What is the Optimal Offsets Discount under a Second-Best
Cap & Trade Policy?

Heather Klemick

Working Paper Series

Working Paper# 12-04
July, 2012

sta^ U.S. Environmental Protection Agency

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* A rc National Center for Environmental Economics


1200 Pennsylvania Avenue, NW (MC 1809)


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* Washington, DC 20460

What is the Optimal Offsets Discount under a Second-Best
Cap & Trade Policy?

Heather Klemick

NCEE Working Paper Series
Working Paper # 12-04
July, 2012


The views expressed in this paper are those of the author(s) and do not necessarily represent those
of the U.S. Environmental Protection Agency. In addition, although the research described in this
paper may have been funded entirely or in part by the U.S. Environmental Protection Agency, it
has not been subjected to the Agency's required peer and policy review. No official Agency
endorsement should be inferred.

What is the Optimal Offsets Discount under a Second-Best Cap & Trade Policy?

Heather Klemick, U.S. EPA National Center for Environmental Economics1
July 2012


Despite concerns about additionality, leakage, permanence, and verification, carbon
offsets have been proposed as a core component of recent cap-and-trade proposals in
order to contain costs, involve uncapped sectors in GHG reduction goals, and build
mitigation capacity in developing countries. Discounting the value of offsets relative to
GHG allowances (i.e., setting a trading ratio less than one) has been suggested as one
approach to protect the integrity of the cap. This paper presents a simple theoretical
model to derive the optimal trading ratio between offsets and allowances when coverage
of emissions by the cap-and-trade and offsets programs is incomplete. I discuss the
relationship between the trading ratio and the GHG cap and offsets baseline, which
jointly determine the stringency of the policy. While a discount for leakage is always
optimal, one notable result is that if "hot air" is introduced by setting either the baseline
cap or the cap too leniently, an extra discount is warranted.

Key words: offsets, additionality, leakage, baseline, cap and trade, second-best theory

Subject Area: Climate Change, Environmental Policy, Pollution Control Options &
Economics Incentives

JEL classification: D62, H23, Q54, Q58

1 Corresponding author contact information: klemick.heather@epa. gov. The views expressed in this paper
are those of the author and do not necessarily reflect the views or policies of the U.S. Environmental
Protection Agency. I would like to thank Ann Wolverton, as well as audience members participating in The
Role of Carbon Offsets in Climate Policy: Theory and Practice (Cornell University, 2011), the Southern
Economics Association 2011 Annual Meeting, and the NCEE Brownbag Series for their helpful comments
and suggestions. All errors are my own.



Carbon offsets are a critical component of cap-and-trade proposals because of their
potential to contain costs, involve uncapped sectors and countries in greenhouse gas
(GHG) reduction goals, and build mitigation capacity in developing countries. However,
many observers are skeptical that offset credits generate emission reductions equivalent
to those from capped sources due to concerns about additionality, leakage, permanence,
measurement, and verification (Lecoq and Ambrosi 2007; Olander 2008). Several design
features have been incorporated into recent U.S. climate legislative proposals to address
these concerns, including extending offset crediting to larger spatial scales, limiting the
number of allowable offsets, and discounting the value of offset credits relative to

This paper addresses the last of these proposals by constructing a simple
theoretical model to determine the socially optimal trading ratio between carbon offsets
and allowances in a second-best world. The scenario I consider is second-best because
coverage of all sectors and countries, whether through mandatory caps or voluntary
offsets, is incomplete, creating potential for leakage—a situation likely to persist even if
several major economies adopt comprehensive climate policies. I also examine the
interaction between the optimal trading ratio and the offsets crediting baseline, which, in
combination with the emissions cap, determines the stringency of the climate policy. The
baseline serves as a proxy for additionality, since offsets credited against a higher-than-
business-as-usual baseline include emissions reductions that would have occurred
without the policy and can be considered non-additional.


