Emission Taxation and Investment in Cleaner Production: The Case of Differentiated Duopoly

Ouchida, Yasunori; Okamura, Makoto; Orito, Yukiko

doi:10.1007/978-981-13-2363-8_3

Yasunori Ouchida⁴,
Makoto Okamura^4,5 &
Yukiko Orito⁴

Part of the book series: New Frontiers in Regional Science: Asian Perspectives ((NFRSASIPER,volume 34))

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Abstract

Ouchida et al. (Cleaner production technology and optimal emission tax, mimeo. 2017) examine the cleaner production technology of the pollution abatement and specify the technology as a log form. They develop the following three-stage game. In the first stage, a government sets a pollution tax rate. In the second stage, duopolistic firms decide its level of abatement investment. In the third stage, the firms engage in Cournot competition in a homogeneous product market. They obtain the explicit solution of the perfect Nash equilibrium of the game. No previous studies have derived the explicit solution of this three-stage game. By incorporating differentiated product markets into the third stage of the game, we generalize Ouchida et al. (Cleaner production technology and optimal emission tax, mimeo. 2017). We derive the explicit equilibrium values of the optimal tax rate, the level of the abatement investment, and the outputs.

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Notes

1.
For example, see Katsoulacos and Xepapadeas (1996), Petrakis and Xepapadeas (1999, 2003, Section 3), Chiou and Hu (2001), and Ben Youssef (2009, 2010, 2011).
2.
Katsoulacos and Xepapadeas (1996, Section 4), Chiou and Hu (2001), Ben Youssef (2009, 2010, 2011), and Tsai et al. (2015) employ a quadratic cost function.
3.
For example, see Chiou and Hu (2001), Petrakis and Xepapadeas (1999, 2003, Section 3), Ben Youssef (2010), Cato (2011), and Hattori (2013).
4.
The assumptions of d = 1 and $\bar {a}=1$ are consistent with Wang and Wang’s (2009) model. In addition, $\bar {a}=1$ is identical to the value used by Ben Youssef (2010), Bárcena-Ruiz and Campo (2012), Ouchida and Goto (2014, 2016a,b), Liu et al. (2015), Pal and Saha (2015), and Moner-Colonques and Rubio (2016).
5.
The second derivatives are $\partial ^{2}\pi _{i}/\partial a_{i}^{2}=\frac {4t^{2}}{(4+b)^{2}(4-b)^{2}}-\frac {1}{(1-a_{i})^{2}}$. Noting that the value of a _i is in [0, 1), if at least t < 15∕2, the second-order condition (SOC) is satisfied.
6.
Consequently, none of output, consumer surplus, or total tax revenue depend on t.
7.
The SOCs are satisfied because $d^{2}W_{1}^{\mathrm {N}}(t)/dt^{2}<0$ and $d^{2}W_{2}^{\mathrm {N}}(t)/dt^{2}<0$ for all b ∈ [0, 1].
8.
Subscript N stands for the non-cooperative environmental investment, and subscripts N1 and N2 express the equilibrium values realized in Region I and Regions (II and III), respectively, in Fig. 3.2. Because $\max \{t_{\mathrm {N1}}, t_{\mathrm {N2}} \}<15/2$ for all b ∈ [0, 1], the SOC at stage 2 is satisfied.
9.
The SOC is $\partial ^{2}\Pi _{i}/\partial a_{i}^{2}=\frac {4t^{2}}{(4+b)^{2}(4-b)^{2}}-\frac {1}{(1-a_{i})^{2}}+4[\partial q_{j}/\partial a_{i}]^{2}<0$. For details, see footnote 13.
10.
In fact, $(1-[(2A-\sqrt {X})/4t])-(1-[(A-\sqrt {Z})/2t])=(\sqrt {X}-2\sqrt {Z})/4t>0$.
11.
None of output, consumer surplus, or tax revenue depends on t.
12.
The SOC is satisfied.
13.
Subscript C represents cooperative environmental investment, and subscripts C1 and C2 the equilibrium values realized in Regions (I and II) and Region III, respectively, in Fig. 3.2. Under precommitment of t _C ∈{t _C1, t _C2}, the SOC at stage 2 is satisfied.

References

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Authors and Affiliations

Department of Economics, Hiroshima University, Higashi-Hiroshima, Japan
Yasunori Ouchida, Makoto Okamura & Yukiko Orito
Economics Department, Gakushuin University, Toshima-ku, Japan
Makoto Okamura

Authors

Yasunori Ouchida
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Makoto Okamura
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Yukiko Orito
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Corresponding author

Correspondence to Yasunori Ouchida .

