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Abstract

Widespread implementation of energy efficiency is a key greenhouse gas emissions mitigation measure, but rebound can “take back” energy savings. However, the absence of solid analytical foundations hinders empirical determination of the size of rebound. A new clarity is needed, one that involves both economics and energy analysis. In this paper (Part I), we advance foundations of a rigorous analytical framework for consumer-sided rebound that starts at the microeconomic level and is approachable for both energy analysts and economists. We develop foundations of a framework that (i) clarifies the energy, expenditure, and consumption aspects of rebound, (ii) combines embodied energy with operations, maintenance, and disposal effects (under a new “emplacement effect”), and (iii) provides the first operationalized link between microeconomic and macroeconomic levels. The framework enables determination of the effects of non-marginal energy service price decreases, satiated demand for the energy service, and reduced economy-wide energy demand.

JEL Classification: O13, Q40, Q43

Keywords

energy efficiency energy rebound energy services microeconomic rebound substitution and income effects macroeconomic rebound

1. Introduction

Energy efficiency is often considered to be the most important means of reducing energy consumption and CO₂ emissions (International Energy Agency 2017, 139: Fig. 3.15). But energy rebound makes energy efficiency less effective at decreasing energy consumption by taking back (or reversing, in the case of “backfire”) energy savings expected from energy efficiency improvements (Sorrell 2009). As such, energy rebound is a threat to a low-carbon future (Brockway et al. 2017; van den Bergh 2017).

Recent evidence shows that rebound is both larger than commonly assumed (Stern 2020) and mostly missing from large energy and climate models (Brockway et al. 2021). Thus, rebound could be an important reason why energy consumption and carbon emissions have never been absolutely decoupled from economic growth (Brockway et al. 2021; Haberl et al. 2020).

1.1. A Short History of Rebound

Famously, the roots of energy rebound trace back to Jevons who said “[i]t is wholly a confusion of ideas to suppose that the economical use of fuel is equivalent to a diminished consumption. The very contrary is the truth” (Jevons 1865, 103, emphasis in original). Less famously, the origins of rebound extend further backward from Jevons to Williams (1840) and Parkes who wrote “[t]he economy of fuel is the secret of the economy of the steam-engine; it is the fountain of its power, and the adopted measure of its effects. Whatever, therefore, conduces to increase the efficiency of coal, and to diminish the cost of its use, directly tends to augment the value of the steam-engine, and to enlarge the field of its operations” (Parkes 1838, 161). For nearly 200 years, then, it has been understood that efficiency gains may be taken back or, paradoxically, cause growth in energy consumption, as Jevons suggested.

The oil crises of the 1970s shone a light back onto energy efficiency, and research into rebound appeared late in the decade (Madlener and Turner 2016; Saunders et al. 2021). A modern debate over the magnitude of energy rebound commenced. On one side, scholars including Brookes (1979, 1990) and Khazzoom (1980) suggested rebound could be large. Others, including Lovins (1988) and Grubb (1990, 1992), claimed rebound was likely to be small. Debate over the size of energy rebound continues today. Advocates of small rebound (less than, say, 50%), suggest “the rebound effect is overplayed” (Gillingham et al. 2013, 475), while others claim (i) that the evidence for large rebound (greater than 50%) is growing (Berner et al. 2022; Saunders 2015) and (ii) that rebound will reduce the effectiveness of energy efficiency to decrease carbon emissions (van den Bergh 2017).

1.2. Absence of Solid Analytical Foundations

Turner contends that the lack of consensus on the magnitude of energy rebound in the modern empirical literature is caused by “a rush to empirical estimation in the absence of solid analytical foundations” (Turner 2013, 25). Progress has been made recently on how price changes affect economy-wide rebound in general equilibrium frameworks (Blackburn and Moreno-Cruz 2020; Fullerton and Ta 2020; Lemoine 2020). And arguments from microeconomics (i.e., at sectoral and individual level) have been used from the outset of the modern debate (e.g., Greening, Greene and Difiglio 2000; Khazzoom 1980), and Borenstein (2015) and Chan and Gillingham (2015) made further progress toward solidifying the microeconomic analytical foundations.

Rebound involves simultaneous changes in energy, expenditure, and consumption aspects—keeping an overview of all aspects is difficult, with no approach to our knowledge documenting all changes in a straightforward and consistent manner. For instance, while the microeconomic categories of substitution and income effects provide analytical clarity about how behavior changes affect energy service consumption, it has been unclear how they could be used for precise numerical rebound calculations. Where previous numerical calculations were made, they tended to approximate the substitution effect from other goods to the cheaper energy service, without maintaining constant utility for the device user. They also used constant price elasticities for non-marginal efficiency improvements, even though constant price elasticities typically provide only approximations of substitution and income effects for small efficiency changes. Further, previous analytical studies have stressed the importance of the cost of buying an upgraded device as well as the energy embodied in the device. Yet, there is no clearly formulated approach for how to incorporate these cost and energy components into rebound calculations. Finally, while recent general equilibrium rebound modeling has led to important insights about the effects of changing prices, dynamic aspects of a macroeconomic rebound have been neglected by these approaches.

In the absence of solid analytical foundations, the wide variety of rebound calculation approaches contributes to a wide range of rebound values, giving the appearance of uncertainty and leading some energy and climate modelers to either (i) use questionable rebound values or (ii) ignore rebound altogether. Insufficient inclusion of rebound in energy and climate models could lead to overly optimistic projections of the capability of energy efficiency to reduce carbon emissions (Brockway et al. 2021). We suggest that improving the conceptual foundations of rebound and solidifying the analytical frameworks will (i) help generate more robust estimates of rebound, (ii) lead to better rebound calculations in energy and climate models, and (iii) provide improved evidence for policymaking around energy efficiency.

But why is there an “absence of solid analytical foundations?” We propose that development of solid analytical frameworks for rebound is hampered by the fact that rebound is a decidedly interdisciplinary topic, involving both economics and energy analysis. Birol and Keppler (2000, 458) note that “different implicit and explicit assumptions of different research communities (‘economists’, ‘engineers’) … have in the past led to vastly differing points of view.”¹ Turner states that “[d]ifferent definitions of energy efficiency will be appropriate in different circumstances. However, … it is often not clear what different authors mean by energy efficiency” (Turner 2013, 237–8). If authors from the two disciplines cannot even agree on the key terms, it is unsurprising that analytical foundations have not yet been fully elucidated. To fully understand rebound, economists need to have an energy analyst’s understanding of energy, and energy analysts need to have an economist’s understanding of finance and human behavior.² Developing the knowledge and skills required to assess and calculate, let alone mitigate, rebound effects is a tall order, indeed.

1.3. New Clarity Is Needed

We contend that new clarity is needed. Specifically, a description of rebound that is (i) consistent across energy, expenditure, and consumption aspects, (ii) technically rigorous, and (iii) approachable from both disciplines (economics and energy analysis) will be a good starting point toward that clarity. In other words, the finance and human behavior aspects of rebound need to be presented in ways energy analysts can understand. And the energy aspects of rebound need to be presented in ways economists can understand.

Summarizing, we surmise that development of effective carbon reduction policies has been hampered, in part, by the fact that rebound is not sufficiently included in energy and climate models. We suspect that one reason rebound is not sufficiently included is the lack of consensus on rebound calculation methods and, hence, rebound magnitude. Building upon Turner (2013), we contend that lack of consensus on rebound magnitude is a symptom of the absence of solid analytical foundations for rebound. We posit that developing solid analytical frameworks is difficult because energy rebound is an inherently interdisciplinary topic. We believe that providing a detailed explication of a rigorous analytical framework for energy rebound, which is approachable by both energy analysts and economists alike, will go some way toward providing additional clarity in the field.

1.4. Objective, Contributions, and Structure

The objective of this paper is to help advance clarity in the field of energy rebound by supporting the development of a rigorous analytical framework, one that (i) starts at the microeconomics of rebound (building especially upon Borenstein 2015) and (ii) is approachable for both energy analysts and economists.³ We strive to keep the framework as simple as possible and limit our attention to a model of consumer demand for energy services, while demonstrating that the approach is transferable to a producer model with few modifications.

The key contributions of this paper are (i) a novel and clear explication of interrelated energy, expenditure, and consumption aspects of energy rebound, (ii) development of a rebound analysis framework that combines embodied energy effects, operations, maintenance, and disposal rebound effects, and exact expressions for substitution and income rebound effects under non-marginal energy efficiency increases and (by implication) non-marginal energy service price decreases, (iii) an operationalized link between rebound effects on microeconomic and macroeconomic levels, and (iv) development of an extension of the framework to an energy price rebound effect.

The remainder of this paper is structured as follows. Section 2 describes the rebound analysis framework. Section 3 discusses this framework relative to previous frameworks and provides an initial assessment of an energy price effect. Section 4 concludes. Results from the application of our framework to energy efficiency upgrades to a car and an electric lamp can be found in Heun et al. (2025).

2. Methods: Development of the Framework

In this section, we develop an energy rebound framework for an individual consumer who upgrades the energy efficiency of a single device (concisely, “the framework,”“this framework,” or “our framework”). We endeavor to help advance clarity in the field of energy rebound by providing sufficient detail to assist energy analysts to understand the economics and economists to understand the energy analysis.

2.1. Rebound Typology

Table 1 shows our typology of rebound effects. We follow others, including Jenkins, Nordhaus and Shellenberger (2011) and Walnum, Aall and Løkke (2014), in identifying and including both direct and indirect rebound effects, which occur at (direct) and beyond (indirect) the level of the device and its user. Again following others, such as Gillingham, Rapson and Wagner (2016), we distinguish between rebound effects at the microeconomic and macroeconomic levels.

