WORKING PAPER SERIESPRINCIPLES FORPARETO EFFICIENTBORDER CARBON
ADJUSTMENT

Michael Keen (Tokyo College, The University of Tokyo)
Christos Kotsogiannis (University of Exeter)

March 16, 2024
TC-E-23-2

Tokyo College, The University of Tokyo

TC-E-21-1

PRINCIPLES FOR PARETO EFFICIENTBORDER CARBON ADJUSTMENT

Michael Keena and Christos Kotsogiannisb

This version: March 16, 2024

Abstract: Border Carbon Adjustment Mechanisms (BCAMs) are becoming reality in the
EU and elsewhere, and recur—in very different form—in U.S. legislative proposals.
But they remain contentious, with features and differences that leave the underlying
welfare rationale and implications unclear. Exploring these, this paper establishes twogeneral principles for Pareto efficient BCAM design: regulatory measures should berecognized symmetrically with explicit carbon prices; and, whatever the ambition of
mitigation in the BCA-imposing country, a general ‘difference-in-differences’ form ofa BCAM is appropriate. These nest, as special cases, the very different approaches to
BCAM design in Europe and the U.S.

Keywords: Environmental taxation; Carbon pricing; Border tax adjustment; International taxation
JEL classification: H21, H23, H87, F18

a Tokyo College, University of Tokyo; CERDI, Université Clermont Auvergne; CESifo;

Center for Business Taxation, University of Oxford; and Institute for Fiscal Studies,
London. E-mail: michael-keen@g.ecc.u-tokyo.ac.jp
b Tax Administration Research Center (TARC), Department of Economics, University of

Exeter Business School and CESifo. E-mail: c.kotsogiannis@exeter.ac.uk

1.

Introduction

After many years of heated debate, Border Carbon Adjustment Mechanisms (BCAMs)—
broadly, schemes that levy a charge on the carbon content embodied in imports that theimporting country regards as in some sense underpriced or otherwise excessive—are
fast becoming reality. The European Union has begun the transition to full adoption of
a BCAM in 2030, with financial implications for affected companies starting in 2025;
the UK is committed to adopt a BCAM in 2027; and Canada is in reflection stage
after a public consultation. In the U.S., some form of BCAM is a recurrent feature oflegislative proposals for climate action.1 But the use and design of BCAMs remains
highly contentious and politically charged, making it especially important to have aclear understanding of the welfare arguments for and against some form of BCAM.
There is indeed a sizable literature on many of the issues raised by the idea of bordercarbon adjustment, including: compatibility (or not) with WTO rules; practical issues of
implementation; sectoral and trade impacts; the extent to which alternative instruments
can replicate the effects of a BCAM; and the impact on developing countries.2 Surprisingly little attention has been paid, however, to what would seem the most fundamental
question underlying (almost) all of the others: Is there a coherent welfare rationale
for some form of BCAM, and if so, precisely what is that form? Exceptions include alargely overlooked analysis in Gros (2009), focused on the effect on global welfare ofa tariff on implicitly imported carbon, Kortum and Weisbach (2021), focused largelyon the treatment of fossil fuels, and Keen and Kotsogiannis (2014), who show that aform of BCAM—which reduces in a special case to something approaching that now
being put into place in Europe—is indeed Pareto efficient when climate policies in othercountries are for some reason constrained at inefficient levels.3 The aim in this shortpaper, building on Keen and Kotsogiannis (2014), is to explore two sets of issues thathave considerable practical significance but whose implications for the desirability anddesign of BCAMs have either not been addressed or, at the least, remain incompletely
articulated. Doing so leads to two general principles to guide the efficient design of
BCAMs.
1 See, respectively European Parliament and Council of the European Union (2023), Government of

the UK (2023), Government of Canada (2021); and, on recent U.S. proposals, Gangotra and Kennedy(2023) and Bistline et al. (2024).
2As an illustrative list, see respectively: Bullock (2018), Parry et al. (2021) and Tsakiris and Vlassis(2022); Kortum and Weisbach (2017) and Cosbey et al. (2019); Fischer, Morgenstern and Richardson(2015); Fischer and Fox (2012) and Böhringer, Rosendahl and Storrøsten (2017); and UNCTAD (2021)
and Lowe (2021). Recent and relatively brief reviews are in Keen, Parry and Roaf (2022), Böhringer et al.
(2022) and Clausing and Wolfram (2023).
3 See in particular their Propositions 2 and 5. Kotsogiannis and Woodland (2013) extend these results
to the case in which the externality enters production rather than consumer welfare.

1The first set of issues relates to the common presumption in the formal analysis of
BCAMs that mitigation (the reduction of emissions) is by carbon pricing.4 In practice,
however, it is becoming increasing clear that many countries with an intention to
mitigate—including, not least, the U.S.—are instead relying on a wide range of othermitigation instruments (such as performance standards, subsidies and tax credits forinvestment in, or energy sourced by, renewables) that are often not equivalent to a
direct price on emissions: the wide range of credits and subsidies made available in the
U.S. by the Inflation Reduction Act of 2022 is the leading example. The question thatthen arises is whether some allowance should be made in calculating the BCAM for
mitigation measures abroad other than explicit carbon pricing. The answer given by the
EU and UK schemes is an uncompromising "No": both have made very clear that the
charge due on imports will be reduced only to reflect explicit carbon prices paid abroad,so entirely ignoring any other form of mitigation.5 In the U.S. the 2023 Market Choice
Act, which proposed a U.S. carbon tax, would also adjust the import charge only for
explicit carbon charges.6 Canada, however, another carbon-pricing country, recognizes
that the right answer is not obvious: “[a] key challenge [is] to...consider whether andhow non-pricing regulatory instruments can be compared to explicit pricing measures,”(Government of Canada, 2021).
In the U.S., a further issue arises. There, proposals that do not include a domestic
carbon price do nonetheless include a form of BCAM—a form, however, that is structurally quite different from that considered in the analytical literature cited above andnow being put in place in Europe. Instead of looking to differences in carbon pricing
at home and abroad, this looks to differences in embodied emissions. The Foreign
Pollution Fee Act of 2023,7 for example, would apply a common charge—the level ofwhich is unspecified, being left to the Secretary of Energy—to the difference in emission
intensity between the U.S. and origin country. Unlike the EU and UK approach, such aform of BCAM would recognize mitigation efforts abroad through means other thancarbon pricing, but would take no account of the financial burden that such pricing
imposes in addition to the direct private cost of reducing emissions.
There are thus a variety of approaches to structuring BCAMs in the presence of mit4 For brevity, we take any explicit carbon pricing to be in the form of taxation rather than cap-and-trade;

