Rémy Dullieux
email: remy.dullieux.2@wanadoo.fr

Lionel Ragot
email: lionel.ragot@univ-evry.fr

Katheline Schubert
email: schubert@univ-paris1.fr

Carbon tax and OPEC's rents under a ceiling constraint

Keywords: global warming, non-renewable resources, di¤erential games, non-linear strategies JEL classi…cation: C73, Q30, Q4

We study the MPNE of a game between oil importing countries seeking to maintain atmospheric carbon concentration under a given ceiling and exporting countries, when the former sets a carbon tax and the latter controls the producer price. We obtain implicit feedback rules and explicit non-linear time paths of extraction, carbon tax and producer price. Consumers are always able to reap some part of the scarcity and monopoly rents, whereas producers partially preempt the carbon tax only if the marginal damage under the ceiling is small. We compare the MPNE to the e¢cient, open loop and cartel without tax equilibria.

I Introduction 1

Three key lessons stand out from the Copenhagen Summit in December 2009. First, the agreement recognizes the need for the temperature rise to stay below 2 degrees Celsius, which is usually associated with a greenhouse gas concentration ceiling of 450 ppm (IPCC (2007)). This shared objective is a signi…cant progress obtained in Copenhagen. The two other lessons are less constructive: the inability of the largest emitting countries to reach even a basic e¤ective agreement on an international policy architecture designed to aim towards this common objective; and OPEC's hostility to any international agreement which would …nally result in a sensible contraction of world oil demand. OPEC repeatedly argues that climate policy is still another excuse to steal the oil rent to …nance public spending in oil importing countries, and even asks for compensation for the reduction of its income.

That OPEC does not contemplate introducing a climate policy, and reacts negatively when oil importing countries do, is not entirely surprising. As [START_REF] Wirl | Leviathan governments and carbon taxes: Costs and potential bene…ts[END_REF] mentioned, an example of such strategic OPEC's reaction is the oil price increase of $4 per barrel in the 1992's …rst-half, matching the …rst step of the EEC proposal of a hybrid energy-carbon tax. The crude oil price increase allows OPEC to make climate policy to a certain extent useless since it may be su¢cient to trigger the demand decrease desired by consuming countries. By doing so, OPEC captures a part of what we can call the carbon or climate rent.

However, oil consuming countries can also behave as a coalition and adopt a carbon tax which allows them to reap some part of the oil rent 2 . A global climate agreement of oil importing countries coordinating their carbon taxation could be interpreted as a consumers' cartel, would it happen to exist 3 . The implementation of a high carbon tax could prompt OPEC to lower its producer price in order to limit the decrease of oil demand.

Since the increases in oil price between 1998 and 2008, and especially the very strong increase of summer 2008, many OECD countries have led the idea that the government must control the consumer price of oil and that, to do so, taxes on energy should be " ‡oating", or "additional", to use two terms coined in France: they should decrease when the producer price increases, and vice versa. This proposal is clearly a bad one, as OECD (2006) points out, for strategic reasons: "If oil importers start to reduce taxes in order to stabilise tax-inclusive fuel prices, oil exporters will know that they can at no risk increase their resource rents by restraining their production and thus increase crude prices further.

Normally such actions would trigger reductions in demand that could reduce the incomes of the oil producers, but the demand reductions will be absent if tax-inclusive user prices are kept stable by tax reductions."

What is the result of the previous two-sided strategic behaviour? Does the producers' cartel capture the climate rent or does the consumers' cartel capture the scarcity and monopoly rents? Is the time path of extraction more or less conservative than what would 2 Besides objectives of public spending …nancing and energy savings, the high levels of fuel taxes in the European countries can be viewed also as an attempt to capture a part of the scarcity and monopoly rents. On average, between one-third and one-half of the total price of unleaded gasoline is excise tax.

3 Some oil consuming countries have already implemented carbon taxes: Finland, Denmark, Norway and Sweden since 1991, Switzerland in 2008, but these policies remain isolated. Cap and trade schemes have the same consequences but even the largest existing system, the EU-ETS, cannot be interpreted as a global agreement on climate policy.

be optimal? What are the consequences of strategic climate policy in terms of welfare and in terms of distribution among the two coalitions? These are the main questions we address in this paper.

They have to some extent already been addressed from a theoretical point of view by [START_REF] Wirl | Pigouvian taxation of energy for stock and ‡ow externalities and strategic, non-competitive pricing[END_REF][START_REF] Wirl | The exploitation of fossil fuels under the threat of global warming and carbon taxes: A dynamic game approach[END_REF], [START_REF] Tahvonen | International CO2 taxation and the dynamics of fossil fuel markets[END_REF][START_REF] Tahvonen | Trade with polluting nonrenewable resources[END_REF], [START_REF] Rubio | Strategic Pigouvian taxation, stock externalities, and polluting nonrenewable resources[END_REF] and [START_REF] Liski | Can carbon tax eat OPEC's rents?[END_REF]. These papers solve a similar di¤erential game between the producers' and the consumers' cartels, in a framework where global warming creates damages to consumers' welfare. They study the Markov Perfect Nash Equilibrium (MPNE) of the game, and compare it to some benchmarks.

Liski and Tahvonen ( 2004) "consolidate, clarify, and extend" the results obtained in these papers. They show that the optimal design of the carbon tax in the presence of twosided strategic interactions generally deviates from a Pigouvian tax that internalizes only the environmental damage. Moreover, considering linear strategies, they solve explicitly the MPNE in the linear-quadratic case, and prove that when the damage is not too severe, the carbon tax shifts more rents than what is necessary to internalize the environmental externality whereas it is the contrary when the damage is large. They also study the time pro…le of the carbon tax and the producer price. If the damage is small, the carbon tax is decreasing with time and the producer price is increasing; if the damage is intermediate, both are increasing; if the damage is large, the carbon tax is increasing and the producer price decreasing. Finally, they conjecture that the sellers' payo¤ is always -whatever the damage is -lower and the buyers' payo¤ higher in the MPNE than in the pre-tax situation.

The aim of the present paper is to delve further into this question. We revisit the di¤erential game between oil consuming and producing countries when the former shape their climate policy to take into account the existence of a physical upper limit on atmospheric carbon concentration, which overtaking would lead to catastrophic consequences. emission budget", which overtaking would trigger temperature rises above 2 C, because of all the uncertainties in the carbon cycle, the climate response to an increase in the atmospheric carbon concentration, and the natural decay. This global carbon budget is exactly equivalent to a cap on carbon atmospheric concentration, provided that natural carbon absorption by sinks is negligible. The precise level of the ceiling is of course dependent on current scienti…c knowledge and likely to evolve. We shall nevertheless consider it as …xed. Finally, imposing a ceiling constraint beyond which damages become in…nite does not mean that there exists no damage before the ceiling. Consistently with the existence of the ceiling, we assume that these damages are relatively small. We then combine in this paper a ceiling on atmospheric carbon concentration and a "small" damage function before (and at) the ceiling.

This ceiling framework was introduced in the theoretical litterature on optimal fossil resources extraction and global warming by Chakravorty, Magné andMoreaux (2006a), (2006b). Its introduction in a di¤erential game raises a number of technical problems.

Whereas in general linear-quadratic speci…cations make it possible to obtain explicitly a solution of the game, the linear one, the ceiling constraint introduces here an intrinsic non-linearity in the problem and we cannot expect to obtain a linear solution. We are nevertheless able to obtain a solution of the MPNE, which is non-linear .

We obtain implicit feedback rules and explicit time paths of extraction, carbon tax and producer price. We show that the producer and consumer prices are monotonically increasing in time, whereas the carbon tax may be …rst decreasing and is always increasing near the ceiling. As far as rent capture is concerned, we show that the carbon tax always shifts more rent than necessary for an environmental motive, meaning that consumers are able to reap some part of the scarcity and monopoly rents, and that producers are able to preempt to some extent the carbon tax only if the marginal damage under the ceiling is small.

