Environmental Policy Instruments for Investments in Backstop Technologies Under Present Bias - An Application to the Building Sector

Arnold, Fabian; Ashour Novirdoust, Amir; Theile, Philipp Article — Published Version Environmental Policy Instruments for Investments in Backstop Technologies Under Present Bias - An Application to the Building Sector Environmental and Resource Economics Provided in Cooperation with: Springer Nature Suggested Citation: Arnold, Fabian; Ashour Novirdoust, Amir; Theile, Philipp (2025) : Environmental Policy Instruments for Investments in Backstop Technologies Under Present Bias - An Application to the Building Sector, Environmental and Resource Economics, ISSN 1573-1502, Springer Netherlands, Dordrecht, Vol. 88, Iss. 4, pp. 1039-1070, https://doi.org/10.1007/s10640-025-00960-8 This Version is available at: https://hdl.handle.net/10419/323357 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Accepted: 9 January 2025 © The Author(s) 2025, corrected publication 2025 F. Arnold, A. Ashour Novirdoust and P. Theile are listed alphabetically by last name and contributed equally to this paper. The original online version of this article was revised due to a retrospective Open Access order. Philipp Theile [email protected] 1 Institute of Energy Economics at the University of Cologne, Vogelsanger Str. 321a, 50827 Cologne, Germany Environmental Policy Instruments for Investments in Backstop Technologies Under Present Bias - An Application to the Building Sector FabianArnold1· AmirAshour Novirdoust1· PhilippTheile1 Environmental and Resource Economics https://doi.org/10.1007/s10640-025-00960-8 Abstract Governments worldwide have set targets to reduce greenhouse gas emissions from the residential sector to zero or close to zero. Policy instruments, such as carbon pricing or subsidies, are being discussed and implemented to achieve these targets. If individuals exhibit present bias, Heutel (2015) has shown that optimal policies targeting investments in externality-producing durable goods consist of two components, one aimed at the externality and one aimed at the present bias. We generalize Heutel’s theoretical model by defining a larger technology set. This allows us to represent the dependence of fuel prices and emission intensities on technologies used to include a zero-emission backstop technology. We examine the implications of this model generalization, and we numerically assess the effect in a stylized case study for a representative building in Germany. We show that as long as social costs of carbon and the corresponding CO2 price are not high enough to make the backstop technology optimal, Heutel’s proposition holds that optimal policies must consist of two components. Generalizing Heutel’s proposition, a single instrument can address present bias, if the social costs of carbon and the CO2 price are high enough. While the level of this single instrument, i.e., a tax or subsidy, depends on the level of present bias, we find that there exists a tax-subsidy combination that is optimal regardless of the level of present bias. Keywords Present bias · Policy · Heating investments · Durable goods · Climate neutrality JEL Classification D15 · D62 · D91 · H23 · Q48 · Q58 1 3 F. Arnold et al. 1 Introduction 1.1 Background and Motivation Governments of many countries have set themselves climate targets, i.e., emission reduction targets. These targets no longer aim at a mere partial reduction of greenhouse gas (GHG) emissions, but rather at a reduction of GHG emissions to zero or close to zero. More than 70 countries pledged to reach net-zero emissions, including the countries of the European Union, China, and the USA (United Nations 2023). Investments must be stimulated and carried out beyond efficiency improvements to achieve these goals. Therefore, in all sectors investments in zero-emission technologies, i.e., backstop technologies, must be made.1 A backstop technology is a process or a technology in which the use of an exhaustible resource can be completely avoided. In this paper, we define backstop technology more narrowly: as a technology that does not emit CO2 during operation. We assume that such a technology exists at finite cost.2 An example is the residential building sector: Global GHG emissions from building operations, i.e., heating and hot water provision, have increased in recent years. In 2021 global direct CO2 emissions from building operations accounted for around 8 % of global energyrelated CO2 emissions (IEA 2022). In the residential building sector, decarbonization needs to be carried out by private households investing in new technologies (e.g., heating systems and refurbishment) and choosing their indoor temperature level. In this paper, the analysis is applied to the residential building sector, although the results are generalizable. The prominent policy from classic economics to reach the first-best outcome in the presence of an environmental externality (i.e., the emitted emissions) is to introduce a price on said externality (i.e., a carbon or CO2 price), internalizing the externality into the decisionmaking rationale of the households, like the Pigouvian tax (Pigou 1920). Empirical literature suggests that individuals do not always behave according to classic rational choice theory. Behavioral issues, such as time-inconsistent discounting (e.g., present bias), could prevent individuals from investing optimally in time. Heutel (2015) has shown that if consumers experience present bias, a Pigouvian tax does not lead to welfare optimal investment decisions for externality-producing durable goods. Instead, the optimal policy mix consists of an instrument to correct the externality and another one aiming at the present bias, constituting an internality. Besides carbon taxation or pricing, these instruments can include subsidies, taxes based on efficiency, or mandates. In his analysis, Heutel (2015) assumes that consumers can invest in technologies with different efficiencies. Sheer improvement of efficiencies in externality-producing durable goods, however, cannot reduce externalities to zero. Thus, by assumption, no backstop technology exists. The author finds that there is a welfare-optimal amount of externalities (i.e., GHG emissions) corresponding to the Pigouvian tax rate, which represents the monetary damage of the externality. The optimal level balances the damage from the externality with the utility derived from the externality-producing good. In contrast, in many countries, the declared political target is to achieve zero or close to zero emissions.3 The implicit 1 Alternatively or additionally, hard-to-avoid emissions can be offset by natural or technical carbon sinks. 2 More details in section 2. 