Consumers’ choices, infringements and market competition

Consumers’ choices, infringements and marketcompetition

Fav Tsoin Lai • Shun-Chiao Chang

Published online: 15 February 2011

� Springer Science+Business Media, LLC 2011

Abstract The study fits the individuals’ characteristics of consumption into an

analysis of the demands for both genuine goods and counterfeit goods. The con-

sumers’ substitutability of genuine goods for counterfeits and attitudes toward a

dispersed consumption of counterfeit varieties are the dimensions that affect the

niche markets for counterfeits. We show that it is not necessary to increase the

competition among counterfeits to reduce the demand for individual counterfeits if

at the margin the variety of counterfeits enhances the value of consuming the good.

The enforcements against counterfeiting deter the number of counterfeit firms, but

encourage the output of individual counterfeits if the market includes a significant

number of counterfeiters. The optimal private enforcement against counterfeiting is

also fully discussed in the model.

Keywords Consumption � Genuine goods � Counterfeits � Enforcements

JEL Classifications F23 � K42 � L53

1 Introduction

In China, more than half of mobile phone users are using products that may

involve counterfeit goods, or patent infringements and that may have a high

proportion of the pricing/function associated with the white-box, gray or black

F. T. Lai (&)

Department of Economics, National Chi-Nan University, Nan-Tau, Taiwan

e-mail: [email protected]

S.-C. Chang (&)

Department of Management Administration, National Taiwan University of Science

and Technology, Taipei, Taiwan

e-mail: [email protected]

123

Eur J Law Econ (2012) 34:77–103

DOI 10.1007/s10657-011-9225-z

markets.1 The products impinge upon the intellectual property rights of genuine

goods as if they were bandits operating in areas where the law regarding property

rights has broken down. The infringements of intellectual property rights now

encompass China’s 3C products, and more and more industries are being affected.

The rapid growth of counterfeiting activities is not confined to China but is also

emerging in the highly outsourced-supply developing countries such as India.2

The infringement of property rights has gone global and now accounts for a large

share of world trade.3

In developed countries, the law heavily condemns the infringement of intellectual

property rights and its strict enforcement deters the direct production of counterfeit

goods. However, because of the price advantages, huge quantities of products are

imported even though it is not pleasant to see people consuming these goods in full

view of the public. Some developing countries conduct a considerable amount of

outsourcing activity, while only loosely enforcing the protection of intellectual

property rights. In these countries, the infringements take advantage of the

technology that is transferred and the associated activities become increasingly

rampant. For example, producing and consuming the goods are usual events in

China, a country that is a key supplier of outsourced goods and also the primary

source of counterfeit goods.4

The activities associated with infringing property rights create a huge product

value for some developing countries, and give rise to very high costs for some

developed countries such as European countries and the U.S., etc.5 Counterfeiting

1 In China, people refer to products like this as ‘‘shanzhai’’, which in Chinese means a bandit stronghold

in the mountain. These products infringe the property rights of branded firms in relation to both software

and hardware. For instance, several shanzhai mobile phones, e.g., Hiphone, Ophone, or Uphone instead of

iphone or NCKIA rather than NOKIA, as well as PC/NB, Netbook, GPS, monitor and TFT-LCD TV, etc.,

give rise to trademark infringements with confusing similarity. The phones also duplicate some of the

functions that are patented by brand firms such as Apple, Nokia and Samsung.2 For example, as the 3C hardware industries become more mature, shanzhai 3C products manufactured

in China have become more popular in the markets in China, India, Pakistan and Russia in the last

4 years. The mayor of Shenzhen city estimates that there were more than 0.2 billion shanzhai mobile

phones produced for nearly 1 million families in 2008 and nearly ninety thousand types of mobile phones

in 2009 in Guangdong province alone. We can expect that shanzhai 3C products could constitute one of

the business innovation models in the developing countries that must be able to significantly affect the

development of the global IT industry.3 OECD (2007) announced that the international trade in counterfeit and pirated products could have

reached $200 billion in 2005. The Business Software Alliance (BSA) and International Data Corporation

(IDC) estimated the total value of software piracy to be $53 billion in 2008 while customers paid $88

billion for the use of copyrighted software in the same year.4 Chaudhry (2006) argued that one of the main reasons why China has become the primary source of

counterfeit goods is the lack of intellectual property rights (IPR) enforcement. If China is willing to

enforce intellectual property rights, the infringements of the right should decrease. For example, the

Chinese government fiercely protected its Olympic logo, and punished the selling of fake Olympic items

much more severely than of other counterfeits. China has thus successfully protected its property rights in

relation to its Olympic logo.5 For example, more than $16 billion worth of counterfeit goods are sold each year inside China. The US

trade representative based in China reckons that the amount that American industries lose to counterfeit

goods ranges from $200 to $250 billion (The Economist 2003/5/17). The IT software firms could lose

nearly $100 billion per year according to KPMG and the Alliance for Gray Market and Counterfeit

Abatement (Bednarz 2006). The International Chamber of Commerce in Geneva estimated that the global

78 Eur J Law Econ (2012) 34:77–103

123

thrives on eroding the profit of the property right owner, and hence it is not a

victimless crime. When buying a counterfeit good, the consumer may or may not

know the participants in the property right infringement activities. In cases where

the consumers buy the counterfeit good not knowing it is a fake, they themselves

become victims of counterfeiting. However, for a large part of the transactions,

there is no deception involved, and the consumer is an accomplice in the product

counterfeiting. Intentionally purchasing a counterfeit good at a greatly discounted

price compared to the price of a genuine product enables the buyer to consume a

different combination of items from what the genuine good can come up with.

One of the most famous realities recently is that nearly 10 percent (4 million

users) of Apple iphone owners have installed Cydia, an unauthorized App store

application. This kind of device enables iphone consumers to access several types

of software that Apple will not allow. Apple’s IPR is infringed as they jailbreak

their smartphones using specific freely available software tools provided by the

hacker group iphone Dev-team (Wired, February 13; August 6, 2009) (Chen

2009).

In this paper, we analyze competition between a genuine branded product and its

counterfeits as a group and competition among the counterfeit goods. A model is

developed from which we derive long-run equilibrium outputs of the genuine

product and the total output of counterfeits based on monopolistic competition

among counterfeit producers. With public enforcement and free entry in counter-

feiting, there is a long-run relationship between the number of counterfeit producers

and exogenous public enforcement (specifically, the level of enforcement and fines).

This relationship feeds into short-run equilibrium reduced form relations between

outputs of the genuine and counterfeit goods and the demand parameters, the

number of counterfeiters and public enforcement variables. A major finding is that

increased public enforcement reduces the number of counterfeiters and the total

supply of counterfeit goods, while increasing the output of each active counterfeiter.

Thus, increased public enforcement has the effect of raising the output and price of

the genuine good though the negative effects on total counterfeit supply. We also

look at the implications of private enforcement here based on the assumption that

the fines are exogenous under two situations. The branded product supplier chooses

the profit-maximizing level of enforcement under two alternative assumptions:

(Eq. 1) the fines are paid to the state; and (Eq. 2) the branded product enforcer

retains the fines. First, we find that a higher degree of product differentiation

between counterfeit varieties pushes the price of the individual counterfeit higher,

pulls the output down and in turn brings about a larger market size for the genuine

good. Increases in the differentiation thus strengthen the property right owner’s

interest in protecting his property and motivate him to invest more in enforcement.

Moreover, we find that awarding the compensation to the property owner affects his

private enforcement based on two considerations. On the one hand increases in the

Footnote 5 continued

value of counterfeit products may exceed $650 billion per year (Kurtenbach 2006). OECD (2007) also

announced that the international trade in counterfeit and pirated products could have been as high as $200

billion in 2005. It is not hard to imagine how huge the total value of products whose property rights have

been violated must be.

Eur J Law Econ (2012) 34:77–103 79

123

enforcement increase the apprehension rate and boost the expected compensation.

On the other hand, bolstering the enforcement deters some of the counterfeit

activities and consequently decreases the expected compensation. The both

considerations have opposite effect on right owner’s private enforcement.

The remainder of this paper is organized as follows. Section 2 provides a

literature review and Sect. 3 presents the main structure of the model. In Sect. 4, we

discuss the responses of the counterfeiters and the firm manufacturing the genuine

products. Section 5 includes the case where the firm producing the genuine good

can exert private or public enforcements against the counterfeiters. The final section

presents the conclusions.

