Carbon Tariffs: Effects in
Settings with Technology
Choice and Foreign
Production Cost Advantage
David F. Drake
Working Paper
13-021
November 17, 2015
Copyright © 2012, 2015 by David F. Drake
Working papers are in draft form. This working paper is distributed for purposes of comment and
discussion only. It may not be reproduced without permission of the copyright holder. Copies of working
papers are available from the author.
Carbon Tariffs: Effects in Settings with Technology
Choice and Foreign Production Cost Advantage
David F. Drake
Harvard Business School, Harvard University, Boston, MA 02163
ddrake@hbs.edu
It is widely believed that carbon leakage—offshoring and foreign entry in response to carbon regulation—
increases global emissions. It is also widely believed that a carbon tariff—imposing carbon costs on imports
entering the emission-regulated region—would eliminate carbon leakage. However, neither of these assertions
necessarily holds. Under a carbon tariff, foreign firms with a production cost advantage adopt clean technology at a lower emissions price than domestic firms, and foreign entry can increase in emissions price when
foreign firms hold this edge. Further, domestic firms conditionally offshore production despite a carbon tariff,
but doing so implies that they adopt cleaner technology. Therefore, carbon leakage can arise under a carbon
tariff but, under mild conditions, it decreases global emissions. Due in part to this clean leakage, imposing a
carbon tariff is shown to decrease global emissions. However, domestic firm profits can increase, decrease, or
remain unchanged due to a carbon tariff. This suggests a carbon tariff’s principal benefit is not to protect
domestic firm profits, as some argue. Rather, it is to improve emissions regulation efficacy, enabling emissions price to be used to reduce global emissions in many settings in which it would otherwise fail to do so.
Revised November 17, 2015
1.
Introduction
With carbon regulation driving projected production cost increases in excess of 40% in some industries (Drake et al. 2010, Ryan 2012), it can endow facilities located outside the regulated region with
a windfall cost advantage. This cost advantage can enable competitors outside the regulated region
(i.e., “foreign” firms) to increase their penetration into the regulated (i.e., “domestic”) region. Carbon regulation can also lead domestic firms to shift facilities offshore to avoid carbon-related costs.
Such foreign entry and offshoring are both sources of carbon leakage—the displacement of production from the regulated region to offshore locations due to an increase in (or implementation of) an
emissions price. Carbon leakage could potentially be mitigated by border adjustments—tariffs on
the carbon emissions of imported goods that would incur carbon-costs if produced domestically.
It is widely reported in the literature that carbon leakage increases global emissions, offsetting
some or all of the regulation’s emission improvements (e.g., Babiker 2005, Demailly and Quirion
2006, Ponssard and Walker 2008, Di Maria and van der Werf 2008, Fowlie 2009, Fowlie et al.
2014). It is also widely perceived that the implementation of a symmetric border adjustment will
eliminate the threat of leakage (e.g., Barry 2009; Barber 2010, Ponssard and Walker 2008, Fowlie
et al. 2014). However, neither of these assertions necessarily holds if firms choose from technologies
that vary in emissions intensity and if production costs are lower in the foreign region.
1
David Drake: Carbon tariffs, technology choice and foreign cost advantage
2
I show that a symmetric border adjustment does not, in general, eliminate carbon leakage from
foreign entry or offshoring. Rather, when foreign firms hold a production cost advantage, they
adopt clean technology at a lower emissions price than domestic firms, and entry conditionally
increases in emissions price intervals in which foreign firms hold this advantage. Further, conditions
are established under which domestic firms shift production offshore despite a border adjustment,
adopting cleaner technology when they do so. As a consequence, when carbon leakage does occur
under a border adjustment, it implies that offshore technology is cleaner than that used domestically. Therefore, leakage conditionally decreases global emissions when it arises under a border
adjustment, contrasting both the assertion that the threat of carbon leakage is eliminated by border
adjustment and the assertion that carbon leakage increases global emissions.
Without a border adjustment, global emissions increase or remain unchanged in emissions price
in any setting in which foreign firms compete or domestic firms offshore. However, under mild
conditions, global emissions decrease in emissions price across all settings with a border adjustment.
As a consequence, imposing a border adjustment decreases global emissions relative to emissions
regulation without a border adjustment. Domestic firms’ profits, on the other hand, can increase,
decrease, or remain unchanged due to border adjustment. This is pertinent because some argue that
border adjustments are protectionist measures disguised as climate policy (Beattie and Hill 2009,
The Economist 2007). These results suggest otherwise: a border adjustment’s principal benefit is
not to protect domestic firms’ profits, but to improve the efficacy of emissions regulation, enabling
it to deliver reduced global emissions in many settings in which it would otherwise fail to do so.
2.
Relation to the Literature
In the Operations Management literature, Krass et al. (2013) and Drake et al. (2015) both consider
technology choice under emissions regulation in noncompetitive settings, and İşlegen and Reichelstein (2011) estimate the adoption threshold for carbon capture and storage in power generation.
However, foreign entry, offshoring, and asymmetric emissions regulation, of primary interest here,
are not considered (or pertinent) in their context. İşlegen et al. (2015) study the impact of emissions
price variability on firms’ facility location and production decisions in a model of international
trade but largely do not address technology choice. Also in a setting without technology choice,
Sunar and Plambeck (2015) explore the impact of emissions allocation rules when products are
co-produced in a fixed ratio. They find that border adjustment can conditionally increase or strictly
decrease global emissions, depending on the allocation rule chosen.
In the Economics literature exploring carbon leakage, most focuses on leakage due only to foreign
entry (e.g., Di Maria and van der Werf 2008, Fowlie 2009). Di Maria and van der Werf (2008) study
David Drake: Carbon tariffs, technology choice and foreign cost advantage
3
leakage through a model of competition between asymmetrically regulated regions, showing that
a regulated region’s ability to change technology attenuates leakage. Fowlie (2009) studies leakage
when firms operate different but exogenous technologies, finding that leakage eliminates two-thirds
of the emissions reduction obtained by a uniform policy. Babiker (2005) considers leakage in terms
of both entry and offshoring in a numerical study, finding that carbon leakage increases global
emissions by 30%. Of these studies, none consider border adjustments or endogenize the number
of foreign entrants and only Di Maria and van der Werf (2008) allow for technology choice. Also
in the economics literature, both Ryan (2012) and Fowlie et al. (2014) study emissions regulation
as it relates to the US cement industry, estimating a structural model that endogenizes firm entry,
exit, and capacity decisions. However, neither incorporates technology choice—which is central to
the present paper—and Ryan (2012) does not explore leakage or border adjustment. Keen and
Kotsogiannis (2014) explore the interaction between climate and trade policy (including border
adjustment), but they do so at the sectoral-, rather than firm-, level and without technology choice.
Within the Policy literature, a key issue relates to the legality of border adjustments as a leakagemitigating mechanism (e.g., Grubb and Neuhoff 2006, van Asselt and Biermann 2007, de Cendra
2006). Most conclude that border adjustments are legal, conditional on the elimination of the free
allocation of allowances (Grubb and Neuhoff 2006, de Cendra 2006). Others conclude that border
adjustments may only be legal for inputs directly incorporated into finished goods, such as clinker
into cement (Biermann and Brohm 2005, van Asselt and Biermann 2007). In terms of border
adjustment design, Grubb and Neuhoff (2006) propose a symmetric tariff so that imports would
incur the same carbon cost that they would have incurred had they been produced domestically.
Ismer and Neuhoff (2007), on the other hand, propose a flat carbon cost based on the emissions
intensity of the “best available technology.”
3.
Firm Decisions and Performance without Border Adjustment
Under existing emissions regulation, domestic production incurs emissions costs while offshore
production does not, altering the competitive balance between domestic and foreign firms.
3.1.
Model development
A regulator imposes an emissions price ε for each unit of emissions generated through domestic
production. Through Cournot competition, nd domestic firms compete with nf foreign firms. The
domestic market is assumed to be mature prior to the implementation of emissions regulation,
with Nd incumbent domestic firms; domestic firms can choose to exit, but no new domestic firms
enter the market due to emissions regulation, i.e., nd ≤ Nd . Each domestic firm chooses a location,
David Drake: Carbon tariffs, technology choice and foreign cost advantage
4
l ∈ {r, o}, where r indicates production in the regulated region and o indicates production offshore.
Domestic firms choosing to offshore incur fixed cost Fo ≥ 0. Foreign firms enter the domestic market
only if they can earn at least operating profit Fe > 0, where Fe represents a fixed entry cost. Each
unit imported into the domestic market incurs transport cost τ > 0.
Both domestic and foreign firms choose a non-proprietary production technology k ∈ {1, 2}, with
unit production and capital recovery cost γk > 0 and emissions intensity αk ≥ 0. Wlog, technology
1 is the more emissions-intensive technology, α1 > α2 . Emissions intensity for a given technology is
assumed to be the same whether it is utilized domestically or offshore, though offshore production
generates an additional ατ > 0 emissions per unit through transport. A discount factor δ represents
the relative production and capital recovery cost in offshore versus domestic regions.
Three empirically-supported assumptions are made with respect to technology costs:
Assumption 1. The production and capital recovery cost for a given technology is less in offshore
regions than in the domestic region; δ ∈ (0, 1) so that δγk < γk , ∀k ∈ {1, 2}.
Assumption 1 states that production costs with a given technology are less in some set of offshore
regions than in the emissions-regulated region. Practical examples support such an assumption. In
cement, for example, Figure 1 illustrates a fully-loaded, structural production and capital recovery
cost advantage for regions likely to be importers to the EU under carbon leakage scenarios.
Cost per tonne (Euros)
80
70
TransportcosttoEU
Productionandcapitalrecoverycost
60
50
40
30
20
10
0
EU
North
Africa
Saudi
Arabia
Turkey
India
China
Production location
Figure 1
Regional production and transport costs per tonne of cement (source: Boston Consulting Group 2008)
Assumption 2. The domestic production cost of the dirty technology is less than the transport
plus offshore production cost of the dirty technology; γ1 < δγ1 + τ .
This assumption is supported by the costs in Figure 1 and implies that domestic firms produce
locally if emissions are unregulated. In general, the assumption is reasonable for carbon-regulated
sectors (domestic regulation of the sector would not be imposed if it operated offshore when ε = 0).
David Drake: Carbon tariffs, technology choice and foreign cost advantage
5
Assumption 3. The production and capital recovery cost of the dirty technology is less than the
production and capital recovery cost of the clean technology; γ1 < γ2 .
Given α1 > α2 , Assumption 3 ensures that technology 1 is not trivially discarded. Returning to
cement as an example, the most promising abatement technology in that sector is carbon capture
and storage (CCS). In accord with Assumption 3, a CCS kiln is more costly in terms of investment
and operating costs than traditional precalciner kilns (European Cement Research Academy 2012).
The total operating cost of domestic firms’ preferred technology will be referred to as cd (l, ε),
with production cost and emissions intensity γd and αd , respectively (with l and ε notation dropped
from γd and αd for brevity). Without a border adjustment, the total operating cost of foreign firms’
preferred technology will be referred to as cf , with production cost and emissions intensity γf and
αf . Table 1 summarizes cost and emissions parameters.
Parameter
ε
Fo , Fe
γd , γf
αd , αf
τ , ατ
δ
cd (l, ε), cf
Table 1
Description
Price per unit of CO2 emissions
Fixed offshoring cost and fixed entry cost, respectively
Domestic and foreign firms’ production and capital recovery cost
Emissions intensity of domestic and foreign firms’ preferred technology
Per-unit transport cost and emissions intensity, respectively
Discount factor for offshore production
Total unit cost of domestic and foreign firms’ preferred technology
Cost and emissions parameters in setting without border adjustment
Each domestic firm produces xl units with its preferred technology/location option. Each foreign
firm produces y units with its preferred technology. The market clears at price P (nd , nf , xl , y) =
A − b (nd xl + nf y), with A > cd (l, ε) to avoid cases where no competitor produces, and with b > 0.
Each domestic firm competing in the market maximizes operating profit by solving
πd (nd , nf , l, y, ε) = max P (nd , nf , xl , y) xl − cd (l, ε)xl ,
xl ≥0
(1)
while each foreign competitor maximizes operating profit by solving
πf (nd , nf , xl ) = max P (nd , nf , xl , y) y − cf y.
y≥0
(2)
All proofs, including the joint concavity of (1) and (2) in xl and y, are given in Appendix B.
3.2.
Equilibrium technology choice
Without border adjustment, foreign firms select technology k to minimize offshore operating cost
δγk + τ . By Assumption 3, this implies that foreign firms operate technology 1, with cf = δγ1 + τ .
David Drake: Carbon tariffs, technology choice and foreign cost advantage
6
If domestic firms operate in the regulated region (i.e., if l = r), they prefer technology k that
minimizes domestic operating costs γk + αk ε, with εd2 = (γ2 − γ1 ) / (α1 − α2 ) representing the breakeven emissions price between technology 1 and 2 in the regulated region. However, if domestic firms
offshore (i.e., if l = o) then, as with foreign firms,
⎧
⎪
⎨γ1 + α1 ε
cd (l, ε) = γ2 + α2 ε
⎪
⎩
δγ1 + τ
they prefer the technology 1. Therefore,
if l = r and ε < εd2
if l = r and ε ≥ εd2
if l = o.
These technology choices imply that, without a border adjustment, the regulator cannot incentivize clean technology adoption if domestic firms offshore or exit at a lower emissions price than
they adopt clean technology. This has practical implications. The threshold at which cement firms
would offshore from Europe, for example, is estimated to be significantly less than the adoption
threshold for CCS cement kilns (US Energy Information Administration 2011, and Boston Consulting Group 2008), preempting the regulator’s ability to incentivize clean technology adoption.
3.3.
Equilibrium location, exit, and entry
The market structure that evolves in equilibrium is determined by three decisions—domestic firms’
location decisions; domestic firms’ exit decisions; and foreign firms’ entry decisions. Emissions price
and fixed offshoring cost thresholds co-determine each of these decisions. Star notation will be used
to indicate equilibrium decisions. For brevity, star notation will also indicate dependency on fixed
offshoring cost and emissions price, so that a∗ implies a∗ (Fo , ε).
Domestic offshoring For domestic firms to offshore in equilibrium, ε must be sufficiently large
and Fo sufficiently small that domestic firms can operate more profitably outside the regulated
region than in it. Where πd,r (·) is each domestic firm’s operating profit when producing in the
regulated region, and πd,o (·) is its operating profit when producing offshore, domestic firms offshore
at εo (Fo ) = {ε|πd,r Nd , n∗f , r, y ∗ , ε = πd,o Nd , n∗f , o, y ∗ , ε − Fo }. Relocation is cost prohibitive if
fixed offshoring costs exceed the operating profit that each domestic firm can earn when operating
outside the regulated region, i.e., if Fo ≥ F o , where F o = {Fo |πd,o Nd , n∗f , o, y ∗ , ε − Fo = 0}. In
equilibrium, domestic firms choose location l∗ = o if ε ≥ εo (Fo ) and Fo < F o , and l∗ = r otherwise.
Domestic exit If Fo ≥ F o , domestic firms abandon the market at the emissions price where they
can no longer operate profitably domestically; εexit = {ε|πd,r Nd , n∗f , r, y ∗ , ε = 0}. Therefore, n∗d = 0
if ε ≥ εexit and Fo ≥ F o , and n∗d = Nd otherwise.
Foreign entry Foreign firms compete operating profits down to the minimum level for entry such
that n∗f = {(nf )+ |πf (n∗d , nf , x∗l ) = Fe }.
David Drake: Carbon tariffs, technology choice and foreign cost advantage
7
Lemma 1. Without border adjustment, the equilibrium number of foreign entrants is
+
A − cf − n∗d (cf − cd (l∗ , ε))
∗
∗
√
nf =
− nd − 1 .
Fe b
Defining the argument of the positive part as h(Fo , ε), the emissions price threshold at which
foreign firms enter is εentry = {ε|h(Fo , ε) = 0}. It should be noted that, without border adjustment,
n∗f is inelastic in ε when domestic firms offshore because cd (l∗ , ε) = cf . Therefore, if Fo < F o and
εo (Fo ) ≤ εentry , foreign entry is preempted by domestic offshoring and will not occur at any ε.
Equilibrium market structure Figures 2a and 2b provide illustrative examples of how these fixed
offshoring cost and emissions price thresholds determine market structure. Figure 2a does so when
there is no foreign entry at ε = 0. This reflects much of the EU cement industry where less than 1.5%
of cement revenues were generated by imports to the EU (European Commission 2009). Figure 2b
depicts equilibrium market structures with significant foreign entry when ε = 0, reflecting the EU
steel industry where imports served 20.4% of sales in 2004 (International Iron and Steel Institute
2006). Parameters for all numerical illustrations are provided in Appendix A.
Fo
F1500
o
ହ
ଶ
Onshore/
Foreign Entry
1200
F1000
o
Domestic Exit/
Foreign entry
ଵ
ସ
Onshore/ No Entry
800
Fixed offshoring cost
Fixed offshoring cost
1400
Offshore/ Foreign Entry
600
ଷ
ߝ‘ (‫)‘ܨ‬
400
Offshore/No Entry
200
ହ
1300
Domestic Exit/
Foreign entry
ଶ
1100
Onshore / Foreign Entry
Fo
900
700
ߝ௢ (‫ܨ‬௢ )
ସ
500
Offshore/
Foreign Entry
300
100
0
7
9
11
13
ߝ௘௡௧௥௬15
Emissions price
Figure 2
17
ߝ௘௫௜௧
19
21
H
Ͳ100
3
4
5
6
7
8
9
10
ߝ௘௫௜௧
11
12
H
13
Emissions price
(a) εentry > 0
(b) εentry ≤ 0
Equilibrium market structure with no entry at ε = 0 (Figure 2a) and with entry at ε = 0 (Figure 2b)
If there is no foreign entry when ε = 0, then five potential equilibrium market structures, Γ1 , ..., Γ5 ,
can result as illustrated in Figure 2a. In Γ1 , ε is sufficiently low that domestic firms produce in the
regulated region and foreign firms do not enter. In Γ2 , ε is still sufficiently low and Fo is sufficiently
great that domestic firms produce in the regulated region. However, ε is great enough to erode
domestic firms’ operating cost advantage sufficiently to attract foreign entry; i.e., ε > εentry . In
Γ3 , fixed offshoring costs are less than F o and emissions price is greater than εo (Fo ), so domestic
firms offshore production. Entry does not arise in Γ3 , even when ε > εentry , because εo (Fo ) ≤ εentry
so domestic offshoring preempts foreign entry (recall that n∗f is independent of ε if cd (l∗ , ε) = cf ).
In Γ4 , domestic firms shift production offshore, however, unlike in Γ3 , εentry < εo (Fo ) so foreign
David Drake: Carbon tariffs, technology choice and foreign cost advantage
8
firms compete in the market. In Γ5 , ε ≥ εexit and offshoring is prohibitively costly with Fo ≥ F o , so
domestic firms exit the market while foreign firms compete (ε > εentry ). If there is significant entry
when ε = 0, the equilibrium market structures that can evolve are depicted in Figure 2b. Note that
there is no partition Γ1 because εentry ≤ 0 and there is no partition Γ3 because εentry < εo (Fo ), ∀Fo .
3.4.
Equilibrium quantities
When location is unimportant, x∗ and y ∗ will denote each domestic and foreign firm’s equilibrium
quantities, respectively. When location is important, x∗l will denote domestic quantities.
Lemma 2. Domestic firm equilibrium quantities, x∗ , and foreign firm equilibrium quantities, y ∗ ,
depend on equilibrium market structure and are characterized by the following table:
Region Domestic decision Entry decision
Γ1
Equilibrium quantities
No entry
Domestic production
Γ2
Entry
Γ3
No entry
x∗ =
x∗ =
Fe
b
Γ5
Entry
Exit
d −αd ε
+ δγ1 +τ −γ
b
x∗ =
Offshore production
Γ4
A−γd −αd ε
b(Nd +1)
Entry only
A−δγ1 −τ
b(Nd +1)
x∗ =
Fe
b
x∗ = 0
y∗ = 0
y ∗ = Fbe
y∗ = 0
y ∗ = Fbe
y ∗ = Fbe
From Lemma 2, domestic quantities clearly decrease in emissions price when domestic firms produce in the regulated region (partitions Γ1 and Γ2 ). In these settings, total domestic output n∗d x∗
decreases in ε at a rate of −αd Nd /b(Nd + 1) without foreign competition (partition Γ1 ), and at a
rate of −αd Nd /b with foreign competition (partition Γ2 ). Total domestic output is independent of
ε if domestic firms offshore or exit (partitions Γ3 , Γ4 , and Γ5 ).
If domestic firms offshore or exit (partitions Γ3 , Γ4 , or Γ5 ), then cd (l∗ , ε) = cf or n∗d = 0, respectively. Therefore, under either domestic offshoring or exit, it follows directly from Lemmas 1 and
2 that total foreign entry n∗f y ∗ does not change in ε. However, per Section 3.2, if domestic firms
produce in the regulated region and compete against foreign entrants (partition Γ2 ), then cd (l∗ , ε) =
γd + αd ε. In such settings, by Lemma 1, the number of foreign entrants increases in ε at a rate of
√
αd Nd / Fe b. With each foreign entrant producing Fe /b units (per Lemma 2), total foreign entry
increases in ε at a rate of αd Nd /b. Increased entry displaces domestic production, which decreases
in ε under the same conditions at a rate of −αd Nd /b as described above. This implies that, if
domestic firms produce in the regulated region and compete against foreign entrants (partition Γ2 ),
David Drake: Carbon tariffs, technology choice and foreign cost advantage
9
increases in emissions price shift market share from domestic to foreign firms. This has important
emissions regulation implications for the US and EU steel industry where foreign firms compete
even when emissions are costless. In these settings, without a border adjustment, carbon regulation
(i.e., any ε > 0) will result in increased foreign entry relative to an unregulated (ε = 0) baseline.
Table 2 summarizes the sensitivity of total domestic output, total foreign entry, and total output
n∗d x∗ + n∗f y ∗ with respect to emissions price in settings without border adjustment.
Region
Γ1
Γ2
Γ3
Γ4
Γ5
Table 2
Change wrt ε in...
Domestic-owned output Total foreign entry
Total output
−αd Nd /b(Nd + 1)
—
−αd Nd /b(Nd + 1)
−αd Nd /b
αd Nd /b
0
0
—
0
0
0
0
—
0
0
Change in domestic-owned output, total foreign entry, and total output wrt ε without border adjustment
Effect of offshoring on quantities In addition to changing continuously in emissions price as
described above, domestic-owned and total output conditionally step-increase due to offshoring.
Lemma 3. Without border adjustment, offshoring increases equilibrium total output by Q(Fo ),
where Q(Fo ) > 0 iff Fo > 0 and h(Fo , εo (Fo )) < 0, and Q(Fo ) = 0 otherwise.
