can piracy lead to higher prices in the music

Author

Copy

Journal of the Operational Research Society (2009) 60, 372 --383 © 2009 Operational Research Society Ltd. All rights reserved. 0160-5682/09

www.palgrave-journals.com/jors/

Can piracy lead to higher prices in the music andmotion picture industries?M Khouja1∗ and HK Rajagopalan2

1The University of North Carolina at Charlotte, Charlotte, NC, USA; and 2Francis Marion University,Florence, SC, USA

Piracy of copyrighted goods has received increased attention in the literature. Much of this research has focusedon pricing policies, protection against piracy, and governmental policies in the software industries. In thispaper, we focus on pricing policies of producers in the music and motion picture industries. Exact analyticalresults are difficult to obtain; therefore, we develop an approximating function of the cumulative demand.This enables us to obtain closed-form expressions for the optimal price. Our results show that the existenceof piracy in these industries and the lack of positive network externalities may cause monopolists to chargehigher prices to optimize profits. These prices increase with increases in the speed of piracy and longer productlifecycles. We demonstrate the accuracy of our demand approximation function using a numerical experiment.We show how a two-price strategy and dual distribution channels may help in reducing the negative effectsof piracy. We perform some numerical sensitivity analysis and provide managerial insights.Journal of the Operational Research Society (2009) 60, 372–383. doi:10.1057/palgrave.jors.2602552Published online 9 January 2008

Keywords: information goods; pricing; software piracy; decision analysis

Introduction

Piracy of information goods is a major problem facingfirms in the software, recorded music, and motion pictureindustries. Past research has mostly focused on softwarepiracy. While most of the losses from piracy did occurin the software industry, losses in the recorded music andmotion picture industries have grown considerably in the lastfew years. According to the Motion Picture Association ofAmerica (MPAA), the motion picture industry has lost $3.5billion in 2004 because of piracy excluding losses due to filesharing.

Monopolists, who obtain monopoly power through copy-right usually dominate industries susceptible to piracy. Likeother monopolies, they are viewed unfavourably becausethey tend to charge higher prices than what would prevailunder competition. Piracy allows consumers to obtain theproduct, albeit illegally, at much lower cost than legitimatepurchase.

While piracy is expected to cause downward pressure onprices, record labels have maintained high CD prices andincreased piracy has not resulted in a decrease in music CDprices (Bishop, 2004). In 2003, major record labels were

∗Correspondence: M Khouja, Business Information Systems and Opera-tions Management Department, The Belk college of Business Administra-tion, The University of North Carolina at Charlotte, 9201 University CityBlvd., Charlotte, NC, USA.E-mail: [email protected]

cited for pressuring retailers to maintain high CD prices. TheFederal Trade Commission found that record companies haveviolated fair trade practices by intimidating retailers not toadvertise CDs below certain prices. This has led to filing ofantitrust suits by 28 states against the big five record labels(Bishop, 2004).

The objective of this paper is to derive the optimal monop-olist price of a product in a market where some level ofpiracy is unavoidable. We show that under a set of real-istic conditions, piracy may lead to an increase in prices.These conditions are characterized by non-instantaneousdemand where consumers, whose reservation prices for theproduct are met, intend to buy the product within severaltime periods. If a chance to pirate the product arises withinthis time, some consumers will pirate the product. We alsoshow that a two-price strategy and dual distribution channelsmay provide effective approaches to reducing the impact ofpiracy.

The paper is organized as follows. We first review the liter-ature. We then develop the profit maximization models fora monopolist assuming instantaneous demand (ie ignoringpiracy) and then non-instantaneous demand with piracy. Wedevelop an approximation function to legitimate demandunder piracy and derive an accurate closed-form approxima-tion to the optimal price in the subsequent section. A two-price strategy and dual channel distributions are then analysedfollowed by numerical experiments, examples, sensitivityanalysis, and management implications. We conclude withsummary and suggestions for future research.

Author

Copy

M Khouja and HK Rajagopalan—Piracy and pricing in music and motion picture industries 373

Literature review

Most piracy models have focused on software piracy. Themodels deal mainly with pricing, copyright protectionstrategies, and government policies. For software, piracyincreases the consumer base for a software and thereforeincreases a consumer’s utility from that software (Nascimentoand Vanhonacker, 1988; Prasad and Mahajan, 2003). Thisincrease in value is referred to as positive network exter-nality. Nascimento and Vanhonacker (1988) found that, in theabsence of piracy, skimming pricing strategies are optimal.Using a diffusion of innovation model, they also found thatcopy protection is recommended only when sales grow fasterthan piracy and the cost of protection does not significantlyincrease the marginal cost. Conner and Rumelt (1991) exam-ined protection strategies and found that in the presence ofpositive network externalities, having no piracy protectioncan result in a lower price and increased profit. Givon et al(1995) supported the argument by Conner and Rumelt (1991)that software protection can harm the producer’s profit whenpositive network externalities are present. They conjecturedthat software protection discourages shadow diffusion insti-gated by piracy, thereby limiting the growth of the user base.Shy and Thisse (1999) extended the model of Conner andRumelt (1991) in a duopoly setting and derived equilibriumprices for different protection strategies in cooperative andnon-cooperative settings. King and Lampe (2003) showedthat the ‘profitable piracy’ claim by previous studies requiresstrong assumptions such as non-existence of price discrim-ination and negative correlation between ability to pirateand customer willingness-to-pay. They showed that for amonopoly to maximize profit, allowing piracy is always infe-rior to price discrimination. When price discrimination is notpossible, the profit-maximizing solution suggests allowingeither no piracy or complete piracy.

Prasad and Mahajan (2003) analysed the relationshipbetween the rate of software diffusion and piracy to deter-mine the price and piracy level that should be tolerated.Haruvy et al (2004) analysed how piracy affects adoptionof subscription software. The authors developed a model todetermine the price and protection level which maximizeprofit. Results indicate that moderate tolerance for piracy canspeed up adoption and enable the producer to charge higherprices.

