Capital Accumulation and Exhaustible Energy Resources: a Special Functions case.

Transcription

1 Capital Accumulation and Exhaustible Energy Resources: a Special Functions case. Agustín Pérez - Barahona st February 26 First draft Abstract This paper studies the possibilities of technical progress to sustain long run growth, when energy is an input to accumulate physical capital. In particular, I study the conditions under which energy-saving technologies can maintain long run growth, although energy is mainly produced by exhaustible natural resources. This model entails some technical difficulties. However, I can give a full analytical characterization of both short and long run dynamics applying the method of Special Functions. Keywords: Nonrenewable resources, Energy saving technical progress, Special Functions. Journal of Economic Literature: C68, O3, O4, Q3, Q43 I would like to thank Raouf Boucekkine for his useful comments. I knowledge the financial support of the Chaire Lhoist Berghmans in Environmental Economics and Management CORE. Any remaining errors are mine. Corresponding author. Département des sciences économiques and CORE, Université catholique de Louvain, 34 voie du Roman Pays, B-348, Louvain-la-Neuve Belgium. Tel.: ; fax: ; perez@core.ucl.ac.be

2 Introduction This paper studies the possibilities of technical progress to sustain long run growth, when energy is an input to accumulate physical capital. In particular, I study the conditions under which energy-saving technologies can maintain long run growth, although energy is mainly produced by exhaustible natural resources. Moreover, I emphasize the role of energy-saving technical progress as a convincing solution to solve the trade-off between economic growth and the usage of non-renewable energy resources such as fossil fuels. The classical literature assumes that natural resources are one of the inputs together with capital and labour to produce final goods. As is usual, the production of final goods is shared between consumption and investment, where investment is capital accumulation. Thus as the classical literature claims if there is enough capital accumulation i.e., investment, one avoids the problem of exhaustible natural resources see Dasgupta and Heal, 974, Dasgupta, 979 and Stiglitz, 974. Nevertheless, my point is that this explanation is not completely satisfactory. Indeed, the classical literature does not consider the possibility of having energy as an input to accumulate physical capital. If energy is produced by exhaustible natural resources, this would limit capital accumulation, as well as the long-run growth of the economy. However, my paper shows that if there is a production function for capital accumulation, where investment and exhaustible resources are inputs for example, the Cobb-Douglas production function, technical progress in particular, energy-saving technical progress plays a crucial role to ensure long run growth. Boucekkine and Pommeret 22 point out that the use of energy-saving technologies is an alternative and/or complement to the solution of renewable energy resources. The idea is of energy efficiency: non-renewable resources produce energy, but unnecessary energy losses are minimised in both the energy production process and the energy usage. According to various studies, the European Union EU could save at least 2% of its present energy consumption EUR 6 billion per year, or the present combined energy consumption of Germany and Finland improving energy-efficiency. Moreover, an average EU household could save between EUR 2 and EUR per year. There are many examples of energy-saving technologies COM25. Indeed, today renewable energy resources generate only 3.4% of the total world energy production. In Europe, the contribution of renewable energies remains relatively low at 6% in 2, and between 8 and % of the total consumption in 2 COM25. 2

3 An energy-saving electric bulb uses five time less current than a standard one. Replacing bulbs can easily save EUR annually for an average household. The energy wasted by the stand-by control for lighting, heating, cooling and electric motors has increased during these days, because more and more appliances offer this feature. Electricity used in stand-by mode can reach between 5 and % of total electricity consumption in the residential sector International Energy Agency IEA. However, there are more efficient sleep modes that could be implemented. Regarding the car usage, the fleet of new passengers cars more energy-saving put on to the market in 28/9 will reduce fuel consumption of around 25% compared to 998. Furthermore, friction between tyres and the road accounts for up to 2% of a vehicle s consumption. Nevertheless, a properly performing tyres could reduce this consumption by 5%. In addition to that, an average driver using right tyres at the right pressure could save EUR of the annual fuel bill IEA. This concern is shared between developed countries, such as the EU, regarding the Environment and Energy Policy. In particular, the European Commission, through the Green Paper Doing more with less COM25, opened the debate about the opportunities of energy-efficiency savings in the EU. My paper applies a general equilibrium model with three sectors final good, capital good, and extraction sector, based on Hartwick 989. The first contribution of my paper is that, in contrast with the classical literature, my model assumes energy as an input to accumulate physical capital. Here, exogenous technical progress plays a crucial role to sustain long-run growth when energy is produced by means of exhaustible resources. In particular, I study the possibilities of energy-saving technologies as solution for the trade-off between economic growth and the usage of non-renewable energy resources. The second contribution of my paper is methodological. This model entails some technical difficulties. In particular, I have to solve an optimal control problem with mixed constraints and two states variables: capital accumulation and non-renewable resource stock. However, I can give a full analytical characterization of both short and long run dynamics applying the method of Special Functions. This is a very well known method in the field of mathematical physics to describe dynamical systems. This tool gives a full analytical characterization of both short and long run dynamics, without using qualitative techniques as phase diagrams. However, this methodology has not been sufficiently explored in Economic Theory. In this, paper, I use a particular family of Special functions called Gauss hypergeometric functions 3

4 see Boucekkine and Ruiz-Tamarit 24 for an application to the Lucas- Uzawa model. The paper is organized as follows. Section 2 describes my economy and its optimal solution. In section 3, I study the balanced growth path equilibrium. The short-run dynamics of my economy is presented in Section 3. Finally, some concluding remarks are considered in section 4. 2 The Model Let us consider a three sector economy with exhaustible energy resources, where the population is constant 2. 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 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 AK technology Y t = AtKt where At is the disembodied technical progress or scale parameter, and Kt represents physical capital which is the only input to produce the final good. The classical literature see for instance Dasgupta 974, 979, Stiglitz 974, Solow 974, Hartwick 989, or Smulders 23 considers energy as a second input for the production of the final good 3. However, my paper points out the role of energy as an input for physical capital accumulation. 2 See Stigliz 974 for an analysis of the growth possibilities open to an economy possessing exhaustible resources and an exponential growing labour force. 3 They consider Y t = F {Kt, Et} Examples: Y t = { Kt σ /σ + 2 Et σ /σ } σ/σ CES Y t = Kt Et 2 Cobb Douglas 4

5 Here, I consider AK technology because it is the simplest set-up. Nevertheless, I could include more inputs such as energy for final good production, or labour. 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 Durable good sector The durable good sector accumulates physical capital Kt, using two inputs. On the one hand, the durable good sector uses 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 = K is given. In this paper, technical progress is considered as an exogenous variable. Notice that I 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, I assume that the provision of a flow of final good It implies the provision of a certain energy flow. 2.3 Extraction sector Energy is directly produced by extracting non-renewable energy resources from a given homogeneous stock St. As Dasgupta and Heal 974, and Hartwick 989, I assume costless extraction 4. The evolution of the stock is 4 Dasgupta & 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 5

6 described by the expression Rt = Ṡt, where S = S is given. 4 Since one can not extract more than the stock, the following restriction should be included St Rt 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 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 lnatkt Itexp ρtdt, with ρ > Kt = θtrt It, with < < Rt = Ṡt St Rt 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 is a path {Ct, Rt, It, St, Kt} that satisfies the following conditions: Kt = θt Rt It 6 Ċt Ct = Atθt Rt It θt θt It It + Ṙt ρ 7 Rt 6

7 Rt Atθt = θt It + It θt It Ṙt Rt 8 The proof is given in Appendix A. AtKt = Ct + It 9 Ṡt = Rt This system has five equations and five unknowns 5. Eq. 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 = gθt, It, Rt, for a general utility function UCt, Eq. 7 could be rewritten as U cc Ċ U c ρ = ġi g I Ag I This is the Ramsey optimal savings relation. In the same way, Eq. 8 is equivalent to the following expression ġ R g R = Atg I, which is the efficient condition for using up the non-renewable resource i.e. a variant of the Hotelling rule. Finally, Eq. 9 is the budget constraint of the economy, and Eq. is the law of motion of the stock of non-renewable energy resource. 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 6. In this model, I 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, I will prove later that 5 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. I will use these two TC conditions in order to obtain the two constants involved in the calculation of the endogenous variables of my dynamical system. Actually, I 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. 6 xt = x expγ x t 7

8 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 my model, along the BGP: Proposition 2. Along the BGP, Y t, Ct, and It grow at rate γ Y = γ C = γ I = γ θ + γ A ρ, the growth rate of the stock of physical capital is γ K = γ θ + γ A ρ, and energy, Rt, and stock of non-renewable energy resource, St, grow at rate γ R = γ S = ρ. The prof is given 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 is the consumer. Then, household prefers 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 consumption. Furthermore, since the stock of energy resource 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, the next proposition establishes the conditions for technical progress in order to achieve positive long run growth rates of the economy: Proposition 3. If γ A + γ θ > ρ then γ Y = γ C = γ I > and γ K > As one can observe from the Propositions 2-3, technical progress is crucial to offset the problem of exhaustible energy resources. This is because technical progress reduces the energy intensity of the economy. If one defines energy intensity as Rt/Y t or Rt/Kt, or Rt/It, it is easy to prove that, in my model, the growth rate of energy intensity, γ R/Y or γ R/K, or γ R/I, is negative. This is consistent with Azomahou, Boucekkine and Nguyen Van 24. 8

9 4 Short run Dynamics In Section 4, I will fully characterize the short run dynamics of my economy. As I explained in Section, the equilibrium of my economy is described by the dynamical system Eq. 6-. Here, I will use a new methodology in Economics called Special Functions 7. This is a very well known technique in the field of Mathematical Physics. The tool of Special Functions provides me a full analytical characterization of the dynamics of my 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 8. Since I am interested in the dynamics towards the BGP, it is useful to detrend my dynamical system 9. Let us define the following change of variable xt = xt expγ xt. In this way, the transformed 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 r Kt = θ RtĨt Ct = Aθ Rt Rt Ĩt Rt Ĩt Ct Ĩt Rt Ĩt Rt Aθ = Ĩt Ĩt Rt γ θ γ A 2 + γ θ + γ A 3 Ct + Ĩt = A Kt 4 Ṡt = Rtexp rt 5 Achieving the analytical solution of this dynamical system, one fully characterizes the dynamics of the economy described in this paper. Moreover, if there is BGP, the variables involved in this dynamical system will be constant along the BGP. However, since γ A and γ θ are exogenous parameters, one can easily recovers the original variables by unmaking the previous change of variable. 7 See Abramowitz and Stegun 97 8 See Boucekkine and Ruiz-Tamarit 24 for an application to the Lucas-Uzawa model. 9 Notice that this is not a compulsory step to apply Special Functions. However, eliminating long-run trends allows one to have less messy expressions. 9

10 In order to obtain the analytical solution of the dynamical system - 5, I will apply the following strategy: Step. Let us define the ratio Xt = Rt Ĩt. Step 2. One can easily observers that Eq.3 is a Bernoulli s Equation. Then, I can obtain the analytical solution for the ratio Xt. Step 3. Taking Xt in Eq.2, one obtains a linear first-order differential equation. Solving this differential equation, the analytical expression for Ct is straightforward. Step 4. Rewriting Eq., one gets Kt + γ θ + γ A r Kt = θ Xt A Kt Ct, Since I 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, I will use a family of Special Functions called Gauss Hypergeometric Functions. Later, I will briefly introduce this methodology. Step 5. From equation 4, Ĩt = A Kt Ct Since I know both Ct Step 3 and Kt Step 4, the previous expresion gives me the analytical solution for Ĩt. Step 6. Since I know Ĩt and Xt, one can easly gets Rt. Step 7. Finally, from Eq.5, St is obtained by solving a linear firstorder differential equation. 4. Step. Let us define the ratio Xt = Rt respect t, one gets Ĩt. Ẋt Rt Ĩt Xt = Rt Ĩt Taking logs and differentiating with 6

11 4.2 Step 2. Applying Eq.6 in Eq.3, one finds Ẋt = γ θ + Eq.7 is a Bernoulli s equation. γ A Xt + Aθ Xt + 7 Following Sydsæter et al. 999, a Bernoulli s equation is defined as ẋt = qtxt + rtxt n, which has the following solution pt xt = exp C + n n n rtexp ptdt, where pt = n qtdt, C is a constant, and n. Then, the solution of the Bernouilli s equation 7 is where Z = which is constant. Xt = exp Zt C + Aθ Z expzt, 8 γθ + γ A, and C is a constant. Moreover, lim Xt = t γθ + γ A Aθ, 4.3 Step 3. Applying Eq.6 and 8 in Eq.2, one finds that Ct Ct = which is a linear first-order differential equation. Aθ expzt C + Aθ expzt Z, 9 A linear first-order differential equation is defined as ẋt + atxt = bt,

12 whose solution is given by the expression Sydsæter et al. 999 xt = xexp t aξdξ + t bτexp t τ aξdξ dτ Applying this result to Eq.9, I find that [ Ct = Cexp lncz + Aθ expzt lncz + Aθ Zt ], 2 where C is the initial condition for consumption. Moreover, which is constant. lim Ct = t C [ ] exp lncz + Aθ, 4.4 Step 4. Rewriting Eq., one gets Kt + Z ρ Aθ Xt Kt = θ Xt Ct 2 Since from the previous steps I know the analytical solution for Xt Eq.8 and Ct Eq.2, Kt is the solution of a linear first-order differential equation described by Eq.2. Applying the previous result about the solution of a linear first-order differential equation to Eq.2, one finds that [ ] Kt = exp lncz + Aθ exp { [ ]} lncz + Aθ expzt Z rt 22 { K + Z Z+r C A where CZ 2F Aθ expzt [ expz+rt 2 F CZ Aθ expzt 2 F CZ Aθ ]}, Z + r = 2 F Z,, + Z + r Z ; CZ Aθ, expzt 2

13 2F CZ Aθ = 2 F Z + r Z and K is the given initial condition for capital. The proof is given in Appendix C.,, + Z + r Z ; CZ Aθ Eq.22 involves Special Functions 2 F. In this paper, I use a family of Special Functions called Gauss hypergeometric functions, which are defined as 2F a, b, c; z = n= x n is the Pochhammer symbol, defined by: a n b n c n z n n!, x n = Γx + n, Γx where Γx is the Gamma function. 4.5 Step 5. Since I have the solution of CtEq.2 and CtEq.22, from the Eq.4 one finds that { [ ]} Ĩt = exp lncz + Aθ expzt lncz + Aθ Zt 23 { AexpZ + rtζt C } where { ζt = K + Z Z + r [ C CZ A expz + rt 2 F Aθ expzt 2 F CZ ] } Aθ 4.6 Step 6. Since Xt is defined as Xt = Rt/Ĩt, taking the solution of Xt Eq.8 and Ĩt Eq.23, one finds the solution for the energy variable See Boucekkine and Ruiz-Tamarit 24 for a quick overview of Gauss hypergeometric functions. 3

14 Rt: Rt = C + Aθ expzt exp { [ ]} lncz + Aθ expzt lncz + Aθ 24 { AexpZ + rtζt C } 4.7 Step 7. Since I have the solution for RtEq.24, one can find the solution for the stock of non-renewable energy resources Rt by solving Eq.5, which is a Linear first-order differential equation. The solution of this equation is St = S + t Rτexp ρτdτ, 25 where S is given. 4.8 Constants C and C Equations 2, and are the analytical solutions for, respectively, Ct, Kt, Ĩt, Rt, and St. However, I still need to determine the value of the constants C and C, in order to finish the calculations. To do this, I will use the two transversality conditions involved by the states variables of my model: lim t ψtkt = and lim λtst =, t where ψt and λt are the corresponding lagrangian multipliers. 5 Concluding remarks The work described in this document is a first draft of the paper Capital Accumulation and Exhaustible Energy Resources: a Special Functions case. In order to terminate this work, I will need to finish the following points: First. In order to obtain the analytical solution for St, I have to solve the integral involved in Eq.25. 4