Pareto efficient BCAM design in the case of the latter is discussed in Keen and Kotsogiannis (2014).
5The EU regulation provides credit against the charge on the carbon content of imported products
only for carbon price that has been effectively paid in the country of origin (European Parliament andCouncil of the European Union (2023), especially Articles 6.2 and 9), and the UK makes clear that “[t]heprice applied by a BCAM will...be set on the basis of the explicit carbon price differential between the
UK and the country where the products were produced,” (Government of the UK, 2023).
6 HR 6665 at https://www.govinfo.gov/content/pkg/BILLS-118hr6665ih/pdf/BILLS-118hr6665ih.
pdf, Section 102.
7 S 3198 at https://www.congress.gov/118/bills/s3198/BILLS-118s3198is.pdf.

2

igation measures other than direct emissions pricing (which for brevity we simply referto as ‘regulatory’ measures)—each with some intuitive appeal, but with no clear analytical guidance as to which, if any, is the most appropriate in any given circumstances.
The result cited earlier that constrained Pareto efficiency requires a (price-based) BCAM
provides no answer, since it assumes that all countries use explicit carbon pricing.
Relaxing this, the answers that emerge from the first general principle for BCAM design
established below is clear-cut: the shadow prices for carbon associated with regulatory
mitigation should be treated in just the same way as explicit carbon prices. Taking thoseshadow prices as metaphors for the strength of regulatory mitigation, the implication isthat, in looking only to explicit prices, the approach of the EU and UK is misconceived.
Moreover, there emerge within this encompassing principle circumstances in which
it is the emissions-based approach of the Foreign Pollution Fee Act, not a price-based
approach, that is constrained Pareto efficient, along with guidance as to what the proper
fee then is.
The second set of issues arises from the presumption underlying previous resultson the efficiency case for an appropriately designed BCAM that the BCA-imposing
countries are completely free in the carbon price they set (or, transposed to the context
here, in their mitigation policy more generally), and so set it at efficient levels. This too
risks being significantly at odds with reality. Countries may, for instance, price carbon
at a level which reflects not global harm but only that which they themselves suffer; andit is all too evident in Europe and elsewhere that political resistance to ambitious carbon
pricing is strong. Even in the EU, the pioneer of BCAMs, the carbon price of around
USD 70 per tonne (in February 2024), which is relatively aggressive by international
standards, is well below the USD 120 per tonne that the Environmental Protection
Agency (2023) gives as its lowest estimate of the social cost of carbon (for 2020). What
becomes of the case for a BCAM in such circumstances? While one might expect
some type of BCAM to remain desirable even with significant under-pricing (explicit
or shadow), its precise form is not immediately apparent. What emerges below—
the second general principle—is that constrained Pareto efficiency then requires a
generalized form of BCAM which looks not to the difference between domestic andforeign (explicit or shadow) carbon prices, but to the difference in how far each deviatesfrom the first best level: a ‘difference-in-differences’ BCAM. Conveniently, it will beseen too that in one not wholly unreasonable special case, this too reduces to theusual mechanical form of BCAM even when the carbon price in the importing country(explicit or shadow) is inefficient. By nesting as special cases the quite different forms of
BCAM observed and proposed, this general characterization of Pareto efficient BCAMs
makes clear the (also quite different) circumstances in which each can be given an3

efficiency rationale.
The welfare perspective adopted here is, almost entirely, that of constrained Pareto
efficiency. Viewed not just as a benchmark for evaluation but as a guide to action,
this perspective requires some degree of benevolence on the part of the unconstrained
country, at least to the extent of preferring not to make others worse off. But withoutsome regard for the well-being of others it is hard to understand why any country
would ever introduce into its tariff structure the recognition of foreign mitigation
policies that is the core element of any BCAM;8 this will be seen when, for comparison,
the case of self-interest is examined briefly9 below.
Section 2 describes the model and analytical approach. Results are in Section 3, andconclusions drawn in Section 4. The formalities are straightforward but involved, andso are for the most part relegated to appendices.

2.

