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Is Shale Gas a Good Bridge to Renewables? An Application to Europe

Abstract

This paper explores whether climate policy justifies developing more shale gas and addresses the question of a potential arbitrage between shale gas development and the transition to clean energy. We construct a Hotelling-like model where electricity may be produced by three perfectly substitutable sources: an abundant dirty resource (coal), a non-renewable less polluting resource (shale gas), and an abundant clean resource (solar). The resources differ by their carbon contents and their unit costs. Shale gas extraction’s technology (fracking) generates local damages. Fixed costs must be paid to develop shale gas and to deploy the clean resource on a large scale. Climate policy takes the form of a carbon budget. We show that, at the optimum, a more stringent climate policy does not always go together with an increase of the quantity of shale gas extracted, and that banning shale gas extraction most often leads to bring forward the development of the clean resource, but not always. We calibrate the model for Europe in order to determine whether shale gas should be extracted and in which amount, and to evaluate the effects of a moratorium on shale gas use.

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Notes

  1. IFOP survey, Sept. 13th, 2012: 74% of the respondents are opposed to shale gas exploitation; BVA survey, Oct. 2nd, 2014: 62%. Note that this is greater than the opposition to nuclear energy, which provides most of France’s electricity.

  2. According to the Economic Report of the President 2013, “... actual 2012 carbon emissions are approximately 17 percent below the “business as usual” baseline. (...) of this reduction, 52 percent was due to the recession (...), 40 percent came from cleaner energy (fuel switching), and 8 percent came from accelerated improvement in energy efficiency (...).” See also Feng et al. (2015).

  3. This argument is widespread, see for instance the speech by Edward Davey, UK Secretary of State, for Energy and Climate Change on shale gas exploration, on september 9th 2013: “Gas will buy us the time we need over the coming decades to get enough low carbon technology (...).” (https://www.gov.uk/government/speeches/the-myths-and-realities-of-shale-gas-exploration) and oil and gas magnate T. Boone Pickens argues on his website, “Natural gas is not a permanent solution to ending our addiction to imported oil. It is a bridge fuel to slash our oil dependence while buying us time to develop new technologies (...).”

  4. The model may be very easily adapted to consider more generally natural gas in all its forms –conventional and unconventional, rather than specifically shale gas. Indeed, the local damage may be seen as a differential cost or local damage caused by gas extraction and combustion compared to coal, and may be negative. Coal extraction is environmentally damaging (isses of land use, waste management, water pollution etc.), and coal combustion as well (local air pollution). Besides, coal mining has been a very dangerous activity in the past, and still remains so in many developing countries. However, the public attention is at the moment focused on local damages due to shale gas extraction. Moreover, one could interpret some of the past coal externalities as now being part of private costs due to regulation.

  5. The findings of Solomon et al. (2009) go in the same direction.

  6. According to BP, World proved coal reserves are currently sufficient to meet 153 years of global production. http://www.bp.com/en/global/corporate/energy-economics/statistical-review-of-world-energy/coal/coal-reserves.html.

  7. This cost is in fact the levelized cost of electricity (LCOE) generated by coal-fired power plants. According to the US Energy Information Administration, it represents the per-kilowatt hour cost (in real dollars) of building and operating a generating plant over an assumed financial life and duty cycle. See EIA (2014).

  8. Notice that whereas the combustion of natural gas is without controverse less \(\hbox {CO}_{2}\) emitting than the combustion of coal, methane leakage from the shale gas supply chain could be high enough to offset the benefits. Heath et al. (2014) do not take into account methane leakage in their analysis because of the wide variability of estimates (0.66–6.2% for unconventional gas, 0.53–4.7% for conventional gas).

  9. This reflects the fact that society explores first, and then can choose to extract gas at a later date. This assumption allows to get rid of problems of concavity of the value function appearing when exploration and exploitation of shale gas reserves are performed at the same date.

  10. We have assumed that society chooses the date at which it pays the fixed cost of developing solar energy, we could have assumed that this fixed cost is also paid at date 0 but that the cost that must be paid for innovation to take place at date t is \(CF(t)=F(t)e^{-\rho t}\). We prefer the first interpretation because we have in mind massive investment expenditures to build equipments.

  11. We do not include electricity generation from renewables before the date at which electricity can be produced with renewables alone. It is straightforward to modify the model to account for the use of renewables before this date, for a given installed capacity, by simply assuming a time varying demand for fossil fuels, which reflects that electricty from renewables before the date of the transition cannot be produced on demand. On the contrary, all gas power plants and most coal power plants have been originally designed or can be modified for flexible output: the ability to “ramp” on an hourly basis to much less than full output, and “cycle” on and off on a daily basis. See Martinot (2016) for flexible coal power plants in Germany.

  12. This would not be the case in case of convexly increasing utilization costs and/or a positive regeneration rate (see Kollenbach 2015).

  13. Of course, as we are considering a central planner problem, the term “price” is used simply but inaccurately to denote the marginal utility of electricity consumption.

  14. The empirical literature shows that this is actually the case. See Alberini et al. (2011, Table 1 pp. 871), for a survey of recent estimates of price elasticities of residential electricity consumption.

  15. Note nevertheless that it leads to the optimal solution in the case where the development of shale gas is actually not optimal, that is when the local damage is large and climate policy lenient (more precisely, \(\overline{Z}>\overline{Z}_{2}\)). In this case, the moratorium is inconsequential.

  16. The 2015 Paris Agreement includes a two-headed temperature goal: “holding the increase in the global average temperature to well below \(2^{\circ }\hbox {C}\) above pre-industrial levels and pursuing efforts to limit the temperature increase to \(1.5^{\circ }\hbox {C}\)”. The impacts associated with these two carbon targets have been studied recently in Schleussner et al. (2016).

  17. Sensitivity analysis around \(\rho =0.02\) and \(\gamma =0.03\) show that the results do not change significantly.

  18. Using the fact that 1 ppmv = 2.13 GtC = 2.13*3.664 Gt\(\hbox {CO}_{2}\) = 7.8 GtCO\(_{2}.\)

  19. http://www.ipcc.ch/ipccreports/sres/emission/index.php?idp=118#533.

  20. www.ademe.fr/sites/assets/documents/rapport100enr_comite.pdf. See Table 4 in the Appendix of the report.

  21. http://ec.europa.eu/eurostat/statistics-explained/index.php/File:Gross-electricity-production-by-fuel-GWh-EU28-2014-TABLE.png.

  22. http://www.eia.gov/naturalgas/weekly/.

  23. Figure 1b p. 4, “Basin Economics for various US plays (single well) shale gas” gives the current HH price for different plays (the Henry Hub price is the pricing point for natural gas futures contracts traded on the New York Mercantile Exchange and the OTC swaps traded on Intercontinental Exchange).

  24. Remember the carbon budget we consider in the simulations corresponds to an average increase of temperature of 3\(^{\circ }\)C. A target of 2\(^{\circ }\)C is already impossible to satisfy.

  25. This change is the change in energy consumption only, recall that we assume a quasilinear utility function in money as the costs enter linearly.

  26. This change is the change in energy consumption only, recall that we assume a quasilinear utility function in money as the costs enter linearly.

  27. \(T_{b}\) such that \(x_{e}(T_{b})=x_{b}\) is the lowest possible \(T_{b}\) satisfying that solar production begins at the date of innovation (see Dasgupta et al. 1982).

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Acknowledgements

We acknowledge financial support from Agence Nationale de la Recherche, France (ANR-16-CE03-001).

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Correspondence to Katheline Schubert.

Appendices

Appendix

A Optimal Switch to Clean Energy

A.1 Large Local Damage

In this case, coal is used first, and then shale gas. Hence gas is used just before the switch to solar. Using the envelope theorem, the marginal benefit of delaying innovation can be written as:

$$\begin{aligned} \frac{\partial V(T_{b})}{\partial T_{b}}e^{\rho T_{b}}&=\left[ u\left( x_{e}(T_{b})\right) -(c_{e}+d+(\lambda _{0}+\theta _{e}\mu _{0})e^{\rho T_{b} })x_{e}(T_{b})\right] \nonumber \\&\quad -\left[ u\left( x_{b}\right) -c_{b}x_{b}\right] -(F^{\prime }(T_{b})-\rho F(T_{b}))\nonumber \\&=\pi _{e}(T_{b})-\pi _{b}+(\rho F(T_{b})-F^{\prime }(T_{b} )) \end{aligned}$$
(26)

Functions \(T_{b}\rightarrow (\rho F(T_{b})-F^{\prime }(T_{b}))\) and \(T_{b}\rightarrow \pi _{e}(T_{b})\) are decreasing with \(T_{b}\). It follows the assumptions made on F(.) for the first one (\(F^{\prime \prime }(.)<0\)). For the second one, we have:

$$\begin{aligned} \pi _{e}^{\prime }(T_{b})&=\left[ u^{\prime }\left( x_{e}(T_{b})\right) -(c_{e}+d+(\lambda _{0}+\theta _{e}\mu _{0})e^{\rho T_{b}})\right] \frac{\partial x_{e}(T_{b})}{\partial T_{b}}-\frac{\partial (\lambda _{0} +\theta _{e}\mu _{0}){e^{\rho T_{b}}}}{\partial T_{b}}x_{e}(T_{b})\\&=-\frac{\partial (\lambda _{0}+\theta _{e}\mu _{0}){e^{\rho T_{b}}}}{\partial T_{b}}x_{e}(T_{b})<0 \end{aligned}$$

as the final price of shale gas \(c_{e}+d+(\lambda _{0}+\theta _{e}\mu _{0})e^{\rho T_{b}}\) increases with \(T_{b}\). Hence \(\frac{\partial V(T_{b} )}{\partial T_{b}}e^{\rho T_{b}}\) is continuous and decreases with \(T_{b}\). It is positive for \(T_{b}\) such that \(x_{e}(T_{b})=x_{b}\)Footnote 27 and strictly negative when \(T_{b}\) goes to \(+\infty \).

