## Abstract

We consider a partial equilibrium model to study the optimal phasing out of polluting goods by green goods. The unit production cost of the green goods involves convexity and learning-by-doing. The total cost for the social planner includes the private cost of production and the social cost of carbon, assumed to be exogenous and growing at the social discount rate. Under these assumptions the optimization problem can be decomposed in two questions: (i) when to launch a given schedule; (ii) at which rate the transition should be completed that is, the design of a transition schedule as such. The first question can be solved using a simple indicator interpreted as the MAC of the whole schedule, possibly non optimal. The case of hydrogen vehicle (Fuel Cell Electric Vehicles) offers an illustration of our results. Using data from the German market we show that the 2015–2050 trajectory foreseen by the industry would be consistent with a carbon price at 52€/t. The transition cost to achieve a 7.5 M car park in 2050 is estimated at 21.6 billion € that is, to JEl 4% discount rate, 115 € annually for each vehicle which would abate 2.18 tCO\(_2\) per year.

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## Change history

### 06 December 2017

The correct affiliation for Anna Creti is: Université Paris-Dauphine, PSL Research University, LEDa, CGEMP, 75016 Paris, France.

## Notes

Partial derivatives are denoted with indexes (except if it could be confusing) for instance \(C_X\) stands for \(\partial C/\partial X\).

See Tol (2014) for a textbook derivation with several objectives. A targeted stabilization of the concentration of carbon in the atmosphere implies a growth rate of the CO\(_2\) price equal to the interest rate plus the decay rate of carbon in the atmosphere (Tol 2014, p. 53, the “shadow price” of CO\(_2\) \(\mu \) should be divided by \(U^\prime (C_t)\) to get the expression of the current price of CO\(_2\)).

Several recent contributions attempt to provide and test rule of thumb pricing. Golosov et al. (2014) propose a simple formula, based on a specified dynamic general equilibrium framework: the price of carbon should be proportional to world gross domestic product see also (Grimaud and Rouge 2014; Rezai and van der Ploeg 2015; Bijgaart et al. 2016).

The launching date and not the ending date should adjust to the change of duration reflecting that the DAC is a cost computed at the end of deployment.

Theoretically, both cases are possible: If the objective is to stabilize the concentration of CO\(_2\) in the atmosphere, the CO\(_2\) price should grow at a higher rate than the interest rate (e.g. Goulder and Mathai 2000). To slow the extraction of fossill exhaustible resource the growth rate of the carbon price should be lower than the intereste rate, with constant extraction costs (e.g. Grimaud and Rouge 2008).

For a complete discussion of our cost assumptions the reader is referred to Creti et al. (2015). The Excel model is available from the authors upon request.

Each cost component is obtained via Table 1 by adding the annualized capital cost, using the discount rate and the life time for the annualized factor, and the maintenance cost, using the capital cost and the percentage for maintenance. The Excel model is available from the authors upon request.

Estimates depend on various assumptions most notably the pure time discount rate and the convexity of the damage function. For instance Nordhaus (2011) computes 12$ /tCO\(_2\) (in 2005 US$) and Stern (2007) 68$/tCO\(_2\) while Golosov et al. (2014) find 15.5 $ and 135$ with their respective discount rate. At the country level, for France to reach its current policy objective, Quinet (2009) and Quinet (2013) suggest around 30 €/t for 2015; for the US, the EPA provides estimates (see https://www.epa.gov/climatechange/social-cost-carbon) the average of which is 56$ in 2015 with a 3% discount rate.

## References

Acemoglu D, Aghion P, Bursztyn L, Hemous D (2012) The environment and directed technical change. Am Econ Rev 102(1):131–66

Allen MR, Frame DJ, Huntingford C, Jones CD, Lowe JA, Meinshausen M, Meinshausen N (2009) Warming caused by cumulative carbon emissions towards the trillionth tonne. Nature 458(7242):1163–1166

Amigues J-P, Ayong Le Kama A, Moreaux M (2015a) Equilibrium transitions from non-renewable energy to renewable energy under capacity constraints. J Econ Dyn Control 55:89–112

Amigues J-P, Lafforgue G, Moreaux M (2015b) Optimal timing of carbon sequestration policies. Econ Bull 35(4):2242–2251

Archsmith J, Kendall A, Rapson D (2015) From cradle to junkyard: assessing the life cycle greenhouse gas benefits of electric vehicles. Res Transp Econ 52:72–90

Beeker E (2014) Y a-t-il une place pour l’hydrogène dans latransition ènergètique? Note d’analyse, France Stratègie,Commissariat gènèral à la Stratègie et à la Prospective . http://www.strategie.gouv.fr/sites/strategie.gouv.fr/files/atoms/files/201-08-06na-fs-hydrogene-hd.pdf

Bramoullé Y, Olson LJ (2005) Allocation of pollution abatement under learning by doing. J Public Econ 89(9):1935–1960

Brunet J, Ponssard J.-P (2016) Policies and deployment for fuel cell electric vehicles an assessment of the normandy project, Int J Hydrog Energy (

**forthcoming**)Creti A, Kotelnikova A, Meunier G, Ponssard J-P (2015) A cost benefit analysis of fuel cell electric vehicles. Report 1:1–41

Golosov M, Hassler J, Krusell P, Tsyvinski A (2014) Optimal taxes on fossil fuel in general equilibrium. Econometrica 82(1):41–88

Goulder LH, Mathai K (2000) Optimal CO\(_2\) abatement in the presence of induced technological change. J Environ Econ Manag 39(1):1–38

Grimaud A, Rouge L (2008) Environment, directed technical change and economic policy. Environ Resour Econ 41(4):439–463

Grimaud A, Rouge L (2014) Carbon sequestration, economic policies and growth. Resour Energy Econ 36(2):307–331

Harrison P (2014) Fueling Europe’s Futur: how auto innovation leads to eu jobs, technical report, Cambridge econometrics. http://www.camecon.com/EnergyEnvironment/EnergyEnvironmentEurope/FuellingEuropesFuture.aspx

IEA (2000) Experience curves for energy technology policy. International Energy Agency, Paris, France

IPCC (2013) Climate Change 2013: the physical science basis. contribution of working group I to the fifth assessment report of the intergovernmental panel on climate change, Cambridge University Press, Cambridge, United Kingdom and New York, USA, chapter Summary for Policymakers

Loulou R (2008) ETSAP-TIAM: the TIMES integrated assessment model. Part II: mathematical formulation. CMS 5(1–2):41–66

Loulou R, Labriet M (2008) ETSAP-TIAM: the TIMES integrated assessment model Part I: model structure. CMS 5(1–2):7–40

Marcantonini C, Ellerman AD (2014) The implicit carbon price of renewable energy incentives in Germany. Robert Schuman Centre for Advanced Studies Research Paper No, RSCAS, p 28

Matthews HD, Caldeira K (2008) Stabilizing climate requires near-zero emissions. Geophys Res Lett 35:L04705. doi:10.1029/2007GL032388

McDonald A, Schrattenholzer L (2001) Learning rates for energy technologies. Energy Policy 29(4):255–261

McKinsey & Company (2010) Portfolio of power-trains for Europe: a fact-based analysis. http://www.fch.europa.eu/node/786

Nordhaus WD (2011) Estimates of the social cost of carbon: background and results from the RICE-2011 model. Technical report, National Bureau of Economic Research

Oshiro K, Masui T (2014) Diffusion of low emission vehicles and their impact on \(\text{CO}_{2}\) emission reduction in Japan. Energy Policy (

**in press**)Quinet A (2009) La valeur tutélaire du carbone. Rapport du Centre d’Analyse Stratégique, La documentation Française

Quinet E (2013) L’évaluation socioéconomique des investissements publics, gènèral à la stratègie et à la prospective

Rezai A, van der Ploeg F (2015) Robustness of a simple rule for the social cost of carbon. Econ Lett

**132**:48–55. http://www.sciencedirect.com/science/article/pii/S0165176515001573Rosen S (1972) Learning by experience as joint production, Q J Econ, pp 366–382

Rösler H, van der Zwaan B, Keppo I, Bruggink J (2014) Electricity versus hydrogen for passenger cars under stringent climate change control. Sustain Energy Technolog Assess 5:106–118

Schennach SM (2000) The economics of pollution permit banking in the context of Title IV of the 1990 Clean Air Act Amendments. J Environ Econ Manag 40(3):189–210

Stern NH (2007) The economics of climate change: the Stern review. Cambridge University Press, Cambridge

Thompson P (2010) Learning by doing. Handb Econ Innov 1:429–476

Tol RS (2014) Climate economics: economic analysis of climate, climate change and climate policy. Edward Elgar Publishing, Cheltenham

Van den Bijgaart I, Gerlagh R, Liski M (2016) A simple formula for the social cost of carbon. J Environ Econ Manag 77:75–94