The issues of leakage and additionality have received considerable attention in the
literature on carbon offsets and other conservation programs. Empirical estimates of
leakage in the forestry and agriculture sectors range widely, from nil to over ninety
percent (Murray et al. 2004; Gan and McCarl 2007; Sohngen and Brown 2004; Wear and
Murray 2004). Estimates of leakage in the energy and industrial sectors are more modest,
and proposals such as output-based rebates for energy-intensive trade-exposed industries
are likely to be effective in minimizing it further (Aldy and Pizer 2009; Fischer and Fox
2009; U.S. Government 2009).

Lack of additionality of project-based offsets credited under the Clean
Development Mechanism and other voluntary opt-in programs has been raised as a
concern due to inflated emissions baselines and adverse selection (Lecoq and Ambrosi
2007; Montero 1999, 2000; Bushnell 2010). Montero (2000) examined use of the offsets
baseline as a policy tool to minimize adverse selection under uncertainty about firm-
specific counterfactual emissions. Crediting offsets against a country, sector, or other
highly aggregate baseline is another proposed solution to reduce leakage and adverse
selection reflected in recent U.S. climate legislation (Plantinga and Richards 2008;
Murray 2009). However, the literature has not seen extensive theoretical work evaluating
optimal offset design to address both additionality and leakage.

Some research has examined the optimal trading ratio between offsets and
allowances for GHGs and other pollutants to address uncertainty about offset
measurement and permanence, particularly from non-point sources (Shortle 1987; Malik
et al. 1993; Marshall 2010; Kurkalova 2005; Ranjan and Shortle 2007; Mignone et al.


2009). This paper does not address this potentially important rationale for discounting
offset credits.

Results from the model illustrate the importance of leakage and stringency when
setting the optimal trading ratio and baseline. When designed optimally, offset credits
should always be discounted in proportion to the emissions leakage to uncovered sectors
or regions, and the crediting baseline should be set jointly with the emissions cap to
achieve a level of mitigation such that marginal abatement costs equal the marginal social
damages from emissions. When the emissions cap and baseline are set exogenously at a
level where marginal benefits of mitigation exceed marginal costs, then an extra discount
is warranted to address the "hot air" in the system. Alternatively, if the trading ratio is set
exogenously and does not account for leakage but the policymaker can choose the offsets
baseline and/or emissions cap, the second-best allowance price is typically lower than the
first-best level.

In the next section, I describe the model and focus on a case in which the
policymaker can choose optimally the offsets trading ratio as well as the emissions cap
and/or baseline. I then turn to cases in which the policymaker is constrained to selecting
either the trading ratio or the GHG cap and/or baseline. Then I summarize the insights
and policy implications gleaned from the model. The last section concludes.


The model starts with a three-sector or three-country economy in which each sector or
country emits greenhouse gases (GHGs) when producing a composite commodity. The
model is generalizable to other pollutants, but GHGs provide a compelling case both


because of their policy significance and because each unit of emissions causes the same
global damages regardless of the sector or location, making leakage particularly
pertinent. I focus on aggregate sector- or country-level emissions to be consistent with
the international offsets provisions of recent climate legislative proposals such as the
2009 American Clean Energy and Security (ACES) Act. It might also be more
reasonable to assume the regulator has better information about aggregate than firm-
specific marginal abatement costs to sidestep the adverse selection problem.

A single representative consumer derives utility from the consumption of three
goods denoted by xl3x2, andx3, with respective emissions intensities gl,g2,aridg3. Total

GHG emissions are given by G = gxxx + g2x2 + g3x3. Each good's production cost is


represented by a convex cost function, c (.r). GHG emissions harm social welfare
according to a damage function, D(G). Overall social welfare can be represented as

W = u(xl,x2,x3)-cl(xl)-c2(x2)-c3(x3)-D(glxl+g2x2+g3x3).

For the sake of simplicity, I do not explicitly include abatement costs in the
model. Rather, sectors can reduce emissions by decreasing production of their respective
goods. The cost to society of reducing emissions is therefore the foregone utility of
consumption (net of production costs), which is a reasonable way of conceptualizing
abatement costs at an aggregate societal level.

2	The model is flexible enough to accommodate alternative interpretations of the three sectors, e.g., they
could represent three individual firms instead of three aggregate production units.