Among the models applicable to most information goods,Gopal and Sanders (1997) studied preventive controls, whichuse technology to make piracy costly and difficult, and deter-rent controls, which use educational and legal campaigns tocurtail piracy. One of their key findings is that when preven-tive controls increase, revenue decreases, whereas when deter-rent controls increase revenue increases. Chen and Png (1999)developed a model that incorporates a penalty for piracy setby government in the absence of network externalities. Themonopolist determines price and piracy monitoring rate. Theauthors show that changes in pricing and monitoring rates

have qualitatively different effects on consumers and froma social welfare perspective, reductions in price are betterthan increases in monitoring. Chen and Png (2003) extendedtheir earlier model to include a copying media and equipmenttax and a government subsidy for purchases. Sundararajan(2004) analysed optimal pricing and technological protectionfor a monopolist using price discrimination among consumerswilling to buy variable quantities of a good. The author showsthat the optimal pricing schedule can be characterized as acombination of a zero-piracy pricing schedule and a piracy-indifferent pricing schedule.

Chellappa and Shivendu (2003) developed a model formotion picture DVDs. The authors analyse the implications ofmaintaining different regional technology standards in DVDplayers on global pricing and piracy. The model analysespiracy within a region and then analyses global piracy whereconsumers pirate products meant for a different region. Theauthors introduce a fixed cost which is convex in qualityin order to represent the infrastructure costs for producinganother DVD quality. Some regions have consumers withhigher marginal willingness to pay compared to other regionswith lower income. The results indicate that when piracy isprevalent, losses from global piracy can be higher than whenthere is only regional piracy. Thus, having separate technologystandards is critical.

Many software piracy models may not be applicable toother information goods due to the lack of positive networkexternalities for other information goods. There is littleevidence that piracy provides any benefits to the record labelsor the motion picture studios. One argument that has beenadvanced (Walsh, 2000) is that piracy allows consumers to‘try before you buy’.

Empirical research on piracy in the recorded music andmovie industries has received much less attention than thesoftware industry. Most of the research on all forms of piracyhave focused on demographics and cultural factors associatedwith piracy (Bhattacharjee et al, 2003; Kwong et al, 2003;Gopal et al, 2004; Papadopoulos, 2004; Chiou et al, 2005;Wang, 2005).

Our model focuses on pricing of information goods in theabsence of positive network externalities. We incorporate theaffects of non-instantaneous demand on piracy and pricing.Demand is realized in a decreasing fashion over time. Thisallows legitimate and pirated products in the market to becomeavailable for copying. This approach allows a better under-standing of piracy which in turn allows for better decisionmaking.

Optimal pricing under piracy

We begin by finding the best price if the producer ignorespiracy and assumes all consumers will either buy the productor go without. We then examine the impact of piracy onpricing.

Author

Copy

374 Journal of the Operational Research Society Vol. 60, No. 3

Cost of ignoring the time dimension for pirated products

We assume that a profit-maximizing monopolist controlsprice. The following notation is used:

v variable cost per unit of the productD quantity demanded, a decreasing function of priceP price per unit at which demand becomes zeroa demand when price per unit is zerob decrease in demand due to a $1 increase in price per unit

Each consumer may buy or pirate only one copy of the productand demand is measured in units. The demand at price P is

D = a − bP (1)

Usually, it is assumed that upon offering the product at a pricePI all consumers whose reservation prices are met purchasethe product instantaneously, Profit is given by

Z = (a − bP)(P − v) (2)

Since d2Z/dP2 = −2b< 0, the optimal price, which ignorespiracy, is

P∗I = a + bv

2b(3)

and the quantity demanded is

D∗I = a − bv

2(4)

Incorporating the time dimension for pirated products

In practice, demand is realized over several periods (days,weeks, etc). If it takes T periods for the demand of DI to berealized in a decreasing fashion, then demand at time t is

Qt = a0 − b0ty (5)

where y�1. A y=1 indicates linearly decreasing demand overtime and y < 1 indicates demand decreases at a decreasingrate over time. Total demand is the integral of Qt over T andtherefore ∫ T

0Qt dt = DI (6)

Substituting from Equations (4) and (5) into Equation (6) andusing QT = a0 − b0T = 0, which implies demand becomeszero by the end of the product lifecycle, gives:

a0 = (a − bP)(1 + y)

T y(7)

and

b0 = a0T y

(8)

Therefore, the cumulative demand up to time period t is

Dt = t (a − bP)[T y(1 + y) − t y]T 1+y

(9)

The demand during period t is the difference between thecumulative demand at the end of period t and the cumulativedemand at the end of period t − 1, which gives:

dt = Dt − Dt−1

= (a − bP)(T y(1 + y) + (t − 1)1+y − t1+y)

yT (1+y)(10)

Since demand is not instantaneous, consumers without theproduct, including those whose reservation prices are met,may have a chance to pirate. To pirate, a consumer mustknow another consumer with the product who is willing toallow piracy and must also have access to the technology forduplication.

In each period, the number of consumers who pirate theproduct depends on the total number of products, legitimateand pirated, available in the market and on the time fromthe release of the product. In addition, a proportion of thelegitimate products become available for pirating in the periodin which they are purchased. Pirated copies are not availablefor piracy until the subsequent period to the one in which theywere pirated. Let

�t proportion of products, legitimate and pirated, that getscopied in period t

� proportion of legitimate products sold in a period whichbecomes available for piracy within the period

c cost of pirating the product which includes the cost ofthe copying media

At quantity of the product, legitimate or pirated, availablefor piracy in the beginning of period t

Wt number of consumers whose reservation prices aregreater than or equal to the selling price and do not owna copy of the product at the beginning of period t

�t number of consumers who pirate the product in period t�t number of consumers out of Wt who pirate the product

in period tUt number of consumers whose reservation prices are less

than the selling price and do not own a copy of theproduct at the beginning of period t

�t number of consumers out of Ut who pirate the productin period t

Lt legitimate product demand in period t

We assume �t is non-increasing in t according to

�t = min{�0 − �t, 0} (11)

Equation (11) reflects the presence of active pirates in themarket who seek ways of obtaining the product illegitimatelyright away and whose numbers will decrease over time.

According to the above assumptions, A1=�L1, �1=��1L1,W1 = DI , and U1 = (a − bc) − DI . The legitimate demandin period t is the regular legitimate demand without piracy(dt ) adjusted for the decrease in the remaining demand due topiracy. The decrease in the remaining demand is proportionalto the ratio of the number of consumers whose reservation

Author

Copy


prices are met and have not purchased or pirated the productyet to the number of the same category of consumers butwithout piracy. Therefore, legitimate demand in period t isgiven by

Lt = dt

(Wt

DI −∑t−1j=1d j

)(12)

The total quantity, legitimate and pirated, available for piracyat the beginning of period t is

At = At−1 + �t−1 + (1 − �)Lt−1 + �Lt (13)

The total quantity of pirated copies made in period t is

�t = �t At (14)

The total quantity of pirated copies made by consumers in Wt

during period t is

�t = �t

(Wt

Wt +Ut

)(15)

The total quantity of pirated copies made by consumers in Ut

during period t is

�t = �t

(Ut

Wt +Ut

)(16)

The number of consumers whose reservation prices are greaterthan or equal to the selling price and do not own a copy ofthe product at the beginning of period t + 1 is

Wt+1 = Wt − Lt − �t (17)

The number of consumers whose reservation prices are lessthan the selling price and do not own a copy of the productat the beginning of period t + 1 is

Ut+1 =Ut − �t (18)

The cumulative legitimate demand up to period t is

DL ,t =t∑

i=1

Li (19)

A plot of cumulative demand with and without piracy fora given price and two piracy rates is shown in Figure 1 fora = 200, 000, b = 4000 and T = 7. In the presence of piracy,all demand is realized faster than in the case of no piracy (ie� = 0). The difference in cumulative demand increases withthe piracy rate and the length of product lifecycle. Figure 2shows the demand per period for P = $30 with � = 0 andwith � = 0.40. As the figure shows, the difference in demandper period between the two cases increases with time becausemore product become available in the market over time.Figure 3 shows demand per period for two prices. Demandper period at P = $20 per unit decreases at faster ratethan for P = $30 because at the lower price, larger quan-tity of the product becomes available in the market earlier

0

10000

20000

30000

40000

50000

60000

70000

80000

0 1 2 3 4 5 6 7

Period, t

Cum

ualti

ve le

gitim

ate

dem

and α=0

α=0.4

Figure 1 Cumulative legitimate demand with and without piracy.

-

5,000

10,000

15,000

20,000

25,000

0 1 2 3 4 5 6 7 8 9

Period, t

Dem

and,

Lt

α = 0.4 α = 0

Figure 2 Demand per period with and without piracy.

-

5,000

10,000

15,000

20,000

25,000

30,000

35,000

0 1 2 3 4 5 6 7

Period, t

Dem

and,

Lt

P = $20

P = $30

Figure 3 Legitimate demand per period at � = 0.4 for twodifferent prices.

which increases piracy and causes legitimate demand perperiod to decrease faster. Figure 4 shows the increasedpiracy per period at P = $20 per unit compared to P = $30per unit.

For a price of P∗I , denote the largest t for which Lt > 0 by

� and let the price which satisfies L� = 0 be denoted by P�.Let Pj be the price for which the legitimate demand in period

Author

Copy


j + 1 vanishes. The profit function can be written as:

Z =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(P − v)J1∑t=1

Lt if v� P� PJ1

. . .

(P − v)�−1∑t=1

Lt if P�−2 < P� P�−1

(P − v)�∑

t=1Lt if P�−1 < P� P�

(P − v)�+1∑t=1

Lt if P� < P� P�+1

. . .

(P − v)J2∑t=1

Lt if PJ2 < P� P0

(20)

where J1 is minimum number of periods before adjusteddemand vanishes at any price above unit production costand J2 is the maximum number of periods before demandvanishes at any price below P0 = a/b.

Equations (12)–(19) can be used to obtain analytical expres-sion for Lt for values of T �4, which are shown in Table 1.

-

20,000

40,000

60,000

80,000

100,000

120,000

0 1 2 3 4 5 6 7

Period, t

Uni

ts p

irate

d, ρ

t

P = $20

P = $30

Figure 4 Quantity pirated per period at �=0.4 for two differentprices.

Table 1 Period demand in the presence of piracy

� Demand d2Z/dP2

1(a − bp)(2T − 1)

T 2

2b(1 − 2T )

T 2< 0

2(a − bP)(2T − 3)

T 2

8b(1 − T )

T 2< 0

3(a−bP)(2T−5)(bP(2T−1)((T−2)2+(T−1)2�)+a(T−1)2(4+T 2+�−2T (2+�))−b(T−2)2T 2c)

(T−2)2T 2(−bp+a(T−1)2+bT (2P − T c))−2b

(1+20�(a−bv)

81(a−bc)

)<0

(if T = 3)

4 ((a − bP)(2T − 7)((T − 2)2(−bP + a(T − 1)2 + bT (2P − cT )) − (a − bP)(T − 1)2(2T − 1)�)

((T − 3)2(a(T − 2)2 + b(4P(T − 1) − cT 2)) − (a − bP)(−25 + T (56 − 33T + 6T 2))� − (a − bP)

(T − 2)2(2T − 1)�2))/((T − 3)2(T − 2)2T 2(−bP + a(T − 1)2) + bT (2P − cT )

(a(4 + T 2 + � − 2T (2 + �)) + b(−cT 2 + P(−4 − � + 2T (2 + �)))))

These expressions can be used to find the cumulative legiti-mate demand at different prices which can be used to computeprofit. The expressions for Lt are not linear in price andnumerical methods must be used to find the breakpoint pricesof Equation (20). Since d2Z/dP2 < 0 for up to three periodsas shown in Table 1, Z is concave when demand vanishes byperiod 4. Therefore, the optimal price for any price range inEquation (20) satisfies dZ/dP = 0 or is at the boundary ofthe range. For larger number of periods, Z is a higher degreepolynomial in price and we are unable to prove concavity.Algorithm 1 uses the Steepest Decent method in Mathematicato find a good solution. The algorithm converged to a singleprice for each price range in Equation (20) for all problemstested in the numerical experiment. Because we are unable toprove concavity of Z, we do not have a proof that the identi-fied solution is globally optimal.

