adverse selection as a policy instrument: unraveling
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Adverse Selection as a Policy Instrument:
Unraveling Climate Change∗
Steve Cicala
Tufts University
David Hemous
University of Zurich
Morten Olsen
University of Copenhagen
Click here for the latest version.
December 3, 2021
Abstract
This paper applies principles of adverse selection to overcome obstacles that prevent theimplementation of Pigouvian policies to internalize externalities. Focusing on negative exter-nalities from production (such as pollution), we evaluate settings in which aggregate emissionsare known, but individual contributions are unobserved by the government. We propose givingfirms the option to pay a tax on their voluntarily and verifiably disclosed emissions, or pay anoutput tax based on the average rate of emissions among the undisclosed firms. The certifica-tion of relatively clean firms raises the output-based tax, setting off a process of unraveling infavor of disclosure. We derive sufficient statistics formulas to calculate the welfare of such aprogram relative to mandatory output or emissions taxes. We find that our mechanism woulddeliver significant gains over output-based taxation in two empirical applications: methaneemissions from oil and gas fields, and carbon emissions from imported steel.
∗We are grateful to Thom Covert, Meredith Fowlie, Michael Greenstone, Suzi Kerr, Gib Metcalf, Mark Omara,Mar Reguant, James Sallee, Joseph Shapiro, Andrei Shleifer, Allison Stashko, Bob Topel and seminar participants atTufts, Universitat Bern, the Environmental Defense Fund, UC San Diego, the Midwest EnergyFest, Yale, Harvard,the Utah Winter Business Economics Conference, UC Santa Barbara, UC Berkeley, and the University of Chicagofor helpful comments. Ivan Higuera-Mendieta provided excellent research assistance. Cicala gratefully acknowledgesfunding from the 1896 Energy and Climate Fund at the University of Chicago, and along with Olsen thanks theUniversity of Zurich for hospitality. All errors remain our own. e-mail: scicala@gmail.com, david.hemous@gmail.com,graugaardolsen@gmail.com
1 Introduction
Uninternalized externalities abound. In spite of the simplicity of economists’ advice when
the magnitude of the harm is known, the obstacles to correcting such market failures are
myriad: political opposition, excessive implementation costs, the presence of havens induced
by competing jurisdictions, among others. In this paper we show the extent to which such
obstacles may be overcome in situations in which damage is caused by heterogenous agents.
We apply results from the literature on mechanism design under asymmetric information as
a policy lever to encourage program participation and voluntary revelation of harm.
We consider situations in which the aggregate level of harm (such as pollution or traffic
congestion) is known by the government, but the exact contributions of specific agents is not.
In such settings it is impossible to levy Pigouvian taxes due to the unobserved sources of
harm. The optimal uniform fee (such as an output tax on producers rather than an emissions
tax) falls short of the first best since the fee does not depend on one’s contribution to the
problem. It also fails to incentivize abatement to reduce damage (Cropper and Oates (1992);
Schmutzler and Goulder (1997); Fullerton et al. (2000); Farrokhi and Lashkaripour (2021)).
We propose creating the option to certify one’s damage, upon which a Pigouvian tax
will be levied, combined with an output-based fee that tracks the average rate of damage
among those choosing not to participate in the certification program.1 This encourages those
who inflict relatively little damage to certify, thus raising the output-based fee paid by non-
participants. This sets off an unraveling in favor of program participation as increasingly
damage-intensive agents seek to separate themselves from the tail of the distribution that
becomes concentrated by adverse selection.
We first develop a closed-economy model in which production is heterogenously associated
with an externality and derive the distance of an optimally-set output tax from the first-best
Pigouvian policy. We approximate with this distance with a sufficient statistics formula that
depends on marginal damages, the slope of the supply curve, variance of emissions, and
monitoring costs. We show how the option to reveal one’s emissions yields welfare objects
that are a linear combination of the outcomes under output and emissions taxes, with weights
equal to the relative variance of emissions under each policy. We also show that, under certain
conditions, the policy maker can achieve the same outcome by only knowing the mean of the
emissions distribution. This is achieved through an algorithm that encourages the gradual
unravelling of the emissions distribution, converging to an equilibrium in which the policy
1This can be calculated because the overall level of harm is observed, and subtracting the contribution ofcertified agents reveals the average contribution among those who remain uncertified.
1
maker has full information on the emissions distribution. We extend the analysis to allow
firms to abate and show that there is a natural complementarity between the two; only
through certification is it worthwhile for firms to abate.
As an empirical application of the closed-economy model, we use our mechanism to
internalize the cost of methane emissions from oil and gas production in the Permian basin
in Texas and New Mexico. The Permian is the source of 30% of U.S. oil production and
10% of natural gas (Administration (2019)). While royalty adders have been proposed to
address the climate externality of embodied carbon in these fuels on federal land (Prest and
Stock (2021)), methane emissions are a thornier problem. Methane is a potent greenhouse
gas that either leaks from the supply chain or is intentionally vented into the atmosphere
in an unmonitored fashion. Alvarez et al. (2018) estimate about 13 million tons of methane
leak from the oil and gas sector annually, for a social cost of about $20B. Nearly 60% of those
emissions occur during production, and in 2018 about 2.7 million metric tons were released
from the Permian basin, for a social cost of $4B (Zhang et al. (2020)). However, emissions
rates across wells are highly heterogeneous (Robertson et al. (2020)). We find that while
a royalty adder levied on production would reduces emissions by about 4%, a tax levied
directly on pollution would reduce emissions by 80%, before abatement. We find that our
sufficient statistics approximations deliver answers that closely follow more data-intensive,
bottom-up calculations. A voluntary emissions tax paired with a rolling output tax would
deliver identical outcomes as a emissions mandatory tax, with about $3B/year in benefits,
even with significant costs of emissions certification. The opt-in policy actually exceeds the
welfare of mandatory taxation as certification costs grow by allowing firms to economize on
the regulatory burden.
We then extend our model to an international setting, focusing on the constraints of
unilateral climate policy. Because greenhouse gases are global pollutants, it is natural that
research on climate policy has focused on international environmental agreements between
sovereign nations who regulate their respective producers.2 Such agreements must overcome
the unilateral incentive to shirk (Barrett (1994)), possibly by punishing countries outside
of the agreement (Nordhaus (2015)). Dynamic considerations also come into play as costly
investments in clean technology create hold-up problems in future negotiations due to their
complementarity with abatement (Beccherle and Tirole (2011); Harstad (2012); Battaglini
and Harstad (2016)). Governments are the key decisionmakers in this paradigm, and only
policies that are individually rational from each country’s perspective are feasible.
2See Chan et al. (2018) for a review.
2
The absence of strong, binding international agreements raises the question of how emis-
sions might be reduced through unilateral policy. The ability of unilateral carbon taxation to
reduce emissions is reduced (and possibly reversed) when production can profitably move to
unregulated jurisdictions, a problem known as ‘leakage’ (Bohm (1993); Copeland and Taylor
(1995); Aldy and Stavins (2012); HA c©mous (2016); Fowlie (2009); Fowlie et al. (2016)).
This prospect strengthens the incentive to shirk on international commitments by simulta-
neously reducing the effectiveness of the tax and increasing the benefits of becoming a haven.
Trade policy in the form of a “Border Carbon Adjustment” (BCA) has been considered the
primary instrument to mitigate the competitive disadvantage caused by taxing one’s own
emissions (Copeland (1996); Metcalf and Weisbach (2009); Elliott et al. (2010, 2013); Larch
and Wanner (2017), see Condon and Ignaciuk (2013) for a literature review). BCAs levy
tariffs based on the average carbon content of production in the country of origin so that
foreign producers (on average) cannot undercut domestic firms. Under such a policy foreign
producers remain effectively outside the reach of the government, as their tax burden is unre-
lated to firm-specific emissions. They face no individual incentives to abate their emissions,
and any pollution reductions depend on price elasticities of demand and supply.3
The goal of our approach in the international setting is to approximate the emissions
reductions that might be achieved with a widely-adopted price on carbon, but without re-
quiring the legally-binding international agreements that have proven elusive to date. To do
so, we focus on the direct interactions between a government and firms whose disclosures of
emissions are voluntary. International sovereignty may restrict what governments can man-
date of foreign firms, but does not foreclose the possibility of creating incentives to shape
their behavior. We do this by providing firms with the option to certify their emissions, and
basing the default rate on the average emissions of uncertified firms. This recasts the problem
of jurisdiction into one of screening, in which clean firms wish to separate themselves from
more intensive polluters (Spence (1973); Stiglitz (1975)). This separation causes the uncer-
tified mean to rise, setting off a process of unraveling that encourages further certification
(Akerlof (1970)).
We show the conditions under which the unraveling mechanism is preferable to a unilat-
eral domestic carbon tax, or a tax combined with a BCA. As an empirical application, we
consider the case of international trade in steel, an energy-intensive, trade-exposed sector
3Markusen (1975) and Hoel (1996) derive the optimal tariff in the presence of transboundary pollution.As highlighted by Keen and Kotsogiannis (2014) and Balistreri, Kaffine and Yonezawa (2019), an optimalenvironmental tariff generally differs from the BCA formula (even if a BCA were able to distinguish betweenthe carbon contents of different imports and even in the absence of terms of trade effects).
3
that is central to environmental trade policy (Miller and Boak (2021)). We consider trade
policy between the OECD and Brazil, a major steel exporter. Using the sufficient statis-
tics formulas we develop, we estimate that an optimally-implemented certification program
would achieve nearly 75% of the welfare gains of a universal carbon tax. However, we also
find that unrestricted access to such a program would create significant ‘backfilling’ wherein
the the dirtiest firms would expand their output to serve the untaxed foreign market. This
reduces the welfare benefits of certification, making the unrestricted program slightly inferior
to a standard border carbon adjustment. This example highlights the countervailing forces
that limit a government’s ability to reduce externalities outside of its jurisdiction, while also
presenting a mechanism to productively expand its reach.
The combination of optional disclosure and a rolling default creates a policy that mimics
the strategy that has been applied in private markets to ensure quality (Jovanovic (1982);
Grossman (1981); Milgrom and Roberts (1986); Milgrom (2008). See Dranove and Jin (2010)
for a review). In these settings firms voluntarily provide warranties or submit to audits in
order to separate themselves from low-quality producers. Even relatively lower-quality firms
become willing to make such disclosures to separate themselves from the absolute worst
offenders when consumers update their beliefs regarding those who decline to disclose (Jin
and Leslie (2003); Jin (2005); Lewis (2011)). It has also been used by firms to improve risk
selection for credit (Einav et al. (2012)), improve safety (Viscusi (1978); Hubbard (2000);
Jin and Vasserman (2019)), and has been suggested to encourage more efficient electricity
consumption (Borenstein (2005, 2013)) and fisheries management (Holzer (2015)). To our
knowledge this is the first paper to apply these principles to overcome obstacles to the
implementation of Pigouvian policies.
The use of screening mechanisms in public policy has been successfully applied to im-
prove the targeting of recipients of public benefits (Alatas et al. (2016); Finkelstein and
Notowidigdo (2019); Deshpande and Li (2019)). In such settings the government creates
hurdles so uptake is limited to those who value benefits more than the ordeal of enrollment.
These policies typically do not entail unraveling as the government is is free to choose the
magnitude of the enrollment ordeal so that the optimal point of separation is achieved im-
mediately (Kleven and Kopczuk (2011); Besley and Coate (1992); Nichols and Zeckhauser
(1982); Nichols et al. (1971)): The costs or benefits of non-participation are not program-
matically adjusted with the extent of participation. In our setting this would be analogous to
the government choosing its preferred carbon content of uncertified imports, which is likely
4
to run afoul of strategic trade considerations.4
There is a long tradition of regulation under asymmetric information in the mechanism
design literature (Baron and Myerson (1982); Laffont and Tirole (1993)). In the pollution
context, the regulator seeks to elicit information on abatement costs (Kwerel (1977); Roberts
and Spence (1976); Dasgupta et al. (1980); Baron (1985); Laffont (1994)) and must design
a policy schedule that elicits truthful revelation. In these settings, as in the context of
non-point source pollution, the lack of verifiability is the key constraint on the regulator
(Segerson (1988); Xepapadeas (1991); Laffont (1994); Xepapadeas (1995), among others.
For a review see Xepapadeas (2011)). While emissions remain unobserved at uncertified
firms, our focus on an optional, verifiable revelation of emissions converts the problem into
a traditional point-source setting in which firms face incentives to abate. Recent work on
voluntary environmental regulation notes the improved enforcement targeting for uncertified
firms, but does not unravel non-participation with changing in audit probabilities (Foster
and Gutierrez (2013, 2016)).
Organizationally, the paper separates the analysis between domestic and international
settings. In each setting we develop a theoretical model, and then apply the results from the
model to an empirical setting. The final section concludes.
2 Unraveling in the Domestic Case
To focus on the central issue of disclosure, we first consider a simplified setting of a closed
economy in which firms differ only in their emission rates. Throughout this section we focus
on a simple partial equilibrium model with an externality, though our approach would also
apply to a broader class of models. The government knows the full distribution of emis-
sions, but not those of individual firms unless they choose to certify. We take as given that
mandatory certification is not possible and characterize the welfare benefits of an optional
certification program.
First, we derive “sufficient statistics” in the sense of approximations to changes in wel-
fare that can be expressed through simple objects such as emissions variances and supply
elasticities. We begin by solving a benchmark model for any level of certification, and then
derive the optimal level of certification. We then weaken the information available to the
regulator and show the program may be implemented knowing only the first moment of the
4At the extreme, the government could simply prohibit imports from firms whose emissions are uncertified.This is not without precedent—the U.S. Food and Drug Administration mandates access for inspectors atfacilities abroad for any firm wishing to sell food or pharmaceuticals in the US (Federal Food, Drug andCosmetic Act, Section 807 (b) as amended by Section 306 of the FDA Food Safety Modernization Act). Thebasis of the jurisdiction problem we address is that such mandates are infeasible with respect to emissions.
5
emissions distribution. We conclude the section with a series of extensions to the benchmark
model: allowing for abatement, heterogeneous productivity, and adjustments that only occur
through entry and exit. These extensions yield minor adjustments to the exact expressions
of the benchmark model, but share a common fundamental structure.
2.1 Baseline model
We consider a closed economy and focus on a specific industry that produces a homogeneous
polluting good under perfect competition. A representative agent has preferences over this
good, represented by the following quasi-linear utility function:
U = C0 + u(C)− vG,
where C is total consumption of the polluting good and C0 is the consumption of an outside
good. The price of the outside good is normalized to 1. G denotes emissions from the
production of good C. The marginal social cost of emissions is v. The outside good does not
pollute.
The polluting good is produced by an (exogenous) mass 1 of firms who operate under
perfect competition. Firms have the same strictly convex cost function c(q) but vary in the
extent to which they pollute. The emissions rate per unit produced is denoted by e and
follows the cdf Ψ(e), with full support and corresponding pdf of ψ(e) > 0, on the domain
[e, e] where e ≥ 0 and e may be infinite. Though the overall distribution of emissions, Ψ,
and the production of each firm is observable, the emissions of an individual firm are private
information (unless the firm is certified as described below).
2.2 Equilibrium with an output tax or emission tax
In the following, we distinguish between a tax on emissions and one on output. First, consider
an output tax, t, which can be implemented even when individual emissions are not observed.
Let the market price be p and solve the firm’s problem in a decentralized equilibrium to get:
p = c′(q) + t. (1)
This defines a supply function q = s(p − t). With a mass 1 of firms this is also total
production, Q. The resulting profit function follows as π(p− t) = (p− t)s(p− t)−c(s(p− t)).The supply curve is upward-sloping by the convexity of the cost function and the profit
function is increasing in p − t. Utility maximization gives: u′(C) = p which together with
Q(p− t) = s(p− t) and C = Q defines an equilibrium price, p, and quantity, Q. Emissions
6
are given by:
G =
∫ e
e
s(p− t)eψ(e)de = s(p− t)E(e) (2)
Next, and in anticipation of the discussion of certification below, we solve for a setup in
which emissions are observable and taxed at τ . This gives an individual supply function of
s(p− τe) and aggregate supply and emissions of:
Q =
∫ e
e
s(p− τe)ψ(e)de = E [s(p− τe)] and G = E [es(p− τe)] (3)
The social planner would set t = vE(e) when emissions are not observed and τ = v when
emissions are observable. In the following, we solve for an equilibrium in which firms can
certify and be taxed based on their individual emissions.
2.3 Equilibrium with certification
We now introduce voluntary certification of emissions, in which a firm can choose between
two tax settings. If the firm chooses to (verifiably) reveal its level of emissions, e, it is
taxed at τe (where we do not necessarily impose that τ equals the social cost of carbon,
v). If the firm chooses not to reveal its level of emissions, it is taxed at the mean level of
emissions of the firms who do not certify t = τE(e|R), where R denotes the set of firms
who have not certified. We will keep this relationship between τ and t as an assumption
throughout most of the paper. We consider it a natural starting point since our interest lies
with the reallocation of taxes based on better information of underlying emissions and not
with changing the overall tax rates. Second, when the price of certification is set optimally
as in Section 2.4, t = τE(e|R) is in fact optimal.
When presenting this choice to firms, the policy maker must calculate R ex ante based on
knowledge of the distribution of emissions, Ψ(e). Here we assume that the policy maker can
do that. The total cost to a firm of certification equals the technical cost of certification, in
the form of a third-party expert, an objective monitoring system etc., F > 0 and a potential
additional tax/subsidy that the government might impose, f ≶ 0. In an equilibrium in which
some firms certify, and others do not, an indifferent firm with emissions level e is defined by:
π(p− τ e)− (F + f) = π(p− t). (4)
Since the left-hand side is decreasing in e, all firms with e < e certify and firms with e > e
do not. Consequently f is a tool to determine e. The resulting tax rate on output for firms
7
who do not certify is:
t = τE[e|e > e],
and where importantly, ∂t/∂e > 0, that is the rate at which uncertified firms are taxed is
increasing in the number of certified firms.
To facilitate the discussion below, we introduce ε, which is equal to the emissions rate at
which a firm is effectively taxed:
ε =
e
E(e|e > e)
if e ≤ e
if e > e,(5)
where E(ε) = E(e). Production by firms who do not certify is s(p − τE(e|e > e)) and for
those who do certify it is s(p− τe) such that total production i
Q =
∫ e
e
s(p− τe)ψ(e)de+ (1−Ψ(e))s(p− τE(e|e > e)) = E(s(p− τε)),
with corresponding emissions of:
G = E [εs(p− τε)] . (6)
The equilibrium price follows from market clearing, C = Q, and utility maximization,
u′(C) = p. We consider sufficient conditions for this equilibrium to be unique in Appendix
A. These include i) E[e|e > e] − e is decreasing in e and ii) s(· ) is weakly convex or τ is
small. The condition on E[e|e > e] is satisfied for most frequently used distributions and
below we will rely on first order approximations in which case ii) is satisfied.
For any variable x we let xV denote its value with certification and xU its value under
the output tax without certification. Comparing equations (6) and (2) gives the difference
in emissions between the two tax systems:
Lemma 1. The difference between emissions under voluntary certification, GV , and the
output tax, GU , is given by:
GV −GU = Cov[ε, s(pV − τε
)]+ E (e)
{E[s(pV − τε
)]− s
(pU − τE (e)
)}, (7)
where pV and pU are the equilibrium prices under certification and the output tax, respectively.
The effect of certification on emissions is generally ambiguous. However, emissions decline
8
when s is weakly convex and es(pV − τe) is concave in e. This is satisfied for linear supply
or small τ .
The equation in Lemma 1 consists of two terms. First, a reduction in emissions from a
reallocation effect: certification allows the reallocation of production from firms with high
emissions to those with low emissions. This is captured by the negative covariance term. The
second term combines two effects: i) a classical rebound effect as certified firms are taxed at a
lower rate and increase production and with it emissions and ii) a price effect in that possible
increases in the equilibrium price could further increase production. The condititions in the
lemma rule out these severe cases of rebound. They are automatically satisfied for linear
supply or small τ , in which case certification lowers total emissions: certification lowers the
tax rate for some firms and raises it for others, keeping the average tax rate constant at
τE(ε) = τE(e). With linear supply curves, total production and hence the market-clearing
price is unaffected by certification (when all firms remain in operation). With a reallocation
towards less polluting firms, but no change in aggregate production, total emissions must
decline.5
The following proposition gives the difference in welfare between the two settings for a
given tax rate, τ :
Proposition 1. The difference between social welfare with certification and the output tax
is given by:
W V −WU = E(π(pV − τε
))− π
(pV − τE (e)
)︸ ︷︷ ︸reallocation effect
+
∫ pV
pU(s (p− τE (e))−D (p)) dp︸ ︷︷ ︸
price effect
− (v − τ)(GV −GU
)︸ ︷︷ ︸untaxed emissions effect
− FΨ (e) . (8)
Where
a) The “reallocation effect” and the “price effect” are always weakly positive.
b) The “untaxed emissions effect” is zero for v = τ , and otherwise depends on whether
rising (falling) emissions are over (under) taxed relative to Pigouvian levels.
