The rebound effect representation in climate and energy models

We review the state-of-the-art and common practice of climate and energy modeling vis-á-vis the rebound literature. In particular, we study how energy system and economy-wide models include and quantify rebound effects—the gap between actual and expected saving or the behavioral adjustment in response to an energy efficiency improvement, in terms of energy or greenhouse gas emissions. First, we explain the interaction between drivers of energy efficiency improvements, energy efficiency policies, and the rebound effect to provide a framework for a general theoretical revision from micro- to macro-economic levels. Using this classification, we analyze rebound effect representations in empirical models by four dimensions: actors (industry or the production side, and private households or the consumption side), the aggregation level (from micro- to macro-economic levels), income level (developed or developing countries), and time (short- and long-run). Furthermore, we review rebound effect studies whose models focus on three drivers of energy efficiency improvements: market-based policies, non-market-based policies, and a costless energy efficiency improvement that holds other attributes constant (zero-cost breakthrough). We find that a clear representation of one or simultaneous drivers of energy efficiency improvements is crucial to target the goals of energy savings, greenhouse gas mitigation, and welfare gains. Under this broader view, the rebound effect is one additional phenomenon to be taken into consideration. This perspective provokes and provides additional policy implications. Reporting rebound effects as a stand-alone percentage is not sufficiently informative for policy considerations and the distinction of the aggregation level is important to asses the scalability of energy efficiency policies. Finally, we identify some ideas and motivations for future research.


Introduction
Under the umbrella of the 17 Sustainable Development Goals of the United Nations (UN 2015), goals such as sustainable economic growth, responsible production and consumption, affordable clean energy, and climate action have promoted the implementation of a cluster of energy efficiency (EE) and climate policies as part of the global agenda. Some examples include the promotion of EE standards, energy savings, sufficiency strategies, greenhouse gas (GHG) emission reductions or renewable energy targets. In particular, due to the existence of the EE gap as a result of market failures (Jaffe andStavins 1994, Gillingham andPalmer 2014), EE policies are often being implemented worldwide as seemingly win-win and cost-effective policies. However, the goals of these policies imply a complex web of non-linear interactions that are not yet well understood (Jenkins et al 2011). Borenstein (2013) and Schmitz and Madlener (2020) argue that a reduction in energy consumption is not the end goal, but reducing fossil fuel and GHG emissions is, while Freire-González (2017b) proposes that either one or both might be ultimate goals. Van den Bergh (2011) concludes that EE improvement (EEI) should not be a stand-alone policy goal, and Azevedo (2014) and Pollitt (2017) introduce a multiobjective trade-off perspective between goals.
EEIs are desired results of an EE policy. Much of the controversy has focused around what level of efficiency we can obtain feasibly with energy efficiency policies, given the existence of rebound effects (REs), as illustrated in (Gillingham et al 2016), "buy a more fuel-efficient car, drive more". Thus, backfire, or the possibility that energy consumption increases by more than the energy saving levels expected from energy efficiency gains, would undermine the effectiveness of energy efficiency policies. However, very often the goal of an energy efficiency policy is not limited to reducing energy consumption, but more generally to producing less GHG (Borenstein 2013). Moreover, its effects on individual and social welfare are of utmost importance (Gillingham et al 2016). Hence, although the rebound effect impacts energy consumption and thereby energy savings, it would have implications for emission reductions and welfare gains as well. EEIs that might result in backfire could correlate or cause positive (or negative) effects with respect to welfare gains and GHG reduction goals, which is more evident at the macro-economic level. This ambiguity makes it difficult to assess the effectiveness of energy efficiency policies, because evaluating energy savings per se would result in an incomplete assessment of the effectiveness of an energy efficiency policy. Beyond the controversy, our paper extends the discussion on the inherent ambiguity. At the micro-economic level, Borenstein (2013) states that backfire is unlikely, while Saunders (1992) and Saunders (2013) 6 find theoretical and historical empirical evidence of backfire on both, the microand macro-economic levels. Nonetheless, Gillingham et al (2013) calls into question the methodological validity of the previous two studies. Likewise, at the macro-economic level, Gillingham et al (2013) state that the RE has been overplayed because even at this level, it is highly probable that EE policies will not backfire. However, Rausch and Schwerin (2016) and Brockway et al (2017) find theoretical and empirical evidence of backfire. Moreover, Lemoine (2018) finds that backfire might occur, theoretically, from improvements in EE of the energy supply sector; however, empirically it might be dampened by increased consumption of non-energy inputs to production, and a size reduction of the supply sector. Gillingham et al (2013) address the possibility of ambiguous effects of EEIs by looking at social welfare effects. Although this view helps to extend the scope of the effectiveness of EE policies, it would still miss the important interaction with the goal of GHG reductions. Thus, in our review we include the interaction of EEIs with GHG emission reductions towards a 6 See Cullenward and Koomey (2016) and Saunders (2017) for additional discussions. more comprehensive assessment of EE policies and a better representation of this interaction in RE studies.
Hence, in response to the observed gaps between the micro-and macro-economic levels in the literature, we conduct a review to describe how drivers of EEIs shape the representation of REs by level of aggregation. We define the level of aggregation as the aggregation of consumers or firms going from local energy systems (micro-economic scale) to the overall level (macro-economic scale) 7 . Furthermore, we identify essential pieces necessary to build an RE model and describe methodologies found in the literature. We present findings in energy and climate models by four dimensions: level of aggregation, actors, income level, and time; taking into account heterogeneity (in terms of households, firms, energy services, goods, products, and attributes 8 ). This allows us to discuss possible directions to extend the understanding of the energy rebound and the so-called "GHG rebound" effect 9 . To this end, we report on three important trade-offs between possible benefits and costs associated with drivers of EEIs: GHG reduction, welfare gains, and energy reduction. Additional impacts, such as energy security, health, labor, and other social impacts (Pollitt 2017), are beyond the scope of this review. To the best of our knowledge no empirical RE study has yet examined the interaction between energy consumption reduction, welfare impact, and GHG emission reduction.
Our article follows this structure. Section 2 defines drivers, dimensions, and effects of EEIs. In addition, it presents a summary of RE typologies and taxonomies. With these concepts at hand we aim to guide the understanding and comparison of empirical studies. Section 3 explains the methodologies and summarizes common results of empirical studies categorized by level of aggregation, actor, income level, and time. Section 4 concludes with a discussion on climate and energy modeling for policy decision making, and section 5 summarizes future research directions and perceived research needs.