If Fo = 0, domestic firms offshore at εo (0), the emissions price at which operating costs in the
regulated region γd + αd ε are equivalent to operating costs offshore δγ1 + τ . If Fo > 0, however,
operating costs in the offshore region are less than those in the regulated region at the threshold
εo (Fo ); relocating domestic firms require greater operating profit offshore to compensate for the
fixed offshoring cost they incur. Therefore, when Fo > 0, domestic firms’ operating costs decrease
if they offshore in equilibrium and, consequently, total domestic-owned output increases. Further,
total foreign entry decreases as a result of domestic firms’ reduced operating costs, which is evident
through Lemma 1. If foreign firms remain in the market when domestic firms offshore (i.e., if
h(Fo , ε(Fo )) > 0), then the decrease in total foreign entry fully offsets the increase in domestic
output, leaving total output unchanged. Otherwise, total output increases by Q(Fo ) as a result of
domestic offshoring—i.e., total output increases if h(Fo , ε) is negative when evaluated at offshoring
threshold εo (Fo ). The magnitude of increase, Q(Fo ), is characterized in the proof to Lemma 3.
Figure 3a provides an example of these quantity results, while Figure 3b provides the corresponding emissions results (discussed in Section 3.5 below), where η ∗ represents total global emissions.
In Figure 3a there is no foreign entry at ε = 0, so domestic firms produce in the regulated
region without foreign entrants at sufficiently low emissions prices (partition Γ1 ), with quantities
David Drake: Carbon tariffs, technology choice and foreign cost advantage
ߝ௢ ‫ܨ‬௢
ߝ௘௡௧௥௬
Dirty domestic
owned output
ߟ‫כ‬
݊ௗ‫ כ ݔ כ‬൅ ݊௙‫כ ݕ כ‬
ߝଶௗ
Clean domestic
owned output
Equilibrium emissions
10
Total output
Dirty foreign
owned output
ܳௗ ‫ܨ‬௢
Emissions tax rate
Related
Partition
Figure 3
*1
*2
ߝ௘௫௜௧
ߝଶௗ
Domestic
emissions
Offshore
emissions
ߝ
*4
ߝ௢ ‫ܨ‬௢
ߝ௘௡௧௥௬
Emissions tax rate
Related
Partition
*1
*2
*4
ߝ
(a)
(b)
Example of equilibrium quantities (Figure 3a) and emissions (Figure 3b) without border adjustment
decreasing in ε. At εd2 , domestic firms adopt clean technology, reducing the rate at which domestic
quantities decrease in ε. Foreign firms enter at εentry , shifting the market structure to Γ2 , with
domestic quantities decreasing, foreign entry increasing, and total output unchanged in ε. In this
example Fo < F o , so domestic firms offshore at εo (Fo ). Because Fo > 0, domestic firms’ production
costs decrease with offshoring and their output increases (indicated by Qd (Fo ) in the figure). With
all firms operating outside of the regulated region (partition Γ4 ), they are unaffected by further
increases in ε. If, in this example, Fo ≥ F o , domestic firms would not offshore and total domestic
quantities would follow the dashed line and exited the market at εexit (partition Γ5 ).
3.5.
Equilibrium emissions
Emissions regulation is intended to abate the driver of anthropogenic climate change: equilibrium
global CO2 emissions, η ∗ . Without border adjustment, equilibrium global emissions are defined as
η∗ =
n∗ x ∗ α
d r d
Domestic emissions
+
n∗d x∗o + n∗f y ∗ (α1 + ατ ) .
(3)
Offshore emissions
The regulator’s ability to reduce global emissions without border adjustment is limited.
Proposition 1. Without border adjustment, equilibrium global emissions: a) decrease in ε if
domestic firms operate in the regulated region and varepsilon ≤ εentry ; b) increase in ε if domestic
firms operate in the regulated region and ε > εentry ; c) are unaffected in ε if domestic firms operate
offshore or exit; and d) increase due to offshoring at εo (Fo ) if Fo < F o .
Regarding Propositions 1a-c, when domestic firms produce in the regulated region without foreign
competition (partition Γ1 ), total output decreases in ε at a rate of −αd Nd /b(Nd + 1) per Table 2.
Global emissions therefore decrease in ε at rate of −αd2 Nd /b(Nd + 1). When domestic firms produce
in the regulated region and foreign firms compete (partition Γ2 ), increases in ε result in carbon
David Drake: Carbon tariffs, technology choice and foreign cost advantage
11
leakage due to entry. Total output does not change in ε under these conditions—increases in foreign
entry are offset by decreases in domestic production—however, each unit of domestic production
displaced by foreign entry increases emissions by α1 + ατ − αd . As a consequence, increasing emissions price increases global emissions in this setting; e.g., see Figure 3b over ε[εentry , εo (Fo )).
Under domestic offshoring (partitions Γ3 and Γ4 ) or exit (partition Γ5 ), the market is served
only by offshore facilities. Therefore, without border adjustment, increases in ε beyond εo (Fo ) if
Fo < F o , or beyond εexit if Fo ≥ F o , do not affect global emissions. Table 3 summarizes these effects.
Region
Γ1
Γ2
Γ3
Γ4
Γ5
Table 3
Change in global emissions wrt ε driven by...
Total output Carbon leakage through entry
−αd2 Nd /b(Nd + 1)
0
0
(α1 + ατ − αd )(αd Nd /b)
0
0
0
0
0
0
Change in global emissions wrt emissions price by source (without border adjustment)
Proposition 1d indicates that carbon leakage due to offshoring, which occurs at εo (Fo ) if Fo < F o ,
increases global emissions in the absence of a border adjustment. As with foreign entry, emissions
intensity increases by α1 + ατ − αd for every unit of domestic production offshored, which results
in greater global emissions even if total output remains unchanged. If total output increases due to
offshoring, per Lemma 3, volume and emissions intensity increases combine to drive greater global
emissions. The issue of an industry offshoring en masse without border adjustment is of practical
concern. Studies of EU cement sector suggest that all production in Italy, Greece, Poland, and the
UK would offshore at an emissions price of 25 Euro per tonne of CO2 , increasing global emissions
by an estimated minimum of 7 million tonnes of CO2 annually (Boston Consulting Group 2008).
In short, without border adjustment, the regulator’s ability to reduce global emissions through
emissions regulation is limited—it is only possible in cases of a domestic oligopoly (partition Γ1 ) or
by incentivizing the adoption of clean technology. Without border adjustment, the regulator can
only achieve the latter if the clean technology adoption threshold is less than the offshoring or exit
threshold (whichever is applicable given fixed offshoring costs). Further, without border adjustment,
the regulator would inadvertently increase global emissions through leakage by imposing emissions
regulation on sectors where domestic firms produce in the regulated region and compete against
foreign entrants (partition Γ2 ), or that results in offshoring (partitions Γ3 and Γ4 ).
David Drake: Carbon tariffs, technology choice and foreign cost advantage
12
4.
Firm Decisions and Performance with Border Adjustment
Much debate related to emissions regulation has centered on border adjustments as a possible
means to prevent the adverse effects of carbon leakage. It is therefore important to understand
how border adjustments impact technology choices, production decisions, and emissions.
4.1.
Model development
The model here resembles that in Section 3 except imports incur a border adjustment βk (ε) =
αk ε, ∀k ∈ {1, 2}. Hat notation distinguishes objectives, decisions, and thresholds with border adjustment from those without it, with ĉd (ˆl, ε) and ĉf (ε) representing the operating cost of domestic and
foreign firms’ preferred technology, respectively. Similarly, γ̂d , α̂d , γ̂f , and α̂f denote the production
cost and emissions intensity of domestic and foreign firms’ preferred technology, respectively (with
ˆl and ε notation dropped from γ̂d and α̂d , and ε notation dropped from γ̂f and α̂f for brevity).
Therefore, with a border adjustment, domestic firms maximize operating profit by solving
π̂d n̂d , n̂f , ˆl, ŷ, ε = max P (n̂d , n̂f , x̂l , ŷ) x̂l − ĉd (ˆl, ε)x̂l .
x̂l ≥0
(4)
Each foreign competitor maximizes operating profit by solving
π̂f (n̂d , n̂f , x̂l , ε) = max P (n̂d , n̂f , x̂l , ŷ) ŷ − ĉf (ε)ŷ.
ŷ≥0
4.2.
(5)
Equilibrium technology choice
With a border adjustment, the regulator can incentivize any firm serving the regulated market—
domestic or foreign; operating in the regulated region or offshore—to adopt clean technology at
some emissions price. Counterintuitively, foreign firms’ technology choices are more sensitive to
emissions price than domestic firms’ technology choices.
Proposition 2. With a border adjustment, foreign firms serving the regulated region adopt clean
technology at a lower emissions price than domestic-owned firms.
The threshold at which foreign firms adopt clean technology ε̂f2 is the break-even emissions price
for technology 1 and 2, where ε̂f2 = δ (γ2 − γ1 ) / (α1 − α2 ). Clean technology is adopted domestically at εd2 = (γ2 − γ1 ) / (α1 − α2 ), identical to the adoption threshold without border adjustment
(therefore hat notation is dropped for εd2 ). Conditional upon entry, foreign firms therefore operate
cleaner technology than locally producing domestic firms when ε ∈ ε̂f2 , εd2 , and operate identi
cal technology to locally producing domestic firms when ε ∈
/ ε̂f2 , εd2 . Further, (as described below
in Section 4.3), if domestic firms offshore in equilibrium, they do so at a threshold greater than
foreign firms’ clean technology adoption threshold. This implies that domestic firms adopt clean
technology, in the domestic region or offshore, at a greater emissions price than foreign firms.
David Drake: Carbon tariffs, technology choice and foreign cost advantage
4.3.
13
Equilibrium offshoring, exit, and entry
As without border adjustment, equilibrium market structure with a border adjustment is determined by domestic firms’ location and exit decisions, and by foreign firms’ entry decisions. As in
Section 3, star notation indicates firms’ equilibrium decisions as well as dependency on Fo and ε.
Domestic offshoring As in the setting without border adjustment, domestic firms do not offshore
at any emissions price if Fo is sufficiently great. Specifically, with π̂d,o (·) noting domestic firm
profits when domestic firms produce offshore, fixed offshoring costs are prohibitive in equilibrium
if Fo ≥ F̂ o where F̂ o = {Fo |π̂d,o (Nd , n̂∗f , o, ŷ ∗ , ε) − Fo = 0}. However, if Fo < F̂ o , then domestic firms
conditionally offshore despite incurring a symmetric border adjustment. This is formalized in the
following proposition where π̂d,r (·) notes domestic firm profits when producing in the regulated
region and ε̂o (Fo ) = {ε|π̂d,r Nd , n̂∗f , r, ŷ ∗ , ε = π̂d,o Nd , n̂∗f , o, ŷ ∗ , ε − Fo }.
Proposition 3. With a border adjustment, domestic firms offshore and adopt clean technology
at emissions price ε̂o (Fo ) iff Fo < F̂ o and ε̂o (Fo ) < εd2 , but otherwise do not offshore at any ε.
Figures 4a and 4b depict the domestic (solid black line) and offshore (solid gray line) operating
cost frontiers wrt emissions price, and are useful in developing the intuition behind Proposition 3.
Regional Cost Frontiers: With Offshoring
Regional Cost Frontiers: No Offshoring
Offshorecostfrontier
Offshorecostfrontier
Total unit cost
Domesticcostfrontier
Total unit cost
Domesticcostfrontier
௙
ߝƸଶ
ߝƸ௢ (0)
Emissions tax rate
1
3
5
Figure 4
7
9
௙
ߝଶௗ
ߝƸଶ
ɸ
ߝଶௗ
Emissions tax rate
ɸ
11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55
(a) With offshoring
(b) No offshoring
Operating cost frontiers when technologies are deployed domestically (black) and offshore (gray)
The gap between the domestic and offshore cost frontiers in Figures 4a and 4b is the per-unit
operating cost difference between locations. Over intervals in which domestic and offshore facilities
prefer the same technology, ε ∈ [0, ε̂f2 ) or ε ≥ εd2 , the frontiers are parallel; both regions have the
same production emissions intensity and therefore equivalent per-unit carbon costs. In emissions
price intervals where offshore facilities operate cleaner technology than facilities in the regulated
region, ε ∈ [ε̂f2 , εd2 ), the slope of the offshore frontier is less than that of the domestic frontier—i.e.,
offshore carbon costs increase at a slower rate in ε than domestic carbon costs.
