ELIAS OLOFSSON

Autumn 2012

Master Thesis, 15 ECTS Master’s Program in Economics, 60 ECTS

Piracy on the market

A theoretical study of the policy tools a government can use when it wants to maximize social

welfare on a market influenced by piracy

Elias Olofsson

Abstract

This paper explores the influence of piracy (the illegal redistribution of digital goods) on an

information good (i.e. music) market. The purpose of the study is to investigate how a

government can use policy instruments to minimize losses on the market and at the same time

correct the behaviour of a monopoly firm. To the author’s knowledge, a study with the aim to

explore the effects of policy instruments on a market influenced by digital piracy has not been

done before. To answer the purpose, a theoretical model is developed consisting of three

agents; a consumer, a firm and a government. The policy instruments available for the

authorities are; introducing a sale tax on the original good, a fine on illegal consumption,

change the probability of being caught, or implementing lump-sum transfers. The welfare-

maximizing objective of the government is complicated by the fact that two types of market

failures influence the information good market simultaneously. The result of this paper shows

that the optimal fine will be larger than zero. However, the lone implementation of a fine

solution will be argued to be suboptimal, as the fine only will correct the behaviour of the

consumer and not the monopoly firm. The conclusions in the end are that the authorities can

either introduce a negative sales tax (subsidy), or use a combination of subsidy and fine to

correct the consumer and firm’s behaviour.

Keywords: Information good, Public good, Piracy, Optimal policy tools

Table of content

1. Introduction 1 1.1 Statement of the problem 3 1.2 Purpose 3 1.4 Previous Studies 3

2. Information as Public good 8

3. Model 10 3.1 The Consumer 10 3.2 The Firm 13 3.3 The government 14 3.4 First order conditions 15

4. Solving for the optimal policy 16 4.1 The simplified problem - The optimal sales tax 17 4.2 The simplified problem - The optimal fine 18 4.3 The simultaneous optimality problem 19

5. Discussion 20

6. Summary and Conclusion 23 6.1 Suggestions for future studies 24

7. Bibliography 25 Appendix 1 28 Appendix 2 34 Appendix 3 37

1

1. Introduction

With the rapid spread of Internet access across the world, ideas and information can be shared

across national boarders at low marginal cost. As traditional telecommunication monopolies

are shattered and new communication networks are introduced, competitive market powers

should lower any prices or costs associated with the sharing of information, and information

goods.1 The cost of one of these goods and the implications of sharing will be explored in this

thesis.

An example of information good is typically digital music products which posses the

characteristics of high fixed costs for the first copy, but low marginal cost for additional

copies.2 There are many goods that posses the characteristics of high fixed cost and low

marginal cost, that may be defined as information goods, however this paper will only focus

on music as the information good. The main problem for information good producing firms

toady is the presence of piracy, which makes the firm’s product available on a black market.

Piracy, the illegal redistribution and downloading of the information goods, combines low

reproduction and distribution costs for additional consumers to obtain the good.3 These

characteristics, low reproduction and distribution costs, also allows piracy to be defined as a

public good.

Although the music industry includes hundreds of different record labels, the industry can still

be described as a monopolistic competition were the music sales of the industry is

concentrated around a few companies.4 This definition is partly due to the economies of scale

that can be observed in the industry, where it is more economically sound to produce a vast

number of different artists to spread out the fixed cost that arise in production. In addition to

cost efficiency, it is possible to argue that artists (and their music) are not perfect substitutes

for one another. This notion implies that a firm has a monopoly on the production and sale of

the specific artist, and allows the firm to price their products above perfect competition levels.

The introduction of piracy on a market defined as operating under monopolistic competition

will challenge the price-setting capabilities of the firms by offering the same (or similar)

1 Kaul, Grunberg, and Stern, Global Public Goods, 318.

2 Belleflamme, Pricing Information goods in the presence of copying, 2.

3 Holm, Can economic theory explain piracy behaviour?, 1.

4 Rob and Waldfogel, Piracy on the high C’s, 4.

2

product, but at a much lower cost for the consumer, a marginal cost (opportunity cost) that

usually is assumed to approach zero. A government contemplating to intervene on a market

influenced by piracy must evaluate the positive effects offered by piracy, as the good is made

available below the monopoly price, against the losses inflicted on the firm’s profits. These

problems, the incomplete market caused by piracy, and monopoly pricing and welfare losses,5

are at the core of what is being explored in this thesis.

The presence of piracy will affect the firms by reducing their potential profits, an effect that in

2008 was estimated to cost the music industry US$40 billion in lost revenue.6 However, this

is a figure that must be viewed with some caution due to the fact that the actual revenue lost

for the industry should only contain consumers that would buy the original good if piracy was

not available.7 Individuals with a low willingness to pay for the original good and who obtain

the information good by illegally downloading it do not constitute a source of lost revenue for

the music industry as they would not have consumed the good if piracy was not available.8

These individuals, with their low valuation of the original information good, therefore

constitute a positive effect on the consumer surplus and will increase the social welfare

function through their actions. However, the growth of piracy networks may have a negative

effect on the firms profit function if the availability of the information good via the Net

changes consumer preferences. That is to say, consumers who initially would have bought the

good change their actions and choose to illegally download the good. Rob and Waldfogel

(2004) found in their sample that for every ten albums illegally downloaded, between one and

two would have been bought legally if piracy were not an option.

There may be many reasons why consumers choose to illegally download an information

good instead of obtaining the good on the legal market. Besides the economic motive of

downloading the information good, the low marginal cost for the consumers, there may be

moral, social or psychological factors behind consumers decision to obtain the information

good on the black market.9 If the consumers are rational, the decision comes down to which

action, buying the good legally or illegally downloading it, yields the greatest expected utility.

However, the (expected) utility of an individual who illegally downloads the good will

5 Stiglitz, Economics of the Public sector, 71-79.

6 Cammaerts and Meng, Creative destruction and copyright protection, 5.

7 Rafael and Waldfogel, Piracy on the high C’s, 2.

8 Cammaerts and Meng, Creative destruction and copyright protection, 5.

9 Marchese, Taxation, black markets, and other unintended consequences. 255-256.

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depend on the likelihood of being caught and the resulting punishment, which are decided by

the government.10

1.1 Statement of the problem

By consuming the illegal good, a consumer can obtain a greater utility than it otherwise would

have been able to. This is due to piracy offering the consumer the information good at a lower

cost. However, if the proportion of illegal consumption becomes too large, the information

good producing firm may not be able to recuperate the cost of production. This may further

reduce output, or a worse case scenario; force the firm to cease all production.

When a market is operating inefficiently, even thought there are well-assigned property rights,

it falls upon the government to decide how to internalize any externality. The problem facing

the government is that by fully protecting the firm’s property rights, it will inflict monopoly

deadweight losses upon society. On the other hand, by not protecting the interest of the firm,

there is a risk that the firm will go bust and the market will disappear.

1.2 Purpose

The purpose of this study is to provide a theoretical model of how a government can use

policy instruments to minimize losses on a market influenced by piracy, and at the same time

correct the behaviour of the monopoly firm. The policy instruments investigated are: sales

tax, fine, the probability of being caught, and lump-sum transfers.

1.4 Previous Studies

The subject of piracy and information goods has been explored and discussed in the economic

literature for a long time. The initial interest of the subject was on physical copies of original

goods (e.g. photocopies of books) but during the late 1990’s and early 2000’s the notion of

digital piracy became a topic of discussion. This was due to the introduction of personal

computers with access to the Internet, where consumers could with a great ease obtain

copyrighted material at a low opportunity cost, instead of obtaining the original good in a

store. The rapid expansion of digital piracy during these years can partly be dedicated to the

suspicion that information good producing firms exhibited towards this new distribution

channel, i.e. Internet.11

In addition to the firms not capitalizing on this new distribution

possibility, the slow pace of national government to implement new more stringent copyright

10

Gopal, et al, A behavioural model of digital music piracy. 11

For example, the iTunes store did not open until 2003.

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law12

and the subsequent ineffectiveness of these laws,13

allowed the concept of piracy to

settle amongst the consumers.

Previous studies of the subject have mainly focused on a consumer-firm perspective.

Empirical studies have tried to convey which parameters explain why consumers choose to

download the good illegally, while theoretical studies have defined the consumer demand and

firm profit under different market scenarios.

Paul Belleflamme (2002) provides a theoretical model to describe the effects copying has on

the producer of information goods. Production is initially set by a monopolist, which has to

take into account that the consumers can obtain the pirated good at a constant cost. The author

describes the market as a strategic pricing game where copies are assumed to be of lower

quality then originals. To investigate the (short- and long-run) welfare effects piracy has on

society, Belleflamme constructs three models, where the baseline model consists of a single

good produced by a monopoly. This model is then elaborated to include multiple goods

produced by multiple firms, and where there is a constant marginal cost of copying but no

fixed cost. And finally, a model with multiple goods and firms, where there is a fixed cost of

copying but no marginal cost. The author defines three reaction possibilities for the firm(s)

when it faces piracy, blocked, deter, or accommodate. The result of Belleflamme’s study is

that the demand and welfare effects of piracy will depend on whether the good is associated

with a fixed cost and no marginal cost, or a constant unit cost and no marginal cost.

Christiansen and Smith (2012) explore the impact of policy instrument to control externalities

caused by consumption behaviour. The authors assume (based on previous studies) that the

use of tax instruments to correct externality problem are imperfect/inadequate policy tools.

This is due to inefficiencies in the taxes and imperfect tax targeting of the government. Based

on this knowledge, Christiansen and Smith focus on inefficiencies caused by insufficient

differentiation and undesired differentiation. Where the first inefficiency occur when a

uniformed tax is implemented, but where the externality differs between the consumption

goods. And the second inefficiency occurs when some consumption goes untaxed due to

evasion or importation. As taxation is an imperfect correction instrument, the authors argue

12

The ”Intellectual Property Rights Enforcement Directive”-law (IPRED) was not enacted into Swedish law

until 2009. 13

Börkdahl, Ingen långsiktigt effekt av IPRED (No long term effect of IPRED).

5

that corrective regulation ought to be implemented as a complement to taxes. To explore the

two tax inefficiencies and the effects of regulation, Christiansen and Smith construct two

theoretical models. Depending on which tax inefficiency is observed, corrective regulation on

the market will be implemented differently. The authors conclude that regulation in

combination with taxation will (in general) lead to a more efficient outcome than only using

imperfect externality taxation or regulations.

Piolatto and Schuett (2012) investigate the effects of piracy when taking into consideration

differences in popularity between artists. Their model consists of firms (artists) producing one

good (i.e. recording) and selling it to a continuum of consumers, and where the artist’s level

of popularity is exogenously given. The illegal good is not subject to congestion, but the

availability and marginal cost of the illegal good is assumed to be decreasing as original sales

increases. The music-producing artist only faces the fixed cost of production and earns

revenue either from record sells or alternative commitments (e.g. live performances). Piolatto

and Schuett argue that popular artist’s products are easier accessible online and will be

(illegally) consumed to a large extent compared to less popular artists, which will entail the

popular artist to lose revenue in terms of record sales. However, the monetary losses inflicted

by piracy are compensated by an increase in alternative revenue sources. The authors find

three possible scenarios for how the artist can react to the notion of piracy, accommodate,

deterrence or blockade, where the artist reaction will be dependent on their popularity.

Popular artist will generally accommodate, while less popular artist will either deter or block

piracy. Piolatto and Schuett conclude that popular artist will experience a benefit from the

presents of piracy, as their side revenues will increase, while less popular artist will only

experience negative (monetary) effects of piracy. On the other hand, the authors argue that the

long run effects of piracy may be a decline in music variety, as less popular artist may hesitate

to continue their production.

In the article Optimal Enforcement and Anti-copying Strategies to counter copyright

infringement published 2008, Banerjee, Banerjee and Raychaudhuri (BBR) explore the anti-

coping investment a firm may consider, together with the policy instruments available for the

government when addressing the issue of commercial piracy. The authors construct a

theoretical model that consists of a monopoly producer (producing the original good) and a

possible entrant on the market by the fake producer (commercial pirate) that produces

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identical copies of the monopoly product. The government is responsible of deterring the fake

producer from entering the market by monitoring and penalizing piracy. The problem BBR

presents is described as a Stackelberg game split into three stages. At a first stage the

government will choose its rate of surveillance of the market. The monopoly producer will

then set the output and the level of anti-coping investment to maximize profits. In the final

stage of the game, the fake producer will react to the government’s and monopolist strategies,

and decide whether to enter the market and if so, the level of (fake) production. The authors

define three possible strategies for the monopoly producer. The monopolist can either invest

in anti-copying so as the fake producer cannot produce the good (No copying), or set price

and output at a level where entrance is not profitable for the fake producer (Aggressive). A

finally strategy for the monopoly producer is to use its first mover advantage and maximizing

profits and assuming that the fake producer will copy the product (Accommodating). BBR

find two different equilibriums to the game. If the government monitors the market for fake

producers, the monopolist’s best strategy is to accommodate piracy. If on the other hand the

government does not monitor the market, either accommodate or no coping, are the best

strategies available for the monopolist.

