Economics of Global Warming: How to Decentralize the Social Optimum in the Presence of an Oil Rent ? Antoine BELGODERE 1 Institut de l’Environnement Project ’Dynamiques des Territoires et D´eveloppement Durable’ Universit´e de Corse - Campus Caraman - 7, avenue Jean Nicoli 20250 CORTE CORSICA (FRANCE) Abstract In this paper, I study the optimal climate policy in the presence of an oil rent. Previous papers show that, in the long run, because of the Hotelling’s rent, the optimal tax must decrease. However, as the full depletion of the polluting non renewable resource may be suboptimal, it is an equilibrium condition for the resource market, if the price is strictly positive. In this situation, an optimal tax would imply the disappearance of the hotelling rent. Thus, the interaction between global warming and oil depletion can be broken, and the tax can be increasing. Key words: Hotelling rent, Global warming, Pigovian tax, Economic growth JEL Classification: H30, Q32, Q40 1 Introduction This paper studies the optimal climate policy in the presence of an oil rent. This study is done by combining the features of the cumulative pollution models and those of the non renewable resources models. Since the seminal papers by Keeler et al. (1971), Plourde (1972) and Forster (1973), a large literature in environmental economics has focused on the problem of cumulative pollution. Concerns about climate change explain this inEmail address: [email protected] (Antoine BELGODERE). The author is grateful to Sjak Smulders, Flora Bellone, Dominique Prunetti and Sauveur Giannoni for their precious comments. 1 terest, as greenhouse effect is caused by the accumulation of carbon in the atmosphere. Optimal control models indicate that an increasing Pigovian tax should be raised on the use of polluting goods, such as fossil fuels. This result matches the intuition. As pollution accumulates, the marginal external cost of polluting is increasing over time, for a given amount of polluting emissions. In a situation where pollution is a flow, the Pigovian tax, at each period, must equate the marginal external cost for the period. But when pollution accumulates into a stock, the Pigovian tax must take into account not only the current external cost of pollution, but also the actual value of the future external cost caused by accumulation. In these conditions, the shadow price of the pollution grows over time, and asymptotically reaches a steady-state value. This result implies that, if an optimal policy were to be enforced by an international agreement on climate change, one should expect the price of the fossil fuels, including a Pigovian tax, to grow over time. Actually, such kind of expectations already exists in the public’s mind, but the main cause is not the global warming. Fossil fuels are a non-renewable resource. Hotelling (1931) showed that, at the market equilibrium, the price of a nonrenewable resource must grow according to a rate that equates the discount rate. This result directly follows from 1) a non-arbitrage condition and 2) the fact that, by definition, the stock of a non-renewable resource can only decrease. A stock of non-renewable resource is an asset, and must yield the same rate of return to its owner than any other asset. The only way a stock of non-renewable resource can yield the same return as other assets, is that its price grows at a rate that equates this common rate of return. Indeed, the limits of the oil stocks, that are rather low from a human scale, may allow to predict an increasing trend of oil prices for the next decades 2 . As cumulative pollution models advocate an increasing Pigovian tax on fossil fuels while nonrenewable resources models predict an increase in the price of fossil fuels, one can ask if a model integrating both features could make the market equilibrium closer to the social optimum. Indeed, the increasing rent can be expected to have the same effect as a Pigovian tax, i.e. to incite users of fossil fuels to save energy and to adopt renewable energy sources, which are assumed not to increase the carbon concentration in the atmosphere. This idea has led to a debate in the early 90’s, engaged by Sinclair (1992) with an article whose provocative title was: ’High does nothing and rising is worse: Carbon taxes should be kept declining to cut harmful emissions’. As the title indicates, his main results are 1) the optimal path of a tax is defined only by its rate of growth, and not by its initial condition that is neutral, and 2) this rate of growth must be negative 3 . As we will see in more detail, 2 Although recent pressures on oil prices can be explained, to a large extend, by short-run phenomena. 3 Schulze (1973), Hoel (1978) and Forster (1980) already dealt with the interaction 2 point 1) is explained by the nil price-elasticity of the resource supply. Point 2) is due to the fact that, under several assumptions, the resource stock is asymptotically exhausted. In this situation, the only effect of an environmental policy is to transfer pollution from the present to the future. At least two arguments advocate this transfer. First, future damages are perceived to be less harmful than present ones because future utility is discounted. Second, while the use of the resource becomes asymptotically nil, the stock of pollution decreases after reaching a top, because of the natural assimilation. Thus, the external cost of carbon emissions becomes also asymptotically nil. Several articles were published in response to Sinclair’s (Ulph and Ulph (1994), Sinclair (1994), Hoel and Kverndokk (1996), Farzin and Tahvonen (1996), among others). Tahvonen (1997) provides a major contribution to this debate. He establishes, in a very general framework, that 11 different kinds of paths can be obtained, depending on the assumptions about the use of backstop technology and the decay process. Schou (2002) takes into account the incentive to invest in knowledge, and finds that no tax is needed to implement the optimum. But Grimaud and Roug´e (2005) show that Schou’s result holds for a Cobb-Douglas utility specification, but is not always valid for more general specifications. This article adds to the debate by describing a regime where the climate policy makes the Hotelling’s rent to disappear. The argument presented here is very similar to the intuitive one made on Berck and Roberts (1996) (p68), but is presented in a complete macroeconomic framework. With no Hotelling’s rent, the link between oil depletion and climate change is broken. In this context, an increasing Pigovian tax is needed. This result is based on two key assumptions: 1) the presence of a backstop technology, and 2) there is some irreversibility in pollution. The paper is organized as follows: section 2 presents the model and characterize the optimal path, section 3 investigates in which way the social optimum can be decentralized, section 4 discusses some issues involved in this model and then section 5 concludes. 2 2.1 Presentation of the model and analysis of the optimal path The model The representative consumer of the economy has an instantaneous utility function U which depends on consumption C(t) and on the stock of pollution M(t), between oil depletion and cumulative pollution, but did not focus on the specific questions discussed here. 3 with 4 ∂U ∂C > 0 and ∂U ∂M < 0. U is assumed to be strictly concave. M is assumed to follow M˙ = ζE with ζ ∈ [0; 1] being a parameter and E the rate of resource extraction. It is assumed here that the stock of pollution is expressed in the same unit as the stock of resource. In the case of global warming, this unit can be the ton of carbon. Every unit of resource extracted will be released into the nature. A fraction (1 − ζ) is absorbed by the ecosystem (afforestation, oceans,...), and a fraction ζ accumulates into the pollution stock. Notice that no decay process is assumed here. This choice needs to be justified. Our point concerns the cases where full asymptotic depletion of a nonrenewable resource is not optimal. With a linear decay function, which is often assumed in the literature, and a non-renewable polluting resource, the stock of pollution will be asymptotically nil. So will be its external cost. In this situation, only an excessive private cost of extraction could explain a partial resource depletion. More complex decay processes, such as that used by Forster (1975) and Withagen and Toman (1998), become nil after a given threshold attained by the stock of pollution. In Farzin and Tahvonen (1996), carbon accumulates into 4 stocks, 3 of which have a strictly positive linear rate of decay, while the fourth has a nil rate of decay. These features account for the limits in the absorptive capacity of the environment. In this case, the stock of pollution can, on an optimal path, satisfy lim M (t) > 0 . However, t→∞ for simplicity, I will assume that decay is nil. The stock of non-renewable resource evolves according to: S˙ = −E (1) With this formulation, for M(0) and S(0) given, M(t) can be expressed as a function of S(t). M (t) = M (0) + ζ [S(0) − S(t)] It follows that M can be replaced by S in the utility function 5 . We then have 2 ∂U > 0 and ∂∂SU2 < 0. ∂S 4 When no confusion is possible, time references are removed. Similar modelings are used in Schulze (1973), Hoel (1978), Forster (1980) and Sinclair (1994). 