5Alternatively, consider a convex supply function, which implies that for a given price, total supply mustincrease with certification, so that the equilibrium price declines (pV < pU ). As a result, es(pV − τe) <es(pU − τe). In addition, when es(pV − τe) is concave in e an application of Jensen’s inequality ensures thatoverall emissions decline.
9
c) W V −WU + FΨ(e) is of ambiguous sign but when v ≥ τ it is positive when supply
curves are linear or for small τ .
Proof. Proof in Appendix 1.
The first term in equation (8) is a reallocation effect and captures the increase in profits
from a reallocation of production from firms with higher taxes to those with lower. ε is the
effective level of emission taxation under certification such that E[π(pV −τε)]−π(pV −τE(e))
is the average gain for firms. This effect is always positive.
Next, consider the untaxed emissions effect which captures the welfare effects of changing
emissions. These are zero when emissions are taxed at the Pigovian level, τ = v, whereas if
taxes are lower than this, there are net welfare gains from emissions reductions, which will
occur depending on the conditions of Lemma 1. That is, if τ < v, and the rebound effect
is not too strong, the untaxed emissions effect will be positive. FΨ(e) captures the fraction
Ψ(e) of firms and the F resources required to certify them.
Figure 1 illustrates how reallocation and price effects depend on the concavity of the
supply function. SV (p) always intersects the y-axis lower than the SU(p) curve because
firms with low emissions face lower taxes. For sufficiently low prices, the number of firms
producing is increasing along SV (p). When the price is high enough for all firms to produce,
the two curves overlap when supply curves are linear, as in Panel (a). In this region, the
positive supply effect for firms with lower taxes is exactly matched by the negative effect for
firms facing a higher tax. Total quantities are unchanged and there is no price effect. The
reallocation effect is measured by the increase in producer surplus, represented by the area
between SV (p) and SU(p).
When supply curves are not linear, the price need not remain constant. Panel b considers
the case of convex supply curves.6 Convexity implies that the firms that face lower taxes
will increase their production by more than the firms who face higher taxes will reduce their
production. Consequently, the supply curve will be to the right. Again, the reallocation
effect is captured by the area between the curves (C), whereas the price effect is captured
by B. The area A is just a reallocation from producers to consumers and does not feature
in aggregate welfare changes.
6Section A.1.5 in the Appendix considers the opposite case in which supply is concave and the priceincreases. Proposition 1 still holds, but the allocation of welfare between producers and consumers is different.
10
Figure 1: Market Equilibria with and without Voluntary Certification
(a) Linear supply curves: The supply curve undercertification, SV (p), starts below SU (p) because theleast emitting firms are willing to supply at a lowerprice when certified. Once price is high enough forall firms to produce the two curves overlay and theequilibrium price will be the same under either set-ting. The area between the two curves captures thereallocation effect.
D(p)
SU(p)
SV(p)
p
Q
pV=pU
(b) Convex supply curves: When supply curvesare convex SV (p) is everywhere below SU (p) andprices must decline under certification. Lower pricesimply increased consumption as well as a transferfrom producers to consumers. The price effect is cap-tured by B and the reallocation effect is captured byC.
SU(p)
SV(p)
pU
pVA B
C
D(p)
Q
p
Note: pV and pU are equilibrium prices with and without voluntary certification, respectively.SU(p) is the aggregate supply curve when firms are not certified, and they all face a tax oft = τE(e). SV (p) denotes the case where some firms are certified, face different taxes, and
consequently, different supply curves.
11
As can be intuited from Panel b, the price effect is of a higher order than the other terms.
Our goal is to establish “sufficient statistics” for changes in emissions and welfare that can be
easily evaluated using readily available data. In this pursuit, and with little loss of generality,
we will consider first-order approximations in τ in much of the analysis to come. This implies
that the price effect is zero and the reallocation effects and untaxed emission effects are both
positive (for τ ≤ v).
Corollary 1 considers such approximations. We assume that taxes, τ , and the social costs
of emissions, v, are small relative to prices. We then conduct Taylor expansions around τ = 0
(where v is of the same order).
Corollary 1. The expression W V −WU in Proposition 1 can be written as:
W V −WU =(v − τ
2
)s′ (p0)V ar (ε) τ − FΨ (e) + o
(τ 2), (9)
with a difference in emissions of:
GV −GU = −s′ (p0) τV ar (ε) + o(τ),
where p0 is the price when τ = 0. It holds that:
a) V ar(ε) = 0 if e = e and ∂V ar(ε)/∂e > 0.
b) W V −WU + FΨ(e) is positive if τ < 2v (to a second order).
c) GV −GU is negative (to a first order).7
d) These expressions hold exactly when supply curves are linear.
Proof. Appendix A.1.2.
Corollary 1 demonstrates that there is no price effect at first order and the primary
driver of the welfare consequences of certification come from the shift in production from
more to less polluting firms. With a constant production but a shift towards firms with
fewer emissions, total emissions are sure to decline. Even if average emissions were already
taxed efficiently, τ = v, total welfare increases because production is reallocated towards less
polluting firms. The size of this reallocation depends on the supply response, s′(p0), and the
variance of the taxed emissions rate, V ar(ε). When τ = v the entire welfare benefit from the
certification program accrues to firms (at first order). This result, however, depends on the
7When τ > v emissions are taxed excessively. The costs of doing so are quadratic in the deviations fromthe optimal tax, implying a quadratic distortion of increasingly high taxes on the dirtier firms. When τ > 2vthis effect dominates and causes negative welfare effects.
12
assumption of an exogenous unit mass of firms. Appendix A.1.6 demonstrates that if we allow
for endogenous entry of firms, all additional effects are of higher order and the expressions
of Corollary 1 remain unchanged, but the entire welfare benefit accrues to consumers.8
A natural question to ask is how far welfare of the voluntary certification in Proposition
1 is from the welfare that would be obtained if firm emission rates were known and taxable
(without necessarily imposing τ = v) (Jacobsen et al., 2020). Labeling such an equilibrium
with W FI for “full information” we find (at second order)
W V =V ar (ε)
V ar (e)W FI +WU
(1− V ar (ε)
V ar (e)
)− FΨ (e) + o(τ 2). (10)
By construction V ar(ε) ≤ V ar(e) so welfare under voluntary certification (gross of cer-
tification costs) is a weighted average of welfare with no certification, WU , and with full
information, W FI . The weight reflects the relative variance of the effectively-taxed emission
rate, ε, and the actual emission rates, e. This means that greater benefits of voluntary certi-
fication accrue as a larger share of the variance of emissions certify. Having additional firms
certify is only beneficial insofar as they represent a higher share of the emissions variance.
2.4 Optimal policy
In the following, we solve for the optimal combination of the three available policy tools: the
emissions tax on certified firms, τ , the output tax on uncertified firms, t, and the subsidy/tax
on certification, f . Appendix A.1.3 shows that these are
τ = v,
t = vE(e|e > e),
f = vE (e− e|e > e) s (p− vE (e|e > e)) > 0. (11)
The condition τ = v and t = vE(e|e > e) recover the standard Pigovian result that
emissions ought to be taxed at their (expected) social cost. Equation (11) shows that cer-
tification is excessive when τ and t are set optimally, and should be taxed. To see why, we
employ the first order condition with respect to f . This can equivalently be written wrt. to
8Specifically, we specify a free entry condition that E(π(ε)) = FE , where a slight abuse of notationpermits us to let π(ε) denote the profits of a firm with emissions rate ε. In this case, the mass of firms N isendogenous. The expressions in Corollary 1 take this into account by replacing s′(p0) with N0s
′(p0) whereN0 is the mass of firms under p0.
13
e as:
π (p− ve)− π (p− vE (e|e > e))− vE (e− e|e > e) s (p− vE (e|e > e))− F = 0. (12)
Increased certification implies higher profits for the marginal firm, e, captured by the first
two terms in the expression. In addition, the quantity tax on uncertified firms goes up, which
has a negative effect on profits for the uncertified firms. The third captures this whereas
the fourth term captures the social cost of certification, F. As τ = v emissions are taxed
correctly, and there is no marginal gain from changes to emissions. Comparing equation
(12) with the indifference condition of when firms certify (equation 4) we get Equation (11).
Since the certifying firms do not internalize the costs they impose on still non-certified firms,
the optimal policy requires a tax.
These considerations relate to the size of F and f . Corollary 1 does not specify the order
of F and f . Consider f = 0 (no tax or subsidy on certification). The gains from certification
are first order, so if F is first order, some firms will certify. Since welfare gains are second
order, equation (9) then delivers negative welfare benefits. If F is second order, almost all
firms certify. Neither of these is efficient, as just discussed. An f > 0 of first order according
to equation (11) ensures the optimal certification.
2.5 An “Unraveling” Algorithm
Having solved for the decentralized equilibrium and the social planner’s allocation, we take
a step back and assess the informational requirements needed to implement such a policy.
Whereas equation (9) gives an intuitive result of the welfare gains based on statistics that
are relatively easily obtained — such as the variance of emission rates and supply elasticities
— the implementation requires complete information on the distribution of e, which is rarely
available. We show the conditions under which a given “algorithm” can achieve a comparable
outcome without complete information on the distribution of e.
We assume that neither firms nor the government knows the distribution of emissions
rates, but they do observe the average emissions rate (through aggregated accounts or changes
in ambient pollution, for example). We further assume for this section that e <∞. Initially,
certification is not available and the government imposes an output tax t0 = τE(e). We
assume that the government introduces certification which allows firms to pay the emission
tax τ at some certification cost F . Since the government does not know the distribution Ψ(e),
it cannot predict the eventual threshold e and therefore cannot implement the equilibrium
described above by immediately announcing a new output tax τE (e|e > e).
14
Consider instead an iterative process where the government allows firms to certify at
increasing levels of emissions rates. That is, the government asks if firms with emissions
rate e want to certify and only those firms are allowed to do so. If they do, the level of
certification increases and the procedure starts again. The government continuously adjusts
the output tax as the emissions distribution is revealed. We show that we obtain a Nash
equilibrium when firms decide on certification as if they were the last ones to certify with
the information available at that point in time.
Assume that all firms with an emissions rate below e have certified and consider the
decision facing a firm with emissions rate e. The firm will certify if:
g (e) ≡ π (p (e)− τ e)− π (p (e)− τE (e|e > e))− F ≥ 0,
where p (e) is the price that would prevail on markets should the certification stop here and
the threshold be e = e. Since the distribution up to e has been revealed publicly and since
E (e) is known, both the government and the firm can compute E (e|e > e). Assuming that
g (e) > 0, then at least some firms will certify. Furthermore, with a bounded distribution,
g (e) = −F so not all firms will certify as long as F > 0. As firms decide sequentially to
certify, the process will continue up to the smallest emissions rate for which g switches sign,
which we denote e (which also corresponds to a laissez-faire equilibrium when the government
knows the full distribution ψ).
This leads to a Nash equilibrium because for firms with a lower emissions rate, e < e,
π (p (e)− τ e)− π (p (e)− τE (e|e > e))− F > g (e) = 0,
so that none of these firms would benefit from deviating from certification at the equilibrium.
Since g becomes negative just after e, firms with emission rate e + de, decide not to certify
and would indeed be worse off otherwise. Firms with a higher emission rate e > e, will not
certify either. Hence, even if the government does not know the distribution of Ψ(e) it can
implement an algorithm where the optimal certification decisions of firms gradually reveal
the shape of the distribution up until e.
The analysis can be straightforwardly extended to the social optimum if the government
continuously adjusts the certification tax f = τ (E (e|e > e)− e) s (p(e)− τE (e|e > e)) with
the information available.
15
2.6 Extensions
The following sections add two dimensions of flexibility to the baseline model: abatement
and heterogeneous productivity (Appendix A.1.6 allows for free entry). The extensions fit
easily in the framework presented so far, and we present them with a focus on the added
terms to emissions and welfare. The full explicit expressions are given in Appendix A.1.6.
Abatement
We keep the same structure as above but allow firms to spend b(a) per unit produced to
reduce their per-unit emissions by a. We require: b′(a) > 0 and b′′(a) > 0 for a > 0. For
simplicity, we further add b′(0) = b(0) = 0. Pre-abatement emissions are still distributed
according to Ψ(e), and a certified firm i pays an emission tax on e(i)− a(i) instead of e(i).
We continue to define ε in equation (5) as the pre-abatement emissions rate for certified
firms and the conditional mean of emissions for uncertified firms. Abatement investments
are not observable and non-certified firms consequently have no economic incentive to abate.
Hence, in an equilibrium without certification, no abatement takes place. Certified firms, in
contrast, do abate a strictly positive amount. They solve the problem:
maxq,apq − c(q)− τ(e− a)q − b(a)q
which leads to a common abatement level, a∗, among all firms that certify of a∗ = b′−1(τ).
We let A(τ) ≡ τa∗(τ) − b(a∗(τ)) > 0 denote the savings per unit of production due to
abatement. If e < a some firms sequester emissions. Individual supply functions of certified
firms with emissions rate e takes the form:
q = s (p− τe+ A(τ)) ,
such that supply is higher under abatement. The expression for changes in total emissions
with abatement takes the structure of Lemma 1 but adds two additional terms:
−a∗Ψ(e)E[s(pV − τe+ A(τ)
)|e < e
]+Ψ(e)E
{e[s(pV − τe+ A(τ)
)− s
(pV − τe
)]|e ≤ e
},
(13)
where (with slight misuse of notation) pV now represents the equilibrium price under cer-
tification with abatement. The first term in expression (13) is the direct impact of a mass
of Ψ(e) certifying and thereby abating their emissions by a∗. At the same time certification
lowers their tax burden, which yields a supply response analogous to a rebound effect on the
16
quantity produced. This additional effect pulls in the direction of higher emissions. If s is
convex and es(pV − τe) is increasing and weakly concave in e, then emissions must decline.
If τ is small emissions must decline as well.
Similarly, we can establish a result on welfare analogous to that of Proposition 1. Em-
bedded in equation (13) are the additional negative effects on emissions which increases the
positive untaxed emissions effect when v > τ . In addition, we get:
Ψ (e)(E(π(pV − τe+ A(τ)
)− π
(pV − τe
)|e ≤ e
)).
This captures the increase in profits to firms that receive a higher net price after abatement.
This benefit accrues only to the share of firms Ψ(e) that certify. For given price pV , profit
maximization ensures that this term is positive.
Finally, we continue our analysis using Taylor expansions and derive the following corol-
lary to Proposition 1:
Corollary 2. To a second-order approximation, the difference in social welfare when moving
to voluntary certification from an output tax in the presence of abatement is
W V −WU = τ(v − τ
2
)(s′(p0)V ar(ε) +
s(p0)Ψ (e)
b′′(0)
)− FΨ (e) + o
(τ 2),
where W V −WU+FΨ(e) is positive.
Corollary 2 takes advantage of the fact that to a first order e remains unchanged with
abatement and consequently we can add a single term to equation (9) from Corollary 1. b′′(0)
captures the curvature in the abatement costs. By assumption, it is always profitable to do
some abatement and a low curvature of the abatement function (b′′(0)), implies that, to a
first order, the optimal abatement level will be higher. Abatement benefits are proportional
to total production by the firms that abate: s (p0) Ψ(e). For this effect to be positive requires
τ < 2v (the reasoning is analogous to that of footnote 7).9
9Analogously to equation (10), to a first order, we can write welfare under certification with abatement asa weighted average of welfare without certification, WU , and welfare with full information under abatement,WFIA as:
WV = WFIAω + (1− ω)WU − FΨ(e) + o(τ2),
where ω =(V ar (ε) + s(p0)
s′(p0)b′′(0)Ψ (e)
)/(V ar (e) + s(p0)
s′(p0)b′′(0)
). Welfare under certification (gross of certi-
fication costs) continues to be a weighted average of the welfare with complete information and with no
information, but with the added term, s(p0)s′(p0)b′′(0)
Ψ(e) which captures (to a first order) the amount of abate-
ment done with certification.
17
Heterogeneous Productivity
In the following we again preclude abatement, but we allow firms to differ in their level of
productivity. In particular, firm i has costs of production of c(q)/ϕi, where c(q) has the
same properties as the cost function in Section 2 but ϕi > 0 differs across firms. We let
Ψ(e, ϕ) denote the joint distribution and allow unrestricted covariance between e and ϕ. We
consider supply functions with constant elasticity such that a firm i that certifies produces
qi = s0 [ϕi(p− τei)]α and the uncertified firms produce qu = s0 [ϕi(p− t)]α, where t is τ
times total emissions of uncertified firms divided by total production by uncertified firms.
This is also the optimal quantity tax rate under no certification.10
In this setup, firms differ both in their productivity and their emissions. Therefore, the
cut-off e of Section 2 is replaced by a cut-off function e(ϕ) which depends positively on
productivity.11 To a first order, the expressions for the change in emissions and welfare of
Corollary 2 take the same form, except V ar(ε) is replaced by the output-weighted variance
of emissions:˜V ar(ε) =
∫ϕ
∫ε
(ε− E(ε))2ψ(ϕ, ε)dεdϕ,
where ψ(ϕ, ε) = ϕαψ(ϕ, ε)/(∫
ϕ
∫εϕαψ(ϕ, ε)dεdϕ
)is a density distribution rescaled by out-
put (proportional to ϕα at price p0) such that E(ε) equals the average emissions per unit
without certification:
E(ε) =
∫ϕ
∫ε
εψ(ϕ, ε)dεdϕ = GU/SU
Intuitively, the reallocation effect is still the driving force behind our results, though firms’
emissions are now weighted by their size.
2.7 Adjustments along the extensive margin
The model presented above considered reallocation on the intensive margin. For complete-
ness, we consider situations in which the only margin of adjustment is whether to produce
or not. To focus on the supply side we consider an exogenous price p such that consumer
welfare (excluding emissions) is constant.12
10Although our approach could also be used with other supply functions, the analysis becomes morecomplicated, in particular because t is no longer the optimal quantity tax.
11Specifically, equation 4 is replaced by: 1ϕπ(ϕ(p − τ e(ϕ))) − (F + f) = 1
ϕπ(ϕ(p − t)) where π() is as
previously defined. This defines e(ϕ) where the cut-off emission rate depends positively on productivitybecause production increases with productivity whereas the certification cost, F + f , does not.
12This implies that we deviate from our assumption of a closed economy. Section 3 focuses explicitly on aninternational case, and here we do not explicitly model alternative consumers or suppliers. This approach is
18
Consider a mass 1 of potential firms each characterized by potential production q, entry
costs c and emissions, e. These are distributed according to Ψ(c, q, e) with a corresponding
pdf of ψ. The domain is [c, c] × [q, q] × [e, e] with weakly positive lower bounds and finite
upper bounds and Ψ has support everywhere. It will further be convenient to define the
unconditional distribution of e, ψe(e), and the two conditional distributions ψc(c|e, q) and
ψq(q|e). In the following we present only results and keep derivations in Appendix A.2.
Firms face a uniform price of p. In laissez-faire, a firm i produces if its individual draw
satisfies pqi ≥ ci. When a uniform quantity tax is imposed, this condition becomes (p−t)qi ≥ci.
Total production and emissions are then given by:
S(p− t) =
∫q1(p−t)q≥cdΨ(c, q, e), (14)
G(p− t) =
∫eq1(p−t)q≥cdΨ(c, q, e),
where 1(p−t)q≥c is the indicator function and it is understood that when not otherwise specified
integrals are over the full support of (c, q, e).
The slope of the aggregate supply curve is:
S ′(p− t) =
∫e
ψe(e)
∫q
q2ψc((p− t)q|e, q)ψq(q|e)dqde. (15)
The slope of the aggregate emission curve, G′(p − t), follows the same formula with eq2
instead of q2. Price or tax changes only affect production by firms on the margin of entry so
that ψc is only evaluated where c = (p − t)q. The “q2” reflects the fact that potential firms
with greater output are also more sensitive to per-unit price or tax changes. It will prove
useful to define
ψe(e) ≡(ψe(e)
∫q
q2ψc(pq|e, q)ψq(q|e)dq)/ (S ′(p)) ,
as the “sensitivity and size”-corrected distribution on e. ψe is the distribution of e scaling by
the size of firms (one q), their sensitivity to tax or price changes (the other q), evaluated at
t = 0 for the marginal firms (c = pq). Using this, the change in emissions (at t = 0) from
exact if i) demand is perfectly elastic at p, ii) there are no externalities associated with production elsewhereor iii) the externalities elsewhere are corrected by a Pigovian tax.
19
price changes is:
G′(p) = Eψe(e)× S′(p),
where Eψe simply denotes the expectation with respect to ψe. One can demonstrate that
(to a first order) the optimal uniform quantity tax is t∗ = vG′(p)/S ′(p) = vEψe(e), average
emissions weighted by q2 along the relevant margin of adjustment (where pq = c) and not
total emissions divided by total supply as was the case for the model on the intensive margin.
The difference arises because i) only firms on the margin of pq = c adjust to marginal price
changes and ii) larger firms are more important for changes to emissions and also more
sensitive to tax changes.
2.7.1 Equilibrium with certification
We set up a certification system along the same lines as for the intensive-margin case. We
assume that the certification cost F is second-order in τ . Firms with e ≤ e certify whereas
those with e > e do not.13 A firm that certifies will pay τe in emission tax and those that
do not will pay according to the average emissions of active non-certified firms:
t = τE (eq|e ≥ e, (p− t)q ≥ c)
E (q|e ≥ e, (p− t)q ≥ c)= τ
∫eq1e>e1(p−t)q≥cdΨ(c, q, e)∫q1e>e1(p−t)q≥cdΨ(c, q, e)
. (16)
This implies that total supply is given by:
S(p, τ, t) =
∫q(1e≤e1(p−τe)q≥c+F + 1e>e1(p−t)q≥c
)dΨ(c, q, e),
where corresponding expressions for total costs C(p, τ, t) and emissions G(p, τ, t) follow the
structure of S(p, τ, t) but replace q with c and eq, respectively. The total mass of firms who
certify is given by M =∫
1e≤e1(p−τe)q≥c+FdΨ(c, q, e). We proceed along the same lines as for
Corrollary 1 to arrive at the following result:
Proposition 2. Consider τ = v. The difference in welfare and emissions between certifica-
tion and no certification is given by:
W V −WU =v2
2S ′(p)×
13In the intensive margin case, it was immaterial whether we chose an exogenous e or a tax f to incentivizethe same level of certification. Here a tax, f , however, will affect both the margins of certification and ofentry and these two setups will not be equivalent. Therefore, we define e as the maximum emission rate thatthe government permits to certify. With F being second order in τ , the constraint binds and e = e.
20
[V arψe(ε) +
(G′(p)
S ′(p)− G(p)
S(p)
)2
−(
1− Ψe(e))(
Eψe(e|e > e)− E (eq|e ≥ e, pq ≥ c)
E (q|e ≥ e, pq ≥ c)
)2]−FM+o
(τ 2),
(17)
where G(p) and S(p) are evaluated at t = τ = 0 and Ψe(e) =∫ eeψe(e)de is the (sensitivity and
size-)corrected mass of firms with e ≤ e. ε is given by ε = e if e ≤ e and ε = Eψe(e|e > e)
otherwise.
Change in emissions is given by:
GV −GU = −τS ′(p)×[V arψe(ε) +
(G′(p)
S ′(p)− G(p)
S(p)
)Eψe(e) + (1− Ψe(e))
(G(p)
S(p)Eψe(e|e > e)− G′(p)
S ′(p)Eψe(e)
)]+o(τ).
(18)
Proof. See Appendix A.2.
Consider first the change in welfare which replicates the structure of welfare gains under
the intensive margin. As in the model with an intensive margin, the first term captures the
reallocation effect of moving from an optimally set uniform tax to an individual tax. However,
as discussed above, t = G(p)/S(p) in equation (16) is not set optimally. The change in welfare
is therefore best thought of as a two-step change: First moving to an optimal uniform tax
of G′(p)/S ′(p) (captured by the second term) and thereafter to the certification program
(captured by the first). The last term captures the fact that the uncertified firms do not pay
the optimal output tax. Naturally, if the uniform tax were already set to G′(p)/S ′(p) only
the reallocation (the first) term would appear.
The change in emissions share similar intuitions. The second term captures the change in
taxation rate for certified firms when moving from a uniform tax of G(p)/S(p) to G′(p)/S ′(p)
and the third term is a “correction”-term for the fact that uncertified firms are not taxed
according to individual emissions.
2.8 Domestic Empirical Application: Methane Emissions in the Permian Basin
A carbon-based economy has dizzying array of emissions sources for potential regulation.
One way to economize on enforcement costs while maximizing coverage is to focus attention
on fuel extraction rather than points of emission (Metcalf and Weisbach (2009)). The close
measurement of fuel production for royalties charges creates a point of taxation with minimal
21
additional regulatory cost. Taxation at the point of extraction also ensures the price of carbon
is reflected throughout the supply chain. Recent studies have evaluated the effects of pricing
coal as a ‘royalty adder’ above existing royalty rates in the Federal Coal Program (Gillingham
et al. (2016); Gerarden et al. (2020)) and oil and gas leasing more broadly (Prest and Stock
(2021)).
In this section we build on these royalty adder proposals to account for unmonitored
methane emissions from oil and gas fields. Methane (CH4) is a powerful greenhouse gas,
with an estimated social cost of $1500/metric ton (on Social Cost of Greenhouse Gases
(2021)), compared to $51/metric ton for carbon dioxide. It is the primary compound in
natural gas, and is released into the atmosphere during production from natural gas wells
due to faulty equipment, or from safety valves to relieve pressure. Methane is created when
longer hydrocarbon chains break down under heat and pressure underground. As a result,
wells drilled to recover oil also co-produce methane to varying degrees. In places without
sufficient infrastructure to deliver methane from oil wells to market, it is a waste product.
Best practices in such settings entail flaring the methane in situ, which converts the gas into
less-harmful carbon dioxide.14
We focus our analysis on the Permian Basin of west Texas and southeast New Mexico,
whose oil-rich shales yield nearly one-third of U.S. oil and 10% of natural gas production
(Administration (2019)). Poorly operated or malfunctioning flares, direct venting, and leaks
from these oil and gas fields are responsible for significant methane emissions. These are
recently estimated to be 2.7 million metric tons (Tg) per year based on the analysis of
satellite data (Zhang et al. (2020)), for a social cost of about $4B/year. Zhang et al. (2020)
calculate these emissions to have roughly similar global warming potential as CO2 emissions
from the entire U.S. residential sector.
The main regulatory challenge for internalizing methane emissions is that exact sources
are currently unobserved and enormously heterogeneous across leases. A random ground-
based sample of oil and gas wells in the Permian found that 70% of emissions came from
15% of the measured sites, while nearly one third of measurements were below detectible
levels (Robertson et al. (2020)). An output-based royalty adder that accounts for average
methane emissions falls short of an actual emissions tax according to the results of Section
2: there is too little output from clean leases and too much from polluting sources due to
14In North Dakota’s oil-rich Bakken Shale, for example, approximately one third of natural gas was flaredin the mid 2010’s, making the sparsely-populated oil fields prominently visible at night from space (Cicala(2015)). The state adopted regulations to reduce flaring, and recent work has found a drop of about 20%through 2016 (Lade and Rudik (2020)).
22
the lack of emissions-targeting.
We seek to answer the following questions: What are the differences in emissions and
welfare between an output-based tax and an emissions tax? How might adverse selection push
outcomes under an output-based tax with voluntary emissions taxation closer to those of a
fully mandatory emissions tax? To answer these questions we combine the complete annual
production data from New Mexico and Texas with emissions estimates from a stratified
random sample of wells (Robertson et al. (2020)). We use this sample to simulate the
distribution of lease-level emissions per barrel of oil equivalent (BOE) for 2019 in the Permian
Basin.
With the variance of emissions in hand, we require only the social cost of pollution
($1500/t) and the slope of the supply curve to approximate the emissions and welfare changes
due to voluntary certification according to Corollary 1. For this last number we use an
elasticity of supply of 0.89 from Newell et al. (2019),15 while limiting production to zero
for the share of leases whose emissions fees would exceed revenue.16 Output prices per
BOE are constructed by weighing spot market prices for fuels from the Energy Information
Administration by the site-level share of production from oil, gas, or natural gas liquids,
respectively. Further details on data construction and summary statistics are provided in
the Data Appendix.
The first column of Table 1 presents quantity and welfare calculations based on observed
outcomes in 2019. We calculate standard errors for objects involving emissions by bootstrap-
ping from the estimates of Zhang et al. (2020) and Robertson et al. (2020). Details of the
procedure are provided in Appendix B. For each bootstrap sample’s estimates of emissions,
we calculate the corresponding quantities and welfare measures. The table reports the means
and standard deviations across 1,000 iterations. When calculating welfare, we assume that
world prices are unchanged, and therefore do not calculate consumer surplus.
To test our approximations, we calculate quantity and welfare changes two different
ways—using aggregate statistics and micro data. The second column applies a uniform tax
per barrel to the aggregate supply function to calculate outcomes. Section 2.1 derives the
difference between output and emissions taxes as a function of market aggregates, allowing
15This is the drilling elasticity based on Anderson et al. (2018), not the short-term change in productionfrom existing wells, whose marginal costs are essentially ignorable. We are interested in the long-term supplyresponse to our program. In steady state, the production elasticity is equal to the drilling elasticity (Hausmanand Kellogg (2015)). Anderson et al. (2018); Newell and Prest (2019) find similar elasticity estimates.
16Corollary 1 assumes that all agents remain active. In Appendix A we derive the corresponding expressionswhen firms have the option to shut down. We assume that leases whose tax bill exceeds revenue will shut in(i.e. pause production), and apply the expressions that account for shutting in.
23
heterogeneity only with respect to emissions rates. We use the empirical analogs of equation
(9) with an adjustment for shut-ins (i.e. zero output), plus the second column to arrive
at the third column, which estimates how outcomes are affected by a mandatory methane
tax. With a mandatory methane tax, we assume certification is costless and compliance is
complete.
Applying aggregate statistics to Corollary (1) is sufficient to approximate the effects of
policy. In the fourth column we show how these approximations compare to a bottom-up
calculation from lease-level data.17 For each bootstrap sample we use linear supply curves,
and each lease’s emissions intensity to calculate production decisions, emissions, producer
surplus, and tax revenues.
We estimate that methane leaks from the Permian basin in 2019 had a social cost of
about $4B, which is about 7% of producer surplus. An optimal output tax based on average
emissions per barrel would be about $1.60, or a 3% tax on oil and an 11% tax on natural gas.
An output tax of this magnitude reduces output by a relatively small amount, and its poor
targeting of the externality means that it is not particularly effective at reducing emissions.
The tax revenue and reduced externality costs exceed the lost producer surplus by about
$80M.
Even without allowing firms to abate their emissions, taxing the externality directly yields
significant welfare improvements. Rather than scaling back output with a uniformly-applied
tax, targeting those with relatively higher emissions causes about 1% of output to shut-in.
This zero bound means that the reduction in output is less under an emissions tax than an
output tax—even though we are using linear supply curves. Emissions fall by nearly 80%,
producer surplus losses are about one quarter of those under an output tax, and net welfare
is about $3B higher per year. These results are essentially unchanged in the fourth column.
Estimates based on calculations using market aggregates and applying Corollary (1) closely
track those from bottom-up lease-level calculations of the effects of an emissions tax.
Figure 2 plots net welfare of three prospective policies as a function of the cost per well
of monitoring methane emissions, or “certification costs.” First, an output tax piggy-backs
on the royalty model of metering output, so no additional regulatory costs are involved. The
net welfare of an output tax is therefore flat with respect to certification costs. The second
prospective policy is a mandatory emissions fee with costly certification. This is depicted as
a dashed blue line in Figure 2. Such a policy loses ground relative to an output tax as the
17Emissions may come from an individual well or equipment at a ‘site’ shared across multiple wells. Sinceproduction and emissions are measured at the lease level, we refer to our unit of measurement as a lease.
24
social cost of enforcement rises. Ultimately there is a point at which the cost of monitoring
emissions is greater than the benefits of reduced pollution, and the output tax becomes
superior. In Figure 2, this occurs around $75,000 per well-year. When certification costs are
zero, the difference between output taxes and emissions fees are the values in Table 1.
Finally, the solid green line plots the net welfare associated with a voluntary emissions
tax combined with an output tax that reflects the average emissions of uncertified firms.
We continue to assume that output prices are unaffected by the policy. We implement the
discretized version of the algorithm in Section 2.5 to numerically solve for the equilibrium. For
certification costs less than $25,000 we find that all producers would be either participating in
the emissions program or shut-in, delivering an identical outcome to a mandatory emissions
tax.
Above $25,000 in certification costs per lease-year, the opt-in emissions tax becomes supe-
rior to both output and mandatory emissions taxes. The voluntary program delivers greater
social value than a mandatory tax by economizing on certification costs. Uncertified firms
are charged based on the conditional mean of emissions, which is closer to actual emissions
than the unconditional mean. Society therefore receives the benefit of fewer emissions (due
to reduced output) from high-emissions firms, while not having to bear significant certifica-
tion costs. We estimate that above $110,000 per lease-year, the voluntary program becomes
inferior to an output tax. At this high certification cost, business stealing largely dominates
the welfare calculation: expanded producer surplus and emissions at certified firms does not
exceed the value of reduced emissions and producer surplus at uncertified firms (facing a
higher output tax) by more than the cost of certification. Alternatively, a policy that in-
cludes an optimally-set certification tax will deliver outcomes that are weakly above both
the emissions and output taxes for all social costs of certification (see subsection 2.4).
Analysis from recently promulgated rules on the oil and gas sector suggests that the
per-site cost of well emissions certification is less than $600 (Agency (2020b)). Though
these calculations appear to focus on inspections rather than equipment costs, there are a
range of existing monitoring technologies that could economically be brought to bear on the
problem. Tamper-proof meters are already widely used to levy royalties, and similar devices
could prospectively used to record flaring and safety valve releases. Internet-ready methane
imaging cameras are widely available for less than $1000 to monitor malfunctioning flares.
It is therefore unlikely that the annual per-site monitoring cost would approach those that
begin to yield a meaningful difference between the voluntary and mandatory emissions taxes.
The upshot is that in the absence of a mandate to tax methane emissions directly, a
25
Table 1: Methane Emissions in the Permian Basin: Annual Effects of Outputand Emissions Taxes
Predicted Change
Observed Output Tax Emissions Tax Emissions Tax(Aggregate Data) (Aggregate Data) (Micro Data)
QuantitiesProduction (Billions BOE) 2.53 -0.09 -0.04 -0.05
(0.02) (0.01) (0.01)Methane Emissions (Tg) 2.70 -0.10 -2.19 -2.22
(0.04) (0.47) (0.47)
WelfareProducer Surplus (Billion USD) 54.74 -3.97 -0.96 -1.08
(0.73) (0.27) (0.28)Tax Revenue (Billion USD) 3.90 0.71 0.71
(0.71) (0.20) (0.20)External Cost (Billion USD) 4.05 -0.15 -3.29 -3.34
(0.76) (0.06) (0.71) (0.71)Total (Billion USD) 50.69 0.08 3.09 2.97
(0.76) (0.03) (0.69) (0.68)
Note: Bootstrapped standard errors in parentheses. World prices are assumed to be invariant to policy,so consumer surplus is not calculated. External costs of methane do not include the social cost of carbondioxide. Aggregate data use total quantities and average price with the elasticity of supply to create a singlemarket supply curve, as well as moments of the emissions distribution. Simulated lease-level emissions andsupply curves are based on micro data on production and fuel mix.
methane royalty adder combined with the option to certify methane emissions may replicate
an identical outcome, with an annual social benefit of $3B/year. These benefits are a lower
bound, as abatement creates the potential for further emissions reduction at lower cost.
3 Unraveling in the International Case
We extend our model to an international setting. A Home policy maker values welfare both
in Home and Foreign but can implement policies only in Home. There is no Foreign policy
maker. This assumption avoids various well-understood results regarding terms-of-trade
manipulation and reflects the interest of relatively rich countries in the global social cost of
carbon.18 We consider a policy design analogous to that of the domestic model in Section 2:
18As a matter of fact, the social cost of carbon used by the U.S. government is supposed to reflect globaldamages.
26
Figure 2: Methane Emissions in the Permian Basin: Annual Net WelfareAgainst Certification Cost per Well
5051
5253
54Ne
w W
elfa
re (B
illion
s US
D)
0 25 50 75 100 125Certification Cost (Thousands USD)
Emissions Tax Output TaxVoluntary Emissions Tax
a uniform import tax on Foreign exporters is replaced with a voluntary certification program
and uncertified firms pay a unit tax of t = E(e|e > e).
In the purely domestic model, certification lowers the tax on the less-polluting firms,
raises it on those that pollute more, and, to a first order, keeps total production constant
and with it consumer prices. The international setting also features a reallocation of tax
rates amongst Foreign producers, but two additional forces are present, both arising from
(untaxed) Foreign consumption. First, whereas in a domestic setting the most polluting
firms are forced to sell in a market where they face higher taxes, in an international set-
ting, they might focus their production entirely on their domestic (Foreign) market, thereby
avoiding the tax entirely. Second, if prices decline in Foreign, consumption there increases.
Since foreign consumption is untaxed and consequently inefficiently high, this increase lowers
overall welfare. We formally derive these two effects below.
We build on the structure of the model in Section 2. Individuals in Home and Foreign
have potentially distinct utility functions of the form:
UH = C0,H + uH(CH)− vG,
UF = C0,F + uF (CF )− vG.
27
These result in demand functions for Home, DH(p), and foreign, DF (p). Consumers in
both countries experience the same negative disutility, v, from global emissions, G. It costs
κ units of the outside good to transport the polluting good between Home and Foreign. It is
free to ship the outside good. The outside good, C0, is produced emissions-free, competitively
and one-for-one with labor. Labor is the only factor of production, and we choose the stock
in both Home and Foreign such that the outside sector is active in both countries. We
normalize the price to 1 in the outside good sector such that wages in both countries equal
1. The identical wages play no role in what follows.
The polluting good is produced competitively by a continuum of mass 1 of firms in Home
and a mass of 1 in Foreign. The emissions per unit produced in Home are distributed
according to ΨH(e) with ΨF (e) describing the Foreign distribution. Within each country,
firms differ only in their emissions and share a common cost function. To focus on the
international aspect and with little impact on the analysis to follow, we assume that all
emissions in Home are observable and emissions are taxed at τH . Firms can abate as described
in Section 2.6, and a Home firm will have a supply curve of sH(p − τHe + AH(τH)), where
AH(τH) ≡ τHaH(τH) − b(aH(τH)) is the net gain in price from abatement and aH(τH) =
b′−1(τH) is the level of abatement (and AF (τF ), analogously for Foreign). To streamline the
presentation, the main text will be devoted to comparing a tax policy analogous to that of
the domestic setting: Firms can choose to certify and be taxed according to emissions and
non-certified firms pay an output tax according to average emissions of uncertified firms.
Other comparisons exist, in particular allowing the output tax t to be set optimally. We will
discuss alternatives towards the end of this section and explore the optimal output tax in
Appendix A.3.3.
3.1 Equilibrium
Since the setting with no certification is a special case of the equilibrium with certification
(where no firm chooses to certify) we will solve for the general case. Consider an equilibrium
in which Home firms are all taxed at the emission rate of τH and certified Foreign firms pay an
emission tax of τF . There exists a cutoff e such that all Foreign firms with e ≤ e are certified
and all other Foreign firms pay a common output tariff on exports of τFEF (e|e > e) where
EF is the expectation operator over ΨF . In what follows, we take e as a policy parameter
determined either by an indifference condition such as equation (4) or by only permitting
firms below some predetermined level e to certify.
There are two subclasses of equilibria. In a pooled equilibrium, uncertified firms are
28
indifferent between selling domestically and exporting and continue to export to Home. In a
separating equilibrium uncertified firms only sell in the Foreign market. In either case, their
production can be evaluated at the foreign price denoted pF . If we define ρ as the difference
in net price between selling in Foreign and Home for an uncertified firm we have:
ρ ≡ pF − (pH − τFEF (e|e > e)− κ), (19)
where ρ = 0 in the pooled equilibrium and ρ ≥ 0 in the separating equilibrium.
The world market clearing condition for the polluting good is:
DH(pH) +DF (pF ) = (20)
EH [sH(pH − τHe+ AH)] + ΨF (e)EF [sF (pH − τF e− κ+ AF )|e < e] + (1−ΨF (e))sF (pF ).
The first line constitutes world demand as a function of the two consumer prices pH and
pF . The first term on the second line is Home production where emissions are always taxed
at τH and all firms abate. Hence, the second term of the second line is the production by
certified Foreign firms who face an emission tax of τF , transportation costs κ, abate at level
aF , and sell in the Home market at price pH . The equilibrium is characterized by the two
endogenous variables (pH , pF ). Equation (20) contributes one equilibrium equation. In the
pooled equilibrium equation (19) binds which adds the second equation of the equilibrium.
In such an equilibrium, equation (19) directly gives that the difference in consumer prices,
pH − pF , is increasing in e: the marginal exporter is more polluting and consequently faces
a higher import tariff which creates a higher price gap. Foreign prices must decline, but if
abatement is strong enough it is possible for prices in Home to decline as well (see details in
Appendix A.3).
The separating equilibrium features ρ ≥ 0 and the second identifying equation is instead
provided by market clearing in the Foreign market:
DF (pF ) = sF (pF )(1−ΨF (e)). (21)
This equation alone pins down pF as a function of the certification cutoff e. The function
is monotone positive: Higher certification implies fewer firms producing for the domestic
market and consequently a higher Foreign price. Conditional on this pF , the Home price pH
is then given by equation (20) and depends negatively on e.19 In the separating equilibrium,
19Specifically, the indifferent Foreign firm is defined as in equation (4), where F are the actual costs of
29
an increase in certification lowers prices in Home and raises them in Foreign.
The equilibrium conditions in the Home market are illustrated in Figure 3 which takes
as its starting point Figure 1 for the domestic case. For clarity and with little loss of
intuition, the figure is drawn for linear supply curves, without abatement, and no production
in Home. Instead of the market supply curve of Figure 3, we use two supply curves: one
representing each type of equilibrium. Consider first the pooling equilibrium (panel a).
Exports from Foreign to Home are given by total production less Foreign demand: SP (pH) =
sF × [pH − τEF (e)− κ] − DF (pH − τE(e|e > e)− κ) , where sF ≥ 0. For given pH total
production is independent of certification for reasons analogous to the domestic setting:
The expansion in output from certified firms exactly offsets the reduction from uncertified
firms. The second term of SP (pH) is consumption in Foreign. For a given pH , an increase in
certification will increase the spread in taxation between Home and Foreign for the marginal
firm (equation 21). This lowers the price in Foreign, increases consumption, and lowers
exports. Consequently, SVP (pH) must be to the left of SUP (pH) and a higher e moves the curve
leftward. This increases the Home price.20
The separating equilbrium is reached when all uncertified production is required to meetForeign demand, DF (pF ) = (1−Ψ(e))sF (pF ). In this case, the supply curve for Home marketis instead given only by certified Foreign firms, SS(pH) = ΨF (e)sF × [pH − τE(e|e < e)− κ].This is illustrated by the red lines in Figure 3(b). In this case a higher e, which increasesthe number of certified firms, will pivot the supply curve clockwise: each additional certifiedfirm sells only in the Home market at the expense of Foreign, reducing pH and raising pF .
Which equilibrium is active depends on the relative sizes of SS(pH) and SP (pH). If
SS(pH) < SP (pH) for a given price pH , certified firms cannot meet Home demand by them-
selves and the economy is in the pooled equilibrium. If the inequality is flipped, the economy
is in the separating equilibrium.
3.2 Welfare
We slightly abuse notation and let W denote world welfare. Under certification it obeys:
W = CSH+CSF+PSH+PSF−[(v − τH)GH + (v − τF )(GF −GF,dom) + vGF,dom]−FΨF (e),
certification and f is a tax or subsidy. Any desired e can be achieved with an appropriate f . In what follows,we will suppose that e is determined in this manner, though nothing would be lost by instead assuming thate is exogenously set.
πF (pH − τF e+AH − κ)− (F + f) = πF (pF ) (22)
20This positive effect of e on pH depends on the absence of abatement. With abatement increased certifi-cation may lead to an overall positive supply effect which could lower Home prices.
30
Figure 3: Equilibrium with and without certification. The figure plots
the equilibrium in the Home country with no local production and linear supply curves in Foreign with
slopes of sF . Demand in the Home market is given by DH(pH). Panel A plots net supply from Foreign in
the pooling equilibrium given by total production net of Foreign demand. SP = sF × [pH − τEF (e)− κ] −
DF (pH − τE(e|e > e)− κ). An increase in e increases the import tax for the marginal firm, drives down
the price in Foreign, encourages Foreign consumption and pushes the SVp curve to the left. In a pooled
equilibrium pH is given by the intersection of DH and SVp and is always increasing in e (when there is no
abatement).This equilibrium is active when the supply of Foreign certified firms is insufficient to meet Home
demand. When this supply, given by SS = ΨF (e)sF × [pH − τEF (e|e < e)− κ] is sufficient, SVS is to the
right of SVP (Panel B) and the economy is characterized by the separating equilibrium. In this equilibrium,
greater e increases supply in Home at the expense of Foreign, so pH falls and pF rises.
(a) Pooling equilibrium
𝐷 (𝑝 )
𝑆 (𝑝 )
𝑆 (𝑝 )
𝑝
𝑝
(b) Separating equilibrium
𝐷 (𝑝 )
𝑆 (𝑝 )
𝑆 (𝑝 )𝑆 (𝑝 )
𝑆 (𝑝 )
𝑝
𝑝
31
where CSH and PSH refer to consumer surplus and producer surplus in Home with corre-
sponding expressions for Foreign. These expressions are standard and the details are dele-
gated to Section A.3 in the appendix. The novel term is GF,dom = EF (e|e > e)DF (pF ). Only
uncertified firms supply the Foreign domestic market and consequently they have an average
emission rate of E(e|e > e) implying that emissions associated with Foreign consumption is
given by GF,dom. Both Home production, GH , and emissions from Foreign production but
exported to Home, (GF −GF,dom) are taxed under Home jurisdiction and can be taxed so as
to eliminate the externality (τH = τF = v). GF,dom is not taxed by the Home policy maker
but is still a function of e and prices.
The following Proposition, analogous to Corollary 1, gives the difference in welfare and
emissions between an equilibrium with voluntary certification and one without. Here we take
approximations in τ and take v and κ to be of the same order:
Proposition 3. To a second-order approximation in (τF , τH , v), the difference between global
welfare under certification for Foreign firms and with a uniform output-based tariff of τFEF (e)
is given by:
W V −WU = s′F τF
(v − 1
2τ
)V arF (ε)︸ ︷︷ ︸
Reallocation Effect
+ τF
(v − 1
2τF
)sFΨF (e)
b′′(0)︸ ︷︷ ︸Abatement Effect
(23)
−[(v − τH)EH(e)s′H + (v − τF )EF (e)s
′
F
]∆pH︸ ︷︷ ︸
Untaxed Average Emissions Effect
−D′F τF(τF (EF (e) + EF (e|e > e))− ρ
2
)∆pF︸ ︷︷ ︸
Consumption Leakage Effect
−s′F(1−ΨF (e)
)(∆pH + ρ
2+ (v − τF )E(e|e > e)
)ρ︸ ︷︷ ︸− FΨF (e)︸ ︷︷ ︸
Cost of Certification
Backfilling Effect
+ o(τ 2),
where ρ ≥ 0 — defined by equation (19) — is the (net of taxes) price premium for uncertified
Foreign firms of selling in Foreign compared with Home. We let ∆pH ≡ pVH − pUH and
∆pF ≡ pVF − pUF denote the changes in prices due to certification in each country. It holds
that
- The Reallocation and Abatement effects are always positive
- The Consumption Leakage effect has the same sign as ∆pF
- The Backfilling Effect is 0 in the pooling equilibrium.
- The total effect is ambiguous.
The corresponding changes in emissions are given by:
32
Corollary 3. To a first-order approximation change in emissions from moving to certifica-
tion is given by:
GVH −GU
H = EH (e) s′
H∆pH + o (τ) ,
GVF −GU
F
=s′F
(EF (e) ∆pH − τFV arF (ε) + ρEF (e|e > e) (1−ΨF (e))
)−ΨF (e) aF sF + o(τ)
.
The Proposition mirrors and extends the analysis in the domestic case.21 The Realloca-
tion and Abatement effects are as described in Section 2, and FΨF (e) continues to be the
cost of certification. We add the Untaxed Average Emissions Effect, which arises since prices
are no longer constant and an increase in home prices pH might encourage global production.
This is harmful to welfare if taxes are lower than v. In addition, we have the two terms with
which we opened this section:
The Consumption Leakage Effect — is new from the domestic case and builds on the
intuition of Figure 1. This is a leakage effect because when Home imposes an output-based
tariff (equal to τFEF (e)) the Foreign price goes down which encourages Foreign consumption
which is not taxed. With voluntary certification, this distortion increases further if the
Foreign price decreases, which always occurs in the pooling equilibrium. In contrast, in the
21Formally, we can use first and second order approximations to get expressions for the price changes. Weuse εD and εS to denote world demand and supply elasticities, which are the sum of share-weighted localelasticities, so εD = εDF θ
DF + εDHθ
DH , and similarly for supply (with θDH = DH/(DF + DH) and analogously
for other θ expressions). In Appendix A.3 we show that such a voluntary certification program will have thefollowing effect on the price at Home and in Foreign:
∆pH =−εDF θDFεS − εD
τF (EF (e|e > e)− EF (e))− (1−ΨF (e)) εSF θSF − εDF θDF
εS − εDρ+ o (τ) , (24)
∆pF =εDHθ
DH − εS
εS − εDτF (EF (e|e > e)− EF (e)) +
εS − εSF θSF (1−ΨF )− εDHθHDεS − εD
ρ+ o(τ), (25)
where elasticites are evaluated at p0 for Home and p0 − κ for Foreign. Further:
∆pH −∆pF = τF [EF (e|e > e)− EF (e)]− ρ. (26)
In the pooling equilibrium ρ = 0 and in the separating equilibrium it is ρ ≥ 0. In this latter case, ρ is givenby combining equations (19) and (22)
ρ = τF [EF (e|e > e)− e]− F + f
sF (p0 − κ)+ o(τ),
33
separating equilibrium, increased certification might reduce the pool of Foreign producers
servicing their own market so much that the Foreign price increases, in which case the
distortion is mitigated. The welfare cost is proportional to the underpricing which (to a first
approximation) equals (τF (EF (e) + EF (e|e > e))− ρ/2), the average of the price gap under
no certification and certification, respectively. This effect is only present when D′F < 0.22
The Backfilling Effect is also new. Whereas Consumption Leakage captures that Foreign
consumption faces a price that is too low, this term captures that the most polluting produc-
ers might receive a price that is “too” high. Recall that the reallocation effect captures that
uncertified firms receive a lower price and correspondingly reduce their production. This
is true in both the Domestic case and in the pooling equilibrium. However, in the sepa-
rating equilibrium where ρ > 0, Foreign firms can divert their sales entirely to the Foreign
market, which is untaxed. Because the relatively clean firms are exporters, those left to
serve demand in Foreign are relatively pollution-intensive. Uncertified Foreign firms conse-
quently receive a price that is too high by ρ. Their price changes by ∆pF and the size of
the distortion depends on the gap between ∆pF and what the price change would have been
had these Foreign firms been forced to export namely τF (EF (e) − EF (e|e > e)). Note that
∆pF − τF (EF (e)− EF (e|e > e)) = ∆pH + ρ. The welfare changes of the backfilling effect is
then given by ρ(∆pH +ρ)/2, where the division by 2 occurs for standard “Harberger” triangle
reasons (for ν = τF ). This effect is amplified when there Foreign emissions are under-taxed.
It is only active in the separating equilibrium. Unlike the Consumption Leakage Effect, it
does not require that D′F < 0 but only DF > 0.
The overall welfare effect is the sum of several terms and is in general ambigious. However,
if we consider a pooling equilbirium (ρ = 0), where taxes are Pigovian (τF = τH = ν),
abatement effects are large compared to fixed costs of certification (ν2
2sFb′′(0)
> F ), then s′Flarge enough relative to D′F is a sufficient condition. The relative size of s
′F and D
′F will play
some role in the empirical application below.
Changes in emissions in Corrolary 3 can be interpreted along similar lines. The Home
price change ∆pH increases emissions for all firms. Emissions in Foreign are reduced through
a reallocation effect and abatement, but increase because of a backfilling effect if ρ > 0.
Consequently, Proposition 3 shows how Foreign demand alters the conclusion from the
domestic model of Section 2. Though this policy continues to reallocate production from the
22An alternative intuition is to think of Foreign consumption as a zero-emission way of increasing inter-national supply. As such this “supply” should be given no tax and consequently be facing a price pH − κ.However, the actual price of Foreign consumption is pF (less than pH − κ) and consumption is consequentlytoo high. This problem grows when the Foreign price declines.
34
more polluting to the less polluting firms, the presence of Foreign consumption poses limits
to the taxing ability of the Home policy maker.
3.3 Alternative policy environments
This naturally raises the question of what the second-best policy is, that is the optimal
program where Home cannot treat Foreign producers differently unless they certify and where
Foreign does not impose any tax. We discuss this formally in Appendix A.3.3. We find that
in an attempt to reduce the Consumption Leakage Effect described above, the social planner
sets the output tax on uncertified firms, t∗, lower than τFE(e|e > e). A similar intuition
explains why in general, a border tariff adjustment (even tailored to the exact emission
rate of the exporter) is not the optimal environmental tariff (Balistreri et al., 2019; Keen
and Kotsogiannis, 2014; Markusen, 1975; Hoel, 1996). The optimal level of certification is
set through a positive tax f . Additional certification in the separating equilibrium creates
convergence in Home and Foreign prices, and divergence in the pooling equilibrium. A
tax on certification allows the government to control this process. Certification allows for
the separation of some Foreign exporters from their domestic market such that they are
essentially under the jurisdiction of the Home country. The emission tax is then set at their
Pigovian level: τF = τH = v.23
More broadly, the optimal policy also permits the policy maker to set taxes such that
Home exports to Foreign, thereby creating bilateral trade in a homogenous product. This
is not optimal when κ is first order but becomes more valuable as transportation costs fall.
In effect, the Home policy maker broadens its scope of taxation since it can tax production
23We compare the welfare under optimal policy W ∗ to that with a Laissez-faire setting, WLF , of no taxeson Foreign (τF = t = 0) and Pigouvian taxes on Home production (τH = v). We find:
W ∗ −WLF = s′
F (p0 − κ)v2
2V ar(ε) + ΨF (e)AF sF (p0 − κ)− FΨF (e)
+D′
F (p0 − κ)t∗
2EF (e|e > e)− s
′
F (p0 − κ)v
2EF (e)
(p∗H − vEF (e)− pLFH
).
t∗ is the optimally set output tax on imports, lower than its Pigovian value, and given by:
t∗ =s′
F (p0 − κ)(1−ΨF (e))
s′F (p0 − κ)(1−ΨF (e))−D′
F (p0 − κ)vE(e|e > e) < vE(e|e > e).
The first three terms on the RHS replicate the reallocation effect, gains from Abatement and the fixed costof certification. The Consumption Leakage effect is rederived as the subsequent element, though contrary toProposition 3 it is always negative because the spread between Home and Foreign consumer prices alwaysincreases in the optimum. The last term is novel to this setting: Since the comparison takes as a startingpoint no output tax, the “first step” is to introduce a uniform output tax at vEF (e). Adding in the associatedchange to prices p∗H − pLFH one gets the welfare change of the last term.
35
if it is either consumed or produced at Home. The result is Home exporting its production
to Foreign (covered under a domestic carbon tax), and importing certified production from
Foreign to Home (covered under the voluntary program). This reduces the Backfilling effect.
Though this is unlikely to be the optimal policy for international trade of steel, which is
relatively costly to transport, it might be relevant in other contexts such as electricity sold
on a grid that spans jurisdictional borders.
In the following section, we conduct a quantitative exercise where we calculate such
welfare gains explicitly for the steel trade.
3.4 International Empirical Application: Brazilian Steel Trade
To illustrate the implications of our program in the international setting, we conduct an
analysis based on our approximation formulas for trade in steel between Brazil and the
OECD. The iron and steel sector is one of the most energy and carbon-intensive sectors
responsible for 10.5% of total CO2 emissions (including indirect emissions from electricity
use, see IEA, 2020). Iron and steel are internationally traded, and it is therefore considered a
key sector for carbon leakage—for instance, the EU puts it on a list of highly exposed sectors
which get a disproportionately large share of free CO2 allowances in the EU cap-and-trade
system. Steel is mostly produced through two different processes. In the blast furnace-basic
oxygen furnace (BF-BOF) process, coke and iron ore are combined at high temperature to
produce liquid steel. This process emits CO2 emissions through the combustion of coke.
Alternatively, steel can be produced with scrap steel and electricity using an electric arc
furnace (EAF), a process which leads to fewer emissions. Within each process, there is still
a high level of heterogeneity in emissions depending on plant’s energy efficiency or on the
fuel used to produce electricity. This makes steel an interesting sector to explore the costs
and benefits of voluntary certification.
We calibrate our model to the Brazilian steel sector in 2019 and consider a two-country
world where OECD countries alone decide to implement carbon tariffs (either output-based
or with voluntary certification) on Brazilian steel. We will use the welfare formula given in
Proposition 3. This exercise requires a handful of key statistics and economic parameters
of supply and demand. We provide a brief overview here, and describe the sources and
calibration strategy in detail in Appendix B.2.
Production, trade and transport costs. Brazil produced 32.6 Mt of steel in 2019, exporting
8.5 Mt (on net) to OECD countries (including 6.1 Mt to the US, the largest net export market
for Brazilian steel) at an average price of $489/t, which we take as the laissez-faire price of
36
steel in Foreign in our calibration (Instituto Aco Brasil, 2021).24 We use data from World
Steel (2020) to determine production of steel in the OECD. We set the transport cost κ at
$50/t (Eckett (2021)).
Emissions rates. World Steel (2020) also reports the share of steel production with the
BF-BOF and EAF processes across countries in 2019. Hasanbeigi and Springer (2019) report
the emission rates per process in 2016 for the countries which produce the most steel, which
includes Brazil and the US. To give some context, the emission rate using BF-BOF is 2.07
tCO2/t steel in Brazil and 1.82 tCO2/t steel in the US, while the corresponding emission rates
using EAF are 0.46 and 0.62 tCO2/t steel. The EAF shares are 22.2 in Brazil and 69.7 in the
US. These numbers highlight that Brazilian steel is on average significantly dirtier than from
the US (or the average OECD country) even though its EAF producers are relatively clean.
We use the same data sources to estimate mean emission rates in the OECD. Of course,
there is still substantial variation in emission rates within each process. To account for this,
we assume that for each process, the emission rate distribution is a (both-sides) bounded
log-normal distribution. We then assume that the standard deviation of log emission rates
within each technology is the same as the standard deviation of log productivity in the basic
metal products sector in Brazil according to Schor (2004).25 See Appendix B.2 for details.
Social cost of carbon, abatement cost and certification cost. We set the social cost of
carbon at $51 (on Social Cost of Greenhouse Gases (2021)). We get an estimate for the
slope of the marginal abatement function b′′(0) using a marginal abatement cost curve for
the iron and steel sector in Brazil from Pinto et al. (2018). To estimate the certification
cost F , we rely on a study of the iron and steel sector in the US by the EPA which aimed
at evaluating the cost of reducing hazardous air pollutants (manganese, lead, benzene, etc.
but not CO2) in that industry Gallaher and Depro (2002). They found that the annualized
cost of monitoring pollutants for one plant was $1.04M in 2001. Assuming a constant fixed
cost to total output ratio, we find that this corresponds to $23M to certify all production in
Brazil (that is a cost of around $0.7 per ton).
24We keep constant net exports of Brazil to other countries. Therefore, in the OECD case, we removethe 2Mt of net exports to other countries. Net exports to the OECD therefore represent 27.7% of Brazilianproduction (excluding net exports to non-OECD countries).
25That is, once we control for the technology type, we consider that emission heterogeneity reflects theaverage productivity heterogeneity in the sector. This assumes implicitly that i) Once one controls forthe process-type, most heterogeneity reflects differences in energy intensity and that those move with TFPdifferences; ii) In the basic metal sector, within subsector heterogeneity dominates heterogeneity across sub-sectors. We adjust the total standard deviation of log productivity for the small productivity premiumenjoyed by the EAF process on average, using estimates for the US from Collard-Wexler and Loecker (2015).
37
Table 2: Emission and Welfare Effects from Environmental Trade Policies
Voluntary Voluntarycertification certification
First Best BCA f = 0 f = f ∗
Welfare
Gains in M USD 1161 714 692 850
% of First Best Gains 100 61.5 59.6 73.2
Emissions
Reduction in Mt 22.5 5.6 5.8 10.5
Note: All gains are calculated relative to a unilateral domestic carbon tax in the OECDwithout border adjustments. f is a tax on certification, with f ∗ denoting the optimum
certification tax.
Elasticities. Finally, we use Fernandez (2018) who derives demand elasticities for steel
in the US (−0.306) and in Brazil (−0.414). For the supply elasticity, we use a single value
of 3.5, which is the supply elasticity used by EPA (2002). Therefore the supply elasticity
is larger in magnitude than the demand elasticity. Data Appendix Table B.1 gives all
parameters and their sources.
Table 2 reports the effects on global welfare and emissions of introducing various taxation
programs. Each of the calculations report the gains relative to a Pigovian emission tax in the
OECD with no border adjustment. As a benchmark for comparison we calculate the welfare
gains that would result from a universal carbon tax covering production in both the OECD
and Brazil. To give some context to these numbers, note that the net export value of steel
from Brazil to the OECD is 4.1B USD. Adding an output-based carbon border adjustment
(i.e. a tariff on Brazilian exports of νEF (e)) increases welfare by 714M USD which is already
61% of the gains that could be achieved by implementing the universal carbon tax.
