(i). A stricter environmental policy enhances growth provided that m <m.˜ Differentiating the growth rate of the economy (12) with respect to φ, gives:
∂g
As the traditional effect which is equal to (χ − 1)∂δ(φ)∂φ , is negative while the satisficing effect which is equal to −κ(χ−1)
∂δ(φ)
∂φ
[(χ−1)δ(φ)]2 is positive, the sign of ∂φ∂g is either positive or negative depending upon the value of m. A stricter environmental policy enhances growth provided that m < 1
1+[(χ−1)δ(φ)]2κ ≡ m,˜ otherwise the opposite effect emerges. Notice that 0<m <˜ 1, which proves
the co-existence of two opposing effects and thus part (i) of Proposition 1.
(ii). A stricter environmental policy reduces pollution.
Let us first compute aggregate pollution. Substituting for xi,t using (5) into (2), pollution in sectori is:
Pi,t =δ(φ), (B.2)
which is constant over time and across sectori. Aggregate pollutionR1 0 Pi,tdi, which we denote by Pt, is thus equal toδ(φ) as well. Differentiating Pt with respect to φ, we get:
∂Pt
∂τ = ∂δ(φ)
∂φ , (B.3)
which is clearly negative (because of ∂δ(φ)∂φ <0) and thus proves part (ii) of Proposition 1. Notice that this result reflects the one in Bianco and Salies (2016, 2017) meaning that a stricter environmental policy permanently re-duces aggregate pollution.
(iii). A stricter environmental policy increases downstream firms’ profits provided that m <m.˜˜
Let us first compute the reduced form of downstream firms’ profits. Starting by substituting for xi,t using (5) into (3) and then using the definition of the adjusted environmental tax rate, i.e., φ≡ Aτi,t
i,t, we obtain:
πt(y) = [(1−α)δ(φ)α]At, (B.4)
whereAt≡R1
0 Ai,tdirepresents the average productivity. Then, substituting
for At using (10), we get:
which in turn can be rewritten with the help of the innovation process using Am,t = γmAm,t−1 and Ac,t = γcAc,t−1 and the assumption of normalization of the productivity parameters at date t −1, i.e., Am,t−1 = Ac,t−1 = 1, as follows:
πt(y) = F(φ) [mγm+ (1−m)γc], (B.6) where F(φ)≡ (1−α)δ(φ)α is a decreasing function of the quality-adjusted environmental tax φ.
Now, differentiating downstream firms’ profits (B.6) with respect to φ, we get:
At this stage, the sign of this derivative is ambiguous. Indeed, a stricter environmental policy, through the growth effect, impacts downstream firms’
profits positively or negatively depending on the parameter m (see part (i) of Proposition 1). In addition to this ambiguous effect, through the direct effect, a stricter environmental policy always reduces downstream firms’ prof-its (because mγm+ (1−m)γc > 0 and ∂F∂φ(φ) = αFδ(φ)(φ)∂δ(φ)∂φ < 0, because of
∂δ(φ)
∂φ <0).
Inserting F(φ) and the derivative of F(φ) with respect to φ in (B.7), we get:
then, using the definition of γc and γm and their derivative with respect to φ simplifies (B.8) as follows:
∂πt(y)
Finally, substituting for γc and γm using (7) and (8) into (B.9), we can demonstrate that (B.9) is positive provided that m < 1
1+1+α1−α[(χ−1)δ(φ)]2κ ≡ m,˜˜
otherwise the opposing effect emerges. This proves part (iii) of Proposition 1. In addition to the sign of the previous derivative, as the denominator is greater than one, it obvious that ˜˜m <1 which proves the co-existence of the two effects described above.
Let us now compare the two values ofm. As we assume that 0< α <1, it can easily be shown that 1+α1
−α >1, which in turn by comparaison with the definition of ˜m proves that ˜˜m <m.˜
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