Alessandro Balestrino (2012) investigates indirect redistributive taxation and corrective

taxation on physical status goods (e.g. designer handbags) when illegal copies are available

on the black market. Balestrino constructs a market with a well-defined illegal segment that

produces high-end fashion items, items that by nature is tax exempt. Classical assumptions

indicate that status goods ought be taxed more heavily than general consumption, due to the

fact that status goods are overconsumed (primarily) by high-income individuals. However, as

the market is influenced by the presence of pirated status goods, the classical theory of heavy

taxation does not hold. Where as by introducing large taxes on the legal status good market,

the government will only encourage the expansion of the black (tax exempt) market.

Balestrino also conclude that corrective taxation will encourage those who buy the (legal)

status good to consume the good at an efficient level, but at the same time increase the size of

the black market. The paper highlights the inefficiency of governmental interaction on a

market that is influenced by piracy.

In the article Does Copyright Enforcement Encourage Piracy? by Harbaugh and Khemka

(2010), the two authors argue that enforcement of copyrights can have adverse effects on

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society. While the aim of such policy instrument is to raise the cost of piracy for all

consumers, the enforcement is most likely to affect the consumer segment with a low

willingness to pay. Harbaugh and Khemka describe a model with a continuum of potential

buyers, which can buy the copyrighted good (software program) from the firm, buy an

inferior pirated copy or not consume the good at all. The authors define the price and output

strategy of the firm as; acting like a monopoly, competing with the pirated good, or extending

the enforcement of the copyrighted good. By only enforcing their copyrights upon high-value

buyers (targeting), the firm will have an incentive to raise the price above monopoly levels

(super-monopoly pricing), which will have the effect of encouraging piracy consumption of

all other segments. By extending the enforcement of the copyright to other consumer

segments, the firm will need to lower the output price (below super-monopoly pricing) so as

to sell the additional goods. The reduction in price will thus raise consumer surplus and be

welfare enhancing. However, Harbaugh and Khemka also note that a more broad-based

enforcement scheme (instead of targeted enforcement) may cause consumer price to rise,

piracy consumption to fall, and consumer surplus to decrease.

In an article published 2003, Gayer and Shy investigate the effects of taxing hardware (e.g.

CD’s, cassettes, etcetera) and transferring the proceedings to information good producers (e.g.

software producer). The authors assume that the government cannot directly enforcement of

property of the producers, and instead have to adopt a second-best solution to combat piracy.

Due to the fact that piracy occurs on a black market, taxing the consumer who are consuming

the illegal good is impossible, instead other methods must be used to force consumers to face

the true cost of their (illegal) consumption and at the same time compensate the producer’s

loss in revenue. To investigate the problem of uniformly taxing all consumption of (possible)

piracy hardware, Gayer and Shy construct a model consisting of an infinite number of

consumers, a monopoly firm (software producer), and a government. The authors assume that

there exists a positive but diminishing relationship between the size of the monopoly firm’s

profits and the hardware tax. As the size of the tax increases, consumer demand for hardware

will decrease, but the demand for the monopoly good will also decrease. The problem facing

the government is that the tax rate that eliminates piracy is above the profit maximizing tax of

the firm. The authors conclude that due to the positive relationship between the taxed

hardware and the monopoly good, second-best governmental interaction is inefficient unless

the market is in danger of failing.

8

In an empirical study of college students valuation of music albums, Rob and Waldfogel

(2004) concludes that an additional downloaded unit will lower sales of the original good by

0.1-0.2 units. That is to say, one in ten (or two in ten) did not buy a specific album, which

they otherwise would have done, due to piracy. In addition, Rob and Waldfogel also conclude

that piracy raises consumer welfare by $70 per capita.

Håkan J. Holm (2003) performs a survey study to investigate if economic theory can explain

piracy, where students at the university of Lund were asked about their preferences for

information goods and individual traits. Holm concludes that if an individual has a low

valuation of the original good, possess computer skills and is a male, he is more likely to

engage in piracy activities. The author also finds suggestions that any campaign aimed in

changing moral opinions associated with piracy will be inefficient.

Christopher Yoo (2007) performs a comprehensive study to explore the properties of

copyrightable goods. The conclusions of Yoo’s study is that, contrary to conventional

assumptions, copyrightable works do not possess pure public goods characteristics, rather

they ought to be viewed as impure public goods.

2. Information as Public good

The definition of a pure public good must start with the clarification of the concepts of non-

rivalrous14

, and non-excludability in consumption, two properties a pure public good must

posses. Non-rivalrous in consumption implies that more than one individual can consume the

good without preventing other individuals from consuming the same good. While non-

excludability is the inability of the provider to control the use of the specific good.15

Goods

that posses both properties are defined as pure public goods, however there are few good that

posses both properties, were classic examples include national defence and lighthouses. More

commonly are goods that possess some of the characteristics of the pure public good, often

referred to as impure public goods, for example parks and motorways, which may be the

subject of exclusion through congestion.16

The inability of controlling the usage of public

goods often causes firms not to produce them, as firms are not able to extract the true cost of

14

Following the terminology used by John Leach in ”A Course in Public Economics”. 15

Leach, A Course in Public Economics, 155. 16

Ibid. 156.

9

production from the consumers. The inability of the market to provide public goods typically

forces the government to provide the public goods that are being consumed.

Information goods can be expressed as possessing some of the characteristics of a public

good. This is due to the ability of consumers to copy and redistribute the good on the black

market, where one individual’s consumption of the good does not hinder additional

consumers from obtaining the same good. Once a consumer has obtained the information

good, i.e. bought a CD or music file online, the consumer can make the good readily available

for other consumers on the Internet without the firm being able to charge additional users the

cost of consuming their product.17

That is, due to piracy the firm cannot exclude consumers

from obtaining and consuming the information good. The information good is said to be non-

rivalrous in consumption due to the fact that the marginal cost of providing additional copies

of the good is zero.18

This means that as the information good is illegally provided on the

Internet, one consumer’s consumption (download) does not stop another consumer from

acquiring the good.

