Payment for Environmental Services and Environmental Tax Under Imperfect Competition

Appendix A: Welfare Function Concavity and Existence of the Solution

If ψ = 0, we construct the Hessian matrix, I(W) from equations (10) and (11):

where

Following our assumptions we can simplify the above matrix to

We have F′ < 0 and G′ < 0. We calculate the determinant of I:

After simplification, we obtain: D e t ( I ) = ( d X 2 d s ) 2 [ G ′ ] ( ∂ X 1 ∂ t ) 2 [ F ′ ] > 0 .

Thus, the welfare function is concave because the determinant is positive while [ d X 2 d s ] 2 [ G ′ ] + [ ∂ X 1 ∂ t ] 2 [ F ′ ] < 0 .

Equation (10) sets Ω₁(s, t) and equation (11) sets Ω₂(s, t). We must establish that at least one pair (s, t) satisfies Ω₁(s, t) = 0 and Ω₂(s, t) = 0. We use Brouwer’s fixed-point theorem. X ₁(t, s) and X ₂(s) are defined on [s _min, s _max] and [t _min, t _max]. Ω₁(s, t) and Ω₂(s, t) are continuous in s and t due to the regularity of the functions P ₁(X ₁), P ₂(X ₂), c ₁(X ₁), c ₂(X ₂), B(Y) and D(X ₁). The convexity and increasing nature of the functions c ₁(X ₁), c ₂(X ₂) and D(X ₁) and concavity of B(Y) guarantee that the equations are well-defined and do not diverge. Thus according to Brouwer’s fixed-point theorem, the solution (s, t) exists.

Because the welfare function is concave, the solution given by equations (10) and (11) is so unique.

If ψ > 0, (17) sets a function h(t). We obtain:

Under our assumptions, we have:

The limit h(t) when t → 0 is positive and the limit h(t) when t → +∞ is −∞ because ∂ X ̄ 1 n u ( t ) ∂ t < 0 and ∂ X ̄ 2 n u ( t ) ∂ t > 0 . According to the intermediate value theorem, it exists t ^nu such as h(t ^nu ) = 0. As h′(t) < 0 the welfare function is concave when ψ > 0. The solution given by equation (17) is so unique.

Appendix B: Welfare Function Concavity and Existence of the Solution with MCF

If ψ = 0, we use (20) and (21) to create the Hessian matrix:

where

and

Thanks to our assumptions, we can simplify the Hessian to:

So the determinant is:

Simplifying, we find:

With A′ < 0 and B′ < 0, we find a positive determinant. And, because ( ∂ X 1 ∂ s ) 2 [ A ′ ] + [ d X 2 d s ] 2 [ B ′ ] + 2 ϵ ( ∂ X 1 ∂ s + d X 2 d s ) < 0 , we have a concave function.

Equation (20) sets ϒ₁(s, t) and equation (21) sets ϒ₂(s, t). We must establish that at least one pair (s, t) satisfies ϒ₁(s, t) = 0 and ϒ₂(s, t) = 0. We use Brouwer’s fixed-point theorem. X ₁(t, s) and X ₂(s) are defined on [s _min, s _max] and [t _min, t _max]. ϒ₁(s, t) and ϒ₂(s, t) are continuous in s and t due to the regularity of the functions P ₁(X ₁), P ₂(X ₂), c ₁(X ₁), c ₂(X ₂), B(Y) and D(X ₁). The convexity and increasing nature of the functions c ₁(X ₁), c ₂(X ₂) and D(X ₁) and concavity of B(Y) guarantee that the equations are well-defined and do not diverge. Thus according to Brouwer’s fixed-point theorem, the solution (s, t) exists.

As the welfare function is concave, the solution given by equations (20) and (21) is unique.

If ψ > 0, equation (23) sets g(t):

where

With our assumptions we can simplify this to:

The limit g(t) when t → 0 is positive and the limit g(t) when t → +∞ is −∞ because ∂ X ̄ 1 n u ( t ) ∂ t < 0 and ∂ X ̄ 2 n u ( t ) ∂ t > 0 . According to the intermediate value theorem, it exists t n u mcf such as g t n u mcf = 0 . As g′(t) < 0, the welfare function is concave when ψ > 0. The solution given by equation (23) is so unique.

Appendix C: Tax and PES Changes with MCF if ψ > 0

According to (22), t and s depend on ϵ. Moreover t and s must satisfy conditions (20) and (21). We set:

Additionally, we know: ∂ X 1 ∂ t = ∂ X 1 ∂ s < 0 .

We differentiate (20) and (21) with respect to ϵ and rearrange the equations into the following matrix form:

where K = i j k l , with:

We multiply each side of the equation by K ⁻¹ to isolate d s d ϵ and d t d ϵ :

where

We calculate det K:

because q < 0 and z < 0, ∂ X 1 ∂ t = ∂ X 1 ∂ s < 0 and ∂ X 2 ∂ s < 0 .

We calculate d s d ϵ , using (A1):

i.e. if ∂ X 1 ∂ t t + X 1 > 0 ⇔ ∂ X 1 ∂ t t X 1 + 1 > 0 ⇔ ∂ X 1 ∂ t t X 1 ︸ e X 1 / t > − 1 .

So ∂ s ∂ ϵ < 0 if e X 1 / t > − 1 .

We calculate d t d ϵ , using (A1):

We know that

− ∂ X 1 ∂ t 2 ∂ X 2 ∂ s t B y y − ∂ X 1 ∂ t ∂ X 2 ∂ s X 1 B y y > 0 ⇔ − ∂ X 1 ∂ t ∂ X 2 ∂ s B y y [ ∂ X 1 ∂ t t − X 1 ] > 0 ⇔ ∂ X 1 ∂ t t X 1 > − 1

− 2 ∂ X 1 ∂ t ∂ X 2 ∂ s t ϵ − 2 ∂ X 2 ∂ s X 1 ϵ > 0 ⇔ − 2 ∂ X 2 ∂ s ϵ [ ∂ X 1 ∂ t t + X 1 ] > 0 ⇔ ∂ X 1 ∂ t t X 1 > − 1

− ∂ X 1 ∂ t ∂ X 2 ∂ s 2 z t − ∂ X 2 ∂ s 2 z X 1 > 0 if − ∂ X 2 ∂ s 2 z [ ∂ X 1 ∂ t t + X 1 ] > 0 ⇔ ∂ X 1 ∂ t t X 1 > − 1

− ∂ X 2 ∂ s 2 X 1 B y y − ∂ X 1 ∂ t ∂ X 2 ∂ s 2 t B y y > 0 if − ∂ X 2 ∂ s 2 B y y [ ∂ X 1 ∂ t t + X 1 ] > 0 ⇔ ∂ X 1 ∂ t t X 1 > − 1

− ∂ X 1 ∂ t ∂ X 2 ∂ s 2 z s + ∂ X 1 ∂ t 2 ∂ X 2 ∂ s q s > 0 ⇔ ∂ X 1 ∂ t ∂ X 2 ∂ s s [ ∂ X 1 ∂ t q − ∂ X 2 ∂ s z ] > 0 ⇔ [ ∂ X 1 ∂ t q − ∂ X 2 ∂ s z ] > 0 ⇔ ∂ X 1 ∂ t / ∂ X 2 ∂ s > z / q ≡ ϖ .

⇒ ∂ t ∂ ϵ > 0 if e X 1 / t > − 1 and ∂ X 1 ∂ t / ∂ X 2 ∂ s > ϖ ( ϵ ) .

Appendix D: Tax and PES Changes with MCF if ψ = 0

We use (23) and we set: J ( t , ϵ ) = d X 2 d t − p 1 ( T − X 2 ) + p 2 ( X 2 ) + c 1 ′ ( T − X 2 n ) − c 2 ′ ( X 2 n ) + D ′ ( T − X 2 ) − ϵ t + ϵ ( T − X 2 ) . Applying the implicit function theorem we find: