ISSN 1403-2473 (Print)  
ISSN 1403-2465 (Online) 
 
 
 
 
 
 
 
 
 
Working Paper in Economics No. 794 
 
 
 
Optimal fuel taxation with suboptimal health 
choices  
 
Simona Sulikova, Inge van den Bijgaart, David Klenert,  
Linus Mattauch 
 
Department of Economics, October 2020 
 
 
Optimal fuel taxation with suboptimal health choices
Simona Sulikova Inge van den Bijgaart David Klenert
Linus Mattauch∗
October 16, 2020
Abstract
Transport has a large number of significant externalities including carbon
emissions, air pollution, accidents, and congestion. Active travel such as cycling
and walking can reduce these externalities. Moreover, public health research
has identified additional social gains from active travel due to health benefits
of increased physical exercise. In fact, on a per mile basis, these benefits
dominate the external social costs from car use by two orders of magnitude.
We introduce health benefits and active travel options into an optimal taxation
model of transport externalities to study appropriate policy responses. We
characterise the optimal second-best fuel tax analytically: when physical exercise
is considered welfare-enhancing, the optimal fuel tax increases. Under central
parameter assumptions it rises by 49% in the US and 36% in the UK. This is
due to the low fuel price elasticity of active travel. We argue that fuel taxes
should be implemented jointly with other policies aimed at increasing the uptake
of active travel to reap its full health benefits.
Keywords: Transport Externalities; Congestion; Active travel; Fuel; Health
Behaviour; Optimal Taxation
JEL Codes: H23, I12, Q58
∗Sulikova: University of Oxford, simona.sulikova@ouce.ox.ac.uk. van den Bijgaart: University of
Gothenburg, inge.van.den.bijgaart@gu.se. Klenert: Joint Research Centre, European Commission,
david.klenert@ec.europa.eu. Mattauch: University of Oxford, linus.mattauch@inet.ox.ac.uk. We
thank seminar audiences in Berlin, Oxford, ISER Osaka, at EAERE, IIPF and Elizabeth Baldwin,
Geir Bjertnaes, Christian Brand, Bart DeFloor, Felix Creutzig, Marc Fleurbaey, Franziska Funke,
Reyer Gerlagh, Cameron Hepburn, Zarko Kalamov, Sebastian Kraus, Ian Parry, Ryan Rafaty, Gregor
Schwerhoff and Jiaxin Zhao for helpful comments.
1 Introduction
Transport policies need to balance the economic gains from vehicle use with a large
number of significant externalities, including air pollution, accidents, congestion, and
climate change. For example, in the US, UK, and France, the transport sector is now the
largest contributor of greenhouse-gas emissions (Hockstad and Hanel, 2018; Gabbatiss,
2018; Lepoutre, 2018). Increased active travel such as cycling and walking – even to
the nearest public transport stop – can reduce these externalities, especially in urban
areas. An additional benefit from active travel is that physical exercise is beneficial
for health, especially given high rates of inactivity and obesity in many populations.
Previous scenario-based modelling in public health has indicated that the health benefits
from active travel due to increased exercise are larger than the benefits from abating
emissions and air pollution of private vehicles (Woodcock et al., 2009; Maizlish et al.,
2017; Wolkinger et al., 2018; De Hartog et al., 2010). For example, Woodcock et al.
(2009) find that an increased active travel scenario would avoid 530 premature deaths
per million population in London annually, while a lower-carbon-emission motor vehicles
scenario would only save 17 through lower air pollution exposure.
Surprisingly, economists have yet to examine the significance of these health benefits
from active travel for optimal regulation of urban transport. Many citizens are not aware
of the full health benefits exercise provides (Fredriksson et al., 2018; San Diego and
Merz, 2020). For instance, the effectiveness of simple interventions such as reminders to
go to the gym (Calzolari and Nardotto, 2017), initial payments (Charness and Gneezy,
2009), and evidence of overspending on gym contracts (DellaVigna and Malmendier,
2006) point to self-control problems and an underappreciation of the health benefits
of exercise, especially before they materialise. Therefore, the health benefits of active
travel make passenger transport an unexamined case of a “behavioural-environmental
second-best problem” (Shogren and Taylor, 2008). It is, however, yet to be determined
whether instruments such as fuel taxes are appropriate to reap the health benefits from
1
increased active travel in addition to mitigating the externalities of car use.
In this article, we examine a novel economic effect by adding an active travel mode
to a model of transport externalities from car use. Households respond to higher fuel
taxes by buying more fuel-efficient cars and reducing car travel, and additionally shift to
alternative models of travel that involve exercise, such as walking or cycling. However,
they do not fully internalize that they get healthier by adopting such active modes
of travel travelling. We confirm that, on a per mile basis, the monetary value of the
health benefits from active travel exceeds the social costs of unregulated externalities of
carbon emissions, air pollution, congestion, and accidents by two orders of magnitude.
Any first-best policy would thus involve a large subsidy to promote active travel. In
the absence of such subsidies, we derive the optimal second-best fuel tax that corrects
for the externalities and the unrealised health benefits. We examine the difference for
the tax rule and quantify the appropriate tax rate both including, and excluding health
benefits from active travel.
Importantly, we find that the optimal tax increases by 49% in the US and 36%
in the UK when health benefits from physical exercise are included. The second-best
optimal fuel tax for the US is $10.13/gal, and $6.81/gal without physical inactivity
costs, while the current rate in the US is $0.55/gal (American Petroleum Institute,
2020). The optimal fuel tax for the UK is $4.54/gal, which is somewhat higher than
the current rate of $4.06/gal (RAC, 2020). Without physical activity costs, the optimal
second-best tax would be $3.35/gal.
Previous work established that the external costs of transport are not fully reflected
in fuel price; fuel taxes are inefficiently low in most European countries (Santos, 2017),
and US states.1 Accounting for the physical health benefits from active travel thus
further increases the gap between actual and optimal fuel taxes. Our sensitivity analysis
shows that the optimal second-best tax varies significantly within the range of realistic
1An exception is Parry and Small (2005), who claim that UK fuel taxes are too high.
2
parameter values, from $6/gal to $13/gal for the US, and from $3.75/gal to $7/gal for
the UK. Hence, neither the UK nor the US current fuel taxes likely exceed the optimal
rate. Varying the Social Cost of Carbon (SCC) results in a range of $9.73-14.3/gal for
the US, and $4.03-8.12/gal for the UK. Varying the physical inactivity cost results in
a range of $8.80-11.72/gal in the US, and $4.31-7.60/gal for the UK.
On a per mile basis, physical inactivity represents the largest social cost of motorised
private transport. Nevertheless, the “active travel adjustment” of the optimal fuel tax
is comparatively small: a fuel tax is a fairly inefficient instrument to address these
high costs because the uptake of active travel is not highly responsive to fuel price. A
greater responsiveness of active travel to fuel taxation would increase the optimal tax
adjustment. Indeed, it is often claimed in urban planning and transportation research
that pricing car use should be complemented with infrastructure re-development to
deliver feasible and effective transport solutions (Banister, 2008; Pucher et al., 2010;
Buehler et al., 2017). The broader insight from our article is therefore that, when it
is acknowledged that individuals’ health decisions are not always welfare-maximising,
this conclusion is supported from an economic point of view.
This manuscript builds on three distinct strands of literature: First, a large body
of literature studies optimal levels of fuel taxes, and which externalities should be
addressed by them (van Essen et al., 2019; Sansom et al., 2001). In addition to
generating government revenue, fuel taxes are typically used for the purpose of reducing
most forms of non-priced costs of transport, e.g. the externalities of carbon dioxide and
particulate matter, or reducing congestion by raising the cost of driving. Parry and
Small (2005) derive the optimal gasoline taxes for the US and Britain, accounting
for congestion, accidents, carbon emissions and air pollution, and Anto´n-Sarabia and
Herna´ndez-Trillo (2014) apply it to Mexico. Sterner (2012) compares the optimality
of fuel taxes in Europe and US, West and Williams III (2007) estimate an optimal
fuel tax and its influence on labour supply in the US, and Hirte and Tscharaktschiew
3
(2015) compare the fuel tax composition across US states and cities. Fuel taxes vary
considerably between countries. In 2017, the fuel tax raised on a gallon of unleaded
gasoline in the US was $0.55, and $3.75 in the UK (American Petroleum Institute, 2020;
RAC, 2020).2 Yet, the optimal fuel tax literature has so far not considered the health
benefits from active travel.
Second, the field of public health, starting with Woodcock et al. (2009), has identified
high social benefits from active travel over and above the benefits from abating emissions
and air pollution of private vehicles (De Hartog et al., 2010; Rabl and De Nazelle, 2012;
Wolkinger et al., 2018). To the majority of the population, increasing physical activity
outweighs the negative impacts of increased exposure to air pollution (Tainio et al.
(2016), Mueller et al. (2015)). This is due to the overwhelmingly sedentary lifestyles
that people in both the UK and US lead, making physical inactivity a leading risk
factor for 6 of the 10 largest causes of death worldwide (WHO, 2018). Most UK adults
do not exercise regularly (37% never, 16% less than once a week, 57% admit they never
do activity strenuous enough to be out of breath, Eurobarometer 2018). This leads to
significant costs including higher rates of disease incidence, lower quality of life, loss
of income, excess healthcare costs, and productivity losses in the workplace. We build
on the valuation methods in public health to quantify the welfare cost of travel that
is inactive. The social costs of inactivity can be derived from quantifying the health
benefits of physical activity and the value of a statistical life, once it is acknowledged
that inactive citizens impose a cost on themselves.
Third, behavioural public economics research has elaborated on the important role
of “internalities” in various domains of public policy (Allcott and Sunstein, 2015). An
“internality” occurs when an individual imposes a significant cost on herself due to
behavioural failures.3 As these private costs are imposed only or mainly on oneself
2For the UK, this is the sum of the fuel and the excise taxes. For the US this is the average over
the different federal and state taxes.
3While this raises intricate issues on diverging normative positions about individual well-being,
in this article we follow the treatment of “internalities” in recent behavioural public economics. An
4
– which is true for lack of physical activity –, they fall outside the definition of an
externality, which is a cost imposed on others. Nonetheless, governments regulate to
increase welfare by reducing internalities, in cases where scientific evidence substantiates
it. Internality taxes have been applied to the market for smoking (Gruber and
Ko˝szegi, 2004), gym memberships and exercise (in the form of subsidies, DellaVigna
and Malmendier, 2006), sugary drinks (Allcott et al., 2019a; Allcott et al., 2019b) and
the energy and automobile market (Allcott and Wozny, 2014; Allcott and Sunstein,
2015), where they also interact with environmental externalities. In the latter case,
the interaction leads to a behavioural-environmental second-best problem (Shogren
and Taylor, 2008). Chetty (2015), Bhargava and Loewenstein (2015), Allcott and
Sunstein (2015), and Allcott et al. (2019b) all provide more extensive discussions of
why regulating internalities is desirable, arguing that the complexity of choices people
face, and large internal costs in e.g. health and energy efficiency, warrant the greater use
of behavioural economics in regulation. Sin taxes, surcharges on prices of goods of which
people consume too much because of internalities, have been modelled as either simple
extensions of a Pigouvian tax (O’Donoghue and Rabin, 2006), or as complex interactions
between taxes and individuals’ heuristics and decisions, to achieve an optimal outcome
in second-best settings (Allcott et al., 2014).
However, this body of literature has not considered the internality of physical
inactivity in urban transport. Walking, cycling, and switching to public transport are
considered ways in which people can achieve “appropriate” levels of physical activity
as prescribed by public health guidelines (Watts et al., 2019; Gibson-Moore, 2019;
OSG, 2015). The public health evidence supporting the health benefits of physical
activity refers to it as a “miracle cure” (Davies et al., 2019) in the UK Chief Medical
Officers’ 2019 Physical Activity Guidelines. Meeting the minimum recommendations
alternative normative viewpoint, which we are not assessing in this article, would be to evaluate how
urban transport policy is conducive to improving health, which seems predominant in the public health
assessments we build on. For what could lead to different welfare evaluations of behavioural failures
see Crisp, 2017; Fleurbaey and Blanchet, 2013; Mattauch and Hepburn, 2016.
5
of 150 minutes of moderate-intensity physical activity per week has significant health
benefits, which include reduced risk of cognitive impairment and dementia (Loprinzi
et al., 2018), almost halving the risk of depression (Catalan-Matamoros et al., 2016), a
6% reduction in the risk of hypertension (Liu et al., 2017), a reduction in the risk of type
2 diabetes (Smith et al., 2016), cardiovascular disease, as well as higher bone mineral
density (Onambele-Pearson et al., 2019). This is true even for exercise carried out in
areas with elevated pollution levels (Tainio et al., 2016). It is well-known that people
under-value the contribution of physical exercise to their long-term health (Zamir and
Teichman, 2014). Even when citizens are aware of the health benefits from exercise, they
most often claim not to have the time for it or not be interested in it (Eurobarometer,
2018). This reinforces the case for building active travel into their commuting routines.