Without any certification tax, our voluntary certification program leads to a high level of
certification. The economy ends up in the separating equilibrium with a price gap between
Home and Foreign which is lower than in the output-based tariff case (105 USD versus 137
USD). As a result, voluntary certification reduces welfare relative to an output-based tariff.
This comes from a large negative “backfilling effect” of −228M USD, while the “reallocation
effect” leads to 150M USD. A tax or other program to limit certification to the optimal
38
level, however, brings 136M USD of additional welfare gains relative to the output-based
tariff. This corresponds to closing close to a third of the gap between the output-based tariff
and the first best policy. With the optimal certification tax, the equilibrium is still in the
separating case but the price gap slightly increases relative to the output-based tariff at 148
USD. This leads to a “backfilling effect” of a much smaller magnitude at −10M USD, while
the reallocation effect remains similar at 150M USD since the certification threshold is close
in both cases.26 The second best policy described in section 3.3 also leads to a separating
equilibrium. In that case, the tariff t∗ on uncertified exporters is irrelevant, so that the
second best policy coincides with the optimal certification tax case reported here.
Finally, note that while the output-based tariff reduces emissions by 5.6Mt of CO2,
the optimal certification program reduces emissions by 10.5Mt of CO2, nearly half of the
reductions of a first best policy which also taxes emissions in Brazil. To give an idea of the
magnitude involved, embodied carbon in the net exports of steel in laissez-faire represents
14.5Mt of CO2.
To illustrate how the benefits of an opt-in emissions taxation program change with the
extent of certification, Figure 4 displays the welfare gains of the certification program for
different values of the certification tax relative to a Pigovian tax in the OECD only. The
certification tax is expressed as a share of revenues in laissez-faire. For comparison, we also
plot the gains of a standard border carbon adjustment. For a sufficiently high certification
tax (corresponding to 15.6% of laissez-faire revenues), no firm certifies and the welfare gains
are the same as with the output based-tariff. As the certification tax decreases, the share
of firms certifying grows quickly, bringing most of the welfare gains from the certification
program. With a certification tax as high as 15% of laissez-faire revenues, 16.3% of firms
certify, close to 3/4 of EAF producers in Brazil. This reflects the presence of a sizable mass
of relatively clean producers in Brazil. The welfare gains remain high (above 100M USD) as
long as the certification tax remains above 5% of revenues, and they only disappear for very
small certification taxes.27
We stress that this exercise is a simple proof-of-concept and that our numbers are indica-
tive of orders of magnitude but not exact values. It shows that there are potentially large
26In the separating equilibrium the mass of certifying firms Ψ(e) is close to the export share in laissez-faire,i.e. Ψ(e) = 1−DF (p0 − κ)/SF (p0 − κ) + o(1). This is the reason why we show welfare gains as a functionof the certification tax instead of e in Figure 4 below.
27The small jump in Figure 4 marks the point where we switch from the separating to the pooling equilib-rium. This jump only occurs because we compute the welfare gains using Taylor approximations and woulddisappear if we were to compute welfare gains at higher orders.
39
Figure 4: Welfare gains relative to a carbon tax at Home only for differentlevels of the certification tax
0 0.05 0.1 0.15
certification tax as a share of LF revenues
600
650
700
750
800
850
900
M U
SD
voluntary certification
output based tariff
40
emission reductions and significant welfare gains from our certification program, though it
may not be desirable to let certification occurs unabated if it leads to a large reduction in the
price gap between the two countries. Fortunately, prevailing price gaps relative to border ad-
justment fees are observable. This makes it possible for policymakers to adjust certification
criteria in order to avoid significant backfilling losses.
4 Conclusion
Settings in which a relatively small set of agents disproportionately contribute to a global
externality perfectly encapsulates the problem of concentrated benefits and diffuse costs
as described by Olson (1965). It should be unsurprising that there is limited appetite for
Pigouvian taxes to internalize such externalities. In this paper we study a mechanism that
counterbalances and ultimately unravels this dynamic.
The counterbalance comes from offering a substantial reduction in the tax burden to those
who contribute little to the externality. Low-emissions agents receive concentrated benefits
when they voluntarily certify their emissions for direct taxation. Increased certification raises
the output tax on uncertified firms, but this marginal increase is dispersed widely. Unraveling
occurs when the default output tax for uncertified firms is updated to reflect the higher mean
emissions of the uncertified group and the cost of certification is not too large. If unraveling
is complete, such a voluntary program achieves the same outcome as the otherwise-infeasible
mandate to tax emissions directly (minus certification costs). We show that the welfare gains
of such a policy scale with the variance of emissions and the slope of supply. The welfare
achieved by a voluntary program is a weighted average of the Pigouvian first-best and the
output-based tax, with the weights reflecting the relative variance of effective emissions
subject to taxation to the variance of emissions in the population.
We apply these results to oil and gas production in the Permian basin of New Mexico and
Texas, where methane emissions are a significant, largely unregulated problem. Coupling a
royalty adder based on the average of uncertified emissions per barrel of oil equivalent with
the option to certify emissions sets off an unraveling that converges on universal taxation,
even with significant implementation costs. This would have an annual social benefit of
about $3B.
In the international setting, our mechanism extends the incentive to abate emissions
beyond the borders of the country adopting a carbon tax. Such a policy is therefore most
attractive for countries whose carbon footprint is most heavily embodied in imports. We
show in addition to the variance of emissions, the elasticity of demand for carbon-intensive
41
goods in non-adopting countries plays a key role in prospective emissions reductions, as
demand responses to lower prices abroad offset reductions from certified firms’ abatement
efforts (consumption leakage). In addition, with too much certification there is a risk that
the most pollution-intensive foreign producers will expand operations to serve the foreign
market (backfilling). We derive the condition that determines whether such a program
would further reduce emissions beyond those achievable with border carbon adjustments.
Applying our sufficient statistics formulas to the Brazilian-OECD steel trade, we find that a
managed certification program could deliver nearly 75% of the welfare gains of a universal
carbon tax.
42
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A Theory Appendix
In the following we first present the model of the domestic setting with an adjustment on
the intensive margin. We then proceed to the setting with an extensive margin and finally
consider the international model
A.1 The Domestic model
A.1.1 Setup and proof of uniqueness, Lemma 1 and Proposition 1.
We solve the model for the general case of certification at level e with the understanding that
the case of no certification can be found at e = e. The supply of uncertified and certified
firms are given by, respectively:
s(p− τE(e|e > e)),
s(p− τe).
An indifference condition determines which firms choose to certify:
π(p− τ e)− F − f = π(p− t),
where firms with e ≥ e certify. Consequently, for any given (t, τ, F ) there is a monotonic
relationship between e and f . We solve the model for given e.
Welfare derived from the operation of this market is the sum of consumer utility, net of
welfare costs from emissions, profits of firms, government revenue, and certification cost:
W = I − pC + u(C)− vG+ Eπ(p− τε) + τG− FΨ(e),
where the representative agent has exogenous income I and D(p) is demand function defined
by u′(D(p)) = p. We continue to employ the definition of ε of:
ε =
e
E(e|e > e)
if e ≤ e
if e > e.
and emissions are given by:
G = E[εs(p− τε)],
where p is determined by:
D(p) = E(s(p− τε)) (27)
51
and we label pU price without certification (e = e) and pV the price with certification.
Proof of uniqueness
.
Proof of Lemma 1
Proof of equation (7)
We use that i) emission without certification is given by GU = E(e)s(pU − τE(e)), ii)
emissions under certification is given by GV = E(εs(pV − τε)) and iii) Cov(ε, s(pV − τε)) =
E(es(pV − τε))− E(e)E(s(pV − τε)), and recover the expression for GV −GU .
The sign of equation (7)
To get conditions for whether GV −GU is positive or negative we proceed in two steps:
i) If s is everywhere convex then pV < pU . If s is everywhere concave then
pV > pU
Proof:
The equilibrium price with certification is given by the market clearing condition:∫ e
e
s(pV − τe
)ψ (e) de+ (1−Ψ (e)) s
(pV − τE (e|e > e)
)= D
(pV),
which we differentiate to get:
dpV
de= −ψ (e)
s(pV − τ e
)− s
(pV − τE (e|e > e)
)− τE (e− e|e > e) s′
(pV − τE (e|e > e)
)∫ e0s′ (pV − τe)ψ (e) de+ (1−Ψ (e)) s′ (pV − τE (e|e > e))−D′ (pV )
,
(28)
where if s is convex, we have that
s(pV − τ e
)− s
(pV − τE (e|e > e)
)− τE (e− e|e > e) s′
(pV − τE (e|e > e)
)> 0
and therefore dpV
de< 0. Since pU is for e = e we must have pV < pU . And similarly if s is
concave we have that pV > pU
ii) dGV
de< 0 for s convex in p and es(pV − τe) concave in e.:
52
dGV
de=
d
deE(εs(pV − τε
))= ψ (e)
[es(pV − τ e
)− E (e|e > e) s
(pV − τE (e|e > e)
)+
(E (e|e > e)− e)(s(pV − τE (e|e > e)
)− τE (e|e > e) s′
(pV − τE (e|e > e)
)) ]
+E(εs′(pV − τε
)) dpVde
= ψ (e)[e(s(pV − τ e
)− s
(pV − τE (e|e > e)
))− (E (e|e > e)− e) τE (e|e > e) s′
(pV − τE (e|e > e)
)]︸ ︷︷ ︸A
+E(εs′(pV − τε
)) dpVde︸ ︷︷ ︸
B
If s is weakly convex we know that dpV
de≤ 0 and term B is negative.
We define h(e) = es(pV − τe) and write:
e(s(pV − τ e
)− s
(pV − τE (e|e > e)
))− (E (e|e > e)− e) τE (e|e > e) s′
(pV − τE (e|e > e)
)= h (e)− h (E (e|e > e))− (e− E (e|e > e))h′ (E (e|e > e)) .
If h is weakly concave in e then we must have
h (E (e|e > e))− h (e) ≥ (E (e|e > e)− e)h′ (E (e|e > e)) ,
such that term A is negative as well. Consequently:
dGV
de< 0 for s convex in p and h concave in e.
iii). Consider linear supply which implies from equation (28) that pV = pU . Second, we
write h(e) = e× s(pV −τe) where s and pV are constants. The second derivative is −2sτ < 0
and consequently part ii is met as well.
Emissions decline for small τ
For general h the second derivative is:
−2τs′(pV − τe) + τ 2es′′(pV − τe)
which is negative for τ → 0. Consequently, for 1small τ emissions decline with certification.
53
Proof of Proposition 1
We use superscript ”V ” for all variables associated with the equilibrium with certification
and ”U” for all variables when not certified. We write:
W V −WU (29)
= u(D(pV ))− u(D(pU)) +[π(pV − τE(e))− π(pU − τE(e))
]− [pVCV − pUCU ]
−(v − τ)[GV −GU
]+ Eπ(pV − τε)− π
(pV − τE(e)
)− FΨ(e)
=
∫ pV
pU[s(p− τE(e))−D(p)] dp−(v−τ)(GV −GU)+Eπ(pV −τε)−π
(pV − τE(e)
)−FΨ(e)
where we use ∂π/∂p = s and the first order condition of consumers: u′ = p.
To establish a) of the proposition, we first note that the price effect is trivially zero if
pV = pU . Assume that pU < pV . Note that s(p−τE(e))−D(p) is increasing in p and takes the
value zero at p = pU by definition of the equilibrium in the uncertified case. Consequently,
the integral must be positive. Conversely, if pU > pV then s(pU − τE(e))−D(pU) < 0 for all
p ∈ [pV , pU). It therefore follows that∫ pVpU
[s(p− τE(e))−D(p)] dp = −∫ pUpV
[s(p− τE(e))−D(p)]dp > 0. The price effect is therefore positive.
The weakly positive sign of the reallocation effect follows from the convexity of the profit
function in ε and Jensen’s inequality.
To establish c) note that for linear supply curves GV − GU < 0 and the price effect is
zero. The reallocation effect is positive and the untaxed emissions effect is weakly positive
such that W V −WU + FΨ(e) > 0. The result for small τ follows from Proof of Corollary 1
below.
A.1.2 Proof of Corollary 1
We establish the proof of the corrollary using first order approximations. Throughout we
take approximations in τ around τ = 0 and consider v to be of the same order. We label p0
the equlibrium price when τ = 0 (identical for both certified and uncertified equilibrium).
Taylor expansions of equilibrium prices
We take a first order expansion of pU using s(pU − τE (e)
)= D
(pU)
to get:
pU − p0 =E (e) s′ (p0)
s′ (p0)−D′ (p0)τ + o (τ) . (30)
54
We take a similar approach with pV using equation (27) which we write explicitly as:∫ e
e
s(pV − τe
)ψ (e) de+ (1−Ψ (e)) s
(pV − τE (e|e > e)
)= D
(pV),
such that:∫ e
e
(s (p0) + s′ (p0)
(pV − τe− p0
))ψ (e) de+(1−Ψ (e))
(s (p0) + s′ (p0)
(pV − τE (e|e > e)− p0
))= D (p0) +D′ (p0)
(pV − p0
)+ o (τ)⇔
s′ (p0)(pV − τE (e)− p0
)= D′ (p0)
(pV − p0
)+ o (τ) ,
which, when reordered returns equation (30) such that we establish that, to a first order,
prices are the same under certification and no certification:
pV = pU + o(τ)
Taylor expansion of welfare change W V −WU
We proceed under the assumption that W V −WU + FΨ(e) is second order in τ and will
demonstrate that this is so.
We take taylor expansions of each element of equation (29). We start with the price effect
and recall that pV − pU = o(τ)
∫ pV
pU[s(p− τE(e))−D(p)] dp
=
∫ pV
pU((s′ (p0) (p− τE (e)− p0)−D′ (p0) (p− p0)) + o (τ)) dp+ o(τ 2)
= (s′ (p0)−D′ (p0))
(pV − p0
)2 −(pU − p0
)2
2− τE (e) s′ (p0) o (τ) + o
(τ 2)
= o(τ 2),
such that the price effect is zero at second order.We proceed with the reallocation effect and note that the integral here is zeroth order
55
and develop profits correspondingly (at second order).
E(π(pV − τε
))− π
(pV − τE (e)
)=
∫ e
e
(π (p0) + s (p0)
(pV − τe− p0
)+s′ (p0)
(pV −τe−p0)2
2
)ψ (e) de
+ (1−Ψ (e))
(π (p0) + s (p0)
(pV − τE (e|e > e)− p0
)+s′ (p0)
(pV −τE(e|e>e)−p0)2
2
)−
(π (p0) + s (p0)
(pV − τE (e)− p0
)+s′ (p0)
(pV −τE(e)−p0)2
2
)+ o (τ)
2
=
(∫ e
e
(pV − τe− p0
)2ψ (e) de+ (1−Ψ (e))
(pV − τE (e|e > e)− p0
)2 − (pV − τE (e)− p0)2) s′ (p0)
2+ o (τ)
2
=
(∫ e
e
τ2e2ψ (e) de+ (1−Ψ (e)) τ2E (e|e > e)2 − τ2E (e)
2
)s′ (p0)
2+ o (τ)
2
=s′ (p0)
2τ2V ar (ε) + o
(τ2)> 0.
We proceed with the untaxed emission term. We take the approximation in τ but note that
our approach supposes that v is of the same order of τ such that we retain the term (v− τ):
(v − τ)E (e) s(pV − τE (e)
)− E
(εs(pV − τε
))= s′ (p0) τ (v − τ)
(∫ e
e
e2ψ (e) de+ (1−Ψ (e))E (e|e > e)2 − E (e)2
)+ o
(τ 2)
= (v − τ) s′ (p0) τV ar (ε) + o(τ 2),
which also delivers GV −GU = −s′ (p0) τV ar (ε) + o(τ). We combine these three terms (and
add the certification costs) to get equation 9:
W V −WU =(v − τ
2
)s′ (p0)V ar (ε) τ − FΨ (e) + o
(τ 2),
which established that the expression W V −WU + FΨ(e) is indeed second order.
Part a) follows directly from the definition of ε (equation 5)
Parts b+c) follow from 2v > τ whenever the distribution of ε is not degenerative.
For part d, we perform analogous operations as above and note that the expressions
become exact.
A.1.3 Optimal policy
We solve for the optimal allocation keeping the constraint that output cannot differ between
firms that are not certified. Consequently, we choose emissions of certified firms qV (e), those
56
that are not certified, qU as well as the level of certification to maximize welfare:
W = maxqV (e),qU ,e
I + u (C)−(∫ e
e
c(qV (e)
)ψ (e) de+ (1−Ψ (e)) c
(qU))− vG− FΨ (e) ,
where consumption and emissions are given by:
C =
∫ e
e
qV (e)ψ (e) de+ qU (1−Ψ (e)) ,
G =
∫ e
e
(e− a) qV (e)ψ (e) de+ qU∫ e
e
eψ (e) de.
We differentiate wrt qV (e) to get:
u′(C)− c′(qV (e))− ev = 0,
wrt qU to get:
u′(C)− c′(qU)− vE(e|e > e) = 0.
By denoting the marginal utility p = u′(C) and the emissions cost τ = v we recover:
qV (e) = s(p− τe) and qU = s(p− τE(e|e > e)),
which delivers that the social planner would want to set τ = v and t = vE(e|e > e). We
solve for the optimal threshold by taking first order conditions:
p(qV (e)− qU
)−(c(qV (e)
)− c
(qU))− τ e
(qV (e)− qU
)− F = 0,
Hence
π (p− τ e)− π (p− τE (e|e > e)) = F + τ (E (e− e|e > e)) s (p− τE(e|e > e)
such that, as stated in equation (12), the tax on certification has to equal:
f = v (E (e− e|e > e)) s (p− vE(e|e > e) > 0
57
Figure A.1: Equilibrium with concave supply functioins.
SU(p)
SV(p)
pVpU
A B
CD(p)
Q
p
DE
A.1.4 Unraveling algorithm
A.1.5 Figure for concave supply functions
In line with Figure 1, we present the equilibrium when the supply function is concave. The
SV (p) curve continues to intersect at a lower point than SU(p) since the least polluting firm
will produce at a lower price. It will, however, intersect the SU(p) such that, when all firms
produce, production SV (p) < SU(p) and therefore pV > pU . Consider the case of τ = v.
Then the welfare effect from moving from SU(p) to SV (p) is C − (B +E). The reallocation
effect counts profits under pV and the difference between the two is therefore (C−B−E−D).
This “overcount” by D which the price effect, D, corrects for. It is apparent that D is always
positive and the analysis above demonstrates that the reallocation effect is as well. The term
A is now a reallocation from consumers to producers.
58
A.1.6 Extensions
Abatement
We add abatement by allowing all firms to reduce their emissions per unit from e to e−a by
spending b(a) of the outside good per unit, q, of production. We add that b(a) = 0 if a ≤ 0,
b′(0) = 0 as well as b′′(a) > 0 if a ≥ 0.
An uncertified firm cannot verify that it has abated and will consequently choose a = 0.
A certified ei chooses:
maxq,apq − c(q)− τ(ei − a)q − b(a)q,
which defines optimal abatement as τ = b′(a∗) which we write as a∗(τ). We define A(τ) =
τa∗(τ)− b(a∗(τ)). This leads to an optimal supply of:
qi = s(p− τei + A(τ)) = c′−1(p− τei − A(τ)).
For some tax on certification of f the equilibrium indifferent firm is then defined as:
π(p− τ e+ A(τ))− π(p− τE(e|e > e)) + F + f,
since uncertified firms are taxed on quantity at t = τE(e|e > e). With abatement it is
perfectly possible that all firms wish to certify even for F + f > 0.
Differences in emissions The difference in emissions between no certification and cer-
tification is given by:
GV −GU
= E (e) s(pU − τE (e)
)−(∫ e
e
(e− a∗) s(pV − τe+ A
)ψ (e) de+ s
(pV − τE (e|e > e)
)E (e|e > e) (1−Ψ (e))
)= E (e) s
(pU − τE (e)
)− E (e) s
(pV − τE (e)
)︸ ︷︷ ︸price effect
+ E (e) s(pV − τE (e)
)− E
(εs(pV − τε
))︸ ︷︷ ︸reallocation effect
+
∫ e
e
[es(pV − τe
)ψ (e)− (e− a∗) s
(pV − τe+ A(τ)
)ψ (e)
]de︸ ︷︷ ︸ .
abatement effect
The sum of the price effect and the reallocation effect replicate Lemma 1. The abatement
effect can be written to give equation (13). Note, that we have slightly misused notation: pV
59
and pU here refer to prices with abatement whereas in Lemma 1 they refer to those without
abatement.
