Global Warming and Border Carbon Adjustments

Proof of Lemma 1.

Summing up the first order conditions of home and foreign countries and reordering them yields

Since v ″ ( q ‾ ) q ‾ + ( 2 n + 1 ) v ′ ( q ‾ ) < 0 , we know that d ( q H + q F ) / d τ i < 0 .

Taking derivative of (5) with respect to τ _i and applying the symmetric condition ( q i = q i c = q ‾ ) yields

Since d ( q H + q F ) / d τ i ≠ 0 , we obtain that v ( q i ) = c + γ + θ γ for each i ∈ { H , F } . ∎

Proof of Lemma 2.

(i) Suppose to the contrary that q _H is non-decreasing in τ _H . Since ( q H + q F ) strictly decreases in τ _H , q _F should strictly decrease in τ _H . Subtracting (7) from (8) and reordering implies that

Equation (A3) implies that q H ≤ q F if and only if τ H ≥ τ F . It also implies that as τ _H increases, − ( 1 / 2 ) v ′ ( q ‾ ) should increase, because ( q F − q H ) strictly decreases in τ _H . In addition, − ( 1 / 2 ) v ′ ( q ‾ ) q H should increase in τ _H . Rewriting the first order conditions of each country and connecting them yields

It implies that as τ _H increases unilaterally, − 1 2 v ′ ( q ‾ ) q H + n τ H should increase as much as − 1 2 v ′ ( q ‾ ) q F + n τ F does. However, since q _F is decreasing in τ _H but q _H is nondecreasing, − 1 2 v ′ ( q ‾ ) q H + n τ H increases further than − 1 2 v ′ ( q ‾ ) q F , which is contradiction.

Proof of Lemma 3.

Without loss of generality, assume that τ H > τ F . Summing up the first order conditions of foreign firms yields

Suppose to the contrary that q F e ≠ 0 , which implies that q F e should solve for  n [ v ( q ‾ ) − c − τ H γ ] + [ q F d + q F e ] v ′ ( q ‾ ) / 2 = 0 . Since τ F < τ H , n [ v ( q ‾ ) − c − τ F γ ] + [ q F d + q F e ] v ′ ( q ‾ ) / 2 > 0 so that q F d = 0 . However, individual foreign firm j obtains larger profit by choosing ( q F j d + ϵ , q F j e − ϵ ) for a sufficiently small ϵ > 0 without affecting the output price, because q F j d incurs less cost than q F j e . It’s contradiction. Therefore, we know that q F e = 0 .

Now, suppose to the contrary that q H e ≠ 0 . Summing up the first order conditions of home firms yields

It implies that ( q F d + q F d ) solves for n [ v ( q ‾ ) − c − τ H γ ] + 1 2 v ′ ( q ‾ ) [ q H d + q H e ] = 0 , whereas ( q F d + q F d ) solves for n [ v ( q ‾ ) − c − τ F γ ] + 1 2 v ′ ( q ‾ ) [ q F d + q F e ] = 0 . Since q F e = 0 and τ H > τ F , we obtain that

Then, it is a contradiction to p F = v ( q F d + q H e ) = v ( q H d ) = p H , that is, the output price in each country cannot be equal. Therefore, we conclude that q F e = q H e = 0 under mutual BCAs.

Proof of Lemma 4.

First, consider the situation with τ H < τ F . Home country’s BCAs are not binding. It is same to the case of ‘No BCAs.’ Then, there is no mutual best response with τ H < τ F . Now, consider the situation with τ H ≥ τ F . Home country’s BCAs are binding. Then, it is same to the case of ‘Mutual BCAs.’ Therefore, the mutual best responses satisfy equation (14) as in Lemma 3. ∎