Balancing the Carbon Budget for Oil: the Distributive
Effects of Alternative Policies
Stephen W. Salant∗
Carolyn Fischer
December 6, 2016
Revised
Abstract
Keeping temperature change below 2 degrees C will require leaving large reserves
of fossil fuels unextracted. We assess alternative policies to achieve this goal in a world
divided into two regions, one regulated with emissions pricing and one unregulated.
Global emissions can be reduced by any one of three policies: (1) increasing the size of
the regulated coalition, (2) raising the tax (or tightening the cap) within the regulated
coalition, and (3) accelerating cost reductions of a clean substitute for fossil fuels. With
notable exceptions, tightening control of regulated consumers results in a contemporaneous expansion of emissions by unregulated consumers (spatial leakage) and price
reductions by foresighted oil extractors (intertemporal leakage). We ask how combinations of the three policies achieving the same cumulative CO2 emissions (the “carbon
budget”) affect the discounted surplus of regulated and unregulated consumers and the
wealth of the extractors. We first evaluate these policy alternatives in a two-pool, theoretical model with high and low-cost extractors and, once tradeoffs are illuminated, in
a calibrated oil market model. In both, a clean backstop becomes cheap enough that
regulated consumers and eventually unregulated consumers switch to it.
∗
Carolyn Fischer is Senior Fellow at Resources for the Future (1616 P Street NW, Washington, DC
20036; [email protected]) and Fellow of the CESifo Research Network. Fischer is grateful for the support of the
European Community’s Marie Sklodowska-Curie International Incoming Fellowship, “STRATECHPOL—
Strategic Clean Technology Policies for Climate Change,” financed under the EC Grant Agreement PIIFGA-2013-623783, and for the hospitality of the Fondazione Eni Enrico Mattei (FEEM). Stephen W. Salant is
Professor Emeritus in the Department of Economics, University of Michigan, Research Professor at University
of Maryland (AREC) and Visiting Scholar at Resources for the Future (RFF); [email protected]. Support
from the Mistra Foundation ENTWINED program is gratefully acknowledged. We would also like to thank
participants at the NBER Summer Institute on Environmental and Energy Economics (July 23, 2013), the
CESifo workshop ”The Green Paradox and Fossil Fuel Markets” (July 2–3, 2015), the Tinbergen Institute
conference ”Combating Climate Change” (April 21–22, 2016) and seminar participants at the Environmental
Defense Fund, European University Institute, Montreal Natural Resources and Environmental Workshop,
Paris Environmental Economics Seminar (PEES), University of Rome—Tor Vergata, University of Rennes,
and University of California at Santa Cruz, University of Guelph, and Virginia Tech for helpful comments
on an earlier draft.
1
1
Introduction
According to the Intergovernmental Panel on Climate Change (IPCC), to hold global warming to 2◦ C with a 50% probability, cumulative emissions cannot exceed 1,000 gigatons of
carbon, or 3,670 tons of carbon dioxide (CO2 ), starting from the industrial revolution. However, more than half that “carbon budget” has already been emitted, and at current rates,
we will reach the limit in the next 30 years (IPCC 2013, Meinshausen et al. 2009). Estimates
of cumulative emissions targets for this century are roughly 1,100 gigatons of CO2 (Meinshausen et al. 2009), but many times this amount is embodied in available fossil fuel reserves
(McGlade and Ekins 2015). Thus, to keep emissions below this limit, dramatic changes must
occur in energy consumption over the next generation and, in the absence of geoengineering
or carbon sequestration, large shares of fossil resources must be left unexploited.
The Paris Agreement of the United Nations Framework Convention on Climate Change
(UNFCCC) Conference of the Parties in 2015 reaffirms the goal to hold the increase in
the global average temperature to well below 2◦ C above pre-industrial levels. However,
emissions mitigation efforts are not coordinated, but rather are pledged unilaterally as intended nationally determined contributions (INDCs), of which only a fraction intend to use
emissions pricing as a central policy mechanism.1 The result is divergent levels of ambition
among the parties, arising in large part due to the principle of “common but differentiated
responsibilities” (CDBR) embedded in the UNFCCC.2
A challenge for reducing GHGs in this setting is that while regulations are subglobal,
fossil fuel markets and oil markets in particular are global. If one region reduces its demand
for fossil fuels by imposing emissions prices, then the inward shift in global demand would
depress the world price of oil, which in turn would stimulate oil demand in parts of the
world where carbon emissions are not regulated. Thus, differential carbon pricing results
in “carbon leakage .” footnoteVarious studies using static computable general equilibrium
models (CGE) models have shown the sensitivity of carbon leakage estimates to fossil fuel
supply elasticities (e.g., Burniaux and Martins 2000; Mattoo et al. 2009).
At the same time, fossil fuels and oil in particular are also scarce resources that are
exploited over time. If climate policies make selling fossil fuels in the future less attractive
1
As of 2016, roughly 13% of global emissions are covered by carbon pricing mechanisms, and the implementation of the proposed Chinese emissions trading system could double this amount. A majority of
signatories to the Paris Agreement, representing 56% of global emissions, are planning or considering some
form of carbon pricing. (World Bank 2016)
2
In the Paris Agreement, this principle was reflected in Article 4.4: Developed country Parties should
continue taking the lead by undertaking economy-wide absolute emission reduction targets. Developing
country Parties should continue enhancing their mitigation efforts, and are encouraged to move over time
towards economy-wide emission reduction or limitation targets in the light of different national circumstances.
2
than current sales, suppliers may prefer to extract more in the present, offsetting future
emissions reductions. Indeed, current oil prices are well above costs for some of the world’s
largest reserves, leaving ample room for price reductions if consumers begin switching from
heavily taxed fossil fuels to clean, increasingly affordable substitutes. Sinn (2008) popularized
this notion of future climate policies accelerating current emissions as the “green paradox.”
This paper assesses climate policy options in a world with both spatial and intertemporal
leakage. We use a transparent, dynamic, partial-equilibrium model inspired by the global
market for liquid fuels. We divide the world into a regulated region and an unregulated
region, as is common in the static models of spatial leakage. Suppliers are price-takers and
choose wealth-maximizing extraction paths. We consider three exogenous policies to reduce
global emissions: (1) raising the emissions tax or tightening the cap imposed on consumers
in the regulated region; (2) expanding (through international negotiation) the size of the regulated region; and (3) speeding up the cost-reducing technical change in the clean backstop.
Although any of these policies can reduce cumulative emissions, they affect extractor wealth
and consumer welfare differentially.3 We show where extractors and regulated consumers
alike would strictly prefer some ways of achieving a given cumulative emissions reduction to
other ways .
To fix intuition at the outset, we first assume that there are only two grades of fossil fuel
in the world, low cost and high cost. We focus on regimes in which the high-cost pool is
incompletely exhausted and explore the trade-offs among our three policies. Subsequently,
we use a calibrated oil market model to resolve theoretical ambiguities and explore the
trade-offs quantitatively.
As we show, given a rate of technological change, an emissions reduction achieved by
imposing a low carbon price on consumers in every country is superior to the same emissions reduction achieved by imposing a carbon price on a subset of the world’s consumers.
Extractors would prefer the global carbon price because their Hotelling rents are higher.
Consumers who remain regulated gain since the lower carbon price outweighs these higher
Hotelling rents, resulting in their paying initially lower prices. Emissions revenues increase
if the given emissions reduction is achieved with a global carbon price. And although consumers who become taxed are injured, their losses can be fully compensated out of the
additional revenue collected from a global price on carbon. Indeed, the allocations in the
competitive equilibrium in this case duplicate what a social planner would do if constrained
to achieve the same reduction in aggregate carbon emissions and constrained to the same
speed of technical change. The situation in which regulated consumers pay a higher price
3
Of course, the tax policy has its limits. Once a tax is so high that regulated consumers switch at the
first instant to the clean backstop, a further increase in the tax does not affect prices or carbon emissions.
3
for the same good as unregulated consumers pay at the same time inevitably involves a loss
in social surplus due to the distributive inefficiency.
However, we also find an important role for clean technology innovation in climate policy,
both from a distributive and feasibility perspective. While global carbon pricing is costeffective, several stakeholder types benefit more from technology policies that encourage
a faster transition to clean fuel sources. Extractor rents are highest with a policy that
emphasizes bringing down the global cost of a future oil substitute, rather than taxing
emissions. Consumers also benefit from lower energy prices over time, although taxpayers
face a higher burden from supporting the cost reductions. In fact, if the coalition committed
to policies to meet a global carbon budget is less than perfect, second-best global policy
involves more inducements for technologies—and lower carbon prices—than the regulating
coalition has incentives to provide. From a global perspective, the technology policy is better
able to address distortions created by emissions leakage, while from a coalition perspective,
the emissions price is more effective at capturing rent away from extractors and benefits
taxpayers. In any case, dramatic reductions in global emissions seem impossible in the
absence of widespread participation in carbon regulation and rapid development of clean
energy alternatives.
Our approach draws on the green paradox literature.4 Early investigations of this type
of “intertemporal leakage” (e.g., Strand 2007; Grafton et al. 2012; Chakravorty et al. 2011)
relied on assumptions implying that all oil would ultimately be extracted . As a result, the
carbon content of the entire underground stock was predicted to end up ultimately in the
atmosphere; the primary focus was, therefore, on the time path of emissions. In models
where these cost reductions do not reduce cumulative carbon emissions, the only effect of
technology policy is to accelerate emissions, leading to a green paradox.
A second generation of intertemporal models took account of heterogeneity in the extraction costs of different fossil fuel reserves.5 Once the assumption that reserves were homogeneous was relaxed, climate policies such as emissions prices were predicted to be effective in
limiting cumulative carbon emissions. Gerlagh (2011), Van der Ploeg and Withagen (2012),
and Fischer and Salant (2014) showed that climate policies can render high-cost pools of fossil fuels too expensive to utilize. This second generation of intertemporal models implicitly
assumed, however, that policies to mitigate emissions would be applied uniformly around
the world, thus eliminating the possibility of spatial leakage.
4
For broader reviews of the green paradox literature, see Jensen et al. (2015) and Van der Ploeg and
Withagen (2015).
5
Chakravorty et al. (1997) formulated the earliest second-generation model. They looked at the potential
role of future cost reductions in solar energy on emissions over time in a global model with multiple fossil
resources of multiple grades with multiple demands and endogenous substitution.
4
Hoel (2011) identified a key shortcoming of these first and second-generation models:
“In almost all of the body of literature referred to above, the economy analyzed is a single
unit; in the context of climate policy, it seems reasonable to interpret this as the whole
world . . . This is in sharp contrast to reality. carbon prices and other climate policies differ
substantially across countries”(p. 848). To rectify this shortcoming, Hoel (2011) initiated a
third generation of Hotelling models that allow for different stringencies of climate regulation among regions of the world (Long 2015). In Hoel’s model, a homogeneous fossil fuel in
fixed global supply is extracted at zero cost, and a perfect substitute is available at constant
marginal cost. Consumers are divided into two regions, each of which imposes stationary
policies—a tax on fuel consumption and/or a subsidy for using the backstop. He considered
the effects of changes in one region’s policy, given an unchanged policy in the other region.