2 Literature background

The economics literature regarding counterfeits focuses on status goods where it is the

label and identifying design characteristics that are of themselves of value to

consumers. Grossman and Shapiro (1988a, b) point out that counterfeits provide the

advantage of function, which unbundles the status and quality features of the branded

or patented ones via the price differentiation between the fake and genuine products.

An individual with low income trades off quality for status; however, an individual

with medium or high income keeps both. For instance, many iphone owners prefer

to install genuine application software from the App store and counterfeit or

unauthorized products from Cydia. Andres (2006) indicates that the consumer’s

income is a key determinant of purchasing counterfeit software. Higgins and Rubin

(1986) explore the aspect that counterfeiting reduces the distinctiveness of the

genuine good, in addition to emphasizing that the individual’s taste for the exclusivity

signaled by the goods hinges on his income level. The special characteristics of the

demand for status goods explain how the substitution of a counterfeit good for a

luxury genuine good could occur. However, it is still possible for a counterfeit good to

be a complement to the status good.6 More importantly, the markets for counterfeits

are no longer confined to status goods but are more diverse and larger in scale.7

Counterfeit goods range from the small scale copying of a luxury good, to a large

scale brand (patent) infringement of low- (high-) tech products. As Chaudhry and

Walsh (1996) point out, providing alternatives to the genuine product is simply what

counterfeiters aspire to achieve.8 Consumers may consume counterfeit goods and

genuine goods as long as neither good is a perfect substitute for the others. Not all

6 For example, Higgins and Rubin (1986) provide data indicating that 27% of the buyers of the

counterfeit goods already owned genuine Rolexes.7 Counterfeit goods have become increasingly rampant over the years: from small scale copying of

luxury goods in the 1980s, to large-scale production of low-tech products in the late 1980s and 1990s, and

to the much larger scale production of high-tech electrical products (Chaudhry 2006). In 2004, the

European Union experienced rapid growth of counterfeits with data on seizures revealing a dramatic

percentage increase in counterfeits in the sectors for electrical equipment (?707%) and computer

equipment (?899%) (EU Taxation and Customs Union 2005). Counterfeit goods account for 12% of

products being sold in the toys and sports marketplace in the UK (Jones and Cline, 2002).8 Counterfeit products can be classified into four distinct types: (1) True counterfeit products that look as

much like the original as possible and use the same brand name. (2) Look-alikes that duplicate the

80 Eur J Law Econ (2012) 34:77–103

123

goods such as status goods have such strong consumption dispersion that consumers

will consume either genuine goods or counterfeit goods. The qualities of both goods

represent their vertical characteristics, and the pattern of substitution between

genuine goods and counterfeit goods is the main content that features their

horizontal characteristics. By consuming counterfeits, people trade the goods’

quality for the advantage of price. They allocate their income so as to consume both

goods, and the amount consumed is based on the consumers’ preferences for the

goods’ characteristics and the prices of the goods.

The emergence of counterfeits poses a challenge to the monopoly power of the

firm that produces the genuine products. The competition between the genuine

product and the counterfeit has a combination of vertical and horizontal features;

both products are substitutes for each other and can be differentiated in terms of

quality. The costs of mimicking the characteristics of the genuine product are

relatively small compared with those where the branded or patented firm invests in

developing the uniqueness of its own products. In this respect, counterfeiters can

compete with the producer of the genuine good while incurring few expenses. What

really protects the monopoly power of the genuine good from being eroded by

counterfeiting activities is the enforcement of intellectual property rights. The

stronger the enforcement is the more the potential counterfeiters can be deterred.

In fact, to encourage the activities of innovation most countries have a law of

intellectual property rights against counterfeiting. The law assigns monopoly rights

to innovators or brand owners, and firms determine the values of the property rights

when investing in R&D. Under the premise that the rights are perfectly protected,

Reiganun (1989) shed light on the firms’ ex ante decision in the R&D process. The

value in applying for a patent is central to the analysis in Aoki and Spiegel (2001).

Klemperer (1990) and Gilbert and Shapiro (1990) explore the optimal breadth of the

patents. Marjit and Shi (2001) investigate how infringement agreements can deal

with the issues of accidental patent violations when firms’ profits are slightly

affected by accident. However, most infringements are intentional, and they are the

concern of the property right owners. The efforts made by the patent-holder to

enforce his property rights are able to deter some of the potential counterfeiters from

infringing their rights (Crampes and Langinier 2002). Intellectual property rights

reward their owner with the right to sue the parties that infringe upon his rights, and

the law also provides public enforcement against the counterfeiters. When the legal

rewards are violated, it is not expected that such rights will be exercised

automatically. For their own interests, the patent and brand holders may need to

expend their resources to nail down those that infringe upon their rights. Moreover,

to encourage innovation, the public enforcement removes the infringements that

hinder the incentive to innovate, and its effectiveness depends on the scale of

resources expended on it. Inadequate expenditure on the enforcement of intellectual

property rights results in imperfect intellectual property rights. Most counterfeit

products can be traced to a few developing countries for which the strong


original and bear a different name, but not a private label of a branded industrial product. (3) Repro-

ductions that are not exact copies. (4) Unconvincing imitations.

Eur J Law Econ (2012) 34:77–103 81

123

enforcement of intellectual property rights is not in their short-term interests

(Maskus 2000; Ordover 1991; Maskus and McDaniel 1999).9

The enforcement of the law against counterfeiting activities involves maximizing

social welfare by trading the consumer’s surplus for the benefits that accrue from

innovation activities (Nordhaus 1969; McCalman 2001; Chaudhuri et al. 2006).10

When a social optimum is achieved, at the margin, a small increase in the

enforcement creates the value that is equal to the cost incurred (Higgins and Rubin

1986; Grossman and Shapiro 1988a, b). The counterfeiters encounter the expected

cost of being caught, and strict law enforcement greatly reduces their profit.

However, the firms that make genuine goods have the advantage of being the first

movers in the market, and a rigorous enforcement strengthens their monopoly

power. The enforcement of intellectual property rights curbs counterfeiting

activities and increases the property right-owner’s market power (Yao 2005;

Andres 2006). The legitimate firm’s rights to the fines for infringement affect the

prices of both goods. In the event that the firm can determine the amount of fine on

the basis of the illegal output, it imposes a tax on the counterfeiters so that the price

discrimination allocation is duplicated (Higgins and Rubin 1986).

It has been documented that the consumption of counterfeits can promote the sale

of genuine goods. On the other hand, the popularity of counterfeits may make their

consumption more convenient or their negative image fade in the short run, and in

turn the counterfeiting activities will become more rampant. The consumption of

counterfeits may give rise to bandwagon effects, in which case the aggregate

consumption induces more individual consumption. The quantity of an individual

counterfeit good demanded by a typical consumer increases in response to the

growth in counterfeit purchases of other consumers.

3 The model

3.1 Consumer preference

A representative consumer has a preference that is defined over a genuine good, a

continuum of differentiated counterfeit varieties indexed by i 2 ½0; n�; and a

numeraire. As in Melitz and Ottaviano (2008), we assume that consumers have

utility functions of the quasi-linear form

9 Ordover (1991) and Maskus and McDaniel (1999) document the evidence that the rapid postwar

industrialization in East Asian countries such as Japan and South Korea was accomplished under

relatively weak IPR system and that a hasty imposition of a strong IPR regime could slow down the

industrial development of today’s developing countries. Maskus(2000) also mentions these issues. Chin

and Grossman (1990), Deardorff (1992), Helpman (1993), Lai (1998), Helpman and Lai (2005) and

Branstetter et al. (2007) theoretically address the question of whether a country with limited capacity to

innovate will benefit from extending IPRs to foreign inventors.10 Nordhaus (1969) argued that the optimal policy render the marginal dynamic benefit equal to the

marginal static efficiency loss. Strengthening intellectual property right provides greater incentives for

innovations and thus the benefit that come from having more and better products.

However, McCalman (2001) and Chaudhuri et al. (2006) stress that a strong IPR also incurs static

welfare losses.

