We consider a special case of Schmutzler's and Goulder''s (1997) analysis of output taxes vs emission taxes as environmental policy instruments. We identify new necessary conditions for the existence of an optimum. We also show that, in this case, it is always optimal to have a mixed tax with positive enforcement effort.

Athanasios Kampas

Laurent Franckx

English

Research Papers in Economics

``The choice between emission taxes and output taxes under
imperfect monitoring": a comment 
Laurent Franckx Athanasios Kampas
Royal Military Academy, Belgium Agricultural University of Athens
Abstract
We consider a special case of Schmutzler's and Goulder's (1997) analysis of output taxes vs
emission taxes as environmental policy instruments. We identify new necessary conditions
for the existence of an optimum. We also show that, in this case, it is always optimal to have
a mixed tax with positive enforcement effort.
The authors would like to thank Katrin Millock for instructive comments on a previous version of this paper.
Citation: Franckx, Laurent and Athanasios Kampas, (2003) "``The choice between emission taxes and output taxes under
imperfect monitoring": a comment." Economics Bulletin, Vol. 17, No. 1 pp. 1−7
Submitted: June 26, 2003.  Accepted: September 1, 2003.
URL: http://www.economicsbulletin.com/2003/volume17/EB−03Q20003A.pdf1 Introduction
In an interesting and stimulating paper, Schmutzler and Goulder (1997) have
compared output taxes with emission taxes as environmental policy instruments.
In the most important section of the paper, they assume that monitoring a
polluting ﬁrm’s output is free. However, for the regulator, the cost of monitoring
polluting emissions is assumed to depend on monitoring intensity.
Our comment is motivated by the following observation. For illustrative pur-
poses, Schmutzler and Goulder start their analysis with a speciﬁc proﬁt function
for the polluting ﬁrm. However, they quickly move on to a more abstract setting.
Their analysis therefore gains in generality but also loses somehow in intuition.
In this model, we stick to (a slightly modiﬁed version of) the illustrative
proﬁt function Schmutzler and Goulder introduced in the beginning of their
paper.
Our results have intuitively appealing properties. Moreover, we obtain sev-
eral surprising new results. First, we identify new necessary conditions for the
optimality of a mixed tax. Second, we show that it is never optimal to have a
pure (output or emission) tax.
We retain all the notational assumptions used by Schmutzler and Goulder.
x is the output level of the polluting ﬁrm, p is the output price, t0 is the unit
output tax, C(e,x) is the cost function, e are real emissions, ed are declared
emissions and te is the unit tax on declared emissions.
We use the following speciﬁc ﬁne function for false self-reports: f(e,ed,m)
(discussed by the authors in footnote 13 of the original article). Thus, the ex-
pected ﬁne depends in a non-speciﬁed, but exogenous way on real emissions,
undeclared emissions and monitoring eﬀort m. 1 Schmutzler and Goulder as-
sume that “without any monitoring, the ﬁrm will behave as if there was no
emission tax at all”. We interpret this assumption as: if m = 0, then te = 0
and f(e,ed,m) = 0 for all e and ed.
The ﬁrm’s expected proﬁts are then:
Π = (p − t0)x − C(e,x) − teed − f(e,ed,m)
To simplify notation, let:
•
∂C(e,x)
∂x = Cx and
∂C(e,x)
∂e = Ce
•
∂f(e,ed,m)
∂e = fe,
∂f(e,ed,m)
∂ed = fd and
∂f(e,ed,m)
∂m = fm
•
∂
2C(e,x)
∂x2 = Cxx,
∂
2C(e,x)
∂x∂e = Cxe and
∂
2C(e,x)
∂e2 = Cee
•
∂
2f(e,ed,m)
∂e2 = fee,
∂
2f(e,ed,m)
∂(ed)2 = fdd,
∂
2f(e,ed,m)
∂ed∂e = fed,
∂
2f(e,ed,m)
∂ed∂m = fdm
and
∂
2f(e,ed,m)
∂m∂e = fme.
1For instance, in Schmutzler and Goulder, “monitoring eﬀort” corresponds to the number
of ﬁrms that are monitored. The regulator is then assumed to observe perfectly the amount
of actual emissions by the ﬁrms that have indeed been monitored.
12 The ﬁrm’s decision problem
It seems reasonable to limit the attention to interior solutions for x and e, but
to consider explicitly the possibility of a corner solution for ed.
The ﬁrm’s FOC are then:
∂Π
∂x
= p − t0 − Cx = 0 (1)
∂Π
∂e
= −Ce − fe = 0 (2)
and:
∂Π
∂ed
= −te − fd ≤ 0 ed ≥ 0 (te + fd)ed = 0 (3)
Now consider the Hessian corresponding to the second-order conditions:

