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energies
Article
Energy Rebound as a Potential Threat to a
Low-Carbon Future: Findings from a New
Exergy-Based National-Level Rebound Approach
Paul E. Brockway 1, *, Harry Saunders 2, Matthew K. Heun 3, Timothy J. Foxon 1,4,
Julia K. Steinberger 1, John R. Barrett 1and Steve Sorrell 4
1
Sustainability Research Institute, School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK;
t.j.foxon@sussex.ac.uk (T.J.F.); j.k.steinberger@leeds.ac.uk (J.K.S.); j.r.barrett@leeds.ac.uk (J.R.B.)
2The Breakthrough Institute and Decision Processes Incorporated, 2308 Saddleback Drive, Danville,
CA 94506, USA; hsaunders@earthlink.net
3Engineering Department, Calvin College, Grand Rapids, MI 49546, USA; mkh2@calvin.edu
4Centre on Innovation and Energy Demand and Sussex Energy Group, Science Policy Research Unit,
University of Sussex, Falmer, Brighton BN1 9QE, UK; s.r.sorrell@sussex.ac.uk
*Correspondence: p.e.brockway@leeds.ac.uk; Tel.: +44-113-343-2846
Academic Editor: Bin Chen
Received: 30 September 2016; Accepted: 19 December 2016; Published: 7 January 2017
Abstract:
150 years ago, Stanley Jevons introduced the concept of energy rebound: that anticipated
energy efficiency savings may be “taken back” by behavioural responses. This is an important
issue today because, if energy rebound is significant, this would hamper the effectiveness of energy
efficiency policies aimed at reducing energy use and associated carbon emissions. However, empirical
studies which estimate national energy rebound are rare and, perhaps as a result, rebound is
largely ignored in energy-economy models and associated policy. A significant difficulty lies in
the components of energy rebound assessed in empirical studies: most examine direct and indirect
rebound in the static economy, excluding potentially significant rebound of the longer term structural
response of the national economy. In response, we develop a novel exergy-based approach to estimate
national energy rebound for the UK and US (1980–2010) and China (1981–2010). Exergy—as “available
energy”—allows a consistent, thermodynamic-based metric for national-level energy efficiency.
We find large energy rebound in China, suggesting that improvements in China’s energy efficiency
may be associated with increased energy consumption (“backfire”). Conversely, we find much lower
(partial) energy rebound for the case of the UK and US. These findings support the hypothesis
that producer-sided economies (such as China) may exhibit large energy rebound, reducing the
effectiveness of energy efficiency, unless other policy measures (e.g., carbon taxes) are implemented.
It also raises the prospect we need to deploy renewable energy sources faster than currently planned,
if (due to rebound) energy efficiency policies cannot deliver the scale of energy reduction envisaged
to meet climate targets.
Keywords:
constant elasticity of substitution (CES) function; aggregate production function (APF);
energy efficiency; energy rebound; exergy efficiency; exergy; macroeconomic rebound; energy policy
1. Introduction: A Low Carbon Future—Under Threat from Energy Rebound
1.1. Concepts: Energy Efficiency and Energy Rebound
Reducing energy-related CO
2
emissions is a key component of energy policies designed to meet
climate targets—given burning of fossil fuels accounts for around 80% [
1
] of global greenhouse gas
(GHG) emissions. As illustrated in Figure 1, the path to a low carbon future is envisaged via two
Energies 2017,10, 51; doi:10.3390/en10010051 www.mdpi.com/journal/energies
Energies 2017,10, 51 2 of 24
key policy-supported measures: the introduction of zero/low carbon energy sources [
2
], and the
deployment of energy efficiency technologies to reduce energy use [
3
]. Efficiency-induced decoupling
of global energy use from economic output (GDP) would mark a significant departure from observed
history where energy-GDP have been tightly linked [
4
], in which case the feasibility and realism of
emissions reduction strategies typical of Figure 1is brought into question. Meanwhile, whilst authors
including Alcott [
5
] advocate a strategy of sufficiency—by capping the production (or consumption) of
products and services—the 5% CO
2
abatement from demand measures in Figure 1suggests this has
little envisaged role at present. This may reflect a simple truth: that reducing energy demand in the
face of rising affluence is a hard task [6].
Energies 2017, 10, 51 2 of 24
policy-supported measures: the introduction of zero/low carbon energy sources [2], and the
deployment of energy efficiency technologies to reduce energy use [3]. Efficiency-induced
decoupling of global energy use from economic output (GDP) would mark a significant departure
from observed history where energy-GDP have been tightly linked [4], in which case the feasibility
and realism of emissions reduction strategies typical of Figure 1 is brought into question. Meanwhile,
whilst authors including Alcott [5] advocate a strategy of sufficiency—by capping the production (or
consumption) of products and services—the 5% CO
2
abatement from demand measures in Figure 1
suggests this has little envisaged role at present. This may reflect a simple truth: that reducing energy
demand in the face of rising affluence is a hard task [6].
Figure 1. Envisaged contribution of different measures to energy-related CO
2
emissions reductions in
the International Energy Agency’s (IEA’s) 450 Scenario [7] (p. 114).
However, “energy rebound” is a potential threat to the success of energy efficiency technologies
to reduce energy use at the scale envisaged. This occurs when “energy-saving innovations induce an
increase in energy consumption that offsets the technology-derived saving” [8] (p. 40). Energy
rebound is not a new concept: it is now more than 150 years since William Stanley Jevons in his book
“The Coal Question” [9] made the counterintuitive claim that the introduction of energy efficient
technologies to the Scottish coal mining industry had led to increases rather than reductions in energy
consumption.
To understand rebound further, we now outline the components of energy rebound. At a
national level, energy rebound is comprised of microeconomic and macroeconomic mechanisms, as
shown in Table 1.
Following Saunders [10] and Wei [11], we formally define energy rebound, , by Equation (1),
where
is the elasticity of energy use () with respect to energy efficiency gain (τ):
=1+
× 100 = 1 + ℎ(%)
(%)× 100 (1)
Behavioural responses to the adoption of energy efficient technologies affect the level of energy
savings actually delivered, leading to five possible states of energy rebound as shown in Table 2. For
example, a 1% efficiency gain and 0.5% reduction in energy use would yield = 50%, which is a
case of partial rebound. Jevons’ paradox [9] in the 1800s referred to the case of
backfire (
> 100%).
Referring to Figure 1, the potentially significant importance of energy rebound to a low-carbon
future is now revealed. On one hand, if national energy rebound is small (say 0%–10%), energy
efficiency policies will be largely unaffected by rebound, and will translate into effective emissions
reductions. On the other hand, if energy rebound is large (
> 50%), failure to account for rebound
effects will lead to an significant overestimate of the effectiveness of energy efficiency policies - with
serious implications for meeting emission reduction targets.
Figure 1.
Envisaged contribution of different measures to energy-related CO
2
emissions reductions in
the International Energy Agency’s (IEA’s) 450 Scenario [7] (p. 114).
However, “energy rebound” is a potential threat to the success of energy efficiency technologies
to reduce energy use at the scale envisaged. This occurs when “energy-saving innovations induce an
increase in energy consumption that offsets the technology-derived saving” [
8
] (p. 40). Energy rebound
is not a new concept: it is now more than 150 years since William Stanley Jevons in his book “The Coal
Question” [
9
] made the counterintuitive claim that the introduction of energy efficient technologies to
the Scottish coal mining industry had led to increases rather than reductions in energy consumption.
To understand rebound further, we now outline the components of energy rebound. At a national
level, energy rebound is comprised of microeconomic and macroeconomic mechanisms, as shown in
Table 1.
Following Saunders [
10
] and Wei [
11
], we formally define energy rebound,
Re
, by Equation (1),
where ηE
τis the elasticity of energy use (E)with respect to energy efficiency gain (τ):
Re =1+ηE
τ×100 =1+Change in energy use (%)
ener gy e f f ic iency gai n (%)×100 (1)
Behavioural responses to the adoption of energy efficient technologies affect the level of energy
savings actually delivered, leading to five possible states of energy rebound as shown in Table 2.
For example, a 1% efficiency gain and 0.5% reduction in energy use would yield
Re
= 50%, which is a
case of partial rebound. Jevons’ paradox [9] in the 1800s referred to the case of backfire (Re > 100%).
Referring to Figure 1, the potentially significant importance of energy rebound to a low-carbon
future is now revealed. On one hand, if national energy rebound is small (say 0%–10%),
energy efficiency policies will be largely unaffected by rebound, and will translate into effective
emissions reductions. On the other hand, if energy rebound is large (
Re
> 50%), failure to account
for rebound effects will lead to an significant overestimate of the effectiveness of energy efficiency
policies—with serious implications for meeting emission reduction targets.