Vogt-Schilb A, Meunier G, Hallegatte S (2012) How inertia and limited potentials affect the timing of sectoral abatements in optimal climate policy. World Bank Policy Research Working Paper (6154)

Zachmann G, Holtermann M, Radeke J, Tam M, Huberty M, Naumenko D, Faye AN (2012) The great transformation: decarbonising Europe’s energy and transport systems. Bruegel Blueprint 16

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## Appendices

### Appendix A: proof of Lemma 1

### Proof

To minimize the total cost (2), let us introduce \(\lambda _t\) the co-state variable associated to the relation (3), and \(\theta _t\) and \(\delta _t\) the Lagrange multipliers associated to the two constraints (4) on \(x_t\): it is positive (\(\theta _t\)) and smaller that the total fleet size (\(\delta _t\)). The first order conditions (together with the complementarity slackness conditions) are:

The main step of the proof consists in proving that \(x_{t}\) is increasing if \(x_{t}\in (0,N)\). This condition ensures that once \(x_{t}>0\) the number of green cars cannot come back to zero, and that \(x_{t}\) does not move when \(x_{t}=N\). If \(x_{t}\) is strictly positive ( \(\theta _{t}=0\)) and lower than the total car fleet (\(\delta _{t}=0\)), Eq. (19) becomes \( C_{x}(X_{t},x_{t})-c_{o} =p_{t}^{CO2}+\lambda _{t} \) and taking the time derivative:

The last term of the right hand side is positive because \(C_{X}(X,x)\) is concave with respect to *x* and \(C_{X}(X,0)=0\) (since \(C(X,0)=0, \forall X\)). Since \(C_{xx}\), \( p^{CO2}_t\) and *r* are all positive, \(\dot{x}\) is also positive so that \(x_t\) is increasing through time.

Then, since the CO\(_{2}\) price increases exponentially, \(x_{t}\) cannot be always null along an optimal trajectory. Then either \(x_{0}=0\) or \(x_{0}>0\). In the latter case \(T_{s}=0\), whereas in the former case \(T_{s}\) is the inf of the dates at which \(x_{t}>0\).

The ending date is finite, \(T_e<+\infty \): From the above proof, when \(x_t\) is positive its time derivative is bounded below by a positive number, so \(x_t\) necessarily reaches *N* in a finite time.

Finally, Eq. (5) is obtained by integrating Eq. (20), between *t* and \(+\infty \) (and using the boundary conditions \(\text {lim}_{t\rightarrow +\infty } e^{-rt}\lambda _t = 0\)):

and injecting this expression into Eq. (19). \(\square \)

### Appendix B: proof of Lemma 2

### Proof

If \(C_{xx}=0\), given that \(C(X,0)=0\), \(C_{X}(X,x)=C_{Xx}(X,x)x\). Then, we resort by reductio ad absurdum assuming \(T_{s}<T_{e}\). Between the two dates the Eq. (5) is satisfied and taking its derivative with respect to *t* gives:

Therefore, using that \(C_{X}(X,x)=C_{Xx}(X,x)x\) in that case,

a contradiction.

Therefore, the number of green cars jumps from 0 to *N* at date \( T_{s}=T_{e}\), and the total discounted cost \(\Gamma \) could be written as a function of the date \(T_{s}\):

Along the optimal trajectory, \(T_{s}\) should minimize this function. Taking the derivative with respect to \(T_{s}\) in the equation above and setting it equal to zero gives the Eq. (6). \(\square \)

### Appendix C: proof of Lemma 3, Proposition 2

From Lemma 1, the optimal trajectory \((x_t^*)_{t\in [0,+\infty )}\) can be described as the launching of a deployment schedule. There is therefore no loss to minimize the cost over the set of trajectories defined as the launching (and ending) of a deployment schedule. The trajectory obtained from the optimal launching of the optimal deployment schedule coincides with the optimal trajectory described in Lemma 3.