Preliminaries

The model
The setting is of a competitive global economy with just two goods and two countries,
each with a single representative consumer.10 One good, taken as numeraire, is clean;
the other is ‘dirty’ in the sense that its production generates harmful emissions.11 The‘home’ country—whose possible deployment of some form of BCAM is the focus ofinterest—is described in lower case letters, and the ‘foreign’ in upper case.
Preferences in the home country are described by expenditure function of the forme( p0 , p, u) + p0 θk, where p0 denotes the price of the numeraire and p the (consumer andproducer) price of the dirty good, while u indicates home welfare, k global emissionsand θ > 0 the damage they cause to the home country.12 In what follows p0 is set
to unity and suppressed as an argument. The price of the dirty good is p = w + t,
8 One might argue that adopting a BCAM can serve a self-interested purpose by encouraging tighter

mitigation abroad. But the direct force of any such incentive seems likely to be limited: even for the EU
the proportion of emissions from the large-emitting countries embodied in imports is modest (Keen,
Parry and Roaf, 2022), and the best targeted response is an offsetting export charge, not a generalized
carbon price. Nordhaus (2015) is notably skeptical as to the effectiveness of BCAM as a sanction.
9This case is more familiar, having been examined, albeit from perspectives somewhat different fromthat here, by, for example, Markusen (1975) and Neary (1985).
10The additional considerations that arise with many countries and goods are much as in Keen andWildasin (2004): the possibility, for example, of using tariff policy to engineer inter-country transfers
when there are more goods than countries.
11We speak of these as emissions from burning fossil fuels, but other interpretations are of course
possible; and it is straightforward to generalize the damage function introduced below to allow for local
damage.
12These preferences imply that compensated demand for the dirty good is independent of damage, which avoids cumbersome feedback effects on tariff revenue along lines discussed in Keen andKotsogiannis (2014).

4

with w being the world price and t the home tariff. For definiteness (but with no
implications for the analytics), in the discussion we shall have it in mind that this ishome’s importable, so that t > 0 indicates an import tariff (while for the ‘foreign,’ upper
case, country, T > 0 means an export subsidy). This dirty commodity may, but neednot be, interpreted as the fossil fuel from whose use damage arises: we characterize andtake up this special case later. Compensated demand for the dirty good is given by13e p ( p, u), with standard concavity properties implying that e pp ( p, u) ≤ 0. Preferences in
the foreign country, analogously, are E( P, U ) + Θk.
Production decisions are described by a revenue functionr ( p, s) ≡ max{ xn + p x − s z : ( xn , x, z) ∈ f } ,xn ,x,z(1)

where xn denotes output of the numeraire, x output of the dirty good, z emissions fromthe burning of carbon (the latter both assumed throughout to be strictly positive), s an
explicit carbon tax charged on those emissions and f (·) the technologically feasible
set. As standard properties,14 r ( p, s) is convex and differentiable in p and −s, withr p = x and rs = −z; for simplicity, we assume strict convexity, so that r pp , rss andr pp rss − (r ps )2 are all strictly positive. It will be assumed too that rsp = −z p < 0, so thatan increase in the price of the dirty good induces higher emissions.15
Countries may, however, choose to mitigate not by carbon taxation but instead,or also, by regulatory measures. Leading to some level of emissions z̄, we take these
to be characterized by a shadow price s̄ such that z̄ = −rs ( p, s̄). Production is thendetermined by whichever of the explicit price s and shadow price s̄ is higher, so that
output and emissions are given byx = r p ( p, s∗ ) ;z = −rs ( p, s∗ ) ,(2)

where s∗ ≡ max{s, s̄}; we refer to s∗ as the ‘binding’ price, since it is this that determines
production decisions. Under regulation, for instance, the binding price is the shadow
price, and emissions are z̄ = −rs ( p, s̄). Net private income from production, however,reflects financial payments that are made only of the explicit carbon price s, not of the
shadow price s̄. When a shadow price s̄ is binding but an explicit price s is neverthelessin place, for example, net revenue is r ( p, s̄) + s̄z̄ − sz̄ = r ( p, s̄) − (s̄ − s)rs ( p, s̄): that is,while producers behave as if they faced a price of s̄ the fact that they do not corresponds
to their receiving a lump sum rebate of s̄z̄; they do, however, face a real cost on their
13With obvious exceptions, subscripts indicate derivatives.
14 See Dixit and Norman (1980) and Woodland (1982).
15 Neary (2006) refers to z

p as an indicator of pollution intensity.

5inputs of sz̄. More generally, net income is given by the adjusted revenue function16g( p, s∗ , s) ≡ r ( p, s∗ ) − (s∗ − s)rs ( p, s∗ ) .

(3)

The description of the foreign country is analogous, though preferences and technologies may be quite different. Aggregate emissions, recalling (2), are thusk ( p, P, s∗ , S∗ ) = −rs ( p, s∗ ) − RS ( P, S∗ ) ,(4)and, denoting home’s (compensated) net import demand by m ( p, s∗ , u) ≡ e p ( p, u) −r p ( p, s∗ ), market clearing requires thatm ( p, s∗ , u) + M ( P, S∗ , U ) = 0 ,(5)where, from the properties noted above, m p < 0, MP < 0.
To focus on the environmental concerns motivating the potential use of BCAMs,
revenue raised by the tariff and any explicit carbon prices are assumed in our central
case to be returned to the representative consumer as a lump sum; we shall thoughconsider the implication of policymakers also having a distinct need to raise revenue.
For now, using (2), the income-expenditure identity for the home country is thene( p, u) + θk = g( p, s∗ , s) − srs ( p, s∗ ) + tm ( p, s∗ , u) − α ,(6)where α denotes a lump sum transfer to the foreign country; the identity for the foreigncountry is analogous, but with α entering positively.17

Constrained efficiency
It is assumed throughout that the foreign country is, for some reason—perhaps reflecting domestic political constraints or views on differential responsibilities to mitigate—
wholly constrained in its tariff and mitigation policies: T, S and S̄ are all taken as
fixed at arbitrary levels. Since Z = − RS ( P, S̄), fixity of the shadow price S̄ serves toallow some responsiveness of emissions there to policies in the unconstrained country,
through the impact, via world prices, on the foreign producer price P = ω + T. The
framework thus captures the looser impact of regulation relative to the absolute limits
set directly by emissions standards or cap-and-trade schemes.
The home country, however, is not so constrained, and the aim in what follows is tocharacterize the Pareto efficient choice of the instruments at its disposal, conditional on
the foreign country achieving welfare of at least Ū. Throughout, we assume the home
16This approach is similar to those of Neary and Roberts (1980) and Neary (1985).
17The combination of (5) and (6) ensures market clearing for the numeraire.