14
David Drake: Carbon tariffs, technology choice and foreign cost advantage
If Fo = 0, then ε̂o (Fo ) is determined by the emissions price at which domestic and offshore
operating costs are equal, represented by the intersection of the cost frontiers in Figure 4a. If
ε̂o (Fo ) < εd2 , then the offshore operating profit advantage π̂d,o (·) − π̂d,r (·) exceeds fixed offshoring
cost Fo at any ε > ε̂o (Fo ), and domestic firms offshore at ε̂o (Fo ) despite the implementation of a
border adjustment. Because offshore operating costs only improve relative to operating costs in the
regulated region over emissions prices between ε̂f2 and εd2 , the threshold ε̂o (Fo ) must fall within the
interval ε ∈ [ε̂f2 , εd2 ) when it exists. Offshoring under border adjustment therefore implies domestic
firms adopt cleaner technology when they relocate than they operated in the regulated region. As
a consequence, domestic firms adopt clean technology at some emissions price êd2 > εf2 , where
ε̂o (Fo ) if εd2 > ε̂o (Fo ) and Fo < F o
êd2 = d
ε2
otherwise.
Domestic exit Theoretically, if fixed offshoring costs were prohibitive, then domestic firms would
exit at the threshold ε̂exit = {ε|π̂d,r Nd , n̂∗f , r, ŷ ∗ , ε = 0}. However, under a border adjustment, it is
straightforward to show that domestic firm profits are independent of ε when foreign firms compete
in the market and domestic and foreign firms operate the same technology.1 Therefore, ε̂exit < εd2
is a necessary condition for domestic firm exit, with ε̂exit < εd2 holding only if offshore production
√
costs were unrealistically small relative to domestic production costs; i.e., if δ < 1 − ( Fe b + τ )/γ2 .
Such a condition is unlikely to be met in practice given the substantial transport costs for leakagethreatened goods. In cement, for example, transport costs are greater than production costs in four
of the five offshore regions in Figure 1, which would require δ < 0 for domestic exit. While quantity
results will be provided for the setting in which domestic firms exit, given the impracticality of
such outcomes, it is assumed that n̂∗d = Nd in the discussion that follows.
Foreign entry and exit The equilibrium number of foreign firms competing in the market is
determined as in the setting without border adjustment: by foreign firms competing operating
profits down to Fe such that n̂∗f = {(n̂f )+ |π̂f (n̂∗d , n̂f , x̂∗l , ε) = Fe }. The characterization of n̂∗f is
similar to the characterization of n∗f provided in Lemma 1 (and the proof is symmetric), with
⎛
⎞+
A − ĉf (ε) − n̂∗d ĉf (ε) − ĉd (ˆl, ε)
√
− n̂∗d − 1⎠ .
(6)
n̂∗f = ⎝
Fe b
The equilibrium number of entrants n̂∗f can be non-monotonic, conditionally increasing over inter
vals in which foreign firms operate cleaner technology than domestic firms, ε ∈ ε̂f2 , êd2 , and strictly
decreasing when domestic and foreign firms operate identical technology, ε ∈
/ ε̂f2 , êd2 .
1
As will be shown, total foreign quantities decrease in such settings at a rate of −α̂f /b while domestic quantities do
not change in ε. Therefore, price increases in ε at a rate of α̂f = α̂d , offsetting domestic firms’ increased carbon costs.
David Drake: Carbon tariffs, technology choice and foreign cost advantage
15
Defining the argument of the positive part statement in (6) as ĥ(Fo , ε), if α1 > α2 (1 + 1/Nd ), then
n̂∗f increases in ε over intervals where foreign firms operate cleaner technology than domestic firms.
In contexts where this inequality holds,2 foreign firms enter the market at ε̂entry = {ε|ĥ(Fo , ε) =
0, ε ∈ ε̂f2 , êd2 } and they exit at ε̂fexit = {ε|ĥ(Fo , ε) = 0, ε ∈
/ ε̂f2 , êd2 }. However, if α1 < α2 (1 + 1/Nd ),
then ĥ(Fo , ε) strictly decreases in ε. In such a case, there is no entry threshold and foreign firms
that competed in the market at ε = 0 would exit at ε̂fexit = {ε|ĥ(Fo , ε) = 0}.
Equilibrium market structure With border adjustment, there are four equilibrium market structures that are of interest, Ω1 , ..., Ω4 , as illustrated through Figures 5a and 5b. Domestic exit is
unlikely to arise with border adjustment, as discussed above, and so is not considered here.
3000
F
o
Fo
Fixed offshoring cost
Fixed offshoring cost
120
Onshore / No Entry
Onshore/
Foreign Entry
2000
80
ଵ
ଶ
2500
ଵ
100
Onshore/
No Entry
௙
ߝƸ௘௫௜௧
1500
60
ߝƸ௢ (‫)‘ܨ‬
40
Offshore / No Entry
15
20
௙
ߝƸ25ଶ
ߝ30ଶௗ
35
40
45
50
H
55
Emissions price
Figure 5
ସ
ଷ
Offshore/
Foreign Entry
500
20
0
ߝƸ௢ (‫)‘ܨ‬
1000
ଷ
0
5
10
௙
ߝƸଶ
15
ߝଶௗ
20
25
Offshore/
No Entry
30
35
40
45
H
50
Emissions price
(a) no entry at ε = 0
(b) ε̂entry ≤ 0
Equilibrium market structure with no entry at ε = 0 (Figure 5a) and with entry at ε = 0 (Figure 5b)
In Figure 5a, there is no foreign entry at ε = 0, similar to European cement. In Ω1 , emissions
price is sufficiently low that domestic firms operate in the regulated region without foreign entry.
In this example, ε̂o (0) < εd2 . Therefore, when Fo is sufficiently small, per Proposition 3, domestic
firms offshore at emissions prices greater than ε̂o (Fo ) despite the presence of a border adjustment
(partition Ω3 ). However, if Fo is sufficiently large, then clean technology adoption in the regulated
region preempts offshoring (recall that domestic and offshore operating cost frontiers are parallel
in ε when the same technology is preferred in both regions). In such a case, market structure
equilibrium Ω1 persists at all emissions prices.
Figure 5b is representative of the steel sector in the US and Europe, where foreign firms compete
in the market when ε = 0. At sufficiently low emissions prices, equilibrium market structure Ω2
results, with domestic firms operating in the regulated region competing against foreign entrants.
2
For example, in cement, CCS technology is expected to eliminate more than 90% of emissions relative standard
precalciner kilns, i.e., α2 ≤ .10α1 (European Cement Research Academy 2009). Therefore, α1 > α2 (1 + 1/Nd ) holds,
requiring only Nd > 1/9 (while Nd ≥ 1 can be assumed in an emissions regulated setting with border adjustment).
David Drake: Carbon tariffs, technology choice and foreign cost advantage
16
Domestic firms offshore, resulting in equilibrium market structure Ω4 , if Fo is sufficiently small that
ε̂(Fo ) < εd2 and emissions price is between the domestic offshoring and foreign exit thresholds, ε̂o (Fo )
and ε̂fexit , respectively. If ε > ε̂fexit , foreign firms exit, resulting in equilibrium market structure Ω3
if domestic firms operate offshore and Ω1 if domestic firms operate in the regulated region.
The parameters underlying Figures 5a and 5b were chosen to illustrate the spectrum of market
structure equilibria that can arise. Scenarios other than those illustrated here can occur, however
these would each result in a combination of the four market structures described above.
4.4.
Equilibrium quantities
As in the setting without border adjustment, x̂∗ and ŷ ∗ denote each domestic and foreign firm’s
equilibrium quantity when location is clear and non-central, and x̂∗r and x̂∗o denote each domestic
firm’s production in the regulated and offshore region, respectively, when location is otherwise
unclear or of central importance.
Lemma 4. With border adjustment, domestic firm equilibrium quantities, x̂∗ , and foreign firm
equilibrium quantities, ŷ ∗ , are characterized by the following table:
Region Domestic decision Entry decision
Ω1
Equilibrium quantities
No entry
Domestic production
Ω2
Entry
Ω3
No entry
Offshore production
Ω4
N/A
Entry
Exit
Entry only
x̂∗ =
x̂∗ =
A−γ̂d −α̂d ε
b(Nd +1)
Fe
b
δγ̂f +α̂f ε+τ −γ̂d −α̂d ε
b
+
x̂∗ =
A−δγ2 −α2 ε−τ
b(Nd +1)
x̂∗ =
Fe
b
x̂∗ = 0
ŷ ∗ = 0
ŷ ∗ = Fbe
ŷ ∗ = 0
ŷ ∗ = Fbe
ŷ ∗ = Fbe
As described in Section 3, without border adjustment, total domestic-owned output n∗d x∗
decreases in ε if domestic firms operate in the regulated region, but does not change in ε if domestic
firms operate offshore. With border adjustment, on the other hand, total domestic output decreases
in ε at a rate of −α̂d Nd /b(Nd + 1) when domestic firms operate without foreign competition (partitions Ω1 and Ω3 ). This is true whether domestic firms produce in the regulated region or offshore
(with α̂d = α2 in the latter case per Proposition 3). Total domestic output also decreases in ε if
foreign firms operate cleaner technology than domestic firms (partition Ω2 if ε ∈ [ε̂f2 , êd2 )), decreasing
at a rate of −(α1 − α2 )Nd /b. However, if domestic and foreign firms operate the same technology,
i.e., if α̂f = α̂d , total domestic output does not change in ε. This is the case whether domestic firms
/ [ε̂f2 , êd2 )) or offshore (partition Ω4 ).
operate in the regulated region (partition Ω2 if ε ∈
David Drake: Carbon tariffs, technology choice and foreign cost advantage
17
Proposition 4. With border adjustment, if α1 > α2 (1 + 1/Nd ) and ε ∈ [ε̂f2 , êd2 ), total foreign
entry increases in ε. Total foreign entry decreases in ε otherwise.
The conventional wisdom is that a border adjustment will eliminate the threat of carbon leakage
(Barry 2009, Barber 2010, Ponssard and Walker 2008, Fowlie et al. 2014). However, as Proposition 4
indicates, foreign entry can increase with emissions price despite the implementation of a border
adjustment. As evident from Lemma 4, when foreign firms compete in the market, their quantities
do not change in emissions price. However, as described following Equation (6), n̂∗f is potentially
non-monotonic in ε. When foreign firms operate cleaner technology than domestic firms, i.e., over
ε ∈ [ε̂f2 , êd2 ), total foreign entry n̂∗f ŷ ∗ changes at a rate of [(α1 − α2 )Nd − α2 ]/b through Equation (6)
and Lemma 4. Therefore, total foreign entry increases over ε ∈ [ε̂f2 , êd2 ) if clean technology offers a
sufficient emissions improvement, i.e., if α1 ≥ α2 (1 + 1/Nd ). As described in footnote 2 above, this
inequality holds trivially in cement, requiring only Nd > 0.11.