Algorithm 1

Step 1: Compute the optimal price without piracy P∗I using

Equation (3)Step 2: Using P∗

I ,find the largest t for which Lt > 0. Denotethis value of t by �. Solve L� = 0 for P and denotethis value as P�.

Step 3: Set j = � + 1Step 4: If j < T find Pj for which Lt= j = 0.

If Pj < P0 go to step 3.Step 5: Set j = � − 1Step 6: If j > 2 find Pj for which Lt= j = 0.

IfPj > v go to step 5.Step 7: Construct the profit function using Equation (20)Step 8: Numerically find the maximum of the profit func-

tion constructed in step 7.

Owing to the complexity of the problem for large values ofT, a simpler method for computing a good price is needed.The analysis in the next section indicate that for reasonable

Author

Copy


Table 2 Goodness of fit for twelve 64-problem groups

R2 Adjusted R2

� y � Average Minimum Maximum Average Minimum Average

0 1.0 0.0 0.99847 0.99487 0.99997 0.99844 0.99482 0.999970 1.0 0.5 0.99910 0.99676 0.99997 0.99908 0.99672 0.999970 0.5 0.0 0.99806 0.99327 0.99995 0.99802 0.99316 0.999950 0.5 0.5 0.99848 0.99462 0.99996 0.99844 0.99451 0.9999610 1.0 0.0 0.99735 0.99345 0.99995 0.99730 0.99319 0.9999510 1.0 0.5 0.99848 0.99462 0.99996 0.99844 0.99451 0.9999610 0.5 0.0 0.99610 0.98631 0.99996 0.99602 0.98578 0.9999610 0.5 0.5 0.99545 0.98136 0.99999 0.99535 0.98064 0.9999920 1.0 0.0 0.99830 0.99510 0.99993 0.99826 0.99501 0.9999220 1.0 0.5 0.99873 0.99670 0.99993 0.99870 0.99665 0.9999220 0.5 0.0 0.99758 0.99307 0.99966 0.99752 0.99295 0.9996520 0.5 0.5 0.99779 0.99453 0.99993 0.99774 0.99447 0.99993

Average 0.99776 0.99271 0.99993 0.99771 0.99251 0.99992Minimum 0.99545 0.98136 0.99966 0.99535 0.98064 0.99965Maximum 0.99910 0.99676 0.99999 0.99908 0.99672 0.99999

problem parameters, a regression model can be used to esti-mate total legitimate demand and the model can then be usedto obtain a closed-form expression for an approximate optimalprice.

A regression approach for estimating legitimate demand

While we are unable to find empirical studies on estimatingtotal music or motion picture piracy rates, Marron and Steel(2000) estimated software piracy rates to be 28% in the USA,36% in the UK, and 40% in Germany for 1994–1997. Morerecent studies put the piracy rate in the US at 23% (Condry,2004). Our analysis indicate that for given values of a, b, T, y,�0, and �, the total legitimate demand is well approximatedby a regression model of the form:

Yi = 0 + 1P3/2i + εi (21)

where Yi is the response variable (ie total legitimate demand)in the ith trial, 0 and 1 are regression parameters, and εiis the error term. The regression model was fitted using thedemand data computed from Equations (12)–(19) for 768problems with parameter combinations a=1, b=0.005, 0.01,0.018, and 0.025, �0=0.1, 0.2, 0.3, and 0.4, �=0, �0/10, and�0/20, y = 0.5 and 1, T = 4, 8, 12, and 16 periods, and �= 0and 0.50. Fitting the regression model to data from the 768problems resulted in an average R2 of 0.99776, a maximumR2 of 0.99999 and a minimum R2 of 0.98136. Table 2 alsoshows the average, the minimum, and the maximum valuesof R2 for each of the twelve 64-problem categories. Resultsin Table 2 shows that the model in Equation (21) can be usedto accurately approximate the total legitimate demand.

Substituting from Equation (21) into Equation (2) anddifferentiating twice w.r.t. P gives d2Z/dP2 = 31(5P −v)/4

√P . For 1 < 0 (ie negatively sloping demand curve),

d2Z/dP2 < 0 for any profitable price. Therefore, the optimal

price is obtained by setting dZ/dP = 0 which (as theAppendix shows) yields:

P∗ =

[21v+

(102

021−6

1v3+√20(54

081−2

0101 v3)

)1/3]2

521

(102

021 − 6

1v3 +

√20(54

081 − 2

0101 v3)

)1/3

(22)

Algorithm 2

Step 1: Using Equation (12)–(19) find the total legitimatedemand for P=v+0.2(P0−v) to P=v+0.9(P0−v)

with increments of 0.05(P0 − v).Step 2: Fit the regression model of Equation (21) to the data

of step 1.Step 3: Compute the optimal price using Equation (22).

If piracy rates are a function of price, then a model similarto Equation (21) may still apply. In this case, � = f (P) andtherefore � and P are correlated. Therefore, Equation (21)becomes:

Yi = 0 + 1P3/2i + 2 f (Pi ) + εi (23)

The simplest case is one in which the piracy rate can beapproximated by �= c1P3/2 and Equation (23) becomes Yi =0 + 3P

3/2i + εi where 3 = 1 + 2c1.

Approaches to limiting piracy

We describe two strategies that may be used to reduce piracy.The first is based on using two prices whereas the second isbased on using dual distribution channels.

Author

Copy


4

5

6

7

8

9

10

11

0 5 10 15

Amount of price increase, P2-P1

Leng

th o

f pro

duct

life

cyc

le, T

y

T1=7

T1=3

Figure 5 The effect of low initial price on the length of theproduct lifecycle.

Two-price strategy to limiting piracy

A two-price strategy is one where a low price is first usedto encourage price-sensitive consumers to buy the productearly. This will decrease the total quantity of pirated productsbecause at a lower price many consumers may not have asbig of an incentive to pirate. Also, because of the shortenedlifecycle, diffusion of the product occurs at a much faster rateresulting in less available time for piracy in spite of increasedlegitimate products in the market. In other words, this strategymay decrease the number of periods for demand to be fullyrealized and may shift the bulk of the demand to earlier timeperiods in the product lifecycle. Furthermore, most consumerswho do not purchase the product prior to the price increaseare price-insensitive and are less likely to pirate. Let

T1 index of the time period of the price changeP1 price per unit for the first T1 − 1 periods of timeT0 length of the lifecycle if a one-price strategy is usedP2 price per unit at the beginning of period T1 P2 − P1 > 0, the amount of the price increase

We assume that the price change occurs at the beginning ofthe period. The number of periods for all demand to be real-ized using the two-price strategy, denoted Ty , is a decreasingfunction of the difference between the prices and of the timeof the price increase according to

Ty = T0

(T0 − T1)�

(24)

where � is an empirically determined constant. Equation (24)implies that the time needed to realize all demand decreasesas the difference in prices increases and as the price changeoccurs earlier in the product lifecycle. Equation (24) impliesthat a consumer whose reservation price is met is more likelyto purchase the product earlier when he/she knows that a priceincrease is coming. The speed of purchase is positively relatedto the price difference and negatively related to the lengthof time until the price increase. Figure 5 shows the product

lifecycle as a function of the price difference for T0 = 10periods and two different times for the price increase.

Using similar analysis to the one-price strategy with y = 1gives the following expressions for the cumulative demandup to time period t and the demand during period t as:

Dt= (a−bP1)t (−t+2T0(T0−T1)(P1−P2)�)(T0−T1)

−2(P1−P2)�

T 20

(25)

and

dt = Dt−Dt−1

= (a−bP1)(1−2t+2T0(T0−T1)(P1−P2)�)(T0−T1)−2(P1−P2)�

T 20

(26)

respectively. Legitimate demand is calculated using Equations(12)–(19) with dt given by Equation (26). The profit usingthe two-price strategy is

ZTP =T1−1∑i=1

P1 max{Li , 0} +Ty∑

i=T1

P2 max{Li , 0} (27)

To examine the profitability of the two-price strategy, wesuggest the following algorithm.

Algorithm 3

Step 1: Select m integer values of T1 over the interval(1, T0). For each value of T1 perform the following.

Step 2: Set P1 to 90% of the best price of the one-pricestrategy.

Step 3: Using increments 10%P1 for , keep increasing P2as long as profit (computed using Equation (27)with Equations (12)–(19) and (26)) is increasing andP2 < a/b.

Step 4: If only one value of P1 has been used, decrease P1by another 10% of the best price of the one-pricestrategy and go to step 3.If profit increased from last value of P1, decreaseP1 by another 10% of the best price of the one-pricestrategy and go to step 3.

Step 5: Compare the profit to the best profit of the one-pricestrategy and select the pricing policy resulting inthe largest profit.

Algorithm 3 is based on a search which fixes the first price inthe two-price strategy to a value lower than the best price ofthe one-price strategy and then searches for the best secondhigher price in increments of 10%. Once the best second priceis found, the first price is fixed again at a lower value witha decrement of 10% and the search continues. The resultsof the algorithm will most likely improve if the increments(decrements) are reduced to smaller value. One simple wayof improving the algorithm is to identify the best prices based

Author

Copy


on 10% increments (decrements) and then restart the searchbeginning at the resulting best first price with smaller incre-ment (decrement) such as 2%. Because we are unable to proveconcavity for ZTP and since the identified solution is not basedon solving the first order conditions, no claim of global orlocal optimality can be made.

Dual distribution channel approach to limiting piracy

Another approach to limiting piracy is to use dual distribu-tion channels. In addition to offering the physical productthrough brick-and-mortar and online retailers, the product canbe downloaded. Two key technologies enable such a distribu-tion mode. The Internet as an infrastructure for downloadingfiles and efficient file formats and compression technolog-ies for storing and transmitting files (Rao, 1999; Clemonsand Lang, 2003; Bockstedt et al, 2005). Examples of onlineretailers include iTunes and Napster. Suppose that the marketcan be divided into two part, a connected segment and a non-connected segment. The connected segment is made up ofconsumers who have the capability to download music on theInternet. The e-channel is one in which the product itself istransmitted via the Internet. Let

o a subscript denoting the connected segment� a subscript denoting the non-connected segment� variable cost per unit of the product sold on the

e-channelF investment required to achieve a variable cost per unit

of � j

The best price and profit for the connected segment canbe computed using the approximation procedure developedearlier. For the e-channel, suppose the variable cost per unitis given by

� ={�1 − 1F

r if 0�F < F1

�2 − 2(F − F1)r if F1�F < F2

�3 − 3(F − F2)r if F�F2

(28)

which implies that there are two technologies for e-distribution, the first requiring a minimum investment of F1,and the second requires additional investment up to F2. Weassume r < 1 which implies that additional investments inthe ranges of (0, F1), (F1, F2), and (F2, +∞) have dimin-ishing returns in terms of reducing the variable cost per unit.The analysis applies to any number of technologies and thechoice of two is for simplicity. The profit function is

Zo =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Zo1= (0 + 1P3/2)(P − �1 + 1F

r ) − Fif 0�F < F1

Zo2= (0 + 1P3/2)[P − �2 + 2(F − F1)

r ] − Fif F1�F < F2

Zo3= (0 + 1P3/2)[P − �3 + 3(F − F2)

r ] − Fif F�F2

(29)

The use of the total legitimate estimated using the regres-sion implies that products sold through the e-channel arealso subject to piracy with a rate that can be different fromthe traditional channel. We anticipate that information goodssold on the e-channel may have higher piracy rates becausethe product is already digitized and consumers using thee-channel are more likely to be technologically savvy. TheAppendix shows if (5P − �i ) > 0 (which is expected to holdfor any reasonable value of �i ), then each Zoi is concave forprices resulting in positive demand and F − Fi−1�0. Let

xi = 1020

41 + 6

1((F − Fi−1)r i − �i )

3 (30)

and

yi =√52

081(5

20 + 2

1((F − Fi−1)r i − �i )

3) (31)

The sufficient condition for the investment in the i th tech-nology to be optimal

F∗ = Fi−1 +(

1

(0 + 1P3/2i )r i

)( 1r−1

)(32)

The sufficient conditions for price to be optimal are:

P∗o,i = (2

1(�i − (F − Fi−1)r i ) + (xi + 2yi )

1/3)2

521(xi + 2yi )

1/3(33)

The following algorithm is used to examine the profitabilityof the e-channel.

Algorithm 4

Step 1: Compute P∗ ,1 and F∗ for Zo1. If F > F1 then no

F < F1 can be optimal. Otherwise keep the P∗o,1

and F∗1 = F∗ for Zo1 as candidates for the optimal

solution.Step 2: Compute P∗

o,2 and F∗ for Zo2. If F∗ > F2 thenno F1 < F < F2 can be optimal. If F∗ < F1 thenF∗2 = F1 is optimal on [F1, F2). Otherwise P∗

o,2 andF∗2 = F∗ are optimal on Zo2.

Step 3: Compute P∗o,3 and F∗ for Zo3. If F∗ < F2 then

F∗3 =F2 is optimal on [F2, +∞). Otherwise P∗

o,3 andF∗3 = F∗ are optimal on Zo3.

Step 4: Compare all local optimal solutions and select thebest.

Experiment, numerical example, and discussion

To test the quality of the solutions of the algorithms,we compared the solutions obtained using Algorithm 1programmed in Mathematica versus solutions from Algo-rithm 2 for 20 randomly generated problems. Problemparameters were generated from a ∼ U [20000, 100000],b ∼ U [0.02a, 0.05a], v ∼ [3, 7], c ∼ [1, 4], T ∼ [4, 8],� ∼ [0.10, 0.40]. The results in Table 3 show that pricescomputed using Equation (22) are very close to the prices

Author

Copy


Table 3 Comparison between the solutions of Algorithm 1 and Algorithm 2

P∗, $ P∗, $ |%| difference Z∗, $1 Z∗, $1 % differencea b v $ c $ T � Algorithm 1 Algorithm 2 in P∗ Algorithm 1 Algorithm 2 in Z∗

86649 3342 7 2 8 0.22 17.20 17.18 0.13 261 148 261 147 0.000481128 3034 3 3 8 0.27 16.08 16.07 0.02 345 826 345 826 0.000099915 3483 4 3 8 0.20 17.51 17.49 0.10 455 240 455 239 0.000281544 3022 4 3 8 0.16 16.35 16.46 0.00 352 946 352 906 0.011372434 2176 5 2 7 0.11 19.78 19.88 0.54 407 869 407 845 0.005921864 851 7 1 7 0.39 17.35 17.30 0.29 61 248 61 246 0.003360566 2002 5 1 7 0.23 18.56 18.69 0.71 279 840 279 808 0.011426818 809 7 2 7 0.22 20.96 21.19 1.14 122 346 122 303 0.035167369 1517 7 2 6 0.30 27.10 27.06 0.15 461 433 461 430 0.000722519 813 7 2 6 0.26 17.92 18.10 0.95 77 735 77 713 0.028344907 1169 5 2 6 0.37 23.37 23.08 1.25 273 651 273 558 0.034024612 1064 5 4 6 0.27 14.74 14.75 0.04 76 235 76 235 0.000088751 2023 4 2 5 0.23 25.26 25.02 0.96 730 351 730 237 0.015657384 2038 4 1 5 0.21 16.73 16.63 0.62 274 448 274 427 0.007777459 2500 4 4 5 0.23 18.46 18.21 1.34 410 788 410 639 0.036366373 3143 3 3 5 0.31 12.60 12.69 0.70 227 055 227 032 0.010133899 1573 3 3 4 0.11 12.50 12.57 0.55 130 763 130 756 0.005460949 2379 3 2 4 0.19 14.36 14.32 0.29 311 414 311 410 0.001339058 1301 6 1 4 0.10 18.20 18.09 0.63 183 213 183 196 0.009336565 1178 3 2 4 0.17 17.52 17.45 0.42 220 073 220 067 0.0027

Maximum 1.34 0.036Minimum 0.00 0.000Average 0.54 0.011

found using Algorithm 1. The maximum absolute differ-ence was 1.34% while the average absolute difference was0.54%. When prices were substituted in Equation (20), themaximum penalty from using the regression approximationwas 0.036% and the average penalty was 0.011%. Therefore,the approximate solution procedure can be used to solveproblems accurately.

Consider a numerical example with a = 200, 000 units,b = 4000, T = 7 weeks, and v = $1.00/unit. In the absenceof piracy, P∗

I = $25.5/unit and Z = $2.401 million. Underpiracy with � = 0, if management ignores piracy, when it ispresent with � = 0.40 and c = $1.00/unit, and keeps price atP∗I = $25.50/unit, Equation (2) gives Z = $1.85497 million.

Algorithm 1 gives P∗=$28.18 for which Equation (20) yieldsZ=$1.87971 million which is an increase of $24, 738.7 fromthe profit if management ignores piracy. Using the approxi-mation results in

∧D f = 119103.128 − 331.971P3/2 (34)

For which Equation (22) gives P∗=$27.81 and Z=$1.87924million, which is less than the Mathematica solution by only$467.24 (or 0.025%). For the same example, if � = 0.5, thenAlgorithm 1 gives P∗=$28.31 for which Equation (22) yieldsZ = $1.73879 million which is an increase of $47378.4 fromthe profit if management ignored piracy. Using the regressionapproximation results in

∧D f = 106839.98 − 286.64 P3/2 (35)

For which Equation (22) gives P∗=$28.52 and Z=$1.73669million, which is less than the profit of the Mathematica solu-tion by $2105.49 (or 0.12%).

