One-Parameter GHG Emission Policy with R&D-Based Growth

Abstract

This document examines the GHG emission policy of regions which use land, labor and emitting inputs in production and enhance their productivity by devoting labor to R&D, but with different endowments and technology. The regions also have different impacts on global pollution. The problem is to organize common emission policy, if the regions cannot form a federation with a common budget and the policy parameters must be uniform for all regions. The results are the following. If a self-interested central planner allocates emission caps in fixed proportion to past emissions (i.e. grandfathering), then it establishes the Pareto optimum, decreasing emissions and promoting R&D and economic growth.

Appendices

Appendix 1

Region j maximizes (21) by (l _j ,m _j ) subject to (6), given m. It is equivalent to maximize

$$E\int^{\infty}_{T}a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta}e^{-\rho ({{t}}-{T})}d{{t}} $$

by (l _j ,m _j ) subject to (6).

Assume for a while that energy input m _j is kept constant. The value of this maximization is

$$\begin{aligned} \varPi^j(\gamma_j,m_j,T) &=\max_{\text{$l_{j}$ s.t. (6)}} E\int ^{\infty}_{T}a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta}e^{-\rho ({{t}}-{T})}d{{t}}. \end{aligned}$$

(32)

Let us denote Π ^j=Π ^j(γ _j ,m _j ,T) and $\widetilde{\varPi}^{j}=\varPi^{j}(\gamma_{j}+1,m_{j},T)$. The Bellman equation corresponding to the optimal program (32) is given by (cf. Dixit and Pindyck 1994)

$$\begin{aligned} &\rho\varPi^j=\max_{l_j,m_j}\varPsi(l_j,m_j,\gamma_j,T),\quad\text {where} \\ &\quad\varPsi(l_j,m_j,\gamma_j,T)=a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta }+\bigl(\widetilde{\varPi^j}-\varPi^j\bigr){\lambda_j}(L_j-l_j). \end{aligned}$$

(33)

Noting (4), this leads to the first-order condition

$$\begin{aligned} \frac{\partial\varPsi}{\partial l_j} &=a_j^{\gamma _j}f^j_l(l_j,m_j)m_j^{-\beta}-{\lambda_j}\bigl(\widetilde{\varPi^j}-\varPi ^j\bigr) \\ &=\frac{1}{l_j}a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta}\biggl[1-\xi ^j\biggl(\frac{l_j}{m_j}\biggr)\biggr]-{\lambda_j}\bigl(\widetilde{\varPi ^j}-\varPi^j\bigr)=0. \end{aligned}$$

(34)

To solve the dynamic program (32), assume that the value of the program, Π ^j, is in fixed proportion ϑ _j >0 to instantaneous utility at the optimum. Noting (4), this implies

$$\begin{aligned} \begin{aligned} &\varPi^j(\gamma_j,m_j,T)={\vartheta_j}a_j^{\gamma _j}f^j(l_j^*,m_j)m_j^{-\beta}\quad\text{with}\\ &\frac{\partial\varPi^j}{\partial m_j}=\varPi^j\biggl[\frac {f_m^j(l_j,m_j)}{f^j(l_j,m_j)}-\frac{\beta}{m_j}\biggr]=\frac{\varPi ^j}{m_j}\biggl[\xi^j\biggl(\frac{l_j}{m_j}\biggr)-\beta\biggr], \end{aligned} \end{aligned}$$

(35)

where $l_{j}^{*}$ is the optimal value of the control variable l _j . This implies

$$\begin{aligned} (\widetilde{\varPi^j}-\varPi^j)/{\varPi^j}={a_j}-1. \end{aligned}$$

(36)

Inserting (35) and (36) into the Bellman equation (33) yields

$$\begin{aligned} 1/{\vartheta_j}=\rho+(1-{a_j}){\lambda_j} (L_j-l_j^*)>0. \end{aligned}$$

(37)

Inserting (35), (36) and (37) into (34), and noting (ξ ^j)′>0 yield (12):

$$\begin{aligned} 0 &=\vartheta_j\frac{l_j}{\varPi^j}\frac{\partial\varPsi}{\partial l_j}=\underbrace{a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta}\frac{\vartheta _j}{\varPi^j}}_{=1}\biggl[{{\alpha_j}}-\xi^j\biggl(\frac{l_j}{m_j}\biggr)\biggr] -\biggl(\underbrace{\frac{\widetilde{\varPi^j}}{\varPi ^j_j}}_{={a_j}}-1\biggr){\lambda_j} l_j\vartheta_j \\ &={{\alpha_j}}-\xi^j\biggl(\frac{l_j}{m_j}\biggr)-\frac {({a_j}-1){\lambda_j} l_j}{\rho+(1-{a_j}){\lambda_j} (L_j-l_j^*)}. \end{aligned}$$