Energies 2017,10, 51 3 of 24
Table 1. Typical components of energy rebound, based on Greening et al. [12] and Jenkins et al. [13].
Component of Energy Rebound Origin/Mechanism
Microeconomic rebound: these
rebound mechanisms occur within
the static economy, based on
responses to the reduction in
implicit price of an energy service.
Direct rebound: describes the
direct response to the energy
efficiency improvement.
Jenkins et al. [13] split into two sub-classes:
•Income/output effects: This is the increasing
demand for that energy service by producers to
expand their output (“an output effect”) or
consumers (an “income effect”).
•Substitution effects: this captures the
substitution of that energy service for the other
goods or services (consumers) or inputs to
production (producers).
Indirect rebound: this
captures the indirect effects
of direct energy rebound.
Jenkins et al. [13] split into two sub-classes:
•Embodied energy effects: The energy
“embodied” in the efficiency improvements
themselves will offset some portion of the energy
savings achieved.
•Re-spending and re-investment effects:
If consumers and firms see net cost savings from
energy efficiency improvements, this may
increase consumer expenditures or investments
in production—increasing demand for goods,
services, and factors of production, which in turn
require energy to produce and support.
Macroeconomic rebound
These mechanisms originate
from the dynamic response
of the economy to reach a
stable equilibrium (between
supply and demand for
goods and energy services).
Greening et al. [12] split these into two sub-classes:
•Economy-wide effects: shorter-term induced
changes in prices and quantities of
goods/services throughout the economy,
to reach a new, stable equilibrium.
•
Transformational effects: these stem from longer
term change to consumers' preferences, social
institutions, and rearrangement of the
organization of production.
Table 2. States of energy rebound (adapted from Saunders [10]) (p. 2197).
State of Energy Rebound, Re (%) ∆E, Change in Energy Use from 1% Efficiency Gain
Super-conservation (Re < 0%) ∆E<−1%
Zero (Re = 0%) ∆E=−1%
Partial (0% < Re < 100%) −1% < ∆E< 0%
Full (Re = 100%) ∆E= 0%
Backfire (Re > 100%) ∆E> 0%
1.2. The Issue: More Empirical National Energy Rebound Studies Are Required
The Intergovernmental Panel on Climate Change (IPCC) acknowledge the potential importance
of rebound to energy policy, stating “by considering the size of the rebound effect, a more-realistic
calculation of energy-efficiency measures can be achieved providing a clearer understanding of their
contribution to climate policy” [
14
] (p. 391). It is then perhaps surprising that energy rebound
appears largely ignored by policymakers—Maxwell et al.’s [
15
] study of rebound for the European
Commission highlighted the UK government’s Department of Energy and Climate Change (DECC)
as the “only case example identified where the direct rebound effect is now recognised and factored
into energy policy development guidance and tools”. Similarly, Font Vivanco et al. [
16
] find only
two other examples—in Ireland and the USA—of policy including rebound effects: For the most part
though, rebound is absent from energy policy: for example there is no mention of rebound in the
UK’s Energy Bill [
17
] or the European Commission’s Energy Directive [
3
]. Thus whilst policymakers
are aware of rebound as a topic [
15
], the current empirical literature has not had sufficient impact to
penetrate energy policy [
16
,
18
–
20
]. This situation arguably contradicts the “Precautionary Principle”,
Energies 2017,10, 51 4 of 24
which “emphasizes anticipation and prevention of future risks, even in the absence of full scientific
certainty about the impacts of climate change” [
14
] (p. 1009). Aligning actions to the Precautionary
Principle would mean nations conservatively accounting for energy rebound in their energy efficiency
policies—given the urgency of required climate mitigation.
The nature of current empirical rebound literature—which serves as an evidence-base to inform
energy policy—provides a potential explanation. On one hand, there is a growing range of empirical
studies—for example refer to review studies by Sorrell and colleagues [
21
–
25
] and Jenkins et al. [
13
].
On the other hand, the empirical literature mainly focusses on part of the whole national energy
rebound—in particular, (referring to Table 1) microeconomic studies of consumer-sided energy rebound.
Such empirical studies range from direct rebound effects in personal transport [
26
,
27
] to broader
consumer/household studies of direct-plus-indirect rebound effects [
28
,
29
]. The IPCC [
1
] concluded
the majority of (empirical) household studies suggested (direct and indirect) rebound in developed
countries may typically erode 20%–45% of potential energy savings.
The narrower focus (of most empirical studies) on consumer-sided direct/indirect rebound is
therefore only part of the picture for national energy rebound. It misses producer-sided (direct and
indirect) rebound, which studies suggest may give higher energy rebound [
30
–
32
], and also excludes
the macroeconomic rebound effects from the long term structural response of the economy. As a result,
empirical studies of national energy rebound—which consider all components in Table 1—are rare.
By considering only part of national energy rebound, studies may be underplaying the true magnitude
(and importance) of energy rebound—for example several recent studies [
32
–
36
] suggest national
energy rebound may over 50%, and in some cases over 100%.
If national energy rebound—i.e., including all components of Table 1—is indeed significant,
then this would have knock-on implications for the design of energy and climate policy. Therefore,
more estimates of national energy rebound are required to strengthen the evidence base to better
inform both energy-economy modelling [37–39] and policy.
1.3. The Response: An Exergy-Based Approach to Estimate National Energy Rebound
Our novel approach to estimating national energy rebound is centred on the inclusion of exergy
efficiency as a national energy efficiency metric. Exergy, a term introduced in 1956 by Rant [
40
],
was defined simply by Reistad [
41
] as the thermodynamic measure of “available energy”, meaning it is
the “usable” part of energy—i.e., available to perform physical work. Largely in response to the energy
crises of the 1970s, several exergy-based studies of energy consumption were completed
[41–44]
,
having considerable synergy with the parallel field of thermo-economics developed largely by
Georgescu-Roegen [
45
–
47
] at the same time. As energy prices fell, interest in national-level exergy
analysis waned, until a resurgence of interest in the 2000’s led by Ayres and Warr [48–50].
To illustrate how energy and exergy are different, let us consider the thermal energy content of the
water molecules in a room full of air, and a 12 V car battery. Both “systems” have the same (first law)
thermal energy content (in Joules), but only the 12 V battery has energy in concentrated, usable form
(i.e., exergy) from which we can extract physical work. Thus exergy is a measure of thermodynamic
energy quality, defined more formally by Ayres and Warr [
50
] (p. 186) as “the maximum work that a
subsystem can do as it approaches thermodynamic equilibrium (reversibly) with its surroundings”.
Unlike energy, which (in first law terms) can be neither created nor destroyed, some exergy is
necessarily destroyed in all real-world conversion processes. As energy flows through a conversion
chain, the usable part reduces in size until it is fully dissipated (the last measurable stage being at the
point of useful exergy) in exchange for energy services, as illustrated in Figure 2.
By considering exergy consumption across all energy end use categories (i.e., heat, electrical end
use, mechanical drive and muscle work), exergy accounts from primary-to-useful stages can be
constructed for entire countries [50–52], as illustrated in Figure 3.
Energies 2017,10, 51 5 of 24
Energies 2017, 10, 51 5 of 24
Figure 2. Primary-to-final-to-useful exergy conversion stages for illustrative lamp (courtesy of T.
Domingos, Instituto Superior Técnico, Lisbon).
Figure 3. Sankey diagram of 2010 UK primary-to-final-to-useful exergy conversion [53].
The aim of such aggregate, national exergy studies to date [50,51,54–56] has been to study trends
over time of aggregate exergy efficiency (as defined in Equation (2)) and useful exergy, or explore the
relationship between energy consumption and economic growth.
,τ = ()
(2)
To date, these exergy-based studies have not been applied to the area of energy rebound. This is
the research gap we seek to explore. In response, we adopt aggregate national exergy efficiency (τ)
as our metric for energy efficiency in Equation (1) and use it to estimate national energy rebound (i.e.,
considering all-components of Table 1). In doing so, we address Patterson’s observation that such
thermodynamic metrics are not in use, despite appearing “the most natural or obvious way to
measure energy efficiency” [57]. (p. 378) Referring to Figure 3—we estimate energy rebound at the
primary stage (i.e., the level of extracted energy), since this is the relevant point for climate
(emissions) policy. We estimate national energy rebound for three countries (US, UK, China), using
two different methods based on aggregate production functions (APFs). Whilst other studies such as
Zhang and Lin [33] consider multiple methods for a single country, and Malpede and Verdolini [58]
Figure 2.
Primary-to-final-to-useful exergy conversion stages for illustrative lamp (courtesy of
T. Domingos, Instituto Superior Técnico, Lisbon).
Energies 2017, 10, 51 5 of 24
Figure 2. Primary-to-final-to-useful exergy conversion stages for illustrative lamp (courtesy of T.
