# Abstract

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.

# Appendix

## 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

Also, under assumption Assumption 1, we obtain that:

Finally, assumption Assumption 2 allows to state that:

## B Proof of Proposition Proposition 2

To get

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

or equivalently with

where

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

Moreover, the budget constraint writes:

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

which implies

with

Moreover, the difference between net profits is:

Replacing with the values obtained for the instruments

Summing over *i* and using eq. (14), we thus get

## 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 *f* as a function of *s*:

When there is homogeneity, *λ* = 0 and we recover

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

or using eq. (15) and the definition of

This allows to compute *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 *τ* and *s* by, respectively,

## E Reallocation of production and pollution between firms

We compare

From eq. (19), we know that

As solution 1 entails overtaxation of emissions (

Now we compare the emission levels:

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

Recall that

Once again, using that

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**Published Online:**2019-01-30

© 2019 Walter de Gruyter GmbH, Berlin/Boston