If α1 ≤ α2 (1 + 1/Nd ), total foreign entry decreases over emissions price intervals in which foreign
firms operate cleaner technology than domestic firms. When foreign and domestic firms operate
/ [ε̂f2 , êd2 ), and partition Ω4 ), total foreign entry decreases at
the same technology (partition Ω2 if ε ∈
a rate of −α̂f /b while domestic output remains unchanged (per Lemma 4). Table 4 summarizes the
effects described above. As evident in Table 4, total output decreases in ε across all possible market
structures, which sharply contrasts the setting without border adjustment where total output only
decreased in the case of a domestic oligopoly (partition Γ1 ).
Change wrt ε in...
Domestic-owned output Total foreign entry
Total output
Ω1
−α̂d Nd /b(Nd + 1)
—
−α̂d Nd /b(Nd + 1)
f
d
−(α1 − α2 )Nd /b
[(α1 − α2 )Nd − α2 ]/b
−α2 /b
Ω2 , ε ∈ [ε̂2 , ê2 )
/ [ε̂f2 , êd2 )
0
−α̂f /b
−α̂f /b
Ω2 , ε ∈
Ω3
−α2 Nd /b(Nd + 1)
—
−α2 Nd /b(Nd + 1)
Ω4
0
−α2 /b
−α2 /b
Region
Table 4
Change in domestic-owned output, total foreign entry, and total output wrt ε with border adjustment
Effect of offshoring on quantities Similar to the setting without border adjustment, when a
border adjustment has been implemented, domestic offshoring potentially increases total output.
Lemma 5. With border adjustment, offshoring increases equilibrium total output by Q̂(Fo ), where
Q̂(Fo ) > 0 iff Fo > 0 and ĥ(Fo , ε̂o (Fo )) < 0, and Q̂(Fo ) = 0 otherwise.
As described in the setting without border adjustment, if Fo > 0 total domestic-owned output
increases with offshoring. This results because, at the offshoring threshold ε̂o (Fo ), unit operating
cost in the regulated region is greater than it is offshore if Fo > 0. As in the setting without border
David Drake: Carbon tariffs, technology choice and foreign cost advantage
18
adjustment, if Fo > 0, foreign entry decreases when domestic firms offshore due to domestic firms’
decreased operating cost (and consequently increased output). If foreign firms remain in the market
despite domestic offshoring, i.e., if ĥ(Fo , ε̂o (Fo )) > 0, then the decrease in foreign entry fully offsets
the increase in domestic output, leaving total output unchanged as a consequence of offshoring.
Otherwise, total output increases by Q̂(Fo ), where Q̂(Fo ) is characterized in the proof to Lemma 5.
Figure 6a provides an example of quantity results, while Figure 6b illustrates the corresponding
Clean domestic
owned output
Dirty foreign
owned output
Clean foreign
owned output
Emissions tax rate
Related
Partition
Figure 6
Dirty domestic
owned output
:2
ߟƸ ‫כ‬
ߝଶௗ
௙
ߝଶௗ
ߝƸଶ
Equilibrium emissions
௙
ߝƸଶ
Total output
݊ොௗ‫ݔ כ‬ො ‫ כ‬+ ݊ො௙‫ݕ כ‬ො ‫כ‬
emissions results (discussed in Section 4.5), where η̂ ∗ represents total global emissions.
ߝ
Domestic emissions
Offshore emissions
Emissions tax rate
Related
Partition
:2
ߝ
(a)
(b)
Example of equilibrium quantities (Figure 6a) and emissions (Figure 6b) with border adjustment
In Figure 6a, domestic and foreign firms operate dirty technology at ε = 0, with foreign entry
decreasing and domestic output constant in ε, per Lemma 4 and Proposition 4. At ε̂f2 , foreign
firms adopt clean technology (at a lower emissions price than domestic firms, per Proposition 2).
Over the interval between ε̂f2 and εd2 , foreign firms utilize cleaner technology than domestic firms.
As a consequence, with α1 > α2 (1 + 1/Nd ) in this example, entry increases and domestic output
decreases in ε (again per Lemma 4 and Proposition 4). If α1 had been less than α2 (1 + 1/Nd ),
then both domestic and foreign production would decrease, but foreign production would decline
at a lesser rate. At εd2 , domestic firms also adopt clean technology, and domestic output does not
change while foreign entry decreases in ε. That foreign entry can decrease in emissions price—
conditionally decreasing when foreign firms operate cleaner technology than domestic firms and
strictly decreasing when they operate similar technology—contrasts the setting without a border
adjustment where total foreign entry increases monotonically in ε. In settings where foreign firms
compete when ε = 0, this implies that emissions regulation combined with a border adjustment can
increase domestic market share relative to the unregulated baseline, which arguably lends credence
to concerns over the potential anti-competitiveness of the mechanism in such a context.
David Drake: Carbon tariffs, technology choice and foreign cost advantage
4.5.
19
Equilibrium emissions
The implementation of a border adjustment improves the efficacy of emissions regulation by improving the regulator’s ability to use emissions price as a mechanism to reduce global emissions. This
is evident through the following proposition, where ι is infinitesimal.
Proposition 5. With border adjustment, global emissions: a) decrease monotonically in ε
unless ĥ(Fo , ε) > 0 and (α1 − α2 − ατ )(α1 − α2 )Nd + α2 (α2 + ατ ) < 0, in which case they increase
in ε ∈ [ε̂f2 , êd2 ) and decrease in ε ∈
/ [ε̂f2 , êd2 ); and b) decrease due to offshoring if α1 /(α2 + ατ ) >
1 + Q̂(Fo )/Nd x∗ (Fo , ε̂o (Fo ) − ι), but otherwise increase due to offshoring.
Under border adustment, when domestic firms compete alone in the market, increases in emissions
price reduce their output (evident through Table 4) and therefore reduce global emissions. Unlike
the case without border adjustment, this is true whether domestic firms operate alone in the
regulated region (partition Ω1 ) or offshore (partition Ω3 ). When domestic firms compete against
foreign firms and operate identical technology (partition Ω2 when ε ∈
/ [ε̂f2 , êd2 ) and partition Ω4 ),
total foreign entry decreases in ε while domestic owned output is unchanged, per Proposition 4 and
Table 4. Global emissions decrease as a result, with domestic firm emissions unchanged and foreign
firm emissions decreasing, which again contrasts the setting without border adjustment where
global emissions are unchanged or increasing with emissions price when foreign firms compete.
If domestic firms operate in the regulated region and foreign firms compete and operate cleaner
technology (partition Ω2 if ε ∈ [ε̂f2 , êd2 )), increases in emissions price decrease total output and
contribute to carbon leakage. Per Table 4, total output decreases in ε at a rate of −α2 /b in such
settings, and foreign production displaces domestic production at a rate of (α1 − α2 )Nd /b. In
contrast to the setting without border adjustment where carbon leakage strictly increased global
emissions, with a border adjustment, each unit of carbon leakage replaces a unit of dirty production
with a unit of clean production and international transport. Accounting for the decrease in total
output, when domestic firms produce in the regulated region and compete against cleaner operating
foreign firms, global emissions therefore decrease in ε if (α1 − α2 − ατ )(α1 − α2 )Nd + α2 (α2 + ατ ) >
0. This inequality holds under the sufficient condition that per-unit transport emissions are less
than twice the emissions intensity reduction achieved by adopting clean technology; i.e., if ατ <
2(α1 − α2 ). This is likely to hold in most practical settings, as it does in cement where α2 ≤ 0.1α1
and, for example, ατ ≤ .1α1 for transport from China to Western Europe (Cement Sustainability
Initiative 2009, and Log-net.com 2014 assuming 22 tonnes of cement per 20 foot container).
Table 5 provides the rate of change in global emissions for each market equilibrium, offering a
stark contrast to the corresponding effect without border adjustment (presented in Table 3).
David Drake: Carbon tariffs, technology choice and foreign cost advantage
20
Change in global emissions wrt ε driven by
Total output
Carbon leakage through entry
2
Ω1
−α̂d Nd /b(Nd + 1)
0
f
d
−α2 (α2 + ατ )/b
−(α1 − α2 − ατ )(α1 − α2 )Nd /b
Ω2 , ε ∈ [ε̂2 , ê2 )
/ [ε̂f2 , êd2 )
−α̂f (α̂f + ατ )/b
0
Ω2 , ε ∈
Ω3
−α2 Nd (α2 + ατ )/b(Nd + 1)
0
Ω4
−α2 (α2 + ατ )/b
0
Region
Table 5
Change in global emissions by source wrt emissions price (with border adjustment)
Regarding Proposition 5b, there is an intensity and volume effect to consider. The emissions
intensity of production decreases by α1 − α2 for each unit of production moved offshore to the
foreign region (per Proposition 3). However, ατ transport emissions are also generated for each unit
imported into the regulated region. Therefore, the emissions intensity of the Nd x∗ (Fo , ε̂o (Fo ) − ι)
units moved offshore decreases if α1 − α2 − ατ > 0. However, total global emissions decrease as
a consequence of offshoring only if emissions intensity decreases sufficiently to offset the α2 + ατ
emissions generated per unit of additional output, Q̂(Fo ), produced as a consequence of offshoring
(per Proposition 3). This gives rise to the condition of Proposition 5b. Returning to cement as an
example, recall that α2 and ατ are each approximately 10% of α1 (Cement Sustainability Initiative
2009, and Log-net.com 2014). Therefore, global emissions would decrease due to offshoring unless
total output were to increase by more than 400% of pre-offshoring domestic production.
In short, the implementation of a border adjustment provides the regulator with substantially
greater ability to use emissions price to reduce global emissions. With a border adjustment: i)
total output strictly decreases in ε, rather than decreasing only if domestic firms produce in the
regulated region without foreign competition; ii) carbon leakage from both offshoring and foreign
entry conditionally decreases emissions, rather than strictly increasing emissions; and, iii) the
regulator can incentivize the adoption of clean technology by both domestic and foreign firms,
rather than only being able to incentivize such adoption among domestic firms, and only if the
adoption threshold is less than the offshoring or exit threshold (whichever is applicable given Fo ).
5.
Effects on Emissions Regulation Efficacy and Welfare Performance
Sections 3 and 4 focused on firms’ equilibrium market entry and exit, technology choice, facility
location, and production quantity decisions, as well as the emissions that result. This section
focuses on how the domestic regulator’s decision whether or not to implement a border adjustment
affects the efficacy of its emissions regulation and social welfare. Throughout this section, border
adjustment costs per unit are βk , where βk = 0 without border adjustment and βk = αk ε with
border adjustment. Superscript w notation is used to denote equilibrium welfare components and
David Drake: Carbon tariffs, technology choice and foreign cost advantage
21
domestic and foreign firm equilibrium decisions given the regulator’s decision whether or not to
implement a border adjustment. For example, equilibrium domestic quantities are:
x∗ if βk = 0
w
x =
x̂∗ if βk = αk ε.
5.1.
The effect of border adjustment on emissions regulation efficacy
The per-ton social cost of emissions, εs ≥ 0, is the monetized climate change damage done by each
ton of carbon dioxide equivalent emissions. These costs have been estimated as beginning at $21.40
in 2010 and increasing to $44.90 by 2050, with both figures based on 2007 dollars (Greenstone
et al. 2011). Such social costs provide the impetus for emissions regulation and are included in the
w
w w
w w
w w
α f + α τ εs .
welfare problem as ξ w = η w εs = nw
d x r α d + nd x o + nf y
Emissions regulation is intended to reduce global emissions and thereby reduce the social cost of
carbon ξ w . If, without a border adjustment, foreign firms would compete in the market or domestic
firms would offshore, a border adjustment improves the regulator’s ability to deliver on this intent.