Table 1.

Rebound Typology for Our Framework.

Rebound level	Direct rebound ( $R e_{dir}$ )	Indirect rebound ( $R e_{indir}$ )
Microeconomic rebound ( $R e_{micro}$ )	Emplacement effect ( $R e_{dempl}$ )	Emplacement effect ( $R e_{iempl}$ )
These mechanisms occur at the single device/user level within a static economy based on responses to the reduction in implicit price of an energy service.	Accounts for performance of the Energy Efficiency Upgrade (EEU) only. No behavior changes occur. The direct energy effect of emplacement of the EEU is expected device-level energy savings. By definition, there is no rebound from direct emplacement effects ( $R e_{dempl} \equiv 0$ ).	Differential energy adjustments beyond the usage of the upgraded device, via (i) the embodied energy associated with the manufacturing phase ( $R e_{emb}$ ) and (ii) the implied energy demand from operations, maintenance, and disposal ( $R e_{OMd}$ ). $R e_{iempl}$ can be $> 0$ or $< 0$ , depending on the characteristics of the EEU.
	Substitution effect ( $R e_{dsub}$ )	Substitution effect ( $R e_{isub}$ )
	Increase in energy service consumption due to its lower prices as a result of the EEU. Excludes, by definition, the effects of freed cash (income effects). $R e_{dsub} > 0$ is typical due to greater consumption of the energy service.	Reduction in other goods consumption due to the relatively higher prices as a result of the EEU. Excludes, by definition, the effects of freed cash (income effects). $R e_{isub} < 0$ is typical due to reduced consumption of other goods and services.
	Income effect ( $R e_{dinc}$ )	Income effect ( $R e_{iinc}$ )
	Spending of some of the freed cash to obtain more of the energy service. $R e_{dinc} > 0$ is typical due to increased consumption of the energy service.	Spending of some of the freed cash on other goods and services. $R e_{iinc} > 0$ is typical due to increased consumption of other goods and services.
Macroeconomic rebound ( $R e_{macro}$ )		Macroeconomic effect ( $R e_{macro}$ )
These mechanisms originate from the dynamic response of the economy to reach a stable equilibrium (between supply and demand for energy services and other goods). These mechanisms combine various short and long run effects.		Increased energy consumption in the broader macroeconomic system, that is, beyondresponses at the microeconomic (device/user)level. $R e_{macro} > 0$ is typical due to sectoralproductivity increases (the EEU) causingadditional economic growth in the widereconomy.

Microeconomic rebound occurs at the level of the single device and its user and in our framework comprises three effects: an emplacement effect, a substitution effect, and an income effect, with direct and indirect partitions for each.

“Emplacement” is a new term we introduce to collect effects associated with installing higher-efficiency devices, including (i) embodied energy of their manufacture (subscript $emb$ ), (ii) operations and maintenance (subscript $OM$ ), and (iii) disposal (subscript $d$ ) activities. Although none of the embodied, operations and maintenance, or disposal effects are new (see Borenstein 2015, 3: footnote 5; 16: footnote 37; Saunders et al. 2021; Sorrell, Dimitropoulos and Sommerville 2009; Sorrell, Gatersleben and Druckman 2020), we separate them from substitution and income microeconomic effects (Table 1) to calculate rebound according to the steps in our framework (see Section 2.5).

The direct rebound effect can be partitioned into a direct emplacement effect, a direct substitution effect, and a direct income effect. At the level of the device, all of the direct rebound effects change the consumption of energy by the device whose efficiency has been upgraded, according to a microeconomic behavioral model of the consumer who responds to the cheaper energy service.

Similarly, the indirect rebound effect can be partitioned into an indirect emplacement effect, an indirect substitution effect, and an indirect income effect. All of the indirect effects change the induced energy consumption beyond the upgraded device, again according to a microeconomic behavioral model. We assume a partial equilibrium response to the energy efficiency upgrade (EEU) at the microeconomic level; other prices in the economy ( $p_{g}$ ) remain unchanged in response to the EEU.

In contrast, macroeconomic rebound is a broader, economy-wide response to the single device upgrade. Like other authors, we recognize many macroeconomic rebound effects, even if we don’t later distinguish among them.⁴ At the macroeconomic level, general equilibrium effects can occur as prices for all goods and services (even energy) may change in response to the EEU. Further treatment of macroeconomic rebound can be found in Section 2.5.4 of this paper (Part I) and in Section 4.1 of Heun et al. (2025). Discussion of an energy price rebound effect can be seen in Section 3.2 below and in Section 4.6 of Heun et al. (2025).

Figure 1 shows rebound effects arranged in the left-to-right order of their discussion in this paper. The left-to-right order does not necessarily represent the progression of rebound effects through time. Rebound symbols are shown above each effect ( $R e_{empl}$ , etc.). Nomenclature for partitions of direct and indirect rebound is shown beneath each effect ( $R e_{dempl}$ , etc.). Decorations for each stage are shown between rebound effects (○, * etc.). Names for the decorations are given at the bottom of the figure (“orig,”“star,” etc.).⁵

Figure 1.

Flowchart of rebound effects and decorations.

2.2. Rebound Relationships

Energy rebound ( $Re$ ) is defined as

Re \equiv 1 - \frac{actual final energy savings rate}{expected final energy savings rate},

(1)

where both actual and expected final energy savings rates are in MJ/yr (megajoules per year) and expected positive. The final energy “takeback” rate is defined as the expected final energy savings rate less the actual final energy savings rate.⁶ Rewriting equation (1) with the definition of takeback gives

Re = 1 - \frac{expected final energy savings rate - takeback rate}{expected final energy savings rate} .

(2)

Simplifying gives

Re = \frac{takeback rate}{expected final energy savings rate} .

(3)

We define rebound at the final energy⁷ stage of the energy conversion chain, because the final energy stage is the point of energy purchase by the device user. To simplify derivations, we choose not to apply final-to-primary energy multipliers to final energy rates in the numerators and denominators of rebound expressions derived from equations (1) and (3); they divide out anyway.⁸ Henceforth, we drop the adjective “final” from the noun “energy,” unless there is reason to indicate a specific stage of the energy conversion chain.

2.3. The Energy Conversion Device and Energy Efficiency Upgrade (EEU)

We assume an energy conversion device (say, a car) that consumes energy (say, gasoline) at a rate ${\overset{\cdot}{E}}^{°}$ (in MJ/yr). We use “rate” to indicate any quantity measured per unit time, such as a flow of energy per year or a flow of income per year. None of the rates in this paper indicate exponential (%/yr) changes. Rates are identified by a single dot above the symbol, a convention adopted from the engineering literature where, for example, $\overset{\cdot}{x}$ often indicates a velocity in m/s (meters per second) and $\overset{\cdot}{E}$ often indicates an energy flow rate in kW (kilowatts). The overdot is an important notational element in this paper, as it distinguishes between stocks (without overdots) and flows (with overdots). For example, $E$ is a quantity of energy in, say, MJ, while $\overset{\cdot}{E}$ is a rate of energy in, say, MJ/yr. We later annualize capital costs ( $C_{cap}$ in $), disposal costs ( $C_{d}$ in $), and energy embodied in the device during its production ( $E_{emb}$ in MJ) to create undiscounted cost rates ( ${\overset{\cdot}{C}}_{cap}$ and ${\overset{\cdot}{C}}_{d}$ in $/yr) and embodied energy rates ( ${\overset{\cdot}{E}}_{emb}$ in MJ/yr). (Cost discounting⁹ is captured by the variables $τ_{α}$ and $τ_{ω}$ . See Appendix B.1 for details.)

Energy is available at price $p_{E}$ (in $/MJ). The original energy conversion device provides a rate of energy service ${\overset{\cdot}{q}}_{s}^{°}$ (in, say, vehicle-km/yr) with final-to-service efficiency $η^{°}$ (in, say, vehicle-km/MJ). An energy efficiency upgrade (EEU) increases final-to-service efficiency such that $η^{°} < η^{*} = \hat{η} = \bar{η} = \tilde{η}$ , as shown in Table B.1. The EEU is not costless, so the upgraded device may be more expensive to purchase than a like-for-like replacement of the original device. We call this increased “capital cost” ( $C_{cap}^{°} < C_{cap}^{*}$ ). It may also be more costly to operate and maintain (subscript $OM$ ) and dispose (subscript $d$ ) of the upgraded device ( ${\overset{\cdot}{C}}_{OM}^{°} < {\overset{\cdot}{C}}_{OM}^{*}$ and ${\overset{\cdot}{C}}_{d}^{°} < {\overset{\cdot}{C}}_{d}^{*}$ ). However, the opposite may hold, too. As final-to-service efficiency increases ( $η^{°} < η^{*}$ ), the price of the energy service declines ( $p_{s}^{°} > p_{s}^{*}$ ). The energy price ( $p_{E}$ ) is assumed exogenous at the microeconomic level ( $p_{E}^{°} = p_{E}^{*} = {\hat{p}}_{E} = {\bar{p}}_{E} = {\tilde{p}}_{E}$ ), so the energy purchaser (the device user) is a price taker.¹⁰ Initially, the device user spends income ( $\overset{\cdot}{M}$ ) on energy for the device ( ${\overset{\cdot}{C}}_{s}^{°} = p_{E}^{°} {\overset{\cdot}{E}}_{s}^{°}$ ), annualized capital costs for the device ( $τ_{α}^{°} {\overset{\cdot}{C}}_{cap}^{°}$ ), annualized costs for operations and maintenance ( ${\overset{\cdot}{C}}_{OM}^{°}$ ) and disposal of the device ( $τ_{ω}^{°} {\overset{\cdot}{C}}_{d}^{°}$ ), and other goods and services ( ${\overset{\cdot}{C}}_{g}^{°}$ ). The budget constraint for the device user is