3 In the following, we abstract from the possibility of carbon sinks to achieve net-zero targets and assume that the goal of the investments under investigation is to reduce emissions to zero. 1 3 Environmental Policy Instruments for Investments in Backstop… assumption when applying a zero-emission target is, that the marginal damage from GHG emissions is higher than corresponding marginal abatement costs, and correspondingly, the optimal amount of GHG emissions is zero. Put differently, the policy maker is interested in target-consistent CO2 pricing and policy measures rather than taxing the externality at the rate of social costs of carbon (Aldy et al. 2021). Building on the work of Heutel (2015), this raises the following questions. First: How can Heutel’s model be generalized to account for the existence of zero emission backstop technologies with finite costs? Second: What does this generalization imply for the main propositions of the model? Third: What are optimal policies under present bias for externality-producing durable goods if the optimal investment decision is the investment in the backstop technology? We generalize the analytical model of Heutel (2015) for investments in externality-producing durable goods under present bias by allowing for a greater technology space. In the generalized model, the investment may be accompanied by the substitution of the fuel used, for example, in the case of heating investments, switching from a gas heating system to an electric heat pump. The integration of fuel substitution into the investment decision allows us to depict the existence of a zero emissions backstop technology. We first examine the effect of the model generalization on Heutel’s main propositions, assuming still that there is a welfare-optimal inner solution, i.e., that the backstop technology is not optimal. We then discuss the implications of the situation when the investment in the backstop technology is optimal. This may be the case if the assumed damage of the externality is high enough so that the backstop technology is welfare-optimal, or due to politically set zero-emission targets. In a stylized case study for a representative building of the German building sector, we assume a politically set zero-emission target. We numerically estimate real-world magnitudes of the present bias effect on heating-related investment and utilization decisions, emissions, policies, and associated deadweight loss. In our analysis, we show that as long as social damage of carbon and the corresponding CO2 price is not high enough to make the backstop technology optimal, households in the optimum will still emit CO2 . In this case, Heutel’s propositions hold that to reach the social optimum, we need two policy instruments, one to address the internality and a second one to address the externality. Generalizing Heutel’s propositions, if the social costs of carbon and the corresponding CO2 price are high enough, a mark-up on the CO2 price can also induce the social optimum. Therefore, present bias can be addressed by a tax or another single instrument when aiming at zero emissions in the presence of a zero-emission backstop technology. In numerical simulations for a representative household in Germany and under the assumption of continuous investment choices, we quantify the target-consistent CO2 price for reaching zero-emissions without present bias at 192€/ tCO2 . Applying this target-consistent CO2 price in the case of present bias leads to a welfare loss. In the case of a present-biased household, a higher CO2 tax exists that reaches the target (in our exemplary building and an assumed present bias of 0.7: 235€/ tCO2 including an internality-mark-up of 43€/ tCO2 ). While the optimal tax rate and subsidy depend on the level of present bias, we find that there exists an optimal tax-subsidy combination that is optimal regardless of the level of present bias. 1 3 F. Arnold et al. 1.2 Related Literature and Contribution Ever since (Strotz 1955) introduced the idea of time-inconsistent discounting with his theory of commitment, it has been recognized that consumers may deviate from the assumption of exponential, thus time-consistent, discounting.4 In line with time-inconsistent discounting, Laibson (1997) coined the concept of present bias, i.e., agents’ preference for immediate benefits over advantages in future periods beyond exponential discounting.5 To represent this behavior, the literature has introduced and applied models of quasi-hyperbolic discounting (Phelps and Pollak 1968; Laibson 1997; O’Donoghue and Rabin 1999). One metric for policy evaluation is welfare. Assuming time-inconsistent preferences implies that preferences change over time, complicating welfare analysis. Economists provide several welfare criteria to overcome this complication. The two most prominent criteria are the Pareto criterion, i.e., considering each period’s perspective in overall utility, and the long-run criterion, i.e., evaluating the "true" utility from a long-run perspective (O’Donoghue and Rabin 2015). O’Donoghue and Rabin (1999) argue that the Pareto criterion is too strong an assumption when applied to intertemporal choice. O’Donoghue and Rabin (2015) claim that both approaches, as well as other thinkable welfare criteria, frequently yield the same conclusions but argue for the usage of the long-run criterion.6 As Heutel (2015) utilizes the long-run criterion in his model, we will also apply it. Applying the long-run criterion deviates from standard social welfare analysis, which relies on revealed preferences as information about the consumer’s true utility. The paternalistic assumption that the consumer’s choices do not optimize her welfare is as critical as it is controversial. Saint-Paul (2011) argues that taxes levied for inducing a particular behavior might only lead to consumers paying higher prices instead of changing behavior, reducing overall welfare. According to Whitman (2006), the justifications of policy interventions for addressing internalities are based on the idea of Pigouvian taxation, ignoring Coase’s theorem (Coase 1960). The theorem states that externalities can be resolved by negotiation between individual parties when transaction costs are low. Since internalities consist of choices within the individual, Whitman (2006) argues that Coase’s theorem is better suited for dealing with internalities. The information required to find the least costly option addressing the damage from time-inconsistent discounting is only available to the individual. Moreover, Krusell et al. (2002) argues that to tackle consumers’ time-inconsistent preferences, only an intervention by a time-consistent social planner is welfare enhancing. Time-consistency of social planners could partly be achieved by avoiding short-term political pressure by establishing credible rules and institutions, enforcing commitment. Examples of such institutions are independent central banks, fiscal rules, and social security systems with automatic adjustments based on demographic or economic changes. We apply our analysis to the case of households’ heating system investment decisions. The empirical literature regarding behavioral biases in energy efficiency decision-making 4 Frederick et al. (2002) includes a critical review of the history and models of time discounting including time-consistent utility discounting models as well as time preferences and (quasi-)hyperbolic discounting models. 