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Transaction Utility of End-User Pirates – An Analysis

  • Paulomi Basu EMAIL logo and Tanmoyee Banerjee

Abstract

The present study hypothesized the concept of transaction utility that consumers receive from end-user piracy and assumes that it is positively associated with the difference between the price of original good and cost of copying the pirated product. We assume two groups of consumers – one enjoying transaction utility from piracy and the other not enjoying the same. The results show that a monopolist producing information-goods will adopt a low-price low-quality strategy in presence of transaction utility. An increase in the quality of the pirated product will also induce the firm to continue with the low-price–low-quality strategy. However, an increase in the proportion of the consumers not receiving transaction utility from piracy will increase the price and quality of the original good. In addition, with an increase in the average and marginal willingness-to-pay parameter of the consumers, the price, quality and profit of the original good will increase. The expected piracy rate is observed to be decreasing in transaction utility parameter when the upper boundary of willingness-to-pay parameter of the consumer is below a critical level.

JEL Classification: D91; L12; L15

Acknowledgements

We are immensely indebted to the two anonymous referees whose valuable comments have helped us to revise the paper. We are also thankful to Prof. Siddhartha Mitra, Department of Economics, Jadavpur University, Kolkata, India for his valuable comments.

Appendix

Proof of Proposition 1

Differentiating eq. (14) with respect to λ, and given the second-order condition of profit maximisation in eq. (17), we have

(18)qλ=1γc21+λ1μ2q31αθˉ21α1μ41+λ1μ2+c21μ4q21α<0

Putting q in eq. (11), we get the equilibrium price p

(19)pλ=θˉ1α211+λ1μqλq1μ1+λ1μ2<0

Putting q in eq. (13), we get the equilibrium profit π

(20)π*=[θ¯q*(1α)+c{1+λ(1μ)}]24q*(1α){1+λ(1μ)}γq*22
(21)dπ*dλ=π*λ+π*q*q*λ=(1μ)[θ¯2q*2(1α)2c2{1+λ(1μ)}2]4q*(1α){1+λ(1μ)}2<0 [asπ*q*=0 andq*>c{1+λ(1μ)}θ¯(1α)]

Proof of Proposition 2

(22)qα=1γc21+λ1μ2q31αθˉ241+λ1μ+c21+λ1μ4q21α2<0

This follows from the condition γ>c21+λ1μ2q31α from the second-order condition of profit maximisation with respect to quality:

(23)pα=θˉ21+λ1μq+1αqα<0asqα<0
(24)dπdα=πα+πqqα=θˉ2q21α2c21+λ1μ24q1α21+λ1μ<0 asπq=0 

Proof of Proposition 3

(25)qμ=1γc21+λ1μ2q31αλθˉ21α41+λ1μ2+λc24q21α>0
(26)pμ=θˉ1α211+λ1μqμ+λq1+λ1μ2>0
(27)dπdμ=πμ+πqqμ=λθˉ2q21α2c21+λ1μ24q1α1+λ1μ2>0Asπq=0

Proof of Proposition 4

(28)qθˉ=1γc21+λ1μ2q31αθˉ1α21+λ1μ>0
(29)pθˉ=1α21+λ1μθˉqθˉ+q>0
(30)πθˉ=πθˉ+πqqθˉ=θˉq1α+c1+λ1μ21+λ1μ>0As πq=0

Proof of Lemma 1

By replacing values for q and p in eqs. (2) and (3), the marginal values of the willingness-to-pay parameter of the consumers for the group not receiving transaction utility from piracy are as follows:

(31)θ1=θˉ21+λ1μc2q1α>0
(32)θ1l*=cαq*>0

By replacing values for q and p in eqs. (6) and (7), the marginal values of the willingness-to-pay parameter of the consumers for the group receiving transaction utility from piracy are as follows:

(33)θ2=θˉ1+λ21+λ1μc1+λ2q1α= 1+λθ1>0 
(34)θ2l*=12α[c(2+λ)q*θ¯λ(1α){1+λ(1μ)}]

