The rate of change of the social cost of carbon and the social planner's hotelling rule

Kögel, Tomas Working Paper The rate of change of the social cost of carbon and the social planner's hotelling rule Economics Discussion Papers, No. 2012-37 Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Kögel, Tomas (2012) : The rate of change of the social cost of carbon and the social planner's hotelling rule, Economics Discussion Papers, No. 2012-37, Kiel Institute for the World Economy (IfW), Kiel This Version is available at: https://hdl.handle.net/10419/60491 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/2.0/de/deed.en The Rate of Change of the Social Cost of Carbon and the Social Planner’s Hotelling Rule Tomas Kögel University of Greifswald Abstract This paper derives the social cost of carbon (SCC) and its rate of change. It does so in a deterministic Ramsey model of optimal economic growth with carbon emissions from burning fossil fuels. It is shown that the determinants of the rate of change of the SCC are substantially almost identical to the determinants in the social planner’s Hotelling rule if a unit of fossil fuel use leads to exactly one unit of carbon emission, while otherwise these formulas differ substantially. As is also shown in this paper, in the special case in which the two formulas are substantially almost identical, a Pigovian tax on fossil fuel use and a Pigovian tax on carbon emissions are both equal to the SCC, while otherwise only a Pigovian tax on carbon emissions equals the SCC. Paper submitted to the special issue The Social Cost of Carbon JEL D61, Q54 Keywords Climate change; social cost of carbon Correspondence Tomas Kögel, Ernst-Moritz-Arndt-University of Greifswald, Friedrich-Loeffler-Str. 70, 17489 Greifswald, Germany. E-mail: [email protected] This paper is a revision of Kögel (2012) submitted to this journal´s special issue on The Social Cost of Carbon. The co-editor decided that the author should place his paper better into the context of existing literature and that this would necessarily constitute a new submission to the journal. The author hereby followed this advice by a slight change of his model assumptions, which allowed him to better contrast his paper results to the results in the existing literature. © Author(s) 2012. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany Discussion Paper N o. 2012-37 | August 14, 2012 | http://www.economics-ejournal.org/economics/discussionpapers/2012-37 1 Introduction The social cost of carbon (SCC) is de…ned as the present value of the marginal damage from carbon emission, where the damage is caused through climate change. It represents an externality that is not considered by market agents in their decision making process. The externality can however be corrected with a Pigovian carbon tax. Complete internalization of the externality requires the tax rate on carbon emissions to equal the SCC at the optimal carbon emission level.1As a consequence, using Pigovian taxation or alternative climate change policies requires understanding of the determinants of the SSC at the optimal carbon emission level. Moreover, Dasgupta and Heal (1995) suggests that only the time path and not the level of a carbon tax a¤ects the resource extraction path. This suggests that knowledge of the rate of change of the SCC is more important than knowledge of the level of the SCC. In turn, the theoretically correct rate of change of the SCC at the optimal carbon emission level can be derived from the optimality conditions of a social planner. The SCC is usually estimated in applied work in integrated assessment models (IAMs), i.e. in simulation models that integrate economic and scienti…c models of global warming. The …rst step in calculating the SCC is to estimate the stream of future relative marginal damages of carbon. The second step in calculating the SCC is then to employ a discount rate (sometimes labeled consumption discount rate) to convert this stream of future relative marginal damages into a present value.2To choose the discount rate, IAMs usually employ a Ramsey rule, i.e. an optimality condition that must be ful…lled on the