): =leakage, =, = =02 and =08 increases in and. The maximum possible level of leakage is. Given a range of parameter values, we can calculate the range of leakage for this approximation. The leakage estimate in equation (8) is proportional. Aichele and Felbermayr (2012) estimate =038 for the economy at large, but the number might be much larger for carbon intensive sectors. The figures below conservatively assume the carbon intensity under BAU is insensitive to the level of output, so =1. If a 10% decrease in the allowable level of emissions leads to a 2% decrease in production of the carbon-intensive commodity, then =02. If the unconstrained elasticity of supply is 25% greater than the elasticity under the constraint, then = 1 =08. With these guesstimates, Figure 6 plots the approximation 125 of leakage evaluated at BAU (the axis) as a function of (the axis) and (the axis). The membership fraction ranges over (01 08) andtheratioofdemandtosupplyelasticity ranges over (05 3). Over most of this range, the estimate of leakage is under 10% (well below the maximum level 20% when =02) even when the membership ratio is small The linear model The linear model uses a linear demand and quadratic cost function, implying a linear supply function. Here I also assume symmetry; insiders and outsiders have the same cost and demand functions. A formula for leakage holds for arbitrary (rather than infinitesimal) 5 The magnitude of the estimate is proportional to. For example, if we think that a 10% decrease in the allowable level of emissions results in a 6% decrease in production of the carbon-intensive good (rather than a 2% decrease as the figure assumes), then the estimate of leakage increases by a factor of 3. However, if we think that a larger scale of production leads to less carbon-intensive methods ( 1), then the estimate of leakage falls. 19

21 reduction in emissions, and the effectofbtasiseasytoexamine. The cost function is =( 0 1 ) + ³ which I further specialize, setting 0 = 0 =0. BAU = 1 ; the BAU emissions intensity equals 1 with respect to output equals =1. By choice of units I set 1 = 1 2 ( ) 2 This specialization implies BAU emissions, and the elasticity of BAU emissions =1. BAU costs equal with. The BAU supply function is =, implying the BAU elasticity of supply with respect to price =1. The demand function is also linear: =. The BAU emissions per dollar of output equals the inverse of the equilibrium BAU price, 1 BAU. The supply elasticity with respect to emissions, evaluated at BAU emissions, is 1. This model has four primitive parameters, all positive. By choice of units I can select two parameters arbitrarily. To facilitate interpretation, I present results using elasticities. Some calculations show price elasticity of demand, evaluated at BAU = ( ) supply elasticity wrt constrained emissions. evaluated at BAU = Choosing units so that the BAU price and emissions both equal 1 provides two more equations, enabling me to write the primitive parameters as functions of. If countries restrict emissions to BAU, then leakage equals 6 (1 ) = 1+ (9) The symmetry assumption implies zero trade at BAU. If the insiders reduce emissions and allow free trade, they import the carbon intensive good. Outsiders increased production creates leakage. Idefine the aggressive BTA as the BTA that charges an outsider a unit import tax equal to the insider s price of carbon (equal to their marginal cost of abatement) times the outsider s carbon intensity. 6 The formula for leakage in equation (8) collapses to the expression in equation (9) using =1= and =1. 20