\overset{\cdot}{M} = τ_{α}^{°} {\overset{\cdot}{C}}_{cap}^{°} + {\overset{\cdot}{C}}_{s}^{°} + {\overset{\cdot}{C}}_{OM}^{°} + τ_{ω}^{°} {\overset{\cdot}{C}}_{d}^{°} + {\overset{\cdot}{C}}_{g}^{°},

(4)

where $τ_{α}^{°}$ and $τ_{ω}^{°}$ account for discounting, and ${\overset{\cdot}{C}}_{cap}^{°}$ and ${\overset{\cdot}{C}}_{d}^{°}$ are undiscounted cost rates given by $C_{cap}^{°} / t_{life}^{°}$ and $C_{d}^{°} / t_{life}^{°}$ . Note that $τ_{α} \geq 1$ , and $τ_{ω} \leq 1$ ; equalities apply when interest rate ( $r$ ) is zero. (See Appendix B.1 for details on discounting.) After substituting the product of energy price ( $p_{E}$ ) and the rate of energy consumption (given by the ratio of the rate of energy service consumption and efficiency, ${\overset{\cdot}{q}}_{s} / η$ ), after substituting the product of price ( $p_{g}$ ) and the rate ( ${\overset{\cdot}{q}}_{g}$ ) of other goods consumption, after substituting ${\overset{\cdot}{C}}_{OMd}^{°} \equiv {\overset{\cdot}{C}}_{OM}^{°} + τ_{ω}^{°} {\overset{\cdot}{C}}_{d}^{°}$ , and after some rearrangement, equation (4) becomes

\overset{\cdot}{M} - τ_{α}^{°} {\overset{\cdot}{C}}_{cap}^{°} - {\overset{\cdot}{C}}_{OMd}^{°} = p_{E}^{°} \frac{{\overset{\cdot}{q}}_{s}^{°}}{η^{°}} + p_{g} {\overset{\cdot}{q}}_{g}^{°},

(5)

which is the usual discounted budget constraint for the microeconomic consumer after subtracting capital, operations and maintenance, and disposal costs.

Later (Sections 2.5.1–2.5.4), we walk through the four rebound effects (emplacement, substitution, income, and macro), deriving rebound expressions for each, but first we show typical energy and cost relationships (Section 2.4).

2.4. Typical Energy and Cost Relationships

With the rebound notation of Appendix A, four typical relationships emerge. First, the consumption rate of the energy service ( ${\overset{\cdot}{q}}_{s}$ ) is the product of final-to-service efficiency ( $η$ ) and the rate of energy consumption by the energy conversion device ( ${\overset{\cdot}{E}}_{s}$ ). Typical units for automotive transport and illumination (the examples in Heun et al. (2025)) are shown beneath each equation.¹¹

\begin{matrix} {\overset{\cdot}{q}}_{s} = η {\overset{\cdot}{E}}_{s} \end{matrix}

(6)

\begin{matrix} [pass \cdot km / yr] = [pass \cdot km / MJ] [MJ / yr] \\ [lm \cdot hr / yr] = [lm \cdot hr / MJ] [MJ / yr] \end{matrix}

Second, the energy service price ( $p_{s}$ ) is the ratio of energy price ( $p_{E}$ ) to the final-to-service efficiency ( $η$ ).

p_{s} = \frac{p_{E}}{η}

(7)

\begin{matrix} [$ / pass \cdot km] = \frac{[\frac{$}{MJ}]}{[pass \cdot km / MJ]} \\ [$ / lm \cdot hr] = \frac{[$ / MJ]}{[lm \cdot hr / MJ]} \end{matrix}

Third, energy service expenditure rates ( ${\overset{\cdot}{C}}_{s}$ ) are the product of energy price ( $p_{E}$ ) and device energy consumption rates ( ${\overset{\cdot}{E}}_{s}$ ).

{\overset{\cdot}{C}}_{s} = p_{E} {\overset{\cdot}{E}}_{s}

(8)

[$ / yr] = [$ / MJ] [MJ / yr]

Fourth, indirect energy rates for operations and maintenance ( ${\overset{\cdot}{E}}_{OM}$ ), disposal ( ${\overset{\cdot}{E}}_{d}$ ), and other goods expenditures ( ${\overset{\cdot}{E}}_{g}$ ) are the product of expenditures rates ( ${\overset{\cdot}{C}}_{OM}$ , $τ_{ω} {\overset{\cdot}{C}}_{d}$ , and ${\overset{\cdot}{C}}_{g}$ ) and the energy intensity of the economy ( $I_{E}$ ).

{\overset{\cdot}{E}}_{OM} = {\overset{\cdot}{C}}_{OM} I_{E}

(9)

{\overset{\cdot}{E}}_{d} = τ_{ω} {\overset{\cdot}{C}}_{d} I_{E}

(10)

{\overset{\cdot}{E}}_{g} = {\overset{\cdot}{C}}_{g} I_{E}

(11)

[MJ / yr] = [$ / yr] [MJ / $]

Note that the indirect energy rate for the disposal effect is obtained from disposal costs that include discounting. (See Appendix B.1 for details on cost discounting.)

2.5. Rebound Effects

The four rebound effects (emplacement, substitution, income, and macro) are discussed in subsections below. In each subsection, we define the effect and show mathematical expressions for rebound ( $Re$ ) caused by the effect. Detailed derivations of all rebound expressions can be found in Appendix B. See, in particular, Tables B.3 to B.6, which provide a parallel structure for energy and financial accounting across all rebound effects. We begin with the emplacement effect.

2.5.1. Emplacement Effect

The emplacement effect accounts for performance changes of the device due to the fact that a higher-efficiency device has been put in service (and will need to be decommissioned at a later date); consumption patterns are assumed unchanged. Behavior adjustments are addressed later, in the substitution and income effects. Any (positive or negative) adjustment in income due to emplacement (measured as net income, ${\overset{\cdot}{N}}^{*}$ ) is added to the freed cash ( $\overset{\cdot}{G}$ ) spent in the income effect.

Direct emplacement effect ( $R e_{dempl}$ ). The direct emplacement effects of the EEU include device energy savings ( ${\overset{\cdot}{S}}_{dev}$ ) and device energy cost savings ( $Δ {\overset{\cdot}{C}}_{s}^{*}$ ). ${\overset{\cdot}{S}}_{dev}$ can be written conveniently as

{\overset{\cdot}{S}}_{dev} = (\frac{η^{*}}{η^{°}} - 1) \frac{η^{°}}{η^{*}} {\overset{\cdot}{E}}_{s}^{°} .

(12)

(See Appendix B.4.1 for the derivation.)

Because the original and upgraded device are assumed to have equal performance¹² and because behavior changes are not considered in the direct emplacement effect, actual and expected energy savings rates are identical, and there is no takeback. By definition, then, the direct emplacement effect causes no rebound. Thus,

R e_{dempl} = 0 .

(13)

Indirect emplacement effects ( $R e_{iempl}$ ). Although the direct emplacement effect does not cause rebound, indirect emplacement effects may indeed cause rebound. Indirect emplacement effects account for the life cycle of the energy conversion device, including (i) changes in the embodied energy rate ( $Δ {\overset{\cdot}{E}}_{emb}^{*}$ ), (ii) changes in the operations and maintenance energy and expenditure rates ( $Δ {\overset{\cdot}{E}}_{OM}^{*}$ and $Δ {\overset{\cdot}{C}}_{OM}^{*}$ ), and (iii) changes in the disposal energy and expenditure rates ( $Δ {\overset{\cdot}{E}}_{d}^{*}$ and $Δ {\overset{\cdot}{C}}_{d}^{*}$ ).

Embodied energy effect ( $R e_{emb}$ ). One of the unique features of this framework is that independent analyses of embodied energy and capital costs of the EEU are required. We note that the different terms (embodied energy rate, ${\overset{\cdot}{E}}_{emb}$ , and capital cost rate, ${\overset{\cdot}{C}}_{cap}$ ) might seem to imply different processes, but they actually refer to the same emplacement effect. Purchasing an upgraded device (which likely leads to ${\overset{\cdot}{C}}_{cap}^{°} \neq {\overset{\cdot}{C}}_{cap}^{*}$ ) will likely mean a changed embodied energy rate ( ${\overset{\cdot}{E}}_{emb}^{°} \neq {\overset{\cdot}{E}}_{emb}^{*}$ ) to provide the same energy service. Our names for these aspects of rebound (embodied energy and capital cost) reflect common usage in the energy and economics fields, respectively.

Consistent with the energy analysis literature, we define embodied energy to be the sum of all energy consumed in the production of the energy conversion device, all the way back to resource extraction.¹³ Energy is embodied in the device within manufacturing and distribution supply chains prior to consumer acquisition of the device. We assume no energy is embodied in the device while in service. The EEU causes the embodied energy of the energy conversion device to change from $E_{emb}^{°}$ to $E_{emb}^{*}$ .