Drivers of EE
A first crucial step of modeling geared towards the representation and calculation of the rebound effect and its components is to clearly identify the driver that might potentially be causing the planned or observed EEI. The conceptualization of the driver might result in different ways to represent an EEI. The main three drivers of EEIs identified in our literature review are: market-based policies, non-market-based policies and zero-cost breakthroughs 10 , see figure 1. And although we isolate a driver or possible cause of an EEI, its causal relationship can be tested only on rare occasions. A second step would be to choose one or several dimensions in which an EEI might result in an effect. Throughout our paper, we distinguish between four dimensions for the study of an EEI: the level of aggregation (of each actor separately or jointly); actors (producers and consumers) income level (e.g. in developing and developed countries); and time (short-and long-run effects). A third step is to analyze correlations (or causal relationships) between relevant effects. The effects that we identify as the most relevant for current policy debate are: energy savings, GHG savings, and social welfare. Finally, after disentangling these concepts, one could estimate the energy or GHG RE. Figure 1 11 illustrates a roadmap to walk the reader through drivers of EEIs, dimensions, and effects.
The way we think about EEIs is at the core of the energy RE representation. After identifying the drivers of EEIs, we now ask ourselves: What kind of energy improvement representation 12 would make our quantitative studies more reliable? 10 A zero-cost breakthrough energy efficiency improvement is a costless exogenous increase in energy efficiency holding other product attributes constant. A market-based policy change in energy efficiency is typically costly, a result of an energy efficiency policy, and bundled with changes in other product attributes (or including heterogeneity), see Gillingham et al (2016). 11 We thank Ken Gillingham for comments on this representation. 12 A summary of energy efficiency and rebound effect formulations at the micro-and macro-economic levels can be found in appendix 1.
A clear definition of the term "energy service" in studies with explicit representations of EEIs is important for reproducibility and to contribute to objective debates on EE policies. To better identify drivers of EEIs in order to model REs, we explain the main three drivers: market-based policy, nonmarket-based policy, and zero-cost breakthrough (shown above in figure 1).
As a potential first driver of an EEI, a marketbased policy is sometimes modeled by means of price-based instruments. It is important to notice that some types of EEIs could arise from marketbased policies, non-market-based policies, or zerocost breakthroughs. This could result in a different formulation of the EEI.
Price-based instruments.-These are instruments that produce a change in relative prices, such as taxes or subsidies for households or/and industries. Taxes imposed on the production side include energy and carbon taxes, whereas on the consumption side, they include taxes on energy-intensive goods (e.g. private transport fuels). Subsidies for the production side could come in the form of R&D investment to foster low-emission technologies and utilitysponsored rebate programs, while for the consumption side these might include subsidies for the adoption of low-pollutant emission devices, e.g. rooftop solar technologies, light bulbs, or electric cars. To the best of our knowledge, market-based policy EEIs including bundles of attributes have not been modeled yet on the production side. In particular, when a change in relative prices is introduced by a tax to promote energy savings, a rebound is no longer a possible effect of concern within the energy domain; however, a tax could still be a cause of rebound with respect to GHG savings and welfare gains or losses.
Change in product attributes.-When we represent EEI induced by a market-based policy, most often the energy service that a unit of energy provides is not only a function of useful work derived from a more energy-efficient device but is also a function of its attributes other than energy conversion efficiency. An attribute is a non-EEI in a characteristic of a product (or energy service), such as size (e.g. computer), comfort, reliability, speed, or acceleration (Sorrell and Dimitropoulos 2008). Examining a household vehicle portfolio, Archsmith et al (2017) found that complementarity and substitution effects between energy and non-energy inputs are not the only causes of lost energy savings; they found that bundles of attributes may also interact in a way that reduces energy savings, eroding as much as 60% of fuel savings from an increase in fuel efficiency, thus compromising the cost-effectiveness of EE policies. In another study, Galvin (2017) examined how average increases in the vehicle-speed attribute (acceleration) can be incorporated into calculations of energy rebounds, showing that the relationship between energy services and energy consumption levels might be non-linear. The main insight was that it is possible to completely expunge EE increases by interactions between both speed and acceleration. Studies in computing services, such as in Galvin and Gubernat (2016), also reveal the importance of representing attribute parameters in models.
Behavioral/societal.-Lifestyle and consumer change of preferences in time, or reprogramming of preference orderings to change a determined habitual behavior (i.e. shift to public transport, healthier diets) could also play a complementary role in meeting energy reduction and climate change targets. A change in consumer patterns might arise from selfor externally-(i.e. commonly attained by policies) imposed rules. In this scenario, a change in preferences is not seen as a potential source of undesirable outcomes (Elster 2000), but is consciously placed in order to achieve desired better outcomes and consistency in time. Using a computable general equilibrium (CGE) model, Duarte et al (2016) found that promoting public transport was a successful economic and environmental policy for Spain. Moreover, Bjelle et al (2018) examined a set of 34 possible behavioral actions to be undertaken in Norwegian households; they found that people could potentially reduce their carbon footprint by 58%. In Sweden, Grabs (2015) calculated that switching to a vegetarian diet can save 16% of energy use and lower GHG emissions by 20% related to their dietary consumption, with corresponding energy RE of 96% and GHG rebounds of 49%. However, this study only focused on income effects. Finally, Chitnis and Sorrell (2015) recommended including a lagged variable in studies to capture inertia in energy prices (habit formation), which can help to mitigate correlation problems and at the same time better reflect behavioral change/consumer behavior.
A second potential driver of EEIs is a non-marketbased policy which could arise as qualitative changes, Command and Control instruments (CaC), change in product attributes, or behavioral/societal changes (the last two as explained previously, without government intervention).
Qualitative changes.-Without the use of a change in prices, the government could intervene by increasing quality or accessibility to information. Moreover, softer interventions include the use of nudges.
Command and Control instruments.-For the production side, these might include technology mandates (i.e. fixed input-output (IO) ratios restricting production flexibility) (Landis and Böhringer 2019), and performance standards on both the producer and consumer side (e.g. minimum EE standards, caps on residential energy use or residential energy intensity (Bye et al 2018).
As a third potential driver, we explore how EEIs are studied as a zero-cost breakthrough.
Technical change.-In general terms, an exogenous zero-cost breakthrough technical change can be modeled as neutral (also referred to as 'unbiased' , i.e. equal reduction of all inputs), or non-neutral (also called "biased", i.e. some inputs are reduced more than others) (Broadstock et al 2007), where an EEI is given at a specific point in time, or as factor-augmenting (assuming a rate of growth of EEI over time). A clear distinction between a neutral technical change or a relative effect on inputs (affecting total factor productivity) or the effect on outputs, might reduce bias in estimations (Du and Lin 2015). Outputs might cause structural changes in the economy (e.g. growth of the share of services in the economy) via substitution of products between energy-intensive and non-energy-intensive sectors (Bibas et al 2015). In Frieling and Madlener (2016), Frieling and Madlener (2017a), and Frieling and Madlener (2017b) technical change is represented as an exogenous constant or linear time trend, while Schmitz and Madlener (2020) explore a quadratic trend. Technical change can be represented using a latent variable approach (market-based policy or zero-cost breakthrough EEI), depending on past energy prices (Hunt et al 2014). Moreover, it can be represented as energy source prices, relative prices, real prices, growth rates, or a reduction in discount rates. It is represented also as a reduction in the costs of technologies or price-diminishing (e.g. labeling and perceived costs) (Löschel 2002, Löschel andSchymura 2013). Representing energy improvements as induced or endogenous technical change might produce a more accurate representation of the overall RE (Löschel 2002, Witajewski-Baltvilks et al 2017. Endogenous technical change has been far less studied in energy system models (Gillingham et al 2016), but is more often considered in economy-wide studies and Integrated Assessment models. Otto et al (2007), Otto et al (2008) and Löschel and Otto (2009) develop and apply an endogenous model of energybiased technical change with knowledge capital stocks and technology externalities in innovation and production. Therefore, an induced technical change as an EEI might be more accurate for the representation of REs on the producer side.
The increasing interest in climate policies leads to a more detailed analysis of energy REs in terms of GHG emissions, whereby the RE triggered by an increase in energy efficiency is converted into GHG emission units (the so-called "GHG rebound"). However, due to the lack of intrinsic value of carbon consumption, the incentive to increase the demand for carbon is quite weak. Thus, strictly speaking, as discussed in Birol and Keppler (2000), there exists to date no RE driven by a reduction of carbon consumption.

RE theory: typology and taxonomy
The analysis of the drivers of EEIs along the aggregation level results in the classification of types of RE through decomposition channels. To this aim, it is useful to systematically de-construct these effects into known components available in the literature. Further motivations to parse the RE involve linking the theoretical point of view to empirical calculations, and exploring causal effects whenever possible. Hence, tables 1 and 2 combine the typology and taxonomy of the RE from two perspectives: that of (1) a producer of energy services, and (2) an enduse consumer; and similarly from a combined perspective, along the aggregation level. These tables have gathered the contributions in the literature about the underpinnings of the RE, traditionally from Khazzoom (1980), Saunders (1992), Greening et al (2000), Berkhout et al (2000), and Birol and Keppler (2000), to more recent contributions from Van den Bergh (2011), Saunders (2013, Borenstein (2013), Azevedo (2014), Gillingham et al (2016), Madlener and Turner (2016), and Santarius (2016).

