A model of non-renewable energy resources as input for physical capital accumulation: a Special functions representation.

Transcription

1 A model of non-renewable energy resources as input for physical capital accumulation: a Special functions representation. Agustín Pérez - Barahona Paris School of Economics Université de Paris Panthéon-Sorbonne December 4, 27 I would like to thank Raouf Boucekkine, Thierry Brechet, Catarina Goulão, Juan de Dios Moreno-Ternero, Ingmar Schumacher, Katheline Schubert, Sjak Smulders, Alfonso Valdesogo-Robles, and the conference participants at SURED 26 Monte Verita, Ascona Switzerland, and the 2nd Atlantic Workshop on Energy and Environmental Economics, A Toxa Spain, for their useful comments. I knowledge the financial support of the Chaire Lhoist Berghmans in Environmental Economics and Management CORE and Marie Curie Intra-european Fellowship. Any remaining errors are ours. Corresponding author. Paris School of economics. Université de Paris Panthéon- Sorbonne. Maison des Sciences Economiques, 6-2 boulevard de l Hôpital, Paris Cedex 3 France. Tel.: ; fax: ; agustin.perezbarahona@univ-paris.fr

2 Abstract In contrast to the standard approach of energy economics, this paper assumes that physical capital accumulation is relatively more energy-intensive than consumption. By means of Gaussian hypergeometric functions, we provide a full analytical characterization of the global dynamics of the model. More precisely, we provide the sufficient conditions for existence and uniqueness of balanced growth path equilibrium BGP. Moreover, we prove that there is asymptotic convergence to the BGP. Finally, we explicitly study the monotonicity of the optimal trajectories of the variables in levels, which is typically unhallowed in the standard approach. Keywords: Non-renewable resources, Energy-saving technical progress, Special Functions. Journal of Economic Literature: C68, O3, O4, Q3, Q43 2

3 Introduction Energy economics widely admits that physical capital accumulation is a key component to offset the constraint on production possibilities due to nonrenewable energy resources. Moreover, the standard approach see for instance Dasgupta and Heal 974, 979, Solow 974a,b, Stiglitz 974, Hartwick 989 and Pezzey and Withagen 998 assumes the same technology for both physical capital accumulation and consumption, which implies among other things that the energy intensity of both sectors is the same. However, data does not support this simplification. Taking the Structural Analysis Database of OECD and the Energy Balances and Energy Prices and Taxes of IEA, Azomahou et al. 26 build an energy intensity measurement ratio between energy consumption and value added of 4 sectors of the economy. One can observe that this ratio is particularly high for activities closely related to physical capital accumulation iron and steel sector.89, transport and storage.85, non-ferrous metals.599 and nonmetallic minerals.57. However, regarding to consumption goods, the energy intensity is lower food and tobacco.34, textile and leather.82 and construction.8. Therefore, data does not confirm the previous hypothesis. Indeed, data seems to support that physical capital accumulation is more energy-intensive than consumption. In this paper, we study the implications on economic growth of assuming a different technology for physical capital accumulation and consumption. More precisely, in an economy where energy is directly produced by extracting non-renewable resources, we assume that physical capital accumulation is relatively more energy-intensive than consumption. We use a general equilibrium model with three sectors: consumption, physical capital and extraction. In addition, this paper gives special attention to energy-saving technologies defined as technologies that reduces energy intensity. Indeed, the importance of this kind of technologies were pointed out in numerous studies see for instance Boucekkine and Pommeret 24, COM 25 and Pérez-Barahona and ou 26a,b. This study focuses on the analysis of balanced growth path equilibrium BGP. By means of Gaussian hypergeometric functions, we provide a closeform representation of the optimal solution paths of our variables in levels. In contrast to the standard approach, the method of Special functions allows for According to COM 25, the European Union could save at least 2% of its energy consumption in 25 by improving energy-efficiency. 3

4 a global description of the dynamics of the economy 2 and does not rely on dimension reduction see Boucekkine and Ruiz-Tamarit 26 for further details. Taking the close-form solutions of the optimal paths, we provide the sufficient conditions for existence and uniqueness of BGP. Moreover, we prove that there is asymptotic convergence to the BGP. Furthermore, we explicitly study the monotonicity of the optimal trajectories of the variables in levels, which is typically unhallowed in the standard approach 3. We get that, in general, the optimal trajectories are non-monotonic. Dasgupta and Heal 974 previously observed this result for the optimal consumption in a model with the same technology for both physical capital accumulation and consumption. However, our paper raises several novelties with respect that study, which are direct consequence of assuming both a physical capital accumulation sector which is relatively more energy intensive than consumption, and a growing technical progress in particular, energy-saving. The paper is organised as follows. Section 2 describes our economy, providing the dynamical system corresponding to the optimal solution. In Section 3, we study the BGP equilibrium. Section 4 solves the dynamical system presented in Section 2 by using Gaussian hypergeometric functions. In Section 5, we provide the analytical expressions of the optimal solution paths. Section 6 studies the monotonicity properties of the optimal trajectories in levels. Finally, in Section 7, we consider some concluding remarks. 2 The model Let us consider a three sector economy with exhaustible energy resources, where the population is constant 4, based on Dasgupta and Heal 974, Hartwick 989 and Pérez-Barahona 26c. The final good sector only produces a non-durable good, which can be assigned to consumption or investment. The final good is produced by means of AK technology, where the physical capital durable good is the only input. The durable good sector accumulates physical capital by means of a technology defined over two inputs: investment and energy. Finally, the extraction sector directly produces 2 Notice that, in general, the standard approach focus on local dynamics, i.e., around the steady state equilibrium. 3 An exception is Dasgupta and Heal 974 where only consumption and extraction can be analyzed in levels. 4 As Pérez-Barahona 26c observes, Stigliz 974 studies an economy with nonrenewable resources and exponential growing labor force. 4

