Technical Appendix (Not for publication) Generic and Brand Advertising Strategies in a Dynamic Duopoly

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1 Tehnal Appendx (Not fo publaton Gene and Band Advetsng Stateges n a Dynam Duopoly Ths Tehnal Appendx povdes supplementay nfomaton to the pape Gene and Band Advetsng Stateges n a Dynam Duopoly. It s dvded nto the followng setons: A. Poofs n geate detal than n the Appendx. B. Poofs of ompaatve stats esults fo asymmet fms. C. Poofs of ompaatve stats esults fo symmet fms. D. Devatons fo the Extenson seton. A. Poofs n Geate Detal than n the Appendx. Poof of Poposton The Hamlton-Jaob-Bellman (HJB equaton fo fm { } s gven by V ps( bp dp3 ( a u ( u S3 3u3 S θ( a a S V max (A u a p V ( 3u3 S u S3 θ3( a a. S3 Fom ths the fst-ode ondtons fo u and a { } yeld espetvely V V V V u ( S a ( θ θ. 3 3 S S3 S S3 (A The fst-ode ondtons fo p and p yeld bp dp 0 d p bp 0. (A3 Solvng the two smultaneous equatons n (A3 we obtan the optmal pe of fm to be p d b 3 4bb dd We an theefoe smplfy the pe tems n (A so that. (A4

2 d b p ( bp dp b( m. Fom ths pont fowad we wll use 3 3 4bb dd m b denote the equlbum magns of the two fms. Substtutng (A and (A5 nto (A yelds d b ( 4bb dd V V V V V m S S and m d b b( 4bb dd ( θ θ3 ( 3 S S3 S S3 3 V V V V 3 V V V V S θ θ θ θ 3 S S S S 3 S S S S ( ( ( (. The lnea value funton V α βs γs3 { } satsfes (A6. The optmal band and gene advetsng desons n (A an now be ewtten as 3 3 (A5 to (A6 u ( β γ S a ( θ β θ γ. (A7 Substtutng V α βs γs3 nto (A6 and smplfyng we have α βs γs3 ms ( θβ θ3γ ( β γ S3 (A8 ( ( ( (. 3 3 β γ β γ S θβ θγ θβ θγ 3 3 Equatng the oeffents of S S3 and the onstant n equaton (A8 esults n the followng smultaneous equatons to solve fo α β and γ : ( ( ( (A9 α 3 3 θ β θ γ θ β θ γ θ β θγ 3 3 β m ( β γ ( β γ (A0 3 γ β γ (. (A Wtng out these equatons expltly fo { } we get ( ( ( (A α θ β θ γ θ β θ γ θ β θ γ

3 β β γ β γ m ( ( (A3 ( (A4 γ β γ ( ( ( (A5 α θ β θ γ θ β θ γ θ β θ γ ( ( (A6 β m β γ β γ (. (A7 γ β γ To solve subtat (A4 fom (A3 and (A7 fom (A6. Let y β γ and z β γ. Upon smplfyng we have the followng smultaneous quadat equatons n y and z : y yz y m 0 (A8 z yz z m 0. (A9 Substtutng fo z n tems of y n equaton (A9 afte solvng fo z n (A8 yelds the followng quat equaton n y : η η η η η (A0 4 3 y y 3y 4y whee η 3 η 4 ( η 4 ( ( m m 4 η 8 m( and η 4 m. 5 3 The oots of equaton (A0 an be omputed usng Mathemata v4.0. Of the fou solutons fo y two ae magnay one s negatve and only one s always eal and postve. Ths s the unque Nash equlbum of the dffeental game. That soluton s gven by η ω y 4η 4 ω ω ω ( 7 ω ω5 ω6 ω ω5 ω6 5 6 (A 3

4 η 3 whee η ω 4η 3 ω η3 3ηη 4 ηη 5 ω3 η3 9ηηη 3 4 7ηη 4 7ηη 5 7 ηηη 3 5 3η ω ( ω ω 4 ω ω 3 ω ω 4 5 ω6 3η ω Knowng y we an ompute the followng: 3 4ηη 3 η 8η 4 and ω7. 3 η η η η γ y β y y z ( m y y γ z β z z. (A y One an see fom (A that β > 0 γ > 0 and β > γ esultng n postve values fo the ontols. Poof of Coollay To solve the smultaneous equatons: 3 α ( β γ (A3 8 β m ( β γ (A4 γ ( β γ (A5 m multply (A5 by and add to (A4. Ths yelds β γ. Substtutng ths nto (A5 esults n the followng quadat equaton n γ : m γ γ γ m γ m (A6 ( (3 0. The two solutons of ths quadat equaton ae 3 m ± ( 6 m γ. (A7 9 To fnd out whh of the two oots to hoose we use the test that γ 0 when m 0. Ths s beause the value funton should be dentally equal to zeo when the goss magn s zeo sne the fm maes zeo poft n ths ase. Cheng wth (A7 t an be seen that 4

5 3 m ( 6 m γ (A8 9 s the only oot that satsfes ths ondton. m Knowng γ we an ompute β and α usng β γ and (A3 espetvely to be 3m ( 6 m β (A ( 3m 8 m (6 m ( 6 m α 3 4. (A30 08 An examnaton of equatons (A8-30 eveals that α > 0 β > 0 γ > 0 and β > γ so the ontols and the value funtons ae postve. Poof of Poposton To deve the optmal sales paths we substtute the esults fom Poposton nto the two state equatons to obtan the followng system of dffeental equatons: S S S S S (A3 S ( S ( S ( ( S (0 S. ( βγ ( β γ θ( ( θβ θγ ( θγ θβ (0 0 β γ β γ θ( θβ θγ θγ θβ 0 Fo expostonal ease denote ψ ( β γ ψ ( β γ ψ ( θβ θγ ( θ γ θβ. (A3 3 The dffeental equatons an now be ewtten as S ψs ψs θψ 3 S(0 S0 S ψ S ψ S θ ψ S (0 S. 3 0 Note fom (A33 that thee s no long-un equlbum n sales.e. S and S need not go to (A33 zeo. Solvng the smultaneous dffeental equatons n (A33 usng Mathemata v4.0 yelds 5