Proposition 6. If i) ε > εentry or ii) ε > εo (Fo ) and Fo < F o , then the equilibrium social cost
of carbon with border adjustment, η̂ εs , is strictly less than without it, η εs .
a) If i) ε > εentry , then η̂ εs < η εs due to reduced total output, and due to cleaner entry if
ĥ(Fo , ε) > 0 and ε ∈ ε̂f2 , ε̂fexit .
b) If ε > εo (Fo ) and Fo < F o , then η̂ εs < η εs due to reduced total output, and due to cleaner
production in the regulated region if εd2 ≤ ε ≤ ε̂o (Fo ) or Fo ≥ F̂ o , or cleaner, offshore production
if ε > ε̂o (Fo ) and Fo < F̂ o .
Proposition 6 states that border adjustment reduces global emissions if, without such a mechanism, foreign firms compete and/or domestic firms offshore. In such settings, border adjustment
improves the regulator’s control over both the volume of total output and the emissions intensity
of production. Thus, border adjustment enables greater control over global emissions.
Without border adjustment, total output n∗d x∗ + n∗f y ∗ decreases in ε only in the case of domestic
oligopoly, partition Γ1 (evident through Table 2). However, as evident through Table 4, total output
decreases in ε across all market structure equilibria with border adjustment. As a consequence,
total output is less under a border adjustment if, without a border adjustment, foreign firms
compete (per the conditions of Proposition 6a), domestic firms offshore (per the conditions of
Proposition 6b), or both. Further, without border adjustment, offshore facilities incur no emissions
costs and therefore produce with type 1 technology. However, if foreign firms compete in the market
under border adjustment (i.e., if ĥ(Fo , ε) > 0), they adopt clean technology if ε ≥ ε̂f2 , reducing the
emissions intensity of foreign-owned production vis-à-vis the setting without border adjustment.
David Drake: Carbon tariffs, technology choice and foreign cost advantage
22
Border adjustment can also enable reduced emissions intensity among domestic-owned firms
if offshoring would occur without such a mechanism (i.e., if ε > εo (Fo ) and Fo < F o ). Border
adjustment decreases the per-unit operating profit domestic-owned firms earn when producing
offshore. As a consequence, ε̂o (Fo ) ≥ εo (Fo ). Therefore, it is possible that border adjustment deters
offshoring. If ε ∈ [εo (Fo ), ε̂o (Fo )) and Fo < F o , domestic firms operate offshore in the absence of
a border adjustment, but operate in the regulated region with one. In such a case, the regulator
incentivizes clean technology adoption by domestic firms in the regulated region if ε ≥ εd2 . On
the other hand, if domestic firms offshore despite border adjustment (i.e., if ε > ε̂o (Fo ) and Fo <
F̂ o ), then they adopt clean technology when doing so, per Proposition 3. In either case, border
adjustment enables the adoption of clean technology among domestic-owned firms in settings within
which they would offshore and operate dirty technology in the absence of a border adjustment.
In short, a border adjustment improves emissions regulation efficacy, enabling the domestic
regulator to use emissions price as a lever to reduce global emissions across settings where it would
be ineffectual without border adjustment.
5.2.
The effect of border adjustment on welfare
When choosing whether or not to implement a border adjustment, the domestic regulator must
consider traditional welfare drivers as well as emissions revenues and emissions-driven social costs.
w
w
Total domestic firm profits in equilibrium are Πw = nw
d πd − Io nd Fo , where Io is 1 if domestic
firms offshore and zero otherwise. Given the demand curve in Sections 3 and 4, P (n̂d , n̂f , x̂l , ŷ) =
w
w w 2
.
A − b (n̂d x̂l + n̂f ŷ), equilibrium consumer surplus is ψ w = (1/2)b nw
d x + nf y
Since domestic and foreign firms use identical technology when producing offshore, imports to the
region incur border adjustment βf when produced by either type of firm. The domestic regulator
w w
w w
w w
βf . The social cost
therefore earns equilibrium emissions revenues, ρw = nw
d x r α d ε + nd x o + nf y
of emissions, ξ w , are defined and discussed in detail above.
The domestic regulator maximizes equilibrium welfare, W , through the following objective:
W=
max
βf ∈{0,α̂f ε}
Πw + ψ w + ρw − ξ w .
(7)
Border adjustment affects welfare drivers in opposing directions and can therefore increase or
decrease welfare. Where Δa = â − a, a border adjustment improves welfare if and only if
Δξ < ΔΠ + Δψ + Δρ.
(8)
As demonstrated by Proposition 6 and its related discussion, border adjustment decreases the
social cost of carbon in any setting in which it would apply (i.e., Δξ is negative). Further, it is
David Drake: Carbon tariffs, technology choice and foreign cost advantage
23
straightforward to demonstrate that border adjustment increases emissions revenues (i.e., Δρ is
positive); unsurprising given that the regulator collects such revenues from both domestic and
offshore production under border adjustment and from only domestic production without such a
mechanism. Both of these effects increase welfare and contribute to the inequality in (8) holding. However, consumer surplus works in the opposite direction. As discussed above, total output
decreases with border adjustment. Consequently, consumer surplus also decreases (i.e., Δψ is negative). As will be discussed in more detail below, border adjustment can increase, decrease, or have
no effect on domestic firms’ profits. As a result, the effect border adjustment has on welfare is
ambiguous. For example, where T = n∗d x∗ + n∗f y ∗ , and T̂ = n̂∗d x̂∗ + n̂∗f ŷ ∗ , if domestic firms offshore
and face foreign competition with and without border adjustment, the inequality in (8) equates to
T̂ (α2 + ατ ) − T (α1 + ατ ) εs < T̂ 2 − T 2 b/2 + T̂ α2 ε.
(9)
The LHS of (9) is negative as T̂ < T and α2 < α1 , while the RHS can be positive or negative as
(T̂ 2 − T 2 )b/2 < 0 and T̂ α2 ε > 0. Whether (9) holds depends, in part, on the per-unit social cost of
carbon (εs ), the magnitude of change in total output (T̂ − T ), and the emissions intensity of firms’
preferred technologies with and without border adjustment (in this case, α2 and α1 , respectively).
Total domestic firm profits can also increase or decrease due to border adjustment.
Proposition 7. If i) ε > εentry or ii) ε > εo (Fo ) and Fo < F o , then domestic firm profit can
increase, decrease, or remain unchanged due to the implementation of a border adjustment.
Proposition 7, under conditions identical to those of Proposition 6, states that domestic firm
profit can increase, decrease, or remain unchanged due to the implementation of a border adjustment. For example, a border adjustment increases domestic firm profit if domestic firms operate
in the regulated region with foreign entry in the absence of such a mechanism. In this case, border
adjustment reduces total foreign entry and domestic profit benefits. However, if domestic firms
would offshore without a border adjustment, then the mechanism can reduce their profits. In this
case, domestic firms incur greater emissions costs due to border adjustment—either directly through
the border adjustment while operating offshore, or by continuing to operate in the regulated region
if offshoring is deterred by border adjustment. Lastly, if domestic firms offshore and foreign firms
participate in the market with and without border adjustment, foreign entry competes operating
profit down to Fe for all participants, domestic firms included. As a consequence, domestic firms
would earn equivalent profit with and without border adjustment.
Considered together, Propositions 6 and 7 suggest that the primary benefit of border adjustment
is not to protect domestic firms’ profits as some argue (e.g., Beattie and Hill 2009, The Economist
David Drake: Carbon tariffs, technology choice and foreign cost advantage
24
2007). Proposition 7 states that, if foreign entry and/or domestic offshoring occur in the absence of a
border adjustment, domestic profits only conditionally benefit—and potentially decrease—from the
implementation of such a mechanism. Proposition 6, on the other hand, states that global emissions
strictly decrease with the implementation of a border adjustment under identical conditions. This
suggests that, rather than protecting domestic profits, the primary benefit of border adjustment
is to improve the efficacy of carbon regulation, enabling emissions regulation to deliver reduced
global emissions across settings where it fails to do so without such a mechanism.
6.
Implications and Conclusions
There are two widely held beliefs related to carbon leakage—that carbon leakage increases global
emissions; and that a border adjustment eliminates the threat of carbon leakage. However, neither
of these beliefs necessarily holds if: i) firms can choose production technologies; and ii) offshore
producers have a production cost advantage. In such settings, foreign firms adopt clean technology
at a lower emissions price than domestic firms (Proposition 2), with entry conditionally increasing
in emissions price over intervals in which foreign firms hold this advantage (Proposition 4). Further, domestic firms may choose to offshore in equilibrium despite a border adjustment, but they
adopt cleaner technology when doing so (Proposition 3). In short, carbon leakage can still occur
under border adjustment. However, if it does, global emissions conditionally decrease as a result
(Proposition 5), which contrasts the strict increase in global emissions that results from carbon
leakage arising in the absence of border adjustment (Proposition 1).
A border adjustment is shown to provide four potential sources of emissions improvement relative
to emissions regulation without border adjustment (Proposition 6). A border adjustment conditionally: 1) reduces total output relative to emissions-regulated settings without border adjustment; 2)
enables cleaner production in the regulated region in settings in which incentivizing clean technology adoption through emissions regulation is not otherwise feasible; 3) results in “clean” leakage
due to foreign entry by incentivizing the adoption of lower-emissions technology by foreign firms;
and 4) results in “clean” leakage due to offshoring by domestic firms. All but the first of these
potential sources of advantage require endogenous technology choice, highlighting the importance
of incorporating this decision in assessments of border adjustment policy.
Domestic firm profits can increase or decrease as a result of border adjustment (Proposition 7).
This, along with the observation that global emissions are less through the inclusion of a border
adjustment in any setting in which it would apply (Proposition 6), suggests that the primary
benefit of the mechanism is not to protect domestic-owned firms as some argue, but rather to
enable carbon regulation to deliver on its intended purpose: the reduction of global emissions.
David Drake: Carbon tariffs, technology choice and foreign cost advantage
25
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David Drake: Carbon tariffs, technology choice and foreign cost advantage
A.
27
Parameters for Numerical Illustrations
Figure(s)
A
b nd
Figure 2a
110 .02 6
Figure 2b
110 .02 3
Figures 3a and 3b 120 .01 4
Figure 4a
Figure 4b
Figure 5a
100 .05 4
Figure 5b
100 .01 3
Figures 6a and 6b 120 .01 4
Table A.1
B.
γ1
75
75
75
50
50
50
55
50
α1
.70
.70
.70
1.0
1.0
.70
.78
.70
γ2
85
85
85
70
70
65
65
65
α2
.20
.20
.20
.20
.20
.20
.28
.20
Fe Fo
1000 1000 1000 600
5000 5000 1000 600
δ
.80
.80
.85
.75
.80
.75
.75
.80
τ
30
30
30
15
15
15
15
15
ατ
.15
.15
.15
.15
.15
.15
Market, cost, and emissions parameters used in numerical illustrations.
Proofs
Proof of joint concavity of firm objectives. FOCs for each firm i ∈ Nd and firm j ∈ Nf are
and
∂πi (nd , nf , l, y, ε)
= A − b(xi,l + (nd − 1)x−i,l + nf y) − bxi,l − cd (l, ε) = 0, ∀i ∈ Nd ,
∂xi,l
(10)
∂πf (nd , nf , xl )
= A − b(nd xl + (nf − 1)y−j + yj ) − byj − δγf − τ = 0, ∀j ∈ Nf .
∂yj
(11)
The joint concavity of firm objectives can be proven directly through the Hessian. Based on the
FOCs in Equations (10) and (11), the second derivative of domestic and offshore objectives are
∂ 2 πi ( · )
= −2b, ∀i ∈ Nd ,
∂x2i,l
and
∂ 2 πj (·)
= −2b, ∀j ∈ Nf ,
∂yj2
while the cross-partials are
∂ 2 πi ( · )
= −b,
∂xi,l ∂y
and
∂ 2 πj ( · )
= −b, ∀i ∈ Nd , ∀j ∈ Nf .