6

tariff t—and hence options regarding a possible BCAM—to be unconstrained. But weallow for the possibility that domestic mitigation policy (regulatory or explicit pricing)is fixed at some arbitrary level, reflecting possible constraints of the kind discussed in
the introduction.
The various problems of constrained Pareto efficiency to be considered18 are encompassed by the Lagrangean

L = u + ϕ(U − Ū )+ λ{e( p, u) + θk( p, P, s∗ , S∗ ) − g( p, s∗ , s) + srs ( p, s∗ ) − tm( p, s∗ , u) + α}
+ Λ{ E( P, U ) + Θk( p, P, s∗ , S∗ ) − G ( P, S∗ , S) + SRS ( P, S∗ ) − TM( P, S∗ , U ) − α}
+ µ{m( p, s∗ , u) + M( P, S∗ , U )},(7)

where ϕ is the multiplier on the constraint that foreign welfare be at least Ū, λ and Λ
those on the home and foreign income-expenditure identities respectively and µ that onmarket-clearing; bearing in mind too that aggregate emissions k ( p, P, s∗ , S∗ ) are givenby (4), while producer prices are given by p = ω + t and P = ω + T.

3.

The Design of Pareto Efficient BCAMs

The analysis addresses the issues raised above by looking in turn at a series of cases.
It begins with the simplest: that in which distributional concerns are eliminated by
supposing the cross-country transfer α to be efficiently chosen. After extending this tolook briefly at the case in which policy is driven not only by environmental concerns but
also by a need for revenue, we then characterize constrained efficient policies when thetransfer α is fixed at some arbitrary level (possibly zero), so that issues of cross-country
equity arise. This latter serves to raise the question, prominent in the policy debate, of
how imports from lower income countries—poorer, with less historical responsibility
and/or higher mitigation costs—should be treated in the design of BCAMs. Finally,and by way of contrast, we address briefly the better-understood case in which policyin the unconstrained country is wholly self-interested.

Efficient BCA in the absence of distributional concerns
We start with the case in which mitigation policy in the potentially BCA-imposing
country (as well as that abroad) is constrained at some arbitrary level, perhaps forreasons of the kind touched on in the Introduction, then turn to that in which it too,along with tariff policy, can be freely chosen.
18 Other than that in which revenue is a distinct concern.

7

Constrained mitigation policy
In this case the problem is thus to choose the home tariff t, world prices w (to ensure
market clearing) and the transfer α to maximize u subject also to U = Ū. For clarity—
and again temporarily—we take T = 0.
For this case, denoting the aggregate damage from a unit of emissions by θ A ≡θ + Θ:
Proposition 1. Whatever its domestic mitigation policy s∗ , and given T = 0, when international transfers are freely available, constrained Pareto efficiency requires a tariff in the homecountry ofrsp ( p, s∗ )t = (s∗ − θ A )m p ( p, s∗ , u)

!

!
∗)
R
(
P,
S
SP
− (S∗ − θ A ).
MP ( P, S∗ , U )(8)

Proof: The circumstances are as in Proposition 5 below, with the addition that thecross-country transfer α is freely chosen, implying from (7) that λ = Λ; the result then□follows from (17) below.

Two lessons follow from Proposition 1.19 First, it is only s∗ and S∗ that enter the rulein (8): all that matters is what the binding price is, not whether it is implemented by
explicit carbon pricing or by regulation. We return to the significance of this later, thepoint being established more completely by considering also, below, the Pareto efficientchoice of home mitigation policy.
The second and perhaps more striking implication of Proposition 1 is that theappropriate BCAM looks not to the difference between home and foreign (explicit or
shadow) prices, s∗ − S∗ , but to the difference between how far each deviates from thefirst best level θ A : a kind of ‘difference-in-differences’ BCAM. These differences (andwe will have in mind that binding prices are below first-best, so that emissions areunderpriced in both countries) are weighted by (positive) terms reflecting behavioral
responses that we look at shortly.20 Loosely speaking, the BCAM of Proposition 1 thus
weighs the effects of a higher home tariff in (a) increasing the domestic producer priceand thereby adversely expanding home production and so aggravating the welfare loss
from domestic emissions that are underpriced by an amount θ A − s∗ > 0 against thatof (b) reducing the producer price abroad and so beneficially reducing under-priced
foreign emissions. It is only if the (behaviourally-weighted) under-pricing abroad is
19 Since (8) applies whatever the direction of trade in the dirty good, it also illustrates a principle

for BCAM design that is well-recognized in theory but widely neglected in practice: that the case forremission on exports mirrors that for charging on imports.
20 Gros (2009), it should be noted, makes a similar observation in passing (around his equation (30)),
for the carbon tax case.

8

greater than that at home that a positive tariff is appropriate.
A more precise intuition follows on noting that—as shown in Appendix A—the‘difference-in-differences’ BCAM in (8) implies that−t

 dZ dmdz= (θ A − S∗ ) −− (θ A − s∗ ) ,
dt
dt
dt

(9)

where the derivatives are conditional on the levels of welfare and binding prices. On
the left of (9) is the marginal welfare loss from the distortionary impact of the tariff
itself (as a positive number), which in the absence of environmental concerns, would
call for the tariff to be set to zero. On the right is the marginal gain from the reductionin underpriced emissions abroad (−dZ/dt > 0) reduced by the marginal loss from theincrease in under-priced domestic emissions (dz/dt > 0). The ‘difference-in-differences’
BCAM simply equates these marginal welfare costs and benefits—-from traditional
production inefficiency and environmental harm—of increasing the home tariff.
Returning to (8), the home weight in the generalized BCAM of Proposition 1 can be
written in perhaps more transparent elasticity form asz
rsp ( p, s∗ )ϵ(z, p)=,m p ( p, s∗ , u)x ϵ(e , p) e p + ϵ( x, p)
p

(10)

x

where ϵ(c, d) denotes the (absolute value of) the elasticity of c with respect to d. Assuming domestic emissions to be under-priced, so that s∗ < θ A , the charge on imports
is thus lower: the greater is the emissions intensity (dirtiness) of production at home(z/x), the more responsive are home emissions to a tariff-induced increase in the priceof the dirty good and the less price responsive are domestic compensated demand andproduction (hence the smaller the impact of the tariff on the foreign producer price)—all
factors that point to environmental benefits from a high tariff that are modest relative tothe associated non-environmental distortions. Conversely, the import charge is greater
the more strongly the same conditions apply abroad.
In general, the generalized Pareto-efficient BCAM in Proposition 1 looks to differences across countries in both binding prices and emissions intensity, mediatedby behavioral responses in consumption and productions. There are circumstances,
however, in which it reduces to simpler forms, and ones directly related to practicalpolicy actions and proposals along the lines of those described in the Introduction.
To see this, suppose that there are fixed coefficients in the production of thedirty good in the sense that emissions are simply proportional to output, so thatz = −rs ( p, s) = br p ( p, s) = bx for some constant b > 0. The most compelling instanceof this is that in which the non-numeraire good is fossil fuel, though one can conceive
too of other goods for which abatement technologies are severely limited. This proves9a particularly instructive special case. It implies (since then r ps = −br pp ) that the homeweight in (8) of Proposition 1 becomesrsp ( p, s∗ )b=,∗m p ( p, s , u)
1 − (e pp /r pp )(11)

and of course similarly abroad. The weights are thus strictly less than unity so long as
there is some price responsiveness in compensated demand. Through the first term in(8), the Pareto efficient BCAM thus does not fully equalize the treatment of imports anddomestic production: it leaves domestic producers at some competitive disadvantage.21
By the same token, however, it also gives less than full credit for the binding price paidby exporters abroad.
Pursuing this special case of fixed emission intensities, suppose further that compensated demand for the dirty good is completely inelastic, so that e pp = EPP = 0. The
general CBAM of Proposition 1 then becomest = (s∗ − θ A )b − (S∗ − θ A ) B .

(12)

From this, the alternative forms of BCAM based either on only differences in binding
prices (along similar lines to the EU and UK, though there of course not only for explicit
prices) or on only differences in technology (as in the U.S. Foreign Pollution Fee Act)
emerge as distinct special cases of the generalized BCAM of Proposition 1:
Proposition 2. Under the conditions of Proposition 1, suppose further that, in both countries,
emissions intensities are technological constants and that compensated demands are perfectly
inelastic. Then:(a) If technologies in each country are the same, constrained Pareto efficiency requires a tariffin the home country oft = (s∗ − S∗ ) B .

(13)(b) If binding prices in each country are the same, at say s′ , constrained Pareto efficienc
yrequires a tariff in the home country oft = s̃( B − b) ,(14)

where s̃ ≡ θ A − s′ .
Part (a) shows that even when mitigation policy in the home country is set at anarbitrary level, if technologies are the same at home and abroad then the efficient
21This is similar to a result of Kortum and Weisbach (2021) to the effect that constrained Pareto

efficiency requires a tax on the home extraction of fossil fuels that is less than fully rebated on export.

10

BCAM has the key feature of those adopted and proposed in Europe, but extended also
to regulatory measures: it charges an amount equal to the excess of the home over theforeign price—explicit or shadow, efficiently set or not—multiplied by a mechanicalindicator of carbon content.
Part (b) shows, in contrast, that—again whatever the forms and levels of mitigation
policies in the two countries—a BCAM based only on differences in emissions intensity,
as under the U.S. Foreign Pollution Fee Act, is appropriate. (And this is so, even if
the foreign country mitigates by explicit carbon taxation.) The result also indicatesthe correspondingly appropriate fee (which the proposal leaves open): a charge equal
to the amount by which the common binding price falls short of the first best carbon
price.22
While showing that both can be rationalized on efficiency grounds, the juxtapositionof parts (a) and (b) of Proposition 2 highlights the differing circumstances in which
BCAMs based on differences in binding prices and those based on differences inemissions intensities are warranted. The former, loosely peaking, is appropriate whentechnologies are sufficiently similar that the key difference that the BCAM needs toadjust for is the ambition of mitigation efforts, as reflected in binding prices. The latter
is appropriate when mitigation efforts are broadly similar, but emissions intensities
differ.

Unconstrained mitigation policy
None of the results above assumes mitigation policy in the BCA-imposing country to
be in any sense optimal. If, however, that mitigation policy is unconstrained then it
straightforward to see that efficiency requires that s∗ be set at the first best level of θ A .
It then follows from Proposition 1 that:
Proposition 3. Under the conditions of Proposition 1, if mitigation in the home country isefficient—whether achieved by price or regulation—then s∗ = θ A , and constrained efficiency
requires a tariff in the home country of
RSP ( P, S∗ )t = (s − S )MP ( P, S∗ , U )
∗

∗




.

(15)

□

Proof: Appendix B.

The appropriate BCAM is thus again the difference in binding prices multiplied by
22 Identical binding prices are of course consistent with differing emissions intensities if technologies

differ.