5 4 In these conditions, our model becomes a cake eating model with a stock effect, close to Krautkraemer (1985). The final output Y is produced using capital K(t) 6 and an energy input N(t). Two energy sources are available. The first is the polluting non-renewable resource E(t). The second is an infinitely but costly available non-polluting resource A(t). So: N (t) = E(t) + A(t) Perfect substitution is assumed here 7 . This is justified by the fact that what are principally expecting energy purchasers are joules. No distinction can be made between joules produced by different energy sources. Of course, complementary services are also required, such as storage and transportation. The substitution is imperfect when one takes into account these features of energy demand. The differences between energy sources from the point of view of these complementary services will be taken into account by the convex cost function of the renewable energy. If oil and wind are not perfect substitutes for a car driver, this is not because the joules produced by the wind are different from those produced by the oil. This is because the joules must be stored and transported. To store and transport wind energy in the same way as oil, one has to transform it into hydrogen, which is is costly. To reflect this, we introduced q(A), the amount of final good devoted to produce A. We assume q 0 (A) ≥ 0, q 0 (0) = 0, and q 00 (A) > 0. ∂Y > 0, We assume ∂N according to: ∂Y ∂K > 0 and the strict concavity of Y. The capital evolves K˙ = Y − C − δK − q(A) (2) Where δ is a depreciation parameter. 2.2 2.2.1 Social optimum Characterization of the optimal path The social planner maximizes 0∞ U e−ρt dt, where ρ is the social discount rate, under (1), (2) and E, C, A ≥ 0. C, E and A are the control variables. The R 6 By seek of simplicity, we will not take into account any other input, such as labor. Thus, K(t) must be thought of as an aggregate of all the inputs but the energy. 7 Such as in Tahvonen (1997) among others. 5 Hamiltonian associated with this problem is: H = U − λE + µ [Y − C − δK − q(A)] Where λ and µ are the co-state variables associated, respectively, with S and K. The maximum principle 8 give the following conditions 9 : UC = µ (3) YE ≤ λ/µ E ≥ 0 E (µYE − λ) = 0 (4) YA = q 0 (A) (5) λ˙ = ρλ − US (6) µ˙ = µ(ρ + δ − YK ) (7) lim e−ρt λS = 0 (8) lim e−ρt µK = 0 (9) t→∞ t→∞ The assumptions made about UC and q(A) allow to look for an interior solution for C and A, but not for E. (3) states that the marginal utility of consumption must equate the shadow price of the capital. In order to interpret (4) when YE = λ/µ 10 , one can use (3) in order to write UC YE = λ. Along an optimal path, UC YE is the marginal utility of the extractions. Indeed, if an extra unit of resource is extracted, then the output will increase by YE . Along an optimal path, this marginal output can be indifferently affected to consumption or to the capital accumulation. If it is consumed, then the utility will increase by UC YE . Thus, (4) states that the marginal utility of the extractions must equate the shadow price of the stock of resource. (5) states that the marginal productivity of the renewable resource must equate its marginal cost. (6) is the modified Hotelling’s rule. It can be interpreted as a non-arbitrage condition. An asset whose value is λ, must yield, in term of utility and from a social point of view, the same as if this value was placed at a rate ρ. What it yields is US , the marginal utility of the stock of resource, which corresponds to the avoided ˙ which is the gain in capital. (7) is the classical Ramseypollution, plus λ, Keynes rule. (8) and (9) are the transversality conditions. Eliminating the costate variables in (3)-(7) provides the two following equations, that describe 8 These conditions are both necessary and sufficient thanks to the conditions on U and Y. 9 I denote by M the derivative of the variable M with respect to N. N 10 That is to say when the non-negativity constraint on E is not binding. 6 the optimal path: d + Yc = ρ − U C E US U C YE (10) d =ρ+δ−Y U C K (11) where a letter with a hat is the growth rate of the variable. (10) and (11) are benchmarks that will be compared to the equilibrium path. The elimination of the co-state variables is necessary to make this comparison, because the co-state variables are not identically equal to the corresponding prices in the market equilibrium. An analysis of the whole optimal path could be provided, probably with some difficulty. Indeed, if the economy starts with a low stock of capital, then one expects the marginal productivity of the resource to be low. But, if the economy also starts with a low stock of pollution (i.e. a hight S), then the marginal external cost of using