82 Eur J Law Econ (2012) 34:77–103

123

UðqB; q0; qðiÞÞ ¼ aBqB �1

2bBq2

B � cBqB

Zn

0

qðiÞdiþ aCðnÞZn

0

qðiÞd

� 1

2bC

Zn

0

qðiÞ2di� 1

2cC

Zn

0

Zn

0

qðiÞqðjÞdidjþ q0;

ð1Þ

where qB, q(i) and q0 are individual consumption levels of the genuine good, each

counterfeit variety i and the numeraire, respectively. The demand parameters aI, cI

and bI are all positive, where I 2 fB;Cg. The utility function gives rise to a linear

demand structure. The inverse demands for the genuine good and the individual

counterfeit i are given by

pB ¼ aB � bBqB � cBQ; ð2ÞpðiÞ ¼ aCðnÞ � bCqðiÞ � cBqB � cCQ; i 2 ½0; n�; ð3Þ

The parameter aB does not only measure the quality of the genuine good in vertical

sense, but indexes the substitution pattern between the genuine good and the

numeraire: increases in aB shift out the demand for the genuine good. The parameter

aC denotes the index quality of the individual counterfeit, and joins cC to index

the substitution pattern between the differentiated counterfeit varieties and the

numeraire. Increases in aC and decreases in cC both shift out the demand for the

counterfeit varieties relative to the numeraire. The parameter bC indexes the degree

of product differentiation between the counterfeit varieties. In the limit when

bC = 0, the counterfeit varieties are perfect substitutes, and the consumers only care

about their consumption levels over the genuine good and all counterfeit varieties,

qB and Q �R n

0qðiÞdi: Since cB [ 0, counterfeit varieties are the substitutes for

the genuine good. When aB = aC and bC = 0, (cB)2/(cC � bB) denotes the degree of

product differentiation between genuine good and counterfeit varieties, ranging

from zero when both are independent to one when both are perfect substitutes. In

this case, cB = cC = bB counterfeit varieties as a whole are a perfect substitute for

the genuine good. In addition we further assume that the degree of the counterfeits’

consumption dispersion is smaller than that of the genuine good, bB [ bC.

Let the diversion ratio from genuine good to counterfeit varieties be expressed as

DB �oQ

opB

�oqB

opB¼ cB

bC=nþ cC

The ratio measures the fraction of sales lost on genuine good that is diverted to

counterfeit varieties following the increase in the price of the genuine good. Without

loss of generality, DB is assumed to be less than one. As n is large, DB is approx-

imated to cB/cC. So, these lead to an indirect assumption cB \ cC. Similar to the

definition of DB, we can have the fraction of the sales lost by the counterfeit

varieties after a price increase that flows back to the genuine good

DC �oqB

opðiÞ

�oQ

opðiÞ ¼cB

bB

Eur J Law Econ (2012) 34:77–103 83

123

Thus, following the argument regarding the substitution between the goods, we

assumed that cB \ bB.

Because the demand for the genuine good and that for the individual counterfeit

are interdependent, a rearrangement of (Eq. 2) and (Eq. 3) yields

pB ¼ aB � ðbB � cBÞqB � cBðQþ qBÞpðiÞ ¼ aCðnÞ � bCqðiÞ � ðcC � cBÞQ� cBðqB þ QÞ

The demand expression is similar to that in Higgins and Rubin (1986) in such a way

that the total output affects prices via the snob demand component cB(Q ? qB). One

of the snob components in the demand for the individual counterfeit is derived from

the total output of both goods, and the other is derived from the output of the

counterfeit varieties. More specifically, their snob effect of the counterfeit on the

consumption of the genuine good is equivalent to, cB, one of the parameters

indexing the substitution pattern between the genuine good and the counterfeit

varieties. The genuine good can be a substitute for the counterfeit varieties, and so

its consumption level affects the demand for the counterfeit varieties. The

assumption that consumers can consume both goods makes our model depart from

the existing literature on counterfeit supplies. However, our modifications fit in our

analysis of the activities of counterfeiting that are no longer confined to infringing

the property rights of the status goods.

The enjoyment derived from consuming the individual counterfeit goods is

reinforced by the variety of counterfeits; that is, a0CðnÞ� 0: The marginal

enhancement that the variety can initiate does not increase with the number n; in

mathematical terms, a00CðnÞ� 0: The quality of the counterfeit good is always

inferior to that of the genuine good; limn!1

aCðnÞ\aB: In the case where the variety

does not affect aC, it is assumed that aC is equal to aC(0).

3.2 Short run equilibrium

The model establishes a two-stage game. The intellectual property right owner

moves first as a Stackelberg leader deciding his output, and the counterfeiters are the

followers. While seeing the outputs of the genuine good, each producer of

counterfeits products supplies his product in an industry that has a monopolistic

structure. Each counterfeiter faces a probability of being convicted, q(E), is subject

to a punishment F and incurs an expected punishment. The probability q(E), is an

increasing function of the resources expended in the enforcement against the

infringement of the intellectual property rights. We further assume that

limE!1

qðEÞ�w� 1; where w is the maximum probability of convicting an individual

counterfeit producer. We also assume that there is a penalty ceiling for those

convicted of counterfeiting activities, F; that is, F�F.

The productions of both genuine goods and counterfeits give rise to an

interaction between the producers of the both goods. Since the property right owner

is a leader in production, he knows that his output affects the outputs of the

counterfeit varieties and in turn takes this into account while supplying the market.

84 Eur J Law Econ (2012) 34:77–103

123

Each producer of counterfeit variety i is a follower in production, and so he treats

the output of the genuine good as given. In addition, the individual counterfeiter

knows that his firm operates in a competitive situation and cannot affect the total

output of the counterfeit producers. It can thus be seen that the individual

counterfeiter does not directly compete with the property right owner but does affect

the production of the genuine good through a residual aggregate demand as shown

by Q.

Given the number of the firms n and the output of the genuine good, the

counterfeiters engage in quantity competition by selling differentiated products that

are highly substitutable for one another but are not perfect substitutes. To make the

analysis tractable, we assume that as long as a producer is convicted of infringing

intellectual property rights he will incur a fine and lose all his revenue. Therefore, in

selling counterfeits each producer i has his expected profit:pðiÞ ¼ ð1� qðEÞÞpðiÞqðiÞ � qðEÞF;where for simplicity we assume that the marginal cost of

production is equal to zero. The first order condition for the maximization of the

profit of a monopolistic firm is

aC � cBqB � 2bC � qðiÞ � cCQ ¼ 0:

By rearranging the above equation, we can connect the equilibrium price of the

individual counterfeit to its quantity by the following equation

pðiÞ ¼ bC � qðiÞ

The number of producers actually participating in counterfeiting is n=ð1� qðEÞÞ;however, there are n counterfeit varieties surviving in the enforcement against the

infringement of intellectual property rights.

The symmetry among counterfeit varieties gives the equilibrium quantity and

price of each variety in the market as functions of the number of active

counterfeiters n:

qðiÞ ¼ aC � cBqB

2bC þ cCnand pðiÞ ¼ bCðaC � cBqBÞ

2bC þ cCn

Thus, given the output of patent holder qB, the total amount of counterfeits in the

market are equal to Q ¼ n � ½ðaC � cBqBÞ=ð2bC þ cCnÞ�. Since we assume that the cost

of producing the genuine good is zero, the property right owner generate a profit, which

is equal to his revenue from producing the genuine good. He maximizes profit:

PB ¼ aB � bBqB � cB n � aC � cBqB

2bC þ cCn

� �� qB ð5Þ

with respect to the output qB of the genuine good subject to market demand (Eq. 3).

The first-order condition gives the equilibrium quantity of the right owner’s output:

qseB ¼

2aBbC þ nðaBcC � cBaCÞ4bCbB þ 2nðbBcC � cBcBÞð Þ: ð6Þ

Since aB [ aC and cC [ cB, the equilibrium quantity qseB [ 0. The marginal change

in the genuine good output with respect to n is

Eur J Law Econ (2012) 34:77–103 85

123

dqseB

dn¼ oqse

B

on� oqse

B

oaC

daC

dn

whereoqse

B

on ¼4cBbCðaBcB�aCbBÞ

4bCbBþ2nðbBcC�cBcBÞð Þ2 andoqse

B

oaC¼ ncB

4bCbBþ2nðbBcC�cBcBÞð Þ.

Thus, if aC=aB [ cB=bB;11 from daC=dn� 0 we have dqse

B

�dn\0. aC=aB can be

a measure used to identify the qualitative difference between the genuine good and

an individual counterfeit. cB/bB is a ratio that measures the difference between both

degrees of product differentiation between the genuine good and the counterfeit

varieties; one is due to the consumer’s snob effect and the other is based on loving

the particular variety. Furthermore, substituting (Eq. 6) into q(i) yields counterfeit

variety i’s equilibrium output

qseðiÞ ¼ 2bCð2aCbB � cBaBÞ þ nfcCðaCbB � cBaBÞ þ aCðcCbB � cBcBÞgð2bC þ ncCÞ½4bCbB þ 2nðbBcC � cBcBÞ�

ð7Þ

and then its equilibrium price

pseðiÞ ¼ bC � qseðiÞ: ð8Þ

The marginal change in the individual counterfeit output with respect to n is

dqseðiÞdn

¼ 1

2bC þ cCn

daC

dn� cB

dqseB

dn

� �� cCðaC � cBqBÞð2bC þ cCnÞ2

: ð9Þ

Equation (9) implies that the increase in the varieties of counterfeits affects the

output of the individual counterfeiter’s output in three ways. First of all, the

varieties of counterfeits have a herd effect, which promotes the consumption of

counterfeits. Second, the increase in the number of counterfeits intensifies the

competition among counterfeiters and discourages the individual counterfeiter from

producing. Moreover, the competition among counterfeiters also reduces the output

of the genuine good, and encourages the output of the individual counterfeiter. We

can more specifically calculate the marginal changes with respect to n that are

included in (Eq. 9) as there are very few counterfeit varieties in the market:

limn!0þ

dqseB

dn¼ cBbCðaBcB�aCbBÞ

4ðbCbBÞ2and lim

n!0þ

cCðaC�cBqBÞð2bCþcCnÞ2 ¼

ð2aCbB�cBaBÞ4bCbB

: According these results,

we have the following conclusion

Proposition 1 In the short run, the increase in the variety of counterfeits does notalways reduce the demand of the individual counterfeit.

Prof. When n is very close to zero, the marginal change in qseðiÞ with respect to

n approximates limn!0þdqseðiÞ

dn¼ 1

2bClim

n!0þdaC

dn� cB

dqseB

dn

� � cC aC�cBqse

Bð Þ

2bC

� �: Thus the

condition that leads to limn!0þ

dqse=dn [ 0 is

11 Given that the individual counterfeit producer is symmetric, the demand for counterfeit varieties is

Q ¼ ðaCbB � aBcBÞ=d� ðbB=dÞ � pþ ðcB=dÞ � pB;where d � bBðbC=nþ cCÞ � ðcBÞ2: Since d[ 0, it

needs ðaCbB � aBcBÞ[ 0 to guarantee that the demand for the counterfeit varieties is not negative

when p and pB are equal to zero.

86 Eur J Law Econ (2012) 34:77–103

123

limn!0þ

daC

dn[

bC cCbB þ ðcBÞ2�

ðaCbB � cBaBÞ þ cCbBbCaCbB

4ðbCbBÞ2

As long as the condition holds, we have the following statement: the demand

functions for the counterfeits and the genuine good are continuous in n, and hence

there exists a range n such that increasing the variety of counterfeits induces more

consumption of individual counterfeit. Q.E.D.This Proposition squares well with the phenomenon that the varieties can

stimulate the desire to consume more counterfeits because more people accept the

similar products varieties. Similar to the Cydia software store, the industrial

clustering of China’s ShanZhai 3C productions provides the products with more

flexible and speedy supply chains and a specific yet complete marketing and sales

system. This has served to boost the variety of counterfeit goods and has in turn

induced further consumption of individual counterfeits even though substitution

patterns between varieties of counterfeits do in fact exist.

4 Market equilibrium and the enforcements of IPR

4.1 Enforcements

The protection of intellectual property rights can take place through either public or

private enforcements. Most of the countries that advocate intellectual property rights

believe that strong enforcement can enhance the industrial development process and

lead to increased worldwide innovation, benefiting consumers everywhere. Optimal

public enforcement is involved both a country’s short- and long- run welfare and is an

important issue that is too broad to be analyzed in this paper.

Therefore, instead of seeking he optimal public enforcement, we analyze the

equilibrium behavior of the property right owner and of the individual counterfeit

producer, as they produce goods within an industry that has a certain degree of

public enforcement. Since a rational producer of a genuine good is a profit-

maximizer, he will choose to enforce his rights in such a way that reflects his best

interests. The property right owner exerts efforts to protect his property right in

responding to the markets within the industry.

We assume that the property right owner will invest resources to detect

infringements of his intellectual property and the government’s efforts are available

to him only after he engages in such activities. Both kinds of efforts to detect the

infringement of intellectual property rights impose an expected cost on the

counterfeit producers. We assume that those counterfeit producers convicted of

infringing the property right will be punished by paying a fine to the government or

through the provision of punitive compensation to the property right owner. The

cost function associated with convicting any counterfeit producer is denoted by

E(q), which is the inverse function of the probability function q(E).12 Thus,

12 As for the general setting of the cost function, the expenditure on detecting the infringement of the

intellectual property rights is convex in the level of the probability of each counterfeiter i being caught.

Eur J Law Econ (2012) 34:77–103 87

123

E denote the level of public or private enforcement. The larger is the level of

enforcement, the higher the probability of a counterfeit producer being convicted.

However, the marginal probability with respect to the resource invested in

diminishes as the amount of resource increases

4.2 Long run equilibrium

4.2.1 Counterfeiter’s decision

Each producer of a counterfeit maximizes his profit, p(i) with respect to its output

q(i) subject to the market demand. The competition between the counterfeit varieties

drives the expected profits of individual counterfeits to zero. Therefore, in

equilibrium, the price of the counterfeit must be equal to the counterfeiter’s average

cost. Given the number of counterfeit producers, the price of counterfeit p(i) is equal

to bC � qðiÞ: The zero profit condition for monopolistic competition yields the

individual counterfeiter’s equilibrium output qle ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqðEÞF=ð1� qðEÞÞbC

pand the

equilibrium price ple ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCqðEÞF=ð1� qðEÞÞ

p.

Notice that the substitution patterns relevant to the demands for counterfeit

varieties do not appear in the equilibrium outputs. Increases in aC; decreases in cC

and curtailment in the outputs of the genuine good qB all shift the demand curves for

the differentiated counterfeit varieties. These not only lead to a larger market but

also induce more entry, to the point where the demand levels faced by an individual

counterfeiter become independent of the market size. The dispersion of consump-

tion or the degree of counterfeit differentiation leads to a reduction in the

equilibrium outputs. In addition, a more stringent enforcement or a higher penalty

for infringing the right should be associated with a higher demand level for an

individual counterfeit. When a manufacture of counterfeit goods earns less or incurs

higher costs, the number of counterfeit producers declines.

Proposition 2 Increases in the public enforcement against the activities ofcounterfeiting lead to increase in the output and price of individual counterfeiters.

Plugging the price ple and the output qle into the demand function of the

individual counterfeiter, in which the output of the genuine goods qB is given, yields

the number of counterfeit producers

nR ¼ðaC � cBqBÞ

cC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqðEÞF=ð1� qðEÞÞbC

p � 2bC

cC

ð10Þ

Increases in the output of the genuine goods hinder the market size of individual

counterfeits and induce less entry if there is no bandwagon externality induced by the

consumption of the counterfeit product. The same applies to greater substitutions

between counterfeit varieties. When the dispersion of consumption is higher,

counterfeit varieties are closer substitutes, and there are fewer counterfeit producers in

the industry.

Remark 1: The property right owner can block the emergence of counterfeiting by

engaging in massive production.

88 Eur J Law Econ (2012) 34:77–103

123

A rearrangement of (Eq. 10) reveals that nR [ 0 if and only if

ac � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcqðEÞF=ð1� qðEÞÞ

p� .cB [ qB:

We define the greatest lower bound for the existence of counterfeiting activities

under law enforcement E and penalty F as

qTBðE;FÞ � ac � 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcqðEÞF=ð1� qðEÞÞ

p� .cB:

Free entry drives the expected profit from counterfeiting to zero and leads to the

equilibrium outputs of individual counterfeits being independent of market size.