Hoel did not consider climate policies sufficient to reduce the cumulative amount of carbon
eventually going into the atmosphere; his focus instead was on the time profile of these emissions. Similarly, Eichner and Pethig (2011) distinguished among regulating, nonregulating,
and fossil fuel-exporting countries. They used a two-period model to assess the effects of
policy timing and rates of intertemporal substitution on the green paradox. They found
that increasing the size of the regulating coalition tends to reduce carbon leakage. However,
they did not include a backstop technology. Recently, Ryszka and Withagen (2016) analyzed
a model motivated by Hoel (2011) and an earlier draft of our current paper (Fischer and
Salant 2013).6 In Ryszka and Withagen (2016), there are two groups of consumers, each of
which has available two exhaustible resources extractable at different constant marginal costs
and its own nontradeable backstop with a stationary, constant marginal cost of production.
Given their assumptions, climate policy cannot reduce cumulative emissions: “all carbon
that was in the crust of the earth initially will eventually end up in the atmosphere.”PAGE
CITE?
Our model differs from previous work in several respects. Unlike Hoel (2011), we assume
that extraction costs are positive. Unlike both Hoel (2011) and Ryszka and Withagen (2016),
we assume that the marginal cost of energy from a clean backstop declines over time because
of exogenous technical change. As a result, some reserves may be left in the ground and
appropriate government policy can reduce cumulative carbon emissions. We also add realism
by assuming that pools differ in extraction costs and, in our simulations and appendix, in
emissions intensities.7
6
An earlier draft of this paper (Fischer and Salant, 2013) included the same theoretical model and simulation model (although calibrated using older data). However, we now focus on the distributive consequences
of alternative policies to achieve the same carbon budget and have changed the title of the paper accordingly.
7
In both the simulation section and the Appendix, we assume that pools with different marginal costs
of extraction may also different emissions factors. Throughout the text, we assume that the regulator taxes
each barrel at the same rate regardless of its source. In the Appendix, however, we show how the model
5
Our focus differs as well. In previous models, technology policy did not reduce cumulative emissions at all and in fact resulted in faster accumulations in the atmosphere. In
contrast, our model identifies the significant benefits that technology-oriented policies can
provide. First, accelerating cost reductions in the backstop induces consumers not just from
the regulated region but also from the unregulated region to reduce emissions.8 Second,
more rapid backstop cost reductions decrease the minimum coalition size required to influence cumulative emissions. Thus, rather than aggravating the problem of climate change,
technology policies permit other climate policies to be relaxed, improving cost-effectiveness
by limiting the dispersion of carbon prices. Finally, technology policies lead to the lowest
loss of wealth among oil extractors remaining active, as well as the largest increase in surplus
among unregulated consumers.
We proceed as follows. In Section 2, we describe our model. The model takes as given
three policy parameters: the fraction of world demand subject to regulation, the emissions
price within that sector, and the rate of cost-reducing technical change in the green backstop.
In Section 3, we determine the equilibrium effects of changing each of these policy parameters
unilaterally. We also deduce qualitatively the consequences for stakeholders of changing pairs
of the policy parameters so as to achieve the same carbon budget. In Section 4, we calibrate
a five-pool generalization of our theoretical model to the oil market example. Simulations
permit us to quantify in a more realistic model the payoff tradeoffs assessed only qualitatively
in the theoretical model. Moreover, they enable us to resolve several ambiguous results from
the theory and determine if the changes in the consumer surplus of regulated consumers
dominate changes in the tax revenue that would be rebated to these consumers and the
costs that would be borne to speed up backstop price reductions. Section 5 concludes the
paper.
should be modified if the tax takes into account the carbon content of each source.
8
When a change in policy not only induces regulated consumers to cut their cumulative emissions but also
induces unregulated consumers to reduce their emissions, the latter reduction has been dubbed “negative
leakage (Baylis et al., 2014). Negative leakage can occur in general equilibrium models through channels of
labor and capital inputs (Baylis et al. 2014). International spillovers of input-saving technical change, if large
enough, can also lead to negative leakage in general equilibrium models with input substitution (Gerlagh and
Kuik 2014). Computable general equilibrium models, however, find that typical climate policies generate
positive leakage. We show that negative leakage can occur in a partial equilibrium setting through resource
market dynamics and technical change in a backstop technology.
6
2
The Two-Pool Model
2.1
Assumptions
In this section, we consider the case of a competitive market for barrels of oil equivalent
(BOE). There are two pools of oil, one low-cost (L) and one high-cost (H), and a backstop
technology. Global demand is apportioned into two regions, one regulated (R) and the other
unregulated (U). The interest rate is assumed exogenous and is denoted r.
2.1.1
Oil Extraction and Emissions
Denote the initial size of the underground stocks as SL and SH . Per-unit extraction costs
are cL < cH . Denote the present value of a barrel of low-cost and high-cost oil in the ground
as λL and λH , respectively. These Hotelling rents are endogenous. Let πi (t) = ci + λi ert
be the cost of acquiring a barrel of type i (i = L, H) oil at time t and bringing it to the
surface. Assume that transportation costs are zero. Then, since we assume that the market
is competitive, πi (t) is also the price that unregulated consumers will pay at time t to obtain
oil from source i. πi (t) weakly increases over time.
Let qi (t) (i = L, H) be the total quantity extracted from pool i in period t. Let µ
denote carbon emissions per barrel, which we assume in this section to be the same for each
pool of oil.9 Carbon emissions can be reduced only by reducing cumulative oil usage. Let
θi ∈ [0, 1], (i = L, H) denote the fraction of the initial stock of pool i that is depleted in the
equilibrium.10 Cumulative emissions from each pool are thus EL = µθL SL and EH = µθH SH ,
and total cumulative emissions are E = EL + EH .11
2.1.2
Backstop Technology
At time t, a carbon-free backstop technology is available in unlimited capacity at constant
marginal cost per unit of output within that period, B(t; z), where z denotes the intensity
of cost-reducing technical change. The backstop is initially too expensive to warrant consideration by consumers: B(0; z) > cH > cL . In the absence of technical change (z = 0), the
marginal cost of the backstop would remain forever higher than the cost of extracting the
more expensive pool: B(t; 0) > cH for all t. However, we assume that because of technical
9
In the Appendix, we relax this assumption and allow the emissions factors and emissions tax burden to
vary by pool.
R∞
10
That is, θi = 0 (qi (t))dt/Si . In our Hotelling framework, θi < 1 implies the stock constraint is not
binding, so in equilibrium (1 − θi )λi = 0 and each factor must be nonnegative.
11
In the five-pool simulation model, we continue to assume that the tax (or pollution permit) applies to
the average carbon content per barrel of oil regardless of source. However, we do distinguish different carbon
intensities of extraction and consumption from different poolswhen computing cumulative emissions.
7
improvements (z > 0), the marginal cost of the backstop declines exogenously over time toward a long-run cost that is lower than the cost of extraction of the cheaper pool: BLR < cL ,
where BLR = limt→∞ B(t; z).12
2.1.3
Demand
We assume that the demand per person is stationary and so is the world population.13 The
world demand function at time t is D(·). We assume the N consumers in the world have
identical demand curves: each consumer has demand D(·)/N . Let α denote the fraction of
the N consumers who are regulated and thus face the price τ (t) on their emissions at time t.
No emissions price is imposed on consumers in the unregulated region. With αN consumers
in the regulated region, demand there is αD(·) and demand in the unregulated region is
(1 − α)D(·). Denote the price consumers pay in region j at time t as pj (t) (j = U, R). Then
the quantity demanded in region R at time t is αD(pR (t)), and demand in the unregulated
region U at time t is (1 − α)D(pU (t)), where α ∈ [0, 1].
Consumers in each region choose at each instant the least-cost energy source. Consumers
in the regulated region pay pR (t) = min(πL (t)+τ (t)µ, πH (t)+τ (t)µ, B(t; z)) while consumers
in the unregulated region pay pU (t) = min(πL (t), πH (t), B(t; z)).
We also focus on the emissions per consumer in each region, EU and ER , such that
per-capita global emissions are αER + (1 − α)EU = E/N .
2.1.4
Policies
We take as our starting point that a global commitment has been made to meet a cumulative
emissions budget, although the exact portfolio of policies is to be determined. Our framing
corresponds to: (1) the commitment made under the Cancun Agreements (UNFCCC COP
16, 2010) to a maximum temperature rise of 2 degrees Celsius above pre-industrial levels;
(2) the corresponding carbon budget discussions by climate scientists (e.g., IPCC 2015); and
(3) the format of the Paris Agreement (UNFCCC COP 21, 2015), which allows countries to
determine their own policy contributions. Furthermore, requiring all policy combinations to
meet the same carbon budget leads to a transparent cost-effectiveness exercise.
One pillar of the policy portfolio is an emissions price regime in the regulated region.
Consistent with our cost-effectiveness framework, we consider an emissions price that rises
at the interest rate; that is, τ (t) = τ ert . This path corresponds to the optimal global tax
12
It can be verified that the marginal-cost function used in the simulations, B(t; z) = BLR + (B(0; z) −
BLR )e−zt , satisfies each of these properties.
13
In the numerical simulations, we take account of population growth.
8
policy for meeting a given cumulative emissions constraint.14 Under this interpretation,
τ (t) is an exogenous tax rising at the rate of interest and αN ER is the resulting cumulative
carbon pollution of the regulated consumers. Alternatively, one can think of these consumers
as regulated by a cap-and-trade program with permit banking. In that case, αN ER is the
exogenous cumulative emissions cap and τ (t) is the resulting permit price, which must rise
at the rate of interest in equilibrium to induce the private sector to bank permits.
A second policy margin is to expand the size of the regulated coalition, α. This strategy
brings a larger share of global demand under the carbon pricing regime.
Third, government policy can intervene to accelerate the rate of technical change. In
the no-policy scenario, we assume that the baseline per-unit cost declines at a rate denoted
z0 > 0 which is slow enough that the two pools of oil are completely exhausted before the
backstop is utilized (but just barely, such that λH = 0). Technology policy increases the
parameter z > z0 and results in faster technical change.
2.1.5
Equilibrium
In equilibrium, prices adjust such that everything extracted at t is purchased contemporaneously by consumers in the two regions: qL (t) + qH (t) = αD(pR (t)) + (1 − α)D(pU (t)). If
at time t the low-cost (respectively, high-cost) oil is the cheapest of the three energy sources
for at least one of the two groups of consumers, then qL (t) > 0 (respectively, qH (t) > 0). We
adopt the multiple demand generalization of Hotelling’s (1931) exhaustible resource model.15
In that model, consumers in different regions may have to pay different prices to purchase oil
extracted from the same source at the same time. In our application, the additional cost is
the emissions price per barrel that must be paid by consumers in the regulated region. Since
we assume cL < cH , pool L has a lower per-unit cost sum than pool H. In the regulated
region, cL + τ µerx < cH + τ µerx .
An important property of the multiple-demand model is that if consumers in a region
use two pools, the pool with the lower per-unit cost sum will be used first. This property
is known in the literature as the “Generalized Herfindahl Principle,” extending Herfindahl’s
(1967) result for the single-demand model that all consumers utilize pools in order of their
per-unit cost sums and completely exhaust one before switching at the same instant to the
14
That is, if the speed of technological change is z, when the global (α = 1) tax τ is chosen to limit global
emissions to a binding carbon budget E∗, then the allocation in the resulting competitive equilibrium will
coincide with what a social planner would choose if he had the same initial stocks of low-cost and highcost oil extractible at the same cost, and was constrained to carbon budget E ∗ . The shadow price on the
carbon-budget constraint in the planners problem would coincide with the emissions tax in the competitive
equilibrium, each rising at the rate of interest.