For the market segment receiving transaction utility from piracy to be fully covered by either the original good or the pirated good, θ2l*<0

This implies  c2+λq<θˉλ1α1+λ1μ

(35)i.e.θˉ>c2+λ1+λ1μλ1αq 
(36)or q>c2+λ1+λ1μλ1αθˉ

Alternatively the sufficient condition of profit maximising quality satisfies,

(37)q>c231+λ1μ132γ131α13

Thus, condition eq. (36) will always be satisfied if c231+λ1μ132γ131α13c2+λ1+λ1μλ1αθˉ

Orθˉ(2γc)131+λ1μ232+λλ1α23

For the sufficient condition θˉ(2γc)131+λ1μ232+λλ1α23, θ2l*<0,the piracy rate is defined as R1= μ(θ1*θ1l*)+(1μ)θ2μ (θ¯θ1l*)+(1μ)θ¯·.

Otherwise when θ2l>0 the piracy rate is defined asR2=μ(θ1*θ1l*)+(1μ)(θ2*θ2l*)μ(θ¯θ1l*)+(1μ)(θ¯θ2l*)

Proof of Proposition 5

For θˉ(2γc)131+λ1μ232+λλ1α23, piracy rate is defined as R1=μ(θ1*θ1l*)+(1μ)θ2*μ(θ¯θ1l*)+(1μ)θ¯  

(38)R1λ=1[μ(θ¯θ1l*)+(1μ)θ¯]2[D.NλN.Dλ]

where N=μ(θ1*θ 1l*)+(1μ)θ2*,D=μ(θ¯θ 1l*)+(1μ)θ¯

(39)Dλ=μθ1lλμcαq2qλ<0
Nλ=μθ1*λμθ1l*λ+(1μ)(1+λ)θ1*λ  [θ2*λ=(1+λ)θ1*λ]

Replacing eq. (39)

(40)Nλ=θ1*λ{1+λ(1μ)}+Dλ  Nλ<0[ θ1*λ=θ¯(1μ)2[1+λ(1μ)]2+c2q*2(1α)q*λ<0]

Hence, replacing the values of eqs. (39) and (40) in eq. (38)

R1λ=1μθˉθ1l+1μθˉ2θ1λD.1+λ1μ+DNDλ<0 Henceproved.

Forθˉ<(2γc)131+λ1μ232+λλ1α23, piracy rate is defined as R2=μθ1θ1l+1μ(θ2θ2l)μθˉθ1l+1μθˉθ2l

(41)R2λ= 1[μ(θ¯θ1l*)+(1μ)(θ¯θ2l*]2 D.NλN.Dλ 

where N= μ(θ1*θ1l*)+(1μ)(θ2*θ2l*),D=μ (θ¯θ 1l*)+(1μ)(θ¯θ2l*)

(42)This reduces R2λ=1[μ(θ¯θ1l*)+(1μ)(θ¯θ2l*]2[D{1+λ(1μ)}θ1*λ+(DN)Dλ]
Dλ=μθ1lλ1μθ2lλ
(43)Dλ=μcαq2+1μ2+λc2αq2qλ1μc2αq+1μ1αθˉ2α1+λ1μ2
ForDλ<0, 
(44)θˉ<c1+λ1μ21αqorq<c1+λ1μ21αθˉ

Alternatively from S.O.C. of quality maximisation, we get eq. (37) or q>c231+λ1μ132γ131α13

A necessary condition to satisfy eq. (33) is c231+λ1μ132γ131α13<c1+λ1μ21αθˉ

Replacing eq. (37) into eq. (44)

(45)θˉ<2γc131+λ1μ531α23
R2λ<0  when  θ¯ <(2γc)13{1+λ(1μ)}53(1α)23(Henceproved)

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Published Online: 2019-05-31
Published in Print: 2019-06-26

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