### 1.1 Appendix C.1: proof of Lemma 3

### Proof

Using the decomposition of the total discounted cost \(\Gamma \) provided by Eq. (11), the schedule \(\xi \) only influences the DAC and no other component of the cost. So that the \(\xi \) that minimizes \(\Gamma \) corresponds to the \(\xi \) that minimizes the DAC. A deployment schedule \((\bar{X},D,\xi )\) and its DAC are only defined for \(\xi \) such that \(\int _0^D \xi _\tau d\tau =\bar{X}\). The optimal \(\xi \), for a given \(\bar{X}\) and *D*, is then the solution of the optimization program:

This result holds for any \(\bar{X}\), *D* and \(T_l\). \(\square \)

It is interesting to write the equations satisfied by \(\xi \) in order to recover the optimality conditions (5) satisfied by the optimal trajectory. Write the Lagrangian:

in which \(\mu _\tau \) is the co-state variable of \(\dot{X}=\xi _\tau \); \(\delta _t\) and \(\theta _t\) are the Lagrange multiplier of \(\xi _\tau \le N\); and \(\xi _\tau \ge 0\) respectively, and \(\alpha \) is the Lagrange multiplier of the constraint \(\int _0^D \xi _\tau d\tau =\bar{X}\). The optimal \(\xi \) satisfies the equations:

Then, \(\xi _\tau \) is increasing (similar reasoning than for the proof of Lemma 1), and integrating the second equation gives

Together with the optimality conditions satisfied by \(\bar{X}^*\), \(D^*\) and \(T_l^*\), to be studied below, these first order conditions will coincide with 5. However, even for suboptimal \(\bar{X}\) and *D* the schedule \(\xi \) should satisfies these equations, which then gives the minimized deployment cost \(I^*(\bar{X},D)\) and the associated \(DAC(\bar{X},D)\). The derivative of the deployment cost are:

### 1.2 Appendix C.2: proof of Proposition 2

There are two possible strategies to prove that \(\bar{X}^*\) and \(D^*\) are independent of the CO\(_2\) price \(p_0\). One is sketched in the main text. The other consists in looking at first order conditions and showing that the optimal duration and accumulated quantity satisfy a pair of equation independent from the CO\(_2\) price.

### Proof

From the expression (11) of the total discounted cost, the optimal \(D^*\), \(\bar{X}^*\) and launching date satisfy the equations:

The first equation corresponds to Eq. (13), the third correspond to (15). And injecting the third into the second gives Eq. (14) satisfied by \(\bar{X}^*\).

The two Eqs. (13) and (14) are independent of the CO\(_2\) price or the launching date, the couple \(\bar{X}^*\) and \(D^*\) is therefore independent of \(p_0\), and so is the optimal deployment schedule \((\xi _\tau ^*)_{\tau \in [0,D]}\) associated to them.

In addition, it is interesting to see how these equations together with Eq. (23) give back the Eq. (5).

so that for \(\xi _{\tau } \in (0,N)\) the Eq. (23) is equivalent to Eq. (5) with \(\xi _\tau =x_{T_s+\tau }\).

Furthermore, one can show that Eq. (14) leads to \(\xi _D=N\) and that Eq. (13) leads to \(\xi _0=0\). This relatively fastidious proof is available upon request.

### Appendix D: growth rate of the CO\(_2\) price

Proof of Corollary 3.

### Proof

The total discounted cost should be written:

which is similar to (8) except that the second line above, the value of interim abatement, replaces \(p_0\bar{X}\). The derivative of the second line with respect to \(T_l\) is (after an integration by parts):

so that the derivative of the discounted cost with respect to \(T_l\) is

This derivative is equal to zero at the optimal launching date (the second order condition is satisfied) which gives Eq. (17). \(\square \)

The following result is proved:

*Result* The ending (resp. launching) CO\(_2\) price of scenario \((\xi _\tau )_{\tau \in [0,D]}\) is increasing (resp. decreasing) with respect to \(\rho \) if \(\xi _0=0\) and \(\xi ^\prime _\tau >0\).

### Proof

We denote \(\xi _\tau ^\prime \) the derivative with respect to time of \(\xi _\tau \).

The ending CO\(_2\) price is given by Eq. (17). The derivative of its denominator with respect to \(\rho \) is:

so the ending CO\(_2\) price is increasing with respect to \(\rho \).

The launching CO\(_2\) price is:

The derivative of the denominator with respect to \(\rho \) is

so, the launching CO\(_2\) price is decreasing with respect to \(\rho \). \(\square \)

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Creti, A., Kotelnikova, A., Meunier, G. *et al.* Defining the Abatement Cost in Presence of Learning-by-Doing: Application to the Fuel Cell Electric Vehicle.
*Environ Resource Econ* **71**, 777–800 (2018). https://doi.org/10.1007/s10640-017-0183-y

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DOI: https://doi.org/10.1007/s10640-017-0183-y