consumption path that maximises life-time utility of a representative household (e.g. in the Ramsey model). The Ramsey rule relates the discount rate to the income growth rate. In turn, since a well-known stylized fact of modern growth of Kaldor is a constant average income growth rate in industrialised countries over periods of at least hundred years (see e.g. Sørensen and Whitta-Jacobsen, 2010), it is standard in IAMs to employ a constant discount rate. It appears that this can also be motivated by the fact that historical data show trendless market rates of return on physical capital (more precisely, returns on risky stocks or risk-free government bonds). As 1See e.g. Nordhaus (2011, p. 2). 2See, among others, the articles in the special edition of this Economics E-Journal on "The Social Cost of Carbon" at www.economics-ejournal.org/special-areas/specialissues/the-social-cost-of-carbon. 2 a consequence, IAMs also usually use in their numerical simulations the historical average market rates of return as the value of the discount rate and use parameter values of the utility function that make the historical average market rates of return consistent with the Ramsey rule, given the historical average income growth rate. Recently, Sinn (2008) shows that environmental policies that aim to decelerate climate change actually do in general just the opposite and accelerate climate change and labels this phenomenon a Green paradox. He acknowledges however that a Pigovian carbon tax would do the job to reduce the market economy’s rate of carbon emissions to the social optimal level (cf. page 383). He nevertheless brushes o¤ the usefulness of an optimal carbon tax, by arguing to the e¤ect that a Pigovian carbon tax must be equal to the SCC on the optimal carbon emission path and that in practise it would be di¢ cult to calculate the theoretically correct time path of the SCC. In contrast to this, Tol (2009, page 7) assesses quantitatively the importance of climate policy by assuming the carbon tax rate (and therefore the SCC) to grow with the discount rate. If appropriate, then one could employ the Ramsey rule to also estimate the growth rate of the SCC. For this reason, calculating the theoretically correct time path of the optimal carbon tax would then not be that di¢ cult. In turn, Kuik (2009, page 9) …nds it reasonable to assume the growth rate of the SCC to be equal to the discount rate, in light of the Hotelling rule of optimal resource extraction. Newbold et al. (2009) argue however that the growth rate of the SCC should be lower than the discount rate.3 The …rst contribution of this paper is a derivation of the rate of change of the SCC. More precisely, the predecessor version of this paper - Kögel (2012) - is the …rst paper that derived this rate of change in a Ramsey model with carbon emissions from burning fossil fuels. It has later also been derived in van der Ploeg and Withagen (2012b), which is an updated version of van der Ploeg and Withagen (2011).4However, contrary to this paper version, Kögel 3The applied climate change literature that employs IAMs uses the wording discount rate and calibrates it with historical average market rates of return. The present paper solves a social planner model. As a consequence, henceforth the paper only uses the wording social discount rate, which is de…ned to be equal to the social marginal product of capital. 4The rate of change of the SCC in van der Ploeg and Withagen (2012b) replaced the rate of change of the Hotelling rent that has been derived in van der Ploeg and Withagen (2011). 3 and van der Ploeg and Withagen (2012b) derived the rate of change of the SCC for the special case in which a unit of fossil fuel use leads to exactly one unit of carbon emission. This simpli…cation does however matter, as this paper version shows. This paper provides also intuitive reasoning for the possible size and sign of the growth rate of the SCC. As I found out after the writing of this paper’s predecessor version, the …rst contribution of this paper is also very closely related to a result in Grimaud et al. (2011), who derive the growth rate of the optimal carbon tax in an endogenous growth model with carbon emissions from burning fossil fuels. Their formula looks much more complicated than this paper’s growth rate of the SCC. However, this paper shows that nevertheless their growth rate of the optimal carbon tax is substantially identical to the present paper’s growth rate of the SCC if the social planner’s optimality conditions are used to simplify their formula. A somewhat similar growth rate of the optimal carbon tax has also been derived in partial equilibrium models, such as Ulph and Ulph (1994).5As this paper shows, the latter literature’s growth rate of the optimal carbon tax can be seen as a special case of the present paper’s growth rate of the optimal carbon tax if instantaneous utility from consumption is linear and additively separable from instantaneous disutility from carbon emissions. Hence, the paper shows that the optimal carbon tax in partial equilibrium models is not too unrealistic, but is also not substantially identical to the one in general equilibrium models, since whether or not instantaneous utility is linear in consumption makes a di¤erence. The second contribution of this paper is to show how the rate of change of the SCC is related to the social planner’s Hotelling rule (i.e. the formula describing the social planner’s optimal resource extraction path). An understanding of this relation is important because the social planner’s Hotelling rule has been derived in earlier work (e.g. in Groth and Schou, 2007, and Sinn, 2008) and is therefore well-known within the literature. The present paper shows that the determinants of the rate of change of the SCC are substantially almost identical to the determinants in the social planner’s Hotelling rule if a unit of fossil fuel use leads to exactly one unit of carbon emission. Otherwise those formulas di¤er substantially. As is also shown in this paper, in the special case in which the two formulas are substantially 5See also, among others, Goulder and Mathai (2000, Section 3), who extend such a model with R&D based technical progress in carbon abatement and van der Ploeg and Withagen (2012a), who extend such a model with backstops. 4 almost identical, a Pigovian tax on fossil fuel use and a Pigovian tax on carbon emissions are both equal to the SCC, while otherwise only a Pigovian tax on carbon emissions equals the SCC. For its derivation, the present paper employs a model that can be viewed as a version that combines elements of, on the one hand, the model of van der Ploeg and Withagen (2012b) with those of, on the other hand, the models of Groth and Schou (2007) and Grimaud et al. (2011).6It shares with van der Ploeg and Withhagen the assumption that there is disutility from carbon emissions and shares with Groth and Schou and Grimaud et al. that there is productivity loss from carbon emsissions. Van der Ploeg and Withagen and Groth and Schou make, contrary to the present paper, the simplifying assumption that a unit of fossil fuel use leads to exactly one unit of carbon emission. On the other hand, van der Ploeg and Withagen is richer than the present paper’s model because it allows for renewable backstops and resource extraction costs, while Groth and Schou is richer than the present paper’s model because it allows for endogenous growth from productive externalities. Grimaud et al. is richer than the present model because it allows for R&D based endogenous growth and for a more realistic climate damage speci…- cation. Moreover, it allows for resource extraction costs, dissipation of the carbon stock in the atmosphere, a renewable backstop and so-called carbon capture and storage (CCS) (i.e. the option to pump carbon in the underground to store it away from the atmosphere). The richer model assumptions in these papers are important for the research questions of those papers. Since these assumptions are however not very relevant for the present paper’s research question, they are dropped from the present paper’s model. In my view, the model of van der Ploeg and Withagen builds on Krautkrämer (1985, second part), in which households enjoy utility from the remaining stock of a non-renewable