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Refunding Emissions Taxes: The Case For A Three-Part Policy

Philippe Bontems

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.

JEL Classification: Q52; Q58; H23

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:

p=c(q)+θaq(q,e)θae(q,e)=t.

Differentiating totally this system and dropping arguments, we obtain:

dq=1Δaqedt+aeedp(aqaeeaeaqe)dθde=1Δ(aecθ(aeaqqaqaqe))dθc+θaqqdtθaqedp

where Δ=aeec+θ(aeeaqqaqe2)>0 under quasi-convexity of total cost c(.)+a(.,.). We obtain that:

qp=1Δaee>0and et=c+θaqqΔ<0.

Also, under assumption Assumption 1, we obtain that:

qt=1Δaqe<0and ep=aqeΔ>0.

Finally, assumption Assumption 2 allows to state that:

qθ=(aqaeeaeaqe)Δ<0and eθ=(aeaqqaqaqe))Δ>0.

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

f=t+sae=(1s)tτ=saq.

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

(11)nfe=nτq+nsa+R

or equivalently with R=(1δ)tne

t+saee=saqq+sa+(1δ)tes=δteaaeeaqq=δteσa

where σ=1εa,eεa,q with εa,e=aee/a<0 and εa,q=aqq/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

σ>δtea

whenever is is positive.

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

πˆ(τ,s,f)π=(p+τ)qc(q)(1s)afepq+c(q)+a+te=τq+sa+(tf)e=fe(1δ)te+(tf)e=δte

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

(12)fi=(1si)tτi=siaiq.

Moreover, the budget constraint writes:

(13)ifiei=iτiqi+isiai+(1δ)tE

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

i(1si)tei=isiaiqqi+isiai+(1δ)tE

which implies

(14)iσiaisi=δtE

with σi=1εai,eεai,q.

Moreover, the difference between net profits is:

πˆiπi=p+τiqici(qi)(1si)ai(qi,ei)fieipqici(qi)ai(qi,ei)tei=τiqi+siai+(tfi)ei

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

πˆiπi=siaiaiqqiaieei=siaiσi

Summing over i and using eq. (14), we thus get ΠˆΠ=iσiaisi=δtE>0.

D Example

From the expression for individual production,

qˆi=p+τfγic

we get by summing over i:

Qˆ=np+τfγˉc

Also from

eˆi=γi(qˆifγik(1s))

we obtain similarly

Eˆ=iniγi(qˆifγik(1s))=iniγi(p+τfγicfγik(1s))Eˆ=nγˉp+τcnf(1c+1k(1s))(σγ2+γˉ2)

Finally, for abatement expenditures we obtain:

(15)Aˆ=inik(qˆieˆi/γi)22=f22k(1s)2iniγi2=nf2(σγ2+γˉ2)2k(1s)2

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

(16)Qˆ=Q
(17)Eˆ=E
(18)fEˆ=τQˆ+sAˆ+(1δ)tE

Equation (16) can be expressed as

np+τfγˉc=nptγˉc

from which we deduce that

(19)τ=(ft)γˉ

Also, equation (17) writes as

nγˉp+τcnf(1c+1k(1s))(σγ2+γˉ2)=nγˉpcnt(1c+1k)(σγ2+γˉ2)

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

γˉ2(ft)cf(1c+1k(1s))(σγ2+γˉ2)=t(1c+1k)(σγ2+γˉ2)

which simplifies into

(20)f1s=t+λ(tf)

where λkcσγ2σγ2+γˉ20. We can deduce f as a function of s:

f=t(1+λ)(1s)1+λ(1s)

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

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

ft(1δ)E=τQ+sAˆft(1δ)E=(ft)γˉQ+sAˆ

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

ft(1δ)E=(ft)γˉQ+snf2(σγ2+γˉ2)2k(1s)2

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

s1s=2kσγ2+γˉ2f(EγˉQ)+t(γˉQ(1δ)E)t+λ(tf)

Hence, it follows that:

11s=1+2kσγ2+γˉ2f(EγˉQ)+t(γˉQ(1δ)E)t+λ(tf)

and replacing in eq. (20):

f1+2kσγ2+γˉ2f(EγˉQ)+t(γˉQ(1δ)E)t+λ(tf)=t+λ(tf)

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:

(21)qˆiqi=p+τ1f1γicptγic=τ1+(tf1)γic

From eq. (19), we know that τ1=(f1t)γˉ and replacing in eq. (21), we get:

qˆiqi=(f1t)(γˉγi)c

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

Now we compare the emission levels:

(22)eˆiei=γi(qˆif1γik(1s1))γi(qitγik)=γi(f1t)(γˉγi)c+γiktf11s1

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

eˆiei=γi(f1t)(γˉγi)cγikλ(tf1)=γi(f1t)γˉγic+γikλ

Recall that λkcσγ2σγ2+γˉ2 so that by rearranging,

eˆiei=γi(f1t)γˉ2c(σγ2+γˉ2)σγ2γˉ+γˉγi

Once again, using that f1>t, it follows that eˆi>eiγi<γˉ+σγ2γˉ.

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Published Online: 2019-01-30

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