The Optimal Taxation of Polluters in Non-Competitive Markets:
Does Regulatory Sequence Matter?
*
Stephan Schott, Carleton University
SPPA Working Paper
May 21, 2008
*
Stephan Schott is an Assistant Professor at Carleton University, School of Public Policy and Administration, 1125
Colonel By Drive, Ottawa, Ontario, K1S 5B6, Canada, E-mail: sschott@connect.carleton.ca, Tel.: 613-520-2600
ext. 2557, Fax: 613-520-2551.
ABSTRACT
The paper examines the first-best use of instruments to control emissions in a
non-competitive market with a social planner and with independent competition
and environmental regulators. We show that it is optimal to combine a first-best
Pigovian tax with one firm-specific ad valorem subsidy per type of firm when
entry is blockaded. This is not only more efficient but also has clear informational
and regulatory advantages to other suggested tax instruments in the literature,
including firm-specific ad valorem taxes that also regulate emissions.
With endogenous entry and a homogenous industry we need to supplement a firstbest Pigovian tax and firm-specific subsidies with an entry correcting lump-sum
fee. The sequence of introduction of instruments has several efficiency impacts.
With blockaded entry it is preferable to introduce the Pigovian tax before a
competition regulator determines the ad valorem subsidies. With endogenous
entry the first-best Pigovian tax leads to excessive entry, unless an optimal entry
correcting ad valorem tax is in place or firms behave competitively after entry.
Competition policy that precedes environmental policy will always lead to
excessive entry, even if a competitive market emerges after entry.
Key words: Pigovian tax, first and second best taxation, imperfect competition,
information, regulation
I. Introduction
There has been an extensive discussion in the economics literature on how to adjust Pigouvian
taxes (Pigou 1932), when the industry that causes pollution behaves non-competitively. The
Pigouvian tax rule under monopoly was first examined by Buchanan (1969), then by Barnett
(1980) and Baumol and Oates (1988), who pointed out that a monopolist should not be taxed
according to the full marginal external damages that are created by the industry. Instead the
authors suggested a second-best tax that should be set below marginal external damages, as it
simultaneously considers distortions in market output as well as marginal external damages.
Shaffer (1995), Simpson (1995), Katsoulacos and Xepapadeas (1995) Xepapadeas (1997) and
Lee (1999) examine the taxation of oligopolistic polluters and also suggest a deviation from
Pigovian taxation. Shaffer suggests the use of firm-specific ad valorem taxes that could be
implicit or explicit subsidies for some firms and taxes for other firms. Simpson (1995) showed
in a blockade entry duopoly model that the second-best Pigovian tax could exceed marginal
damages in order to induce the more efficient firm to produce relatively more than the less
efficient firm. Katsoulacos and Xepapadeas, Xepapadeas and Lee examine second-best emission
taxes with endogenous entry and demonstrate that the Pigovian tax could exceed or fall below
Pigovian taxation because of a “business-stealing” effect. The latter is an externality between
firms, which induces firms to reduce their output as the number of entering firms increases.
Because firms do not internalize this effect there is excessive entry, which could cause the
second-best tax to overinternalize the marginal damages stemming from emissions. Katsoulacos
and Xepapadeas (1995) and Xepapadeas (1997) furthermore show how the use of an entry fee
and a second-best under-internalising emissions tax can improve social welfare compared to a
second-best over-internalizing emissions tax.
2
The literature on the optimal taxation of imperfectly competitive firms leaves us with a
number of unresolved issues. First none of the papers derive the first-best combination of ad
valorem taxes, emission taxes and entry fees/subsidies that simultaneously control entry,
emissions and imperfect pricing. Secondly, papers either focus on second-best Pigovian taxation
without considering the impact of ad valorem taxes on emissions, output and abatement (Lee,
Simpson, Buchanan, Katsoulacos and Xepapadeas, Xepapadeas, Barnett), or they ignore the
substitution between abatement and output adjustments for emission reductions (Lee, Shaffer).
Only Baumol and Oates (1998) mention (but do not derive) the use of an ad valorem tax in a
monopoly context. Finally none of the papers evaluate suggested tax instruments in the context
of a decentralized regulatory environment. Most regulatory environments consist of separate
environmental, finance and competition regulators that either have no mandate to interfere in or
lack information about other regulatory areas. It, therefore, seems crucial to assess instruments
not just under the assumption of a social planner. A social planner presumably would always
choose the first-best set of instruments and introduce them simultaneously. That would,
therefore, rule out second-best suggestions in the literature. We, therefore, derive the first-best
set of instruments and then evaluate the more realistic situation of having a separate competition
and environmental regulator, as well as a central agency that could use ad valorem taxes strictly
for revenue purposes.
We first show that it is optimal to use the first-best Pigovian tax in combination with
firm-specific ad valorem subsidies with blockaded entry. This is consistent with second-best
results in the literature that suggest an under-internalizing (implicitly subsidizing) emission tax
or firm-specific ad valorem taxes/subsidies that also regulate emissions. A second-best tax or
firm-specific ad valorem taxes alone would, however, not allow for the independent regulation of
3
emissions and imperfect competition by separate regulators. Firm-specific ad valorem taxes that
also regulate emissions are furthermore shown to require very detailed information; particularly
with abatement activity. With endogenous entry it remains optimal to use the first-best Pigovian
tax and firm-specific ad valorem subsidies, but we need to supplement them with entry fees in
order to correct the entry distortion effect created by ad valorem subsidies. When regulators
cannot control competitive behaviour after entry, we can control entry either with ad valorem
taxes (and not subsidies as with blockaded entry) or with entry fees or subsidies. First-best
instruments are shown to not just have efficiency advantages but also informational advantages.
They allow for regulatory independence and with entry we do not need to know the impact of
entry on output and abatement adjustments of firms. We find that introducing regulatory
instruments in sequence will create distortions but could also provide information for a
competition regulator who need to know social marginal cost in order to set ad valorem taxes.
With endogenous entry competition policy will always lead to excessive entry in a polluting
industry, even when a competitive market emerges. This is not necessarily the case when a firstbest Pigovian tax is introduced first.
The paper proceeds as follows: We start with a blockaded entry model, derive the firstbest combination of instruments, and re-examine the use of firm-specific ad valorem taxes in lieu
of Pigovian taxes when emissions are a function of output and abatement. We evaluate
informational requirements and the sequential introduction of instruments. We then focus on
endogenous entry of identical firms with sunk costs. First we compare free entry implications to
the efficient number of firms in the industry when no policy instruments are used. Next we
analyze the impact on entry when a Pigovian tax is introduced, and then derive the optimal use of
entry-controlling instruments. Finally we derive the optimal combination of instruments that
4
regulate entry, emissions and firm behaviour after entry. We contrast the informational
requirements of the derived policy instruments to other suggestions in the literature and evaluate
the impact of sequential introduction of regulatory instruments on the number of firms entering
the industry. We conclude the paper with policy recommendations and future research
suggestions.
II. Optimal Taxation with Blockaded Entry
We start with a blockaded entry model in which we assume that the number of firms in the
market is fixed.
1.Social welfare maximization:
A comprehensive model is used, that allows for the use of all instruments and explicitly models
emissions as a function of output levels (xi) and abatement activities of each firm (ai). As in
Barnett’s and Baumol and Oates’ model, firm i’s cost of production (ci) is a function of output
(xi) and abatement activity (ai), where ci'(xi)>0, ci''(xi)≥0, ci'(ai)>0 and ci''(ai)≥0. Total external
k
damages (D) are a function of aggregate emissions of all ni firms of k types ( E = ∑ ni ei (xi , ai ) ),
i =1
and firm’s emissions depend on their output and abatement levels, where ei'(xi)≥0 and ei'(ai)≤0.
We start our analysis with a single regulator that tries to choose the optimal firm-specific output
and abatement level, which maximizes social welfare. The latter is defined as the unweighted
sum of consumers’ and producers’ surplus less environmental damages of firms and less
aggregate fixed costs:
k
∑ ni xi
i=1
max .W =
xi ,ai
∫
0
k
k
k
i =1
i =1
i =1
p(u)du − ∑ ni ci (xi , ai ) − D(∑ ni ei (xi , ai )) − ∑ ni Fi
(1)
5
Social welfare maximization results in three first-order conditions when the number of firms of
each type (ni) is exogenously fixed:
∂W
∂c
∂e
= p(X) − i − D '(E) i = 0
∂ xi
∂ xi
∂ xi
(2)
∂W
∂c
∂e
= − i − D '(E) i = 0
∂ ai
∂ ai
∂ ai
(3)
The first necessary condition states that we should produce output until the market price of the
product equals the marginal social cost of one additional unit of output. The second first-order
condition says that marginal abatement cost should equal the marginal benefits from emission
reduction due to abatement activity.
2. Firm Behaviour
One solution to induce firms to act in the socially optimal way is to impose a firm-specific ad
valorem tax (τi) and an emissions tax (t). Each firm’s objective then is to maximise the
following profit function with respect to xi and ai :
max.(1 − τ i ) p( X ) x i − ci ( xi , ai ) − tei ( xi , ai )
xi ,a i
(4)
We assume that firms do not try to strategically influence the choice of τi. Each firm then follows
two necessary conditions:
1.
∂π i
∂c
∂e
= (1 − τ i )[ p(X) + (1 + ri )p '(X)xi ] − i − t i = 0
∂ xi
∂ xi
∂ xi
(5)
2.
6
∂π i
∂c
∂e
= − i −t i = 0
∂ ai
∂ ai
∂ ai
(6)
Equation (5) shows that firms produce until their after tax marginal revenue equals the sum of
marginal production costs and emissions tax costs that result from the last unit of output
produced. The second profit maximization condition states that firms abate until marginal
abatement costs are equal to the marginal reduction of tax payments due to an additional unit of
abatement activity. We can then derive the optimal set of taxes by solving equation (6) for
and substituting for
∂c i
,
∂ai
∂c i
in equation (3), which yields a first-best emissions tax (t*) that equals
∂ai
marginal damages (D'(E)). After substituting for t* in equation (5) and for p(X) from equation
(2), it is possible to derive the optimal ad valorem tax (τi*). The combination of taxes that are
required to achieve the optimal solution under blockaded entry, therefore, are:
t * = D '(E)
τ i* =
p '(X)xi
p '(X)xi + p(X)
(7)
(8)
We can express equation (8) in terms of the price-elasticity of demand ( ηi =
τ i* =
1
.
1i + ηi
∂ xi p
):
∂ p xi
(9)
The optimal ad valorem tax is always negative, and, therefore, an explicit subsidy for
noncompetitive industries, because firms will operate at an output level with an elasticity smaller
7
than –1 (as long as c>0).
Result 1: The optimal taxation of firms with blockaded entry requires the first-best
Pigovian tax and a firm-specific ad valorem subsidy for each type of noncompetitive
industry.
The optimal combination of taxes consists of the first-best Pigovian tax and an ad valorem
subsidy that depends on firm’s market elasticity of demand. The ad valorem subsidy must be
firm-specific, unless firms are identical (as in a monopoly or oligopoly with homogenous firms).
A Pigovian tax and one ad valorem subsidy per type of firm are required to achieve a first-best
solution. The Pigovian tax has two welfare-enhancing effects: (1) It causes firms to fully
internalize social marginal cost (SMCt) of production and (2) It reduces social marginal cost to
SMCt because it induces firms to choose the optimal combination of abatement and output
adjustments for emission reductions. The ad valorem subsidy has the function of increasing
firms’ output to the optimal production level. Shaffer (1995) has suggested a first-best ad
valorem tax that also controls for the emission externality. In Shaffer’s model emission are
directly related to output and cannot be abated. With abatement a firm-specific ad valorem tax
would need to satisfy both welfare conditions and take into consideration both necessary
conditions for each firm. We could derive a firm-specific tax that would simultaneously satisfy
welfare conditions (2) and (3). Due to the absence of an emissions tax firms would set marginal
abatement costs equal to zero (
∂ci
= 0 ). From the social-welfare maximization conditions
∂ai
∂e
∂c
∂c ∂x 1
(equation (2) and (3)) we can substitute p +
for i in equation (5) and set it equal to
∂xi
∂a ∂e
∂a
8
∂ci
(both equal 0 according to the firm’s necessary conditions). We can then solve for the
∂ai
optimal firm-specific ad valorem tax with abatement activity:
τ =
*
i
∂c
p ' xi − i (1 +
∂ai
p ' xi + p
∂ei
∂xi
∂ei
∂ai
)
(10)
Notice that the optimal firm-specific ad valorem tax with abatement is definitely a subsidy as
long as
∂ei
∂e
> i.
∂ai ∂xi
Result 2: A firm-specific ad valorem tax that controls both emissions and imperfect
competition when firms can invest into abatement activity is a subsidy as long as
∂ei
∂e
> i ; otherwise it could be either a subsidy or tax.
∂ai ∂xi
We could still use firm-specific ad valorem taxes, even when firms have the option to engage in
abatement activity. Regulators would, however, need to have very specific information about
each firm’s substitution of emission reduction for abatement activity, as well as firm-specific
marginal abatement costs. This would unnecessarily increase the informational burden compared
to using two separate instruments. Firms automatically choose the right mix of abatement and
output reductions when we use Pigovian taxes.
Apart from the efficiency aspects of using the optimal set of first-best tax instruments,
1
After substituting −
∂ci
∂ai
∂ei
∂ai
for D′(E).
9
there are informational and regulatory advantages of separating emissions control and the
regulation of imperfect competition. We often make the assumption of a social planner, which
allows us to get around the discussion of regulatory coordination since a social planner is
assumed to have simultaneous control over all regulatory sectors. It is, however, common to
have a number of ministries/departments in charge of different areas such as health, finance,
environment, transportation, competition and industrial organization. Furthermore policy
instruments are rarely introduced at the same time, i.e. competition policy often precedes
environmental policy. It, therefore, makes sense to evaluate to what extent regulators need to
consult with other Departments when they determine policy instruments, and what impact
sequential introduction of policies has on industry output, prices, emissions and social marginal
production costs.
The first observation is that an environmental regulator can always choose the first best
Pigovian tax without consulting with any competition regulator. The environmental regulator
will choose a tax equal to marginal damages of an additional unit of emissions.2 A competition
regulator would not choose the right ad valorem tax if she ignored marginal emissions cost and
the substitution of abatement for output adjustments. The right ad valorem tax depends on the
difference between price and marginal damages (p’xi) at the optimal output and price level. By
ignoring social marginal cost a competition regulator would induce too much production unless
firms would choose to fully abate emissions without any change in output once a Pigovian tax is
introduced. We can summarize this in the following proposition:
2
The tax could either be constant if marginal damages of an additional unit of emissions were always the same or
could vary with emissions if marginal damages vary with the aggregate level of emissions.
10
Proposition 1: Competition policy that ignores environmental cost of production only
leads to the optimal output if environmental taxation is simultaneously introduced and
firms choose to fully abate emissions so that the internalized private marginal production
cost equals the social marginal cost of production.
Proposition 1 implies that competition policy should not precede environmental policy in a
blockaded industry situation with at least some of the firms being polluters. An ad valorem tax
would otherwise be chosen that would induce too much production in many cases, which causes
tremendous marginal social cost, as firms have no incentive to invest into the optimal abatement
scheme. When environmental taxation is introduced first, social marginal cost of production is
reduced and firms produce less (unless they choose to abate rather than decrease production).
Monopolists or oligopolists would, however, now underproduce. This could then be
counteracted with the appropriate ad valorem tax. Competition regulators could use the
information from the emissions tax to determine the appropriate ad valorem tax that will lead to
the optimal output and market price.
Next we will examine how the results for blockaded entry change when entry is
endogenous and how sequential introduction of policy instruments influences the number of
firms in the industry and their pricing and emission behaviour after entry.
III. Endogenous Entry with sunk costs
Endogenous entry complicates decision-making of a social planner because she also needs to
regulate the number of firms entering the industry. Entry, on the other hand, might lead to a
more competitive market structure. Since firms could differ with respect to fixed costs,
emissions or cost functions and could have different degrees of market power, a careful
distinction between markets with homogenous and heterogeneous technologies is necessary. We
11
will, therefore, focus in this paper on the simplest model with identical firms. The conventional
assumption is adopted that firms enter at a fixed cost ‘F’≥0, as long as they can make
nonnegative profit. Fixed costs are assumed to be sunk costs after firms have entered. Notice
that the regulator has an additional instrument with endogenous entry: it can impose an entry
subsidy or fee, which we will refer to as ‘S’. The additional condition for ‘n’ incoming firms,
therefore, is:
π = (1 − τ )p(nx)x − c(x, a) − te(x, a) − F + S = 0
(11)
With entry we, therefore, have the following three-stage game3:
(1) First regulators choose an ad valorem tax/subsidy, a lump-sum entry fee/subsidy and an
emissions tax.