resources. Krautkrämer does however not addressed the issue of global warming, not to speak about that his model allows for renewable backstops. Groth and Schou and Grimaud et al. in turn build on Stiglitz (1974) and in my view also on Dasgupta and Heal (1974) and Solow (1974, Appendix C), which introduced non-renewable resources into the Ramsey model, but did not allow for a negative external e¤ect from pollution. The present paper’s model is also related to Golosov et al. (2011). Golosov et al. make however speci…c assumptions on the utility function, the production function and on the carbon accumulation process that deliver a constant 6See also Sinn (2008) for a model within the second group of models. 5 consumption–output ratio, consistent with stylized fact of modern growth. Since the latter issue is not addressed in the present paper, a discussion of the assumption of Golosov et al. lies outside of the scope of this paper. Section 2 of the present paper presents the social planner model structure. Section 3 derives the social planner model results. Section 4 derives Pigovian tax rates in a regulated market economy. Section 5 shows how this paper’s growth rate of the optimal carbon tax is related to the one in Grimaud et al. and in partial equilibrium models, such as Ulph and Ulph. Finally, section 6 concludes. 2 The Model To derive the SSC analytically, the paper assumes a social planner with perfect foresight, who maximizes life-time utility, W(O), of an in…nitely lived representative household subject to the economy’s resource constraints. The social planner solution can be replicated in a regulated market economy with climate policy such as Pigovian taxation. Following Krautkrämer (1985), but adapted to climate damage, life-time utility is assumed to be:7 W(0) = Z1 0 U(C; P)etdt; (1) where U(C; P)represents instantaneous utility, Cdenotes consumption and Pdenotes the stock of carbon in the atmosphere.8It is assumed that UC>0; UP<0and that UCC <0:No assumptions are made on the signs of UCP and UP P :9Literature often assumes UCP = 0 and UP P 0, i.e. marginal damages from pollution to be non-decreasing in the pollution stock (e.g. Stokey, 1998). However, as emphasised in Hoel and Kverndokk (1996), the carbon stock in the atmosphere a¤ects utility only indirectly by increasing the world temperature. As literature assumes the world temperature to be logarithmic in the carbon stock in the atmosphere, while marginal damages from rising world temperature might be non-decreasing, climate damage might be decreasing in the stock of carbon in the atmosphere (see also van der Ploeg 7The time index t is for most part of this paper omitted. 8Van der Ploeg and Withagen (2012b) assume a similar life-time utility function with U(C; P ) = V(C)D(P), i.e. instantaneous utility from consumption, V(C), to be additively separable from instantaneous disutility from the stock of carbon, D(P). 9Note that symmetry requires that UP C =UCP : 6 and Withagen, 2012b). Moreover, denotes the pure rate of time preference. For simplicity, the number of household in the economy is normalised to one. Note that it is therefore abstracted from population growth, which seems not to be too unrealistic for the very long-run, as population growth in industrialised countries is low and world population growth is predicted to slow down in the distant future. For simplicity, it is also abstracted from uncertainty, leaving its consideration to future research. As is standard in growth models with a non-renewable resource, the stock of fossil fuel in the ground evolves according to:10 S(t) = S(0) Z1 0 R(t)dt )_ S=R; (2) where Sdenotes the stock of fossil fuel left in the ground and Rdenotes the use of fossil fuel in output production. Moreover, the stock of carbon in the atmosphere is assumed to evolve according to:11 _ P=M(R);with MR>0;(3) where Mdenotes the ‡ow of carbon emissions. As mentioned in the introduction, the speci…cation in (3) di¤ers from van der Ploeg and Withagen (2012b) and Groth and Schou (2007), who assume P=R; i.e. who assume that a unit of fossil fuel use leads to exactly one unit of carbon emission. The exact functional form of M(R)is left open. We could follow Ulph and Ulph (1994) and assume M(R) = R; where is a constant. Alternatively, we could for example assume that M(R) = R, where MR=R1, which is rising in Rif  > 1and is declining in Rif  < 1. Finally, we could follow Grimaud et al. (2009) and assume that M(R) = RQ, with Q=R, where Qrepresents abatement and 0<  < 1:12 Following van der Ploeg and Withagen (2012b) and Groth and Schou (2007), it is in (3) for simplicity abstracted from dissipation of the carbon stock in the atmosphere.13 10 Cf. Perman et al. (2003, page 489). A hat on a variable represents the rate of change (or in other words the time derivative) of that variable. 11 See also Perman et al. (2003, Chapter 16) for such a possibly non-linear e¤ect from fossil fuel use on accumulation of the carbon stock. 12 More precisely, Grimaud et al. actually assume that M(R) = hR Qand Q= (hR)(LQ)1, if LQ< hR, and Q=hR, if LQhR, where his constant and LQdenotes labour used for abatement. 13 Sinclair (1994) argues that the speed by which the stock of carbon is dissipated is slow enough to be ignored as a …rst-order approximation. 7 Similar to Groth and Schou (2007) and Grimaud et al. (2011) production of output, Y, takes place according to the following aggregate production function:14 Y=F(K; R; P; t);(4) where Krepresents the stock of physical capital and Fdepends on tto allow for exogenous technical progress.15 For simplicity it is abstracted from use of labor in output production. It is assumed that FK>0; FR>0and FP<0: Moreover, FKK <0and FRR <0:For the same reasons as in case of the utility function, no assumptions are made on the signs of FKR; FKP and FRP , as well as on the sign of FP P :16 Finally, capital accumulation is assumed to evolve according to the following di¤erential equation: _ K=YC; (5) where it is for simplicity abstracted from capital depreciation. More importantly, in (5) it is also abstracted from costs for extraction of fossil fuel (see similarly, Groth and Schou, 2007).17 3 Results Combining (1)-(5) the present value Hamiltonian that the social planner maximises is:18 14 Contrary to this production function, Groth and Schou assume emissions to reduce environmental quality, where environmental quality is an input into output production. The authors also assunme a Cobb-Douglas production function. As mentioned in the introduction, this paper’s production function abstracts, for simplicity but without loss of generality, from various rich elements in the production functions of Groth and Schou and Grimaud et al. 15 Sinn (2008) suggests interpreting productivity loss from the carbon stock to include output loss from devoting output to mitigate climate change that is therefore not available for consumption or capital accumulation. 16 The latter is in contrast to Sinn (2008), who assumes to the e¤ect that output loss from the carbon stock is rising in the stock of carbon. 17 Sinclair (1994), who apparently identi…es fossil fuels mainly with oil, justi…es abstracting from extraction costs with the argument that marginal extraction costs for oil represent a very modest fraction of the price of oil. 18 In the Hamiltonian, the minus in front of Pensures Pto be positive, which allows it to be interpreted as a (positive) shadow price. 8 5 Relation to Previous literature As was mentioned in the introduction, Grimaud et al. (2011) derive the growth rate of the optimal carbon tax rate in an endogenous growth model with carbon emissions from burning fossil fuels. Their formula looks much more complicated than the present paper’s growth rate of the SCC. It remains more complicated even if we abstract from their more realistic but also more complicated climate damage speci…cation. Using the present paper’s simpler climate damage speci…cation and using the assumption UP= 0, which Grimaud et al. make, then their growth rate of the optimal tax on carbon emissions becomes:27 ^M=r+2 6 6 4 FPUC Z1 t FP(z)UC(z)e(zt)dz: 3 7 7 5:(21) It is shown in Appendix E that using the social planner’s optimality conditions, then (21) reduces to the present paper’s optimal tax on carbon emisssions, which can be shown from