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, envi-

ronmental 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, email: karp@berkeley.edu

1 Introduction

Stricter climate policies in one region may cause countries with weaker environmental regu-

lation 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 mod-

els, 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

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 equilibriummodels, 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

from 25% with IRTS and differentiated 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%. Demailly 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

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 non-

signatory’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

, which can take any

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, , both using capital

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

p

x

y

( )D p

( ; )S p t

( ; ')S p t

'p

Figure 1: The relative supply and demand curves under two levels of tax, 0 . The higher

tax increases the autarkic relative price to 0.

is CRTS and preferences homothetic.

These assumptions imply that the relative demand and supply () are functions of (Fig-

ure 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 leak-

age 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 do-

mestic 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

x

y

a

bc

'a

( )IEP p

'b'c( ; )BOP p t

( , ')BOP p t

d

Figure 2: The balance of payments constraint before, (; ), and after, (; 0),the increase in emissions tax. Corresponding consumption points are and 0. The ini-

tial 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 equilibriummodels 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

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 theinitial 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 environ-

ment. The new consumption point therefore lies on a lower BOP curve, e.g. (; 0).

Holding constant, consumption occurs at point 0. By construction, triangles ∆000 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 highertax 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

, 1yc w r

, ,xc w r t p

, , 'xc w r t p

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 is the amount of factor ∈ {capital, labor, emissions} used to produce one unit ofsector output, ∈ { }. The zero profit conditions are

( ) = 1 and ( ) =

Figure 3 graphs the zero profit conditions for a given emissions tax, , and commodity

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

effect, operating through changed factor prices, moderates the partial equilibrium effect of

a higher tax. At point , 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 equi-

librium 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 , for a fixed 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

price is for 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

, and Remark 1 tell us how a

higher tax affects the import demand function.

Holding the commodity price fixed and differentiating with respect to the tax, I obtain

= + with (1)

≡

∙ −

( − )2

¸µ

−

¶≤ 0 and

8

≡∙

( − )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, , causing the

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 = ( ( )), with the function 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

1h

2h

1

w

r 2

w

r

capital

labor

a b

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.

Figure 4 shows: (i) two “ isoquants”, 2 1; (ii) the equilibrium ratios,¡

¢1and

¡

¢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 con-

dition for positive leakage.

1A 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 ; denote that

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 , the movement

from to capital’s proportional increase exceeds labor’s.

10

If production is not separable, then the general equilibrium effect may oppose and over-

whelm 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.

2This 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

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 advan-

tage in the dirty good, leading to negative leakage. In other cases, the general equilibrium

effect possibly reinforces the partial equilibrium effect, unlike in the HOS model above.

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

,Ac w r

,xc w r p

, 1yc w r

w

r

a

b

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

0 profits in each sector:

= ( ) 1 = ( ) − = ( ) (5)

System (5) contains three equations in three unknowns, and ; under incomplete spe-

cialization, 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

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 = 1 curve 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 manipu-

late the result to obtain the expression for the change in the price needed to retain incomplete

specialization:

= 1−

µ

¶ −

− (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 effect if and only

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:

= 1 +

1.

2.3 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

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’ emis-

sions (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 cal-

culations of the magnitude of leakage. This information is useful when confronting CGE

models, where the relation between assumptions and outcomes is not transparent.

15

The previous section shows general equilibrium effects always moderate the partial equi-

librium effects in the HOS setting, and “often” in the Ricardian setting. A partial equilib-

rium 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

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 , so they have

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

. These

expressions, the definition of elasticity of supply with respect to price, =

, the

elasticity of BAU emissions with respect to output, =

, and notation in Table 1, yield

the approximation of leakage:3

= (1− )

+ (1− ) +

= (1− )

+

+ (1− )

³1−

´ If demand is more elastic than supply, then

1. The ability to choose emissions freely

3Both 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

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 , equals 1. If outsiders are more

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

1 +

µ((−)(1−))

+

¶ which is less than 1 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 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

, = and ̄ 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 =

Ã1−

+ (1− ) +

! (8)

Leakage decreases in the membership ratio, , and in the elasticity ratios

and

, and

4If insiders’ and outsiders’ cost functions are different, this inequality need not hold.

18

1

0.05

0.00

y

32

0.10

z

0.20.4

x0.6

0.8

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

125= 08. With these guesstimates, Figure 6 plots the approximation

of leakage evaluated at BAU (the axis) as a function of (the axis) and

(the axis).

The membership fraction ranges over (01 08) and the ratio of demand to supply elasticity

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.5

3.2 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)

5The 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

reduction in emissions, and the effect of BTAs is easy to examine .

The cost function is

= (0 − 1)+

22 +

³22 − 0

´

which I further specialize, setting 0 = 0 = 0. This specialization implies BAU emissions

BAU = 1

; the BAU emissions intensity equals 1

, and the elasticity of BAU emissions

with respect to output equals = 1. By choice of units I set 1= 1. BAU costs equal

=1

2(− )2

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, 1BAU. 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 equa-

tions, enabling me to write the primitive parameters as functions of .

If countries restrict emissions to BAU, then leakage equals6

=(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.

I define 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.

6The formula for leakage in equation (8) collapses to the expression in equation (9) using = 1 = and

= 1− .

20

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.

To verify this Remark, note that for arbitrary , 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

0.12

0.10

0.08z 0.06

0.04

0.02

0.000.5

y

0.60.2 0.90.80.70.4

x0.6

0.8

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 ) both reduce the

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) to 50% ( = 05) the ad valorem

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 ; it equals

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

7The 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

1.02.5 3.02.0

y

1.5

-0.15

-0.10

-0.05

0.000.5

0.05

z

0.10

0.2

x

0.4 0.6 0.8

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−(1−+) . This tax is independent of the membership ratio

. 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 = 02 and = 15 and 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 in-

efficiencies 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

or moderate, but neither the theoretical nor empirical basis exists for asserting this with

confidence. Most non-economists who have considered leakage appear to accept its impor-

tance. People opposed to climate policy for other reasons may also use leakage to justify

inaction. In my view, climate change presents a major risk, while the risk to the trading

system appears small. If that view is correct, economists should accept that non-global

climate policy requires BTAs, and concentrate on designing these to do little harm.

Better estimates of parameters needed to calibrate simple models would be particularly

useful. These parameters include, for partial equilibrium models, price elasticities of supply

and demand, output elasticities with respect to constrained emissions, and unconstrained

emissions with respect to output. Some of these estimates can be obtained econometrically,

and others likely require engineering models. It is also important to remember the differences

between partial and general equilibrium estimates, and be wary of hidden assumptions (e.g.

separability) that determine conclusions in general equilibrium settings.

24

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