However, each counterfeit variety is only able to survive in such a market if the

producer can sell the product at a price not lower than the average cost. In planning

to produce a substitute for the genuine good, a potential entrant should take the

consumption levels of the genuine good into account. If counterfeiting exists,

summing up the output of the individual counterfeits over the number of counterfeit

varieties yields the total output (QR(qB)) of counterfeits, for which the output of the

genuine good is given as:

QRðqBÞ � ðaC � cBqBÞ � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcqðEÞF=ð1� qðEÞÞ

p� .cC:

The property right owner has the first mover advantage in production, and if his

product is a substitute for any counterfeit, then both factors may deter the

production of counterfeits. A large supply of the genuine good leads to a low price

of the good and makes counterfeiting an unprofitable business. However, it may not

be the property right owner’s profit-maximization choice. By substituting the total

output into (Eq. 3) the following revenue arrangement should yield the property

right owner’s profit:

qB aB � bBqB � cB

ðaC � cBqBÞ � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCqðEÞF=ð1� qðEÞÞ

p� �cC

!

Solving the first-order condition of the property right owner’s optimization with

respect to his output qB gives the equilibrium output of the genuine good

qleB ¼

aBcC � cB aC � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCqðEÞF=ð1� qðEÞÞ

p� �2ðbBcC � cBcBÞð Þ :

If a0Cð�Þ ¼ 0; substituting the equilibrium output of the genuine good qleB

into (Eq. 10) obtains the equilibrium number of counterfeit varieties. Then,

by incorporating the individual counterfeiter’s equilibrium outputs qle ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqðEÞF=ð1� qðEÞÞbC

pand prices ple ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCqðEÞF=ð1� qðEÞÞ

pinto it, we can

present the equilibrium number of the counterfeit varieties in a more neat

expression13

13 In the long run, the equilibrium number of the counterfeit varieties is derived from the zero profit

condition for the production of individual counterfeit that makes the output and price of the individual

counterfeit be the constants qle and ple. The output of the genuine good affects the market size of the

counterfeit varieties, and hence the firm with the genuine good takes this into consideration to maximize

Eur J Law Econ (2012) 34:77–103 89

123

nleR ¼

2bBcC � cBcBð Þ � aC � 2ple� �

� aBcBcC

2ðbBcC � cBcBÞð Þ � cC � qle:

Rearranging the above equation yields the equilibrium aggregate output of

counterfeits

QleR ¼

2bBcC � cBcBð Þ aC � 2ple� �

� aBcBcC

cC � 2ðbBcC � cBcBÞð Þ

Substituting all the equilibrium outputs into (Eq. 3) the inverse demand of genuine

good yields the equilibrium price of genuine good

pleB ¼

aBcC � cBðaC � 2pleÞ2cC

In the equilibrium, the property right owner has the revenue function Rle(E)that is

presented as

RleðEÞ ffi pleB � qle

B ¼aBcC � cBðaC � 2ple� �2

4cC bBcC � cBcBð Þ ð11Þ

If there exist counterfeits, the equilibrium output of the genuine good qleB must be

less than the greatest lower bound qTBðE;FÞ or nle

R [ 0.

If the counterfeit varieties do not enhance the consumption of the individual

counterfeit; aC is a constant, and then the inequality qTBðE;FÞ[ qle

B holds if

ac � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCqðEÞF=ð1� qðEÞÞ

paB

[cCcB

2 � bBcC � cBcB

: ð12Þ

Based on the assumptions that bB [ cB and cC [ cB, it is not hard for us to infer that

cB

bB

[cBcC

bBcC þ ðbBcC � cBcBÞ:

The sufficient condition determining which counterfeiting is active can be more

restricted if it is changed toac�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCqðEÞF=ð1�qðEÞÞp

aB[ cB

bB:.Obviously, if a0Cð�Þ ¼ 0; then

oqleB

�oE [ 0, ope

B

�oE [ 0 and oQle

R

�oE\0. Consequently, we have the following

proposition


its profit. So, the first order condition of profit-maximization for the production of the genuine good

includes the effect of the output of the genuine good on the number of the counterfeit varieties. However,

in the short run, the output of the genuine good does not affect the number of the counterfeit varies, but

does affect the output and price of individual counterfeit. So, equating the short run outputs qse and the

long run outputs qle of individual counterfeit and solving for n should not arrive at the long run equi-

librium number of the counterfeit varieties nleR . This can be seen from the equalization qse ¼ qle that is

specified as

2bCð2aCbB�cBaBÞþnfcCðaCbB�cBaBÞþaCðcCbB�cBcBÞgð2bCþncCÞ½4bCbBþ2nðbBcC�cBcBÞ�

� ¼


p. The solution for n in this

equation is not equal to the long run equilibrium number of the counterfeit varieties nleR .

90 Eur J Law Econ (2012) 34:77–103

123

Proposition 3: Suppose that only the snob component is the externality inconsuming counterfeits. Increases in the public enforcement against the activities ofcounterfeiting increase the equilibrium outputs and the equilibrium prices of thegenuine good, but decrease the equilibrium number of counterfeit varieties.

An increase in public enforcement against the activities of counterfeiting shifts

the demand for the genuine good and pushes up its price and output. Thus,

strengthening the enforcement pushes up the property right owner’s profit.

Proposition 3 supplies a reason why so many international enterprises strongly

urge countries, in which the counterfeiting activities are rampant, to enforce

intellectual property laws more intensively. For example, multinational 3C firms

such as Apple, Nokia and Microsoft have asked the Chinese government to fight

against counterfeiting.

In the case where the varieties of counterfeits induces the consumer to value

individual counterfeit more a0Cð�Þ[ 0, the inequality qTBðE;FÞ[ qle

B holds if

ac � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcqðEÞF=ð1� qðEÞÞ

paB

[cC

bBcC þ bBcC � cBcB þ ðcBa0Cð�Þ � dnR=dqBÞ� �:

Obviously, there is an additional term cBa0Cð�Þ � dnR=dqB in the denominator of

the right hand side of this inequality, comparing with (12). Thus, we have the

following statement.

Remark 2: Suppose that the variety of counterfeits enhances the consumption of

individual counterfeit. The bandwagon effect in consuming counterfeits can

decreases the sales of the genuine good as long as the marginal externality of the

counterfeit varieties is unnoticeable in equilibrium.

The mathematical illustration of Remark 2 is presented in the ‘‘Appendix’’.

The values of the counterfeit varieties in the vertical sense should be higher

when the bandwagon effects of consuming counterfeits prevail than when the

effects do not. Even though increases in the counterfeit variety may give rise to

little extra externality in equilibrium, the market size of individual counterfeit is

increased in response to the growth of counterfeit varieties. This induces more

entries of counterfeit producers and in turn squeezes the market size of the

genuine good. Although the equilibrium outputs of individual counterfeiters are

independent of market size, the increases in the number of counterfeit varieties

lead to increases in the total output of counterfeits. The market size of the

genuine good shrinks as long as the marginal bandwagon effects are unnotice-

able in equilibrium.

In this case the equilibrium price of the genuine good is lower when the

bandwagon effects exist than when the effects do not exist. It should also be

noted that since the genuine good is a substitute for the counterfeit varieties,

increases in its output deter some producers from engaging in the manufacture

of the counterfeit variety and reduce the externality involved in the

consumption of the counterfeits and reduce the externality from consuming

counterfeits.

Eur J Law Econ (2012) 34:77–103 91

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This can be identified by the term cBa0Cð�Þ � ðdnR=dqBÞ14 that is presented in the

equilibrium price of genuine good. This term does not exist when the counterfeit

variety does not enhance the consumption of individual counterfeit. Thus, the

bandwagon effects on the consumption level of the genuine good are ambiguous

when the property right owner considers generating a large output as a mean of

curbing the externality from consuming counterfeits. However, if at the margin an

increase in the output of genuine good does little to change the bandwagon effect in

consuming the counterfeits, in equilibrium the externality reduces the output of the

genuine good.

4.2.2 Property right owner’s enforcements and static comparison

To encourage firms to invest in the activities of innovation, most governments have

established an intellectual property laws to curb the infringement of intellectual

property rights. However, the developing countries may lack the enthusiasm to

enforce intellectual property rights because stringent enforcements do not favor

their short-term interest. This leads increasingly leads to multinational firms joining

together in an industry or forming local temporary alliances to deal with the issue,

and bring pressures to bear on the government either by lobbying directly or through

indirect means to enforce the law more effectively. These firms exert efforts to

protect their property, and hence the level of the enforcement of IPR can be their

choice. To simplify the illustration as to how the characteristics of the demand for

the goods affects the equilibrium private enforcements, in this section we focus on

the case that aC is a constant.

Here we use the same notations are deployed as that presented in the previous

section to denote the private enforcement E and the probability of convicting an

individual counterfeit producer q(E), while treating the level of public enforcement

as given. To protect his right, the producer of the genuine good can invest resource,

which is directly inverted into the enforcement, and the enforcement of IPR is

endogenous. The processes of demonstrating the equilibrium prices and outputs can

be the same in both cases, with the enforcement being private in one case and pubic

in the other. As indicated in the previous section, the analysis remains the feature: a

long-run relationship subsuming the zero profit assumption as to the production of

individual counterfeits.

When only the government spends resource to enforce intellectual property

rights, the protection of intellectual property right is exogenous to the property

owner’s objective function, and the producer merely chooses the levels of output to

maximize his profits. However, when the property owner can put effort into

protecting his right, the enforcement is endogenous to his objective function. He

chooses not only the level of his production but also the amount of resources to

maximize his interest. The resources directly invested into the enforcement, i.e., the

cost of private enforcement amounts to E. We assume that the producer of the

genuine good expends resources on protecting his property right before choosing his

14 peB ¼ aB

2þ cBð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcqðEÞF=ð1�qðEÞÞp

�aCÞ2cC

� �1þ cBa0Cð�ÞdnR=dqB

2 2ðbBcC�cBcBÞþcBa0Cð�ÞdnR=dqBð Þ

� �

92 Eur J Law Econ (2012) 34:77–103

123

production level. After the output of genuine good is chosen, the individual

counterfeit producers enter the market until their normal profits are exhausted.

Apart from the need for the property owner to choose the level of enforcement at

the beginning of the game, in this section all the algorithm for the choices made by

the producers are the same as the ones in previous section. Thus, we can directly use

the revenue function shown in (Eq. 11) to construct the property right owner’s profit

function. The producer of the genuine good has his profit evaluated at a fixed level

of enforcements E in the initial stage

YB

¼aBcC � cBðaC � 2ple� �2

4cC bBcC � cBcBð Þ � E;

where ple ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCqðEÞF=ð1� qðEÞÞ

pÞ. The property owner’s profit simply is simply

his revenue minus the resources expended on protecting his property. A profit-

maximizing firm enforces its property rights to such a level that, with respect to the

resource E, its marginal revenue equals its marginal cost. The enforcement drives up

the price of the individual counterfeit, and in turn the property right owner benefits

from the substitution of his products for counterfeits. At the margin, the increases in

enforcements lead the revenue to increase by

aBcC � cBðaC � 2pleÞ� �

cC bBcC � cBcBð Þ cBople�oE

and the cost to increase by one. Of course, if there exists an internal optimal private

enforcement, the property right owner’s revenue function must be concave in the

resources E that are invested in it. To illustrate how the substitution patterns and the

degrees of the product differentiation affect the property right owner’s investment in

enforcing his right, we use a special form of the apprehension rate q(E).

Case*: In this case, we assume that w\1 and the probability ratio

ð1� wÞqðEÞ=w 1� qðEÞð Þ is the exponential distribution function of enforcements

E, and G(E) is denoted by

GðEÞ � ð1� wÞqðEÞ=w 1� qðEÞð Þ:G(E) is increasing in q(E), and meanwhile they are both one-to-one correspon-

dence. Thus, by being transformed from q(E), G(E) can be perceived as a

probability model for the enforcement of the conviction of an individual

counterfeiter. As a random variable, E has its density function gðEÞ ¼ le�lE:A rearrangement indicates that l ¼ gðEÞ=ð1� GðEÞÞ is the hazard rate, and that a

producer of individual counterfeit good will be convicted under enforcement E.

Furthermore, based on the probability ratio G(E), we can rewrite the probability that

the conviction of the property right infringer will have occurred because of some Eas qðEÞ ¼ w � GðEÞ=ð1� wð1� GðEÞÞÞ: qðEÞ is also increasing and concave in E.15

This result obviously coincides with our assumption regarding the probability

function q(E) that limE!1 qðEÞ ¼ w: The firm with the genuine good has the

revenue function

15 q00 ðEÞ ¼ wð1�wÞ

1�wð1�GðEÞð Þ2G00 ðEÞ � 2

w2ð1�wÞ1�wð1�GðEÞð Þ3ðG

0 ðEÞÞ2

Eur J Law Econ (2012) 34:77–103 93

123

RncG ðEÞ �

ðaBcC � cBðaC � 2pgeÞÞ2

4cCðbBcC � cBcBÞ:

where pge ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCw � F=ð1� wÞ

p ffiffiffiffiffiffiffiffiffiffiffiGðEÞ

p: In addition, the revenue function is con-

cave in E; that is, Rnc00G ðEÞ� 0:16 Thus, there exists an interior optimal resource for

the property right owner to invest in enforcing his property right. With respect to the

enforcement the marginal revenue equals the marginal cost. The first order condi-

tion of the profit-maximization with respect to E requires that the marginal revenue

is equal to the marginal cost: Rnc0G ðEÞ ¼ 1. This equalization of marginal revenue

and marginal cost yields the optimal choice Enc and the equilibrium output

qnceB ¼ aBcC � cB aC � 2pnceð Þ

2ðbBcC � cBcBÞð Þ

where pnce �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCw � F=ð1� wÞ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiGðEncÞ

p: As the property right owner’s resource

expended on protecting his property rights from infringement is large, the property

right owner’s marginal revenue of private enforcement is close to

MBnc � limE!1

Rnc0

G ð�Þ ¼cB � aBcC � cB aC � 2pmaxð Þð Þ

2cC bBcC � cBcBð Þ pmax;

where pmax �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCw � F=ð1� wÞ

p: In addition, a calculation yields limE!0þ

Rnc0

G ð�Þ ¼ 1:The marginal cost MCE and the marginal revenue MBE with respect

to E are presented in Fig. 1. The changes in the parameters of the demand functions

will shift out or shift in the marginal revenue and hence will alter the optimal

enforcements.

Since the property right owner’s revenue is concave in terms of the resource

invested in protecting his property, the equilibrium private enforcement must be

unique. Thus, we can easily identify several static comparatives regarding the

equilibrium enforcements, Enc (all the derivations are shown in the Appendix).

Increases in the public punishment F on infringing intellectual property dampen the

output of counterfeits and increase the market size of the genuine good. To protect

the higher revenue created by a severer punishment, the property right owner

increases his private enforcement. This can be seen by oEnc=oF [ 0.

A higher degree of product differentiation between counterfeit varieties pushes

the price of individual counterfeit goods higher and causes the output to fall. A

lower output and a higher price of counterfeits leads to a larger market size for the

genuine good, and in turn the property right owner invests more in private

enforcement to protect his property. Thus, oEnc=obC [ 0:In the vertical sense, the quality of the genuine good reinforces the property right

owner’s decision to invest resources in enforcing his property right. The increases in

the quality cause the property right owner to spend more to protect his right. It

should therefore be straight forward to see that oEnc=oaB [ 0: By contrast, the

higher the quality of counterfeit goods the easier it is for counterfeits to be a

16 Rnc00G ðEÞ ¼

2cB

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcw�F=ð1�wÞp

l2 1�GðEÞð Þ4cC bBcC�cBcBð Þ

ffiffiffiffiffiffiffiffiGðEÞp

� � � accBð Þ 1þGðEÞ

2ð Þ�2cB

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcw�F=ð1�wÞp

GðEÞ3=2

GðEÞ

� �

94 Eur J Law Econ (2012) 34:77–103

123

substitute for the genuine good and consequently the lower the profit that the

property right owner has to protect. Thus, when the quality of the counterfeits is

increasing, the firm expends less on private enforcement against counterfeiting,

oEnc=oaC\0.

The consumption dispersion of the genuine goods dampens the market size of the

genuine goods, and at the margin it decreases the revenue increment that the

increase in the enforcement can bring about. Consequently, the higher the degree of

product differentiation between the genuine good and the counterfeit varieties, the

less the property right owner invests in private enforcement, oEnc=obB\0:The substitutability of counterfeits for genuine goods represents the same

meaning as in the case where the substitutability of the genuine goods for

counterfeits does the demands for the goods. The substitutability does not have a

definite effect on the supply of genuine good. However, the genuine good is under a

fierce competition when the good has a close substitute. This leads to the situation

where the higher the substitutability, the lower the price of the genuine good.

Furthermore, the sign of oEnc=ocB is ambiguous since the effects of the

substitutability between the genuine good and the counterfeit goods varies with

the equilibrium output of the genuine good. Furthermore, if the substitution pattern

among the counterfeit varieties is significant, the competitions among the

counterfeit producers will be fierce. The higher the substitution is, the lower will

be the number of counterfeit producers. A larger substitution pattern makes the

decrease in the number of counterfeits smaller while the enforcement is increasing.

Increases in enforcement cause the supplies and the prices of the genuine good to

also increase; the increments in output are decreasing but the increments in prices

E

(E)MBE

ncE

EMC1

ncMB

Marginal benefit Marginal cost

Fig. 1 Illustration of private optimal enforcement

Eur J Law Econ (2012) 34:77–103 95

123

are increasing in the substitution pattern. Thus, the sign of oEnc=ocB is ambiguous

(Table 1).

5 Private enforcement and punitive compensation

The property right owner exerts his private enforcement against the infringement of

his intellectual property rights, and meanwhile for his right being violated he may

collect punitive compensation from the convicted counterfeit producers. To simplify

the analysis, in this section, our discussion only includes snob and hence aC is a

constant. Each individual counterfeiter producer has the chance 1 - q(E) of not

being caught and survives in the market. To make a comparison with the case where

the property right owner is not awarded compensation, we assume that the

probability q(E) has the same characteristic as that of Case*. The number of

survived counterfeit varieties is nR and hence there are nR= 1� qðEÞð Þ½ � counter-

feiting varieties participating in infringing the intellectual property rights at the

beginning in the second stage. The expected number of violators convicted is

nR= 1� qðEÞð Þ½ � � qðEÞ. Thus, the property owner can collect the expected amount

of compensation CPS, which is the expected number of convicted counterfeiters

multiplied by the fine F;

CPS � nR= 1� qðEÞð Þ½ � � qðEÞ � FWith respect to the quantities involved, the firm with the genuine goods

maximizes its expected profit (PB) that consists of a private enforcement (cost) E, a

revenue RCG and an award CPS compensated by the convicted

PB ¼ RCGðqBÞ þ CPSðE;F; qBÞ � E ð13Þ

where RCGðqBÞ � pBqB; CPSðE;F; qBÞ ¼ ðaC � cBqBÞpe � 2ðpgeÞ2

� .cC: Using

backward induction, we can solve the property right owner’s optimal choices of

output and the price of genuine good when he exerts an enforcement E:

Table 1 Expected signs of the parameters in static comparisons

Parameters Static comparisons Sign

Punishment F oEnc=oF. ?

The product differentiation between counterfeit varieties oEnc=obC ?

The quality of the genuine good oEnc=oaB ?

The quality of individual counterfeits oEnc=oaC -

The product differentiation between the genuine good

and counterfeit varieties

oEnc=obB -

The substitution pattern between the counterfeit varieties

and the genuine good

oEnc=ocB ?

The substitution pattern between counterfeit varieties oEnc=ocC ?

96 Eur J Law Econ (2012) 34:77–103

123

qBðEÞ ¼aBcC � cB aC � pgeð Þ

2ðbBcC � cBcBÞð Þ ; pBðEÞ ¼aB

2þ cBðpge � aCÞ

2cC

:

In addition, substituting the price and output of the genuine good qB(E), pB(E) into

RCGðqBÞ; and CPS(E, F, qB) yields the market revenue of the genuine good

RCGðEÞ ¼

aBcC � cB aC � pgeð Þð Þ2

4 bBcC � cBcBð ÞcC

and the award compensated by the convicted

CPSðEÞ ¼ ð2bBcC � cBcBÞaC � aBcBcCð Þpge � ð4bBcC � 3cBcBÞðpgeÞ2

2ðbBcC � cBcBÞ

All the other notations have the same definition as before. The marginal benefit with

respect to private enforcement is RC0G ðEÞ þ CPS0ðEÞ ¼ ðaC � 2pgeÞ=cCð Þ�

opge=oE:17 Then, the first-order condition in the right owner’s best interests with

respect to his private enforcement is RC0

G ðEÞ þ CPS0ðEÞ ¼ 1; and yields the optimal

choice of enforcement EC. In equilibrium, the output of the genuine good qCeB is

qCeB � aBcC � cB aC � pCe

� �� 2ðbBcC � cBcBÞð Þ:

where the price of individual counterfeit pCe �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCw � F=ð1� wÞ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiGðECÞ

p: When

the property owner expends a large amount of resources in protecting his right, his

marginal benefit of enforcement is close to

MBC � limE!1

RC0G ðEÞ þ CPS0ðEÞ ¼ ðaC � 2pmaxÞ=cCð Þ � pmax=2:

A comparison of MBC and MBnc concludes that MBnc [ MBC. Since

o2pge�o2E\0; a sufficient condition forRC

GðqBÞ þ CPSðEÞ being concave in E is

aC � 2pge [ 0: For this condition, the vertical value of the individual counterfeit

should be much greater than its price that is positively related to the degree of

differentiation between the counterfeit varieties and the severity of punishment.

If there exists an interior solution for Rc0GðEÞ þ CPS0ðEÞ � Rnc

G ðEÞ ¼ 0 then the

enforcement must be equal to

E � G�1 bBaC � aBcBð Þ2bBaC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCw � F=ð1� wÞ

p !2

0@

1A:

The accumulation function G(E) is monotonic with E. In addition, limE!0þ RC0G ðEÞ þ

CPS0ðEÞ � RncG ðEÞ[ 0: Thus, there exists an interior solution for RC0

G ðEÞ þCPS

0 ðEÞ � RncG ðEÞ ¼ 0; and it must be unique. The solution E* is such that

a critical enforcement of RC0G ðEÞ þ CPS0ðEÞ[ Rnc

G ðEÞ for E \ E* and of

17 The marginal revenue and compensation with respect to private enforcement are

RC0

G ðEÞ ¼cB aBcC�cB aC�pgeð Þð Þ

4cC bBcC�cBcBð Þ 2 opge

oE ;CPS0ðEÞ ¼ 2 2bBcC�cBcBð ÞaC�2aBcBcC� 4bBcC�3cBcBð Þ2pge

4ðbBcC�cBcBÞð ÞcC

opge

oE :

Eur J Law Econ (2012) 34:77–103 97

123

RC0G ðEÞ þ CPS0ðEÞ\Rnc

G ðEÞ for E [ E*exists. These situations are illustrated in

Fig. 2. Thus, we have the following results induced.

Proposition 4: If RC0G ðEÞ þ CPS0ðEÞ[ 1; the property right owner puts more

effort into enforcing his right when he can collect lump sum compensation from theconvicted than when he can not.

There are two cases in which the punitive compensation motivates the firm

producing the genuine good to expend resources to protect its intellectual property.

In this case, the number of counterfeit producers is a relatively significant factor that

affects the amount of compensation collected from the convicted. Increases in

private enforcement deter counterfeiting and in turn reduce the number of the

convicted; as a result, the compensation decreases. In the other case, the property

right owner sees the apprehension rate as a main measure to increase the

compensation. Being able to collect the punitive compensation from the convicted

provides an extra incentive for the firm with genuine goods to oppose the

infringement in additional to taking into consideration its revenue. Both consid-

erations are traded off and make the equilibrium outputs and prices in the regime of

punitive compensation different from those in the one with public fine.

Lemma 1: If RC0G ðEÞ þ CPS0ðEÞ\1, awarding punitive compensation loosens

the property owner’s private enforcement and hence the output (price) of thegenuine good is less (lower) compared with that where the owner is notcompensated from the convicted.

)(')(' ERER CcG

)(' ERncG

CE ncEE

Marginal benefit

Marginal cost

CMB

ncMB

1

*E

EMC

Fig. 2 Illustration of private optimal enforcement comparison

98 Eur J Law Econ (2012) 34:77–103

123

If RC0G ðEÞ þ CPS0ðEÞ\1; thenEC\Enc; and hence GðECÞ\GðEncÞ: The

equilibrium price of the individual counterfeit under the enforcement EC is lower

that the one under Enc;

pCe ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCw � F=ð1� wÞ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiGðECÞ

p\pnce ¼


p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiGðEncÞ

p:

In addition, the output of the genuine good when the property right owner is

rewarded for being infringed is smaller than when he is not rewarded;

qCeB ¼

aBcC � cB aC � pCeð Þ2ðbBcC � cBcBÞð Þ \qnce ¼ aBcC � cB aC � 2pnceð Þ

2ðbBcC � cBcBÞð Þ :

This further implies that pCcB \pnce

B if the compensation has the same value as the

public fine.