11

a term reflecting induced emission responses abroad.23 This resembles the result in
Proposition 2: the key difference is that the result there requires identical technologies
while Proposition 3 instead requires efficient mitigation in the BCA-imposing country.
Taken together, Propositions 1 and 3 make clear that—whatever the nature of the
mitigation measures taken at home and abroad—the Pareto efficient BCAM treats
explicit carbon prices and regulatory shadow prices associated with mitigation policies
in exactly the same way, taking account only of whichever is binding on production
decisions. In this broad sense, it is thus the approach typically found in draft U.S.
legislation, which gives credit for non-price mitigation abroad, that is in line withconstrained Pareto efficiency, not that in the EU Directive and rules proposed in the
U.K., which gives credit only for explicit carbon prices. Indeed there is no trace inthese results of the ‘leveling the playing field’ argument for imposing a financial charge
on imports to match that on domestic production. Such a charge—in line with the’destination principle’ that is the worldwide standard for commodity taxation—serves
production efficiency by tilting consumption and hence production towards firms withlower private marginal costs; in the climate context, however, the wider efficiency
perspective requires balancing such concerns against environmental harm.

When revenue matters
An obvious difference between carbon pricing and direct regulatory measures, suchas non-tradable performance standards, is that the former collects tax revenue while
the latter do not, a potentially significant difference when policy makers have not only
environmental concerns but also a need to finance valued public expenditures. In
that case, one might expect the appropriate form of BCAM to differ between the two
approaches; here we look briefly24 at precisely how.
Suppose then that the marginal value of public funds in the home country is 1 + v,
with v > 0, meaning that, at the margin, a dollar transferred from the public sector
conveys a social gain of v. For this case, it is shown in Appendix C that:
Proposition 4. Under the conditions of Proposition 1, but now valuing home revenue fromthe tariff and any carbon tax by the marginal value of public funds 1 + v, constrained Pareto
23This generalizes Proposition 3 of Keen and Kotsogiannis (2014), replacing explicit prices there bybinding prices.
24The brevity is partly because the starkness of the difference between the two is some way fromreality: many prominent mitigation measures involving other than the direct pricing of emissions—suchas the tax credits of the 2022 Inflation Reduction Act of 2022—also have revenue consequences. Dealing
with that, however, would require adding additional structure to the simple setting above.

12

efficiency requires—whatever the mitigation policy is there25 —that the home tariff be1t=(1 + v )(∗s + vs − θ

A



rsp
mp

!



∗

− S −θ

A



RSP
MP

!)

v
dlnp
+
ϵ(m, p)
1+v
dt

! −1
.
(16)

Comparing this with (8) of Proposition 1, there are three adjustments to be made.
First, the ‘difference-in-differences’ BCAM term is scaled down by the factor 1 + v:
intuitively, the core rationale for the BCAM is to prevent emissions reductions at homebeing offset by increased emissions abroad—but that is less of a concern when domestic
emissions contribute valuable revenue. Second, the domestic under-pricing is taken
to be reduced by vs: this reflects the adverse distortionary impact of the carbon tax
in reducing the wider domestic tax base.26 Third, an additional term appears that isessentially a familiar Ramsey-type inverse elasticity capturing the pure revenue-raisingrole of the tariff that would remain even in the absence of environmental concerns. This
last points to higher tariff if the dirty good is the importable, and to a lower export
subsidy if it is the exportable; the latter case carries echoes of the feature of the EU
BCAM, otherwise hard to rationalize as a matter of principle, that no carbon rebates
are provided to exports.27

Cross-country distributional concerns
To allow for distributional concerns, we now relax the assumption that the internationaltransfer α is freely available, and denote by σ the ratio Λ/λ: thus σ > 1, for example,
means that welfare is increased by a transfer from the unconstrained home to theconstrained foreign country. This case thus captures the concern which, as noted at theoutset, has been prominent in the BCAM debate, as to the proper treatment of imports
from lower income countries: the U.S. Market Choice Act, for instance, would exclude
least developed countries.28 In the absence of income effects for the dirty good andnormalizing eu = EU , it is easily seen29 that σ = ϕ, so that σ can also be thought of,somewhat loosely, as the welfare weight attached to the foreign country.
25 Because of the value attached to tax revenue, carbon taxation is readily shown to welfare-dominate

regulatory measures in this setting (given a stability condition, eu − te pu , along the lines of Hatta (1977)
and Neary (2006)).
26This becomes clearer on supposing taxation to be the binding instrument: with s∗ = s the part ofthe BCAM relating to domestic underpricing becomes s − θ A /(1 + v), which corresponds to the resultof Sandmo (1975) that when environmental taxes distort they are optimally set with an environmental
component correspondingly below the Pigovian level.
27 Strikingly, the U.S. Market Choice Act would remit on exports.
28 Section 1994.
29This follows from the first order conditions on u and U from (7): with e
pu = EPU = 0, these are
1 + λeu = 0 and ϕ + ΛEU = 0. Non-zero income effects complicate the link with ϕ by introducing
feedback effect from international transfers to tariff revenues and producer prices.

13

Taking again the case in which the home country’s mitigation policy is fixed at somearbitrary level, and now allowing also for T ̸= 0, efficient tariff and BCAM designed ischaracterized in:
Proposition 5. Whatever its domestic mitigation policy s∗ , constrained Pareto efficiency
requires a tariff in the home country of






rsp
M
RSP
∗
AA
∗
AA
+ (s − θ )t = σT − (σ − 1),
− (σS − θ )
MP
mp
MP
|
{z
}

(17)

BCA term

where θ AA ≡ θ + σΘ.
Proof: Appendix D.
There are thus two additional terms, and some recasting of the BCAM, relative to thespecial case in Proposition 1. The first of these, σT, reflects the possibility of a directtransfer, albeit distortionary, between the two countries. Suppose for instance that T <0, meaning that the foreign country sets an export tax. If σ = 1, so that distributional
concerns vanish, constrained efficiency would require (absent environmental concerns)
setting t = T < 0, meaning an import subsidy that exactly offsets that export tax, andeliminates the production inefficiency that it otherwise causes abroad. If, on the other
hand, σ > 1, capturing the case in which the foreign country is of relatively low income,then Pareto efficiency calls for a home import subsidy that more than offsets foreign’sexport tax: the rationale is that this effectively transfers revenue from the higher income
country (through the subsidy it offers) to the lower income country (through theadditional export tax revenue it collects from consequently increased demand for thoseexports), more than offsetting, in welfare terms, the induced production inefficiency.
The second additional term comes into play only when σ ̸= 1. With σ > 1, it calls(given M, MP < 0), all else equal, for t < 0: an import subsidy in the home country.
The intuition is simply that by this means the high income home country engineers aterms of trade effect that favors the low income country.
The appropriate BCAM term itself alters in two ways. The first is that the damagelevel taken as reference in calculating the difference terms is no longer the simple sumθ A = θ + Θ but instead an adjusted aggregate θ AA = θ + σΘ, the adjustment being to
attach greater weight to the damage suffered by the low income country. The second
is that additional credit is given to mitigation efforts in the low income country, thesebeing treated as if the binding price were not S∗ but σS∗ . The implications are especiallyclear in the circumstances of Proposition 2 (a)—identical, constant emissions intensities

14and inelastic compensated demand—as the BCAM term then becomes(s∗ − σS∗ ) B .

(18)

The appropriate BCAM thus takes the form of applying the difference in binding prices
to embodied emissions, with additional credit being given for mitigation efforts in low
income countries.30

Self-interest
The focus so far on Pareto efficiency does not presume that the unconstrained country
must somehow entirely set aside its own self-interest. The implication of Proposition5, for example, is that if (17) does not hold then the home country could change its
tariff policy to make itself better off—with the added feature that it could do so without
making the foreign country any worse off, even without any international transfers.
It may of course be, however, that the unconstrained country simply attaches noweight to the well-being of the other country, in which case it will be able to do evenbetter for itself than by following Proposition 5. This is the special case of the general
structure above in which ϕ = 0. Simplifying for clarity by supposing that there are
fixed emissions intensities in both countries, it follows immediately from Proposition 5
that:
Proposition 6. If wholly self-interested, the unconstrained country, whatever its domestic
mitigation policy—and assuming there to be fixed emissions intensities in both countries31 —will
set a tariff oft = M ( MP )−1 + θB − (θ − s∗ )b .

(19)

Pure self-interest thus calls, beyond a standard optimal tariff, for charging the carbon
content of imports at a rate which reflects the damage that production abroad causes
at home, reducing this to the extent that damage from production at home is underpriced.32 Comparing with the corresponding Pareto efficient BCAM in (12), and with
the EU- and U.S.-style special cases of Proposition 2, the key difference (beyond accountbeing taken only of damage at home, not globally) is that no credit is given for mitigation
costs abroad, S∗ B.
30 For completeness: it is straightforward to show—by the same steps as proving Proposition 3(Appendix B)—that when mitigation policy is freely variable Pareto efficiency requires, as one would
expect, that s∗ = θ AA .
31We naturally assume too that the home country cannot transfer resources to itself.
32The final term corresponds to the result of Neary (2006), implicitly setting B = 0: imports should besubsidised if damage is underpriced domestically.

15

4.

Concluding Remarks

The adoption of BCAMs opens a new frontier in efforts to use of tax-like tools to
address climate change—efforts that so far have been, to put it mildly, disappointing.
Understanding how they are to be best designed is thus of some importance. In thatspirit, the results here suggest two principles for the desirability and structure of Pareto
efficient BCAMs.
The first, and conceptually more straightforward, is that efficiency arguments for a
BCAM and results on its appropriate structure apply with perfect symmetry to explicit
carbon prices and shadow prices associated with mitigating regulatory measures. In
this sense—interpreted, if not literally, at least as signaling a need to reflect all forms of
mitigation adopted abroad—it is the proposals in the U.S., not the actions in the EU
and UK, that emerge as the more defensible on efficiency grounds.
The second principle is that a generalized ‘difference-in-differences’ form of BCAM
is required for Pareto efficiency: whatever the ambition and nature of the mitigation
policies adopted, in the BCA-imposing country or abroad, efficiency calls for a BCAM
based not on the difference between (explicit or shadow) prices at home and abroadbut on the difference between how far each deviates from the first-best level.
Taken together, these two principles provide a unifying framework within whichalternative forms of BCAM can be evaluated and, potentially, designed. An approachbased on applying the difference in prices across countries to embodied imports, forinstance—as in the EU and proposed for the UK, but, importantly, extended to allow for
shadow prices—is constrained Pareto efficient if differences in technology are negligible;
and this is so whatever the level of ambition of domestic mitigation policies. The very
different approach of applying a single charge to differences in the carbon content ofdomestic and foreign products, on the other hand—as in the Foreign Pollution Fee
proposed in the U.S.—is constrained efficient if mitigation efforts abroad are comparable
to those in the BCA-imposing country, with guidance also emerging here as to theappropriate level of that charge (as the excess of marginal climate damage over thecommon shadow price). Not the least benefit of encompassing these very different
approaches within a common framework, may, in due course, be in seeing how best tolink the variety of BCAM schemes towards which the world may well be headed.
The analysis above is of course subject to many limitations. It abstracts, for instance,from considerations of imperfect competition and (within-country) firm heterogeneity,and from the diversity and complexity, in design and coverage, of the mitigation
instruments used in practice. It sets aside too the slew of informational and otherpotential obstacles to implementation of any form of BCAM, including in relation to16

explicit carbon prices33 but no doubt still greater in terms of other forms of mitigation.
These problems, however, are by no means evidently insoluble. And the place to begin
the design of BCAMs and address the practical problems they pose is surely with clarity
on the principles it is intended to apply.

33This requires taking some view, for example, as to when a coal tax or routine excise of fuels is to be

counted as a carbon tax.

17

Appendices
Appendix A. Derivation of equation (9)Multiplying (8) of Proposition 1 by m p (dp/dt) = dm/dt and recalling that rsp = −z p
gives




dp
∗
A dz
∗
A
tm p
= − s −θ− S − θ RSP
dt
dt



mp
MP



dp
.
dt

(A.1)

For the second term on the right of (A.1), note from the market-clearing condition (5)
that



mp
dP
dω
,
=
=−
dt
dt
m p + MP

(A.2)

dp
dω
MP
=
+1 =
.
dt
dt
m p + MP(A.3)

and hence, since p = w + t,

Substituting (A.3) into that second term gives


 
 



mp
MP
dP
∗
A
∗
A
− S − θ RSP,= S − θ RSP
MP
m p + MPdt

(A.4)

use also having been made of (A.2). Recalling that RSP = − ZP , rearranging gives□

equation (9).

Appendix B. Proof of Proposition 3
Since (15) is immediate given s∗ = θ A , it suffices to verify that s∗ will indeed be set
equal to θ A whether the unconstrained country uses explicit pricing or regulation.
Take first the case in which explicit pricing is used at the margin. Then s∗ becomes s
everywhere in the Lagrangean (7) and g( p, s∗ , s) in (3) becomes simply r ( p, s). The firstorder condition on s is then, after canceling terms in rs , dividing by λ (=Λ, transfers
being freely chosen) and collecting terms in rssµA(s − θ )rss + t −r ps = 0 .
λ

(B.1)

From (D.4) in Appendix D below (with σ there set to unity, and so θ AA = θ A ),
t−

rsp
µ
= (s − θ A )
,
λ
mp

18

(B.2)

substituting which into (B.1) and collecting terms givesh(rsp )2 i(s − θ A ) rss +
= 0.
mp

(B.3)

Since m p = e pp − r pp < 0, the term in square brackets can be written as [rss e pp −(r pp rss − (rsp )2 )]/m p ; concavity of the expenditure function and strict convexity ofr ( p, s) imply that rss e pp ≤ 0 and r pp rss − (rsp )2 > 0, and it then follows from (B.3) thats = θ A.
Turning to the case in which the home country uses regulation at the margin, so thats∗ in (7) becomes s̄, noting that thengs∗ ( p.s̄, s) = −(s̄ − s)rss ( p, s̄) ,(B.4)

it is readily seen that the first order condition on s̄ takes the same form as (B.1) but withs replaced by s̄. It follows that efficiency in this case requires s̄ = θ A as claimed.

□

Appendix C. Proof of Proposition 4
The Lagrangean for the problem described in the text is given by

L = u + ϕ(U − Ū )+ λ{e( p, u) + θk − g( p, s∗ , s) + (1 + v)srs ( p, s∗ ) − t(1 + v)m p ( p, s∗ , u)
+ E( P, U ) + Θk − G ( P, S∗ , S) + SRS ( P, S∗ )}
+ µ m p ( p, s∗ , u) + MP ( P, S∗ , U ) .

(C.1)

The first order condition on ω is
λ{e p − θ (rsp + RSP ) − g p + (1 + v)srsp − (1 + v)tm p + EP − Θ(rsp + RSP ) − GP + SRSP }+µ(m p + MP ) = 0 .

(C.2)

Using (5) to cancel terms, and using also (D.2) from Appendix D below and its foreign
analogue, this becomes, on collecting terms,
n



o
λ
s∗ + vs − θ A rsp + S∗ − θ A RSP − (1 + v)tm p + µ(m p + MP ) = 0 .

(C.3)

The first order condition on t, using (D.2) and collecting terms, is
n

o
∗
A
λ
s + vs − θ rsp − v m − (1 + v) tm p + µm p = 0 .

(C.4)

19

Solving (C.4) for µ, substituting into (C.3) and noting from (A.3) thatm(m p + MP )=m p MPdlnp
ϵ(m, p)
dt

! −1
,

(C.5)

□

the result simplifies to (16) of Proposition 4.

Appendix D. Proof of Proposition 5
From (7), the first order condition on ω is
λ{m + (s∗ − θ )rsp − θRSP − tm p } + Λ{ M + (S∗ − Θ) RSP − Θrsp − TMP }+ µ(m p + MP ) = 0 ,(D.1)

use having been made of m ≡ e p − r p and M = EP − R P together with the implicationof (3) thatg p = r p − (s∗ − s)rsp ,(D.2)(and similarly abroad). Dividing by λ, recalling that θ AA ≡ θ + σΘ, and collecting
terms in rsp and RSP , (D.1) becomesm + (s∗ − θ AA )rsp − tm p + σM + (σS∗ − θ AA ) RSP − σTMP +µm p + MP = 0 . (D.3)
λ

The first order condition on t, again using (D.2), dividing by λ and collecting terms in
rsp is(s∗ − θ AA )rsp − tm p +µ

mp = 0 .
(D.4)
λ
Solving (D.4) for µ/λ and substituting into (D.3), equation (17) of Proposition 5 follows
after some simplification, dividing by MP and recalling from (5) that m = − M.

20□

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