15
See Gaudet and Salant (2015) for an expository survey of this literature.
9
next pool. In the multiple-demand generalization, consumers in the regulated region need
not utilize the same sequence of pools as consumers in the unregulated region.
Since the backstop becomes cheaper over time, the only temporal sequences for each
region consistent with the Generalized Herfindahl Principle are L → H → B or L → B or
H → B or B. We denote the dates when regulated and unregulated consumers switch to the
U
backstop as xR
B and xB , respectively. We denote the date when the unregulated consumers
switch from the low-cost pool to the high-cost pool as xH . If xH < xR
B , the regulated
consumers also utilize the high-cost pool and switch to it from the low-cost pool at the same
date as the unregulated consumers.
Regardless of the number of pools, we can distinguish two types of equilibria. In the
first, all pools that are utilized are fully exhausted and the scarcity rent on each of them
is therefore positive. In the second, part of the highest-cost pool that is utilized is left in
the ground and its scarcity rent is zero. Since marginal changes in policy affect cumulative
emissions in the second but not in the first type of equilibria, we focus here on the second
type of equilibria.16
2.2
Equilibria with Incomplete Extraction
In the equilibria of interest, the initial reserves (SL > 0) in pool L are fully exhausted and
pool H is partially exhausted (λL > 0, θL = 1, λH = 0, θH ∈ (0, 1]). Given the Generalized
Herfindahl Principle, our assumptions that B(0, z) > cH > cL and that θH > 0, the unregulated consumers must always use the low-cost pool, then the high-cost pool, and finally the
backstop: LHB. The regulated consumers use these resources in one of the following three
orders: (1) LHB, (2) LB, or (3) B. We denote the sequence of pools utilized by the unregulated consumers as the first component of a two-tuple and the sequence of pools utilized
LB ). As the
by the regulated consumers as its second component; for example, (LHB
| {z }, |{z}
U
R
regulation becomes more stringent, the equilibrium would shift from case (1) to case (2) to
case (3).
In each of the three cases, the following five equations uniquely determine the five en16
The first type has been well studied in the above-cited green paradox literature, including Hoel (2011),
although not fully generalized to our case of multiple pools and emissions factors. See also Fischer and Salant
(2012).
10
U
dogenous variables (λL , θH , xH , xR
B , xB ) given our policy variables (τ, α, z):
Z
min(xH ,xR
B)
Z
rt
xH
D(cL + e [λL + τ µ])dt + (1 − α)
SL = α
t=0
xR
B
D(cL + λL ert )dt (1)
t=0
Z
D(cH + ert τ µ)dt + (1 − α)
θH SH = α
t=min(xH ,xR
B)
Z
xU
B
D(cH )dt
U
B(xUB ; z) = min(cH , cL + λL exB )
rxR
B
B(xR
B ; z) ≤ min{cL + e
(2)
t=xH
(3)
[λL + τ µ], cH + e
rxR
B
τ µ}, xR
B ≥ 0, (comp. slack.)
cH = (cL + λL erxH )
(4)
(5)
Subsequently, we will set a carbon budget E ∗ that defines total allowable cumulative
emissions, and therefore the share of the high cost pool that can be extracted:
E ∗ = µ(SL + θH SH ).
(6)
Adding this sixth equation to the system will pin down one policy variable, given the other
two.
The transition dates exhibit distinctive patterns. If the regulated consumers use the
high-cost pool (LHB, LHB), they will switch to it at the same time (xH ) as the unregulated
consumers. At xH , the unregulated consumers find for the first time that oil from the pool
with the high extraction cost but no rent is weakly cheaper than oil from the other pool
(cH = cL + λL erxH ). At the same time, the regulated consumers reach the same conclusion
about which pool is cheaper since they pay a tax per barrel (µτ erxH ) which is the same for
the two pools. The lower the rent (λL ) on the low-cost pool, the later will be the transition
to the high-cost pool. That is dxH /dλL < 0.
When the regulated consumers eventually switch to the backstop (xR
B > 0), the unreguR
U
lated consumers will still be using fossil fuel (xB < xB ). This is because at the instant when
the cost of the backstop is equal to the cost of the taxed fossil fuel, consumers not obliged
to pay an emissions price will find the fossil fuel cheaper than the backstop.
3
3.1
Comparative Statics
Individual Policies and Cumulative Emissions
To clarify the consequences of achieving the same carbon budget with different combinations
of policies, we first consider how a unilateral increase in each of our three policy variables
(τ, α, z) affects the equilibrium in our model. For each policy, we first consider the case
11
Figure 1: Equal Emissions Factors and (LHB, LHB)
$ / BOE
Backstop cost
Price path,
Regulateds
Price path,
Unregulateds
Time
xH
T
where consumers in the regulated region, like those in the unregulated one, use both the
low-cost and the high-cost pools before switching to the backstop (LHB, LHB). Then we
consider the case where only the unregulated consumers use the high-cost pool; the regulated
consumers switch directly from the low-cost pool to the backstop (LHB, LB). Each policy
has the potential to reduce cumulative emissions, but they have different implications for the
surplus of regulated and unregulated consumers (before any lump-sum taxes or subsidies),
the wealth of low and high cost extractors, and other variables of interest.
When the regulated consumers do use the high-cost pool (LHB, LHB), they must switch
to it at the same time as the unregulated consumers: xH < xR
B . Moreover, from (5),
xH = xH (λL ), a strictly decreasing function. Equation (1) therefore reduces to:
Z
xH (λL )
Z
rt
xH (λL )
D(cL + e [λL + τ µ])dt + (1 − α)
α
t=0
D(cL + λL ert )dt = SL .
(7)
t=0
When the policies are more stringent, the regulated consumers switch directly from the
low-cost pool to the backstop without using the high-cost pool (LHB, LB). Hence, xR
B < xH .
R
From (4), xR
B = xB (λL + τ µ, z), a strictly decreasing of each argument. Since the policy
variables will be more stringent in the (LHB, LB) regime, λL and the other endogenous
variables will be different. Nonetheless the function xH (λL ) defined by (5) is unchanged.
12
Equation (1) therefore reduces to:
Z
xR
B (λL +τ µ,z)
Z
rt
xH (λL )
D(cL + e [λL + τ µ])dt + (1 − α)
α
t=0
D(cL + λL ert )dt = SL .
(70 )
t=0
These two equations will be useful in our comparative-static analysis. Each equation is
an α-weighted average of two terms. The first term is smaller than the second since, by
assumption, D0 (·) < 0 and, in (70 ), xR
B < xH in the (LHB,LB) regime.
We summarize our comparative-static results in Tables 1 and 2 and justify each row of
these tables in the proofs of Propositions 1–3 below.
Table 1: Comparative Statics: (LHB, LHB)
Prices
Switch Dates
Emissions
Surplus
R
U
Policy λL (λL + µτ ) xH xB xB E ER EU CSU CSR
τ
−
+
+ −
0 − −
+
+
−
z
0
0
0
− − − −
−
+
+
α
−
−
+
0
0 − +
+
+
+
Table 2: Comparative Statics: (LHB, LB)
Prices
Switch Dates
Emissions
Surplus
U
E
E
E
CS
CSR
x
Policy λL (λL + µτ ) xH xR
R
U
U
B
B
τ
−
+
+ −
0 − −
+
+
−
z
−
−
+ − − − −
?
+
+
α
−
−
+ +
0 − +
+
+
+
3.1.1
An Increase in the Emissions Price
Proposition 1. A unilateral increase in the emissions price (τ ) decreases cumulative emissions (E). The emissions price increase raises uniformly the price path faced by the regulated
consumers and lowers their emissions while lowering uniformly the price path faced by the
unregulated consumers and raising their emissions. Consumer surplus of the regulated consumers falls; consumer surplus of the unregulated consumers rises; and the wealth of the
low-cost extractors falls, while that of high-cost extractors remains zero. These results hold
in both the (LHB, LHB) case and the (LHB, LB) case. In both cases, the unregulated consumers switch later (xH ) to the high-cost pool and switch to the backstop at an unchanged
date (xUB ). In both cases, the regulated consumers switch sooner (xR
B ) to the backstop; in
the (LHB, LHB) case, the emissions price increase induces the regulated consumers to delay
their switch (xH ) to the high-cost pool.
Proof. When the emissions price increases, τ µ rises. In response, λL must fall and λL + τ µ
must rise for (7) to hold in the (LHB, LHB) case and for (70 ) to hold in the (LHB, LB)
13
case.17 Therefore, the price path faced by the unregulated (resp. regulated) consumers is
uniformly lower (resp. higher). Consequently consumer surplus of the unregulated (resp.,
regulated) consumers rises (resp. falls). Since cumulative oil consumption falls (resp. rises)
for the regulated (resp. unregulated) consumers, emissions fall for the regulated consumers
and rise for the unregulated ones. Overall, emissions must fall since emissions from the lowcost pool remain µSL but there is less extraction from the high-cost pool so µθH SH declines:
the carbon price increase does induce leakage in the unregulated sector, but it only partially
offsets the emissions reduction in the regulated sector. Finally, the wealth of the low-cost
extractors (λL SL ) declines while the wealth of the high-cost extractors remains zero.18
3.1.2
An Increase in the Speed of Cost-Reducing Technical Change in the Backstop
Proposition 2. A unilateral increase in the rate of technical change in the backstop (z)
decreases cumulative emissions (E). If both types of consumers utilize the high-cost pool
(LHB, LHB), the increase in z affects neither the initial prices that consumers of either
type pay for oil nor the date when both switch to high-cost oil. However, since each type
of consumer switches to the clean backstop sooner, each has a higher consumer surplus and
lower emissions. The increase in the rate of technical change has no effect on the wealth of
either type of extractor. On the other hand, if regulated consumers use no high-cost oil (LHB,
LB), then the increase in z does reduce the initial prices that both regulated and unregulated
consumers pay for oil and does delay the date when the unregulated consumers switch to
the high-cost pool. The consumer surplus of each group increases. The wealth of low-cost
extractors decreases while the wealth of the high-cost extractors remain zero. Total emissions
decline as do the emissions of the regulated consumers.
Proof. In the case of (LHB, LHB), λL (and therefore λL + µτ ) cannot change or (7), which
does not contain z, would no longer hold. Since λL does not change, both types of consumers
begin to utilize the high-cost pool at an unchanged date (xH ).19 Consequently, both the regulated and the unregulated consumers deplete the low-cost pool at unchanged rates and
17
In the Appendix, we show that the rents of the low-cost resource can increase with the emissions price if
the tax distinguishes the carbon content of each source and their emissions factors are sufficiently different.
This occurs because, if the carbon content of the high-cost resource is much higher, a carbon price increase
will raise its retail price by much more than it raises that of the low-cost resource, and the desire of regulated
consumers to avoid these higher taxes by consuming more of the low-cost resource will bid up its rent.
18
In the equilibrium, the low-cost extractor strictly prefers to sell his oil before xH but otherwise would
earn the same wealth no matter when he sells it. If he sold all SL barrels immediately, the present value of
his wealth would be (pU (0) − cL )SL = λL SL .
19
This result easily generalizes if there are more than two pools. In a multiple-pool model, any increase
in the speed of technical change (z) that induces both regulated and unregulated consumers to reduce their
consumption of the same zero-rent resource will have no effect on the rents or wealth of lower-cost resources,
14
switch simultaneously at an unchanged time to the high-cost pool; each group also consumes
that pool at the same rate as before, but each group switches earlier to the backstop. Consumer surplus of each group must increase and their emissions must decrease.20 Cumulative
emissions fall since all the reserves of the low-cost pool are extracted but a smaller fraction
of the high-cost, equally dirty reserves are consumed. The wealth of the low-cost extractors
remains λL SL and of the high-cost extractors remains zero.
In the case of (LHB, LB) the exogenous, unilateral increase in z must cause λL to strictly
decrease or (70 ) which does contain z, would be violated. The smaller rent implies that
the oil prices paid by regulated and unregulated consumers are uniformly lower. This in
turn implies that the consumer surplus of each group rises and the wealth of the low-cost
extractors declines. Because λL decreases, xH increases and the unregulated consumers use
more of the low-cost stock. Since regulated consumers also use the low-cost stock at a faster
rate and its reserves have not increased, regulated consumers must switch to the backstop
sooner than before in order for their cumulative use of the low-cost stock to decrease. In
the (LHB, LB) case, only the unregulated consumers use the high-cost pool. They reduce
their aggregate usage of this stock since they use it at an unchanged rate over a shorter time
interval. Hence total emissions must fall, as must the emissions of the regulated consumers.
Since unregulated consumers expand their use of the low-cost pool and contract their use of
the high-cost pool, their cumulative emissions may increase or decrease.21
3.1.3
An Increase in the Coalition Share
Proposition 3. A unilateral increase in the coalition size (α) decreases cumulative emissions (E), even though the emissions of each consumer who remains unregulated (EU ) rises,
as do the emissions of each consumer who remains regulated (ER ); the emissions reductions
come from those who become regulated when the coalition expands. Increasing the coalition
size reduces uniformly the price path facing each type of consumer, raising their consumer
surplus. The wealth of low-cost extractors (λL SL ) declines and the wealth of high-cost extractors remains zero. The foregoing qualitative changes occur whether or not the regulated
which are exhausted even if these pools differ in their emissions factors. Intuitively, the per-unit cost of
extraction of the zero-rent resource imposes a ceiling on the prices of every resource that is less costly to
extract. Since each of these resources is exhausted before the price hits this ceiling, their rents and thus
price and consumption paths do not change when the technology policy tightens.
20
This is one of many cases where a more stringent policy results in reduced emissions of the unregulated
consumers. Such behavior has been labelled “negative leakage,” as recently popularized by Fullerton (xxx).
His study and subsequent studies used static, general equilibrium models to show how price adjustments in
labor or capital markets can in theory reduce emissions among unregulated sectors. We show that similar
reductions in the emissions of unregulated consumers can arise in our dynamic, partial equilibrium model.
21
If, however, most consumers were unregulated, then the fact that cumulative emissions (E) contract
implies that emissions of the unregulated consumers also contract, another example of negative leakage.
15
consumers use high-cost oil before switching to the backstop. The only difference between
the (LHB, LB) and (LHB, LHB) cases is in the effect of increasing the coalition size on the
time (xR
B ) when the regulateds switch to the backstop. It increases in the former case and is
unchanged in the latter case.
Proof. An increase in α puts more weight on the smaller definite integral in (7) and on the
smaller definite integral in (70 ). For each of these equations to continue to hold, λL must
decline. The decline in λL in both the (LHB, LHB) case and the (LHB, LB) case means that
both regulated and unregulated consumers face uniformly lower price paths and therefore
enjoy an increase in consumer surplus. In both cases, the wealth of the low-cost extractors
(λL SL ) declines while the wealth of the high-cost extractors remains zero.
In the (LHB, LHB) case, the decline in λL means that both groups switch later to the
high-cost pool (xH increases). Since rents on the high-cost pool remain zero, each group
switches to the backstop at the same dates as before. Since the price path for each type of
consumer is uniformly lower, each type consumes more and emits more cumulatively before
switching to the backstop. However, total emissions from the low-cost pool remain µSL . and
since each group uses the high-cost pool at unchanged rates and for a shorter time interval,
and since a greater share of world demand comes from the sector with the smaller demand,
total emissions (E) must fall.
In the case of (LHB, LB), only the unregulated consumers utilize the high-cost pool.
Since there are fewer unregulated consumers and each of them utilizes the high-cost pool at
an unchanged rate but over a shorter interval, cumulative emissions (E) must decline.
3.2
Policy Tradeoffs for the Same Cumulative Emissions Target
Since the three policy variables are substitutes, we can achieve the same emissions target in
different ways. In this subsection, we fix one policy and consider the tradeoffs of achieving
the same carbon budget E ∗ by tightening a second policy and loosening the third.
∗
Let θH
be the equilibrium share of the high cost pool that can be exploited and still
∗
satisfy the carbon budget E ∗ from (6): θH
= (E ∗ − µSL )/(µSH ). We can substitute this into
(2), eliminating θH . We then have five equations in λL , xH , xUB , xR
B and one policy variable
as a function of the other two policy variables, one of which we hold fixed and the other of
which we vary.
In analyzing these tradeoffs, we distinguish between the cases of (LHB, LHB) and (LHB,
LB).22
22
The boundary of this region occurs where xH = xR
B . If this equation replaces equation (4), the new
system of 5 equations defines the emissions price for a given policy pair z and α that is exactly on the
boundary of this region.
16
3.2.1
A Lower Emissions Price for a Larger Coalition of Consumers
First, we fix the speed of technical change and consider the effects of imposing a lower
emissions price on a larger subset of consumers. We summarize our results in Table 3.1.
Proposition 4. In an (LHB, LB) equilibrium, an increase in α with a reduction in τ sufficient to keep cumulative emissions constant despite an unchanged z must increase extractor
wealth (λL SL ). Consumers who remain unregulated pay uniformly higher prices while regulated consumers pay uniformly lower prices. Consequently, the consumer surplus and emissions of each regulated consumer increase while those of each unregulated consumer decrease.
Proof. Since neither cH nor the speed of technological change varies when α is increased,
unregulated consumers do not change the date when they switch to the backstop (dxUB /dα =
0, from equation (3)). Since cumulative emissions from the low-cost pool will continue to be
µSL , in order for aggregate emissions from the two pools to remain E ∗ , emissions from the
high-cost pool cannot change. In an (LHB, LB) equilibrium only the unregulated consumers
utilize the high-cost pool; hence, this group must deplete an unchanged amount of the highcost pool. Since fewer consumers are assumed to be unregulated, and each of them depletes
the high-cost pool at an unchanged rate (D(cH )) for an interval that ends at an unchanged
date (xUB ), each of them must begin depleting the high-cost pool sooner. For the unregulated
consumers to switch to the high-cost pool at an earlier date, λL must increase. Hence,
dλL /dα > 0, implying dxH /dα < 0.
Expanding the coalition while lowering the tax affects other variables of interest. Before
analyzing equations (7) and (70 ), it is helpful to make the substitution γ = λL + µτ . For any
γ and λL we can recover τ since τ = (γ −λL )/µ. Note that γ > λL since τ > 0. Furthermore,
R xUH
let QUL = t=0
D(cL + λL ert )dt/N be the cumulative demand from the low-cost pool by an
R xRB
unregulated consumer and QR
=
D(cL + γL ert )dt/N be that of the regulated consumer.
L
t=0
Substituting into (70 ) and (4), we obtain:
R
rxB
B(xR
B ; z) = cL + γe
SL /N = (1 − α)QUL + αQR
L.
(8)
(9)
Totally differentiating (9) and rearranging, we have
U
R
(QUL − QR
L )dα = (1 − α)dQL + αdQL .
(10)
With (LHB, LB), regulateds switch to the backstop before unregulateds switch to the
U
R
high-cost pool. Since xR
B < xH and pR (t) > pU (t) ∀t, QL −QL > 0. Since we have established
17
that dλL /dα > 0 and dxH /dα < 0, then we have dQUL /dα < 0. Therefore, from equation
(10), it must be that dQR
L /dα > 0; i.e., individual as well as total regulated consumption
from the low-cost pool must increase. The properties of cumulative demand for the low-cost
R
R
pool imply ∂QR
L /∂xB > 0 and ∂QL /∂γ < 0, so either a delay in the switch to the backstop
or a fall in the regulated price would increase the cumulative consumption of regulated users.
23
Equation (8) implies that xR
Intuitively, if the
B and γ must move in opposite directions.
transition to the backstop occurred later, the backstop marginal cost on the right-hand side
of equation (8) would be lower; for the left-hand side to decrease as well, γ must decrease.
We therefore conclude that dγ/dα < 0 and dxR
B /dα > 0.
Proposition 5. In an (LHB, LHB) equilibrium, an increase in α with a reduction in τ
sufficient to keep cumulative emissions unchanged must uniformly lower the price path faced
by regulated consumers, raising their surplus and emissions.
Proof. These results follow from Table 1. When the size of the regulated sector (α) increases
and the emissions price (τ ) falls, both changes reduce the initial price (λL + τ µ) faced
by regulated consumers. Moreover, the reduction in the emissions price (τ ) delays their
transition to the backstop (xR
B ). Therefore, the regulated consumers face a uniformly lower
price path and their consumer surplus and cumulative emissions must be higher.
In the (LHB, LHB) equilibrium, unregulated consumers do not change the date when
they switch to the backstop (dxUB /dα = 0, from equation (3)). Nothing more can be said in
general about the unregulated consumers.24
Table 3.1: Holding Cumulative Emissions Constant with a Broader Coalition
and a Lower Emissions Price (α ↑, τ ↓)
Prices
Switch Dates
Emissions
Surplus
U
Equilibrium λL λL + µτ xH xR
x
E
E
E
CS
CSR
R
U
U
B
B
(LHB,LB)
+
−
− +
0
0 +
−
−
+
(LHB,LHB) ?
−
?
+
0
0 +
?
?
+
3.2.2
A Lower Emissions Price Supplemented by Faster Technical Change
Next, we fix the coalition size. We show that extractors and regulated consumers both
benefit if the carbon price is reduced while technical change in the backstop is increased
enough to satisfy the same carbon budget. We summarize our conclusions in Table 3.2.
R
R
In particular, dγ/dα = −[γr − e−rxB B1 (xR
B ; z)]dxB /dα, where B1 denotes partial differentiation with
respect to the first argument. The term in square brackets is positive.
24
For the functional forms used in our simulations, however, we have established that an increase in the
coalition size raises λL and reduces xH . Unregulated consumers face a uniformly higher price path and hence
have smaller consumer surplus and cumulative emissions.
23
18
Proposition 6. An increase in z with a reduction in τ sufficient to keep cumulative emissions constant despite an unchanged α must increase extractor wealth (λL SL ). Regulated
consumers face uniformly lower prices and therefore receive higher consumer surplus. Emissions of the regulated consumers rise while emissions of the unregulated consumers fall.
Proof. In the (LHB, LB) regime, only the unregulated consumers use the high-cost pool.
Since, in response to faster technological change, each of the unchanged number of unregulated consumers would still deplete the high-cost pool at the same rate (D(cH )) but would
switch to the backstop earlier (lower xUB ), target emissions could only be maintained by beginning to exploit the high-cost pool sooner (lower xH ). But for unregulated consumers to
abandon the low-cost pool sooner, its price must reach cH sooner, and that requires a higher
λL .
As a result, over time the cumulative demand of the unregulated consumers for the
low-cost reserves must decrease. Since the low-cost pool will still be exhausted in the new
equilibrium, the regulated consumers must increase their cumulative demand, and this requires that λL + µτ decline. Hence, the emissions price would have to decline enough that
regulated consumers faced lower prices for low-cost oil even though the pre-tax cost of that
oil increases
The results for the (LHB,LHB) equilibrium follow clearly from Table 1: λL increases
and λL + τ µ decreases when τ is reduced and both are unaffected when z increases. The
wealth of the low-cost extractors must therefore increase and, since regulated consumers face
uniformly lower prices, their consumer surplus must rise. Unregulated consumers face higher
prices but switch to the backstop sooner, after which they face lower prices. So the change
in their consumer surplus is indeterminate. Facing higher oil prices and an earlier switch to
the backstop, emissions of unregulated consumers fall; since the size of the coalition (α) is
unchanged, emissions and consumer surplus of a regulated consumer must rise.
Table 3.2: Holding Cumulative Emissions Constant with Faster Technical
Change and a Lower Emissions Price (z ↑, τ ↓)
Prices
Switch Dates
Emissions
Surplus
U
R
Equilibrium λL λL + µτ xH xB xB E ER EU CSU CSR
(LHB, LB)
+
−
−
?
− 0 +
−
?
+
(LHB, LHB) +
−
−
?
− 0 +
−
?
+
3.2.3
Faster Technological Change with a Smaller Coalition
Finally, we fix the emissions price and consider the effect of a shift to a smaller coalition
with a faster rate of technical progress in the backstop. We summarize our findings in Table
3.3.
Proposition 7. In an (LHB, LHB) equilibrium, an increase in z with a reduction in α
sufficient to keep cumulative emissions unchanged must raise λL and thereby the wealth of
the low-cost extractors and the initial prices of both regulated and unregulated consumers.
Emissions per consumer of each type fall.
Proof. In an (LHB, LHB) equilibrium, the results follow from Table 1: an increase in z
R
lowers xR
B while the decrease in α has no effect on xB . λL and λL + τ µ both increase when α
19
is reduced and are unaffected when z increases.The wealth of the low-cost extractors must
therefore increase. Emissions per consumer of each type will fall, but the carbon budget is
nonetheless maintainted because more consumers are unregulated.
Proposition 8. In an (LHB, LB) equilibrium, an increase in z with a reduction in α sufficient to keep cumulative emissions unchanged must lower emissions per regulated consumer.
Proof. The result follows from Table 2: both an increase in z and a decrease in α cause
regulated consumers to switch earlier to the backstop and reduce their emissions.
In an (LHB, LB) equilibrium, the high-cost pool is consumed exclusively by the unregulated consumers. An increase in z causes them to switch to the backstop sooner. Whether
the increase in z and simultaneous reduction in α raises or lowers λL depends on the size of
the regulated coalition. If virtually every consumer is regulated, an increase in z will raise
λL . In that case, the wealth of the low-cost extractors increases and the cumulative emissions
of each type of consumer decline. If virtually none of the consumers is regulated, however,
an increase in z, will lower λL . In that case, the wealth of the low-cost extractors falls, and
the consumer surplus of each type of consumer increases.25
Table 3.3: Holding Cumulative Emissions Constant with a Smaller Coalition
and a Higher Rate of Technical Change (z ↑, α ↓)
Prices
Switch Dates
Emissions
Surplus
R
U
Equilibrium λL λL + µτ xH xB xB E ER EU CSU CSR
(LHB,LB)
?
?
?
− − 0 −
?
?
?
(LHB,LHB) +
+
− − − 0 −
−
?
?
4
Calibrated Oil Market Model
In this section, we use simulations to quantify phenomena that previously we could discuss
only qualitatively. In addition, while the theory identifies tradeoffs in rents, extractor wealth,
consumer surplus, and emissions, quantitative analysis can also take account of recycled tax
revenues and subsidy costs in assessing the net and relative welfare effects on different types
of stakeholders of climate policy portfolios.
We generalize our model to take account of multiple types of exhaustible resources and
calibrate it using available oil data.26 On the demand side, we draw on empirical estimates
of the price elasticity and projections of demand growth over time. Throughout, we assume
an annual interest rate (r) of 2%.
25
An example and an analytic proof of this claim is available upon request.
We focus on emissions from the extraction and use of oil because this fuel arguably has the greatest
potential for the rent adjustments. GHG emissions from coal are potentially much larger, but the resource
is also considered much less (or non-) scarce.
26
20
4.1
4.1.1
Calibration
Supply
Our multiple-pool model includes the five major types of oil: low-cost Middle East and
North African (“MENA”) oil, other conventional oil with mid-range costs (“Other”), enhanced oil recovery and deep-water drilling (“EOR”), heavy oil bitumen (including oil sands)
(“Heavy”), and oil shales (“Shale”).27 In Table 5, we summarize for each pool its reserves,
its extraction cost, and its emissions factor relative to that of conventional oil.28 To convert
to CO2 emissions (right column of Table 5), we assume that a barrel of conventional oil
contributes 0.43 tons of CO2 .29 We adjust for the fact that different unconventional sources
have larger emissions factors relative to conventional oil.30
Table 4: Reserves and Cost Assumptions
Relative
Cost emissions
Oil reserve source
BBOE ($/BOE)
factor
MENA conventional
1124
$18
1
Other conventional
1217
$40
1
EOR, deep water
750
$58
1.105
Heavy oil, oil sands
1471
$69
1.27
Oil shale
1309
$79
2
Biofuels, backstop technology Unlimited
$115
0
CO2
(Gt)
483
523
356
803
1126
0
We assume that the backstop energy source is nonemitting, in keeping with the theoretical analysis and following common representations in the climate literature (starting
27
Coal to liquids is estimated to be higher cost and more emitting than Shale, and given our baseline
assumptions, it would not come into play before the clean backstop out-competes it.
28
underlying data from the 2013 World Energy Outlook from the International Energy Agency (IEA 2013,
Chapter 17), which gives a range of production costs and available (technically recoverable) reserves by
oil type. The Energy Information Administration (EIA 2016) currently estimates global proven reserves
to be about 1,660 billion barrels, including conventional and some unconventional, like Canadian oil sands
(https://www.eia.gov/cfapps/ipdbproject/IEDIndex3.cfm?tid=5&pid=57&aid=6). However, we acknowledge that estimates of exploitable oil reserves and costs vary widely. The IEA data we use imply larger
reserves and thus emissions than the reserves defined in some other studies, like McGlade and Ekins (2015),
who distinguish between proven reserves and remaining ultimately recoverable resources. On the other hand,
Kharecha and Hansen (2008) report reserves estimates ranging from 1,000 to 2,100 BBOE of conventional oil
and 1,300 to 8,500 BBOE of unconventional oil. Aguilera et al. (2009) include projections of future reserve
growth, leading to estimates of conventional oil reserves of 6,000 to 7,000 billion barrels available at prices
as low as $5 a barrel, heavy oil reserves of 4,000 billion barrels at $15 per BOE, oil sands reserves of 5,000
billion barrels at $25 per BOE, and up to 14,000 billion barrels of oil shale that could be tapped at $35 per
BOE. We find that changing our baseline reserve assumptions affects price and policy levels, but it does not
affect the relative trade-offs among policies.
29
This assumption follows the suggestion of the US Environmental Protection Agency, EPA.
http://www.epa.gov/grnpower/pubs/calcmeth.htm.
30
See Table 3-2 of the California technical analysis of the low-carbon fuel standard (Farrell et al. 2007).
21
with Nordhaus 1973).31 The assumed initial backstop per-unit cost is drawn from a range
of common estimates of advanced biofuels, in line with the IEA estimates.32 Although we
draw on biofuels in making these cost estimates, we recognize that future backstops could
include other alternatives, such as hydrogen or clean electricity for plug-in vehicles.33 We
assume that the backstop cost is initially $115/BOE but, because of cost-reducing technical change, will ultimately approach $10/BOE. That is, it will eventually be cheaper to
produce a unit of the backstop than to extract an equivalent amount of conventional oil:
B(t; z) = 10 + (115 − 10)e−zt .Hence, the difference between the per-unit cost of the backstop
at time t and in the long run declines exponentially at rate z. Our baseline rate of technical
change is calibrated on the assumption that all resources would be exhausted in the absence
of policy intervention although the most expensive resource (shale) would have a Hotelling
rent of zero.
4.1.2
Demand
We assume that the demand depends linearly on the contemporaneous price. According to
the 2014 International Energy Outlook (IEO) by the Energy Information Administration
(EIA), annual global oil consumption in 2012 was 32.3 billion barrels.34 As time elapses, the
demand curve is assumed to shift out because of economic growth. EIA (2014, Table A2)
projects that global demand will increase at an annual rate of 1.07% between 2010 and 2040,
primarily due to economic growth in developing countries; demand from the countries of
the Organization for Economic Cooperation and Development (OECD) is actually expected
to decline slightly during this period. We assume that the slope of the demand curve does
not change over time. To calibrate this slope, we assume a current effective price elasticity
of −0.25. This value roughly corresponds to the median estimate of the price elasticity of
global oil demand from Kilian and Murphy (2014).35
31
The actual emissions factors for biofuels, particularly those associated with land-use change, are admittedly controversial.
32
Conventional biofuels like sugarcane ethanol are currently cheaper, but the second-generation fuels like
cellulosic ethanol, drop-in biofuels, and biodiesel—which have greater potential for the larger-scale supplies
needed to function as backstop technologies—have higher costs. In 2007, the US Department of Agriculture
estimated cellulosic ethanol production costs at $2.65 per gallon, compared with $1.65 for corn-based ethanol.
33
Hydrogen or electricification would have zero emissions if produced in conjunction with renewable energy
or carbon capture and storage (IPCC 2010). Of course, synthetic fuels derived from coal or natural gas could
also be substitutes, but we assume fuel-based backstops are precluded.
34
http://tonto.eia.doe.gov/cfapps/ipdbproject/IEDIndex3.cfm?tid=5&pid=54&aid=2.
35
Surveys of earlier estimates of the price elasticity of demand for gasoline (primarily in the United States)
find short-term demand elasticities of about −0.25 and long-run elasticities of about −0.6 (Espey 1996;
Goodwin et al. 2004). On the other hand, Cooper (2003) and Dargay and Gately (2010) find much lower
price elasticities of demand (−0.15 and smaller) when considering a broader array of countries, particularly
non-OECD countries, and more recent time periods. Hughes et al. (2008) find that short-run elasticities
22
We thus assume that in year t, the demand curve is D(p(t), t) = 40.4e.0107t − 0.2p(t) per
year. Given this demand curve, our baseline calibration leads to the following: (1) the initial
price in the equilibrium ($41/BOE) induces an annual quantity demanded of 32.3 BBOE; (2)
the price elasticity at that point is 0.25; and (3) in the “no policy” case, complete exhaustion
of all resources occurs even though shale oil has zero rent.
The baseline rate of technical change consistent with this equilibrium is z0 = 0.0042.3637
As another indicator of the speed of technical progress, we denote the number of years
required for the backstop to become competitive with a given reserve of oil.38 The baseline
model calibration predicts resource exhaustion and a switch to the backstop by the end of
the century, when the backstop becomes cheaper than shale oil (tShale = 99).
4.2
Welfare calculations
The global welfare cost of a given reduction in cumulative emissions is the sum of changes
in the following: (1) discounted profits from each resource pool; (2) discounted emissions
revenues raised in the regulated region net of technology support costs; and (3) discounted
consumer surplus in the regulated and unregulated regions We measure discounted profits
as rent per barrel multiplied by the share θi extracted from stock i (λi θi Si ). Discounted
emissions revenues equal the embodied carbon cost (τ µ) multiplied by the cumulative consumption by the regulated region. Consumer surplus in each year is measured as the area
under the annual demand curve above the price. The present discounted value of this surplus
is integrated to year T , the year when the backstop at the baseline rate of change (z0 ) becomes cheaper than the last resource extracted; in the case of our 60% emissions reduction,
that resource will be EOR (with a marginal cost of $58 per barrel) and the transition to the
backstop occurs at T = 185 years.
have declined over time. Kilian and Murphy (2014) warn that most studies of such elasticities using dynamic
models do not account for price endogeneity. Our sensitivity analysis finds that the precise elasticity estimate
does not affect our results significantly.
36
To offer some context, our baseline rate of technical change sees backstop costs falling roughly 7% over
the next two decades. EIA’s 2013 Annual Energy Outlook forecasts similar rates of cost reduction for wind
generation, while solar costs are expected to fall 29% over this timeframe. Of course, those technologies
are more relevant if transportation sector moves toward electrification. Advanced biofuels have been more
disappointing in their cost evolution; although EIA forecasts a roughly 25% drop in production costs for
cellulosic ethanol over that timeframe, recent expectations have not been met, causing EPA to loosen the
corresponding requirements in the renewable fuel standard.
37
The calibrated value would be higher if we allowed technical change in fossil resources or assumed that
some of the fossil resources included by IEA would be left unexploited.
LR
38
Recall that B(t; z) = BLR + (B(0; z) − BLR )e−zt or, equivalently z(t) = ln B(0;z)−B
B(t;z)−BLR . Denote by ti
the time when the backstop marginal cost reaches the cost of extraction of pool i: that is, B(ti ; z) = ci . At
LR
that time, z(ti ) = ln B(0;z)−B
ci −BLR . Since we assume B(0; z) = 115, ln z, BLR = 10, and cEOR = 58, tEOR =
1
105
z ln 48 = .783/z. That is, tEOR is inversely proportional to z with their product being 0.783.
23
Increasing z above the baseline rate (z0 ) requires costly policy intervention such as investment in innovation. Lacking a reliable way to calibrate the cost of such investment,
we construct an upper bound for the cost of that policy using subsidies. Even at baseline technical change, governments can subsidize consumption of the clean backstop so
as to duplicate the price consumers would have paid at each moment if instead technical
change had increased to z. The present value of the per-unit subsidy at time t is [B(t; z0 ) −
B(t; z)]e−rt . The subsidies are assumed to be applied to all sales of the backstop, and thus
are offered to regulated and unregulated consumers alike. The cumulative gross technology
R xU
subsidy payments, G(z), are therefore G(z) = xRB (B(t; z0 ) − B(t; z)) αD(B(z, t))e−rt dt +
B
RT
−rt
(B(t;
z
)
−
B(t;
z))
D(B(z,
t))e
dt.
No
subsidies
would need to be paid after T to inU
0
xB
duce any consumer to use the backstop since, even with baseline technical change (z0 ) it
would be the cheapest alternative. Throughout we assume that all revenues from emissions
pricing and all subsidy costs accrue to the regulated region as lump-sum transfers or taxes.
G(z) equals the present value of the subsidy payments required to induce the same
behaviors from extractors and consumers as would have occurred if the speed of technical
change had increased from z0 to z. Of course, policymakers may reduce the cost of achieving
the same path of the backstop supply price by investing in innovation. Thus, the cost of
these subsidies tends to overestimate the cost of achieving a given technology price path and
therefore underestimates the net welfare benefits of the technology policy.
5
Results
In this section we report results from simulations of the parameterized five-pool model,
focusing on a cumulative emissions reduction target of 60%. This target would leave all
shale oil, all heavy oil, and 13% of EOR oil untouched. Since many policy combinations can
achieve this target, it is useful to define some reference policies:
• a global CO2 price, τ̂ , as the price that achieves the target with a global coalition
(α = 1) and no additional technical change in the backstop technology (z = z0 ); and
• a technology-only policy, ẑ, that meets the target by accelerating cost reductions in the
backstop without any carbon pricing (τ = 0). Any α can be associated with meeting
the emissions target, although its choice influences the distribution of the subsidy cost
burden.
According to the model, a cumulative emissions reduction of 60% could be achieved with
a global CO2 price of $20/ton initially: τ̂ = 20. Alternatively, the same reduction could be
24
achieved by roughly tripling the speed of technical change of the backstop (an acceleration
of 196%) so that it will become competitive with EOR oil after 63 years, 122 years sooner
than in the base case: ẑ = 0.0125.
Let us also define αmin (z) as the smallest coalition that can achieve the target given
the rate of technical change (z) in the backstop. The minimum coalition imposes such
a draconian emissions price on regulated consumers that they switch immediately to the
backstop to avoid paying any taxes. An initial price of $226/ton of CO2 (or $97/BOE) is
a sufficient deterrent in all cases.39 For example, at least 87% of consumers would have to
be regulated to achieve a 60% cumulative emissions reduction with carbon pricing alone:
αmin (z0 ) = .87.40
These extreme policies set some bounds on intermediate cases to explore.
5.1
Effects of Policy Portfolios on Oil Rents and Welfare Metrics
In the theoretical analysis, we identified some clear tradeoffs for the response of extractor
wealth, as well as regulated and unregulated consumer surplus, to adjustments in the policy
portfolio. We also identified some ambiguous tradeoffs. Using the parameterized model, we
can quantify these tradeoffs.
In every simulation, the unregulated consumers first use MENA oil, then Other conventional, and finally use EOR before switching to the backstop. We denote this sequence as
MOEB, the analog of LHB in the theory section. We distinguish between simulations in
which regulated consumers draw from all pools (MOEB) and simulations where they skip
the high-cost pool(s), resulting in the sequence MOB or MB. These sequences are the analog
of LB in the theory section.
We focus on the change in total extractor wealth and in consumer surplus for regulated
and unregulated consumers, expressing these changes in percentage terms relative to a no
policy scenario with no reductions.41 Changes in oil rents of each pool follow the same
patterns as changes in extractor wealth. In our calibrated simulations, we are able to take
account of net subsidy expenditures and emissions tax revenues.
39
This tax is calculated as the difference between the initial backstop price (B(0; z) = 115) and the
extraction cost of MENA oil ($18), divided by the emissions content (0.43), ensuring that even without any
rent for the lowest-cost resource, regulated consumers would immediately switch to the backstop.
40
The International Energy Outlook of 2014 predicts that while energy demand will rise by one-third from
2010 to 2040, the share attributed to the OECD will decline from roughly 53% to 37%. In other words, we
find that an OECD-sized coalition cannot achieve the 60% reduction target without faster technical change.
41
I.e., (Π − Π0 )/Π0 , where Π = λMENA SMENA + λOther SOther , and (CSj − CSj0 )/CSj0 .
25
5.1.1
Tradeoff between α and τ , given z
Proposition 4 showed that, when the regulated consumers do not use the last pool, extractor
wealth rises as the coalition size increases while the CO2 price falls to maintain the same
cumulative emissions. Proposition 5 and subsequent analysis found that when regulated
consumers draw on some of each utilized pool, the effects on extractor wealth could be
ambiguous. We have proved, however, that with the functional forms used in our simulations
(any linear demand and any backstop cost function with the gap between the current and long
run unit cost declining exponentially), extractor wealth continues to be strictly increasing
with coalition size in the two-pool model.42 Consequently, the emissions (and surplus) of
the unregulated consumers declines as the coalition size increases, while those of unregulated
consumers increase.
This result is illustrated in the multi-pool model in Figure 2, where it is assumed that
z = 2z0 . The figure depicts the change in total extractor wealth as a function of α. Starting
from the minimum coalition that can achieve the target at this rate of technical change
(αmin (2z0 ) = 46%), we see that extractor wealth strictly increases as the coalition size
increases.43 Extractor wealth with a global CO2 price is still lower than with the technologyonly policy (z = ẑ).
Figure 2: Achieving 60% Cumulative Emissions Reductions with a
Broader Coalition and Lower CO2 Price: Effects on Extractor Wealth
( z = 2 z0 )
% Change
50
60
70
80
90
100
Coalition size (%)
– 25
Technology policy
– 30
– 35
– 40
– 45
– 50
– 55
(MB)
(MOEB)
(MOB)
In our simulations, unregulated consumer surplus declines monotonically with coalition
42
One can show analytically that this result holds under these functional form restrictions.
We observe kinks in the oil rent function as the equilibrium moves from MB to MOB to MOEB, becoming
flatter each time.
43
26
size, illustrating the result derived for the two-pool model. Furthermore, as shown in the
propositions, regulated consumer surplus increases monotonically. Their net surplus (assuming they receive lump sum the tax revenue net of subsidy costs) also increases monotonically.
Intuitively, each regulated consumer benefits from having the costs of the technology subsidies shared by more consumers.
5.1.2
Tradeoff between z and τ , given α
Proposition 6 showed that given a coalition size, an increase in τ with a reduction in z
sufficient to keep cumulative emissions constant must decrease extractor wealth. Figure 3
illustrates this relationship for a coalition size of 50%: extractor wealth decreases monotonically from its value with the technology-only policy (ẑ) as the CO2 price increases and z
falls. As the equilibrium changes from MOEB to MOB to MB, the decline in wealth kinks
and becomes steeper.
Proposition 6 stated that the effect on the present value of consumer surplus for the unregulateds was indeterminate whether or not the regulated consumers skip the pool partially
used by the unregulated consumers. Figure 4 illustrates this nonmontonicity. The surplus of
an unregulated consumer declines at first but then increases as the given coalition relies more
on CO2 pricing and less on technical change. Regulated consumer surplus falls monotonically
as it does in the two-pool model. However, after accounting for the lump-sum receipt of tax
revenue net of subsidy costs, we see that the net surplus of the regulated consumers initially
rises and then falls.
5.1.3
Tradeoff between z and α, given τ
Proposition 7 showed that when regulated consumers draw from all pools, an increase in α
combined with a reduction in z sufficient to keep cumulative emissions unchanged must lower
extractor wealth. However, Proposition 8 indicated this relationship could be ambiguous
when regulated consumers skip the last pool, in this case EOR. Both of these results are
illustrated by Figure 5, which shows the percentage change in extractor wealth as a function
of α, holding the CO2 price fixed at τ = τ̂ /2. Starting with a coalition of zero, increasing
α lowers z from ẑ and extractor wealth falls initially. In this phase, technical change is fast
enough that the CO2 price induces regulated consumers to switch from Other conventional oil
to the backstop without consuming EOR (an MOB equilibrium). However, as the coalition
size approaches the level to support a MOEB equilibrium, extractor wealth begin to turn
up, meaning such wealth does not have a monotonic relationship in a MOB equilibrium. At
the switch point, extractor wealth jumps up discontinuously. Throughout the MOEB phase,
27
Figure 3: Achieving 60% Cumulative Emissions Reductions with a
Higher CO2 Price and Slower Technical Change: Effects on Extractor Wealth
(α = 50%)
% Change
20
40
60
80
CO2 price
−25
Technology Policy
− 30
− 35
− 40
− 45
− 50
(MOEB)
(MB)
(MOB)
Figure 4: Achieving 60% Cumulative Emissions Reductions with a
Higher CO2 Price and Slower Technical Change: Effects on Consumer Surpluses and Net
Welfare of the Regulated Consumers
(α = 50%)
% Change
10
er
Unregulated consumer
Regulated
consumer
5
CO2 price
20
40
60
80
Regulated consumer
+ taxpayer
–5
– 10
(MOEB)
(MB)
(MOB)
28
extractor wealth declines monotonically, illustrating Proposition 7.