the resource is low. While the low stock of capital advocates a low rate of extraction, the low pollution stock allows a hight rate of extraction. As the economy grows, one expects both stocks (i.e. capital and pollution stock) to increase, which have an ambiguous effect on the evolution of E(t). In particular, a path in which the non negativity constraint on E is binding for a period, and then non binding for a latter period cannot be excluded. However, the main purpose of this paper, is to focus on the behavior of the system when t tends toward infinity, because the difficulty to decentralize the optimum with a tax arises from asymptotic conditions. That is why the next sub-section deals with the steady state of the model. 2.2.2 Variables in steady state In order to search the steady-state, we introduce the following specifications of U, Y and q: U (C, S) = S 1− C 1−σ +Λ 1−σ 1− Y (K, E) = ΩK α (A + E)1−α q(A) = 0.5βA2 Where σ > 0, > 0, α ∈]0; 1[, Λ > 0, Ω > 0 and β > 0 are parameters. ˙ = µ∗ ˙ = K∗ ˙ = 0. I note with an I am looking for a steady-state, where λ∗ ˙ = S∗ asterisk the steady-state values. The co-state variables are now reintroduced 7 for computational purpose. Simple calculations give: E∗ = 0 α A∗ = ψχ α−1 α+1 K∗ = ψχ α−1 2α Y ∗ = Ωψχ α−1 2α C∗ = ψχ α−1 ζ −σ α 2α ρ α−1 S∗ = (1 − α)Ωχ α−1 ψχ ζ Λ 2α −σ 2α −σ µ∗ = ψχ α−1 ζ λ∗ = ψχ α−1 ζ Where ψ ≡ Ω(1−α) β − 1 ε α (1 − α)Ωχ α−1 > 0, χ ≡ ρ+δ αΩ > 0 and ζ ≡ Ω − δχ−1 − 1−α Ω 2 > 0. = ∞. It means With this specification, S∗ > 0. That is because lim ∂U S→0 ∂S that there is a threshold in the pollution stock, from which pollution has ¯ = M (0) + catastrophic consequences. It is only for simplicity that I chose M S(0) as particular value for this threshold. The qualitative result of this model would be the same with lim ∂U = ∞ with φ ≥ 0. ∂S S→φ From a comparative statics point of view, one can remark that S*, the asymptotic remaining oil stock, is negatively related to β and positively related to Λ. The interpretation is rather trivial. The costlier are renewable resources, the more oil will be used. The more weight is put on the environment in the utility function, the less oil will be used. Surprisingly, A* does not depend on Λ. This is because, on steady state, no oil is used, and then, using renewable resource is not related, on the long run, to a trade off between clean and polluting energy, but only to a cost-advantage analysis of the use of energy. It can be shown 11 that the steady state is a saddle path for certain values of the parameters, among which the set of reasonable values given in table 1 12 . In this case, the variables tend asymptotically to their steady-state values. In particular, I can state: lim S = S∗ > 0 (12) t→∞ 11 12 See appendix A for the demonstration. See appendix B for a justification of these values. 8 α β δ σ ρ Ω Λ 0.95 3.22 × 10−11 0.1 1.75 0.9 0.03 0.0038 1 Table 1 Parameters values This equation is important for our purpose, as it states that the stock of resource will not be fully exhausted along an optimal path. (12) will be compared to the equilibrium condition for the resource market in the next section. Furthermore, transversality conditions (8) and (9) are fulfilled, because µ, λ, S and K tend asymptotically to a constant steady-state value. 3 Implementation of the optimum 3.1 Agents’ behavior The economy is composed of 3 representative agents, plus the social planner: a representative household, a representative firm in the final good sector, and representative firm that manages the stock of nonrenewable resource. 3.1.1 The household The household chooses a consumption path in order to maximize his intertemR poral utility 0∞ U e−ρt dt. The constraint he faces is the evolution of its assets X, given by X˙ = rX + T − C, where r is the interest rate, and T are transfers from the government. X is a mix of equities in the firm of the final sector and of equities of the firm that manages the stock of resource. As risk is absent in this model, in the market equilibrium both assets yield the same rate of return r. This is a standard Ramsey model, from which one gets: d =ρ−r U C 3.1.2 (13) The final good sector In the final good sector, the firm maximizes its profit π = Y − (r + δ)K − P τ E −q(A), where P is the price of the resource, the final good being taken as the numeraire, and τ is an ad-valorem tax on extraction plus one. This yields: YE = P τ ⇔ YcE = Pb + τb (14) YK − δ = r (15) 9 YA = q 0 (A) (16) The tax revenue and the profit 13 are redistributed lump-sum to the consumer: T = P E(τ − 1) + π, which does not affect the consumer’s behavior. 