22 Remark 6 For the symmetric linear model with the parameter restrictions 0 = 0 =0,the aggressive BTA is an export subsidy, and leads to negative leakage. ToverifythisRemark,notethatforarbitrary, the outsider s equilibrium supply, 0, satisfies =( ) 0. Given, and the insider s carbon constraint, BAU,the insider s equilibrium supply,,satisfies + = 1,or = 1. The parameter restrictions 0 = 0 =0(and the normalization 1 =1) imply the outsider s carbon intensity equals 1, so the aggressive BTA equals the insider s marginal abatement cost: = = + 1 = ( ). The insider s equilibrium condition therefore satisfies = 1 ( ) = ( ). Thus, for any, the aggressive BTA implies that insiders and outsiders supply the same amount. However, the outsider consumers face price, whereas insider consumers face price +. By the symmetry assumption, insider demand is lower than outsider demand; because the two supplies are equal, insiders export the good. Relative to BAU (where trade is 0), domestic price in the outsider countries falls, so production and thus emissions also falls in the outsider countries. Remark 6 relies on symmetry, linearity, and the parameter restrictions 0 = 0 =0,so the result is not general. However, the argument shows the forces at work. The BTA offsets the cost disadvantage of the emissions constraint, and decreases insider demand, thus discouraging imports, or encouraging exports by the carbon-constrained country. Absent the symmetry assumption, trade under BAU is non-zero; Remark 6 suggests the likely direction of change in trade. If insiders import the carbon intensive good both before and after they institute the policy (e.g. because their demand is high relative to outsiders ), they obtain terms of trade benefits from the lower outsider price. The BTA reduces outsider welfare, apart from environmental considerations. If insiders export after they institute the policy, they subsidize outsider consumption; insiders lose and outsiders gain (ignoring environmental considerations). Denote the insiders constrained level of emission as a fraction of the BAU level as 1. Smaller values of entail stricter carbon limits. There is a simple relation between the aggressive BTA, expressed as an ad valorem tax = rather than the unit tax, and. The aggressive ad valorem tax, equals = = ( +1) 1 (1 + )+1 (10) It is straightforward to calculate the comparative statics of this expression with respect to the parameters and the policy variable,. For example, 0 and 0. A weaker policy 21

23 z x y Figure 7: The aggressive ad valorem BTA (the axis) for membership fraction (01 08) (the axis) and policy variable (05 095) with =02 and =15 intervention (a larger value of ) or a larger level of membership (larger ) bothreducethe ad valorem aggressive BTA. Figure 7 graphs the aggressive ad valorem BTA, as a function of the membership fraction (01 08) (the axis) and the policy variable (05 095) (the axis) for =02 and =15. 7 As insiders cut emissions from 5% ( =095) to50%( =05) theadvalorem tariff rises from a negligible level to approximately 12%. The tax is insensitive to the level of membership. Leakage under the aggressive BTA is independent of the policy variable ; itequals aggressive 1 = ( +1)(1 )+ 0 and is negative by Remark 6. Insider producers use a less carbon intensive process, and outsider producers continue to use the original process, but they produce less. Consumption shifts from the insider countries to the outsider countries. Figure 8 shows the two graphs of leakage (the axis), with and without the aggressive 7 The elasticity of demand for carbon-intensive goods is likely less than 1. However, the analysis of the more general model shows that what matters is the ratio of elasticities of supply and demand. In the linear model here, the elasticity of supply at BAU is fixed to 1 by my decision to set 0 = 0 =0. If we think that demand is more elastic than supply, then 1. However, the graph of the BTA under =08 is almost identical to the graph in Figure 7. 22

24 z x 1.5 y Figure 8: Leakage (the axis) under the aggressive BTA (for 0) and under free trade (for 0) as a function of (01 08) (the axis) and (05 3) (the axis). BTA, as a function of membership (01 08) (the axis) and the demand elasticity (05 3) (the axis); the top orthant ( 0) shows leakage under free trade, and the bottom orthant ( 0) shows leakage under the aggressive BTA. In both cases, it is less than 10% in absolute value over most of the parameter space shown in the figure. A modest BTA that eliminates leakage leaves the world price unchanged, thus leaving emissions in the unconstrained countries fixed at their BAU level. The ad valorem BTA that eliminates leakage is = 1. This tax is independent of the membership ratio (1 +). If each of the other insiders uses a BTA that leaves their equilibrium excess demand unchanged relative to BAU, then the level of the BTA that a particular insider must choose to achieve the same goal, does not depend on the fraction of insiders. Not surprisingly, the modest BTA that eliminates leakage is always less than the aggressive BTA. For example, for =02and =15and over the range of graphed in Figure 7, the modest ad valorem BTA is between 35% and 39% of the aggressive BTA. If =08, the modest BTA is between 50% and 55% of the aggressive BTA. 4 Discussion If leakage creates small environmental or economic consequences, while BTAs create large inefficiencies and camouflage protectionists, climate legislation should not include trade policy. If leakage leads to large environmental and economic costs, sub-global climate agreements require BTAs. We lack reliable estimates of leakage. My best guess is that it will be small 23