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The Optimal Tax on Products
with Pirates is Low
Vesa Kanniainen
CESifo GmbH
Poschingerstr. 5
81679 Munich
Germany
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+49 (0) 89 9224-1410
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The Optimal Tax on Products with Pirates is
Low
Vesa Kanniainen
University of Helsinki, CESifo and HECER¤
February 1, 2009
Abstract
Pirate products reduce the monopoly power of innovators providing access to low price varieties. They, however, reduce the incentives
for innovations, threaten the governments’ tax revenues and provide
means for false signalling within social groups. The optimal tax on
products with pirates is shown to be low for these reasons but should
take account of the issue of how much to invest in detecting the illegal
producers. A government who cares for people’s morality invests more
in detection but collects more tax revenue even when it is willing to
collect less.
Keyords: pirates, tax revenue, optimal tax
1
Introduction
Pirate products press consumer prices. From the point of view of low income
consumers, this is welcome as it helps to control for the market power of
monopoly producers. However, such a disciplinary impact weakens the pro…ts
of the original innovator-producer destroying the incentives to innovate in
the …rst place. Moreover, tax revenue is reduced if many people buy those
products. There is also a morality issue. Many people refuse to buy pirate
¤
Department of Economics, P.O.Box 17, FI-00014 University of Helsinki, phone:
+358919128725, fax: +358919128736, e-mail: vesa.kanniainen@helsinki.…. A grant from
the Yrjö Jahnsson Foundation is gratefully acknowledged.
1
products for the reason that it is against their moral values. However, if
the reputation of being a moral person has value in private interactions,
immoral people may pretend to be moral ones creating a wrong signal of
their type in the social contexts. For those reasons hence, pirate products
are condemned in the society. The question is how a society should invest
resources to combat against them. Second, should the tax on the product
which is threatened by a pirate to have a special treatment?
Though there are reasons to control for and catch the producers of pirates,
the issue of imitation is more complex. After all, all current innovations
are built on existing information and previous innovations. In a sense, and
throughout the human history, all current economic activity is based on
imitating the ideas of the chain of the previous generations. Nobody can
truly innovate from scratch! All innovations are fundamentally incremental.
This fact would favor a norm of free imitation as no innovator develops the
full value-added without the past knowledge. However, there is a social cost
in terms of reduced incentives to innovate if imitators can freely enter. In
the case of literature and music, the societies have come to the conclusion
that temporary copyright protection is a workable social norm. As imitation
often improves the quality of products, imitation must be allowed in the
end. However, copying without any improvement should - perhaps - not be
accepted as a norm.
The theory of optimal commodity taxation when products are the subject
of potential imitation has not studied the tax e¤ects on the innovation incentive in the …rst place not to mention adjustment for the cost of detection the
producers of the pirate products. As the theory of optimal commodity taxation appears to have missed this point, this paper will focus on this problem.
The availability of pirates has an impact on the demand elasticity of the legal
products and the optimal commodity tax theory captures this mechanism.
This is not su¢cient, however.
In democracies, government however often are short-sighted and do not
pay attention to the policy e¤ects beyond their period they are in power.
The innovation incentives are precisely of such type as innovations are typically not created in a short period. Innovations mature over long terms.
Consequently, a revenue-maximizing tax rate tends to be too high. Such
government may not have incentives to invest in a costly catching technology which reduces the resources available to them. A long-sighted government, in contrast, pays attention also to the economic e¤ects in the long
run. In particular, it recognizes that the innovation intensity tends to have
2
a favorable impact on the tax revenue over the longer term. Therefore, the
revenue-maximizing tax rate tends to be lower. Moreover, it desires to limit
the imitation intensities recognizing that the imitation activities reduce the
innovation e¤orts in the …rst place. Thus, such a government tends to invest
resources in catching technologies.
We also address the questions how morality a¤ects the optimal tax rate.
Are morality and investment in detections substitutes or complements?
2
Model
2.1
Firms
There is one innovator and one imitator. There are two stages, an innovation stage and the market stage. The model is solved by backward induction.