3.1.3 The resource sector The stock owner maximizes the sum of the actual value of its future rent Z∞ P Ee− Rt 0 r(u)du dt 0 under (1). As perfect competition prevails, P does not depend on E, from the firm’s point of view. In equilibrium, the two following equalities hold: P˙ = rP lim e− (17) Rt 0 r(u)du t→∞ P (t)S(t) = 0 (18) (17) is the Hotelling ’r-rule’, and (18) is the classical equilibrium condition that states that the value of the exceeding demand must be nil. Indeed, from anRinter-temporal point of view, the resource supply is S(0), and the demand is 0∞ E(t)dt. Notice that lim S(t) = S(0) − Z∞ t→∞ E(t)dt 0 So, lim S(t) that appears on (18) is the opposite of the excess demand. t→∞ So, the resource market will be in equilibrium if lim S(t) = 0 or if − lim e Rt 0 t→∞ r(u)du t→∞ P (t) = 0. (17) and (18) lead to P (0) lim S(t) = 0 (19) t→∞ (19) is respected only for lim S(t) = 0 or for P(0)=0. t→∞ 13 The profit is not nil because of the strict convexity of q(A). 10 3.2 Optimal policy tool 3.2.1 The optimal ad-valorem tax (13) and (15) are identical to (11). No economic policy is needed at this stage. (13)-(17) imply: d + Yc = ρ + τb U C E (20) Comparing (10) to (20) indicates the rate of growth that must follow the ad-valorem tax in order to decentralizes the optimum: τb = − US <0 U C YE (21) At first glance, (21) seems to confirm Sinclair’s result of a decreasing tax, only defined by its rate of growth and not by its level. But, actually, I did not prove that an ad-valorem following (21) enables to decentralize the optimum. Such a tax only ensures that the variables in the decentralized economy have the same rates of growth that in the social optimum. But in order to be identical, two paths need not only to have the same rates of growth, but also to have one common point. In the context of the present model, the latter condition is not respected. Indeed, the optimal steady-state value for S, (12) does not match the equilibrium condition (19). If P (0) > 0, then they are not compatible, because in the market equilibrium, it implies lim S(t) = 0 6= S∗. In these t→∞ conditions, the optimum is not decentralized. If P(0)=0, then it follows from (14) and from our assumptions about the properties of Y(.) that the firm’s demand for E(0) will be infinite. Of course, only S(0) can be supplied, and then S(t) = 0∀t > 0, which does not correspond to the optimum. This result partly depends on the modeling of the tax as an ad-valorem tax. With a tax whose amount per unit is fixed by the social planner, the optimum can be decentralized 14 . 3.2.2 The optimal per-unit tax With a per-unit tax, if the price perceived by the stock owner is nil, the price paid by the firm, including the tax, can be strictly positive. But this does not affect the equilibrium conditions for the resource market. Thus, if one considers a lump sum per-unit tax, and if this tax enables to impose lim S = S∗, the price perceived by the owner of the stock of resource is nil all t→∞ 14 Alternatively, a constant cost of extraction could be introduced. 11 over the planning horizon. Let’s call θ this tax per unit of resource. As P=0, the profit maximization in the final good sector becomes: YE = θ ⇔ YcE = θb (22) In these conditions, the optimum is decentralized if: θ= λ µ (23) Indeed, (6), (7), (13), (15), (22) and (23) are identical to the benchmark (10). As the tax equates the marginal productivity of the resource, and provided K(0) < K∗, it will be continuously rising, and reach asymptotically ∂F (K∗,0+A∗) . Thus, the necessity to maintain a strictly positive stock in the ∂E steady-state breaks the interaction between the cumulative pollution and the resource depletion, when the policy tool is a tax. The problem then becomes a classical model `a la Plourde. To understand this result, let’s transpose it into a discrete, finite-time model. Assume that, for some environmental reason, the regulator imposes that the stock is not fully depleted at t*, which is the end of the planning horizon. Then, E(t∗) < S(t∗), so P (t∗) = 0. In this situation, what can be an equilibrium price at t*-1 ? Obviously, P (t ∗ −1) = 0, because nobody will invest in an asset whose value will be nil at the next period. By recurrence, one understands that P (t) = 0 for every t if E(t∗) < S(t∗). 