Impact of fossil fuel transition and population expansion on economic growth

Abstract

The existing works in endogenous growth have focused on technology and rarely on population impacts. In contrast, the research on environmental degradation caused by fossil fuel utilization has relied mostly on exogenous technology and population growth. Building upon the previous literature, I propose a dynamic growth model that allows the interaction between an economy and energy consumption of renewable and nonrenewable and the transitional path from one to another. I also allow endogenous population growth, where the population is affected by living standards and industrialization and indirectly natural resources through production, considering the trade-off between nonrenewable energy reserves and renewable resources. By creating a feedback loop from the population to the level of industrialization and GDP in this setup, GDP per capita’s growth rate is lower under endogenous population scenario relative to exogenous population growth. This particular outcome conveys that many projections for future energy use might overestimate our energy use, hence the economic and environmental costs. Firms utilize nonrenewable energy more intensively in a decentralized model since they do not fully internalize the negative externalities that arise from using nonrenewable energy, unlike the social planner approach. Imposing carbon-tax elements on the energy producers’ profit would accelerate clean energy adaptation and sustain the fossil fuel resources for a more extended period while increasing the total welfare by 3%. It would also increase the individuals’ long-term total consumption.

Appendices

Appendix 1: Data calibration

Since this research’s model is newly introduced and some of the equations are new, I have to structurally estimate all parameters (new and old) using the same model setup. Even though some of the parameters have been carefully estimated before using different structural models, as this model has been extended in various aspects to develop a new one, the results/values might not be consistent and valid for this new composition.

To estimate the parameters of Eq. (7), I used the time series for the US data. The main equation according to the model is: ${\text{ED}}_{t} = 1 - \left( {\frac{{{\text{FE}}_{t} }}{\varphi }} \right)^{\vartheta }$. Therefore, the estimating equation is given by:

$$\log \left( {y_{t} } \right) = \beta_{1} + \beta_{2} \log ({\text{FE}}_{t} ) + \varepsilon_{t}$$

(39)

where $y_{t} = 1 - {\text{ED}}_{t}$, $\beta_{2} = \vartheta$ and $\beta_{1} = - \vartheta \log \left( \varphi \right)$, $\varepsilon_{t} = \rho \varepsilon_{t - 1} + \varepsilon_{t}$.

The table below shows the results. The error terms are serially correlated. To estimate the above model, I used the generalized least-squares method to estimate the parameters in a linear regression analysis in which the errors are serially correlated. Specifically, the errors are assumed to follow a first-order autoregressive process. Based on the above estimation, we get the below values for the estimated parameters: $\vartheta =1.151$ and $\varphi =202{,}864.2$ (Table 3).

Table 3 Estimating the parameters in Eq. (7)

To derive the values of parameters in Eq. (8) [${\text{AC}}_{t + 1} = {\text{AC}}_{0} {\text{AC}}_{t}^{\theta } \left( {{\text{TL}}_{t} {\text{TY}}_{t} } \right)^{\omega }$], we can use the following values for $\theta \,\mathrm{ and }\,\omega$ based on Jones’ (2002) calibration: $\theta =0.94$ and ω $=0.015$. However, I changed the model by entering the interaction of financing the technology; therefore, it is advantageous to estimate it as follows (Table 4):

$$\log \left( {{\text{AC}}_{t + 1} } \right) = \log \left( {{\text{AC}}_{0} } \right) + \theta \log \left( {{\text{AC}}_{t} } \right) + \omega \log \left( {{\text{TL}}_{t} *{\text{TY}}_{t} } \right) + \varepsilon_{t}$$

(40)

Table 4 Estimating the parameters in Eq. (8)

Based on the above estimation, we get the below values for the estimated parameters: $\theta =0.87$ and ω $=0.02$ which is close to the Jones’ original calibration.

To estimate Eq. (15) parameters, I need to use time series again. The main equation according to the model is: $L_{t + 1} = L_{t} + L_{0} \left( {\frac{{Y_{t} }}{{L_{t} }}} \right)^{{\varepsilon_{1} }} \left( {\frac{{L_{t} }}{{K_{t} }}} \right)^{{\varepsilon_{2} }}$.

Therefore, the estimating equation is defined as:

$$\log \left( {g_{{L_{t} }} } \right) = \varepsilon_{0} + \varepsilon_{1} \log \left( {\frac{{Y_{t} }}{{L_{t} }}} \right) + \varepsilon_{2} \log \left( {\frac{{L_{t} }}{{K_{t} }}} \right) + \varepsilon_{t}$$

(41)

where $\varepsilon_{0} = \log \left( {L_{0} } \right)$ and g is the growth rate, rearrange the above equation for L, we get:

$$\log \left( {g_{{L_{t} }} } \right) = \varepsilon_{0} + \varepsilon_{1} \log \left( {Y_{t} } \right) + \varepsilon^{\prime}_{2} \log \left( {K_{t} } \right) + \varepsilon ^{\prime\prime}\log \left( {L_{t} } \right) + \varepsilon_{t}$$

(42)

where $\varepsilon^{\prime}_{2} = - \varepsilon_{2}$ and $\varepsilon^{\prime\prime} = \varepsilon_{2} - \varepsilon_{1}$ (Table 5).

Table 5 Estimating the parameters in Eq. (15)

Based on the above regression, the estimated parameters are: ${\varepsilon }_{1}=1.72$, ${\varepsilon }_{2}=2.17$.

Appendix 2: Explaining the used method to solve the model

To solve the model, first we can simplify the constraints by substituting Eq. (6) into (5), and then substitute back the new equation (total energy production) and (8) (environmental degradation) into the production function (Eq. 4), yielding Eq. (43) (section “Solving the social planner's F.O.C [exogenous population]” in Appendix). Then, we substitute the modified production function and the price for fossil fuel energy (Eq. 9) into the income allocation function (Eq. 3) to get Eq. (44). Then, substitute Eqs. (12) and (13) into (43) and (8), respectively, for PL and TL, to get the two constraints (Eqs. 45, 46) for the Lagrangian. Now we can establish the Lagrangian, in which households are maximizing their utility over infinite time, for the base model in which the population growth is exogenous Eq. (16).

Considering the three equations for income allocation (Eq. 3), production (Eq. 4),¹⁵ and technological production for renewable energy (Eq. 8), and the Euler equations (Eqs. 52, 53, 54, 55, 56), derived from the F.O.Cs, I can solve for this path using the actual values of the variables for the initial year (t = 0)—which are shown in Table 6—and then update the variables based on the above equations. Therefore, I directly use the law of motions (by forward iteration)¹⁶ to obtain next period values based on the previously driven values. Thus, there is an implicit uncertainty about the ending period of fossil fuel energy at the starting point.¹⁷ To select these values, I use 2018 as a reference year, extracted the values for the U.S., and then normalize it by million. The amount of clean energy (including nuclear) is set to be 22% of the total energy consumption.

Table 6 Initial values of the variables in the model

The only issue we have to derive the growth path, using the law of motions is to define the value of C₀ which is demonstrated in the footnote.¹⁸ Having the above values as initial conditions (and defining C₀ as it has been explained), we can compute the level of production from Eq. (4), the next period required technology for the clean energy from Eq. (8), and the cost of extracting the fossil fuel (CEX) from Eq. (9). Now, utilizing the budget constraint (Eq. 3), we can calculate the next period physical capital (K_t+1), knowing all values for the current (t = 0) state. Now, we can update the labor force using Eq. (14) for the exogenous case. The next period technological progress in the production process (A) can be achieved from Eq. (10). Therefore, we can use Eq. (54) to get the required fossil fuel energy (FE) for the next period. At this time, we can use Eq. (52) (intertemporal consumption decision) to compute the level of consumption for the next period as well. Now, the only unknown variable for the next period would be the required resources for financing the clean energy technology (TY). Using the last Euler Eq. (55), we can calculate the amount of this element. Repeating the same process, we can update all values for each period moving forward.

It must be noted that the approach I develop in this study is not a standard computational method. A social planner is not predicting the growth path. The planner maximizes the utility each period due to the existing, present resources. Therefore, the backward induction method has not been used since the exact time of depletion of natural resources is unknown. This form of set up is the real uncertainty of the model, implicitly implemented in the solving process. However, the issue of the discoveries uncertainty or the exact time of running out of fossil fuel, in the starting point, has not been studied explicitly within this framework since the current setup is deterministic, not stochastic. It is also worth mentioning that the social planner does not account for the nonrenewable resources constraint in the optimization problem in the beginning but tries to deal with it while there are not enough resources left to utilize. The main reason I use a non-conventional method to solve this model is that the ideology of this research is built on. There is no end time for resources (fossil fuel can be replaced by renewable energy); thus, the values of the transversality conditions for both physical capital and investment on renewable energy are unknown. The proposed approach does not sound quite appealing, but it saves the day.