As the coalition size increases and less technical change in the backstop is needed to
achieve the same emissions reduction, we were unable to determine how the two consumer
surpluses change in the two-pool model (Propositions 7 and 8). However, given our functional
forms and particular calibration, the effects on consumer surplus are monotonic for τ =
ˆ
tau/2:
consumer surplus of both types falls as illustrated in Figure 6. Intuitively any fall
in oil rents is more than offset by the reduced surplus from switching later to the backstop.
However, for the regulated consumers having to pay for the technology policy, net surplus
increases monotonically as those costs fall and are shared across a larger coalition of actors.
5.2
Policy Preferences of Stakeholder Groups
We now determine which policy combination reducing emissions by 60% best serves the
interests of each group of stakeholders: extractors, unregulated consumers, regulated consumers, and the world as a whole. If a given group of stakeholders could choose all three
policy variables, each has a clear preference:
• From the extractor perspective, the technology-only policy (ẑ) preserves the most
wealth of any combination.44
• An unregulated consumer also prefers the technology-only policy (ẑ) to all other combinations.
• In contrast, a regulated consumer would instead want every consumer (α = 1) to have
to pay the global CO2 price (τ̂ ). Intuitively, this least-cost solution transfers rents from
the extractors while requiring no subsidy burden (z = z0 ).
• The globally optimal policy combination is also the global CO2 price, τ = τ̂ (with
α = 1, and z = z0 ). Since all consumers pay the same price at the same time, this
solution eliminates distributive inefficiencies at any time. Moreover, in the absence
of technological spillovers, the private choice of backstop technological change (z0 ) is
socially optimal.45
However, individual stakeholders cannot directly choose the coalition size (α). Rather, a
broader political process determines which consumers should be regulated. 46 Conditional
44
As long as z = ẑ and τ = 0, the size of the coalition (α) does not matter to the extractors.
If unregulated consumers were required to shoulder some of the subsidy costs or to receive some of the
tax revenue lump sum, that would not affect aggregate social welfare.
46
For example, developing and emerging economies are not expected to carry the same burdens for regulating emissions under the UNFCCC principle of common but differentiated responsibilities and respective
capabilities (CBDR–RC).
45
29
Figure 5: Achieving 60% Cumulative Emissions Reductions with a
Broader Coalition and Slower Technical Change: Effects on Extractor Wealth
(τ = τˆ / 2)
% Change
20
40
60
80
100
Coalition size (%)
– 26
Technology Policy
– 27
– 28
– 29
(MOB)
(MOEB)
Figure 6: Achieving 60% Cumulative Emissions Reductions with a
Broader Coalition and Slower Technical Change: Effects on Consumer Surplus
(τ = τˆ / 2)
% Change
Unregulated consumer
10
5
Regulated consumer
20
40
60
–5
– 10
80
100
Coalition size (%)
Regulated consumer
+ taxpayer
– 15
30
on that agreement, the regulating coalition makes the commitment to achieve the global
target, but does so in a manner that maximizes its members’ welfare. Below we determine
the optimal policy pair (τ, z) achieving the 60% emissions reduction that would be most
preferred by regulated consumers and compare it to the policy pair that maximizes social
welfare.
The two panels of Figure 7 portray the optimal policy pair (τ, (z − z0 )/z0 ) for regulated
consumers (solid line) and for society as a whole (dashed line) as the coalition size increases.
As the coalition size increases, it is socially optimal to increase the CO2 price and reduce the
reliance on technology policy In the socially optimal solution, regulated consumers always
use some of the last resource (MOEB).
In contrast, as the coalition size increases, the preferred policy pair of the regulated
consumers is not monotonic: as the regulated share increases from 0% in the regime where
regulated consumers skip the last pool (MOB), the optimal CO2 price initially decreases and
then increases; when the regime shifts (MOEB), the price jumps down. As the coalition
size approaches 100%, the CO2 price approaches τ̂ . The investment in accelerating the cost
reduction decreases monotonically as the coalition share increases in the MOB regime; when
the regime shifts to MOEB it jumps up; it decreases as the coalition size expands further
until z = z0 .
The strategically optimal technology policy for the regulated region has the subglobal
coalition always investing less in reducing the cost of the backstop than a global planner
would like. Instead, the coalition relies more on carbon pricing, which is more effective at
shifting rents. As a result, the emissions budget set by the coalition for itself is strictly
smaller than the cumulative cap the global planner would set for the regulated region. In
essence, the coalition is more concerned about capturing rents, while the global planner
is more concerned about emissions leakage and the inefficiency of divergent carbon prices
between regulated and unregulated consumers.
The coalition preference for stricter pricing policies imposes costs on the other stakeholders. Figure 8 shows that impacts on extractors are particularly large for mid-range coalition
shares. Extractor wealth decreases modestly as the coalition size increases if the policy pair
is set to maximize social welfare. However, if regulated consumers choose their optimal policy pair, extractor wealth decreases sharply as the coalition size increases when the coalition
is small, but regains much of the losses when the coalition reaches sufficient size such that
regulated consumers use all available pools. Technology-only policy is the preferred option
of extractors, but barring that, they would prefer a large (but not quite complete) coalition.
Figure 9 shows that unregulated consumers are also best off with technology-only policy.
While their welfare declines monotonically as the coalition size increases, and it is uniformly
31
Figure 7: Optimized Policies for a 60% Cumulative Emissions Reduction
CO2 price (τ)
Coalition policy (MOB)
25
τˆ
20
Coalition policy (MOEB)
15
10
5
Planner policy (MOEB)
20
40
60
80
100
Coalition size (%)
Acceleration of technical change ((z – z0)/z0)
2.0
Technology policy
1.5
Planner policy
(MOEB)
Coalition policy
(MOB)
1.0
Coalition policy
(MOEB)
0.5
20
40
60
80
100
Coalition size (%)
32
Figure 8: Effects on Extractors of Preferred Policy Combinations
% Change
Coalition size (%)
20
40
60
− 26
80
Technology policy
Planner policy (MOEB)
− 28
Coalition policy
(MOEB)
− 30
− 32
100
Coalition policy
(MOB)
− 34
− 36
− 38
Figure 9: Effects on Consumers of Preferred Policy Combinations
Change in welfare per unregulated consumer (%)
10
5
20
40
60
80
100
–5
- - Global policy
— Coalition policy
–10
–15
Change in welfare per regulated consumer/taxpayer (%)
33
lower with the coalitions preferred policy mix than with the socially preferred one, the
difference is not so large, as they also benefit from lower resource rents.
On the other hand, regulated consumer welfare (which accounts for taxpayer impacts)
increases monotonically with coalition size (with the exception of a discontinuity at the
point of regime change from MOB to MOEB). It is uniformly higher (by definition) with
their preferred policy mix than with the socially preferred policy pair. Interestingly, when
the coalition is large enough, regulated consumer welfare increases relative to no policy.
Intuitively, the carbon revenues offset the costs to consumers from the CO2 price, while they
enjoy lower pre-tax oil prices. But despite this boon, unregulated consumers are still better
off than regulated ones, reflecting a cost to committing to the coalition. 47
Given some form of global commitment to the climate goal, the gap between the regulating region’s preferred policy mix and the cost-effective one indicates some room for
bargaining over technology policy. For example, if 50% of consumer-taxpayers are in the
regulated region, their cumulative spending on subsidies is $45 trillion, compared to $53 trillion in the cost-effective policy combination. Extractors would be $0.4 trillion better off with
the cost-effective policy and $0.5 trillion better off with the technology-only policy than with
the preferred policy of the coalition. Unregulated consumers would be $3.4 trillion better
off with the cost-effective policy and $5.6 trillion better off with the technology-only policy.
Thus, facing a committed coalition, both of these stakeholder groups should be willing at
least to help finance more cost reductions in the backstop technology and encourage lower
carbon prices in the regulating region.
6
Conclusion
The Paris Agreement reaffirms a global commitment to meeting a carbon budget, and although all parties to the Agreement have agreed to contribute some effort, only a subset
are approaching that effort with carbon pricing. In this paper, we present a stylized model
of the world oil market in order to analyze the policy choices the parties face in reducing
global carbon emissions. We divide the world’s consumers into two groups, those whose
emissions are regulated and those whose emissions are unregulated. We assume regulated
consumers pay an emissions price that rises at the rate of interest, such as may arise under
a cap-and-trade system with bankable permits. We assume various liquid fuels are perfect
substitutes and hence consumers purchase at any time the cheapest one then available. In
the theory section, consumers have three choices, two exhaustible and one inexhaustible.
47
The problem of free-riding is known to be endemic in climate policy. See., e.g. Nordhaus (2015), Barrett
(2005).
34
In the calibrated simulation section, consumers have six choices, five exhaustible and one
inexhaustible. The inexhaustible backstop is clean and its marginal cost declines over time
because of technical change. The other sources are exhaustible and differ in their reserves
and their carbon content. Although initially the clean backstop costs more per unit to produce than even the most costly exhaustible, eventually the clean backstop becomes cheaper
to produce than even the lowest-cost exhaustible.
We investigate the effects of three policies: expanding the size of the regulated sector,
increasing the tax imposed on the regulated sector, and speeding up cost reductions of the
clean backstop. The three policies are substitutes in curbing cumulative carbon emissions.
Hence, if one can achieve a cumulative emissions target with one set of policies, one can also
achieve that same target by reducing the emissions price and increasing sufficiently either
the size of the regulated sector or the speed of technical change. We point out, however, that
although such policy substitutions keep emissions at the target level, agents in the model
prefer different combinations of policies. Extractors prefer greater reliance on technical
change in the backstop than on carbon pricing. And given a rate of technical change, they
prefer broader coalition to a smaller one with higher carbon prices. Consumers, all else equal,
benefit from lower rents, which should conflict with the preferences of extractors; however,
consumers also benefit from cost reductions in the backstop technology and taxpayers benefit
from emissions revenues.
The parameterized model based on real-world reserves and consumption data reinforces
and augments our theoretical results and highlights the challenges of staying within a carbon
budget in a global policy framework such as that set out in the Paris Agreement. Global
carbon pricing is the least-cost solution to the problem. Regulated consumers benefit, since
the latter actually gain if they are the beneficiaries of the tax revenue. However, members
outside the coalition still stand to gain more, as they would enjoy lower oil prices without
having to restrict their consumption. Even with a global consensus to address climate change,
the heterogeneous circumstances of countries in the real world, as well as the principle of
CBDR–RC, mean we cannot expect a global carbon price to emerge in the timeframe needed
to manage the global carbon budget. As a result, if one believes the carbon budget will be
met, it will have to be met with some combination that includes technology policy.