Differences in welfare Writing the difference in welfare between certification and no
certification we get:
W V −WU =
∫ pV
pU(s (p− τE (e))−D (p)) dp︸ ︷︷ ︸
price effects
+ E(π(pV − τε
))− π
(pV − τE (e)
)︸ ︷︷ ︸reallocation gains
−(v − τ)(GV −GU
)︸ ︷︷ ︸untaxed emissions
+ Ψ (e)(E(π(pV − τe+ A(τ)
)− π
(pV − τe
)|e < e
))︸ ︷︷ ︸abatement gains
− FΨ (e) + o(τ 2),
where the abatement gains are what we add to the expression in the main text.
Taylor expansion. We follow an approach analogous to Section A.1.2 and let p0 be the
price at τ = 0. We start out by taking a Taylor expansion of the optimal level of abatement
b′(a) = τ to get:
b′(0) + ab′′(a) = τ + o(τ).
We exploit that b′(0) = 0 to write:
a =τ
b′′(0)+ o(τ),
which implies that:
A(τ) = τa− b(a) =τ 2
b′′(0)−(b′(0)a+
1
2b′′(0)a2
)+ o(τ 2) =
1
2
1
b′′(0)τ 2 + o(τ 2).
Since A(τ) is second order we recover the result that:
pV = pU + o(τ) = p0 + τE(e)s′(p0)
s′(p0)−D′(p0)+ o(τ).
We therefore recover the results of the price effect and the reallocation effect as:∫ pV
pU(s (p− τE (e))−D (p)) dp = o
(τ 2)
E(π(pV − τε
))− π
(pV − τE (e)
)=s′ (p0)
2τ 2V ar (ε) + o
(τ 2).
60
The abatement effect becomes:
Ψ (e)(E(π(pV − τe+ A(τ)
)− π
(pV − τe
)|e < e
))=
∫ e
e
(s (p0)
((pV − τe+ A(τ)− p0
)−(pV − τe− p0
))+ s′(p0)
2
((pV − τe+ A(τ)− p0
)2 −(pV − τe− p0
)2)
+ o (τ 2)
)de
= s (p0)τ 2
2b′′ (0)Ψ (e) + o
(τ 2).
Finally, the change in pollution is given by
GU −GV
= E (e) s(pU − τE (e)
)− E (e) s
(pV − τE (e)
)+ E (e) s
(pV − τE (e)
)− E
(εs(pV − τε
))+∫ e
e
[es(pV − τe
)ψ (e)− (e− a∗) s
(pV − τ (e− a∗)− b (a∗)
)]ψ (e) de+ o(τ)
= s′ (p0) τV ar (ε)
+
∫ e
e
e(s (p0) + s′ (p0)
(pV − τe− p0
))−e(s (p0) + s′ (p0)
(pV − τ (e− a∗)− b (a∗)− p0
))+a∗s (p0) + o (τ)
ψ (e) de+ o (τ)
= s′ (p0) τV ar (ε) +
∫ e
e
(−es′ (p0)
(τ 2
2b′′ (0)
)+
τ
b′′ (0)s (p0) + o (τ)
)ψ (e) de+ o (τ)
= s′ (p0) τV ar (ε) + Ψ (e)τ
b′′ (0)s (p0) + o (τ) .
We add these terms up (and multiply (GU −GV ) with (v − τ)) as well as FΨ(e) :
W V −WU = (v − τ
2)τ
[s′ (p0)V ar (ε) +
Ψ (e) s (p0)
b′′(0)
]− FΨ(e) + o(τ 2).
Heterogenous Productivity
We revert to a setting without abatement, but firms have heterogenous productivity ϕ > 0
such that costs of firm i is c(q)/ϕi. The resulting supply function of an (untaxed) firm is
s(ϕip). In the following, we will consider the special case of iso-elastic supply curve such that
supply of certified and uncertified firms, respectively, are:
qi = s0 [ϕi(p− τe)]α ,
61
qi = s0 [ϕi(p− t)]α ,
where α > 0. We let the (unconditional) distribution of productivity be given by ψϕ(ϕ) and
the conditional distribution of emissions on productivity be given by ψe(e|ϕ).
The indifferent firm e(ϕ) must be conditioned on productivity through:
1
ϕπ (ϕ (p− τ e(ϕ)))− 1
ϕπ (ϕ (p− t)) = F + f,
where e(ϕ) is monotonically increasing in ϕ.
The output tax is given by total emissions of uncertified firms divided by total output of
uncertified firms:
t = τ
∫ϕψϕ (ϕ)
∫∞eϕes (ϕ (p− t))ψe (e|ϕ) dedϕ∫
ϕψϕ (ϕ) s (ϕ (p− t)) (1−Ψe (e(ϕ)|ϕ)) dϕ
= τ
∫ϕψϕ (ϕ)
∫∞eϕeϕαψe (e|ϕ) dedϕ∫
ϕψϕ (ϕ)ϕα (1−Ψe (eϕ|ϕ)) dϕ
= τ
∫ϕE(eϕα|ϕ, e > e(ϕ))ψϕ(ϕ)dϕ∫
ϕϕαE(e|ϕ, e > e(ϕ))ψϕ(ϕ)dϕ
,
where we let tU denote the tax when no certification is in place (e(ϕ) = e for all ϕ) and tV
the tax on uncertified firm when certification is in place.
The market clearing condition is given by:
D (p) =
∫ϕ
(∫ e(ϕ)
e
s0 [ϕ (p− τe)]α ψ (e|ϕ) de+ s0 [ϕ (p− t)]α (1−Ψ (e(ϕ)))
)ψϕ(ϕ)dϕ.
(31)
We can collect welfare and emissions effects and write:
W V −WU (32)
=
∫ pV
pU
[(∫ϕ
s(ϕ(p− tU
))ψϕ (ϕ)
)dϕ−D (p)
]dp︸ ︷︷ ︸
price effects
+
∫ϕ
1
ϕ
( ∫ e(ϕ)
eπ(ϕ(pV − τe
))ψe (e|ϕ) de
+ (1−Ψe (e(ϕ)|ϕ))π(ϕ(pV − t
))− π
(ϕ(pV − tU
)) )ψϕ (ϕ) dϕ︸ ︷︷ ︸reallocation gains
−(v − τ)(GV −GU
)︸ ︷︷ ︸untaxed emissions
− F∫ϕ
ψϕ (ϕ) Ψϕ (eϕ|ϕ) dϕ,
62
where:
GV −GU
=
∫ϕ
ψϕ (ϕ)(s0
(ϕ(pU − tU
))α − s0
(ϕ(pV − tU
))α)E (e|ϕ) dϕ︸ ︷︷ ︸
price effect
+
∫ϕ
ψϕ (ϕ) dϕ
E (e|ϕ) s
(ϕ(pV − tV
))−[∫ e(ϕ)
es(ϕ(pV − τe
))eψe (e|ϕ) de
+ (1−Ψe (e(ϕ)|ϕ)) s0
(ϕ(pV − tV
))αE (e|e > e(ϕ), ϕ)
]
︸ ︷︷ ︸reallocation
.
Differentiating equation (31) returns:
pV = pU + o(τ)
and the elements of equation (32) are given by
Price Effects = o(τ 2),
Reallocation gains = τ 2S(p0) 12αp0V ar (ε) + o(τ 2),
where S(p0) is total production under p0:
S (p0) = D (p0) =
∫ϕ
s (ϕp0)ψϕ (ϕ) dϕ = s0pα0
∫ϕ
ϕαi ψϕ(ϕ)dϕ.
and S ′(p0) = αS(p0)/p0, such that we replicate the slope of aggregate supply from Corollary
1.
Furthermore:
V ar(ε) =
∫ϕ
∫ε
(ε− E(ε)
)2
ψ(ϕ, ε)dεdϕ,
where:
ψ(ϕ, ε) =ϕαψe(e|ϕ)ψϕ(ϕ)∫ϕ
∫εϕαψ(ϕ, ε)dεdϕ
is the output-scaled joint distribution of ϕ and ε. The expectation is defined analogously:
63
E(ε) =∫ϕ
∫εεψ(ϕ, ε)dεdϕ, such that:
E(ε) =
∫ϕ
∫εεϕαψe(e|ϕ)ψϕ(ϕ)∫
ϕ
∫εϕαψ(ϕ, ε)dεdϕ
dεdϕ =
∫ϕ
∫εε [ϕp0]α ψe(e|ϕ)ψϕ(ϕ)∫
ϕ
∫ε[ϕp0]α ψ(ϕ, ε)dεdϕ
dεdϕ
=
∫ϕ
[∫e>e(ϕ)
eψe(e|ϕ) +∫e<e(ϕ)
E(e|e > e(ϕ), ϕ)]pα0ϕ
αψϕ(ϕ)∫ϕ
∫ε[ϕp0]α ψ(ϕ, ε)dεdϕ
dεdϕ =GU
SU,
where GU and SU are emissions and total supply without certification.
Change in emissions are given by:
GV −GU = −τS (p0) αp0V ar (ε) ,
which then delivers:
W V −WU = τ(ν − τ
2
) αS (p0)
p0
V ar (ε)− FEϕΨϕ(e(ϕ)) + o(τ 2).
Free Entry
In the following we allow for free entry. In particular, firms pay an entry cost FE before
drawing an emission rate. This implies an endogenous mass of firms N . Consequently,
market clearing is given by:
D(p) = N
(∫ e
e
s(p− τe)ψ(e)de+ (1−Ψ(e))s(p− t)), (33)
where N is endogenously given by requiring entry costs to equal expected profits prior to
drawing the emission rate.∫ e
e
π(p− τe)ψ(e)de+ π(p− t)(1−Ψ(e)) = FE. (34)
We note that free entry implies that total profits equal 0. Consequently, we can take an
approach analogous to Section A.1.2 and find the change in welfare as
W V −WU = −∫ pV
pUD(p)dp− (v − τ)
(GV −GU
)−NV FΨ(e), (35)
64
where we note that profits equal zero due to free entry.
Emissions are given by:
GU −GV
= NV[E (e) s
(pU − τE (e)
)− E (e) s
(pV − τE (e)
)]︸ ︷︷ ︸price effect
+(NU −NV
)E (e) s
(pU − τE (e)
)︸ ︷︷ ︸entry effect
+NV
[E (e) s
(pV − τE (e)
)−[∫ e
e
es (p− τe)ψ (e) de+ s (p− t)E (e|e > e) (1−Ψ (e))
]]︸ ︷︷ ︸
reallocation effect
+ o(τ).
Taylor expansion
We take a Taylor expansion in equations (33) and (34) to find:
pV = pU + o (τ) = p0 + τE (e) + o (τ) ,
NV = NU + o (τ) = N0 + τD′ (p0)E (e)
s (p0)+ o(τ 2),
that is, at first order, the number of firms and the prices are the same in either equilibrium.
Consequently, at first order the welfare changes given by equation (35) excluding the certifi-
cation costs are zero. Since −∫ pVpU
D(p)dp is of the same order as (pV −pU) we therefore need
to develop pV − pU further. Analogous reasoning but taking a second order approximation
implies:
pV − pU = −τ2
2
s′ (p0)
s (p0)V ar (ε) + o
(τ 2),
which shows that the price will be lower with certification. The change in the number of
firms is given by:
NV −NU =τ 2
2
(−D′ (p0)
s′ (p0)
s (p0)+N0
((s′ (p0))2
s (p0)− s′′ (p0)
))V ar (ε) + o
(τ 2).
We then expand the first term of the welfare change to get:∫ pV
pU(−D (p)) dp = D (p0)
τ 2
2
s′ (p0)
s (p0)V ar (ε) + o
(τ 2)
=τ 2
2N0s
′ (p0)V ar (ε) ,
65
where we use that D(p0) = N0s(p0) at τ = 0.
And we proceed with the untaxed emissions term to get:
GV −GU = N0s′ (p0) τV ar (ε) + o (τ) ,
such that when τ is small:
W V −WU =(v − τ
2
)N0s
′ (p0) τV ar (ε)−NV FΨ (e) + o(τ 2).
The welfare expression is the same as in Corollary1 (regardless of whether τ = v or not).
Since profits are zero for producers the whole welfare effect (besides changes in emissions)
must accrue to consumers.
A.2 Adjustment on the extensive margin
A.2.1 Equilibrium without taxes
We consider a mass 1 of potential firms heterogenous in quantity they can produce, q, cost
of entry, c, and emissions per unit produced, e. All firms face the same price p which is
exogenous. The cost c has to be paid upfront and the firm faces to intensive margin of
construction. Consequently, without taxes, if pq ≥ c, the firm produces q and emits eq. The
joint distribution of (c, q, e) is Ψ(c, q, e). c has domain [c, c] and analogouosly for q and e.
The lower bound is positive for all. Ψ has full support. The aggregate supply function is
given by:
S =
∫ c
c
∫ q
q
∫ e
e
q1pq≥cdΨ(c, q, e). (36)
with corresponding emissions of:
G =
∫ c
c
∫ q
q
∫ e
e
eq1pq≥cdΨ(c, q, e). (37)
1pq≥c is an indicator function as is not differentiable at pq = c. Consequently, we employ
the Direct Delta Function and briefly remind the reader how this function operates. Strictly
speaking it is a functional with the properties that
δt−a = 0, for t 6= a,∫ a+ε
a−εf(t)δt−adt = f(a), for ε > 0,
66
that is a function that “picks out” the point t = a such that the integral around still sums
to 1.
For some vector (x, y) distributed according to Ψ(x, y) with pdf of ψ(x, y) and ψx(x) and
ψy(y|x). We consider the function:
F (a) =
∫x
∫y
1ax≥yf(y, x, a)dΨ(x, y),
for some constant a. We can differentiate wrt a to get:
F ′(a) =
∫x
∫y
∂g
∂a(x, y, a)δax=yf(y, x, a)dΨ(x, y) +
∫x
∫b
1ax≥y∂f
∂a(y, x, a)dΨ(x, y).
The second term is standard. The first term can be written as:∫x
∫y
xδax=yf(y, x, a)dΨ(x, y) =
∫x
x
∫y
δax=yf(y, x, a)ψy(y|x)ψx(x)dydx
=
∫x
xf(ax, x, a)ψy(ax|x)ψx(x)dydx.
We will be repeatedly exploiting this property in what follows.
In particular we differentiate equations (36) and (37) delivers:
S ′(p) =
∫c,q,e
q2δpq=cψc(c|e, q)ψq(q|e)ψe(e)dcdqde (38)
=
∫e
ψe(e)
∫q
q2ψc(pq|e, q)ψq(q|e)dqde,
G′(p) =
∫e
ψe(e)
∫q
eq2ψc(pq|e, q)ψq(q|e)dqde,
which gives equation (15) in the main text.
A.2.2 Equilibrium under certification
We consider a tax system in which, when certified a firm pays τ per unit emission for a total
tax of τeq. Instead of imposing a tax of f , we permit firms to certify to the point of e. Firms
still pay F . Consequently a firm with e > e will produce if:
(p− t)q ≥ 0,
67
and a firm with e ≤ e will produce if:
max{(p− τe)q − c− F, (p− t)q − c} ≥ 0.
Below we will use approximations. For reasons analoguous to the model on the intensive
margin welfare effects these welfare effects will only be positive if F is second order. Conse-
quently, any firm with e ≤ e with certify such that e = e and we can take e as an exogenous
variable.
t is defined (implicitly) as using the average emissions for uncertified firms:
t = τ
∫eq1e>e1(p−t)q≥cdΨ(c, q, e)∫q1e>e1(p−t)q≥cdΨ(c, q, e)
,
We write (where the integral is over (c, q, e) when not otherwise specified):
S(p, τ, t) =
∫q(1e≤e1(p−τe)q≥c+F + 1e>e1(p−t)q≥c
)dΨ(c, q, e),
G(p, τ, t) =
∫eq(1e≤e1(p−τe)q≥c+F + 1e>e1(p−t)q≥c
)dΨ(c, q, e),
where the mass of firms that certify is given by:
M =
∫1e≤e1(p−τe)q≥c+FdΨ(c, q, e).
We write welfare as:
W = pS − C − vG− FM
=
∫ [(pq − c− veq − F ) 1e≤e1(p−τe)q≥c+F + (pq − c− veq) 1e>e1(p−t)q≥c
]dΨ(c, q, e)
We seek Taylor approximations to W in τ but scale F and t accordingly. That is we define
F (τ) and t(τ) and differentiate wrt τ :
dW
dτ=
∫ [− (pq − c− veq − F ) (eq + F ′(τ)) 1e≤eδ(p−τe)q=c+F − (pq − c− veq) t′(τ)q1e>eδ(p−t)q=c
]dΨ(c, q, e)
(39)
−∫F ′(τ)1e≤e1(p−τe)q≥c+FdΨ(c, q, e).
Which, evaluated at τ = 0 and using that F is second order in τ and consequently F ′(0) = 0
68
as well as t(τ) = 0:
dW
dτ|τ=0 = −
∫[q (pq − c− veq) e1e≤eδpq=c + (pq − c− veq) 1e>et
′(0)δpq=c] dΨ(c, q, e). (40)
We define ψq(q) as the unconditional, ψe(e|q), and ψc(c|e, q) as the pdfs of Ψ. We use this
and substitute for c to get:
dW
dτ|τ=0 =
∫veq2 [e1e≤e + 1e>et
′(0)1e>e] δpq=cdΨ(c, q, e)
= v
∫q
q2ψq(q)
[∫e≤e
e2ψc(pq|e, q)ψe(e|q)de+
∫e>e
et′(0)ψc(pq|e, q)ψe(e|q)de]dq.
To get the second order derivative we rewrite equation (39) as:
dW
dτ=
∫q
ψq(q)q
[∫e≤e
(v − τ) eq (eq + F ′(τ))ψe(e|q)ψc((p− τe)q − F |e, q)de+
∫e>e
(ve− t(τ)) t′(τ)q2ψe(e|q)ψc((p− t)q|e, q)]dq
−∫q
ψq(q)F′(τ)
∫e≤e
ψe(e|q)Ψc((p− τe)q − F |e, q)dedq,
where Ψc(c|e, q) is the cdf of c conditional on e and q.
We differentiate again:
dW 2
dτ 2=
∫q
ψq(q)
[∫e≤e
ψe(e|q) (v − τ) eq (eq + F ′(τ)) (−eq − F ′(τ))ψ′c((p− τe)q − F |e, q) +
∫e>e
ψe(e|q) (ve− t) q2t′(τ) (−t′(τ)q)ψ′c((p− t)q|e, q)de]dq
+
∫q
ψq(q)
[∫e≤e
ψe(e|q) (v − τ) eqF ′′(τ)ψc ((p− τe)q − F |e, q) de+
∫e>e
ψe(e|q) (ve− t) q2t′′(τ)ψc ((p− t)q|e, q) de]dq
+
∫q
ψq(q)
[∫e≤e−ψe(e|q)eq (eq + F ′(τ))ψc ((p− τe)q − F |e, q) de+
∫e>e
−ψe(e|q)q2 (t′(τ))2ψc ((p− t)q|e, q) de
]dq
−∫ψq(q)F
′(τ)
∫e≤e
ψe(e|q) (−eq − F ′(τ))ψc ((p− τe)q − F |e, q) dedq
−∫q
ψq(q)F′′(τ)
∫e≤e
ψe(e|q)Ψc ((p− τe)q − F |e, q) dedq.
69
We evaluate this expression at τ = 0:
dW 2
dτ 2|τ=0 =
∫q
ψq(q)
(−∫e≤e
ψe(e|q)v (eq)3 ψ′c(pq|e, q)de−∫e>e
ψe(e|q)veq3 (t′(0))2ψ′c(pq|e, q)de
)dq
(41)
+
∫q
ψq(q)
(∫e≤e
ψe(e|q)veqF ′′(0)ψ′c(pq|e, q)de+
∫e>e
ψe(e|q)veq2t′′(0)ψc(pq|e, q)de)dq
+
∫q
ψq(q)
(∫e≤e−ψe(e|q) (eq)2 ψc(pq|e, q)de+
∫e>e
−ψe(e|q)q2 (t′(0))2ψc(pq|e, q)de
)dq
−∫q
ψq(q)F′′(0)
∫e≤e
ψe(e|q)Ψc(pq|e, q)dedq.
We will use this to perform a Taylor expansion of the welfare:
W (τ, v) = W (0, v) +dW
dτ|τ=0τ +
d2W
dτ 2|τ=0
τ 2
2+ o(τ 2),
and use equations (40) and (41) while noting that many terms are of higher than second
order. We further misuse notation and let ψe(e) be the unconditional distribution of e with
ψq(q|e) and ψc(c|e, q) denoting corresponding conditional distributions.