5 energy by extracting non-renewable energy resources from a given stock. 2. Final good sector The final good sector produces a non-durable good Y t by means of the following AK technology: Y t = AtKt, where At is the disembodied technical progress or scale parameter, and Kt represents the physical capital, which is the only input to produce the final good. Here, we consider AK technology because it is the simplest setup 5. Final good is used either to consume, Ct, or to invest in physical capital, It, verifying the budget constraint of the economy: Y t = Ct + It. 2 The standard literature on non-renewable energy resources see for instance Dasgupta and Heal 974, 979, Stiglitz 974, Solow 974, Hartwick 989, or Smulders and Nooij 23 assumes the same technology for physical capital accumulation, Kt, and consumption 6. However, as we will see in Section 2.2, our paper assumes that physical capital accumulation is a function of investment and non-renewable energy resources, Rt, i.e., Kt = g[it, Rt]. This implies that physical capital accumulation is relatively more energy-intensive than consumption. 2.2 Durable good sector The durable good sector accumulates physical capital, Kt, by using two inputs. On the one hand, the durable good sector takes the fraction of final good devoted to investment. And on the other, this sector uses the energy, Rt, produced by the extraction sector. The technology for physical capital accumulation is represented by the following Cobb-Douglas function: Kt = [θtrt] It, with < <, 3 where θt denotes the embodied energy-saving technical progress, and K is given. In this paper, technical progress is considered as an exogenous variable. 5 Nevertheless, one could include more inputs such as energy for final good production, or labor. 6 They consider Y t = F [Kt, Rt], where Y t = Ct + It and Kt = It. 5

6 Two observations should be made at this point. First, notice that we assume substitutability between energy and investment. Indeed, there is a very well known debate about substitutability vs. complementarity. If one considers the idea of minimum energy requiring to use a machine, the assumption of complementarity should be chosen. However, following the argument of Dasgupta 979, one can assume substitutability if It is interpreted as final good service. Following that, we assume that the provision of a flow of final good It implies the provision of a certain energy flow. As second observation, one can notice that the main different between Pérez-Barahona 26c and our model is that Pérez-Barahona 26c considers physical capital as a flow variable instead of a stock. As this author points out, this is a simplification which assumes total capital depreciation for each period. However, our model eliminates that simplification, considering physical capital as a stock variable. The main consequence of our assumption is the following. The model of Pérez-Barahona 26c has not dynamical transition, i.e., the economy is always in the BGP. However, as we will prove later, considering physical capital as a stock generates dynamical transition towards the BGP. Moreover, despite of having the same long-run growth rate as Pérez-Barahona 26c, our BGP levels are different because they are affected by the growth rates of technical progress. 2.3 Extraction sector Energy is directly produced by extracting non-renewable energy resources, Rt, from a given homogeneous stock St. As Dasgupta and Heal 974, and Hartwick 989, we assume costless extraction 7. The law of motion of the stock of non-renewable energy resources is described by the expression Rt = Ṡt, where S is given. 4 Since one can not extract more than the available stock, the following restriction should be included St. 5 7 Dasgupta and Heal 974: Extraction cost do not appear to introduce any great problem, provided that we assume any non-convexities. Indeed, one can easily introduce extraction cost ECt by modifying the budget constraint of the economy: Y t = Ct + It + ECR, S, where EC/ R > and EC/ S. 6

7 2.4 Optimal solution The central planner optimal solution maximizes the instantaneous utility function of the representative household: max W = ln[ct] exp ρtdt, with ρ >, subject to the equations -5, where ρ is the time preference parameter it is assumed to be a positive discount factor. One can easily rewrite the problem as optimal control problem with mixed constraints: subject to: max ln[atkt It] exp ρtdt, with ρ > Kt = [θtrt] It, with < <, Rt = Ṡt, St, with K and S given, where Kt and St are the state variables, and It and Rt are the control variables. Following Sydsæter et al. 999, the sufficient condition for this problem is a dynamical system described by the next proposition: Proposition. Given the initial conditions K and S, the optimal solution of our optimal control problem is a path {Ct, Rt, It, St, Kt} that satisfies the following conditions: Kt = θt Rt It, 6 Ċt Rt θt = Atθt Ct It θt It It + Ṙt ρ, 7 Rt Rt Atθt = θt It + It θt It Ṙt, 8 Rt The proof is provided in Appendix A AtKt = Ct + It, 9 Ṡt = Rt. 7

8 This system has five equations and five unknowns 8. Equation 6 is the accumulation law of physical capital. Notice that, in this model, energy is an input to accumulate physical capital. If one denotes Kt = g[θt, It, Rt], for a general utility function U[Ct], equation 7 could be rewritten as U cc Ċ U c ρ = ġi g I Ag I. This is the Ramsey optimal savings relation. Similarly, equation 8 is equivalent to the following expression: Atg I = ġr g R, which is the efficient condition for using up the non-renewable resource, i.e., a variant of the Hotelling rule. Finally, equation 9 is the budget constraint of the economy, and equation is the law of motion of the stock of nonrenewable energy resources. 3 Balanced growth path equilibrium Let us define balanced growth path equilibrium BGP as the situation where all the endogenous variables grow at constant rate, i.e., xt = x expγ x t. Following Solow 974a, we assume that At = A expγ A t and θt = θ expγ θ t for all t, where A, θ, γ A, γ θ are strictly positive and exogenous parameters. Indeed, we will prove later that both kind of exogenous technical progress should grow at a high enough rate in order to have BGP with positive growth rates. The following proposition summarizes the growth rates of the endogenous variables of our model, along the BGP necessary condition: Proposition 2. Along the BGP, Y t, Ct, and It grow at rate γ Y = γ C = γ I = γ θ + γ A ρ, 8 Indeed, there are two more equations: one transversality condition TC for each state variable: lim ψtkt = and lim λtst =, t t where ψt and λt are the corresponding lagrangian multipliers. We will use these two TC conditions in order to obtain the two constants involved in the calculation of the endogenous variables of the dynamical system. Actually, one could write a system of seven equations the five previous equations and these two TC and seven unknowns the preceding unknowns and the two constants related with the state variables. 8