5 See the reviews Frederick et al. (2002) and DellaVigna (2009) for empirical estimates for present bias in various circumstances and Imai et al. (2021) and Cheung et al. (2021) for recent meta studies of papers reporting present bias estimates. 6 Kang (2015) shows that improvements in the Pareto criterion are also welfare-improving from the long-run perspective. 1 3 Environmental Policy Instruments for Investments in Backstop… is limited (Gillingham et al. 2009). Schleich et al. (2019) investigated the role of present bias and other behavioral aspects in adopting energy-efficient technologies within different countries in the European Union. They provide evidence for the significance of present bias in reducing investments in energy-efficient appliances and building retrofitting. Werthschulte and Löschel (2021) find that present bias increases power consumption. Therefore, as households undervalue energy costs, price-based policies might fail to reduce household energy consumption. Furthermore, in the specific case of investment in household appliances in India, Fuerst and Singh (2018) find that present bias becomes more significant the larger the purchase object investigated. This finding is relevant to our work, as heating system replacement represents a particularly large investment decision for households. Overall, there is not yet a comprehensive empirical view on the effect of present bias on heating system investments. We account for this lack of estimates by considering a range of present bias factors in our numerical simulation. This paper focuses on the consequence of present bias in agents’ decision-making on policies for decarbonization. The model from Heutel (2015) constitutes the basis of our analysis. A detailed description of the model for analyzing optimal policy instruments for externality-producing durable goods under present bias can be found in Section 2.1. Heutel (2015) considers a technology space with efficiency and investment costs as dimensions. We expand this space by allowing technologies to differ in emission intensity and fuel price. As we will see, this generalization enables us to discuss the subject of zero-emission backstop technologies. Other researchers have also addressed the question of how to design policy with externalities and internalities such as present bias (Alcott et al. 2012; Allcott and Sunstein 2015). Allcott and Sunstein (2015) discuss principles for regulating internalities in the field of energy. They find that internalities such as present bias can justify government intervention, given that "true preferences" of individuals can be identified in contrast to revealed preferences. Alcott et al. (2012) find that when households undervalue long-term energy costs in their investment decisions for durable goods, an externality tax such as a GHG tax yields a double dividend, since it also addresses the internality. They also find, that optimal policy mixes addressing both externalities and internalities depend on unknown information about levels of internalities, private to households. Our results deviate from this finding due to the presence of a zero-emission backstop technology. Since Heutel (2015), recent work has deepened the understanding of present bias in economic policy design and welfare analysis (Drugeon and Wigniolle 2021; Kotsogiannis and Schwager 2022; Kang 2022; Bar-Gill and Hayashi 2021; Lades et al. 2021; Chan and Globus-Harris 2023). Bar-Gill and Hayashi (2021) discuss the investment decisions for durable goods by present-biased agents. In contrast to our work, they focus on the effect of purchase financing. They find countervailing effects of present bias on the valuing of the benefits of an investment and the costs of financing said investment and derive recommendations for credit regulation. Since they discuss general durable goods, they do not consider the emission externalities from using energy technologies. Lades et al. (2021) examine investments from present-biased households in energy efficiency technologies. They illustrate particularly how administrative burden can reduce these investments. Similar to our work, they apply a theoretical model and a simulation with exemplary building data. Chan and Globus-Harris (2023) discuss incentivization of energy-efficient appliances such as air conditioners and refrigerators. They find that efficiency incentives on their own such as subsidies do not directly address externalities and thus distort consumer decision-making. They show that under certain circumstances, 1 3 F. Arnold et al. efficiency subsides may lead to more energy use overall due to the rebound effect. As we will see, our key point of departure from Heutel (2015), Lades et al. (2021), and Chan and Globus-Harris (2023) is that we consider policies reaching zero emissions combined with the availability of a backstop technology. While there is literature on policies in the context of present bias, to the best of our knowledge, there is no literature addressing the subject of policies for externality-producing durable goods aiming at zero emissions. In the present work, we aim to close this gap by (i) generalizing the model from Heutel to more complex technologies also differing in emission intensity and fuel price to be able to account for zero emission backstop technologies, (ii) analyzing the consequences of the existence of an optimal backstop technology, and (iii) illustrating the consequence of such policies in the residential building sector numerically. 2 Analytical Model In this Section, we first describe the representative agent model for investments in externality-producing durable goods under present bias from Heutel (2015). Then we generalize the model and apply it to the building sector. By defining a larger technology set, we are able to represent technologies running on different fuels and thus zero emission backstop technologies. Based on the generalized model, we discuss two different cases: First, the case that the backstop technology is not optimal. Second, the case that the backstop technology is the optimal technology choice. 2.1 A Representative Agent Model for Investments in Externality-producing Durable Goods Under Present Bias Heutel (2015) describes the investment and operation problem for externality-producing durable goods under present bias in a representative agent model. We present the model based on nomenclature for residential heating. The investment decision is made in the initial period ( t=0 ) and the good lasts T periods. In each period after the investment ( t=1 through t=T ), the household decides on the operating intensity of the good: the generated heat or indoor temperature. The model is defined by the household’s problem and the social planner’s problem. In the household’s problem, future utility and costs are discounted using quasi-hyperbolic discounting. Quasi-hyperbolic discounting is a method for modeling the behavior of households who experience present bias, i.e., prefer immediate payoffs and undervalue future costs and payoffs.7 To this end, two discount factors are introduced. δ is called the “longrun” discount factor, and β represents the “present bias”. If a household experiences present bias, then β<1 . The present-biased household perspective is contrasted with the social planner’s problem. Present bias is a behavioral anomaly that a social planner does not experience due to fully rational behavior. One way to solve the social planner’s optimization problem is to directly apply the long-run criterion while disregarding the household’s present bias, i.e., setting β=1 . The approach assumes that the household’s utility maximization deviates 7 Technically speaking, present-biased households discount utility and costs in the near future at a higher implicit discount rate than in the distant future (Laibson 1997). 