combining Proposition 2(ii) with (14) and imposing UP= 0 to be: ^M=r+FP M ;(22) Equation (22) clearly looks much simpler than (21). As was also mentioned in the introduction, a somewhat similar growth rate of the optimal carbon tax as in the present paper has been derived in partial equilibrium models, such as Ulph and Ulph (1994). In that model a social planner maximises life-time utility W(0) = Z1 0 [B(R)D(P)] ertdt subject to the constraints _ S=Rand _ P=R, where B0(R)>0and D0(P)>0and the notation remains unchanged.28 From this optimisation problem, Ulph and Ulph derive the growth rate of the optimal tax on carbon emissions as:29 27 Cf. Grimaud et al (2011, equation (48)). 28 In this summary of the results of Ulph and Ulph, I abstract for comparability from extraction costs and from dissipation of the carbon stock in the atmosphere, which are considered in Ulph and Ulph. 29 Cf. Ulph and Ulph (1994, equation (8)). 15 ^M=rD0(P) M :(23) Combining Proposition 2(ii) with (14) and imposing FP= 0, we …nd the present paper’s growth rate of the optimal tax on carbon emissions to be: ^M=r+UP=UC M :(24) It is straightforward to see that (24) is identical to (23) if we assume this paper’s instantaneous utility function to be: U(C; P) = CD(P);(25) that is, if we assume instantaneous utility from consumption to be linear and additively separable from instantaneous disutility from carbon emissions. Hence, (23) can be seen as a special case of (24). Therefore, on the one hand, the optimal carbon tax in partial equilibrium models is not too unrealistic. On the other hand, it is also not substantially identical to the one in general equilibrium models. This is so because whether or not instantaneous utility is linear in consumption makes a di¤erence. 6 Conclusion This paper derived the rate of change of the SCC in a Ramsey model with emissions from burning fossil fuel. It has been shown that the determinants of the rate of change of the SCC are substantially almost identical to the determinants in the social planner’s Hotelling rule if a unit of fossil fuel use leads to exactly one unit of carbon emission, while otherwise those formulas di¤er substantially. The paper has also shown that in the special case in which the two formulas are substantially almost identical, then a Pigovian tax on fossil fuel use and a Pigovian tax on carbon emissions are both equal to the SCC. Otherwise only a Pigovian tax on carbon emissions equals the SCC. Future research might examine the time path of the SCC and therefore of the optimal carbon tax. Grimaud et al. (2011) estimated their paper’s growth rate of the optimal carbon tax with a cap on carbon and without it. They did the estimations with use of functional forms and calibrated parameters from the latest version of the DICE model (Nordhaus, 2008). In 16 the case without a carbon cap, they found the growth rate of the optimal carbon tax to be insigni…cantly di¤erent from the social discount rate and to remain so over time. Climate change pessimists however might argue that the DICE model makes assumptions that ensure ful…llment of many of the conditions listed in section 3 of this paper and ful…llment of those conditions can be questioned. However, calibrated parameters for the case in which many of the conditions listed in section 3 are not ful…lled are not readily available because getting calibrated parameters in such a scenario requires to account for climate externalities, which is not an easy task. Clearly, this task lies outside of the scope of the present paper. Appendix A: Derivation of the Ramsey Rule, i.e. of (11) in the Text Taking time derivatives of (6) we obtain: UCC _ Cet +UCP _ Pet UCet =_ : (26) Upon substituting (6) in (8) we get: _ =UCetFK:(27) Substituting (27) in (26) yields: UCC _ Cet +UCP _ Pet UCet =UCetFK_ : (28) Rearranging (28) and using the de…nition rFKgives rise to equation (11) in the text. Appendix B: Derivation of the Social Planner’s Hotelling Rule, i.e. of (12) in the Text De…ne !=, where we used the further de…nition S+pMR. Taking natural logarithm and then time derivatives of the de…nition of !, we …nd: ^!