(38)

From (8), (32) and (37) it follows that

$$\begin{aligned} \varUpsilon^j &=\max_{\text{$l_{j}$ s.t. (6)}}E\int^{\infty }_{T}a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta}m^{-\delta}e^{-\rho(\theta -{T})}d\theta \\ &=m^{-\delta}E\int^{\infty}_{T}a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta }e^{-\rho(\theta-{T})}d\theta=m^{-\delta}\varPi^j(\gamma_j,m_j,T). \end{aligned}$$

(39)

Results (35), (38) and (39) lead to Proposition 1.

Appendix 2

Given (1), (3), (4) and (12), it then holds true that

$$\begin{aligned} \begin{aligned} &\rho+(\underbrace{1-{a_j}}_{-}){\lambda_j} (\underbrace {L_j-l_j^{L}}_{+})\underbrace{\xi^j}_{\in(0,1)}>\rho+(1-{a_j}){\lambda _j} (L_j-l_j^{L})>0,\\ &\frac{({a_j}-1){\lambda_j} l_j^{L}}{\rho+(1-{a_j}){\lambda_j} (L_j-l_j^{L})}<\alpha_j-\beta<{{\alpha_j}}<1,\quad\rho +(1-{a_j}){\lambda_j} L_j>0. \end{aligned} \end{aligned}$$

(40)

Noting (1), (12) and (40) yield

$$\begin{aligned} &\frac{d}{dl_j^L}\log\biggl[\frac{({a_j}-1){\lambda_j} l_j^L}{\rho +(1-{a_j}){\lambda_j} (L_j-l_j^L)}\biggr]=\frac{1}{l_j^L}\biggl[1-\smash {\underbrace{\frac{({a_j}-1){\lambda_j} l_j^L}{\rho+(1-{a_j}){\lambda _j} (L_j-l_j^L)}}_{\in(0,1)}}\biggr]>0 \end{aligned}$$

and

$$\begin{aligned} &\frac{d}{dl_j^L}\biggl[\frac{({a_j}-1){\lambda_j} l_j^L}{\rho +(1-{a_j}){\lambda_j} (L_j-l_j^L)}\biggr]>0. \end{aligned}$$

Noting this and differentiating the left-hand equation in (12), one obtains

$$\begin{aligned} \underbrace{\frac{d}{dl_j^L}\biggl[\frac{({a_j}-1){\lambda_j} l_j^L}{\rho+(1-{a_j}){\lambda_j} (L_j-l_j^L)}\biggr]}_{+}dl_j^L+d\beta=0, \end{aligned}$$

and $dl_{j}^{L}/d\beta<0$. Given (1), this implies $dz_{j}^{L}/d\beta =-dl_{j}^{L}/d\beta>0$. Finally, differentiating the right-hand equation in (12), and noting (12), one obtains

$$\begin{aligned} &\frac{dm_j^L}{d\beta}=\frac{m_j^L}{l_j^L}\biggl[\underbrace{\frac {dl_j^L}{d\beta}}_{-}-\underbrace{\frac{m_j^L}{{{(\xi^j)'}}}}_{+}\biggr]<0. \end{aligned}$$

Appendix 3

Region j maximizes (21) by l _j subject to (6), given (m,m _j ,R,R _j ). It is equivalent to maximize the expected value of the flow of output for region j,

$$E\int^{\infty}_{T}a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta}e^{-\rho(\theta -{T})}d\theta, $$

by l _j subject to (6), given m _j . The value of this maximization is

$$\begin{aligned} &\varGamma_j^j(\gamma_j,m_j,T)=\max_{\text{$l_{j}$ s.t. (6)}} E\int^{\infty}_{T}a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta}e^{-\rho(\theta -{T})}d\theta. \end{aligned}$$

(41)

Denote Γ ^j=Γ ^j(γ _j ,m _j ,T) and $\widetilde{\varGamma}^{j}=\varGamma^{j}(\gamma_{j}+1,m_{j},T)$. The Bellman equation corresponding to the optimal program (41) is

$$\begin{aligned} &\rho\varGamma^j=\max_{l_j}\varPsi(l_j,\gamma_j,m_j,R-R_j,T),\quad\text {where} \\ &\quad\varPsi(l_j,\gamma_j,m_j,T)=a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta }+{\lambda_j} (L_j-l_j)\bigl(\widetilde{\varGamma}^j-\varGamma^j\bigr). \end{aligned}$$

(42)

Noting (4), this leads to the first-order condition

$$\begin{aligned} \frac{\partial\varPsi}{\partial l_j} &=a_j^{\gamma _j}f^j_l(l_j,m_j)m_j^{-\beta}-{\lambda_j}\bigl(\widetilde{\varGamma }^j-\varGamma^j\bigr) \\ &=\frac{1}{l_j}a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta}\biggl[{\alpha _j}-\xi^j\biggl(\frac{l_j}{m_j}\biggr)\biggr] -{\lambda_j}\bigl(\widetilde{\varGamma}^j-\varGamma^j\bigr)=0. \end{aligned}$$

(43)

To solve the dynamic program (41), assume that the value of the program, Γ ^j, is in fixed proportion ϑ _j >0 to instantaneous utility:

$$\begin{aligned} \varGamma^j(\gamma_j,m_j,T)={\vartheta_j}a_j^{\gamma _j}f^j(l_{j}^{*},m_j)m_j^{-\beta}, \end{aligned}$$

(44)

where $l_{j}^{*}$ is the optimal value of the control variable l _j . This implies

$$\begin{aligned} &(\widetilde{\varGamma}^j-\varGamma^j)/{\varGamma^j}={a_j}-1. \end{aligned}$$

(45)

Inserting (44) and (45) into the Bellman equation (42) yields

$$\begin{aligned} 1/{\vartheta_j}=\rho+(1-{a_j}){\lambda_j} (L_j-l_j)>0. \end{aligned}$$

(46)

Plugging this (46) into (44), one obtains

$$\begin{aligned} \varGamma^j(\gamma_j,m_j,T)=\frac{a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta }}{\rho+(1-{a_j}){\lambda_j} (L_j-l_j^*)}, \end{aligned}$$

(47)

where $l_{j}^{*}$—the optimal value of the control variable l _j —is taken as given.

Inserting (47), (45) and (46) into (43), one obtains (23):

$$\begin{aligned} 0 &=\vartheta_j\frac{l_j}{\varGamma^j}\frac{\partial\varPsi}{\partial l_j}=\underbrace{a_j^{\gamma_j}f^j(l_j,m_j)m_j^{-\beta}\frac{\vartheta _j}{\varGamma^j}}_{=1} \biggl[{{\alpha_j}}-\xi^j\biggl(\frac{l_j}{m_j}\biggr)\biggr] -\biggl(\underbrace{\frac{\widetilde{\varGamma}^j}{\varGamma ^j}}_{={a_j}}-1\biggr){\lambda_j} l_j\vartheta_j \\ &={{\alpha_j}}-\xi^j\biggl(\frac{l_j}{m_j}\biggr)-\frac {({a_j}-1){\lambda_j} l_j}{\rho+(1-{a_j}){\lambda_j} (L_j-l_j)}. \end{aligned}$$

Noting (41) and (47), the expected utility (21) becomes (22):

$$\begin{aligned} &\varTheta(m_j,m,R_j,R) \\ &\quad=m^{-\delta}E\int^{\infty}_{T}\bigl[a_j^{\gamma _j}f^j(l_j,m_j)m_j^{-\beta}+R-R_j\bigr]e^{-\rho(\theta-{T})}d\theta \\ &\quad=m^{-\delta}\biggl[E\int^{\infty}_{T}a_j^{\gamma _j}f^j(l_j,m_j)m_j^{-\beta}e^{-\rho(\theta-{T})}d\theta +\int^{\infty}_{T}(R-R_j)e^{-\rho(\theta-{T})}d\theta\biggr] \\ &\quad=m^{-\delta}\biggl[E\int^{\infty}_{T}a_j^{\gamma _j}f^j(l_j,m_j)m_j^{-\beta}e^{-\rho(\theta-{T})}d\theta+\frac {R-R_j}{\rho}\biggr] \\ &\quad=m^{-\delta}\bigl[\varGamma^j(\gamma_j,m_j,T)+(R-R_j)/{\rho}\bigr], \\ &\frac{\partial\varTheta}{\partial m_j} =\frac{\varGamma^j}{m^{\delta }}\biggl[\frac{f^j_m(l_j,m_j)}{f^j(l_j,m_j)}-\frac{\beta}{m_j}\biggr] =\frac{\varGamma^j}{m^{\delta}m_j}\biggl[\xi^j\biggl(\frac {l_j}{m_j}\biggr)-\beta\biggr], \\ &{\partial\varTheta}/{\partial M} =-\delta m^{-\delta-1}\bigl[\varGamma ^j+(R-R_j)/{\rho}\bigr],\qquad-{\partial\varTheta}/{\partial R_j}={\partial\varTheta}/{\partial R}={m^{-\delta}}/{\rho}. \end{aligned}$$