Domingos, Instituto Superior Técnico, Lisbon).
Figure 3. Sankey diagram of 2010 UK primary-to-final-to-useful exergy conversion [53].
The aim of such aggregate, national exergy studies to date [50,51,54–56] has been to study trends
over time of aggregate exergy efficiency (as defined in Equation (2)) and useful exergy, or explore the
relationship between energy consumption and economic growth.
,τ = ()
(2)
To date, these exergy-based studies have not been applied to the area of energy rebound. This is
the research gap we seek to explore. In response, we adopt aggregate national exergy efficiency (τ)
as our metric for energy efficiency in Equation (1) and use it to estimate national energy rebound (i.e.,
considering all-components of Table 1). In doing so, we address Patterson’s observation that such
thermodynamic metrics are not in use, despite appearing “the most natural or obvious way to
measure energy efficiency” [57]. (p. 378) Referring to Figure 3—we estimate energy rebound at the
primary stage (i.e., the level of extracted energy), since this is the relevant point for climate
(emissions) policy. We estimate national energy rebound for three countries (US, UK, China), using
two different methods based on aggregate production functions (APFs). Whilst other studies such as
Zhang and Lin [33] consider multiple methods for a single country, and Malpede and Verdolini [58]
Figure 3. Sankey diagram of 2010 UK primary-to-final-to-useful exergy conversion [53].
The aim of such aggregate, national exergy studies to date [
50
,
51
,
54
–
56
] has been to study trends
over time of aggregate exergy efficiency (as defined in Equation (2)) and useful exergy, or explore the
relationship between energy consumption and economic growth.
Exergy e f f ic iency,τ=output (us e f ul)exergy
input exergy (2)
To date, these exergy-based studies have not been applied to the area of energy rebound. This is
the research gap we seek to explore. In response, we adopt aggregate national exergy efficiency (
τ
)
as our metric for energy efficiency in Equation (1) and use it to estimate national energy rebound
(i.e., considering all-components of Table 1). In doing so, we address Patterson’s observation that
such thermodynamic metrics are not in use, despite appearing “the most natural or obvious way to
measure energy efficiency” [
57
]. (p. 378) Referring to Figure 3—we estimate energy rebound at the
primary stage (i.e., the level of extracted energy), since this is the relevant point for climate (emissions)
policy. We estimate national energy rebound for three countries (US, UK, China), using two different
methods based on aggregate production functions (APFs). Whilst other studies such as Zhang and
Energies 2017,10, 51 6 of 24
Lin [
33
] consider multiple methods for a single country, and Malpede and Verdolini [
58
] consider
multiple countries with a single method, this is believed to be the first combined multi-country
multi-method empirical study. Our approach is intentional, to allow comparisons, and expand the
rebound evidence base.
Finally, a note on boundaries and terminology, which are designed to match the aggregate national
level of climate (emissions) policy. First, the estimates are strictly those of primary exergy rebound,
since the efficiency metric (
τ
) records the efficiency of energy conversion from primary-to-useful
stages, measured in exergy (as available energy) terms. However, as primary exergy-to-energy ratios
are 1.04–1.08 for fossil fuels [
59
,
60
], aggregate primary energy and primary exergy values (in Joules)
are close [
53
] in fossil-dominated economies. This means primary “exergy” or “energy” rebound
values will be also very close. Therefore, to link to existing rebound literature, we use the term
“energy rebound” from this point. Second, our study uses a territorial boundary of national energy
use (and thus rebound), thereby accounting for the energy used within the geographic country
boundaries—as opposed to a consumption-based assessment which would include trade flows of
products with embodied energy. Third, our use of “aggregate” in “aggregate” production functions
means at the national level, rather than at a more granular (e.g., industry sector or firm-level) scale.
2. Methods and Data
2.1. Step 1: Selecting the Aggregate Production Function
APFs are the basis for our estimation of national energy rebound. They seek to explain economic
“production” or “output” (typically measured at the national scale as GDP) through a series of
input “factors of production”. Lloyd [
61
] and Mishra [
62
] provide excellent papers on the history
of APFs. Today, the two most common APFs in use [
63
] are the Cobb-Douglas (C-D) function and
the constant elasticity of substitution (CES) function. APFs are widely used in the fields of growth
accounting [
64
–
66
], macroeconomic models [
67
–
71
] and importantly for us—the estimation of energy
rebound [
11
,
38
,
72
,
73
]. Therefore, APFs as the starting point for our exergy-based method makes sense.
An example of the C-D function is given in Equation (3), where economic output
(Yt)
is related to
capital (Kt) and labour (Lt)inputs:
Yt=θeλtKα
tLβ
t(3)
where
α
,
β
are the elasticities of output (
Yt
) with respect to capital and labour respectively;
θ
is scale
parameter (for base-year);
eλt
is the Solow residual—the share of output not explained by
Kt
and
Lt
;
t
is time relative to the base year; and
λ
is exogenous growth, equal to the rate of change in the
Solow residual.
To select our APF, we next make three choices. The first (and most obvious) is to add energy (E) as
a third factor of production, in order to explore and estimate energy rebound. The second is to use the
CES function, which allows a broader range of possible values for the unknown parameters compared
to the C-D function, including non-unity elasticity of substitution between inputs. The third choice is
to “nest” the inputs, since as Sorrell [
74
] notes, the alternative non-nested CES function—where all
elasticities of substitution are equal—provides a very restricted (and unlikely) scenario. Our preference
is for a two-level KL-Efunctional structure, where capital-labour are in an inner nest, and energy
is in the outer nest. This matches the structure of the most common macroeconomic models used
to inform climate policy [
32
,
75
], and as Saunders also reported [
10
] (p. 2199) KL-Eis the most
“rebound flexible” CES function structure—permitting all rebound states in Table 2except the (unlikely)
super-conservation scenario. It also matches the nesting structure of other recent empirical rebound
studies [32,33].
The resulting KL-ECES function is shown in Equation (4):
Yt=θeλt[δ1h(δK−ρ1
t+(1−δ)L−ρ1
tiρ/ρ1+(1−δ1)E−ρ
t]
−1
ρ
(4)
Energies 2017,10, 51 7 of 24
where
ρ
,
ρ1
are substitution parameters which indicate the ease of substituting one input for
another, and are used to calculate Hicks Elasticities of Substitution (HES) between capital and
labour (via
σ1=1/(1+ρ1)
) and between capital-labour and energy (via
σ=
1
/(1+ρ)
);
δ
,
δ1
are output share parameters which specify the weight of contribution from each input (values between
0 and 1) to economic output (note that output share parameters are different from output elasticities,
which measure the change in output from a change of a single input factor of production).
2.2. Step 2: Specifying and Estimating the Exergy-Based CES Function Parameters
2.2.1. The Exergy-Based CES Function
Our aim is to econometrically obtain fitted values for the unknown parameters of an exergy-based
CES function, in order to use these parameter values in the derived rebound equations (Section 2.3).
To do this, we need to define the exergy-based CES function. We start by adopting useful exergy (U) as
the input energy factor of production, which translates Equation (4) to the exergy-based CES function
given in Equation (5):
Yt=θeλt[δ1h(δK−ρ1
t+(1−δ)L−ρ1
tiρ/ρ1+(1−δ1)U−ρ
t]
−1
ρ
(5)
Including
U
as the energy input has support in the wider economic literature—various researchers
including Ayres and Warr [
50
,
76
] and Voudouris et al. [
77
] claim that useful exergy—not primary
energy or final energy—provides the energy “input” which is most closely linked to economic growth.
Next, we normalise the inputs (K,L,E) and output (Y) to a starting (base) year, in line with
recommendations by Temple [
78
] and Klump et al. [
69
]. This overcomes a key criticism of empirical
APFs, that aggregate variables with differing units cannot be combined. Our input data is normalised
to base years of 1980 (UK and US) and 1981 (China). By convention, the aggregate variables
(Y,K,L,U)
become lower case (
y
,
k
,
l
,
u)
when normalised, such that
yt=Yt/Y0
;
kt=Kt/K0
;
lt=Lt/L0
;
ut=Ut/U0
; which modifies Equation (5) to become Equation (6), which is the final CES form
econometrically fitted:
yt=θeλt[δ1h(δk−ρ1
t+(1−δ)l−ρ1
tiρ/ρ1+(1−δ1)u−ρ
t]
−1
ρ
(6)
2.2.2. Input Data
Now that the CES function (Equation) is obtained, we assemble input data (y,k,l,u) for each
country, before econometrically estimating the six unknown parameters:
θ
,
λ
,
δ
,
δ1
,
ρ
,
ρ1
. The factors
of production and economic output (prior to normalisation) are annual time-series of Y,K,L, and U,
for the UK (1980–2010), US (1980–2010), and China (1981–2010). The output measure (Y) is taken
as aggregate GDP in 2005US$ constant prices from the Penn World Tables (PWT) 8.1 [
79
]. Capital,
labour and energy data are all quality-adjusted, meaning they seek to better account for the productive
effect of raw capital (stock), labour (workhours) and (primary) energy. Quality-adjusted inputs are
now commonly used in growth accounting studies [80–82].
For labour, quality-adjusted total hours are obtained via human capital indices from Barro
and Lee [
83
], multiplied by average workhours for the US and UK from PWT8.1 [
79
], and for
China from Wu [
84
]. Quality-adjusted capital is taken as capital services: a measure of the flow
of productive services from a capital asset. Capital service data for 1980–2010 was sourced for the
UK [
85
] and the US [
86
,
87
]. For China, capital service data was obtained for 1981–2010 from Wu [
88
].
The limitations in the availability of capital services data constrains the time-period for the study.
Previously calculated values of useful exergy (as quality-adjusted energy) for the UK, US and China
are taken from Brockway et al. [53,89].
The normalised input datasets (y,k,l,u) are shown in Figures 4–7:
Energies 2017,10, 51 8 of 24
Energies 2017, 10, 51 8 of 24
Figure 4. UK, US, China—GDP (y).
Figure 5. UK, US, China—capital services (k).
Figure 6. UK, US, China—quality-adjusted labour (l).
Figure 7. UK, US, China: useful exergy (u).
Figure 4. UK, US, China—GDP (y).
Energies 2017, 10, 51 8 of 24
Figure 4. UK, US, China—GDP (y).
Figure 5. UK, US, China—capital services (k).
Figure 6. UK, US, China—quality-adjusted labour (l).
Figure 7. UK, US, China: useful exergy (u).
Figure 5. UK, US, China—capital services (k).
Energies 2017, 10, 51 8 of 24
Figure 4. UK, US, China—GDP (y).
Figure 5. UK, US, China—capital services (k).
Figure 6. UK, US, China—quality-adjusted labour (l).
Figure 7. UK, US, China: useful exergy (u).
Figure 6. UK, US, China—quality-adjusted labour (l).
Energies 2017, 10, 51 8 of 24
Figure 4. UK, US, China—GDP (y).
Figure 5. UK, US, China—capital services (k).
Figure 6. UK, US, China—quality-adjusted labour (l).
Figure 7. UK, US, China: useful exergy (u).
Figure 7. UK, US, China: useful exergy (u).
Energies 2017,10, 51 9 of 24
2.2.3. Econometric Fitting of the CES Aggregate Production Function
Last, we obtain values for the six unknown parameters:
θ
,
λ
,
δ
,
δ1
,
ρ
,
ρ1
, by econometrically
fitting Equation (6) using a customised version of the iterative, non-linear technique developed by
Henningsen and Henningsen [
90
,
91
]. Two steps are completed. First, we determine “base-fit” values
for the six parameters by fitting to the historical data. Secondly, to provide an indication of the precision
we can attach to values of the CES parameters (and later, overall rebound), we use a “bootstrapping”
resampling technique on the CES equation to resample the residuals, following the detailed description
in Heun et al. [
92
]. Whilst bootstrapping is becoming more commonly used in APF-based growth
accounting studies [
70
,
80
,
81
], to our knowledge this is the first use in an empirical rebound study.
We resample the residual values 1000 times and re-fit to obtain resample estimates of CES parameters.
We report the 2.5% and 97.5% values (i.e., those of the 25th and 976th ranking in order of magnitude)
to provide an indication of the precision with which the original parameter estimates are known.
2.3. Step 3: Derive Equations for Estimation of National Energy Rebound
With the function and the estimation process in place, we now present two approaches (which use
the fitted CES parameter values) to estimate overall values for national energy rebound. Both methods
have been previously used for the estimation of energy rebound [
33
,
73
,
93
–
95
] and so provide a
suitable place to introduce new (exergy efficiency and useful exergy) datasets. Whilst being based
on the parameters determined from the same econometrically fitted CES function, they adopt very
different independent variables (and thus routes) for the estimation of rebound effects: the first
adopts factor-neutral technical change (
λ
) whilst the second uses the exergy efficiency metric (
ε
).
Further discussion of the two methods is given in Section 4.3.
2.3.1. Method 1: Ratio of Actual to Potential Energy Savings (AES/PES)
The method is based on estimating the ratio of actual energy savings (AES) to potential energy
savings (PES), such that rebound is defined as Equation (7). Hence if AES equals PES, then Re = 0
(zero rebound), whilst if AES is zero, then Re = 1.0.
Re =1−Actual Energy Saved (AES)
Potential Energy Saved (PES)=(PES)−(AES)
(PES)(7)
We follow the approach of Zhang and Lin [
33
] and Shao et al. [
73
], who completed empirical
rebound studies by translating Equation (7) into a rebound equation which is based on the CES
function parameters. Their logical derivation starts by finding an expression for the denominator
(PES). Taking
Yt
as GDP in year t, and
EIt
as energy intensity (
Et
/
Yt
) in year t, the energy use in year t
is thus
Yt×EIt
, whilst in year t+ 1 it is
Yt+1×EIt+1
. However, if no energy efficiency gains occurred
from year tto t+ 1, energy intensity remains unaltered (
EIt
), and the energy use in year t+ 1 would be
Yt+1×EIt. Therefore, the PES term is given by Equation (8):
PES =(Yt+1×EIt−Yt+1×EIt+1) = Yt+1×(EIt−EIt+1)(8)
The expression for the numerator is based on the central assumption that “take-back” of energy
consumption (PES-AES) in year t+ 1 is seen via
λt+1
, i.e., exogenous (all-factor) technical progress.
In other words, it assigns the rebound of energy (from energy efficiency) to exogenous growth
λ
,
meaning if
λ
= 0 then there is no energy rebound. In aggregate energy terms, the rebound energy
consumption (PES-AES) is given by the fraction of economic growth attributable to exogenous growth
[λt+1×(Yt+1−Yt)] multiplied by the energy intensity in year t+1(EIt+1), as shown in Equation (9):
PES −AES =[λt+1×(Yt+1−Yt)] ×(EIt+1)(9)
The resultant overall expression for energy rebound (from year tto t+ 1) is given in Equation (10):
Energies 2017,10, 51 10 of 24
Ret+1=λt+1(Yt+1−Yt)(E It+1)
Yt+1(EIt−EIt−1)(10)
The input data sources are primary energy consumption (
Et)
from the IEA [
96
]: GDP
(Yt
) from
PWT8.1 [
79
]; and yearly values for the parameter
λt+1
from the econometric fitting of the CES function
in Equation. The values obtained for
Ret+1
at each year (t+ 1, t+ 2, etc.) then enable the estimation
of national energy rebound for the UK, US and China by taking the mean value of
Ret+1
from all the
values obtained.
2.3.2. Method 2: Elasticity of Energy Use with Respect to Efficiency (EEE)
This method is based on deriving an expression for
ηE
τ
from the CES function, the elasticity of
energy use (in our case primary exergy, E) with respect to exergy efficiency,
τ
. The expression for
ηE
τ
can then be inserted back into Equation (1) (
Re = (1+ηE
τ)×
100
)
which can then be evaluated to
given an empirical estimate for national energy rebound.
To do this, we proceed in several stages. First, we insert
Ut=τtEt
into Equation (5), where
τt
is
exergy efficiency and Etis primary exergy, yielding Equation (11):
Yt=θeλt[δ1h(δK−ρ1
t+(1−δ)L−ρ1
tiρ/ρ1+(1−δ1)(τtEt)−ρ]
−1
ρ
(11)
There are close similarities with the APFs proposed for the estimation of aggregate energy
rebound by Saunders [
10
] in Equation (12) and Wei [
11
] in Equation (13) which both combine
τ
(as an
engineering efficiency parameter) with final energy consumption. Fundamentally, this shares our
assumption that it is the energy use at the end of energy conversion stages (in our case via useful
exergy) that serves as the appropriate input to the production function, not primary energy.
Y=ha(K∝L1−∝)ρ+b(τE)ρi1
ρ(12)
Y=fKd,τEd(13)
Similarities aside, the CES function in Equation (11) has greater flexibility than Saunders’s
use of the Hogan-Manne [
97
] CES function (where KL is a C-D function) and Wei’s C-D function.
Our approach therefore provides a larger range of possible values for the fitted parameters, which is
desirable to allow the best possible estimate of rebound.
Second, we derive an expression for
ηE
τ
based on the CES function. Saunders [
10
] previously
derived such an expression based on the more restrictive Hogan-Manne CES function given in
Equation (12). Thus, we must derive a new expression for
ηE
τ
, since our APF is in the extended
CES format of Equation (11). It follows a very similar process outlined by Saunders [
10
] for the CES
(Solow) production function. A full derivation is given in the Supplementary Information, with the
summary logic explained now.
To start we recap our aim: to assess how primary exergy use (
E)
responds to changes in the
energy efficiency gain parameter (τ), as given in Equation (14):
Re =1+ηE
τ=1+τ
E
∂E
∂τ(14)
We cannot obtain
τ
E∂E
∂τ
directly from the CES function, so it is obtained indirectly via the Implicit
Function Theorem. First we set up three implicit functions (i.e., of the form
g(x,y)=
0) using the
production function itself and two first-order economic theory conditions, whereby energy and capital
supply are adjusted in the long term such that the real price of energy (
∂Y
∂E
=
pE
c
) and capital (
∂Y
∂K
=
pK
c
)
are fixed and equal to their marginal productivities. The implicit functions are thereby of the form
shown in Equation (15):
Energies 2017,10, 51 11 of 24
ψ1=g(Y,f(K,L,τE)=0
ψ2=hpF
c,∂f(K,L,τE
∂E=0
ψ3=kpK
c,∂f(K,L,τE
∂K=0
(15)
From the vector (
Ψ= (ψ1
,
ψ2
,
ψ3)
), a Jacobian matrix assembling their partial derivatives is then
constructed in the form
J="∂ψi(Y0,E0,τ0)
∂Xj#
, where
Xj=Y
,
E
,
K
. With the Jacobian matrix now
in place, we apply the Implicit Function Theorem, which allows us to implicitly (i.e., rather than
explicitly) derive expressions (as shown in Equation (16)) for
∂Y
∂τ
;
∂E
∂τ
;
∂K
∂τ
, i.e., equations which show
how the endogenous variables (Y,E, and K) change when the efficiency gain parameter
(τ)
changes
(which remember is our goal):
∂Y
∂τ
∂E
∂τ
∂K
∂τ
=−J−1
∂ψ1
∂τ
∂ψ2
∂τ
∂ψ3
∂τ
(16)
In our case we find the following result in Equation (17):
∂Y
∂τ
∂E
∂τ
∂K
∂τ
=−c2sEsK
pEpK(1+sE+sK)
pE
csE
pK
csK
pE
c
pK
csK
pK
c
pE
csE
pK
csK
−pK
c1+sK
sKpK
c
pE
csE
pE
c−pE
c1+sE
sE
−sEY
τ
−ρ
1+ρ
pE
csE
E
τ
0
(17)
sE
and
sK
are the cost shares of energy and capital, and as shown in Equation (18), are equal to the
price of a factor (
pE
or
pK
) multiplied by its quantity
(Eor K)
, and then divided by the price of output
(c) times the output quantity (Y):
sE=pEE
cY sK=pKK
cY (18)
Now we can extract the required elasticity term as Equation (19):
⇒τ
E
∂E
∂τ=1
(1+sE+sK)ρ(sE−sK−1)+sE
(1+ρ)(19)
Thus the long-term energy rebound equation is given by Equation (20):
Re =1+τ
E
∂E
∂τ=(1+sE+sK)(1+ρ)+(ρ(sE−sK−1)+sE)
(1+sE+sK)(1+ρ)(20)
Equation (20) is interesting as energy rebound is only related to cost shares (
sE
,
sK
) and the
substitution parameter (
ρ
) between the capital-labour composite and energy. Rebound is thereby
independent of ρ1—the substitution parameter between capital and labour.
Values for the substitution parameter (
ρ
) are determined by the econometric fitting of the CES
function in Equation (6). The energy cost share
pE
c
over the period analysed for the UK and US are
taken as the average ratio 0.08 [
98
,
99
], and for China, the energy cost share ratio is taken as slightly
higher (0.10), based on the assumption that the economy is less competitive, causing energy to be
relatively more expensive compared to the UK and US (with also labour cheaper). The sensitivity
(to rebound estimate) of this assumption for China is later evaluated (in Section 4.2), by comparing
Energies 2017,10, 51 12 of 24
our estimates of rebound for energy cost shares ranging from 0% to 20%, and is found to make little
difference to the rebound estimate.
The capital cost share is indirectly obtained via the labour cost share data, which is more readily
available. The average capital/labour cost share for UK and US was taken as a 30%/70% split of total
GDP, based on data from Schneider [100] shown in Figure 8:
Energies 2017, 10, 51 12 of 24
estimates of rebound for energy cost shares ranging from 0% to 20%, and is found to make little
difference to the rebound estimate.
The capital cost share is indirectly obtained via the labour cost share data, which is more readily
available. The average capital/labour cost share for UK and US was taken as a 30%/70% split of total
GDP, based on data from Schneider [100] shown in Figure 8:
Figure 8. Labour cost shares (payments to labour as % of GDP) for selected countries (Schneider, [100]
(p. 4)).
For China, the average capital/labour cost share was taken as a 50%/50% split, based on the
average for labour compensation/GDP found by Qi [101], as shown in Figure 9 (dark line).
Figure 9. Labour cost shares (payments to labour as % of GDP) for China [101].
At this point we note that energy is taken by economists as an intermediate good, and therefore
the cost shares shown in Figures 8 and 9 exclude energy costs. Therefore, to account for the cost of
energy, the cost shares of energy, labour and capital are amended to add to unity. This slightly
reduces the capital (and labour) costs shares derived from Figures 8 and 9 (i.e., to make space for
energy), as given in Table 3, which gives the assumed cost share values taken for the estimation of
energy rebound in Equation (20):
Table 3. Adopted cost shares, sE and sK.
Country Study Time-Scale Cost Shares
sEsK
UK 1980–2010 0.08 0.28
US 1980–2010 0.08 0.28
China 1981–2010 0.10 0.45
Figure 8.
Labour cost shares (payments to labour as % of GDP) for selected countries
(Schneider, [100] (p. 4)).
For China, the average capital/labour cost share was taken as a 50%/50% split, based on the
average for labour compensation/GDP found by Qi [101], as shown in Figure 9(dark line).
Energies 2017, 10, 51 12 of 24
estimates of rebound for energy cost shares ranging from 0% to 20%, and is found to make little
difference to the rebound estimate.
The capital cost share is indirectly obtained via the labour cost share data, which is more readily
available. The average capital/labour cost share for UK and US was taken as a 30%/70% split of total
GDP, based on data from Schneider [100] shown in Figure 8:
Figure 8. Labour cost shares (payments to labour as % of GDP) for selected countries (Schneider, [100]
(p. 4)).
For China, the average capital/labour cost share was taken as a 50%/50% split, based on the
average for labour compensation/GDP found by Qi [101], as shown in Figure 9 (dark line).
Figure 9. Labour cost shares (payments to labour as % of GDP) for China [101].
At this point we note that energy is taken by economists as an intermediate good, and therefore
the cost shares shown in Figures 8 and 9 exclude energy costs. Therefore, to account for the cost of
energy, the cost shares of energy, labour and capital are amended to add to unity. This slightly
reduces the capital (and labour) costs shares derived from Figures 8 and 9 (i.e., to make space for
energy), as given in Table 3, which gives the assumed cost share values taken for the estimation of
energy rebound in Equation (20):
Table 3. Adopted cost shares, sE and sK.
Country Study Time-Scale Cost Shares
sEsK
UK 1980–2010 0.08 0.28
US 1980–2010 0.08 0.28
China 1981–2010 0.10 0.45
Figure 9. Labour cost shares (payments to labour as % of GDP) for China [101].
At this point we note that energy is taken by economists as an intermediate good, and therefore
the cost shares shown in Figures 8and 9exclude energy costs. Therefore, to account for the cost of
energy, the cost shares of energy, labour and capital are amended to add to unity. This slightly reduces
the capital (and labour) costs shares derived from Figures 8and 9(i.e., to make space for energy),
as given in Table 3, which gives the assumed cost share values taken for the estimation of energy
rebound in Equation (20):
Table 3. Adopted cost shares, sEand sK.
Country Study Time-Scale Cost Shares
sEsK
UK 1980–2010 0.08 0.28
US 1980–2010 0.08 0.28
China 1981–2010 0.10 0.45
Energies 2017,10, 51 13 of 24
3. Results
3.1. The CES Aggregate Production Function Results
Table 4summarises the base-fit values of the six unknown parameters
(θ
,
λ
,
δ
,
θ1
,
ρ
,
ρ1
) obtained
from the non-linear fitting of historical data to Equation (6), together with their 2.5%/97.5% resampled
(confidence interval proxy) values. Table 4also gives goodness of fit (R
2
) for the overall fitted function,
and elasticities of substitution
(σ
,
σ1
, from
σ=
1
/(
1
+ρ
)) are also included as they add interpretive
value. The base-fit value of factor-neutral technical progress (
λ
) is input to Method 1 (AES/PES)
Equation (10) whilst the substitution parameter (
ρ
) value is input to the Method 2 (EEE) Equation (20).