Innovation e¤ort requires an investment but is risky. The success probability
the choice variable of the innovator but leads to increasing cost,
¡ 1 ¢ is
2
0 Imitation is without cost requiring no investment. How2
imitator has to invest in a hiding e¤ort,
at an increasing cost,
¡ever,
¢ the
1
2
This
is
because
the
imitator
has
to
organize
the sales through the
2
shadow markets. The success of the imitator is conditional on the success of
the innovator. The variable is the government’t investment in a atching
technology.
If the innovation is successful and is not succesfully imitated, the innovator is entitled to monopoly pro…t
in the market stage. The government
is able to collect a turnover tax only on the legal product. If the imitation
is successfully carried out, the …rms compete as a duopoly in the consumer
market with duopoly pro…ts denoted by 1 and 2 The illegal …rm pays
no tax.1
2.2
Innovation stage
The innovator’s expected pro…t over the both stages is
µ ¶
1
¦ = [ ¡( ¡ 1 ) ]¡
2
1
2
In an earlier paper, Kanniainen and Stenbacka (2000) analyzed innovations and legal
imitation. They abstracted from illegal pirates and the issue of optimal taxation.
3
where 1 is the pro…t of the innovation in the market stage. His …rst-order
condition for the innovation e¤ort, the reaction function
¡(
=
¡
1
)
The imitator’s pro…t is conditional on the innovator’s success and is
µ ¶
1
2
¦ =
2 ¡
2
where 2 is the imitator’s pro…t in the market stage. His …rst-order condition
for the hiding e¤ort, the reaction function
2
=
Thus, the catching probability operates as a discipline for the imitator. Natural conditions introduced are
·1
·1
Nash equilibrium in the innovation stage The innovation e¤ort is
given by
=
=
¡(
¡
+(
¡
1
1
)
)
2
2
The imitator’s investment in hiding is given by
2
=
2
=
¡
+(
1
)
2
Taking the derivative,
=
=
[
2
¡
+(
¡
1
2
4
)
[ ]2
0
1
2
]¡
2
The positive sign is guarenteed when
=
is su¢ciently high. Moreover,
2
0
1
The innovator is hence sensitive to the imitator’s success. The greater is
the monopoly pro…t,
and the higher is the duopoly pro…t of the competitor, 1 the more is he willing to invest.
The government policy ( ) a¤ects both producers. Interestingly, there
are two opposite e¤ect from on the innovator. Why?
With perfect copyright protection, the innovator has access to monopoly
pro…t choosing the innovation e¤ort as
µ ¶
1
2
¦ =
¡
2
or
=
¡(
¡
1
)
A guaranteed monopoly thus provides stronger innovation incentive. The
comparative statics on
is straightforward.
2.3
Consumers2
People di¤er in their attitudes towards illegal products. Some may …nd it
valuable to misrepresent their hidden preferences for a confused signal to
their co-citizens. There are two types of consumers. An -type has moral
preferences; a -type does not.3 The mass of -type consumers is in the
2
The idea of consumer signalling introduced here is discussed in another paper by
Glazer, Kanniainen and Poutvaara (2008) but in another context.
3
The origin of ethical preference lies beyond our scope. A natural source is that the
preferences are created by evolutionary mechanisms among human beings becoming integrated into a social contract, cf. Binmore (1998). It is appropriate to think that the
ability to commit to a social norm and the option to participate in a boycott develop like
a social meme introduced by Dawkins (1976) and elaborated by Blackmore (1999).
5
model; the mass of -type consumers is scaled to 1.4 Each consumer buys
at most one unit of the good. In the two groups, consumers are indexed in
decreasing order on [0 ] and [0 1] with respect to their basic willingness
to pay for the product. Consumers = 0 and = 0 have the highest basic
willingness to pay for the product, say in each group; consumers =
and = 1 have zero willingness to pay for it. The willingness to pay by the
remaining consumers is uniformly distributed on (0 1) in both groups. To
illustrate, and ignoring moral and reputational e¤ects for the moment, the
utility from consumption by consumers, say and are given by indirect
utility functions
= ( ¡ ) ¡ and = (1 ¡ ) ¡ where is the market
price.
The products become di¤erentiated if one product is a pirate while the
other is a legal one. In the social context, inviduals may view it important
to be considered moral so as to avoid exclusion from particular social groups,
loss of friendship, and even barriers in the marriage market. We let
0
denote the cost imposed on a consumer who buys a pirate product. This can
be thought of as a social pressure.
2.4
Equilibrium
We consider now the case where one of the …rms is succesful in innovating,
thereby incurring an innovation cost, while the other imitates. The products
of the …rms, though perfect substitutes in consumption, di¤er with di¤erent
images. Some consumers will refuse to buy the pirate product, of the imitating …rm say , and consider buying only from the …rm which innovates,
say . More speci…cally, the high-moral consumers abstain from buying at
…rm and buy only at …rm . Since in equilibrium not all may buy, we
denote the number of active high-moral buyers by . The number of lowmoral consumers who stay at …rm is denoted by
Some of the low types,
however, buy the legal product at the …rm in order to (falsely) signal high
morality. Their number is denoted by
.
A tax is collected on the legal product but not on the illegal one. We
denote the resulting producer price at the legal …rm by
and the producer
price at the non-ethical pirate producer by . The resulting market equilibrium has the following structure. With the tax rate on legal products
4
The population hence consists of a mixture of individuals of homo moralis and homo
oeconomicus types.
6
and from the de…nition of the marginal moral consumer
the equilibrium price at …rm
satis…es
(1 ¡
)=
5
+
, we know that
(1)
where (1 ¡ ) is the willingness to pay by the marginal moral consumer
and
+ is the consumer price of the legal product. Thus, the producer
price is
= (1 ¡ ) ¡
We can of course have a market equilibrium where no low-moral consumer
buys at …rm . However, to make the analysis interesting, we assume that
the bene…t from signalling is su¢cient so that some, i.e.
0 do. The
marginal low-moral consumer must be indi¤erent between the two markets.
Thus, consumer prices must satisfy
(1 ¡
where, to recall,
Therefore,
)¡(
+ ) = (1 ¡
)¡ ¡
(2)
is the social pressure when a consumer buys at …rm L.
Lemma 1. The di¤erence in the consumer prises arises from the cost of
social pressure adjusted for the commodity tax,
¡
= ¡
6
(3)
There is another marginal low-moral consumer (with an index =
+ )
who is indi¤erent between buying at …rm or buying nothing. Thus, his net
utility is
(1 ¡
¡ )¡ =
(4)
5
Therefore, not all moral consumers buy. They all buy only if
= , making
+ =
0 Otherwise,
6
We notice that all low-moral types are indi¤erent between the two markets as the
social cost of pressure just matches the price di¤erence adjusted fo the tax rate. Each …rm
chooses its output knowing the consumers’ behavior.
7
2.5
Equilibrium in the Market Stage
In the Cournot model, …rms decide on their outputs allowing the prices to
adjust.7 We denote the outputs of the innovator and the imitator by
and
Then,
= +
=
(5)
Thus, the number of active moral buyers is
=
¡
To solve for
the prices, we …rst determine the number of signalling consumers. Using
¡
= ¡
(1 ¡
) ¡ ¡ (1 ¡
¡
)+ = ¡
or
(1 ¡
+
) ¡ (1 ¡
¡
) = 0
we have that
¡
+ (1 + )
+
=0
This gives for the number of signalling consumers (noticing
=
¡
1+
=
)
(6)
With the producer price
= (1 ¡
)¡
with
¡
+
1+
=
=
7
The behavior of …rms in duopolistic markets has been subject to some debate, cf.
Kreps and Sheinkman (1983). Güth (1993) shows how quantity competition can be justi…ed without the complexities discussed by the earlier literature.
8
we obtain the producer prices
=
(1 ¡
=
(1 ¡
)¡
+
1+
)¡
and
¡ +
=
=
+
1+
(1 ¡
)¡
Production cost is taken to be zero. Then, the pro…ts after a successful
innovation and imitation are
=
·
=
=
·
=
+
1+
¡
¡
+
1+
¡
¡
(7)
¸
(8)
(9)
¸
Reaction functions are,
+
1+
=
¡
=
¡ ¡2
µ
¡
¡ +
1+
¡
+
1+
=
¡
=
¡ ¡2
1+
1+
¶
=0
1
1+
¡ ¡
¡
1
1+
1+
=0
To solve for the Nash-Cournot equilibrium, the …rst of those conditions
gives:
µ
¶
2
=
1¡
¡
1+
1+
9
or
µ
1+
=
2
1+
=
2
1¡
¡
2
1+
¡
¶
¡
(1 + )
2
1+
2
The second gives
2
=
1+
¡ ¡
1+
(1 + )( ¡ )
=
¡
2
2
Inserting,
1+
(1 + )( ¡ )
1+
¡
+
¡
2
4
4
2
(1 + ) ( + ¡ 2 )
=
3
=
gives
=
(1 + ) ( + ¡ 2 )
3
=
(1 + )( ¡ )
¡
2
2
Similarly,
(1 + )( ¡ )
¡
2
2
·
¸
¡2 +
= (1 + )
3
=
We then obtain the solutions for the outputs:
Proposition 1. The Cournot-Nash equilibrium is given by the market
shares
(1 + )( ¡ 2 + )
=
3
(1 + )( + ¡ 2 )
=
3
10
Prices Then, the total output in the market will be
+
=
(1 + )( ¡ 2 + ) (1 + )( + ¡ 2 )
+
3
3
(1 + )
[2 ¡ 2 ¡ ]
3
Then prices can be solved,
=
=
=
=
Price e¤ects of the tax
+
1+
+2¡2
3
(1 ¡
)¡
+
)¡
1+
+2+ ¡3
3
(1 ¡
Taking derivatives we …nd
= ¡2 3
= 1 3
Thus, we …nd that a tax policy has a greater impact on the price of the
legal producer than on the illegal producer. For some reason thus,
Lemma 1. Pirates are more isolated from the tax e¤ect than are the legal
products.
Market pro…ts As
=
=
we obtain
=
=
(1 + )( + ¡ 2 ) ( + 2 ¡ 2 )
9
11
=
=
(1 + )( ¡ 2 + ) ( + 2 + ¡ 3 )
9
Morality and signalling The amount of false signalling is given by
¡
1+
= ¡ (1 ¡ 2 ¡ )
=
(10)
(11)
Propositio 1 Societies with a large moral population have less those
immoral who send a false signal of being moral.
The intuition for this unexpected result has to be worked out. Moreover,
Proposition 2 Societies with a high tax rate have less those immoral who
send a false signal of being moral.
Proof. Solving
µ ¶
1
=
[(1 ¡ ) + (1 + 2 ) ¡ (2 ¡ )]
3
Thus,
= ¡(2 ¡ )
0
Proposition 3 Societies with a strong moral pressure have more those
immoral who send a false signal of being moral.
Proof. From above,
= (1 + 2 ) 0
Monopoly pro…t Suppose that the imitator is caught for producing illegal private products (which are destroyed). The innovator then is the sole
producer in the market. Both types of consumers must visit him. In both
consumer classes, there is a marginal one who is indi¤erent between buying
at the market price
and not bying.8 We denote the marginal moral consumer by and the marginal immoral consumer by
Then, the willingness
to pay by the marginal moral consumer must satisfy
(1 ¡
8
)=
The demand curve becomes now ‡atter.
12
+
(12)
This gives
1¡
+
=
µ
1¡
=
¶
+
The willingness to pay by the marginal non-moral consumer is
(1 ¡
)=
+
Total output then is
=
+
=
¡
or
Therefore, it holds
(1 ¡ (
or
¡
)) =
= (1 ¡ (
Inserting
(1 ¡
Ã
=
µ
1¡
=
¡
+
)) ¡
µ
¡
1¡
1+
¡
+
(1 + )
1+
!