As a result, \(V(T_{b})\) has a unique maximum \(T_{b}^{*}\) which satisfies:

$$\begin{aligned} \pi _{b}-\pi _{e}(T_{b}^{*})=\rho F(T_{b}^{*})-F^{\prime }(T_{b}^{*}) \end{aligned}$$

Functions \(T_{b}\rightarrow (\rho F(T_{b})-F^{\prime }(T_{b}))\) being decreasing with \(T_{b}\) and function \(T_{b}\rightarrow \pi _{b}-\pi _{e}(T_{b})\) increasing with \(T_{b}\), \(T_{b}^{*}\ge 0\) if and only

$$\begin{aligned} \rho F(0)-F^{\prime }(0)\ge \pi _{b}-\pi _{e}(0) \end{aligned}$$

A.2 Small Local Damage

The same reasoning applies, except that in this case, shale gas is used first, then coal. Hence coal is used just before the switch to solar occurs. \(V(T_{b})\) has a unique maximum \(T_{b}^{*}\) which satisfies:

$$\begin{aligned} \pi _{b}-\pi _{d}(T_{b}^{*})=\rho F(T_{b}^{*})-F^{\prime }(T_{b}^{*})>0 \end{aligned}$$

\(T_{b}^{*}\ge 0\) if and only

$$\begin{aligned} \rho F(0)-F^{\prime }(0)\ge \pi _{b}-\pi _{d}(0) \end{aligned}$$

B Thresholds

B.1 Large Local Damage

If shale gas is used alone, and coal is left under the ground, then the values of \(\lambda _{0},\)\(\mu _{0},\)\(T_{b}\) and \(X_{e}\) must solve the system composed of Eqs. (1), (11), (17) and

$$\begin{aligned} \theta _{e}X_{e}=\overline{Z}-Z_{0} \end{aligned}$$
(27)

which replaces (2). Moreover, to ensure that there exists no incentive to introduce coal at date 0,  the initial price of shale gas \(p_{e}(0)\) must be below the initial price of coal, \(p_{d}(0),\) i.e. we must have

$$\begin{aligned} (\theta _{d}-\theta _{e})\mu _{0}\ge c_{e}+d-c_{d}+E^{\prime }(X_{e}) \end{aligned}$$
(28)

If the solution of the above system is such that this condition is satisfied, then shale gas is used alone to get to the ceiling. There exists a threshold value of the ceiling \(\overline{Z}_{1}\) under which only shale gas is used. It is solution of the system composed of Eqs. (1), (11), (17), (27) and (28), this last equation being taken as an equality.

If coal is used alone to get to the ceiling, then the values of \(\mu _{0}\) and \(T_{b}\) must solve the following system:

$$\begin{aligned}&\displaystyle \theta _{d}\int _{0}^{T_{b}}x_{d}(t)dt =\overline{Z}-Z_{0} \end{aligned}$$
(29)
$$\begin{aligned}&\displaystyle \quad \left[ u\left( x_{b}\right) -c_{b}x_{b}\right] -\left[ u\left( x_{d}(T_{b})\right) -(c_{d}+\theta _{d}\mu _{0}e^{\rho T_{b}})x_{d} (T_{b})\right] =\rho F(T_{b})-F^{\prime }(T_{b}) \qquad \end{aligned}$$
(30)

where Eq. (29) is the combination of Eqs. (1 ) and (2) for \(X_{e}=0,\) and Eq. (30) is Eq. (17) in the case \(X_{e}=0.\) Moreover, we must make sure that there is no incentive to extract shale gas: the final price of coal \(p_{d}(T_{b})\) must be lower than the price of the first unit of shale gas that could be extracted at date \(T_{b}\), \(c_{e}+d+\theta _{e}\mu _{0}e^{\rho T_{b}}.\) Hence we must have:

$$\begin{aligned} (\theta _{d}-\theta _{e})\mu _{0}e^{\rho T_{b}}\le c_{e}+d-c_{d} \end{aligned}$$
(31)

meaning that the marginal gain in terms of pollution of switching from coal to shale gas, evaluated at the carbon value at date \(T_{b}\), is smaller than the marginal cost of the switch. If the solution of the above system is such that this condition is satisfied, then shale gas is never extracted. There exists a threshold value of the ceiling \(\overline{Z}_{2}\), such that if \(\overline{Z}\ge \overline{Z}_{2}\) shale gas is not developed. \(\overline{Z}_{2}\) is solution of the system composed of Eqs. (29), (30) and (31), this last equation being written as an equality.

For an intermediate ceiling \(\overline{Z}\) such that \(\overline{Z} _{1}<\overline{Z}<\overline{Z}_{2}\), the three phases exist.

Note that these two thresholds cannot coincide, except if \(T_b=0\), which cannot be the case, by assumption.

B.2 Small Local Damage

If shale gas is used alone to get to the ceiling, then \(\lambda _{0},\)\(\mu _{0},\)\(T_{b}\) and \(X_{e}\) must solve the system composed of Eqs. (1), (27), (11) and:

$$\begin{aligned} \left[ u\left( x_{b}\right) -c_{b}x_{b}\right] -\left[ u\left( x_{e}(T_{b})\right) -(c_{e}+d+(\lambda _{0}+\theta _{e}\mu _{0})e^{\rho T_{b} })x_{e}(T_{b})\right] =\rho F(T_{b})-F^{\prime }(T_{b}) \end{aligned}$$
(32)

Moreover, the final price of shale gas \(p_{e}(T_{b})\) must be lower than the price of the first unit of coal that could be extracted at date \(T_{b},\)\(p_{d}(T_{b}),\) i.e. we must have:

$$\begin{aligned} (\theta _{d}-\theta _{e})\mu _{0}e^{\rho T_{b}}>c_{e}+d-c_{d}+E^{\prime } (X_{e})e^{\rho T_{b}} \end{aligned}$$
(33)

meaning that the cost in terms of pollution of switching to coal instead of going directly to solar is higher than the advantage in terms of production costs. It happens for values of the ceiling below \(\overline{Z}_{3}\) defined by (1), (27), (11), (32) and (33) taken as an equality.

For \(\overline{Z}>\overline{Z}_{3},\) the three resources are used.

C The Effects of a More Stringent Climate Policy

C.1 Large Local Damage

In this case, Eqs. (1) and (2) may be written as:

$$\begin{aligned}&\displaystyle \int _{T_{e}}^{T_{b}}x_{e}(t)dt =X_{e}\nonumber \\&\displaystyle \int _{0}^{T_{e}}\theta _{d}x_{d}(t)dt+\int _{T_{e}}^{T_{b}}\theta _{e} x_{e}(t)dt=\overline{Z}-Z_{0} \end{aligned}$$
(34)

Using (34), this last equation reads:

$$\begin{aligned} \int _{0}^{T_{e}}x_{d}(t)dt=\frac{1}{\theta _{d}}\left( \overline{Z} -Z_{0}-\theta _{e}X_{e}\right) \end{aligned}$$
(35)

Totally differentiating system (34), (35), (14), (17) and (11) yields:

$$\begin{aligned}&x_{e}(T_{b})dT_{b}-x_{e}(T_{e})dT_{e}+\int _{T_{e}}^{T_{b}}dx_{e}(t)dt=dX_{e}\\&x_{d}(T_{e})dT_{e}+\int _{0}^{T_{e}}dx_{d}(t)dt=\frac{1}{\theta _{d}}\left( d\overline{Z}-\theta _{e}dX_{e}\right) \\&\left[ \theta _{d}\mu _{0}-(\lambda _{0}+\theta _{e}\mu _{0})\right] \rho dT_{e}+(\theta _{d}-\theta _{e})d\mu _{0}-d\lambda _{0}=0\\&\qquad -\left[ u^{\prime }\left( x_{e}(T_{b})\right) -(c_{e}+d+(\lambda _{0}+\theta _{e}\mu _{0})e^{\rho T_{b}}\right] dx_{e}(T_{b})+((d\lambda _{0}+\theta _{e}d\mu _{0})\\&\qquad +(\lambda _{0}+\theta _{e}\mu _{0})\rho dT_{b})e^{\rho T_{b}}x_{e}(T_{b}) =\left( \rho F^{\prime }(T_{b})-F^{\prime \prime }(T_{b})\right) dT_{b}\\&\qquad \qquad \qquad \qquad d\lambda _{0}=E^{\prime \prime }(X_{e})dX_{e} \end{aligned}$$

As

$$\begin{aligned} x_{d}(t)&=D(p_{d}(t))\Rightarrow dx_{d}(t)=D^{\prime }(p_{d}(t))dp_{d} (t)=D^{\prime }(p_{d}(t))\theta _{d}e^{\rho t}d\mu _{0}\\ x_{e}(t)&=D(p_{e}(t))\Rightarrow dx_{e}(t)=D^{\prime }(p_{e}(t))dp_{e} (t)=D^{\prime }(p_{e}(t))e^{\rho t}\left( d\lambda _{0}+\theta _{e}d\mu _{0}\right) \end{aligned}$$

the first two equations read equivalently:

$$\begin{aligned} x_{e}(T_{b})dT_{b}-x_{e}(T_{e})dT_{e}+\left[ \int _{T_{e}}^{T_{b}}D^{\prime }(p_{e}(t))e^{\rho t}dt\right] \left( d\lambda _{0}+\theta _{e}d\mu _{0}\right)= & {} dX_{e}\\ x_{d}(T_{e})dT_{e}+\left[ \int _{0}^{T_{e}}D^{\prime }(p_{d}(t))e^{\rho t}dt\right] \theta _{d}d\mu _{0}= & {} \frac{1}{\theta _{d}}\left( d\overline{Z}-\theta _{e}dX_{e}\right) \end{aligned}$$