Although a group of high-ambition countries can help the world meet deep reduction
targets, without rapid technical change in alternative energy sources, that coalition still
needs to be very large—more than just the developed countries—and that effort comes at a
significant cost to coalition members and extractors alike. Both would likely prefer to invest
in technology policy rather than implement prohibitively high carbon pricing. Unregulated
consumers also benefit from lower-cost technologies in the future, which offsets the smaller
35
benefits from oil price adjustments in the near term. These dynamics may help explain why
we observe greater reliance on technology-oriented policies than on carbon pricing, despite
prior concerns about the green paradox.
36
Appendix: Comparative Statics When Extraction Is Incomplete and Emission Factors Are Unequal
In reality, resources that are more costly to extract often have higher emissions factors. If the
difference in these factors is trivial, the results in Section 3 still hold. But if the difference
is sufficiently large, strikingly different results emerge. As we will show, when emissions
intensities differ, we observe the possibility of negative leakage with all policy options.48
In an (LHB, LHB) equilibrium, from equations (??) and (5), unregulated consumers
U
switch to the high-cost resource when cL + λL erxH = cH , while regulated consumers switch
R
R
when cL + λL erxH = cH + τ (µH − µL )erxH ; thus, when µH > µL , regulated consumers switch
later to the higher-cost, higher-emissions resource, as depicted in Figure 10.
In an (LHB, LB) equilibrium, on the other hand, the regulated consumers may switch
to the backstop before the unregulated consumers switch to the high-cost pool, as seen in
Figure 11.
A.1 An Increase in the Emissions Price
When regulated consumers use some of the high-cost pool (LHB, LHB), an increase in the
emissions price would, in the absence of a change in λL , raise what the regulated consumers
must pay for oil from the high-cost pool by more than it raises the price of oil from the lowcost pool. As a result, regulated consumers would use the low-cost pool longer, although they
would consume it uniformly more slowly because it is more expensive. In this circumstance,
it is possible for the cumulative demand of the regulated consumers for the low-cost oil to
increase.49
Whenever an increase in the emissions price causes regulated consumers to demand more
low-cost oil, λL will increase, raising the price for the unregulated consumers as well. They
will then use it more slowly, switching earlier to the high-cost pool (since it continues to
sell for cH ). Since they switch to the backstop at the same time as before but commence
consuming the high-cost pool sooner, unregulated consumption of the higher intensity pool
increases. The net effect on emissions from unregulated consumers is ambiguous, since they
consume less low-cost oil but more high-cost oil; indeed, if the relative intensity of the latter
48
We have not yet been able to rule out the theoretical possibility that greater policy stringency (when it
causes rents on the low-cost oil to rise) may increase total emissions.
49
To see this, consider the extreme case in which the low-cost pool has a zero emissions factor (µH > µL =
0). Consumers in both regions would then pay the same rising price of low-cost oil, but regulated consumers
would continue to use it after the unregulated consumers had switched to the high-cost oil. In an absence
of a change in the Hotelling rent on the low-cost pool, a tax increase would further delay the time when
the regulated consumers switched to the high-cost pool and their cumulative demand for low-cost oil would
increase. Since the size of the low-cost reserve is unchanged, the rent on it would therefore have to rise.
37
Figure 10: Unequal Emissions Factors and (LHB, LHB)
$ BOE
Backstop cost
Price path,
Regulateds
Price path,
Unregulateds
xUH
xHR
Time
Figure 11: Unequal Emissions Factors and (LHB, LB)
$ BOE
Backstop cost
Price path,
Regulateds
Price path,
Unregulateds
Time
38
is high enough, their emissions will increase. Regulated consumers extract more of the lowcost oil but at a slower rate, switching later to the high-cost oil and then earlier to the
backstop; they necessarily reduce their emissions. Cumulative consumption of the low-cost
oil is unaffected, but now cumulative consumption of the high-cost pool is ambiguous.
We are investigating whether cumulative emissions can ever increase in response to an
increase in the emissions price. In the (LHB, LB) case, an increase in the emissions price
can no longer raise the rent on the low-cost reserve: if it did, both types of consumers would
utilize the low-cost reserve less intensively and would abandon it sooner, meaning cumulative
demand for the low-cost reserve would no longer equal the unchanged stock. Similarly, the
rent on the low-cost reserve cannot fall so much that it outweighs the increase in the tax per
barrel (τ µL ): a reduction in λL + τ µL would raise cumulative demand above the unchanged
stock. Hence, after the increase in the emissions price, the Hotelling rent must fall, but the
reduction must not outweigh the increase in the tax per barrel. In response, the unregulated
consumers deplete the resource at a more rapid rate and switch to the high-cost resource at a
later date. Since the timing of their switch to the backstop is unchanged, they consume less
of the high-intensity resource. Meanwhile, because the price paid by regulated consumers
for the low-cost oil is uniformly higher than before the tax increase, the regulated consumers
switch to the backstop sooner. Since the low-cost resource is exhausted but less of the
high-cost resource is utilized, aggregate carbon emissions fall.
It is possible, however, that the unregulated users themselves reduce their cumulative
emissions. To see that this is possible, note that µH > µL and can be scaled up to any
extent without altering either group’s consumption behavior, since in the (LHB, LB) case,
the regulated consumers use no high-cost oil. The tax increase causes the unregulated sector
to increase its cumulative depletion of the low-cost oil but to reduce its depletion of the highcost oil. The emissions factor of the high-cost oil can be so large that the unregulated users
reduce their cumulative emissions from the high-cost pool by more than they increase their
emissions from the low-cost pool. Whenever that occurs, the response of the unregulated
consumers reinforces— rather than undermines—the attempts of the regulated region to
reduce aggregate carbon emissions. This represents another example of negative leakage.
A.2 An Increase in the Speed of technical change in the Backstop
When the regulated consumers use some of the high-cost resource (LHB, LHB), speeding
up cost-reducing technical change does not affect the rents on the low-cost pool (λL ). Every
consumer in each group switches to the backstop sooner and consumes less of the high-cost
reserve. Since cumulative emissions from the low-cost pool are unchanged, total emissions
fall. Once again, the unregulated consumers cause negative leakage, which is enhanced be39
cause of the unequal emissions factors. When the regulated consumers switch from the
low-cost pool directly to the backstop (LHB, LB), speeding up technical change would, in
the absence of a change in the Hotelling rent, decrease cumulative demand for the low-cost
resource. The Hotelling rent must therefore fall, and this causes unregulated consumers to
deplete the low-cost pool at a more rapid rate and to switch later to the high-cost resource.
Since they use more of the low-cost oil, the regulated consumers must use less of it. Regulated consumers accomplish this by using low-cost oil for a shorter interval, albeit at a faster
rate. Emissions reductions from the regulated consumers are exactly offset by the regulated
consumers’ increase in emissions from the low-cost pool. Since, however, the unregulated
consumers reduce their emissions from the high-cost pool, total emissions fall. If the emissions factor of the high-cost pool is sufficiently large, another case of negative leakage would
occur.
A.3 An Increase in the Coalition Share
When regulated consumers use part of the high-cost reserve (LHB, LHB), they deplete the
low-cost resource at a (weakly) slower pace but for a longer time. Expanding the coalition of
those subject to the emissions price may therefore raise the cumulative demand for the lowcost resource just as in the case of the emissions price.50 Whenever that occurs, expanding
the size of this group will drive up the Hotelling rent (λL ). Consumers in both the regulated
and the unregulated regions will deplete the low-cost pool at slower rates and for shorter
intervals before switching (albeit at different times) to the high-cost resource. Hence, the
per person cumulative consumption of low-cost oil falls in both regions. However, since more
consumers are now in the region with the higher cumulative demand, cumulative demand
for the low-cost oil continues to match the unchanged stock, and emissions from the lowcost pool remain µL SL . Meanwhile, each consumer who was previously taxed increases his
emissions from the high-cost pool, as does every person who remains untaxed. This does
not mean, however, that cumulative emissions necessarily increase, since consumers who
are now taxed but who were not previously will (nonmarginally) cut their emissions from
the high-cost pool. (In the numerical examples we have explored, this last effect always
dominates and cumulative emissions fall. We conjecture that this is true in general and are
in the process of investigating.)
Now consider the case in which regulated consumers switch from the low-cost pool to
the green backstop without ever utilizing the high-cost reserves (LHB, LB). Increasing the
50
To see that this is possible, suppose again that only the high-cost resource is dirty (µH > µL = 0). In
that case, regulated consumers use the low-cost resource at the same rate as the unregulated consumers but
switch later to the high-cost resource and hence have a larger cumulative demand for the low-cost resource.
40
coalition share now has different competing effects. On the one hand, since prices are higher,
regulated users consume the low-cost resource at a lower rate than their unregulated counterparts On the other hand, regulated users may abandon the low-cost pool (for the backstop)
sooner or later than the unregulated users abandon it (for the high-cost pool). As a result,
expanding the coalition may increase or decrease cumulative demand for the low-intensity
resource. The effect on unregulated emissions depends on the net effect on the Hotelling rent
(λL ) and the relative emissions intensities. If this rent decreases, the unregulated consumers
will use the low-cost resource at a more rapid rate but will switch to the high-cost resource
at a later date. If the emissions factor on the high-cost pool is sufficiently large, negative
leakage will occur.51 On the other hand, if λL increases, the unregulated consumers use less
of the low-cost resource and will switch to the high-cost resource sooner. Positive leakage
results if the emissions factor on the high-cost pool is sufficiently large. We are investigating
whether increasing the share of consumers subject to regulation can ever raise emissions
because the increase in the consumption of the high-cost pool by the unregulated consumers
who remain may outweigh the reduction in consumption by consumers who switch to the
regulated coalition.
A.4 Numerical Results with Differentiated CO2 prices
We previously considered a standard CO2 price based on the carbon content of oil. However,
one can also envision a differentiated CO2 price that would impose a higher burden on
resources with higher emissions factors, as with a life-cycle approach.52 Such a mechanism
would include the upstream emissions from extraction, as well as the downstream emissions
from utilization. In the Appendix, we discuss how the equilibria are affected by a CO2 price
that takes into account different emissions factors among oil from different sources. Here,
we use the numerical model to compare the standard and differentiated CO2 prices; both
are constant in present-value terms, as in the theory.
Does differentiating among emissions factors matter? We find that the choice of basis for
the CO2 charge does not affect global welfare, as measured by total surplus. In effect, the
cumulative emissions constraint is still tantamount to a cumulative stock of consumption
constraint, taking emissions factors into account. The tax choice does, however, affect the
rents of the lower-cost resource owners. By discriminating against higher-emitting EOR
oil, the differentiated CO2 price puts less downward pressure on conventional oil rents. For
51
Recall that when emissions intensities are equal this scenario leads to higher emissions among the unregulated consumers.
52
Life-cycle analysis is used in some low-carbon fuel standards, including California’s, which assigns higher
emissions intensities to unconventional oil sources. In practice, fuel taxes and downstream emissions charges
are much more common than life-cycle taxes on oil.
41
example, with a global coalition, conventional oil rents they fall 25% with a differentiated CO2
price and 30% with a standard CO2 price. This $2 trillion present-value difference is gained by
taxpayers, who raise more revenues, and consumers, who benefit from lower initial oil prices
under the standard CO2 price. In the case of global taxes, where there are no unregulated
consumers, regulated consumers are held harmless, as the change in tax and rental rate
fully offset each other, and tax revenues increase. Thus, while conventional oil owners might
prefer a differentiated CO2 price, the implementing region prefers an undifferentiated one,
and the choice is neutral from a global perspective.
7
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