If a two-price strategy with � = 0.005 is used, then Algo-rithm 3 gives T1 = 5 periods, P1 = $25.36, P2 = $32.97, andZ = $1.95685 millions which is an increase of $77,140 (4%)over the single-price strategy. A summary of the numericalexample results is shown in Table 4.

Suppose dual channels can be used with both consumersegments having parameters of a = 100, 000 units and b =2000. To illustrate a case in which investment in an e-channelis profitable, suppose that v=$3/unit for the traditional distri-bution channel. For the e-channel, T = 5 periods because ofmore accessibility to the product, r = 0.5, �1 = 3, �2 = 2,�3=1, F1=$30, 000, F2=$60, 000, 1=0.001, 2=0.0005,and 3 = 0.0001. Without the e-channel then Algorithm 2gives P∗ = $28.71 and Z∗ = 1.74311 million. If managementdecides to take advantage of the e-channel, then Algorithm2 yieldsP∗

� = $28.71 and Z∗� = $0.87155 millions and Algo-

rithm 4 yields P∗o =$27.43, F∗=$60, 004 and Z∗

o =0.973224millions for a total profit of Z∗ = $1.84478 millions. There-fore, the use of dual channels increased profit by $101,666.

We use the above numerical example to conduct numericalsensitivity analysis. To investigate the effect of the length ofthe product lifecycle, we compute the prices, shown in Figure6, and profit, shown in Figure 7, for values of T from 4 to 8periods for �=0.10 and 0.40. From Figure 6, as the length ofthe product lifecycle (T ) increases, the price also increases.The reason is that by increasing the price, the number of

Author

Copy


Table 4 Summary of numerical example

Price ($/unit) Profit� Strategy (using approximation) (millions of $)

NA Ignore piracy 25.50 1.854970 One-price 27.81 1.879240.5 One-price 28.52 1.736690 Two-price P1 = $25.36, P2 = $32.97 1.95685

$25.50

$26.00

$26.50

$27.00

$27.50

$28.00

$28.50

4 5 6 7 8

Number of Time Periods, T

Pric

e

α=0.4

α=0.1

Figure 6 Best price as a function of the length of the productlifecycle.

$1,500,000

$1,700,000

$1,900,000

$2,100,000

$2,300,000

$2,500,000

4 5 6 7 8

Number of Time Periods, T

Pro

fit

α=0.4

α=0.1

Figure 7 Best profit as a function of the length of the productlifecycle.

copies available in the market, decreases, which mitigates theeffects of piracy especially in the initial few periods. Figure 7shows that as the length of the product lifecycle increases, theprofit decreases since this longer sales period gives consumerslonger time to pirate. For small a piracy rate such as � = 0.1,the effects of the increased length of the product lifecycle areminor compared to � = 0.4.

The proposed model has two implications for managementdecisions. First, the length of the product lifecycle has a nega-tive relationship to profit. The longer it takes for demand tobe realized, the more chances consumers get to pirate andthe lower the total legitimate demand. Therefore, encouragingconsumers to purchase early can prove to be profitable. Thismay be accomplished by providing a discount only early inthe product lifecycle. Such an approach should be accompa-nied by a promotional campaign to inform consumers of theimpending expiration of this discount opportunity. Anotherpossibility for moving purchases earlier in the product life-cycle is to offer something extra at no charge for purchases

in the first few periods. For music CDs it can be a coupon forfuture purchases or free song singles. Second is that the useof technology to exploit the differences in consumer segmentsby using dual distribution channels may be an effective wayto reduce piracy. For technologically savvy consumers whoare more likely to find pirating easy, an online distributionchannel with low price may be best since it may cause manyof them to purchase instead of pirate. This channel can beserved through online retailers such as Apple’s online iTunesstore (Condry, 2004). For the remaining less technologicallysavvy segment, the brick-and-mortar channel with a higherprice may provide the best distribution alternative since manyof consumers in this segment may lack the ability to pirate.

Conclusions and suggestions for future research

In this paper, we examined the pricing policy of a monopo-list in a market where some piracy is unavoidable. Demandis not instantaneous and some duration of time is needed forconsumers to purchase the product. In every period, a certainproportion of products in the market are pirated. We use aregression approximation for the total legitimate demand andthen develop a closed form solution for identifying the optimalprice. We test the goodness of fit of the regression and thequality of the solution using the regression approximationagainst the solutions from the steepest descent method. Theresults show that both the regression and the resulting approx-imation to the best price are quite accurate for reasonableproblem parameters.

Incorporating of piracy may cause the monopolist toincrease the price, which may not be in line with the conven-tional wisdom in the literature but supports some of theobserved pricing practices of the record labels. The increasein price due to piracy is more significant the longer productlifecycle and the higher the piracy rate. High piracy ratesincrease the number of products that become quickly avail-able in the market. When this acceleration is compounded bya long product lifecycle causing many late buyers to pirateinstead of purchase, a strong upward pressure on price iscreated.

Some possible ways of reducing the negative effects ofpiracy are the use of a two-price strategy and dual distribu-tion channels. The two-price strategy is successful when animpending price increase causes consumers to buy the productearly. A dual channel approach is successful when the market

Author

Copy


can be segmented into a two groups and the second channeltakes advantage of information technology to enable quickproduct distribution at low cost.

There are several possible extensions to the model such asusing a skimming strategy with decreasing prices. In this case,to discourage piracy, the monopolist may drop the price aftera few periods. Another extension is the effect of using sometechnological controls to decrease the piracy rate (Gopal andSanders, 1997). These controls may enable the monopolist todecrease piracy rates for at least the early part of the productlifecycle.

References

Bhattacharjee S, Gopal RD and Sanders GL (2003). Digital music andonline sharing: Software piracy 2.0? Commun ACM 46: 107–111.

Bishop J (2004). Who are the pirates? The politics of piracy, poverty,and greed in a globalized music market. Pop Music Soc 27:101–106.

Bockstedt J, Kauffman RJ and Riggins FJ (2005). The move to artist-led online music distribution: Explaining structural changes in thedigital music market. In: Sprague Jr RH (ed). Proceedings ofThirty-Eighth Hawaii International Conference on System Sciences.IEEE Computer Society Press: Los Alamitos, CA.

Chellappa RK and Shivendu S (2003). Economic implications ofvariable technology standards for movie piracy in a global context.J Mngt Inform Syst 20: 137–168.

Chen Y and Png I (1999). Software pricing and copyright enforcement:Private profit vis-a-vis social welfare. In: De P and de Gross P (eds).Proceedings of the 20th International Conference on InformationSystems. ACM, Charlotte, North Carolina, USA, pp. 119–123.

Chen Y and Png I (2003). Information goods pricing and copyrightenforcement: Welfare analysis. Inform Syst Res 14: 107–123.

Chiou J-S, Huang C-Y and Lee H-H (2005). The antecedents of musicpiracy attitudes and intentions. J Bus Ethics 57: 161–174.

Clemons EK and Lang KR (2003). The decoupling of value creationfrom revenue: A strategic analysis of the markets for pureinformation goods. Inform Technol Manage 4: 259–287.

Condry I (2004). Cultures of music piracy: An ethnographiccomparison of the US and Japan. Int J Cult St 7: 343–363.

Conner KR and Rumelt RP (1991). Software piracy: An analysis ofprotection strategies. Mngt Sci 37: 125–139.

Givon M, Mahajan V and Muller E (1995). Software piracy:Estimation of lost sales and the impact on software diffusion.J Market 59: 29–37.

Gopal RD and Sanders GL (1997). Preventive and deterrent controlsfor software piracy. J Mngt Inform Syst 13: 29–47.

Gopal RD, Sanders GL, Bhattacharjee S, Agrawal M and Wagner SC(2004). A behavioral model of digital music piracy. J Org CompElect Com 14: 89–105.

Haruvy E, Mahajan V and Prasad A (2004). The effect of piracyon the market penetration of subscription software. J Bus Ethics77: S81–S108.

King SP and Lampe R (2003). Network externalities, pricediscrimination and profitable piracy. Inf Econ Policy 15: 271–290.

Kwong KK, Yau OHM, Lee JSY, Sin LYM and Tse ACB (2003).The effects of attitudinal and demographic factors on intention tobuy pirated CDs: The case of Chinese consumers. J Bus Ethics47: 223–235.

Marron DB and Steel DG (2000). Which countries protect intellectualproperty? The case of software piracy. Econ Inq 38: 159–174.

Nascimento F and Vanhonacker WR (1988). Optimal strategic pricingof reproducible consumer products. Mngt Sci 34: 921–937.

Papadopoulos T (2004). Pricing strategy and practice: Pricing andpirate product market formation. J Prod Brand Mngt 13: 56–63.

Prasad A and Mahajan V (2003). How many pirates should a softwarefirm tolerate? An analysis of piracy protection on the diffusion ofsoftware. Int J Res Mark 20: 337–353.

Rao B (1999). The Internet and the revolution in distribution: A cross-industry examination. Technol Soc 21: 287–306.

Shy O and Thisse J-F (1999). A strategic approach to softwareprotection. J Econ Mgnt Strat 8: 163–190.

Sundararajan A (2004). Managing digital piracy: Pricing andprotection. Inform Syst Res 15: 287–308.

Walsh EO (2000). Young Customers Embrace Napster, Buy MoreCDs. Forrester Research Inc., 20 June 2000.

Wang C-C (2005). Factors that influence the piracy of DVD/VCDmotion pictures. J Am Acad Bus 6: 231–237.

Appendix

The first derivative of Z after Equation (22) is used inEquation (2) w.r.t. P is

dZ/dP = 0 + 1C√P/2 − 51P

3/2/2 (A.1)

Since 1C√P/2 is strictly increasing and −51P

3/2/2 isstrictly decreasing for P > 0 and any value of 0, there isa single value of P , given by Equation (22), which satisfiesdZ/dP = 0.

Derivatives and Hessian matrix of profit function for dualchannel strategy

For a given investment function, the profit for the e-channelpolicy is given by

Zoi = (0 + 1P3/2)[P − �i + i (F − Fi )

r ] − F (A.2)

The first partial derivatives of Zoi with respect for P and F are

�Zoi

�P= 0 + 1P

3/2 + 3

21

√P(P − �i + 3 i (F − Fi )

r )

(A.3)

and

�Zoi

�F= (0 + 1P

3/2)r i (F − Fi )(r−1) − 1 (A.4)

Setting (A.3)–(A.4) to zero gives Equations (32)–(33).Elements of Hessian Matrix of Zoi are

a11 = 31

[√P + (P − �i + i (F − Fi )

(r−1))

4√P

](A.5)

a12 = a21 = 321

√Pr i (F − Fi )

(r−1) (A.6)

a22 = (0 + 1P3/2)r(r − 1) i (F − Fi )

(r−2) (A.7)

Since 1 < 0, a11 < 0 for any price larger than the cost perunit. Zoi is concave if determinant of the second principalminor of the Hessian matrix D2 is positive.

Author

Copy


D2 = 1

4(F − Fi )

(r−2)r i

×(31(0 + 1P

3/2)(r − 1)(5P − �i + i (F − Fi )r )√

P

−921(F − Fi )

r P i r)

(A.8)

Let �Fi = F − Fi , then D2 is positive if

31(0 + 1P3/2)(r − 1)(5P − �i + i�Fr

i )

− 921�Fr

i P3/2 i r > 0 (A.9)

Setting (A.9) to zero gives

� fi =[− (1 − r)(0 + 1P

3/2)(5P − �i )[(1 − r)0 + 1P3/2(1 + 2r)] i

]1/r(A.10)

If �Fi >� fi then D2 > 0. Since r < 1, � fi �0 for any Pwhich results in positive demand if (5P−�i ) > 0 which holdsfor reasonable value of �i . Thus, � fi �0 and for any F −Fi �0, Zoi is concave.

Received January 2007;accepted October 2007 after one revision

can piracy lead to higher prices in the music

Documents