 
 

−Cxx −Cxe 0
−Cxe −Cee − fee −fed
0 −fed −fdd

 
 

Let |Dij,kl| be the minor with upper-left element ij and lower-right element
kl.
The SOC are satisﬁed if the three principal minors alternate in sign:
• |D11,11| = −Cxx < 0,
• |D11,22| =

 

−Cxx −Cxe
−Cxe −Cee − fee

 
 > 0
• |D11,33| < 0, or (fed)2Cxx − fdd|D11,22| < 0.
We shall from now on assume that these conditions are fulﬁlled. Note that
the second SOC can only be satisﬁed if Cee + fee > 0. The third SOC requires
fdd > 0. This implies that the ﬁrm’s SOC not only impose restrictions on the
ﬁrm’s cost function, but also on the ﬁne for undeclared emissions!
3 The regulator’s problem
Following Schmutzler and Goulder, suppose that the regulator’s objective func-
tion takes the following form:
B(x,e) − C(x,e) − Cµ(m) (4)
where B(x,e) are the social beneﬁts of the produced good and environmental
quality, and Cµ(m) is the cost of monitoring with intensity m.
From Conditions 1, 2 and 3, we see that t0, te and m determine x, e and ed.
We will now have to distinguish between interior and corner solutions for
declared emissions.
24 Interior solution for declared emissions
Remember that if m = 0, then te = 0 and f(e,ed,m) = 0 for all e and ed.
Therefore, ed > 0 requires both m > 0 and te > 0.
If there is an interior solution for ed, then Condition 3 reduces to −fd = te.
This condition can only be fulﬁlled if fd < 0. This condition requires that,
in the optimum, and for given real emissions and monitoring eﬀort, an increase
in declared emissions leads to a decrease in the expected ﬁne. It has not been
identiﬁed in the original paper, but it is clearly a necessary condition for the
existence of an interior solution. As the functional relation between the ﬁne and
undeclared emissions is exogenous in the model, there is no compelling reason
why this condition should be fulﬁlled.
We can therefore reformulate the problem as follows: the decision problem
for the regulator is to choose the values of x, e, ed and m in order to maximize
Expression 4 subject to Equation 2. Solving Equations 1 and 3 then gives the
desired values of t0 and te.2
The Lagrangian is therefore (where µ is the Lagrange multiplier):
L = B(x,e) − C(x,e) − Cµ(m) + µ(−fe − Ce) (5)
This implies immediately that the FOC with respect to x, e and ed are given
by:
∂L
∂x
= Bx − Cx − µCxe = 0 (6)
∂L
∂e
= Be − Ce − µ(fee + Cee) = 0 (7)
∂L
∂ed
= −µfed = 0 (8)
This implies immediately that an interior solution for ed requires that there
exist an e and an ed such that fed = 0. Again, we obtain here a necessary
condition for the existence of an interior solution that has not been identiﬁed
in the original paper either.
Now remember that the third SOC for the ﬁrm, |D11,33| < 0, can only be
fulﬁlled if fdd > 0. Thus, if the expected ﬁne would take the speciﬁc form
proposed in Equation 3 in the Schmutzler and Goulder analysis, then fdd =
−fed. Thus, in that particular formulation of the problem, the SOC are not
satisﬁed.
2Alternatively, we could also add t0 and te as choice variable. However, it is straightforward
to verify that, in this alternative formulation of the problem, Equations 1 and 3 do not bind
as constraints. The reason for this is that the regulator does not care about the tax rates
themselves: they are merely instruments that induce the ﬁrm to choose the optimal output
and emission levels.
3However, suppose the FOC are fulﬁlled. Then fed = 0 implies that the
ﬁrm should declare emissions up to the point where they minimize the expected
marginal (relative to real emissions) ﬁne.
From Equations 6 and 7, we see that Bx−Cx
Cxe = Be−Ce
fee+Cee. Schmutzler and
Goulder assume that Cxe < 0. We have seen above that the ﬁrm’s SOC can
only be satisﬁed if fee + Cee > 0. Thus, we obtain that, in the equilibrium,
Bx − Cx and Be − Ce have opposite signs.
The regulator’s FOC with respect to m are given by (remember that we
require m > 0):
∂L
∂m
= −
∂Cµ
∂m
− µfem = 0 (9)
Schmutzler and Goulder assume that ∂C
µ
∂m > 0. Therefore, an interior solu-
tion for m is only possible if µfem < 0.
Suppose ﬁrst that, in the optimum, fem < 0. This condition requires that, in
equilibrium, the marginal impact of an increase in real emissions on the expected
ﬁne decreases when monitoring eﬀort increases. A possible interpretation of
this condition is that an increase in monitoring eﬀort leads to a decrease in the
probability that the regulator overestimates the ﬁrm’s emissions.