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Editors and Affiliations

Faculty of Economics, Chukyo University, Nagoya, Aichi, Japan
Keiko Nakayama
Toyohashi, Aichi, Japan
Yuzuru Miyata

Appendices

Appendices provide supplementary explanations of Sects. 3.3.2 and 3.4.2.

1.1 II Appendix A

The following results show the subgame equilibrium of output, consumer surplus, profit, total emissions, total tax revenue, and environmental damage.

$$\displaystyle \begin{aligned} \begin{array}{rcl} q(t)&=& \left\{ \begin{array}{l} \displaystyle \frac{A-t}{4+b}~~~~~~~~~~if~~\displaystyle X<0~\mathrm{or}~t \le \frac{2A-\sqrt{X}}{4},~(X\ge 0)\\ \displaystyle \frac{2A+\sqrt{X}}{4(4+b)}~~~~if~~\displaystyle t > \frac{2A-\sqrt{X}}{4},~(X\ge 0), {} \end{array} \right. \end{array} \end{aligned} $$

(3.20)

$$\displaystyle \begin{aligned} \begin{array}{rcl} CS(t)&=& \left\{ \begin{array}{l} \displaystyle (1+b)\Biggl( \frac{A-t}{4+b} \Biggr)^{2}~~~~~~~~~~if~~\displaystyle X<0~\mathrm{or}~t \le \frac{2A-\sqrt{X}}{4},~(X\ge 0)\\ \displaystyle (1+b)\Biggl( \frac{2A+\sqrt{X}}{4(4+b)} \Biggr)^{2}~~~~if~~\displaystyle t > \frac{2A-\sqrt{X}}{4},~(X\ge 0), {} \end{array} \right. \end{array} \end{aligned} $$

(3.21)

$$\displaystyle \begin{aligned} \begin{array}{rcl}\qquad\pi(t)&=& \left\{ \begin{array}{l} \displaystyle 2\Biggl( \frac{A-t}{4+b} \Biggr)^{2}~~~~~~~~~~if~~\displaystyle X<0~\mathrm{or}~t \le \frac{2A-\sqrt{X}}{4},~(X\ge 0)\\ \displaystyle 2\Biggl(\frac{2A+\sqrt{X}}{4(4+b)} \Biggr)^2+\log\Biggl(\frac{2A-\sqrt{X}}{4t} \Biggr)~~~~if~~\displaystyle t > \frac{2A-\sqrt{X}}{4},~(X\ge 0), {} \end{array} \right. \end{array} \end{aligned} $$

(3.22)

$$\displaystyle \begin{aligned} \begin{array}{rcl} E(t)&=& \left\{ \begin{array}{l} \displaystyle \frac{2(A-t)}{4+b}~~~~~~~~~~~if~~\displaystyle X<0~\mathrm{or}~t \le \frac{2A-\sqrt{X}}{4},~(X\ge 0)\\ \displaystyle \frac{(4-b)(4+b)}{8t}~~~~if~~\displaystyle t > \frac{2A-\sqrt{X}}{4},~(X\ge 0), {} \end{array} \right. \end{array} \end{aligned} $$

(3.23)

$$\displaystyle \begin{aligned} \begin{array}{rcl} T(t)&=&tE(t)= \left\{ \begin{array}{l} \displaystyle \frac{2t(A-t)}{4+b}~~~~~~~~~~if~~\displaystyle X<0~\mathrm{or}~t \le \frac{2A-\sqrt{X}}{4},~(X\ge 0)\\ \displaystyle \frac{(4-b)(4+b)}{8}~~~~if~~\displaystyle t > \frac{2A-\sqrt{X}}{4},~(X\ge 0), \end{array} \right. \end{array} \end{aligned} $$

(3.24)

$$\displaystyle \begin{aligned} \begin{array}{rcl} D(E(t))&=& \left\{ \begin{array}{l} \displaystyle 2\Biggl( \frac{A-t}{4+b} \Biggr)^{2}~~~~~~~~~~if~~\displaystyle X<0~\mathrm{or}~t \le \frac{2A-\sqrt{X}}{4},~(X\ge 0)\\ \displaystyle \frac{(4-b)^{2}(4+b)^{2}}{128t^{2}}~~~~if~~\displaystyle t > \frac{2A-\sqrt{X}}{4},~(X\ge 0). {} \end{array} \right. \end{array} \end{aligned} $$

(3.25)

1.2 II Appendix B

The following results denote the subgame equilibrium of output, consumer surplus, profit, total emissions, total tax revenue, and environmental damage.

$$\displaystyle \begin{aligned} \begin{array}{rcl} q(t)&=& \left\{ \begin{array}{l} \displaystyle \frac{A-t}{4+b}~~~~~~~if~~\displaystyle Z<0~\mathrm{or}~t \le \frac{A-\sqrt{Z}}{2},~(Z\ge 0)\\ \displaystyle \frac{A+\sqrt{Z}}{2(4+b)}~~~~if~~\displaystyle t > \frac{A-\sqrt{Z}}{2},~(Z\ge 0), {} \end{array} \right. \end{array} \end{aligned} $$

(3.26)

$$\displaystyle \begin{aligned} \begin{array}{rcl} CS(t)&=& \left\{ \begin{array}{l} \displaystyle (1+b)\Biggl( \frac{A-t}{4+b} \Biggr)^{2}~~~~~~~if~~\displaystyle Z<0~\mathrm{or}~t \le \frac{A-\sqrt{Z}}{2},~(Z\ge 0)\\ \displaystyle (1+b)\Biggl( \frac{A+\sqrt{Z}}{2(4+b)} \Biggr)^{2}~~~~if~~\displaystyle t > \frac{A-\sqrt{Z}}{2},~(Z\ge 0), {} \end{array} \right. \end{array} \end{aligned} $$

(3.27)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \pi(t)&=& \left\{ \begin{array}{l} \displaystyle 2\Biggl( \frac{A-t}{4+b} \Biggr)^{2}~~~~~~~~~~if~~\displaystyle Z<0~\mathrm{or}~t \le \frac{A-\sqrt{Z}}{2},~(Z\ge 0)\\ \displaystyle 2\Biggl(\frac{2A+\sqrt{Z}}{2(4+b)} \Biggr)^2+\log\Biggl(\frac{A-\sqrt{Z}}{2t} \Biggr)~~~~if~~\displaystyle t > \frac{A-\sqrt{Z}}{2},~(Z\ge 0), {} \end{array} \right. \end{array} \end{aligned} $$

(3.28)

$$\displaystyle \begin{aligned} \begin{array}{rcl} E(t)&=& \left\{ \begin{array}{l} \displaystyle \frac{2(A-t)}{4+b}~~~~~~~if~~\displaystyle Z<0~\mathrm{or}~t \le \frac{A-\sqrt{Z}}{2},~(Z\ge 0)\\ \displaystyle \frac{(4+b)}{2t}~~~~if~~\displaystyle t > \frac{A-\sqrt{Z}}{2},~(Z\ge 0), {} \end{array} \right. \end{array} \end{aligned} $$

(3.29)

$$\displaystyle \begin{aligned} \begin{array}{rcl} T(t)&=&tE(t)= \left\{ \begin{array}{l} \displaystyle \frac{2t(A-t)}{4+b}~~~~~if~~\displaystyle Z<0~\mathrm{or}~t \le \frac{A-\sqrt{Z}}{2},~(Z\ge 0)\\ \displaystyle \frac{(4+b)}{2}~~~~~~if~~\displaystyle t > \frac{A-\sqrt{Z}}{2},~(Z\ge 0), \end{array} \right. \end{array} \end{aligned} $$

(3.30)

$$\displaystyle \begin{aligned} \begin{array}{rcl} D(E(t))&=& \left\{ \begin{array}{l} \displaystyle 2\Biggl( \frac{A-t}{4+b} \Biggr)^{2}~~~~~if~~\displaystyle Z<0~\mathrm{or}~t \le \frac{A-\sqrt{Z}}{2},~(Z\ge 0)\\ \displaystyle \frac{(4+b)^{2}}{8t^{2}}~~~~~if~~\displaystyle t > \frac{A-\sqrt{Z}}{2},~(Z\ge 0). {} \end{array} \right. \end{array} \end{aligned} $$

(3.31)

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Ouchida, Y., Okamura, M., Orito, Y. (2019). Emission Taxation and Investment in Cleaner Production: The Case of Differentiated Duopoly. In: Nakayama, K., Miyata, Y. (eds) Theoretical and Empirical Analysis in Environmental Economics. New Frontiers in Regional Science: Asian Perspectives, vol 34. Springer, Singapore. https://doi.org/10.1007/978-981-13-2363-8_3

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DOI: https://doi.org/10.1007/978-981-13-2363-8_3
Published: 18 May 2019
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-2362-1
Online ISBN: 978-981-13-2363-8
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Emission Taxation and Investment in Cleaner Production: The Case of Differentiated Duopoly

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Appendices

1.1 II Appendix A

1.2 II Appendix B

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