This paper examines theoretically whether by combining both output-based refunding and abatement expenditure-based refunding, it is possible to limit the negative consequences that a pollution tax implies for a polluting industry. We show that this is indeed the case by using a three-part policy where emissions are subject to a fee and where output and abatement expenditures are subsidized. When the industry is homogenous, it is possible to replicate the standard emission tax outcome by inducing a polluting firm to choose the production and emission levels obtained under any emission tax, without departing from budget balance. By construction, any polluter earns strictly more than under the standard tax alone without rebate, making this proposal more acceptable to the industry. When firms are heterogeneous, the refunding policy needed to replicate the standard emission tax outcome is personalized in the sense that at least the output subsidy should be type dependent and it is strictly preferred only from the industry’s point of view to a standard environmental tax. We also explore the implications of uniform three-part refunding policies for a heterogeneous industry.
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 under quasi-convexity of total cost . 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
Also, the subsidy s is given by the budget constraint eq. (3) that now writes:
or equivalently with
where with and and recalling that from eq. (1). Observe that 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
Moreover, the budget constraint writes:
Moreover, the difference between net profits is:
Replacing with the values obtained for the instruments and and recalling that , we obtain:
Summing over i and using eq. (14), we thus get
From the expression for individual production,
we get by summing over i:
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 . We can deduce f as a function of s:
When there is homogeneity, and thus λ = 0 and we recover as in Proposition 2.
or using eq. (15) and the definition of and :
This allows to compute 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 and . And the corresponding values of τ and s by, respectively, , , and .
E Reallocation of production and pollution between firms
We compare and taken at the solution 1 () with their counterparts and in the standard tax benchmark. Let us start with quantities:
As solution 1 entails overtaxation of emissions (), we obtain that .
Now we compare the emission levels:
Recall that so that by rearranging,
Once again, using that , it follows that .
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