Music Downloads and the Flip Side of Digital Rights Manageme(15)

Vernik,Purohit,andDesai:MusicDownloadsandtheFlipSideofDRM

MarketingScience,ArticlesinAdvance,pp.1–17,©2011INFORMS

15

¯ < 2.Indeed,itiseasytoseethatNextweshowthat

R¯ ;hence,itis U / isequalto0with = RLRL

122¯ < ,i.e.,theroot isoutsideofthethecasethat <

feasiblesetofparameters.

R1

Thus URL RL≥0forallfeasible = .Inotherwords,when < 1 ,therecordlabel’spro tdecreaseswitheliminationofDRM,andwhen = 1 ,therecordlabel’spro tincreaseswitheliminationofDRM.

ers.The lemaynotbeermissions@.

Nowcompare(B4)and(B6).NotethatiftheoverallweightedimpactofDRM, ,ispositive,thenourmodelformulationisequivalenttothemodelwhereallconsumersderiveutility R= fromaDRM-restrictedproductwhere = DRM 2 1 .TheweightedimpactofDRM, = DRM 2 1 ,ispositiveif >0 5.

AppendixC.AllowingDirectCompetitionBetweenPiratedDigitalandLegalTraditionalFormats

Let = 1 and = 1 1 .Considerthedif-ferencebetweenrecordlabel’spro tinthecasewhenitsellsDRM-restricteddigitalmusicandthecasewhenitsellsAppendixD.ProofsofPropositions

Proposition1A.With =0,thedifferenceSU SRis

bpircoDRM-freedigitalmusic:

tsbyuc URLLsil RR=A 2

+B +C

oopwhere

tesilbhA= eT 1 +2t eT 1+2 2tal8t +e3+ 1 2 +4t

igTt T nvaiadrB= 2t +eT 1 4eMt +eT 1 eT + 3t+eM

ea +e

Tt 8tT 3+õõ 1

dgaermand

ssnioiC= 1

htsciehu

128e2Mt4

4 e6T 2

1 4 2 9+6

2

wq+16eTe2Mt3

1+ 3 1 3+5 ,ynn 2e3Tt 1 2 2eMt 3 2

+3 1+ 1

oaisrdn 3e2M 1+ 1 3+2 + 2

veese+2t2 24 25 52 + 2+ 24 1

censa+4e2T

t2 2

1 2 46t2 6eMt 3 vaedlAP+e2

M 2 1+ 2 3+ + 9+12 +13 2 ne.iti 2e5T 1 3 eM 2 3 1

sselc’sr+2t 6+ + 9 3 1 2 8 2+2 3

itorAhtu+e4T 1 2 e2M 1 2 2 9+ 2

saih+2eMt 2 3 4 + 2 2 3+2 3

teohtt+2t2 4 11 7 12+ 2 1 18+6 8 1tghngi+ 4 19 2+7 3 idryul /

4eT 2t +eT 1 pc· 3eT+8t eT 2+ 2t +eT 1

onicse,· 3eT +8t eT 2+ dtil

osThedifferenceisquadraticin hbeA<0.

,anditiseasytoseethat

SwMSolvetheequation UrRehtworootsas 1RL R

RL=0for anddenotethe

2

OtFoNy 1B √and :

BB+√BIn a= and 2

:thngoSinceA<0, the2A =difference Uir islocatedbetweenthetwo 2A

RL

RrootsRL

ispositiveifandonlyifofequation;i.e.,

ydpetos UCoRLp RRL≥0 ∈ 1 2

However,thefeasibilitycondition

D

RD

≥0impliesthat

4e + 3t+e =

Mt +eT 1 eTM

≤ ¯

equalto

2SU SR=

1 1 9e2

T eMTeMt 1+ 1 32eMt 1 2

3e3e

TT+8t 1 T +8t 1

ItiseasytoseethatSU<SRaslongas

>

4eMt 3eTT 3eT t eM 4eMt 1

= S

Proposition1B.Thedifference

SU SR= 3e2

T 3eM 1 3t 1 +2

+4eTt 3eM 1 2 + 4+3 1 32t2 eM 1 + 1 2 1 · 2eT 3eT+8t 1 3eT +8t 1 1

islinearanddecreasingin thecutoffpoint

.SolvingSUresultsin SR=0for

2 S 1 9e2

=

T eMTeMt 1+ 1 +32eMt2 1 t 1 e

T +8TProposition2A.Considerthedifference+2t 1

betweentherecordlabel’spro tinthecasewhenitsellsDRM-restricteddigitalmusicandthecasewhenitsellsDRM-freedigitalmusicwith =0:

URL RRL

= 9e4T 2 24e3

Tt 1+ 1 +e2T 9e2

M 18eMt 46t2 1 2

+12eTe2Mt 1+ 1 3+32e2Mt2

1 4·Itiseasyto 4eT 3eT+8t 1 3eT +8t 1 1 seethatthisdifferencehasthesamesign

1

asthenumeratoroftheexpressionabove,whichinturnisquadraticinwithpositivequadraticcoef cient.Hence

URL R

RL<0when ∈ 1 2 ,where

1 A2+48eTe2

Mt 1 3eT +8t 1 2 A1=6eT 3eT+8t 1 and

1 A2 +48eTeMt 1 3eT+8t 1 2 A

2=

2

6e

T 3eT+8t 1

whereA=224e2Tt eT 9e2

M 18eMt 46t2 1 12e2Mt 1

Itiseasytodemonstratethat 1isalwaysnegative;hence, URL> RRLiff > 2.Let 2= .