We then compare the MPNE and three benchmarks. The comparison with the e¢cient equilibrium allows us to assess to what extent the carbon tax of the MPNE departs from the tax of the e¢cient equilibrium, designed to correct the environmental problem only.

The comparison with the open loop equilibrium highlights the impact of the producers' and consumers' feedback strategies. The comparison with the cartel without carbon tax equilibrium which, as argued by [START_REF] Liski | Can carbon tax eat OPEC's rents?[END_REF] is the proper benchmark since it is the pre-tax situation, allows us to see whether consumers gain in terms of welfare when they adopt a common environmental policy.

The paper is organized as follows. Section 2 presents the assumptions of the model, solves the Markov-perfect Nash equilibrium and studies its properties. Section 3 studies the three benchmarks against which the properties of the MPNE will be assessed. The equilibria are compared in Section 4. Section 5 concludes.

II Markov-perfect Nash Equilibrium

We study the strategic interactions between a cartel of fossil fuel producers and a coalition of consumers coordinating their carbon emission taxation to …ght global warming, in a di¤erential games framework of analysis. Whereas producers set the fossil fuel price, consumers set the carbon tax in order to meet an atmospheric carbon concentration constraint based on scienti…c knowledge and taken as given, and to internalize the damage caused by the rise in temperature.

Consumers' area

In the consumers' area, the utility of the representative consumer is derived from the use of the fossil resource. The utility function is denoted u (x t ) ; where x is the consumed resource ‡ow (let's say oil). It is assumed to be quadratic and concave:

u (x) = ax b 2
x 2 ; a > 0; b > 0:

(1) u 0 (0) = a is the choke price, for which demand becomes nil.

The initial resource stock is X 0 ; the stock still in the ground at date t is X t : The additional (to the pre-industrial level) atmospheric carbon concentration at date t is Z t :

We assume that natural carbon absorption is nil, so that the additional atmospheric carbon concentration at date t is strictly equal to the stock yet extracted and burnt at this date: Z t = X 0 X t . This assumption has two justi…cations: …rstly, natural absorption by sinks (oceans, forests) is uncertain and likely to decrease while carbon concentration increases; secondly and more technically, it allows us to consider only one stock in the problem and obtain tractable solutions.

Climate policy takes the form of a ceiling constraint: X 0 X t Z where Z is the ceiling on the additional carbon stock. Z is a physical constraint, and can be seen as the global carbon budget allowing humanity to contain the rise of temperature under 2 C.

But this does not mean of course that the rise of temperature does not cause any damage before the ceiling is reached. To be consistent with the choice of the value of the ceiling, it can be assumed that these damages are small -otherwise the ceiling would have been ill-chosen. We then introduce the damage that indeed appears before the ceiling, and,

following [START_REF] Amigues | Optimal use of a polluting non-renewable resource generating both manageable and catastrophic damages[END_REF], we suppose this damage linear:

D(X t ) = d(X 0 X t ); d > 0: (2)
The buyers' regulator sets a unit carbon tax for the area to control the pollution accumulation due to the resource use, taking into account the demand function of the representative consumer, given by:

u 0 (x) = p + ; (3)
where p is the producer price and the carbon tax. The regulator maximizes on the whole horizon the discounted net surplus, di¤erence between the consumers' utility and the amount paid to producers, subject to the law of carbon accumulation in the atmosphere and the ceiling constraint. Tax revenues are reimbursed as lump-sum transfers to consumers, so they do not appear into the net surplus.

The two players act simultaneously, and the game is played with feedback (Markovian)

strategies. The buyers' regulator, when choosing the carbon tax , takes into account the fact that the producer price depends on the resource stock, that is p (X).

The buyers' regulator problem is 4 :

V m c (X 0 ) = max t Z 1 0 e t [u (x (p (X t ) + t )) p t x (p (X t ) + t ) D (X t )] dt s.t. _ X t = x (p (X t ) + t ) X 0 X t Z X 0 ; Z given. ( 4
)
Let c be the shadow price of the resource and ! the Lagrange multiplier associated to the ceiling constraint. First order optimality conditions and the complementarity slackness condition read 5 :

t = ct (5) _ ct = ct + p 0 (X t )x t + D 0 (X t ) ! t (6) lim t!1 e t ct X t = 0 (7) ! t 0; X t X 0 + Z 0; ! t (X t X 0 + Z) = 0: (8)
Let T m be the date at which the ceiling becomes binding. Given the structure of the problem, if the initial atmospheric carbon concentration is lower than the ceiling, which 4 Superscript m for Markov. We will use in what follows the superscripts o for optimum/e¢cient equilibrium, ol for open loop equilibrium and c for cartel without tax equilibrium. 5 Conventionally, Hamilton-Jacobi-Bellman equations are used to solve the MPNE because they yield directly strategies depending on the state variable, here X: We have prefered to use here the maximum principle, because the value function is potentially non-di¤erentiable at the ceiling, and consequently discontinuities of the feedback rule at the ceiling are possible. The use of the maximum principle allows us to overcome this di¢culty. Notice that the non-di¤erentiability of the value function at the ceiling is equivalent to the discontinuity of the stock shadow price at the ceiling. We study this potential discontinuity in the on-line Appendix B and show that indeed it exists. We solve the same problem using

Hamilton-Jacobi-Bellman equations in the on-line Appendix E, to convince the reader that we actually obtain the same strategies.

we assume, the ceiling will not be binding before T m (! t = 0 8t < T m ), and will remain binding forever after T m (! t > 0 8t T m ). After the ceiling is reached, the resource consumer price, sum of the producer price and the carbon tax, remains equal to u 0 (0); the price at which demand is choked, and x = 0:

Before the ceiling i.e. 8t < T m ; equations ( 5) and ( 6) yield the following evolution of the carbon tax:

_ t = t + p 0 (X t )x t + D 0 (X t ): (9)
(9) integrates into:

t = e Tm Tm e t Z Tm t e (s t) D 0 (X s )ds Z Tm t e (s t) p 0 (X s )x s ds: (10)
The carbon tax is the sum of three terms.

The …rst term on the right-hand side of equation ( 10) is what we may name the "pure Hotellinian tax". It comes from the existence of the ceiling constraint and grows at the discount rate. The second term represent the "pure Pigouvian tax" of [START_REF] Liski | Can carbon tax eat OPEC's rents?[END_REF]. It is due to the existence of damages before the ceiling and is the usual discounted sum of future marginal damages. Without ceiling and small damage, T m would tend to in…nity and both terms would be nil6 . There would then exist no environmental motive for taxation.

The third term on the right-hand side of ( 10) is the strategic component of the carbon tax. It represents the "pure import tari¤" of [START_REF] Liski | Can carbon tax eat OPEC's rents?[END_REF] if p 0 (:) < 0;

and their "pure import subsidy" if p 0 (:) > 0. In the …rst case, the carbon tax is positive even absent any damage. Consumers tax oil more heavily than the environmental motive would require, and are thus able to reap to some extent the scarcity and monopoly rents of producers. In the second case, consumers subsidize oil consumption to correct the monopoly distorsion.

After the ceiling i.e. 8t T m ; equations ( 5) and ( 6) yield the following evolution of the carbon tax:

_ t = t + D 0 (X t ) ! t : (11)
Using the transversality condition ( 7), ( 11) integrates into:

t = Z 1 t e (t s) ! s ds Z 1 t e (s t) D 0 (X s )ds: (12)

Producers' area

Producers face a unit extraction cost depending on the resource stock still in the ground.