∂yj ∂xl
Elements composing the main diagonal of the Hessian are equal to −2b while all other elements
are equal to −b. As a consequence, all odd-ordered leading principal minors are strictly negative
and all even-ordered leading principal minors are positive, thereby implying strict concavity.
Proof of Lemma 1. The following equilibrium quantities are required to prove Lemma 1:
Lemma 6. With the number of domestic and foreign competitors fixed at nd and nf > 0, domestic
and foreign firms, respectively, produce at equilibrium quantities
x∗l =
A − cd (l, ε)
nf (cf − cd (l, ε))
+
,
b (nd + nf + 1)
b (nd + nf + 1)
and
y∗ =
nd (cf − cd (l, ε))
A − cf
−
.
b (nd + nf + 1)
b (nd + nf + 1)
Proof of Lemma 6. Since the problem is symmetric for all domestic firms and is likewise
symmetric for all foreign firms, solving Equation (11) for y ∗ yields
y∗ =
A − cf − bnd xl
, ∀j ∈ N f .
b(nf + 1)
(12)
David Drake: Carbon tariffs, technology choice and foreign cost advantage
28
Substituting (12) into Equation (10) and then solving for xl yields
x∗l =
A − cd (l, ε)
nf (cf − cd (l, ε))
+
, ∀i ∈ N d ,
b (nd + nf + 1)
b (nd + nf + 1)
(13)
which, by substituting into Equation (12) yields
y∗ =
nd (cf − cd (l, ε))
A − cf
−
, ∀j ∈ N f .
b (nd + nf + 1)
b (nd + nf + 1)
(14)
The number of offshore entrants follows directly from its definition,
max πf (nd , nf , xl ) = max [P (nd , nf , xl , y) y − cf y] = Fe , ∀j ∈ Nf .
y
y
⇒ [A − b (nd xl + nf y)] y − cf y = Fe , ∀j ∈ Nf .
The result follows from the quantities in Equations (13) and (14), and the constraint that nf ≥ 0.
A − cf − nd (cf − cd (l, ε))
∗
√
nf = max 0,
− nd − 1 . (15)
Fe b
Proof of Lemma 2 When domestic firms produce in the regulated region cd (l, ε) = γd + αd ε,
and when domestic firms produce offshore cd (l, ε) = δγ1 + τ . Domestic quantity solutions for Γ1
follow directly from Equation (13) when cd (l, ε) = δγ1 + τ , and n∗f = y ∗ = 0 (because Γ1 is partially
defined by the absence of foreign entry). Similarly, domestic quantity solutions for Γ3 follow from
Equation (13) when cd (l, ε) = δγ1 + τ and n∗f = y ∗ = 0. For Γ2 and Γ4 , n∗f > 0, and therefore is equal
to the righthand argument of the maximum statement given in Equation (15), and cf = δγ1 + τ .
Domestic quantities then follow directly from Equation (13) and foreign quantities follow directly
from Equation (14). For Γ5 , foreign quantities are determined directly from Equation (14) when
n∗f is again equal to the righthand argument of Equation (15) and n∗d = x∗l = 0.
Proof of Lemma 3 Define Q(Fo ) as the change in total output resulting from offshoring.
Where ι > 0 is infinitesimal and Fo ≤ F o ,
Q(Fo ) = n∗d x∗r (Fo , εo (Fo ) − ι) + n∗f y ∗ (Fo , εo (Fo ) − ι) − n∗d x∗r (Fo , εo (Fo )) − n∗f y ∗ (Fo , εo (Fo )).
If Fo ≤ F o , then without entry (i.e., when n∗f = y ∗ = 0), the unique solution to εo (Fo ) is
cf − γd A − cf − (A − cf )2 − Fo b(Nd + 1)2
εo (Fo ) =
+
,
αd
αd
while, with entry (i.e., n∗f > 0), the unique solution is
√ √
√
b( Fe − Fe − Fo )
cf − γd
+
.
εo (Fo ) =
αd
αd
(16)
(17)
(18)
David Drake: Carbon tariffs, technology choice and foreign cost advantage
29
If Fo = 0, then cd (r, ε(Fo )) = cf . Wrt Fo , cd (r, ε(Fo )) > 0. Therefore, if 0 < Fo ≤ F o , then
h(Fo , εo (Fo )) < h(Fo , εo (Fo ) − ι) because cf = cd (l∗ , ε(Fo )) < cd (l∗ , ε(Fo ) − ι). Substituting the appropriate values of x∗l and y ∗ from Lemma 2, n∗f from Lemma 1, and εo (Fo ) from (17) and (18),
⎧
√
Nd (A−cf − (A−cf )2 −Fo b(Nd +1)2 )
⎪
⎪
if h(Fo , εo (Fo ) − ι) ≤ 0
⎨
b(Nd +1)
√
((N
+1)
F
b−A+c
)
e
d
f
Q(Fo ) =
(19)
if h(Fo , εo (Fo )) < 0 < h(Fo , εo (Fo ) − ι)
b
⎪
⎪
⎩0
if h(F , ε (F )) ≥ 0.
o
o
o
√
By Lemma 1, h(Fo , εo (Fo )) < 0 ⇔ ((Nd + 1) Fe b − A + cf )/b > 0. Therefore, when 0 < Fo ≤ F o ,
Q(Fo ) > 0 iff h(Fo , εo (Fo )) < 0, and Q(Fo ) = 0 otherwise. If Fo = 0, then at the limit as ι →
√
0, h(Fo , εo (Fo )) = h(Fo , εo (Fo ) − ι) with both equal to (A − cf )/ Fe b − Nd − 1 per Lemma 1.
Consequently, per (19), the interval (h(0, εo (0)), h(0, εo (0) − ι)) is empty and Q(0) = 0.
Proof of Proposition 1 From (3) and Lemma 2, ∂η ∗ /∂ε = −αd2 Nd /b(Nd + 1) when l∗ = r and
n∗f = 0. When l∗ = r and n∗f > 0, ∂n∗d x∗r αd /∂ε = −αd2 Nd /b, while ∂n∗f y ∗ (α1 + ατ )/∂ε = αd Nd (α1 +
ατ )/b through Lemma 1 and Lemma 2; therefore ∂η ∗ /∂ε = (α1 + ατ − αd )(αi Nd /b). If domestic
firms offshore or exit, ∂η ∗ /∂ε = 0 because cd (l∗ , ε) = δγ1 + τ = cf or n∗d = 0, respectively. Related
to Proposition 1d, the change in global emissions due to offshoring is εΔ (n∗d x∗r (Fo , εo (Fo ) − ι)) +
Q(Fo )(α1 + ατ ), where εΔ is the change in emissions intensity due to offshoring, and Q(Fo ) is the
change in total output due to offshoring defined in Lemma 3. Emissions intensity increases due to
offshoring by at least αt and, per Lemma 3, total output is non-decreasing due to offshoring.
Proof of Proposition 2 Offshore firms prefer type 2 if δγ2 + α2 ε + τ ≤ δγ1 + α1 ε + τ . Therefore
the offshore adoption threshold is ε̂f2 = δ(γ2 − γ1 )/(α1 − α2 ). Symmetrically, the domestic adoption
threshold in the regulated region is εd2 = (γ2 − γ1 )/(α1 − α2 ). δ ∈ (0, 1) implies ε̂f2 < εd2 if domestic
firms produce in the regulation region. Further, where α̂r and α̂o are the emissions intensity of
domestic firms’ preferred type within and outside the regulated region, respectively, with entry
√ ∂ (π̂d,r (·) − π̂d,o (·)) 2 (α̂r − α̂o ) ε − Fe b (αr − αo )
=
,
(20)
∂ε
b
which is negative only if α̂r = α1 and α̂o = α2 (α̂r = α2 and α̂o = α1 is infeasible due to the adoption
thresholds discussed above). Without entry, the number of competitors is constant at Nd and all
firms make symmetric decisions, so it is sufficient to observe that γ̂1 + α̂1 ε < δγ̂1 + α̂1 ε+τ ∀ε ∈ [0, ε̂f2 ].
Therefore, if domestic firms offshore, they do so only at an emissions price ε > ε̂f2 .
Proof of Proposition 3 Without entry, the unique solution to ε̂o (Fo ) is
((A − δγ̂o − τ )α̂r − (A − γ̂r )α̂o )2 − Fo b(Nd + 1)2 (α̂r2 − α̂o2 )
(A − γ̂r )α̂r − (A − δγ̂o − τ )α̂o
−
,
ε̂o (Fo ) =
α̂r2 − α̂o2
α̂r2 − α̂o2
(21)
David Drake: Carbon tariffs, technology choice and foreign cost advantage
30
while the unique solution with entry is
δγ̂o + τ − γ̂r
ε̂o (Fo ) =
+
α̂r − α̂o
√ √
√
b( Fe − Fe − Fo )
.
α̂r − α̂o
(22)
Assume α̂r > α̂o and Fo = 0. Then per (21) and (22), ε̂o (Fo ) = (δγ2 + τ − γ1 )/(α1 − α2 ). α̂r < α̂o
is not feasible due per the discussion in the proof of Proposition 2. Likewise, α̂r = α̂o is infeasible
through (21) and (22). Further, without and with entry, respectively, π̂d,o (·) is
2
π̂d,o (·) =
(A − (δγ̂o + τ + α̂o ε))
,
2
(Nd + 1) b
and
π̂d,o (·) = Fe .
If Fo > F̂ o = π̂d,o (·), then ε̂o (Fo ) is infeasible through (21) and (22). (In the case without entry,
ε̂o (Fo ) is infeasible because (21) simplifies to ε̂o (Fo ) = (A − γ̂r )/α̂r when Fo > F̂ o = π̂d,o (·), implying
A − γ̂r − α̂r ε < 0, which violates the profitability condition A − γ̂r − α̂r ε ≥ 0). Therefore, there exits
a unique solution to ε̂o (Fo ) iff ε̂o (Fo ) ∈ [ε̂f2 , εd2 ) and Fo ≤ F̂ o , otherwise ε̂o (Fo ) is infeasible.
Proof of Lemma 4 The proof for Lemma 4 is symmetric to that of Lemma 2 except that
ĉd (ˆl, ε) = γ̂d + α̂d ε when domestic firms produce in the regulated region, ĉd (ˆl, ε) = δγ̂d + α̂d ε + τ
when they produce offshore, and ĉf (ε) = δγ̂f + α̂f ε + τ .
Proof of Proposition 4 ∂ ŷ ∗ /∂ε = 0 under all market structures, which is made obvious
through Lemma 4. Changes in total entry n̂∗f ŷ ∗ wrt ε are therefore driven through changes in n̂∗f .
√
By (6), ∂ n̂∗f /∂ε = −αf / Fe b if foreign and domestic firms operate identical technology. When foreign firms operate type 2 technology and domestic firms operate type 1, ∂ n̂∗f /∂ε = [Nd α1 − (Nd +
√
1)α2 ]/ Fe b, which is positive iff α1 > α2 (1 + 1/Nd ). Proof of Lemma 5 The proof to Lemma 5 is symmetric to that of Lemma 3 except that the
unique solution to ε̂o (Fo ) without entry is given by (21), while it is given by (22) with entry, and
where Δ = (A − δγ2 − τ )α1 − (A − γ1 )α2 ,
√
⎧
2
Nd (Δ− Δ2 −Fo b(Nd +1)2 (α2
1 −α2 ))
⎪
⎪
⎨
(Nd +1)(α1 +α2 )b √
√
Q̂(Fo ) = ((Nd +1)α1 −Nd α2 ) Fe b−α2 (Fe −Fo )b−Δ
⎪
(α1 −α2 )b
⎪
⎩
0
if h(Fo , εo (Fo ) − ι) ≤ 0
if h(Fo , εo (Fo )) < 0 < h(Fo , εo (Fo ) − ι)
if h(Fo , εo (Fo )) ≥ 0.