9 the growth rate of the stock of physical capital is γ K = γ θ + γ A ρ, and extraction, Rt, and stock of non-renewable energy resources, St, grow at rate γ R = γ S = ρ. The proof is provided in Appendix B Parameter ρ is the positive discount rate, which represents the degree of impatience of the household. Indeed, the greater ρ the more impatient the household. Then, household prefer a higher level of present consumption than to postpone it. As one can observe from the previous proposition, the greater ρ the greater decreasing rate of the stock of non-renewable energy resources. This is because, since household does not want to postpone consumption, he prefers to extract a higher amount of energy resource in order to have greater present consumption. Furthermore, since the stock of non-renewable energy resources is lower, this reduces the long-run growth rates of the economy for a given growth rate of technical progress. However, this negative effect of impatience could be compensated by greater growth rates of both disembodied technical progress γ A and energy-saving technical progress γ θ. Indeed, from Proposition 2, if γ A +γ θ > ρ, then γ Y = γ C = γ I > and γ K > if BGP exists. The reason is that technical progress in particular, energy-saving technical progress reduces the energy intensity of the economy i.e., increases energy efficiency. If one defines energy intensity as the ratio Rt/Y t, it is easy to prove that, in our model, the growth rate of energy intensity along the BGP is 9 γ R/Y = γ θ + γ A, which is negative for γ A > and γ θ >. This is consistent with empirical works, such as Azomatou et al Notice that we can consider two alternatives definitions of energy intensity, Rt/Kt or Rt/It. For these definitions, the corresponding BGP growth rates are γ R/K = γ θ + γ A and γr/i = γ θ + γ A. 9

10 4 Analytical solution using Gauss Hypergeometric Functions In Section 4, we will fully characterize the dynamics of our economy. As we explain in Section, the equilibrium of our economy is described by the dynamical system 6-. Here, we will use a new technique in Economics called Special Functions. This is a very well known technique in the field of Mathematical Physics. The tool of Special Functions provides us a full analytical characterization of the dynamics of our system, without using qualitative techniques such as phase diagrams. However, this methodology has not been sufficiently explored in Economic Theory and, in particular, in Growth models. Since we are interested in the dynamics towards the BGP, it is useful to detrend our dynamical system 2. Let us define the following change of variable xt = xt/expγ x t. Then, the transformed detrended variables will be constant along the BGP if it exists. Applying this change of variable, one obtains the following detrended dynamical system with five equations and five unknowns K, C, R, Ĩ, S: Kt + γ θ + γ A ρ Kt = θ RtĨt, Ct Ct = Aθ Aθ Rt Ĩt Rt Ĩt = Rt Ĩt Rt Ĩt Ĩt Rt Ĩt Rt γ θ γ A, 2 + γ θ + γ A, 3 Ct + Ĩt = A Kt, 4 Ṡt = Rt exp ρt. 5 Achieving the analytical solution of this dynamical system, one fully characterizes the dynamics of the economy described by in this paper. Moreover, if there is BGP, the variables involved in this detrended dynamical system will be constant along the BGP. Furthermore, since γ A and γ θ are exogenous parameters, one can easily recovers the original variables by retrieving the change of variable. See Abramowitz and Stegun 97. See Boucekkine and Ruiz-Tamarit 24 for an application to the Lucas-Uzawa model. 2 Notice that this is not a compulsory step to apply Special Functions. However, eliminating the long-run trends allows us to have less messy expressions.

11 In order to obtain the analytical solution of the dynamical system - 5, we will apply the following strategy: Step. Let us define the ratio Xt = Rt Ĩt. Step 2. One can easily observes that equation 3 is a Bernoulli s equation see Section 4.2. Then, we can obtain the analytical solution for the ratio Xt. Step 3. Taking Xt into equation 2, one obtains a linear first-order differential equation. Solving this differential equation, the analytical expression for Ct is straightforward. Step 4. Rewriting equation, one gets Kt + γ θ + γ A r Kt = θ Xt [A Kt Ct]. Since we have the analytical solution for Ct Step 3, the previous expression is a linear first-order differential equation. By solving it, one gets the analytical solution for Kt. Notice that, this step will involve the application of Special Functions. In particular, we will use a family of Special Functions called Gauss Hypergeometric functions. Later, we will provide a briefly introduction to these fuctions. Step 5. From equation 4, Ĩt = A Kt Ct. Since we know both Ct Step 3 and Kt Step 4, the previous expression gives us the analytical solution for Ĩt. Step 6. Since we know Ĩt and Xt, one can easly gets Rt. Step 7. Finally, from equation 5, St is obtained by solving a linear first-order differential equation. 4. Step. Let us define the ratio Xt = Rt respect to t, one gets Ĩt. Taking logs and differentiating with Ẋt Xt = Rt Rt Ĩt Ĩt. 6

12 4.2 Step 2. Applying equation 6 into equation 3, one finds γ A Ẋt = γ θ + Xt + Aθ Xt +. 7 One can observe that equation 7 is a Bernoulli s equation. Following Sydsæter et al. 999, a Bernoulli s equation is defined as ẋt = qtxt + rtxt n, which has the following solution: pt xt = exp ϕ + n n n rt exp ptdt, where pt = n qtdt, ϕ is a constant, and n. Then, the solution of the Bernouilli s equation 7 is where = 4.3 Step 3. Xt = exp t ϕ + Aθ expt, 8 γθ + γ A, and ϕ is a constant which should be calculated. Applying equations 6 and 8 into equation 2, one finds that Ct Ct = which is a linear first-order differential equation. Aθ expt ϕ + Aθ expt, 9 A linear first-order differential equation is defined as ẋt + atxt = bt, whose solution is given by the expression Sydsæter et al. 999 t t t xt = x exp aξdξ + bτ exp aξdξ dτ. Applying this result into equation 9, we find that [ Ct = C exp lnϕ + Aθ expt lnϕ + Aθ t ], 2 where C= C is the initial condition for consumption, which should be determined. 2 τ

13 4.4 Step 4. Rewriting equation, one gets Kt + ρ Aθ Xt Kt = θ Xt Ct 2 Since from the previous steps we know the analytical solution for Xt equation 8 and Ct equation 2, Kt is the solution of the linear firstorder differential equation 2. Applying the previous result about the solution of a linear first-order differential equation, from equation 2 one finds that [ Kt = exp ] lnϕ + Aθ { [ ] } exp lnϕ + Aθ expt ρt K + C 2 ϕ F Aθ expt 2 F ϕ + ρ A exp + ρt Aθ, 22 where ϕ 2F Aθ = 2 F, + ρ expt 2F ϕ Aθ = 2 F, + ρ, + + ρ ; ϕ, + + ρ ; ϕ Aθ and K= K is the given initial condition for capital. Aθ expt, As one can observe, equation 22 involves Special Functions 2 F. In this paper, we use a family of Special Functions called Gauss Hypergeometric functions 3, which are defined as pf q a; b; z = n= a n a 2 n... a p n z n b n b 2 n... b q n n!, where a = a, a 2,..., a p, b = b, b 2,..., b q, and x n is the Pochhammer symbol, defined by: Γx + n x n =, Γx where Γx is the Gamma function. The proof of equation 22 is provided in Appendix C 3 See Boucekkine and Ruiz-Tamarit 24 for a quick overview of Gauss Hypergeometric functions. 3,

14 4.5 Step 5. Since we have the solution of Ctequation 2 and Ktequation 22, from equation 4 one finds that { [ ] } Ĩt = exp lnϕ + Aθ expzt lnϕ + Aθ t { A exp + ρtζt C }, 23 where { K + + ρ ζt = [ C ϕ A exp + ρt 2 F Aθ expt 2 F ϕ ] } Aθ. 4.6 Step 6. Since Xt is defined as Xt = Rt/Ĩt, taking the solution of Xt equation 8 and Ĩt equation 23, one finds the solution for the energy flow Rt: [ Rt = exp ] lnϕ + Aθ exp t 4.7 Step 7. { { } A exp ρ t ζt C exp } t. 24 Since we have the solution for Rt equation 24, one can find the solution for the stock of non-renewable energy resources, St, by solving equation 5, which is a Linear first-order differential equation. The solution of this equation is St = S + t Rτ exp ρτdτ, 25 where S is given. The corresponding integral is equal to t Rτ exp ρτdτ = [] + [2] + [3] + [4], 4

15 where [] = ϕ + Aθ AK 2 ρ [3F 2 a; b; ϕ Aθ exp with 2F ϕ Aθ [3] = A 2 2 ρ { [ ] } exp 2 ρ t, [2] = A 2 + Ω + ρ + Ω + ρ t ϕ 3F 2 a; b; Aθ expt [4] = ϕ + Aθ C ρ { [ ] } exp 2 ρ t, exp ρt, + A 2 = ϕ + Aθ C + ρ, Ω = 2ρ 2, + ρ a =, + ρ, + ρ + Ω, b = + + ρ, + + ρ and ϕ Aθ <. The proof is provided in Appendix D + Ω 4.8 Computation of the constants ϕ and C, ], Equations 2, and are the analytical solutions for, respectively, Ct, Kt, Ĩt, Rt, and St. However, we still need to determine the value of the constants ϕ and C= C, in order to finish the calculations. To do this, we use the two transversality conditions TC involved by the states variables of our model: lim ψtkt =, 26 t lim t λtst =, 27 where ψt and λt are the corresponding Lagrangian multipliers. In addition, the TCs will also imply restrictions on the parameters, which allow us to complete the full characterization of the optimal trajectories. 5

16 4.8. TC for the stock of physical capital Kt From equation A. see Appendix A, we obtain the shadow price of physical capital, ψt: ψt = ϕ + Aθ θ C ϕ + Aθ expt. 28 Taking equation 22, one gets 4 { K + C + ρ A ψtkt = ϕ + Aθ expt θ C ϕ [exp + ρt 2 F Aθ expt Then, the TC for Kt equation 26 implies that where K is given. C = A 2F ϕ Aθ ]} ϕ 2 F Aθ. + ρ K, 29 Equation 29 establishes a first relationship between ϕ and C. In order to have a system of two equations and two unknowns, we can obtain the second equation from the TC for the non-renewable energy resources equation TC for the stock of non-renewable energy resource St Since λt = λ for all t see Appendix A 5, equation 27 implies that lim St =. 3 t The transversality condition for St implies that the stock of non-renewable energy resources can not infinitely increase i.e., as in Boucekkine and Ruiz- Tamarit 24, non-explosive solution and it should be depleted in the infinite this is because we assume that this kind of energy resources are essential and there are not alternative resources, such as renewable energy resources. 4 Notice that Kt = Kt exp [ ρ t ]. 5 Since qt =, λ is easily obtained taking equation 28 into equation A.2 in Appendix A. 6

17 Taking equation 29 into equation 25, we get that St = S ϕ + Aθ C exp ρt A 2 ρ + Ω + ρ [3F 2 ϕ Aθ exp + Ω + ρ ] t ϕ 3F 2 Aθ, 3 expt where 3F 2 ϕ Aθ = 3 F 2 a; b; ϕ Aθ, ϕ ϕ 3F 2 Aθ = 3 F 2 a; b; expt Aθ. expt From equation 3, if the term Ω + +ρ is negative 6, then St will infinitely increases. Hence, Ω + +ρ should be greater than zero in order to avoid explosive solutions. Since ρ and are positive, we have to ensure that Ω + is positive too 7. Then, from the definition of Ω equation 25, we can establish the following proposition: Proposition 3. Any particular non-explosive solution to the dynamical system 6- has to satisfy the following condition: ρ > 2 γ θ + 3 γ A. Moreover, Proposition 2 establishes that the growth rate of technical progress should be high enough to ensure BGP equilibrium. Indeed, we can consider both conditions in the following proposition: Proposition 4. Any particular BGP solution to the dynamical system 6- has to satisfy the following condition: 2 γ θ + 3 γ A < ρ < γ θ + γ A. 32 The interpretation of Proposition 4 is the following. As we observe in Section 3, ρ represents the household s impatience degree. Indeed, the greater is ρ 6 3 F 2 a; b; = for a and b different than zero. 7 Notice that, from Proposition 2 and the definition of Ω equation 25, Ω should be less or equal to in order to have BGP. 7

18 the less important is future for the household i.e., the more impatient is the household. Proposition 2 establishes that technical progress should be high enough γ θ + γ A > ρ to guarantee BGP, which is the right hand side RHS of condition 32. However, the left hand side LHS of condition 32 establishes that the growth rate of technical progress should be not too high 2γ 3 θ + γ A < ρ. If this is not the case, an explosive solution will appear see Proposition 3. Nevertheless, an explosive solution is not possible because of the transversality condition for St. After eliminating the possibility of explosive solution, we can establish a condition on the parameters to deplete the stock of non-renewable energy resources in the infinite. From equation 3 and condition 32, it is clear that lim St = t S + C ρ ϕ + Aθ A2 + Ω Ω + ρ 3 F 2 ϕ Aθ. 33 From equation 3, lim t St =. Then, we obtain the following condition replacing A 2 in equation 33 and equalizing to zero: C ϕ + Aθ where S is given. S = [ 2 + ρ + Ω 3 F 2 ϕ ] Aθ, 34 ρ Equation 34 is the second relationship between ϕ and C. Then, equations 29 and 34 is a non-linear system of two equations and two unknowns. Solving this system, ϕ and C will be determined. 5 The optimal solution paths Section 4 provided the analytical solution of the dynamical system -5 using Gauss Hypergeometric Functions. But this is the solution of the detrended dynamical system. However, one can easily recover the solution of the original dynamical system 6- by multiplying each detrended variable by its corresponding BGP growth rate, i.e, xt = xt expγ x t. Moreover, Section 4 also established the parameters constraints to have BGP solution. In Section 5, we put together all the previous conditions, providing the optimal solution paths of our economy. 8