Modeling the RE
In a similar vein as in Varian (2016), in sections 1 and 2 we identified some essential pieces necessary to build an RE model. From possible causes or drivers of EE to existing rebound formulations, in this section we now turn to describe common methodologies found in the literature, used to model the RE.
In general, modelers seek to get a closer look at how energy is being consumed in real settings by collecting data to use in models, and/or studying treatment effects (i.e. of energy efficiency policies). They decide on (1) the representation of an EEI, (2) a mathematical representation of the RE, and in most cases (3) the economic theory, assuming a choice faced by a representative consumer (utility maximization), a producer (profit maximization), or a consumer-producer ("prosumer", householdfactory) that integrates production and consumption (a household produces energy services by minimizing costs in order to maximize utility derived from those energy services) (Becker 1965, Scott 1980, and (4) to include a degree of heterogeneity of actors (households or firms), energy services, goods, products, or attributes.
Our review has grouped energy-and economywide studies under the following categories: Structural models, Econometric studies, Simulation studies, and Integrated Assessment models. We present general assumptions for each type of model 13 , report on EEIs as drivers for RE representations, and show results of empirical studies between 2016 and 2018. Using the tables depicted in appendix B, we categorize the EE driver or RE channel used in studies according to the discussion provided in section 2.

Structural models of neoclassical economic growth
Structural models have been the most common means to calculate direct REs as represented in appendix 1, equations (A1) to (A3). They include preferences and technology using observed past behavior (a characteristic of ex-post studies, often econometric studies) to estimate fundamental parameters.

Energy system structural models
The approach with these types of models is to adopt an industrial (or household) production functional form of first-or second-order of approximation or, alternatively, a derived cost function, such as, Leontief, generalized Leontief, Cobb-Douglas, CES (Solow), nested CES (Solow), generalized Barnett, generalized McFadden, Gallant, Fourier function (Saunders 2008, Saunders 2015, the Rotterdam model, or the translog function (Saunders 2013, Mishra 2011, Frieling and Madlener 2016, Frieling and Madlener 2017a, and Frieling and Madlener 2017b. To identify the substitution (output) effect and the income effect for consumption (production), it is common to use decomposition methods to calculate elasticities, such as the implicit function theorem. Other sets of structural models represent household demand, and allow to compute direct and indirect REs. Some examples include almost ideal demands (AIDs) (Deaton and Muellbauer 1980) or linearized AIDs with multi-stage budgets (Thomas andAzevedo 2013a, Schmitz andMadlener 2020), linear expenditure systems (Lin and Liu 2015), direct addilog, indirect addilog (Thomas 2012), double-log systems (Freire-González 2017a). Parameters are obtained 13 There might be some overlap between structural models and econometric studies; however, our criteria for categorization is based on the degree of flexibility allowed by each type. 1.1 Substitution (+) Own/price elasticity of demand, substitution to consume more of good 0 due to price reduction.
1.2 Income c (+) Free income used to consume more of good 0 due to price reduction.
(2) Compensating cross-elasticies b 2.1 Fixed income (-) Expenditure on good 0 takes away expenditure on other goods with energy content.
(3) Indirect rebound 3.1. Substitution (-) Cross-price elasticity of demand for other goods, substitution to consume less of other goods due to more consumption of good 0.
3.2 Income c (+) Consuming more of other goods due to savings on good 0 (re-spending effect).

Embodied energy (+)
Energy or emissions associated with the lifecycle of an energy service.
3.4 Behavioral (+) Indirect rebounds not caused by EE improvement, but by changes in consumption behaviors.
(4) Time savings 4.5 Time d Available time that individuals have to spend on other activities that use energy.

Output (+)
Free expenditure (savings) to use more energy input 0 due to cost reduction resulting in increased production.
(6) Indirect rebound 6.1 Factor substitution (-) Substitution to use less of other inputs due to cost reduction.
6.2 Output (+) Free expenditure (savings) to use more of other inputs due to cost reduction resulting in increased production (Re-investment effect).
6.3 Embodied energy (+) Investments in energy efficiency technologies increase demand for energy (Grey energy).
(7) Complementary rebound 7.1 Redesign(+) Ex-ante expected cost savings for consumers lead producers to invest in redesigning the original product.
a There also exist the less-studied transformational rebound effects (Greening et al 2000), and motivational psychological rebound effects (Santarius 2016 in developed countries, in which a decrease in working time might lead to an increase in energy-intensive leisure activities with high carbon footprints. Increased aggregate energy demand due to reduction in the market price of energy services, leading to a decrease in the demand for a particular fuel. Reinforcing effect from market price on the consumer side income effect. Interplay from a firm, sector or numerous individual households up to the level of a sector or market. 8.2 Disinvestment (-) Direct and derived demands are not sufficiently elastic to prevent falling market prices of energy, leading to a decline in revenue, profitability and return on capital in domestic energy supply sectors.

Composition (+)
Reduction in market price favors energy-intensive sectors of the economy, reducing the price of energy-intensive goods and services and thus causing the increase of their demand, altering the composition of the economy's portfolio of goods.
8.4 Economies of scale (+) Income and market effects causing an increase in demand for energy services or goods, leading to firm expansion that reinforces falling prices, whose impact reduces along the level of production.
8.5 Rising labor income (+) Firms using additional income from energy efficiency of production process to raise worker's wages.
(9) Macroeconomic 9.1 Price The adjustment of consumers and producers following a shift to the left of the market demand curve (Economy-wide).
9.2 Growth: Sectoral allocation Change in efficiency of energy inputs in an energy-intensive sector may lead to this sector's grow relative to others (Equal to the composition effect but causing economic growth).

Growth: Induced innovation
Spillover effects of an energy improvement in one sector, attributable to an improvement in another one.
9.4 Growth: Fiscal multiplier Freed money previously spent on energy used in new economic activity that utilizes previously idle resources. Long-term debt associated with fiscal stimulus (Multiplier effect).
9.5 Labor supply (-) c Consumers adjust their labor supply to the extent that EE has an impact on real wages. It depends on the elasticity of substitution between leisure and consumption.
a Further, Saunders (2013) includes so-called "frontier effects", enabling new product applications or services. In the short term, RE models include changes in energy service demands while holding capital or investments constant; in the long term, they can incorporate laws of motion for capital costs, savings, scrappage, crowding-out effects, and/or increasing market saturation of appliances (Thomas and Azevedo 2013b) in order to capture consumer responses to price changes (Gillingham More recent cases include the use or purchase of heavy units or units with more functions/services and consequently using more energy (e.g. proof of work in block-chains for microgrids (Hittinger and Jaramillo 2019)).
c At the macro-economic level, REs are more ambiguous than at the micro-economic level. However, Böhringer and Rivers (2018) found that the elasticity of substitution between leisure and consumption is directly related to the labor supply elasticity, which is low across the economy as a whole; thus it is likely that the RE due to this channel is (-). It is closely related to the rising labor income effect channel (8.5).
using linear or non-linear econometric methodologies (i.e. ordinary least squares, dynamic ordinary least squares, feasible generalized least squares, non-linear least squares). Usual inputs are energy (or energy commodities, services), capital, labor, and materials. Recent studies have focused on the mesoeconomic RE to study production-side sectoral, and interactive REs (e.g. market effects) (Santarius 2016). Tables 5 and 6 in appendix B show in detail the review of selected structural models from the production and consumption sides, and their respective RE magnitudes as percentage figures.

Economy-wide structural models
Aggregated production functions (APFs) using Solow's residual can also be used to approximate total energy and GHG REs at national levels, as represented in appendix 1, equations (A4) and (A5). These models assume that parameters remain unchanged, in order to predict the responses to possible economic system changes, including those that have never happened before. Therefore, they can conveniently be used to conduct welfare calculations (Nevo and Whinston 2010). Nonetheless, the major concern is that the use of an "elaborate superstructure" will provide results driven by the model rather than the data (Angrist and Pischke 2010). Table 7 in appendix B shows a review of selected structural models.

Econometric studies
To avoid restrictions imposed by ex-post structural forms as in section 3.1, empirical modelers usually turn to reduced-form statistical ex-post estimations. Additionally, Nevo and Whinston (2010) argue that welfare calculations using this methodology would be less credible, due to the variety of economic environmental change parameters possible to be estimated. Econometric studies represent the RE in two broad categories, which vary according to the aggregation level of study. The first category includes energy systems that compute the direct RE, whereas the second category contains economy-wide contexts to calculate a total national or sectoral RE. However, Acemoglu (2010) and Lemoine (2018) argue that reducedform models should not be used as stand-alone tools to evaluate the development of policies.

Energy system econometric estimations
Models in this section are categorized as ex-post estimations and calculated using regression analysis (e.g. at the less-studied meso-economic level; Wang et al (2016), e.g. uses a double-logarithmic model to study factors affecting electricity consumption; generalized linear models, ARIMA, vector autoregression, and cointegration models. Data used to solve these models include time-series data, cross-section analysis, panel data, and stochastic frontier functions. Less common are panel instrumental variable (IV) estimators, difference-in-difference estimators, and field quasi-experimental methods. More recently, machine learning (artificial intelligence algorithms) is being used in econometric estimations as well, see table 8 in appendix B. The advantage of these types of studies is that they might demonstrate causality and derive more robust results, but exogenous variables should be carefully controlled. Reducing the scope of the model to focus on a specific energy service could provide significant insights. Though Jacobsen and Van Benthem (2015) investigate the Gruenspecht effect 14 , this study is a good example of the direction that RE studies might take. This is due to several reasons: they demonstrate causality using an IV estimator to calculate an elasticity of vehicle scrapping (i.e. using gasoline prices and vehicle prices), study the change in prices due to a fuel policy, and consider heterogeneity. Finally, quasi-experimental ex-post studies could provide more realistic insights about specific EE program performance and effectiveness.

Macro-econometric models
Despite of the difficulties in attaining a good degree of identification with reality, these post-Keynesian ex-ante models might perform useful forecasting and policy analysis (if an effective existing rule prevails, Sims 1980). Barker et al (2009) was the first to study the global RE using a macro-econometric model, the so-called E3ME. The E3ME (or E3ME variant) and non-equilibrium models have been used to assess co-benefits and trade-offs of policy scenarios in European economies using multiple sets of computable econometric equations. In the E3ME model, the RE is modeled in two parts: the direct RE (equation (A2) in appendix 1) is taken from the PRIMES bottom-up model (an energy system model), and this is then used to calculate the endogenous indirect RE and the economy-wide RE (equation (A4) in appendix 1), derived from the IO structure of the model (Pollitt 2017). Inputs of the model are shared with other models such as PROMETHEUS (fossil fuels and import prices) and GEM-E3 (macroeconomic and sectoral projections) (E3MLab and IIASA 2016). The main assumption with regard to EE is that rising fuel prices will stimulate technological innovation and boost growth of the world economy, thus the endogenous representation of technological change also has implications for the calculation of the RE. The model allows varying returns of scale and non-linear substitution, and it avoids the representative agent assumption. Nonetheless, it does not allow substitution between cheaper energy services and other inputs within production and embodied energy representation. The E3ME has focused on representing, from a macro-economic point of view, the price and growth effect (sectoral allocation channel). To see a comprehensive formulation on how the RE has been disentangled into partial and general effects (Barker et al 2009, Pollitt 2017, please refer to appendix A.1. Main results highlight the importance of capital formation modeling to account for crowding out effects (Pollitt 2016). Table 9 in appendix B shows the review of macro-econometric studies.

Simulation models 3.3.1. Energy system simulation models: IO models and environmentally-extended input-output models (EEIO)
The most comprehensive studies applying this methodology use estimates of direct REs as inputs. These ex-post static models allow the calculation of indirect REs as cross-price elasticities for n goods (or n services). Following this estimation, total REs are computed as represented in equation (A4) in appendix 1. Most studies have focused on studying indirect REs on the consumption side. These models assume that constant returns to scale, sectors producing homogeneous goods and services, and outputs are created with constant and fixed proportions of inputs (linear representation, Miller and Blair 2009). Moreover, cross-price elasticities of other goods are modeled as constant, and re-spending to be proportional in each good and service. Widely used data inputs include Consumer Expenditure Surveys, Eora Global MRIO data, EXIOBASE, the Global Trade and Analysis Project (GTAP), and the World Input-Output Database (WIOD), see table 10 in appendix B for studies on the consumption side. Modeling RE with an EEIO model, Thomas and Azevedo (2013b) found that indirect rebound effects (IREs) are inversely proportional to direct rebound effects (DREs) and are bounded by consumers' budget constraints. Freire-González (2017b) developed risk and vulnerability indicators for REs. Böhringer and Löschel (2006), Allan et al (2007) and Turner and Figus (2016) provide comprenhensive reviews on these ex-ante "what-if " neo-classical models and their applicability to model energy-economyenvironment interdependencies for exploring tradeoffs and co-benefits. Known models used to parse the RE include GTAP-E, WARM, SCREEN, MSG-6, ENVI-UK, ORANI-G, REMES, SNOW-NO, CEPE, WIOD-CGE, and climate models such as GRACE which could potentially be used for rebound studies (Aaheim et al 2018). EEIs in this review are modeled as exogenous autonomous EEI and energyaugmenting, or as endogenous technical change using a latent variable approach of a market-based policy type (taxes or subsidies on production or consumption). However, induced technical change, as in Witajewski-Baltvilks et al (2017) and Lemoine (2018), and the implications of diffusion effects remain to be further studied. RE is calculated using equations (7) and (8). Advances in the analysis of RE tractability have also been applied, namely the decomposition of energy and GHG REs from partial to general equilibrium, as described in section 2.5. To parse the RE in direct and indirect partial equilibrium components, as described in tables 2-5 (i.e. substitution and income effects), modelers set all prices fixed except for the energy sector or service of interest in their analysis. To calculate the general equilibrium component, commonly used channels are: price, growth (sectoral allocation), labor supply Rivers 2018, Chang et al 2018), and growth (fiscal stimulus) (Figus et al 2019). Finally, the total RE is obtained summing up the partial equilibrium components and general equilibrium component (or the economy-wide component, as discussed in section 3.2.2). Sensitivity analyses are more common, thus providing robust estimates mainly on the upper bound of the spectrum. Moreover, studies have investigated the influence of RE on macro-economic parameters such as GDP, employment, etc (Madlener and Turner 2016) and on welfare (Gillingham et al 2016). Birol and Keppler (2000) discuss the importance of modeling real world energy markets which are far from perfect competition; bridging the gap of theoretical and actual EE levels. Along these lines, we checked the adaptation and tailoring of models for relevant interactions (e.g. imperfect markets, substitution effects, reversibility or dynamic frameworks) that might potentially impact calculations of energy and GHG rebounds (Turner and Figus 2016):
(2) Most models do not integrate adjustment of capital/labor growth (or decline) with regard to EEI. (3) Revised models assumed perfect competition, except Figus et al (2017), Figus et al (2018). For (4) and (5), capital that flows freely between national sectors, investments, and labor increases gradually. (6) Recent models are not only dynamic, but also capture consumer's responsiveness (Figus et al 2017, Figus et al 2018, Chang et al 2018, Bye et al 2018, Duarte et al 2018, including consumer response to price changes in time, but are often also regional-specific (or spatial CGE models) (Helgesen et al 2018). (7) To represent energy and non-energy goods, CES or Cobb douglas functions are commonly used, and inputs in the energy sector are modeled as Leontief composites, with no possibility of substitution. (8) While EEI in total factor productivity has not commonly been modeled, it has been included from one consumer aggregate with no possibility of substitution or CES/Klein-Rubin utility preferences, to bottom-up representations that capture consumer heterogeneity and distributional impacts (Bye et al 2018, Landis andBöhringer 2019). Tables 11-15 in appendix B show recent studies for production and consumption.