3.2.3 The optimal subsidy The representative household assumption does not directly allow to adopt a distributive point of view, as Amundsen and Schob (1999) do in an international trade framework. However, one can guess that if the owners of the resource stock are different from the rest of the population, the distributive impact of the tax is important. This is especially true if these owners earn the main part of their income from the resource exploitation, such as OPEC countries do. In the Amundsen and Schobs’ paper, the rent is partly transferred to the other countries by the environmental policy. In the framework of the present model, this rent would be fully transferred. However, another policy tool can be used to implement the optimum, with an opposite distributive effect: a subsidy on the remaining resource stock. A subsidy s is paid at each point of time to the firm of the resource sector for each unit of the stock of resource. In this context, the equilibrium conditions in the resource market become: s rP = P˙ + s ⇔ Pb = r − P (24) 12 and (18), which is unchanged. (24) is the modified Hotelling rule. Now, the investment in the resource stock yields not only P˙ , but also s. Thus, P will grow, on the equilibrium, slower than without subsidy. (13) is not modified by the introduction of the subsidy. Now, T can be negative 15 . The subsidy is financed by a lump-sum tax. As no more tax is assumed, (14) gives: YcE = Pb (25) (13), (15), (24) and (25) give: d + Yc = ρ − U C E s YE (26) that is identical to the benchmark (0.9) if: s= US UC (27) Of course, such a subsidy will create an incentive to find new resource stocks, that is not taken into account in the present framework, where the size of the stock is exogenously given. 4 Discussion 4.1 Has global Warming killed the Hotelling’s rent ? If one takes the Hotelling rule in the strict sense of an increase of the resource price at the interest rate, then this rule is not compatible with the observed long run trends of oil prices. Actually, empirical tests of the non renewable resources theory rely on more complex models 16 than the genuine Hotelling model. These models, by taking into account the presence of extraction costs and technical progress, make compatible a non-increasing trend of prices and the arbitrage assumption that underlies the ’r-rule’ theory. However, these tests are not consensual about the validity of the Hotelling ’r-rule. One can advocate that the this rule holds for a given size of the known stocks. As new stocks are discovered, a new path initiates for the rent, still 15 16 T is negative if the subvention is greater than the profit in the final good sector. See Chermak and Patrick (2002) for a classification. 13 increasing, but starting from a lower level. Actually, such an explanation is far from convincing, as it assumes very myopic expectations. The model presented in the previous sections gives a potential explanation for the non-increasing of the oil rent. If the owners of the oil stocks anticipate that, on the long run, it will be socially optimal not to deplete the stocks, and that some politic agreement will implement this optimum, then the market equilibrium implies a non-growing price. Notice that the reason why such expectations might exist need not to be the global warming, that was not a big concern 25 years ago. Any other reason would produce the same effect, such as the political wish of the countries to be energetically independent. While there is some doubt about the existence of an increasing scarcity rent, some oil rents actually exist, shared by oil-producing firms and oil-exporting countries. These rents seem to be in contradiction with the model discussed here. But the presence of oil rents does not mean that these are Hotelling rents. They can simply be oligopoly rents. Indeed, oil extraction requires high investments, and implies high sunk costs. That advocates an imperfectly competitive market. Moreover, as extraction capacities, at a given point of time, are limited and take time to expend, booms in oil demand have an impact on prices. For technical reasons, these short-run effects can last for a rather long time. So, big profits in the oil sector is not incompatible with a model that predicts no Hotelling rent. 4.2 Empirical implication of the model Slight changes in the model presented here could enable to implement an insightful empirical test. Shortly, assume that a political decision about global warming has to be taken at finite time tdec 17 . From 0 to tdec , the decision is uncertain. The decision consists in choosing between the adoption or the rejection of an environmental policy. This policy implies the hotelling rent to disappear. The uncertainty that prevails in the first sub-period would alter the ’r-rule’, but would not make it to disappear. The decision taken in tdec , by changing the state of knowledge, must change the path of the rent. In particular, if the policy is adopted, the rent becomes nil. A simple empirical test would consist in testing for the constancy of the rent and of its rate of growth in the two sub-periods. 17 In this short, informal presentation, I assume only two sub-periods, whereas the argument can be generalized to a higher number of sub-periods. 