W (τ, v) = W (0, v) + τ
[ ∫e≤e
(v − τ
2
)ψe(e)e
2∫qψq(q|e)ψc(pq|e, q)dqde
+∫e>e
(ve− τ
2t′(0)
)t′(0)ψe(e)
∫qq2ψq(q|e)ψc(pq|e, q)dqde
]
+F
∫e≤e
ψe(e)
∫q
ψq(q|e)Ψc(pq|e, q)dedq + o(τ 2)
We use the definition of t(τ) to get:
t = τ
∫eq1e>e1(p−t)q≥cdΨ(c, q, e)∫q1e>e1(p−t)q≥cdΨ(c, q, e)
t′(τ) =
∫eq1e>e1(p−t)q≥cdΨ(c, q, e)∫q1e>e1(p−t)q≥cdΨ(c, q, e)
+ τ∂
∂τ
∫eq1e>e1(p−t)q≥cdΨ(c, q, e)∫q1e>e1(p−t)q≥cdΨ(c, q, e)
such that:
t = t(0) + t′(0)τ + o(τ 2) =
∫eq1e>e1pq≥cdΨ(c, q, e)∫q1e>e1pq≥cdΨ(c, q, e)
τ + o(τ 2).
70
We further define:
ψe (e) =ψe (e)
∫qψq (q|e) q2ψc (pq|e, q) dq∫
e′ψe (e′)
∫qψq (q|e′) q2ψc (pq|e′, q) dqde′
=ψe (e)
∫qψq (q|e) q2ψc (pq|e, q) dq
dSdp|τ=0
,
where dS/dp|τ=0 is given by equation 38. We then write:
W (τ, v) = W (0, v) +dS
dp|τ=0
[∫ e
0
τ(v − τ
2
)ψ(e)e2de+
∫e>e
(ve− t
2
)tψ(e)de
]
−F∫e≤e
ψe(e)
∫q
ψq(q|e)Ψc(pq|e, q)dedq + o(τ 2).
We introduce ε defined as:
ε =
e
E|ψe(e|e > e)
if e ≤ e
if e > e,
where E|ψe is the expection operator over ψe: E|ψe(e|e > e) =∫e>e eγe(e)de∫e>e γe(e)de
. Note this is
not the average emission rate above e > e since it is weighted by q2. We can also define
V ar|ψe(ε) = E|ψee2 − (E|ψee)
2. This gives:
W (τ, v)
= W (0, v) +dS
dp|τ=0
[τ(v − τ
2
)V ar|ψe (ε) + τ
(v − τ
2
) (E|ψe (e)
)2
+(
1− Ψe (e)) (t− τE|ψe (e|e > e)
) ((v − τ
2
)E|ψe (e|e > e)− t
2
) ]
−F∫e≤e
ψe(e)
∫q
ψq(q|e)Ψc(pq|e, q)dedq + o(τ 2),
where 1− Ψ(e) =∫e>e
ψ(e)de is the “share” of firms that don’t certify but weighted by q2.
We consider the welfare expression for τ = v to get:
W V (ν, ν)
= W (0, ν) +v2
2S ′(p)|τ=0
[V ar|ψe (ε) +
(E|ψe (e)
)2
−(
1− Ψe (e))(E|ψ (eq|e > e, pq > c)
E|ψ (q|e > e, pq > c)− E|ψe (e|e > e)
)2]
−F∫e<e
ψe (e)
∫q
ψq (q|e) Ψc (pq|e, q) dedq + o(τ 2),
71
where we make explicit that this is welfare for certification, W V , in anticipation for
welfare under no certification. We can make analoguous calculations as above to find that a
Taylor expansion of welfare under no certification is:
WU(t, v) = W (0, v) + t
∫q2δpq=c
[ve− 1
2t
]dΨ(q, c, e) + o(t2)
= W (0, v) + S ′(p)|τ=0E|ψe
(ve− 1
2τG (0)
S (0)
)τG (0)
S (0)+ o
(t2),
where G(0) and S(0) are emissions and supply under τ = 0. Then we can write:
W V−WU = S ′(p)|τ=0
v2
2V ar|ψe(ε) +
v2
2
(dGdp
dSdp
− G(0)
S(0)
)2
−(
1− Ψe(e)) v2
2
(E|ψe(e|e > e)− E|ψ(eq|e > e, pq > c)
E|ψ(q|e > e, pq > c)
)2
−F∫e≤e
ψe(e)
∫q
ψq(q|e)Ψc(pq|e, q)dedq + o(τ 2).
Change in emissions. Emissions are given by:
GV =
∫eq(1e≤e1(p−τe)q≥c+F + 1e>e1(p−tS)q≥c
)dΨ(c, q, e),
and
GU =
∫eq1(p−tU )q≥cdΨ(c, q, e),
where these refer taxes under Steve and uncertified.
We differentiate:
∂GV
∂τ= −
∫eq (eq + F ′(τ))
(1e≤eδ(p−τe)q=c+F
)dΨ(c, q, e)
−∫eq(tS ′q
) (1e>eδ(p−tS)q=c
)dΨ(c, q, e)
Evaluated at zero:
∂GV
∂τ|τ=0 = −
∫e2q2 (1e≤eδpq=c+F ) dΨ(c, q, e)−
∫eq2(tS ′(0)
)(1e>eδpq=c) dΨ(c, q, e)
= −∫e≤e
e2
∫q
q2ψc(pq|q, e)ψq(q|e)ψe(e)dqde− tS ′(0)
∫e>e
e
∫q
q2ψc(pq|q, e)ψq(q|e)ψe(e)dqde
72
= −S ′(p)∫e≤e
e2ψe(e)de− tV ′(0)S ′(p)
∫e>e
eψe(e)de
Further:∂GU
∂τ= −
∫eq2tU ′δ(p−tU )q=cdΨ(c, q, e)
such that:∂GU
∂τ|τ=0 = −S ′(p)tU ′(0)
∫eψe(e)de,
Let’s just look at
tU = τ
∫eq1(p−t)q≥cdΨ(c, q, e)∫q1(p−t)q≥cdΨ(c, q, e)
,
and we will get:tU
t=
∫eq1pq≥cdΨ(c, q, e)∫q1pq≥cdΨ(c, q, e)
+ o(τ)
tV
τ=
∫eq1e>e1pq≥cdΨ(c, q, e)∫q1e>e1pq≥cdΨ(c, q, e)
+ o(τ)
We use this for a first order taylor expansion in each and subtract the two:
GV−GU = −τS ′(p)[∫
e≤ee2ψe(e)de+
∫e>e
(E|ψe
)2
ψe(e)de−∫e>e
(E|ψe
)2
ψe(e)de+ tS ′(0)
∫e>e
eψe(e)de− tU ′(0)
∫eq2ψe(e)de
]
= −τS ′(p)[∫
e
ε2ψe(e)de−∫e>e
(E|ψe
)2
ψe(e)de+
∫eq1e>e1pq≥cdΨ(c, q, e)∫q1e>e1pq≥cdΨ(c, q, e)
∫e>e
eψe(e)de− tU ′(0)
∫eψe(e)de
]Change in emissions
We seek to establish what the change in emissions in given by:
GV =
∫eq(1e≤e1(p−τe)q≥c+F + 1e>e1(p−tV )q≥c
)dΨ(c, q, e),
and
GU =
∫eq1(p−tU )q≥cdΨ(c, q, e),
where these refer to t under certification and without. We wish to perform first-order Taylor-
expansions so we first find:
∂GV
∂τ|τ=0 = −S ′(p)
[V arψeε+
(E|ψee
)2Ψ(e) +
∫eq1e>e1pq≥cdΨ(c, q, e)∫q1e>e1pq≥cdΨ(c, q, e)
(1− Ψ(e))E|ψ(e)
],
73
∂GU
∂τ|τ=0 = −S ′(p)
∫eq1pq≥cdΨ(c, q, e)∫q1pq≥cdΨ(c, q, e)
E|ψ(e),
where we have used that
tV ′(0) =tV (0)
τ=
∫eq1e>e1pq≥cdΨ(c, q, e)∫q1e>e1pq≥cdΨ(c, q, e)
+ o(τ),
and likewise for tU .We write G(τ) = G(0) + ∂G/∂τ |τ=0 + o(τ) for both GV and GU and subtract to find:
GV−GU = −S′(p)τ[V arψe
(ε) +
(G′(p)
S′(p)− G(p)
S(p)
)Eψe
(e) + (1− Ψe(e))
(G(p)
S(p)Eψe
(e|e > e)− G′(p)
S′(p)Eψe
(e)
)]+o(τ),
which is equation (18) in the main text.
A.2.3 Special case for application to the Permian Basin
In the following we make two alterations to the expressions above. First, we let prices vary
across firms such that each firm is de+scribed by a vector (c, q, e, p) with an associated Ψ.
A firm receives a price per unit of p+ ρ, where ρ is a common element of prices.
Supply is given by:
S =
∫q(1e≤e1(p+ρ−τe)q>c+F + 1e>e1(p+ρ−t)q>c
)dΨ (c, q, e, p) .
and consequently:∂S
∂ρ|ρ=0,τ=0 =
∫c,q,e,p
q21pq>cdΨ(c, q, e, p).
We perform calculations in line with the section above to find:
W V (v, v)−WU (t (v) , v)
=dS
dρ|ρ=0,τ=0
v2
2V ar|ψe (ε) +
v2
2
(dGdρ
dSdρ
− G (0)
S (0)
)2
−(
1− Ψ (e)) v2
2
(E|ψe (e|e > e)− E|ψ (eq|e > e, pq > c)
E|ψ (q|e > e, pq > c)
)2
−FM + o(τ 2),
where:
ψe(e) =ψe (e)
∫pψp (p|e)
∫qψq (q|e, p) q2ψc (pq|e, q, p) dqdp
dSdρ|τ=0
,
and other expressions are done analoguously.
74
For our empirical investigation we impose:
c = upq,
where u is independent of (p, q, e). We use the distribution Γ(u, q, e, p) as the full distribu-
tion and defining the marginal distribution of u as Γu(u|q, e, p) = Γu(u). Further, we let
Γq,e,p(q, e, p) denote the distribution of (q, e, p). That is: Γ(u, q, e, p) = Γu(u)× Γq,e,p(q, e, p)
S =
∫u,q,e,p
q (1e≤e11≥u + 1e>e11≥u) dΓ (u, q, e, p) =
∫u
ψu(u)
∫q,e,p
qdΓq,e,p (q, e, p)
= Γu(1)
∫q,e,p
qdΓq,e,p (q, e, p) .
There is a mass 1 of potential firm so S denotes total production as well as average
production of potential firms.
Consequently:
E(q) =S
Γu(1)=
∫q,e,p
qdΓq,e,p (q, e, p)
is observable in the data where E(q) =∑
i qi/N is just the observed average in the pre-tax
equilibrium with N existing firms. Analogous calculations deliver:
E(q2) =
∫q,e,p
q2dΓq,e,p (q, e, p) ,
where E(q2) =∑
i q2i /N is the sample average of q2.
The expression for V arψe(ε) is:
V arψe (ε) = E
(ε2 q2
E (q2)
)−
(E
(eq2
E (q2)
))2
,
which naturally depends on the choice of e.
The remaining expressions are:
G (0)
S (0)=Edata (eq)
Edata (q).
Ψ (e) =Σ (q2|e > e)
Σ (q2),
75
E|ψ (eq|e > e, pq > c)
E|ψ (q|e > e, pq > c)=E (eq|e > e)
E (q|e > e),
M = Ndata (e ≤ e)
where∑
is just the sum in the data and |e > e only counts wells with e above the cutoff
e. Finally, M counts the number of firms with e ≤ e. Combining these terms gives equation
(??) in the main text.
A.3 The International Model
A.3.1 Baseline setting
We reproduce equations (20) and (19) in the pooling quilibrium (ρ = 0)
DH(pH) +DF (pF ) = (42)
EH [sH(pH − τHε+ AH)] +
∫ e
e
[sF (pH − τF e− κ+ AF )ψF (e)de] + (1−ΨF (e))sF (pF ).
pF = (pH − τFEF (e|e > e)− κ). (43)
And note that differentiating pH wrt. e gives:
dpH
de=
(D′F − (1−ΨF (e))s′F (pF )
)τF
∂EF (e|e>e)∂e
+ [sF (pF +AF )− sF (pF )]ψF (e){D′H(pH) +D′F (pF )−
∫ ee
[s′F (pH − τF e− κ+AF )ψF (e)de
]− (1−ΨF (e))s′F (pF )
} ,
dpF
de=−D′H(pH) +
∫ ee
[s′F (pH − τF e− κ+AF )ψF (e)de
]τF
∂EF (e|e>e)∂e
+ [sF (pF +AF )− sF (pF )]ψF (e){D′H(pH) +D′F (pF )−
∫ ee
[s′F (pH − τF e− κ+AF )ψF (e)de
]− (1−ΨF (e))s′F (pF )
} < 0,
where ∂E(e|e > e)/∂e > 0. Without abatement (AF = 0), pH must increase following an
increase in certification.
Proof of Proposition 3
The Welfare Expressions
We proceed under the assumption that W V −WU + FΨ(e) is second order in τ and will
demonstrate that this is so. W V is welfare under certification and WU is without certification.
W V is given by:
W V = CSH + CSF︸ ︷︷ ︸consumer surpluses
+ PSH + PSF︸ ︷︷ ︸producer surpluses
− [(v − τH)GH + (v − τF )GF + τFGF,dom]︸ ︷︷ ︸non-internalized emissions
− FΨF (e) ,
76
with
CSH = uH (CH)− pHCH and CSF = uF (CF )− pFCF .
PSH =
∫ ∞e
(pH − τHe+ AH) sH (pH − τHe+ AH)ψH (e) de−∫ ∞e
cH (sH (pH − τHe+ AH))ψH (e) de,
PSF =
∫ e
e
(pH − τF e+ AF − κ) sF (pH − τF e+ AF − κ)ψF (e) de
−∫ e
0
cF (sF (pH − τF e+ AF ))ψF (e) de+ (pF sF (pF )− cF (sF (pF ))) (1−ΨF (e)) ,
where cF and cH are the cost functions associated with Foreign and Home supply functions.
Further:
GH =
∫ e
e
(e− aH) sH (pH − τHe+ AH)ψH (e) de,
GF =
∫ e
e
(e− aF ) sF (pH − τF e+ AF − κ)ψF (e) de+ sF (pF )E (e|e > e) (1−ΨF (e)) ,
GF,dom = τFEF (e|e > e)DF (pF ) .
Taylor Approxiomations of price changes
We start out by taking a Taylor-approximation of the equilibrium without certification
(“U”) (equation (45 with e = e). Let p0 denote the prie in an equilibrium without taxes and
Foreign exports to Home. In such a setting p0 − κ is the price in Foreign. Letting the price
in the uncertified equilibrium be pUH this delivers:
pUH − p0 =s′H (p0) τHEH (e) + s
′F (p0 − κ) τFEF (e)−D′F (p0 − κ) τFEF (e)
s′H (p0) + s
′F (p0 − κ)−D′H (p0)−D′F (p0 − κ)
+ o (τ) . (44)
We derive approximations for the separating equilibrium with the understanding that
analogous derivations will deliver the expression for the pooling equilibrium.
It must always hold that:
DH(pH) +DF (pF )
=
∫ e
e
sH(pH − τHe+AH)ψH(e) +
∫ e
e
sF (pH − τF e+AF − κ)ψF (e)de+ sF (pF ) (1−ΨF (e))) ,
To a first order the expression above delivers:
D′H(p0)(pH − p0) +D′
F (p0 − κ)(pF − p0 + κ) (45)
77
= s′
H(p0)(pH−τHEH(e)−p0)+s′
F (p0−κ) (ΨF (e)(pH − τFEF (e|e < e) + (1−ΨF (e))(pF + κ)− p0)+o(τ),
where we use that:
AF = o(τ), AH = o(τ).
Further, for a separating equilibrium it must hold that at zeroth order:
sF (p0 − κ)(1−ΨF (e)) = DF (p0 − κ), (46)
which pins down e at zeroth order:
e = e0 + e+ o(τ), where e0 = Ψ−1F (1−DF (p0 − κ)/sF (p0 − κ)) ,
and e is the “first order” term of e. If e is such that equation (46) holds with “>” we are in
the pooling equilibrium, if it holds with “<” certified firms will not want to export. Note, e
and e0 differ only to a first order and consequently whether we evaluate functions at e or e0
will be equivalent in our Taylor expansions. For ease of exposition we will use e both here
and in equation (23) in the main text. This is correct, though a more stringent adherence
to convention would have us evaluate at e0 for the separating equilibrium.
Alternatively, Home firms export to Foreign. We preclude this here, but briefly discuss
the implications below. In effect this pins down e (at zeroth order).
From the certification condition, one gets (to a second order):
(pH − τF e−κ−pF )
(sF (p0 − κ) + s
′
F (p0 − κ)
(pH − τF e+ κ+ pF
2− p0
))= F + f + o(τ 2),
where again we use that AF = o(τ).(pH−τF e+κ+pF
2− p0
)is first order such that when (F+f)
is first order we get:
pH − τF e− pF = κ+
(F + f
sF (p0 − κ)
)+ o (τ) (47)
Note that this requires:
0 < τF (EF (e|e > e)− e)− F + f
sF (p0 − κ)< 2κ,
where both inequality use equation (47) and the first one ensures that profits from selling do-
78
mestically are strictly higher for uncertified Foreign firms and the right hand side guarantees
that Home firms would not want to export toWe use equation (47) along with equation (45) to get:
pVH − p0 =
s′H(p0)τHEH(e) + s
′F (p0 − κ)
(ΨF (e0)τFEF (e|e < e0) + (1−ΨF (e0))
(τF e0 + F+f
sF (p0−κ)
))−D′
F (p0 − κ)(τF e0 + F+f
sF (p0−κ)
) s′H(p0) + s
′F (p0 − κ)−D′
H(p0)−D′F (p0 − κ)
+ o(τ),
and combine this with equation (44) to get
pVH − pUH =
=
(1−ΨF (e0)) s′F (p0 − κ)
((τF e0 + F+f
sF (p0−κ)
)− τFEF (e|e > e0)
)−D′F (p0 − κ)
(τF e0 + F+f
s(p0−κ)− τFEF (e)
) s′H (p0) + s′H (p0 − κ)−D′H (p0)−D′F (p0 − κ)
+ o (τ) , (48)
and
pVF − pUF =
s′H (p0)
(τFEF (e)−
(τF e0 + f
sF (p0)
))+s
′F (p0 − κ)
(τFEF (e)− (1−ΨF (e)) τFEF (e|e > e0)−ΨF (e)
(τF e+ f
sF (p0−κ)
))−(τFEF (e)−
(τF e0 + f
sF (p0−κ)
))D
′H (p0)
s′H (p0) + s
′F (p0 − κ)−D′
H (p0)−D′F (p0 − κ)
+ o(τ) (49)
We define εDF = D′F (p0 − κ)(p0 − κ)/DF (p0 − κ) = D′F (p0 − κ)p0/DF (p − κ) + o(τ)
since κ is of the same order as τ . We further define Home share in demand θDF = DF (p0 −κ)/ (DF (p0 − κ) +DH(p0)). Analogous versions of ε and θ exist for Home and supply and
we let εD = θDF εDF + θDHε
DH . We then write:
pVH − pUH =
=
(1−ΨF (e0)) θSF εSF
((τF e0 + F+f
sF (p0−κ)
)− τFEF (e|e > e0)
)−εDF θDF
(τF e0 + F+f
s(p0−κ)− τFEF (e)
) εS − εD
+ o (τ) .
We then use the definition of ρ = pF−(pH−τFEF (e|e > e)−κ) = −(τF (e0 − EF (e|e > e0))+
79
(F+f
sF (p0−κ)
)) to write:
pVH − pUH =−εDF θDFεS − εD
τF (EF (e|e > e0)− EF (e))− (1−ΨF (e0)) εSF θSF − εDF θDF
εS − εDρ+ o (τ) .
Analgous derivations for pVF−pUF as well as the expressions for the pooling equilibrium delivers
the expressions in footnote 21.
Taylor approximations for welfare changes
We separate expressions from the welfare change. First, we write consumer and producer
surplus terms as:
CSVH + CSVF + PSVH + PSVF −(CSUH + CSUF + PSUH + PSUF
)=
EH
(∫ pVHpUH
sH (p− τHe+ AH) dp)
+ EF
(∫ pVHpUH
sF (p− τFEF (e)− κ) dp)
−(∫ pVF
pUFDH (p) dp+
∫ pVFpUF
DF (p) dp)
︸ ︷︷ ︸price effect
+(1−ΨF (e)) [πF (pF )− πF (pH − τFEF (e|e > e)− κ)]︸ ︷︷ ︸adjustment term
+ΨF (e)EF ((πF (pH − τF e+ AF − κ)− πF (pH − τF e− κ)) |e < e)︸ ︷︷ ︸abatement gains
+E (πF (pH − τF ε− κ))− πF (pH − τFEF (e)− κ)︸ ︷︷ ︸reallocation term
.
and handle the terms in order:
price effects
= EH
(∫ pVH
pUH
sH (p− τHe+ AH) dp
)+ EF
(∫ pVH
pUH
sF (p− τFEF (e)− κ) dp
)
−
(∫ pVF
pUF
DH (p) dp+
∫ pVF
pUF
DF (p) dp
)= −DF (p0 − κ)
(pVF − pVH + τFEF (e) + κ
)+(pVH − pUH
) s′H (p0)
(pVH+pUH
2− τHEH (e)− p0
)+sF ′ (p0 − κ)
(pVH+pUH
2− τFEF (e)− p0
)−D′H (p0)
(pVH+pUH
2− p0
) −D′F (p0 − κ)
(pVF − pUF
)(pVF + pUF2
− p0 + κ
).