The smaller this stock the higher the marginal extraction cost: the last drop of oil is very costly to extract. More precisely, the unit extraction cost is c(X t ); with c(X) > 0;

c 0 (X) < 0: We use the following linear speci…cation:

c (X) = c 1 c 2 X; c 1 > 0; c 2 > 0: (13)
We make the assumption that initial extraction is pro…table:

c(X 0 ) < u 0 (0) , X 0 > c 1 a c 2 : ( 14
)
With a constant marginal extraction cost, scarcity is purely physical. Here, scarcity can be economic, in the sense that the marginal cost of extraction of the last drop of oil can be higher than the choke price. Following [START_REF] Liski | Can carbon tax eat OPEC's rents?[END_REF], we therefore assume that economic scarcity is binding:

c(0) > u 0 (0) , c 1 > a: (15)
Then the last drop will never be extracted; producers will stop extraction before and leave some oil in the ground. Without any environmental constraint -neither ceiling nor damage in our framework, they will leave in the ground a stock X 1 de…ned by:

c(X 1 ) = u 0 (0) , X 1 = c 1 a c 2 : (16)
We assume that producers do not intend to adopt any climate policy, but are perfectly aware that consumers do.

The sellers' regulator, when choosing the producer price, maximizes on the whole horizon its discounted pro…ts, subject to the law of evolution of the resource stock:

V m p (X 0 ) = max pt Z 1 0 e t (p t c(X t )) x (p t + (X t )) dt s.t. _ X t = x (p t + (X t )) X 0 given. (17)
Producers are assumed to be aware of the reaction function of buyers to the state variable, (X).

The …rst order conditions give the producers' price strategy and the evolution of the shadow price of the resource p ; together with the transversality condition:

p t = c(X t ) x (p t + (X t )) x 0 (p t + (X t )) + pt (18)
_ pt = pt + (c 0 (X t ) + 0 (X t ))x (p t + (X t )) (19) lim t!1 e t pt X t = 0: (20)
In equation ( 18), c(X) x (:) =x 0 (:) is the static component of the monopoly price.

The scarcity rent must be added to this static component.

Integrating equation ( 19) forward before the ceiling (t < T m ) yields:

pt = e Tm pTm e t Z Tm t e (s t) (c 0 (X s ) + 0 (X s ))x s ds: (21)
The scarcity rent exhibits a Hotelling component, and a second component expressing the fact that a marginal amount of stock extracted at a given date s a¤ects future pro…ts by c 0 (X s )+ 0 (X s ); because of increased extraction costs and the variation of the consumers tax rate, which must be re ‡ected in the current price. While the extraction cost e¤ect clearly reduces future pro…ts and increases the scarcity rent and today's price, the tax e¤ect is at this stage ambiguous.

With the speci…cation (1) adopted for the utility function, the demand function is:

x(p + ) = a (p + ) b ; (22)
and ( 18) reads before the ceiling:

p = 1 2 [c(X) + a (X) + p ] : (23)
This formulation highlights the two e¤ects of the carbon tax on the producer price: a negative static e¤ect through the monopoly price (when increases, the cartel decreases p to support demand, and vice versa), and a dynamic e¤ect through the scarcity rent. The sign of this last e¤ect is at this stage indeterminate. If it is positive, it can be interpreted as a preemptive behaviour of the cartel: when the cartel extracts one unit of oil it knows that it will increase the atmospheric carbon concentration and that consequently the consumers' coalition will increase the carbon tax, and it reacts by increasing the producer price.

Consider now what happens after the ceiling is reached. Integrating (19) forward after T m and using the transversality condition (20) shows that the scarcity rent is nil after the ceiling, which makes sense since the resource won't be extracted anymore. Then ( 18)

yields p t = c(X 0 Z) = p; t T m : Then, since p t + t = u 0 (0) t T m ; t = u 0 (0) p = ; t T m :
Notice that nothing insures neither that the scarcity rent is continuous at the ceiling, i.e. that pTm = 0; nor that the producer price and the carbon tax are.

The non-linear solution

In general, the Markov Perfect Nash Equilibria of di¤erential games are studied under the assumptions of quadratic objective functions (utility, pro…t) and linear cost functions.

Linear solutions are then the only ones that can be computed throughout analytically, even if other solutions may exist. But here, the ceiling constraint introduces an intrinsic non-linearity in the problem. Hence it is natural that the linear solution holds only if this constraint is not binding, that is if the ceiling is su¢ciently high, so that the increase in unit extraction cost triggers the stop of extraction for economic reasons before the ceiling is reached. We make the opposite assumption that the ceiling is su¢ciently low, so that it requires to stop extraction before the date at which it would have stopped spontaneously, and to leave more fossil fuel in the ground. The study of the linear solution, valid under the assumption of a high ceiling, is relegated to the on-line Appendix A.

The equilibrium of the game is characterized by equations ( 5) and ( 23) giving the carbon tax and the producer price depending on the shadow price of the resource respectively for consumers and producers, and equations ( 6) and ( 19) giving the evolution of these shadow prices.

Considering the shadow prices c and p as functions of the resource stock, notice that )x; and that the same holds for _ p : Then the sum of equations ( 9)

_ c = 0 c (X) _ X = 0 c (X
giving the evolution of the carbon tax before the ceiling and ( 19) giving the evolution of the scarcity rent yields, at the equilibrium:

( (X) + p (X)) + (p 0 (X) + c 0 (X) + 2 0 (X) + 0 p (X))x + D 0 (X) = 0: (24) (X) + p (X) + D 0 (X) + 3 2 (c 0 (X) + 0 (X) + 0 p (X))x = 0: (25)
Wirl and [START_REF] Wirl | Leviathan governments and carbon taxes: Costs and potential bene…ts[END_REF] suggests (in a somewhat di¤erent framework) to introduce a proper sum of the relevant variables in order to help solving the problem. One possibility here turns out to be:

(X) = c(X) + (X) + p (X) + D 0 (X) : (26)
Equation ( 25) now reads:

( (X) c(X)) + 3 2 0 (X)x = 0; (27)
while, from (23) and using the assumption of a constant marginal damage (2),

p(X) + (X) = 1 2 a + d + (X) (28)
which can also be written:

a bx = 1 2 a + d + (X) : (29)
By elimination of x; (27) and ( 29) yield a non-linear di¤erential equation in X:

0 (X) = 4 b 3 0 @ (X) c (X) (X) a d 1 A : (30)
We assume that a > d= which means that at date 0 the choke price is higher than the discounted value of the marginal damage or, to put it di¤erently, that the marginal damage before the ceiling is small.

We show in the on-line Appendix A that a linear solution to this equation exists, but that it is valid only in the case where the ceiling constraint is not binding. We want to …nd a solution in the case of a su¢ciently stringent ceiling, in the sense that the oil stock that is left in the ground with the ceiling constraint is larger than the one that would be left without it (but taking into account the existence of the damage):

X 0 Z > c 1 a d c 2 = e X: (31)
This solution has to be non-linear. Moreover, it is unique. Indeed, as shown by [START_REF] Tsutsui | Nonlinear strategies in dynamic competition with sticky prices[END_REF], [START_REF] Dockner | International pollution control: Cooperative versus non-cooperative strategies[END_REF], [START_REF] Rowat | Non-Linear strategies in a linear quadratic di¤erential game[END_REF] and [START_REF] Wirl | Do multiple Nash equilibria in Markov strategies mitigate the tragedy of the commons[END_REF], when the ceiling is not binding, (uncountably many) non-linear strategies coexist with the linear strategy. They all admit as a steady state an atmospheric carbon concentration lower than the one reached asymptotically by the Markov linear strategy, which amounts to say that the oil stock ultimately left in the ground is higher than e X. These non-linear strategies are Pareto-inferior to the linear one [START_REF] Wirl | Leviathan governments and carbon taxes: Costs and potential bene…ts[END_REF]). But in our game the boundary condition (the ceiling) allows us to pin down the unique solution, among the non-linear ones, which admits as a steady state an atmospheric carbon concentration equal to Z: The boundary condition rules multiple equilibria out.

It is not possible to …nd (X) analytically, so we resort to a phase diagram (Figure 1).

< Figure 1 about here > On the phase diagram, 0 (X) = 0 , (X) = c(X); the admissible zone is under the line (X) = a d= , since p + = (a + d= + (X)) =2 < a ) (X) < a d= ; 0 (X) < 0 when (X) > c(X); and 0 (X) > 0 when (X) < c(X); …nally, 0 (X) ! 1 , (X) = a d= : Starting from (X 0 ) > c(X 0 ); the unique stable arm is travelled along from the right to the left, towards the equilibrium (X 0 Z) = a d= : X decreases, and as 0 (X) < 0; (X) increases.