Table 4. Constant elasticity of substitution (CES) function fitted parameter values and diagnostics.
Country Value Fitted Parameter Value
θ λ δ1δ ρ1ρ σ1σR2
UK
2.5% resampled 0.996 0.0120 0.020 0.000 −1.000 22.87 ∞0.042
0.998
Base-fit 1.014 0.0129 0.053 0.012 −1.000 65.16 ∞0.015
97.5% resampled 1.029 0.0137 0.859 0.771 171.2 1290 0.006 0.001
US
2.5% resampled 0.974 0.0034 0.262 0.675 −1.000 −1.00 ∞ ∞
0.999
Base-fit 0.958 0.0093 0.338 1.000 −1.000 84.78 ∞0.012
97.5% resampled 0.994 0.0110 1.000 1.000 16.51 113.3 0.057 0.009
China
2.5% resampled 0.959 0.0462 0.029 0.310 −1.000 −1.00 ∞ ∞
0.999
Base-fit 0.980 0.0559 1.000 0.532 228.1 −0.52 0.004 2.082
97.5% resampled 1.024 0.0606 1.000 0.724 548.5 1.07 0.002 0.484
Table 4contains some interesting features. First, the overall fit of the function is very good
(as measured by R
2
= 0.998
−
0.999), which is a common feature of empirical APF studies. Part of
the exceptional fit comes from the exogenous growth term (
λ
), so a very good fit (R
2
= 0.999) does
not mean the input factors of production (K,L,U) explain 99.9% of economic output—refer to Table 5
where λis over 50% of economic growth for the UK and China, and over 30% for the US.
Second, the resampled values show that
λ
is fitted with greater precision than
ρ
, which has
subsequent implications for the relative precision of the rebound estimates for the two methods
(see Sections 3.2 and 3.3)—accepting that other differences may also be important. The resampled
values of
λ
and (in particular)
ρ
suggest highly asymmetric intervals of precision: for example,
the rounded UK values for
ρ
are 23 (2.5% lower bound), 65 (base-fit), and 1290 (97.5% upper bound).
Third, when the values of
ρ
or
ρ1
are equal to
−
1, the elasticity of substitution parameter (
σ
or
σ1
)
equals infinity (
∞
), meaning the inputs are perfect substitutes at this point (i.e., reductions in one input
are compensated by increases in the other to maintain output).
3.2. Method 1 (AES/PES): Results
The rebound Equation (10) is split into three components as in Equation (21), to help the
decomposition of the rebound value:
Ret+1=λt+1×(Yt+1−Yt)
Yt+1
×(EIt+1)
(EIt−E It−1)(21)
The rebound results of the AES/PES method are then shown in Table 5. The results suggest that
the UK and US experienced national (base-fit) rebound effects in the range 40%–50% over this period,
whilst China experienced higher rebound (77%), but not backfire. Owing to the tighter resampling
banding of
λ
established in Section 3.1, the 2.5% and 97.5% resampled values are very similar to the
basefit values.
Table 5decompose the results. For the UK and US, the output (B) and energy intensity (C) values
were very similar, meaning the higher rebound in the UK can be attributed to their greater proportion
Energies 2017,10, 51 14 of 24
of economic growth from the Solow residual term
(eλt
) —over 50% versus the US (30%). For China,
all components of the rebound equation (A), (B), (C) are larger than the UK and US, leading to much
higher rebound overall than the other two countries.
Table 5. Ratio of Actual to Potential Energy Savings (AES/PES) method—energy rebound results.
Rebound Equation Component UK (1980–2010) US (1980–2010) China (1981–2010)
(A1)λt+1(2.5% resam ple)0.512 0.121 0.452
(A2)λt=1(base f i t)0.551 0.332 0.546
(A3)λt+1(97.5% resam ple)0.585 0.390 0.592
(B) Yt+1−Yt
Yt+10.022 0.026 0.093
(C) EIt−E It+1
EIt+10.023 0.022 0.066
Re2.5% resample =A1×B
C∗100
150% 15% 64%
Reba se f it =A2×B
C∗100
154% 40% 77%
Re97.5% resample =A3×B
C∗100
157% 47% 83%
3.3. Method 2 (EEE): Results
From the fitted values of
ρ
given in Table 4, we estimate (from Equation (20)) the values of primary
energy rebound (Re) shown in Table 6. The base-fit results suggest partial national energy rebound for
the UK and US is (Re = 13%), whereas China exhibits backfire (Re = 208%). The UK results have a very
tight banding between resampled values, indicating greater confidence in the precision of the base-fit
rebound estimate. In contrast, both the US and China have highly asymmetric resampled values,
particularly their (infinite) upper bound values—which stems from the value of
ρ
=
−
1, meaning the
denominator in Equation (20) is zero. In this case, savings in energy can be entirely substituted without
any restraint by capital-labour. As infinite rebound is obviously not possible—i.e., energy efficiency
cannot lead to infinite energy use—we may view this result as suggestive of backfire (
Re >
100%),
but also a limitation of the method that arises out of statistical imprecision in estimating the elasticity
of substitution between KL and U (
σ
). For China, the best we can say is that the economy appears to
be in a state of “backfire” (
Re >
100%), although large rebound (but below 100%) cannot be ruled out.
Rebound is small for the UK and US, with less certainty for the US. The results indicate that rebound is
very likely to be higher in China than either the US or UK.
Table 6.
Elasticity of Energy Use with Respect to Efficiency (EEE) method—total energy rebound results.
Rebound Value UK (1980–2010) US (1980–2010) China (1981–2010)
Re2.5% resample 12% 13% 58%
Reba se f it 13% 13% 208%
Re97.5% resample 16% Infinity (∞) Infinity (∞)
4. Discussion
4.1. Comparison to Previous Studies
A summary of our base-fit rebound estimates is shown in Figure 10. From Figure 10, we can see
that both methods give broadly similar results: partial rebound (Re~13%–50%) for the UK and US,
but much higher rebound (Re~80%–210%) for China. These findings are at least partly supported
by the literature. For the UK and US, the estimates are similar to the 25%–40% range suggested by
Energies 2017,10, 51 15 of 24
Jenkins et al. [
13
] for developed countries, and the national rebound estimates of 25%–70% for OECD
economies suggested by Barker et al. [102] and Saunders [32].
Energies 2017, 10, 51 15 of 24
Figure 10. Summary of base-fit rebound results.
For China, Table 7 summarises six national rebound estimates found in the literature using
AES/PES or EEE methods, plus our own base-fit results. The six AES/PES studies in Table 7 have
broadly consistent results: estimating national energy rebound for China to be in the range 37% to
77%, with our value the highest (77%). If we assume that all AES/PES studies (including ours) contain
similar aggregate energy intensity (E/GDP) values, it suggests our estimate of (exogenous growth
term) is higher than the other studies. Certainly, differences would be expected, given all studies
obtain values for using differing APFs and techniques. Only one other study uses the EEE
method—Zhang and Lin [33], however, their estimated value (52%) is for short-term rebound. Whilst
Saunders [10] (p. 2208) suggests long term rebound is slightly higher, our EEE-based estimate (208%)
is significantly higher, due to a very high elasticity of substitution between KL and E (σ~2.0).
Table 7. National energy rebound estimates for China.
Source (Reference) Time-Series Method Estimate of National Rebound
Shao et al. [73] 1954–2010 AES/PES 37%
Zhang and Lin [33] 1979–2004 AES/PES 41%
1981–2009 EEE 52% (short term)
Lin and Liu [93] 1981–2009 AES/PES 53%
Li and Lin [94) 1985–2008 AES/PES 67%
Li and Han [95] 1997–2009 AES/PES 74%
Brockway et al. (this study) 1981–2010 AES/PES 77%
Brockway et al. (this study) 1981–2010 EEE 208%
4.2. Interpretation
The key divergence in our results between countries lies in the finding of partial rebound (13%–
50%) for UK-US, versus much higher rebound (close to, or above 100%) for China. One explanation
may be that 1980–2010 covers the period when China rapidly industrialised—so energy efficiency
measures were largely on the producer side, leading to larger rebound—versus the deindustrialising
process to more service-based economies of the UK and US. This concept is shown graphically in
Figure 11.
Figure 10. Summary of base-fit rebound results.
For China, Table 7summarises six national rebound estimates found in the literature using
AES/PES or EEE methods, plus our own base-fit results. The six AES/PES studies in Table 7have
broadly consistent results: estimating national energy rebound for China to be in the range 37% to
77%, with our value the highest (77%). If we assume that all AES/PES studies (including ours) contain
similar aggregate energy intensity (E/GDP) values, it suggests our estimate of
λ
(exogenous growth
term) is higher than the other studies. Certainly, differences would be expected, given all studies obtain
values for
λ
using differing APFs and techniques. Only one other study uses the EEE method—Zhang
and Lin [
33
], however, their estimated value (52%) is for short-term rebound. Whilst Saunders [
10
]
(p. 2208) suggests long term rebound is slightly higher, our EEE-based estimate (208%) is significantly
higher, due to a very high elasticity of substitution between KL and E(σ~2.0).
Table 7. National energy rebound estimates for China.
Source (Reference) Time-Series Method Estimate of National Rebound
Shao et al. [73] 1954–2010 AES/PES 37%
Zhang and Lin [33]1979–2004 AES/PES 41%
1981–2009 EEE 52% (short term)
Lin and Liu [93] 1981–2009 AES/PES 53%
Li and Lin [94] 1985–2008 AES/PES 67%
Li and Han [95] 1997–2009 AES/PES 74%
Brockway et al. (this study) 1981–2010 AES/PES 77%
Brockway et al. (this study) 1981–2010 EEE 208%
4.2. Interpretation
The key divergence in our results between countries lies in the finding of partial rebound
(13%–50%) for UK-US, versus much higher rebound (close to, or above 100%) for China.
One explanation may be that 1980–2010 covers the period when China rapidly industrialised—so
energy efficiency measures were largely on the producer side, leading to larger rebound—versus the
deindustrialising process to more service-based economies of the UK and US. This concept is shown
graphically in Figure 11.
Energies 2017,10, 51 16 of 24
Energies 2017, 10, 51 16 of 24
Figure 11. Conditions under which rebound may be large or small [103].
There is support in the literature for this view: the IPCC suggests consumer-sided rebound
(which we may expect to dominate in a service-based/deindustrialised economy) may be of the order
of 20%–45% [104], similar to our (UK-US) results. Meanwhile, Stern [8] describes how producer-sided
rebound may be higher as producer responses (i.e., increasing production) are not constrained by a
fixed nominal income (as in the case for consumers). Van den Bergh [105] concurs, suggesting
developing (or in China’s case—industrialising) countries would have higher rebound than a
developed economy due to four factors: higher growth rates; highly intensive energy use; higher cost
of energy; and lack of saturation in key energy services such as lighting. All are true of China. Ouyang
et al. [106] also highlight the lack of energy service saturation as a key reason for China’s higher
energy rebound.
Our fitted values of Hicks elasticity of substitution (σ) between the capital-labour composite and
energy also offers some support for this interpretation: σ values were very small for the UK (0.02)
and US (0.01), but large for China (2.08). Economic theory suggests where σ is low, energy is not
easily substituted for capital-labour, meaning that energy savings (at low σ) would stay largely
within the energy sector, yielding smaller rebound. Conversely for larger σ, energy savings are easily
replaced by increases in the composite capital-labour inputs, which in turn increases energy use,
yielding higher rebound.
Figure 12 examines the influence of the energy cost share (national energy spend/GDP) on the
estimates of rebound:
Figure 12. EEE method: sensitivity of base-fit results to energy cost share.
It suggests that China’s rebound is essentially independent of the energy cost share, whereas the
UK and US are influenced by the cost-share. The elasticity of substitution value is likely the driving
variable, so this finding is reliant on the σ values obtained. Figure 12 also suggests when the energy
cost share approaches zero, energy rebound (Re) tends to the value of the elasticity of substitution, σ.
This makes sense as, if sE = 0, then Equation (20) reduces to Equation (22):
Figure 11. Conditions under which rebound may be large or small [103].
There is support in the literature for this view: the IPCC suggests consumer-sided rebound (which
we may expect to dominate in a service-based/deindustrialised economy) may be of the order of
20%–45% [
104
], similar to our (UK-US) results. Meanwhile, Stern [
8
] describes how producer-sided
rebound may be higher as producer responses (i.e., increasing production) are not constrained by a fixed
nominal income (as in the case for consumers). Van den Bergh [
105
] concurs, suggesting developing
(or in China’s case—industrialising) countries would have higher rebound than a developed economy
due to four factors: higher growth rates; highly intensive energy use; higher cost of energy; and lack
of saturation in key energy services such as lighting. All are true of China. Ouyang et al. [
106
] also
highlight the lack of energy service saturation as a key reason for China’s higher energy rebound.
Our fitted values of Hicks elasticity of substitution (
σ
) between the capital-labour composite
and energy also offers some support for this interpretation:
σ
values were very small for the UK
(0.02) and US (0.01), but large for China (2.08). Economic theory suggests where
σ
is low, energy is
not easily substituted for capital-labour, meaning that energy savings (at low
σ
) would stay largely
within the energy sector, yielding smaller rebound. Conversely for larger
σ
, energy savings are easily
replaced by increases in the composite capital-labour inputs, which in turn increases energy use,
yielding higher rebound.
Figure 12 examines the influence of the energy cost share (national energy spend/GDP) on the
estimates of rebound:
Energies 2017, 10, 51 16 of 24
Figure 11. Conditions under which rebound may be large or small [103].
There is support in the literature for this view: the IPCC suggests consumer-sided rebound
(which we may expect to dominate in a service-based/deindustrialised economy) may be of the order
of 20%–45% [104], similar to our (UK-US) results. Meanwhile, Stern [8] describes how producer-sided
rebound may be higher as producer responses (i.e., increasing production) are not constrained by a
fixed nominal income (as in the case for consumers). Van den Bergh [105] concurs, suggesting
developing (or in China’s case—industrialising) countries would have higher rebound than a
developed economy due to four factors: higher growth rates; highly intensive energy use; higher cost
of energy; and lack of saturation in key energy services such as lighting. All are true of China. Ouyang
et al. [106] also highlight the lack of energy service saturation as a key reason for China’s higher
energy rebound.
Our fitted values of Hicks elasticity of substitution (σ) between the capital-labour composite and
energy also offers some support for this interpretation: σ values were very small for the UK (0.02)
and US (0.01), but large for China (2.08). Economic theory suggests where σ is low, energy is not
easily substituted for capital-labour, meaning that energy savings (at low σ) would stay largely
within the energy sector, yielding smaller rebound. Conversely for larger σ, energy savings are easily
replaced by increases in the composite capital-labour inputs, which in turn increases energy use,
yielding higher rebound.
Figure 12 examines the influence of the energy cost share (national energy spend/GDP) on the
estimates of rebound:
Figure 12. EEE method: sensitivity of base-fit results to energy cost share.
It suggests that China’s rebound is essentially independent of the energy cost share, whereas the
UK and US are influenced by the cost-share. The elasticity of substitution value is likely the driving
variable, so this finding is reliant on the σ values obtained. Figure 12 also suggests when the energy
cost share approaches zero, energy rebound (Re) tends to the value of the elasticity of substitution, σ.
This makes sense as, if sE = 0, then Equation (20) reduces to Equation (22):
Figure 12. EEE method: sensitivity of base-fit results to energy cost share.
It suggests that China’s rebound is essentially independent of the energy cost share, whereas the
UK and US are influenced by the cost-share. The elasticity of substitution value is likely the driving
variable, so this finding is reliant on the
σ
values obtained. Figure 12 also suggests when the energy
cost share approaches zero, energy rebound (Re) tends to the value of the elasticity of substitution,
σ
.
This makes sense as, if sE= 0, then Equation (20) reduces to Equation (22):
Energies 2017,10, 51 17 of 24
Re =1
(1+ρ)=σ(22)
Further interpretation is found by decomposing Method 2 (EEE) rebound into two parts using
Equation (23) (the derivation is in Supplementary Information)—first done by Saunders [
10
] (p. 2197)
In the first part,
ηEOutput
τ
depicts the relative change in energy use from changes in output, with energy
intensity (E/Y) held constant, whilst
ηESubsti tution
τ
captures the substitution (or intensity) effect, i.e.,
the relative change in energy use from input substitution, with output (Y) held constant.
ηE
τ=ηEOutput
τ+ηESubsti tution
τ(23)
Table 8presents the rebound results in this split format. Table 8considers rebound as occurring in
two stages. First, is rebound from the substitution effect (i.e., rebound holding output constant). In this
case we find negligible rebound for the UK and US (Re
1
~1%), versus very high rebound (backfire)
for China (Re
1
= 208%). The result for China follows Saunders [
10
] (p. 2197), who suggested that
whereby rebound is governed by substitution effects at high elasticities of substitution. The second
(i.e., output effect) component is negligible for China—which needs further exploration, but causes
nearly all rebound for the UK and US.
Table 8. EEE method rebound—output and substitution components.
Country Substitution Effect
ηFSubstitution
τ
Re1=1+ηFSubstitution
τ
(as Decimal Value) Output Effect ηFOutput
τ
Re =1+ηFSubstitution
τ+ηFOutput
τ
As Decimal Value As %
UK −0.98 0.01 0.12 0.13 13%
US −0.99 0.01 0.12 0.13 13%
China 1.08 2.08 0.00 2.08 208%
The results also raises the prospect that developed countries who have “offshored”
energy-intensive manufacturing industries and thus carbon emissions [
107
,
108
] may implicitly also
be “offshoring” energy rebound. In our case study, China’s higher rebound (assumed from its
producer-sided economy) is therefore due in significant part to the export demands for products
manufactured in China from countries including the UK and US. So, in a rebound study based on a
consumption-based national rebound approach, we might expect the UK and US rebound to increase,
and China’s to reduce.
4.3. Reflections of the Exergy-Based Approaches
This paper has sought to develop and test novel exergy-based approaches to estimate national
energy rebound. This is an important aspiration, since national-level rebound studies are rare, and so
contributions (as set out in Section 5) are welcome. Reflecting on the exergy-based approaches
themselves, whilst sharing a common CES-based framework, it is important to note the two methods
are actually very different. This is because they use different independent variables for the estimate
of rebound effects—namely endogenous exergy efficiency (
τ
) and exogenous growth (
λ
). The former
is a direct measure of exergy efficiency improvements, the latter is an estimate of factor-neutral
technical change.
There are particular concerns about the ability of the Solow residual (
eλt
) to capture the rebound
effect from energy efficiency alone, since as Shao et al. [
73
] (p. 239) assert, the Solow residual is
“unable to accurately reflect practical technological contribution to economic growth as it contains
factors which are too broad”. This is correct: the Solow residual cannot distinguish between the
various possible sources of technological change, which may be driven by better managerial skills,
augmenting of efficiency units of labour, better matching between capital and labour, or the advance
of the technology itself. Thus, energy efficiency may only be part of the rebound picture. On the
Energies 2017,10, 51 18 of 24
other hand, “factor-neutral” energy rebound—i.e., capturing the energy rebound from the broad
technological change that has actually occurred, whatever its provenance—may actually be desirable,
and complementary to “energy efficiency” only rebound. For example, it potentially allows the
comparison of “factor-neutral” and “energy efficiency” rebound—similar to Saunders [
32
]—and by
subtraction may be able to isolate the non “energy efficiency” rebound components. Such seemingly
divergent approaches may therefore be seen as in a more complementary light.
In addition, our exergy-based approaches have several caveats in common. The first relates to
the use of useful exergy as a new approach to the estimation of national energy rebound. There are
conceptual reasons for its inclusion (e.g., useful exergy is closer to production processes that are the
foundation of economic activity). But there may also reasons for its exclusion—for example if you
consider useful exergy is an output of capital equipment (it is thereby not really a primary input) or
that it is too closely linked with the adoption of technical change in the production function. In short,
further studies and more effort are required to explore its use and potential merit within an energy
rebound context.
Second, the exergy efficiency and useful exergy datasets used as inputs to the rebound analysis
are based on an approach which lacks a universal, consistent methodology. This is discussed
by Sousa et al. [
109
] and Miller et al. [
110
], who both highlight several areas for improvement,
which—once addressed—will strengthen the approach and provide more robust exergy efficiency
datasets. Therefore, the estimation of the CES function parameters (and hence energy rebound) will be
affected by any methodological flaws in the exergy-based datasets.
Third, both AES/PES and EEE methods are founded on the econometric estimation of CES-based
APFs. Despite their prevalent use in energy-economic modelling to inform policy, the use of APFs can
also be viewed as a risk, given there remains a longstanding debate about whether they are meaningful
at all [
111
,
112
]. Also, though we chose the most rebound flexible APF structure, any limitations of the
functional form may propagate to the rebound estimates. For example, amending the approach to
account for general equilibrium effects along the lines of Wei [
11
] may alter the CES parameter values
and rebound estimates.
5. Conclusions
Several important steps have been made in this paper. This is the first time that calculated values
of aggregate exergy efficiency and useful exergy have been employed in a study of national rebound
effects, and the resulting estimates are plausible. An exergy-based approach may help to address
Madlener and Alcott’s [
113
] (p. 374) demand that “some physical metric or metrics enabling a rigorous
definition and measurement of macro-level energy efficiency change (e.g., at the national or global
level) must be found”. Also, the provision of a multi-method, multi-country empirical study of national
energy rebound is in itself noteworthy, as it increases analytical depth and comparability, which serves
to broaden the rebound research field. Indeed, to our knowledge only one other KL(U) empirical
CES-based study exists [
92
], so in itself this contributes to other areas such as growth accounting.
In addition, the EEE-based energy rebound equation advances Saunders’ [
10
] existing CES-based
rebound equation, as the CES function we adopt is more flexible. The use of resampling also provides
an important sense for the precision with which we can determine the CES parameters and estimate of
national energy rebound. This is particularly useful given the highly asymmetric resampling intervals
obtained for the fitted parameters values, meaning the standard convention of multiplying standard
errors by a factor to obtain a symmetric uncertainty interval is not applicable in our case.
From the analysis, we find large energy rebound in China, suggesting that improvements
in China’s energy efficiency may be associated with increased energy consumption (“backfire”).
Conversely, we find much lower (partial) energy rebound for the case of the UK and US. These findings
support the hypothesis that producer-sided economies (such as China) may exist in states of high
rebound. This is problematic for climate policy, since if energy efficiency policies are not as effective as
planned, this would significantly hamper efforts for energy demand reductions via energy efficiency.
Energies 2017,10, 51 19 of 24
Shao et al. [
73
] suggested that China—in the face of large rebound effects—should further liberalise
energy markets coupled to increases in energy taxes. Jenkins et al. [
13
] (p. 53) concur, advocating taxes
should be “sufficient to keep the final price of energy services constant despite improvements in energy
efficiency, eliminating any net productivity gains from the efficiency measures”. Returning to the
carbon reduction “wedges” of Figure 1, we see how, this would place greater importance on deploying
renewable energy sources more rapidly than currently planned, in order to meet climate targets.
Our results also highlight how in higher rebound countries (i.e., China), energy rebound (as the
response to energy efficiency measures) may also act as a key component of economic growth.
This would follow the advocation of Ayres and Warr [
114
], who saw energy efficiency (and rebound) as
the engine of economic growth. This leads to a potential trade-off, where future economic growth may
be restricted by climate-based policies (e.g., carbon taxes) seeking to reduce energy rebound effects.
Supplementary Materials: The following are available online at www.mdpi.com/1996-1073/10/1/51/s1.
Data Repository:
A complete set of input and results datasets for the methods described in this paper to estimate
rebound, has been deposited at the University of Leeds Data Repository at https://doi.org/10.5518/137.
Acknowledgments:
We gratefully acknowledge the support of Engineering and Physical Sciences Research
Council (EPSRC) and Arup for contributing to the PhD CASE (Collaborative Award in Science and Engineering)
scholarship under grant reference EP/K504440/1 of Paul Brockway. Paul Brockway’s time was also funded in
the latter stages of this work as part of the research programme of the UK Energy Research Centre (UKERC),
supported by the UK Research Councils under EPSRC award EP/L024756/1. The support of the Economic and
Social Research Council (ESRC) is also gratefully acknowledged. The work contributes to the programme of
the ESRC Centre for Climate Change Economics and Policy. We also gratefully acknowledge the data supplied
by Professor Harry Wu for China’s capital services and labour data. Finally, we would also like to thank the
comments from the anonymous reviewers, who comments and subsequent revisions have improved this paper.
Author Contributions:
Paul E. Brockway conceived and designed the research, with contributions from
Timothy J. Foxon
, Julia K. Steinberger and John R. Barrett, who also provided conceptual and technical feedback
throughout the duration of the research. Harry Saunders completed the CES-based derivation of energy rebound
given the Supplementary Information, which was later amended by Paul E. Brockway. Matthew K. Heun
undertook the CES function analysis (including statistical resampling) which obtained estimates of the CES
parameters used by Paul E. Brockway in the rebound estimation. Paul E. Brockway then write the majority of the
paper. All co-authors (Harry Saunders, Matthew K. Heun, Timothy J. Foxon, Julia K. Steinberger, John R. Barrett
and Steve Sorrell) provided significant comments and revised text to the manuscript draft versions.
Conflicts of Interest:
The authors declare no conflict of interest. The funding sponsors had no role in the design
of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the
decision to publish the results.
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