Transaction Utility of End-User Pirates – An Analysis

Paulomi Basu; Tanmoyee Banerjee

doi:10.1515/roms-2018-0042

Published by De Gruyter May 31, 2019

Transaction Utility of End-User Pirates – An Analysis

Paulomi Basu and Tanmoyee Banerjee

From the journal Review of Marketing Science

https://doi.org/10.1515/roms-2018-0042

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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.

Keywords: transaction utility; behavioural economics; information good; consumer behaviour; end-user piracy; piracy rate

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)∂q∗∂λ=−1γ−c21+λ1−μ2q∗31−αθˉ21−α1−μ41+λ1−μ2+c21−μ4q∗21−α<0

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

(19)∂p∗∂λ=θˉ1−α211+λ1−μ∂q∗∂λ−q∗1−μ1+λ1−μ2<0

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

(20)π*=[θ¯q*(1−α)+c{1+λ(1−μ)}]24q*(1−α){1+λ(1−μ)}−γq*22

(21)dπ*dλ=∂π*∂λ+∂π*∂q*∂q*∂λ=−(1−μ)[θ¯2q*2(1−α)2−c2{1+λ(1−μ)}2]4q*(1−α){1+λ(1−μ)}2 <0 [as ∂π*∂q*=0 and q*>c{1+λ(1−μ)}θ¯(1−α)]

Proof of Proposition 2

(22)∂q∗∂α=−1γ−c21+λ1−μ2q∗31−αθˉ241+λ1−μ+c21+λ1−μ4q∗21−α2<0

This follows from the condition γ>c21+λ1−μ2q∗31−α from the second-order condition of profit maximisation with respect to quality:

(23)∂p∗∂α=θˉ21+λ1−μ−q∗+1−α∂q∗∂α<0as∂q∗∂α<0

(24)dπ∗dα=∂π∗∂α+∂π∗∂q∗∂q∗∂α=−θˉ2q∗21−α2−c21+λ1−μ24q∗1−α21+λ1−μ<0 as∂π∗∂q∗=0

Proof of Proposition 3

(25)∂q∗∂μ=1γ−c21+λ1−μ2q∗31−αλθˉ21−α41+λ1−μ2+λc24q∗21−α>0

(26)∂p∗∂μ=θˉ1−α211+λ1−μ∂q∗∂μ+λq∗1+λ1−μ2>0

(27)dπ∗dμ=∂π∗∂μ+∂π∗∂q∗∂q∗∂μ=λθˉ2q∗21−α2−c21+λ1−μ24q∗1−α1+λ1−μ2>0As∂π∗∂q∗=0

Proof of Proposition 4

(28)∂q∗∂θˉ=1γ−c21+λ1−μ2q∗31−αθˉ1−α21+λ1−μ>0

(29)∂p∗∂θˉ=1−α21+λ1−μθˉ∂q∗∂θˉ+q∗>0

(30)∂π∗∂θˉ=∂π∗∂θˉ+∂π∗∂q∗∂q∗∂θˉ=θˉq∗1−α+c1+λ1−μ21+λ1−μ>0As ∂π∗∂q∗=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+λ1−μ−c2q∗1−α>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+λ1−μ−c1+λ2q∗1−α= 1+λθ1∗>0

(34)θ2l*=12α[c(2+λ)q*−θ¯λ(1−α){1+λ(1−μ)}]

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∗<θˉλ1−α1+λ1−μ

(35)i.e.θˉ>c2+λ1+λ1−μλ1−αq∗

(36)or q∗>c2+λ1+λ1−μλ1−αθˉ

Alternatively the sufficient condition of profit maximising quality satisfies,

(37)q∗>c231+λ1−μ132γ131−α13

Thus, condition eq. (36) will always be satisfied if c231+λ1−μ132γ131−α13≥c2+λ1+λ1−μλ1−αθˉ

Orθˉ≥(2γc)131+λ1−μ232+λλ1−α23

For the sufficient condition θˉ≥(2γc)131+λ1−μ232+λλ1−α23, θ2l*<0, the piracy rate is defined as R1= μ(θ1*−θ1l*)+(1−μ)θ2μ (θ¯−θ1l*)+(1−μ)θ¯·.

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

Proof of Proposition 5

For θˉ≥(2γc)131+λ1−μ232+λλ1−α23, piracy rate is defined as R1=μ(θ1*−θ1l*)+(1−μ)θ2*μ(θ¯−θ1l*)+(1−μ)θ¯

(38)⇒∂R1∂λ=1[μ(θ¯−θ1l*)+(1−μ)θ¯]2[D.∂N∂λ−N.∂D∂λ]

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

(39)∂D∂λ=−μ∂θ1l∗∂λ⇒μcαq∗2∂q∗∂λ<0

∂N∂λ=μ∂θ1*∂λ−μ∂θ1l*∂λ+(1−μ)(1+λ)∂θ1*∂λ [∵∂θ2*∂λ=(1+λ)∂θ1*∂λ]

Replacing eq. (39)

(40)⇒∂N∂λ=∂θ1*∂λ{1+λ(1−μ)}+∂D∂λ ⇒∂N∂λ<0[ ∵∂θ1*∂λ=−θ¯(1−μ)2[1+λ(1−μ)]2+c2q*2(1−α)∂q*∂λ<0]

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

⇒∂R1∂λ=1μθˉ−θ1l∗+1−μθˉ2∂θ1∗∂λD.1+λ1−μ+D−N∂D∂λ<0 Henceproved.

Forθˉ<(2γc)131+λ1−μ232+λλ1−α23, piracy rate is defined as R2=μθ1∗−θ1l∗+1−μ(θ2∗−θ2l∗)μθˉ−θ1l∗+1−μθˉ−θ2l∗

(41)⇒∂R2∂λ= 1[μ(θ¯−θ1l*)+(1−μ)(θ¯−θ2l*]2 D.∂N∂λ−N.∂D∂λ

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

(42)This reduces ⇒∂R2∂λ=1[μ(θ¯−θ1l*)+(1−μ)(θ¯−θ2l*]2 [D{1+λ(1−μ)}∂θ1*∂λ+(D−N)∂D∂λ]

∂D∂λ=−μ∂θ1l∗∂λ−1−μ∂θ2l∗∂λ

(43)⇒∂D∂λ=μcαq∗2+1−μ2+λc2αq∗2∂q∗∂λ−1−μc2αq∗+1−μ1−αθˉ2α1+λ1−μ2

For∂D∂λ<0,

(44)θˉ<c1+λ1−μ21−αq∗orq∗<c1+λ1−μ21−αθˉ

Alternatively from S.O.C. of quality maximisation, we get eq. (37) or q∗>c231+λ1−μ132γ131−α13

A necessary condition to satisfy eq. (33) is c231+λ1−μ132γ131−α13<c1+λ1−μ21−αθˉ

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

(45)θˉ<2γc131+λ1−μ531−α23

∴∂R2∂λ < 0 when θ¯ <(2γc)13{1+λ(1−μ)}53(1−α)23 (Hence proved)

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Published Online: 2019-05-31

Published in Print: 2019-06-26

Transaction Utility of End-User Pirates – An Analysis

Abstract

Acknowledgements

Appendix

Proof of Proposition 1

Proof of Proposition 2

Proof of Proposition 3

Proof of Proposition 4

Proof of Lemma 1

Proof of Proposition 5

References

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