19 5. Computation of the constant ϕ Taking equation 29 into equation 34, we reduce the previous non-linear system of two equations 29 and 34 and two unknowns ϕ and C to the following single equation for ϕ, where C is directly found equation 29 once ϕ is determined:. + ρ + R S K = A 3F 2 ϕ Aθ 2F ϕ Aθ ϕ + Aθ + ρ ρ 2F ϕ Aθ, 35 where K and S are given. Since equation 25 requires ϕ Aθ <, we only consider parameterizations with solution for equation 35 such that 8 ϕ Aθ, Aθ Optimal solution paths for the ratio Xt From equation 8, rearranging terms, we get the following proposition for the optimal solution path for Xt: Proposition 5. Under the conditions 32 and 36, if equation 35 admits a unique solution, then i it does exist a unique path for Xt, starting from X =, such that Xt = ϕ exp t + Aθ ϕ+aθ ; 37 ii this equilibrium path shows transitional dynamics, approaching asymptotically to the constant X BGP =. Aθ 8 If some particular parametrization generates a solution for the equation 35 out the interval 36, we should study the continuation formulas of the Gauss Hypergeometric representation for St. However, this is beyond of the objective of this paper. 9

20 Moreover, it is easy to see that condition 36 implies that the starting point, the transitional dynamics, and the BGP of Xt are always positive: Corollary. Under the conditions 32 and 36, the optimal solution path for Xt is positive. 5.3 Optimal solution paths for the consumption Ct From equation 2, we recover Ct by multiplying Ct by expγ C t see Proposition 2. Then, rearranging terms, we establish the following proposition: Proposition 6. Under the conditions 32 and 36, if equation 35 admits a unique solution, then i it does exist a unique path for Ct, starting from such that Ct = C C = A 2F ϕ Aθ + ρ K, [ ] ϕ Aθ ϕ + Aθ exp t + ϕ + Aθ { exp γ θ + } γ A ρ t ; 38 ii this equilibrium path shows transitional dynamics, approaching asymptotically to the unique BGP C BGP t = C where K is given. Aθ { ϕ + Aθ exp γ θ + } γ A ρ t ; Regarding to the sigh of the optimal solution path, for C >, condition 36 implies positive starting point, transitional dynamics and BGP. From Proposition 6, the condition for C > is that 2 F ϕ >. Aθ 2

21 Then, we can establish the following corollary: Corollary 2. Under the conditions 32, 36, and 2 F ϕ the optimal solution path for Ct is positive. Aθ >, 5.4 Optimal solution paths for the physical capital Kt Taking equation 29 into equation 22 and rearranging terms, we obtain the optimal solution path for the detrended physical capital. Then, multiplying Kt by expγ K t, one can establish the Proposition 7: Proposition 7. Under the conditions 32 and 36, if equation 35 admits a unique solution, then i it does exist a unique path for Kt, starting from K, such that ϕ 2F Kt = Kϕ + Aθ Aθ expt ϕ 2F ϕ expt + Aθ Aθ { exp γ θ + γ A ρ } t 39; ii this equilibrium path shows transitional dynamics, approaching asymptotically to the unique BGP K where K is given. Aθ ϕ + Aθ K BGP t = { exp 2F ϕ Aθ γ θ + γ A ρ Furthermore, as in the previous case, we can establish the following corollary for positive optimal solution path: Corollary 3. Under the conditions 32, 36, and 2 F ϕ the optimal solution path for Kt is positive, for a given K >. The proof is provided in Appendix E 2 Aθ } t ; >,

22 5.5 Optimal solution paths for the investment It Applying equation 29 into equation 23, one gets the optimal solution path for the detrended investment. After multiplying Ĩt by expγ It and rearranging terms, we find the following proposition: Proposition 8. Under the conditions 32 and 36, if equation 35 admits a unique solution, then i it does exist a unique path for It, starting from I = AK + ρ, 2F ϕ Aθ such that [2F AK ϕ + Aθ expt It = 2F ϕ ϕ + Aθ expt Aθ ϕ Aθ expt + ρ ] { exp γ θ + } γ A ρ t ; 4 ii this equilibrium path shows transitional dynamics, approaching asymptotically to the unique BGP AK Aθ I BGP t = 2F ϕ ϕ + Aθ + ρ Aθ where K is given.. exp { γ θ + γ A ρ } t ; Aθ expt Aθ Aθ It is easy to prove that condition 32 implies > +ρ. If 2F ϕ > ϕ, then I >. Since 2 F > 2 F ϕ > see Appendix E, equation 4 will be positive too. Then, we establish the following corollary: Corollary 4. Under the conditions 32, 36, and 2 F ϕ the optimal solution path for It is positive, for a given K >. 22 Aθ >,

23 5.6 Optimal solution paths for the output Y t Since Y t = Ct + It, Propositions 6 and 8 provides the optimal solution path for the output: Proposition 9. Under the conditions 32 and 36, if equation 35 admits a unique solution, then i it does exist a unique path for Y t, starting from Y = AK, such that ϕ + Aθ ϕ expt 2 F Aθ expt Y t = AK ϕ + Aθ expt 2F ϕ Aθ { exp γ θ + } γ A ρ t ; 4 ii this equilibrium path shows transitional dynamics, approaching asymptotically to the unique BGP Y BGP t = Y t = AK Aθ ϕ + Aθ { exp γ θ + } γ A ρ t ; 2F ϕ Aθ where K is given. Concerning to the sign of the optimal solution path, we proceed as in Corollary 3: Corollary 5. Under the conditions 32, 36, and 2 F ϕ the optimal solution path for Y t is positive, for a given K >. Aθ >, 5.7 Optimal solution paths for the extraction of nonrenewable resources Rt Since Rt = XtĨt, we get the optimal solution path for the detrended non-renewable resources from Proposition 5 and equations 23 and 29 23

24 notice that C = C. Considering γ R = ρ, we recover the optimal solution path for Rt. Then, we establish the following proposition: Proposition. Under the conditions 32 and 36, if equation 35 admits a unique solution, then i it does exist a unique path for Rt, starting from R = AK + ρ such that ϕ + Aθ ϕ + Aθ Rt = AK ϕ [2F 2F ϕ Aθ expt Aθ 2F ϕ Aθ, + ρ ] exp ρt; 42 ii this equilibrium path shows transitional dynamics, approaching asymptotically to the unique BGP R BGP t = where K is given. ϕ + Aθ exp ρt; AK 2F ϕ Aθ [ + ρ ] Following the same reasoning as in Corollary 4, we can establish the following condition for positive optimal solution path for Rt: Corollary 6. Under the conditions 32, 36, and 2 F ϕ the optimal solution path for Rt is positive, for a given K >. Aθ >, 5.8 Optimal solution paths for the stock of non-renewable resources St Taking equation 3 into equation 25, rearranging terms, we get the optimal solution path for the stock of non-renewable energy resources: 24

25 Proposition. Under the conditions 32 and 36, if equation 35 admits a unique solution, then i it does exist a unique path for St, starting from S, such that exp 2 +ρ +Ω St = C + Ω +ρ t 3 F 2 ϕ + Aθ ϕ Aθ expt ρ expρt exp ρt; 43 ii this equilibrium path shows transitional dynamics, approaching asymptotically to zero; where K and S are given. Regarding to the sigh of the optimal solution path for the stock of nonrenewable resources, for C >, it is enough to require that expρt increases faster than exp + Ω +ρ t : Corollary 7. Under the conditions 32, 36, 2 F ϕ Aθ >, and ρ > + Ω +ρ, the optimal solution path for St is positive, for K > and S > given 9. Finally, we conclude this section with a corollary of all the conditions to guarantee positive optimal solution paths of our economy: Corollary 8. Under the conditions 32, 36, 2 F ϕ >, ρ > Aθ +Ω +ρ, 3F 2 ϕ > and, the optimal solution paths for Xt, Ct, Aθ Kt, It, Y t, Rt, and St, are positive, for K > and S > given. 9 Notice that + Ω +ρ > see Section Moreover, the condition 2F ϕ > is required to guarantee C >. Aθ 25

26 6 On the monotonicity of the optimal solution paths Applying Gauss Hypergeometric representation, Section 5 provided the closedform solution of the optimal paths of our variables in level. An important outcome of these closed-form solutions is that we can explicitly study the monotonicity of the optimal trajectories of the variables in level. These kinds of studies are typically unallowed in standard approaches The relationship between ϕ and S/K We will see later that the sign of ϕ is important to determine the monotonicity properties of our optimal solutions paths. Therefore, we have to study the sign of ϕ before dealing with the monotonicity issue. From condition 35, taking all parameters as given, one observes that the ratio S/K decides ϕ and, in particular, the sign of ϕ 2. Let us define S/K as the ratio corresponding to ϕ =. Then, S = A K Aθ + ρ + Ω + ρ ρ From 35, applying the implicit function theorem, one obtains that ϕ S/K <. The proof is provided in Appendix F Hence, for S/K > <S/K, ϕ is negative positive Monotonicity of the ratio Xt From equation 37, we easily obtains that [ ] dxt Xt = ϕ dt ϕ + Aθ expt Taking Corollary 8, one can observe that the term between the square brackets is positive. Then, the sign of the derivative is determined by the sign of 2 An exception is Dasgupta and Heal 974, where only consumption and extraction can be analyzed in level. 2 Notice that Corollary 8 implies that both left and right hand side of 35 are positive. 26

27 ϕ. If ϕ < >, then dxt/dt < > for all t. Moreover, if ϕ =, then Xt is constant for all t. Since the sign of ϕ is determined by the ratio S/K see Section 6., we can establish the following proposition: Proposition 2. If S/K > <S/K, then Xt decreases increases monotonically for all t. 2 If S/K = S/K, then for all t. Xt = Aθ See Appendix H, Figure, for an illustration. 6.3 Monotonicity of consumption Equation 38 is the optimal solution path for consumption. First, let us study the detrended consumption Ct = C [ ] ϕ Aθ ϕ + Aθ exp t + ϕ + Aθ. 45 From equation 45, we get d Ct = dt exp t ϕ exp t + Aθ ϕ Ct ϕ + Aθ ϕ + Aθ 46 As for Xt, the expression between braces is positive. Then, the sign of ϕ determines the sign of the derivative. Hence, we can establish the following proposition: Proposition 3. If S/K > <S/K, then Ct increases decreases monotonically for all t. 27

28 2 If S/K = S/K, then for all t. Ct = A + ρ K See Appendix H, Figure 2, for an illustration. Knowing the monotonicity properties of Ct, let us study Ct. As we established before, Ct = Ct expγ C t, where γ C = γ θ + γ A ρ. Then, it is straightforward that dct dt = expγ C t d Ct + dt Ctγ C. 47 Taking Corollary 8, one can observe that all terms in equation 47 are positive, except d Ct/dt. However, we can study this derivative from the monotonicity properties of Ct. If S/K > S/K, then d Ct/dt > for all t. Consequently, from equation 47, dct/dt > for all t. If S/K = S/K, then d Ct/dt = for all t. Nevertheless, from equation 47, we conclude that dct/dt > for all t. However, the difficult case arises for S/K < S/K because d Ct/dt < for all t. Replacing equation 46 into equation 47, and rearranging terms, it yields [ ] dct = expγ C t dt Ct ϕ γ C ϕ + Aθ. 48 expt As in the previous section, all terms of equation 48 are positive. Since expt increases with time, the term between brackets achieves its maximum value when t =. Thus, if γ C ϕ ϕ + Aθ γ C, then, dct/dt > for all t > 22. However, if γ C < γc, there exist a strictly positive t such that dct/dt < for all < t < t, dct/dt = for t = t, and dct/dt > for all t > t. It is easy to prove that t = [ ln ϕ Aθ ] γ C γ C 22 If γ C = γc, then dct/dt = for t =, and dct/dt > for all t >. 28.

