Online Appendix for

“Environmental Innovation and Policy Harmonization in International Oligopoly”∗ Keisuke Hattori†

Appendix: Convex environmental damages In this appendix, we extend the model by considering convex environmental damages and compare firm incentives for environmental R&D. In our main paper, we assume that the environmental damages are linear in emissions. We show that the results obtained in the main paper qualitatively hold for the convex environmental damages. The net benefit of country i is modified as follows:   πi + ti (e − i )yi − δi [(e − i )yi + γ (e − j ) yj ]2     π − δ (a + γ a )2 i i i j Wi =  ¯ πi + t (e − i ) yi − δi [(e − i )yi + γ (e − j ) yj ]2     πi − δi (¯ a+γa ¯)2

under regime NT, under regime NQ, under regime CT, under regime CQ,

In the above formulation, environmental damages are quadratic in the total emissions including transboundary emissions from the foreign country. The parameter δi indicates the intensity of environmental damages in country i. In the case of quadratic environmental damages, the mathematical derivation of equilibrium and its comparisons are too complex. Therefore, we simply display variables in a symmetric equilibrium and present graphical comparison results of firm incentives in each regime.1 The model settings are the same as those in the main paper except for the formulation of environmental damages. Subsequently, the equilibrium levels of emission tax, emission cap, ∗

Keisuke Hattori (2013). Environmental innovation and policy harmonization in international oligopoly. Environment and Development Economics, 18, pp 162-183. Please see the acknowledgements in the main paper. † Faculty of Economics, Osaka University of Economics, 2-2-8, Osumi, Higashiyodogawa-ku, Osaka 5338533, Japan. Email: [email protected] 1 The results and graphs presented here are generated by the mathematical software Mathematica 7. The detailed calculations and the accompanying Mathematica code are available upon request.

1

output and R&D incentives of firms in each regime, evaluated at a symmetric equilibrium, are derived by 2 (e − )2 δ (1 + γ) (2 − γ) − 1 2 [ ], y N T = , 2 2 (e − ) 2δ (e − ) (1 + γ) (2 − γ) + 5 5 + 2δ (e − ) (1 + γ) (2 − γ) 16eδ [12 + γ + 4γ 2 + 2e2 δ (2 − γ) (1 + γ) (4 − γ (3 − 2γ))] F INT = , [1 + 2e2 δ (2 − γ) (1 − γ)] [5 + 2e2 δ (2 − γ) (1 + γ)]3 e− 1 aN Q = , yN Q = , 2 3 + 2δ(e − )2 (1 + γ) 3 + 2 (e − ) δ (1 + γ) 2eδ [2 − γ + 4e2 δ (1 + γ) (2 + γ + e2 δ(2 − γ)(1 + γ))] , F INQ = [1 + 2e2 δ(1 − γ)] [3 + 2e2 δ (1 + γ)]3 1 + 2δ(e − )2 (1 + γ)2 1 CT [ ], t = y CT = [ 2] , 2 2 (e − ) 2 + δ(e − ) (1 + γ) 2 2 + δ (e − )2 (1 + γ)2 2 + 7e2 δ(1 + γ)2 + 2e4 δ 2 (1 + γ)4 F I CT = , 4e [2 + e2 δ(1 + γ)2 ]3 e− 1 ], aCQ = [ y CQ = [ 2 2 ], 2 2 + δ (e − ) (1 + γ) 2 2 + δ (e − )2 (1 + γ)2 [ ] 2 + 3e2 δ(1 + γ)2 1 + e2 δ(1 + γ)2 F I CQ = . [ ]3 4e 2 + e2 δ (1 + γ)2 tN T =

Note that the equilibrium output in each regime is necessarily positive. Unlike the case of linear environmental damages, in the case of convex environmental damages, marginal environmental damage becomes small as output decreases. Therefore, the government has no incentive to impose very stringent tax or quantity regulations that restrain production to zero. The four panels of Figure 3 depict the comparison of firm incentives under different regimes.2 The comparison results in the case of convex damages are qualitatively equivalent to those in the case of linear damages, except for the comparison of firm incentives between non-cooperative and cooperative tax regulations (the bottom-left panel of Figure 3). We can observe from the figure that F I CT > F I N T holds for both extremely small δ and large δ. The intuition of the former is similar to the case of linear environmental damages, but that of the latter needs to be explained. When δ is significant, the cooperative tax rate is higher and thus a direct cost-reducing effect of innovation is also more significant than those under a non-cooperative tax regime. For significant environmental damages, the direct effect dominates the strategic effect of innovation, which leads to F I N T < F I CT . In the case of linear damages, for significant environmental damages, the cooperative tax is so stringent 2

In the figure, ζi can be obtained by equating firm incentives when γ = 0 that is to be compared. For

example, ζ1 is the value of δ such that F I N T |γ=0 = F I N Q |γ=0 .

2

Figure 3: Comparison of firm incentives in the case of convex environmental damages

that it restrains production to zero, which is represented by the area “ruled out” in the left panel of Figure 2 in the main paper.

Proof of Proposition 4 From (6) and (12), we have F I N T − F I CT = ηd2 + κd + µ, where √ e (−71 + 66γ + 41γ 2 ) 20 10 − 33 η= R 0 ⇔ γ R γˆ ≡ ≈ 0.738, 100 41 309 + 53γ 1 κ= > 0, µ = − < 0. 400 16e

3

(A1)

In the case of γ ≥ γˆ (i.e., η ≥ 0), the condition for (A1)> 0 is d < dS or d > dL , where dS and dL are the smaller and the larger solutions for (A1)= 0. dS and dL are derived as follows: 50 ], dS = [ √ 2 e 309 + 53γ − (259) + 3γ (19718 + 6403γ) 50 ], dL = [ √ 2 e 309 + 53γ + (259) + 3γ (19718 + 6403γ) where dS < 0 < dL < 1/e holds for γ > γˆ . Thus, in the case of γ > γˆ , d > dL ensures (A1)> 0. In the case of γ < γˆ (i.e., η < 0), the condition for (A1)> 0 is d0S < d < d0L , where d0S and d0L are the smaller and larger solutions for (A1)= 0. 50 ], d0S = [ √ 2 e 309 + 53γ + (259) + 3γ (19718 + 6403γ) 50 ], d0L = [ √ 2 e 309 + 53γ − (259) + 3γ (19718 + 6403γ) where 0 < d0S < 1/e < d0L holds for γ < γˆ . From Assumption CT, we exclude 1/e < d. Thus, in the case of γ < γˆ , d > d0S ensures (A1)> 0. The fact dL = d0S ≡ Ψ proves F I N T R F I CT ⇔ d R Ψ. From the fact that Ψ is strictly decreasing in γ, we can prove that F I N T > F I CT is more likely to hold as γ becomes large.

Proof of Proposition 5 From (9) and (15), we obtain F I N Q − F I CQ = νd2 + ξd + $,

(A2)

where ν=

e [71 − 9γ (2 + γ)] > 0, 144

ξ=

1 > 0, 9

$ = −

1 < 0. 16e

Thus, the condition for (A2)> 0 is d < d00S or d > d00L , where d00S and d00L are the smaller and the larger solutions for (A2) = 0. √ 8 + (19 − 9γ) (37 + 9γ) d00S = − < 0, e [71 − 9γ (2 + γ)]

√ d00L = 4

(19 − 9γ) (37 + 9γ) − 8 > 0, e [71 − 9γ (2 + γ)]

where 0 < d00L ≡ Ω < 1/[(1 + γ)e] for all γ ∈ [0, 1]. Thus, we have F I N Q R F I CQ ⇔ d R Ω. Differentiating Ω in γ yields 729 (1 + γ) dΩ = > 0, dγ eΛ (8 + Λ)2 where Λ ≡

√

(19 − 9γ)(37 + 9γ) > 0. Thus, we can prove that F I N Q > F I CQ is more

likely to hold as γ becomes small.

5