When discussing public goods, and piracy’s effect on social welfare, it is important to present

the concept of “free riding”. Due to the fact that the public good is non-excludable, there is no

one that can hinder an individual from consuming the good, and also, there is no one on the

market that can force the individual to pay the costs of providing the good.19

Whether or not

piracy constitutes a positive externality or a free riding problem depends on the market

scenario and which actor’s objective is taken into consideration. The free riding problem will

only become problematic for society if the extent of the free riding causes the whole market to

break down, that is to say that all individuals decide to download the information good.

The government’s reaction to the notion of the information good as a public good depends on

the circumstances of the market and does not automatically imply that the government must

intervene on the market. Instead of the government providing the information good according

to the Samuelson condition for optimal provision of a public good, with all the implication

this entails,20

the government can use its policy instruments to create and enforce the

17

Yoo, Copyright and Public Good Economics, 10-11. 18

Ibid. 11. 19

Except the authorities if they catch the individual consuming the illegal good. 20

See Chapter 10 in A Course in Pubic Economics by Johan Leach.

10

existence of the intellectual property rights for the producing firms. Following Yoo’s (2007)

argument that information goods does not possess the characteristics of pure public goods, the

notion that the government should intervene and provide the good by using the Samuelson

condition becomes unsuitable. While the government will not be able to change the non-

rivalry property of the information good, it is possible to make the information good more

excludable by writing and enforcing the rule of law.21

However, the mere existence of the

firm’s intellectual property rights does not guarantee that all consumers will obey the law and

buy the information good in the legal market.22

Through the allocation of funds to antipiracy

activities the government can secure the existence of the information good market.

3. Model

Three agents populate the economy described below: consumers, producer and a government.

There are n identical consumers normalized to one, one monopoly firm, and a government

that wants to maximize society’s welfare. The economy operates over one time period so

there are no intertemporal decisions being made by any of the agents.

3.1 The Consumer

The consumer’s utility can be described by a general utility function

[( ) ( )] (1)

where x is the information good the individual consumes, ( ) is the amount of legal

information good consumed and the amount of illegal information good, c is a numeraire

good, f is the total amount of time available for the consumer and ( ) is the opportunity

cost of time spent downloading illegal information good. The consumer will experience

different utility depending on whether the good is the original- or illegal good, due to the

assumption that there is a quality difference between the goods. The proportion of illegal

consumption is in the interval of 0 and 1, were indicates that all consumption of the

information good is provided on the illegal market and that all consumption is provided

on the legal market. ( ) can be thought of as time spent devoted to searching and

downloading the illegal information good. That is to say, if the consumer increases the

21

Varian, Markets for Information Goods, 7. 22

Belleflamme, Pricing information goods in the presence of copying, 25.

11

proportion of illegal consumption, a, or increases the total amount of information goods

consumed, x, he/she will need to allocate more time to obtain the illegal goods.23

The consumer’s budget restriction will differ depending on if the consumer is caught or not

caught downloading the information good.

( ) (2)

( ) (3)

The left hand side (LHS) of the consumer’s budget restriction when caught, equation (2),

consists of the disposable income level m, assumed to be fixed, and a lump-sum transfer, .

For the consumer’s budget to balance, LHS must equal the cost of the numeraire good, c,

where prices are normalized to one, the cost of consuming the legal information good,

( ) ,24

and the fine imposed by the government on caught consumers, .

When the consumer is not caught, equation (3), the LHS of the budget restriction will equal

the cost of the numeraire good and the cost of legal consumption.

The consumption of the illegal good will result in some uncertainty for the consumer, as it is

not known whether the individual will experience an increase in welfare from the illegal good

or be forced to pay the piracy fine imposed by the government. This notion results in that the

utility of the consumer is expressed as an expected utility function.

[ ] ( ) [ ] ( ( )) [ ] (4)

The expected utility function is the sum of the utility the consumer obtained when caught

multiplied with the probability of being caught, and the utility the consumer obtains when not

caught multiplied with the probability of not being caught.25

The probability of being caught

is a function of R, an institutional parameter referring to antipiracy activities from law

enforcement agencies and court proceedings. If the individual does not obtain the information

23

That is and

. 24

Where is the consumer price and can be expressed as , which is the producer price plus the

sales tax. 25

Varian, Microeconomic Analysis, 174.

12

good illegally, , the utility is known with certainty and will be: [ ]. The direct

utilities are:

[ ] [( ) ( )] (5)

[ ] [( ) ( ) ] (6)

where the differences in utility between equation (5) and (6) is that the consumer will

experience an increase in welfare from illegal consumption, , when not caught. On the

other hand when caught, the consumer must allocate funds to cover the expense of the fine, as

expressed in equation (2). This allocation will entail that the consumer will be able to afford

less of all goods and subsequently obtain a lower utility. An assumption is made that a

consumer is only caught after all illegal information goods is downloaded. As a consequence,

the consumer will pay the marginal opportunity cost, ( ), independent if caught or not.

By combining equations (2) to (6) it is possible to express the consumers expected utility

when caught- or not caught illegally downloading the information good.

[ ] ( ) [( ) ( ) ( )]

( ( )) [( ) ( ) ( ) ] (7)

To obtain the first order conditions for the consumer demands, equation (7) is differentiated

with respect to x and a:

[ ]

( ) ( ( ) ( ( ) ) (

))

( ( )) ( ( ) ( ( )) (

) ( ))

(8)

[ ]

( ) ( ( ) ( ) (

))

( ( )) ( ( ) ( ) (

) ( ))

(9)

13

which expresses that the consumer will make an optimal choice when the marginal utility of

consumption equals the consumer’s marginal cost. From equations (8) and (9) it is also

possible to stated that the consumer demand for x is a function ( ( ) ) and the

consumer demand for a is a function ( ( ) ).

3.2 The Firm

Turning to the firm, its profit is determined by whether the consumer buys the original good,

but since piracy occurs on the market, the firm does not make a profit on all goods produced.

That is to say, the firm produces more information goods then they sell, but still have to pay

the cost associated with this production. An assumption is made that all the information goods

produced will be available on the illegal information good market. The firms profit function

can be described as:

( ) (10)

where the firm earns revenue from each unit of x sold on the legal information good market,

but due to piracy the firm does not sell all goods produced, ( ) . Associated with

the production of the information good is a fixed cost, F, normalized to zero, and a marginal

cost of producing additional units of the information good, . The firm may earn additional

revenue from a lump-sum transfer from the government, .

By assuming that the firm faces a negative demand curve on the market, an increase in

production of the good must be met by a decrease in price to sell the additional output. This

results in that the output price of the firm is a function of the x. However unlike normal

monopoly’s, the information good producing firm faces an externality, as some proportion of

output is lost through piracy. By using equations (8) and (9), it is possible to obtain

expressions for and a as functions of x.

( ( ( ) ) ( ) )

( ( ( ) ) ( ) )

(11)

By substitution the expressions in equation (11) into equation (10) and differentiating the

function with respect to x, the first order condition of the firm obtained. By rearranging the

expression, the monopoly firm’s profit maximization condition is obtained:

14

( ) ( )

(12)

where the monopoly firm’s output choice will entail that the marginal revenue, the LHS, must

at optimum equal marginal cost, the RHS. The solution in equation (12) does however differs

from the profit maximization condition for a firm operating on a market not influenced by

piracy,

. That is to say, when the firm ears a profit on all goods

produced. However when the market is influenced by piracy, , the firm does not earn a

profit on all goods produced as some consumption is lost for the firm. There is also an

additional cost of production to the firm. By differentiating equation (10) with respect to a,

the marginal damage of piracy for the firm is obtained.26

The last term on LHS in equation

(12) represents the marginal damage of piracy, multiplied with the change in illegal demand

as the firm increases output. By assuming that

, any increase in output will result in the

firm increasing the marginal damage inflicted upon itself.

The notion that the firm is the only producer of the information good will entail that the price

and output is set different to social optimal and the market can be said not to operate

efficiently.27

This result, monopoly pricing and welfare losses, will influence the level of the

optimal policy tools later on.

3.3 The government

The government’s budget constraint is given by:

( ) ( ) (13)

where ( ) is the revenue from the sales taxation for the legal information good,

is the fine an individual who is caught illegally downloading the information good has

to pay, which is dependent on the probability of being caught ( ) . The probability

parameter will appear in the governments budget constraint due to the fact that the authorities

initially does not know whether the consumer will be caught and made to pay the fine. R is

26

27 Rosen and Gayer. Public Finance, 46.

15

the cost of counter piracy actions from the government. As stated above, the probability of

getting caught p is a function of R, were the governments counter piracy actions can be

thought of as allocation of funds for police and court proceedings, which increase the

probability of catching consumers who illegally download the information good. is the

lump-sum transfer paid by the consumer, and is the lump-sum transfer received by the

firm.

By using equation (7) and equation (10), it is possible to express the social welfare function

as:

[ ] (14)

where the utilitarian social welfare function is the expected utility of the consumer, and the

profit of the firm. The utilitarian welfare function is used in this instance so as the government

will treat a change to the expected utility the same way as a change in the firms profit. The

objective of the government is to maximize the social welfare function, subject to its budget

restriction. This objective is complicated by the fact that the market experiences two market

failure, monopoly and incomplete markets. To provide an expression for the governments

maximizing problem, a Lagrange function is constructed using equation (14) as the objective,

subject to equation (13), where and corresponds to the optimal choice made by the firm,

and is the Lagrange multiplier.

[ ] [ ( ) ( ) ] (15)

3.4 First order conditions

Presented below are the first order conditions for the government’s maximization problems.

These are obtained by differentiating equation (15) with respect to the sales tax, t, the fine, T,

the allocation of funds for police and court proceedings (changing the probability of being

caught), R, and the lump-sum transfers (consumer) and (firm). In Appendix 1, the

mathematics for equations (16) to (20) is provided. The result from the equations will then be

used to obtain an expression for the optimal sales tax and the optimal fine.

16

(( ) )

( )

[( ) ( )

( )

( )

]

(16)

( ) ( )

( )

[ ( )

( ) ( )

( )

]

(17)

( )

( )

( )

[ ( )

( )

( )

]

(18)

( )

( )

[ ( )

( )

( )

]

(19)

(20)

4. Solving for the optimal policy

By combining the results from equations (16), (17), (19) and (20) it is possible to provide an

expression for the optimal sales tax and the optimal fine. To solve the problem, equation (19)

is multiplied by a given expression and the new equation is subtracted equation (16) and (17),

respectively. The equation is the solved for the specific parameter, and provides an expression

for the optimal policy tool. The mathematical procedure of solve the optimal policy problem

is provided in Appendix 2.

The problem will be solved in two stages, first by assuming that one of the policies is

unavailable, i.e. equal to zero. The result for the optimal policy will then be analysed, and the

sign of the expression will be decided. This is followed by a discussion of the effects and

implications of using both policy tools simultaneously. The equations of the simultaneous

17

policy tools will not be presented in the text below, due to the fact that the sign of the

expressions cannot be determined. However, the equations will be provided in Appendix 3.

4.1 The simplified problem - The optimal sales tax

Under the assumption that the use of the fine is unavailable, , the authorities must

combat two market failures by the implementation of an optimal sales tax. To solve the

simplified optimality problem, equation (19) is multiplying by ( ) and then the new

equation is subtracted from equation (17).28

By using compensated demand functions29

and

solving for parameter t, the equation will sum to:

[

( )

]((

)

( ) ) (21)

where the sales tax can be shown to be smaller than zero, . That is to say, the optimal

policy tool for the government when combating the market failures, and no other instrument

is available, is to implement a subsidy. In addition to trying to correct the behaviour of the

consumer, the government also wants to correct the monopoly behaviour of the firm. Where

as by introducing a subsidy the government can influence the firm to increase output to (or

closer to) social optimal levels. From equation (20) it can be observed that the Lagrange

multiplier (the government’s shadow price) equals one. This notion indicates that there is no

additional cost for the government in implementing any of the policy tools. The denominator

of equations (21) will be negative. This is due to the fact that a small change in the sales tax

will decrease the compensated demand for the original good,

, and at the same time

increase the compensated demand of illegal consumption,

. The first term in the

nominator corresponds to the marginal damage of piracy for the firm, a concept discussed in

section 3.2. The expression indicates that an increase in the sales tax will increase

compensated demand of illegal consumption and inflict the firm loss with an additional loss in

revenue. The second term in the nominator corresponds to the change in the compensated

price function from an increase in the sales tax. As the sales tax increases, the firm will need

28

Where both expressions are adjusted to the notion of T=0. 29

For example,

expresses the difference in illegal demand after a change in the sales tax and the adjusted

lump-sum transfer. That is, [

( ) ].

18

to decrease output price to keep consumption constant. That is to say,

. The result of

the denominator and nominator indicates thus that the government will set the sales tax below

zero.

4.2 The simplified problem - The optimal fine

Assuming that the sales tax is not available for the government, , the authorities must

use a fine scheme to combat the two market failures. The appeal of using a fine imposed on

the caught individual is that it offers the government a cost efficient and passive policy

instrument of correcting the market failures. As long as the probability of being caught is

greater than zero (true by assumption), the government will rely on the market to adjust itself

after the introduction of the fine. To obtain an expression for the optimal fine, equation (19) is

multiplied by ( )( ) , and then the expression is subtracted from equation (17).30

By

solving for T and using the expressions for the compensated demand functions, the equation

will sum to:

(

( ) [

])((

)

( ) ) (22)

where the optimal fine will be greater than zero. The denominator of equation (22) will be

assumed to be smaller than zero. The compensated demand for the original good will be

assumed to increase from a small change to the fine,

, at the same time as the

compensated demand of illegal consumption will decrease,

. However, the

compensated demand for the original good is scaled with the proportion of illegal

consumption, a, which is interval 0 and 1. The compensated demand of illegal consumption is

multiplied with the total amount of information goods produced, x, which is (by assumption) a

real number greater than one. This results in the assumption that the bracketed term in the

denominator is smaller than zero. In addition to the compensated demands, the denominator is

also determined by the probability of being caught, ( ), which is indirectly selected by the

government through the allocation of funds for police and court proceedings, and is by

assumption greater than zero.31

The conclusion of the positive sign of the probability

30

As in the previous example the equations are adjusted for . 31

If the probability of being caught were zero, there would be no risk for the consumer of downloading the

illegal good. However, as consumption of the illegal good entails some uncertainty, the probability of being

caught must be greater than zero. ( ) .

19

parameter and the negative sign of the bracketed term is that the denominator sums to a

negative expression. The first term of the nominator corresponds, as for equation (21), to the

marginal damage inflicted on the firm by piracy. However the expression in equation (22) is

multiplied with the change in compensated demand of illegal consumption as the fine is

increased. As stated above, the compensated demand for illegal consumption is decreasing as

the fine increases. This results in that the first expression of the nominator is negative,

implying that firm will experience an increase in revenue (or an decrease in the marginal

damage) as the fine is increased. The second term in the nominator, the effect to the

compensated price function as the fine is changed, can be assumed to be greater than zero,

. That is to say, as the fine increases, illegal consumption will decrease, allowing the

firm to raise the output price without loosing consumption to piracy. These notions, the

effects to the marginal damage and the compensated price, results in the nominator summing

to a negative expression. The negative denominator and nominator will does entail that the

optimal policy available for the government when it wants to correct for two market failures

is to introduce a fine that is greater than zero.

4.3 The simultaneous optimality problem

Dropping the assumption that the government can only use one policy tool at a time to correct

the market failures, the authorities can solve the optimality problem by setting both policy

tools simultaneously. However unlike for the simplified problems, equations (21) and (22),

the sign of the expressions (positive or negative) for when the government sets both policy

tools simultaneously cannot be determined with certainty. Due to this implication the

equations for the simultaneous policy tools will not be presented in the text below,32

instead a

discussion of the effects of setting co-ordinated policy tools will follow.

By assuming that the simultaneous optimality tools have the same sign as for the simplified

solutions, and

, a question can then be stated; “Why would the authorities

introduce a second policy tool if they could fully correct both market failures using only

one”? If the government set the sales tax optimally (introduces a subsidy), why introduce the

fine? One possible answer may be that a subsidy scheme does not fully correct the behaviour

of both agents. Where it is possible that illegal consumption may not decrease to such an

extent that the authorities expected, due to the fact that the marginal opportunity cost of piracy

32

The mathematics are however presented in Appendix 3.

20

can be assumed to be lower than the subsidy compensated consumer price. A second possible

answer may be that the cost of implementing an optimal subsidy will be too large for society

to bear.

If, on the other hand, the authorities introduced the optimal fine initially, what would be the

incentive to introduce a subsidy scheme? From the nominator of equation (22) it can be

observed that the introduction of the optimal fine will mainly act in the interest of the firm by

increasing its price-setting capability and at the same time decrease illegal comsumption. The

consumer on the other hand will experience a decrease in its expected utility, as the disutility

of being caught will increase as the fine is raised, causing the consumer to decrease illegal

consumption. The consumer will experience additional losses in welfare as the price of the

legal good increases. The introduction of a subsidy scheme will then both boost consumer

utility (lowering consumer price) and firm profits (increasing production and output price),

and consequently raise social welfare.

The implication of setting the policy instruments simultaneously will in the end (probably) be

that the both tools are set at lower levels compared to the simplified solutions.33

That is to say,

and

. This is due to the fact that both tools will decrease the amount of market

failure the other tool needs to correct for.

5. Discussion

In the following section, the implications of using the optimal policy instruments and their

effects on the social welfare function will be discussed. To the author’s knowledge, a study

with the aim to investigate the effects of governmental policy instruments on a market

influenced by digital piracy has not been done before.

The government’s maximization objective is impeded due to two market failures that are

observed and were the two failures affected the agents in opposing ways. If the market had

been characterized by perfect competition, any presence of piracy had been deemed welfare

deteriorating, as the market would have been operating at the social optimal level. However,

just because there exists market failures does not justify the government to intervene.34;35

On

33

Also argued in Christiansen and Smith, Externality-Correcting Taxes and Regulation. 34

Brent, Applied Cost-Benefit Analysis, 109.

21

the other hand, by actively engaging the marketplace the government can force the market to

operate closer to the social optimal point and advert the possibility of the market breaking

down completely or a decrease in variety of goods produced36

.

The result from equation (21) states that the optimal sales tax should be set smaller then zero.

That is to say, the authorities will introduce a subsidy.37

A subsidy scheme will have the effect

of offering the consumer a price closer to perfect competition levels, which will increase legal

consumption and at the same time decrease illegal consumption. As the consumer can afford

more of the original good at a given level of income, the expected utility of the consumer will

increase. The firm on the other hand will face an output price greater than the consumer price,

were as by increasing output the firm can raise its profits and production will move closer to

social optimal levels. By introducing a subsidy scheme the authorities can counteract the two

market failure, increase production and raise social welfare. However, if the government aims

to cover all consumer demand for the original good, the size of the subsidy may in the end

become too large for society to bear, where the cost of the subsidy outweighs the increase in

welfare.38

Even at perfect competition price and output levels there may be an incentive for

the consumer to consume the illegal good, as the marginal opportunity cost of piracy

approaches zero. Under such a scenario, piracy can be argued to be welfare deteriorating as

the market is operating (by assumption) at the social optimal point.

While a subsidy solution both boosted legal consumption (increasing output and decreasing

the consumer price) and decrease the proportion of consumption lost to the firm (increasing

profits), a fine solution will only directly correct the behaviour of the consumer (the aspect of

incomplete markets). The introduction of an optimal fine can be view as a passive policy

instrument by the government, where as long as the probability of being caught is greater than

zero, any increase in the fine will influence the information good market and does not entail

any additional interaction from the authorities.39

The instrument will however decrease the

proportion of illegal consumption by increasing the disutility of being caught, and

subsequently lowering the expected utility of the consumer. That is to say, the potential of

having to pay the fine is a sufficient deterrent to correct the effects of the incomplete market.

35

Ibid, “Just because an externality exists in a private market, it doesn’t necessarily mean that the equilibrium

output is not socially optimal”, 128. 36

Piolatto and Schuett, Music piracy, 39. 37

A result also noted in Auerbach and Hines, Taxation and Economic Efficienty, 59. 38

Harbaugh and Khemka, Does Copyright Enforcement Encourage Piracy?, 314. 39

True under the assumption that the government does not pay any court or policing cost to get the consumer

convicted of piracy.

22

The positive compensated price function in equation (22) also indicates that any increase in

the fine will offer the firm the possibility of raising the output price closer to monopoly levels.

By implementing a sufficiently large fine, the government will only act in the interest of the

firm, increasing the market inefficiency and imposing monopoly losses on the society.

However the result from equation (22) only states that the optimal fine is set greater than zero.

This notion, , allows the government to adjust the market failures by

implementing a fine that is sufficiently large to decrease some of the illegal consumption but

not so large as to impose additional welfare losses. This result indicates that the optimal fine

may be set relatively close to zero so as to not cause monopoly pricing and welfare losses.

Due to the fact that a fine solution does not correct the monopoly behaviour of the firm, it is

possible to argue that the policy instrument produces a suboptimal solution. This notion

entails that any stable market equilibrium (where the firm does not inflict any substantial

monopoly losses upon society and piracy does not jeopardize the existence of the market or

the variety of production) will only be reached by chance.

Finally the government can combat the market failure by implementing both policy

instruments simultaneously. As explored in section 4.3, the simultaneously optimality

solutions will most likely entail that the optimal levels are set lower than in the simplified

solution. This is due to the government counteracts the failures on two fronts, by offering a

subsidy (a carrot) and threatening the consumers with a fine (a whip). By using both policy

instruments simultaneously it is possible that the authorities can obtain a market equilibrium

similar to that of perfect competition, where production is social optimal and piracy is viewed

as welfare damaging. The effects facing the consumer will then be an increase in utility by

offering the information good at a lower price, which is offset by an increase in the fine that

raises the disutility of being caught. As the proportion of illegal consumption decreases, the

firm will be able to use its market powers to raise output price closer to monopoly levels. At

the same time as the firm raises the output price it will increase production to face the

increased demand caused by the subsidy. The effect of the government’s interaction may in

the end be to raise social welfare as both the firm and consumer are (arguably) made be better

of than before.40

40

It is important to note that this has not been proven in the text as the size of the consumer surplus and social

welfare has not been computed.

23

6. Summary and Conclusion

The purpose of this study was to provide a theoretical model on how a government could use

policy instruments to minimize losses on a market influenced by piracy (incomplete market),

and at the same time correct the behaviour of a monopoly firm (monopoly pricing and welfare

loss from monopoly). This was done by the defining and characterizing the actors in chapter

3. All consumers were normalized to one, and the demand for the legal and illegal good was

determined. Due to the fact that the illegal consumption included an aspect on uncertainty, the

consumer utility was defined by an expected utility function, where the probability of being

caught (or not caught) influenced the consumer’s behaviour. The firm was assumed to be a

profit-maximizing monopolist, but where some proportion of revenue was lost to piracy. By

using this notion, it was possible to determine the marginal damage of piracy, which stated

the adverse effects piracy had on the firm’s profit function. The government could then use

the expected utility function of the consumer and the profit function of the firm to construct a

utilitarian welfare function, which was then used to construct a Lagrange function, subject to

the authorities budget constraint. By using the government’s Lagrange function it was

possible to obtain first order conditions following a change in the sales tax, fine, probability

of being caught, and two types of lump-sum transfers. From these first order conditions it was

then possible to extrapolate expressions for optimal policy instruments.

The optimal fine was determined to be greater than zero, however the use of a fine on a

market with two types of market failures was argued to produce a suboptimal solution. This

was due to the fact that the fine was unable to correct the behaviour of the monopolist, and the

government would thus rely on market to adjust itself to a stable equilibrium.41

The optimal sales tax (subsidy) was on the other hand able to correct both market failures and

had the potential of raise social welfare. However, it was possible that the use of the subsidy

scheme did not provide adequate incentive for the consumer to decrease illegal consumption

to a more desirable level.42

By introducing both policy instruments simultaneously, the

government could correct the behaviour of the agents and (possibly) reach an equilibrium

similar to that of perfect competition.

41

Where a stable equilibrium in this case referred to the scenario when the firm does not inflict any substantial

losses on society and piracy does not jeopardize the existence of the market or the variety of goods produced. 42

As the marginal opportunity cost of piracy was assumed to approach zero.

24

6.1 Suggestions for future studies

By adding a preference parameter to the consumer demand, as done in Belleflamme, a model

depicting a more realistic market would emerge. The preference parameter would allow the

model to accommodate the notion that all piracy is not revenue damage, a notion that was

empirically observed by Rob and Waldfogel, and theoretically modelled (in another context)

by Piolatto and Schuett. The problem of setting an optimal policy tools would be complicated

further, but the model would provide an interesting foundation for further discussion of the

subject. It is also possible to add the notion that the firm can affect the initial cost of pirating,

as explored by Banerjee, Banerjee and Raychaudhuri, and Harbaugh and Khemka, by

investing in anti-copying technology. Where it is possible to argue that such an addition

would have implications for the firm, the government, and the pirating consumer.

25

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28

Appendix 1 Expected utility:

[ ] ( ) [( ) ( )]

( ( )) [( ) ( ) ]

Budget constraint caught ( )

Budget constraint not caught ( )

Rearranging [ ] ( ) [( ) ( ) ( )]

( ( )) [( ) ( ) ( ) ]

Profit function ( )

Government budget constraint ( ) ( )

Consumer: FOC w.r.t. x

[ ]

( ) ( ( ) ( ( ) ) (

))

( ( )) ( ( ) ( ( )) (

) ( ))

FOC w.r.t. a

[ ]

( ) ( ( ) ( ) (

))

( ( )) ( ( ) ( ) (

) ( ))

( ( ) ). ( ( ) ) Firm:

( ( ) ) ( ( ) ( ) ) ( ( ) ) FOC w.r.t. x

( ) ( )

29

Government Lagrange function

[ ] [ ( ) ( ) ] Differentiate with respect to t, the sales tax

( ) ( (( )

)

( ( ) ( )

)

(

))

( ( ))( (( )

)

( ( ) ( )

)

(

) (

))

Rearranging

(( ( )( ( ) ( ( ) ) (

))

( ( )) ( ( ) ( ( )) (

) ( )))

( ( )( ( ) ( ) (

))

( ( )) ( ( ) ( ( ) ) (

) ( )))

( ) ( ( ) ) ( ( )) ( ( ) ))

Sums to: (( ) )

Firms profit:

( )

( ) ( ( ) )

Sums to:

( )

Government Constraint:

30

[( ) ( )

( )

( )

]

Summing expressions:

(( ) )

( )

[( ) ( )

( )

( )

]

Differentiate with respect to T, the increased fine

( )( (( )

)

( ( )

)

(

))

( ( ))( (( )

)

( ( )

) (

)

(

))

31

Rearranging

(

( ( )( ( ) ( ( ) ) (

))

( ( ))( (( )

) ( ( )) (

)

( )))

( ( )( ( ) ( ) (

))

( ( ))( ( ) ( ) (

) ( )))

( ) ( )

)

( ) ( )

( )

( ) ( ( ) )

( )

[ ( )

( ) ( )

( )

]

Sums to:

( ) ( )

( )

[ ( )

( ) ( )

( )

]

I

32

ncreasing the likelihood of being caught

(

( )

( )

( )( (( )

)

( ( )

) (

))

( ( ))( (( )

)

( ( )

)

(

) (

)))

Rearranging

( ( )( ( ) ( ( ) ) (

))

( ( ))( ( ) ( ( )) (

) ( ))

( )( ( ) ( ) (

))

( ( ))( ( ) ( ) (

) ( ))

( )

( ))

( )

( )

( )

( ) ( ( ) )

( )

33

[ ( )

( )

( )

]

( )

( )

( )

[ ( )

( )

( )

]

Lump-sum transfer consumer

( )

( (( )

)

( ( )

)

(

)) ( ( ))

( (( )

)

( ( )

)

(

) (

))

( )

( )

( ) ( ( ) )

( )

[ ( )

( )

( )

]

34

( )

( )

[ ( )

( )

( )

]

Lump-sum firm

[ ]

Appendix 2

For T=0.

[ ] [ ( ) ]

(( ) )

( )

[( ) ( )

]

( )

( )

[ ( )

]

( )

(( ) )

( )

( )

[ ( )

] ( )

( )

(( ) )

( )

[( ) ( )

]

( (( ) )

( )

( )

[ ( )

] ( ) )

35

( )

( )

( )

( ( )) [

( ) ]

( ) [

( ) ]

(

( ) ) (

( ) ) ( )

( ) [

( )

]

[

( )

]((

)

( ) )

Signing:

⏟

[

( )

]⏟

((

)

⏟

( )

⏟

⏞

)

For t=0

[ ] [ ( ) ]

( ) ( )

( )

[ ( ) ( )

( )

]

( )

( )

[ ( )

( )

]

( ) ( )

( ) ( ) ( )

( ) ( )

[ ( )

( )

]

( ) ( )

36

( ) ( )

( ) ( )

( )

[ ( ) ( )

( )

]

( ( ) ( ) ( )

( ) ( )

[ ( )

( )

] ( ) ( ))

( ) ( )

( )

( )

[ ( ) ( )

( )

]

[ ( )

( )

] ( ) ( )

[

( )

] ( ) [

( )

]

( ( ) ) [

( )

]

( ( ) ) [

( )

]

( )

( ( ) ) [

]

(

( ) [

]) ((

)

( ) )

Signing:

Assumption

⏟

(

( ) [

]⏟

)

((

)

⏞

( )

⏞

)

37

Appendix 3 Solve for the sales tax

( )

(( ) )

( )

( )

[ ( )

( )

( )

]

( )

( ) (( ) )

( )

[( ) ( )

( )

( )

]

(

(( ) )

( )

( )

[ ( )

( )

( )

]

( )

)

( )

( )

( )

( )

( ( )

( )

( )

( ) ( )

( ) )

(

( )

( ) ( )

)

38

[

( ) ] [

( ) ] ( )

( ( ) ) [

( ) ]

( ( ) ) [

( ) ]

( ( )) [

( ) ]

( ) [

( ) ]

( ) ( ( ) )

( ( ) )

( ( ))

( )

( ) ( ( ) ) [

]

( ) [

( )

]

[

( )

](

( )

( ( ) ) [

])

Solve for fine: Multiply the lump-sum function with ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

[ ( )

( )

( )

]

( )( )

39

( ) ( )

( ) ( )

( )

[ ( )

( ) ( )

( )

]

(

( ) ( ) ( )

( ) ( )

[ ( )

( )

( )

]

( )( )

)

( ) ( )

( )

( ) ( )

( )

[ ( )

( ) ( )

( )

]

[ ( ) ( )

( )

( )

( )

( ) ( )

]

( ) [

( )

] [

( )

] ( )

( ( )) [

( )

]

( ) [

( )

]

( ( ) ) [

( )

]

( ( ) ) [

( )

]

( ) ( ) [

( )

]

( ( ) ) [

]

40

(

( ) [

])

(

( ) [

( )

])

Setting up the Matrix

( ) ( ) [

( )

]

( ( ) ) [

]

( ) ( ) [

( )

]

( ( ) ) [

]

[[

( )

] [

]

[

( )

] [

]

] [( )

( ( ) )]

[

( )

( )

]

| | ([

( )

] [

]

[

] [

( )

])

Solve for t

( )

|

( ) [

]

( ) [

]|

| |

( )

(

( ) ) [

] (

( ) ) [

]

| |

(

( ) ) [

] (

( ) ) [

]

([

( )

] [

] [

] [

( )

])

41

Solve for T

( ( ) )

|[

( )

]

( )

[

( )

]

( )

|

| |

( ( ) )

(

( ) ) [

( )

] (

( ) ) [

( )

]

| |

(

( ) ) [

( )

] (

( ) ) [

( )

]

( ) ([

( )

] [

] [

] [

( )

])