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Differentiated Carbon Prices and the Economic Cost of Decarbonization


Employing a numerical general equilibrium model with multiple fuels, end-use sectors, heterogeneous households, and transport externalities, this paper examines three motives for differentiated carbon pricing in the context of Swiss climate policy: fiscal interactions with the existing tax code, non-\(\hbox {CO}_2\) related transport externalities, and social equity concerns. Interaction effects with mineral oil taxes reduce carbon taxes on motor fuels and transport externalities increase them. We show that the cost-effective overall carbon tax on motor fuels should be lower than the one on thermal fuels. This is found in spite of the fact that pre-existing taxes on motor fuels are well below our estimate of the transport externality per unit of transport fuel consumption. Differentiating taxes in favor of motor fuels yields only slightly more equitable incidence effects among households, suggesting that equity considerations play a minor role when designing differentiated carbon pricing policies.

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  1. 1.

    Other recent assessments of Swiss climate policy include Bretschger et al. (2011), Sceia et al. (2009), Sceia et al. (2012), which do not explicitly consider differentiated carbon prices. Sceia et al. (2009) and Sceia et al. (2012) analyze transportation-specific emission reduction targets and find them to be an inefficient deviation from economy-wide uniform carbon pricing.

  2. 2.

    Bovenberg and Goulder (1996) analyze optimal taxation of a polluting activity in presence of distortionary labor taxes. Their argument applies to our setting if one views the problem of taxing motor fuels more or less than thermal fuels as correcting the pre-existing motor fuel taxes to optimally manage the transport externality in presence of a distortionary \(\hbox {CO}_{2}\) tax. From a Swiss perspective, the \(\hbox {CO}_{2}\) tax is distortionary because, presumably, it makes emitters internalize global effects of climatic change of which Switzerland internalizes only a very small fraction.

  3. 3.

    Boeters (2014) finds that the most important drivers for carbon price differentiation are market power in export markets and taxes on consumption, intermediate inputs, and domestic outputs. At the same time, he warns that his model views taxes as distortive inefficiencies and shows that his case for carbon price differentiation weakens if his model channels tax revenues on motor fuels to road construction and maintenance and assumes that these have to be provided in proportion to motor fuel consumption.

  4. 4.

    The remaining difference between income and expenditure of households was attributed to direct transfers to or from the government. In our model, we index government transfers to the consumer price index thereby effectively assuming that households’ transfer income (in real terms) is unaffected by carbon taxation.

  5. 5.

    This is obviously a simplifying assumption. For example, people who have been in traffic accidents are likely to change their consumption of health services and people who experience changes in traffic noise in their neighborhood are likely to change their investment behavior in sound insulation. As this paper is not focused on analyzing transport externalities per se, we leave for future research to investigate the implications of relaxing this assumption—which, if addressed in a general equilibrium context, would call for an in-depth analysis going beyond the scope of this paper (see, for example, Carbone and Smith 2008). Moreover, welfare does not include the benefits from averted climate change due to reducing \(\hbox {CO}_2\) emissions in Switzerland which would be negligible due Switzerland’s tiny share in global emissions.

  6. 6.

    In equilibrium, \(Y_i\) and \(C_h\) are determined by the zero-profit conditions (3) and (4), respectively, shown in “Appendix 1”.

  7. 7.

    Consistently with Parry and Small (2005), \(\beta \) can be interpreted as the ratio between the elasticity of V with respect to the consumer fuel price (\(\eta _{{\textit{MF}}}\)) and the own-price elasticity of demand for fuel (\(\eta _{{\textit{FF}}}\)), i.e. \(\beta =\eta _{{\textit{MF}}}/\eta _{{\textit{FF}}}\). If fuel efficiency were fixed, then vehicle kilometers traveled would change in proportion to fuel use, so that \(\eta _{{\textit{MF}}}=\eta _{{\textit{FF}}}\).

  8. 8.

    We assume that the marginal and the average welfare loss from the transport externality per transport activity are the same. This appears appropriate as fuel taxes will reduce traffic in remote as well as in congested areas, in noisy as well as in silent places, and in accident prone locations as well as on straight, fenced roads.

  9. 9.

    A study of the Swiss Federal Office for Spatial Development (ARE 2014) has estimated the external cost from private transportation (including traffic congestion, air pollution, climate-related costs, and others) to be CHF 6.6 billion in 2010 of which 81.2% were not related to climate. According to this study, the total external costs per person kilometer are CHF 0.060 and the non-climate part of this is CHF 0.049 per person kilometer. In 2010, new private vehicles in Switzerland on average carried 1.6 persons and used 6.4 liters of gasoline per 100 kilometers (BFS 2013b). Thus, private transportation delivers at least 25 person kilometers per liter of gasoline. The costs of non-\(\hbox {CO}_2\) related transport externalities can thus be estimated to be about CHF 1.225 per liter of gasoline.

  10. 10.

    It is hard to quantify the range of uncertainty about estimates for the transport externalities. Based on findings from previous literature, Parry and Small (2005) suggest confidence intervals of size up to around (\(0.2\cdot \mu \), \(2.5\cdot \mu \)) for single components of non-climate related transport externalities, where \(\mu \) is the best guess estimate for the given component. Given that the Swiss estimate for transport externalities is rather high compared to the range of estimates cited by Parry and Small (2005), we choose the upper bound to the confidence interval of only 1.8 times the central estimate: (\(0.266\cdot \mu \), \(1.8\cdot \mu \)).

  11. 11.

    This type of forward calibration procedure has been used, for example, in Böhringer and Rutherford (2002).

  12. 12.

    See \(\hbox {``CO}_2\) Verordnung” (Anhang 8 zu Art. 45 Abs. 1) which regulates the Swiss ETS until 2020. We assume the same rate of change after 2020.

  13. 13.

    The cost-effective ETS cap is determined endogenously to minimize the welfare cost in Switzerland for achieving the given emissions reduction target.

  14. 14.

    The target is formulated in terms of all greenhouse gas (GHG) emissions and requires a reduction of at least 50% by 2030 relative to 1990 of which at least 30% should come from domestic reductions. Reducing all GHGs domestically by 30% corresponds to 40% domestic reduction of \(\hbox {CO}_2\) (Federal Council 2015a).

  15. 15.

    We here employ the central case estimates for the size of the transport externalities from Sect. 1.3. Section 4 then explores how our results are affected by using lower and upper bound estimates of parameters related to transport externalities.

  16. 16.

    Focusing on the case of Switzerland as a small open economy, we exclude by construction any arguments for tax differentiation stemming from market power on international markets.

  17. 17.

    While the magnitude of MWC difference between the transportation and non-transportation sectors depends of course on the specific technology assumptions for modelling the transportation sector, the finding that marginal abatement costs in the transportation sector tend to be higher in the transportation sector is in line with previous studies (see, for example, Paltsev et al. 2005a; Abrell 2010; Karplus et al. 2013).

  18. 18.

    As of 2015, the mineral oil tax on motor fuels is CHF 0.7422 per liter while the tax on thermal fuels is only CHF 0.003 per liter.

  19. 19.

    Moreover, the numerical model failed to produce a solution for reaching reduction targets of 40% and higher in scenarios Transport or Thermal. While this does not rule out that a solution exists, it illustrates the difficulty of reaching the targets by taxing one fuel type only.

  20. 20.

    To prove this point, we run the model \(\beta = 1\), thus assuming that any change in motor fuel demand is directly proportional to changes in vehicle distance traveled. For a 30% overall reduction target, the cost-effective \(\hbox {CO}_{2}\) taxes would then be CHF 279.6/t\(\hbox {CO}_{2}\) on motor fuel, and CHF 195.7/t\(\hbox {CO}_{2}\) for thermal fuels. The difference of CHF 83.9/t\(\hbox {CO}_{2}\) corresponds to CHF 0.22 per liter of motor fuel and in combination with the mineral oil tax, motor fuels are taxed CHF 0.96 more per liter than they would be if they were used as thermal fuels. This is lower than the CHF 1.23 per liter that would be the efficient extra tax on motor fuels in absence of tax interaction effects. As the stringency of climate policy and thus the \(\hbox {CO}_{2}\) taxes increase, the tax interaction effect becomes stronger: For a 50% overall \(\hbox {CO}_{2}\) reduction target, cost effective \(\hbox {CO}_{2}\) taxes on motor and thermal fuels are the same (at CHF 766.5/t\(\hbox {CO}_{2}\)). Thus, motor fuels are only taxed CHF 0.74 more per liter than they would be if they were used as thermal fuels.

  21. 21.

    Note that following the definition of social welfare in (1), we assume that the impacts of transport externalities a distributed on a per-capita basis.

