A Comparative statics

Dropping the firm’s index for the sake of clarity and introducing a positive scale parameter θ for abatement cost function, the system eq. (1) rewrites as:

Differentiating totally this system and dropping arguments, we obtain:

where Δ=aeec′′+θ(aeeaqq−aqe2)>0 under quasi-convexity of total cost c(.)+a(.,.). We obtain that:

Also, under assumption Assumption 1, we obtain that:

Finally, assumption Assumption 2 allows to state that:

B Proof of Proposition Proposition 2

To get qˆ(τ,s,f)=q∗ and eˆ(τ,s,f)=e∗, in view of the FOCs eqs. (2) and (5), it is sufficient to take

Also, the subsidy s is given by the budget constraint eq. (3) that now writes:

or equivalently with R=(1−δ)tne∗

where σ∗=1−εa,e∗−εa,q∗ with εa,e∗=ae∗e∗/a∗<0 and εa,q∗=aq∗q∗/a∗>0 and recalling that t=−ae∗ from eq. (1). Observe that σ∗=0 needs to be ruled out otherwise the budget equality constraint cannot hold.

Also, to preserve quasi-concavity of the polluter’s program, we need s < 1 which amounts to assume that

whenever is is positive.

Finally, by construction, the difference in terms of net profit is

by using the budget constraint.

C Proof of Proposition Proposition 5

To get qˆi=q∗ and eˆi=e∗, in view of the FOCs eqs. (1) and (7), it is sufficient to take

Moreover, the budget constraint writes:

Using eq. (12) and replacing in eq. (13), we get

which implies

with σi∗=1−εai,e∗−εai,q∗.

Moreover, the difference between net profits is:

Replacing with the values obtained for the instruments τi and fi and recalling that t=−aie∗, we obtain:

Summing over i and using eq. (14), we thus get Πˆ−Π∗=∑iσi∗ai∗si=δtE∗>0.

D Example

From the expression for individual production,

we get by summing over i:

Also from

we obtain similarly

Finally, for abatement expenditures we obtain:

The system to be solved in (τ,s,f) rewrites as follows:

Equation (16) can be expressed as

from which we deduce that

Also, equation (17) writes as

and replacing τ using eq. (19) allows to obtain

which simplifies into

where λ≡kcσγ2σγ2+γˉ2≥0. We can deduce f as a function of s:

When there is homogeneity, σγ2=0 and thus λ = 0 and we recover f=(1−s)t as in Proposition 2.

Last, using eqs. (16) and (17), the budget constraint eq. (18) rewrites simply as

or using eq. (15) and the definition of E∗ and Q∗:

This allows to compute s/(1−s) as a function of f:

Hence, it follows that:

and replacing in eq. (20):

which amounts to solve a polynomial equation of degree 2 in f. We denote the two solutions f1 and f2. And the corresponding values of τ and s by, respectively, τ1, τ2, s1 and s2.

E Reallocation of production and pollution between firms

We compare qˆi and eˆi taken at the solution 1 (f1,τ1,s1) with their counterparts qi∗ and ei∗ in the standard tax benchmark. Let us start with quantities:

From eq. (19), we know that τ1=(f1−t)γˉ and replacing in eq. (21), we get:

As solution 1 entails overtaxation of emissions (f1>t), we obtain that qˆi>qi∗⇔γi<γˉ.

Now we compare the emission levels:

Using eq. (20) and replacing in eq. (22), we obtain:

Recall that λ≡kcσγ2σγ2+γˉ2 so that by rearranging,

Once again, using that f1>t, it follows that eˆi>ei∗⇔γi<γˉ+σγ2γˉ.