29 The proof is provided in Appendix G Moreover, it is easy to see that dt dρ = γ C γ C. From Proposition 2, taking the definition of equation 8, one concludes that dt /dρ >. This result implies that the more impatient is the household i.e., the greater is ρ the greater is the number of periods where consumption decreases t 23. We summarize the monotonicity properties of Ct in the following proposition: Proposition 4. If S/K S/K, then Ct increases monotonically for all t. 2 If S/K < S/K, then 2. if γ C γc then Ct increases monotonically for all t > ; 2.2 if γ C < γc then Ct decreases monotonically for all < t < t, and increases monotonically for all t > t. See Appendix H, Figure 3, for an illustration. Moreover, Figure 4 considers an example of initially decreasing consumption, but after t = periods, consumption increases monotonically. The economic interpretation of Propositions 3 and 4 is the following. If the proportion of non-renewable energy resources with respect to physical capital is high enough i.e., S/K > S/K, then the detrended consumption Ct increases monotonically until its BGP level. Moreover, this effect is reinforced by the growth rates of technical progress through the long-run growth rate of consumption. Therefore, the consumption Ct increases monotonically for all t >. If S/K = S/K, the detrended consumption is constant its BGP level for all t >. However, as in the previous case, technical progress allows for a consumption increasing monotonically for all t >. Finally, if the proportion of non-renewable 23 Unfortunately, we can not provide a general conclusion of the effect of technical progress on t. 29

30 energy resources is too low i.e., S/K < S/K, the detrended consumption decreases monotonically for all t >. However, if the growth rates of technical progress are high enough to achieve a high enough long-run growth rate of consumption i.e., γ c γc, then the consumption increases monotonically for all t >. Nevertheless, if the growth rates of technical progress are not so high i.e., γ c < γc, then the consumption decreases monotonically until t. And then, consumption increases monotonically for all t > t because the technical progress effect is now high enough notice that technical progress increases with time. An observation should be made at this moment. Taking a standard model of non-renewable energy resources with same technology for both physical capital and consumption, Dasgupta and Heal 974 also described a nonmonotonic behavior. However, we find two main differences between Dasgupta and Heal s result and our case. First, notice that what matters for Dasgupta and Heal s result is the size of both S and K. However, since our model assumes that physical capital accumulation is more energy intensive that consumption, the key element for our result is the proportion of S with respect to K. Second, according to Dasgupta and Heal 974, if the initial stocks of S and K are high enough, the consumption initially rises and, after some periods, it will decreases. However, in our model, the non-monotonic behavior appears when the proportion of S with respect to K is not high enough. Indeed, as we explained before, if the growth rates of technical progress are high enough γ c γc the negative effect of a low proportion of S with respect to K is compensated. Otherwise, if the growth rates of technical progress are not so high γ c < γc, the consumption decreases during an initial period. But, since technical progress rises with time, consumption increases after some periods. 6.4 Monotonicity of physical capital, investment and output Equation 39 provides the optimal solution path for Kt. Let us study the detrended capital: ϕ 2F Kt = Kϕ + Aθ Aθ expt ϕ. 49 2F ϕ expt + Aθ Aθ 3

31 From equation 49, taking the formula E. from Appendix E, we get d Kt dt + ρ 2F a, b, c; 2 + ρ Aθ 2F = Kt ϕ 2 expt Aθ, 5 + ϕ expt ϕ Aθ expt ϕ Aθ expt Aθ expt where a = 2; b = + + ρ ; c = b +. ϕ Since 2 F >, applying the Euler integral representation Aθ expt ϕ as in Appendix D, one verifies that 2 F a, b, c; >. Therefore, following Corollary 8, all terms in equation 5 are positive. However, we can neither establish the sign of d Kt/dt nor conditions on the parameters to find out that sign. Nevertheless, we can conclude that Kt is not necessarily monotonic. Moreover, since Kt = Kt expγ K t, where γ K = γ θ + γ A ρ, Kt is not necessarily monotonic neither: Proposition 5. Kt is not necessarily monotonic. 2 If S/K = S/K, then Kt = K for all t. 3 Kt is not necessarily monotonic. 4 If S/K = S/K, then Kt increases monotonically for all t. See Appendix H, Figure 5, for an illustration. Let us study the monotonicity of investment. Since Ĩt = A Kt Ct, then dĩ/dt = Ad Kt/dt d Ct/dt. Hence, from Sections 6.3 and 6.4, we conclude that the detrended investment is not necessarily monotonic 24. Moreover, since It = Ĩt expγ It, where γ I = γ θ + γ A ρ, the investment is not necessarily monotonic neither: Proposition We achieve the same conclusion by differentiating equation 4 with respect to t. 3

32 Ĩt is not necessarily monotonic. 2 If S/K = S/K, then for all t. Ĩt = AK 3 It is not necessarily monotonic. + ρ 4 If S/K = S/K, then It increases monotonically for all t. See Appendix H, Figure 6, for an illustration. Since the detrended output Ỹ t is equal to A Kt, then dỹ t/dt = Ad Kt/dt. From section Proposition 5, we know that Kt is not necessarily monotonic, but we can not establish a general condition for this non-monotonicity. Hence, we can only conclude that Ỹ is not necessarily monotonic 25. Since Y t = Ỹ t expγ Y t, where γ Y = γ θ + γ A ρ, we conclude that Y t is not necessarily monotonic neither: Proposition 7. Ỹ t is not necessarily monotonic. 2 If S/K = S/K, then Ỹ t = AK for all t. 3 Y t is not necessarily monotonic. 4 If S/K = S/K, then Y t increases monotonically for all t. See Appendix H, Figure 7, for an illustration. 6.5 Monotonicity of extraction and stock of non-renewable energy resources Let us conclude Section 6 studying the monotonicity properties of the extraction Rt and stock St of non-renewable energy resources. From 25 As for investment, we get the same conclusion by differentiating equation 4 with respect to t. 32

33 Equation 42, we obtain the detrended Rt: ϕ + Aθ Rt = AK ϕ [2F 2F ϕ Aθ expt Aθ Taking logs and differentiating with respect to t, it yields d Rt dt = + ρ Rt 2 2F a, b, c; 2 + ρ Aθ expt ϕ 2F Aθ expt ϕ Aθ expt + r ] +ρ 5 ϕ, 52 where a = 2; b = + + ρ ; c = b +. Following Corollary 8, we observe that the terms between square brackets in equation 52 are positive. Then, the sign of d Rt/dt is determined by the sigh of ϕ. Hence, we establish the following proposition: Proposition 8. If S/K > S/K, then Rt decreases monotonically for all t. 2 If S/K = S/K, then 26 Rt = AK Aθ + ρ 3 If S/K < S/K, then Rt increases monotonically for all t. See Appendix H, Figure 8, for an illustration. Since Rt = Rt exp ρt, then drt/dt = exp ρtd Rt/dt ρ Rt. Hence, applying equation 52, we obtain. drt dt = Rt exp ρt 26 Notice that condition 32 implies that > +ρ. 33