Similar to Allcott et al. (2019b), one can identify two behavioural biases influencing
people’s decisions not to exercise sufficiently: imperfect information and insufficient
self-control. These lead to insufficient levels of exercise and further health impacts.4
Our contribution to the literature is threefold: First, and most importantly, we
introduce a physical activity-related health internality into an established framework
of transport decisions (Parry and Small, 2005), and use this behavioural-environmental
framework to provide an analytical solution for the optimal second-best fuel tax.
Second, we provide an updated quantification of the external costs of travel provided
by Parry and Small (2005), considering recent research and global climate policy goals,
and complement this with a quantification of the health benefits of active travel. For
example, updating the carbon price estimates increases the contribution of fuel pollution
to the optimal fuel tax by an order of magnitude; conversely, the relative contribution of
4Allcott et al. (2019a), Handel and Kolstad (2015) and Bronnenberg et al. (2013) all follow a
similar set-up. Allcott et al. (2019a) describe their approach as one where they assume a hypothetical
“counterfactual normative consumer,” who has a sophisticated level of knowledge and no self-control
problem, and assume that only their choices are welfare relevant. For the case at hand, this would
translate to an individual who knows about the health benefits of physical activity and has sufficient
self-control to exercise. Given enough time, she thus would carry out at least the minimum weekly
recommended amounts of exercise according to WHO guidelines.
6
air pollutants to the fuel tax falls. Third, in terms of policy implications, we contribute
to evaluating the potential use of a fuel tax as opposed to other travel management
policies. We confirm that raising the propensity of consumers to switch to active
travel modes can greatly impact the appropriate fuel tax: the demand for vehicle miles
travelled (VMT) is so inelastic that increasing appropriate elasticities to their upper
bound found in the literature raises the fuel tax by up to 62% for the UK and 155%
for the US.5
With increasing congestion, air pollution, and awareness of climate change, a number
of cities worldwide have declared zero emissions zones, congestion charges, number-plate
policies, or bans of certain types of vehicles in inner city areas – but while such local
policies are attractive for addressing specific transport externalities, we focus on fuel
taxes as they have some distinct advantages. First, since most countries already have
fuel taxes in place and levels would only have to be adjusted accordingly, they can
be implemented with relatively small administrative costs compared to infrastructure
subsidies, physical exercise subsidies, or congestion fees. Second, fuel taxes and
pricing instruments have a proven track record of reducing carbon emissions (Brand
et al., 2013; Bretschger and Grieg, 2020; OECD, 2019; Sterner, 2012). Third, they
generate government revenue, which could be used either for green spending on projects
that further reduce emissions, including in low-carbon transport infrastructure, or for
compensating low- and middle-income households that are especially affected by the
tax. Both measures could make the public more supportive of fuel taxation (Klenert
et al., 2018).
The remainder of the paper is organised as follows. Section 2 describes the model,
and our analytical result for the optimal fuel tax. Section 3 explains our choice of
parameterisation. Section 4 presents the quantitative results and Section 5 discusses
5It may be argued that, especially in the US case, realising such an increase is politically unrealistic
in the foreseeable future. Still, our result indicates that current fuel tax rates are further below their
preferred levels than previously thought, which, as we discuss in Subsection 4.2, implies fuel tax
increases have greater benefits.
7
the policy implications. Section 6 concludes.
2 Analytical Framework
2.1 Model
To explore how the fuel tax might be optimally adjusted to account for health benefits
of public transport, we consider a stylised framework of travel decisions. We take
advantage of the fact that in certain settings, internalities can be treated as extensions
of externalities (O’Donoghue and Rabin, 2006). This allows us to base our framework
on Parry and Small (2005), and extend it to account for active travel decisions and
associated health benefits.
We consider a representative agent with the utility function
U = u(ψ(C,M, T in, T ac, G), N)− ϕ(P )− δ(A) + ξ(Q), (1)
where C is the quantity of numeraire consumption, M total distance travelled, T in and
T ac is total time travelled using active and inactive modes respectively, G exogenous
government spending, and N leisure, with UC , UM , UG, UN > 0, and UT in and UTac < 0,
with the subscript denoting a partial derivative. The level of pollution is denoted by
P , A captures accidents, and health is denoted by Q. As Parry and Small (2005), we
assume u(·) and ψ(·) are quasi-concave, and ϕ(·) and δ(·) are convex. The functions
ϕ(·) and δ(·) capture the dis-utility from pollution and accidents, respectively. We add
the concave function ξ(·), which captures the positive utility from health Q.6
Total travel M can be separated into two components, inactive travel M in and active
travel Mac.
M = M in +Mac. (2)
6Equation (1) models the utility from health and leisure as separable. As a consequence, any
improvement in health will leave the labour-leisure trade-off unaffected.
8
Inactive travel denotes travel using modes that require very little physical activity,
most importantly using the car. Active travel instead captures walking and cycling.
We also consider public transport as an active mode of travel, as it typically requires
the individual to walk or bike to the bus stop, tram stop, or train station (Rissel et al.,
2012), in some cases providing up to 30% of daily exercise recommendations (Besser
and Dannenberg, 2005). As such, active travel requires spending S, which will be
further specified below. Inactive travel distance M in requires fuel F and other travel
inputs H: M in = χ(F,H). In line with Parry and Small (2005), we assume that M in
is homogeneous of degree one with respect to its inputs. This specification allows for
multiple channels of substitution. For instance, as fuel prices increase, the consumer
can decide to i) reduce total distance travelled, M , ii) spend more on other inactive
travel inputs, H, such as purchasing a vehicle with higher fuel economy, or iii) increase
active travel distance, Mac.
The agent spends time T in in inactive travel. For a given distance M in, this time
is increasing in the amount of congestion on roads, which we take as an increasing
function of the population average inactive miles travelled, M¯ in:
T in = piin(M¯ in)M in, (3)
where piinM¯ in > 0, with the subscript denoting a partial derivative, and the bar indicating
that we consider a population average. piin is equal to the inverse of speed of inactive
travel, which we assume the agent takes as exogenous. In equilibrium, M¯ in = M in.
For active travel we abstract from congestion,7 and model time travelled as directly
proportional to distance:
T ac = piacMac, (4)
with piac the inverse of speed from active mobility. Only inactive travel contributes to
7Even though public transport can get congested, this does not typically increase travel time.
Bicycle paths do not generally get congested to the extent that travel time increases.
9
pollution, both in the form of carbon dioxide emissions, and local air pollution. CO2
emissions are directly proportional to fuel use. To capture local air pollution effects,
inactive miles travelled offer a better proxy (Hitchcock et al., 2014).8 This allows us to
write
P = P f (F¯ ) + Pm(M¯ in), (5)
with P fF¯ > 0 and P
m
M¯ in > 0. Similar to congestion, we assume the agent will take
pollution as given; she will not internalise the effect of travel decisions on the population
averages F¯ and M¯ in.
Both active and inactive travel are subject to accident risk. We separate accident
costs associated to active and inactive travel. For inactive travel, accident costs are
increasing with the amount of travel. As travel increases, the agent also imposes an
“accident externality” upon other users: the higher average travel, M¯ in, the more likely
a road user will be involved in an accident. For active travel, we similarly assume
that higher travel increases the number of, and thereby costs of, accidents. Yet, roads
that are busier with cars tend to be more dangerous to both cyclists and pedestrians.
Conversely, there exists a so-called “safety in numbers” effect: more cyclists on the
road tend to make cycling safer overall (Elvik and Bjørnskau (2017), Kahlmeier et al.
(2017)). Hence, we assume that the accident costs associated with active travel are
increasing in the average amount of inactive travel, M¯ in, and decreasing in average
active travel, M¯ac. This gives
A = Ain(M in, M¯ in) + Aac(Mac, M¯ in, M¯ac), (6)
with AinM in > 0 and A
in
M¯ in > 0. Likewise, A
ac
Mac > 0, and A
ac
M¯ in > 0, while A
ac
M¯ac < 0.
We assume active travel is conducive to health. To capture this we write health as
8Substantial emissions of particulate matter from transport are due to tyre, brake, and road
abrasion, rather than fuel consumption. Fuel emissions contribute mostly to noxious gas emissions
such as NOx and ozone.
10
a function of active travel:
Q = Q(Mac, O), (7)
where O are other forms of exercise,9 with QMac > 0 and QO > 0. We assume
that the agent considers only a constant share ω ∈ [0, 1] of Q as relevant in her
optimisation problem. More specifically, instead of considering actual health Q, she
considers “perceived health”, Qper:
Qper = ωQ+ Q˜, (8)
where the agent considers Q˜ as outside of her control, while in reality, Q˜ = (1− ω)Q.
Whenever ω < 1, (8) represents the notion that the individual underestimates the effect
of exercise on health. This underestimation is consistent with substantive evidence that
individuals do not fully appreciate the positive effects of activity-related health.10 If
ω = 0, health is considered as fully exogenous and unaffected by exercise, while if ω = 1,
the agent accurately perceives health and the corresponding benefits of active travel.
With Equation (8) we adopt a specification of limited attention proposed by
DellaVigna (2009), which assumes that the benefit of completing travel (in active mode)
is “visible”, while the health benefit from active travel is “opaque”. This seems justified
as many citizens are largely unaware of the high health benefits of even short walks
(Steinhilber, 2017; Fredriksson et al., 2018; Bennett et al., 2009). Alternatively, the
unrealised health benefits from active travel could represent a case of time-inconsistent
preferences (Laibson, 1997; O’Donoghue and Rabin, 1999), where citizens highly
value their health, but repeatedly postpone undertaking exercise. DellaVigna and
Malmendier (2006) find evidence for the latter in the case for exercise in gyms. This
can be captured by an equivalent formulation of Equation (8) in our static model,
9Note that any adverse effect of pollution on health is already subsumed in ϕ(P ).
10Additionally, publicly financed healthcare systems and moral hazard in health insurance imply
that individuals may not bear the full cost of unhealthy decisions.
11
as the assumption of time-inconsistent preferences implies an activity level less than
desirable in the long term is pursued at any point in time. This holds in the absence
of commitment devices, which arguably do not exist for active travel.
The agent’s budget constraint is given by
C + (pf + tf )F + phH + poO + S = w
(
1− tl
)
L, (9)
where pf +tf is the consumer price of fuel, ph is the price of other inactive travel inputs,
and po is the price of other forms of exercise. In addition, active travel requires the
consumer to spend on items such as a bicycle or public transport. We denote by S any
such spending on active travel (with normalised price), with S = S(Mac), S(0) = 0
and SMac > 0. Finally, we denote the gross wage rate by w, and the labour tax rate
is given by tl. The total amount of time available is given by L¯, which is allocated to
labour L, leisure, N , and time spent travelling T in and T ac, such that
L+N + T in + T ac = L¯. (10)
In the remainder of this paper, we assume that all prices are exogenous and constant.
The fuel tax tf , will be set by the policymaker. The proceeds of the fuel tax will be
used to fund government spending G. The labour tax will in turn be set such that the
government budget constraint is binding:
G = tfF + tlwL. (11)
Throughout, we assume that there exists an unique and interior equilibrium, where the
consumer chooses strictly positive levels of C, F , H, Mac, O and L, and that G is such
that tl > 0.
12
2.2 Second-Best Fuel Tax
In the above setup, an increase in fuel use is associated with carbon emissions.
Additionally, higher fuel use increases the number of miles travelled, which increases
local pollution, and also congestion and accident risk. All these effects are not
internalised by the representative consumer; she takes the level of pollution, speed
of travel, and marginal accident risk as given. On their own, these externalities already
justify the introduction of a positive “externality tax” on fuel. Such a tax will be
welfare-improving, as it forces the agent to internalise (part of) the externality. In
addition to the externalities, the framework also features an “internality”: whenever
ω < 1, the agent underestimates the extent to which higher levels of active travel deliver
positive health benefits. Consequently, the choices of Mac and O, and resulting Q, may
be suboptimally low.
Our aim is to quantify how the consideration of these health benefits of active
travel affects the welfare-maximising (optimal) fuel tax. For this purpose, we derive
the solution for the optimal fuel tax, tf∗, and calibrate its value. We present the full
derivation of tf∗ in Appendix A, where we obtain the following result:
tf∗ = ZPF¯ +
[
ZPM¯in + ZC + ZAM¯in
]
(
−dM in
dtf
/
−dF
dtf
)
+ ZAM¯ac
(
−dMac
dtf
/
−dF
dtf
)
− (1− ω) Z˜Q
(
−dQ
dtf
/
−dF
dtf
)
− wtl
(
−dL
dtf
/
−dF
dtf
)
. (12)
Here we define
ZPF¯ ≡
ϕP
µI
P fF¯ ; Z
PM¯in ≡
ϕP
µI
PmM¯ in ; Z
C ≡ ΓinpiinM¯ inM
in,
and
ZAM¯in ≡
δA
µI
[
AinM¯ in + A
ac
M¯ in
]
; ZAM¯ac ≡
δA
µI
AacM¯ac ; Z˜
Q ≡
ξQ
µI
,
with Γin ≡ w
(
1− tl
)
− ψTinψC .
13
Equation (12) characterises the optimal fuel tax. This tax is equal to the sum
of un-internalised costs associated with fuel use. The first term in (12), ZPF¯ , is the
direct pollution externality of fuel use. It is equal to the marginal cost of pollution,
ϕP , multiplied by the effect of additional fuel use on pollution, P
f
F¯ , and converted to
consumption units using the shadow value of income, µI .
Next, higher fuel use is associated with more inactive miles travelled. The term
ZPM¯in + ZC + ZAM¯in captures the marginal externality cost of inactive miles travelled,
with ZPM¯in , ZC and ZAM¯in the cost associated with increased air pollution, congestion,
and accidents, respectively.11 The contribution of these costs to the magnitude of the
optimal fuel tax depends on the extent to which fuel taxes reduce miles travelled vis-
a`-vis fuel use. If the reduction in fuel use due to higher fuel taxes is associated with
a small reduction in miles travelled (small
(
−dM in
dtf /
−dF
dtf
)
), then only a small portion
of the externality costs associated with miles travelled can (implicitly) be attributed
to fuel use. Vice versa, if
(
−dM in
dtf /
−dF
dtf
)
is large, then the externality costs associated
primarily with miles travelled greatly contribute to the magnitude of the optimal fuel
tax.
Likewise, fuel taxes may lead to changes in active travel distance, which is associated
with accident externalities, with cost ZAM¯ac . The contribution of those costs to the
optimal fuel tax then depends on the relative response of active travel to fuel taxes:
(
−dM¯ac
dtf /
−dF
dtf
)
.
Our main effect of interest is (1− ω) Z˜Q
( dQ
dtf /
−dF
dtf
)
: the adjustment of the optimal
fuel tax to the health internality. Z˜Q is the marginal value of additional health, with
1 − ω the un-internalised portion, see sections 3.1.6 and 3.1.7. As can be seen from
(12), a high value of Z˜Q does not automatically imply that, once health internalities
are accounted for, the optimal fuel tax is adjusted much; this is only the case if the
fuel tax is an effective tool to increase health Q. Following (7), fuel taxes can affect
11The term Γin in ZC captures the notion that congestion is costly for two reasons: it creates a
direct disutility (see (1)) and reduces time available to allocate to labour (see (10)).
14
health through two channels: by changing active travel Mac, or through other forms of
exercise O. In the remainder of this article, and consistent with the empirical literature
(Martin et al., 2012), we will assume that the effect of fuel taxes on other forms of
exercise O is negligible. This implies we set dO/dtf = 0, and focus on changes in active
travel as the primary channel through which fuel taxation affects health.
The interpretation of the final remaining term is similar. Fuel taxes may also affect
labour supply. Even though the agent takes into account that higher labour supply
increases income, she does not internalise the positive effect of increased labour on the
government budget. This effect is equal to the wage, multiplied by the labour tax rate,
wtl. The contribution of this effect to the optimal fuel tax is larger the larger the
increase in labour supply in response to higher fuel taxes.
In the next section we quantify the optimal tf and the effect of considering the
health benefits of active travel thereon. To facilitate this quantification, we further
manipulate Equation (12) to
tf∗ =
MEC
1 +MEB
+
tl
(
pf + tf
)
1− tl
εcLL
(
1− ηM
inI
)
−ηFF
+
tl
1− tl
[
εLL − ε
c
LL
(
1− ηM
inI
)]
ZCβM
inM in
F
,
(13)
with εcLL and εLL the compensated and uncompensated labour supply elasticity. Akin
to Parry and Small (2005), this optimal tax is separated in three components. The
first component is the “adjusted Pigouvian tax,” equal to the marginal external cost
associated to fuel use, corrected by the marginal excess burden of labour taxation. The
marginal external cost of fuel use is given by
MEC ≡ ZPF¯ +
[
ZC + ZAM¯in + ZPM¯in
]
βM
inM in
F
+
[
ZAM¯ac − (1− ω)ZQ
]
βM
acMac
F
,
(14)
with ZQ ≡ ξQµIQMac the marginal value of active travel through induced changes in
health. We use the following ratios of fuel and income price elasticities to capture
15
the indirect benefits of fuel taxes through inactive and active distance travelled and
health12:
βM
in
≡
ηM
inF
ηFF
; βM
ac
≡
ηM
acF
ηFF
, (15)
with ηXF the fuel price elasticity of X ∈ {F,M in,Mac}, and ηM
inI the income elasticity
of inactive travel. The marginal excess burden MEB is commonly defined as
MEB ≡
tl
1−tl εLL
1− t
l
1−tl εLL
. (16)
The second component is the “Ramsey tax”: fuel taxes raise revenues, which are used
to finance government spending. This component commands a positive tax on fuel even
in the absence of external costs. The third component is the “congestion feedback”: a
reduction in travel due to fuel taxation reducing congestion, freeing up time for labour.
This creates a positive welfare effect as long as labour is taxed at a positive rate (tl > 0),
and as such increases the optimal fuel tax.
3 Fuel tax components
In this section we explain how we chose the parameter values for the quantification of the
optimal fuel tax. We specify a central value and a plausible range. Table 1 summarises
the main parameter values, and Figure 1 provides a graphical representation of the
social costs. For comparability, where relevant, we adjust all values to year 2017 US
dollar prices and US gallons. Finally, we state the implications for first-best policy.
In Figure 1, we converted CO2 emissions pollution per gallon into per mile units
using baseline fuel efficiency as presented in Table 1, such that all social costs are
expressed in per mile units. It shows that the per mile benefits of using an active mode
12The β’s are equal to the response of miles travelled to fuel taxes relative to the response in
fuel taxes. As such, they capture the relative effectiveness of fuel taxes in inducing changes in miles
travelled, both active and inactive.
16
Table 1: Parameter values used for optimal fuel tax calculation and sensitivity analyses
US UK
Parameter Central value Range Central value Range
Baseline fuel efficiency, M in,0/F 0 24 28
Fuel pollution (CO2), per gallon, ZPF¯ 91 [41, 405] 86 [38, 380]
Distance pollution (air), per mile, ZPM 4.5 [1, 9] 3.6 [1, 8]
Congestion, per mile, ZC 10 [3, 14] 5 [0.1, 7.3]
Accidents, inactive, per mile, ZAM¯in 6.4 [2, 18] 1.6 [1, 2.3]
Accidents, active, per mile, ZAM¯ac 5.3 [1.5, 15] 1.6 [1.2, 2.6]
Inactivity, per mile, ZQ 691 [403, 999] 244 [146, 683]
Rate of health internalisation, ω 0.5 [0.3, 0.7] 0.5 [0.3, 0.7]
Fuel price elasticity, ηFF -0.36 [-0.21, -0.75] -0.48 [-0.3, -0.9]
VMT-fuel price elasticity, ηM inF -0.25 [-0.05, -0.3] -0.35 [-0.2, -0.5]
Income elasticity of inactive travel, ηM inI 0.4 [0.02, 0.6] 0.605 [0.3, 0.8]
Cross-elasticity of active travel, ηMacF 0.18 [0.17, 0.25] 0.18 [0.17, 0.25]
Current tax rate on gasoline*, t0f 55 n.a. 406 n.a.
*Includes VAT for the UK but not US, and is provided in US gallons. All values are provided in
USD cents at end 2017 prices, using the end 2017 exchange rate 1 GBP = 1.351 USD. = 1.1251 EUR,
and either per mile or per US gallon. M in,0 and F 0 denote intensive miles travelled and fuel used at
the initial gasoline tax rate. Entries in the upper half of the table correspond to the social costs of
transport and justification for the values is given in subsections 3.1.2 through 3.1.7.
of travel are two orders of magnitude larger than most other social costs. This difference
is partly due to the difference in time it takes to travel one mile by motor vehicle as
opposed to travelling one mile by foot or by bicycle.
Figure 1: Social costs of personal car travel in US cents and on a per-mile basis.
17
3.1 Parameterisation
3.1.1 Baseline fuel efficiency and elasticities
Baseline fuel efficiency The US average fuel efficiency was 24 miles/gallon in
2016 (Federal Highway Administration, 2018), which we use to parameterise the fuel
efficiency at current tax levels, M in,0/F 0. In the UK, fuel efficiency of personal cars is
28 miles/gallon (DfT, 2018b; DfT, 2018c). The difference in fuel efficiency is due to a
smaller average size of the UK private vehicle fleet, and a higher proportion of diesel
cars, which have a higher average fuel economy.
Fuel price elasticities Based on papers reviewed by Litman (2019), Dieler et al.
(2015), and Litman (2013), we choose a fuel price elasticity, ηFF , of -0.38 for the US as
a central estimate. This is slightly less elastic than the UK value of -0.45, where wider
public transit offers alternatives to car use. The elasticities of inactive miles travelled
(VMT) with respect to fuel price, ηM
inF , are calibrated at -0.25 in US, and -0.35 in the
UK.13
To our knowledge, few direct estimates of the cross-elasticity of active travel (walking
and cycling) with respect to the fuel price, ηM
acF , exist. Instead, we primarily utilise
estimates of the cross-elasticity of public transport with respect to fuel price (Blanchard,
2009; Lane, 2008; Maghelal, 2011). The use of public transit requires getting to and
from stations, often done on foot or by an alternative active mode, and certain public
transit investments have been found to be an effective way of increasing active travel
(Reis et al., 2016). We adopt a range of 0.17 - 0.25 for ηM
acF , with a central value of
0.18 for both the US and the UK.
13The lower elasticity of miles travelled vis-a`-vis fuel use is explained by the fact that the most
significant response of an increase in fuel price is typically not reduction in the distance travelled by
people, but rather upgrading to a higher fuel economy car (Coglianese et al., 2017).
18
Income elasticity of inactive travel The elasticity of inactive travel demand with
respect to income, ηM
inI , is calibrated at 0.4 (0.02-0.6) for the US, and 0.605 (0.3-0.8)
in the UK. This is more conservative than older estimates, which sometimes exceed
unity (Graham and Glaister, 2002), and more aligned with newer estimates, such as
Small, Van Dender, et al. (2007).
We use long-term elasticities where possible to allow for mode shifts and other
behaviour changes. Further details and a full list of references on fuel price and income
elasticities can be found in Appendix B.2.
Labour supply elasticities For the compensated and uncompensated labour supply
elasticities, εcLL and εLL, we adopt the same values as Parry and Small (2005). These
estimates fall in between the more recent estimates by e.g. Bargain et al. (2011) and
Erosa et al. (2016).
3.1.2 External cost of CO2 emissions (fuel pollution), ZPF¯
The external costs associated to fuel use are the cost of carbon emissions and associated
climate damages. We derive our central estimate and range of plausible values from a
large body of literature (see Appendix B.3 for a full overview). We multiply the value of
climate change costs per tonne of CO2 emitted by the amount of CO2 emitted per gallon
of fuel burnt, weighted by fuel type consumption in both countries (diesel/gasoline) to
derive an average value of fuel pollution costs per gallon of fuel.14 With a social cost
of carbon of $90/tCO2 and a range $40 - 400/tCO2, the central estimate for the US is
91 cents/gallon, with a range of 41-405 cents/gallon. The central estimate for the UK
is 86 cents/gallon, with a range of 38-380 cents/gallon. Throughout, we abstract from
any effects of fuel taxation on the cost of carbon.
14As fuel mix is almost exclusively gasoline in the US but approximately half and half gasoline
and diesel in the UK, the marginal cost of climate damages per gallon of fuel is not the same in both
countries.
19
3.1.3 External cost of air pollution (distance pollution), ZPM¯in
Local air pollution is caused by car tyre and break wear emissions of PM2.5 and PM10,
which are approximately proportional to miles travelled (Grigoratos and Martini, 2014),
and by gases from incomplete fuel combustion processes.
For the US, estimates of the cost of air pollution of about $0.03 per vehicle mile
travelled are considered reasonable (Mashayekh et al., 2011).15,16 The OECD (2015)
report provides a high estimate of $0.089/mile, and Parry et al. (2014) find a value of
$0.06/mile for all road transport. Conversely, a CGE analysis of the marginal damage
of local pollutants emitted by all US economic sectors done by Muller et al. (2011)
reveals cost of passenger road transport air pollution of only around $0.007/mile. To
reflect this uncertainty, we adopt values of $0.01, $0.045 and $0.09 as the low, central,
and high estimates for the US, respectively.
For the UK, national project evaluations use a value of $0.036/mile (Hitchcock
et al., 2014). The OECD (2015) report on transport air pollution costs based on a
number of stated preference surveys yields a high value of $0.08/mile travelled, while
the average per-mile damage of air pollution for a passenger car in the EU is estimated
at $0.009/mile for gasoline and $0.033/mile for diesel cars. Reflecting the 55-45% split
between gasoline and diesel cars in the UK and updated costs of air pollution (Birchby
et al., 2019), we adopt values of $0.01, $0.036 and $0.08 as the low, central, and high
estimates.