We can now express the feedback rules for extraction, the scarcity rent, the carbon tax and the producer price. Equation ( 29) is equivalent to

x = 1 2b a d (X) : (32)
From ( 19), p (X) + 0 (X)x = 0:

i.e., with ( 32) and (30),

p (X) = 1 2 b (X) a + d 0 (X) = 2 3 [ (X) c (X)] : (34)
From ( 25), using ( 32), ( 34) and ( 30),

(X) = 1 3 [ (X) c (X)] + d : (35)
Finally, from (28), using (35),

p(X) = 1 6 (X) + 1 3 c(X) + 1 2 a d : (36)
Notice that absent any damage before the ceiling (d = 0), the carbon tax is equal to half the scarcity rent ( (X) = p (X)=2 8X): This implies that the dynamic strategic e¤ect of the carbon tax on the producer price is stronger than the static monopoly e¤ect (see equation ( 23)), and that consequently the producers' cartel is able to preempt to some extent the carbon tax, as explained by [START_REF] Wirl | The exploitation of fossil fuels under the threat of global warming and carbon taxes: A dynamic game approach[END_REF]. This is not true anymore with damage: we now have (X) = p (X)=2 + d= ; and preemption is all the less likely since d is large. On the phase diagram, the carbon tax can be read as d= plus one third of the vertical distance between the curve (X) and the line 0 (X) = 0 (see equation ( 35)).

At the ceiling, x is nil, X is equal to X 0 Z; and we have shown that, 8t T m :

p t = p = c X 0 Z (37) t = = a c X 0 Z (38) ! t = ! = d: (39)
The producer price at the ceiling is equal to the unit extraction cost. The level of the carbon tax is then such that demand is totally choked, since the consumer price is equal to the choke price a. The scarcity rent is nil.

Proposition 1 MPNE (i) Before the ceiling, the consumer and producer prices are monotonically increasing;

the carbon tax may be …rst decreasing and then increasing, and is always increasing near the ceiling. The carbon tax includes an import tari¤ element.

(ii) The carbon tax and the producer price are not continuous at the ceiling, whereas the consumer price is. When reaching the ceiling, the carbon tax jumps upwards and the producer price jumps downwards to the marginal extraction cost.

Proof. Equation ( 28) shows that the consumer price is monotonically increasing in time, since 0 (X) < 0 and X is monotonically decreasing in time. Moreover, ( 28) and ( 37)-( 38)

show that the consumer price is continuous at the juncture, and equal to the choke price

u 0 (0) = a.
Equation (35) yields 0 (X) = ( 0 (X) c 0 (X)) =3: As 0 (X) < 0 and c 0 (X) < 0; the sign of 0 (X) is indeterminate. However, lim X!X 0 Z 0 (X) = 1; and, by continuity, 0 (X) < 0 and the carbon tax is increasing in time near the ceiling.

Equation (36) yields p 0 (X) = 0 (X)=6 + c 0 (X)=3 < 0: Hence the producer price is increasing with time.

This proves part (i) of the proposition.

When X ! X 0 Z; (35) and (36) show that (X) ! (a d= ) c X 0 Z =3 + d= and p(X) ! 2 (a d= ) + c X 0 Z =3: However at the ceiling, according to (38) and

(37), t = = a c X 0 Z and p t = p = c X 0 Z : This proves part (ii) of the proposition.

The important result lies in the …rst part of the Proposition. It concerns the time pro…le of the carbon tax and the producer price. Whereas the price increases along the whole trajectory, the carbon tax may be …rst decreasing, but always ends up increasing.

Moreover, the carbon tax always includes an import tari¤, allowing consuming countries to reap a part of the oil and monopoly rents. This result challenges the robustness of [START_REF] Liski | Can carbon tax eat OPEC's rents?[END_REF] …ndings, namely that the carbon tax includes an import tari¤ when the damage is not too severe, whereas it includes an import subsidy when the damage is large.

Though the feedback rules cannot be expressed as closed-form functions of X; it is possible to obtain the time paths of extraction, carbon tax and producer price implied by the MPNE.

Di¤erentiating (32) w.r.t. time and using (30) yields:

_ x = 2 3 0 @ (X) c (X) (X) a d 1 A _ X: (40)
Notice that _ x = • X and that, from ( 32), (X) (a d= ) = 2bx = 2b _ X: Equation (40) then reads:

• X = 3b ( (X) c (X)) = 3b 2b _ X + a d c (X) :
Replacing c(X) by its linear speci…cation (13), we obtain a second order di¤erential equa-

tion in X : 3b • X 2 b _ X c 2 X + c 1 a d = 0: (41)
The solution of this di¤erential equation is:

X t = 1 e v 1 t + 2 e v 2 t + c 1 a d c 2 ; ( 42
)
with:

v 1 = 3 1 + r 1 + 3c 2 b > 0 and v 2 = 3 1 r 1 + 3c 2 b < 0: (43)
If the ceiling constraint never binds, that is if X 0 Z e X; the v 1 root can be ruled out as X t can neither become negative (if 1 < 0) nor increase (if 1 > 0). In such a case, the solution is the linear one, where X converges asymptotically to e X. We have excluded this case. Under the opposite assumption, the two roots must be conserved and we obtain the non-linear solution.

Three equations are needed to determine implicitly the three unknown 1 ; 2 ; and T m : the initial condition X 0 , the condition at the ceiling X Tm = X 0 Z; and the fact that extraction becomes nil at the ceiling7 , x Tm = 0: These conditions read:

X 0 = 1 + 2 + c 1 a d c 2 (44) X 0 Z = 1 e v 1 Tm + 2 e v 2 Tm + c 1 a d c 2 (45) 0 = v 1 1 e v 1 Tm + v 2 2 e v 2 Tm : (46)
Equations ( 45) and ( 46) yield:

1 = v 2 v 1 v 2 X 0 Z e X e v 1 Tm > 0 (47) 2 = v 1 v 1 v 2 X 0 Z e X e v 2 Tm > 0: (48)
Then ( 44) yields

v 1 e v 2 Tm v 2 e v 1 Tm v 1 v 2 = X 0 e X X 0 Z e X ; (49)
which gives T m implicitly. It is an increasing function of Z: It is also an increasing function of d. The existence of a small damage puts forward the juncture date: indeed, because of this damage, at each date oil consumption is lowered and the ceiling is reached later.

We can now obtain the time paths of the extraction, consumer price (from (3)), scarcity rent (from ( 33) and ( 40)), carbon tax and producer price:

x t = _ X t = v 1 1 e v 1 t v 2 2 e v 2 t ( 50
)
p t + t = a bx t = a + b(v 1 1 e v 1 t + v 2 2 e v 2 t ) (51) pt = 2b _ x t = 2b v 2 1 1 e v 1 t + v 2 2 2 e v 2 t (52) t = 1 2 pt + d = b v 2 1 1 e v 1 t + v 2 2 2 e v 2 t + d ( 53
)
p t = a d + b v 1 1 ( v 1 ) e v 1 t + v 2 2 ( v 2 ) e v 2 t : (54)

III Benchmarks

Three possible benchmarks are studied, against which the properties of the MPNE will be assessed: the e¢cient equilibrium, the open loop equilibrium of the game, and the cartel without carbon tax equilibrium. The …rst one allows us to assess whether the monopoly power of the producers' cartel and the strategic behaviour of the two players lead to too much or too little extraction, compared to what is optimal. The second one allows us to assess the e¤ect of the feedback strategies on the producer price and the carbon tax.

The last one is the proper benchmark, as argued by [START_REF] Liski | Can carbon tax eat OPEC's rents?[END_REF], since it is the pre-tax situation. The comparison of the pre-tax and the MPNE outcomes for consumers shows whether consumers bene…t in terms of welfare from the implementation of the carbon tax.

Optimum and e¢cient equilibrium

The world central planner's problem reads8 :

V (X 0 ) = max xt Z 1 0 e t [u (x t ) c(X t )x t D (X t )] dt s.t. _ X t = x t X 0 X t Z X 0 given. ( 55
)
Denoting by t the shadow price of the resource stock and ! t the Lagrange multiplier associated to the ceiling constraint, …rst order optimality conditions and the complementarity slackness condition are:

u 0 (x t ) = c(X t ) + t ( 56
)
_ t = t + c 0 (X t )x t + D 0 (X t ) ! t (57) lim t!1 e t t X t = 0 (58) ! t 0; X t X 0 + Z 0; ! t (X t X 0 + Z) = 0: (59)
Before the ceiling, the marginal utility on the optimal path is the sum of the marginal extraction cost and of a rent t , encompassing the scarcity rent and the carbon shadow value, in this simpli…ed framework where the same stock characterizes the fossil resource stock in the ground and the atmospheric carbon concentration.

It is possible to show that extraction x is continuous at the juncture, in the same line as for the MPNE. Then, from (56), the costate is also continuous at the juncture.

Di¤erentiating (56) w.r.t. time and using (57) yields:

u 00 (x) _ x = c 0 (X) _ X + ( + c 0 (X)x + D 0 (X) !) = + D 0 (X) !:
Before the ceiling, ! = 0 and we get, using (56) again, the optimal extraction path:

u 00 (x) _ x = (u 0 (x) c(X)) + D 0 (X): (60)
With the speci…cations adopted for the utility, damage and extraction cost functions,

(60) reads: b • X b _ X c 2 X + c 1 a d = 0; (61)
a linear di¤erential equation of the second order, as in the MPNE, but with di¤erent coe¢cients.

The solution is:

X t = 1 e u 1 t + 2 e u 2 t + c 1 a d ; (62)
with

u 1 = 2 1 + r 1 + 4c 2 b > 0 and u 2 = 2 1 r 1 + 4c 2 b < 0: (63)
As in the case of the MPNE, three boundary conditions allow us to obtain the unknown 1 ; 2 and T o : the initial condition X 0 , the condition at the ceiling X To = X 0 Z; and the fact that extraction becomes nil at the ceiling, x To = 0: With the same argument as in the MPNE, it is possible to show that 1 ; 2 > 0:

At date T o ; the ceiling is reached, and we have, 8t

T o : 8 > > > > > > > > > > < > > > > > > > > > > : X t = X 0 Z x t = 0 t = u 0 (0) c(X 0 Z) ! t = t d: (64)
The decentralization of the optimum leads to an e¢cient competitive equilibrium, provided that the right environmental tax -redistributed by lump-sum transfers to consumersis implemented in the consumers' area.

The demand function of the representative consumer is given by u 0 (x) = p + :

On the producer side:

V o p (X 0 ) = max xt Z 1 0 e t [p t c (X t )] x t dt (65) s.t. _ X t = x t ; X 0 given.
Denoting by p the scarcity rent, the …rst order conditions read:

p t = c(X t ) + pt (66)
_ pt = pt + c 0 (X t )x t (67) lim t!1 e t pt X t = 0: (68)
The equilibrium is then de…ned by:

u 0 (x t ) = c(X t ) + pt + t : (69)
Di¤erentiating equation ( 69) w.r.t. time and using (67) yields: b

• X b _ X c 2 X + (c 1 a) = _ : (70)
Comparing ( 61) and ( 70) shows that for the equilibrium to be an optimum the carbon tax before the ceiling must be such that _ t t = d; which integrates into t = To e To e t + d 1 e (To t) ; 8t 0:

The optimal carbon tax is the sum of the Hotellinian tax and the pure Pigouvian tax.

Extraction x and the marginal extraction cost c 0 (X) being continuous at the ceiling, (67) implies that _ p p is continuous, and yields, integrating forward and using the transversality condition (68) 9 , pt = Z 1 t e (s t) c 0 (X s )x s ds; 8t 0:

Then, as x s = 0; s T o ; pTo = 0: The scarcity rent is continuous at the juncture, and equal to 0. Then the carbon tax is also continuous, and it is equal to given by ( 38).

This allows us to obtain the initial level of the carbon tax: 0 = e To + d= 1 e To :

Moreover, it is straightforward to show that the carbon tax is monotonically increasing i¤ > d= which, according to the de…nition (38) of ; is true when the ceiling is binding.

Equations ( 66) and ( 67) show that the producer price is monotonically increasing before the ceiling. Moreover, by (66), the producer price is continuous at the juncture and equal to p given by (37).

These results are summarized in the following proposition.

Proposition 2 E¢cient equilibrium (i) Before the ceiling, the carbon tax is the sum of a pure Hotellinian tax and a pure Pigouvian tax, monotonically increasing, and the producer price is also monotonically increasing.

(ii) The carbon tax and the producer price are continuous at the ceiling.

Open loop equilibrium

In this case, the players base their strategies on time alone.

The consumers' regulator problem is similar to problem (4), but for the fact that he takes the producer price as given. Equally, the producers' regulator problem is similar to (17), but for the fact that he takes the carbon tax as given.

9 Notice that the same reasoning cannot be made in the case of the MPNE, since we are not sure that 0 (X) is continuous.

The …rst order optimality conditions are, on the consumers' side:

t = ct (73)
_ ct = ct + D 0 (X t ) ! t (74) lim t!1 e t ct X t = 0; (75)
which shows that before the ceiling the carbon tax is the sum of the Hotellinian tax and the Pigouvian tax.

On the producer side, the …rst order optimality conditions read:

p t = 1 2 [c(X t ) + a + pt t ] (76)
_ pt = pt + c 0 (X t )x t (77) lim t!1 e t pt X t = 0: (78)
The equilibrium before the ceiling is characterized by:

u 0 (x) = p + () a bx = 1 2 [c(X) + a + + p ] : (79)
Di¤erentiating (79) w.r.t. time and using ( 74) and (77) yields:

2b • X 2 b _ X c 2 X + c 1 a d = 0; (80)
again a linear di¤erential equation of the second order, but with di¤erent coe¢cients.

The solution has the same form as in the MPNE and the e¢cient equilibrium:

X t = 1 e w 1 t + 2 e w 2 t + c 1 a d c 2 ; (81)
with

w 1 = 2 1 + r 1 + 2c 2 b > 0 and w 2 = 2 1 r 1 + 2c 2 b < 0: (82)
As in the cases of the MPNE and the e¢cient equilibrium, three boundary conditions allow us to obtain the unknown 1 ; 2 and T ol ; the date at which the ceiling is reached:

the initial condition X 0 , the condition at the ceiling X T ol = X 0 Z; and the fact that extraction becomes nil at the ceiling, x T ol = 0: It is possible to show that 1 ; 2 > 0:

Proposition 3 Open loop equilibrium (i) Before the ceiling, the carbon tax is the sum of a Hotellinian tax and a Pigouvian tax and is always increasing, whereas the producer price may be …rst increasing and then decreasing, and is decreasing when approaching the ceiling.

(ii) The carbon tax and the producer price are continuous at the ceiling.

Proof. To prove part (ii), remark that p is continuous at the juncture by the same argument as in the e¢cient equilibrium case. Its continuity implies that of the carbon tax (from ( 79)) and the producer price.

To prove part (i) note …rst that the consumer price is increasing as

p + = a + b _ X ) d(p + )=dt = b • X > 0:
As far as the carbon tax before the ceiling is concerned, ( 73) and ( 74) show that it has the same expression than in the e¢cient equilibrium (equation ( 71)), hence has the same properties. Finally, the sign of _ p is indeterminate. However at the juncture at date T ol ; using ( 76) and (77), _ p T ol = _ T ol =2 < 0: This proves part (i) of the proposition.

The discontinuity of the carbon tax and the producer price at the juncture at the MPNE is then the consequence of the feedback strategies of the players, since this discontinuity does not exist at the open loop equilibrium.

Cartel equilibrium without carbon tax

This equilibrium can be seen as the present situation, where everybody is aware of the existence of the physical limit Z to atmospheric carbon concentration but ignores it.

The buyers' demand for oil is simply given by u 0 (x) = p: The sellers' regulator solves the same problem as in the open loop game, but for the fact that now the carbon tax is nil. The equilibrium is characterized by equation ( 80), as in the open loop game; but now producers behave as if no constraint could prevent them from extracting all what is economically pro…table. They intend to leave asymptotically X 1 in the ground, and choose the extraction path accordingly. In the solution of the linear di¤erential equation ( 80), the positive exponential has to be ruled out. Therefore the time path of oil stock and extraction are:

X t = (X 0 X 1 ) e w 2 t + X 1 ( 83
)
x t = w 2 (X t X 1 ) (84)
with w 2 given in (82).

But ignoring the ceiling does not make it disappear. Once the ceiling is reached, the damage from consuming more oil becomes in…nite whereas the marginal utility of consumption remains …nite, and therefore consuming countries are not willing to buy oil any more, even under a zero tax. Extraction drops to zero in …nite time (at date T c ), while a stock of oil X Tc = X 0 Z is left forever in the ground. From (83) we get:

T c = 1 w 2 ln X 0 Z X 1 X 0 X 1 : (85)
Finally, the producer price is given by:

p t = a + bw 2 (X 0 X 1 ) e w 2 t ; t T c ; (86)
from which we deduce p Tc = a + bw 2 X 0 Z X 1 < a:

These results are summarized in the following proposition:

Proposition 4 Cartel without tax equilibrium Producers and consumers behave as if the ceiling did not exist. The resource price is an increasing and concave function of time. It is lower than the choke price at the ceiling, where it jumps upwards to the choke price while extraction jumps downwards to zero. (ii) The ranking of the payo¤s for the producers' and the consumers' area respectively are:

IV Comparison of equilibria

V ol p (X 0 ) > V m p (X 0 ) > V o p (X 0 ) = 0 and V o c (X 0 ) > V ol c (X 0 ) > V m c (X 0 ):
When the marginal damage is small enough, the consumers' area gets a higher payo¤ in the MPNE than in the cartel without tax case if the ceiling is high; it is the contrary if it is low.

Proof. (i) We prove in the on-line Appendix C that T m > T ol > T o ; T ol > T c , and that

x o 0 > x ol 0 > x m 0 ; x c 0 > x ol 0 :
Notice that we cannot deduce the ranking of initial extractions from the ranking of the dates at which the ceiling is reached and the fact that total extraction is the same in the three equilibria, because in some of the equilibria extraction can be convex-concave. It may be the case for instance in the MPNE since t can be …rst decreasing and then increasing and

• x t = =b: _ t .
The comparison of T c and T o on the one hand, x c 0 and x o 0 on the other hand is also relegated to the on-line Appendix C.

(ii) The ranking of payo¤s is deduced from the ranking of initial extractions (except for the producers' payo¤ at the e¢cient equilibrium and both payo¤s at the cartel without tax equilibrium), since

V m p (X 0 ) = b (x m 0 ) 2 ; V ol p (X 0 ) = b= : x ol 0 2 ; V o c (X 0 ) = b=(2 ): (x o 0 ) 2 ; V m c (X 0 ) = b=(2 ): (x m 0 ) 2 ; V ol c (X 0 ) = b=(2 ): x ol 0
2 ; see the on-line Appendix E. The producers' payo¤ at the e¢cient equilibrium is nil.

For the cartel without tax equilibrium the Hamilton-Jacobi-Bellman equation cannot be used since consumers do not take into account the evolution of the stock in their problem (the damage is an externality). The consumers' payo¤ must be computed directly. We get

V c c (X 0 ) = b 2 Z 1 0 e t x 2 t dt d Z 1 0 e t (X 0 X t )dt; (87)
x t and X t being given by ( 83) and ( 84). We compare in the on-line Appendix D the consumers' payo¤s V m c (X 0 ) and V c c (X 0 ) in the case d = 0 and prove the result stated in the Proposition. This result can be extended by continuity to d small. As V m c (X 0 ) and V c c (X 0 ) both are decreasing functions of d; it is not possible to obtain analytically a ranking of the payo¤s for any value of d:

This Proposition contains at least three strong results.

First of all, when the two players act strategically, the sellers win. They get a higher payo¤ than at the e¢cient equilibrium, whereas the buyers' payo¤ is reduced.

Secondly, consumers and producers are both better o¤ in the open loop equilibrium than in the MPNE. In this sense, playing feedback strategies is a lose-lose situation, both parties ending up being worse o¤. There exists in this game a commitment value.

Lastly, when the ceiling is not too stringent and the marginal damage small enough, consumers gain in the MPNE with respect to the pre-tax case. To put it di¤erently, consumers are better o¤ with the carbon tax than without it if the global warming problem is not too severe. In this case indeed, they do not su¤er from a too drastic reduction of their oil consumption whereas they bene…t from the reduction of damages. Conversely, consumers may lose, and we are sure that this is the case if the marginal damage is small enough and the ceiling very stringent.

V Conclusion

Studying the MPNE of a game between two coalitions of oil producing and oil consuming countries, [START_REF] Liski | Can carbon tax eat OPEC's rents?[END_REF] show, within the damage function approach, that the carbon tax is not purely Pigouvian. If the damage is not too severe, it includes an import tari¤ element and exceeds the present value of marginal damages, allowing oil consuming countries to reap resource rents from the cartel of oil producers, whereas for a serious damage this element is an import subsidy and the strategic tax falls short of the Pigouvian one. The optimal design of the strategic tax (import subsidy or import tari¤, tax increasing or decreasing in time) depends on the value of the parameter of the quadratic damage function, featuring the severity of the damage. This severity also determines the temporal pro…le of the strategic producer price: increasing when the damage is not too severe, decreasing otherwise. In terms of payo¤s, Liski and Tahvonen conjecture that the strategic tax reduces the producers' payo¤ and enhances the consumers' payo¤, compared to the pre-tax case, whatever the severity of the damage.

We revisit this game within the ceiling approach. We obtain a monotonically increasing strategic producer price before the ceiling, and a carbon tax which may be decreasing or increasing at the beginning of the planning horizon, but is always increasing near the ceiling, and this independently on the stringency of the ceiling and the severity of the "small" linear damage before the ceiling. Moreover, in this framework, the strategic tax includes an import tari¤ element whatever the stringency of the ceiling. These results challenge the robustness of the conclusions of the existing literature.

Compared to the open loop solution, behaving strategically is a lose-lose situation, both parties ending up being worse o¤. Compared to the pre-tax situation (the cartel without tax equilibrium), we prove that when the ceiling is tight and the "small" marginal damage small enough the consumers are worse o¤ in the MPNE, whereas when the ceiling is relatively high they are better o¤. We do not con…rm here the conjecture of [START_REF] Liski | Can carbon tax eat OPEC's rents?[END_REF], which is that consumers always gain from introducing the carbon tax.

The practical discussions about the introduction of a carbon tax very often concentrate on the distributive consequences of the tax within each country and between countries

  The level of this ceiling is based on scienti…c evidence. It has been set at 450 parts per million carbon dioxide equivalent by the Intergovernmental Panel on Climate Change in 2007, in its Fourth Assessment Report (IPCC (2007)), target which would provide a "reasonable chance" of averting warming beyond 2 C above pre-industrial temperature. Following the work of Allen et al. (2009) and Meinshausen et al. (2009), it has been recently argued that the correct way of thinking about global warming is in terms of a global "CO 2

  The comparison of the MPNE and the e¢cient equilibrium allows us to assess to what extent the carbon tax of the MPNE departs from the Pigouvian tax of the e¢cient equilibrium, designed to correct the environmental problem only, and to see if the game is more or less conservative than what is optimal. The comparison of the MPNE and the open loop equilibrium highlights the impact of the producers' and consumers' feedback strategies. Finally, the comparison of the MPNE and the cartel without carbon tax equilibrium allows us to see whether consumers gain in terms of welfare when they adopt a common environmental policy.Technically, the comparison is made easier by the fact that all equilibria reach in …nite time the same state, where a stock of oil X 0 Z is left in the ground, atmospheric carbon concentration is at the ceiling and oil consumption is nil. What changes in the di¤erent equilibria is the intertemporal allocation of extraction, driven by di¤erent producer prices and carbon taxes, and thus producers and consumers' payo¤s.Proposition 5 (i) The MPNE is more conservative than the open loop equilibrium, in the sense that initial extraction is lower. Both are excessively conservative, compared to what is e¢cient, and are also more conservative than the cartel without tax equilibrium.The ceiling is reached later in the MPNE than in the open loop equilibrium, and later in the open loop equilibrium than in the e¢cient equilibrium and than in the cartel without tax equilibrium. The cartel without tax equilibrium is less conservative than the e¢cient equilibrium and the ceiling is reached sooner if it is low; it is the contrary if the ceiling is high.

The authors acknowledge the support of the French National Research Agency (ANR) under the CLEANER project (ANR_NT09_505778). We are grateful to two anonymous referees for helpful suggestions and comments.

This comes from the transversality condition (7), since X t would tend in the long run to a strictly positive value because of assumption (15) on the extraction cost function.

The proof is relegated to the on-line Appendix B.

Notice that it is exactly equivalent to the problem without ceiling but with an initial resource stock X 0 Z:

adopting the environmental policy and countries refusing to do so, without considering a central actor in the climate change game, namely fossil fuel producers. We have in this paper contributed to …ll this gap, in a two-zones framework. But a lot remains to be done. Some very recent papers open new directions of research in this area. For instance [START_REF] Fujiwara | Welfare implications of leadership in a resource market under bilateral monopoly[END_REF] consider a game with Stackelberg leadership, where the leader can be the oil producing area or the oil consuming area, and wonder whether being the Stackelberg leader is better than being the follower or not. Compared to [START_REF] Rubio | Strategic Pigouvian taxation, stock externalities, and polluting nonrenewable resources[END_REF] they study two varieties of Stackelberg leadership, the global and the stagewise ones, but do not introduce the climate motive for oil taxation. Another path is opened by [START_REF] Wei | The fossil endgame: strategic oil price discrimination and carbon taxation[END_REF], which consider that oil producing countries also consume oil, and can counteract climate policy by using a strategy of price discrimination, subsidizing the oil they consume. They only study the open loop equilibrium of the game.

In the same spirit, it would be very useful to distinguish between two di¤erent zones of oil consuming countries, a "Kyoto zone", setting a common carbon tax, and a "non-Kyoto zone" refusing to do so. In this three players game, the oil cartel's power should be enhanced, but the consequences of the unilateral climate policy on Kyoto and non-Kyoto countries is not trivial and deserves further research.


Cedric Payan

T J Ulrich

P Y Le Bas

M Griffa

P Schuetz

M C Remillieux

T A Saleh

Probing material nonlinearity at various depths by time reversal mirrors

In this letter, the time reversal mirror is used to focus elastic energy at a prescribed location and to analyze the amplitude dependence of the focus signal, thus providing the nonlinearity of the medium.

By varying the frequency content of the focused waveforms, the technique can be used to probe the surface, by penetrating to a depth defined by the wavelength of the focused waves. The validity of this concept is shown in the presence of gradual and distributed damage in concrete by comparing actual results with a reference nonlinear measurement and X ray tomography images.

The principle of time reversal acoustics is based on a simple idea. In any medium, send a pulse from a source. That pulse propagates into the medium. The pulse is eventually reflected many times at the boundaries and by other scatterers. The resulting signal is recorded at a defined location by a receiver. If the recorded signal is time reversed and sent back from the receiver, the wave will play this propagation history backward (as a movie played backward). The wave energy will focus at the precise source location, at a given time (namely the focal time). Thanks to the reciprocity principle [1], the same scenario can be achieved even if the time reversed signal is sent back from the initial source.

In this case, the focus will occur at the receiver location. While this is true with a single emitter, using multiple emitters allows for a proportionately higher amplitude to be obtained at the focal time.

This physical principle has been under study for many years and has been largely developed by Fink [2] with most of the applications in liquids or biological tissues for the medical field.

Applications to the field of nonlinear elasticity in solids were developed with the goal of using the high energy focus to extract some nonlinear properties of solids. It has been successfully applied to locate and image cracks in a metal component [3], to evaluate the quality of diffusion bonds [4], and to probe the interior of a solid [5]. To probe the nonlinearity of the medium with varying amplitude time reversal experiments, so far most of the studies have employed the harmonic content [3] or a scaled subtraction method [6,7]. However, these methods are dedicated to locate or size a single or multiple nonlinear scatterer embedded in a linear medium. They proved very efficient for nonlinear source localization. Here we explore the applicability to volumetrically distributed cracks in a nonlinear material.

The aim of this letter is to propose a physics-based method to measure the nonlinearity of the medium and to evaluate the nonlinearity at various depths. To provide a reliable nonlinear parameter, we make use of concrete samples whose nonlinear elastic behavior was assessed by a quantitative Nonlinear Resonant Ultrasound Spectroscopy (NRUS) measurement. The same samples are in this work also inspected by X-ray tomography to search for evidence of damage, usually considered as the source of the nonlinear behavior [8]. The reader is invited to refer to [8] for details about samples and NRUS results. To probe the nonlinearity at various depths, the frequency content of the transmitted signals is modified. It results in a variable focal spot size which depends on the wavelength.

The experiments are conducted on four concrete samples of size 10×10×6 cm 3 . The first sample remains undamaged while the others are thermally damaged at 120, 250 and 400°C, respectively. A plexiglass sample with the same geometry is also used as a linear elastic reference sample. The experimental protocol is based on reciprocal time reversal (abbreviated "TR" hereafter).

The sample is placed onto a reverberant cavity which is a simple aluminum block with 8 piezoelectric discs (emitters) bonded to the surface at various locations. This cavity allows multiple reflections to occur, delaying the information available over time. It has been shown that both multiple reflections at the boundaries and multiple scattering improve the efficiency of the time reversal process [9]. A laser vibrometer records the out-of-plane particle velocity at the top of the sample. An 8-channel 14-bit generator/digitizer system is used for signal generation and acquisition. The generator is connected to an 8-channel amplifier, which drives the emitters up to a 100Vpp voltage. A computer controls the TR experiments and allows moving the sample with a synchronized motion controller.

Five frequencies (f=100, 150, 200, 250 and 300kHz) corresponding to various wavelengths are selected. Note that the wavelength varies as a function of speed of sound c as λ=c/f. The size of the focus, i.e., the penetration depth, is known to correspond to λ/2 [1,6]. Over the full set of samples and frequencies, the wavelengths range from 9mm to 48mm. A chirp signal (sinusoid with frequency varying in a given range) with a 50kHz bandwidth is sent to one emitter. The signal is recorded by the laser vibrometer, cross-correlated with the initial chirp signal (this operation allows obtaining the impulse response of the sample in the selected frequency range) and recorded by the system. The same chirp is emitted from another emitter and the corresponding propagated signal is recorded as well. This process is executed for each channel. When the 8 impulse responses are recorded, a signal containing 4 successive amplitudes (scaled by 1, 0.9, 0.8 and 0.7 respectively) is built for each channel. All of them are then time reversed and sent back from their initial emitters simultaneously. The laser then records the four resulting focused waves, repeated each 50ms to average 64 times. This process is performed for 4 increasing voltages (12.5, 25, 50 and 100Vpp respectively) producing a total of 16 amplitudes. The full protocol is repeated at 2 other locations on the sample, 2cm away from each other's, in order to average the nonlinearity over the surface.

In materials such as rocks and concrete, so called nonlinear mesoscopic elastic materials [START_REF] Guyer | Nonlinear Mescopic Elasticity[END_REF], several nonlinear and non-equilibrium phenomena occur. The phenomenon of interest in this work is related to the material softening subjected to high strain waves (~ 10 -6 ), also called fast dynamics. NRUS experiments [8]. The softening of the material is evaluated as the relative velocity change Δc/c 0 with c 0 being the low amplitude speed of sound. The time delay Δt between signals is estimated by the cross-correlation of the low amplitude signal with higher amplitudes ones, similarly to the methods used by Rivière et al. [START_REF] Rivière | [END_REF] and Tournat et al. [12], for assessing nonlinearity. For more accuracy, the maximum position of the cross correlation is obtained by fitting the peak region with an oversampled second order polynomial function. The delay measured for the 400°C sample [Fig. 1(a)] is 7ns for the highest amplitude. Over all measurements, the delay ranges from 20ps. to 140 ns.

As most of the energy is concentrated into a λ/2 region, the hypothesis is that the focused waves reach the softening nonlinear regime [13] only in this region. Assuming in first approximation the retro-focused wave field as a standing wave in a volume extended in depth for a length of λ/2, the relative velocity change can be approximated as Δc/c 0 = -Δt/t 0 ≈ -2f Δt, where f is the central frequency of the chirp source signal and t 0 the time of flight for a λ/2 path. Similar to NRUS experiments [8], the nonlinearity parameter α is extracted from the slope of the relative velocity change versus the strain amplitude Δc/c 0 = α Δε [solid lines Fig. 1(b)]. The relative velocity change is averaged over the three measurements points at the surface of the sample. As an approximation, the strain amplitude is evaluated as Δε zz ≈ Δv zz /c 0 where Δv zz is the out-of-plane particle velocity amplitude recorded by the laser. Due to the dispersive nature of concrete, the speed of sound c 0 is measured by time of flight for each frequency. Imperfections in the linear fit Fig. 1(b) are explained by experimental considerations associated with the coupling between nonlinearity and conditioning [14]. In each curve plotted in Fig. 1(b), considering a group of four points, the successive 4 amplitudes focused waves, averaged 64 times, make the material conditioned at the highest amplitude. Therefore, for the lower amplitudes, the material does not have time to entirely recover. Due to experimental conditions, the other groups of points are recorded about one minute later, leaving time to the material to recover. By considering only the highest amplitude signals at each voltage (suppressing the cited effect) the results are affected by less than 2%. However, as for NRUS, time delay is evaluated under fully conditioned conditions. It is also noticeable that in some cases Fig. 1(b), the fit could also apply with a quadratic function. Over the entire data set, there is no general trend allowing to draw a conclusion. However, this nonlinear trend can be explained by the strain estimation which relates more the effect of pressure components here. The complex interplay of compressional and shear components at the focus is not accounted, thus depending on the orientation of the microcracks into the focal spot, it is likely that the nonlinearity could answer more to shear than pressure waves or inversely. The full set of results is presented and discussed in the next section. Fig. 2 relates the full set of results and validations. Figure 2(a) shows the nonlinearity as a function of penetration depth for the full set of samples. The linear reference, i.e., the plexiglass sample, is one order of magnitude less nonlinear than the concrete. It provides the Signal-to-Noise Ratio (SNR) of this experiment. The SNR is more important in this measurement than for the NRUS measurement performed on the same samples [8]. Due to the homogeneity of thermal damage in concrete [15], the 20,120 and 250°C samples exhibit roughly constant nonlinearity. For the full set of concrete samples, the evolution of the average nonlinearity matches very well the reference NRUS measurements [See [8] and Fig. 2(c)]. The underestimation of the nonlinearity in the present study is due to the strain evaluation which is approximated by Δε zz measured at the surface instead of the volumetric strain evaluated in [8]. Janssen et al. [6] have shown that the focal shape looks like a pear shape with a maximum volumetric stress at the surface. Considering the interaction zone as the whole focal spot, the average strain amplitude should be lower than the one measured at the surface. This leads to an overestimation of the strain amplitude, which tends to decrease the nonlinear parameter.

Figure 2(a) shows that the 400°C sample does not exhibit a constant nonlinearity with the depth of the material, with a clear increase of the nonlinearity in the near surface (~1cm). To understand this result, X-ray tomography measurements are performed on the reference concrete sample (20°C) and on the 400°C one. The details of such measurements are reported in the Supplementary Materials [15]. During thermal loading of concrete, the main damage (thus nonlinearity) source consists of debonding at the interface between the cement paste and the aggregates [8,16]. Figure 2(b) shows two vertical, digital cross-sections extracted from two X-ray tomographic datasets, one dataset for the reference sample at 20°C (top image), the other dataset for the sample thermally loaded at 400°C (bottom image). The arrows overlaid on top of the two images point to regions of potential debonding between the aggregate and the cement paste. Despite the low spatial resolution (about 100 m) compared with the debonding size, it is clear from the two sample images that such regions of potential debonding are more frequent and much more widespread in the thermally loaded sample. A semi-quantitative, 2D image analysis of such regions is presented in the Supplementary Materials [15] and it confirms that the thermally loaded sample is characterized by aggregate-to-cement paste boundary regions with more debonding features. The bottom image in Fig. 2(b) clearly shows that the debonding runs all along the boundaries between a large aggregate and the surrounding cement paste. Signs of debonding with such a spatial extension are present in other parts of the sample. However, their concentration is larger within the first centimeter from the surface. This is due to aggregate segregation by size, occurring during mixing, leading to a higher number of large aggregates closer to the surface than in depth. Larger aggregates correspond to smaller inter-aggregate distances, which then leads to larger, localized thermal stresses in the cement paste, thus more frequent debonding [16]. In addition, debonding at the aggregate-cement paste interface is also driven by drying shrinkage, which occurs more easily closer to the surface [17], especially at elevated temperature such as 400°C. We show in this letter the feasibility of using time reversal at various frequencies to probe the nonlinearity of materials at various depths. The correlation of the results with reference measurements from the literature, along with X-ray CT images which explain the results for the 400°C sample, validates the concept. The time delay observed in the experiments, a signature of nonlinear mesoscopic elasticity, suggests a physically based method to quantitatively evaluate the nonlinearity. This approach could be extended to the classical nonlinearity by monitoring the harmonic content with increasing amplitude or slow dynamics by looking at the slow recovery of initial focal time. A better estimation of strain amplitude is under study by further numerical simulations associated with three component laser measurements. It should ultimately allow precise and local nonlinearity measurements using a time reversal mirror and it has the potential to be easily applied in situ for practical applications, such as inspection of concrete infrastructures.

Figure 1 (

 1 Figure 1(a) shows an example of a recorded retro focal time series zoomed at focal time. It is noticeable [refer to the dotted circles] that the zero crossing occurs at the same time for each amplitude in the linear Plexiglas sample while a time delay is clearly visible in the 400°C sample. The focal time appears more delayed, due to the decreasing speed of sound with increasing amplitude, a signature of the material softening. This time shift refers to the relative resonance frequency shift measured in

Figure 1 (

 1 b) shows results at 150 kHz. As expected, the Plexiglas sample is less nonlinear than concrete ones and the nonlinearity increases with thermal damage in the concrete samples.

FIG. 1 .

 1 FIG. 1. (Color online) (a) Zoom at focal time for the linear plexiglass sample and the most damaged sample at 150kHz. For clarity, the time delay is magnified. (b) Relative velocity change at 150kHz for the full set of sample.

FIG. 2 .

 2 FIG. 2. (Color online) (a) Compilation of the results. Solid lines are the average nonlinearity for each sample. (b) Xray CT images of the 20°C sample (top) and the 400°C sample (bottom). (c) Comparison of the average nonlinearity noted TR with NRUS data from [8].

Acknowledgments

Authors thank the US DOE Used Fuel Disposition (Storage) campaign and the Electrical Power Research Institute for supporting this study and Payan's visit to LANL. Payan was also supported by Aix Marseille Univ. We would also like to thank James Ten Cate, Paul Johnson, and Robert Guyer for useful discussion during the course of this work.