Besides,

$$\begin{aligned}&\dot{D}(p_{d}(t))=D^{\prime }(p_{d}(t))\dot{p}_{d}(t)=D^{\prime } (p_{d}(t))\theta _{d}\mu _{0}\rho e^{\rho t}\\&\quad \Rightarrow \int _{0}^{T_{e}}D^{\prime }(p_{d}(t))e^{\rho t}dt=\frac{1}{\theta _{d}\mu _{0}\rho }\int _{0}^{T_{e}}\dot{D}(p_{d}(t)dt=\frac{1}{\theta _{d}\mu _{0}\rho }\left[ D(p_{d}(T_{e}))-D(p_{d}(0)\right] \\&\quad =\frac{x_{d}(T_{e})-x_{d}(0)}{\theta _{d}\mu _{0}\rho } \end{aligned}$$

and

$$\begin{aligned} \int _{T_{e}}^{T_{b}}D^{\prime }(p_{e}(t))e^{\rho t}dt=\frac{x_{e}(T_{b} )-x_{e}(T_{e})}{(\lambda _{0}+\theta _{e}\mu _{0})\rho } \end{aligned}$$

Hence the first two equations read:

$$\begin{aligned} -x_{e}(T_{e})dT_{e}+x_{e}(T_{b})dT_{b}-dX_{e}+\frac{x_{e}(T_{b})-x_{e}(T_{e} )}{(\lambda _{0}+\theta _{e}\mu _{0})\rho }\left( d\lambda _{0}+\theta _{e}d\mu _{0}\right)= & {} 0\\ x_{d}(T_{e})dT_{e}+\frac{\theta _{e}}{\theta _{d}}dX_{e}+\frac{x_{d} (T_{e})-x_{d}(0)}{\mu _{0}\rho }d\mu _{0}= & {} \frac{1}{\theta _{d}}d\overline{Z} \end{aligned}$$

Using the equality between marginal utilities, the fourth equation simplifies, and we obtain easily:

$$\begin{aligned} A\times \left( \begin{array}{c} dT_{e}\\ dT_{b}\\ dX_{e}\\ d\lambda _{0}\\ d\mu _{0} \end{array} \right) =\left( \begin{array}{c} 0\\ \frac{1}{\theta _{d}}\\ 0\\ 0\\ 0 \end{array} \right) d\overline{Z} \end{aligned}$$

with

$$\begin{aligned} A=\left( \begin{array}{ccccc} -x_{e}(T_{e}) &{} x_{e}(T_{b}) &{} -1 &{} \frac{x_{e}(T_{b})-x_{e}(T_{e})}{(\lambda _{0}+\theta _{e}\mu _{0})\rho } &{} \theta _{e}\frac{x_{e}(T_{b} )-x_{e}(T_{e})}{(\lambda _{0}+\theta _{e}\mu _{0})\rho }\\ x_{e}(T_{e}) &{} 0 &{} \frac{\theta _{e}}{\theta _{d}} &{} 0 &{} \frac{x_{e} (T_{e})-x_{d}(0)}{\mu _{0}\rho }\\ \left[ \lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0}\right] \rho &{} 0 &{} 0 &{} 1 &{} \theta _{e}-\theta _{d}\\ 0 &{} (\lambda _{0}+\theta _{e}\mu _{0})\rho x_{e}(T_{b})+z_{1} &{} 0 &{} x_{e} (T_{b}) &{} \theta _{e}x_{e}(T_{b})\\ 0 &{} 0 &{} -z_{2} &{} 1 &{} 0 \end{array} \right) \end{aligned}$$

where

$$\begin{aligned} z_{1}&=-\left( \rho F^{\prime }(T_{b})-F^{\prime \prime }(T_{b})\right) e^{-\rho T_{b}}>0\\ z_{2}&=E^{\prime \prime }(X_{e})>0 \end{aligned}$$

Hence:

$$\begin{aligned}&\rho \theta _{d}\mu _{0}(\lambda _{0}+\theta _{e}\mu _{0})\det A\\&\quad =\theta _{d}\left[ \underbrace{\left( x_{e}(T_{e})-x_{e}(T_{b})\right) }_{>0}x_{d}(0)\theta _{d}\mu _{0}+\underbrace{\left( x_{d}(0)-x_{e} (T_{e})\right) }_{>0}x_{e}(T_{b})\left( \lambda _{0}+\theta _{e}\mu _{0}\right) \right] z_{1}z_{2}\\&\qquad +\rho \left\{ \left[ \underbrace{\left( \theta _{e}x_{e}(T_{b})-\theta _{d}x_{d}(0)\right) }_{<0}\theta _{e}\mu _{0}-x_{d}(0)\theta _{d}\lambda _{0}\right] \underbrace{\left( \lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0}\right) }_{<0}+x_{e}(T_{e})\theta _{d}\lambda _{0}^{2}\right\} z_{1}\\&\qquad +\rho \theta _{d}x_{d}(0)x_{e}(T_{e})x_{e}(T_{b})\theta _{d}\mu _{0} (\lambda _{0}+\theta _{e}\mu _{0})z_{2}\\&\qquad +\rho ^{2}\theta _{d}(\lambda _{0}+\theta _{e}\mu _{0})x_{e}(T_{b})\left[ x_{e}(T_{e})\lambda _{0}^{2}-x_{d}(0)(\lambda _{0}+\theta _{e}\mu _{0} )\underbrace{\left( \lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0}\right) } _{<0}\right] \end{aligned}$$

i.e. \(\det A>0.\)

$$\begin{aligned}&A^{-1}\times \left( \begin{array}{c} 0\\ \frac{1}{\theta _{d}}\\ 0\\ 0\\ 0 \end{array} \right) =\frac{1}{\rho \theta _{d}\mu _{0}(\lambda _{0}+\theta _{e}\mu _{0})\det A}\times \\&\quad \left( \begin{array}{c} \mu _{0}\left( \lambda _{0}+\theta _{e}\mu _{0}\right) \left[ \frac{\theta _{d} }{\lambda _{0}+\theta _{e}\mu _{0}}\left( x_{e}(T_{e})-x_{e}(T_{b})\right) z_{1}z_{2}+\rho z_{1}(\theta _{d}-\theta _{e})+\rho x_{e}(T_{b})\left( x_{e}(T_{e})z_{2}\theta _{d}+\rho (\theta _{d}-\theta _{e})\left( \lambda _{0}+\theta _{e}\mu _{0}\right) \right) \right] \\ -\, \rho x_{e}(T_{b})\mu _{0}(\lambda _{0}+\theta _{e}\mu _{0})\left[ -x_{e} (T_{e})\theta _{d}z_{2}+\rho \theta _{e}\underbrace{\left( \lambda _{0} +(\theta _{e}-\theta _{d})\mu _{0}\right) }_{<0}\right] \\ -\, \rho \mu _{0}\left[ -x_{e}(T_{b})z_{1}\theta _{e}\underbrace{\left( \lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0}\right) }_{<0}+x_{e}(T_{e} )\theta _{d}\lambda _{0}\left( z_{1}+\rho x_{e}(T_{b})(\lambda _{0}+\theta _{e}\mu _{0})\right) \right] \\ -\, z_{2}\rho \mu _{0}\left[ -x_{e}(T_{b})z_{1}\theta _{e}\left( \lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0}\right) +x_{e}(T_{e})\theta _{d}\lambda _{0}\left( z_{1}+\rho x_{e}(T_{b})(\lambda _{0}+\theta _{e}\mu _{0})\right) \right] \\ -\, \rho \mu _{0}\left( \lambda _{0}+\theta _{e}\mu _{0}\right) \left[ \begin{array}{c} \frac{\theta _{d}\mu _{0}}{\lambda _{0}+\theta _{e}\mu _{0}}(x_{e}(T_{e} )-x_{e}(T_{b}))z_{1}z_{2}+x_{e}(T_{b})z_{1}z_{2}-\rho z_{1}\left( \lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0}\right) \\ -\rho x_{e}(T_{b})\left[ -x_{e}(T_{e})\theta _{d}\mu _{0}z_{2}+\rho (\lambda _{0}+\theta _{e}\mu _{0})\left( \lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0}\right) \right] \end{array} \right] \end{array} \right) \end{aligned}$$

As \(\det A>0,\) we deduce:

$$\begin{aligned} \frac{\partial T_{e}}{\partial \overline{Z}}>0,\quad \frac{\partial T_{b} }{\partial \overline{Z}}>0,\quad \frac{\partial X_{e}}{\partial \overline{Z} }<0,\quad \frac{\partial \lambda _{0}}{\partial \overline{Z}}<0,\quad \frac{\partial \mu _{0}}{\partial \overline{Z}}<0 \end{aligned}$$

C.2 Small Local Damage

In this case, Eqs. (1) and (2) may be written as:

$$\begin{aligned} \int _{0}^{T_{d}}x_{e}(t)dt= & {} X_{e} \end{aligned}$$
(36)
$$\begin{aligned} \int _{T_{d}}^{T_{b}}x_{d}(t)dt= & {} \frac{1}{\theta _{d}}\left( \overline{Z} -Z_{0}-\theta _{e}X_{e}\right) \end{aligned}$$
(37)

Totally differentiating system (36), (37), (16), (21) and (11) yields:

$$\begin{aligned}&\displaystyle x_{e}(T_{d})dT_{d}+\frac{x_{e}(T_{d})-x_{e}(0)}{(\lambda _{0}+\theta _{e}\mu _{0})\rho }=dX_{e}\\&\displaystyle x_{d}(T_{b})dT_{b}-x_{d}(T_{d})dT_{d}+\frac{x_{d}(T_{b})-x_{d}(T_{d})}{\theta _{d}\mu _{0}\rho }=\frac{1}{\theta _{d}}\left( d\overline{Z}-\theta _{e}dX_{e}\right) \\&\displaystyle -((d\lambda _{0}+\theta _{e}d\mu _{0})+(\lambda _{0}+\theta _{e}\mu _{0})\rho dT_{d})e^{\rho T_{d}}x_{e}(T_{d})+\theta _{d}(d\mu _{0}+\mu _{0}\rho dT_{d})e^{\rho T_{d}}x_{d}(T_{d})=0\\&\displaystyle \theta _{d}(d\mu _{0}+\rho dT_{b})e^{\rho T_{b}}x_{d}(T_{b})=\left( \rho F^{\prime }(T_{b})-F^{\prime \prime }(T_{b})\right) dT_{b}\\&\displaystyle d\lambda _{0}=E^{\prime \prime }(X_{e})dX_{e} \end{aligned}$$

Using \(x_{e}(T_{d})=x_{d}(T_{d}),\) we obtain:

$$\begin{aligned} A\times \left( \begin{array}{c} dT_{d}\\ dT_{b}\\ dX_{e}\\ d\lambda _{0}\\ d\mu _{0} \end{array} \right) =\left( \begin{array}{c} 0\\ \frac{1}{\theta _{d}}\\ 0\\ 0\\ 0 \end{array} \right) d\overline{Z} \end{aligned}$$

with

$$\begin{aligned} A=\left( \begin{array}{ccccc} x_{d}(T_{d}) &{} 0 &{} -1 &{} \frac{x_{d}(T_{d})-x_{e}(0)}{(\lambda _{0}+\theta _{e}\mu _{0})\rho } &{} \theta _{e}\frac{x_{d}(T_{d})-x_{e}(0)}{(\lambda _{0} +\theta _{e}\mu _{0})\rho }\\ -x_{d}(T_{d}) &{} x_{d}(T_{b}) &{} \frac{\theta _{e}}{\theta _{d}} &{} 0 &{} \frac{x_{d}(T_{b})-x_{d}(T_{d})}{\mu _{0}\rho }\\ \left[ -\theta _{d}\mu _{0}+(\lambda _{0}+\theta _{e}\mu _{0})\right] \rho &{} 0 &{} 0 &{} 1 &{} -(\theta _{d}-\theta _{e})\\ 0 &{} y_{1} &{} 0 &{} 0 &{} \theta _{d}x_{d}(T_{b})\\ 0 &{} 0 &{} -E^{\prime \prime }(X_{e}) &{} 1 &{} 0 \end{array} \right) \end{aligned}$$

where

$$\begin{aligned} y_{1}=-\left( \rho F^{\prime }(T_{b})-F^{\prime \prime }(T_{b})\right) e^{-\rho T_{b}}+\rho x_{d}(T_{b})\theta _{d}\mu _{0}>0 \end{aligned}$$

Let’s denote

$$\begin{aligned} y_{2}=E^{\prime \prime }(X_{e})\left[ x_{d}(T_{d})\theta _{d}\mu _{0} +x_{e}(0)\left( \lambda _{0}+\left( \theta _{e}-\theta _{d}\right) \mu _{0}\right) \right] \end{aligned}$$

According to (16), we have:

$$\begin{aligned} \lambda _{0}+\left( \theta _{e}-\theta _{d}\right) \mu _{0}=\left( c_{d} -(c_{e}+d)\right) e^{-\rho T_{d}}>0 \end{aligned}$$

which implies that \(y_{2}\) is also positive.

We have

$$\begin{aligned}&-\rho \theta _{d}\mu _{0}(\lambda _{0}+\theta _{e}\mu _{0})\det A\\&\quad =\rho x_{d}(T_{b})^{2}\theta _{d}^{2}\mu _{0}\left\{ \rho (\lambda _{0} +\theta _{e}\mu _{0})(\lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0})+E^{^{\prime \prime }}(X_{e})\left[ x_{d}(T_{d})\theta _{d}\mu _{0}\right. \right. \\&\qquad \left. \left. +\,x_{e}(0)(\lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0})\right] \right\} \\&\qquad +\,y_{1}\rho \left\{ x_{d}(T_{d})\theta _{d}\lambda _{0}^{2}+x_{e}(0)\theta _{e}^{2}\mu _{0}(\lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0})-x_{d}(T_{b} )\theta _{d}(\lambda _{0}+\theta _{e}\mu _{0})(\lambda _{0}\right. \\&\qquad \left. +\,(\theta _{e}-\theta _{d})\mu _{0})\right\} \\&\qquad +\,y_{1}E^{^{\prime \prime }}(X_{e})\theta _{d}\left\{ x_{e}(0)(\lambda _{0}+\theta _{e}\mu _{0})\left( x_{d}(T_{d})-x_{d}(T_{b})\right) +x_{d} (T_{b})\theta _{d}\mu _{0}(x_{e}(0)-x_{d}(T_{d}))\right\} \end{aligned}$$

It is straightforward that the terms of the first and third lines are positive. Let look at the term of the second line:

$$\begin{aligned}&y_{1}\rho \big \{x_{d}(T_{d})\theta _{d}\lambda _{0}^{2}+x_{e}(0)\theta _{e} ^{2}\mu _{0}(\lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0})\\&\qquad -\,x_{d}(T_{b})\theta _{d}(\lambda _{0}+\theta _{e}\mu _{0})(\lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0})\big \} \end{aligned}$$

Dividing by \(y_{1}\rho >0\), it has the sign of:

$$\begin{aligned}&\lambda _{0}^{2}(\theta _{d}x_{d}(T_{d})-\theta _{d}x_{d}(T_{b}))\\&\quad +\lambda _{0}\mu _{0}(\theta _{e}^{2}x_{e}(0)+\theta _{d}^{2}x_{d} (T_{b})-2\theta _{e}\theta _{d}x_{d}(T_{b}))\\&\quad +\mu _{0}^{2}\theta _{e}(\theta _{e}^{2}x_{e}(0)+\theta _{d}^{2}x_{d} (T_{b})-\theta _{e}\theta _{d}x_{d}(T_{b})-\theta _{e}\theta _{d}x_{e}(0)) \end{aligned}$$

It is straightforward that \(\lambda _{0}^{2}(\theta _{d}x_{d}(T_{d})-\theta _{d}x_{d}(T_{b}))>0\). Moreover

$$\begin{aligned} \lambda _{0}\mu _{0}(\theta _{e}^{2}x_{e}(0)+\theta _{d}^{2}x_{d}(T_{b} )-2\theta _{e}\theta _{d}x_{d}(T_{b}))=\lambda _{0}\mu _{0}x_{d}(T_{b})(\theta _{d}-\theta _{e})^{2}+\lambda _{0}\mu _{0}\theta _{e}^{2}(x_{e}(0)-x_{d}(T_{b})) \end{aligned}$$
(38)

and

$$\begin{aligned} \mu _{0}^{2}\theta _{e}(\theta _{e}^{2}x_{e}(0)+\theta _{d}^{2}x_{d}(T_{b} )-\theta _{e}\theta _{d}x_{d}(T_{b})-\theta _{e}\theta _{d}x_{e}(0))=\mu _{0} ^{2}\theta _{e}(\theta _{d}-\theta _{e})(\theta _{d}x_{d}(T_{b})-\theta _{e} x_{e}(0)) \end{aligned}$$
(39)

so that regrouping the last two terms (38) and (39), one gets :

$$\begin{aligned}&\lambda _{0}\mu _{0}\left( \theta _{e}^{2}x_{e}(0)+\theta _{d}^{2}x_{d} (T_{b})-2\theta _{e}\theta _{d}x_{d}(T_{b})\right) \\&\qquad +\mu _{0}^{2}\theta _{e}\left( \theta _{e}^{2}x_{e}(0)+\theta _{d}^{2}x_{d}(T_{b})-\theta _{e} \theta _{d}x_{d}(T_{b})-\theta _{e}\theta _{d}x_{e}(0)\right) \\&\quad =\lambda _{0}\mu _{0}x_{d}(T_{b})(\theta _{d}-\theta _{e})^{2}+\lambda _{0} \mu _{0}\theta _{e}^{2}(x_{e}(0)-x_{d}(T_{b}))\\&\qquad +\mu _{0}^{2}\theta _{e}(\theta _{d}-\theta _{e})(\theta _{d}x_{d}(T_{b})-\theta _{e}x_{e}(0))\\&\quad =\lambda _{0}\mu _{0}x_{d}(T_{b})(\theta _{d}-\theta _{e})^{2}+\lambda _{0} \mu _{0}\theta _{e}^{2}(x_{e}(0)-x_{d}(T_{b}))\\&\qquad +\mu _{0}^{2}\theta _{e}(\theta _{d}-\theta _{e})((\theta _{d}-\theta _{e})x_{d}(T_{b})-\theta _{e}(x_{e} (0)-x_{d}(T_{b})))\\&\quad =\lambda _{0}\mu _{0}x_{d}(T_{b})(\theta _{d}-\theta _{e})^{2}+\lambda _{0} \mu _{0}\theta _{e}^{2}(x_{e}(0)-x_{d}(T_{b}))+\mu _{0}^{2}\theta _{e}(\theta _{d}-\theta _{e})^{2}x_{d}(T_{b})\\&\qquad -\mu _{0}^{2}\theta _{e}^{2}(\theta _{d} -\theta _{e})(x_{e}(0)-x_{d}(T_{b}))\\&\quad =\mu _{0}x_{d}(T_{b})(\theta _{d}-\theta _{e})^{2}(\lambda _{0}+\theta _{e} \mu _{0})+\mu _{0}\theta _{e}^{2}(x_{e}(0)-x_{d}(T_{b}))(\lambda _{0}+\mu _{0}(\theta _{e}-\theta _{d})) \end{aligned}$$

which is positive. As a result:

$$\begin{aligned} \det A<0 \end{aligned}$$

We also obtain:

$$\begin{aligned} A^{-1}\times & {} \left( \begin{array}{c} 0\\ \frac{1}{\theta _{d}}\\ 0\\ 0\\ 0 \end{array} \right) =\frac{1}{\theta _{d}(\lambda _{0}+\theta _{e}\mu _{0})\det A}\\&\left( \begin{array}{c} y_{1}\left[ E^{\prime \prime }(X_{e})(x_{e}(0)-x_{d}(T_{d}))\theta _{d} +\rho \left( \theta _{d}-\theta _{e}\right) (\lambda _{0}+\theta _{e}\mu _{0})\right] /\rho \\ -x_{d}(T_{b})\theta _{d}\left[ \rho (\lambda _{0}+\theta _{e}\mu _{0})\left( \lambda _{0}+\left( \theta _{e}-\theta _{d}\right) \mu _{0}\right) +y_{2}\right] \\ y_{1}\left[ x_{d}(T_{d})\theta _{d}\lambda _{0}-x_{e}(0)\theta _{e}\left( \lambda _{0}+\left( \theta _{e}-\theta _{d}\right) \mu _{0}\right) \right] \\ y_{1}E^{\prime \prime }(X_{e})\left[ x_{d}(T_{d})\theta _{d}\lambda _{0} -x_{e}(0)\theta _{e}\left( \lambda _{0}+\left( \theta _{e}-\theta _{d}\right) \mu _{0}\right) \right] \\ y_{1}\left[ \rho (\lambda _{0}+\theta _{e}\mu _{0})\left( \lambda _{0}+\left( \theta _{e}-\theta _{d}\right) \mu _{0}\right) +y_{2}\right] \end{array} \right) \end{aligned}$$

As \(\det A<0,\) we deduce:

$$\begin{aligned} \frac{\partial T_{d}}{\partial \overline{Z}}<0,\quad \frac{\partial T_{b} }{\partial \overline{Z}}>0,\quad \frac{\partial X_{e}}{\partial \overline{Z} }\text { ambiguous},\quad \frac{\partial \lambda _{0}}{\partial \overline{Z}}\text { ambiguous},\quad \frac{\partial \mu _{0}}{\partial \overline{Z}}<0 \end{aligned}$$

\(\frac{\partial X_{e}}{\partial \overline{Z}}\) and \(\frac{\partial \lambda _{0} }{\partial \overline{Z}}\) have the same sign as \(x_{e}(0)\theta _{e}\left( \lambda _{0}+\left( \theta _{e}-\theta _{d}\right) \mu _{0}\right) -x_{d} (T_{d})\theta _{d}\lambda _{0}.\) It is negative when \(\theta _{e}=0,\) and positive when \(\theta _{e}=\theta _{d}.\)

D Low Price Elasticity of Demand

Step 1. Expenditure pD(p) is continuous and increasing with p. From Lagrange theorem, denoting \(p_{T_{b}}\equiv p(T_{b})\) and \(x_{T_{b}}=D(p(T_{b}))\) there exists a price \(p_{i}\in ]c_{b},p_{T_{b}}[\) such that:

$$\begin{aligned} p_{T_{b}}x_{T_{b}}=c_{b}x_{b}+(D(p_{i})+p_{i}D^{\prime }(p_{i}))(p_{T_{b} }-c_{b}) \end{aligned}$$

The elasticity of demand at price \(p_{i}\) is \(\epsilon _{i}=-\frac{p_{i}D^{\prime }(p_{i})}{D(p_{i})}\) so that the above equation can be rewritten as:

$$\begin{aligned} \frac{x_{T_{b}}}{D(p_{i})}=\frac{c_{b}x_{b}}{p_{T_{b}}D(p_{i})}+(1-\epsilon _{i})\left( 1-\frac{c_{b}}{p_{T_{b}}}\right) \end{aligned}$$

or equivalently:

$$\begin{aligned} \frac{x_{T_{b}}}{D(p_{i})}-1=\frac{c_{b}}{p_{T_{b}}}\left( \frac{x_{b} }{D(p_{i})}-1\right) -\epsilon _{i}\left( 1-\frac{c_{b}}{p_{T_{b}}}\right) \end{aligned}$$

As \(\frac{x_{T_{b}}}{D(p_{i})}-1<0\) and \(\frac{c_{b}}{p_{T_{b}}}\left( \frac{x_{b}}{D(p_{i})}-1\right) >0\), denoting \(\epsilon =\max _{i}(\epsilon _{i})\), it comes that:

$$\begin{aligned} \frac{x_{T_{b}}}{D(p_{i})}-1&=O(\epsilon ) \end{aligned}$$
(40)
$$\begin{aligned} \frac{x_{b}}{D(p_{i})}-1&=O(\epsilon )\frac{p_{T_{b}}}{c_{b}} \end{aligned}$$
(41)

Similarly, using Lagrange theorem between prices \(c_{e}\) and \(c_{b}\), one gets, with \(p_{j}\in ]c_{e},c_{b}[\):

$$\begin{aligned} \frac{x_{b}}{D(p_{j})}-1&=O(\epsilon ) \end{aligned}$$
(42)
$$\begin{aligned} \frac{x_{c_{e}}}{D(p_{j})}-1&=O(\epsilon )\frac{c_{b}}{c_{e}} \end{aligned}$$
(43)

So that, if the price elasticity of demand is such that \(\epsilon \frac{cb}{c_{e}}=O(\zeta )\), then \(\frac{x_{b}}{D(c_{e})}-1=O(\zeta )\).

Step 2. Recall that:

$$\begin{aligned} (u(x_{b})-c_{b}x_{b})-(u(x_{T_{b}})-p_{T_{b}}x_{T_{b}})=\rho F(T_{b} )-F^{\prime }(T_{b}) \end{aligned}$$
(44)

\(\rho F(T_{b})-F^{\prime }(T_{b})\) is decreasing with \(T_{b}\) as \(F^{\prime \prime }>0,\) so that \(\rho F(T_{b})-F^{\prime }(T_{b})<\rho F\left( \frac{\overline{Z}}{\theta _{d}x_{c_{e}}}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}x_{c_{e}}}\right) .\) Using Eq. (43), it comes that \(\forall c_{b},c_{e}\), there exists \(\epsilon \) such that \(\rho F(T_{b})-F^{\prime }(T_{b})\le \rho F\left( \frac{\overline{Z} }{\theta _{d}x_{b}}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) \). Equation (44) thus implies that:

$$\begin{aligned} (u(x_{b})-c_{b}x_{b})-(u(x_{T_{b}})-p_{T_{b}}x_{T_{b}})\le \rho F\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) \end{aligned}$$

so that

$$\begin{aligned} 0\le p_{T_{b}}x_{T_{b}}-c_{b}x_{b}\le \rho F\left( \frac{\overline{Z} }{\theta _{d}x_{b}}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) \end{aligned}$$

so that

$$\begin{aligned} 0\le \frac{p_{T_{b}}x_{T_{b}}}{c_{b}x_{b}}-1\le \frac{\rho F\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) }{c_{b}x_{b}} \end{aligned}$$

and thus

$$\begin{aligned} 1\le \frac{p_{T_{b}}}{c_{b}}\le \left[ 1+\frac{\rho F\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) -F^{\prime }\left( \frac{\overline{Z} }{\theta _{d}x_{b}}\right) }{c_{b}x_{b}}\right] \frac{x_{b}}{x_{T_{b}}} \end{aligned}$$

Substituting the equation above in Eq. (41), it comes that:

$$\begin{aligned} \frac{x_{b}}{D(p_{i})}-1\le O(\epsilon )\left[ 1+\frac{\rho F\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) }{c_{b}x_{b}}\right] \frac{x_{b} }{x_{T_{b}}} \end{aligned}$$

which can be rewritten, multiplying both sides by \(\frac{x_{T_{b}}}{x_{b}}\):

$$\begin{aligned} \frac{x_{T_{b}}}{D(p_{i})}-\frac{x_{T_{b}}}{x_{b}}\le O(\epsilon )\left[ 1+\frac{\rho F\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) }{c_{b}x_{b}}\right] \end{aligned}$$

For an arbitrarily small \(\zeta \), one can find \(\epsilon \) such that \(\epsilon \left[ 1+\frac{\rho F\left( \frac{\overline{Z}}{\theta _{d}x_{b} }\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) }{c_{b}x_{b}}\right] \le \zeta \). As a result, if \(\epsilon \left[ 1+\frac{\rho F\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}x_{b}}\right) }{c_{b}x_{b}}\right] =O(\zeta )\), then, using Eq. (40): \(\frac{x_{T_{b}}}{x_{b}} =\frac{x_{T_{b}}}{D(p_{i})}+O(\zeta )=1+O(\zeta )\).

So that, \(\forall \zeta ,c_{e},c_{b},x_{b},\overline{Z}\), there exists \(\epsilon \) such that, if the elasticity of demand is always below \(\epsilon \) then \(\forall p\in [c_{e},p_{T_{b}}],\):

$$\begin{aligned} \frac{x_{p}}{x_{e}}=1+O(\zeta ) \end{aligned}$$

For a small local damage, we have shown in Appendix C.2 that \(\frac{dX_{e}}{d\overline{Z}}\) has the sign of \(x_{e}(0)\theta _{e}(\lambda _{0}+\theta _{e}-\theta _{d})\mu _{0}-x_{d}(T_{d})\theta _{d}\lambda _{0}\). Using that, for a sufficiently low elasticity of demand \(x_{e}(0)=x_{d} (T_{d})+O(x_{e}(0)\zeta )\), it comes that \(\frac{dX_{e}}{d\overline{Z}}\) has the sign of \(-x_{e}(0)((\theta _{d}-\theta _{e})(\lambda _{0}+\theta _{e}\mu _{0})+O(\zeta \theta _{d}\lambda _{0}))<0\).

E Cheaper Renewable Energy

E.1 Large Local Damage

Using the same steps as above one can define:

$$\begin{aligned} A\times \left( \begin{array}{c} dT_{e}\\ dX_{e}\\ d\lambda _{0}\\ d\mu _{0} \end{array} \right) =\left( \begin{array}{c} -x_e(T_b)\\ 0\\ 0\\ 0 \end{array} \right) dT_b \end{aligned}$$

with

$$\begin{aligned} A=\left( \begin{array}{ccccc} -x_{e}(T_{e}) &{} -1 &{} \frac{x_{e}(T_{b})-x_{e}(T_{e})}{(\lambda _{0}+\theta _{e}\mu _{0})\rho } &{} \theta _{e}\frac{x_{e}(T_{b} )-x_{e}(T_{e})}{(\lambda _{0}+\theta _{e}\mu _{0})\rho }\\ x_{e}(T_{e}) &{} \frac{\theta _{e}}{\theta _{d}} &{} 0 &{} \frac{x_{e} (T_{e})-x_{d}(0)}{\mu _{0}\rho }\\ \left[ \lambda _{0}+(\theta _{e}-\theta _{d})\mu _{0}\right] \rho &{} 0 &{} 1 &{} \theta _{e}-\theta _{d}\\ 0 &{} -z_{2} &{} 1 &{} 0 \end{array} \right) \end{aligned}$$

where

\(z_{2} =E^{\prime \prime }(X_{e})>0\)

Hence:

\(\frac{dX_e}{dT_b}\) has the sign of \(x_e(T_e)\lambda _0 + x_d(0)(\theta _d \mu _0-(\lambda _0+\theta _e \mu _0))>0\)

E.2 Small Local Damage

$$\begin{aligned} A\times \left( \begin{array}{c} dT_{d}\\ dX_{e}\\ d\lambda _{0}\\ d\mu _{0} \end{array} \right) =\left( \begin{array}{c} 0\\ -x_{d}(T_{b})\\ 0\\ 0 \end{array} \right) dT_b \end{aligned}$$

with

$$\begin{aligned} A=\left( \begin{array}{ccccc} x_{d}(T_{d}) &{} -1 &{} \frac{x_{d}(T_{d})-x_{e}(0)}{(\lambda _{0}+\theta _{e}\mu _{0})\rho } &{} \theta _{e}\frac{x_{d}(T_{d})-x_{e}(0)}{(\lambda _{0} +\theta _{e}\mu _{0})\rho }\\ -x_{d}(T_{d}) &{} \frac{\theta _{e}}{\theta _{d}} &{} 0 &{} \frac{x_{d}(T_{b})-x_{d}(T_{d})}{\mu _{0}\rho }\\ \left[ -\theta _{d}\mu _{0}+(\lambda _{0}+\theta _{e}\mu _{0})\right] \rho &{} 0 &{} 1 &{} -(\theta _{d}-\theta _{e})\\ 0 &{} -E^{\prime \prime }(X_{e}) &{} 1 &{} 0 \end{array} \right) \end{aligned}$$

Hence:

\(\frac{dX_e}{dT_b}\) has the sign of \(\theta _dx_d(T_b)\lambda _0 + \theta _ex_e(0)(\theta _d \mu _0-(\lambda _0+\theta _e \mu _0))\)

F Switch to Clean Energy in the Moratorium Case

Using the envelope theorem:

$$\begin{aligned} \frac{\partial \widetilde{V}(T_{b})}{\partial T_{b}}e^{\rho T_{b}}=\left[ u\left( \widetilde{x}_{d}(T_{b})\right) -(c_{d}+\theta _{d}\widetilde{\mu }_{0}e^{\rho T_{b}})\widetilde{x}_{d}(T_{b})\right] -\pi _{b}-(F^{\prime }(T_{b})-\rho F(T_{b}))\quad \end{aligned}$$
(45)

F.1 Large Local Damage

To avoid confusions between the optimum and the moratorium case, let us denote in this Appendix \(x_{e}^{*}(t)\) the optimal extraction of shale gas, \({\mu }_{0}^{*}\) the optimal initial shadow price of carbon and \(\lambda _{0}^{*}\) the optimal initial scarcity rent of shale gas (for a date of the switch to solar \(T_{b}^{*}\) by definition of the optimum); remember that the variables in the moratorium case are denoted with a \(\widetilde{};\) and denote for instance by \(\widetilde{{\mu }}_{0,T_{b}}\) the initial shadow price of carbon in the moratorium case for a date of the switch to solar \(T_{b}\).

According to Eq. (17), in the case of a large local damage, \(T_{b}^{*}\) is such that:

$$\begin{aligned} \pi _{b}-\left[ u\left( x_{e}^{*}(T_{b}^{*})\right) -(c_{e} +d+(\lambda _{0}^{*}+\theta _{e}\mu _{0}^{*})e^{\rho T_{b}^{*}} )x_{e}^{*}(T_{b}^{*})\right] =\rho F(T_{b}^{*})-F^{\prime }({T} _{b}^{*}) \end{aligned}$$
(46)

Introducing Eq. (46) in Eq. (45), it comes that:

$$\begin{aligned} \left. \frac{\partial V(T_{b})}{\partial T_{b}}\right| _{T_{b}^{*} }e^{\rho T_{b}^{*}}&=\left[ u\left( \widetilde{x}_{d,T_{b}^{*} }(T_{b}^{*})\right) -(c_{d}+\theta _{d}\widetilde{\mu }_{0,T_{b}^{*} }e^{\rho T_{b}^{*}})\widetilde{x}_{d,T_{b}^{*}}(T_{b}^{*})\right] \\&-\left[ u\left( x_{e}^{*}(T_{b}^{*})\right) -(c_{e}+d+(\lambda _{0}^{*}+\theta _{e}\mu _{0}^{*})e^{\rho T_{b}^{*}})x_{e}^{*} (T_{b}^{*})\right] \end{aligned}$$

This expression is strictly positive if and only if:

$$\begin{aligned} c_{e}+d+(\lambda _{0}^{*}+\theta _{e}\mu _{0}^{*})e^{\rho T_{b}^{*} }>c_{d}+\theta _{d}\tilde{\mu }_{0,T_{b}^{*}}e^{\rho T_{b}^{*}} \end{aligned}$$
(47)

At the date \(T_{e}^{*}\) of the switch from coal to gas at the optimum, coal and gas price are equal. Hence:

$$\begin{aligned} c_{e}+d-c_{d}=\left( \theta _{d}{\mu }_{0}^{*}-(\lambda _{0}^{*} +\theta _{e}\mu _{0}^{*})\right) e^{\rho T_{e}^{*}}>0 \end{aligned}$$
(48)

As \(T_{b}^{*}>T_{e}^{*}\), it implies that \(c_{e}+d-c_{d}<\left( \theta _{d}{\mu }_{0}^{*}-(\lambda _{0}^{*}+\theta _{e}\mu _{0}^{*})\right) e^{\rho T_{b}^{*}}\). Using this inequality and assuming that inequality (47) holds, we get that

$$\begin{aligned} {\mu }_{0}^{*}>\widetilde{\mu }_{0,T_{b}^{*}} \end{aligned}$$
(49)

This last inequlity implies that more coal is extracted between dates 0 and \(T_{e}^{*}\) in the moratorium case than in first best. Equality (48) and inequality (49) imply that

$$\begin{aligned} c_{e}+d+(\lambda _{0}^{*}+\theta _{e}\mu _{0}^{*})e^{\rho T_{e}^{*} }>c_{d}+\theta _{d}\widetilde{\mu }_{0,T_{b}^{*}}e^{\rho T_{e}^{*}} \end{aligned}$$
(50)

Together with Assumption (47), inequality (50) implies that for all t in \(\left[ T_{e}^{*},T_{b}^{*}\right] \), \(c_{e}+d+(\lambda _{0}^{*}+\theta _{e}\mu _{0}^{*})e^{\rho t}>c_{d}+\theta _{d}\widetilde{\mu }_{0,T_{b}^{*}}e^{\rho t}\). This last equation implies that extraction of coal in the moratorium case is higher than extraction of gas at the optimum between dates \(T_{e}^{*}\) and \(T_{b}^{*}\), implying more pollution between 0 and \(T_{e}^{*}\) also in the moratorium case than in the optimum. There is a contradiction: emissions are higher at all dates in the moratorium case than at the optimum, which contradicts the fact that the ceiling \(\overline{Z}\) should not be violated in both cases.

F.2 Small Local Damage

Call \(x_{d}^{*}(t)\) optimal extraction path of coal and \({\mu }_{0}^{*}\) the optimal shadow price of carbon. By definition \(T_{b}^{*}\) is such that (small local damage):

$$\begin{aligned} \pi _{b}-\left[ u\left( {x}_{d}^{*}({T}_{b}^{*})\right) -(c_{d} +\theta _{d}{\mu }_{0}^{*}e^{\rho {T}_{b}^{*}}){x}_{d}^{*}(T_{b}^{*})\right] =\rho F(T_{b}^{*})-F^{\prime }({T}_{b}^{*}) \end{aligned}$$
(51)

As a result, introducing (51) in (45), it comes that:

$$\begin{aligned} \left. \frac{\partial V(T_{b},\overline{Z})}{\partial T_{b}}\right| _{T_{b}^{*}}= & {} \left[ u\left( \widetilde{x}_{d,T_{b}^{*}}(T_{b}^{*})\right) -(c_{d}+\theta _{d}\widetilde{\mu }_{0,T_{b}^{*}}e^{\rho T_{b}^{*}})\widetilde{x}_{d,T_{b}^{*}}(T_{b}^{*})\right] \\&-\left[ u\left( {x}_{d}^{*}({T}_{b}^{*})\right) -(c_{d}+\theta _{d}{\mu } _{0}^{*}e^{\rho {T}_{b}^{*}}){x}_{d}^{*}(T_{b}^{*})\right] \end{aligned}$$

so that:

$$\begin{aligned} \left. \frac{\partial V(T_{b},\overline{Z})}{\partial T_{b}}\right| _{T_{b}^{*}}>0\Leftrightarrow \widetilde{\mu }_{0,T_{b}^{*}}<{\mu } _{0}^{*} \end{aligned}$$

F.3 Low Elasticity of Demand

Assume that \(\widetilde{\mu }_{0,{T}_{b}^{*}}<\mu _{0}^{*}\). Then more coal is extracted between dates \(T_{d}^{*}\) and \(T_{b}^{*}\) in the moratorium case than in first best. Before \(T_{d}^{*}\), only gas in extracted in first best and coal in the moratorium. For the ceiling constraint to be satisfied in both cases, it must be the case that:

$$\begin{aligned} \theta _{e}\int _{0}^{T_{d}^{*}}D\left( c_{e}+d+(\lambda _{0}^{*} +\theta _{e}\mu _{0}^{*})e^{\rho t}\right) dt>\theta _{d}\int _{0} ^{T_{d}^{*}}D\left( c_{d}+\theta _{d}\widetilde{\mu }_{0,T_{b}^{*} }e^{\rho t}\right) dt \end{aligned}$$
(52)

We have:

$$\begin{aligned} D(c_{d})\ge D\left( c_{d}+\theta _{d}\widetilde{\mu }_{0,T_{b}^{*}}e^{\rho t}\right) \ge D\left( c_{d}+\theta _{d}\widetilde{\mu }_{0,T_{b}^{*} }e^{\rho T_{b}^{*}}\right) \equiv x(\widetilde{p}(T_{b}^{*})) \end{aligned}$$

Step 1. Expenditure pD(p) is continuous and increasing with p. From Lagrange theorem, there exists a price \(p_{i}\in ]c_{b},\widetilde{p}({T_{b}^{*}})[\) such that:

$$\begin{aligned} \widetilde{p}({T_{b}^{*}})D(\widetilde{p}({T_{b}^{*}}))=c_{b} x_{b}+(D(p_{i})+p_{i}D^{\prime }(p_{i}))(\widetilde{p}({T_{b}^{*}})-c_{b}) \end{aligned}$$

The elasticity of demand at price \(p_{i}\) is \(\epsilon _{i}=-\frac{p_{i}D^{\prime }(p_{i})}{D(p_{i})}\) so that the above equation can be rewritten as:

$$\begin{aligned} \frac{D(\widetilde{p}({T_{b}^{*}}))}{D(p_{i})}=\frac{c_{b}x_{b}}{\widetilde{p}({T_{b}^{*}})D(p_{i})}+(1-\epsilon _{i})\left( 1-\frac{c_{b} }{\widetilde{p}({T_{b}^{*}})}\right) \end{aligned}$$

or:

$$\begin{aligned} \frac{D(\widetilde{p}({T_{b}^{*}}))}{D(p_{i})}-1=\frac{c_{b}}{\widetilde{p}({T_{b}^{*}})}\left( \frac{x_{b}}{D(p_{i})}-1\right) -\epsilon _{i}\left( 1-\frac{c_{b}}{\widetilde{p}({T_{b}^{*}})}\right) \end{aligned}$$

As \(\frac{D(\widetilde{p}({T_{b}^{*}}))}{D(p_{i})}-1<0\) and \(\frac{c_{b} }{\widetilde{p}({T_{b}^{*}})}\left( \frac{x_{b}}{D(p_{i})}-1\right) >0\), denoting \(\epsilon =\max _{i}(\epsilon _{i})\), it comes that:

$$\begin{aligned} -\epsilon<-\epsilon \left( 1-\frac{c_{b}}{\widetilde{p}({T_{b}^{*}} )}\right)<\frac{D(\widetilde{p}({T_{b}^{*}}))}{D(p_{i})}-1<0 \end{aligned}$$

so that:

$$\begin{aligned} \frac{D(\widetilde{p}({T_{b}^{*}}))}{D(p_{i})}-1&=O(\epsilon ) \end{aligned}$$
(53)
$$\begin{aligned} \frac{x_{b}}{D(p_{i})}-1&=O(\epsilon )\frac{\widetilde{p}({T_{b}^{*}} )}{c_{b}} \end{aligned}$$
(54)

Similarly, using Lagrange theorem between prices \(c_{d}\) and \(c_{b}\), one gets, with \(p_{j}\in ]c_{d},c_{b}[\):

$$\begin{aligned} \frac{x_{b}}{D(p_{j})}-1&=O(\epsilon ) \end{aligned}$$
(55)
$$\begin{aligned} \frac{D({c_{d})}}{D(p_{j})}-1&=O(\epsilon )\frac{c_{b}}{c_{d}} \end{aligned}$$
(56)

So that for any arbitrarily small \(\zeta \), if the price elasticity of demand is such that \(\epsilon \frac{c_{b}}{c_{d}}=O(\zeta )\), then:

$$\begin{aligned} \frac{x_{b}}{D(c_{d})}-1=O(\zeta ) \end{aligned}$$
(57)

Step 2. Recall that:

$$\begin{aligned} (u(x_{b})-c_{b}x_{b})-(u(D(\widetilde{p}(T_{b}^{*}))-\widetilde{p} (T_{b}^{*})D(\widetilde{p}(T_{b}^{*}))= \end{aligned}$$
(58)

But \(\rho F(T_{b}^{*})-F^{\prime }(T_{b}^{*})\) is decreasing with \(T_{b}^{*}\) (as \(F^{\prime \prime }>0\)), so that \(\rho F(T_{b}^{*})-F^{\prime }(T_{b}^{*})\le \rho F\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d} D(c_{d})}\right) \). Equation (58) thus implies that:

$$\begin{aligned} (u(x_{b})-c_{b}x_{b})-(u(D(\widetilde{p}(T_{b}^{*}))-\widetilde{p} (T_{b}^{*})D(\widetilde{p}(T_{b}^{*}))\le \rho F\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) \end{aligned}$$

so that:

$$\begin{aligned} 0\le \widetilde{p}(T_{b}^{*})D(\widetilde{p}(T_{b}^{*}))-c_{b}x_{b} \le \rho F\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) \end{aligned}$$

so that:

$$\begin{aligned} 0\le \frac{\widetilde{p}(T_{b}^{*})D(\widetilde{p}(T_{b}^{*}))}{c_{b}x_{b}}-1\le \frac{\rho F\left( \frac{\overline{Z}}{\theta _{d}D(c_{d} )}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) }{c_{b}x_{b}} \end{aligned}$$

and thus:

$$\begin{aligned} 1\le \frac{\widetilde{p}(T_{b}^{*})}{c_{b}}\le \left[ 1+\frac{\rho F\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) }{c_{b}x_{b}}\right] \frac{x_{b}}{D(\widetilde{p}(T_{b}^{*}))} \end{aligned}$$

Substituting the equation above in Eq. (54), it comes that:

$$\begin{aligned} \frac{x_{b}}{D(p_{i})}-1\le O(\epsilon )\left[ 1+\frac{\rho F\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) }{c_{b}x_{b}}\right] \frac{x_{b}}{D(\widetilde{p}(T_{b}^{*}))} \end{aligned}$$

which implies, multiplying both sides by \(\frac{D(\widetilde{p}(T_{b}^{*}))}{x_{b}}\):

$$\begin{aligned} \frac{D{(\widetilde{p}(T_{b}^{*}))}}{D(p_{i})}-\frac{D{(\widetilde{p} (T_{b}^{*}))}}{x_{b}}\le O(\epsilon )\left[ 1+\frac{\rho F\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) }{c_{d}D(c_{d})}\right] \end{aligned}$$