Solving the first-order conditions, I can follow the same process as it has been done for the previous case to derive the Euler equations. Deriving the first-order conditions in the endogenous model is shown section “Solving the social planner’s F.O.C [endogenous population]” in Appendix. Having the Euler equations beside the constraints, I am able to follow the same process in the exogenous population scenario to update the next period values with some minor adjustments. First, I am going to use Eq. (15) instead of 14 to update the next year’s total labor force. And second, I need to solve (Eqs. 67, 69, 70) simultaneously to get the next period values for C, FE, and TY.

Appendix 3: Solving the F.O.Cs for both social planner and market-based approaches

3.1 Solving the social planner's F.O.C [exogenous population]

$$Y_{t} = \left( {1 - \frac{{{\text{FE}}_{t} }}{\varphi }^{\vartheta } } \right)\left[ {A_{t} K_{t}^{\alpha } {\text{PL}}_{t}^{{1 - \alpha - \gamma }} \left[ {\left( {{\text{AC}}_{t} {\text{CE}}} \right)^{\rho } + {\text{FE}}_{t}^{\rho } } \right]^{{\frac{\gamma }{\rho }}} } \right]$$

(43)

$$\begin{aligned} & \left( {1 - \frac{{{\text{FE}}_{t} }}{\varphi }^{\vartheta } } \right)\left( {A_{t} K_{t}^{\alpha } {\text{PL}}_{t}^{{1 - \alpha - \gamma }} } \right)\left( {\left( {{\text{AC}}_{t} {\text{CE}}} \right)^{\rho } + {\text{FE}}_{t}^{\rho } } \right)^{{\frac{\gamma }{\rho }}} \\ & \quad = C_{t} + K_{{t + 1}} - \left( {1 - \delta } \right)K_{t} + \left( {P_{0} + P_{1} \left( {\frac{{\mathop \sum \nolimits_{{i = 1}}^{t} {\text{FE}}_{i} }}{{\overline{{{\text{FE}}}} }}} \right)^{{P_{2} }} + P_{3} \left( {\frac{{{\text{FE}}_{t} }}{{\overline{{{\text{CTE}}}} }}} \right)^{{P_{4} }} } \right){\text{FE}}_{t} + {\text{TY}}_{t} \\ \end{aligned}$$

(44)

$$\begin{aligned} & \left( {1 - \frac{{{\text{FE}}_{t} }}{\varphi }^{\vartheta } } \right)\left( {A_{t} K_{t}^{\alpha } \left( {l_{\rm PL} L_{t} } \right)^{1 - \alpha - \gamma } } \right)\left( {\left( {{\text{AC}}_{t} {\text{CE}}} \right)^{\rho } + {\text{FE}}_{t}^{\rho } } \right)^{{\frac{\gamma }{\rho }}} = C_{t} + K_{t + 1} - \left( {1 - \delta } \right)K_{t} \\ & \quad + \left( {P_{0} + P_{1} \left( {\frac{{\mathop \sum \nolimits_{i = 1}^{t} {\text{FE}}_{i} }}{{\overline{{{\text{FE}}}} }}} \right)^{{P_{2} }} + P_{3} \left( {\frac{{{\text{FE}}_{t} }}{{\overline{{{\text{CTE}}}} }}} \right)^{{P_{4} }} } \right){\text{FE}}_{t} + {\text{TY}}_{t} \\ \end{aligned}$$

(45)

$${\text{AC}}_{t + 1} = {\text{AC}}_{0} {\text{AC}}_{t}^{\theta } \left( {l_{{{\text{PL}}}} L_{t} {\text{TY}}_{t} } \right)^{\omega }$$

(46)

Solving the Euler equations for the social planner when the population is exogenous.

First-order conditions are:

$$\left\{ {C_{t} } \right\}: \beta^{t} \frac{{C_{t}^{ - \sigma } }}{{L_{t}^{1 - \sigma } }} = \lambda_{1t}$$

(47)

$$\left\{ {K_{t + 1} } \right\}: \lambda_{1t + 1} \left\{ {\alpha \left( {1 - \frac{{{\text{FE}}_{t + 1} }}{\varphi }^{\vartheta } } \right)\left( {A_{t + 1} K_{t + 1}^{\alpha - 1} \left( {l_{{{\text{PL}}}} L_{t + 1} } \right)^{1 - \alpha - \gamma } } \right)\left( {\left( {{\text{AC}}_{t + 1} {\text{CE}}} \right)^{\rho } + {\text{FE}}_{t + 1}^{\rho } } \right)^{{\frac{\gamma }{\rho }}} + 1 - \delta } \right\} = \lambda_{1t}$$

(48)

$$\begin{aligned} & \left\{ {{\rm FE}_{t} } \right\}: \left( {1 - \frac{{{\rm FE}_{t} }}{\varphi }^{\vartheta } } \right)\left( {A_{t} K_{t}^{\alpha } \left( {l_{\rm PL} L_{t} } \right)^{1 - \alpha - \gamma } } \right)\gamma {\rm FE}_{t}^{\rho - 1} \left( {\left( {{\rm AC}_{t} {\rm CE}} \right)^{\rho } + {\rm FE}_{t}^{\rho } } \right)^{{\frac{\gamma}{\rho} - 1}} \\ & \quad - \frac{\vartheta }{\varphi }\frac{{{\rm FE}_{t} }}{\varphi }^{\vartheta - 1} \left( {A_{t} K_{t}^{\alpha } \left( {l_{\rm PL} L_{t} } \right)^{1 - \alpha - \gamma } } \right)\left( {\left( {{\rm AC}_{t} {\rm CE}} \right)^{\rho } + {\rm FE}_{t}^{\rho } } \right)^{{\frac{\gamma}{\rho}}} \\ & \quad - \left( {P_{0} + P_{1} \left( {\frac{{\mathop \sum \nolimits_{i = 1}^{t} {\rm FE}_{i} }}{{\overline{{\rm FE}} }}} \right)^{{P_{2} }} + P_{3} \left( {\frac{{{\rm FE}_{t} }}{{\overline{{\rm CTE}} }}} \right)^{{P_{4} }} + \left( {\frac{{P_{1} P_{2} }}{{\overline{{\rm FE}} }}\left( {\frac{{\mathop \sum \nolimits_{i = 1}^{t} {\rm FE}_{i} }}{{\overline{{\rm FE}} }}} \right)^{{P_{2} - 1}} + \frac{{P_{3} P_{4} }}{{\overline{{\rm CTE}} }}\left( {\frac{{{\rm FE}_{t} }}{{\overline{{\rm CTE}} }}} \right)^{{P_{4} - 1}} } \right){\rm FE}_{t} } \right) = 0 \\ \end{aligned}$$

(49)

$$\begin{aligned} & \left\{ {{\rm AC}_{t + 1} } \right\}:\lambda_{2t + 1} \left\{ {\theta {\rm AC}_{0} {\rm AC}_{t + 1}^{\theta - 1} \left( {l_{\rm TL} L_{t + 1} {\rm TY}_{t + 1} } \right)^{\omega } } \right\} \\ & \quad + \lambda_{1t + 1} \left\{ {\left( {1 - \frac{{{\rm FE}_{t + 1} }}{\varphi }^{\vartheta } } \right)A_{t + 1} K_{t + 1}^{\alpha } \left( {l_{\rm PL} L_{t + 1} } \right)^{1 - \alpha - \gamma } \gamma {\rm CE}^{\rho } {\rm AC}_{t + 1}^{\rho - 1} \left( {\left( {{\rm AC}_{t + 1} {\rm CE}} \right)^{\rho } + {\rm FE}_{t + 1}^{\rho } } \right)^{{\frac{\gamma}{\rho} - 1}} } \right\} = \lambda_{2t} \\ \end{aligned}$$

(50)

$$\left\{ {{\rm TY}_{t} } \right\}: \lambda_{1t} = \lambda_{2t} \left[ {\omega {\rm AC}_{0} {\rm AC}_{t}^{\theta } \left( {l_{\rm TL} L_{t} } \right)^{\omega } {\rm TY}_{t}^{\omega - 1} } \right]$$

(51)

Substituting Eqs. (47) and (49) (and the updated forms of them) in the above F.O.Cs, we get the following Euler equations:

$$\beta \frac{{C_{t + 1}^{ - \sigma } }}{{L_{t + 1}^{1 - \sigma } }}\left\{ {\alpha \left( {1 - \frac{{{\rm FE}_{t + 1} }}{\varphi }^{\vartheta } } \right)\left( {A_{t + 1} K_{t + 1}^{\alpha - 1} \left( {l_{\rm PL} L_{t + 1} } \right)^{1 - \alpha - \gamma } } \right)\left( {\left( {{\rm AC}_{t + 1} {\rm CE}} \right)^{\rho } + {\rm FE}_{t + 1}^{\rho } } \right)^{{\frac{\gamma}{\rho}}} + 1 - \delta } \right\} = \frac{{C_{t}^{ - \sigma } }}{{L_{t}^{1 - \sigma } }}$$

(52)

And the above equation, when there is no fossil fuel energy left to use, is:

$$\beta \frac{{C_{t + 1}^{ - \sigma } }}{{L_{t + 1}^{1 - \sigma } }}\left\{ {\alpha \left( {A_{t + 1} K_{t + 1}^{\alpha - 1} \left( {l_{\rm PL} L_{t + 1} } \right)^{1 - \alpha - \gamma } } \right)\left( {{\rm AC}_{t + 1} {\rm CE}} \right)^{\gamma } + 1 - \delta } \right\} = \frac{{C_{t}^{ - \sigma } }}{{L_{t}^{1 - \sigma } }}$$

(53)

$$\begin{aligned} & \left( {1 - \frac{{{\text{FE}}_{{t + 1}} }}{\varphi }^{\vartheta } } \right)\left( {A_{{t + 1}} K_{{t + 1}}^{\alpha } \left( {l_{{{\text{PL}}}} L_{{t + 1}} } \right)^{{1 - \alpha - \gamma }} } \right)\gamma {\text{FE}}_{{t + 1}}^{{\rho - 1}} \left( {\left( {{\text{AC}}_{{t + 1}} {\text{CE}}} \right)^{\rho } + {\text{FE}}_{{t + 1}}^{\rho } } \right)^{{\frac{\gamma }{\rho } - 1}} \\ & \quad = \frac{\vartheta }{\varphi }\frac{{{\text{FE}}_{{t + 1}} }}{\varphi }^{{\vartheta - 1}} \left( {A_{{t + 1}} K_{{t + 1}}^{\alpha } \left( {l_{{{\text{PL}}}} L_{{t + 1}} } \right)^{{1 - \alpha - \gamma }} } \right)\left( {\left( {{\text{AC}}_{{t + 1}} {\text{CE}}} \right)^{\rho } + {\text{FE}}_{{t + 1}}^{\rho } } \right)^{{\frac{\gamma }{\rho }}} \\ & \quad \quad + P_{0} + P_{1} \left( {\frac{{\mathop \sum \nolimits_{{i = 1}}^{{t + 1}} {\text{FE}}_{i} }}{{\overline{{{\text{FE}}}} }}} \right)^{{P_{2} }} + P_{3} \left( {\frac{{{\text{FE}}_{t} }}{{\overline{{{\text{CTE}}}} }}} \right)^{{P_{4} }} \\ & \quad \quad + \left( {\frac{{P_{1} P_{2} }}{{\overline{{{\text{FE}}}} }}\left( {\frac{{\mathop \sum \nolimits_{{i = 1}}^{{t + 1}} {\text{FE}}_{i} }}{{\overline{{{\text{FE}}}} }}} \right)^{{P_{2} - 1}} + \frac{{P_{3} P_{4} }}{{\overline{{{\text{CTE}}}} }}\left( {\frac{{{\text{FE}}_{{t + 1}} }}{{\overline{{{\text{CTE}}}} }}} \right)^{{P_{4} - 1}} } \right){\text{FE}}_{{t + 1}} \\ \end{aligned}$$

(54)

$$\begin{aligned} & \frac{{{\rm B}\frac{{C_{t + 1}^{ - \sigma } }}{{L_{t + 1}^{1 - \sigma } }}}}{{\omega {\rm AC}_{0} {\rm AC}_{t + 1}^{\theta } \left( {l_{\rm TL} L_{t + 1} } \right)^{\omega } {\rm TY}_{t + 1}^{\omega - 1} }}\left\{ {\theta {\rm AC}_{0} {\rm AC}_{t + 1}^{\theta - 1} \left( {l_{\rm TL} L_{t + 1} {\rm TY}_{t + 1} } \right)^{\omega } } \right\} \\ & \quad + \beta \frac{{C_{t + 1}^{ - \sigma } }}{{L_{t + 1}^{1 - \sigma } }}\left\{ {\left( {1 - \frac{{{\rm FE}_{t + 1} }}{\varphi }^{\vartheta } } \right)A_{t + 1} K_{t + 1}^{\alpha } \left( {l_{\rm PL} L_{t + 1} } \right)^{1 - \alpha - \gamma } \gamma {\rm CE}^{\rho } {\rm AC}_{t + 1}^{\rho - 1} \left( {\left( {{\rm AC}_{t + 1} {\rm CE}} \right)^{\rho } + {\rm FE}_{t + 1}^{\rho } } \right)^{{\frac{\gamma}{\rho} - 1}} } \right\} \\ & \quad \quad = \frac{{\frac{{C_{t}^{ - \sigma } }}{{L_{t}^{1 - \sigma } }}}}{{\omega {\rm AC}_{0} {\rm AC}_{t}^{\theta } \left( {l_{\rm TL} L_{t} } \right)^{\omega } {\rm TY}_{t}^{\omega - 1} }} \\ \end{aligned}$$

(55)

And the above equation, when no fossil fuel energy remains, is:

$$\begin{aligned} & \frac{{\beta \frac{{C_{t + 1}^{ - \sigma } }}{{L_{t + 1}^{1 - \sigma } }}}}{{\omega {\rm AC}_{0} {\rm AC}_{t + 1}^{\theta } \left( {l_{\rm TL} L_{t + 1} } \right)^{\omega } {\rm TY}_{t + 1}^{\omega - 1} }}\left\{ {\theta {\rm AC}_{0} {\rm AC}_{t + 1}^{\theta - 1} \left( {l_{\rm TL} L_{t + 1} {\rm TY}_{t + 1} } \right)^{\omega } } \right\} \\ & \quad + \beta \frac{{C_{t + 1}^{ - \sigma } }}{{L_{t + 1}^{1 - \sigma } }}\left\{ {A_{t + 1} K_{t + 1}^{\alpha } \left( {l_{\rm PL} L_{t + 1} } \right)^{1 - \alpha - \gamma } \gamma {\rm CE}\left( {{\rm AC}_{t + 1} {\rm CE}} \right)^{\gamma - 1} } \right\} = \frac{{\frac{{C_{t}^{ - \sigma } }}{{L_{t}^{1 - \sigma } }}}}{{\omega {\rm AC}_{0} {\rm AC}_{t}^{\theta } \left( {l_{\rm TL} L_{t} } \right)^{\omega } {\rm TY}_{t}^{\omega - 1} }} \\ \end{aligned}$$

(56)

3.2 Solving the social planner’s F.O.C [endogenous population]

By solving the Euler equations for the social planner when the population is endogenous, the first-order conditions are:

$$\left\{ {C_{t} } \right\}: \beta^{t} \frac{{C_{t}^{ - \sigma } }}{{L_{t}^{1 - \sigma } }} = \lambda_{1t}$$

(57)

$$\begin{aligned} & \left\{ {L_{t + 1} } \right\}: - \beta^{t + 1} C_{t + 1}^{1 - \sigma } L_{t + 1}^{\sigma - 2} + \lambda_{1t + 1} \left\{ {\left( {1 - \alpha - \gamma } \right)\left( {1 - \frac{{{\rm FE}_{t + 1} }}{\varphi }^{\vartheta } } \right)\left( {A_{t + 1} K_{t + 1}^{\alpha } \left( {l_{\rm PL} } \right)^{1 - \alpha - \gamma } \left( {L_{t + 1} } \right)^{ - \alpha - \gamma } } \right)\left( {\left( {{\rm AC}_{t + 1} {\rm CE}} \right)^{\rho } + {\rm FE}_{t + 1}^{\rho } } \right)^{{\frac{\gamma}{\rho}}} } \right\} \\ & \quad + \lambda_{2t + 1} \left\{ {\omega {\rm AC}_{0} {\rm AC}_{t + 1}^{\theta } \left( {l_{\rm TL} {\rm TY}_{t + 1} } \right)^{\omega } L_{t + 1}^{\omega - 1} } \right\} \\ & \quad + \lambda_{3t + 1} \left\{ {L_{0} \left( {\varepsilon_{2} - \alpha \varepsilon_{1} - \gamma \varepsilon_{1} } \right)\left( {1 - \frac{{{\rm FE}_{t + 1} }}{\varphi }^{\vartheta } } \right)^{{\varepsilon_{1} }} A_{t + 1}^{{\varepsilon_{1} }} K_{t + 1}^{{\alpha \varepsilon_{1} - \varepsilon_{2} }} l_{\rm PL}^{{\varepsilon_{1} - \alpha \varepsilon_{1} - \varepsilon_{1} \gamma }} L_{t + 1}^{{\varepsilon_{2} - \alpha \varepsilon_{1} - \gamma \varepsilon_{1} - 1}} \left( {\left( {{\rm AC}_{t + 1} {\rm CE}} \right)^{\rho } + {\rm FE}_{t + 1}^{\rho } } \right)^{{{\raise0.7ex\hbox{${\gamma \varepsilon_{1} }$} \!\mathord{\left/ {\vphantom {{\gamma \varepsilon_{1} } \rho }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\rho $}}}} + 1} \right\} = \lambda_{3t} \\ \end{aligned}$$

(58)

$$\begin{aligned} & \left\{ {K_{t + 1} } \right\}: \lambda_{1t + 1} \left\{ {\left( {1 - \frac{{{\rm FE}_{t + 1} }}{\varphi }^{\vartheta } } \right)\alpha \left( {A_{t + 1} K_{t + 1}^{\alpha - 1} \left( {l_{\rm PL} L_{t + 1} } \right)^{1 - \alpha - \gamma } } \right)\left( {\left( {{\rm AC}_{t + 1} {\rm CE}} \right)^{\rho } + {\rm FE}_{t + 1}^{\rho } } \right)^{{\frac{\gamma}{\rho}}} + 1 - \delta } \right\} \\ & \quad + \lambda_{3t + 1} \{ \left( {\alpha \varepsilon_{1} - \varepsilon_{2} } \right)L_{0} \left( {1 - \frac{{{\rm FE}_{t + 1} }}{\varphi }^{\vartheta } } \right)^{{\varepsilon_{1} }} A_{t + 1}^{{\varepsilon_{1} }} K_{t + 1}^{{\alpha \varepsilon_{1} - \varepsilon_{2} - 1}} l_{\rm PL}^{{\varepsilon_{1} - \alpha \varepsilon_{1} - \varepsilon_{1} \gamma }} L_{t + 1}^{{\varepsilon_{2} - \alpha \varepsilon_{1} - \gamma \varepsilon_{1} }} \left( {\left( {{\rm AC}_{t + 1} {\rm CE}} \right)^{\rho } + {\rm FE}_{t + 1}^{\rho } } \right)^{{{\raise0.7ex\hbox{${\gamma \varepsilon_{1} }$} \!\mathord{\left/ {\vphantom {{\gamma \varepsilon_{1} } \rho }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\rho $}}}} = \lambda_{1t} \\ \end{aligned}$$

(59)

$$\left\{{\mathrm{FE}}_{t}\right\}: {\lambda }_{1{t}}\left\{\left(1-{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta }\right)\left({A}_{t}{{K}}_{t}^{\alpha }{\left({I}_{\mathrm{PL}}{L}_{t}\right)}^{1-\alpha -\gamma }\right)\gamma {\mathrm{FE}}_{t}^{\rho -1}{\left({\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{t}^{\rho }\right)}^{\frac{\gamma }{\rho }-1}- \frac{\vartheta }{\varphi }{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta -1}\left({A}_{t}{{K}}_{t}^{\alpha }{\left({I}_{\mathrm{PL}}{L}_{t}\right)}^{1-\alpha -\gamma }\right){\left({\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{t}^{\rho }\right)}^{\frac{\gamma }{\rho }}-\left({\mathrm{P}}_{0}+{\mathrm{P}}_{1}{\left(\sum_{\mathrm{i}=1}^{t}\frac{{\mathrm{FE}}_{\mathrm{i}}}{\overline{\mathrm{FE}} }\right)}^{{\mathrm{P}}_{2}}+{\mathrm{P}}_{3}{\left(\frac{{\mathrm{FE}}_{t}}{\overline{\mathrm{CTE}} }\right)}^{{\mathrm{P}}_{4}}+\left(\frac{{\mathrm{P}}_{1}{\mathrm{P}}_{2}}{\overline{\mathrm{FE}} }{\left(\sum_{\mathrm{i}=1}^{t}\frac{{\mathrm{FE}}_{\mathrm{i}}}{\overline{\mathrm{FE}} }\right)}^{{\mathrm{P}}_{2}-1}+\frac{{\mathrm{P}}_{3}{\mathrm{P}}_{4}}{\overline{\mathrm{CTE}} }{\left(\frac{{\mathrm{FE}}_{t}}{\overline{\mathrm{CTE}} }\right)}^{{\mathrm{P}}_{4}-1}\right){\mathrm{FE}}_{t}\right)\right\}+ {\lambda }_{3{t}}\left\{{L}_{0}{\left(1-{ \frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}}{A}_{t}^{{\varepsilon }_{1}}{{K}}_{t}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{t}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}\gamma {\varepsilon }_{1}{\mathrm{FE}}_{t}^{\rho -1}{\left({\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{t}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }-1}- {L}_{0}\frac{\vartheta {\varepsilon }_{1}}{\varphi }{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta -1}{\left(1-{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}-1}{A}_{t}^{{\varepsilon }_{1}}{{K}}_{t}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{t}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}+1}{\left({\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{t}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }}\right\}=0$$

(60)

$$\begin{aligned} \left\{ {{\rm AC}_{t + 1} } \right\} & :\lambda_{1t + 1} \left\{ {\left( {1 - \frac{{{\rm FE}_{t + 1} }}{\varphi }^{\vartheta } } \right)A_{t + 1} K_{t + 1}^{\alpha } \left( {l_{\rm PL} L_{t + 1} } \right)^{1 - \alpha - \gamma } \gamma {\rm CE}^{\rho } {\rm AC}_{t + 1}^{\rho - 1} \left( {\left( {{\rm AC}_{t + 1} {\rm CE}} \right)^{\rho } + {\rm FE}_{t + 1}^{\rho } } \right)^{{\frac{\gamma }{\rho } - 1}} } \right\} \\ & \quad + \lambda_{2t + 1} \left\{ {\theta {\rm AC}_{0} {\rm AC}_{t + 1}^{\theta - 1} \left( {l_{\rm TL} L_{t + 1} {\rm TY}_{t + 1} } \right)^{\omega } } \right\} \\ & \quad + \lambda_{3t + 1} \left\{ {\gamma \varepsilon_{1} {\rm CE}^{\rho } {\rm AC}_{t + 1}^{\rho - 1} L_{0} \left( {1 - \frac{{{\rm FE}_{t + 1} }}{\varphi }^{\vartheta } } \right)^{{\varepsilon_{1} }} A_{t + 1}^{{\varepsilon_{1} }} K_{t + 1}^{{\alpha \varepsilon_{1} - \varepsilon_{2} }} l_{\rm PL}^{{\varepsilon_{1} - \alpha \varepsilon_{1} - \varepsilon_{1} \gamma }} L_{t + 1}^{{\varepsilon_{2} - \alpha \varepsilon_{1} - \gamma \varepsilon_{1} }} \left( {\left( {{\rm AC}_{t + 1} {\rm CE}} \right)^{\rho } + {\rm FE}_{t + 1}^{\rho } } \right)^{{\frac{{\gamma \varepsilon_{1} }}{\rho } - 1}} } \right\} = \lambda_{2t} \\ \end{aligned}$$

(61)

$$\left\{ {{\rm TY}_{t} } \right\}: \lambda_{1t} = \lambda_{2t} \left[ {\omega {\rm AC}_{0} {\rm AC}_{t}^{\theta } \left( {l_{\rm TL} L_{t} } \right)^{\omega } {\rm TY}_{t}^{\omega - 1} } \right]$$

(62)

Substituting Eqs. (57) and (60) (and the updated forms of them) in the above F.O.Cs, we get the following equations:

$$-{\beta }^{{t}+1}{C}_{{t}+1}^{1-\sigma }{L}_{{t}+1}^{\sigma -2}+ {\beta }^{{t}+1}\frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}\left\{\left(1-\alpha -\gamma \right)\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)\left({A}_{{t}+1}{{K}}_{{t}+1}^{\alpha }{\left({I}_{\mathrm{PL}}\right)}^{1-\alpha -\gamma }{\left({L}_{{t}+1}\right)}^{-\alpha -\gamma }\right){\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma }{\rho }}\right\}+\frac{{\beta }^{{t}+1}\frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}}{\omega {\mathrm{AC}}_{0}{\mathrm{AC}}_{{t}+1}^{\theta }{\left({I}_{\mathrm{TL}}{L}_{{t}+1}\right)}^{\omega }{\mathrm{TY}}_{{t}+1}^{\omega -1}}\left\{\omega {\mathrm{AC}}_{0}{\mathrm{AC}}_{{t}+1}^{\theta }{\left({I}_{\mathrm{TL}}{\mathrm{TY}}_{{t}+1}\right)}^{\omega }{L}_{{t}+1}^{\omega -1}\right\}+{\lambda }_{3{t}+1}\left\{{L}_{0}\left({\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}\right){\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}}{A}_{{t}+1}^{{\varepsilon }_{1}}{{K}}_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}-1}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }}+1\right\}={\lambda }_{3{t}}$$

(63)

$${\beta }^{{t}+1}\frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}\left\{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)\alpha \left({A}_{{t}+1}{{K}}_{{t}+1}^{\alpha -1}{\left({I}_{\mathrm{PL}}{L}_{{t}+1}\right)}^{1-\alpha -\gamma }\right){\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma }{\rho }}+1-\updelta \right\}+{\lambda }_{3{t}+1}\{(\alpha {\varepsilon }_{1}-{\varepsilon }_{2}){L}_{0}{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}}{A}_{{t}+1}^{{\varepsilon }_{1}}{{K}}_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}-1}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }}={\beta }^{t}\frac{{C}_{t}^{-\sigma }}{{L}_{t}^{1-\sigma }}$$

(64)

$${\beta }^{t}\frac{{C}_{t}^{-\sigma }}{{L}_{t}^{1-\sigma }}\left\{\left(1-{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta }\right)\left({A}_{t}{{K}}_{t}^{\alpha }{\left({I}_{\mathrm{PL}}{L}_{t}\right)}^{1-\alpha -\gamma }\right)\gamma {\mathrm{FE}}_{t}^{\rho -1}{\left({\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{t}^{\rho }\right)}^{\frac{\gamma }{\rho }-1}-\frac{\vartheta }{\varphi }{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta -1}\left({A}_{t}{{K}}_{t}^{\alpha }{\left({I}_{\mathrm{PL}}{L}_{t}\right)}^{1-\alpha -\gamma }\right){\left({\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{t}^{\rho }\right)}^{\frac{\gamma }{\rho }}-\left({\mathrm{P}}_{0}+{\mathrm{P}}_{1}{\left(\sum_{\mathrm{i}=1}^{t}\frac{{\mathrm{FE}}_{\mathrm{i}}}{\overline{\mathrm{FE}} }\right)}^{{\mathrm{P}}_{2}}+{\mathrm{P}}_{3}{\left(\frac{{\mathrm{FE}}_{t}}{\overline{\mathrm{CTE}} }\right)}^{{\mathrm{P}}_{4}}+(\frac{{\mathrm{P}}_{1}{\mathrm{P}}_{2}}{\overline{\mathrm{FE}} }{\left(\sum_{\mathrm{i}=1}^{t}\frac{{\mathrm{FE}}_{\mathrm{i}}}{\overline{\mathrm{FE}} }\right)}^{{\mathrm{P}}_{2}-1}+\frac{{\mathrm{P}}_{3}{\mathrm{P}}_{4}}{\overline{\mathrm{CTE}} }{\left(\frac{{\mathrm{FE}}_{t}}{\overline{\mathrm{CTE}} }\right)}^{{\mathrm{P}}_{4}-1}){\mathrm{FE}}_{t}{\varepsilon }_{1}\right)\right\}+ {\lambda }_{3{t}}\{{L}_{0}{\left(1-{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}}{A}_{t}^{{\varepsilon }_{1}}{{K}}_{t}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{t}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}\gamma {\varepsilon }_{1}{\mathrm{FE}}_{t}^{\rho -1}{\left({\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{t}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }-1}-{L}_{0}\frac{\vartheta {\varepsilon }_{1}}{\varphi }{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta -1}{\left(1-{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}-1}{A}_{t}^{{\varepsilon }_{1}}{{K}}_{t}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{t}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}{\left({\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{t}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }}\}=0$$

(65)

$${\beta }^{{t}+1}\frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}\left\{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right){A}_{{t}+1}{{K}}_{{t}+1}^{\alpha }{\left({I}_{\mathrm{PL}}{L}_{{t}+1}\right)}^{1-\alpha -\gamma }\gamma {\mathrm{CE}}^{\rho }{\mathrm{AC}}_{{t}+1}^{\rho -1}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma }{\rho }-1}\right\}+\frac{{\beta }^{{t}+1}\frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}}{\omega {\mathrm{AC}}_{0}{\mathrm{AC}}_{{t}+1}^{\theta }{\left({I}_{\mathrm{TL}}{L}_{{t}+1}\right)}^{\omega }{\mathrm{TY}}_{{t}+1}^{\omega -1}}\left\{\theta {\mathrm{AC}}_{0}{\mathrm{AC}}_{{t}+1}^{\theta -1}{\left({I}_{\mathrm{TL}}{L}_{{t}+1}{\mathrm{TY}}_{{t}+1}\right)}^{\omega }\right\}+{\lambda }_{3{t}+1}\{\gamma {\varepsilon }_{1}{\mathrm{CE}}^{\rho }{\mathrm{AC}}_{{t}+1}^{\rho -1}{L}_{0}{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}}{A}_{{t}+1}^{{\varepsilon }_{1}}{{K}}_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }-1}\}=\frac{{\beta }^{t}\frac{{C}_{t}^{-\sigma }}{{L}_{t}^{1-\sigma }}}{\omega {\mathrm{AC}}_{0}{\mathrm{AC}}_{t}^{\theta }{\left({I}_{\mathrm{TL}}{L}_{t}\right)}^{\omega }{\mathrm{TY}}_{t}^{\omega -1}}$$

(66)

Now, we rearrange Eq. (65) for ${\lambda }_{3t}$, update it to get ${\lambda }_{3t+1}$, and replace it back into Eqs. (63, 64, 66), having our three Euler equations as follows:

$$-\beta C_{{t}+1}^{1-\sigma }L_{{t}+1}^{\sigma -2}+ \beta \frac{C_{{t}+1}^{-\sigma }}{L_{{t}+1}^{1-\sigma }}\left\{\left(1-\alpha -\gamma \right)\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)\left(A_{{t}+1}K_{{t}+1}^{\alpha }{\left({{l}}_{\mathrm{PL}}\right)}^{1-\alpha -\gamma }{\left(L_{{t}+1}\right)}^{-\alpha -\gamma }\right){\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma }{\rho }}\right\}+\frac{\beta \frac{C_{{t}+1}^{-\sigma }}{L_{{t}+1}^{1-\sigma }}}{\omega {\mathrm{AC}}_{0}{\mathrm{AC}}_{{t}+1}^{\theta }{\left({{l}}_{\mathrm{TL}}L_{{t}+1}\right)}^{\omega }{\mathrm{TY}}_{{t}+1}^{\omega -1}}\left\{\omega {\mathrm{AC}}_{0}{\mathrm{AC}}_{{t}+1}^{\theta }{\left({{l}}_{\mathrm{TL}}{\mathrm{TY}}_{{t}+1}\right)}^{\omega }L_{{t}+1}^{\omega -1}\right\}+\frac{\mathrm{AA}}{\mathrm{BB}}*\left\{L_{0}\left({\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}\right){\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}}A_{{t}+1}^{{\varepsilon }_{1}}K_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{{l}}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }L_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}-1}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }}+1\right\}=\frac{\mathrm{CC}}{\mathrm{DD}}$$

(67)

$$\mathrm{AA} =-\beta \frac{C_{{t}+1}^{-\sigma }}{L_{{t}+1}^{1-\sigma }}\left\{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)\left(A_{{t}+1}K_{{t}+1}^{\alpha }{\left({{l}}_{\mathrm{PL}}L_{{t}+1}\right)}^{1-\alpha -\gamma }\right)\gamma {\mathrm{FE}}_{{t}+1}^{\rho -1}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma }{\rho }-1}-\frac{\vartheta }{\varphi }{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta -1}\left(A_{{t}+1}K_{{t}+1}^{\alpha }{\left({{l}}_{\mathrm{PL}}L_{{t}+1}\right)}^{1-\alpha -\gamma }\right){\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma }{\rho }}-\left({\mathrm{P}}_{0}+{\mathrm{P}}_{1}{\left(\sum_{{i}=1}^{{t}+1}\frac{{\mathrm{FE}}_{{i}}}{\overline{\mathrm{FE}} }\right)}^{{\mathrm{P}}_{2}}+{\mathrm{P}}_{3}{\left(\frac{{\mathrm{FE}}_{{t}+1}}{\overline{\mathrm{CTE}} }\right)}^{{\mathrm{P}}_{4}}+(\frac{{\mathrm{P}}_{1}{\mathrm{P}}_{2}}{\overline{\mathrm{FE}} }{\left(\sum_{{i}=1}^{{t}+1}\frac{{\mathrm{FE}}_{{i}}}{\overline{\mathrm{FE}} }\right)}^{{\mathrm{P}}_{2}-1}+\frac{{\mathrm{P}}_{3}{\mathrm{P}}_{4}}{\overline{\mathrm{CTE}} }{(\frac{{\mathrm{FE}}_{{t}+1}}{\overline{\mathrm{CTE}} })}^{{\mathrm{P}}_{4}-1}){\mathrm{FE}}_{{t}+1}\right)\right\}$$

$$\mathrm{BB}=\left\{L_{0}{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}}A_{{t}+1}^{{\varepsilon }_{1}}K_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{{l}}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }L_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}\gamma {\varepsilon }_{1}{\mathrm{FE}}_{{t}+1}^{\rho -1}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }-1}-L_{0}\frac{\vartheta {\varepsilon }_{1}}{\varphi }{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta -1}{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}-1}A_{{t}+1}^{{\varepsilon }_{1}}K_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{{l}}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }L_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }}\right\}$$

$$\mathrm{CC}=-\frac{C_{t}^{-\sigma }}{L_{t}^{1-\sigma }}\left\{\left(1-{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta }\right)\left(A_{t}K_{t}^{\alpha }{\left({{l}}_{\mathrm{PL}}L_{t}\right)}^{1-\alpha -\gamma }\right)\gamma {\mathrm{FE}}_{t}^{\rho -1}{\left({\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{t}^{\rho }\right)}^{\frac{\gamma }{\rho }-1}-\frac{\vartheta }{\varphi }{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta -1}\left(A_{t}K_{t}^{\alpha }{\left({{l}}_{\mathrm{PL}}L_{t}\right)}^{1-\alpha -\gamma }\right){\left({\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{t}^{\rho }\right)}^{\frac{\gamma }{\rho }}-\left({\mathrm{P}}_{0}+{\mathrm{P}}_{1}{\left(\sum_{{i}=1}^{{t}}\frac{{\mathrm{FE}}_{{i}}}{\overline{\mathrm{FE}} }\right)}^{{\mathrm{P}}_{2}}+{\mathrm{P}}_{3}{\left(\frac{{\mathrm{FE}}_{t}}{\overline{\mathrm{CTE}} }\right)}^{{\mathrm{P}}_{4}}+(\frac{{\mathrm{P}}_{1}{\mathrm{P}}_{2}}{\overline{\mathrm{FE}} }{\left(\sum_{{i}=1}^{{t}}\frac{{\mathrm{FE}}_{{i}}}{\overline{\mathrm{FE}} }\right)}^{{\mathrm{P}}_{2}-1}+\frac{{\mathrm{P}}_{3}{\mathrm{P}}_{4}}{\overline{\mathrm{CTE}} }{\left(\frac{{\mathrm{FE}}_{t}}{\overline{\mathrm{CTE}} }\right)}^{{\mathrm{P}}_{4}-1}){\mathrm{FE}}_{t}\right)\right\}$$

$$\mathrm{DD}=\left\{L_{0}{\left(1-{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}}A_{t}^{{\varepsilon }_{1}}K_{t}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{{l}}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }L_{t}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}\gamma {\varepsilon }_{1}{\mathrm{FE}}_{t}^{\rho -1}{\left({\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{t}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }-1}-L_{0}\frac{\vartheta {\varepsilon }_{1}}{\varphi }{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta -1}{\left(1-{\frac{{\mathrm{FE}}_{t}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}-1}A_{t}^{{\varepsilon }_{1}}K_{t}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{{l}}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }L_{t}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}\right\}$$

And the above equation, when there is no fossil fuel energy left to use, is:

$$-{\beta }^{2}{C}_{{t}+1}^{1-\sigma }{L}_{{t}+1}^{\sigma -2}+ {\beta }^{2}\frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}\left\{\left(1-\alpha -\gamma \right)\left({A}_{{t}+1}{{K}}_{{t}+1}^{\alpha }{\left({I}_{\mathrm{PL}}\right)}^{1-\alpha -\gamma }{\left({L}_{{t}+1}\right)}^{-\alpha -\gamma }\right){\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\gamma }\right\}+ \frac{{\beta }^{2}\frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}}{\omega {\mathrm{AC}}_{0}{\mathrm{AC}}_{{t}+1}^{\theta }{\left({I}_{\mathrm{TL}}{L}_{{t}+1}\right)}^{\omega }{\mathrm{TY}}_{{t}+1}^{\omega -1}}\left\{\omega {\mathrm{AC}}_{0}{\mathrm{AC}}_{{t}+1}^{\theta }{\left({I}_{\mathrm{TL}}{\mathrm{TY}}_{{t}+1}\right)}^{\omega }{L}_{{t}+1}^{\omega -1}\right\}+\frac{\beta \frac{{C}_{t}^{-\sigma }}{{L}_{t}^{1-\sigma }}- {\beta }^{2}\frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}(\alpha \left({A}_{{t}+1}{{K}}_{{t}+1}^{\alpha -1}{\left({I}_{\mathrm{PL}}{L}_{{t}+1}\right)}^{1-\alpha -\gamma }\right){\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\gamma }+1-\updelta )}{(\alpha {\varepsilon }_{1}-{\varepsilon }_{2}){L}_{0}{A}_{{t}+1}^{{\varepsilon }_{1}}{{K}}_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}-1}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}{\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\gamma {\varepsilon }_{1}}}*\left\{{L}_{0}\left({\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}\right){A}_{{t}+1}^{{\varepsilon }_{1}}{{K}}_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}-1}{\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\gamma {\varepsilon }_{1}}+1\right\}=\frac{ \frac{{C}_{{t}-1}^{-\sigma }}{{L}_{{t}-1}^{1-\sigma }}- \beta \frac{{C}_{t}^{-\sigma }}{{L}_{t}^{1-\sigma }}(\alpha \left({A}_{t}{{K}}_{t}^{\alpha -1}{\left({I}_{\mathrm{PL}}{L}_{t}\right)}^{1-\alpha -\gamma }\right){\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\gamma }+1-\updelta )}{(\alpha {\varepsilon }_{1}-{\varepsilon }_{2}){L}_{0}{A}_{t}^{{\varepsilon }_{1}}{{K}}_{t}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}-1}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{t}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}{\left({\mathrm{AC}}_{t}\mathrm{CE}\right)}^{\gamma {\varepsilon }_{1}}}$$

(68)

$$\beta \frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}\left\{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)\alpha \left({A}_{{t}+1}{{K}}_{{t}+1}^{\alpha -1}{\left({I}_{\mathrm{PL}}{L}_{{t}+1}\right)}^{1-\alpha -\gamma }\right){\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma }{\rho }}+1-\updelta \right\}+\left(\frac{\mathrm{EE}}{\mathrm{FF}}\right)\{(\alpha {\varepsilon }_{1}-{\varepsilon }_{2}){L}_{0}{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}}{A}_{{t}+1}^{{\varepsilon }_{1}}{{K}}_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}-1}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }}=\frac{{C}_{t}^{-\sigma }}{{L}_{t}^{1-\sigma }}$$

(69)

$$\mathrm{EE}=-\beta \frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}\left\{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)\left({A}_{{t}+1}{{K}}_{{t}+1}^{\alpha }{\left({I}_{\mathrm{PL}}{L}_{{t}+1}\right)}^{1-\alpha -\gamma }\right)\gamma {\mathrm{FE}}_{{t}+1}^{\rho -1}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma }{\rho }-1}-\frac{\vartheta }{\varphi }{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta -1}\left({A}_{{t}+1}{{K}}_{{t}+1}^{\alpha }{\left({I}_{\mathrm{PL}}{L}_{{t}+1}\right)}^{1-\alpha -\gamma }\right){\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma }{\rho }}-\left({\mathrm{P}}_{0}+{\mathrm{P}}_{1}{\left(\sum_{\mathrm{i}=1}^{{t}+1}\frac{{\mathrm{FE}}_{\mathrm{i}}}{\overline{\mathrm{FE}} }\right)}^{{\mathrm{P}}_{2}}+{\mathrm{P}}_{3}{\left(\frac{{\mathrm{FE}}_{t}}{\overline{\mathrm{CTE}} }\right)}^{{\mathrm{P}}_{4}}+(\frac{{\mathrm{P}}_{1}{\mathrm{P}}_{2}}{\overline{\mathrm{FE}} }{\left(\sum_{\mathrm{i}=1}^{{t}+1}\frac{{\mathrm{FE}}_{\mathrm{i}}}{\overline{\mathrm{FE}} }\right)}^{{\mathrm{P}}_{2}-1}+\frac{{\mathrm{P}}_{3}{\mathrm{P}}_{4}}{\overline{\mathrm{CTE}} }{\left(\frac{{\mathrm{FE}}_{t}}{\overline{\mathrm{CTE}} }\right)}^{{\mathrm{P}}_{4}-1}){\mathrm{FE}}_{{t}+1}\right)\right\}$$

$$\mathrm{FF}=\left\{{L}_{0}{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}}{A}_{{t}+1}^{{\varepsilon }_{1}}{{K}}_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}\gamma {\varepsilon }_{1}{\mathrm{FE}}_{{t}+1}^{\rho -1}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }-1}-{L}_{0}\frac{\vartheta {\varepsilon }_{1}}{\varphi }{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta -1}{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}-1}{A}_{{t}+1}^{{\varepsilon }_{1}}{{K}}_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }}\right\}$$

$$\beta \frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}\left\{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right){A}_{{t}+1}{{K}}_{{t}+1}^{\alpha }{\left({I}_{\mathrm{PL}}{L}_{{t}+1}\right)}^{1-\alpha -\gamma }\gamma {\mathrm{CE}}^{\rho }{\mathrm{AC}}_{{t}+1}^{\rho -1}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma }{\rho }-1}\right\}+\frac{\beta \frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}}{\omega {\mathrm{AC}}_{0}{\mathrm{AC}}_{{t}+1}^{\theta }{\left({I}_{\mathrm{TL}}{L}_{{t}+1}\right)}^{\omega }{\mathrm{TY}}_{{t}+1}^{\omega -1}}\left\{\theta {\mathrm{AC}}_{0}{\mathrm{AC}}_{{t}+1}^{\theta -1}{\left({I}_{\mathrm{TL}}{L}_{{t}+1}{\mathrm{TY}}_{{t}+1}\right)}^{\omega }\right\}+\left(\frac{\mathrm{GG}}{\mathrm{HH}}\right)\{\gamma {\varepsilon }_{1}{\mathrm{CE}}^{\rho }{\mathrm{AC}}_{{t}+1}^{\rho -1}{L}_{0}{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}}{A}_{{t}+1}^{{\varepsilon }_{1}}{{K}}_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }-1}\}=\frac{\frac{{C}_{t}^{-\sigma }}{{L}_{t}^{1-\sigma }}}{\omega {\mathrm{AC}}_{0}{\mathrm{AC}}_{t}^{\theta }{\left({I}_{\mathrm{TL}}{L}_{t}\right)}^{\omega }{\mathrm{TY}}_{t}^{\omega -1}}$$

(70)

$$\mathrm{GG}=-\beta \frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}\left\{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)\left({A}_{{t}+1}{{K}}_{{t}+1}^{\alpha }{\left({I}_{\mathrm{PL}}{L}_{{t}+1}\right)}^{1-\alpha -\gamma }\right)\gamma {\mathrm{FE}}_{{t}+1}^{\rho -1}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {}_{1}}{\rho }-1}-\frac{\vartheta }{\varphi }{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta -1}\left({A}_{{t}+1}{{K}}_{{t}+1}^{\alpha }{\left({I}_{\mathrm{PL}}{L}_{{t}+1}\right)}^{1-\alpha -\gamma }\right){\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {}_{1}}{\rho }}-\left({\mathrm{P}}_{0}+{\mathrm{P}}_{1}{\left(\sum_{\mathrm{i}=1}^{{t}+1}\frac{{\mathrm{FE}}_{\mathrm{i}}}{\overline{\mathrm{FE}} }\right)}^{{\mathrm{P}}_{2}}+{\mathrm{P}}_{3}{\left(\frac{{\mathrm{FE}}_{t}}{\overline{\mathrm{CTE}} }\right)}^{{\mathrm{P}}_{4}}+(\frac{{\mathrm{P}}_{1}{\mathrm{P}}_{2}}{\overline{\mathrm{FE}} }{\left(\sum_{\mathrm{i}=1}^{{t}+1}\frac{{\mathrm{FE}}_{\mathrm{i}}}{\overline{\mathrm{FE}} }\right)}^{{\mathrm{P}}_{2}-1}+\frac{{\mathrm{P}}_{3}{\mathrm{P}}_{4}}{\overline{\mathrm{CTE}} }{\left(\frac{{\mathrm{FE}}_{t}}{\overline{\mathrm{CTE}} }\right)}^{{\mathrm{P}}_{4}-1}){\mathrm{FE}}_{{t}+1}\right)\right\}$$

$$\mathrm{HH}=\left\{{L}_{0}{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}}{A}_{{t}+1}^{{\varepsilon }_{1}}{{K}}_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}\gamma {\varepsilon }_{1}{\mathrm{FE}}_{{t}+1}^{\rho -1}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }-1}-{L}_{0}\frac{\vartheta {\varepsilon }_{1}}{\varphi }{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta -1}{\left(1-{\frac{{\mathrm{FE}}_{{t}+1}}{\varphi }}^{\vartheta }\right)}^{{\varepsilon }_{1}-1}{A}_{{t}+1}^{{\varepsilon }_{1}}{{K}}_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}{\left({\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\rho }+{\mathrm{FE}}_{{t}+1}^{\rho }\right)}^{\frac{\gamma {\varepsilon }_{1}}{\rho }}\right\}$$

And the above equation, when there is no fossil fuel energy left to use, will be:

$$\beta \frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}\left\{{A}_{{t}+1}{{K}}_{{t}+1}^{\alpha }{\left({I}_{\mathrm{PL}}{L}_{{t}+1}\right)}^{1-\alpha -\gamma }{\gamma {\rm CE}}{\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\gamma -1}\right\}+\frac{\beta \frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}}{\omega {\mathrm{AC}}_{0}{\mathrm{AC}}_{{t}+1}^{\theta }{\left({I}_{\mathrm{TL}}{L}_{{t}+1}\right)}^{\omega }{\mathrm{TY}}_{{t}+1}^{\omega -1}}\left\{\theta {\mathrm{AC}}_{0}{\mathrm{AC}}_{{t}+1}^{\theta -1}{\left({I}_{\mathrm{TL}}{L}_{{t}+1}{\mathrm{TY}}_{{t}+1}\right)}^{\omega }\right\}+\frac{\beta \frac{{C}_{t}^{-\sigma }}{{L}_{t}^{1-\sigma }}- {\beta }^{2}\frac{{C}_{{t}+1}^{-\sigma }}{{L}_{{t}+1}^{1-\sigma }}(\alpha \left({A}_{{t}+1}{{K}}_{{t}+1}^{\alpha -1}{\left({I}_{\mathrm{PL}}{L}_{{t}+1}\right)}^{1-\alpha -\gamma }\right){\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\gamma }+1-\updelta )}{(\alpha {\varepsilon }_{1}-{\varepsilon }_{2}){L}_{0}{A}_{{t}+1}^{{\varepsilon }_{1}}{{K}}_{{t}+1}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}-1}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{{t}+1}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}{\left({\mathrm{AC}}_{{t}+1}\mathrm{CE}\right)}^{\gamma {\varepsilon }_{1}}}*(\gamma {\varepsilon }_{1}{L}_{0}{A}_{t}^{{\varepsilon }_{1}}{{K}}_{t}^{\alpha {\varepsilon }_{1}-{\varepsilon }_{2}}{I}_{\mathrm{PL}}^{{\varepsilon }_{1}-\alpha {\varepsilon }_{1}-{\varepsilon }_{1}\gamma }{L}_{t}^{{\varepsilon }_{2}-\alpha {\varepsilon }_{1}-\gamma {\varepsilon }_{1}}{\mathrm{CE}}^{\gamma {\varepsilon }_{1}}{\mathrm{AC}}_{t}^{\gamma {\varepsilon }_{1}-1})=\frac{\frac{{C}_{t}^{-\sigma }}{{L}_{t}^{1-\sigma }}}{\omega {\mathrm{AC}}_{0}{\mathrm{AC}}_{t}^{\theta }{\left({I}_{\mathrm{TL}}{L}_{t}\right)}^{\omega }{\mathrm{TY}}_{t}^{\omega -1}}$$

(71)

3.3 Solving the market-based F.O.C

Solving the first-order conditions for households, we get:

$$\left\{ {c_{t} } \right\}: \beta^{t} \frac{{C_{t}^{ - \sigma } }}{{L_{t}^{1 - \sigma } }} = \lambda_{1t}$$

(72)

$$\left\{ {K_{t + 1} } \right\}: \lambda_{1t} = \left( {1 + r_{t + 1} } \right)\lambda_{1t + 1}$$

(73)

$$\left\{ {L_{t + 1} } \right\}: \beta^{t + 1} \frac{{C_{t + 1}^{1 - \sigma } }}{{L_{t + 1}^{2 - \sigma } }} = \lambda_{1t + 1} w_{t + 1} - \lambda_{2t} + \lambda_{2t + 1} \left( {1 + \overline{L}} \right)$$

(74)

And when the population grows endogenously, we have the below F.O.C:

$$\left\{ {K_{t + 1} } \right\}: \lambda_{1t} = \left( {1 + r_{t + 1} } \right)\lambda_{1t + 1} - \lambda_{2t + 1} \varepsilon_{2} L_{0} L_{t + 1}^{{\varepsilon_{2} - \varepsilon_{1} }} Y_{t + 1}^{{\varepsilon_{1} }} K_{t + 1}^{{ - \varepsilon_{2} - 1}}$$

(75)

$$\left\{ {L_{t + 1} } \right\}: \beta^{t + 1} \frac{{C_{t + 1}^{1 - \sigma } }}{{L_{t + 1}^{2 - \sigma } }} = \lambda_{1t + 1} w_{t + 1} - \lambda_{2t} + \lambda_{2t + 1} \left( {1 + L_{0} \left( {\varepsilon_{2} - \varepsilon_{1} } \right)Y_{t + 1}^{{\varepsilon_{1} }} K_{t + 1}^{{ - \varepsilon_{2} }} L_{t + 1}^{{\varepsilon_{2} - \varepsilon_{1} - 1}} } \right)$$

(76)

Updating and substituting Eqs. (72) in (73), (74), (75), (76) we derive the Euler equations for households for both cases:

$$\frac{{C_{t}^{ - \sigma } }}{{L_{t}^{1 - \sigma } }} = \beta \left( {1 + r_{t + 1} } \right)\frac{{C_{t + 1}^{ - \sigma } }}{{L_{t + 1}^{1 - \sigma } }}\,\, \left( {{\text{Population}}\,{\text{ is}}\,{\text{ exogenous}}} \right)$$

(77)

$$\begin{aligned} \beta^{2} \frac{{C_{t + 1}^{1 - \sigma } }}{{L_{t + 1}^{2 - \sigma } }} & = \beta^{2} \frac{{C_{t + 1}^{ - \sigma } }}{{L_{t + 1}^{1 - \sigma } }}w_{t + 1} - \frac{{\left[ {\left( {1 + r_{t} } \right)\beta \frac{{C_{t}^{ - \sigma } }}{{L_{t}^{1 - \sigma } }} - \frac{{C_{t - 1}^{ - \sigma } }}{{L_{t - 1}^{1 - \sigma } }}} \right]}}{{\left[ {\varepsilon_{2} L_{0} L_{t}^{{\varepsilon_{2} - \varepsilon_{1} }} Y_{t}^{{\varepsilon_{1} }} K_{t}^{{ - \varepsilon_{2} - 1}} } \right]}} \\ & \,\,\,\,\,\,\, + \frac{{\left[ {\left( {1 + r_{t + 1} } \right)\beta^{2} \frac{{C_{t + 1}^{ - \sigma } }}{{L_{t + 1}^{1 - \sigma } }} - \beta \frac{{C_{t}^{ - \sigma } }}{{L_{t}^{1 - \sigma } }}} \right]}}{{\left[ {\varepsilon_{2} L_{0} L_{t + 1}^{{\varepsilon_{2} - \varepsilon_{1} }} Y_{t + 1}^{{\varepsilon_{1} }} K_{t + 1}^{{ - \varepsilon_{2} - 1}} } \right]}}\left( {1 + L_{0} \left( {\varepsilon_{2} - \varepsilon_{1} } \right)Y_{t + 1}^{{\varepsilon_{1} }} K_{t + 1}^{{ - \varepsilon_{2} }} L_{t + 1}^{{\varepsilon_{2} - \varepsilon_{1} - 1}} } \right) \\ & \,\,\,\,\,\,\,\left( {{\text{Population}}\,{\text{is}}\,{\text{endogenous}}} \right) \\ \end{aligned}$$

(78)

Solving the first-order conditions for the final good market, we have:

$$\left\{ {{\rm KY}_{t} } \right\}: \alpha {\rm EDA}_{t} {\rm KY}_{t}^{\alpha - 1} {\rm PL}_{t}^{1 - \alpha - \gamma } \left( {{\rm CE}_{t}^{\rho } + {\rm FE}_{t}^{\rho } } \right)^{{\frac{\gamma }{\rho }}} = r_{t} + \delta$$

(79)

$$\left\{ {{\rm PL}_{t} } \right\}:\left( {1 - \alpha - \gamma } \right){\rm EDA}_{t} {\rm KY}_{t}^{\alpha } {\rm PL}_{t}^{ - \alpha - \gamma } \left( {{\rm CE}_{t}^{\rho } + {\rm FE}_{t}^{\rho } } \right)^{{\frac{\gamma }{\rho }}} = w_{t}$$

(80)

$$\left\{ {{\rm FE}_{t} } \right\}:\gamma {\rm EDA}_{t} {\rm KY}_{t}^{\alpha } {\rm PL}_{t}^{1 - \alpha - \gamma } {\rm FE}_{t}^{\rho - 1} \left( {{\rm CE}_{t}^{\rho } + {\rm FE}_{t}^{\rho } } \right)^{{\frac{\gamma }{\rho } - 1}} = P_{{{\text{FE}}_{t} }}$$

(81)

$$\left\{ {{\rm CE}_{t} } \right\}:\gamma {\rm EDA}_{t} {\rm KY}_{t}^{\alpha } {\rm PL}_{t}^{1 - \alpha - \gamma } {\rm CE}_{t}^{\rho - 1} \left( {{\rm CE}_{t}^{\rho } + {\rm FE}_{t}^{\rho } } \right)^{{\frac{\gamma }{\rho } - 1}} = P_{{{\text{CE}}_{t} }}$$

(82)

In the end, the F.O.Cs for the energy sector are:

$$\begin{aligned} \left\{ {{\rm FE}_{t} } \right\} & : P_{0} + P_{1} \left( {\frac{{\mathop \sum \nolimits_{i = 1}^{t} {\rm FE}_{i} }}{{\overline{{\rm FE}} }}} \right)^{{P_{2} }} + P_{3} \left( {\frac{{{\rm FE}_{t} }}{{\overline{{\rm CTE}} }}} \right)^{{P_{4} }} \\ & \,\,\,\,\, + \left( {\frac{{P_{1} P_{2} }}{{\overline{{\rm FE}} }}\left( {\frac{{\mathop \sum \nolimits_{i = 1}^{t} {\rm FE}_{i} }}{{\overline{{\rm FE}} }}} \right)^{{P_{2} - 1}} + \frac{{P_{3} P_{4} }}{{\overline{{\rm CTE}} }}\left( {\frac{{{\rm FE}_{t} }}{{\overline{{\rm CTE}} }}} \right)^{{P_{4} - 1}} } \right){\rm FE}_{t} = P_{FEt} \\ \end{aligned}$$

(83)

$$\left\{ {{\rm TL}_{t} } \right\}: \lambda_{t} \omega {\rm AC}_{0} {\rm AC}_{t}^{\theta } {\rm TY}_{t} \left( {{\rm TL}_{t} {\rm TY}_{t} } \right)^{\omega - 1} = \beta^{t} w_{t}$$

(84)

$$\left\{ {{\rm TY}_{t} } \right\}: \lambda_{t} \omega {\rm AC}_{0} {\rm AC}_{t}^{\theta } {\rm TL}_{t} \left( {{\rm TL}_{t} {\rm TY}_{t} } \right)^{\omega - 1} = \beta^{t} r_{t}$$

(85)

$$\left\{ {{\rm AC}_{t + 1} } \right\}: \lambda_{t} - \lambda_{t + 1} {\rm AC}_{0} {\rm AC}_{t}^{\theta } \left( {{\rm TL}_{t} {\rm TY}_{t} } \right)^{\omega } = \beta^{t + 1} {\rm CE}*P_{{\rm CE}_{t + 1}}$$

(86)

Combining Eqs. (84) and (85), we get:

$${\rm TY}_{t} = \frac{{w_{t} }}{{r_{t} }}{\rm TL}_{t}$$

(87)

Updating and substituting Eqs. (87) into (86), we get another Euler equation for the Energy sector:

$$\frac{{w_{t} }}{{\omega {\rm AC}_{0} {\rm AC}_{t}^{\theta } {\rm TY}_{t} \left( {{\rm TL}_{t} {\rm TY}_{t} } \right)^{\omega - 1} }} - \beta {\rm CE}*P_{{\rm CE}_{t + 1}} = \frac{{\beta w_{t + 1} }}{{\omega {\rm AC}_{t + 1}^{\theta } {\rm TY}_{t + 1} \left( {{\rm TL}_{t + 1} {\rm TY}_{t + 1} } \right)^{\omega - 1} }}{\rm AC}_{t}^{\theta } \left( {{\rm TL}_{t} {\rm TY}_{t} } \right)^{\omega }$$

(88)

Using Eqs. (81) and (83), we derive the price and the amount of fossil fuel energy.

Appendix 4: Altering the population growth

One would argue that the U.S. population grows around 1%, whereas, in the proposed model it converges to zero. Because of this, I have used different parameterization for Eq. (15) $\left( {L_{t + 1} = L_{t} + L_{0} \left( {\frac{{Y_{t} }}{{L_{t} }}} \right)^{{\varepsilon_{1} }} \left( {\frac{{L_{t} }}{{K_{t} }}} \right)^{{\varepsilon_{2} }} } \right)$, to see if 1% growth rate in population is achievable using the current setup. As it is shown in figure below, a population can grow faster in the observed period’s early stages; however, it tends to drop toward the end. While the proposed model is well-fitted in Japan and Western European countries, we need to change the value of the parameters in Eq. (15) ($\varepsilon_{1} = 1.72 \to 1.8$, $\varepsilon_{2} = 2.18 \to 2.1$ and $L_{0} = 2.35 \to 12.35$) to capture the growth rate in population for US.

For the case of the US, I can think of a plausible argument. If we deduct US immigration rate (including immigrants’ descendants, although they might be the U.S.-born), the population growth would be much lower than the current rate. Although the counter argument would be that they still participate in the economy, however, they are not born in that economy but are brought in. The proposed model shows a high growth rate in early stages, so we can think about the entrance of immigrants with the high rate of population which tends to converge to its steady state (Fig.

Fig. 7

Population and economic growth for two different scenarios of population growth. The blue lines show the base model scenario in which population grows at 0.02% rate. The red lines match the U.S. population growth which is about 0.6% on average. The social planner solution has been applied for both models

7).