1 3 Environmental Policy Instruments for Investments in Backstop… from optimal welfare even from the household’s perspective. Thus, the household "makes a mistake" and does not optimize its "true utility".8 In the initial period, the household chooses the heating system’s ratio of fuel input and generated heat, the so-called effort coefficient fph (fuel per heat), representing the investment decision for the durable good. In the subsequent periods, the heat generated in each period ht (t) is chosen, which translates into indoor temperature. U(ht) , where U′>0 and U′′ <0 , describes the utility from generated heat in monetary terms. The costs per kWh of fuel are calculated as the sum of the time-dependent fuel cost ( pt ) and a tax per kWh of fuel ( τt ). This fuel tax, hereinafter referred to as carbon tax, is intended to put a price on the GHG emissions. When choosing a level of fph, the household faces investment costs of c(fph). It is assumed that c′<0 , meaning that less efficient goods (heating systems) are less expensive, and c′′ >0 . The household’s problem is thus described in Equation (1): max fph, { ht } T t=1 −c(fph)+β· [ T ∑ t=1 δt· [ U(ht)−[pt+τt]·fph ·ht ]] (1) The social planner’s problem is characterized by including the externality of fuel consumption. The external damage from fuel consumption, i.e., damage from GHG emissions, depends on ht , the kWh of fuel used in each period t, times fph, the fuel used for producing the heat. The damage is denoted as d(ht·fph) , where d(0) = 0 , d′>0 and d′′ =0 . The corresponding social planner’s problem, using the long-run criterion for discounting and including external damages, is described in Equation (2): max fph, { ht } T t=1 −c(fph)+ T ∑ t=1 δt·[U(ht)−pt·fph ·ht−d(ht·fph)] (2) 2.2 Model Generalization In the model described in the previous section, consumers invest in one technology and can decide on its efficiency. By assumption, no backstop technology exists because efficiency improvements cannot reduce externalities to zero, and fuel cost differences between technologies used are neglected. We extend the technology set by allowing technologies running on different fuels. Therefore the investment decision affects fuel costs and GHG emissions per unit of generated heat. This enables us to analyze how optimal investment decisions depend on fuel cost ratios and to include a zero emission backstop technology. An example for a zero emission backstop technology in the building sector would be the switch to renewably generated heat from solar thermal energy or to electric heating powered by renewably generated electricity. 8 Alternative welfare criteria in the case of time-inconsistent discounting include the Paretian approach (e.g. Bhattacharya and Lakdawalla (2004)), or the "dictatorship of the present" approach discussed in Gruber and Köszegi (2004) or Laibson (1997), which prioritizes the preferences of the current self over the preferences of all future selves. Analogous to the approach in Heutel’s basic model and following the arguments of O’Donoghue and Rabin (1999), we apply the long-run criterion. 1 3 F. Arnold et al. By allowing technologies to vary in fuel price and emission intensity (down to zero), we extend the technology set and generalize the model. In this generalized model, the fuel price pt(fph) and the CO2 factor of the heating system epf(fph) are represented as functions of the effort coefficient fph.9 The functional form of pt(fph) is ambiguous: it is conceivable that the change to a more efficient heating system, e.g., from a gas boiler to an electric heat pump, is accompanied by decreasing fuel prices, in € per kWhfuel , but also that the fuel price increases, if, for example, electricity is more expensive than gas.10 We incorporate a backstop technology with finite costs fphBS by assuming that the emission function epf(fph) equals zero for all fph <=fphBS , and epf′>0 for fph >=fphBS . This means that when investing in the reduction of fph, epf(fph) decreases linearly until the backstop technology fphBS is reached, where emission intensity is zero. Further investments in reducing fph cannot further reduce the emission intensity. The household’s problem, including quasi-hyperbolic discounting as defined in Section 2.1, is thus described as follows11: max fph, { ht } T t=1 −c(fph)+β· [ T ∑ t=1 δt· [ U(ht)−[pt(fph)+epf(fph)·τt]·fph ·ht ]] (3) The household’s problem differs from Heutel (2015), since the investment decision fph depends on pt and the newly introduced CO2 factor epf. This yields first-order conditions for fph and each ht . Assume that there exists a unique interior solution.12 The solutions to the household’s problem are called fph∗ and h∗ t . −c ′( fph ∗) −β· T ∑ t=1 δt·h∗ t·[pt(fph∗)+epf(fph∗)·τt] −β· T ∑ t=1 δt·h∗ t· [ [p′ t(fph∗)+epf′(fph∗)·τt]·fph∗ ]=0 (4) U′( h ∗ t)−[ p t( fph ∗)+ epf ( fph ∗) τ t]· fph ∗=0 , ∀ t (5) In Eq. 4, considering the negative sign, the first term −c′(fph∗) is positive. The term represents the benefit of a marginal increase in fph. Since c′>0 , it is cheaper to choose a system with higher fph and hence, lower efficiency. Similar to Heutel (2015), the first sum represents the discounted cost of a marginal increase in fph due to the decrease in efficiency: the utility in each future period decreases as heating costs increase. The second sum adds the changes in fuel prices p′ t(fph) and changes in emission costs epf′(fph)·τt . While epf′ t is 9 We model the choice of fph, the investment costs, the change in fuel price, and CO2 factor as continuous. This serves the theoretical tractability of the model. 10 By including pt(fph) as a continuous function we do not consider explicitly the case of a backstoptechnology without fuel costs ( pt(fph) equals zero for all fph <=fphBS ). An example for that could be self-sufficiency using solar energy. The results of our analysis apply for that case as well. 11 Appendix A shows the isocost curves of the household’s decision problem for illustration. 12 It is assumed that lim ht→ 0 U′(h t )=∞ to ensure a unique interior solution. 1 3 Environmental Policy Instruments for Investments in Backstop… real heat demand (Mertesacker 2021; Loga et al. 2012). To account for this overprediction, we assume an adaptation factor of 0.8 for our representative building, based on Loga et al. (2012) and IWU (2016). The choice of this factor is consistent with the results of Mertesacker (2021), who estimates lower adaptation factors, but does not take into account domestic hot water generation. The heat demand in our model h [kWh] is then approximated as a linear function, depending on the chosen temperature T [ ◦ C]: h =0.8·147.1m2· ( 11.61 kWh m2 ·◦ C·T−19.85 kWh m2 ) (12) To evaluate a continuous investment choice of households, we estimate functions for investment costs, CO2 emissions, and fuel prices based on real data (Danish Energy Agency 2021; Pickert et al. 2022; BAFA 2021). The heating technologies include oil and gas condensing boilers with and without solar thermal support and an air-source heat pump. As in the theoretical model, the functions are formulated concerning the heating technology’s energy intensity level fph. We assume a system lifetime and an assessment period of 20 years. Table 1 shows the resulting technology functions.22 It should be noted that fuel prices incorporate consumer taxes, including value-added tax, as well as electricity and gas taxes. Additionally, electricity prices include the costs of emission certificates derived from the European Emission Trading System. When interpreting the numerical results, one should keep in mind that these intricacies introduce distortions in optimal policy instruments and deadweight loss.23 3.2 Results 3.2.1 Continuous Model From the negative gradient of the fuel price in Table 1 follows that p′(fph)<0 . Considering Proposition 2 from Sect. 2, p′(fph)<0 means that a CO2 price (or a subsidy) has to 22 Appendix C.3 presents the underlying data and the computation of the technology functions. 23 One distortion, for example, is that since the electricity sector in Germany has not been fully decarbonised, electricity prices today do not reflect the price of zero-emission electricity. These could be higher, and the results would change accordingly. Table 1 Estimated continuous functions of investment costs, CO2 emissions, and variable costs Unit Function Data Investment costs € c(fph)=14,100e−1.019·f ph Fitted function illustrated in the left plot in Figure 10. Underlying fph data from column 3 and costs from columns 4 and 5 in Table 3. CO2 emissions kg/kWh epf(fph)=−0.110 + 0.358 ·fph Fitted function illustrated in the middle plot in Figure 10. Underlying fph data from column 3 in Table 3 and emissions derived from column 3 of Table 4. Fuel price € /kWh p(fph)=0.394 −0.331 ·fph Fitted function illustrated in the right plot in Figure 10. Underlying fph data from column 3 in Table 3 and prices derived from column 2 of Table 4. 1 3 F. Arnold et al. at least offset the decrease of the heating system’s variable fuel price induced by increasing fph. For such a CO2 price, present bias leads to under-investment and, consequently, under-consumption of thermal energy. The numerical results replicate this finding on the relationship between present bias, investment, and consumption choices as illustrated in Fig. 2. In case of no present bias, β=1.0 , there is no investment up to a CO2 price of 137€/ tCO2 . The chosen indoor temperature at this price is 17.8°C. With an increasing CO2 price, the investments in lower fph increase. As heating costs decrease, the indoor temperature increases, which is commonly referred to as the rebound effect. At a CO2 price of 192€/ tCO2 , the household invests in zero-emission heating technology and reaches the corresponding indoor temperature of 18.2°C. As described in Sect. 3.1.1, we interpret this carbon tax rate τneu t as the implied emission damage to evaluate social welfare and consequently deadweight loss. In the presence of present bias, the household invests less and chooses a lower indoor temperature. Figure 3 shows that the total CO2 emissions over the 20 years of heating system lifetime follow the household’s investment and consumption choices. As discussed in Sect. 2.3.1, the investment in lower fph impacts emissions more than decreasing indoor temperature. In case of β=0.7 , emissions decrease from 81 tCO2 to 75 tCO2 for a CO2 price increase Fig. 3 Total emissions and deadweight loss over the heating system’s lifetime of 20 years depending on the CO2 price for present biases of 1.0, 0.9, 0.8, and 0.7 Fig. 2 The chosen fph and indoor temperature levels depending on the CO2 price for present biases of 1.0, 0.9, 0.8, and 0.7 1 3 Environmental Policy Instruments for Investments in Backstop… from 0€/ tCO2 to 157€/ tCO2 , due to the decrease in temperature. The emission decline turns more significant once the investments in lower fph start at 158€/ tCO2 . At an emission price of 235€/ tCO2 , the household invests in the zero-emission technology so that total CO2 emissions are 0. The deadweight loss over the heating system’s lifetime of 20 years is illustrated in Fig. 3. It is based on the two-step procedure described in Sect. 3.1.1 and follows the chosen fph level. The deadweight loss constitutes the difference to the case of an investment in the zero-emission technology in investment costs, heating costs, gained utility from indoor temperature, and emission damage. Whereby the emission damage is calculated applying the minimal target-consistent carbon tax rate inducing investments suitable for zero-emission goals. The resulting deadweight loss is mainly driven by the emission damage reduced by benefits through lower investment and heating costs. Without a CO2 price, the household invests in the option with the highest fph, leading to a deadweight loss above 3,000€ due to the emissions. With an increasing CO2 price, the indoor temperature first decreases slighty and with it consequently the emissions. Once the household invests in lower fph, the deadweight loss decreases convexly. Without present bias, the carbon tax rate τneu t of 192€/ tCO2 is sufficient to incentivize investment in the zero-emission technology. As Proposition 2 in Sect. 2.3.1 suggests, present bias leads to a deadweight loss caused by under-investment and, consequently, under-consumption. For β=0.9 , β=0.8 and β=0.7 the deadweight loss at τneu t is 58€ , 252€ , and 613€ , respectively. The loss results from the present bias internality as the τneu t addresses the emission externality. In Sect. 2.3.2, we argue that under a zero-emission target regime, a mark-up on top of the CO2 price which addresses the externality can address the internality and reach the zero-emission technology. The required mark-ups in the case study for β=0.9 , β=0.8 and β=0.7 are 11€/ tCO2 , 25€/ tCO2 , and 43€/ tCO2 , respectively. According to Proposition 3 in Sect. 2.3.1, a subsidy is an alternative to a mark-up on the carbon tax. If the subsidy is high enough to induce investments in heat pumps, no CO2 price is needed since subsequent heating does not emit CO2 . Consequently, a negative relationship exists between the two policies. All policy combinations that lead to the social optimum are illustrated in Fig. 4. The function’s slope describing the relationship between policies depends on the level of present bias and is lower for a high present bias. The slope differences originate from the differing times at which subsidies and CO2 prices affect the household. Present bias hinders households from fully considering the CO2 price in their optimal choice problem. Subsidies take effect directly at the time of the