Optimal Extraction of a Polluting Nonrenewable Resource with R&D toward a Clean Backstop Technology

OPTIMAL EXTRACTION OF A POLLUTING NONRENEWABLE

RESOURCE WITH R&D TOWARD A CLEAN BACKSTOP

TECHNOLOGY

FANNY HENRIETParis School of Economics and Banque de France (DGEI-DEMS)

AbstractWe study the optimal extraction of a polluting nonrenew-able resource within the following framework: environmen-tal regulation is imposed in the form of a ceiling on thestock of pollution and a clean unlimited backstop technol-ogy can be developed by research and development. Morespecifically, the time taken to develop a new technology de-pends on the amount spent on R&D. A surprising result isthat the stringency of the ceiling and the size of the initialstock of the polluting nonrenewable resource have a bear-ing on whether environmental regulation speeds up the op-timal arrival date of this new technology. Compared to ascenario with no environmental externalities, stringent en-vironmental regulation drives up the optimal R&D invest-ment and brings forward the optimal backstop arrival dateonly in the case of a large initial resource stock. Otherwise,if the initial resource stock is small, regulation reduces opti-mal R&D and postpones the optimal backstop arrival date.These results are explained by the two roles played by thebackstop technology. First, the backstop serves to replaceoil once it has been exhausted. As extraction is slowed downby regulation, the exhaustion of the nonrenewable resourceis postponed and the long-run gains of innovation are

Fanny Henriet, Paris School of Economics, Paris, France; and Banque de France (DGEI-DEMS), Paris, France ([email protected]).

The author would like to thank Roger Guesnerie and four anonymous referees for theiruseful comments and suggestions on an earlier version of this paper. The views expressedherein are those of the author and not necessarily those of the Banque de France.

Received October 15, 2010; Accepted December 2, 2011.

C© 2012 Wiley Periodicals, Inc.Journal of Public Economic Theory, 14 (2), 2012, pp. 311–347.

311

312 Journal of Public Economic Theory

lowered. Second, environmental regulation raises the short-run gains of innovation by increasing the cost of consumingjust oil.

1. Introduction

New energy technologies, such as wind, solar, or hydro energy, are promotedto tackle climate change. Public R&D is conducted to improve these tech-nologies and lower their costs. However, according to Sinn (2008), bringingforward the date when these new technologies will be broadly available, ata relatively cheap cost, may have detrimental effects on climate change. Inthe absence of a carbon tax, the anticipation of a cheap nonpolluting renew-able backstop resource leads fossil fuels owners to increase extraction and,thus, brings forward the date at which fossil fuels are exhausted. This aggra-vates global warming. The second-best outcome, in the absence of a carbontax, would thus involve decreased R&D toward backstop technologies. This“green paradox” argument has recently received a lot of attention and hasbeen studied, among others, by Chakravorty, Leach, and Moreaux (2011)and Van der Ploeg and Withagen (2010).

When the first best outcome can be reached, thanks to a carbon tax thatcorrects the externality and R&D to replace oil, no green paradox occurs, bydefinition. Nevertheless, it is not straightforward whether this first best out-come involves more or less R&D, aiming to make new technologies available,than in a case without externality. In this paper, we account for the fact thata backstop technology would be developed, regardless of pollution, becauseof the exhaustibility of fossil fuels. We study whether adding environmentalconcern makes it more (or less) profitable to spend R&D toward a backstop,in first best. More precisely, we focus on the effect of imposing a ceiling onthe stock of pollution on the optimal extraction path of a polluting nonre-newable resource and on the optimal R&D toward an alternative backstoptechnology, compared to a case with no externality. To do so, we combinetwo strands of the literature.

The first strand of the literature uses the Hotelling textbook model ofexhaustible resources (Hotelling 1931) to determine the optimal path of ex-traction of a polluting exhaustible resource, and thus the optimal time pathfor a carbon tax, defined as the shadow cost of pollution. The fact that pollut-ing resources are nonrenewable is critical for the design of the optimal timepath of a carbon tax. Chakravorty, Magne, and Moreaux (2006) characterizethe effect of environmental regulation, in the form of a ceiling on the stockof pollution, on the dynamics of the transition from an exhaustible resourceto an existing clean backstop technology. We adopt the same modeling but,rather than assuming that a clean renewable resource is already available ata given cost, we assume that R&D investment at date 0 determines the date

Optimal Extraction with R&D 313

at which a new technology is available, based on a deterministic innovationprocess as in Dasgupta, Gilbert, and Stiglitz (1982).

The second strand of the literature that we refer to deals with the opti-mal timing of R&D toward a backstop technology aiming to replace a nonre-newable resource, without pollution constraint. We assume, as in Dasgupta,Gilbert, and Stiglitz (1982), that the amount devoted to R&D determinesthe date at which a backstop is developed. We compare the optimal back-stop technology arrival date when there is a ceiling on the stock of pollution,with the optimal backstop technology arrival date without any pollution con-straint, as determined by Dasgupta, Gilbert, and Stiglitz (1982).

This paper addresses the following questions: Does the optimal R&Dinvestment toward an alternative backstop technology increase or decreasebecause of environmental concern? We find that the answer is not straight-forward because taking into account the environmental constraint has twodivergent effects. The first is that it postpones the date of fossil fuel (sayoil) exhaustion, lengthening the time for which a potential backstop com-petes with oil. From this point of view, R&D toward an alternative tech-nology and carbon taxation are substitutes: the energy used by future gen-erations can be oil that has been saved due to the carbon tax or analternative backstop. The environmental constraint, by postponing oil ex-haustion, tends to decrease the benefit of developing a backstop. But thesecond effect is that environmental regulation increases the marginal costof consuming fossil fuels and gives incentives to develop the backstop. Itcan be used to complement fossil fuels, when their consumption is limitedby the environmental constraint. We find that, when the initial stock of oilis large, the second effect is larger: taking into account the environmen-tal constraint should lead to increase R&D toward a backstop technology,whereas, when the stock of oil is low, the first effect is larger and imposinga ceiling on the stock of pollution leads to postpone the backstop arrivaldate.

A large literature (see, for instance, Popp 2006 or Chakravorty,Leach, and Moreaux 2009) accounts for the potential for technologicalchange to determine the optimal carbon price. But the way the environmen-tal concern should modify the optimal R&D spending toward a backstoptechnology, compared to a case with no externality, has not received a lot ofattention.

Section 2 introduces the model. Section 3 recalls (see Dasgupta, Gilbert,and Stiglitz 1982) the solution to the planning program when there is noconstraint on the stock of pollution. Then, in Section 4, we move on to thecase where there is a ceiling on the stock of pollution. In order to gain intu-ition, we first consider the case where the initial stock of pollution is alreadyat the ceiling in Subsection 4.1. Then we study the general case with a ceil-ing in Subsection 4.2. Finally, we give some insights on possible extensions(in Subsection 4.3) and possible policy implications (in Subsection 4.4).


2. Model with Pollution and Endogenous Innovation

2.1. Assumptions and Notations

We consider the energy commodity market. The corresponding gross sur-plus associated with its consumption is given by a twice continuously differ-entiable function u, strictly increasing and strictly concave. We denote thecorresponding demand (u′)−1(p ) by D(p ).

Energy can be generated from a nonrenewable resource such as oil. Thenonrenewable resource is extracted from a stock Q0 known with certaintyfrom the outset. Let xt the rate of extraction of the nonrenewable resource.The residual stock Qt at date t is hence:

Qt = Q0 −∫ t

0xτ dτ. (1)

The cost of extraction of this nonrenewable resource is zero.1 The burningof this resource increases the stock of CO2 denoted Zt , in the atmosphere.We assume natural decay at a constant rate α such that:

Zt = xt − αZt . (2)

The stock of pollution at date t is thus (Z0 given):

Zt = e −αt(

Z0 +∫ t

0e αuxudu

).

The environmental constraint is that the stock of pollution should not ex-ceed a limit Z : ∀t , Zt ≤ Z . This ceiling Z may be exogenously fixed by aregulatory institution, or a bang-bang damage function. Once the stock ofpollution has reached the ceiling, the flow of the nonrenewable resourcethat can be burnt cannot exceed the natural decay, i.e., xt ≤ αZ ≡ x. Thecorresponding energy minimal price is given by p = u′(x).

A backstop technology can be developed by R&D. The supply of the en-ergy from the backstop technology is perfectly elastic at price q once it hasbeen developed. We call yt the consumption of backstop energy at date t .The total energy consumed at date t is hence xt + yt . The date at which thisinnovation occurs depends on the amount spent on R&D. As in Dasgupta,Gilbert, and Stiglitz (1982), we assume that the necessary investment is re-duced (in date 0 value) by postponing the date of innovation. Let T be thedate of invention; the R&D cost function is c(T), with c ′(T) ≤ 0. We makethe following assumptions:

1 ASSUMPTION 1: The marginal research cost, expressed at date T , of advanc-ing innovation −c ′(T)e rT is decreasing with T . This assumption ensures thatsocial benefits is concave in the date of innovation T .

1 Choosing a low constant extraction cost would not change the results.


2 ASSUMPTION 2: limT→0 −e rT c ′(T) = +∞, so that for all Q0 > 0 andbackstop cost q > 0 it is never optimal to innovate at date 0.

3 ASSUMPTION 3: limT→∞ −e rT c ′(T) < u(D(q )) − q D(q ) (otherwiseperforming no R&D at all would be optimal, irrespective of Q0).

4 ASSUMPTION 4: q < p . At the ceiling, it is cheaper to use the backstop ifany, than the oil alone.2

2.2. The Social Planner Problem

The social planner maximizes the net surplus by choosing extraction ratext , innovation date T , and rate of use of the backstop yt for t ≥ T , whichmaximize: ∫ T

0u(xt )e −rtdt +

∫ +∞

T(u(xt + yt ) − q yt )e −rtdt − c(T),

subject to: ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Qt = −xt(λT

t

)Zt = −αZt + xt

(μT

t

)Zt ≤ Z

(νT

t

)xt , yt ≥ 0

(aT

t ≥ 0, bTt ≥ 0

)Q0, Z0 given

limt→+∞ Qt ≥ 0.

In the above program we have denoted the Lagrange multipliers in brackets.We denote by bT

t the value of any variable b at date t for a backstop arrivaldate T .

2.3. Solving the Planning Problem

For sake of clarity we first derive the solution for T given. We will, in a sec-ond step, determine the optimal date of innovation. Denote by H (s), theindicator function of R

+. The first order conditions are:

u′(xt + H (t − T)yt ) − λTt + μT

t + aTt = 0,

u′(xt + H (t − T)yt ) − q + bTt = 0.

Together with the complementary slackness conditions:

νTt ≥ 0 and νT

t (Z − Zt ) = 0,

aTt ≥ 0 and aT

t xt = 0,

bTt ≥ 0 and bT

t yt = 0.

2 If q > p , the only role of the backstop is to replace oil once exhausted, innovation isalways postponed by the ceiling constraint.


