It is well known that under Cournot conjectures, each firm believes that its counterpart will hold its output fixed while the output level of that firm changes.¹⁰ In this context, profit functions for each of the firms take the following form:

π 1 q 1 , q 2 = P Q q 1 - F (1)

for the domestic firm, and

π 2 q 1 , q 2 , t = P Q q 2 - t q 2 (2)

for the foreign firm.¹¹

Where F stands for the fixed cost of adopting the cleaner technology, and 𝑡 as the specific trade tax. Reaction functions are obtained directly from: ∂ π i / ∂ q i = π q i = 0, for any i= 1, 2.

π q 1 q 1 , q 2 = P Q + q 1 P ' Q = 0 (3)

π q 2 q 1 , q 2 , t = P Q + q 2 P ' Q - t = 0 (4)

With second order conditions ∂ / ∂ q i ∂ π i / ∂ q i = π q i q i < 0 , for any i = 1, 2, and ∂ / ∂ q i ∂ π i / ∂ q j = π q i q j < 0 for any i ≠ j.

π q 1 q 1 = 2 P ' Q + q 1 P ' ' Q < 0 (5)

π q 2 q 2 = 2 P ' Q + q 2 P ' ' ( Q ) < 0 (6)

In order to know the effect of the trade tax 𝑡 on the production levels q₁ and q₂, I totally differentiate eqs. (3) and (4) to obtain the signs for (dq₁) / dt and (dq₂) / dt:

π q 1 q 1 d q 1 + π q 1 q 2 d q 2 + π q 1 t d t = 0 (7)

π q 2 q 1 d q 1 + π q 2 q 2 d q 2 + π q 2 t d t = 0 (8)

Since π q 1 t = 0 and π q 2 t = - 1 , the system can be simultaneously solved to obtain:

d q 1 / d t = - π q 1 q 2 ) / Δ > 0 (9)

d q 2 / d t = π q 1 q 1 / Δ < 0 (10)

where Δ = π q 1 q 1 π q 2 q 2 - π q 2 q 1 π q 1 q 2 > 0 .¹² Provided the Routh-Hurwitz condition for reaction function stability is in place, eqs. (9) and (10) clearly show that an increase in the trade tax t, leads to a domestic output increase, and to a foreign output decrease.

At this point that the home government is the only policy active, to choose the optimal trade tax, it maximizes the following social welfare function:

W t , θ = C S + P S + T R - P C (11)

where CS stands for consumer surplus, PS for producer surplus, TR for tariff revenue and PC for pollution costs. In other words, the social welfare function takes the next explicit form:

W t , θ = ∫ 0 P Q Q d Q - P Q Q + π 1 q 1 , q 2 + t q 2 - θ q 1 (12)

As is usual, eq. (12) considers consumer and producer surpluses and tariff revenues net of social costs of pollution ( t q 2 - θ q 1 ) .

For a given level of the technological parameter, the domestic government performs d W t , θ / d t = 0 .¹³ As I expected, the trade tax has the very well-known negative effect on the consumer surplus:

d / d t ∫ 0 P Q Q d Q - P Q Q = - Q P ' π q 1 q 1 - π q 1 q 2 / Δ < 0 (13)

a positive effect on the producer surplus.¹⁴

d π 1 q 1 , q 2 / d t = q 1 P ' π q 1 q 1 / Δ > 0 (14)

and a positive effect on the tariff revenue net of pollution costs as well:

d t q 2 - θ q 1 / d t = t π q 1 q 1 / Δ + q 2 - q 1 ( d θ / d t ) > 0 (15)

as long as dθ / dt < 0, the effect of the trade tax on the tariff revenue net of pollution costs is positive as well, and if the government’s commitment is credible and the trade tax provides the sufficient incentives to fully adopt the cleaner technology dθ / dt = 0.¹⁵

With these inputs, the government’s tax choice affects the balance between these positive and negative effects: should positive effects more than compensate the negative effect on the consumer surplus, the effect of the managed trade using a trade tax on the social welfare would clearly be positive.¹⁶

In any case, the optimal tax will be a function of the indexed parameter θ, i.e., t o p t = t θ where θ ∈ 0,1 .