3	A single aggregate cost function could instead be used to represent a case in which all three sectors use a
common intermediate input that generates emissions (such as a fossil fuel). In this situation, leakage to non-
regulated firms is even more likely because input costs for non-regulated firms decline when regulated
firms reduce their consumption. The results and intuition that follow are also applicable to the common-
polluting-input case.


In this paper, the policymaker is armed with a limited toolkit to address the social
harm caused by GHG emissions. Specifically, she can implement a cap-and-trade
program with mandatory coverage of emissions from sector 1, while the other sectors can
voluntarily opt to earn offset credits from greenhouse gas reduction below a baseline.
Assume for now that sector 2 opts into the offsets program, while sector 3 does not. (I
return to the issue of endogeneity of participation in the offsets program later.) In other
words, production in sector 1 is subject to an emissions constraint G = gxxx - a(b - g2x2),

where G is the cap, b is the emissions baseline used for offset crediting, and a is the
trading ratio between offset credits and GHG allowances, which is expected to be
bounded by zero and one.

Assuming that firm profits from all three sectors accrue to a single representative
consumer, a single private-sector maximization problem can be written as

u(xi, Vs)" ci Oi)"c2(x2)"c3(x3) s t. G = g1x1-a(b-g2x2).

The first order conditions for this maximization problem are

ul-cly-glX = 0	(1)

u2 - c2g2aA = 0	(2)

u3-c3' = 0	(3)

G-glxl+a(b-g2x2) = 0	(4)

Here, X is the shadow value of loosening the constraint—in other words, the marginal
utility associated with consuming goods that result in an additional unit of emissions. It
also, not coincidentally, represents the allowance price for GHG emissions in sector 1,
serving as a signal of the social cost of emission reductions to the capped sector.


The first condition states that good 1 is produced at a level to equate the marginal
utility of consumption with the marginal production cost plus the marginal value of the
GHG allowances that must be surrendered, which equals the GHG allowance price times
the good's emissions intensity. Likewise, the second condition shows that production of
good two is determined by equating the marginal utility of consumption with the
marginal production cost plus the marginal opportunity cost of the offset credits foregone
by producing good 2. This opportunity cost is given by the allowance price times the
emissions intensity, weighted by the trading ratio, a . While offsets provide sector 2 with
an incentive to reduce GHG emissions, the incentive is less powerful than in sector 1 if
the trading ratio is less than one. The third condition indicates that production in sector
three is set by equating the marginal utility of consumption with the marginal production
cost, absent any incentive to curtail production to mitigate GHGs. The final condition
simply reiterates the emissions constraint.

Under a first-best climate policy, an emissions cap or tax would be set to equate
the marginal utility of consumption of each good minus its marginal cost and divided by
its emissions intensity across all three sectors, such that

U\ ~ C1 _ U2 ~ C2 _ ^3 ~ C3

#2	ft

However, the first-order conditions show that under the incentives created by the cap-
and-trade policy, the private sector instead equates

u,-c,' Wt —Cr.'	, „

-!	- = —	— = u3-c3+A..	(6)

ft g2a

This equality is the crux of the second-best world confronted by the policymaker when
determining how to optimally set the parameters of the GHG constraint.

Case 1: Policymaker can choose trading ratio and cap/baseline

The policymaker's objective is to maximize social welfare, W, with respect to the policy
tools at hand—G, a, and b. To highlight the impact of these parameters on the social
welfare function, I substitute out x1 using the constraint imposed by the cap-and-trade
program (equation (4)), rewriting the objective function as


W = u

G + a(b-g2x2)


^ fG + a(b-g2x2




-D(G + ab + g2 (1 - a)x2 + g3x3)

The first-order conditions of the policymaker's problem are

dG gl 1 1	dG

+^[={u2-c2-g2D}) = Q

(u2 -c2(Wj -c,')-g2( 1 -a)D'


dW (b-g2x2)






02 - c2') -(Mi - Cj') - g2 (1 - a)D'



+ -7±(U3-C3-g3Df) = 0

db g1


(u2 - c2') - (//, - Cj') - g2 0 - a)D'



(u3-c3'-g3D') = 0

Each of these conditions reflects the direct effect of the respective parameter on
welfare, in addition to its indirect impact through its influence on the production of
goods x2 and x3. Using conditions (1) - (3) from the private sector problem allows these

equations to be simplified as follows.