3.1.4 External cost of congestion, ZC
Congestion is defined as the travel delay due to crowding of roads. Its cost is generally
calculated using the value of time, typically proxied by the median hourly wage, extra
15Estimates are based on an analysis involving the 86 largest US metropolitan areas.
16Air pollution exposure is approximately the same for active and inactive modes of transport;
cyclists and pedestrians breathe less filtered air, but often travel on a separate lane or road, meaning
they breathe less contaminated air (de Nazelle et al., 2017).
20
fuel expenditures, and the ratio of peak travel time compared to free-flow.
Ellis and Glover (2019) obtain a cost of congestion in the US of about $1,009 per
commuter ($0.08/mile). We adopt the values by Inrix (2018), who provide a central
value of $0.1/mile, and a range of $0.03- $0.14/mile. In the UK, we follow Inrix (2019)
and set the per-mile congestion costs at $0.05/mile, with a lower and upper bound of
$0.001/mile and $0.073/mile, respectively.17
3.1.5 External cost of accidents, ZAM¯in and ZAM¯ac
There are two components to accident costs: the internalised cost of knowing and
accounting for the risk of getting into a crash; and the external cost of the increased
risk of causing an accident imposed on others by travelling. Hence, using the full cost
of accidents per mile driven would overestimate the size of the accident externality.
Instead, we adopt the approach by Lemp and Kockelman (2008), who estimate the
external costs of transport in the US and assume that 50% of accident costs are external.
To determine the external cost of accidents associated to inactive travel, we consider
all costs associated to car-on-car and car-on-pedestrian accidents, as well as 50% of the
car-on-cyclist accident costs. Similarly for active travel, we include pedestrian only and
cyclist-on-pedestrian accident costs and the remaining half of the car-on-cyclist accident
costs.18
Using accident data and cost estimates from the report by Blincoe et al. (2015), we
find that accident costs attributable to inactive modes of transport, ZAM¯in , amount to
$0.064/mile, with a range of $0.02-$0.18/mile for the US. Accident costs attributable
to active modes of transport, ZAM¯ac amount to $0.053/mile, with a range of $0.015-
17Parry and Small (2005) already noted that the values available in empirical studies in
transportation are VMT-weighted, but congestion costs enter the optimal fuel tax equation as being
both VMT-weighted and fuel-price elasticity weighted. This is because demand for travel is more
inelastic in more congested times and people internalise more of the waiting time cost. Adjusting for
this reduces the marginal cost of congestion.
18In the US, many states legally assign collision fault to the motor vehicle, but some do not follow
this rule, and in the UK collision fault is not automatically assigned to the driver. This is why we
assign “blame” equally to both modes.
21
$0.15/mile. This is due to the proportionately higher death rate per accident and per
mile travelled for active modes.
In the UK, ZAM¯in amounts to $0.015/mile, with a range of $0.01-$0.023/mile.
Accident costs attributable to active modes of transport, ZAM¯ac , amount to $0.016/mile
as well, with a range of $0.012-$0.026/mile (DfT, 2018a). In order to arrive at the high
estimate of accident costs, we used the WHO value of statistical life (VSL) to value the
cost of a death or injury, instead of the UK Department for Transport values.19
3.1.6 Cost of inactivity, ZQ
The health benefits of exercise are well-known to be the most substantial health-related
impact of active travel, dwarfing air pollution or accident effects. For example, De
Hartog et al. (2010) estimate that people shifting from car to bicycle for short trips
(daily commute of 4.7 miles) lose 7 days of life due to traffic accidents, 21 days of life
due to air pollution, but gain 8 months of life due to physical activity. Our analysis
requires translating such benefits into monetary values.
Health benefits of exercise, or conversely the health costs of inactivity, are due
to a number of different effects, listed in Figure 2. First and foremost, these
health benefits comprise all mortality- and morbidity-reducing effects. Second, there
may be productivity benefits, due to a reduction in absenteeism (taking sick leave),
and presenteeism (being at work but having lower productivity due to illness).
Third, greater health reduces the (public) health system costs. Depending on the
characteristics of the health system (and the extent to which the individual bears the
cost of absenteeism), the costs can be labelled as private or external. In the remainder
of the analysis, we focus on the value of unrealised private health benefits only, as they
are much larger than the direct productivity gains to the economy.
In order to calculate the marginal value of the private health benefits from physical
19VSL is most commonly used in policy or investment appraisal in health, environment, and
transport projects, and differs across disciplines.
22
Figure 2: Some of the barriers to, and benefits of, physical activity.
activity for the UK and US, we used the Health Economic Assessment Tool (HEAT)
developed by the World Health Organisation (Kahlmeier et al., 2017; Nieuwenhuijsen
and Khreis, 2018). HEAT calculates the value of the changes in mortality arising from
a specified change in walking and cycling for travel purposes.20
HEAT also requires inputs on the population considered, the baseline mortality
rate per 100,000, and the relevant value of statistical life (VSL) with which to value the
deaths avoided.21 The central value of VSL used for the US is $9.26 million, with a range
of $5.4m-13.4 million (Moran and Monje, 2016). The central value of VSL for the UK
is more conservative, $4.36 million (WHO 2015), with a range of $2.2m-12.6 million
(Thomas, 2020). HEAT assumes a linear relationship between exercise and health
benefits for sedentary societies (Kahlmeier et al., 2017), meaning the baseline level
of physical activity does not affect the marginal value of additional activity. Further
details regarding the HEAT model, inputs and corresponding data sources can be found
in Appendix B.4.
We converted the HEAT output to an estimate of the health benefit per mile of
active travel. We obtained a central ZQ value of $6.91/mile for the US, with a range of
20HEAT is designed as an easy-to-use practitioner-oriented tool for health impact assessments.
More complex health impact assessments would quantify the effect of exercise on morbidity as well as
mortality. Our results are therefore likely to be conservative estimates of the health benefits of physical
activity.
21In transport, environment, and medical fields, health is often quantified using stated or revealed
preference methods, in order to arrive at the VSL. This method values everyone’s life equally, regardless
of their health or age.
23
$4.03-9.99/mile. For the UK, this value is $2.44/mile, with a range of $1.46-6.83/mile.
The UK-US discrepancy has two sources: higher VSL estimates for the US, and higher
US baseline mortality rates for younger members of the population.
3.1.7 Rate of health internalisation, ω
The extent to which individuals are aware of the health benefits of exercise and active
travel are captured by the parameter ω. High numbers of respondents (50-85%) in
attitudinal transport surveys cite health reasons as one of their top three reasons for
walking or cycling for travel (Useche et al., 2019; Fu and Farber, 2017; Bo¨rjesson and
Eliasson, 2012; De Souza et al., 2014). However, in general exercise knowledge surveys,
only about half of respondents are knowledgeable of the amounts of physical activity
required for health, and about 20-30% are capable of identifying the approximate odds
of developing diseases without physical activity (Fredriksson et al., 2018; San Diego
and Merz, 2020; Bennett et al., 2009). We therefore chose a central value of 0.5 and a
range of 0.3-0.7 for ω.
3.2 Implications for first-best policy
In the next section, we use the parameter values specified in Table 1 to quantify the
optimal second-best fuel tax. Table 1, nonetheless, already permits a quantitative
conclusion about first-best policy. In a first-best world, there exist appropriate policy
instruments to address all market failures, as well as non-distortive (e. g. lump-sum)
taxes to generate government revenues. One can verify that under these assumptions,
the optimal carbon (fuel) tax is equal to the cost of fuel pollution, ZPF¯ . Similarly, the
socially optimal level of the price instruments for all other externalities (and internality,
by analogy) are at their respective Pigouvian levels. For the internality, this Pigouvian
level is equal to the social cost of inactivity, multiplied by the uninternalised share
(1-ω). Importantly, this means that, on a per mile basis, a first-best subsidy paid to
24
individuals for incentivising active travel modes would be at a much higher level than
any of the tax levels for the externalities, or indeed, the sum of all other externality
taxes.22
4 Quantitative Results
4.1 Optimal second-best fuel tax rates
We use Equations (13)-(15) and the parameter estimates provided in Table 1 to calculate
the optimal fuel tax.23 An increase in fuel taxation will reduce fuel use and inactive
miles travelled, which will in turn affect the optimal tax through (14). To account
for this, we follow Parry and Small (2005), and endogenise F , M in, and Mac in our
numerical solution. Further details can be found in Appendix C.1.
We find an optimal fuel tax of $4.54/gallon of fuel in the UK, which is slightly higher
than the current fuel tax.24 In the US, the optimal fuel tax amounts to $10.13/gallon
of fuel, which is more than ten times the current (population-weighted) average fuel
tax across the fifty states. Table 2 lists the optimal tax levels and their decomposition.
This decomposition shows that costs associated to congestion and physical inactivity
are the main contributors to the fuel tax, albeit this is somewhat reduced due to the
compensation for the marginal excess burden of labour taxation. Ramsey taxes are
substantial, especially in the US; the congestion feedback does not significantly influence
the optimal fuel tax rate.
Including physical activity increases the UK fuel tax by 36%, and the US fuel tax
by 49%. Although this increase is significantly smaller than the pure per-mile social
cost of physical inactivity, the inactivity component is still the largest contributor to
22For comparison to second-best see Section 4, and for the policy implications of that comparison
see Section 5.
23We used R for all computations. Code available from authors on request.
24The total tax on UK fuel includes VAT (110 cents/gallon) and fuel duty (296 cents/gallon). In
the United States, value added or indirect taxes are not levied on fuel.
25
Table 2: Central calculations of the optimal fuel tax rate
Cost, USD cents/gallon United States United Kingdom
Fuel efficiency, M in/F 28.6 28.3
Adjusted Pigouvian tax:
Pollution, fuel-related, ZPF¯ 91 86
Pollution, distance-related, ZPM¯inβM
in
M in/F 90 74
Congestion, ZCβM
in
M in/F 199 103
Accidents inactive, ZAM¯inβM
in
M in/F 127 33
Accidents active, ZAM¯acβM
ac
Mac/F -4 -1
Physical inactivity, (1− ω) βM
ac
Mac/F 256 109
Adjustment to MEC for excess burden -71 -36
Ramsey tax 326 83
Congestion feedback -1 3
Optimal fuel tax rate with physical activity, tf∗1 1013 454
Optimal fuel tax rate without physical activity 681 335
Na¨ıve fuel tax rate 593 372
Actual (2017) tax rate 55 406
Based on Equation (13) and (14), the optimal rate is the adjusted Pigouvian tax, adjustments for the
excess burden, the Ramsey tax, and the congestion feedback, combined. The na¨ıve rate is given by
MECF (excluding the health internality) from Equation (14) with M in/F = M in,0/F 0 and all β’s
equal to 1.
the MEC part of the tax.
Consistent with both Parry and Small (2005) and Santos (2017), we find that
the second largest externality component of the second-best optimal fuel tax for both
countries is congestion. In London, congestion impacts are 28 times higher than the
EU average (Cookson, 2016), which greatly influences the congestion costs for the UK,
even though the value of travel time estimates for the US are higher than for the UK.
This is followed by inactive accidents in the United States. In the US, traffic
accidents are associated with a far higher per-mile cost, even though the rates of traffic
injuries are very similar in both countries. This is explained by a higher nominal value
that is attached to human life in the US. Contrary to Parry and Small (2005), air
pollution costs for both countries contribute less to the fuel tax than carbon emissions.
This is the result of both increasingly stringent fuel air pollutant emissions standards,25
25Such as Corporate Average Fuel Economy (CAFE) standards.
26
and growing consensus that the social cost of carbon is higher (see Subsection 3.1.2.
In addition to the second-best optimal tax, we compute “na¨ıve” tax rates. This
na¨ıve rate is calculated using three assumptions. First, that our βs are equal to 1; i.e.,
it assumes that both active and inactive miles travelled are equally responsive to fuel
taxation as fuel use. Second, it ignores the feedback of tax-induced changes in fuel use
and miles travelled to the tax rate. Third, the na¨ıve rate abstracts from the interactions
of the fuel tax with the labour tax, as well as the Ramsey component. Instead, the only
relevant components to the tax are the external effects of car use. The na¨ıve rate as
such mimics common practice in transport and cost-benefit analysis evaluations. 26 In
our central calculations βM
in
= 0.69 (US) or 0.73 (UK), and βM
ac
= −0.50 (US) and
-0.38 (UK). The low ηM
inF reflects the very inelastic demand for vehicle-miles travelled
(VMT), meaning that most reduction in fuel use comes from increases in fuel economy
of driving and the vehicle fleet, not reductions in distances covered in cars. Thus,
mileage-related externalities (air pollution, congestion, and accidents) are all inflated
in the na¨ıve fuel tax calculation.
Treating fuel efficiency, fuel consumption and distance travelled as endogenous,
rather than exogenous, in the second-best optimal fuel tax calculation causes fuel
consumption to fall by 43.9% and 3.7% in the US and UK, respectively.27 Inactive
travel M in falls slightly less, by 33.1% in the US, and this is more than compensated
for by an increase of 33.5% in active travel. In the UK, inactive travel also falls by
less than fuel consumption, by 1.7%. However, as active travel increases only by 1.4%,
total travel in the UK falls. The overall tax increases by 81.1% in the US and 3% in
the UK. The change is more dramatic in the US because of the low fuel efficiency of
26In applied transportation research, the difference between the responsiveness of fuel consumption
to fuel prices and miles travelled to fuel prices are often disregarded, and assumed to be unitary.
Multiplying externalities only by fuel efficiency, and not by the responsiveness of VMT to fuel price,
is considered the na¨ıve approach in literature, and can sometimes lead to a doubling of the optimal
fuel tax estimation (Newbery et al., 1995).
27Results for the second-best optimal fuel tax with these variables treated as fixed is presented in
Appendix C.1, Table C.1.
27
motor vehicles, and higher contingent valuation of people’s time and lives, resulting in
a bigger Pigouvian tax. In the UK, the current tax level is very close to the optimal
level. The fuel efficiency of motor vehicles in the UK therefore does not change much in
response to moving to the optimal level, and the endogenous solution does not change
the optimum level significantly.
4.2 Welfare Effects
The welfare gain of implementing the second-best optimal fuel tax is presented in Table
3. We use the current tax rate as a benchmark, and consider a fuel tax that does, and
does not, take into account active travel benefits. All gains are expressed as a share of
current fuel expenditure. The analytical derivation of the welfare benefit is discussed
in Appendix C.2. The welfare gain of implementing the second-best optimal tax that
accounts for the health internality is 129% for the US, but only 0.16% for the UK. The
net welfare gain in the UK is rather small, and becomes a loss either side of the optimal
tax rate. In the US, any increase in the fuel tax yields significant welfare improvements.
This difference is primarily due to the very low current US fuel tax, while the UK fuel
tax is already close to the optimal rate.
We additionally present the changes in active and inactive miles travelled following
the change in the fuel tax. In the UK, the small change in fuel taxes results in relatively
small changes in distance travelled. In the US, however, the fuel price changes are large,
and induce a substantial shift from inactive to active miles.28
Finally, we compute the effect of the tax increase on mortality through increased
active travel. For this purpose, we use the HEAT tool (Kahlmeier et al., 2017), described
in Section 3.1.6 and Appendix B.4. Increases in active travel primarily save lives by
improving health from increasing exercise. HEAT then corrects this value for the
28However, as we used constant fuel price elasticities to calculate these changes, these results should
be interpreted with caution. It is unlikely that a 20-fold increase in the fuel tax in the US would induce
the same rate of response as a 20% increase in the tax.
28
Table 3: Welfare effects of fuel taxation
US
Fuel tax Rate (c/gal) Welfare change M in change Mac change Annual lives
tf∗ 1013 129% -33.04% 33.47% 6266 saved
tf , excluding health benefits 681 124% -27.39% 25.92% 4701 saved
UK
Fuel tax type Rate (c/gal) Welfare change M in change Mac change Annual lives
tf∗ 454 0.16% -2.67% 1.40% 39 saved
tf , excluding health benefits 335 -0.99% 4.58% -2.28% 120 lost
Welfare changes are calculated relative to the current rate, and expressed as a percentage of current
fuel expenditure, which is equal to approximately $1800 in the US and $1500 per person per year in
the UK, according to household expenditure surveys (BLS, 2019; ONS, 2019). The current US fuel
tax rate is $0.55/gal, and the current UK fuel tax rate is $4.06/gal.
lives lost due to greater air pollution exposure, and increase in accident fatalities of
pedestrians and cyclists.29,30 Due to a larger tax increase and larger population size,
lives saved are greatest in the US: setting the fuel tax at its optimal level saves 6266
lives each year.
4.3 Sensitivity Analysis
In Figures 3 and 4, we illustrate the sensitivity of the optimal second-best fuel tax with
respect to each of the non-priced cost parameters and the elasticity parameters, keeping
all other parameters at their central values (denoted X in the graphs). Further details
and figures can be found in Appendix C.4.
For both the UK and US, the fuel tax is most sensitive to ηFF , the fuel price
elasticity. This elasticity directly affects the optimal fuel tax through βM
in
and βM
ac
in the MEC component (see (14)), and indirectly by governing the response of fuel use
F to the introduction of the fuel tax. Jointly, as shown in the figure, this results in a
positive relationship between the fuel price elasticity and the optimal fuel tax. Using
29Further details regarding this computation can be found in Appendix C.3.
30HEAT only computes the lives saved due to an increase in active travel. Higher fuel taxation
also reduces inactive miles travelled, which reduces air pollution and vehicle traffic fatalities. As we
do not capture these effects, the values reported in Table 3 can be considered a lower bound for the
lives saved due to the tax increase.
29
xx
x
x
x
x
x
x
x
x
4.Income el. of inactive travel 5.Health internalisation rate
1.Fuel price elasticity 2.VMT−fuel price elasticity 3.Cross elasticity of active travel
0.0 0.2 0.4 0.6 0.3 0.4 0.5 0.6 0.7
−0.8 −0.6 −0.4 −0.2−0.5 −0.4 −0.3 −0.2 −0.1 0.17 0.19 0.21 0.23 0.25
500
1000
1500
500
1000
1500
Value
Op
tim
al 
tax
, c
en
ts/
ga
llon
Country
a
a
UK
US
Figure 3: Sensitivity of the fuel tax to relevant elasticities and the rate of health
internalisation
the upper or lower bound of the fuel price elasticity, as opposed to the central value, can
either increase or decrease the fuel tax by as much as 50 percent. Conversely, the net
effect of the elasticity of inactive travel (VMT) with respect to the fuel prices, ηM
inF ,
on the fuel tax is negative. Using the upper bound instead of the central value for
ηM
inF , reduces the fuel tax by up to 50 percent; using the lower bound instead has a
noticeably smaller effect. The fuel tax is relatively insensitive to the cross elasticity
of active travel, ηM
acF and the rate of health internalisation. The income elasticity of
inactive travel affects the fuel tax especially for the US. This effect materialises via the
Ramsey tax component, which is comparatively large for the US to begin with (see
Table 2).
Regarding social costs, the fuel tax is most sensitive to ZPF¯ , fuel pollution, i.e. the
social cost of carbon. This is attributed to the fact that ZPF¯ has a very high right-hand
30
xx
x
x
x
x
x
x
x
x
x
x
4.Inactivity 5.Pollution (air) 6.Congestion
1.Pollution (CO2) 2.Accidents, inactive 3.Accidents, active
200 400 600 800 2.5 5.0 7.5 0 5 10
100 200 300 5 10 15 5 10 15
300
600
900
1200
300
600
900
1200
US cents
Op
tim
al 
tax
, c
en
ts/
ga
llon
Country
a
a
UK
US
Figure 4: The sensitivity of the fuel tax to the social cost parameters.
tail, which follows from high estimates for the social cost of carbon under a business-
as-usual scenario for global carbon emissions in the relevant literature (see also Table
B.6 in Appendix B.3). Increasing ZPF¯ to this upper bound increases the optimal fuel
taxes by more than $4/gal for both the UK and US. We find a similar sensitivity of the
fuel tax to the calibration of ZAM¯in for the US, while the UK fuel tax increases by only
a little when moving from the central value to the upper bound for ZAM¯in .
Finally, the fuel tax shows similarly strong sensitivity to the remaining social cost
parameters, with the exception of active accident costs, ZAM¯ac . Higher social costs ZQ,
ZPM¯in and ZC result in higher levels of optimal fuel taxation, with the upper (lower)
bound increasing the optimal fuel tax by around $1.50-2.50/gal.
Figure 5 presents the results of a Latin Hypercube parameter sensitivity analysis
using the pse package in R (Chalom and de Prado, 2015). We varied parameters for
31
external costs and the elasticities using 20 values drawn at random from a uniform
distribution, and allowed fuel use and mileage to be generated endogenously within
the function. The function was run 200 times, with 50 bootstrap replicates. For the
United States, the optimal fuel tax is less than 0.01% likely to be below the current
55 cents/gallon. For the UK, the optimal fuel tax is below current fuel tax of 406
cents/gallon with a 15% probability.
Figure 5: Cumulative distribution function of optimal fuel tax values for both US and
UK.
5 Discussion
This article shows that it may be very costly to societies not to reap the health
benefits from choosing travel modes that lead to more physical activity. Including these
benefits in an analysis of optimal fuel taxation shows that the optimal fuel tax increases
32
significantly, although to a lesser extent than one might expect based on assessments of
the benefits of active travel measured in health. Here we discuss how our findings align
with current thought on optimal transport policy, and subsequently discuss limitations
of our approach.
Experts on urban transport policy have long argued for a mix of “push” and
“pull” factors to efficiently reduce societal costs from car use (Pucher and Buehler,
2007; Creutzig and He, 2009; Sterner, 2012; Brand et al., 2013). “Push measures”
discourage car use and include fuel or road pricing and parking fees; “pull measures”
encourage uptake of other forms of transport by making them more attractive. Our
study only explicitly considers a fuel tax, but a number of conclusions about other
policy instruments may be drawn. First, both active and inactive miles travelled are
relatively inelastic to the fuel tax. This not only affects the second-best optimal level,
but also implies that a fuel tax on its own is not an ideal instrument for reducing
mileage-related externalities. Parry and Small (2005) suggest that using a VMT tax
for all externalities other than carbon emissions would be preferable where feasible.
Indeed, there is renewed academic interest in alternative road pricing mechanisms,
such as congestion or GPS-based charging, cordon pricing, optimal toll pricing, real
time road pricing, or “superblocks” of pedestrian-only traffic (see Parry et al., 2007;
Guo et al., 2017; Cramton et al., 2018; Bjertnæs, 2019; Mueller et al., 2019). However,
ideal externality-correcting mechanisms have not been widely implemented and most
plans to introduce congestion charging are advancing slowly (Reid, 2019).31 Still, the
benefit of the fuel tax over a congestion charge is that a fuel tax can, at least indirectly,
address all large externalities caused by driving concurrently, and be used to generate a
broader revenue base for the government (Rietveld and van Woudenberg, 2005). It also
encourages improvements in the fuel economy of the fleet by encouraging the purchase of
31Amongst the cities that are debating a congestion charge are Bath, Birmingham, Bristol,
Cambridge, Cardiff, Coventry, Edinburgh (UK), Toronto (Canada), and Sydney (Australia).
Congestion charges have been successfully implemented in London, Milan, New York, and Stockholm,
for example.
33
more efficient new cars, and in some cases, more fuel efficient driving patterns (Dhondt
et al., 2013; Bjertnæs, 2019).
Second, our study has further implications for policies encouraging active transport.
Akin to congestion charges, a straightforward public finance approach could involve a
direct subsidy for active modes of travel (Wardman et al., 2007; Regeringskansliet, 2017;
EBS, 2020) or indirect subsidies in the form of suitable cycling and washing facilities
for employees that would improve comfort of cycling relative to driving (Tilahun et al.,
2007).
Appropriate active travel infrastructure may however be a more important “pull”
measure to make walking and cycling attractive.32 Car-free city centres, bicycle lane
networks and improvements in public transport provision will increase the mode share
of active travel (Pucher and Buehler, 2007; Buehler and Pucher, 2012; Go¨ssling, 2013),
and are actively pursued policies.33 For example, in a study of 167 European cities,
Mueller et al. (2018) find that increasing bicycle lane infrastructure in urban areas up
to 315km/100,000 people increases the mode share of cycling up to 24.7%. Further, the
lack of appropriate street lighting, and badly maintained roads and cycle lanes have all
been identified as factors impeding active commuting (Yang et al., 2017; NHTS, 2019).
These policies do not only increase the perceived attractiveness of active travel relative
to other modes of transport (Schepers et al., 2015), but also create opportunities to
adopt active travel modes to begin with. As such, they have the potential to make the
demand for both active miles travelled, and miles by car more responsive to fuel prices.
Investing in cycling infrastructure has not been considered a significant investment
32Standalone marginal increases in fuel taxes are unlikely to spur significant behaviour change in
drivers, because the response to a fuel tax is to alter fuel consumption, but not to reduce the amount or
distances travelled by car – the demand for VMT is more inelastic than the demand for fuel (Gillingham
and Munk-Nielsen, 2019).
33For example, Paris is considering developing a 15-minute city plan, where all facilities should be
available to every inhabitant within a 15 minute walk (Reid, 2020). Barcelona has begun introducing
“superblocks” of 500m2 that are fully pedestrianised (Mueller et al., 2020). In addition, many cities
are focusing on cycling as an alternative to public transport during the COVID-19 pandemic (Sui and
Prapavessis, 2020).
34
strategy until recently, and neither have infrastructure cost assessments been including
it in their analyses (Van Essen et al., 2011; Knudsen and Rich, 2013).34
Our optimal tax result signals that the extent to which an increase in fuel taxes
is welfare enhancing is constrained by the presence of viable low-carbon alternatives
(high β’s). This result was also demonstrated by Martens (2016), and (in part) justifies
the grievances of the Gilets Jaunes movement in France. Conversely, this implies
that improvements in infrastructure designed for active travel and public transport
as discussed above could further support increased fuel taxation.35 While greater
willingness of citizens to switch to active travel in particular would manifest itself in a
higher responsiveness of active travel to fuel taxation, ηM
acF , it would likely also result
in higher active miles travelled Mac, lower fuel use F , and higher (absolute) elasticities
ηM
inF . From (13)-(15), all of these effects would justify higher fuel taxes.
A full characterisation of the “pull effect,” would include the public good
characteristics of infrastructure – everyone benefits from safer and more comfortable
cycling infrastructure – , which will lead individuals to derive more utility from active
travel and therefore “pull” them into these modes. This channel is not considered
in our model and yet could lead to either higher or lower second-best optimal fuel
taxes (Siegmeier, 2016), once infrastructure changes are seen as an additional policy
instrument rather than an exogenous change in elasticities (similar to Bovenberg and
van der Ploeg, 1994). In other words, as improved public transport infrastructure moves
active travel decisions closer to optimal decisions, it may reduce the need for and value
of higher fuel taxes as captured through ZQ. We believe this is a crucial area for further
34Health benefits have long been considered the domain of health departments and ministries only,
leading to a lack of funding for health improvements in transport and urban planning departments
(The European Cyclists’ Federation, personal communication).
35Future research could investigate how cost-benefit analysis of public transport infrastructure
project changes once health benefits are taking into account. Such an analysis should note that
the health benefits are unevenly distributed across the population.
35
work.36.37
We note a number of limitations to our study: health manifests itself only as an
internality within the utility function, though evidence exists to suggest that health-
labour feedback loops, specifically between physical activity and productivity (Van den
Heuvel et al., 2005; Proper et al., 2006), and presenteeism (Hemp, 2004; Pereira et al.,
2015) do exist. We abstract from these effects in our analysis; if anything, including
such effects would strengthen the case for including health benefits from active travel
in transport policy assessments. Further, we do not consider distributional concerns
of the policy instruments, especially related to income or to race (see Bento et al.,
2009, Jephcote et al., 2016, and Tessum et al., 2019). In our context importantly
this also means we do not consider heterogeneities in space within a country, although
spatial heterogeneity of transport infrastructure matters for the effectiveness of pricing
instruments (Creutzig et al., 2020) . We further do not explicitly model pre-existing
regulation such as fuel efficiency standards, which are an important element in current
transport regulation and limit firms’ production possibilities (Greene, 2011). Finally,
we combine public transport with walking and cycling as an active mode of transport,
and so omit recent trends in urban travel such as ride-hailing and sharing apps such as
Uber, or the increasing popularity of e-scooters.
36Further still, norms, manifestations of (unwritten) rules of a specific culture and information
provision can play a relevant role in increasing the effectiveness of fuel taxes and policies aimed
at increasing active travel. Cialdini (2003) and Cialdini (2007) suggests using messages with
injunctive norms (perceptions about socially- approved behaviours) when the intention is to change an
environmentally undesirable but socially-extended behaviour; and using descriptive norms (perceptions
about behaviours socially-extended) to reinforce pro-environmental behaviour that is sufficiently
present (Nyborg et al., 2016). For further relevance of such norm changes to urban transport, a
case of “endogenous preferences or endogenous beliefs,” see Mattauch et al. (2016), Mattauch and
Hepburn (2016) and Mattauch et al. (2018).
37Evidence suggests that the salience of a tax or price change matters and is determined, in part,
by its messaging (Dudley, 2013; Uskul and Oyserman, 2010). In the case of fuel taxes (Li et al., 2014)
and climate policy (Hirte and Tscharaktschiew, 2015), messaging may increase sensitivity to a price
change by up to 70%.
36
6 Conclusion
This article shows that optimal fuel tax rates increase significantly if the health benefits
from increased active travel, such as reduced rates of diabetes, cardiovascular diseases,
dementia, and depression are not fully internalised by citizens. Building on the
established model of optimal fuel taxation developed by Parry and Small (2005), we
present an assessment of optimal fuel taxation when an internality through physical
inactivity is also considered in the tax design.
We confirm the main conclusion of a large body of research in public health that,
per mile travelled, the social costs of inactivity dominate the social costs from transport
externalities by two orders of magnitude. We examine how this fact changes the
appropriate second-best optimal fuel tax, which targets active travel health benefits
only indirectly. We conclude that the second-best optimal fuel tax increases from $3.35
to $4.54/gallon in the UK and $6.81 to $10.13/gallon in the US. Due to the inelastic
demand for vehicle miles travelled and cross-elasticity of active travel, the tax rate
increases by less than the value of the per-mile internality.
In contrast to Parry and Small (2005), we find that the fuel tax rate in the
United Kingdom is close to the optimal when the health benefits from active travel
are accounted for, else it would be too high. We confirm that, even without these
benefits, in the United States fuel is significantly undertaxed: it exceeds the current
average fuel tax rate across the 50 states, as well as the previous estimate by Parry and
Small (2005). In fact, the economic cost of damages to human health (air pollution and
accident externalities) have risen significantly; more time is being spent in congestion
on US roads, and the value of time has also risen faster than inflation. Further, the
social cost of carbon estimates we derived from the literature ($40-400/tCO2) are several
times higher than the values Parry and Small used in 2005 ($6.8/tCO2, $0.2-27/tCO2).
Whereas pollution linked to CO2 emissions contributed the least to their Pigouvian
tax component, it is the third largest component according to our estimates. It has
37
the most significant influence on the fuel tax as the cost of carbon damages increases.
Climate change and urban transport problems have become worse over the past two
decades.
Including the significant health benefits from active travel means that fuel taxes
should be increased, as they are an established instrument for addressing all social
costs of transport. The modest relevance of a fuel tax on an individual’s decision to
walk or cycle indicates, however, that the fuel tax may not be the most appropriate
policy instrument to encourage active travel. This is further reinforced as the optimal
US fuel tax can be deemed politically unrealistic. Instead, more targeted measures
to increase the relative price of car travel such as congestion charges, and measures
aimed at reducing barriers to other modes of transport such as building better active
travel infrastructure, will permit societies to reap the high health benefits. Congestion
charging would be effective in both the UK and the US for this purpose (Cramton et al.,
2018). The US specifically would benefit from more public transit infrastructure that
would result in more active trips as people reach the transit stops by foot or bicycle.
The UK, characterised by denser cities, would especially benefit from improved urban
infrastructure for walking and cycling. Nonetheless, without an associated change in the
price signal in the form of a fuel tax rise or congestion charge, infrastructure investment
is unlikely to lead to sufficient changes in travel decisions on its own (Buehler et al.,
2017; Pucher et al., 2010). Different policies at multiple levels are needed to realise
meaningful change in transport.
38
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51
A Mathematical Appendix.
A.1 Derivation of the optimal fuel tax tf
We derive the optimal fuel tax in two steps. First, we consider the consumer’s
optimisation problem. We define V as the consumer’s maximised value function, and
determine the first-order conditions corresponding to the solution of this value function.
Next, we determine the optimal fuel tax by computing the total derivative of V with
respect to the tax on fuel, dV/dtf . Setting this term equal to zero then allows us to
obtain Equation (13) as detailed below.
A.1.1 Consumer optimisation problem
The consumer maximises (1) with respect to (2)-(10). As described in the text, it takes
pollution P , measures for congestion pi and a large part of the accident risk as given.
Instead of considering actual health (7), it considers perceived health (8). This allows
us to define the following maximised value function:
V ≡MaxC,F,H,Mac,O,L{u(ψ(C,M, T
in, T ac, G), N)− ϕ(P )− δ(A) + ξ(Qper)
+ µM
[
M in +Mac −M
]
+ µM in
[
χ (F,H)−M in
]
+ µT in
[
piin(M¯ in)M in − T in
]
+ µTac [pi
acMac − T ac] + µA
[
Ain(M¯ in,M in) + Aac(Mac, M¯ in, M¯ac)− A
]
+µQper
[
ωQ(Mac, O) + Q¯−Qper
]
+µI
[
w
(
1− tl
)
L− C − (pf + tf )F − phH − poO − Sac
]
+ µS [S(M
ac)− Sac] + µL
[
L¯− L−N − T in − T ac
]
}, (A.1)
52
with corresponding first-order conditions
uψψC − µI = 0, (A.2)
uψψM − µM = 0, (A.3)
−δA − µA = 0, (A.4)
µM − µM in + µT inpi
in + µAA
in
M in = 0, (A.5)
µM + µTacpi
ac + µAA
ac
Mac + µQωQMac + µSSMac = 0, (A.6)
uψψT in − µT in − µL = 0, (A.7)
uψψTac − µTac − µL = 0, (A.8)
µM inχF − µI(p
f + tf ) = 0, (A.9)
µM inχH − µIp
h = 0, (A.10)
ξQ − µQ = 0, (A.11)
µQωQO − µIp
o = 0, (A.12)
−µI − µS = 0, (A.13)
uN − µL = 0, (A.14)
−µL + µIw
(
1− tl
)
= 0. (A.15)
A.1.2 Optimal taxation
The optimal fuel tax is implicitly determined by dV/dtf = 0, taking into account that
tl is determined through (11), and that in equilibrium M¯ in = M in, M¯ac = Mac, F¯ = F ,
and Q˜ = (1− ω)Q. From (A.1) and (A.2)-(A.15) we then obtain
dV
dtf
=
[
µT inpi
in
M¯ inχFM
in − δA
[
AinM¯ in + A
ac
M¯ in
]
χF − ϕP
[
P fF¯ + P
m
M¯ inχF
]
+ µIt
f
] dF
dtf
+
[
µT inpi
in
M¯ inM
inχH − δA
[
AinM¯ in + A
ac
M¯ in
]
χH − ϕPP
m
M¯ inχH
] dH
dtf
+ [−δAA
ac
M¯ac + ξQ (1− ω)QMac ]
dMac
dtf
+ ξQ (1− ω)QO
dO
dtf
+ µIwt
l dL
dtf
.
Further rearranging this term and using dM
in
dtf = χF
dF
dtf + χH
dH
dtf and
dQ
dtf = QMac
dMac
dtf +
QO dOdtf , we obtain
1
µI
dV
dtf
=
[
tf −
ϕP
µI
P fF¯
]
dF
dtf
+
[
µT in
µI
piinM¯ inM
in −
δA
µI
[
AinM¯ in + A
ac
M¯ in
]
−
ϕP
µI
PmM¯ in
]
dM in
dtf
+
[
−
δA
µI
AacM¯ac
]
dMac
dtf
+ (1− ω)
ξQ
µI
dQ
dtf
+ wtl
dL
dtf
. (A.16)
53
Equation (A.16) is a generalisation of Equation (2.9) in Parry and Small (2004), taking
into account active travel decisions. To find the optimal fuel tax we then set (A.16)
equal to zero, and isolate tf . We obtain
tf∗ = ZPF¯ +
[
ZC + ZAM¯in + ZPM¯in
]
(
−dM in
dtf
/
−dF
dtf
)
+ ZAM¯ac
(
−dMac
dtf
/
−dF
dtf
)
− (1− ω) Z˜Q
(
−dQ
dtf
/
−dF
dtf
)
− wtl
(
−dL
dtf
/
−dF
dtf
)
, (A.17)
where we define
ZPF¯ ≡
ϕP
µI
P fF¯ ; Z
PM¯in ≡
ϕP
µI
PmM¯ in ; Z
AM¯in ≡
δA
µI
[
AinM¯ in + A
ac
M¯ in
]
;
ZAM¯ac ≡
δA
µI
AacM¯ac ; Z˜
Q ≡
ξQ
µI
, and ZC ≡ ΓinpiinM¯ inM
in.
To obtain ZC , we use (A.7) and (A.15), and define Γin ≡ w
(
1− tl
)
−ψT in/ψC . Equation
(A.17) can in turn be expressed as
tf∗ = MEC − wtl
(
−dL
dtf
/
−dF
dtf
)
, (A.18)
with
MEC ≡ ZPF¯ +
[
ZC + ZAM¯in + ZPM¯in
]
βM
inM in
F
+
[
ZAM¯ac − (1− ω)ZQ
]
βM
acMac
F
,
βM
in
≡
ηM
inF
ηFF
; βM
ac
≡
ηM
acF
ηFF
; ZQ ≡
ξQ
µI
QMac ,
where ηXF is the fuel price elasticity of X ∈ {M in,Mac}, and we impose dO/dtf = 0.
By linear homogeneity, M in = χFF + χHH. Then, using (A.3)-(A.5), (A.7), (A.9),
(A.10) and (A.15) we obtain
uM
µI
= pM
in
, (A.19)
with uM ≡ uψψM , pM
in
≡
[
(pf + tf )αF + phαH
]
+ Γinpiin + Λin, αF ≡ F/M in, αH ≡
H/M in, and Λin ≡
δAAinMin
uψψC
. Similarly, from (A.2), (A.3), (A.4), (A.6), (A.8), (A.11),
(A.13) and (A.15) we can write
uM
µI
= pM
ac
, (A.20)
with pM
ac
≡ SMac + Γacpiac + Λac − ΩQ, Γac ≡ w
(
1− tl
)
− ψTacψC , Λ
ac ≡ δAA
ac
Mac
uψψC
, and
ΩQ ≡ ξQuψψCωQMac . From the perspective of the consumer, p
M in and pM
ac
denote the
54
(minimised) effective price of inactive and active travel, respectively. As in Parry and
Small (2004), the homogeneity assumption will ensure that we can write αF (tf ) and
αH(tf ). Repeated substitutions using (3), (9), (10), the first order conditions and (A.19)
and (A.20) then allow us to write the following demand functions
C = C(pM
in
, tl); L = L(pM
in
, tl);
M in = M in(pM
in
, tl); Mac = Mac(pM
in
, tl);
F = F
(
pM
in
, tl, tf
)
= αF
(
tf
)
M in(pM
in
, tl);
H = H
(
pM
in
, tl, tf
)
= αH
(
tf
)
M in(pM
in
, tl).
(A.21)
where the last two follow from the definition of αF and αH . For prices we can similarly
write
pM
in
= pM
in (
tf , piin, tl
)
; pM
ac
= pM
ac (
tf , piin, tl
)
. (A.22)
The demand and price functions (A.21) and (A.22) are equivalent to expressions
(B3a) and (B3b) in Parry and Small (2004). The remainder of the derivations then
similarly follows Parry and Small (2004). First, from (A.21) and (A.22), we can write
L(tf , piin, tl). Then
dL
dtf
=
∂L
∂tf
+
∂L
∂piin
dpiin
dtf
+
∂L
∂tl
dtl
dtf
. (A.23)
Now from (11), we have wLdtl + wtldL+ Fdtf + tfdF = 0, which gives
dtl
dtf
= −
F + tf dFdtf + wt
l dL
dtf
wL
. (A.24)
Combining (A.24) and (A.23) then allows us to write
dtl
dtf
= −
F + tf dFdtf + wt
l
[
∂L
∂tf +
∂L
∂piin
dpiin
dtf
]
w
[
L+ tl ∂L∂tl
] . (A.25)
Substituting (A.25) into (A.23) and multiplying by tl then yields
tl
dL
dtf
=
−tl ∂L∂tl
w
[
L+ tl ∂L∂tl
]
[
tf
dF
dtf
]
+
tl
w
[
L+ tl ∂L∂tl
]
[
w
∂L
∂tf
L−
∂L
∂tl
F + wL
∂L
∂piin
dpiin
dtf
]
.
(A.26)
Now define MEB ≡
− ∂L
∂tl
tl
L
1+ ∂L
∂tl
tl
L
, and let εLL denote the uncompensated labour supply
elasticity: εLL ≡ ∂L∂[(1−tl)w]
(1−tl)w
L =
∂L
∂tl
∂tl
∂(1−tl)w
(1−tl)w
L . Then MEBL =
tl
1−tl
εLL
1− t
l
1−tl
εLL
,
55
which we use in (A.26) to obtain
wtl
dL
dtf
= MEB
[
tf
dF
dtf
]
−
MEB
∂L/∂tl
[
w
∂L
∂tf
L−
∂L
∂tl
F + wL
∂L
∂piin
dpiin
dtf
]
. (A.27)
Next consider the second term in brackets in (A.27). Then tf and pi only affect L
through pM
in
(consider (A.21) and (A.22)). So
∂L
∂tf
=
∂L
∂pM in
∂pM
in
∂tf
=
∂L
∂pM in
αF , (A.28)
and
∂L
∂piin
=
∂L
∂pM in
∂pM
in
∂piin
=
∂L
∂pM in
Γin. (A.29)
Next, from (3),
dpiin
dtf
=
∂piin
∂M in
dM in
dtf
. (A.30)
In turn, ∂L
c
∂pMin
= ∂L
∂pMin
+ ∂L∂I
∂I
∂pMin
, where I = w(1 − tl)L denotes income and the
superscript c denotes a compensated coefficient. Then we obtain from (A.21):
∂L
∂pM in
=
∂Lc
∂pM in
−
∂L
∂I
M in;
∂L
∂tl
=
∂Lc
∂tl
−
∂L
∂I
wL. (A.31)
The Slutsky symmetry property gives ∂L
c
∂pMin
= − ∂M
in,c
∂(1−tl)w
. Note again that ∂M
in,c
∂(1−tl)w
=
∂M in,c
∂tl
∂tl
∂(1−tl)w
= −∂M
in,c
∂tl w
−1, so
∂Lc
∂pM in
=
∂M in,c
∂tl
w−1. (A.32)
Next, leisure is weakly separable in utility. So when tl changes, it affects consumption
and demand only through disposable income. This gives
∂M in,c
∂tl
=
∂M in,c
∂I
w
(
1− tl
) ∂Lc
∂tl
, (A.33)
where w
(
1− tl
)
∂Lc/∂tl is the change in disposable income following a compensated
increase in the labour tax. Then we can use (A.28)-(A.33) to find
w
∂L
∂tf
L−
∂L
∂tl
F =
(
ηMI − 1
) ∂Lc
∂tl
F, (A.34)
56
and
wL
∂L
∂piin
dpiin
dtf
=
[
ηM
inI ∂L
c
∂tl
− wL
∂L
∂I
]
ZC
dM in
dtf
, (A.35)
with ηMI ≡ ∂M
in,c
∂I
I
M in . Substituting (A.34) and (A.35) back in (A.27) then gives
wtl
dL
dtf
= MEB
[
tf
dF
dtf
]
−
MEB
εLL
[
εcLL
(
ηM
inI − 1
)
F +
[
εLL − ε
c
LL
(
1− ηM
inI
)]
ZC
dM in
dtf
]
,
(A.36)
where we use the compensated labour elasticity εcLL = −
∂Lc
∂tl
1−tl
L = −
∂Lc
∂tl
tl
L
1−tl
tl , and,
from the Slutsky equation, εLL = εcLL + ηLI . We then substitute (A.36) into (A.18) to
obtain (13).
B Parametrisation, further details
B.1 Additional parameters
Table B.1 lists additional values used in our calculations of Equation (13), but not
previously specified in Table 1.
Table B.1: Remaining parameter values
US UK
Parameter Central value Central value
Number of active miles travelled per person per year, Mac 267 740
Number of vehicle miles travelled per person per year, M in 10,307 9124
Uncompensated labour supply, εLL 0.2 0.2
Compensated labour supply, εcLL 0.35 0.35
Producer price of gasoline, pf 186 186
Tax on labour, tL 0.318 0.31
B.2 Elasticities
Tables B.2, B.3 and B.4 provide an overview of the elasticity of fuel, inactive, and
active miles travelled with respect to the fuel price established in the literature. These
elasticities capture the extent to which fuel consumption, inactive miles travelled
(VMT), and active miles travelled change in response to fuel prices. We use these
values to parametrise our model as explained in Section 3.1.1. Similarly, Table B.5 lists
the literature estimates for the income elasticity of inactive travel, which captures the
response of inactive miles travelled to income changes.
57
Table B.2: Fuel price elasticity, ηFF
Country Value Range Notes Source
OECD -0.3 -0.6 to -0.8 short run and long run, resp. Graham and Glaister (2002)
UK and US -0.55 -0.3 to -0.9 Parry and Small (2005)
US -0.21 -0.21 to -0.75 short run Hughes et al. (2006)
US -0.43 Small, Van Dender, et al. (2007)
US -0.46 -0.1 Davis and Kilian (2011)
US -0.3 Dahl (2012)
UK -0.33 Dahl (2012)
UK -0.6 Fouquet (2012)
US -0.31 Havranek et al. (2012)
Europe -0.82 response to change in tax size Dieler et al. (2015)
US -0.37 Coglianese et al. (2017)
US -0.27 -0.27 to -0.35 Levin et al. (2017)
Denmark -0.3 up to -0.87 Gillingham and Munk-Nielsen (2019)
The only direct estimate of a cross-elasticity of active modes of travel (walking and
cycling) with respect to fuel price, ηM
acF , we identified is a 1999 report summarising the
research of several European studies (Hague Publishing et al., 1999). To our knowledge,
no cross-elasticity estimates for the US exist in the literature. This poses two problems:
first, the data on cross-elasticities is old, and cultural differences may mean that people’s
behaviour is more (less) elastic towards other modes of travel, including bike sharing,
ride-hailing services such as Uber, or electric non-active modes such as e-scooters (see
e.g. Shipman (2019)). Second, the data is context-specific to higher density European
cities.
To adjust for this gap in the literature, we additionally consider the cross-elasticity
of public transport with respect to the fuel price. Although we recognise that the
relationship between active travel, public transport, and car use is likely to be different
in the UK and US, we use the same values of ηM
acF for both countries. This is because
of the limited number of studies estimating this value.
For the income elasticity of inactive travel, ηM
inI , we consider the following. Direct
estimates of ηM
inI can be found in Small, Van Dender, et al. (2007), Santos and
Catchesides (2005) and Fouquet (2012). Additionally, and akin to Parry and Small
(2005), we consider estimates for the income elasticity of fuel use (Mattioli et al.,
2018; West and Williams III, 2007). This approach is justified by Johansson and
Schipper (1997), who found that the income elasticity of fuel use and miles travelled
were approximately equal.
58
Table B.3: Fuel price elasticity of inactive miles travelled (VMT), ηM
inF
Country Value Range Notes Source
OECD -0.3 -0.15 to -0.3 short run and long run, resp. Graham and Glaister (2002)
UK and US -0.4 0.2-0.6 Parry and Small (2005)
US -0.1 Small, Van Dender, et al. (2007)
Germany -0.45 Frondel and Vance (2013)
California -0.147 0.041 to -0.288 Knittel and Sandler (2013)
Denmark -0.32 -0.32 to -0.45 De Borger et al. (2016)
UK -0.301 -0.1803 to -0.417 Cerruti et al. (2019)
US -0.3 -0.05 to -0.3 Gillingham and Munk-Nielsen (2019)
Table B.4: Cross-elasticity of active travel and public transport (PT) use, ηM
acF
Type Country Value Notes Source
PT Australia 0.104 to 0.291 Hensher and King (1998)
Walking and cycling Europe 0.13 Hague Publishing et al. (1999)
PT Europe 0.14 Hague Publishing et al. (1999)
PT - transit general US 0.12 Currie and Phung (2007)
PT - light rail US 0.27 to 0.38 Currie and Phung (2007)
PT - buses US 0.04 Currie and Phung (2007)
PT US 0.24 Haire and Machemehl (2007)
PT US 0.4 short term Holmgren (2007)
PT Australia 0.22 Currie and Phung (2008)
PT US 0.366 Upward trend in elasticities Lane (2008)
PT US 0.08 to 0.16 (medium-)small cities only Mattson (2008)
PT South Korea 0.32 Lee et al. (2009)
PT - rail, regional US 0.27 to 0.38 Maley and Weinberger (2009)
PT - local, bus US 0.15 to 0.23 Maley and Weinberger (2009)
PT - commuter rail 218 US cities -0.012 to 0.213 Upward trend in elasticities 2002-2008 Blanchard (2009)
PT - light rail 218 US cities -0.103 to 0.507 elasticities increased 2002-2008 Blanchard (2009)
PT 0.116 Iseki, Ali, et al. (2014)
Table B.5: Income elasticity of inactive travel, ηM
inI
Country Value Range Source
UK 0.8 0.4-1.2 Parry and Small (2005)
US 0.6 0.3-0.9 Parry and Small (2005)
UK 0.4 0.0681 to 0.6335 Santos and Catchesides (2005)
US 0.02 West and Williams III (2007)
US 0.53 Small, Van Dender, et al. (2007)
UK 0.8 Fouquet (2012)
UK high income 0.62 0.54 to 0.7 Mattioli et al. (2018)
UK low income 0.56 0.36 to 0.75 Mattioli et al. (2018)
59
B.3 External cost of fuel (CO2) pollution, ZPF¯
Table B.6 provides an overview of literature estimates for the social cost of carbon. We
assign the greatest weight to the recent studies by Pindyck (2019) and Ha¨nsel et al.
(2020), which both rely on interviews conducted with experts in the field and their view
regarding the appropriate value of the SCC.