In this case, an interior solution for m is thus only possible if µ > 0. This
implies:
Proposition 4.1 An equilibrium with a strictly positive enforcement eﬀort where
fem < 0 is only possible if, in that equilibrium, Cx > Bx and that Be > Ce.
Suppose next that fem > 0. This condition requires that, in equilibrium, the
marginal impact of an increase in real emissions on the expected ﬁne increases
when monitoring eﬀort increases. In this case, a possible interpretation is that
an increase in monitoring eﬀort leads to a decrease in the probability that the
regulator underestimates the ﬁrm’s emissions. An interior solution for m is then
only possible if µ < 0. Following the same argument as above, µ < 0 implies:
Proposition 4.2 An equilibrium with a strictly positive enforcement eﬀort where
fem > 0 is only possible if, in that equilibrium, Bx > Cx and that Ce > Be.
Let us address the problem whether it could ever be optimal to have a pure
emission tax.
If this were the case, then t0 = 0 and Equation 1 requires that, in the
optimum, p = Cx. If, as Schmutzler and Goulder assume, the “price always
adjusts in such a way that it equals the marginal beneﬁts of output” (formally,
Cx = Bx), then Propositions 4.1 and 4.2 imply:
Proposition 4.3 In an equilibrium with strictly positive enforcement eﬀort, it
is always optimal to have a mixed tax.
4Compare this with Proposition 5 in the Schmutzler and Goulder paper. In
that proposition, the authors had identiﬁed a suﬃcient condition for this result.
Our analysis shows that the result always hold with the speciﬁc proﬁt func-
tion that has been used here if the necessary conditions we have identiﬁed are
fulﬁlled.
Essentially, the point is that in order to induce the ﬁrm to declare any
emissions at all, the regulator must always exert a positive monitoring eﬀort.
This always creates distortions compared to the ﬁrst-best solution where Bx =
Cx and that Ce = Be. Introducing an output tax then leads to a marginal
decrease in the total distortion.
5 Corner solution for declared emissions
Consider now the possibility that it might be optimal to induce a corner solution
for declared emissions: ed = 0.
We see again that, for any desired value of x and e, solving Equations 1 and
3 gives the value of t0 and the minimal required value of te.
We can therefore reformulate the problem as follows: the decision problem
for the regulator is to choose the values of x, e and m in order to maximize
Expression 4 subject to ed = 0.
The Lagrangian is therefore (where µ is the Lagrange multiplier):
L = B(x,e) − C(x,e) − Cµ(m) − µed (10)
Therefore, the FOC with respect to x and e are given by:
∂L
∂x
= Bx − Cx = 0 (11)
∂L
∂e
= Be − Ce = 0 (12)
The regulator’s FOC with respect to m are given by:
∂L
∂m
= −
∂Cµ
∂m
≤ 0 m ≥ 0
∂L
∂m
m = 0 (13)
We see immediately that ∂C
µ
∂m > 0 implies that there is no interior solution
for m.
Now remember that if m = 0, then te = 0 and f(e,ed,m) = 0 for all e and
ed. There is then no t0 such that the regulator’s FOC with respect to e are
compatible with the ﬁrm’s FOC. We thus obtain:
Proposition 5.1 It is impossible to have an optimum with zero declared emis-
sions.
56 Conclusion
From Proposition 4.3 and 5.1, we see that there is only one possible equilibrium:
a mixed tax with positive enforcement eﬀort. A pure tax is thus never optimal.
In order for such a mixed tax to be optimal, the following conditions must
be satisﬁed in the optimum: fd < 0, fdd > 0 and fed = 0. These conditions have
not been identiﬁed in the original paper, but have appealing intuitive properties.
However, as these conditions are related to the (exogenous) ﬁne function, there
is no guarantee that they are fulﬁlled.
Moreover, with a mixed tax, there exists a one-to-one relationship between
the sign of fem and the sign of Bx − Cx and Be − Ce.
References
Schmutzler, A. and L.H. Goulder (1997), The Choice between Emission Taxes
and Output Taxes under Imperfect Monitoring, Journal of Environmental Eco-
nomics and Management 32, 51-64
6

The Choice between Emission Taxes and Output Taxes under Imperfect Monitoring,

``The choice between emission taxes and output taxes under imperfect monitoring": a comment

http://www.accessecon.com/pubs/EB/2003/Volume17/EB-03Q20003A.pdf

``The choice between emission taxes and output taxes under imperfect monitoring": a comment

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics