Music Downloads and the Flip Side of Digital Rights Manageme(16)

16

Proposition2B.Considerthedifferencebetweentherecordlabel’spro tinthecasewhenitsellsDRM-restricteddigitalmusicandthecasewhenitsellsDRM-freedigitalmusic:

R

URL RL 23

4

=64eMt 1 1 4 18eTt 1 2

Vernik,Purohit,andDesai:MusicDownloadsandtheFlipSideofDRM

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

However,thefeasibilitycondition

DRD=

ers.The lemaynotbe

ermissions@.

eTT 3t +4t eM 1

≥0

T 3eTt 3eTM 1 ¯

≤

eT+4t 1

impliesthat

=

2222

48e3Tt 1 1 +2eTt 1 1

22223· 9e2 18et 46t +24eet 1 1 MTMM 1·8eTt 1 3eT+8t 1 3 eT+8t 1

¯ < 2.Indeed,noticethatthemid-Nextweshowthat

dlepointbetweentworootsofEquation(D1),givenby

2

1+ /2,isequalto

bpircotsb+ 18e3T 2t 1 +12e2T t 1 2

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