= ^^ =_S +_p MR+p _ MR^ : (29) Upon substitution of (9) and (10), use of (8) and rearranging, (29) becomes: ^!=FK+UPet +FP MR+p _ MR:(30) 17 Using the de…nition S+pMRin (8) leads to: FR=: (31) Substituting (31) in (30) and using the de…nition !Pp=, we obtain: ^!=FK+UPet FR +FP FRMR+!p FR _ MR:(32) Upon combining (6) with (32), we get: ^!=FK+UP=UC+FP FRMR+!p FR _ MR:(33) Rearranging (31), using the de…nition !=, taking natural logarithms and then time derivatives, gives rise to: ^ FR= ^!: (34) Finally, substituting (34) in (33), using the de…nition rFKand multiplying both sides of the resulting expression by FRyields (12) in the text. Appendix C: Derivation of the SCC, i.e. of (13) in the Text Taking natural logarithms and then time derivatives of the de…nition !p p=, we obtain: ^!P= ^p^ : (35) Combining (10) and (8) with (35) and rearranging gives: ^!P=FK+UPet +FP p:(36) Upon use of !pp= (36) becomes: ^!P=FK+UPet !p +FP !p:(37) Combining (6) with (37), using the de…nition rFKand multiplying both sides of the resulting expression by !p, we …nd: 18 _!P=r!P+UP UC +FP:(38) Solving (38) gives the following general solution:30 !P=eRr(t)dt ~!P+ZeRr(t)dt UP(t) UC(t)+FP(t)dt;(39) where Rf(x)dx is called an inde…nite integral.31 Solving (39) forward by …xing the terminal condition !P(T), we obtain the particular solution as: !P=eRT tr(z)dz!P(T)ZT t eRz tr(u)du UP(z) UC(z)+FP(z)dz: (40) If we let in (40) Tgo to in…nite, then (40) becomes:32 !P=Z1 tUP(z) UC(z)+FP(z)eRz tr(u)dudz + lim !P(T) T!1 eRT tr(z)dz (41) A similar equation to (41) can be found in the asset pricing literature for the stock price.33 Upon use of the analogy to that literature, one can say that (41) has an in…nite number of solutions unless one imposes in (41) the restriction that lim !P(T) T!1 eRT tr(z)dz = 0, which implies that !Pcannot inde…nitely grow faster than r and become in…nitely large according to (41) (cf. Sørensen and Whitta-Jacobsen, 2010, page 395). If one imposes this restriction, then (41) becomes (13) in the text and then !Pequals its so-called "fundamental" part of (41) only. Appendix D: Derivation of the Social Scarcity Rent of Fossil Fuel and of the Price of a Unit of Fossil Fuel De…ning !SS=, taking natural logarithms and then time derivatives yields: 30 See Wälde (2011, pages 94-95) or Sydsaeter et al. (2005, page 200) for the mathematical approach to solve (38). 31 Cf. Sydsaetter and Hammond (1985, page 326). 32 See Wälde (2011, pages 100-101) and Sydsaeter et al. (2005, pages 201-202) for the mathematical approaches to derive (40) and (41). 33 See e.g. Naoui (2011, p. 125). 19 _!S !S =_S S _  :(42) Upon combining (42) with (9) and (8) and using the de…nition rFKand rearranging, we …nd: _!S=r!S:(43) Solving (43) gives the particular solution as:34 !S=!S(0)eRt 0r(z)dz:(44) Next, assume a continuum of identical resource owners with total mass one, which maximise their discounted pro…ts V(0) = R1 0pR(t)R(t)eRt 0i(z)dzdt subject to their constraint _ S=R: This optimisation problem gives rise to the following present value Hamiltonian: ~ H=pR(t)R(t)eRt 0i(z)dz +~ [R]: The …rst order conditions from maximisation of ~ Hare: @~ H @R = 0 )pReRt 0i(z)dz =~ , (45) @~ H @S =_ ~ )_ ~ = 0:(46) Taking the time derivative of (45) gives: _ ~ = _pReRt 0i(z)dz pRieRt 0i(z)dz:(47) Upon substituting (46) in (47) and rearranging we obtain the familiar Hotelling rule: _pR=ipR:(48) 34 See Sydsaetter and Hammond, 1985, page 767, example 21.6. 20 Solving (48) in the same way in which we solved (43) gives the particular solution as:35 pR=pR(0)eRt 0i(z)dz:(49) Appendix E: Derivation that (21) reduces to (22) Rearranging (21) yields: ^M=r+2 6 6 4 FPUCet Z1 t FP(z)UC(z)ezdz: 3 7 7 5:(50) Setting in (6) t=zimplies: UC(z)ez =(z):(51) Using (6) and (51) in (50), we …nd: ^M=r+2 6 6 4 FP Z1 t (z)FP(z)dz: 3 7 7 5:(52) Next, assuming UP= 0, then (10) becomes: _P=FP:(53) Upon ordinary integration of (53) we obtain: P=Z1 t (z)FP(z)dz: (54) Substituting (54) in (52) and using the de…nition !Pp=, gives rise to: ^M=r+FP !p :(55) 35 The maximisation of the Hamiltonian and the derivation of (49) follows Faucheux and Noel (2001, page 149-150). 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