For simplicity, we spread all embodied energy evenly over the lifetime of the device which gives a constant embodied energy rate ( ${\overset{\cdot}{E}}_{emb}$ ). Thus, we allocate embodied energy over the life of the original and upgraded devices ( $t_{life}^{°}$ and $t_{life}^{*}$ , respectively) without discounting to obtain embodied energy rates, such that ${\overset{\cdot}{E}}_{emb}^{°} = E_{emb}^{°} / t_{life}^{°}$ and ${\overset{\cdot}{E}}_{emb}^{*} = E_{emb}^{*} / t_{life}^{*}$ . The change in embodied final energy due to the EEU (expressed as a rate) is given by $Δ {\overset{\cdot}{E}}_{emb}^{*} = {\overset{\cdot}{E}}_{emb}^{*} - {\overset{\cdot}{E}}_{emb}^{°}$ . The expression for embodied energy rebound is

R e_{emb} = \frac{(\frac{E_{emb}^{*}}{E_{emb}^{°}} \frac{t_{life}^{°}}{t_{life}^{*}} - 1) {\overset{\cdot}{E}}_{emb}^{°}}{{\overset{\cdot}{S}}_{dev}} .

(14)

(See Appendix B.4.2 for details of the derivation.)

Embodied energy rebound ( $R e_{emb}$ ) can be either positive or negative, depending on the sign of the term $(E_{emb}^{*} / E_{emb}^{°}) (t_{life}^{°} / t_{life}^{*}) - 1$ . Rising energy efficiency can be associated with increased device complexity, additional energy consumption in manufacturing, and more embodied energy, such that $E_{emb}^{°} < E_{emb}^{*}$ and $R e_{emb} > 0$ , all other things being equal. However, if the upgraded device has longer life than the original device ( $t_{life}^{*} > t_{life}^{°}$ ), ${\overset{\cdot}{E}}_{emb}^{*} - {\overset{\cdot}{E}}_{emb}^{°}$ could be negative, meaning that the upgraded device has a lower embodied energy rate than the original device.

Operations, maintenance, and disposal effects ( $R e_{OMd}$ ). In addition to embodied energy, indirect emplacement effect rebound accounts for energy demanded by operations and maintenance (subscript $OM$ ) and disposal (subscript $d$ ) activities. Operations and maintenance expenditures are typically modeled as a per-year expense, a rate (e.g., ${\overset{\cdot}{C}}_{OM}^{*}$ ). On the other hand, disposal costs (e.g., $C_{d}^{°}$ ) are incurred at the end of the useful life of the energy conversion device (subscript $ω$ ). We annualize disposal costs (with discounting) across the lifetime of the original and upgraded devices ( $t_{life}^{°}$ and $t_{life}^{*}$ , respectively) to form discounted expenditure rates such that ${\overset{\cdot}{C}}_{OMd}^{°} = {\overset{\cdot}{C}}_{OM}^{°} + τ_{ω}^{°} {\overset{\cdot}{C}}_{d}^{°}$ and ${\overset{\cdot}{C}}_{OMd}^{*} = {\overset{\cdot}{C}}_{OM}^{*} + τ_{ω}^{*} {\overset{\cdot}{C}}_{d}^{*}$ .

For simplicity, we assume that operations, maintenance, and disposal expenditures imply energy consumption elsewhere in the economy at its overall energy intensity ( $I_{E}$ ). Therefore, the change in energy consumption rate caused by a change in maintenance and disposal expenditures is given by $Δ {\overset{\cdot}{C}}_{OMd}^{*} I_{E} = ({\overset{\cdot}{C}}_{OMd}^{*} - {\overset{\cdot}{C}}_{OMd}^{°}) I_{E}$ . Rebound from operations, maintenance, and disposal activities is given by

R e_{OMd} = \frac{(\frac{{\overset{\cdot}{C}}_{OMd}^{*}}{{\overset{\cdot}{C}}_{OMd}^{°}} - 1) {\overset{\cdot}{C}}_{OMd}^{°} I_{E}}{{\overset{\cdot}{S}}_{dev}} .

(15)

(See Appendix B.4.2 for details of the derivation.)

2.5.2. Substitution Effect

Neoclassical economic theory determines consumer behavior through utility maximization. It decomposes price-induced behavior change into (i) substituting energy service consumption for other goods consumption due to the lower post-EEU price of the energy service (the substitution effect) and (ii) spending of the higher real income (the income effect).¹⁴ This section develops mathematical expressions for substitution effect rebound ( $R e_{sub}$ ), thereby accepting the standard neoclassical microeconomic assumptions about consumer behavior.¹⁵ (The next section addresses income effect rebound, $R e_{inc}$ .) The substitution effect determines compensated demand, which is the demand for the expenditure-minimizing consumption bundle that maintains utility at the pre-EEU level, given the new prices. Compensated demand is a technical term for a thought experiment from welfare economics: the device user’s budget is altered so that the user is “compensated” for the change in price so as to maintain the same level of utility as before. In the case of an EEU, this implies the budget is reduced because the energy service price has fallen, so that it becomes cheaper to maintain a given level of utility. The change in the budget is called “compensating variation” (CV). The substitution effect involves (i) an increase in consumption of the energy service, the direct substitution effect (subscript $dsub$ ) and (ii) a decrease in consumption of other goods, the indirect substitution effect (subscript $isub$ ). Thus, two terms comprise substitution effect rebound: direct substitution rebound ( $R e_{dsub}$ ) and indirect substitution rebound ( $R e_{isub}$ ).

After emplacement of the more efficient device (but before the substitution effect), the price of the energy service decreases ( $p_{s}^{°} > p_{s}^{*}$ ). After compensating variation tightens the budget constraint, consumption at the new energy service price ( $p_{s}^{*}$ ) yields utility at the same level as prior to the EEU by consuming more of the now-lower-cost energy service (subscript $s$ ) and less of the now-relatively-more-expensive other goods (subscript $g$ ).

A constant price elasticity (CPE) utility model is often used in the literature (e.g., see Borenstein 2015, 17: footnote 43) for determining post-substitution effect consumption and therefore $R e_{dsub}$ and $R e_{isub}$ (see Appendix B.4.3). However, the CPE utility model can deliver only an approximation of the substitution effect for two reasons. First, because it is a reduced form model and only uncompensated elasticities are observed, the CPE utility model reports the sum of direct substitution effect and direct income effect rebound ( $R e_{dsub} + R e_{dinc}$ ). Second, price elasticities typically change as consumption bundles change, whereas the CPE price elasticity remains constant by definition. Typically, constant price elasticities (as in the CPE utility model) are approximations that are applicable only to marginal price changes. As shown in Heun et al. (2025), these approximations can lead to small or large errors depending on the case, relative to the exact model, which we introduce next. Appendix C derives changes in price elasticities for non-CPE models.

Here, we present a constant elasticity of substitution (CES) utility model that allows all of the uncompensated own price elasticity ( $ε_{{\overset{\cdot}{q}}_{s}, p_{s}}$ ), the uncompensated cross price elasticity ( $ε_{{\overset{\cdot}{q}}_{g}, p_{s}}$ ), the compensated own price elasticity ( $ε_{{\overset{\cdot}{q}}_{s}, p_{s}, c}$ ), and the compensated cross price elasticity ( $ε_{{\overset{\cdot}{q}}_{g}, p_{s}, c}$ ) to vary along an indifference curve, thereby enabling numerically precise analysis of non-marginal energy service price changes ( $p_{s}^{°} >> p_{s}^{*}$ ). The CES utility model allows the direct calculation of the utility-maximizing consumption bundle for any constraint, describing the device user’s behavior as

\frac{\overset{\cdot}{u}}{{\overset{\cdot}{u}}^{°}} = {[f_{{\overset{\cdot}{C}}_{s}}^{°} {(\frac{{\overset{\cdot}{q}}_{s}}{{\overset{\cdot}{q}}_{s}^{°}})}^{ρ} + (1 - f_{{\overset{\cdot}{C}}_{s}}^{°}) {(\frac{{\overset{\cdot}{C}}_{g}}{{\overset{\cdot}{C}}_{g}^{°}})}^{ρ}]}^{(1 / ρ)} .

(16)

The device user’s utility rate (relative to the original condition, ${\overset{\cdot}{u}}^{°}$ ) is determined by the consumption rate of the energy service ( ${\overset{\cdot}{q}}_{s}$ ) and the consumption rate of other goods and services ( ${\overset{\cdot}{C}}_{g}$ ). The share parameter ( $f_{{\overset{\cdot}{C}}_{s}}^{°}$ ) between ${\overset{\cdot}{q}}_{s}$ and ${\overset{\cdot}{C}}_{g}$ is taken from the original (pre-EEU) consumption basket. The exponent $ρ$ is calculated from the (constant) elasticity of substitution ( $σ$ ) as $ρ \equiv (σ - 1) / σ$ . All quantities are normalized to pre-EEU values so that the cost share of other goods can be used straightforwardly in empirical applications rather than having to construct quantity and price indices. The normalized specification is commonly used in empirical CES production function applications (Gechert et al. 2021; Klump, Mcadam and Willman 2012; Temple 2012). See Appendix C for further details of the CES utility model.

Direct substitution effect rebound ( $R e_{dsub}$ ) is

R e_{dsub} = \frac{Δ {\hat{\overset{\cdot}{E}}}_{s}}{{\overset{\cdot}{S}}_{dev}},

(17)

which can be rearranged to

R e_{dsub} = \frac{\frac{{\hat{\overset{\cdot}{q}}}_{s}}{{\overset{\cdot}{q}}_{s}^{°}} - 1}{\frac{\hat{η}}{η^{°}} - 1} .

(18)

Indirect substitution effect rebound ( $R e_{isub}$ ) is given by

R e_{isub} = \frac{Δ {\hat{\overset{\cdot}{C}}}_{g} I_{E}}{{\overset{\cdot}{S}}_{dev}},

(19)

which can be rearranged to

R e_{isub} = \frac{\frac{{\hat{\overset{\cdot}{C}}}_{g}}{{\overset{\cdot}{C}}_{g}^{°}} - 1}{\frac{\hat{η}}{η^{°}} - 1} \frac{\hat{η}}{η^{°}} \frac{{\overset{\cdot}{C}}_{g}^{°} I_{E}}{{\overset{\cdot}{E}}_{s}^{°}} .

(20)

To find the post-substitution effect point (∧), we solve for the location on the indifference curve where its slope is equal to the slope of the post-EEU expenditure line, assuming the CES utility model.¹⁶ The results are

\frac{{\hat{\overset{\cdot}{q}}}_{s}}{{\overset{\cdot}{q}}_{s}^{°}} = {f_{{\overset{\cdot}{C}}_{s}}^{°} + (1 - f_{{\overset{\cdot}{C}}_{s}}^{°}) {[(\frac{1 - f_{{\overset{\cdot}{C}}_{s}}^{°}}{f_{{\overset{\cdot}{C}}_{s}}^{°}}) \frac{p_{s}^{*} {\overset{\cdot}{q}}_{s}^{°}}{{\overset{\cdot}{C}}_{g}^{°}}]}^{ρ / (1 - ρ)}}^{- 1 / ρ}

(21)

and

\frac{{\hat{\overset{\cdot}{C}}}_{g}}{{\overset{\cdot}{C}}_{g}^{°}} = {(1 + f_{{\overset{\cdot}{C}}_{s}}^{°} {{[(\frac{1 - f_{{\overset{\cdot}{C}}_{s}}^{°}}{f_{{\overset{\cdot}{C}}_{s}}^{°}}) \frac{p_{s}^{*} {\overset{\cdot}{q}}_{s}^{°}}{{\overset{\cdot}{C}}_{g}^{°}}]}^{ρ / (ρ - 1)} - 1})}^{- 1 / ρ} .

(22)

Equation (21) can be substituted directly into equation (18) to obtain an expression for direct substitution rebound ( $R e_{dsub}$ ) via the CES utility model.

R e_{dsub} = \frac{{f_{{\overset{\cdot}{C}}_{s}}^{°} + (1 - f_{{\overset{\cdot}{C}}_{s}}^{°}) {[(\frac{1 - f_{{\overset{\cdot}{C}}_{s}}^{°}}{f_{{\overset{\cdot}{C}}_{s}}^{°}}) \frac{p_{s}^{*} {\overset{\cdot}{q}}_{s}^{°}}{{\overset{\cdot}{C}}_{g}^{°}}]}^{ρ / (1 - ρ)}}^{- 1 / ρ} - 1}{\frac{\hat{η}}{η^{°}} - 1}

(23)

Equation (22) can be substituted directly into equation (20) to obtain an expression for indirect substitution rebound ( $R e_{isub}$ ) via the CES utility model.

R e_{isub} = \frac{{(1 + f_{{\overset{\cdot}{C}}_{s}}^{°} {{[(\frac{1 - f_{{\overset{\cdot}{C}}_{s}}^{°}}{f_{{\overset{\cdot}{C}}_{s}}^{°}}) \frac{p_{s}^{*} {\overset{\cdot}{q}}_{s}^{°}}{{\overset{\cdot}{C}}_{g}^{°}}]}^{ρ / (ρ - 1)} - 1})}^{- 1 / ρ} - 1}{\frac{\hat{η}}{η^{°}} - 1} \frac{\hat{η}}{η^{°}} \frac{{\overset{\cdot}{C}}_{g}^{°} I_{E}}{{\overset{\cdot}{E}}_{s}^{°}}

(24)

(See Appendix B.4.3 for details of the derivations of equations (18), (20), and (21)–(24).)

2.5.3. Income Effect

The monetary income rate of the device user ( $\overset{\cdot}{M}$ ) remains unchanged across the rebound effects. Thanks to the energy service price decline, real income rises, and freed cash from the EEU is given as $\overset{\cdot}{G} = p_{E} {\overset{\cdot}{S}}_{dev}$ . (See equation (90) in Appendix B.3.) Emplacement effect adjustments and compensating variation modify freed cash to leave the device user with net savings ( $\hat{\overset{\cdot}{N}}$ ) from the EEU, as shown in equation (100) in Appendix B.3. (Derivations of expressions for freed cash from the emplacement effect ( $\overset{\cdot}{G}$ ) and net savings after the substitution effect ( $\hat{\overset{\cdot}{N}}$ ) are presented in Tables B.3 and B.4.) Rebound from the income effect quantifies the rate of additional energy demand that arises when the energy conversion device user spends net savings from the EEU.

Additional energy demand from the income effect is determined by several constraints. The income effect under utility maximization satisfies the budget constraint, so that net savings are zero after the income effect ( $\bar{\overset{\cdot}{N}} = 0$ ). (See Appendix D for a mathematical proof that the income preference equations below (equations (25) and (29)) satisfy the budget constraint.)

A second constraint is that net savings are spent completely on (i) additional consumption of the energy service ( ${\hat{\overset{\cdot}{q}}}_{s} < {\bar{\overset{\cdot}{q}}}_{s}$ ) and (ii) additional consumption of other goods ( ${\hat{\overset{\cdot}{q}}}_{g} < {\bar{\overset{\cdot}{q}}}_{g}$ ). The proportions in which income-effect spending is allocated depends on the utility model, which prescribes the income expansion path for consumption. Given post-EEU prices, maximized CES utility means spending in the same proportion on the energy service and other goods across the income effect, a property known as homotheticity. This constraint is satisfied by construction below, particularly via an effective income term ( $\hat{\overset{\cdot}{M}}'$ ).

However, this framework could accommodate non-homothetic preferences for spending across the income effect (turning the income expansion path into a more general curve instead of a line). Demand for certain energy services could satiate as consumers become more affluent, implying income elasticities of the energy service of less than one (Greening, Greene and Difiglio 2000). At the lower bound, the consumer spends all income after the substitution effect on other goods (subscript $g$ ) and none on the energy service (subscript $s$ ), choices that serve to reduce rebound due to typically lower energy intensity of other goods compared to the energy service.¹⁷

We next show expressions for direct and indirect income effect rebound.

Direct income effect ( $R e_{dinc}$ ). The income elasticity of energy service demand ( $ε_{{\overset{\cdot}{q}}_{s}, \overset{\cdot}{M}}$ ) quantifies the amount of net savings spent on more of the energy service ( ${\hat{\overset{\cdot}{q}}}_{s} < {\bar{\overset{\cdot}{q}}}_{s}$ ). (See Appendix C for additional information about elasticities.) Spending of net savings on additional energy service consumption leads to direct income effect rebound ( $R e_{dinc}$ ).

The ratio of rates of energy service consumed across the income effect is given by

\frac{{\bar{\overset{\cdot}{q}}}_{s}}{{\hat{\overset{\cdot}{q}}}_{s}} = {(1 + \frac{\hat{\overset{\cdot}{N}}}{\hat{\overset{\cdot}{M}}'})}^{ε_{{\overset{\cdot}{q}}_{s}, \overset{\cdot}{M}}} .

(25)

Under the CES utility model, homotheticity means that $ε_{{\overset{\cdot}{q}}_{s}, \overset{\cdot}{M}} = 1$ .

Effective income ( $\hat{\overset{\cdot}{M}}'$ ) is given by

\hat{\overset{\cdot}{M}}' \equiv \overset{\cdot}{M} - τ_{α}^{*} {\overset{\cdot}{C}}_{cap}^{*} - {\overset{\cdot}{C}}_{OMd}^{*} - \hat{\overset{\cdot}{N}} .

(26)

For the purposes of the income effect, effective income (equation (26)) adjusts original income ( ${\overset{\cdot}{M}}^{°}$ ) to account for sunk costs ( $τ_{α}^{*} {\overset{\cdot}{C}}_{cap}^{*}$ and ${\overset{\cdot}{C}}_{OMd}^{*}$ ) and net savings ( $\hat{\overset{\cdot}{N}}$ ).

Direct income rebound is defined as

R e_{dinc} \equiv \frac{Δ {\bar{\overset{\cdot}{E}}}_{s}}{{\overset{\cdot}{S}}_{dev}} .

(27)

(See Table B.5.) After substitution, rearranging, and canceling of terms (Appendix B.4.4), the expression for direct income rebound under the CES utility model is

R e_{dinc} = \frac{{(1 + \frac{\hat{\overset{\cdot}{N}}}{\hat{\overset{\cdot}{M}}'})}^{ε_{{\overset{\cdot}{q}}_{s}, \overset{\cdot}{M}}} - 1}{\frac{η^{*}}{η^{°}} - 1} {f_{{\overset{\cdot}{C}}_{s}}^{°} + (1 - f_{{\overset{\cdot}{C}}_{s}}^{°}) {[(\frac{1 - f_{{\overset{\cdot}{C}}_{s}}^{°}}{f_{{\overset{\cdot}{C}}_{s}}^{°}}) \frac{p_{s}^{*} {\overset{\cdot}{q}}_{s}^{°}}{{\overset{\cdot}{C}}_{g}^{°}}]}^{ρ / (1 - ρ)}}^{- 1 / ρ} .