In this freely managed trade regime, the protection will be stronger, the lower the parameter θ is. Therefore, for each technological choice within the interval [0, 1] there is an optimal trade tax, which according to the assumptions made, it will grant its highest protection when θ=0, therefore, the optimal trade tax, will be in the range: t o p t ∈ m a x t θ = 1 , t θ = 0 .

In the general case, we have:

( d W ( t , θ ) ) / d t = 0 ⇒

t o p t = Q P ' π q 1 q 1 - π q 1 q 2 - q 1 P ' + q 2 Δ / π q 1 q 1 - θ π q 1 q 2 / π q 1 q 1 ≥ 0 (16)

which clearly is positive when θ=0, i.e., when the domestic firm has fully adopted the cleaner technology.

Adopting the environment-friendly technology has naturally to be assessed in two ways: from the firm’s perspective and from the social-welfare point of view. Firstly, the net gain of adopting the cleaner technology must be non-negative:

π 1 q 1 , q 2 ; θ = 0 - π 1 ( q 1 , q 2 ; θ = 1 ) ≥ F (17)

Equation (17) compares the net gain of adopting the environment-friendly technology: the greater the granted managed trade protection through a higher trade tax associated with a low pollution parameter θ, the higher the possibility that it would lead to an adoption choice for a given sunk cost F, provided the net gain function be non-negative.

Secondly, if the technology adoption parameter is indexed and θ∈[0,1], the trade tax would be zero if the domestic firm does not adopt the clean technology. Therefore, as Regibeau and Gallegos (2004) pointed out, a free-trade policy would be harmful for the environment as long as there would be no incentives for the domestic firm to adopt the cleaner technology.

Naturally, at this point I will look for the level of the optimal trade tax and the resultant tariff revenue that compensates for the welfare loss on the consumer surplus due to the application of the trade tax. The answer to this question is strongly linked with the value the pollution parameter θ takes, and the level of imports from the foreign country q₂. Strategically, if the credible threat of granting managed trade protection is taken for acknowledged by the domestic firm and it fully adopts the cleaner technology, the trade tax chosen by the government, is expected to provide the highest protection.¹⁷

In the context of the managed trade protection proposed here, the combination of these two effects, points to a result where an optimal trade tax must be higher than that obtained from a non-cooperative game solution where the domestic firm does not necessarily adopt the cleaner technology. The explanation is straightforward: if the mechanism linking pollution and trade does not provide the incentive to adopt the cleaner technology through the credible threat, there is no reason to adopt it.¹⁸ The limited commitment of the government to provide protection is acknowledged by the domestic firm when trade protection is administered to finally adopt the cleaner technology, but not when the local government does not have such a mechanism. In that case, it might be unfavorable to adopt the cleaner technology at the sunk cost F, because there would be insufficient incentives to adopt it. In other words, the optimal trade tax between the managed trade regime and the non-cooperative solution is in the range:

t o p t θ ∈ m a x t θ = 1 , t θ = 0   f o r   θ ∈ 0,1 (18)

where t(θ =1) is the optimal tax when the domestic firm does not adopt (non-cooperative solution), whereas t(θ =0) is the level of trade tax that emerges from the adoption. In the first case, t=0, whereas in the second case t is at its highest level.¹⁹

Proposition 1: In this model, managed trade provides the strongest ex-post protection. The noncooperative solution including any possible trade liberalization lowers the domestic firm's incentives to adopt the cleaner technology.