= (b ~ g2x2)(l-D') -^g2(l-a)D<-^-g D' = 0
da	da	da



These simplified equations show that the direct effect of loosening the GHG
constraint is given by the allowance value less the marginal social damages from GHG
emissions. The direct effect of increasing the trading ratio is equal to the allowance price
minus marginal climate damages times the total number of offsets. The impact of adding
hot air to the program by increasing the baseline is equal to the allowance price minus
marginal climate damages times the trading ratio. The final two terms in each of these
three equations capture the indirect effects of changing these parameters on production
covered under the offsets program and that outside of the cap.

This gives three equations with three unknowns: A, a, and A . While there is no

closed-form solution forG or b without imposing additional structure on the utility, cost,
and climate damage functions, X can be viewed as a proxy for the stringency of the
constraint, and the policymaker can implicitly choose the carbon price in lieu of the cap
and baseline. Using equations (10) and (11) yields the following solutions for A and a :

g2dx 2

These results show that the optimal allowance price is simply the marginal
damage from each unit of emissions. The policymaker can set a cap that yields this
allowance price, given sufficient knowledge of preferences, production technology, and


A* = D



the climate damage function. For example, EPA's (2010) $20 core estimate of allowance
prices in 2020 under the ACES Act is the same order of magnitude as the U.S.
government's central estimate of the social cost of carbon (SCC) used in regulatory
analysis for the same year of $25 (Interagency Working Group 2010).4

The optimal trading ratio is equal to one plus the change in emissions in sector 3
in response to a one-unit increase in emissions in sector 2, which is expected to be
negative if the two composite goods are substitutes. In other words, the second term
represents GHG emissions leakage from sector 2 to sector 3. Perhaps unsurprisingly, the
optimal discount on offsets relative to allowances is exactly equal to the leakage to the
uncovered sector anticipated to result from each offset credit. Estimates of leakage in
response to a climate or conservation policy range widely and vary by sector and scale of
the project, suggesting that it may be optimal to vary the trading ratio by offset type.

It might initially seem counterintuitive that the leakage rate from sector 1 to sector

3	does not appear directly in the optimal trading ratio. This absence is deceptive because
of the fixed relationship between x1 and x2 under a cap-and-trade program. The effect of

Xj on emissions in sector 3 is already reflected implicitly; indeed, equation (14) is

equivalent to setting the trading ratio such X)\dLtgldxl = ~(g2dx2 +g3dx3), i.e., a one unit

increase in emissions from sector 1 is exactly offset by a one unit decrease in emissions
from both sectors 2 and 3.

While the optimal trading ratio is expected to fall between zero and one in most
circumstances, it is theoretically possible for it to fall outside of these bounds. For
instance, if goods 2 and 3 are perfect substitutes and good 3 is more emissions intensive,

4	Values reported in 2005 dollars.


then leakage rates could exceed 100%, yielding a negative trading ratio (i.e., a penalty for
generating offsets). If the two goods are strong complements, emissions from good 3
could actually decline in response to offsets in sector 2, prompting a trading ratio greater
than one. These two extreme cases are outside the realm of most policy discussions.

While the first-order equations produce straightforward results for X and a, there
is no unique solution for b ; rather, b and G must be set jointly to achieve the optimal
allowance price. (In fact, it is possible to show that equations (10) and (12) are exactly
equivalent.) Raising eitherG or b would lower allowance prices, and there are infinitely
many combinations of the two parameters that could achieve an allowance price equal to
marginal damages. The division of the GHG emission reduction target between
G and b is ultimately a distributional question rather than one of efficiency. If offsets are
not additional—in other words, if they are credited against an overly high baseline—the
best response is to tighten the cap on the regulated sector just enough to offset the hot air
introduced into the system, as noted by Montero (2000). If the emissions cap is
considered fixed, then setting an efficient baseline is a matter of selecting a target that
achieves the right overall level of abatement based on climate damages and abatement
costs, rather than a question of accurately predicting business-as-usual emissions in the
offset sector.5 If this outcome is infeasible, then the policymaker's options are more
limited, as discussed in the next section.

Case 2: Policymaker can only choose trading ratio or cap/ baseline

5 This point is similar to Bushnell's (2010) argument that it can still be advantageous to allow offsets from
sectors following a "surprisingly clean development path" whose business-as-usual emissions are lower
than originally estimated, though my results suggest that the total abatement target should be more
ambitious in such a case. Of course, projected business-as-usual emissions can provide useful information
to policymakers about likely abatement costs.


The equalities above represent the optimal allowance price (and implicitly, the optimal
GHG cap and offsets baseline combination) and trading ratio when the policymaker can
choose both of these parameters. Different solutions to these parameters could result
when the policymaker can only choose one or the other, but not both. This situation
might arise, for example, due to political constraints, or because information is
incomplete when one parameter is set but improves before the other parameter is chosen.

First, I assume that the policymaker faces an exogenous trading ratio and selects
an allowance price (by way of setting a cap and offsets baseline) taking this ratio as a

given. Solving equation (10) for X, the second-best allowance price, gives

X = D<

dx2 dx3

\ + g2(\-a)^ + g,^

2 dG 3dG


This expression indicates that the second-best allowance price is determined by
the marginal damages from emissions, scaled by any change in emissions that occurs in
sectors 2 and 3 as a result of altering the cap on regulated firms. In other words, when
the trading ratio does not appropriately discount the value of offsets for leakage, the
remaining leakage should be reflected in the allowance price. To further illustrate how
the optimal allowance price changes in response to the trading ratio, this expression can
be differentiated with respect to a holding marginal climate damages constant for the
sake of tractability, yielding

dX jj,g2dx2( i /-I \ dx2 S^x2 dx

da dG

-1 + (1 -a)^+63 3 3 .	(16)

da g2dx2 da

This equation shows that the effect of the trading ratio on the optimal allowance
price is ambiguous. The sign of this expression depends on the degree of substitutability


between the three sectors, their relative GHG intensities, and the trading ratio itself.6
Since many combinations could result, I focus on a scenario that assumes some degree of
substitutability between all three sectors, as is likely if they represent large, aggregate
world regions or sectors of the economy. In this case, the term in brackets is expected to
be negative for all a < 1, but the sign of the first term—the effect of the regulated sector
cap on sector 2's emissions—remains ambiguous. This is because lowering the
allowance price has a direct effect of encouraging production in sector 2, but if offsets are
credited at a lower rate than regulated emissions, then there is also an indirect leakage
effect in which production can shift from sector 2 to sector 1. Equations (15) and (16)
together indicate that as a approaches zero—an extreme case in which offsets are not
allowed into the cap and trade program—the allowance price should decline to account
for leakage to both sectors 2 and 3. In order to achieve this lower allowance price, the
regulator would need to set a looser cap.

In the opposite case of a = 1, when offset credits are treated as fully equivalent to
capped-sector emission reductions, leakage between sectors 1 and 2 is not an issue, but
the allowance price should still fall below the marginal climate damage rate to account
for leakage to sector 3. This is not to say that the allowance price should drop
dramatically; rather, it should fall just enough to counteract the remaining leakage in the
overall economy. Instead of loosening the cap as in the previous case, the regulator
might have to tighten the cap relative to the optimum to achieve this outcome. Loosening
the cap could cause allowance prices to plummet as overvalued offset credits flood the
market. Because the offsets sector receives credit for emission reductions that are partly
eroded by leakage, a tighter cap is needed to counterbalance the excess credits. This

6 The Appendix provides relevant comparative statics derivations.


tighter cap is not inconsistent with a slight fall in the allowance price because the offsets
sector generates more emission reductions as a result of the higher offset price, allowing


the regulated sector to do less. The overall relationship between the trading ratio and the
second-best allowance price follows an inverted-U shape passing through/)', the optimal
allowance price.8

The relationship between the allowance price and the trading ratio suggests that
proposals for a "price collar" (Burtraw et al. 2009) could be advantageous to minimize
the impact of an improperly set offsets trading ratio. A price collar, which releases extra
allowances if prices rise to a predetermined ceiling and withholds allowances if prices
sink to a certain price floor, would prop up allowance prices if excess allowances flood
the market due to an overly high trading ratio, and conversely, would contain allowance
prices if offsets are too limited. Equation (15) suggests an allowance price floor equal to
marginal climate damages discounted by the leakage rate under the most extreme
scenario of no participation by non-regulated sectors in offsets markets, while the price
ceiling should not rise much higher than the marginal climate damage rate. The general
results echo the findings of other research noting the complementarity between offsets
and a price collar (Fell et al. 2010).

Turning to the case when the policymaker can't choose the emissions cap/offsets
baseline combination but can choose the trading ratio, and again holding marginal

7	This situation illustrates the moderating effect that offsets have on allowance prices. With a tighter cap, an
influx of offsets can sharply curtail the potential rise in allowance prices, as was shown in EPA's (2009,
2010) analyses of recent climate bills.

8	Note that the preceding discussion assumes constant marginal climate damages. When marginal climate
damages are increasing in emissions (D" > 0), there is an additional effect: Because total emissions fall
with a lower trading ratio (rise with a higher trading ratio), marginal climate damages are lower (higher),
justifying a lower (higher) allowance price. Thus, the function will be flatter when this effect is


climate damages constant, the second-best trading ratio is given by solving equation (11)
for a:

^ _ | g3^3 ! 0 - A/P %b -g2x2)	(17)

g2dx 2	g2dx2/da

The trading ratio is again equal to one discounted by the emissions leakage that occurs
when production in sector 2 decreases to generate offsets, but with an additional third
term that depends on the discrepancy between the allowance price and marginal
emissions damages, the offsets baseline, and the effect of the trading ratio on emissions
from sector 2.

When the allowance price is set lower than optimally, and sector 2 generates a
positive quantity of offsets—the situation of greatest policy relevance—this term is
expected to be negative, indicating that an additional discount is warranted when the
baseline is set too high.9 This discount increases in size the bigger the gap between the
allowance price and marginal climate damages and the higher the offsets baseline. The
lower trading ratio helps to compensate for the introduction of "hot air" because each
emission reduction foregone in the regulated sector is countered by more than one unit of
reductions in sector 2 (even after accounting for leakage). Thus, the trading ratio

9 Under these conditions, the sign of this third term depends on (^x- , which is likely negative because


increasing the trading ratio should increase incentives for abatement in sector 2. However, there is an
indirect positive effect since raising the trading ratio effectively relaxes the total GHG abatement target,
hence lowering the allowance price. In the unlikely event that the lower allowance price more than offsets
the higher trading ratio, it is theoretically possible that the net effect could be to increase sector 2
emissions, reversing the sign of the expression.


provides the policymaker with another lever to reduce emissions when the baseline is too
lax, functioning as an "addititionality discount."10

The opposite result also holds: If the allowance price is set higher than marginal
climate damages but sector 2 still generates a positive quantity of offsets, a trading ratio
higher than a * is optimal. In this case, it is desirable to introduce hot air into the
program to reduce abatement costs because the target is too stringent. It is also worth
noting that if the allowance price is higher than marginal climate damages but sector 2's
emissions exceed the baseline (implying that the sector would have to purchase emission
credits from the regulated sector), the second-best trading ratio again drops below a *.
The lower trading ratio reduces the net tax on emissions in sector 2, helping to
compensate for the excessively high carbon price. This unlikely situation could only
occur under a non-voluntary offsets program.

This additionality discount is entirely additive to the leakage discount. Indeed,
discounting offsets using a trading ratio remains an appropriate second-best policy
instrument even when leakage is eliminated through complete coverage of all non-
regulated emissions in an offsets program, as long as the baseline is not sufficiently
stringent. Discounting is still effective as a way to "tighten" the cap and increase the
actual level of abatement.

It is worth highlighting that this discount is warranted regardless of the source of
the hot air—whether from non-additionality of offsets or from a regulated sector
emissions cap that is too loose. Thus, it could be misleading to refer to the reduction in
the trading ratio as an additionality discount. This is a somewhat counterintuitive result

10 The support for a higher trading ratio is further reinforced when marginal climate damages are increasing
in emissions; when the cap and baseline are too loose, marginal climate damages are increasing, justifying
more abatement.


that suggests that even if offsets are entirely "real"—meaning that they are additional to
business-as-usual emission reductions and cause no leakage—it could still be optimal to
discount them relative to allowances if the covered sector cap is too loose. The offsets
discount essentially allows the regulator to price discriminate as a means to achieve more
aggressive emission reductions. This result provides a justification for a blanket discount
on offsets by observers who consider the climate policy targets included in recent U.S.
legislation to be insufficiently stringent.

Discussion: The Endogenous Opt-In Decision

The discussion thus far has assumed exogenous participation by sector 2 in the offsets
regime and non-participation by sector 3. The focus in the existing offsets literature on
adverse selection (e.g., Montero 2000, Bushnell 2010) highlights the importance of the
baseline in the decision of a firm or entity to participate in a voluntary offsets program.
This paper attempts to sidestep the issue of adverse selection at the individual firm level
by focusing on sectoral or national crediting programs, which have been emphasized in
recent U.S. climate legislation. Aggregate crediting programs have become still more
plausible as major developing economies have proposed climate initiatives involving
regulatory and market-based programs at the sectoral level, such as China's pilot cap-
and-trade scheme for energy emissions (Reuters 2011).

However, it is still worth considering how the analysis might change if the
endogeneity of the participation decision is considered. For example, Montero (2000)
found that when regulators face a fixed pollution cap and uncertainty about firm-level
abatement costs, the second-best offsets baseline should be set lower than expected
emissions to extract additional information. That analysis did not consider the trading


ratio as an additional policy lever to improve the efficiency of a second-best cap-and-
trade program with offsets.

Sectors or countries will only opt in to an offsets program if offset net revenues
more than compensate them for the decreased profits from producing emissions-intensive
goods. The baseline and trading ratio are both important elements of this decision.
Because the offsets baseline can be altered with no loss of efficiency as long as the
emissions cap can be shifted in the opposite direction, it follows that the efficient
baseline, while not unique, must be large enough to incentivize the maximum feasible
level of participation in the offsets regime. (There could be some sectors for which
participation is infeasible regardless of the payoff.) As the literature on leakage suggests,
this level is not necessarily equal to pre-climate-policy business-as-usual emissions, since
reduced production in the regulated sectors could raise the opportunity cost of foregone
production among potential offsets participants.

If it is not politically possible to set the emissions cap low enough to
counterbalance a baseline sufficiently high to maximize participation, then the overall
level of abatement will be less than socially optimal. In either case, the optimal trading
ratio should not change in response to the endogeneity of the participation decision; it
should reflect leakage if some sectors remain uncovered, as well as any hot air introduced
if the baseline and cap are jointly set too high.

Policy Implications

The preceding cases illustrate the roles of leakage and the stringency of the overall policy
in setting the optimal trading ratio between offsets and allowances under cap-and-trade.


The model confirms the intuitive result that offsets should always be discounted in
proportion to any emissions leakage to the uncapped sector. This result suggests that
different trading ratios could be warranted to account for varying leakage rates across
different offset types, and that the trading ratio should rise closer to unity as more
participants opt into the system. Thus, an across-the-board discount like that applied to
post-2017 international offsets in the ACES Act might not be the ideal way to implement
a trading ratio.

The results also emphasize the linked nature of the policymaker's choices about
abatement targets and offset trading ratios. Although the hot-air and leakage discounts
are additive, they can act as backstops for each other. If hot air is introduced by either an
inflated baseline or cap, offsets should be discounted as a correction against the lower
total level of mitigation that would otherwise result. Conversely, an offsets baseline that
increases the stringency of the GHG abatement target beyond the level where the
allowance price equals marginal damages can induce additional leakage, reducing the
marginal effectiveness of the policy. The trading ratio can be a useful lever to effectively
relax the policy in this case.

The scenario in which policymakers face an exogenous trading ratio that deviates
from the optimal solution but exercise some control over the emissions target is perhaps
less likely to happen in practice. In the event that it does, then it becomes desirable to
reduce the stringency of the overall policy to account for leakage. It may also make
sense to support a price collar that moderates any adverse impacts from an improperly set
trading ratio.


It is also worth emphasizing the caveat that these results do not account for any
further adjustments to the trading ratio that might be justified to address the risks posed
by measurement uncertainty and permanence. This model suggests that if the baseline is
set correctly and all uncovered sectors participate in the offsets program, mitigating any
leakage concerns, then the trading ratio should be equal to one; however, a discount could
still be warranted to address the measurement and permanence issues.


This paper examines optimal and second-best offset design to account for the joint effects
of leakage and hot air. While previous research has examined the issue of setting the
baseline optimally in combination with a regulated sector cap, this paper incorporates the
trading ratio between offsets and regulated emission reductions as an additional policy
lever, in accordance with recent cap-and-trade legislative proposals. It demonstrates that
adjusting the trading ratio between emission credits and offsets is indeed an appropriate
second-best tool to address not only leakage occurring when coverage of emitting sectors
is incomplete, but also hot air introduced when the combined abatement target is set sub-
optimally—whether or not this hot air is introduced by non-additional offsets. Of course,
complete coverage of all emitting sectors and countries under a mandatory cap-and-trade
or voluntary offsets program with targets that equate the marginal benefits and costs of
abatement remains the first-best solution to achieving climate goals efficiently rather than
tweaking the allowance price and trading ratio.

The results also highlight the importance of accurate projections of leakage,
marginal climate damages, and abatement costs for setting offset trading ratios and


baselines. Empirical evidence to date suggests that leakage can vary widely depending
on the offset type and scale of crediting, but existing estimates, particularly in the land-
use and forestry sectors, may not be sufficiently robust to support their use as inputs into
key policy parameters like the allowance price and trading ratio. Furthermore, estimates
of the marginal social damages from GHG emissions differ by more than an order of
magnitude (National Research Council 2009). Thus, more research is needed on the
likely distribution of these parameters to support efficient offset design.



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Appendix: Comparative Statics Derivations

The private sector problem can be rewritten using the constraint to concentrate the
problem in terms of sectors 2 and 3 as

max„ „ u

G + g2a(b-x 2)

^ rG + g2a(b-x 2)^




The first-order conditions for this problem are



-(u1-c1') + u2-c2' = 0

U\ _ u2 c2

w3 - c3' = 0



and second-order conditions are given by

U22 =

f V

\ Si j

2 g2a

U\\	un + u22


f V

\ 81 j

cx"-c2 "<0

C/33 = w33 - c3" < 0

f V


v 81 j


U\2 +w22

f V


\ 8\ j






u23		—w,


u33 — c3


These conditions yield the following comparative statics for the effect of lowering the
regulated sector cap, G, on production in sectors 2 and 3.




82a ,

ul2	)




u23 —u





W33 — c3


\ 1


(i/j j Cj ) (u33 c3 ) + 2




The sign of —=ir is ambiguous. When goods produced in all three sectors are substitutes,

it becomes positive as a moves close to 1 and the GHG intensity of sector 2 increases
relative to sector 1. The sign becomes negative if a = 0.





f f y

\ Si j



U\2 +U 22

f V

\ Si j




K S\ J


S 2a

u23	u









( n Sia \

^13^22 ^2	^12)


f Si®}

K Si J

^23(^11 ^1	"12)

Again assuming that goods produced in all three sectors are substitutes, this expression is
negative, indicating that when the cap is loosened, production falls in the uncovered

Turning to the effect of the trading ratio on production in sector 2, the comparative static
result is given by







On -Cj")



u23		—u,





w33 — c3

sign\^\ = sign\^-(ux -cx ')033 -c3")-—(w33 -c3")

I da





un-^^(un -Cj")





The net effect of the trading ratio on production in the offsets sector is determined by two
countervailing forces. A direct negative effect means that incentives for abatement
increase with the trading ratio. However, increasing the value of offsets to firms in sector
2 also loosens the overall abatement target, which could reduce pressure to limit
production. In practice, the first effect is expected to dominate, leading to an overall
negative sign.