Table B.6: Social cost of carbon estimates literature overview
USD 2017, per tonne CO2 Plausible SCC range Source
91.8 2.6 - 367 Parry and Small (2005)
78.8 0 - 297.6 Tol (2011)
900 up to 1500 Ackerman and Stanton (2012)
49.7 14.2 - 73.3 Environmental Protection Agency (2016)
40 17 - 84 van den Bijgaart et al. (2016) and DICE
as calculated by OECD (2018)
517 5274.4 Adler et al. (2017)
92.4 46.2 - 140 BEIS (2018)
41.3 25.4-157.5 Nordhaus (2018)
417 177 - 805 Ricke et al. (2018)
319.3 253.2 - 385.4 Cai and Lontzek (2019)
90 80 - 100 Pindyck (2019)
96 16.2 - 494.4 Ha¨nsel et al. (2020)
A selection of the social cost of carbon estimates found in the literature. BEIS (2018) is a UK specific
value.
To obtain the externality cost per gallon of fuel used, we need to account for
the different fuel CO2 emission intensities, as well as the country-specific average fuel
composition. The average emissions intensity of diesel is 8.7 kgCO2/gallon, and 10.1
kgCO2/gallon for gasoline. The US fuel mix is 98.5% gasoline, while the UK fuel mix
is 55% gasoline and 45% diesel. Using a central value for the social cost of carbon of 90
$/tCO2, and a low (high) value of 40 (400) $/tCO2, we obtain the estimates for ZPF¯
as specified in Table 1.
B.4 Marginal value of health through active travel ZQ (HEAT)
To determine the marginal value of health through active travel increases, ZQ, we use
estimates from the WHO Health Economic Assessment Tool (HEAT). HEAT is an open-
access online tool for conducting economic assessments of active transport (changes)
and their impact on health benefits from physical activity, air pollution, accidents,
and effects on carbon emissions.38 Importantly, the tool relies on international expert
38The tool is available on www.heatwalkingcycling.org.
60
consensus and the methodology is regularly updated to reflect new research evidence
and data. The tool can be used for cost-benefit analysis, and requires inputs on baseline
levels of active travel, the assessed change in active travel, the time needed to achieve
those changes, as well as assumptions regarding substitution away from other forms
of exercise. Other (optional) inputs are the discount rate, the value of statistical life
(VSL), and the source the information supplied to the model comes from (count data,
population survey, modelled data, hypothetical scenario). In its computations, HEAT
assumes a linear relationship between active travel increases and health benefits. More
details can be found in Section 3.4 of Kahlmeier et al. (2017).39
While the tool is not designed for use outside of Europe,40 it has been applied to US
settings. Examples are Colorado (BBC Consulting, 2017), Arkansas (BBC Consulting,
2018) and Boston (James et al., 2014). We therefore used HEAT for both the UK and
the US, and used country-specific values to calibrate the model.
Further details regarding the data inputs we used are given below. Baseline average
levels of walking and cycling of 4.8 and 1.2 minutes per day in the UK, and 5.6 and
0.4 for the US. We consider an increase of 5 minutes of activity per person per day
for the population aged 20-74, split between the walking and cycling according to
current proportions of the two modes. We assume changes materialize over a three year
adjustment period, and denote an average of lives saved per year over a 10 year period
as our outcome variable of interest. We use 2017 population data from Murphy et al.
(2018) for the US, and Patel (2017) for the UK. HEAT requires the user to specify
the degree to which increases in active travel crowd out other forms of activity. As
evidence suggests that exercise through active travel is additive, rather than crowding
out other forms of exercise (Foley et al., 2015; Laeremans et al., 2017; Dons et al., 2018;
Castro et al., 2019), we assume that all increases in physical activity are additional.
Finally, we maintain the HEAT default parameter values for “mortality relative risk
reductions” associated with walking (0.89) and cycling (0.90), which affect the number
of lives saved due to a given increase in active travel.
HEAT focuses on mortality reductions due to active travel. A more accurate
representation of the health effects of active travel would also account for quality of
life effects. Adopting a healthy lifestyle can, for instance, increase the disease-free
39Discussions of the strengths and limitations of HEAT, and the best methods for economic
evaluations of active travel, can be found in Fishman et al. (2015) and Deenihan and Caulfield (2014).
40US cities are less dense and have too small shares of walking and cycling (about 0.2% of total
travel) to make a country-wide assessment about the impact of active travel on health. Conversely,
Asian cities, though often high-density, have such elevated levels of air pollution exposure that the
physical activity relative risk reduction coefficients HEAT uses are not appropriate.
61
lifetime of an adult by up to 10 years (Li et al., 2020). As HEAT does not include such
benefits, we consider it a conservative estimate of the overall health benefits of active
travel per mile travelled.
The main output from HEAT is the number of deaths saved through the specified
increase in active travel. We use the VSL, which reflects the value society places on
a life and is commonly used in cost-benefit analyses, to convert these lives saved to
$ amounts. In line with OECD (2012) and updated to end 2017 values, we adopt a
US VSL of $9.26m; for the UK we use $4.36m (OECD, 2012; Kahlmeier et al., 2017).
This approach gives the value of the total number of lives saved due to an increase in
active travel by 5 minutes. We finally convert this value to the value of health gained
per additional mile of active travel, by dividing by the total increase in active travel,
assuming a speed of 15 minutes per mile. This gives a central value of ZQ of $6.91 for
the US, and $2.44 for the UK.
C Quantification of second-best results, further
details
C.1 Obtaining the numerical results
Our quantification of the optimal fuel tax takes into account that the fuel consumption
F and miles travelled M in and Mac depend on the fuel tax level. In other words, we
approximate Z/F in (14) by
Y
F
=
Z0
F 0
(
pf + tf∗1
pf + tf0
)ηZF−ηFF
,
with Y ∈ {M in,Mac}, and where tf0 and Y 0 denote the baseline level of tf and Y .41
We take labour supply L and labour taxes tl as constant throughout the quantification.
While through Equations (10) and (11), changes in time spent travelling and fuel tax
revenues may affect labour supply and taxes, this effect is likely minor, and abstracting
from this interaction substantially simplifies computations.
We additionally compute a “na¨ıve” tax rate, where we ignore the endogeneity of fuel
consumption and miles travelled, fixing those variables at their baseline levels instead.
The results of this exercise are presented in Table C.1, and show that calculating the
optimal fuel tax at baseline levels lowers the tax level.
41See also Parry and Small (2005).
62
Table C.1: Optimal fuel tax rate, fixed fuel consumption and miles travelled
Cost, USD cents/gallon United States United Kingdom
Adjusted Pigouvian tax:
Pollution, fuel-related, ZPF¯ 91 86
Pollution, distance-related, ZPM¯inβM inM inF 75 74
Congestion, ZCβM inM inF 167 102
Accidents inactive, ZAM¯inβM inM inF 107 33
Accidents active, ZAM¯acβMacMacF -1.7 -1.4
Physical inactivity, (1− ω)ZQβMacMacF 107 104
Adjustment to MECF for excess burden -51 -35
Ramsey tax 66 77
Congestion feedback -0.8 2.8
Optimal fuel tax rate with physical activity, t∗f 560 442
Optimal fuel tax rate without physical activity 462 346
Optimal fuel tax rate with physical activity, main specification 1013 454
Current (2017) tax rate 406 55
The second-best optimal fuel tax that does not allow for fuel or mileage related adjustments (fourth
row from the bottom) is far lower than the second-best optimal fuel tax with endogenously determined
fuel consumption, fuel efficiency, and active and inactive miles travelled (second row from the bottom).
C.2 Quantifying welfare effects
We follow Parry and Small (2005) and Parry and Small (2004) to obtain a formula for
the welfare gain of a marginal tax increase. First, we use the definitions for Z to rewrite
(A.16) as
1
µI
dV
dtf
=
[
ZPF¯ − tf
] −dF
dtf
+
[
ZC + ZAM¯in + ZPM¯in
] −dM in
dtf
+
[
ZAM¯ac
] −dMac
dtf
− (1− ω) Z˜Q
−dQ
dtf
− wtl
−dL
dtf
.
Next, assuming that the effect of tf on Q runs through Mac alone, we can write
1
µI
dV
dtf
=
[
ZPM¯in − tf
] −dF
dtf
+
[
ZC + ZAM¯in + ZPM¯in
] −dM in
dtf
+
[
ZAM¯ac − (1− ω)ZQ
] −dMac
dtf
− wtl
−dL
dtf
.
Now use the the definitions of ηFF ≡ dFdpF
pF
F , η
M inF ≡ dM
in
dpF
pF
M in , η
MacF ≡ dM
ac
dpF
pF
Mac ,
βM
in
= ηM
inF/ηFF and βM
ac
= ηM
acF/ηFF . Then −dFdtf = −
dF
dpF
pF
F
F
pF
= −ηFF FpF and,
63
using a similar approach for −dM
in
dtf , we can write
1
µI
dV
dtf
=
[
ZPM¯in − tf
]
(
−ηFF
F
pf + tf
)
+
[
ZC + ZAM¯in + ZPM¯in
]
(
−ηM
inF M
in
pf + tf
)
+
[
ZAM¯ac − (1− ω)ZQ
]
(
−ηM
acF M
ac
pf + tf
)
− wtl
−dL
dtf
.
Next use (14) to find
1
µI
dV
dtf
=
[
MEC − tf
]
(
−
FηFF
pf + tf
)
+ wtl
dL
dtf
. (C.1)
Note that in Parry and Small (2005), wages w are normalised to 1. In addition, the
(negative) value of ηFF is expressed in absolute terms. Finally, they define pF as the
tax-inclusive fuel price, which in our setting is given by pf + tf . In sum, this means that
Equation (D1) in Parry and Small (2004) and (C.1) and all subsequent expressions in
the derivations are equivalent solutions.
Finally, further substitutions along the lines suggested in Appendix D in Parry and
Small (2004) allow us to further rewrite (C.1), as
1
µI
dV
dtf
=
(
−
Fηff
pf + tf
)
(1 +MEBL)
[
tf∗ − tf
]
.
C.3 Quantifying changes in mortality
To quantify the number of lives lost or gained as discussed in Section 4, we used
HEAT and followed instructions by Kahlmeier et al. (2017). We parameterise HEAT
as described in Section B.4. Further conditions that were specifically calibrated to the
US and UK contexts are listed in Table C.2 below.
Table C.2: Country-specific HEAT inputs into welfare change analysis
US UK
Input Value Source Value Source
Traffic conditions 35km/h HEAT def., some congestion 32km/h HEAT def., EU av.
Air quality, PM2.5 µg/m3 7.5 EPA (2017) 10.5 HEAT def., UK av.
Fatalities/100m km, walking 4.7 Buehler and Pucher (2017) 2.2945 HEAT def., UK av.
Fatalities/100m km, cycling 9.7 Buehler and Pucher (2017) 2.1377 HEAT def., UK av.
Additional inputs that calibrate the calculations HEAT does to the context of the US and UK. Values
labelled “HEAT def.” are default values provided by the HEAT tool and are average values that apply
specifically to the UK or European context.
HEAT provides risk-specific mortality changes, which we aggregated in to a single
64
value in the main text. Table C.3 shows the disaggregation of this value for the change
from the current tax levels to the second-best optimal fuel tax levels for both countries.
Table C.3: Country and cause-specific mortality changes, as provided by HEAT
US UK
Cause of death Value Value
Physical activity -6457 -40
Air pollution exposure 83 0.9
Crash risk 108 0.3
Number of lives gained (negative value) or lost (positive value) per country per year, following a change
from current to second-best optimal fuel tax levels, tf .
C.4 Sensitivity analysis
The sensitivity analysis was done by applying Latin Hypercube algorithms,
implemented by the pse package in R (Chalom and de Prado, 2015). Latin Hypercube
sampling is a method of parameter space exploration and optimisation, and is described
in more detail (and compared against other methods, such as individual parameter
disturbance and Monte Carlo) in Chalom and de Prado (2015). The results are depicted
in Figure 5, which specifies the probabilities that the second-best optimal fuel taxes lie
below a pre-specified value.
Figures C.1a and C.1b additionally show the relative influence of each parameter on
the optimal fuel tax. More specifically, each Figure shows the partial rank correlation
coefficient, defined as the effect of changing one parameter, ceteris paribus, relative to
changing any other parameter, ceteris paribus, on the fuel tax. A negative value means
that a larger parameter value reduces the size of the optimal fuel tax, and vice versa.
From the figure, the external cost of CO2, ZPF¯ and the fuel price elasticity, ηFF have
the greatest effect on the optimal tax, both for the US and the UK. The effect of the
external cost of accidents is comparatively small.
65
PRCC
Partial rank correlation coefficient
etaMI
ZPF
ZC
ZAMin
ZPM
etaMinF
etaFF
ZAMac
omega
ZQ
etaMacF
−1.0 −0.5 0.0 0.5 1.0
(a) PRCC plot for the US.
PRCC
Partial rank correlation coefficient
etaMI
ZPF
ZC
ZAMin
ZPM
etaMinF
etaFF
ZAMac
omega
ZQ
etaMacF
−1.0 −0.5 0.0 0.5 1.0
(b) PRCC plot for the UK.
Figure C.1: Partial rank correlation coefficient (PRCC) plots showing the relative influence of
each parameter on the optimal fuel tax, obtained using Latin Hypercube sensitivity analysis.
66