Integrated assessment models
There are two main types of ex-ante Integrated Assessment models (IAMs) for climate policy analysis. In a broad sense, these can be classified as detailed process (DP) IAMs and benefit-cost (BC) IAMs 15 . The main difference is the way they model climate change impacts. DP IAMs are more disaggregated models that use economic valuation or physical projections to provide forecasts of climate change impacts at detailed sectoral or regional levels. In contrast, BC IAMs represent sectoral (or regional) aggregation functions and climate change mitigation costs into a single economic metric, whose main goal is to analyze potentially optimal climate policies. For a detailed overview of IAMs and their applications, see Weyant (2017). Widely used models include DICE, RICE, FUND, PAGE, IWG (which has focused on EE), MESSAGEix-GLOBIOM, IMACLIM-R, IMAGE, AIM, GCAM4, REMIND-MAgPIE, WITCH, etc Allowing flexibility about the achievement of GHG emission reductions results in lower mitigation costs across all economic assumptions; however, too much flexibility can also be detrimental to the uselfulness of models (Pindyck 2017). Moreover, delays in implementing mitigation policies would result in increases in total discounted costs of meeting particular global GHG concentrations. DP IAMs identify and directly measure impacts on sectors, regions and ecosystems in more detail, providing insights on trade-offs between mitigation and adaptation strategies on global scales, which is useful for international negotiators, and national and/or regional decision makers. Aggregated BC IAMs might help to understand the costeffectiveness of climate policies considering mitigation and adaptation strategies. These models highlight critical cost issues (i.e. including discount rates, risks, damages, social cost curve calculations), while incorporating new scientific findings into projections (Weyant 2017). Controversy around the use of physical or economic units is also found in these types of studies. On the current development of IAM models, Pindyck (2017) finds that these models are at an early stage of development, add much noise, and would require sensitivity analysis on key parameters. Moreover, considering the time pressure exerted by climate change, he concluded that simple models to calculate upper bounds would also be useful. Moreover, Riahi et al (2015) and Rogelj et al (2018) suggest that the proportion of successful IAM scenarios could be used as an indicator of infeasibility risk. Studies included in this overview, and summarized in tables 16 and 17 in appendix B, have included drivers of EEIs as zero-cost breakthrough, marketbased policy, non-marked-based policy, or a combination of the previous ones. These drivers are represented as exogenous or endogenous shocks, through equations and (or) parameters that calibrate IAMs. After selecting drivers to study, models include channels that result in REs (e.g. substitution, income, price effects, etc). However, these studies do not show what would the impact of the RE channel's representation be (i.e. potentially how much energy consumption reduction will not be feasible due to these impacts). Though some studies have found increasing evidence of demand saturation in activity levels (Grubler et al 2018), RE magnitudes might also be used as parameters to run sensitivity scenario cases.

Conclusions
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. (A. Einstein)

Model identification: a trade-off between theory and reality
Overall, the diverse nature of empirical models reviewed in this study contribute to the understanding of the RE from the production and consumption side. Moreover, given the tension between theory and reality, to reach 'reasonable' level of identification, we think it is good practice to have a clear picture about the motivation behind modeling, similar to what Blanchard (2018) presented. We can think of single models or combined models that cover theory without much emphasis on reality; policy (or zero-cost breakthrough) with emphasis in reality; toy models to add pedagogical insights; and forecasting models with emphasis on advanced statistical tools to reduce errors in projections. Other good practices include reporting standard deviations and robustness of results and performing sensitivity analyses on key parameters.
We carried out an extensive review of 118 studies on the RE along different aggregation levels, out of which 61 were empirical studies from the years 2016-2018 and the rest theoretical papers to develop sections 1 and 2. From this review, 25 studies computed and reported energy or GHG RE magnitudes which we summarize in table 3. From this sample of studies we can see that choosing a structural model might increase the uncertainty of RE calculations. Furthermore, there are fewer studies examining the RE on the production side. An important caveat to consider, when looking at this table, is the diverse nature of energy services under study. Combining previous, recent, and future studies on RE magnitudes could provide more data to increase the analytical power of RE estimates. A future meta-analysis study of the RE or the use of crowdsourcing data analysis strategies, as presented in Silberzahn et al (2018), could reveal further insights. We highlight the equal importance and complementarity of ex-ante and ex-post studies given the observed symmetry between models and the computation of REs, which requires the calculation of both expected and realized energy savings. Although magnitudes presented in table 3 are informative, a main take-away is that depending on which EE driver is represented in models, including the study of environmental and welfare effects to the study of the energy RE (a specific phenomena of energy consumption reduction), results in a broader and different extent of policy implications. Therefore, reporting REs as a stand-alone percentage is not sufficiently informative for policy considerations. Additionally, it is important to perform a cost-benefit analysis to understand the effectiveness of legislations within the context of the introduction of EE policies.

Ex-post studies
We find that structural functions are the most often used methodology for modeling the production side in both energy systems and economy-wide models. Although there are clearly several limitations imposed by structural forms and assumptions (Gillingham et al 2016), and these types of models have been criticized for ignoring heterogenous capital at aggregate levels (Burmeister 2000); Saunders (2008) recommends the use of Gallant (Fourier) or the generalized Leontief/Symmetric generalized Barnett cost functions due to their flexibility to model REs. Moreover, on the consumption side, Schmitz and Madlener (2020) similarly found that the magnitude of the RE is sensitive to time and model specification, and they recommend modeling energy services in a system as an alternative to energy commodity models. The distinction between consumption and production direct REs is relevant, as the latter captures two thirds of total energy consumption (Santarius 2016).
While recent econometric models on energy systems (section 3.2.1) have evolved to include data from field experiments, use randomized controlled trials, and study causal effects on the consumption side, there have been fewer studies on the production side (i.e. exploring technology choices and R&D investment) using these up-to-date methodologies. Although the aforementioned studies are computationally expensive, and their results are difficult to scale up due to their specific nature, they provide valuable insights on the effectiveness of EE policies and on the RE. Wang et al (2016) recommends studying final energy consumption habits across a plethora of household appliances.
Ex-post studies that put emphasis on reality depiction (policy and/or zero-cost breakthrough) are of high importance in providing empirical evidence. They serve as an input for ex-ante studies, in order to feed accurate parameters to ex-ante studies. Figures 2  and 3 show that ex-post studies in our review estimate either energy or GHG RE separately, while welfare effects are not computed. The circle shape indicates studies of developed countries (DE), while the triangle shape indicates studies of developing countries (DG). The colors red or blue distinguish ex-ante from ex-post studies, respectively. From 26 RE calculations performed in the studies we review, the magnitudes of the energy RE have a median of 31%, with a maximum of 334% and a minimum of -22%. GHG REs have a median of -30%, with a maximum of 78% and a minimum of -161%.

Ex-ante studies
Similar to ex-post studies, ex-ante studies also rely on structural forms or econometric estimates for the representation of consumer or producer choices. On the production side, Koesler et al (2016) and Brockway et al (2017) propose to review the adequacy of CES functions for representing the nested production function, and to better match the energyaugmenting technical progress paradigm. With regard to the elasticity parameter at macro-economic levels, Lemoine (2018) does not discard backfire  theoretically at the macro-economic level; however, empirically he finds that a 65 percent point reduction in energy savings occurs in the energy supply sector in the US. In addition, he finds that backfire can occur even for small elasticities between energy and nonenergy goods occurring at the least efficient (or most energy-intensive) sectors. Nonetheless, there is a need for ex-post empirical evidence on fossil fuel supply elasticities at the micro-economic level (Böhringer and Rivers 2018). Moreover, Böhringer and Rivers (2018) also find that a large elasticity of substitution between capital and labor would reduce the magnitude of the energy RE. In addition, the larger size of the other sectors not affected by EEIs could also increase the RE magnitude (Böhringer and Rivers 2018), and the substitution effect would dominate (Zhou et al 2018). Another topic to examine more closely is the impact of EEIs on primary energy, which could benefit the expansion of energy services (intermediate energy) (Lu et al 2017). With regard to growth expansion, Ryan et al (2017) recommend examining trade-offs between economic expansion and EEI. Finally, investigating RE behavior over time is of importance, as it is theoretically possible that long-run elasticities are lower than short-run elasticities (Wei 2010), while on empirical grounds, Turner et al (2009) finds super-conservation and Lu et al (2017) finds a diminishing long-run energy RE. On the consumption side, studies find that large elasticity of substitution between energy and non-energy goods determines a larger partial equilibrium component (Gillingham et al 2016) which dominates the general equilibrium component (Böhringer and Rivers 2018). On the other hand, if the aforementioned parameter tends to have a low elasticity of substitution, it would result in low magnitudes of the energy RE due to consumer price unresponsiveness. More recently, heterogeneity has played an important role in studies disaggregating specific energyintensive and less energy-intensive energy services (e.g. public vs. private transport or fossil fuel-vs. renewable-sourced heating), and including the representation of durable goods/investments within energy service sectors could provide more precise policy advice (Ryan et al 2017, Figus et al 2018. Figure 2 above shows energy RE magnitudes obtained from the ex-ante studies examined in this review. Joint estimations of energy rebound and welfare effects have been carried out, while the GHG RE has not been computed, see figure 3. From 19 rebound effect calculations performed in studies shown in appendix B, the magnitudes of the energy RE have a median of 51%, with a maximum of 98% and a minimum of -0.1%. Welfare effects have a median of 0.4% of GDP, with a maximum of 2.25% and a minimum of -1%. Jointly, there can be high energy REs associated with high positive welfare effects (2.25%) but also low ones (0.05%). In our overview, REs from ex-ante studies show both lower median values. From 22 developed country studies along the level of aggregation, shown in the tables in in appendix B, the magnitudes of the energy RE have a median of 50%. Welfare effects have a median of 0.0% of GDP. Jointly, there can be high energy REs associated with high welfare effects (2.25%) but also moderate ones (0.32%). There is no clear link between the magnitude of rebound and welfare effects. For 16 developing country studies along the level of aggregation, joint estimations of energy rebound and welfare effects have been carried out, while the GHG RE has not been computed. The magnitudes of the energy RE have a median of 34%. Welfare effects have a median of 0% of GDP. Jointly, there can be high energy REs associated with moderate welfare effects (0.5%) but also low ones (0.05%). Similar to studies on developed countries, there is no clear link between the magnitude of rebound and welfare effects. In our review, RE studies (along the level of aggregation) from developed countries show both lower median magnitudes than studies from developing countries. Welfare effects from developed country studies show lower median magnitudes.

Combined insights
Taking both sides into account, studies validating elasticities with historical data and the use of more sophisticated methods (i.e. causal identification) and sensitivity analyses would improve the reliability of studies (Saunders 2013 should be checked against assumptions of the year when technical EEI is introduced, to take into account not only innovation phases but also diffusion and approximation to saturation. If policies are already in place, this should be modeled because high initial levels of EEIs in place could result in higher GHG rebounds. Furthermore, the dynamics of the incorporating of EEIs in primary and/or secondary energy would provide further insights (Zhou et al 2018). Another branch of the RE study includes the calculation of REs in terms of GHG emissions (e.g. pollution effects). Chang et al (2018) found that ignoring calculation in terms of GHG emissions (considering only energy REs) could result in underestimation of the energy RE magnitude, though bringing positive welfare effects. In general, models could include locational aspects (e.g. multi-area), temporal aspects (i.e. different consumption or production patterns in summer and winter; Wang et al (2016), and group targeting (low/high income households, owners/tenants (Madlener and Hauertmann 2011), high/low energy intensive and/or high/low GHG emission industries Madlener and Turner 2016, Wang et al 2016 to check distributional effects when price is endogenous (Ghoddusi and Roy 2017). Furthermore, we consider that the analysis of cyclical fluctuations in the energy industry for specific energy services or resources could improve the understanding of EEI adoption and RE in time, both using ex-post and ex-ante studies. Overall, the potential effect of EEIs and REs on the economy would be higher for industry than households; however, we find mixed results. Ex-ante studies can also be used to monitor REs in the economy, not just for forecasting (e.g. using now-casting or back-casting methods in CGE models). The calculation of REs has two components, one expected (or ex-ante), and another real (or ex-post). The expected component shows significant variability depending on how energy reductions are assumed to be realized. Thus, we suggest that GHG reductions and energy savings would be more direct quantitative indicators for policy assessment. Finally, all figures imply that there is a correlation between welfare, GHG reductions and energy savings.

EEIs on consumption and production
Studies included in this review have shed light on the inclusion of EEIs as technical change and preferences on energy systems more often than on economywide models. Few IAM studies have been found to consider EEIs simultaneously on both sides. In particular, less common so far are studies that study the RE as described in section 2.2, complementary RE (7), composition REs (

Heterogeneity
On the production side, and considering the GHG emissions reduction goal, Lemoine (2018) indicates that EEI policies should target energy-efficient sectors with low elasticity of substitution between energy and non-energy inputs and less energy-intensive sectors; however, this study does not include the representation of inter-fuel substitution, long-run effects or impacts of heterogeneity on the consumption side. Likewise, in Norway, Helgesen et al (2018) found that a 50% reduction in GHG emissions through technology investments are achievable by 2030 but at a cost of 6.3% reduction of GDP; however, this study assumes that energy intensity remains constant. Moreover, in developing countries such as China, policies on the supply side should encourage resource-specific technological progress in energyintensive sectors (e.g. industry and manufacturing) (Zhang et al 2017b). On the consumption side, similar to the production side, Ryan et al (2017) suggests that the policy focus should expand to consider not only improvements in EE in energy-intensive sectors, but also how these improvements interact with less energy-intensive sectors. For China, Wang et al (2016) found that in residential electricity consumption, investment should be promoted in energysaving technologies. Moreover, it is common to consider heterogeneity in energy services and attributes in energy system approaches, Bye et al (2018) found that modeling EEI in a specific sector (i.e. the electricity sector), instead of considering EEIs on all energy uses in an economy, could result in economic distortions that may lead to welfare loss, even though the electricity supply in Norway is mainly produced from renewable energy sources. Thus, the question here would be to what degree and for what cases is heterogeneity relevant for policy analysis.

Long-run vs short-run
A clearer distinction of estimates in ex-post and exante studies between the results obtained in the short and long run would improve the insights of the models. For example, Brockway et al (2017) concluded for China that the deployment of renewable energy sources should occur more rapidly than planned. However, Herring and Roy (2007) state that this would make little difference in the long term in order to reduce carbon emissions. Pui and Othman (2017) found that a double dividend in GHG emission reductions and welfare maximization is gained in the short-run with autonomous EEIs, but EEI policies should be accompanied by taxes to control and levelup price reductions. In contrast, Lu et al (2017) found that policies should target the efficiency of EEI policies in the long run, where REs diminish. In that vein, Frieling and Madlener (2017b) concluded from a comparison of production in a structural partial equilibrium model with factor-augmenting inputs for Germany, the US and the UK, that energy consumption is relatively immutable in the short run. It remains to be further analyzed how the RE affects GHG emissions, in scenario cases where the earth warms more than 1.5 degrees.

Uncertainty due to expectations and the counterfactual
Engineering estimates on energy savings found in actual EE policy programs are reported to be much higher than actual savings. Thus improving modeling on both sides, using ex-post and ex-ante studies (e.g. using machine learning to compute counterfactual scenarios), could help to reduce uncertainty in calculations. Furthermore, Frondel and Vance (2018) use an IV estimator to resolve endogeneity between EE and energy services thereby recovering causality. By using this method they find higher upper-bound RE estimates compared to estimations in studies that assume a linear relationship of efficiency between energy and energy services. Ghoddusi and Roy (2017) found that modeling stochastic demand and supply could also increase control for uncertainty in energy RE estimates.

EE up-front costs
More market-based policy studies including EE investment costs such as Burlig et al (2017) and Fowlie et al (2018), at the micro-economic level and Bye et al (2018) at the macro-economic level, could give a more complete picture regarding the costeffectiveness of EE policies. With respect to CGE and IAM models on the producer side, it would be useful to track down how managers' behavior might impact the balance between investments and savings in the long run (the closing rule) and how this mirrors on their inter-temporal decisions (e.g. sunk costs, adjusted cost functions, etc). On the consumer side, CGE and IAM models that represent consumer behavior towards their investment in durables and nondurables and how this could impact different generations, considering their death probabilities, might also help to understand the reasons behind a particular result regarding the effectiveness of EE and climate policies (Conrad 2001).

Imperfect markets, externalities and imperfect regulations
Most of the studies reviewed in this survey propose local, national or global regulations to solve to externalities and imperfect markets. For cases where studies are informative to policy decision-making, we reflect on study insights in this section to raise awareness and promote discussion about how methodological considerations (and limitations) might shape conclusions and their applicability. For the production side in China, Yang and Li (2017) arrive at the conclusion that in power generation, ad valorem taxation on energy input prices (i.e. fossil fuels) could help to better reflect fossil fuel scarcity and environmental costs. Furthermore, they recommend a parallel lift of feed-in tariffs to promote clean energy. Meanwhile, in developed countries like Switzerland, Landis and Böhringer (2019) found that the economic costs of EE CaC policies (Promotion) are five times more expensive than the use of taxes (Steering) combined with per capita rebates. Moreover, there exist trade-offs between cost-effectiveness and distributional impacts of policies. However, this study did not take into account environmental benefits or externalities (which could reduce the gap between both instruments) resulting in an upper-bound estimate. On the consumption side, Bye et al (2018) found that EEI policies for dwellings (i.e. a cap on residential use and energy intensity) are highly costly even when including CO 2 taxes; therefore, these policies would be inefficient to abate CO 2 emissions. Whereas Pollitt (2017) found that EEIs for buildings in Europe would yield all three co-benefits: GHG reductions, welfare increase and energy savings on climate change models, Van den Bergh (2017) found cap-and-trade to be the best approach to manage global and international energy, and more importantly, the GHG RE. Furthermore, energy-saving policies are usually modeled in IAMs, as the common strategy in mitigation scenarios, but transition pathways that can meet such targets are less commonly studied. From six IAMs and five shared socio-economic pathways, Rogelj et al (2018) found that scenarios characterized by a rapid shift away from fossil fuels toward largescale low-carbon energy supplies, reduced energy use and carbon removal successfully reached the target of a temperature rise below +1.5 • C by 2100; while scenarios with scattered short-term climate policy, strong inequalities in socio-economic pathways, and high baseline fossil fuel use, missed it. Gidden et al (2018) analyzed 13 scenarios with open-access and reproducible higher gridding spatial resolution (aneris python library), comparing SSPs to representative concentration pathways, and recommended that the assessment of the role of uncertainty is carried out not only between scenarios, but also between model results for a certain scenario, such as fluorinated gases trajectories. Additionally, carbon dioxide and methane gases are well-known climate forcers that have a higher impact from a political rather than physical perspective, thus adding spatial detail would provide more meaningful insights for policy analysis.

Targeting and distributional concerns
For the case of the transport sector, studying the interaction between carbon taxes, equity effects and investments in infrastructure (i.e. public transport) could shed light on fuel efficiency policies. IAMs find mitigation efforts on the transportation, industry and buildings sectors of particular importance (Méjean et al 2018, Rogelj et al 2018. Taking into account that heterogeneity of attributes is also relevant for policies targeting the transport sector, as described in Galvin (2017), the interaction between speed and acceleration becomes crucial to investigate the efficiency of electric vehicles.

Understanding consumer preferences and changes
Another branch of research to inform policy development includes changes in behavior and lifestyle (Herring and Roy 2007), as well as field experiments and surveys to better approximate, in a more realistic manner, end-user discount rates and preferences. Understanding how to move from bad habits to good habits, in accordance with consumer's preferences, could contribute to reduce energy consumption in the short or medium run. We find that more studies that include heterogeneity of actors (household or firm) would help to shed light on the distributional impacts of EE policies. Chang et al (2018) found for the production side that pollution-minimizing policies are less costly than welfare-maximizing increases in EEIs on green technologies, describing a U-shaped environmental Kuznets curve. In general terms, to reduce global emissions and energy use in the long term, EEI policies on both the demand and supply side could help illustrate existing trade-offs/co-benefits between economic growth, social welfare, reduction of GHG emissions, and total energy use (Wei and Liu 2017). Brockway et al (2017) conclude that because EE and rebound may act as engines of economic growth (Ayres 2010), there might be a potential trade-off between climate and economic growth policies. Although carbon taxes would be better than command-and-control policies to reduce rebound while allowing for economic growth, distributional impacts have to be considered carefully to account for energy poverty and energy climate justice. This could improve the social acceptance of policies. Thus, a better understanding of interactions between energy consumption, energy savings, GHG emissions and economic growth would provide a more comprehensive understanding at macro-economic levels. This could be help identifying adequate policy strategies to target different dimensions, such as the level of aggregation, actors, income level, and time.

Interactions between energy consumption, GHG emissions reductions and welfare
Policies that encourage EEIs should be clear about about the trade-offs and be more explicit about the required level of detail regarding the modeling of the most pressing issues to solve: securing economic growth, reducing GHG emissions, and (or) increasing fossil fuel energy savings. Within the study of these interactions, the RE is only one aspect to consider. Furthermore, BC analysis would be equally necessary to foster well-informed decisions. Future large shifts in policy will require answers and solutions to many open questions regarding complex interactions, to understand how EE and energy saving interacts with low-carbon economies, sustainability, socio-technical (Geels et al 2018) and psychological aspects. Moreover, better knowledge of social transitions is required (van Vuuren et al 2018, Rogelj et al 2018. Although policy strategies must identify clearly their targets among multiple dimensions; they should find common ground at the global level. Studies on spillover effects and strategic alliances between regions could also shed light on feasible futures. To reach national or sectoral policy objectives in a costeffective fashion, we require a comprehensive understanding of the RE from both theoretical and empirical grounds. This has the potential to better guide policy decisions in the future.

Acknowledgments
Andreas Löschel acknowledges support from the 111 Project [grant number B18014] by the Chinese Ministry of Education and the State Administration of Foreign Experts Affairs. Gloria Colmenares acknowledges support from the Katholischer Akademischer Ausländer-Dienst (KAAD). We thank four anonymous reviewers for providing helpful comments on earlier drafts of the manuscript. The authors declare no conflict of interest.

A.1. Energy efficiency improvement formulations
The easiest representations of energy efficiency improvements conceptualize the change as deriving exclusively from energy supply and use (Birol and Keppler 2000).
An explicit representation of EEIs at the microeconomic level, as specified by Hunt et al (2014), defines efficiency as the ratio of useful energy outputs to energy inputs of an energy system, or as units of the energy service (ES) produced per unit of the energy source (E) used, the term energy service 16 in equation A1 17 is sometimes taken as a physical indicator (e.g. vehicle kilometers in transportation), or an explicit thermodynamic measure where heat content is represented (e.g. joules of heat in water heating inside a closed energy system). More recently energy service has been defined as exergy, the usable energy to perform physical work, or the effective energy available for end-use consumption (Brockway et al 2017). Fell (2017) finds 27 definitions of "energy service". A clear definition of the term energy service in studies where there are explicit representations of energy efficiency improvements is important for reproducibility and to contribute to objective debates on energy efficiency policies. Moreover, depending on the type of study, an energy efficiency improvement formulation might be influenced also by a utility or production function, which represents the choice made by the consumer or producer. According to Hunt et al (2014), energy efficiency improvements should be explicitly modeled to avoid bias, but Frondel and Vance (2018) find similar results (though with high standard errors) when comparing an explicit representation of energy efficiency improvement with an implicit representation in their own study. Therefore, a simplified model might be preferred in cases where additional complexity in models leads to robust results. Along these lines, we recommend avoiding the following three representations of energy efficiency improvements; they would entangle increases in energy efficiency with other factors (e.g capital), and therefore be biased: (a) Implicit representation of energy efficiency, not using equation (A1). In these cases, the ownprice elasticity of energy demand is taken as a proxy for the rebound effect (i.e. historical studies of fuel consumption), see equation (A7). (b) Energy intensity as an equivalent measure to energy efficiency (e.g. total energy consumption/GDP). This might be true for one unit of production under unbiased technical change, but not when the level of aggregation is scaled up (Birol and Keppler 2000). (c) Considering energy efficiency improvements as the ratio of the price of an energy service to energy as equal or linear to the ratio of the demand for energy services to energy consumption.
More realistic representations such as in Adeyemi et al (2010) model historical trends of increases and decreases in price 18 . Other studies use energy efficiency improvement indices, where a past maximum price is followed by price recoveries and decreases (using price decomposition methods) (Ang et al 2010).
Although in zero-cost breakthrough studies it is impossible for this condition to happen in the case of partial equilibrium (Lemoine 2018), it is theoretically possible for it to occur when large externalities are modeled (e.g. in studies that model market-based policy improvements). Moreover, depending on the functional form of the production function, this can cause a "disinvestment effect" in the long-run (Turner et al 2009).    This result is likely driven by overly optimistic ex ante predictions or rebound.   Reported only inter-fuel substitution scenario.

A.2. Rebound effect formulations
Conceptual clarity leads to more accurate formulations. After showing how possible causes of energy efficiency improvements might translate into rebound effect components, we now revise available rebound effect formulations in the literature. Thus, formulations that are less prone to bias include: the direct energy rebound effect (DRE) (Berkhout et al 2000), where η ε (ES) is the energy services elasticity of demand with respect to its energy efficiency. But data to calculate the DRE in this form is scant, thus an alternative formulation is; where AES is actual energy savings and PES is potential or expected energy savings in the absence of rebound effects, holding prices constant (Berkhout et al 2000). IREs can be computed using cross-price elasticities, income elasticities, and expenditure elasticities between energy and other goods or energy inputs or non-energy inputs (η PEG,NEG or η PEI,NEI , respectively). IREs can also arise from behavioral changes, not just energy efficiency improvements (Druckman et al 2010).
In the case of a macro-economic rebound calculation, a household productivity shock is usually applied to the model for calculating the difference between AES and PES corresponding to general equilibrium measures (Guerra and Sancho 2010). Notice that for economic growth models, it is also common practice to obtain two scenarios, one assuming engineering savings, and the other represented with a law of motion of capital, to quantify the rebound effect, as in (Turner et al 2009): where γ is the efficiency elasticity of energy, usually represented as an autonomous (or exogenous zerocost breakthrough) energy efficiency improvement, and α = 1 for economy-wide rebound, or takes the value of α = E i /E, modeled for the production or consumption side (sector) of country i, and E is the value of energy in physical or economic units (value share). The rebound effect can also be expressed in terms of GHG emissions: where ∆Q is the net change in GHG emissions and ∆H is the change in emissions without behavioral response (Chitnis and Sorrell 2015). At the economy-wide level, when using a theoretical welfare maximization CGE model, as in Wei (2010), the rebound effect can be expressed as: where R s is global rebound in the short term, and where R l is global rebound in the long term. σ s is the price elasticity of energy supply, σ e e is the energy own elasticity of marginal product with respect to energy input in the welfare function, σ d is the price elasticity of demand, and θ is the own-price elasticity of capital supply and demand, as cross-price elasticity of marginal product with respect to capital and energy inputs in the production of welfare. This theoretical framework is simplified to account for only one non-energy good, and the analysis of elasticities are only for comparison purposes between the microand macro-economic levels. Lemoine (2018) gives a word of caution about the reliability on magnitudes of elasticities of substitution to guide the likelihood of backfire at the macro-economic level, due to the existence of sectoral interactions that need to be taken into account.
Formulations from (i) to (vii) summarize additional formulations that link the micro-economic level to the macro-economic level. It can serve as a guide to further explore additional insightful interactions that are less intuitive due to the complexity of the RE phenomena.
(a) Macro-economic RE ≡ 'indirect rebound effect' + 'economy-wide rebound effect' 19 ; (b) Total rebound effect ≡ 'macro-economic rebound effect' + 'direct rebound effect'; (c) Gross energy savings from IEA energy efficiency policies ≡ 'net energy savings (taken as exogenous in E3MG)' + 'direct rebound energy use'; (d) Change in macro-economic energy use from energy efficiency policies from E3MG ≡ 'energy use simulated from E3MG after the imposed exogenous net energy savings' -'energy use simulated from E3MG before the imposed exogenous net energy savings'; (e) Total rebound effect as % ≡ 100 times the 'change in macro-economic energy use from energy efficiency policies from E3MG'/'gross energy savings from IEA energy-efficiency policies'; (f) Direct rebound effect as % ≡ 100 times 'direct rebound energy use'/'gross energy savings from IEA energy-efficiency policies'; (g) Macro-economic rebound effect as % ≡ 'total rebound effect as %' -'direct rebound effect as %' .
We used these mathematical representations to summarize and classify the existing rebound effect types we found in the literature. This is relevant to organize the rebound effect within the four dimensions discussed in section 2.1, 2.2, and tables in appendix B. Table 4 shows five types of rebound effects and their respective elasticity domains.
In contrast to formulations A2 to A8, the following might lead to upward-biased estimates. These relate to the representations of energy efficiency that we recommend to avoid, explained in section 2.1 Though these conceptions were helpful to study the rebound effect initially, we do not recommend them for future studies, because they do not disentangle changes in relative prices due to an energy efficiency policy from exogenous technical change: where η ε (E) is the energy elasticity of demand (of energy output for the consumer side, or input for the producer side e.g. fuel) with respect to efficiency; where η PE (E) is the own-price elasticity of energy demand for the relevant energy service (of energy commodities on the consumer side, or fuel on the producer side). This only holds when the price of energy (in physical units) remains constant, so that any change in energy efficiency is reflected in the effective price of energy (Guerra and Sancho 2010) (meaning that efficiency is not influenced by other changes in energy prices), and when the reaction to a price decrease equals the reaction to an energy efficiency improvement (Madlener and Hauertmann 2011). Moreover, rebound effects can arise from marginal and non-marginal pricing (Borenstein 2013); and: where η PES (ES) is the own-price elasticity of the energy service. However, this formulation is also subject to bias unless an explicit formulation of efficiency improvement is introduced in the definition of the energy service, in demand or supply functions (or choices), since this approximation also assumes that one source of energy is exclusively used in the production of one energy service (Hunt et al 2014).