(2) Firms then enter simultaneously.
(3) Firms decide how much to produce and how much to abate emissions (if an emissions tax
is being utilized).
Our model is consistent with the literature on endogenous entry of polluters, i.e. entry is
simultaneous, and there will, therefore, be no incumbents that can behave strategically. The
result will be a Cournot Nash equilibrium if n>1. In the case of sequential entry incumbents
could try to deter entry or accommodate entry and we will end up with a different Nash
equilibrium and that involves a Stackelberg leader (Tirole, 1988).
A single regulator would take entry/exit decisions and final output and abatement
behaviour into consideration when determining the taxes. Since regulators now also need to be
concerned about the optimal number of firms or technologies that enter the market, we need to
3
We could consider a 4-stage model in which incumbents in the second stage decide on strategic capacity
investment in order to deter entry in the third stage. In this paper we are not explicitly modeling this setup. A more
detailed model of entry and capacity investment would be best dealt with in a separate paper with a dynamic entry
model.
12
differentiate the social welfare function (equation (1)) with respect to n:
∂W
∂x
∂c ∂x
∂c ∂a
∂e ∂x ∂e ∂a
= p(nx)[n + x] − c − n
−n
− D '[n(
+
) + e] − F = 0 (12)
∂n
∂n
∂x ∂n
∂a ∂n
∂x ∂n ∂a ∂n
Equation (12) determines the optimal number of firms in the industry.4 The actual number of
firms in the industry is determined by equation (11). Usually the entrant’s incentive to enter
deviates from the socially optimal entry. Mankiw and Whinston (1986) have shown that entry
can be excessive (in a model without externalities) because of a “business stealing effect” or
entry can be insufficient if a “business-augmenting” effect prevails. The latter can be ruled out
under the assumption that marginal revenue of each firm is steeper than market demand (see
Seade 1980 and Lee 1999). Firms’ entry contributes to additional social surplus (equal to the
firm’s profit and the social value of its output), but it also changes the output of other firms,
which entrants do not internalize and which diminishes social welfare when firms do not behave
competitively after entry.
We can now extend the model with endogenous entry to polluting industries that can
substitute output changes for abatement. In the symmetric homogenous model all the firms
pollute.5 We can rewrite the optimal entry condition (equation (12)) by referring to πn as
equilibrium profits for firms (as defined by equation (11):
∂W
∂e ∂x
∂c ∂a
∂e ∂a
= π n + (t − D ')e − S + τ px + n( p − c'− D ' ) − n(
+ D'
)
∂n
∂x ∂n
∂a ∂n
∂a ∂n
(13)
The discrepancy between W'(n) and πn depends on the use of τ, t and S. We know that W'(n)=0,
4
Although this is a standard assumption in the literature it has been pointed out by McKitrick and Collinge (2000)
that the optimal number of firms (n*) is not necessarily a continuous variable. We, therefore, need to assume
diminishing returns to additional entry and increasing damages in order to have an interior solution.
5
In an industry with heterogeneous technologies, it is possible that polluting and nonpolluting firms could co-exist.
A good example are electricity generating firms, some of which are relatively small in size and are based on
renewable energy sources, and some are rather large and based on fossil fuel burning or nuclear technologies.
n
13
n*
and when W'(n)≤ π , it must be true that the optimal entry inducing profit is larger than zero (π
≥0)6. The opposite holds when W'(n) ≥ πn, i.e. πn*≤ 0. In a free entry equilibrium, equilibrium
profits (πne) are, however, exactly equal to zero. This means that either too many or too little
firms are in the industry, depending on the change in firms’ profits as the number of firms (n) in
the industry changes. By taking the first partial derivative of firms’ profit function with respect
to n we can evaluate how the number of firms in the market affects firms’ profit:
∂π
∂c ∂x
∂nx ∂c ∂a
∂e ∂x ∂e ∂a
= [(1 − τ )p − ] + (1 − τ )p ' x
−
− t[
+
].
∂n
∂x ∂n
∂n ∂a ∂n
∂x ∂n ∂a ∂n
(14)
From the first-order conditions for profit-maximization of firms we can substitute
(1 − τ )[ p + p ' x] for
∂c
∂e
∂e
∂c
and t
for −
and simplify (14) to:
+t
∂x
∂x
∂a
∂a
∂π
∂nx ∂x
= (1 − τ )p ' x[
− ]
∂n
∂n
n
(15)
Profits will always decline as n increases as long as we can rule out any “business-augmenting”
effect and there is a strong “business-stealing” effect, i.e. that incumbents will reduce their
output but overall output increases. A strong “business-stealing” effect implies
∂π
< 0 and
∂n
∂nx
> 0 , while a weak “business-stealing” effect causes profits to rise with entry as total output
∂n
declines sufficiently to raise profits of all firms. The interesting insight is that this result is
independent of τ, t.
Result 3: Profits decrease with the number of firms entering the market. This is
independent of the use of Pigovian taxes, ad valorem taxes and the impact of entry on the
change in firms’ output and abatement decisions, as long as entry does not reduce
6
Mankiw and Whinston first pointed this out in a model without emissions and without the use of any policy
instruments. The authors, therefore, only considered the case of W'(n)≤ πn.
aggregate industry output.
14
7
Result 3 implies that ne≥n* when W'(n)≤ πn and that ne ≤ n* when W'(n) ≥ πn.
1.Free entry without influencing pricing and emissions of firms
The first question to be explored is if entry is more or less excessive with pollution than without
it when no regulatory instruments are used. In order to tackle this question we can assess the
equilibrium without any use of regulatory tools or by setting τ=t=S=0. Without any emissionscontrolling instrument there is no incentive to invest into abatement since emissions are a free
production input, and consequently the number of firms should have no impact on the change of
abatement activity. We can, therefore, rewrite equation (13):
∂W
∂e ∂x
= π n − D 'e + n( p − c'− D ' )
∂n
∂x ∂n
(18)
As opposed to the findings by Mankiw and Whinston for entry into imperfect markets without
externalities, we now have an ambiguous result. The second term on the right hand side of
equation (16) is definitely negative, but the sign of the last term is ambiguous. The last term on
the right hand side of equation (16) is positive with a “business-stealing” effect if price is smaller
than social marginal cost, and it then depends on the balance of both terms if W’(n) is larger,
smaller or equal to πn. When price equals or exceeds social marginal cost, we definitely have
excessive entry. This implies that even if a competitive market emerges after entry, there will be
too many firms entering.
The first divergence between optimal entry and firms’ incentives to enter is due to
uncontrolled emissions. From a social point of view too many firms enter, and this causes too
7
This is similar to Lemma 2 in Lee (1999). The latter Lemma is in the context of a polluting
not invest into abatement activity and is not regulated with any kind of ad valorem taxes.
industry that does
15
much aggregate external damages. This negative effect is counteracted by the positive effect of
reducing the emissions of other firms due to the “business-stealing” effect. The latter, therefore,
has a positive and negative impact with polluting firms. The more firms enter, the less individual
firms supply, which is the negative impact of entry due to “business-stealing”. A reduction in
firm’s output, however also reduces emissions that are linked to output. If the social value of the
“emission reduction effect” outweighed the “output-reduction” effect and the additional marginal
damages caused by entering firms, then there would be too little entry.
Result 4: Endogenous entry with polluting firms and no regulation of emissions is
excessive as long as price equals or exceeds social marginal cost; otherwise it is
ambiguous if too many or insufficient firms enter.
2. Uncontrolled entry and pollution control with a Pigovian tax
We continue our analysis with unregulated entry and no control over pricing behaviour after
entry, but control of external damages with a Pigovian tax. This implies that τ is either
exogenously determined for example by a central agency such as the Finance Ministry or set
equal to 0. The use of the first-best Pigovian tax ensures that firms optimally substitute between
abatement and emissions, and it guarantees that firms set after tax marginal revenue equal to
marginal social cost of production. After replacing
∂c
∂e
∂e
with (1 − τ )[ p + p ' x] , −t
for
+t
∂x
∂x
∂a
∂c
∂x
∂nx
from the firm’s necessary conditions, setting t=D'(E) and writing x + n
as
we can
∂a
∂n
∂n
derive the following relationship from equation (13):
∂W
∂nx
∂x
= πn +τ p
− n((1 − τ )p ' x)
∂n
∂n
∂n
(17)
The gap between the efficient and unregulated number of firms is now a function of τ and we can
16
make three predictions conditional on τ:
(1) No ad valorem tax is in place, τ = 0 :
∂W
∂π
≤ π n , and since
< 0 excessive entry occurs, i.e. the number of firms in the industry is
∂n
∂n
too large. This is consistent with Mankiw and Whinston, because the Pigovian tax corrects
for the emission externality so that we end up with the same entry problem.
(2) An ad valorem tax ( τ > 0 ):
We now have an ambiguous result. A positive ad valorem tax creates a disincentive to enter,
while the “business-stealing” effect distorts entry in the other direction, because it causes
excessive entry. Entry could, therefore be excessive, optimal or too small.
(3) An ad valorem subsidy ( τ < 0 ):
An ad valorem subsidy would cause excessive entry, so does a “business-stealing” effect.
Both effects have the same kind of impact. Entry is, therefore, excessive.
Excessive entry occurs as long as there is no positive ad valorem tax in place. If a competition
regulator would, for example, use an ad valorem subsidy to correct for imperfect pricing, too
many firms would enter. A positive ad valorem tax, on the other hand could lead to the optimal
number of firms in the industry. The optimal entry controlling ad valorem tax (which ensures
that W'(n)=#n=0) can be derived from equation (17):
*
τ entry
=
np ' x
∂x
∂n
∂nx
∂x
p
+ np ' x
∂n
∂n
(18)
Result5: The optimal entry controlling ad valorem tax in imperfectly competitive markets is
∂nx
positive as long as
> 0 and a strong “business-stealing” effect prevails.
∂n
Result 5 is exactly the opposite of the blockaded entry result, i.e. that we want to tax imperfect
industries despite noncompetitive output levels and despite the imposition of the first-best
17
Pigovian tax according to full marginal damages. It furthermore reassures us that exogenously
determined ad valorem taxes by a central agency might not be as distortionary as commonly
perceived when markets are imperfect and entry is endogenous.
3. Controlling Entry with Entry Fees or Subsidies
The positive ad valorem tax does, of course, not control imperfectly competitive
behaviour after entry; it just prevents excessive startup investment. When we are only concerned
about entry and emissions regulation we could also employ an entry subsidy or fee in lieu of
τentry*. The latter alternative is especially relevant when ad valorem taxes are exogenously
determined by a central agency and are not set at the optimal entry correcting level. It might, for
example, not be feasible to change a currently existing ad valorem tax because of financial
reasons, or because any changes in the ad valorem tax for one sector might distort the allocation
of resources in the entire economy. The optimal entry fee (S*) that controls for the disparity
between firms’ equilibrium profit and W'(n) can be computed from equation (13), writing
x+n
∂x
∂nx
as
and the two first-order conditions for firms (i.e. as a function of τ and t):
∂n
∂n
S * (τ ,t) = (t − D ')[e +
∂e ∂x
∂nx
∂x
∂e ∂a
]+τ p
− n(1 − τ )p ' x + n[(t − D ')
].
∂x ∂n
∂n
∂n
∂a ∂n
(19)
From equation (19) we can establish a number of interesting insights for the optimal entry fee.
We assume that with no emission tax there is no incentive for abatement, which implies that
∂e ∂a
=0, and equation (19) can be simplified to:
∂a ∂n
S * (τ ,t = 0) = −D '[e +
∂e ∂x
∂nx
∂x
]+τ p
− n(1 − τ )p ' x
∂x ∂n
∂n
∂n
(20)
18
First when both t=0 and τ=0, S* is definitely a fee as long as e +
∂e ∂x
>0. The latter should hold
∂x ∂n
as long as total output (nx) increases with entry, and, therefore, emissions should increase with
entry as well, particularly since there is no incentive for abatement when t=0. The sign of S* is
ambiguous when τ>0 and t=0 or t=t*.8 The first and last term on the right hand side in equation
(20) is always negative under the conventional assumption of “business stealing” and emissions
not declining with total output in the industry. The second term on the right hand side depends
on the sign of τ. We will need an entry fee (S*<0) as long as τ≤0. Only when τ>0 is it
ambiguous what the sign of S* should be. If τ=τentry* and t=t* then S*=0, otherwise S* could
either be a fee or subsidy. We can summarize our findings in the following lemma and result 6:
Lemma 1:
S*(τ, t)<0 (a fee)
if
τ≤0
*
S (τ, t)>0 (a subsidy) if
τ>τentry* and t=t*.
S*(τ, t) ambiguous for all other cases.
Result 6: To guarantee the optimal number of firms in an imperfectly competitive market
with polluters we need to deter entry with an entry fee as long as the ad valorem tax is
negative. An optimal entry subsidy is required when an ad valorem tax is in place that is
larger than the optimal entry adjusting ad valorem tax and the first-best Pigovian tax is
used.
Finally we can discuss the case studied by Katsoulacos and Xepapadeas (1995) who suggested
using an entry fee in combination with a second-best Pigovian tax that underinternalizes the
externality (t<D’). Their suggestion can be evaluated by examining S*(τ=0, t<D’):
S * (τ = 0,t < D ') = (t − D ')[e +
8
When t=t*, equation (19) simplifies to:
∂e ∂x
∂x
∂e ∂a
] − np ' x + n[(t − D ')
]
∂x ∂n
∂n
∂a ∂n
S* (τ ,t * )= τ p
∂nx
∂x
− n(1− τ ) p ' x
∂n
∂n
.
(21)
The optimal entry correcting S* is definitely a fee as long as
∂a
≤ 0 . The latter is a plausible
∂n
19
argument if all firms are identical and a “business-stealing” effect cannot be ruled out. The
larger the number of firms entering, the smaller is an individual firm’s output, and, therefore, the
smaller is the need to abate emissions. With more than one type of technology, however, it
might very well be in the interest of some firms to increase abatement in order to maintain a
larger market share after entry.
4. Controlling entry and pollution with influence over pricing behaviour
Entry by itself is not a sufficient condition for a perfectly competitive market equilibrium. Firms
could collude after entry or an oligopolistic market structure with Cournot conjectures might
emerge for example. It, therefore, might be necessary to control firm behaviour with an ad
valorem tax/subsidy after entry. The optimal ad valorem tax is still determined as in
equation (8) but price is now exogenously determined by the zero profit condition and, therefore,
is itself a function of τ and S. The optimal entry fee/subsidy now only has to control for the
effect of the ad valorem tax on entry, because the ad valorem instrument ensures competitive
pricing.9 An ad valorem subsidy, for example, would attract excessive entry and would require
an entry fee in order to achieve the first-best number of firms in the industry. The optimal
combination of instruments with entry, therefore, consist of:
1. t*=D'(E*)
2. τ * =
9
p '(X)x
p '(X)x + p(X)
Excessive or insufficient entry due to the “business-stealing” or “business-augmenting” effect is only a problem if
perfectly competitive pricing does not emerge after entry and a first-best Pigovian tax is not used.
20
3. S = τ px
*
*
We can observe a direct relationship between the optimal ad valorem subsidy and entry fee. An
ad valorem subsidy increases the incentive for firms to enter, and must be counteracted with an
entry fee.
Result 7: The optimal regulation of homogenous polluters with endogenous simultaneous
entry requires a first-best Pigovian emissions tax, an ad valorem subsidy and a lump-sum
entry fee.
We cannot, as Shaffer (1995) proposes, choose between an entry subsidy/fee and an ad valorem
tax. Both are necessary because an ad valorem tax creates a distortion between the equilibrium
number of firms in the industry and the optimal number of firms.10 The results are consistent
with Katsoulacos and Xepapadeas (1995) and Xepapadeas (1997), which suggest a license fee
and a second-best under-internalizing emissions tax in lieu of a single emissions tax that exceeds
marginal damages. The second-best tax results, however, only hold under the assumption of a
strong “business-stealing” effect, and could call for an entry subsidy in combination with an ad
valorem tax if, for example, a weak “business-stealing” effect occurs or a “business-augmenting”
effect arises.
The important advantage of using an ad valorem subsidy and an entry fee is that we do
not need to predict if a “business-stealing” or “business-augmenting” effect will occur. The
latter are irrelevant if we have an instrument that guarantees competitive pricing after entry. The
first-best combination of taxes furthermore consists of simple taxes that are relatively easy to
derive. Second-best taxes are rather complex and rely on predictions of how the choice of t
affects entry and how entry affects output of individual firms (see for example Lee (1999)).
10
Shaffer’s model is missing the social welfare condition for the optimal number of firms in the industry. This
allows substituting entry fees/subsidies for ad valorem taxes/subsidies.
21
5. Regulatory Coordination and Sequential Introduction of Policy Instruments
The sequence regulators introduce their instruments and the authority over ad valorem taxes will
also have different impacts on the number of firms entering the industry, equilibrium prices,
output and emissions. Next we discuss implications under four different scenarios:
1. Entry and pricing after entry is determined first by a competition regulator that ignores
any pollution effects (either because it lacks a mandate or environmental information).
2. An environmental regulator introduces a Pigovian tax without considering any effects on
imperfect entry or pricing.
3. A competition regulator controls entry but not firm’s pricing or emissions after entry.
4. An environmental regulator introduces a Pigovian tax first and a third regulator
exogenously imposes ad valorem taxes.
5.1 Ad valorem subsidies and entry fees are first determined by a competition/industrial
regulator
The competition regulator would set τ=τ* and S*=τ*px. Since there is no Pigovian tax in place
firms will produce according to p=c', but not according to the socially optimal pricing of
p = c'+ D '
∂e
. This will lead to a suboptimally high level of production (unless firms will
∂x
choose to abate all of the emissions without output adjustments once a Pigovian tax is in place)
and excessive social marginal costs (see blockaded entry results). In order to determine the
discrepancy between the equilibrium and optimal number of firms entering we can derive the
following relationship between equilibrium profits under entry and the social welfare conditions
for entry (equation (15)):
∂W
∂e ∂x
= π n − D 'e − nD '
∂n
∂x ∂n
(22)
Equation (22) indicates that entry will be excessive (with a “business-stealing effect”) as long as:
e
∂e ∂x
.
>−
n
∂x ∂n
22
The entry of an additional firm causes other firms to reduce output, which in effect reduces
emissions. Too many firms will, however, enter if average emissions exceed the emission
reduction effect from “business-stealing”. Insufficient entry occurs, if total emissions are reduced
more than the emissions that are added by an additional firm entering the industry. With a strong
business-stealing effect, however, total output increases, and therefore total emissions should
increase more than the emission reductions due to entry of an additional firm, particularly since
there is no incentive for abatement. Competition policy without a Pigovian tax will, therefore,
lead to excessive entry.
5.2 Environmental regulator moves first (no ad valorem tax in place):
An environmental regulator without knowledge or mandate about imperfect competition would
choose the first-best Pigovian tax. Firms now set p + p ' x = c'+ D '
too little. There also will be excessive entry since
∂e
, and, therefore produce
∂x
∂W
∂x
= π n − np ' x . This example shows
∂n
∂n
that the clear advantages of introducing the Pigovian tax first when entry is blockaded are at least
partially offset by excessive entry with endogenous entry of firms. If, however, entry leads to a
competitive market structure, then it is better to introduce environmental policy first as
competition policy that disregards the effect of entry on emissions results in excessive entry,
while environmental policy does not.
5.3. Ad valorem subsidies are exogenously determined and competition regulator moves first:
A competition regulator could now only use entry subsidies/fees as a policy instrument. Since
she would disregard environmental damages, she would choose SComp to just control for the
distortion created by the ad valorem tax and the “business-stealing” effect. We can derive SComp
23
from equation (19) by ignoring any entry distortions due to excessive emissions:
SComp = τ px + n[τ p − (1 − τ )p ' x]
∂x
∂n
The latter is optimal as long as an environmental regulator also imposed the first-best Pigovian
tax. When the competition regulator moves first, however, the optimal entry subsidy/fee for an
exogenous τˆ needs to be determined by equation (20). The discrepancy between the subsidy that
ensures that the efficient number of firms enter and the subsidy chosen by the competition
regulator (SComp) is the same as with an ad valorem subsidy chosen by the competition/industrial
regulator, i.e. S * − SComp = −D '(e + n
∂e ∂x
) and will lead to excessive entry. This result is not
∂x ∂n
surprising because S is a function of τ, and it does not matter if τ=τ* or not. The competition
regulator always corrects the entry distortions caused by any ad valorem tax, but misses the
effects of entry on additional emissions in the industry and emission reductions per firm due to
“business-stealing”, and, therefore chooses the wrong entry subsidy or tax.
5.4. Exogenously determined ad valorem taxes and environmental regulator moves first:
An environmental regulator would ignore the effects of the ad valorem tax on emissions and the
substitution between abatement and output changes. A first best Pigovian tax would be chosen
with an ambiguous effect on entry as we can see from equation (19). A positive ad valorem tax
would now, however, act as an entry deterrent. Consequently there would only be excessive
entry as long as τ<τentry*. When τ>τentry* there could even be insufficient entry, and when
τ=τentry*, entry would be optimal, even when the environmental regulator moves first.
24
Proposition 2: Environmental regulation that precedes competition regulation
undoubtedly will lead to excessive entry under the conventional assumption of a strong
“business-stealing” effect as long as ad valorem taxes are not exogenously imposed and
entry will not lead to a competitive market structure. Competition policy that precedes
environmental regulation will always lead to excessive entry as long as a strong
“business-stealing” effect occurs.
IV. Concluding Remarks
Many environmental economists have praised the advantages of the Pigovian tax. It allows for
the optimal substitution between abatement and output reduction while at the same time making
firms internalize social marginal cost in their production decision. The nicest feature of the
Pigovian tax is that we don’t need information about marginal abatement cost as long as the tax
can vary with total emissions (if marginal damages are not constant for example). Second-best
Pigovian taxes or ad valorem taxes that also try to regulate emissions are not only less efficient,
they also require significantly more information such as output-emission relationships for each
firm, the magnitude of the “business-stealing effect” on firms output reductions and how
abatement per firm changes as the number of firms increase.
We often cannot predict how many firms will enter and how competitive firms will
behave after entry. Entry by itself might lead to a competitive market structure, particularly
when n* is relatively large or the efficient scale of firms is small. When a competitive market
emerges after entry, any Pigovian tax but the first-best tax will lead to suboptimal entry. The
same is true for competition policy that precedes environmental regulation with endogenous
entry. It will always cause suboptimal entry unless Pigovian taxes are simultaneously
introduced.
When entry is blockaded it is recommendable not to introduce ad valorem subsidies
before emission taxes because it would result in overproduction, and, therefore, excessive
25
capacity and excessive social marginal cost. A Pigovian tax would result in suboptimally low
production, but in a reduction of social marginal cost per unit of output. The exact welfare
effects of each distortion are an empirical issue. It is, however, more difficult to reduce excess
capacity (particularly of dirtier firms) than to subsidize production after the introduction of the
Pigovian tax.
In many regulatory situations we either have exogenously determined ad valorem taxes
(mostly for revenue reasons) or a lack of political and financial support for explicit subsidies.
This is less of a concern as long as entry is endogenous and our standard assumption of a strong
“business-stealing” effect holds. An ad valorem tax or an entry fee is then required to control
excessive entry in combination with a first-best Pigovian tax. It might be recommendable to
introduce a first-best Pigovian tax in the first stage and keep positive ad valorem taxes in order to
control excessive entry. In the second stage we could then either completely replace the ad
valorem tax by the first-best emissions tax if a competitive industry emerges, or adjust the ad
valorem tax to a second-best level that controls entry and imperfect pricing at the same time.
This would also have the positive effect of at least partially replacing a distortionary tax with an
efficient tax, while maintaining government revenues at the same time.
Future extensions could investigate the use of second-best ad valorem taxes in
combination with first-best Pigovian taxes, and explore the choice of regulatory instruments
analyzed in this paper in the context of strategic capacity investments by incumbents to deter
entry. The literature on the regulation of polluter in noncompetitive markets (including this
paper) has so far only focused on either blockaded entry without examining the exact reasons for
entry barriers or on simultaneous endogenous entry models. The case of sequential entry and
sequential introduction of regulatory instruments seems very relevant to many industries and
26
should receive more attention in future research.
27
References
Barnett, A.H. (1980). The Pigovian Tax Rule Under Monopoly. American Economic Review,
70, 1037-1041.
Baumol, W.J. and Oates, W.E. (1988). The Theory of Environmental Policy (second edition),
Cambridge University Press, Cambridge.
Buchanan, J.M. (1969). External Diseconomies, Corrective Taxes, and Market Structure.
American Economic Review, 59, 174-177.
Dixit, A (1980). The Role of Investment in Entry-Deterrence. The Economic Journal, Vol. 90,
No. 357, pp. 95-106.
Hart, O.D. (1979). Monopolistic Competition in a Large Economy with Differentiated
Commodities. Review of Economic Studies, Vol. 46, 1-30.
Katsoulacos, Y. and Xepapadeas, A. (1995). Environmental Policy under Oligopoly with
Endogenous Market Structure. Scandinavian Journal of Economics, 97 (3), 411-420.
Lee, Sang-Ho. Optimal Taxation for Polluting Oligopolists with Endogenous Market Structure,
Journal of Regulatory Economics, 15, 85-100 (1999).
Mankiw, N.G. and Whinston, M.D. (1986). Free entry and social inefficiency, Rand Journal of
Economics, Vol. 17, No.1, Spring 1986, 48-58.
McKitrick, R. and Collinge, R.A. (2000). Linear Pigovian taxes and the optimal size of a
polluting industry, Canadian Journal of Economics, Vol. 33, No.4, 1106-1119.
Novshek, W. (1980). Cournot Equilibrium with Free Entry. Review of Economic Studies, Vol.
47, 473-487.
Pigou, A.C. (1932). The Economics of Welfare (fourth edition), MacMillan, London.
Seade, J. (1980). On the Effects of Entry. Econometrica, Vol. 4, 479-490.
Shaffer, S. (1995). Optimal Linear Taxation of Polluting Oligopolists. Journal of Regulatory
Economics, 7, 85-100.
Simpson, R.D. (1995). Optimal Pollution Taxation in a Cournot Duopoly. Environmental and
Resource Economics, 6, 359-369.
28
Spence, M. (1977). Entry, Capacity, Investment and Oligopolistic Pricing. The Bell Journal of
Economics, Vol. 8, No. 2, pp. 534-544.
Tirole, J., 1988. The Theory of Industrial Organization. The MIT Press: Cambridge,
Massachusetts.
Xepapadeas, A. (1997). Advanced Principles in Environmental Policy, chapter 5, pages 170-190,
Edward Elgar Publishing: Northhampton, Massachusetts.