Chapter 3 The Model
3.4 Comparative Static Analysis
This section applies comparative static analysis to discuss the effects impacted by the governments’ environmental policy and R&D spillover rate under non-cooperative R&D and cooperative R&D regimes.
3.4.1 The Comparative Static Analysis with Respect to Environmental Tax
The two governments Ni cooperatively impose the same environmental tax e on Fi. This subsection will discuss the effects on quantity outputs, profits, and emission abatement levels of the two firms, the market price and consumer surplus of the consumption nation, and social welfare of the two nations for changing the environmental tax under non-cooperation R&D and cooperation R&D regimes.
Case 1. Non-cooperative R&D Regime
otherwise
23 employed to discuss how the change of environmental tax will affect social welfare in the regime of non-cooperative R&D.
3 first (second) term of numerator on the right hand side equals to zero, then
)]
24
Compare Eqs. (30.1) and (30.2), we have d1NL d1NR.
Ⅰ. If 0d1N d1NL, SW1N/e0.
In this case, the relatively small pollution damage d1N will cause both the first term and the second term of numerator on the right hand side in Eq. (28) to be negative, therefore, SW1N/e0 . The government of N1 should reduce the environmental tax on F1 to raise the social welfare.
Ⅱ. If d1NL d1N d1NR, the result of SW1N/e is ambiguous.
In this case, the pollution damage to the environment is intermediate, d1N will cause the first term of numerator on the right hand side in Eq. (28) to be negative on the one hand, but the second term to be positive on the other hand. The sign of
e SWN
1 / cannot be determined exactly.
Ⅲ. The case of d1N d1NR does not existed.
Compare Eq. (30.1) with Eq. (13), it is easy to check that d1NR d1Nmax and then
max 1 1
N
N d
d . This high pollution damage parameter will actually deteriorate the social welfare, therefore, the government of N1 does not allow F1 to produce that kind of product.
Similarly, because the first term of numerator on the right hand side in Eq. (29) is obviously positive, let d2NL be the pollution damage parameter that causes the second term equals to zero, then
) )(1 (1
3
) 2 ( 2
2
m
dNL e (31)
Ⅰ. If 0d2N d2NL, the result of SW2N/e is ambiguous.
In this case, the first term of numerator on the right hand side in Eq. (29) is always positive, but smaller d2N will cause the second term to be negative. The sign
25 Case2. Cooperative R&D Regime
otherwise
26 employed to discuss how the change of environmental tax will affect social welfare in the regime of cooperative R&D.
3 first (second) term of numerator on the right hand side equals to zero, then
]
In this case, the relatively small pollution damage d1C will cause both the first term and the second term of numerator on the right hand side in Eq. (33) to be negative, therefore, SW1C/e0 . The government of N1 should reduce the
27
environmental tax on F1 to raise the social welfare.
Ⅱ. If d1CL d1C d1CR, the result of SW1C/e is ambiguous.
In this case, the pollution damage to the environment is intermediate, d1C will cause the first term of numerator on the right hand side in Eq. (33) to be negative on the one hand, but the second term to be positive on the other hand. The sign of
e SWC
1 / cannot be determined exactly.
Ⅲ. The case of d1C d1CR does not existed.
Compare Eq. (35.1) with Eq. (14), it is easy to check that d1CR d1Cmax and then
max 1 1
C
C d
d . This high pollution damage parameter will actually deteriorate the social welfare, therefore, the government of N1 does not allow F1 to produce that kind of product.
Similarly, because the first term of numerator on the right hand side in Eq. (34) is obviously positive, let d2CL is the pollution damage parameter that causes the second term equals to zero, then
) 1 ( 3
2
2 m
dCL e
(36)
Ⅰ. If 0d2C d2CL, the result of SW2C/e is ambiguous.
In this case, the first term of numerator on the right hand side in Eq. (34) is always positive, but smaller d2N will cause the second term to be negative. The sign of SW2C/e cannot be determined exactly.
Ⅱ. If d2CL d2C d2Cmax, SW2C/e0.
In this case, the second term of numerator on the right hand side in Eq. (34) is positive, SW2C/e0. It means when the pollution damage of the environment is intermediate, the government of N2 may impose higher environmental tax on F2
28
to raise the social welfare.
Ⅲ. The case of d2C d2Cmax does not existed.
Since d2Cmax is the maximum pollution damage limit for N2 to allow F2 to produce that kind of product, once the actual pollution damage is beyond the limit,
F2 will not be allowed to produce in N2.
According to the above discussion, the results can be concluded as Propositions 2 and 3.
Proposition 2.
In both of the non-cooperative R&D and cooperative R&D regimes, if the environmental tax lifts (falls),
1. the quantity outputs and profits of the two firms, and the consumer surplus of the two nations will decrease (increase) when the R&D investment amount is large enough for the two firms. But the results reverse when the R&D investment amount is relatively small.
2. the levels of emission abatement always increase (decrease), independent to the amount of R&D investment.
Proposition 3.
In both of the non-cooperative R&D and cooperative R&D regimes, if the environmental tax lifts (falls),
1. For domestic nation
(1) the social welfare will deteriorate (improve) when the damage of pollution is relatively small;
(2) the impact on social welfare cannot be determined exactly when the damage of pollution is intermediate;
(3) the government does not allow the firm in its domain to produce that kind of product when the damage of pollution is high enough.
2. For foreign nation
(1) the impact on social welfare cannot be determined exactly when the damage of pollution is relatively small;
29
(2) the social welfare will improve (deteriorate) when the damage of pollution is intermediate;
(3) the government does not allow the firm in its domain to produce that kind of product when the damage of pollution is high enough.
3.4.2 The Comparative Static Analysis with Respect to Spillover
When the two firms engage in emission abatement R&D, part of the achieved emission abatement technology will inevitably spill over to each other. This subsection will discuss the effects on quantity outputs, profits, and emission abatement levels of the two firms, and social welfare of the two nations, when the spillover rate changes under non-cooperation R&D and cooperation R&D regimes.
Case1. Non-cooperative R&D Regime
1
30 two firms will produce the optimal amount of outputs when 1/2. If the spillover rate is relatively large (1/2 1), the increasing (decreasing) spillover rate will result in decreasing (increasing) outputs for both of the two firms. By Eq. (37.2), the effect of spillover rate impact on price is reversed as the case of quantities.
Eq. (37.3) indicates that the increasing (decreasing) spillover rate will increase (decrease) the profits of the two firms in almost all the range of spillover rate (0 1). When is very close to 1 ( 1), the two firms will receive the optimal profits. Unexpectedly, when the two firms fully spill over their emission abatement technologies ( 1), their profits decline to the optimal below. For the amount of emission abatement, Eq. (37.4) exposes that an increasing (decreasing) spillover rate will reduce (lift) the emission abatement levels.
Observe Eqs. (37.5) and (37.6), when 0 1/2, the increasing (decreasing) spillover rate will increase (decrease) the social welfares of the two nations (SWiN / 0, i1,2). But if 1/2, the sign of SWiN / cannot be decided immediately. Only if diN is large enough, SWiN / 0 can have the possibility to occur.
31
According to the discussion in this subsection, the results can be concluded as Proposition 4.
Proposition 4.
In the non-cooperative R&D regime,
1. when 0 1/2, the two firms will produce more (less) outputs and the market price will lower (higher) while the is increasing (decreasing). But these results will reverse when 1/2 1. When 1/2, the two firms produce optimal outputs and result in the lowest market price.
2. when 0 1 , the two firms will earn higher (lower) profits if increases(decreases). When 1, the profits reach maximum.
3. the increasing (decreasing) will induce the two firms to accrue less (more) amount of emission abatement.
4. when 0 1/2, the social welfares of the two nations will improve (deteriorate) if increases(decreases), but will be ambiguous when 1/2 1.
Case2. Cooperative R&D Regime
) 0 ( ) 1 ( 3
2
* 2 2
*
1
T
e a e
q
qC C
(38.1)
) 0 )(
(1 6
2
* 2
T
e a e
pC
(38.2)
) 0 )(
(1
2
2
* 2 2
*
1
T
e a
C e
C
(38.3)
] 0 ) 1 ( )[2
[ ( 2
2
* 2
T
T e
e a
C e
(38.4)
32 From Eqs. (38.1) ~ (38.6) , it is easy to see that, under cooperative R&D regime, when the spillover rate increases (decreases), the outputs, profits, emission abatement levels of the two firms and social welfares of the two nations will increase (decrease). But, the price will decrease (increase).
The above results are summarized as Proposition 5.
Proposition 5.
In the cooperative R&D regime, the increasing (decreasing) spillover rate will induce the two firms to produce more (less) outputs, earn higher (lower) profits, accrue more (less) amount of emission abatement, and lower (raise) the market price; The social welfares of the two nations will also improve (deteriorate).
3.4.3 The Comparative Static Analysis with Respect to Pollution Diffusing Level Because the damage of crossing border pollution focuses only on the social welfares of both of the two nations, this subsection will discuss the pollution diffusing effects on social welfare of the two nations under non-cooperation R&D and cooperation R&D regimes.
Case1. Non-cooperative R&D Regime
2 0
33
Case2. Cooperative R&D Regime
2 0 decrease (increase) under non-cooperative and cooperative R&D regime. We conclude these results as Proposition 6.
Proposition 6.
In both of the non-cooperative R&D and cooperative R&D regimes, the increasing (decreasing) diffusing level of pollution will always negatively (positively) affect the social welfares of the two nations.