investment. The higher the level of present bias, the more the CO2 price must increase to reduce the required subsidy. At a CO2 price of 89€/t and a subsidy of 8,000€ , there is an intersection of the functions for the different levels of present biases. Thus, at this combination of CO2 price and subsidy, the required policy for the social optimum is independent of the level of present bias. The policy combination’s CO2 price creates parity between the variable costs of all technology options. In other words, the variable costs become independent of the chosen fph. We identify this intersection in Proposition 2 in Sect. 2.3.1 by stating that present bias leads to under-investment and, consequently, under-consumption as long as total future discounted heating costs, including fuel and emission costs, for a marginal increase in fph are greater or equal to zero. If this is not the case, i.e., less efficient heating systems have lower future heating costs, present bias will lead to over-investment. At the intersection between both cases, when future discounted heating costs are equal for all fph, the investment costs 1 3 F. Arnold et al. determine the investment choice. As present bias affects the household’s weighting between marginal changes in investment costs and marginal changes in total future discounted costs, it does not affect the household’s decision for equal future discounted costs. 89€ /t is the CO2 price, which offsets the differences in the fuel costs. In this case, the subsidy must compensate households for the difference in investment costs between CO2 -emitting and zeroemission technologies. This subsidy is 8,000€ . As a result, the policy mix at the intersection of the functions is optimal, independent of the level of present bias.24 The utility function of households is a critical assumption. As shown in Appendix C.1, we identify different valuation levels for indoor temperature. Figure 5 shows the fph level and chosen indoor temperature over CO2 price for three different valuation factors given a 24 The values of the optimal policy mix depend on the assumptions fed into the model, like fuel prices, heating efficiencies, and the utility function. Fig. 5 The chosen fph and indoor temperature levels depending on the CO2 price for valuation factors of 15€/ ∆T2 , 25€/ ∆T2 , and 35€/ ∆T2 Fig. 4 Combinations of CO2 price and subsidy that lead to the social optimum for present biases of 1.0, 0.9, 0.8, and 0.7 1 3 Environmental Policy Instruments for Investments in Backstop… present-bias of β=0.8 . A higher valuation factor implies a lower necessary CO2 price to incentivize investments in fph. In the case of a low valuation factor, the household reacts first with decreasing indoor temperature, as this yields lower utility loss compared to the additional costs of investing, as is shown in the right part of Fig. 5. As soon as investments in more efficient technologies are profitable, efficiency increases, and the household increases the indoor temperature until the installation of the zero-emission backstop technology. The CO2 emissions and deadweight losses over the heating system’s lifetime of 20 years illustrated in Fig. 6 show a slight decline until the start of investments in lower fph followed by a convex decline until the investment into the zero-emission technology. Before investments in more efficient technologies start, the deadweight loss is the highest for the high valuation factor since the under-consumption of indoor temperature weighs the most. The same logic also applies to why households with a high valuation factor start investing in more efficient heating systems at lower CO2 prices than households with lower valuation factors. As a higher level of investments decreases the deadweight loss not only through increased efficiency but also through reduced fuel costs, they exhibit a quadratic effect on the deadweight loss. Thus, the decline in welfare is less significant for lower valuation households, which still react by decreasing indoor temperature. 3.2.2 Discrete Model So far, the presented theoretical and numerical results assume continuous technology options so that all fph levels are feasible between the zero-emission and the least efficient option. In reality, there is only a limited set of heating technologies. Figure 7 illustrates the household’s investment and consumption choices, given a discrete technology set, including an oil condensing boiler, a gas condensing boiler, both boiler combined with solar thermal, and an air-to-water heat pump. We define the set of technologies as the available fph levels from Sect. 3.1 and choose the cost and emission levels according to the functions from the continuous model (see Appendix C.3). For each present bias level, there are four break-even CO2 prices that lead to a technology switch. In case of no present bias, i.e., β=1.0 , the household invests in the highest fph of 1.09, i.e., the oil condensing boiler, until a CO2 price of 139€/ tCO2 . For higher prices, the household chooses a fph of 1.02, i.e., the gas condensing boiler. The break-even points Fig. 6 Total emissions and the deadweight loss over the heating system’s lifetime of 20 years depending on the CO2 price for valuation factors of 15€ / ∆T2 , 25€ / ∆T2 , and 35€ / ∆T2 and a present bias of 0.8 1 3 F. Arnold et al. for investing in the oil condensing boiler combined with solar thermal and the gas condensing boiler combined with solar thermal are at 145€/ tCO2 and 150€/ tCO2 respectively. At a CO2 price of 170€/ tCO2 , the household invests in the heat pump. Thus, in the discrete case 170€/ tCO2 is the carbon tax rate τneu t that induces investment in the zero-emission technology. The τneu t is lower in the discrete case than in the continuous case because the CO2 price only has to create a break-even between the gas condensing boiler with solar thermal and the heat pump, and not between an infinitesimal less efficient heating technology and the zero-emission backstop technology. Analogously to the continuous case, present bias leads to under-investment and under-consumption. The total CO2 emissions over the heating system’s lifetime of 20 years in Fig. 8 mirror the step function of fph. There is a nearly linear decrease in CO2 emissions following the household’s temperature decreases and a more significant step whenever the CO2 price causes a switch between two heating technologies. At the carbon tax rate τneu t , there are zero CO2 emissions in case of no present bias. The under-investment, due to present bias, leads to CO2 emissions increases. These increases are for present biases of β=0.9 , β=0.8 , and β=0.7 , 49 tCO2 , 49 tCO2 , and 54 tCO2 . This step is significantly higher than in the continuous case as the next available technology is a gas condensing boiler with solar thermal Fig. 8 The total emissions and deadweight loss over the heating system’s lifetime of 20 years in case of discrete technology options depending on the CO2 price for present biases of 1.0, 0.9, 0.8, and 0.7 Fig. 7 The chosen fph and indoor temperature levels in case of discrete technology options depending on the CO2 price for present biases of 1.0, 0.9, 0.8, and 0.7 1 3 Environmental Policy Instruments for Investments in Backstop… compared to a technology with infinitesimal higher emission intensity. A mark-up on the CO2 price can address the internality and incentivize investment in the heat pump as stated in Sect. 2.3.2. For β=0.9 , β=0.8 , and β=0.7 , the mark-up is 10€/ tCO2 , 21€/ tCO2 , and 36€/ tCO2 , respectively. Following the two-step procedure described in Sect. 3.1.1, the implied damage from CO2 emissions τneu t is lower than in the continuous case, naturally resulting in a lower total level of deadweight loss. The deadweight loss due to present bias from under-investment and under-consumption is 101€ for a present bias of β=0.7 . The household chooses the oil condensing boiler with solar thermal instead of a heat pump. For present biases of β=0.9 and β=0.8 the household chooses a gas condensing boiler with solar thermal, which leads to neglectable deadweight loss since τneu t is defined as the necessarily implied damage to break-even between the two heating technologies. 4 Discussion In our stylized model, we find that single-instrument policies can be welfare optimal and target-consistent even if the household is present biased. We implicitly assume that all households, their valuation factors, and their level of present bias are homogeneous. Accounting for household heterogeneity, however, implications of policy instruments can differ, especially in distributional effects. According to our analysis, the lower the valuation for heat, the higher the CO2 price must be to induce investment in the zero-emission backstop technology. Assuming the policymaker introduces a CO2 price sufficient for incentivizing investment into the zero-emission backstop technology for a household with an average valuation factor, low-valuation households would not invest in the zero-emission technology. Instead, they would pay the CO2 price and heat less, while high-valuation households invest in the zero-emission backstop technology. Similarly, if instead of a CO2 price, the policymaker sets a target-consistent subsidy for average households, low-valuation households will not invest sufficiently. For households with higher valuations, however, the subsidy is not only sufficient but too high: they receive more money from the state than would have been necessary to stimulate the investment. The empirical literature suggests that high-income households have a higher valuation for thermal energy than low-income households (Cayla et al. 2011; Mertesacker 2021). This would imply that a single, uniform subsidy, which aims to reach households with low valuation factors as well, favors high-income households. Households also show heterogeneity with respect to their level of present bias. Assume the policymaker sets a CO2 price that is target-consistent for households with average present bias. As shown in Sect. 3.2.1, households with stronger present bias ( β<¯ β ) would underinvest and pay the CO2 price in future periods, heating less than optimal. Literature estimations of the correlation between income and the present bias level range between no correlation and a negative correlation, suggesting that low-income households experience higher levels of present bias (Meier and Sprenger 2010; Can and Erdem 2013; Filippini et al. 2021). Based on the above, it might therefore make sense for the policy maker to set a CO2 price that is target-consistent for households with the highest present bias to have a target consistent CO2 price for all households. However, there is another real-world issue which our stylized model does not consider. In reality, investment distortions exist, hindering house1 3 F. Arnold et al. holds from investing. Possible distortions and, thus, obstacles to investment include budget constraints, lack of access to capital, technological or regional circumstances, or split incentives between landlords and tenants. In cases where households cannot invest, otherwise target-consistent CO2 prices may lead to high costs. And these cost increase with the height of the chosen price. Subsidies can help to overcome budget constraints and lack of access to capital. Suppose a subsidy is introduced as a single instrument. In that case, there is no price signal to at least partially internalize the externalities of households that cannot invest and whose heating is still associated with GHG externalities. According to the Tinbergen rule Tinbergen (1952), each political goal needs an own political measure. For instance, income effects of certain political interventions can be addressed more efficiently and more consistently by non-linear income taxation. Nonetheless, the distortionary effects and the investment barriers discussed above could partly be addressed by targeted subsidies or non-linear taxes on energy consumption. Targeted subsidies may incentivize certain households to invest in zero-emission heating technologies, either otherwise being unable to invest or having too low of a valuation for heat. Nonlinear taxation may alleviate burdens to lower income households, in the case that the CO2 price does not lead to the investment into zero-emission technologies and heating still produces emissions. Both these options are difficult to implement and need robust empirical evidence, which we therefore do not further elaborate on. We show in Sect. 3.2.1 that an optimal policy combination of a CO2 price and subsidy exists that can account for different (unknown) levels of present bias. The policy instruments also differ regardless of households’ heterogeneity. In the case of a singular implementation of a CO2 price or bans on GHG emitting technologies, the households bear the full costs. With (supplementary) subsidies, by contrast, the state pays (part of) the costs. The latter may make sense from a social justice point of view or to increase acceptance among the population. Our assumption that there is a backstop technology at finite costs available may be over simplified when considering real-world applications. Even within one sector, the costs of backstop technologies can vary between households and in time (Acemoglu et al. 2012). The differences might become even more pronounced when comparing across sectors, such as heating and aviation. Policy mechanisms must account for the fact that a uniform crosssectoral CO2 price could incentivize backstop technology adoption in one sector while leaving it uneconomical in another. For example, while a CO2 price alone might suffice to drive adoption in one sector, another sector may require additional, targeted technology-specific investment subsidies to achieve similar outcomes. Furthermore, governments themselves could potentially be present biased, which could affect the effectiveness of measures to correct internalities. These aspects of government were excluded in this study, which assumed a social planner with a complete long-term orientation. However, elected officials are often under pressure to implement short-term solutions such as temporary tax cuts or spending increases before elections, complicating the design of policies for long-term welfare improvements. As mentioned in the literature review, one approach to addressing governments own potential present bias is to establish institutions like independent central banks, fiscal rules or automatically adjusting spending for social security systems and other areas of government. Central banks, for example, often operate under rules designed to take a long-term view. While these rules are not infallible 1 3 Environmental Policy Instruments for Investments in Backstop… and may sometimes be broken, they provide a framework that can help mitigate the shortterm political pressures prevalent in political decision-making. Given the heterogeneity of households and their potential investment constraints, as well as the possible desirability of distributing costs between households and the state, there are arguments in favor of combining taxes (or bans) with subsidies. By distinguishing the effects of policies on investment and utilization decisions, our analysis can support a nuanced discussion of appropriate policy mixes. 5 Conclusion The present paper examines the impact of present bias on optimal environmental policies aimed at achieving zero emissions. The study generalizes Heutel’s model for policy design for externality-producing durable goods when internalities are present. Besides increasing efficiency, investments in a new heating system may substitute the fuel used. Accounting for this substitution adds the dimensions of fuel price and emission intensity to our technology space. The generalization allows us to include a backstop technology with finite cost and analyze policy choices that reach zero emissions. This work contributes to the scientific literature in three ways. First, we generalize Heutel’s model by allowing technologies to differ in fuel price and emission intensity. Second, we introduce a model framework for developing target-consistent environmental policies given a backstop technology. It can serve as one element within a toolbox for welfare analysis given political targets beyond externality pricing. Third, we apply the model framework to the case of decarbonization in the German heating sector of private households under present bias and derive numerical magnitudes of the present bias effects. We find that, generalizing Heutel’s propositions, one instrument can be sufficient to address both externality and internality. Still, a combination of subsidies and taxes can be advantageous, as we show that there exists a tax-subsidy combination that is optimal regardless of the present bias level. This finding can be applied to comparable investment decisions in externality (GHG emission) producing durable goods, such as private mobility investments. The existence of the optimal policy mix is particularly relevant because the level of present bias is private information unknown to the policymaker and heterogeneous among households. Policymakers could avoid distributional effects by utilizing the present bias agnostic optimal policy mix. There are further arguments supporting policy mixes that fall short in our stylized model, including heterogeneity in the valuation of heating, investment distortions, and the costs’ distribution between households and the state. Based on our analysis, there remains room for further research. In contrast to our greenfield analysis with constant prices, in reality, households already own heating systems, and the heating system stock’s age structure is heterogeneous. Therefore, households are faced not only with the question of which technology to invest in, but also whether it is worth investing in a new heating system early on before the existing one breaks down. This raises questions about the timing of policy instruments, e.g., concerning the interdependencies of price paths of CO2 taxes or fuel prices over time. Here, as well, the question arises as to what constitutes target-consistent policy instruments. The issue could prove complicated, as it is difficult to determine under which circumstances early heating systems replacement is required to achieve climate targets. Further, we discussed the role of household heterogene1 3 F. Arnold et al. ity in our findings qualitatively. Households differ in their level of present bias, their current heating systems, and their financial capabilities. A more detailed examination of these properties could, in addition to theoretical analyses, e.g., concerning optimal policy mixes across households, also quantify effects at the level of the entire German building stock. Appendix A: Isocost Curves Figure 9 illustrates the isocost curves of the household’s decision problem. For illustrative purpose, the plots are based on the data of the numerical example, although the plots shall only provide an intuition about the properties of the household’s decision context. The costs consist of the investment costs, fuel costs, and the chosen fph. Moreover, emission taxes or subsidies would affect the costs. The indifference curves would be horizontal lines, since the household’s utility depends solely on the heat. In the left illustration of Fig. 9, there are the basic isocost curves. The darker the color, the higher the costs. Without any policy intervention, higher fph can provide heat at lower costs, as the isocost curves are convex in fph. Emission taxes and subsidies can turn the course of the isocost curves so that they are decreasing with fph, and lower fphs can provide heat at lower costs. Without any policy intervention, for all temperature levels, higher fphs are the option with the lowest cost. The isocost curves show a steeper increase towards high fphs so that there is in any case a high fph alternative for each temperature level. Introducing an emission tax, decreases the slope of the isocost curves at the end of higher fphs, here additional cost for paying the emission tax are added to the total costs. With that, the slope at lower fph is higher so that for all temperature levels a lower fph constitutes the technology with the lowest costs. Analogously, a subsidy depending on the fph decreases the slope of the isocost curves at high fph. In contrast to the case of emission taxes, the effect does not scale with the chosen temperature level, so that for high temperature levels the shape remains similar to the case without subsidy, as the relative impact is lower. At low-temperature levels, the impact of the subsidy, however, is higher. The household can take the subsidy profit from the higher efficiency and reach higher temperature levels, or waive the subsidy and remain on lower temperature levels. Fig. 9 Illustration of the isocost curves of the household’s decision problem. The darker the color, the higher the costs. The left plot shows the isocost curves of the original problem, the middle the isocosts under an emission tax rate, and the plot to the right the isocosts given a technology subsidy. The plots are based on the inputs of the numerical case study 1 3 Environmental Policy Instruments for Investments in Backstop… DellaVigna S (2009) Psychology and economics: Evidence from the field. 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