34 + ρ 2 2F a, b, c; 2 + ρ Aθ expt ϕ 2F Aθ expt ϕ Aθ expt +ρ ϕ ρ. 53 Since the terms between square brackets are positives, we obtain the monotonicity properties of Rt: Proposition 9. If S/K S/K, then Rt decreases monotonically for all t. 2 If S/K < S/K, then Rt is not necessarily monotonic. See Appendix H, Figure 9, for an illustration. As Proposition 9 establishes, if the proportion of non-renewable energy resources with respect to physical capital is too low, the extraction of non-renewable energy resources is not necessarily monotonic. Indeed, it is possible to generate an example of non-monotonic behavior of Rt see Figure. But unfortunately, we can not establish an analytical condition for this case. Figure shows a non-monotonic case of the extraction of non-renewable resources, where Rt increases during an initial period. And then, Rt decreases monotonically until zero. The economic interpretation of this case is the following. As we observe in Section 2.4, the Hotelling rule is the efficient condition for using up the non-renewable resources. Indeed, equation 8 is the Hotelling rule for our economy. For this particular case, Figure shows that It always increases. Then, for a fixed Rt, the RHS of equation 8 i.e., the growth rate of the marginal productivity of extraction of non-renewable energy resources rises. Since the RHS of equation 8 i.e., the marginal productivity of investment equals the LHS, then LHS has to increase too. Hence, Rt should rise, increasing the LHS and reducing the RHS. However, technical progress increases with t. Then, after some periods, technical progress can increase the LHS even if Rt decreases 27. Finally, we point out that this result contrasts with Dasgupta and Heal 974. These authors obtained that Rt decreases monotonically for all t. However, our model allows for a non-monotonic behavior of Rt. Indeed, this possibility rises the importance of considering physical capital production as more energy-intensive than consumption sector, and the role of technical progress in particular, 27 Notice that energy-saving technical progress also affects the RHS, through its growth rate γ θ. However, we assumed that γ θ is constant. Then, the effect of energy-saving technical progress on the RHS is constant with t. Nevertheless, the effect of technical progress in particular, energy-saving technical progress on the LHS increases with t. 34

35 energy-saving technical progress. To conclude, we study the monotonic properties of the stock of nonrenewable energy resources. Equation establishes that Ṡt = Rt. Then, dst/dt = Rt. Hence, since Rt > for all t, dst/dt < for all t: Proposition 2. St decreases monotonically for all t. See Appendix H, Figure, for an illustration. 7 Concluding remarks In this paper, we studied the implications of assuming that physical capital accumulation is relatively more energy-intensive than consumption. This study focused on the analysis of BGP. By means of Gaussian hypergeometric functions, we provided a close-form representation of the optimal solution paths of our variables in levels. We proved that there is asymptotic convergence to the BGP. Furthermore, we explicitly studied the monotonicity of the optimal trajectories of the variables in levels, which is typically unhallowed in the standard approach. We got that, in general, the optimal trajectories are non-monotonic. Dasgupta and Heal 974 already pointed out this result for the optimal consumption in a model with the same technology for both physical capital accumulation and consumption However, our paper introduced four novelties with respect that study. First, our sufficient condition for nonmonotonicity involves the proportion of initial endowment of non-renewable resources S with respect to the initial endowment of physical capital K, i.e., the ratio S/K. However, what matters for Dasgupta and Heal 974 s non-monotonicity result is the size of S and K, i.e., not necessarily the size of the ratio S/K. Second, our sufficient condition entails both the ratio S/K and the growth rate of technical progress. This contrasts with Dasgupta and Heal 974 since their sufficient condition does not involve the technical progress. Third, Dasgupta and Heal 974 shows that if the size of S and K are large enough then consumption initially rises and, after some periods, it will decrease. However, our model raised the possibility of an opposite behaviour of consumption. Indeed, if both S/K and the growth rate of technical progress are low enough then consumption depicts a U-shape. Moreover, we analytically proved that 35

36 the more impatient is the household the greater is the number of periods where consumption decreases. Finally, our model showed that the resource extraction is not necessarily monotonic, which contrasts with Dasguptan and Heal 974 where it decreases monotonically. Our model has several limitations. Among of them, we point out the following two. First, as Sollow 974a, we assumed unlimited exogenous technical progress. Regarding to our disembodied technical progress At, this assumption captures the idea of unlimited growth of knowledge. However, unlimited energy-saving technical progress θt could arise physical incompatibility problems related to the thermodynamic laws. In the literature of exogenous energy-saving technical progress, the assumption of unlimited growth of energy efficiency is widely applied see for instance Boucekkine and Pommeret 24, Azomahou et al. 24, and Pérez-Barahona and ou 26a,b. Indeed, these authors assume that the limit of improving energy efficiency is very far away. In our model, we take that point of view. A second limitation of our analysis is that we assumed exogenous technical progress. The reason of taking that assumption is the following. The aim of this paper is to study the conditions under which technical progress can offset the trade-off between economic growth and the usage of non-renewable energy resources. Then, we do not analyze how the economy can achieve these conditions. However, a very interesting extension to our model is to assume technical progress in particular, energy-saving technical as an endogenous variable. Indeed, as we observed in Proposition 2, our BGP growth rates are not affected by the stock of non-renewable energy resources. However, the scarcity of energy resources can induce new investments in energy-saving technologies see for instance, Newell et al. 999, Buonanno et al. 23 and Boucekkine and Pommeret 24, establishing an effect of the stock of non-renewable energy resource on the BGP growth rates. As it is suggested in Pérez-Barahona 26c, a possibility to endogenize energy-saving technical progress could be by means of an R&D sector for energy-saving technologies following Aghion and Howitt 992 and Grossman and Helpman 99a,b. References Abramowitz M. and Stegun A., 97. Handbook of Mathematical Functions. Dover Publications, INC., New York. 36