(28)

If there are no net savings after the substitution effect ( $\hat{\overset{\cdot}{N}} = 0$ ), direct income effect rebound is zero ( $R e_{dinc} = 0$ ), as expected.¹⁸

Under a non-homothetic utility model, the bounding condition is satiated consumption of the energy service such that as the device owner becomes richer, none of the net income ( $\hat{\overset{\cdot}{N}}$ ) is spent on more of the energy service, and thus $R e_{dinc} = 0$ would occur.

Indirect income effect ( $R e_{iinc}$ ). Not all net savings ( $\hat{\overset{\cdot}{N}}$ ) are spent on more energy for the energy conversion device. The income elasticity of other goods demand ( $ε_{{\overset{\cdot}{q}}_{g}, \overset{\cdot}{M}}$ ) quantifies the amount of net savings spent on additional other goods ( ${\hat{\overset{\cdot}{q}}}_{s} < {\bar{\overset{\cdot}{q}}}_{s}$ ). Spending of net savings on additional other goods and services leads to indirect income effect rebound ( $R e_{iinc}$ ).

The ratio of rates of other goods consumed across the income effect is given by

\frac{{\bar{\overset{\cdot}{q}}}_{g}}{{\hat{\overset{\cdot}{q}}}_{g}} = {(1 + \frac{\hat{\overset{\cdot}{N}}}{\hat{\overset{\cdot}{M}}'})}^{ε_{{\overset{\cdot}{q}}_{g}, \overset{\cdot}{M}}} .

(29)

Under the assumption that prices of other goods are exogenous (see Appendix E), the ratio of rates of other goods consumption ( ${\bar{\overset{\cdot}{q}}}_{g} / {\hat{\overset{\cdot}{q}}}_{g}$ ) is equal to the ratio of rates of other goods expenditures ( ${\bar{\overset{\cdot}{C}}}_{g} / {\hat{\overset{\cdot}{C}}}_{g}$ ) such that

\frac{{\bar{\overset{\cdot}{C}}}_{g}}{{\hat{\overset{\cdot}{C}}}_{g}} = {(1 + \frac{\hat{\overset{\cdot}{N}}}{\hat{\overset{\cdot}{M}}'})}^{ε_{{\overset{\cdot}{q}}_{g}, \overset{\cdot}{M}}} .

(30)

Homotheticity means that $ε_{{\overset{\cdot}{q}}_{g}, \overset{\cdot}{M}} = 1$ . As shown in Table B.5, indirect income rebound is defined as

R e_{iinc} \equiv \frac{Δ {\bar{\overset{\cdot}{C}}}_{g} I_{E}}{{\overset{\cdot}{S}}_{dev}} .

(31)

After substitution, rearranging, and canceling of terms, the expression for indirect income rebound under the CES utility model is

R e_{iinc} = \frac{{(1 + \frac{\hat{\overset{\cdot}{N}}}{\hat{\overset{\cdot}{M}}'})}^{ε_{{\overset{\cdot}{q}}_{g}, \overset{\cdot}{M}}} - 1}{\frac{η^{*}}{η^{°}} - 1} (\frac{η^{*}}{η^{°}}) \frac{{\overset{\cdot}{C}}_{g}^{°} I_{E}}{{\overset{\cdot}{E}}_{s}^{°}} \times {(1 + f_{{\overset{\cdot}{C}}_{s}}^{°} {{[(\frac{1 - f_{{\overset{\cdot}{C}}_{s}}^{°}}{f_{{\overset{\cdot}{C}}_{s}}^{°}}) \frac{p_{s}^{*} {\overset{\cdot}{q}}_{s}^{°}}{{\overset{\cdot}{C}}_{g}^{°}}]}^{ρ / (ρ - 1)} - 1})}^{- 1 / ρ} .

(32)

(See Appendix B.4.4 for details of the derivation of direct and indirect income effect rebound.)

Under the bounding satiated utility model, all net income ( $\hat{\overset{\cdot}{N}}$ ) is spent on other goods, and indirect rebound becomes simply $R e_{iinc} = \frac{\hat{\overset{\cdot}{N}} I_{E}}{{\overset{\cdot}{S}}_{dev}}$ .

2.5.4. Macro Effect

The previous rebound effects (emplacement effect, substitution effect, and income effect) occur at the microeconomic level. However, changes at the microeconomic level can have important impacts at the macroeconomic or economy-wide level.

It is one of the basic tenets of economics that productivity gains have been the main long-run driver of economic growth in the last couple of centuries (Marx 1867; Smith 1776; Solow 1957). Interest in the impact of individual sectors on the whole economy reaches arguably even farther back (Quesnay 1759) and continues to the present (Leontief 1986). Recent work revived interest in firm- and sector-specific shocks on aggregate output and demonstrates that due to interlinkages between firms and sectors, productivity shocks in a firm or sector can have larger macroeconomic consequences than the original shock (Acemoglu et al. 2012; Baqaee and Farhi 2019; Gabaix 2011). Foerster et al. (2022) estimate that 3/4 of long-run U.S. growth since 1950 can be attributed to sector-specific (as opposed to aggregate) trend factors. Because the EEU represents a positive, sector-specific productivity shock, the same principles apply. These kinds of rebounds can be captured by a general equilibrium model (Stern 2020), but we propose a simple rule for incorporating this macroeconomic effect of productivity growth into our partial equilibrium framework.

Before establishing a formalism for $R e_{macro}$ , we clarify the link between consumer theory and economic growth. Turner (2013) cautions that when households see the productivity of their non-market activities increase, GDP remains unchanged.¹⁹ That may be true in the short run. But the question over longer periods is whether the more productive household energy services do not also feed through into economic growth accounted for by GDP. People in affluent countries spend about as much time on unpaid (i.e., non-market) work as on paid work (Folbre 2021). Therefore productivity improvements in unpaid work can spill over into paid work, which enters GDP. One channel could be time-saving. If the EEU saves time, then saved time could be spent on more paid work or on increasing human capital (Gautham and Folbre 2024; Sorrell and Dimitropoulos 2008). If the EEU saves money (but no time), then the freed cash could be spent to create additional demand for products that translate into higher GDP and possibly faster productivity growth (Magacho and McCombie 2018). The freed cash could also be spent on more effective (and more costly) human capital-increasing activities or even be used to start a venture. In all cases, it would be rash to conclude that just because some EEUs lead to productivity increases not captured directly by GDP, they do not eventually lead to additional economic growth.²⁰

Borenstein also addressed these macro effects from consumer behavior noting that “income effect rebound will be larger economy-wide than would be inferred from evaluating only the direct income gain from the end user’s transaction” (Borenstein 2015, 11) and likened it to a macroeconomic multiplier.²¹ The sectoral growth shock literature also uses multipliers to conceptualize the impacts of sectoral productivity shocks on aggregate output (Buera and Trachter 2024; Foerster et al. 2022). Using multipliers has the advantage that they can be directly linked to the income effect (minus compensating variation) and its consequence for macroeconomic rebound. Borenstein also notes that scaling from net savings ( ${\overset{\cdot}{N}}^{*}$ ) at the device level to productivity-driven growth at the macro level is unexplored territory.

We operationalize the macro rebound multiplier idea by noting that higher productivity makes the device cheaper to operate (and possibly purchase), which allows consumers to purchase a larger bundle of goods and services. If the overall expansion of the economy is a multiple of the direct increase in productivity expressed as productivity gains in other sectors, then the macro effect can simply be represented as a multiple of the (indirect) emplacement effect at the post-emplacement stage (*) of Figure 1, a multiplier that we represent by a macro factor ( $k$ ).²²

The macro factor ( $k$ ) represents respending in the broader economy after the emplacement effect has occurred and is not tied to any particular EEU or economic sector. $k \geq 0$ is expected. $k = 0$ means there is no macroeconomic effect resulting from the energy efficiency upgrade. $k > 0$ means that productivity-driven macroeconomic growth has occurred with consequent implications for additional energy consumption in the wider economy.

We assume as a first approximation (following Antal and van den Bergh 2014; Borenstein 2015) that macro effect responding implies energy consumption according to the average energy intensity of the economy ( $I_{E}$ ). Macro rebound is therefore given by

R e_{macro} = \frac{k {\overset{\cdot}{N}}^{*} I_{E}}{{\overset{\cdot}{S}}_{dev}} .

(33)

(See Table B.6.) After some algebra (Appendix B.4.5), we arrive at an expression for macro effect rebound:

R e_{macro} = k (p_{E} I_{E} - R e_{cap} - R e_{OMd}) .

(34)

Another macroeconomic rebound could arise from the energy price, which could fall due to lower demand (Borenstein 2015; Gillingham, Rapson and Wagner 2016). The size of the energy price effect depends on the size of the energy savings from the EEU relative to the energy demand in the economy. Therefore, calculating the energy price effect requires additional assumptions about how many households adopt the new device, which we consider to be outside the scope of our core framework. However, we show how it could be incorporated by adding an assumption about EEU adoption shares and a model of the energy market to derive a rebound expression for the energy price effect in Section 3.2 and Appendix F.

2.6. Rebound Sum

The sum of all rebound emerges from the four rebound effects (emplacement effect, substitution effect, income effect, and macro effect). Macro effect rebound ( $R e_{macro}$ in equation (34)) is expressed in terms of other rebound effects. (Derivation details can be found in Appendix B.4.6.) After algebra and canceling of terms, we find

\begin{matrix} R e_{tot} = R e_{emb} + k (p_{E} I_{E} - R e_{cap}) + (1 - k) R e_{OMd} + \\ R e_{dsub} + R e_{isub} + R e_{dinc} + R e_{iinc} . \end{matrix}

(35)

3. Discussion

3.1. Comparison to Other Rebound Frameworks

We developed above a rebound framework for consumers. We note that many of its components are similar to those for a producer-sided framework due to symmetries between neoclassical microeconomic producer and consumer theory. Ours is a partial equilibrium framework at the microeconomic level that provides a detailed assessment of individual EEUs with tractable, easy-to-understand mathematics. Partial equilibrium frameworks are easier to understand, in part, because they constrain price variation to the energy service only; all other prices remain constant (at least at the microeconomic level).²³ In our framework, general equilibrium effects and other dynamic effects at the macroeconomic level are captured by a simplified, one-dimensional rebound effect discussed in Section 2.5.4.

We are not the first to develop a rebound analysis framework, so it is worthwhile to compare our framework to others for key features: analysis of all rebound effects; analysis of energy, expenditure, and consumption aspects of rebound; level of detail in the consumer preference model; allowance for non-marginal energy efficiency changes; and empirical application. When all of the above characteristics are present, a fuller picture of rebound can emerge.²⁴ Table 2 shows our assessment of selected previous partial equilibrium frameworks (in columns) relative to the characteristics discussed above (in rows).

Table 2.

Comparison Among Relevant Rebound Analysis Frameworks.

Note. Empty (white) circles indicate no treatment of a subject by a framework. Partly and fully filled circles indicate partial and comprehensive treatment of a subject by a framework.

Because all frameworks evaluate the expected decrease in direct energy consumption from the EEU, the “Direct emplacement effect” row contains • in all columns. Three early papers (Nässén and Holmberg 2009; Thomas and Azevedo 2013a, 2013b) estimate rebound quantitatively, earning high marks (•) in the “Empirical application” row. Both Nässén and Holmberg and Thomas and Azevedo motivate their frameworks at least partially with microeconomic theory (consumer preferences and substitution and income effects) but use simple linear demand functions in their empirical analyses. Thus, the connection between economic theory and empirics is tenuous, leading to intermediate ratings (◒ or less) in the “substitution effects,”“income effects,” and “Detailed model of consumer preferences” rows. More recently, Chan and Gillingham (2015) and Wang, Yu and Liu (2021) anchor the rebound effect firmly in consumer theory, earning high ratings (•) in the “substitution effects,”“income effects,” and “Detailed model of consumer preferences” rows. They extend their frameworks to advanced topics that our framework does not presently incorporate, such as multiple fuels, energy services, and nested utility functions with intermediate inputs. However, neither Chan and Gillingham nor Wang et al. provide empirical applications, earning ○ in the last row of Table 2. In the middle of the table (and between the other studies in time), the framework by Borenstein (2015) touches on nearly all important characteristics. However, the Borenstein framework cannot separate substitution and income effects cleanly in empirical analysis, reverting to partial analyses of both, leading to a ◒ rating in the “Detailed model of consumer preferences” and “Empirical application” rows.

No previous framework engages fully with either the differential financial effects or the differential energetic effects of the upfront purchase of the upgraded device, leading to low ratings across all previous frameworks in the “Capital cost and embodied energy effect” row. In fact, except for Nässén and Holmberg (2009), no framework engages with capital costs, although all note its importance. (Nässén and Holmberg note that capital costs and embodied energy can have very strong effects on rebound.) Thomas and Azevedo (2013a, 2013b) provide the only framework that traces embodied energy effects of every consumer good using input-output methods, but they do not analyze embodied energy of the upgraded device. Borenstein (2015) notes the embodied energy of the upgraded device and the embodied energy of other goods but does not integrate embodied energy or financing costs into the framework for empirical analysis. Borenstein is, however, the only author to treat the financial side of embodied energy or maintenance and disposal effects. Borenstein (2015) postulates the macro effect, but does not operationalize the link between micro and macro levels, earning ◒ in the “Macro effect” row. No other framework even discusses the link between macro and micro rebound effects, leading to ° in the “Macro effect” row for all previous frameworks (apart from Borenstein 2015). Our framework operationalizes the link between micro and macro levels, via the macro factor ( $k$ ), but more work can be done in this area. Thus, “This paper (2025)” earns ◒ in the “Macro effect” row. Finally, all previous frameworks assume constant price elasticities and implicitly marginal or small improvements in efficiency, excluding the numerically precise analysis of important non-incremental upgrades where price elasticities are likely to vary. Therefore, all previous frameworks earn ○ in the “Non-marginal energy service price changes” row.

Table 2 shows that previous frameworks contain many key pieces, providing starting points from which to develop our rebound analysis framework. A left-to-right reading of the table demonstrates that previous frameworks start from microeconomic consumer theory and move towards more rigorous theoretical treatment over time, with recent frameworks making important advanced theoretical contributions at the expense of empirical applicability. In the end, no previous rebound analysis framework combines all rebound effects across energy, expenditure, and consumption aspects with a detailed model of consumer preferences, non-marginal energy service price changes, and empirical applicability for the simplest case (understandable across disciplines) of a single fuel and a single energy service. In particular, assessing the rebound implications of differential capital costs, non-marginal price changes, and the macro effect required conceptual development as in Section 2.5.4 and Appendix B.4.5. (Development of empirical applications is left for Heun et al. (2025).) This paper addresses most of the gaps in Table 2; hence we fill the “This paper (2025)” column with filled circles (•) in nearly all rows. By so doing, we help advance clarity in the field of energy rebound.

3.2. Notes on an Energy Price Rebound Effect

The income effect (Section 2.5.3) captures the energy and rebound implications of expanding real income at the level of the upgraded device. The partial equilibrium framework described herein enables calculation of income effect rebound ( $R e_{inc}$ ) without regard to changes in energy price ( $p_{E}$ ), because the energy price is assumed exogenous.

But there are other effects at work beyond the device level and outside the boundaries of a partial equilibrium analysis. One of those effects is an energy price effect. This section (and Appendix F) shows that our partial equilibrium framework can be extended to obtain an initial estimate of the rebound implications of an energy price effect ( $R e_{p_{E}}$ ) with an analysis that remains short of full equilibrium.

The energy price effect can lead to rebound when EEUs are applied to energy conversion devices at a scale that is substantial relative to the economy-wide use of energy. Examples of conditions under which the energy price effect could be significant include replacing all cars in the economy by hybrids and replacing all domestic electric lamps in the economy by LEDs, to use the examples from Heun et al. (2025). With reduced energy demand throughout the economy, an energy price reduction can be expected ( $p_{E}^{°} > {\bar{p}}_{E}$ ) as the lower energy price leads to rebalancing of supply and demand. With the now-lower energy price ( ${\bar{p}}_{E}$ ), the device owner has additional freed cash ( ${\overset{\cdot}{G}}_{p_{E}}$ ) to spend, in addition to the adjustments described by the substitution and income effects (see Sections 2.5.2 and 2.5.3).

A complete analysis of the price effect would amount to introducing a full model of the energy market and involve solving a system of simultaneous equations for the new economy-wide energy demand, the new energy price, and a new consumption bundle. But in this instance, as we desire a simple estimate of energy price rebound, we conservatively assume the device owner spends the additional freed cash (the result of the lower energy price) exclusively on other goods, with energy implications at the energy intensity of the economy ( $I_{E}$ ). Under these assumptions, Appendix F derives an expression for rebound from the energy price effect as

R e_{p_{E}} = \frac{{\overset{\cdot}{G}}_{p_{E}} I_{E}}{{\overset{\cdot}{S}}_{dev}},

(36)

where ${\overset{\cdot}{G}}_{p_{E}}$ is the freed cash arising from the reduction in energy price due to widespread adopotion of the EEU throughout the economy.

4. Conclusions

In this paper (Part I), we developed foundations of a rigorous analytical framework that includes all rebound effects across energy, expenditure, and consumption aspects with a detailed model of consumer preferences and non-marginal energy service price changes in an operational manner linking micro and macro effects for the simplest case of a single fuel and a single energy service. Furthermore, we presented approaches for exploring consumer satiation of energy service demand and for analyzing the effect of reduced energy demand on energy price to create energy price rebound. With careful explication of rebound effects and clear derivation of rebound expressions, we help advance the analytical foundations for empirical analyses and facilitate interdisciplinary understanding of rebound phenomena toward the goal of enhancing clarity in the field of energy rebound and enabling more robust rebound calculations for sound energy and climate policy.

Future work could be pursued in several areas. (i) Other utility models (besides the CES utility model, but not a Cobb-Douglas utility model) could be explored for the substitution effect. (ii) Although this is a consumer-sided framework, we demonstrated that it could be extended to effects affecting producers such as the energy price rebound effect. Further work could explore additional extensions to other producer-sided energy rebound effects. Moreover, a neoclassical producer framework, in analogy to the consumer framework, could be derived due to the substantial symmetry between neoclassical consumer and producer models (Fine 2016; Varian 1992). (iii) This framework could be extended to include some of the advanced topics in Chan and Gillingham (2015) and Wang, Yu and Liu (2021), such as multiple fuels or energy services, more than one other consumption good, and nested utility functions with intermediate inputs. (iv) This framework could be extended to include fuel-switching EEUs, wherein the upgraded device uses a different fuel from the original device. (v) The greenhouse gas emissions implications of energy rebound could be evaluated using this framework, provided that the primary energy associated with final energy purchases were available. Borenstein (2015) went some way to analyzing emissions and could provide a starting point for such work. The capability to analyze fuel-switching EEUs (discussed in the previous item) will be important for analyzing the greenhouse gas emissions implications of many EEUs that involve electrification, such as the transition to all-electric vehicles and the conversion of natural gas and oil furnaces to heat pumps for home heating.

In Heun et al. (2025), we further help advance clarity in rebound analysis in three ways. First, we develop a way to visualize the energy, expenditure, and consumption aspects of rebound effects. Second, we apply the framework to two EEUs: an upgraded car and an upgraded electric lamp. Finally, we provide results of rebound calculations for the two examples.

Footnotes

Appendices

Acknowledgements

The authors benefited from discussions with Daniele Girardi (University of Massachusetts Amherst) and Christopher Blackburn (Bureau of Economic Analysis). The authors are grateful for comments from internal reviewers Becky Haney and Jeremy Van Antwerp (Calvin University); Nathan Chan (University of Massachusetts Amherst); and Zeke Marshall (University of Leeds). The authors appreciate the many constructive comments on a working paper version of this article from Jeroen C. J. M. van den Bergh (Vrije Universiteit Amsterdam), Harry Saunders (Carnegie Institution for Science), and David Stern (Australian National University). Finally, the authors thank the students of MKH’s Fall 2019 Thermal Systems Design course (ENGR333) at Calvin University who studied energy rebound for many energy conversion devices using an early version of this framework.

The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.

Author Contributions

Matthew Kuperus Heun: Conceptualization, Methodology, Validation, Investigation, Resources, Writing—Original Draft, Writing—Review & Editing, Supervision, Project Administration. Gregor Semieniuk: Conceptualization, Methodology, Investigation, Resources, Writing—Original Draft, Writing—Review & Editing. Paul E. Brockway: Methodology, Validation, Resources, Writing—Review & Editing, Funding Acquisition.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Paul Brockway’s time was funded by the UK Research and Innovation (UKRI) Council, supported under EPSRC Fellowship award EP/R024251/1, and latterly by Leverhulme Trust Research Project Grant RPG-2024-357.

1

We prefer the term “energy analysts” over “engineers,” because “energy analysts” better describes the group of people engaged in “energy analysis.” For this paper, we define “energy analysis” to be the study of energy transformations from stocks to flows and wastes along society’s energy conversion chain for the purpose of generating energy services, economic activity, and human well-being.

2

Indeed, this is why the authors for these papers come from the disciplines of energy analysis (MKH, PEB) and economics (GS).

3

This objective may mean that some aspects of the development of the framework will seem obvious to energy analysts while other aspects will seem obvious to economists.

4

For example, Sorrell (2009) sets out five macroeconomic rebound effects: embodied energy effects, responding effects, output effects, energy market effects, and composition effects. (We place the embodied energy effect at the microeconomic level.) Santarius (2016) and Lange et al. (2021) introduce meso (i.e., sectoral) level rebound between the micro and macro levels. distinguishes 14 types of rebound, providing, perhaps, the greatest complexity.

5

Note that the vocabulary and mathematical notation for rebound effects is important; and Appendix A provide guides to elements used throughout this paper, including symbols, Greek letters, abbreviations, decorations, and subscripts. The notational elements can be mixed to provide a rich and expressive symbolic “language” for energy rebound. As the goal of this paper is to bridge disciplines, the nomenclature will necessarily have unfamiliar elements to each discipline involved.

6

Note that the takeback rate can be negative, indicating that the actual final energy savings rate is greater than the expected final energy savings rate, a condition called hyperconservation.

7

Conventionally, stages of the energy conversion chain are primary energy (e.g., coal, oil, natural gas, wind, and solar), final energy (e.g., electricity and refined petroleum), useful energy (e.g., heat, light, and mechanical drive), and energy services (e.g., transport, illumination, and space heating). See for an introduction to societal energy and exergy accounting.

8

Primary energy may be important when the upgraded device consumes a different final energy carrier compared to the original device, that is, when fuel-switching occurs ().

9

We discount money because interest changes the available amount of money over time. In contrast, we do not discount energy, because there is no temporal variation in the ability of energy to effect changes (via heat or work) in the physical world. We thank an anonymous reviewer for the insight that, in principle, the carbon content of energy could also be discounted if one assumes that near term emissions are worse than later emissions.

10

Relaxing the exogenous energy price assumption would require a general equilibrium model that is beyond the scope of this paper. However, see Section 3.2 where we discuss an energy price rebound effect as an extension of this framework.

11

Note that “pass” is short for “passenger,” and “lm” is the SI notation for the lumen, a unit of lighting energy rate.

12

Of course, it is often the case that the original and upgraded devices have small performance differences. For example, a high-efficiency LED lamp may have slightly greater or slightly lesser lumen output than the incandescent lamp it replaces. For the purpose of explicating this framework, we assume that the performance of the upgraded device can be matched closely enough to the performance of the original device such that the differences are immaterial to the user.

13

We take an energy approach here, consistent with the literature on energy rebound. One could use an alternative quantification of energy, such as exergy, the work potential of energy (Sciubba and Wall 2007) or emergy, the solar content of energy ().

14

For the original development of the decomposition see Slutsky (1915) and Allen (1936). For a modern introduction see .

15

Alternative assumptions on behavior would arise from, for example, adopting a behavioral economic framework (Dorner 2019; Dütschke et al. 2018) or an informational entropy-constrained economic framework ().

16

Other utility models could be used; however, the Cobb-Douglas utility model is inappropriate for this framework, because it assumes that the sum of substitution and income rebound is 100% always. Regardless of the utility model, expressions for ${\hat{\overset{\cdot}{q}}}_{s} / {\overset{\cdot}{q}}_{s}^{°}$ and ${\hat{\overset{\cdot}{C}}}_{g} / {\overset{\cdot}{C}}_{g}^{°}$ must be determined and substituted into equations (18) and (), respectively.

17

In principle, the energy service could be an “inferior good” whose consumption declines as incomes rise. However, energy service elasticities of income have been estimated to be positive over the long run, so we do not expect the inferior good case to be relevant ().

18

Zero net savings ( $\hat{\overset{\cdot}{N}} = 0$ ) could occur if increases in the capital cost rate ( $Δ {\overset{\cdot}{C}}_{cap}^{*}$ ) and/or the operations, maintenance, and disposal cost rate ( $Δ {\overset{\cdot}{C}}_{OMd}^{*}$ ) consume all freed cash ( $\overset{\cdot}{G}$ ) plus savings from the compensating variation.

19

To appreciate the difference between production for the market and production for the household, consider the example of increased fuel efficiency. In one case, the household upgrades its own car and takes more trips (direct rebound), without effect on GDP. In the other case, the household buys the energy service (transport) directly from a taxi company that upgrades its cars. Here, the taxi company lowers the price but gains more customers, leading immediately to growth in inflation-adjusted (i.e., real) GDP, as more driving services are produced. Yet, the physical change of more car trips is the same in both cases.

20

Nevertheless, as long as the energy efficiency improvement (in this example, an upgraded car) is the only technological progress in the economy, further output growth may be constrained to the extent that the other inputs into production remain constrained at their original levels and substituting energy for the other inputs to production is limited by the prevailing technology.

21

It is important to distinguish this multiplier from an autonomous expansion of expenditure, a demand-side shock, in an otherwise unchanged economy, that is, the Keynesian multiplier (Kahn 1931; Keynes 1936), that risks crowding out other economic activity (Gillingham, Rapson and Wagner 2016). Our energy productivity improvement is a supply-side shock. After the EEU, it takes less energy (and therefore less energy cost) to generate the same economic activity, because energy efficiency has improved, so the concept of crowding-out as defined by macroeconomics does not apply.

22

The macro factor ( $k$ ) appears unitless, but its units are actually $ of economy-wide expansion created per $ of net savings gained by the device user in the emplacement effect ( ${\overset{\cdot}{N}}^{*}$ ).

23

General equilibrium frameworks provide detail and precision on economy-wide price adjustments, but they give up specificity about individual device upgrades, make assumptions during calibration, and lose simplicity of exposition.

24

See Section 2.2 of for literal pictures of rebound in energy, expenditure, and consumption planes.

25

In principle, calculated arc elasticities could describe the relationship between price and quantity changes for any EEU by representing the percentage price and quantity changes between any two known consumption bundles (). However, we do not know the new consumption bundle and instead determine it with the CES utility function whose price elasticities vary along the indifference curve.

26

In the international trade literature, where the CES utility model is often used, the elasticity of substitution is also called the Armington elasticity ().

27

For $σ = 1$ , $m_{s} = 0$ , and the derivative is zero: the Cobb-Douglas special case.

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Energetic and Economic Aspects of Rebound,Part I: Foundations of a Rigorous Analytical Framework

Abstract

Keywords

1. Introduction

1.1. A Short History of Rebound

1.2. Absence of Solid Analytical Foundations

1.3. New Clarity Is Needed

1.4. Objective, Contributions, and Structure

2. Methods: Development of the Framework

2.1. Rebound Typology

2.2. Rebound Relationships

2.3. The Energy Conversion Device and Energy Efficiency Upgrade (EEU)

2.4. Typical Energy and Cost Relationships

2.5. Rebound Effects

2.5.1. Emplacement Effect

2.5.2. Substitution Effect

2.5.3. Income Effect

2.5.4. Macro Effect

2.6. Rebound Sum

3. Discussion

3.1. Comparison to Other Rebound Frameworks

3.2. Notes on an Energy Price Rebound Effect

4. Conclusions

Footnotes

Appendices

Acknowledgements

Author Contributions

Declaration of Conflicting Interests

Funding

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References