80
Taylor approximations of the market clearing under certification and no certification, respec-
tively gives:
D′
H (p0)(pVH − p0
)+D
′
F (p0 − κ)(pVF − p0 + κ
)+ o (τ)
= s′
H (p0)(pVH − τHEH (e)− p0
)+s
′
F (p0 − κ)(ΨF (e)
(pVH − τFEF (e|e < e)
)+ (1−ΨF (e))
(pVF + κ
)− p0
)and
D′
H (p0)(pUH − p0
)+D
′
F (p0 − κ)(pUF − p0 + κ
)+ o (τ)
= s′
H (p0)(pUH − τHEH (e)− p0
)+ s
′
F (p0 − κ)(pUF + κ− p0
)summing up the two we have
s′
H (p0)
(pVH + pUH
2− τHEH (e)− p0
)+s
′
F (p0 − κ)
(ΨF (e)
(pVH − τFEF (e|e < e)
)+ (1−ΨF (e))
(pVF + κ
)+ pUH − τFEF (e)
2− p0
)
−D′H (p0)
(pVH + pUH
2− p0
)−D′F (p0 − κ)
(pVF + pUF
2− p0 + κ
)= 0,
which we use to get:
price effects
= −DF (p0 − κ) (pF − pH + τFEF (e) + κ)
+(pVH − pUH
)s′
F (p0 − κ)(1−ΨF (e))
(pVH − pVF − κ− τFEF
(e|e > eF
))2
+D′
F (p0 − κ)(pVH − pVF − τFEF (e)− κ
)(pVF + pUF2
− p0 + κ
)We continue with the reallocation term:
81
reallocation term
= E(πF(pVH − τF ε− κ
))− πF (pH − τFEF (e)− κ)
=s′F (p0 − κ)
2(τF )2 V arF (ε) + o
(τ 2)
and the abatement term:
abatement term = ΨF (e) sF (p0 − κ)AF + o(τ 2),
and
adjustment term
= (1−ΨF (e))(pVF + κ− pVH + τFE (e|e > e)
) [ sF (p0 − κ)
+s′F (p0 − κ)
[pVF+κ+pVH−τFE(e|e>e)
2− p0
] ]+ o(τ 2).
We add the terms with emissions:
(v − τH)(GVH −GU
H
)+ (v − τF )
(GVF −GU
F
)+ τF
(Gdom,VF −Gdom,U
F
)which we take in turn:
(v − τH)(GVH −GU
H
)= −
(ν − τH
)EH (e) sH′ (p0)
(pH,LB − pH
)+ o (τ)
(ν − τF )(GVF −GU
F
)= − (ν − τF )
s′F (p0 − κ)
( (pUH − pVH
)EF (e) + τFV arF (ε)
+(pVH − τFEF (e|e > e)− pVF − κ
)EF (e|e > e) (1−ΨF (e))
)+ΨF (e) aF sF (p0 − κ)
82
and further:
τF
(Gdom,VF −Gdom,U
F
)= −τF (EF (e)− EF (e|e > e))DF (p0 − κ)
−D′F (p0 − κ)[τFEF (e)
(pUF − p0 + κ
)− τFEF (e|e > e)
(pVF − p0 + κ
)]+ o
(τ 2)
We combine these terms with equations (48) and the corresponding expression for pF to get
(using that F is second order whereas f is (potentially) first order).
W V −WU (50)
= −s′F (p0 − κ)
2(1−ΨF (e))
(τEF (e|e > e)−
(τF e+
f
sF (p0 − κ)
))(pVF − pUF + τF (EF (e|e > e)− EF (e))
)−D
F ′ (p0 − κ)
2
(pVF − pUF
)(τFEF (e) + τF e+
f
sF (p0 − κ)
)+s′F (p0 − κ)
2(τF )2 V arF (ε) + ΨF (e) sF (p0 − κ)AF
+ (v − τH)(GUH −GV
H
)+ (v − τF )
(GUF −GV
F
)− FΨF (e) ,
and changes in emissions are given by:
GVH −GU
H = EH (e) s′
H (p0)(pVH − pUH
)+ o (τ) ,
GVF −GU
F
= −
s′F (p0 − κ)
( (pVH − pUH
)EF (e) + τFV arF (ε)
+(τF e+ f
sF (p0−κ)− τFE (e|e > e)
)EF (e|e > e) (1−ΨF (e))
)+ΨF (e) aF sF (p0 − κ)
+o(τ).
Combine the definition of the indifferent firms:
pVH − pVF = τF e+ κ+f
sF (p0 − κ)+ o (τ) , (51)
with the definition of ρ to get:
ρ = −(τF e+
(f
sF (p0 − κ)
)− τFE(e|e > e)
).
83
Further note that :
∆pH + ρ = ∆pF + τF [EF (e|e > e)− EF (e)] . (52)
Use these expressions as well as GVH − GU
H and GVF − GU
F in equation (50) to get equation
(23) in the main text.
∆pH + ρ > 0 and the Encouraged Foreign Production Effect is always (weakly)
positive
Note, that ∆pH + ρ = ∆pF + τF [EF (e|e > e)− EF (e)] such that:
∆pF + τF [EF (e|e > e)− EF (e)]
=
[sH′ (p0) + s
′FΨF −D
′H
]sH′ (p0) + sF ′ (p0 − κ)−DH′ (p0)−DF ′ (p0 − κ)
ρ
− τFD′ [EF (e|e > e0)− EF (e)]
sH′ (p0) + sF ′ (p0 − κ)−DH′ (p0)−DF ′ (p0 − κ),
where both ρ and [EF (e|e > e0)− EF (e)] are positive. Consequently ∆pH + ρ > 0 and the
Encouraged Foreign Production Effect is always positive in the separating equilibrium. In
the pooled equilibrium it is zero (since ρ = 0).
The sign of the Untaxed Foreign Consumption Effect is ambigious
First, we prove that τF [EF (e) + EF (e|e > e)]− ρ > 0
Recall that:
ρ = pF − (pH − τFEF (e|e > e)− κ),
and
pH − τF e− pF = κ+
(F + f
sF (p0 − κ)
)+ o (τ) .
such that:
τF (EF (e) + EF (e|e > e0))− ρ = τEF (e) +
(F + f
sF (p0 − κ)
)+ τF e > 0
Second, we prove that the sign ∆pF is ambigious. From footnote 21 we recall that:
∆pF =εDHθ
DH − εS
εS − εDτF (EF (e|e > e)− EF (e)) +
εS − εSF θSF (1−ΨF )− εDHθHDεS − εD
ρ,
84
where the first term is negative and the second term is positive. Clearly ∆pF ≤ 0 in the
pooling equilibrium (when ρ = 0) and by continuity for some parameters in the separating
equilibrium as well. e0 is defined by:
sF (p0 − κ)(1−ΨF (e0)) = DF (p0 − κ)
such that if DF (p0 − κ)/sF (p0 − κ) is less than but close to 1 we will have e0 ≈ e. Consider
a case where e = 0 and F + f ≈ 0. Then EF (e|e > e)− EF (e) ≈ 0 and:
pH − pF ≈ κ+ o (τ) .
and consequently:
ρ ≈ pF − (pH − τFEF (e|e > e)− κ) = τFEF (e|e > e) + o(τ) > 0,
and consequently ∆pF < 0.
A.3.2 The change in emissions
Calculations as above can be written as:
GH,LB −GH =(ν − τH
)EH (e) sH′ (p0)
(pH,LB − pH
)+ o (τ)
GF,LB −GF
=
sF ′ (p0 − κ)
( (pH,LB − pH
)EF (e) + τFV ar (ε)
+(τF eF + F+f
sF (p0−κ)− τFE
(e|e > eF
))E(e|e > eF
) (1−ΨF
(eF)) )
+ΨF(eF)aF sF (p0 − κ)
GH,LB −GH = EH (e) sH′ (p0)DF ′ (p0 − κ)
(τFE
(e|e > eF
)− τFEF (e)
)sH′ (p0) + sF ′ (p0 − κ)−DH′ (p0)−DF ′ (p0 − κ)
GF,LB −GH =
sF ′ (p0 − κ)
(DF ′(p0−κ)(τFE(e|e>eF )−τFEF (e))EF (e)
sH′(p0)+sF ′(p0−κ)−DH′(p0)−DF ′(p0−κ)+ τFV ar (ε)
)+ΨF
(eF)aF sF (p0 − κ)
we have
pVH − pUH =
85
=
(1−ΨF (e0)) s′F (p0 − κ)
((τF e0 + F+f
sF (p0−κ)
)− τFEF (e|e > e0)
)−D′F (p0 − κ)
(τF e0 + F+f
s(p0−κ)− τFEF (e)
) s′H (p0) + s′H (p0 − κ)−D′H (p0)−D′F (p0 − κ)
+ o (τ) , (53)
A.3.3 Optimal policy
In the following, we look at the optimal policy. A social planner offers a price pRE−τF (e−a) to
(Foreign) firms who reveal their e and undertake abatement a. It offers pUE to exporters who
do not reveal. The social planner can also pick a certification tax. Further, we potentially
allow for the social planner to choose exports from Home to Foreign (bilateral trade). The
social planner is constrained by market forces in Foreign and must take the behaviors of
foreign firms, inluding whether they choose to certify, and the resulting Foreign prices pF as
a given. Uncertified Foreign firms will export if:
pUE − κ = pF ,
and not if pUE − κ < pF . We preclude pUE − κ > pF by assuming that there is always some
Foreign demand. Uncertified exporting foreign firms, if they exist, must then make the
same profits as foreign firms selling for the domestic market. An indifferent firm exists with
πF(pRE − τF (e− a)− b (a)− κ
)− F − f = πF (pF ) , which must hold whether there are
uncertified exporters or not. f plays no other role than here and the social planner can
equivalently choose e.
The Foreign domestic market clears with:
u′F (CF ) = pF ∧ CF = DF (pF ) ,
qU = sF (pF ) ,
CF = sF (pF ) (1−ΨF (e))−M,
where qU is production by uncertified Foreign firms. M is the import from Home of uncertified
exports (from Foreign). M = 0 in the separating equilibrium and M > 0 in the pooling.
86
We write the objective function (world welfare) as:
W (54)
= uH
(∫ ∞0
qH (e)ψH (e) de+
∫ e
0
sF(pRE − τF (e− aF )− bF (aF )− κ
)ψF (e) de+MU
)−∫ ∞
0
(cH (qH (e)) + bH (aH (e)) qH (e))ψH (e) de
+uF (DF (pF ))−∫ e
0
(cF(sF(pRE − τF e+ AF − κ
))+ (bF (aF ) + κ) sF
(pRE − τF e+ AF − κ
))ψF (e) de− cF (sF (pF )) (1−ΨF (e))
−κ |M | − v(∫ ∞
0
(e− aH (e)) qH (e)ψH (e) de+
∫ e
0
(e− aF ) sF(pRE − τF (e− aF )− bF (aF )− κ
)ψF (e) de+ EF (e|e > e) sF (pF ) (1−ΨF (e))
)−FΨF (e) ,
with the constraint that Foreign markets clear: D(pF)
= sF(pF) (
1−ΨF(eF))−M . We
solve the corresponding Lagrange problem and let λ be the lagrange multiplier. The social
planner can choose [qH (e)]e≤e, [aH (e)]e≤e, pRE, τF , M and pF . We note that pUE is irrelevant
here: either M > 0 and it is equal to pF + κ or M ≤ 0 and the exact value does not matter
(because their are no uncertified exporting firms). We then derive first order conditions and
subsequently check the three cases of M (< 0, = 0 and > 0).
The first order conditions of this problem are:
wrt. qH(e): u′H (CH)− c′H (qH (e))− bH (aH (e))− v (e− aH (e)) = 0
wrt. aH(e) : b′H (aH (e)) = ν
so this is consistent with firms paying a tax τH = ν at home, undertaking optimal abatement
and facing a price pH = u′H (CH).
wrt. pRE :
∫ e
0
(pH − pRE + (τF − v) (e− aF )
)s′F(pRE − τF (e− aF )− bF (aF )− κ
)ψF (e) de = 0,
(55)
wrt. pF :D′F (pF ) pF−s′F (pF ) (pF + vEF (e|e > e)) (1−ΨF (e))+λ (s′F (pF ) (1−ΨF (e))−D′F (pF )) = 0,
87
FOC wrt to M :
λ = pH − κ if M > 0,
λ = pH + κ if M < 0,
or we have DF (pF ) = sF (pF ) (1−ΨF (e)) if M = 0.
FOC wrt τF leads to:∫ e
0
[ (pRE − pH + (ν − τF ) (e− aF )
)s′F(pRE − τF (e− aF )− bF (aF )− κ
)(− (e− aF ))
+daFdτF
(b′F (aF )− ν)(sF(pRE − τF (e− aF )− bF (aF )− κ
)) ]ψF (e) de(56)
= 0
FOC wrt eF
pHsF(pRE − τF
(eF − aF
)− bF (aF )− κ
)− cF
(sF(pRE − τF e+ AF − κ
))− (bF (aF ) + κ) aF
(pRE − τF e+ AF − κ
)+cF (sF (pF ))− v
((e− aF ) sF
(pRE − τF (e− aF )− bF (aF )− κ
)− esF (pF )
)− F − λaF (pF )
= 0.
Equations (55) and (56), give us that
pH = pRE and τF = τH = ν.
This leaves
pFD′F (pF )−s′F (pF ) (pF + vEF (e|e > e)) (1−ΨF (e))+λ (s′F (pF ) (1−ΨF (e))−D′F (pF )) = 0
λ = pH − κ if M > 0
λ = pH + κ if M < 0
πF (pH − τF (e− aF )− bF (aF )− κ)− F = (λ− ve) sF (pF )− cF (sF (pF )) .
88
Solving for these equations return:
t = vs′F (pF )EF (e|e > e) (1−ΨF (e))
s′F (pF ) (1−ΨF (e))−D′F (pF ),
and
πF (pH − τF (e− aF )− bF (aF )− κ)− F − f = πF (pF ) ,
with a certification tax
f = (t− ve) sF (pF ) .
Where if M > 0 (pooling equilibrium) we have λ = pH − κ and:
pF = pH − t− κ,
and if M < 0 (Home exports) we have λ = pH + κ and
pF = pH − t+ κ,
and if M = 0 (separating equilibrium) pF is defined by:
sF (pF ) (1−ΨF (e)) = DF (pF ) .
Which one of these equilibria is optimal depends on prices and transportation costs. This
requires:
pH − pF − κ < t < pH − pF + κ.
We compare welfare in equation (54) to that of a setting with no Home taxes on Foreign
exports, τF = t = 0, to find. We label this WLF (for laissaz-faire). We perform calculations
of a Taylor expansion along the lines of those in previous sections (details omitted) to find:
W ∗ −WLF = s′
F (p0 − κ)v2
2V ar(ε) + ΨF (e)AF sF (p0 − κ)− FΨF (e)
+D′
F (p0 − κ)t∗
2EF (e|e > e)− s′F (p0 − κ)
v
2EF (e)
(p∗H − vEF (e)− pLFH
),
where:
t∗ = vs′F (pF )EF (e|e > e) (1−ΨF (e))
s′F (pF ) (1−ΨF (e))−D′F (pF ),
is the optimal tax on uncertified exports. The welfare expression holds in all three cases,
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though strictly speaking t∗ only applies as a tax in the pooling equilibrium.
B Data Appendix
B.1 Domestic Application
Lease-level annual production in the Permian basin is pulled from the online portals of the
states of Texas and New Mexico. In 2019, daily production was about 4.9 million barrels per
day of oil and 18 million barrels of oil-equivalent of natural gas. There were approximately
75,000 leases actively producing, two thirds of which were in Texas. Texas wells were more
productive on average, yielding 75-80% of the basin’s oil and gas.
The combination of a steep decline in production over the life of the well, and heteroge-
neous skill and deposit resource (Covert (2018)) means that there is enormous heterogeneity
in productivity across wells. This is plotted as a kernel density on a natural log scale in
Figure B.1a.
While 20% of wells produce only gas, and 13% only oil, the overwhemling majority of
output comes from wells that produce a mix of the two. In Figure B.1b this is presented as a
kernel density, in which the oil share is weighted by well production. Most output comes from
wells producing about 80% oil, which in the absence of sufficient collection infrastructure,
may make flaring an attractive option for economically disposing of methane.
To estimate emissions rates, we use the joined emissions-production data from the Sup-
plementary Information to Robertson et al. (2020). The oil share and whether the site was
‘simple’ or ‘complex’ is also reported. Complex sites have some combination of storage tanks
and compressors, disproportionate sources of emissions, and was the basis of stratification
in the study: nearly 90% of simple sites had emissions below detectible limits (BDL), which
was 0.036 kg/hr, while 27% of complex sites were BDL. Among those sites with detectible
emissions, the mean [sd] at simple sites was 1.35 [0.99] while at complex sites it was 6.78
[8.23].
In line with the prior literature on methane leaks from the oil and gas sector, Robertson
et al. (2020) finds emissions follow a log-normal distribution within site type. When draw-
ing bootstrap samples from the original emissions data to reflect the wider composition of
strata,28 we regress log emissions on log daily production separately by simple versus complex
site type. For each site type we draw annual emissions values for each site in the production
data using the regression coefficients and estimated standard deviations as log-normal dis-
28Robertson et al. (2020) use satellite images of sites in New Mexico to determine that approximately onethird are complex.
90
tribution parameters. We then classify whether a site is simple or complex by drawing from
a linear probability model in which site type is the dependent variable, and the share oil
is the independent variable in the bootstrap sample. Applying the predicted site type and
the corresponding draw from the emissions distribution yields a site-level value of annual
methane emissions. We scale emissions by the ratio of 2.7 (0.5) million BOE to the sum
of emissions in the bootstrap sample, so that aggregate emissions match the estimates from
satellite data in Zhang et al. (2020). In this way we are using Robertson et al. (2020) to
simulate the across-site heterogeneity in emissions that add up to the basin-level estimates
of Zhang et al. (2020).
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Figure B.1: Permian Basin Lease Production and Oil Share
(a) Log(BOE per day)
0.0
5.1
.15
.2De
nsity
-5 0 5 10Log(BOE per day)
(b) Oil Share
01
23
4De
nsity
0 .2 .4 .6 .8 1Share of BOE
92
Table B.1: Parameters and Sources for the Steel Numerical Example
Parameter Value Source
OECD production 480.5 Mt World Steel (2020)Brazil consumption 22.1 Mt Instituo Aco Brazil (2021)Brazil net exports to OECD 8.5 Mt Instituo Aco Brazil (2021)Price for Brazil exports to OECD 489 USD USGSShare of EAF in Brazil 0.222 World Steel (2020)Share of EAF in OECD 0.454 World Steel (2020)EAF av. emission rate (Brazil) 0.46 t CO2 / t Hasanbeigi and Springer (2019)BOF av. emission rate (Brazil) 2.07 t CO2 / t HS 2019EAF av. emission rate (OECD) 0.66 t CO2 / t HS 2019BOF av. emission rate (OECD) 2.02 t CO2 / t HS 2019Minimal EAF rate 2/3*0.32 t CO2 / t 2/3 of France’s rate (HS 2019)Minimal BOF rate 2/3*1.46 t CO2 / t 2/3 of Canada’s rate (HS 2019)Maximal EAF rate 3/2*1.62 t CO2 / t 3/2 of India’s rate (HS 2019)Maximal EAF rate 3/2*2.80 t CO2 / t 3/2 of India’s rate (HS 2019)St. dev. of ln(prod.)in metal sector in Brazil 0.409 Schor (2004)Log prod. premium of EAF 0.074 Collard-Wexler and de Loecker (2015)Social cost of carbon 51 USD per ton US administrationTransport cost of carbon 50 USD per tonSlope of M.A.C curve, b′′(0) 490 USD per t CO2
2 Pinto et al.(2018) + own computationcertification cost (all economy) 23 M USD EPA (2002) + own computationOECD (US) demand elasticity -0.306 Fernandez (2018)Brazil demand elasticity -0.414 Fernandez (2018)Supply elasticity 3.5 EPA (2002)
94
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