The dynamics of the costate variables are determined by:

λTt = r λT

t ⇐⇒ λTt = λT

0 e r t ,

μTt = (r + α)μT

t + νTt .

Remark that the costate variable μTt is nonpositive. If Zt < Z over some time

interval, then νTt = 0 and μT

t = μT0 e (r +α)t over that interval. The transversal-

ity conditions at infinity are moreover given by:

limt→+∞ e −rtλT

t Qt = λT0

(lim

t→+∞ Qt

)= 0

and

limt→+∞ e −rtμT

t Zt = 0.

The solution is denoted by starred letters. As long as the nonrenewable re-source is used (a∗T

t = 0), its full marginal cost (equal to the marginal utilityof consumption) is given by p ∗T

t = λ∗Tt − μ∗T

t . This is the sum of the scarcityrent λ∗T

t = λ∗T0 e r t and the shadow cost of pollution −μ∗T

t . At the ceiling, theprice of the resource is p ∗T

t = p if the backstop has not yet arrived (i.e., ift < T). In this case x∗T

t = αZ and y ∗Tt = 0. If the ceiling is binding and the

backstop technology has arrived (i.e., if t ≥ T), then p ∗Tt = q , x∗T

t = αZ , andu′(x∗T

t + y ∗Tt ) = q , so that y ∗T

t = D(q ) − αZ . If the scarcity rent is above p ,the ceiling constraint is not binding (ν∗T

t = 0) and the shadow price followsa Hotelling path: −μ∗T

t = 0 and u′(x∗Tt ) = λ∗T

0 e r t .We can write the following proposition characterizing the solution (with

T fixed).

PROPOSITION 1: With T fixed, calling T ∗e the date of oil exhaustion, the solution

(x∗Tt , y ∗T

t , λ∗T0 , μ∗T

0 , μ∗Tt , T ∗

e ) of the optimization problem is given by3:

1 u′(x∗Tt + y ∗T

t ) = (λ∗T0 e r t − μ∗T

t )H (T − t) + min(q , λ∗T0 e r t − μ∗T

t )H (t − T)

2 −μ∗Tt = max[min(p H (T − t) + q H (t − T) − λ∗T

0 e r t ,−μ∗T0 e (r+α)t ), 0]

3 y ∗Tt = ((D(q ) − x)H (T ∗

e − t) + D(q )H (t − T ∗e ))H (t − T)

4 maxt (e −αt (Z0 + ∫ t0 e αux∗T

u du)) = Z5 T ∗

e = max(T, 1r ln( q

λ∗T0

))

6∫ T ∗

e0 x∗T

u du = Q0

The problem is very similar to that studied by Chakravorty, Magne, andMoreaux (2006). However, unlike them, the backstop technology cannot beused before arrival date T . The solution exposed in Chakravorty, Magne, and

3 If it exists, otherwise the ceiling is not binding and the solution is in Dasgupta, Gilbert,and Stiglitz (1982).


Moreaux (2006) would be the solution to the problem presented here if theR&D cost was zero for all T .

In Proposition 1, the second item shows that several configurations arepossible, depending on arrival date T . For instance if the backstop arriveswhile the ceiling is binding and before exhaustion, then the shadow priceu′(x∗T

t + y ∗Tt H (t − T)) falls from p to q at the very date of innovation. If

the binding date coincides with T then, before innovation date, the shadowprice is equal to the sum of the scarcity rent λ∗T

0 e r t and the shadow price ofpollution −μ∗T

t = −μ∗T0 e (r +α)t , with λ∗T

0 e rT − μ∗T0 e (r +α)T ≤ p , and falls to q

after. In the sequel we shall see all the possible configurations of the optimalshadow prices paths.

At this point of the presentation one can already do several importantremarks. The scarcity rent λ∗T

0 e r t is indeed the price that would be set byoil owners in perfect competition if there was a carbon tax −μ∗T

t , and if thebackstop was planned to arrive at date T . We therefore refer to −μ∗T

t as the“carbon tax” even though there are several possible carbon taxes. Indeed, inthis model, any carbon tax of the form: λe r t − μ∗T

t with λ ≤ λ∗T0 would lead

to the same optimal extraction path, it would only impact the sharing of therent between the tax levier and the oil owners. It is important for the timepath of the carbon tax to depend on the arrival date of the backstop. Forinstance, if the arrival date of the backstop was advanced whereas the carbontax had already been implemented optimally for a later arrival date, it wouldresult in accelerated extraction and more pollution, thereby generating a“green paradox.”

Let denote V (Q0, Z0, Z , T) the value of the program of the social plan-ner, for arrival date T . The second step consists in finding the optimal inno-vation arrival date, which maximizes V (Q0, Z0, Z , T) − c(T).

In the following, we compute the optimal extraction rate and optimalbackstop arrival date. We first compute the optimal price path for each pos-sible innovation date. We derive from the optimal price path for arrival dateT the marginal cost of delaying innovation at this arrival date, and we in-fer optimal arrival date. The next section is devoted to the solution of theproblem in the no externality case.

3. Optimal Innovation Date and Extraction Path: NoExternality Case

The no externality case is a special case of the problem presented above,when Z = +∞. This case is studied in Dasgupta, Gilbert, and Stiglitz (1982).The backstop’s only role is to replace the nonrenewable resource when thereis no oil left. We denote with a˜on top the solution of the problem withoutexternality.

The optimal price path is represented in Figure 1. The main results aresummarized in Proposition 2.


q

t

pt*T = λ0

*Tert

λ0*T

T

Figure 1: Optimal price path and arrival date, no externality case (Z = +∞).

PROPOSITION 2: As shown in Dasgupta, Gilbert, and Stiglitz (1982) in the noexternality case, at the optimal innovation date T :

−c ′(T)e r T = (u(D(q )) − q D(q )) − (u(x∗T

T

) − p ∗TT x∗T

T

)(3)

And the main results are:

• Result 0: Before innovation, the price of the nonrenewable resource rises at therate of interest r , p ∗T

t = λ∗T0 e r t .

• Result 1: The nonrenewable resource is exhausted exactly at the date ofinnovation.

• Result 2: The price of the nonrenewable resource at optimal arrival date T ishigher than the exogenous price of the renewable resource p ∗T

T> q .

• Result 3: The optimal introduction date for the backstop technologyis increasing with the initial stock of the nonrenewable resource: T(Q0)is an increasing function of Q0.

• Result 4: The final price of the nonrenewable resource decreases with the ini-tial stock of the nonrenewable resource.

Result 0 is the standard Hotelling rule. To understand Equation (3), notethat, if innovation is delayed:

• the marginal loss from not consuming the backstop at date T writes:(u(D(q )) − q D(q ))e−r T ,

• the marginal gain from consuming oil instead at date T writes:u(x∗T

T)e −r T ,


• the marginal loss from reduced oil consumption at all dates beforeT , because of delayed innovation, writes: p ∗T

Te −r T x∗T

T. Indeed, it costs

the discounted marginal utility of consumption, p ∗Tt e −rt which is con-

stant along the Hotelling path (and thus equal to p ∗TT

e −r T ), times the

amount of consumption transferred to be consumed at date T, x∗TT

.

• and the marginal benefit from delaying innovation, because of R&Dsaved, writes: −c ′(T)

So that, summing all these terms with the right sign, we get that, at the op-timal arrival date, the marginal cost of delaying innovation (right hand sideof Equation (3)) is equal to the marginal benefit of delaying innovation, be-cause of R&D saved (left hand side of Equation (3)).

Result 1 is a classical result: the backstop and oil are not used simultane-ously. Result 2 is straightforward: the left hand side (LHS) of Equation (3)is positive, the right hand side (RHS) must be positive also at the optimalinnovation date. Result 3 is intuitive. The final scarcity rent, for a given ar-rival date T , decreases with the initial stock Q0, as extraction paths for differ-ent resource stocks do not cross. Hence, the RHS of Equation (3) decreaseswith the initial stock Q0, for a given arrival date T . Then, the optimal in-novation date increases with the initial stock. Result 4 comes from the factthat −e rT c ′(T) decreases with T . Then, the larger Q0, the later is T and thelower is −e rT c ′(T) (LHS of Equation (3)). Then, the RHS of Equation (3)decreases also with Q0, at the optimal arrival date. Hence, the larger is Q0,the lower is the final scarcity rent. This is represented on Figure 2.

From results 3 and 4, the initial scarcity rent decreases with the ini-tial stock Q0. Therefore, the extraction rate is increasing with the initialstock of the nonrenewable resource, as is the stock of pollution in the at-mosphere. For a given ceiling constraint on the stock of pollution Z , thereexists a threshold level for the nonrenewable resource such that, if the initialstock of the nonrenewable resource is over this threshold, the unconstrainedstock of pollution becomes greater than the constrained stock of pollu-tion: ∀Z , ∃Qmin(Z) s.t. Q > Qmin(Z) ⇒ Zt > Z for some t . (Formal proof inAppendix F.)

The same modeling approach is taken in the next section, but the ex-ternality caused by the burning of fossil fuels is taken into account. All thenotations corresponding to the case with no externalities are indicated witha tilde ˜ on top. In the following, we denote N (T) = (u(D(q )) − q D(q )) −(u(x∗T

T ) − p ∗TT x∗T

T ).

4. Optimal Innovation Date and Extraction Path: ExternalityCase

In the following, we recall that we assume αZ < D(q ). This is equivalent toassuming q < p (Assumption (4) of Section 2.1), with q being the price for


q

0Q

tT~

0'0 QQ >

'~T

Figure 2: Optimal price paths and arrival dates, no externality case, for stocks Q0

and Q′0 > Q0.

the backstop once it has been developed. When the ceiling is binding, eitheroil can be consumed at price p or oil and the backstop can be consumedtogether at price q if the backstop has been developed. This second optionis preferable, because q < p . The backstop is not only useful to replaceoil when it is exhausted, but also to lower energy cost when the ceiling isbinding.

4.1. Starting from the Ceiling Z0 = Z

In order to build an intuition and understand how environmental regulationaffects innovation and extraction, we assume first that the initial stock ofpollution is at the ceiling Z0 = Z , this implies that, for all t , xt ≤ αZ ≡ x.

4.1.1. Optimal Price Path for a Given Arrival DateLet denote t0(Q0) the date when oil would be exhausted if xt = αZ for all t :t0(Q0) ≡ Q0

αZ.

If arrival date T is prior to t0(Q0), extraction is limited to αZ at each dateuntil exhaustion at date t0(Q0).

• Before arrival, the ceiling is binding, then ν∗Tt > 0 and p ∗T

t = p . Upto the innovation date, the price is equal to p .

• At the arrival date T , oil is not exhausted and the price falls from pto the backstop price q . As μ∗T

t is nonpositive, item 2 of Proposition 1


pt*T = λ0

*T er t − μt

*T

λ0*T ert

T t0 (Q0 ) t

−μ t*T

p

q

λ0*T

−μ t*T−μ t

*T

Figure 3: Energy price path (bold solid line) for arrival date before t0(Q0) (i.e., oil

is not exhausted at arrival date) in the externality case.

implies that the scarcity rent at arrival date λ∗T0 e rt is thus lower than

q and the shadow cost of pollution falls from −μ∗TT− = p − λ∗T

0 e rT

to −μ∗TT+ = q − λ∗T

0 e rT .

• Then, as long as there is oil left, the nonrenewable resource and thebackstop are used jointly at price q . Exactly x∗T

t = αZ of oil is extractedat each date, and y ∗T

t = D(q ) − αZ at each date until t0(Q0). At datet0(Q0), the scarcity rent reaches the backstop price λ∗T

0 e r t0(Q0) = q andthe oil is exhausted.

• From this date t0(Q0), only energy supplied by the backstop is con-sumed, at price q .

On this price path, as long as there is oil left, the stock of pollution remainsat the ceiling. On the oil extraction path, exactly αZ is extracted at each dateuntil date t0(Q0) ≡ Q0

αZ. The price path is illustrated on Figure 3.

If innovation arrives at date T later than t0(Q0):

• Initially, as the ceiling is binding, then ν∗Tt > 0 and p ∗T

t = p .

• But the scarcity rent λ∗T0 e r t reaches p before t0(Q0). From this date,

as x∗Tt < αZ the ceiling ceases to be binding, the shadow price of


Tt0 (Q0 ) t

p

pt*T = λ0

*T ert − μ t*T

λ0*T

−μ t*T

λ0*Tert

q

p

−μ t*T

Figure 4: Energy price path (bold solid line) for arrival date after t0(Q0) (i.e., oil is

exhausted exactly at arrival date) in the externality case.

pollution −μ∗Tt is zero and resource use follows a Hotelling path

through to exhaustion. During this phase, what determines oil priceis the constraint on the stock of oil (scarcity) and not the constrainton the stock of pollution (ceiling). The resource price is equal to thescarcity rent.

• The final price of oil is over p . Oil is exhausted at the arrival date T .

• From date T , only the backstop is used, at price q .

The price path is illustrated on Figure 4 From these price paths, com-puted for any arrival date, we deduce the optimal arrival date.

4.1.2. Optimal Arrival DateThe value of innovating at date T < t0(Q0) can be written (denotingx ≡ αZ):

V (Q0, Z , Z0, T) =∫ T

0u(x)e −rtdt +

∫ t0(Q0)

T(u(D(q )) − q (D(q ) − x))e −rtdt

+∫ ∞

t0(Q0)(u(D(q )) − q D(q ))e −rtdt.


So that the marginal cost N (T) of delaying innovation expressed at date T ,writes, for all T < t0(Q0):

N (T) = e rT ∂V∂T

= (u(D(q )) − q D(q )) − (u(x) − q x). (4)

Note that this marginal cost of delaying innovation can be broken down intotwo terms.

N (T) = ((u(D(q )) − q D(q )) − (u(x) − p x))︸︷︷︸increased oil price increases N

− x(p − q ).︸︷︷︸oil left at arrival date decreases N

(5)

• The first term in Equation (5) looks like the marginal cost inthe no externality case (right hand side of Equation (3)), N (T) =(u(D(q )) − q D(q )) − (u(x∗T

T ) − p ∗TT x∗T

T ). If for some arrival date T <

t0(Q0), final price in the no externality case satisfies p ∗TT < p , then this

first term is greater than the marginal cost of delaying innovation inthe no externality case, for the same arrival date.

• However, this is not the only driver of innovation. The second term re-duces the marginal cost of delaying innovation. Indeed, as T < t0(Q0),the oil is not exhausted at arrival date T . Hence, delaying innovationdoes not lead to reduce consumption before T . Then, the marginalcost of delaying innovation in the externality case is lower than themarginal cost in the no externality case, for a given price drop fromp to q at arrival date, because extraction is postponed by the ceilingconstraint.

The value of innovating at T > t0(Q) writes, if we denote h(T) the end ofthe ceiling period:

V (Q0, Z , Z0, T) =∫ h(T)

0u(x)e −rtdt +

∫ T

h(T)u(x∗T

t

)e −rtdt

+∫ ∞

T(u(D(q )) − q D(q ))e −rtdt.

As in the no externality case, the marginal cost of delaying innovation can bewritten:

N (T) = (u(D(q )) − q D(q )) − (u(x∗T

T

) − p ∗TT x∗T

T

). (6)

If T > t0(Q0), oil is exhausted at arrival date T . Note that, regulation slowsdown oil extraction initially, the final price p ∗T

T is consequently less than thefinal price in the no externality case for the same arrival date p ∗T

T .


The following proposition holds:

PROPOSITION 3: Starting from the ceiling, the optimal arrival date T ∗ isgiven by4:

−c ′(T ∗)e r T ∗ = N (T ∗).

With, denoting t0(Q0) = Q0

αZ:

N (T) ={

(u(D(q )) − q D(q )) − (u(x) − q x) if T < t0(Q0),

(u(D(q )) − q D(q )) − (u(x∗T

T

) − p ∗TT x∗T

T

)T ≥ t0(Q0

).

N (T) is increasing and −e rT c ′(T) is decreasing so that there is a singlemaximum.

4.1.3. Comparison with the No Externality CaseWe can compare the optimal arrival date in the externality and in the noexternality cases. The following proposition holds:

PROPOSITION 4: Starting from the ceiling, there exists an initial stock of oil Q l

such that:

• For all Q0 ≤ Ql , innovation is postponed by regulation compared to the casewith no externality: T ∗ > T

• For all Q0 > Ql , innovation is advanced by regulation compared to the casewith no externality: T ∗ ≤ T

If limT→+∞ −e rT c ′(T) ≤ u(D(q )) − q D(q ) − (u(x) − q x), then Ql < +∞.

To understand the intuition, note that, in the externality case, the ex-traction rate is limited to x, in order to stay below the ceiling. Even if Q0 isvery large, the oil price before innovation is never below p and the marginalcost of delaying innovation N (T) is bounded below by u(D(q )) − q D(q ) −(u(x) − q x). On the other hand, in the no externality case, the marginalcost of delaying innovation at the optimal arrival date, N (T), depends onthe drop in the prices at the optimal arrival date and decreases with theinitial stock (recall Result 4 of Proposition 2). For Q0 sufficiently large, thismarginal cost of delaying innovation at the optimal innovation date is smallerthan u(D(q )) − q D(q ) − (u(x) − q x). As a result, if Q0 is large enough, inno-vation is profitable sooner in the externality case than in the no externalitycase. This type of case is represented in Figure 5 : at the optimal arrival date

4 V (Q0, Z0, Z , T) is continuous but not derivable in t0(Q0). HoweverlimT→T∗− (e rt ∂V ((Q0 ,Z0 ,Z ,T)

∂T) increases with T ∗, so that V (Q0, Z0, Z , T) − c(T) has a unique

maximum. For ease of notation, we write that, at optimal arrival date −e r T∗ c ′(T ∗) =e r T∗ ∂V (Q0 ,T)

∂Talso if limT→T∗− (e rT ∂V ((Q0 ,Z0 ,Z ,T)

∂T) ≤ −e r T∗ c ′(T ∗) and limT→T∗+ (e rT ∂V ((Q0 ,Z0 ,Z ,T)

∂T) ≥

−e r T∗ c ′(T ∗).


t

q

T

p

λ0*T

λ0*T

p

T

t0(Q0 )

Figure 5: Energy price paths for arrival date T (optimal arrival date in the no

externality case), for Q0 > Ql in the externality case (solid line) and in the no

externality case (dotted line).

of the no externality case T , for Q0 large enough, the price drop is low in theno externality case (see the dotted price path of Figure 5), whereas the pricedrops from p to q in the externality case (see the solid price path of Figure 5)for the same arrival date, innovation is profitable sooner in the externalitycase.

But, if Q0 is small enough, the final price in the no externality case is highand the marginal cost of delaying innovation at the optimal innovation dateT , N (T), is larger than u(D(q )) − q D(q ) − (u(x) − q x). In addition, for Q0

small enough, the mean extraction rate on the optimal price path in the noexternality case is less than x (because the mean extraction rate increaseswith Q0, see Result 4 of Proposition 2)). Then, for the same arrival date T ,oil is exhausted also in the externality case (as, then, T >

Q0

x ≡ t0(Q0)). Butthe final price, at arrival date T , in the externality case is then below thefinal price in the no externality case. Indeed, extraction has been postponedby the ceiling constraint, there is more oil left at arrival date and thus thescarcity rent is lowered. As a result, innovation is profitable sooner in the noexternality case than in the externality case. This type of case is representedin Figure 6: at the optimal arrival date of the no externality case T , forQ0 small enough, oil is exhausted in both the externality case and the no


p

p

T~

t

q

T~

T~

*0

~λT~

*0λ

)( 00 Qt

Figure 6: Energy price paths for arrival date T (optimal arrival date in the no

externality case), for Q0 < Ql in the externality case (solid line) and in the no

externality case (dotted line).

externality case, final price is lower in the externality case, innovation is lessprofitable.

The formal proof is presented below:

Proof : We first look for an initial stock Ql such that the optimal innovationdate is the same in both the externality case and the no externality case.Consider the price p l satisfying

u(x(p l )) − p l x(p l ) = u(x) − q x. (7)

Define Tl such that5:

(u(D(q )) − q D(q )) − (u(x(p l )) − p l x(p l )) = −e r T lc ′(Tl ) (8)

and (Ql , λl ) solution of : ⎧⎪⎨⎪⎩

Ql =∫ Tl

0D(λl e r t )dt,

λl e r T l = p l .

(7)

Then, by construction, for initial stock Ql , the final price in the no externalitycase is p l . And Tl is the optimal date of innovation for initial stock Ql in theno externality case.

5 if T → −e rt c ′(T) is always above (u(D(q )) − q D(q )) − (u(x) − q x), then Q l definedhereafter is equal to +∞.


Note that Equation (7) implies that: u(x(p l )) − p l x(p l ) ≥ u(x) − p x.Then it is straightforward to see that p > p l . For initial stock Ql , the entireoptimal price path is below p l , and then below p , in the no externality case.The quantity extracted at each date on this price path is thus greater than x.

So that, Tl x <∫ Tl

0 x∗Tl

T l dt ≡ Ql , then Tl <Ql

x ≡ t0(Ql ). The marginal cost ofdelaying innovation at date Tl , for this initial oil stock Ql , in the externalitycase, thus satisfies:

N (Tl ) = (u(D(q )) − q D(q )) − (u(x) − q x) (because Tl < t0(Ql ) and

using Proposition 3)

= (u(D(q )) − q D(q )) − (u(x(p l )) (by definition of p l ,

− p l x(p l )) see Equation (7))

= N (Tl )

= −e r T lc ′(Tl ) (by definition of Tl ,

see Equation (8))

So that, for initial stock Ql , optimal arrival date is Tl for both the externalitycase and the no externality case.

For all Q0 > Ql , the optimal arrival date in the no externality case T sat-isfies T > Tl (Result 3 of Proposition 2). And, for Q0 > Ql , t0(Q0) > Tl (be-

cause t0(Q0) = Q0

x >Ql

x = t0(Ql ) > Tl ). So that, for Q0 > Ql , from Proposi-tion 3:

N (Tl ) = u(D(q )) − q D(q )) − (u(x) − q x) = −e −r T lc ′(Tl )

The marginal cost of delaying innovation is unchanged, in the externalitycase, at date Tl , for all Q0 > Ql . So that, for initial stock Q0 > Ql , the opti-mal arrival date in the externality case is Tl whereas it is T > Tl in the noexternality case: the optimal arrival date is advanced by the constraint on thestock of pollution.

For Q0 < Ql , the optimal arrival date T in the no externality case isbefore Tl , and the final price satisfies p ∗T

T> p l . The mean extraction rate

( Q0

T) decreases with Q0 (because of Result 3 and 4 of Proposition 2). As long

as the mean extraction rate is above x, then T <Q0

x ≡ t0(Q0), and in the ex-ternality case N (T) = u(D(q )) − q D(q ) − (u(x) − q x) = N (Tl ) < N (T) =−e r T c ′(T), So that N (T) < −e r T c ′(T). It is less costly in the externality caseto delay innovation: innovation arrives later in the externality case than inthe no externality case. If Q0 is even smaller, the mean extraction rate isless than x on the optimal extraction path of the no externality case, thenT >

Q0

x ≡ t0(Q0). Hence, in the externality case N (T) = u(D(q )) − q D(q ) −(u(x∗T

T) − x∗T

Tp ∗T

T). And the final price p ∗T

Tis below the final price in the no

externality case, for arrival date T : p ∗TT

< p ∗TT

(see Figure 6). Otherwise, less


would be consumed between date 0 and T in the externality case than inthe no externality case. So that oil would not be exhausted at date T in theexternality case, which contradicts the fact that T > t0(Q0). As a result,

N (T) < N (T) = −e r T c ′(T).

Innovation is postponed by the pollution constraint. �The comparison between the marginal cost of delaying the arrival date

in the externality case and in the no externality case depends on the relativemagnitude of two diverging effects: a higher oil price due to the ceiling con-straint and less scarcity of oil because extraction is postponed. The larger theinitial stock of nonrenewable resource Q0, the lower the oil price at optimalarrival date in the no externality case. If Q0 is large, the gains from reducingoil price, from p to q , in the externality case, are sufficiently high comparedto the gain from reducing the oil price in the no externality case to offset thefact that, because of the ceiling constraint, the backstop is in competitionwith oil until oil exhaustion.

4.2. Starting from Below the Ceiling: Z0 < Z

Starting from below the ceiling, it is never useful to innovate before theceiling has been reached. It would result in a dormant technology, whichhas been developed but is not used because cheap oil can be used as longas the ceiling is not binding. This result echoes the Chakravorty, Magne,and Moreaux (2006) finding. In their case, the backstop is never used be-fore the ceiling is reached. We restrict our attention, in the following, tothe case when extraction is slowed down by regulation, for arrival at T(Assumption 5). This seems a reasonable assumption, which is obviously sat-isfied if, for a given arrival date, regulation initially increases oil price. Forall (Z0, Z), a sufficient condition is that Z is not too large compared to Z0

(Then ∀Q0, at the optimal date T of the no externality case, either oil is notexhausted in the externality case or p ∗T

T≤ p ∗T

T, see Appendix C.1).6 In this

case there are two diverging effect of the externality on optimal backstoparrival date: First, the externality makes innovation useful, as it allows forthe consumption of cheap energy despite the externality. Second, regulationpostpones the exhaustion of the oil. In the future, energy from the backstopwill be in competition with oil that has been saved by regulation.

6 If Z − Z0 is large, it might be the case that the proportional pollution dilution makes itprofitable to accelerate extraction (and not to postpone it) so that the final price in theno externality case is below the final price in the externality case. We do not consider thiscase here, but it could be studied in further work. If instead of assuming proportionalnatural decay, we had assumed that α decreases inversely with the stock of pollution in theatmosphere Zt , then this assumption would always be satisfied, irrespective of Z0. Stock de-pendent decreasing natural decay would be more accurate, according to Joos et al. (1996).See a calibration in Appendix C.1.


4.2.1. Optimal Price Path for a Given Arrival DateIt is first necessary to characterize the optimal price path for any arrival dateto compute the marginal cost of delaying innovation at each date. The ex-pression of the marginal cost of delaying innovation depends on the arrivaldate and this expression changes at some pivotal dates that we define below.

The date t1 is the start of the ceiling period when there is backstop avail-able at price q from the outset. If arrival occurs before t1, i.e., while theceiling constraint is nonbinding, the price path is the same as if the backstopwas available from the outset (it is described in Chakravorty, Magne, andMoreaux 2006). The oil price p ∗T

t = λ∗T0 e r t − μ∗T

0 e (r +α)t reaches, along theoptimal path, price q at the start of ceiling period t1. From this date on, thebackstop and the oil are used jointly at price q until oil exhaustion at dateθ1. Date t1 is such that (λ∗t1

0 , μ∗t10 , t1, θ1) satisfy:⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

Zt1 = Z ,

λ∗t10 e r t1 − μ

∗t10 e (r +α)t1 = q ,

λ∗t10 e r θ1 = q ,∫ θ1

0x∗t1

t dt + (θ1 − t1)αZ = Q .

(9)

These price and extraction paths are represented in Figure 7. If the arrivaldate is after t1, then the resource price at the arrival date is greater than q .We define date t2, such that, when the arrival date is between t1 and t2, theceiling is reached exactly at T , at price p ∗T

T between q and p . Before T , theceiling is nonbinding and the shadow price of pollution rises at rate r + α,for t ≤ T , p ∗T


0 e (r +α)t . At arrival date, the price falls from p ∗TT

to q . After the arrival date, the backstop and the nonrenewable resource areused at price q until exhaustion at date h(T). (λ∗T

0 , μ∗T0 , h(T)) satisfy:⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

ZT = Z ,∫ h(T)

0x∗T

t dt = Q ,

λ∗T0 e r h(T) = q .

(10)

The price and extraction paths for arrival date t2 are represented in Figure 8.The oil is exhausted at date t3. If the arrival date is equal to t2 or happensafter t2, the oil price remains equal to p from date t2 until the arrival date oruntil the scarcity rent reaches p . Date t3 is such that, for any arrival date Tbetween t2 and t3, extraction follows the (same) following path: the ceilingis nonbinding in the first phase ending at date t2 at price p ∗T

t2 = p , duringthis first phase p ∗T


0 e (r +α)t ; then, from date t2 until date t3, theoil is used in quantity x at each date until oil exhaustion at date t3, when thescarcity rent reaches q , λ∗T

0 e r t3 = q . Between arrival date T and exhaustion


1*0tλ

11 *0

*0

tt μλ −

q

p

trtrtt ee )(*0

*0

11 αμλ +−rtt e1*

0λ

)(1 Qt )(1 Qθ t

x

)(qD

t)(1 Qt )(1 Qθ

Figure 7: Price and extraction paths for arrival date t1(Q).

t3, both the backstop and oil are used, at price q , exactly x of oil is consumedat each date. For arrival date between t2 and t3, λ∗T

0 = λ∗t20 , μ∗T

0 = μ∗t20 , and

x∗Tt = x∗t2

t . Dates t2 and t3 are such that (λ∗T0 , μ∗T

0 , t2, t3) solve:⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Zt2 = Z ,

λ∗T0 e r t2 − μT

0 e (r +α)t2 = p ,

λ∗T0 e r t3 = q ,∫ t2

0x∗T

t dt + (t3 − t2)αZ = Q .

(11)

For any arrival date between t2 and t3, the extraction path is the same as inFigure 8.

If the arrival date is after t3, the scarcity rent is higher than q at the arrivaldate and the oil is exhausted precisely at the arrival date. We define t4 such


q

p

22 *0

*0

tt μλ −

trtrtt ee )(*0

*0

22 αμλ +−rtt e2*

0λ

2*0tλ

)(2 Qt )(3 Qt t

)(2 Qt )(3 Qt

)(qD

p

x

t)(1 Qθ)(1 Qt

Figure 8: Price path for arrival date t2. The oil is exhausted at date t3 for any

arrival date between t2 and t3.

that for any arrival date between t3 and t4, the scarcity rent at the innovationdate is between q and p . At the arrival date between t3 and t4, the ceilingconstraint is tight and the price falls from p to q , and the oil is exhausted.Denoting g(T) as the start of the ceiling period, for t ≤ g(T), p ∗T

t = λ∗T0 e r t −

μ∗T0 e (r +α)t then, for g(T) ≤ t ≤ T , p ∗T

t = p . With (λ∗T0 , μ∗T

0 , g(T)) solving:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Zg(T) = Z ,∫ T

0x∗T

t dt = Q ,

λ∗T0 e r g(T) − μ∗T

0 e (r +α)g(T) = p .

(12)

For innovation date t4, the ceiling is reached at date θ2. Before that date,p ∗T


0 e (r +α)t , with p ∗Tθ2

= p . From θ2 to t4, price equals p . Oil isexhausted at t4, with final scarcity rent p . The price path and extraction pathfor arrival date t4 are represented in Figure 9 . By definition, (λ∗t4

0 , μ∗t40 , θ2, t4)


)(Q

p

q

)(Qθ

trtrtt ee )(*0

*0

44 αμλ +− rtt e4*0λ

44 *0

*0

tt μλ −4*

0tλ

)(4 Qt)(2 Qθ

x

)(qD

)(2 Qθ )(4 Qt)(2 Qt )(3 Qt)(1 Qθ)(1 Qt

Figure 9: Price path for arrival date t4(Q). The oil is exhausted at the arrival date.

is the solution of: ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Zθ3 = Z ,

λ∗t40 e r θ2 − μ

∗t40 e (r +α)θ2 = p ,

λ∗t40 e r t4 = p ,∫ θ3

0x∗t4

t dt + (t4 − θ2)αZ = Q .

(13)

If the arrival date is after t4, the scarcity rent is higher than p at the arrivaldate, the ceiling constraint is nonbinding at the arrival date and the pricefollows a “Hotelling path”; the oil is exhausted precisely at this date. Callingf1(T) the start of the ceiling period and f2(T) the end of the ceiling pe-riod, for t ≤ f1(T), p ∗T


0 e (r +α)t ; then, for f1(T) ≤ t ≤ f2(T),


p t = p and finally, for t ≥ f2(T), p ∗Tt = λ∗T

0 e r t . With:⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Z f1(T) = Z ,∫ T

0x∗T

t dt = Q ,

λ∗T0 e r f1(T) − μ∗T

0 e (r +α) f1(T) = p ,

λ∗T0 e r f2(T) = p .

(14)

4.2.2. Optimal Arrival DateAll dates (t1, t2, t3, t4) depend on Q and Z .

PROPOSITION 5: The optimal arrival date T ∗ is characterized by:

e −r T ∗c ′(T ∗) = N (T ∗).

With:

N (T) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

0 T ≤ t1,

(u(D(q )) − q D(q )) − (u(x∗T

T

) − x∗TT p ∗T

T

) − x(p ∗T

T − q)

t1 < T ≤ t2,

(u(D(q )) − q D(q )) − (u(x) − q x) t2 < T ≤ t3,

(u(D(q )) − q D(q )) − (u(x) − λ∗T

0 e r t x)

t3 < T ≤ t4,

(u(D(q )) − q D(q )) − (u(x∗T

T

) − p ∗TT x∗T

T

)T > t4.

N(T) is computed in Appendix D.1.

The oil price p ∗TT at arrival date between t1 and t2 increases with T (see

Appendix B.1.1.), and by definition p ∗t1t1 = q and p ∗t2

t2 = p ; the final scarcityrent increases with T between t3 and t4 (see Appendix B.1.2.) and by defini-tion, λ

∗t30 e r t3 = q and λ

∗t40 e r t4 = p . The marginal cost of delaying innovation

is thus increasing with T and continuous at dates t1, t2, t3, and t4. So thatthere is a single solution to the maximization problem. The exhaustion dateis increasing with the innovation arrival date. For any arrival date, the energyprice is first increasing then decreasing at the arrival date. If we assumedlearning by doing in the backstop (as in Chakravorty, Leach, and Moreaux2009), or positive spillovers in the clean sector (as in Acemoglu et al. (forth-coming)), regulation might further speed up the optimal arrival of the cleansubstitute. Assuming an increasing unit cost of clean energy in each periodmay lead to more complicated patterns for the price paths in our case as inChakravorty, Leach, and Moreaux (2009).

The marginal cost of delaying innovation expressed at arrival date,N (T), must be positive at the arrival date: p ∗T ∗

T ∗ > q , so that Result 2 of Propo-sition 2 remains true with environmental regulation. Moreover, Result 3 ofProposition 2 continues to hold. The proof is presented in Appendix E.1.Increasing the stock of the nonrenewable resource reduces the incentive to


innovate. However, if Q0 is sufficiently large, it is as if the supply of Q0 wereinfinite. In the no externality case, if the stock of oil were infinite, innova-tion would be useless, whereas in the externality case, because regulationincreases the price of consuming oil, it is useful to innovate at some dateeven when the stock of oil is infinite. Moreover, there is a cut-off date T lim

such that arrival is never after this date in the externality case (see AppendixE.1).

4.2.3. Comparison with the No Externality CaseThe following proposition holds:

PROPOSITION 6: There exists a stock Ql (Z0, Z)of the nonrenewable resource,such that:

• ∀Q ≥ Ql , regulation advances the optimal date of innovation compared to acase with no externality.

• ∀Q < Ql , regulation postpones the optimal innovation date compared to thecase with no externality.

This result expands on the intuition in the previous section (proof in Ap-pendix F.1). The intuition is the following: for Q0 large enough, optimalfinal price in the no externality case is low enough such that the gain fromreducing marginal consumption cost in the externality case is much higherthan the gain from reducing the marginal consumption cost in the no ex-ternality case. This effect is sufficiently large to offset the fact that, becauseof regulation, the backstop is in competition with oil until oil exhaustion. Ifthe oil stock is sufficiently large, regulation advances the optimal innovationdate. Result 1 of Proposition 2 no longer holds. The oil might not be ex-hausted at the optimal arrival date. This result parallels Chakravorty, Magne,and Moreaux (2006): they show that both the nonrenewable resource andthe renewable resource can be used jointly. However, whether both resourcesare used simultaneously depends on the initial stock of the nonrenewable re-source and not only on the price of energy supplied with the new technology.As the ceiling can be expected to become tighter over time, it is interestingto see how the optimal R&D investment varies with the stringency of theceiling.7

PROPOSITION 7: If, for initial oil stock Q0and ceiling Z l , regulation strictlyadvances innovation, then increasing the stringency of the ceiling from Z l to Z < Z l

also advances the optimal arrival date of the backstop, compared to the no externalitycase. (Proof in Appendix G.1)

7 For values of the ceiling satisfying Assumption 5.


COROLLARY: The smallest initial amount of oil such that regulation advancesinnovation arrival, Q l (Z0, Z ) is decreasing with the stringency of the ceiling, i.e., isincreasing with Z .

The central planner should thus increase R&D if the ceiling is lowered.Note that, in a decentralized economy, the market outcome of lowering theceiling would depend on the structure of the R&D sector. For a private sectorengaging in R&D, there would be also two effects of carbon regulation: thedate at which the backstop is used alone is postponed, so that possible fullmonopoly profits are postponed by regulation, but on the other hand, thebackstop can be used earlier, in combination with oil.

4.3. Insights on Possible Extensions

If the social planner can also conduct additional R&D to decrease the back-stop price, the problem becomes more complicated. The optimal cost q ofthe backstop, as well as its arrival date, depends on the characteristics of thecost function. If, for instance, this cost is separable in its two arguments, qand T , then the aim of the central planner is to find (q , T), which maximizes:

maxq ,T

(V (q , T) − C(q ) − C(T)) = maxq

((maxT

(V (q , T) − C(T)) − C(q )).

The optimal arrival date is T ∗(q ), i.e., the solution of: maxT (V (q , T) −C(T)). If we assume, as before, that regulation postpones extraction for allpossible values of q , we have the following result:

PROPOSITION 8: If the R&D cost function c(T, q ) is separable in its two argu-ments, optimal innovation arrival date T ∗(q )is increasing with the cost q of energysupplied by the backstop. (Proof in Appendix H.1)

The marginal cost of delaying the arrival date decreases with the costof energy supplied by the backstop. If the backstop cost is high, the dropin energy price for a given arrival date is lower and innovation is less desir-able. Thus, the social planner makes a trade-off between developing a cheapbackstop early or an expensive backstop later, as in a case without external-ity. More precise demand and cost function assumptions may be necessarybefore a complete characterization of the solution can be made, and for acomparison between the externality and the no-externality cases, which isbeyond the scope of this paper.

If we assumed positive extraction costs ce greater than the backstop priceq , oil would not be used after innovation so that regulation would only in-crease the marginal cost of consuming oil, but would not lengthen the periodduring which the backstop competes with oil. Starting from the ceiling, thelimit price p l

c is such that: u(x(p lc )) − p l

c x(p lc ) = u(x) − ce x ≤ u(x) − q x, so

that p lc > p l . The corresponding initial oil stock such that regulation leads

to advance innovation is thus smaller: Qlc < Ql . Regulation would advance


optimal arrival date in this case, even for Q0 smaller than Ql defined inSection 3.2.1.

4.4. Policy Implications

If the nonrenewable resource owners are in perfect competition and if thecentral planner chooses the R&D strategy and can put a carbon tax in place,then the optimal carbon tax is equal to the shadow cost of pollution of theplanned economy. It is first increasing then decreasing, as in Chakravorty,Magne, and Moreaux (2006).

Moreover, if the initial stock of oil is large, placing a stringent ceilingon the stock of pollution should drive up the effort toward an alternativebackstop technology, and not to push back the optimal arrival date, contraryto the intuition of the green paradox. On the other hand, imposing a ceilingon the stock of pollution would reduce research into an alternative backstoptechnology compared to the no externality case if the initial stock of oil issmall.

These results do not imply, however, when research is conducted pri-vately, that there should be subsidies to green innovation if the initial oilstock is large. Indeed, it would depend on the market structure associatedwith R&D.

5. Conclusion

We consider the effect of a cap on the stock of pollution on optimal R&Dtoward a clean backstop technology when the effect of R&D is to speed upthe arrival of the backstop. We show that if the price of the backstop tech-nology is such that the environmental cap can be met at a relatively low cost,innovation arrives earlier compared to the case with no externalities whenthe stock of the nonrenewable resource is large. If the stock is small, onthe other hand, the optimal arrival date of the backstop is postponed. Thiscomes from two divergent effects. The backstop fulfills two roles: it is usedto meet the constraint on the stock of pollution, without limiting energyconsumption too much, and it is also used to replace the nonrenewable re-source once it is exhausted. When the initial stock of oil is small, only thesecond role is played by the backstop and regulation postpones the optimalinnovation date for the backstop.

We have examined a case with zero extraction costs, implicitly assumingexogenous technological progress in extraction costs. We have not take intoaccount learning by doing in the clean substitute. One avenue for researchwould be to consider this kind of technological progress.

Another avenue for future research would be to allow for imperfect sub-stitution between the nonrenewable resource and the backstop. Also, wecould discard the hypothesis of the deterministic feature of R&D, the intu-itions would remain the same. Optimal environmental regulation has two ef-fect on the desirability of increasing the probability of arrival of the backstop:


first it postpones exhaustion, so that the backstop is less desirable; second,it increases the price of oil, making the development of a cheap substitutemore desirable. However, introducing probabilistic R&D makes the compu-tation much harder and the optimal environmental regulation would not beas simple as it is in the deterministic model.

More importantly, it would be interesting to study this problem in asecond-best setting, where R&D is conducted privately. Yet the solution of thisproblem cannot easily be decentralized if R&D is conducted by the privatesector (for instance, with a monopoly engaging in R&D at date 0) withoutbinding contracts. The reason for this is that ex ante incentives (lowering thebackstop price in order to accelerate exhaustion) then enter into conflictwith ex post incentives (setting the monopoly price once oil is exhausted).The monopoly price itself depends on the quantity of oil left at the arrivaldate. How the transition to a clean substitute would be affected by regulationwould depend on the market structure associated with R&D.

Appendix A

A.1. For Q0 Large Enough, Zt ≥ Z for some t on the NoExternality Price Path

If there exists Qmin such that the stock of pollution in the atmosphere is aboveZ at some date in the no externality case with endogenous innovation, then,as the optimal initial scarcity rent decreases with Q0, for all Q0 ≥ Qmin, thestock of pollution also exceeds the ceiling at some date. Let find one valueof Q0 such that the stock of pollution in the atmosphere exceeds the ceilingon the no externality price path. Consider (λ, T0) such that p t = λe r t andthe ceiling is reached when the p t reaches q :

e −αT0

(Z0 +

∫ T0

0e αt xt dt

)= Z ,

λe r T0 = q .

For this λ, consider the price path p t = λe r t . There exists a date T , suchthat: e rt c ′(T) = u(x(p T )) − p T x(p T ) − (u(D(q ) − q D(q )). Indeed the func-tion T → −e rt c ′(T) + u(x(p t )) − p t x(p t ) − (u(D(q ) − q D(q )) is continu-ous (as a composition of continuous functions) and is positive, equal to−e r T0 c ′(T0) at date T0, and negative for T sufficiently high such that−e rt c ′(T) + u(x(p t )) − p t x(p t ) − (u(D(q ) − q D(q )) →t→+∞ −α < 0. TakeTmin such that −e r Tmin c ′(Tmin) + u(x(p t )) − p t x(p t ) − (u(D(q ) − q D(q )) =0 and λ defined above, take the stock of pollution Q0, such that if the price isequal to λe r t , Q0 is exhausted at date Tmin. For this Q0, by construction, theceiling is reached at some date on the price path of the no externality case.


Appendix B

B.1. Intermediate Results (IR)

B.1.1. IR 1: Price p TT , Increases with T between t1 and t2

If arrival date is T between t1 and t2, differentiating the system 9 with respectto T :

dλ∗T0

=−

(∫ T

0

∂x∗Tt

∂p ∗Tt

e (r +α)t dt) (

x∗TT − x

)r q dT

xe r h(T)

∫ T

0

∂x∗Tt

∂p ∗Tt

e (r +2α)t +(( ∫ T

0

∂x∗Tt

∂p ∗Tt

e (r +α)t dt)2

−( ∫ T

0

∂x∗Tt

∂p ∗Tt

e r t dt)( ∫ T

0

∂x∗Tt

∂p ∗Tt

e (r +2α)t dt)) ≤ 0

and

dμ∗T0

=

((∫ T

0

∂x∗Tt

∂p ∗Tt

e r t dt)

r q − xe r h(T)

) (x∗T

T − x)

dT

xe r h(T)

∫ T

0

∂x∗Tt

∂p ∗Tt

e (r +2α)t +((∫ T

0

∂x∗Tt

∂p ∗Tt

e (r +α)t dt)2

−(∫ T

0

∂x∗Tt

∂p ∗Tt

e r t dt)( ∫ T

0

∂x∗Tt

∂p ∗Tt

e (r +2α)t dt)) ≥ 0.

As dλ∗T0 ≤ 0 and dμ∗T

0 ≥ 0, it must be the case that dλ∗TT + dμT

∗T ≥ 0, oth-erwise the price between 0 and T would be decreased at all dates when Tincreases and the stock of pollution would exceed the ceiling at some date.

B.1.2. IR 2: Final Scarcity Rent λ∗T0 e rtIncreases with T between t3 and t4

If arrival date is T between t3 and t4, differentiating the system 12 with respectto T :

dλ∗T0 =

(∫ T

0

∂x∗Tt

∂p ∗Tt

e (r +2α)t dt)

xdt(( ∫ T

0

∂x∗Tt

∂p ∗Tt

e (r +α)t dt)2

−( ∫ T

0

∂x∗Tt

∂p ∗Tt

e r t dt)( ∫ T

0

∂x∗Tt

∂p ∗Tt

e (r +2α)t dt)) ≥ 0

B.1.3. IR 3: Date t1(Q) Decreases with QConsider two stocks Q ′ > Q . We call λ (resp. λ′) and −μ (resp. −μ′) thescarcity rent and carbon tax associated with Q(resp. Q ′) when there is abackstop from the outset. It is first straightforward that λ > λ′ ⇔ −μ < −μ′.Otherwise, the pollution stock would exceed the ceiling (or remain strictlybelow) for one of the stocks. Assume moreover that λ > λ′. We denotext (resp. x ′

t ) the extraction at date t . from equation, it must be that :e −αt ′

1 (Z0 + ∫ t ′1

0 e αt xt dt) = Z and e −αt ′1 (Z0 + ∫ t ′

10 e αt x ′(t)dt) = Z If λ < λ′, then

xt − x ′t is decreasing with t , It is first positive and then negative. We call θ1

the date at which price paths cross. It follows that∫ θ1

0 e αt (xt − x ′t )dt = ∫ t ′

1θ1

e αt


(x ′t − xt )dt > 0. Then,

∫ θ1

0 e αt (xt − x ′t )dt <

∫ θ1

0 e αθ1 (xt − x ′t ) and

∫ t ′1

θ1e αt (x ′

t −xt )dt >

∫ θ1

0 e αθ1 (x ′t − xt ), and as a result

∫ t ′1

0 (xt − x ′t )dt > 0. More oil is con-

sumed between 0 and t ′1 with Q than Q ′. But as λ < λ′, the date of oil ex-

haustion is also later with Q than Q ′, so that more oil is consumed, which isin contradiction with Q < Q ′. So that λ > λ′ and −μ < −μ′, implying thatt1(Q) > t1(Q ′)

B.1.4. IR 4: t2(Q) Decreases with Q and t3(Q)Increases with QDate t2 and t3 are defined by system 11. Differentiating this system w.r.t.Q , we get: ((

∫ t20 e (α+r )t x ′(p ∗t2

t )dt)2 − ∫ t20 e (2α+r )t x ′(p ∗t2

t )dt∫ t2

0 e (r )t x ′(p ∗t2t )dt)

r λ∗t20 dλ

∗t20 + x(

∫ t20 e (2α+r )t x ′(p ∗t2

t )dt)dλ∗t20 = −r λ

∗t20

∫ t20 e (α+r )t x ′(p ∗t2

t )dtdQ . Us-ing the Cauchy Schwartz inequality, it appears that dλ

∗t20 < 0, the scarcity rent

decreases with Q . It implies that dt3 > 0. We also find that −dμ∗t20 e (r +α)t2 =

−∫ t2

0 e αt2 e (α+r )t x ′(p∗t2t )dt)

(∫ t2

0 e (2α+r )t x ′(p∗t2t ))

dλ∗t20 e r t2 > −dλ

∗t20 e r t2 . Then: dt2 < 0.

B.1.5. IR 5: For T with t1(Q) < T < t2(Q), Energy Price Increaseswith QWe take Q ′ > Q . Assume that λ′ > λ, then −μ′ < −μ (for ease of notation,we write λ ≡ λ∗T

0 etc), as the prices must cross at some date (otherwise,pollution would exceed the ceiling) . As λ < λ′, then xt − x ′

t is decreasingwith t , It is first positive until date θ1 and then negative. It follows that :∫ θ1

0 e αt (xt − x ′t )dt = ∫ T

θ1e αt (x ′

t − xt )dt > 0. But, because xt − x ′t is decreasing

with t , It is first positive until date θ1 and then negative. So that∫ θ1

0 e αt (xt −x ′

t )dt <∫ θ1

0 e αθ1 (xt − x ′t )dt and

∫ Tθ1

e αt (x ′t − xt )dt >

∫ Tθ1

e αθ1 (x ′t − xt )dt , and as

a result∫ T

0 (xt − x ′t )dt > 0. But as λ < λ′, more oil is consumed after innova-

tion when the stock is Q than when the stock is Q ′, which is a contradiction(more oil is consumed during all the price path). So that Q ′ > Q impliesthat λ > λ′ and −μ < −μ′, so that, at the date of innovation T , p ′

T > p T

B.1.6. IR 6: The Final Scarcity Rent Increases with Q between t3(Q)and t4(Q)For arrival date T between t3(Q) and t4(Q), calling g(T) the date at whichthe ceiling is reached, (g(T), λ∗T

0 , μ∗T0 ) satisfy the set of Equations (12). For

ease of notation, we write λ ≡ λ∗T0 etc). If T is also between t3(Q + dQ) and

t4(Q + dQ), differentiating w.r.t. Q , we get that:

dλ =

∫ g(T)

0e (2α+r )t ∂xt

∂p tdt∫ g(T)

0e (α+r )t ∂xt

∂p tdt

dμ


and

dμ

(∫ g(T)

0e (α+r )t ∂xt

∂p tdt

)2

−∫ g(T)

0e (2α+r )t ∂xt

∂p tdt

∫ g(T)

0e (r )t ∂xt

∂p tdt∫ g(T)

0e (α+r )t ∂xt

∂p tdt

= dQ .

So that, using the Cauchy Schwartz inequality, −dμ > 0 if dQ > 0 anddλ < 0.

Appendix C

C.1. Regulation Postpones Extraction if Z Is Not too LargeCompared to Z0

We first show that for any Z , there is a Z 0, with Z 0 < Z , such that ifZ0 > Z 0, regulation slows extraction down. Assume that Q0 = ∞ and thereis no backstop. The price path with externality is p t = −μ∞e (r +α)t untilp t reaches p at date T Z

∞ when the stock of pollution reaches Z . Fromthis date T Z

∞, the price equals p , we call this price path the infinite pricepath. Define Q Z

∞ the total amount of the nonrenewable resource consumedin this case between dates 0 and date T Z

∞. This date T Z∞(Z0) is decreas-

ing with Z0 and Q Z∞(Z0) is decreasing in Z0. Indeed (μ∞, T Z

∞) are implic-

itly defined by: e −αT Z∞(Z0 + ∫ T Z

∞0 e αt x(p t )dt) = Z and −μe (r +α)T Z

∞ = p . Differ-

entiating w.r.t. Z0, dT Z∞ = e−αTZ∞

−μ(r +α)∫ TZ∞

0 e (r +2α)t x ′(p t )dtdZ0 < 0. As

∫ T Z∞0 x(p t )dt =

Q Z∞. And −μ increases with Z0, then dQ Z

∞ < 0 and dQ Z∞ − dT Z

∞ x = −dμ

∫ T Z∞

0 x ′(p t )e (r +α)t dt = −∫ TZ∞

0 x ′(p t )e (r +α)t dt∫ TZ∞0 x ′(p t )e (r +2α)t dt

e −αT Z∞dZ0 < 0. Consider t1 and λ

solutions of λe r t1 = p and∫ t1

0 x(λe r t )dt = Q Z∞ + (t1 − T Z

∞)x. The functiont1 → ∫ t1

0 x(p e −r (t1−t))dt − t1 x is continuous and increasing with t1, it is equalto zero at t1 = 0 and tends to +∞ when t1 tends to +∞. So that we can findt1 that solves the equation. It cannot8 be the case that t1 ≤ T Z

∞. The quan-tity consumed on the infinite price path between 0 and t1 is the same as thequantity consumed on the Hotelling price path ending at date t1 at price p .

8 Assume that it is the case, then λ > −μ∞ and the quantity consumed between 0 and t1

is less than the quantity consumed between these two dates on the infinite price path.But as the price on the infinite price path is below p between t1 and T Z

∞, and that thetotal quantity consumed on this price path ending at date T Z

∞ is Q Z∞, then the quantity

consumed between 0 and t1 on the infinite price path is strictly less than Q Z∞ − (T Z

∞ −t1)x = Q Z

∞ + (t1 − T Z∞)x. So that the quantity consumed between 0 and t1 on the λe r t price

path is also strictly less than Q Z∞ + (t1 − T Z

∞)x, which contradicts the fact that λ, t1 solve theabove system of equation.


The date t1 decreases with Z0 as∫ t1

0 x(p e −r (t1−t))dt − t1 x is increasing with t1,whereas Q Z

∞ − T Z∞ x is decreasing with Z0. For Z0 high enough, it is then the

case that −e r t1 c ′(t1) ≥ (u(D(q )) − q D(q )) − (u(x) − p x). If this is satisfied,then for Q ∗ = Q Z

∞ + (t1 − T Z∞)x, Q ∗ is sufficiently small so that the final op-

timal price in the no externality case for Q ∗ is above p . If it is the case, thenwe can show that for all Q such that the regulation is binding, regulationpostpones extraction (i.e., either oil is not exhausted at arrival date T in theexternality case, or p T

T≤ p T

T).

Indeed, if final price in the no externality case for arrival date T isabove p then, as the scarcity rent is lowered by the constraint, it is straight-forward that p T

T≤ p T

T. If p T

T≤ p , then Q > Q ∗. We first show that inno-

vation arrives necessarily after date t1 if oil is exhausted in the external-ity case. Assume that innovation arrives at date before t1 defined aboveat final price (in the externality case) lower than p , then less than Q ∗

is consumed between 0 and t1. Indeed, if λ > 0 then −μ ≤ −μ∞. Butthe infinite price path and the t1 price path must cross once before dateT Z

∞ (otherwise if the price path remains below the infinite price path be-fore T Z

∞ the ceiling constraint is not satisfied, and if it is above always,then less than Q ∗ is consumed). Between 0 and t1, it is the case that∫ t1

0 e αt xt dt ≤ ∫ t10 e αt x∞

t dt = Z . As prices cross once, there exists θ , such thatfor t < θ , the infinite price is below and for t > θ1, infinite price is above. Sothat

∫ θ

0 e αt (x∞t − xt )dt ≥ ∫ t1

θe αt (xt − x∞

t )dt ≥ 0, so that e αθ∫ θ

0 (x∞t − xt )dt >∫ θ

0 e αt (xt − x∞t )dt ≥ ∫ t1

θe αt (xt − x∞

t )dt > e αθ∫ t1θ

(xt − x∞t )dt . So that more is

consumed on the infinite price path between 0 and t1. So that less than Q ∗

can be consumed between 0 and t1, so that T > t1 necessarily. If T > t1, atdate t1, as Q > Q ∗ and oil is exhausted at price below p , then price of theno externality case is below p . Then the quantity consumed between 0 andt1 is more than Q ∗. On the other hand, on the externality price path, wehave seen that the quantity consumed increases with Q , so that between 0and t1, less is consumed on the externality price path than if Q is infinite,i.e., less than Q ∗ is consumed. But the externality price after t1 is p , then lessis consumed also on the externality path between t1 and T than on the noexternality price path, which contradicts the hypothesis that oil is exhaustedat T in both cases.

Similarly, we show that for any initial stock of pollution Z0, there isa Z ∗ > Z0 such that if Z ≤ Z ∗, regulation slows extraction down. Take(Q∞, T∞, λ, t1) defined above. It is straightforward that λ decreases with Z .Call t2(Z) such that λe r t2 = p 0. For Z = Z0, t2 = t1 = 0. The date t2 is con-tinuous and increasing with Z . We can find t∗

2 such that, for all T ≤ t∗2 ,

−e rt c ′(T) > u(D(q )) − q D(q ) − (u(x0) − p 0 x0). Denote Z ∗ the value of theceiling corresponding to t∗

2 . Then Z ∗ < Z0 as t∗2 > 0. For all value of the ceil-

ing such that Z ∗ < Z < Z0, the optimal final price of the no externality case,for Q0 = Q∞ + (t1 − T∞)αZ , is above p . The end of the demonstration is thesame as above.


Remark: One can find a backstop cost q ∗ such that Assumption 4 is al-ways true for all (Z0, Z) satisfying D(q ) > αZ > αZ 0, as long as q > q ∗. Us-ing data on price elasticity in Chakravorty, Magne, and Moreaux (2009),data on sectoral energy demands and CO2 content of fossil fuel inCoulomb and Henriet (2010). We compute Q ∗ defined above. We findthat if the backstop was to arrive at price below p for this Q ∗, it wouldbe necessary that the backstop arrives before 7.8 years, which is highlyunlikely.

Appendix D

D.1. Marginal Cost of Delaying Innovation

For T ≤ t1, the marginal change in surplus is zero. If innovation arrives atdate T between date t1 and date t2, the price path is such that the ceiling isreached exactly at date T , the scarcity rent λ (for ease of notation, we denoteλ ≡ λ∗T

0 etc.) the shadow cost of pollution −μ, exhaustion date h(T) satisfy:

e −αT(

Z0 +∫ T

0e αt xt dt

)= Z , (D1)

∫ T

0xt dt + (h(T) − T)αZ = Q , (D2)

λe r h(T) = q . (D3)

And the surplus can be written:∫ T

0 u(xt )e −rtdt + ∫ h(T)T (u(D(q )) − q (D(q ) −

x)e −rtdt + ∫ ∞h(T)(u(D(q )) − q D(q ))e −rtdt. The marginal cost of delaying in-

novation, expressed at date T , is:

N (T) = −e rt ∂

∂T

( ∫ T

0u(xt )e −rtdt +

∫ h(T)

T(u(D(q )) − q (D(q ) − x)e −rtdt

+∫ ∞

h(T)(u(D(q )) − q D(q ))e −rtdt

). (D4)

Developing Equation (D4): N (T) = −u(xT ) − e rt∫ T

0 p t∂xt∂T e −rtdt −

∂h(T)∂T e r (T−h(T))q x + (u(D(q )) − q (D(q ) − x)). And we know that between

the dates 0 et T , p t = (λ − μe αt )e r t , then∫ T

0p t

∂xt

∂Te −rtdt = λ

∫ T

0

∂xt

∂T−

∫ T

0μ

∂xt

∂Te αt dt. (D5)

Differentiating Equation (D1), we obtain: −αZ + xT + e −αT∫ T

0 e αt ∂xt∂T = 0 so

that :∫ T

0 e αt ∂xt∂T = (αZ − xT )e αT . Differentiating Equation (D2), we obtain:∫ T

0∂xt∂T = (αZ − xT ) − ∂h(T)

∂T αZ so that Equation (D5) can be rewritten:


∫ T0 p t

∂xt∂T e −rtdt = λ(αZ − xT ) − μ(αZ − xT )e αT − λ

∂h(T)∂T x. Moreover, using

Equation (D3) and the fact that p T = λe rt − μe (r +α)T , and x = αZ :∫ T0 p t

∂xt∂T e −rtdt = (x − xT )p T e −rt − q e −r h(T) ∂h(T)

∂T x. Delaying the arrival dateof innovation induces a marginal loss in welfare, at the arrival date:

N (T) = −(u(xT ) − xT p T ) − x(p T − q ) + (u(D(q )) − q D(q )).

If innovation occurs between t2 et t3, the price is p t = λe r t − μe (r +α)t untildate t2, p between t2 and T , oil and the backstop are used jointly at priceq until date t3, when oil is exhausted, the backstop is used alone at price q .During this period, λ, μ, the date at which the ceiling is reached t2 and ex-haustion date t3 do not depend on T . The surplus writes:∫ t2

0u(xt )e −rtdt +

∫ T

t2(u(x)e −rtdt +

∫ t3

T(u(D(q )) − q (D(q ) − x)e −rtdt

+∫ ∞

t3(u(D(q )) − q D(q ))e −rtdt. (D6)

Deriving Equation (D6), we find that delaying the innovation date inducescost, expressed at date T :

N (T) = −(u(x) − q x) + (u(D(q )) − q (D(q )).

If innovation occurs between t3 and t4, then the price path is first p t = λe r t −μe (r +α)t until the ceiling is reached at price p , then during the ceiling period,price equals p until exhaustion. At exhaustion date, the scarcity rent is aboveq and below p , it is increasing with T . The scarcity rent λ, the shadow cost−μ, and the date g(T) when the ceiling is reached satisfy:

e −αg(T)(

Z0 +∫ g(T)

0e αt xt dt

)= Z , (D7)

∫ g(T)

0xt dt + (T − g(T))αZ = Q , (D8)

λe rt − μe (r +α)T = p . (D9)

The surplus writes:∫ g(T)

0 u(xt )e −rtdt + ∫ Tg(T) u(x)e −rtdt + ∫ ∞

T (u(D(q )) −q D(q ))e −rtdt. The marginal cost of delaying innovation, expressed at date T ,writes: N (T) = −e rt ∂

∂T (∫ g(T)

0 u(xt )e −rtdt + ∫ Tg(T) u(x)e −rtdt + ∫ ∞

T (u(D(q )) −q D(q ))e −rtdt) = u(D(q )) − q D(q ) − e rt

∫ g(T)0 p t

∂xt∂p t

e −rtdt − u(x). Using that

p t = λe r t − μe (r +α)t ,∫ g(T)

0 p t∂xt∂p t

e −rtdt = λ∫ g(T)

0∂xt∂p t

dt − μ∫ g(T)

0∂xt∂p t

e αt dt. De-

riving Equation (D8): λ∫ g(T)

0∂xt∂p t

dt = −λx. Deriving Equation (D7):

λ∫ g(T)

0 e αt ∂xt∂p t

dt = 0. So that: N (T) = u(D(q )) − q D(q ) − (u(x) − λe rt x).


If innovation occurs after t4, oil price follows a Hotelling path at ar-rival date, and oil is exhausted at this date. The surplus can be writ-ten:

∫ T0 u(xt )e −rt + ∫ ∞

T (u(D(q )) − q D(q ))e −rtdt. So that the marginal costof delaying innovation, at arrival date, writes: N (T) = (u(D(q )) − q D(q )) −e rtu(xT )e −rt − e rt

∫ T0 p t

∂xt∂T dt. Using that p t = λe r t after date T , we get

N (T) = (u(D(q )) − q D(q )) − u(xT ) − λe rt∫ T

0∂xt∂T dt. Oil is exhausted at date

T , so that∫ T

0 xt = Q , implying that∫ T

0∂xt∂T dt = −xT . The marginal cost of de-

laying innovation, expressed at the arrival date, writes: N (T) = (u(D(q )) −q D(q )) − (u(xT ) − p T xT ).

Appendix E

E.1. Proof That Result 3 of Proposition 2 Continues to Hold

In order to demonstrate that Result 3 of Proposition 2 continues to hold, weadopt a recursive reasoning. We show first that for all Q 1

0 such that innova-tion arrives at date T1 after t2(Q 1

0 ), then for any Q 20 > Q 1

0 innovation arrivesat date T2 > T1, satisfying T2 > t2(Q 2

0 ). Denote N1(T) the marginal cost ofdelaying innovation for Q0 = Q 1

0 and N2(T) the marginal cost of delaying in-novation for Q0 = Q 2

0 . Denote λT0 (Q i

0) the initial scarcity rent for arrival dateT and Q0 = Q i

0. Notice first that T1 > t2(Q 20 ) as t2(Q0) is decreasing with Q0

(see Appendix B.1.4.)If T1 satisfies t2(Q 1

0 ) ≤ T1 ≤ t3(Q 10 ), then t2(Q 2

0 ) ≤ T1 ≤ t3(Q 20 ), so

N1(T1) = N2(T1), as t3(Q0) increases with Q0 (see Appendix B.1.4.).If T1 satisfies t3(Q 1

0 ) < T1 ≤ min(t3(Q 20 ), t4(Q 1

0 )), then N1(T1) =u(D(q )) − q D(q ) − (u(x) − λ

T10 (Q 1

0 )e r T1 x) ≥ u(D(q )) − q D(q ) − (u(x) −q x) = N2(T1). If t3(Q 2

0 ) < t4(Q 10 ), and innovation arrives between t3(Q 2

0 )and t4(Q 1

0 ), as the final scarcity rent λTT decreases with Q0 (see Appendix

B.1.6.), then it is the case that N1(T1) = u(D(q )) − q D(q ) − (u(x) −λ

T10 (Q1)e r T1 x) ≥ u(D(q )) − q D(q ) − (u(x) − λ

T10 (Q2)(T1)e r T1 e r T1 x) = N2

(T1). If t3(Q 20 ) > t4(Q 1

0 ), and innovation arrives between t4(Q 10 ) and t3(Q 2

0 ),N1(T1)≥u(D(q ))−q D(q )−(u(x)−p x) ≥ u(D(q ))−q D(q )−(u(x) − q x) =N2(T1).

Between max(t3(Q 20 ), t4(Q 1

0 )) and t4(Q 10 ), N1(T1) ≥ u(D(q )) − q D(q ) −

(u(x) − p x) ≥ u(D(q )) − q D(q ) − (u(x) − λT10 (Q2)(T1)e r T1 x) = N2(T1).

After t4(Q 10 ), the bigger Q0 the lower the scarcity rent and so N1(T1) ≥

N2(T2). So that, at T1, −e r T1 C ′(T1) ≥ N2(T1), so that optimal arrival for Q 20

is after T1. We find that T2 > T1 > t2(Q 20 ).

We show now that for Qmin, optimal arrival is after t2(Qmin). We knowthat final price p min at optimal arrival date for Q0 = Qmin satisfies p min > p(see Appendix C.1). We call θ2 the date at which price p is reached alongthe optimal price path for Q0 = Qmin. As for t > θ2 p t > p , the stock of pollu-tion in the atmosphere declines, for t > θ2 whereas while t ≤ θ2, the stock ofpollution in the atmosphere increases, the ceiling is reached exactly at date


θ2. By definition, t2(Qmin) is the lowest possible date such that the ceiling isreached, when Q0 = Qmin, at price p . So that, θ2 < t2(Qmin) and as final priceis above p min > p , optimal arrival date is at Tmin ≥ θ2 > t2(Q). So that forQ0 = Qmin, optimal arrival date is after t2(Qmin), so that by recursive reason-ing, optimal innovation for increases with Q0.

Corollary: Arrival date is always after t2(Q0).

Appendix F

F.1. Proof of Proposition 6

We first show that, if there exists a Ql such that regulation advances inno-vation, then for all Q > Ql , regulation also advances innovation. We assumethat there is a Ql such that at optimal innovation date Tl in the no external-ity case, N (Ql , Tl ) > N (Ql , Tl ). Notice first that, if N (Ql , Tl ) > N (Ql , Tl )then oil is not exhausted at this date. If it was the case, then it would bethe case that p ∗Tl

T l > p ∗Tl

T l , which is not possible if N (Ql , Tl ) > N (Ql , Tl ).Indeed, one can verify that N (Ql , Tl ) ≤ (u(D(q )) − q D(q )) − (u(x∗Tl

T l ) −p ∗Tl

T l x∗Tl

T l ), so that if p ∗Tl

T l > p ∗Tl

T l , then N (Ql , Tl ) ≤ N (Ql , Tl ). So: if inno-vation is advanced for a given Ql , it must be the case that, under the pol-lution constraint, oil is not exhausted at the optimal innovation date ofthe no externality case, and as a result, that Tl < t3(Ql ). We have shownthat t3(Q0) increases with Q0 (see B.1.4.), so that ∀Q0 > Ql , Tl < t3(Q0).let denote T ∗ the optimal innovation date in the externality case for Q0 >

Ql . We know that, for Q0 > Ql , t1(Q0) < t1(Ql )(see B.1.3), then if the T ∗

is between t1(Q0) and t1(Ql ), then N (T ∗, Q0) > N (T ∗, Ql ) = 0. So thatT ∗ < Tl . We differentiate u(x∗T

T ∗ ) − p ∗T ∗T ∗ x∗T ∗

T ∗ + x(p ∗T ∗T ∗ − q ) w.r.t. Q0, we ob-

tain (x − x∗T ∗T ∗ ) ∂p ∗T∗

T∂Q0

. But we know that ∂p ∗T∗T∗

∂Q0> 0 (see Appendix B.1.5.), so

that if T ∗ is between dates t1(Ql ) and t2(Q0) (t2(Q0) < t2(Ql ) see B.1.3),then N (T ∗, Q0) > N (T ∗, Ql ), so that T ∗ < Tl . Between dates t2(Q0) andt2(Ql ), it is straightforward that N (T ∗, Q0) > N (T ∗, Ql ) as, N (T ∗, Q0) ≥N (t2(Q0), Q0) = N (t2(Ql ), Ql ) > N (T ∗, Ql ), so that T ∗ < Tl . If T ∗ is be-tween t2(Ql ) and t3(Ql ), then N (T ∗, Q0) = N (T ∗, Ql ), so that T ∗ = Tl . Sothat it is always the case T ∗ ≤ Tl . On the other hand, if Q0 > Ql , we know thatT > Tl , so that for Q0 > Ql , we have that T ∗ < T . So that the implication:⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

Q > Ql ,

N (Ql , Tl ) > N (Ql , Tl ),

N (Ql , Tl ) = −e r T lc ′(Tl ),

N (Q , T) = −e rt c ′(T),

⇒ N (Q , T) > N (Q , T),

is true. We show now that there exists an initial stock Ql of the nonre-newable resource such that, at the optimum, innovation arrives earlier inthe externality case than in the no externality case. Consider the date Tl


satisfying (u(D(q )) − q D(q )) − (u(x) − q x) = −e r T lc ′(Tl ). Take Q1 such

that innovation arrives strictly after t2(Qmin) in the no externality case,and such that N1(Tl ) < −e r T l

c ′(Tl ) ((N1(Tl ) decreases with Q0 and tendsto 0). Then, as t2(Q0) decreases with Q0, optimal no externality arrivaldate T > t2(Q1). We have then that N1(T) ≥ (u(D(q )) − q D(q )) − (u(x) −q x) = −e r T ∗

c ′(Tl ) ≥ N1(Tl ) ≥ N1(T).

Appendix G

G.1. Proof of Proposition 7

Take Z1 < Z ∗. According to Appendix E.1, arrival date for Z ∗ isafter t2(Q , Z ∗). Moreover, as we have assumed that regulation postponesinnovation for Z ∗, then it must be the case that optimal arrival date for Z

∗

is prior to t3(Q , ¯Z ∗). We can show now that t3 increases when Z decreases.Indeed, using the set of equation defining t3, we get that:

dt3 =−dZ e r t3

(∫ t2

0x ′

t e(r +α)t dt +

∫ t2

0x ′

t e(r +2α)t dt(t3 − t2)α

)

xe r t3 intt20 x ′(t)e (r +2α)t dt + r λe r t3

((∫ t2

0x ′

t e(r +α)t dt

)2

−∫ t2

0x ′

t e(r +2α)t dt

∫ t2

0x ′

t er dt

) .

So that we dZ < 0, we get dt3 > 0. If dt2 < 0, as u(x∗) − q x∗ > u(x) − q x,then it is straightforward that more stringent ceiling advances innovation.If dt2 > 0, we find that dλ < 0 (cf dt3 > 0) and then necessarily −dμ > 0.As a result, the price at arrival date t2(Z ∗) in the case where the ceilingis Z1 is greater then p ∗. Then: u(xt2(Z ∗)) − p t2(Z ∗)xt2(Z ∗) + x1(p t2(Z ∗) − q ) =u(xt2(Z ∗)) − q xt2(Z ∗) − (p t2(Z ∗) − q )(xt2(Z ∗) − x1) ≤ u(xt2(Z ∗)) − q xt2(Z ∗) ≤u(x∗) − q x∗ As a result, if arrival date for Z ∗ is between t∗

2 and t2(Z1),innovation is also advanced by a more stringent ceiling.

Appendix H

H.1. Proof of Proposition 8

We first show that t2(Q) increases with q . Indeed, taking equations definingt2(Q) and differentiating w.r.t. q , we get:

dt2 =x

(e r t2

∫ t2

0e (r +α)t x ′(p t )(e αt1 − e αt )dt

)dq

(r λe r t2 + (r + α)μe (r +α)t2 )

(xe r θ2

∫ t2

0x ′

(e (r +2α)t dt + r q

((∫ t2

0x ′(e (r +α)t dt)

)2

−∫ t2

0x ′

(e (r +2α)t dt

∫ t2

0x ′e r t dt

)) .

Using the Cauchy Schwartz inequality, we get that dt2 > 0. Innovation alwaysarrives after t2 (see corollary in E.1). We can show that dt3 > 0, indeed, weget:

dt3 =x

(∫ t2

0e (r +α)t x ′(p t )dt

)dq

(r λe r t2 + (r + α)μe (r +α)t2 )(

xe r θ2

∫ t2

0x ′

(e (r +2α)t dt + r q

((∫ t2

0x ′(e (r +α)t dt))2 −

∫ t2

0x ′

(e (r +2α)t dt

∫ t2

0x ′e r t dt

)) .


It is clear that dt4 = 0. Consider q1 < q2, at date t2(q2),N (t2(q2), q1) =u(D(q1)) − q1D(q1) − (u(x) − q1 x) > u(D(q2)) − q2D(q2) −(u(x) − q2 x) = N (t2(q2), q2). So that N (t2(q2), q1) > N (t2(q2), q2). For allT satisfying t3(q2) > T > t2(q2), N (T, q1) = N (t2(q2), q1) > N (t2(q2), q2) =N (T, q2). Between t3(q1) and t3(q2), N (T, q1) > u(D(q1)) − q1D(q1) −(u(x) − q1 x) > u(D(q2)) − q2D(q2) − (u(x) − q2 x) = N (t2(q2), q2). Aftert3(q2), the price path, in the externality case, is the same for both cases butas u(D(q1)) − q1D(q1) > u(D(q2)) − q2D(q2), N (T, q1) > N (t2(q2), q2). Sothat the higher q (below p ), the later innovation.

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