15 Second. To close the analytical solution of the endogenous variables of my model, I need to determine the constants C and C. To do that, I will use the two transversality conditions involved by the states variables. Third. To conclude, I will interpret the dynamics described by this model. Moreover, I will point out the asymptotic properties of my dynamical system. In particular, I will study the dynamics towards the BGP. 5

16 References Abramowitz M. and Stegun A., 97. Handbook of Mathematical Functions. Dover Publications, INC., New York. Boucekkine R. and Ruiz-Tamarit, 24. Special functions for the study of economic dynamics: the case of the Lucas-Uzawa model. CORE Discussion Papers. October. COM, 25. Doing more with less. Green Papers. Official Documents. European Union. 265, June. Dasgupta P.S. and Heal G.M., 974. The Optimal Depletion of Exhaustible Resources. Review of Economic Studies, Special Number, Dasgupta P.S. and Heal G.M., 979. Economic Theory and Exhaustible Resources. James Nisbet and Cambridge University Press, 979, pp.-5. Hartwick J.M., 989. Non- renewable Resources: Extraction Programs and Markets. London: Harwood Academic. Smulders S., Michiel de Nooij, 23. The impact of energy conservation on technology and economic growth, Resource and Energy Economics 25, Solow, Robert M, 974. The Economics of Resources or the Resources of Economics, American Economic Review, American Economic Association, vol. 642, pages -4. Stiglitz J., 974. Growth with Exhaustible Natural Resources: Efficient and Optimal Growth Paths, Review of Economic Studies, Special Number, Sydsæter K., Strøm A. and Berck P, 2. Economists Mathematical Manual, Spinger. 6

17 Appendix A Following Sydsæter et al. 999, pages 9-, the Lagrangian associated with my optimal control problem with mixed constraints is L = UAt Itexp ρt+ψtgθt, It, Rt λtrt+qtst Rt where ψt and λt are the Lagrangian multipliers, U is a general utility function, and g is a general production function for physical capital accumulation. The corresponding first order conditions FOC are: lim t L It = ; L Rt = ; L Kt = ψt; L St = λt; qt = if St > Rt; ψtkt = and lim λtst = Transvarsality conditions From the FOC, one obtains t ψtg I = U c exp ρt A. ψtg R = λt + qt ψt = U c Atexp ρt qt = λt A.2 A.3 A.4 Since it is not optimal to completely deplete the stock of non-renewable energy resource in a finite t i.e. St Rt > for all t, qt =. Taking Eq. A.4, this implies λt = λ for all t. Applying Eq. A. in Eq. A.3, one gets ψt ψt = Atg I A.5 Taking logs in Eq. A.2, and applying Eq. A.5, one obtains the Hotelling rule ġ R g R = Atg I Taking the functional forms of my model, Eq. 8 is easily obtained from the Hotelling rule. 7

18 Taking logs in Eq. A., and applying Eq. A.5, one finds the Ramsey saving rule U cc Ċ U c ρ = ġi g I Ag I Applying the functional forms of my model, one gets the Eq. 7. Finally, Eq. 6 is the technology for physical capital accumulation, Eq. 9 is the budget constraint of the economy, and Eq. is the law of motion of the stock of non-renewable energy resource. Appendix B Along the BGP, all endogenous variables grow at constant rate i.e. xt = xexpγ x t. Moreover, I 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. By writing Eq. 7 along the BGP, one obtains γ A + γ θ = γ I γ R, B. which implies that γ C = Aθ R I + γ A ρ B.2 Since Y t = Ct + It, γ Y = γ I = γ C. Moreover, from the final good technology, Y t = AtKt, one gets that γ K = γ Y γ A. Then, γ K = Aθ R I B.3 From Eq. 8 along the BGP, it is easy to obtain that R = γ θ + γ I γ R I Aθ B.4 Applying Eq. B.4 into Eq. B.3, one gets the value of the ratio R = γ θ + γ A I Aθ B.5 Replacing this ratio in Eq. B.2 and Eq. B.3, one obtains γ Y = γ C = γ I = γ θ + γ A ρ 8

19 and γ K = γ θ + γ A ρ. From Eq. B. and the two previous expressions, one finds that long run growth rate of the energy is γ R = ρ. Finally, since Ṡt = Rt, γ S = γ R = ρ. Appendix C Eq. 2 can be rewritten as a standard linear first-order differential equation: Kt + at Kt = bt C. where and K= K is given. at = Z ρ Aθ Xt, bt = θ Xt Ct, Since the previous steps provide the solutions for Xt Eq.8 and Ct Eq.2, one can solve the differential equation C. applying the following result Sydsæter et al. 999: Kt = Kexp t aξdξ + t Taking Eq.8 and Eq.2, one obtains that bτexp t at = Z ρ ZAθ CZ + expzt Aθ τ aξdξ dτ After some tedious calculations, one finds aξdξ = Z ρ ξ + ln CZ CZ + Aθ expzξ C.2 C.3 C.4 Evaluating the integral C.4 in t and, and after rearranging terms, one obtains the first part of Eq.C.2: t Kexp aξdξ = Kexp lncz + Aθ exp lncz + Aθ Z ρt 9 C.3

20 In order to calculate the second part of Eq.C.2, I first obtain t bτexp aξdξ = Zθ Cexp τ lncz + Aθ expzt lncz + Aθ exp Z + ρ t exp ρτ, CZ+Aθ expzτ by evaluating the integral C.4 in τ and t. Then, t t bτexp aξdξ dτ = Zθ Cexp τ lncz + Aθ expzt lncz + Aθ exp Z + ρ t C.4 C.5 t exp ρτ CZ + Aθ expzτ dτ In order to calculate Eq.C.5, I need to obtain t exp ρτ CZ + Aθ expzτ dτ C.6 To do this, I will follow Boucekkine and Ruiz-Tamarit 24. Eq.C.6 is equal to t CZexpρτ + Aθ expz + ρτ dτ Doing the following change of variable V = exp ρ + Zτ, t CZexpρτ + Aθ expz + ρτ dτ = exp ρ+zt Recalling y = exp ρ + Zt, A = rewritten as y + CZ Aθ V Z Z+ρ dv C.7 CZ, and β =, Eq.C.7 can be Aθ + A V Z Z+ρ β dv C.8 2

21 Applying the binomial theorem: Eq.C.8 equals which is equal to n= Appying the property + ax b c = n= n= c n ax b n n! β y n A n V Z Z+ρ n dv, n! β n A n n! X = X+n y Z n + Z+ρ X n +X n, Eq.C.9 equals a n β n A y + a n n! n= n= y Z Z+ρ n y. C.9 Z Z+ρ n a n β n A n + a n n! Taking the definition of Gauss hypergeometric function a n b n z n 2F a, b, c; z = c n n!, n= C. one finds that Eq.C.7 equals CZ exp ρ + Zt 2 F Aθ 2 F CZ expzt Aθ, C. where CZ 2F Aθ expzt 2F CZ Z + r Aθ = 2 F Z Hence, unmanking the change of varibles, = Aθ Z+ρ t Z + r = 2 F Z,, + Z + r exp ρτ CZ + Aθ expzτ dτ [2F CZ Aθ expρ+zt 2 F 2 Z ; CZ Aθ expzt,, + Z + r Z ; CZ Aθ CZ Aθ expzt ], C.2

22 Finally, putting together the equations C.3, C.5, and C.2, and rearranging terms, one can easily obtain Eq