  22. 22.

    In this case, we assume that government spending is increased by an equal amount.

  23. 23.

    Even when revenues are recycled back to households in a lump-sum fashion, i.e. without distorting relative prices, lump-sum transfers have direct redistributive effects. Intuitively, giving the same amount of money under a per-capita recycling scheme to a poor and a rich household creates a relatively larger gain in utility for the poor household.

  24. 24.

    We have carried out additional sensitivity analysis which independently varies \(t_{{\text {avg}}}\) and \(\beta \) to analyze the relative contributions of the uncertainty of the two parameters. We find that the uncertainty associated with each parameter has a similar effect on the overall uncertainty.

  25. 25.

    A characteristic of many economic models is that they can be cast as a complementary problem. Mathiesen (1985) and Rutherford (1995) have shown that a complementary-based approach is convenient, robust, and efficient. The complementarity format embodies weak inequalities and complementary slackness, relevant features for models that are not integrable. contain bounds on specific variables, for example, activity levels which cannot a priori be assumed to operate at positive intensity. Such features are not easily handled with alternative solution methods.

  26. 26.

    While the carbon price for the non-ETS sectors is exogenously defined, the carbon price for the ETS industries results from the cap \(e_{max}^{{ ETS}}\) of the emission trading system.

  27. 27.

    For ease of notation we suppress the fact that taxes and carbon coefficients are differentiated by agent.


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We gratefully acknowledge financial support by the Swiss Competence Center for Energy Research, Competence Center for Research in Energy, Society and Transition (SCCER-CREST) and the Commission for Technology and Innovation (CTI). We further gratefully acknowledge financial support by the SNF under grant number 407140_153710. We thank Renger van Nieuwkoop for his support with data preparation. We would like to thank Jan Abrell, Pierre-Alain Bruchez, André Müller, and two anonymous referees, for helpful comments.

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Correspondence to Sebastian Rausch.


Appendix 1: MCP Equilibrium Conditions for Numerical General Equilibrium Model

We formulate the model as a system of nonlinear inequalities and characterize the economic equilibrium as a mixed complementary problem (MCP) (Mathiesen 1985; Rutherford 1995)Footnote 25 consisting of two classes of conditions: zero profit and market clearance. Zero-profit conditions exhibit complementarity with respect to activity variables (quantities) and market clearance conditions exhibit complementarity with respect to price variables. We use the \(\perp \) operate to indicate complementarity between equilibrium conditions and variables. Model variables and parameters are defined in Tables 4, 5, and 6. We formulate the problem in GAMS and use the mathematical programming system MPSGE (Rutherford 1999) and the PATH solver (Dirkse and Ferris 1995) to solve for non-negative prices and quantities.

Table 4 Sets, and price and quantity variables
Table 5 Model parameters
Table 6 Parameter values for substitution elasticities in production and consumption

Zero-profit conditions for the model are given by:

$$\begin{aligned} c^Y_{i}&\ge (1-to_i)r_{i}\quad&\bot \quad&Y_{i} \ge 0&\forall i \end{aligned}$$
$$\begin{aligned} c^C_{hh}&\ge { PC}_{hh}\quad&\bot \quad&C_{hh} \ge 0&\forall hh \end{aligned}$$
$$\begin{aligned} c^G&\ge { PG} \quad&\bot \quad&G \ge 0&\end{aligned}$$
$$\begin{aligned} c^I&\ge { PI} \quad&\bot \quad&I \ge 0&\end{aligned}$$
$$\begin{aligned} c^A_{i}&\ge { PA}_{i}\quad&\bot \quad&A_{i} \ge 0&\forall i \end{aligned}$$

where \(Y_i\), \(A_i\), \(C_{hh}\), G, I denote domestic and Armington production, household and government consumption, and investment, respectively. \(to_i\) is the output tax imposed on sector i and \({ PC}_{hh}\), \({ PG}\), \({ PI}\) are the private and public consumption as well as investment price index. c denotes a cost function, r a revenue function. According to the nesting structures shown in Fig. 7a, the unit cost functions for production activities are given as:

$$\begin{aligned} c^{Y}_{i}&:= \left[ \theta ^{{ TOP}}_{i} (c^{TR}_{i})^{1-\sigma _{i}^{{ TOP}}} + \left( 1-\theta ^{{ TOP}}_{i}\right) (c_{i}^{{ NTR}})^{1-\sigma _{i}^{{ TOP}}} \right] ^{\frac{1}{1-\sigma _{i}^{{ TOP}}}} \end{aligned}$$


$$\begin{aligned} c^{TR}_{i}&:= \left[ \sum _{j\in benz}\theta ^{TR}_{ji} \left( \frac{{\textit{PAS}}_{j}}{\overline{{ pas}_{j}}}\right) ^{1-\sigma _{i}^{TR}} + \left( 1-\sum _{j\in benz}\theta ^{TR}_{ji}\right) (c^{{ PUB}}_{i})^{1-\sigma _{i}^{TR}} \right] ^{\frac{1}{1-\sigma _{i}^{TR}}}\\ c^{{ PUB}}_{i}&:= \left[ \sum _{j\in pub}\theta ^{{ PUB}}_{ji}\left( \frac{{\textit{PAS}}_{j}}{\overline{{ pas}_{j}}}\right) ^{1-\sigma _{i}^{{ PUB}}}\right] ^{\frac{1}{^{1-\sigma _{i}^{{ PUB}}}}}\\ c^{{ NTR}}_{i}&:= \left[ \theta ^{{ NTR}}_{i}\left( c^{{ MAT}}_{i}\right) ^{1-\sigma _{i}^{{ NTR}}} + \left( 1-\theta ^{{ NTR}}_{i}\right) (c^{{ VAE}}_{i})^{1-\sigma _{i}^{{ NTR}}}\right] ^{\frac{1}{1-\sigma _{i}^{{ NTR}}}}\\ c^{{ MAT}}_{i}&:= \left[ \sum _{j\in mat}\theta ^{{ MAT}}_{ji} \left( \frac{{\textit{PAS}}_{j}}{\overline{{ pas}_{j}}}\right) ^{1-\sigma _{i}^{{ MAT}}} \right] ^{\frac{1}{1-\sigma _{i}^{{ MAT}}}}\\ c^{{ VAE}}_{i}&:= \left[ \theta ^{{ VAE}}_{i} (c^{VA}_{i})^{1-\sigma _{i}^{{ VAE}}} + \left( 1-\theta ^{{ VAE}}_{i}\right) (c_{i}^{EN})^{1-\sigma _{i}^{{ VAE}}} \right] ^{\frac{1}{1-\sigma _{i}^{{ VAE}}}}\\ c^{VA}_{i}&:= \left[ \theta ^{VA}_{i} \left( \frac{(1+tl_{i}){ PL}}{\overline{pl_{i}}}\right) ^{1-\sigma _{i}^{VA}} + \left( 1-\theta ^{VA}_{i}\right) \left( \frac{(1+tk_{i}){ PK}}{\overline{pk_{i}}}\right) ^{1-\sigma _{i}^{VA}} \right] ^{\frac{1}{1-\sigma _{i}^{VA}}}\\ c^{EN}_{i}&:= \left[ \sum _{j\in edt}\theta ^{EN}_{ji} \left( \frac{{\textit{PAS}}_{j}}{\overline{{ pas}_{j}}}\right) ^{1-\sigma _{i}^{EN}} + \left( 1-\sum _{j\in edt}\theta ^{EN}_{ji}\right) (c^{FF}_{i})^{1-\sigma _{i}^{EN}} \right] ^{\frac{1}{1-\sigma _{i}^{EN}}}\\ c^{FF}_{i}&:= \left[ \sum _{j\in coa}\theta ^{FF}_{ji} \left( \frac{{\textit{PAS}}_{j}}{\overline{{ pas}_{j}}}\right) ^{1-\sigma _{i}^{FF}} + \left( 1-\sum _{j\in coa}\theta ^{FF}_{ji}\right) (c^{LQ}_{i})^{1-\sigma _{i}^{FF}} \right] ^{\frac{1}{1-\sigma _{i}^{FF}}}\\ c^{LQ}_{i}&:= \left[ \sum _{j\in lqd}\theta ^{LQ}_{ji}\left( \frac{{\textit{PAS}}_{j}}{\overline{{ pas}_{j}}}\right) ^{1-\sigma _{i}^{LQ}}\right] ^{\frac{1}{^{1-\sigma _{i}^{LQ}}}} \end{aligned}$$

\(\theta \) refers to share parameters, \(\sigma \) denotes elasticities of substitution. \(tl_i\), \(tk_i\), \({ PK}\) and \({ PL}\) are labour and capital taxes and prices, respectively. Prices denoted with an upper bar generally refer to tax-inclusive baseline prices observed in the benchmark equilibrium.

\({\textit{PAS}}_{i}\) denotes the tax and carbon price inclusive Armington prices, where \(ti_{i}\) is the intermediate input tax and \({ PA}_i\) the Aarmington composite price of commodity i. Carbon prices differ between ETS \(ets \in i\) and non-ETS \({ nets} \in i\) sectors. \(p_{CO_2}^{{ NETS}}\) and \(P_{CO_2}^{{ ETS}}\) denote the carbon prices for ETS and non-ETS industries,Footnote 26 respectively, and \(\phi _{i}\) the carbon coefficient. The price of non-ETS industries additionally includes the mineral oil tax \(pmo_i\). Hence, Armington prices for ETS and non-ETS sectors are defined asFootnote 27:

$$\begin{aligned} {{\textit{PAS}}}_{i}&:= \left( 1+ti_{i}\right) { PA}_{i} + \phi _i p_{CO_2}^{{ NETS}} + pmo_i&\forall i \in { nets} \\ {{\textit{PAS}}}_{i}&:= \left( 1+ti_{i}\right) { PA}_{i} + \phi _i P_{CO_2}^{{ ETS}}&\forall i \in ets \end{aligned}$$

On the output side, producers differentiate between supply to the domestic and supply to export market using a constant elasticity of transformation function. Denoting the domestic product price by \({\textit{PD}}_i\) and the exchange rate by \({ PFX}\) the unit revenue function is defined as:

$$\begin{aligned} r_i := \left[ \theta ^D_i\left( {\textit{PD}}_i\right) ^{1+\sigma ^T_i}+ \left( 1-\theta ^D_i\right) \left( { PFX}\right) ^{1+\sigma ^T_i}\right] ^{\frac{1}{1+\sigma ^T_i}} \end{aligned}$$

Trade is modelled via the Armington approach using a CES function between domestically produced an imported commodities. Denoting \(tm_i\) as import tax, the cost function of the Armington aggregation becomes:

$$\begin{aligned} c^A_i := \left[ \theta _i^D\left( {\textit{PD}}_i\right) ^{1-\sigma ^A} + \left( 1-\theta _i^D\right) \left( \frac{\left( 1+tm_i\right) { PFX}}{\overline{pm}_i}\right) ^{1-\sigma ^A}\right] ^{\frac{1}{1-\sigma ^A}} \end{aligned}$$

According to the nesting structures shown in Fig. 7b, the unit cost functions for production activities are given as:

$$\begin{aligned} c^{C}_{hh}&:= \left[ \theta ^{{ CTOP}}_{i,hh} (c^{{ CTR}}_{hh})^{1-\sigma ^{{ CTOP}}} + \left( 1-\theta ^{{ CTOP}}_{i,hh}\right) (c_{hh}^{{ CNTR}})^{1-\sigma ^{{ CTOP}}} \right] ^{\frac{1}{1-\sigma ^{{ CTOP}}}} \end{aligned}$$


$$\begin{aligned} c^{{ CTR}}_{hh}&:= \left[ \sum _{i\in benz}\theta ^{{ CTR}}_{i,hh} \left( \frac{{{\textit{PAS}}}_{i}}{\overline{{ pas}_{i}}}\right) ^{1-\sigma ^{{ CTR}}} + \left( 1-\sum _{i\in benz}\theta ^{{ CTR}}_{i,hh}\right) (c^{{ CPUB}}_{hh})^{1-\sigma ^{{ CTR}}} \right] ^{\frac{1}{1-\sigma ^{{ CTR}}}}\\ c^{{ CPUB}}_{hh}&:= \left[ \sum _{i\in publ}\theta ^{{ CPUB}}_{i,hh}\left( \frac{{ PAS}_{i}}{\overline{{ pas}_{i}}}\right) ^{1-\sigma ^{{ CPUB}}}\right] ^{\frac{1}{^{1-\sigma ^{{ CPUB}}}}}\,.\\ c^{{ CNTR}}_{hh}&:= \left[ \theta ^{{ CNTR}}_{hh}\left( c^{{ CMAT}}_{hh}\right) ^{1-\sigma ^{{ CNTR}}} + \left( 1-\theta ^{{ CNTR}}_{hh}\right) (c^{{ CEN}}_{hh})^{1-\sigma ^{{ CNTR}}} \right] ^{\frac{1}{1-\sigma ^{{ CNTR}}}}\\ c^{{ CMAT}}_{hh}&:= \left[ \sum _{i\in mat}\theta ^{{ CMAT}}_{i,hh} \left( \frac{{ PAS}_{i}}{\overline{{ pas}_{i}}}\right) ^{1-\sigma ^{{ CMAT}}} \right] ^{\frac{1}{1-\sigma ^{{ CMAT}}}}\\ c^{{ CEN}}_{hh}&:= \left[ \sum _{i\in edt}\theta ^{{ CEN}}_{i,hh} \left( \frac{{ PAS}_{i}}{\overline{{ pas}_{i}}}\right) ^{1-\sigma ^{{ CEN}}} + \left( 1-\sum _{i\in edt}\theta ^{{ CEN}}_{i,hh}\right) (c^{{ CFF}}_{hh})^{1-\sigma ^{{ CEN}}} \right] ^{\frac{1}{1-\sigma ^{{ CEN}}}}\\ c^{{ CFF}}_{hh}&:= \left[ \sum _{i\in coa}\theta ^{{ CFF}}_{i,hh} \left( \frac{{ PAS}_{i}}{\overline{{ pas}_{i}}}\right) ^{1-\sigma ^{{ CFF}}} + \left( 1-\sum _{i\in coa}\theta ^{{ CFF}}_{i,hh}\right) (c^{{ CLQ}}_{hh})^{1-\sigma ^{{ CFF}}} \right] ^{\frac{1}{1-\sigma ^{{ CFF}}}}\\ c^{{ CLQ}}_{hh}&:= \left[ \sum _{i\in lq}\theta ^{{ CLQ}}_{i,hh}\left( \frac{{ PAS}_{i}}{\overline{{ pas}_{i}}}\right) ^{1-\sigma ^{{ CLQ}}}\right] ^{\frac{1}{^{1-\sigma ^{{ CLQ}}}}} \end{aligned}$$

For government and investment consumption fixed shares are assumed:

$$\begin{aligned} c^{G}&:= \sum _{i} \theta ^{{ GTOP}}_{i}\frac{{ PAS}_{i}}{\overline{{ pas}_{i}}} \\ c^{I}&:= \sum _{i} \theta ^{{ ITOP}}_{i}\frac{{ PAS}_{i}}{\overline{{ pas}_{i}}} \end{aligned}$$

Denoting each households initial endowments of labor and capital as \(\overline{ls}_{hh}\) and \(\overline{ks}_{hh}\), respectively, \({ INC}^C_{hh}\) and \({ INC}^G\) as consumer and government income and using Shephard’s lemma, market clearing equations become:

$$\begin{aligned} A_{i}&\ge \sum _j\frac{\partial c_{j}}{\partial { PA}_{i}}Y_{j} + \sum _{hh} \frac{\partial c^C_{hh}}{\partial { PA}_{i}}C \nonumber \\&\quad + \frac{\partial c^G}{\partial { PA}_{i}}G + \frac{\partial c^I}{\partial { PA}_{i}}I \quad&\bot \quad&{ PA}_{i} \ge 0&\forall i \end{aligned}$$
$$\begin{aligned} \frac{\partial r_{i}}{\partial {\textit{PD}}_{i}} Y_{i}&\ge \frac{\partial c^A_{i}}{\partial {\textit{PD}}_{i}}A_{i} \quad&\bot \quad&{\textit{PD}}_{i} \ge 0&\forall i \end{aligned}$$
$$\begin{aligned} \sum _{hh} \overline{ls}_{hh}&\ge \sum _i\frac{\partial c_{i}}{\partial { PL}}Y_{i} \quad&\bot \quad&{ PL} \ge 0&\end{aligned}$$
$$\begin{aligned} \sum _{hh} \overline{ks}_{hh}&\ge \sum _i\frac{\partial c_{i}}{\partial { PK}}Y_{i} \quad&\bot \quad&{ PK} \ge 0&\end{aligned}$$
$$\begin{aligned} I&\ge \sum _{hh} \overline{i}_{hh} \quad&\bot \quad&{ PI} \ge 0&\end{aligned}$$
$$\begin{aligned} C_{hh}&\ge \frac{{ INC}_{hh}^C}{{ PC}_{hh}}\quad&\bot \quad&{ PC}_{hh} \ge 0&\forall hh \end{aligned}$$
$$\begin{aligned} G&\ge \frac{{ INC}^G}{{ PG}}\quad&\bot \quad&{ PG} \ge 0&\end{aligned}$$
$$\begin{aligned} \sum _i\frac{\partial r_i}{\partial { PFX}} Y_i&\ge \sum _i\frac{\partial c^A_i}{\partial { PFX}} A_i + \overline{bop}\quad&\bot \quad&{ PFX} \ge 0 \quad \end{aligned}$$
$$\begin{aligned} e_{max}^{{ ETS}}&\ge \sum _i \phi _{i}^{{ ETS}} \sum _{j \in ets} \frac{\partial c_j}{\partial { PA}_i} Y_j \quad&\bot \quad&P_{CO_2}^{{ ETS}} \ge 0 \end{aligned}$$

Carbon emissions of non-ETS industries are given by:

$$\begin{aligned} E^{{ NETS}} :=&\sum _i \phi _{i}^{{ NETS}} \left[ \sum _{j \in { nets}} \frac{\partial c_j}{\partial { PA}_i} Y_j \right. \nonumber \\&\left. + \sum _{hh} \frac{\partial c_{hh}^C}{\partial { PA}_i} C_{hh} + \frac{\partial c^G}{\partial { PA}_i} G + \frac{\partial c^I}{\partial { PA}_i} I \right] \end{aligned}$$

Private income is given as factor income net of investment expenditure and a lumpsum or direct tax payment to the local government. Public income is given as the sum of all tax revenues:

$$\begin{aligned} { INC}^C_{hh} :=&{ PL} \overline{ls}_{hh} + { PK} \overline{ks}_{hh} - { PI}\overline{i}_{hh} - { PC}_{hh} \overline{{ htax}} { LSM} \end{aligned}$$
$$\begin{aligned} { INC}^G :=&\sum _i to_i\left( {\textit{PD}}_i\frac{\partial r_i}{\partial {\textit{PD}}_i}Y_i + { PFX}\frac{\partial r_i}{\partial { PFX}} Y_i\right) \nonumber \\&+ \sum _i ti_{i}{} { PA}_{i}\left[ \sum _{j}\frac{\partial c_{j}}{\partial { PA}_{i}}Y_{j} + \frac{\partial c^C}{\partial { PA}_{i}}C + \frac{\partial c^G}{\partial { PA}_{i}}G + \frac{\partial c^I}{\partial { PA}_{i}}I\right] \nonumber \\&+ \sum _i tm_i{ PFX}\frac{\partial c^A_i}{\partial { PFX}} A_i \nonumber \\&+ \sum _i Y_{i}\left[ tlPL\frac{\partial c_{i}}{\partial { PL}} + tkPK\frac{\partial c_{i}}{\partial { PK}}\right] \nonumber \\&+ { PC}\overline{{ htax}}{} { LSM} + { PFX} \overline{bop}\nonumber \\&+ p_{CO_2}^{{ NETS}}E^{{ NETS}} \nonumber \\&+ P_{CO_2}^{{ ETS}}e^{{ ETS}}_{max} \nonumber \\&+ \sum _i \overline{pmo} \mu _i Y_i \end{aligned}$$

\(\overline{{ htax}}\) is a lumpsum tax on the representative household, i.e. a lumpsum payment from the household to the government. The multiplier \({ LSM}\) is used to implement revenue recycling in a lumpsum manner and determine by:

$$\begin{aligned} G = 1 \quad \bot \quad {\textit{LSM}} {\text { free}} \end{aligned}$$

If revenues are not recycled but change government purchases, the multiplier is fixed and the preceding is dropped.

Appendix 2: Additional Tables and Figures

See Figs. 7, 8 and Table 7.

Fig. 7

Nested structure for a production and b consumption activities

Fig. 8

Benchmark energy-related a expenditure and b income shares by household group

Table 7 Assumptions underlying forward calibration to year 2030 in the “business-as-usual” scenario

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Landis, F., Rausch, S. & Kosch, M. Differentiated Carbon Prices and the Economic Cost of Decarbonization. Environ Resource Econ 70, 483–516 (2018).

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  • Differentiated carbon taxes
  • Decarbonization
  • Fiscal interactions
  • Transport externalities
  • Heterogeneous households