¶¶
)¡
The monopoly pro…t is
=
=
Ã
1¡
1+
13
¡
(1 + )
1+
!
Taking the derivative, we obtain the pro…t-maximizing monopoly output
Ã
!
(1 + )
=
1¡
¡
¡
=0
1+
1+
1+
1¡
1+
¡
(1 + )
1+
(1 + ) ¡ 2
¡
= 0
1+
¡ (1 +
) = 0
(1 + ) ¡ (1 + )
=
Recalling the monopoly price
=
=
Ã
µ
2
1¡
1+
¡
1 1 1+
¡
2 2 1+
The monopoly pro…t thus is
(1 + )
¶
1+
!
=
h
i
(1 + ) ¡ (1 + ) µ 1 1 1 + ¶
=
¡
2
2 2 1+
µ
¶2
1+
= (1 + )
1¡
4
1+
It is expexted that
However, if the moral e¤ect
proaces the monopoly pro…t.
3
is strong enough, the duopoly pro…t ap-
Short-Sighted Government, Nash Equilibrium
Governments are often short-sighted in democracies. To examine the e¤ects
of tax and detection policies on pirates, we consider …rst the case of a shortsighted government who does not care of the morality in the society. We
14
view the government hence as maximizing its expected tax revenue net of
the detection cost,
µ ¶
1
2
[ ]=
(1 ¡ )
¡
1 +
2
A short-sighted government maximizes T but does not take into account
the impact of its tax policy on the innovation incentives. The resulting
equilibrium is a Nash equilibrium. Recall the pro…ts in the market stage,
1
=
(1 + )( + ¡ 2 ) ( + 2 ¡ 2 )
9
µ
¶2
1+
= (1 + )
1¡
4
1+
The tax thus reduces the pro…ts of the innovator in both cases,
0
1
0
However, the investment in detection only operates as as pre-emptive measure
on the imitator. A short-sighted government hence abstracts from it and
does not care of long-term innovations either. Taking derivatives
the
optimal policy satis…es
[ ]
=
=
1
(
1
+ (1 ¡ )
+
+ (1 ¡ )
)+
1
µ
+
1
(1 ¡ )
+ (1 ¡ )
=0
¶
and
[ ]
We hasten to state,
=¡
0
Lemma 2. The short-sighted government does not invest in detection.
We plan to prove,
Lemma 3. There is a positive optimal tax rate.
15
Proof. Two conditions are needed, (i)
[ ]
0 (ii) [ ] is convace
2
[ ]
0 Note that (i) is satis…ed because the second term is zero at
the origin. (ii) Now
2
=
1
(1 + )( + ¡ 2 ) ( + 2 ¡ 2 )
9
= (1 + )
µ
4
Their derivatives are
1
¶2
1+
1¡
1+
(1 + )
[¡2 ( + 2 ¡ 2 ) ¡ 2( + ¡ 2 )]
9
2(1 + )
= ¡
[2 + 2 + ¡ 4 ] 0
9
=
= ¡ (1 + )
2
The second derivatives are
2
µ
1
2
1¡
2
= (1 + )
µ
2
1+
1+
1+
¶
0
8(1 + )
9
=
2
¶µ
1+
1+
1+
Su¢cient conditions for concavity
2
[ ]
1
2
0
2
¶µ
1+
1+
¶
0
are
2
1
+
2
+
2
0
To elaborate,
()
1
2
+
1
2
8(1 + )
2(1 + )
[2 + 2 + ¡ 4 ] +
9
9
= ¡4 ¡ 4 ¡ 2 + 16
= ¡
16
Take an example,
=1
= 0 5; then this expression is
¡8 ¡ 2 + 8 = ¡2
0
Similarly,
2
( )
+
2
= ¡ (1 + )
2
µ
+ (1 + )
= ¡1 + 2
Now, ¡1 + 2
3.1
0 if
2
1+
1+
¶µ
1+
1¡
1+
1+
µ
¶µ
¶
1+
1+
1+
¶
1+
1+
0 5 This complets the concavity analysis.
Expected Tax Revenue
With a short-sighted government, nothing is invested in catching, = 0
Then, it is optimal for the imitator to choose = 1 and the imitators pro…t
becomes certain, ¦ =
0 With such a high imitation intensity, the
2
innovator’s e¤ort is given by
¡(
=
=
¡
1
)
1
which is the bene…t-cost ratio of a successful innovation. The expected revenue of a short-sighted government is
[ ]=
3.2
2
1
Long-Sighted Government: Stackelberg
We now consider a long-sighted government who chooses the tax rate and the
catching probability of pirate producers to maximize its tax revenue but who
understands that innovation incentives are important also for its tax revenue.
We also assume that the government actually cares about the morality in the
17
society. This will be modelled by the idea that the government overprices
every tax penny but underprices every penny which is invested in detection.
Below, these e¤ects are introduced by parameters 1 1 2 1
The desire to have large tax revenue may arise from the desire to distribute income transfers or from the Leviathan motive. The government,
however has to take into account the threat of the pirate products and a
loss in tax revenue. It pays to invest part of tax revenue to catch the illegal
production and so better control the tax revenue. The government operates
as a Stackelberg leader, choosing simultaneusly the tax rate,
and the investment in the catching technology, As the latter is subject of inceasing
2
the tax should be chosen optimally jointly with the investment
cost 12
in the catching technology.
The government’s objective now is
µ ¶
1
2
[ ] = 1[
(1 ¡ )
]¡ 2
1 +
2
In the …rst-order conditions of the long-sighted government, we have some
extra terms.
[ ]
µ
= 1[
µ
+
+
1
(1 ¡ )
¶
1
¡
= 0
[ ]
=
= 0
1
·µ
1
+
1
+
¶
¶
+
1
+
+ (1 ¡ )
µ
(1 ¡ )
as the pro…ts are
1
=
(1 + )( + ¡ 2 ) ( + 2 ¡ 2 )
9
= (1 + )
4
µ
18
1¡
1+
1+
¶2
1
+
+ (1 ¡ )
¡
¶ ¸
]
¡
2
Rewriting the terms in the …rst-order conditions of the long-sighted government, the marginal increase tax in the expected tax revenue when the
adjustment of the innovation and imitation incentives are appreciated, is
µµ ¶
¶
µ
¶
+
(1 ¡ ) ¡
1 +
1
µ ¶
µ ¶
=
( 1 + (1 ¡ ) ) ¡
( ¡ 1 )
0
¡ ¢
¡ ¢
This is negative as
0
0 and ( ¡ 1 )
0 We have
proved
Proposition 5 A long-sighted government always chooses a lower tax
rate than a short-sighted government.
The intuition is that the government pays attention to the expact of tax
on the innovation incentive in the …rst place. But how much to invest in
detection? Reorganizing the terms in the …rst-order condition with respect
to we obtain
1
=
1
·µ
·µ
1
¶
(
+
1
1
¶
+
+ (1 ¡ )
µ
) ¡
(1 ¡ )
µ ¶
¶ ¸
¡
(
¡
1
)
¸
0
³ ´
0 That a long-sighted government chooses a posThis follows from
itive detection rate is equivalent to the condition
³ ´
³ ´
µ ¶
( 1 + (1 ¡ ) ) ¡
( ¡ 1 )
1
=
2
In this case, the optimal size of depends negatively on . One implication
is that with
0, the innovation intensity is increased. A long-sighted
government invests in detection if
µ ¶
µ ¶
( 1 + (1 ¡ ) ) ¡
( ¡ 1 )
0
We can also prove,
Proposition 6 A government who cares about morality of people invests
more in detection. It also manages to collect more tax revenue.
³ ´
1
Proof. The result follows from 12
19
4
Concluding remarks
Pirates should be condemned as they reduce the innovation incentive in the
…rst place. However, it is not possible to have a perfect shelter as increasing
the catching probability means increasing costs. The loss of tax revenue must
be weighted against the bene…ts of additional detection.
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