Proof of Proposition 5 Per Lemma 4 and Equation (6), the change in domestic and offshore
emissions in ε are summarized in the following table:
It is clear that η̂ ∗ strictly decreases in ε in all cases except Ω2 when ε ∈ [ε̂f2 , êd2 ) where η̂ ∗ decreases
iff (α1 − α2 − ατ )(α1 − α2 )Nd + α2 (α2 + ατ ) < 0. The change in global emissions due to offshoring
is ε̂Δ [n̂∗d x̂∗r (Fo , ε̂o (Fo ) − ι)] + Q̂(Fo )(α2 + ατ ), where ε̂Δ is the change in emissions intensity due to
offshoring which is −(α1 − α2 − ατ ). The condition of Proposition 5b follows directly.
David Drake: Carbon tariffs, technology choice and foreign cost advantage
∂ n̂∗d x̂∗r α̂d /∂ε
−α̂d2 Nd /b(Nd + 1)
∂(n̂∗d x̂∗o
31
+ n̂∗f ŷ ∗ )(α̂f
Region
+ ατ )/∂ε
Ω1
0
Ω2 , ε ∈ [ε̂f2 , êd2 ) −(α1 − α2 )Nd α1 /b ((α1 − α2 )Nd − α2 )(α2 + ατ )/b
/ [ε̂f2 , êd2 )
0
−α̂f (α̂f + ατ )/b
Ω2 , ε ∈
Ω3
0
−α2 Nd (α2 + ατ )/b(Nd + 1)
Ω4
0
−α2 (α2 + ατ )/b
Ω5
0
−α̂f (α̂f + ατ )/b
Proof of Proposition 6 The following lemma is needed to prove the result.
Lemma 7. If ε > εentry and/or ε > εo (Fo ) and Fo < F o , then total output with a border adjustment T̂ = n̂∗d x̂∗ + n̂∗f ŷ ∗ is less than total output without border adjustment T = n∗d x∗ + n∗f y ∗ .
Proof of Lemma 7 Per Lemmas 1 and 2,
Nd (A − δγ1 − τ )/b(Nd + 1) if ε > εo (Fo ), εo (Fo ) ≤ εentry and Fo < F o
√
T=
(A − Fe b − δγ1 − τ )/b
if ε > εentry and either εentry < εo (Fo ) or Fo ≥ F o .
Further, through (6) and Lemma 4,
Nd (A − δγ2 − α2 ε − τ )/b(Nd + 1)
T̂ =
√
(A − Fe b − δγ̂f − α̂f ε − τ )/b
if ε > ε̂o (Fo ), ε̂o (Fo ) ≤ ε̂entry and Fo < F̂ o
if ε > ε̂entry and either ε̂entry < ε̂o (Fo ) or Fo ≥ F̂ o .
(23)
(24)
Define Tz and T̂z , z ∈ {1, 2}, as the z th case of (23) and (24), respectively. T1 > T̂1 and T2 > T̂2
√
because δγ1 + τ < δγ̂f + α̂f ε + τ , ∀ε > 0 by Assumption 3. Further, T1 > T2 iff (Nd + 1) Fe b >
A − δγ1 − τ , which holds iff the conditions for T1 apply (in turn implying n∗f > 0 per Lemma 1 and
l∗ = o). Therefore, if T = T1 , then T1 > T2 , and T > T̂ by transitivity whether T̂ = T̂1 or T̂ = T̂2 .
Likewise, if T = T2 , then T2 > T1 , and T > T̂ by transitivity whether T̂ = T̂1 or T̂ = T̂2 . Greater ξ w without border adjustment than with it implies
(n∗d x∗r ) αd + n∗d x∗o + n∗f y ∗ (αf + ατ ) εs > (n̂∗d x̂∗r ) α̂d + n̂∗d x̂∗o + n̂∗f ŷ ∗ (α̂f + ατ ) εs
(25)
The conditions of Proposition 6a imply that, without border adjustment, potential market equilibria are restricted to Γ2 , Γ4 , and Γ5 per Lemma 2, implying n∗f > 0. Over ε ∈ [0, min{εo (Fo ), εd2 }),
αd = α̂d = α1 , while if εd2 < εo (Fo ) then over ε ∈ [εd2 , εo (Fo )), αd = α̂d = α2 . If ε > εo (Fo ) and Fo < F o
then αd = α1 . Per Proposition 3, α̂d = α2 if domestic firms offshore under border adjustment.
By Proposition 2, αd ≥ α̂f . αf = α1 always. Therefore, we have ordering αf ≥ αd ≥ α̂d ≥ α̂f . By
Lemma 7, T > T̂ under the conditions of Proposition 6. Therefore, n∗d x∗ αd + n∗f y ∗ αf > n̂∗d x̂∗ α̂d +
√ √
n̂∗f ŷf∗ α̂f . It is evident from (18) and (21) that ∂(ε̂o (Fo ) − εo (Fo ))/∂Fo = αo b/2 Fe − Fo (αr −
√
α̂o )αr > 0, ∀Fo ∈ [0, F o ], and that ∂(ε̂o (Fo ) − εo (Fo ))/∂ 2 Fo = αo b/4(Fe − Fo )3/2 (αr − α̂o )αr > 0,
∀Fo ∈ [0, F o ].3 Therefore, ε̂o (Fo ) − εo (Fo ) is convex increasing. It is also evident from (18) and (21)
3
Recall that subscript r in (18) and (21) denotes domestic firms’ technology choice in the regulated region at εo (Fo )
without border adjustment and at ε̂o (Fo ) with border adjustment. Since εd2 = ε̂d2 , αr = α̂r . Likewise, recall that α̂o is
the emissions intensity of the technology domestic firms choose if offshore at ε̂o (Fo ) under border adjustment.
David Drake: Carbon tariffs, technology choice and foreign cost advantage
32
that ε̂o (0) = εo (0). Therefore, ε̂o (Fo ) > εo (Fo ), ∀Fo ∈ (0, F o ] and l∗ = r implies ˆl∗ = r; equivalently,
xo = 0 implies x̂o = 0. Since εd2 = ε̂d2 , l∗ = r also implies cd (l∗ , ε) = ĉd (ˆl∗ , ε). Therefore, per Lemma 1,
and (6), if l∗ = r and n∗d = Nd , then n∗f > n̂∗f iff (Nd + 1)(δγ̂f + α̂f ε − τ ) − (Nd + 1)(δγ1 + τ ) > 0,
which holds by Assumption 3. If n∗d = 0, then T = n∗f y ∗ . T > T̂ = n̂∗d x̂∗ + n̂∗f ŷ ∗ per Lemma 7. Therefore, if l∗ = r and n∗d = 0, n∗f y ∗ > n̂∗f ŷ ∗ . Therefore, l∗ = r implies n∗f y ∗ > n̂∗f ŷ ∗ . As a consequence,
l∗ = r implies (n∗d x∗o + n∗f y ∗ )ατ > (n̂∗d x̂∗o + n̂∗f ŷ ∗ )ατ . Further, if l∗ = o then x∗ = x∗o , and T > T̂ , per
Lemma 7 implies (n∗d x∗o + n∗f y ∗ )ατ > (n̂∗d x̂∗o + n̂∗f ŷ ∗ )ατ .
Because n∗d x∗ αd + n∗f y ∗ αf > n̂∗d x̂∗ α̂d + n̂∗f ŷ ∗ α̂f and (n∗d x∗o + n∗f y ∗ )ατ > (n̂∗d x̂∗o + n̂∗f ŷ ∗ )ατ , (25)
holds. Further, if ĥ(Fo , ε) > 0 and ε ∈ ε̂f2 , ε̂fexit , then n̂∗f > 0 and α̂f = α2 < α1 = αf , with
∗ ∗
(n̂d x̂r ) α̂d + n̂∗d x̂∗o + n̂∗f ŷ ∗ (α1 + ατ ) εs > (n̂∗d x̂∗r ) α̂d + n̂∗d x̂∗o + n̂∗f ŷ ∗ (α2 + ατ ) εs .
The conditions of Proposition 6b imply that, without border adjustment, potential market equilibria are restricted to Γ3 and Γ4 per Lemma 2. Under market equilibria Γ3 and Γ4 , x∗r = 0 and
αd = αf = α1 . α1 ≥ α̂d and α1 ≥ α̂f . By Lemma 7, T > T̂ under the conditions of Proposition 6.
Therefore, n∗d x∗ αd + n∗f y ∗ αf > n̂∗d x̂∗ α̂d + n̂∗f ŷ ∗ α̂f . With x∗r = 0, lower total output under border
adjustment implies n∗d x∗o + n∗f y ∗ ατ > n̂∗d x̂∗o + n̂∗f ŷ ∗ ατ . Therefore, (25) holds. If Fo ≥ F̂ o , ˆl∗ = r.
Therefore, if ε ≥ εd2 , α̂d = α2 , and n̂∗d x̂∗r α̂d + (n̂∗d x̂∗o + n̂∗f ŷ ∗ )(α̂f + ατ ) > n̂∗d x̂∗r α2 + n̂∗f ŷ ∗ (α̂f + ατ ). Lastly,
if ε > ε̂o (Fo ) and Fo < F̂ o . Per Propositions 2 and 3, ˆl∗ = o, α̂d = α̂f = α2 , and (n̂∗d x̂∗r + n̂∗f ŷ ∗ )(α1 +
ατ ) > (n̂∗d x̂∗o + n̂∗f ŷ ∗ )(α2 + ατ ).
Proof of Proposition 7 It is sufficient to prove that, under the conditions of the proposition,
there exists an interval in ε in which Π̂ > Π, another in which Π̂ < Π, and another in which Π̂ = Π.
From Lemma 1 and Equation (6), ε̂entry > εentry . As demonstrated in the proof of Proposition 6,
ε̂o (Fo ) > εo (Fo ). Assume ε̂entry < ε < εo (Fo ), or ε̂entry < ε < εexit and Fo > max{F o , F̂ o }. Then
l∗ = ˆl = r, n∗f > 0, and n̂∗f > 0. By Lemmas 1, 2 and 4 and Equations (1), (4), and (6),
Π = Nd (
Fe b + δγ1 + τ − γd − αd ε)2 /b
while
Π̂ = Nd (
Fe b + δγ̂f + α̂f ε + τ − γ̂d − α̂d ε)2 /b.
Since εd2 = ε̂d2 , γd + αd ε = γ̂d + α̂d ε. Therefore, for any ε > 0, Π̂ > Π by Assumption 3 (i.e., γ2 > γ1 ).
Assume ε̂o (Fo ) ≤ ε ≤ εentry and Fo < min{F o , F̂ o }. Then l∗ = ˆl∗ = o and n∗f = n̂∗f = 0. By Lemmas 1, 2 and 4 and Equations (1), (4), and (6),
Π=
Nd (A − δγ1 − τ )2
− N d Fo
(Nd + 1)2 b
while
Π̂ =
Nd (A − δγ2 − α2 ε − τ )2
− N d Fo .
(Nd + 1)2 b
Therefore, Π̂ < Π by Assumption 3.
Assume ε̂etnry < ε̂o (Fo ) ≤ ε and Fo < min{F o , F̂ o }. Then l∗ = ˆl∗ = o, n∗f > 0, and n̂f > 0. By
Lemmas 1, 2 and 4 and Equations (1), (4), and (6) Π = Π̂ = Nd (Fe − Fo ).