Proof:

For the small open economy case, the non-cooperative solution for an optimal trade tax is: t o p t = 0 ,

At this point, the question that naturally arises is if tariff revenues net of pollution costs do really compensate for the welfare loss associated with the negative effect of the trade tax on the consumer surplus. To answer to this question, the appropriate measure to compare is the change in tariff revenue net of pollution costs plus the change in producer surplus vis-à-vis the change in consumer surplus, i.e.,

Δ P S + T R - P C ≥ ∆ C S (19)

as long as the change in the left-hand side of the eq. (19), more than compensates the change in the right-hand side, the tariff revenue net of pollution costs plays its role.

Proposition 2: Managed trade protection increases the domestic price and the domestic profit, reduces the foreign profit, maximizes the trade tax, and increases tariff revenues net of pollution costs after adoption.

Proof:

i)The increase in the price is given by the following expression:

d P / d t = P ' d q 1 / d t + d q 2 / d t = P ' π q 1 , q 1 - π q 1 , q 2 ⁄ ∆ ) > 0 ;

ii) The increase in the domestic profit follows from:

d π 1 / d t = ( ∂ π 1 / d q 1 ) d q 1 / d t + ( ∂ π 1 / ∂ q 2 ) d q 2 / d t = q 1 P ' π q 1 , q 1 / ∆ > 0 ;

iii) The reduction in the foreign profit from:

d π 2 / d t = q 2 P ' d q 1 / d t - q 2 = - q 2 1 + P ' π q 1 q 2 / ∆ < 0 ;

iv) The maximization of the trade tax from adopting or not the cleaner technology, where:

t o p t ∈ m a x t θ = 1 , t θ = 0   f o r   e a c h     θ ∈ 0,1 .

v) The increase in tariff revenues net of pollution costs:

d t q 2 - θ q 1 / d t = t π q 1 q 1 / Δ + q 2 - q 1 d θ / d t > 0 .

Analyzing the firm’s incentive to adopt the cleaner technology and the social incentives to adopt it naturally leads us to ask if there is a threshold where the negative effect of the administered protection on the consumer surplus could more than compensate for the remaining positive effects considering there is a sunk cost of adoption as well. As I have already mentioned, the incentive domestic private firms have to adopt the environmental-friendly technology can be obtained by simply comparing the firm’s benefits against the sunk cost F.

Φ 1 = π 1 θ = 0 - π 1 ( θ = 1 ) ≥ F (20)

In this way, the private incentive for the domestic firm, i.e., the difference between the benefits of adopting and not adopting, should be no less than the sunk cost F.

I will assume that this is the case, and then I will compare the change in the social welfare of adopting the cleaner technology net of the private incentives. Therefore, the change in the social welfare net of the private incentives becomes:

∆ W - Φ 1 - F = Δ C S + Δ ( T R - P C ) (21)

The first term on the right-hand side of eq. (21) is the change in the consumer surplus. This term is clearly negative since adopting the cleaner technology means to face higher domestic prices reflecting the administered protection given by the government. The second term on the right-hand side is the change in the tariff revenue net of pollution costs. This term has two elements: one that not always is an increasing function of θ, i.e., the tariff revenue ∆TR, because it depends on the value the pollution parameter θ takes.²⁰ The second one is the change in the cost associated with the pollution externality, ∆PC. The change in this latter term is always positive, because dθ / dt ≥ 0 and the full adoption makes the pollution parameter equal to zero, i.e., θ = 0 ⇒ P C = 0 .

Notice that the change in social welfare will be positive if the change in the tariff revenue net of pollution costs more than compensates for the negative effect of managed protection on the consumer surplus. However, in that case, private incentives to adopt the cleaner technology could be insufficient from the social welfare point of view. This means that eq. (21) is positive for the values the indexed θ can take, but there are some values for the sunk cost F at which adoption, though desirable, does not necessarily take place.²¹

In this section I will examine the main effects of changing the number of domestic and foreign firms, upon the optimal trade tariff, the private incentives and the social welfare of adopting the cleaner technology. Moreover, I will consider a subcase in which I am determining endogenously the number of home adopting firms. For that purpose, I will have δ domestic firms adopting the high technology and n - δ not adopting it. These n domestic firms will compete in the third stage with m foreign firms in the domestic market. The timing of the game is like the general case. I will analyze the subgame perfect equilibrium in each of the three stages.

