2eM +t 4 +3
yucsiloeTeMt2 4+3 1 3
op+96eT t3 1 3+16tesi+128eM t3 1 4 lbhtal· 8eTt 1 3eT+8t 1 3 eT+8t 1 1 ignvaiadr 23 3eT 2 1 +2e2T t 4+9 1
2
eadgae+24eT t2 2+ 1 3+64t3 1 4
rms· 8eTt 1 3eT+8t 1 3 eT+8t 1 1
snioiThisdifferenceisquadraticin htscthequadraticequation
withanegativequadraticcoef cient.Solveiehuwq URL,ynn R
RL=0
(D1)
oaifor s anddenotethetworootsas 1 and 2 ,where 1
rdn 2 < :
veese 1ce =nsavae t 3eT+4eM 1 3eT +8t 1
dl· eT+2t 1 1 t 1 3eT +8t 1
APne.· 3e2
T+14eTt 1 +16t2 1 2 itisse× e3Tt 24 2 16+ 1 2e2T 1 2lc’srito· t2 14 23 +6eMt 1 3 3e2M 1
rAhtusa+8eTeMt 1 3 eM 1+ +6t +32e2Mt2 1 4
ihte 6e4
T 1 2 1/2
ohtttghn×
1
gi 3eT +8t 1 eTid+6eTt 1 +8t2 1 2 1 ryulpc 2 on=ic t 3eT+4eM 1 3eT +8t 1 se,· eT+2t 1 1 + t 1 3eT +8t 1
dtiloshbe· 3e2
T+14eTt 1 +16t2 1 2 SwMr×e3Tt 24 2 16+ 1 2e2T 1 2RehOt· t2 14 23 +6eMt 1 3 3e2M 1
FoNy+8eTeMt 1 3 eM 1+ +6t +32e2Mt2
1 4
Ina:thn 6e4
T 1 2 1/2goiryd×
1
pe 3etT +8t 1 eT
osCopSincetheEquation(D1)has+6ae1 +8t2 1 2 1
Tt negativequadraticcoef cient
for ,theleft-handsideof(D1)ispositiveifandonlyif islocatedbetweenthetworootsofequation(D1);i.e.,
URL
RRL
≥0
∈ 1 2
1 2
=
1
3eT +8t 1 eT× t 3e+6eTt 1 +8t2 1 2 1 T=
t 3e+4eM 1 3eT +8t 1 eT+2t 1 1
TeM 1 T= ¯
Thusitisthecasethat 1 < ¯ < 2 ;i.e.,theroot 2outsideofthefeasiblesetofparameters.
is
Thus URLwords, RRL≥when0for1
all <feasible 1 ≥ .Inother ,therecordlabel’spro tdecreaseswitheliminationofDRM,andwhen recordlabel’spro tincreaseswitheliminationof DRM.
≥ 1 ,theDenotethiscutoffpointas 1
Itiseasyto=see .
Proposition3.thatthedifference S
bothpositiveandnegative.Therefore,itcan be
the canbecasethat ≤ S or S
Consider≥the .Proposition4.difference
UD R
D
=
t2 3eT +4eM 1 2 eT +2t 1
Tt 3eT +8t 1 2
eTT 3t 4t eM 1 2
8e
Tt 3eT+8t 1 2
ItiseasytoseethatitisquadraticineMwithapositive
quadraticcoef cient.Theequation UlowingtworootswhensolvedforeDM
R
D=0hasthefol-e1t 3eM=
Te+4M 1
t 1
t 3eT +4eM 1
T· 3eT+8t 1 · 3eT +8t 1
eTT +2t 1 e2
Tt 3e+6eTt 1 +8t2 1 2 1
e2M=
TM 1
e+4t 1
+ t 3eT +4eM 1
T· 3eT+8t 1
eT+2t 1 eT +2t 1
· 3eT +8t 1 e2
T+6eTt 1 +8t2 1 2 1
Hence UThefeasibilityD RD<0ifandonlyifeconditionxUM∈12
L<xU
eM eM .
Himpliesthat
e3e2
TT 11t+2 1 +4t 1 2
M≥ 3eT=M
Vernik,Purohit,andDesai:MusicDownloadsandtheFlipSideofDRM
MarketingScience,ArticlesinAdvance,pp.1–17,©2011INFORMS
2R
ItiseasytoseethateM<M;hence, UD D>0forallfeasiblevaluesofeM.
17
andpro ts.Workingpaper,UniversityofSouthCarolina,Columbia.
Givon,M.,V.Mahajan,E.Muller.1995.Softwarepiracy:Estimation
oflostsalesandtheimpactonsoftwarediffusion.J.Marketing59(1)29–37.
Hennig-Thurau,T.,V.Henning,H.Sattler.2007.Consumer leshar-ingofmotionpictures.J.Marketing71(4)1–18.
Hotelling,H.1929.Stabilityincompetition.Econom.J.39(153)
41–57.
IDC.2002.Musicdownloadsandconsumerperception:Hype,
skepticism,andthegenerationgap.IDCResearchReportB22217,IDC,Framingham,MA.
IFPI.2002.Musicpiracyreport.Technicalreport,InternationalFed-ers.The lemaynotbeermissions@.
Proposition5.Itiseasytoseethat URL/ eM>0,
U
whereas RL/ eTcanbegreaterorsmallerthanzero.Hence,therecordlabel’spro tsalwaysincreasewhenthemoralcostofpiratingincreasebutcanincreaseordecreasewhenthetechnicalcostofpiracygoesup.
Proposition6.Thedifferenceinpiracyvolumes,SU SR= eM 1 + 1 / 2eT ,demonstratesthatSU<SRalways.
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