Differentiated Carbon Prices and the Economic Cost of Decarbonization

Abstract

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.

Appendices

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)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:

$$\begin{aligned} c^Y_{i}&\ge (1-to_i)r_{i}\quad&\bot \quad&Y_{i} \ge 0&\forall i \end{aligned}$$

(3)

$$\begin{aligned} c^C_{hh}&\ge { PC}_{hh}\quad&\bot \quad&C_{hh} \ge 0&\forall hh \end{aligned}$$

(4)

$$\begin{aligned} c^G&\ge { PG} \quad&\bot \quad&G \ge 0&\end{aligned}$$

(5)

$$\begin{aligned} c^I&\ge { PI} \quad&\bot \quad&I \ge 0&\end{aligned}$$

(6)

$$\begin{aligned} c^A_{i}&\ge { PA}_{i}\quad&\bot \quad&A_{i} \ge 0&\forall i \end{aligned}$$

(7)

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:

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

where

$$\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,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 as²⁷:

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

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:

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

where

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

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:

$$\begin{aligned} A_{i}&\ge \sum _j\frac{\partial c_{j}}{\partial { PA}_{i}}Y_{j} + \sum _{hh} \frac{\partial c^C_{hh}}{\partial { PA}_{i}}C \nonumber \\&\quad + \frac{\partial c^G}{\partial { PA}_{i}}G + \frac{\partial c^I}{\partial { PA}_{i}}I \quad&\bot \quad&{ PA}_{i} \ge 0&\forall i \end{aligned}$$

(8)

$$\begin{aligned} \frac{\partial r_{i}}{\partial {\textit{PD}}_{i}} Y_{i}&\ge \frac{\partial c^A_{i}}{\partial {\textit{PD}}_{i}}A_{i} \quad&\bot \quad&{\textit{PD}}_{i} \ge 0&\forall i \end{aligned}$$

(9)

$$\begin{aligned} \sum _{hh} \overline{ls}_{hh}&\ge \sum _i\frac{\partial c_{i}}{\partial { PL}}Y_{i} \quad&\bot \quad&{ PL} \ge 0&\end{aligned}$$

(10)

$$\begin{aligned} \sum _{hh} \overline{ks}_{hh}&\ge \sum _i\frac{\partial c_{i}}{\partial { PK}}Y_{i} \quad&\bot \quad&{ PK} \ge 0&\end{aligned}$$

(11)

$$\begin{aligned} I&\ge \sum _{hh} \overline{i}_{hh} \quad&\bot \quad&{ PI} \ge 0&\end{aligned}$$

(12)

$$\begin{aligned} C_{hh}&\ge \frac{{ INC}_{hh}^C}{{ PC}_{hh}}\quad&\bot \quad&{ PC}_{hh} \ge 0&\forall hh \end{aligned}$$

(13)

$$\begin{aligned} G&\ge \frac{{ INC}^G}{{ PG}}\quad&\bot \quad&{ PG} \ge 0&\end{aligned}$$

(14)

$$\begin{aligned} \sum _i\frac{\partial r_i}{\partial { PFX}} Y_i&\ge \sum _i\frac{\partial c^A_i}{\partial { PFX}} A_i + \overline{bop}\quad&\bot \quad&{ PFX} \ge 0 \quad \end{aligned}$$

(15)

$$\begin{aligned} e_{max}^{{ ETS}}&\ge \sum _i \phi _{i}^{{ ETS}} \sum _{j \in ets} \frac{\partial c_j}{\partial { PA}_i} Y_j \quad&\bot \quad&P_{CO_2}^{{ ETS}} \ge 0 \end{aligned}$$

(16)

Carbon emissions of non-ETS industries are given by:

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

(17)

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:

$$\begin{aligned} { INC}^C_{hh} :=&{ PL} \overline{ls}_{hh} + { PK} \overline{ks}_{hh} - { PI}\overline{i}_{hh} - { PC}_{hh} \overline{{ htax}} { LSM} \end{aligned}$$

(18)

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

(19)

$\overline{{ htax}}$ 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:

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

(20)

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