6 Conclusion remarks

A large part of the business of counterfeiting is directly involved in the usual

exchanges between buyers and sellers and deception is not relevant to the practice.

In addition, counterfeiting activities are not confined to status goods but covers a

diverse range of products. Based on these phenomena, our analysis includes several

specific consumption characteristics to explore what legal institutions can do to

affect the supplies of counterfeits and genuine goods. By doing along with the

development of China’s shanzhai products and unauthorized Applications store for

the iphone, Cydia, we create a two-stage game, in which the firm producing the

genuine good is the leader followed by producers of the counterfeit varieties, which

are assumed to be monopolistically competitive firms. The analysis provides the

insights regarding the infringements of property rights and the legal activities of the

property right owner while competing with the counterfeit varieties.

First of all, the increase in the variety of counterfeits does not always reduce the

demand for individual counterfeits as long as there exists a positive externality from

consuming counterfeits. Second, the producer of individual counterfeits increases

his outputs and prices to cover the higher expected costs that results from the more

stringent enforcement leveled against his infringement. However, a more severe

punishment decreases the size of market for the counterfeits and in turn reduces the

variety of counterfeits. This leads to the total output of the counterfeits decreasing in

terms of enforcement.

Moreover, in the case where the property right owner can expend private

resources to strengthen the enforcement and the government collects the fine

imposed on the convicted violator, the consumption characteristics affect the

property right owner’s private enforcement. The larger the difference in the quality

between the genuine good and the counterfeit varieties is, as perceived by

consumers, the more the property right owner is willing to expend on the private

enforcement.

The difference in quality helps the genuine good to expand its market size and

generate profit for the property right owner, and it gives him an incentive to expend

Eur J Law Econ (2012) 34:77–103 99

123

more resources to enforce his right. The less the product differentiation between the

genuine good and counterfeit varieties, the smaller the size of the market for the

genuine good, so that the property right owner is discouraged from expending more

resources on private enforcement.

By contrast, the more the product differentiation between the individual

counterfeiter and the counterfeit varieties, the more the property right owner is

encouraged to protect his right. The substitution patterns between the genuine good

and the counterfeit varieties or among the counterfeit varieties have the opposite

effect on the marginal price and on the marginal output with respect to the private

enforcement. Thus, the signs by which they affect the expenditure on the private

enforcement are ambiguous.

The property right owner may expend more on private enforcement when he is

able to receive compensations from the producer convicted of counterfeiting

activities than when he is not able to do so. Increases in the enforcement can

increase the probability of convicting the counterfeiter and also deter such illegal

activities. So, in strengthening the private enforcement, the property right owner

does not necessarily receive more compensation. Other things being equal, if at the

margin strengthening the enforcement can bring about an increment in the revenue

that exceeds the decease in the expected level of compensation, the property right

owner that would be compensated will exert more efforts than the one that would

not be. In addition, in this case, the property right owner that receives compensation

sets a higher price and supplies more than the owner that does not.

The analysis could be extended to explore how trade policies affect the supplies

of genuine goods and counterfeit products. For example, lower trade barriers can

expand the market sizes of multinational firms and curb the supplies of counterfeits.

Thus, if the private enforcements are very costly, a move toward lobbying the host

country to lower its trade barriers can be an alternative approach for multinational

firms to protect their intellectual property rights. Furthermore, if the measures

adopted to prevent the infringements of intellectual property rights increase the

marginal cost of producing counterfeit goods, it can be predicted that the output of

the individual counterfeiter will be reduced. This should be different from the setup

in which the enforcement against infringement is equivalent to a fixed cost of

producing individual counterfeits. This issue can be analyzed by extending the

model presented in this paper.

Appendix

Illustration of remark 2

Similarly, if a0Cð�Þ[ 0; the equilibrium outputs are

qleB ¼

aBcC � cB aC � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCqðEÞF=ð1� qðEÞÞ

p� �2ðbBcC � cBcBÞ þ cBa0Cð�ÞdnR=dqB

� �

100 Eur J Law Econ (2012) 34:77–103

123

QRðqleB Þ ¼

aC

2cC

þaCðbB þ 1

2cBa

0Cð�ÞdnR=dqBÞ � cBaB

2ðbBcC � cBcBÞ þ cBa0Cð�ÞdnR=dqB

� �

� ð2bBcC � cBcB þ cBa0Cð�ÞdnR=dqBÞ � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcqðEÞF=ð1� qðEÞÞ

pcC 2ðbBcC � cBcBÞ þ cBa0Cð�ÞdnR=dqB

� �

the equilibrium number of the counterfeit varieties is

nleR ¼

2bBcC � cBcB þ cBa0Cð�ÞdnR=dqB

� �ac � 2


p� �� aBcBcC

2ðbBcC � cBcBÞ þ cBa0Cð�ÞdnR=dqB

� �cC


p

and the equilibrium price of the genuine good is

pleB ¼

aB

2þ cBð2


p� aCÞ

2cC

!

1þ cBa0Cð�ÞdnR=dqB

2 2ðbBcC � cBcBÞ þ cBa0Cð�ÞdnR=dqB

� � !

where dnR=dqB ¼ �cB

�ðcC


p� a0Cð�ÞÞ: If a0Cð�Þ is close to

zero in equilibrium, the equilibrium total output of counterfeits is approaching to

QleR ¼

2bBcC � cBcBð Þ aC � 2peð Þ � aBcBcC

cC � 2ðbBcC � cBcBÞð Þ ;

and the equilibrium price is very close to

pleB ¼

aB

2þ cBð2


p� aCÞ

2cC

!:

Illustration of Table 1

Rnc0G ðEÞ

¼cB aBcC�cB ac�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcw �F=ð1�wÞ


p� �� 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcw �F=ð1�wÞ

plð1�GðEÞÞð Þ

4cC bBcC�cBcBð ÞffiffiffiffiffiffiffiffiffiffiffiGðEÞ

pTo simplify the notations, we define the following value functions:

XffiffiffiffiffiffiffiffiffiffiffiGðEÞ

p� � cB aBcC � cB ac � 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcw � F=ð1� wÞ


p� �

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcw � F=ð1� wÞ

plð1� GðEÞÞð Þ

HðffiffiffiffiffiffiffiffiffiffiffiGðEÞ

pÞ � 4cC bBcC � cBcBð Þ

ffiffiffiffiffiffiffiffiffiffiffiGðEÞ

p:

From the first-order condition, we can derive the static comparisons.

Eur J Law Econ (2012) 34:77–103 101

123

oEnc

oF¼ �

XffiffiffiffiffiffiffiffiffiffiffiGðEÞ

p� �þ 4ðcBÞ2 bCw � F=ð1� wÞð Þ



2Rnc00G ðEÞHð


pÞF

[ 0

oEnc

obC

¼ �X


p� �þ 4ðcBÞ2 bCw � F=ð1� wÞð Þ



2Rnc00G ðEÞHð


pÞbC

[ 0

oEnc

oaB¼ �cBcC2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcw � F=ð1� wÞ


Rnc00G ðEÞHð


p [ 0:

oEnc

oaC¼ cBð Þ22



Rnc00G ðEÞHð


pÞ

\0

oEnc

obB

¼ Rnc0G ðEÞ4c2

C


pRnc00

G ðEÞHffiffiffiffiffiffiffiffiffiffiffiGðEÞ

p� �\0

oEnc

ocB

¼�X


p� �2H


p� ��cB þ 8cBcC


p� �R00GðEÞ H


p� �� 2

þaBcC2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcw � F=ð1� wÞ


R00GðEÞ HffiffiffiffiffiffiffiffiffiffiffiGðEÞ

p� �� 2

oEnc

ocC

¼4cCbBX


p� � ffiffiffiffiffiffiffiffiffiffiffiGðEÞ

pR00GðEÞ H


p� �� 2

�H


p� �ðcBÞ

2.

cC ac�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcw�F=ð1�wÞ


p� �2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibcw�F=ð1�wÞ

plð1�GðEÞÞð Þ

R00GðEÞ HffiffiffiffiffiffiffiffiffiffiffiGðEÞ

p� �� 2

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Consumers’ choices, infringements and market competition