14 4.3 Scope of the paper and further research perspectives The scope of this paper was not to give a definitive answer to the debate about the interaction between oil depletion and climate change. The strong assumptions that I made about the decay process and the cost of extraction obviously don’t allow for such a claim. In addition to the empirical test mentioned above, further researches on the problematic discussed here should take into account the three following points: • More complex decay processes. A nil decay process is probably not a necessary condition to make the Hotelling rent to disappear, and then to break the link between oil depletion and global warming. • Innovation on renewable clean energies should be taken into account. In this context, an endogenous R&D activity would, for instance, reduce the parameter β. • Strategical interactions should be modeled through a dynamic game. Indeed, both the stock of carbon and the stock of knowledge on the backstop technology have public good features. 5 Conclusion In a model integrating both a cumulative pollution and a non renewable resource, I showed that the optimal Pigovian tax can cause the disappearance of the Hotelling’s rent. This breaks the interaction between resource depletion and climate change. This result is obtained by assuming no natural assimilation of the pollutant and the presence of a backstop technology. 15 Appendix A Stability analysis of the steady state in the specified model The theorem used by Tahvonen (1991) enables to prove that a steady state has a saddle-path stability. To use it, one has to define the so-called modified hamiltonian dynamic system of the social planner’s problem. The three control variables can be expressed as explicit functions of the stock variables and the co-state variables, along an optimal path. λ E = E (S, K, λ, µ) = K µ (1 − α) Ω A = A (S, K, λ, µ) = !− 1 α − λ µβ λ µβ 1 C = C (S, K, λ, µ) = µ− σ In these conditions, the optimal path is a dynamic system with 4 differential equations and 4 variables. " λ λ −K S˙ = f1 (S, K, λ, µ) = µβ µ (1 − α) Ω " (1 − α) µ Ω K˙ = f2 (S, K, λ, µ) = K λ #− 1 α # 1−α α − µ1/σ − δ K − 1/2 λ2 β µ2 λ˙ = g1 (S, K, λ, µ) = ρ λ − Λ S − " (1 − α) µ Ω µ˙ = g2 (S, K, λ, µ) = ρ µ − µ α Ω λ # 1−α α −δ f1 , f2 , g1 and g2 correspond to (1), (2), (6) and (7) with the control variables being replaced by E (S, K, λ, µ), A (S, K, λ, µ) and C (S, K, λ, µ). I define J (∞) as the jacobian matrix of the modified hamiltonian system of the problem computed on steady state. Furthermore, I define Z in the following way: Z(t) ≡ det ∂f1 ∂f1 ∂S ∂λ ∂g1 ∂g1 ∂S ∂λ + det ∂f2 ∂f2 ∂K ∂µ ∂g2 ∂g2 ∂K ∂µ 16 + 2 det ∂f1 ∂f1 ∂K ∂µ ∂g1 ∂g1 ∂K ∂µ which takes the value Z(∞) on steady-state. Tahvonen (1991) showed that the steady state is a saddle path if the two following conditions are respected: det [J(∞)] > 0 Z(∞) < 0 I first show that Z(∞) is negative for a range of parameters values. It can be computed 18 that Z(∞) has the same sign as the following expression: Θ − (σ 2 −1)(1−α) σα 1−σ 2 Θ σ β + −δ ρ + δ − ΩΘ σ 2 −1 χΘ σα ! (σ 2 −1)(1−α) σα h α i1 (A.1) α i1 (A.2) ρ(1 − α)Ω ρΘ−σ (1 − α)Ωχ α−1 αβ ε where 2α Θ ≡ ψχ α−1 ζ > 0 which is negative, provided: Θ < (σ 2 −1)(1−α) σα 1−σ 2 Θ σ β + −δ ρ + δ − ΩΘ σ 2 −1 χΘ σα αβ ! (σ 2 −1)(1−α) σα h ρ(1 − α)Ω ρΘ−σ (1 − α)Ωχ α−1 ε Which is true for a range of parameters values, among which those in table 1 Finding a condition on parameters in order to fulfill det [J(∞)] involves very complex expressions 19 . However, it can be shown that a range of parameters respect both this condition and that given by (A.2), among which those in table 1. It should be noted that det [J(∞)] is very sensitive to the value of σ. B Calibration of the specified model Take an aggregate stock of capital of 6’500’000 billion dollars, and an energy consumption of 8’600’000 kt oil-equivalent, half of which being with no green18 The computation is complex, and was made with the help of a mathematical software. 19 The Maple file is available on demand from the author. 17 house effect gas emissions. Then, with the values of α, β and Ω given in table 1, the world GDP would be of 24’750 billion dollars. Gross production would be 25’048 billion dollars, and the cost of the renewable energies would be 298 billion dollars. As inputs are paid at their marginal productivity, the capital earning is 23’796 billion dollars, which represents a 0.961 share of the world GDP. This latter figure is different from α because the GDP is not Y but Y-q(A). The remaining 0.039, that is to say 964.65 billion dollars, is the oil rent, in the case where no climate policy is enforced. 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