At this stage of the game, government chooses the optimal trade tax maximizing a social welfare function that includes, as is usual, the consumer surplus, profits of the domestic firms, the trade tax revenue and the costs associated with pollution emissions. It is important to recall that if domestic firms adopt the cleaner technology, it has been assumed that pollution emissions are reduced to zero, i.e., θq_i = 0 for any i = 1,…, δ. Then the social welfare function takes the following functional form:

W t , θ = ∫ 0 Q P Q d Q - P Q Q + δ π i 0 + n - δ π j θ + t m q f - θ ( n - δ ) q d L (34)

After totally differentiating eq. (34) with respect to t for a given θ, i.e., Dw (t, θ) / dt = 0, we obtain:

t o p t = ( 1 - 2 n - θ n - δ 1 + m + n ) / ( 2 1 + n 2 + m ) (35)

Aiming at having an interior solution the degree of adoption must take the value θ < ( 1 + 2 n ) / ( n - δ ) ( 1 + m + n ) , which naturally occurs because θ ∈ [ 0,1 ] . Therefore, the optimal tax is clearly positive.

Proposition 3: The larger the number of adopting firms δ, the higher the level of the optimal trade tax will be.

Proof: By taking the partial derivative of the optimal trade tax with respect to parameter δ:

∂ t o p t / ∂ δ = θ ( 1 + m + n ) / [ 2 1 + n 2 + m ] > 0 .

Proposition 4: The larger the number of not adopting firms (n - δ), the lower the level of the optimal trade tax will be.

Proof: By taking the partial derivative of the optimal trade tax with respect to parameter (n - δ):

∂ t o p t / ∂ ( n - δ ) = - θ ( 1 + m + n ) / [ 2 1 + m 2 + m ] < 0 .

On the other hand, comparing the domestic firm’s incentives to invest in the non-polluting technology, any single domestic firm compares the net benefit of full adoption versus not adopting at all the cleaner technology:

Φ i = π i θ = 0 - π i θ = 1 ≥ F ,   f o r   a n y   i = 1 ,   … ,   n . (36)

Therefore, when comparing this private benefit with the sunk cost F, there is a threshold for adoption: the lower the sunk cost F is, the higher the degree of adoption the domestic firm will take. Naturally, the i-th firm will invest in the cleaner technology if and only if F < ∅ i .

Comparing the change in the domestic welfare net of the net producer surplus from adopting versus not adopting the cleaner technology:

∆ W - Φ 1 - F = Δ C S + Δ ( T R - P C ) (37)

Equation (37) for the specific case where n = 1 = m allows us to see the expected result:

Δ W - Φ 1 - F = 7 θ ( 8 - θ ) / 162 (38)

which is no negative given that θ ∈ [ 0,1 ] .

Proposition 5: Private incentives are insufficient to adopt the cleaner technology from the point of view of the domestic welfare, specifically when the sunk cost of adoption F lies in the range ( Φ i , Δ W )

for a given ϑ.

Finally, assuming free entry to get the endogenous number of adopting firms, I analyze the benefits of any adopting firm net of sunk costs for a given indexed value of θ, when benefits for this firm are equal to zero to get the maximum number of adopting firms, i.e.,

π i = 2 1 + n 2 + m + m 1 + 2 n - θ n - δ 1 + m + n / 2 1 + n 2 + m 1 + m + n 2 - F = 0 ,     f o r   a n y   i = 1,2 , … , δ . (39)

which implies that the number of adopting firms obtained is:

δ = 2 1 + n 2 + m 1 + m + n F - 1 - m ( 1 + 2 n - θ n ) ) / θ m ( 1 + m + n ) ,   f o r   a n y   θ ∈ ( 0,1 ) . (40)