For an arbitrarily small \(\zeta \), one can find \(\epsilon \) such that \(\epsilon \left[ 1+\frac{\rho F\left( \frac{\overline{Z}}{\theta _{d}D(c_{d} )}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) }{c_{d}D(c_{d})}\right] \le \zeta \). As a result, if \(\epsilon \left[ 1+\frac{\rho F\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) -F^{\prime }\left( \frac{\overline{Z}}{\theta _{d}D(c_{d})}\right) }{c_{d}D(c_{d})}\right] =O(\zeta )\), then, using Eq. (53):

$$\begin{aligned} \frac{D{(\widetilde{p}(T_{b}^{*}))}}{x_{b}}=\frac{D{(\widetilde{p} (T_{b}^{*}))}}{D(p_{i})}+O(\zeta )=1+O(\zeta ) \end{aligned}$$

so that, \(\forall \zeta ,c_{d},c_{b},D(c_{d}),\overline{Z}\), there exists \(\epsilon \) such that, if the elasticity of demand is always below \(\epsilon ,\) then \(\forall p\in [c_{d},\widetilde{p}(T_{b}^{*})],\):

$$\begin{aligned} \frac{D(p)}{D(c_{d})}=1+O(\zeta ) \end{aligned}$$

The exact same reasoning gives that: \(\forall p\in [c_{e},{p} (T_{b}^{*})]\):

$$\begin{aligned} \frac{D(p)}{D(c_{d})}=1+O(\zeta ) \end{aligned}$$

A necessary condition for inequality (52) to hold is that:

$$\begin{aligned} \theta _{e}D\left( c_{e}+d+(\lambda _{0}^{*}+\theta _{e}\mu _{0}^{*})e^{\rho T_{b}^{*}}\right) >\theta _{d}D(c_{d}) \end{aligned}$$

But for any arbitrarily small \(\zeta \), if \(\epsilon \) small enough,

$$\begin{aligned} \theta _{e}D(c_{e}+d+(\lambda _{0}^{*}+\theta _{e}\mu _{0}^{*})e^{\rho T_{b}^{*}})=\theta _{e}D(c_{d})+O(\zeta ) \end{aligned}$$

For \(\theta _{d}>\theta _{e}\), one can choose \(\zeta \) such that \(\theta _{e}D(c_{d})+O(\zeta )<\theta _{d}D(c_{d})\) so that inequality (52) does not hold for \(\epsilon \) small enough.

G Optimal Ceiling: Moratorium Versus Optimum

We consider the case of a large local damage.

For ease of notation, call \(p_{b}=\widetilde{p}(\widetilde{T}_{b})\) the optimal final price of coal under the moratorium constraint. We first show that if this final price is given, and \(X_{e}\) is exogenous, the value of \(\mu _{0}\) decreases with \(X_{e}\). At \(X_{e},p_{b}\) given, the price path is defined by the following system of equations:

$$\begin{aligned} \theta _{d}\int _{0}^{T_{e}}D(c_{d}+\theta _{d}\mu _{0}e^{\rho t})dt&=\overline{Z}-Z_{0}-\theta _{e}X_{e} \end{aligned}$$
(59)
$$\begin{aligned} \int _{T_{e}}^{T_{d}}D(c_{e}+(\lambda +\theta _{e}\mu _{0})e^{\rho t})dt&=X_{e}\end{aligned}$$
(60)
$$\begin{aligned} c_{d}+\theta _{d}\mu _{0}e^{\rho T_{e}}&=c_{e}+(\lambda _{0}+\theta _{e}\mu _{0})e^{\rho T_{e}}\end{aligned}$$
(61)
$$\begin{aligned} c_{e}+d+(\lambda _{0}+\theta _{e}\mu _{0})e^{\rho T_{b}}&=p_{b} \end{aligned}$$
(62)

For ease of notation, we denote \(p_{T_{e}}=c_{e}+d+(\lambda _{0}+\theta _{e} \mu _{0})e^{\rho T_{e}}\) and \(p_{0}=c_{d}+\theta _{d}\mu _{0}\). We make the substitution \(u=T_{b}-t\) in the integral of Eq. (60). We get:

$$\begin{aligned} \int _{T_{b}-T_{e}}^{0}D(c_{e}+d+(p_{b}-c_{e}-d)e^{-\rho t})dt=X_{e} \end{aligned}$$
(63)

Differentiating Eq. (63) gives:

$$\begin{aligned} d(T_{b}-T_{e})x(T_{e})=dX_{e} \end{aligned}$$
(64)

Equations (61) and (62) give \(p_{0}=c_{d}+(p_{T_{e} }-c_{d})e^{-\rho T_{e}},\) so that:

$$\begin{aligned} dp_{0}=dp_{T_{e}}e^{-\rho T_{e}}+(p_{T_{e}}-c_{d})e^{-\rho T_{e}}(-\rho e^{-\rho T_{e}}) \end{aligned}$$

so that:

$$\begin{aligned} \frac{dp_{0}e^{\rho T_{e}}}{\rho (p_{T_{e}}-c_{d})}=\frac{dp_{T_{e}}}{\rho (p_{T_{e}}-c_{d})}-dT_{e} \end{aligned}$$
(65)

We make the substitution \(u=T_{e}-t\) in the integral of Eq. (59). We get:

$$\begin{aligned} -\,\theta _{d}\int _{T_{e}}^{0}D(c_{d}+\theta _{d}\mu _{0}e^{\rho (T_{e} -u)})du=\overline{Z}-Z_{0}-\theta _{e}X_{e} \end{aligned}$$

which can be rewritten as:

$$\begin{aligned} \theta _{d}\int _{0}^{T_{e}}D(c_{d}+(p_{T_{e}}-c_{d})e^{\rho (T_{e} -u)})du=\overline{Z}-Z_{0}-\theta _{e}X_{e} \end{aligned}$$
(66)

Differentiating Eq. (66), one gets:

$$\begin{aligned} \theta _{d}x(0)dT_{e}+\left( \theta _{d}\int _{0}^{T_{e}}\frac{dD(c_{d} +(p_{T_{e}}-c_{d})e^{\rho (T_{e}-u)})}{dp}e^{-\rho u}du\right) dp_{T_{e} }=-\,\theta _{e}dX_{e} \end{aligned}$$

which rewrites:

$$\begin{aligned} \theta _{d}x(0)dT_{e}-\left( \theta _{d}\int _{0}^{T_{e}}\frac{dD(c_{d} +(p_{T_{e}}-c_{d})e^{\rho (T_{e}-u)})}{du}du\right) \frac{dp_{T_{e}}}{\rho (p_{T_{e}}-c_{d})}=-\,\theta _{e}dX_{e} \end{aligned}$$

This gives:

$$\begin{aligned} \theta _{d}x(0)\left[ dT_{e}-\frac{dp_{T_{e}}}{\rho (p_{T_{e}}-c_{d})}\right] +\frac{\theta _{d}x(T_{e})dp_{T_{e}}}{\rho (p_{T_{e}}-c_{d})}=-\,\theta _{e} X_{e} \end{aligned}$$
(67)

Using Eqs. (65) and (67), we get:

$$\begin{aligned} \frac{dp_{0}e^{\rho T_{e}}}{\rho (p_{T_{e}}-c_{d})}=\frac{\theta _{d} x(T_{e})dp_{T_{e}}}{\rho (p_{T_{e}}-c_{d})}+\theta _{e}dX_{e} \end{aligned}$$
(68)

But \(p_{T_{e}}=c_{e}+d+p_{b}+(p_{b}-c_{e}-d)e^{-\rho (T_{b}-T_{e})}\), so that:

$$\begin{aligned} dp_{T_{e}}=-(p_{b}-c_{e}-d)e^{-\rho (T_{b}-T_{e})}\rho d(T_{b}-T_{e}) \end{aligned}$$

Using Eq. (64):

$$\begin{aligned} dp_{T_{e}}=-(p_{b}-c_{e}-d)e^{-\rho (T_{b}-T_{e})}\rho \frac{dX_{e}}{x(T_{e} )} \end{aligned}$$
(69)

So that Eq. (68) can be rewritten as:

$$\begin{aligned} \frac{dp_{0}e^{\rho T_{e}}}{\rho (p_{T_{e}}-c_{d})}= & {} \left[ \frac{\theta _{d}(p_{b}-c_{e}-d)e^{-\rho (T_{b}-T_{e})})}{\rho (p_{T_{e}}-c_{d})}+\theta _{e}\right] dX_{e}\equiv \left[ -\theta _{d}\frac{\lambda _{0}+\theta _{e} \mu _{0}}{\theta _{d}\mu _{0}}+\theta _{e}\right] dX_{e}\\\equiv & {} -\frac{\lambda _{0} }{\mu _{0}} \end{aligned}$$

As a result \(\frac{dp_{0}}{dX_{e}}<0\), which gives that:

$$\begin{aligned} \frac{d\mu _{0}}{dX_{e}}<0 \end{aligned}$$

So that for a given final price \(p_{b}\), \(\mu _{0}\) is higher with a moratorium than without. This is a sufficient condition to prove that the initial shadow cost of pollution is higher in the moratorium case than at the optimum, as we showed that the final price in the moratorium case is in fact higher than in the optimum. As a result, the optimal damage is higher in the moratorium case than at the optimum.

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Henriet, F., Schubert, K. Is Shale Gas a Good Bridge to Renewables? An Application to Europe. Environ Resource Econ 72, 721–762 (2019). https://doi.org/10.1007/s10640-018-0223-2

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  • DOI: https://doi.org/10.1007/s10640-018-0223-2

Keywords

  • Shale gas
  • Global warming
  • Non-renewable resources
  • Energy transition

JEL Classification

  • H50
  • Q31
  • Q35
  • Q41
  • Q42
  • Q54