6 e S ( t ( ( S ( e ( S ( e (A34 ( ψ ψ t ( ψ ψ t ( ψ ψ t ψ ψ ψ 0 ψ ψ 0 ψ ψ ( ψ ψ ψ ( θψ θψ ( ψ ψ θ ψ ψ θ t t e ( ψ ψ t 3 ( ( e S ( t ( ( S ( e ( S ( e (A35 ( ψ ψ t ( ψ ψ t ( ψ ψ t ψ ψ ψ 0 ψ ψ 0 ψ ψ ( ψ ψ ψ ( θψ θ ψ ( ψ ψ θ ψ ψ θ t t e ( ψ ψ t 3 ( (. The long-un equlbum maet shaes ae gven by S( t S( t x lm x lm. t S( t S( t t S( t S( t Smplfyng and tang the lmts we have (A36 x. (A37 ( ( ( ( ( βγ ( β γ x β γ β γ β γ β γ Poof of Poposton 3 If the fm owns both bands ts deson poblem s max u( t a( t u( t a( t p( t p( t ( t e p( t S(( t bp( t dp( t p( t S(( t dp( t bp( t V (A38 0 ( a( t u( t ( a( t u( t dt s.t. S ( t u( t S( t u( t S( t θ( a( t a( t S(0 S0 (A39 S ( t u ( t S ( t u ( t S ( t θ ( a ( t a ( t S (0 S 0 whee V s the value funton of the fm a and u ae the gene and band advetsng desons espetvely of band S s the sales of band s the advetsng ost paamete and m the magn of band and s the fm s dsount ate. The HJB equaton s 6

7 p S( bp dp ps( dpbp ( u a ( u a V V max ( u S u S θ( a a u a p u a p S V ( u S u S θ( a a. S The fst-ode ondtons fo the optmal advetsng desons yeld V V V V u ( S a ( θ θ S S S S (A40 (A4 V V V V u ( S a ( θ θ. S S S S As befoe substtutng the solutons fom (A4-4 nto (A40 suggests a lnea value funton V α β S γ S wll solve the esultng patal dffeental equaton. The optmal m m m advetsng desons an now be ewtten as 0 Note fom (A43-44 that ethe m m m m (A4 u max{0 ( β γ S } a ( θ β θ γ (A43 u max{0 ( γ β S } a ( θ β θ γ. (A44 m m m m u o u s always postve. If β > γ m m > 0 u and u sne band s moe poftable. If β < γ the opposte s tue. Theefoe n a m monopoly total band advetsng need not be zeo. The monopolst an hoose the optmal advetsng desons to ensue the value funton n the monopoly ase s neve less than that n the ase of ompetton.e. V V V whee V and V ae the pofts n the ompettve ase. We theefoe have m α β γ α β γ α γ β (A45. m ms ms S S S S Equaton (A45 an be ewtten to yeld α ( α α ( β ( β γ S ( γ ( β γ S 0. (A46 m m m Sne equaton (A46 holds S 0 S 0 t must be the ase that α ( α α β β γ γ β γ (A47 m m m whee eah of the above oeffents s non-negatve. 7

8 Fom equaton (A7 the total gene advetsng n the ompettve ase s ( θ β θ γ ( θ γ θ β (A48 whle that n the monopoly ase s fom equatons (A43-44 ( θ β θ γ ( θ β θ γ. (A49 m m m m Subtatng equaton (A48 fom (A49 the dffeene between the total gene advetsng n the monopoly ase and that n the ompettve ase s ( θ ( β β θ ( γ γ ( θ ( β γ θ ( γ β (A50 m m m m whh fom equaton (A47 s geate than zeo. Theefoe the monopolst s total gene advetsng s geate than that unde ompetton. B. Poofs of Compaatve Stats Results fo Asymmet Fms. We epodue the ompaatve stats table fo onvenene. Table : Compaatve Stats Results fo the Asymmet Case Vaables 3 m m θ α????????? β?? γ u a? V????????? x Legend: nease; deease; unhanged;? ambguous. 8

9 To obtan the ompaatve stats fo γ and β we fst ty to obtan the esults fo y and z. Defne F( y z m y yz y 0 F ( y z m z yz z 0. Fo any paamete φ we an use the mplt funton theoem y F F F F F F φ y z φ z z φ z F F F F F F φ y z φ y y φ to detemne the ompaatve stats esult. Afte some smplfaton ths an be ewtten as y F y z y φ φ z F z y z φ φ (B (B (B3 whee the detemnant y z y > 0. z y z We now detemne the ompaatve stats wth espet to eah of the model paametes. y y z y y z m z 0 (B4 z y z z m y y z y y m 0 z (B5 z y z y z m 9

10 y y z y y( y z y z 0 (B6 z y z yz y y z y yz 0 z z (B7 z y z z( y z y y z y y z z y z yz y ( y z? yz( y z y y z y yz z z y z z? yz( y z z ( y z y y z y y z z y z yz y ( y z? yz( y z (B8 (B9 (B0 0

11 Theefoe y y z y yz z z y z z yz( y z.? z ( y z y z y z y z > 0 < 0 < 0 > 0 < 0 > 0 m m m m y z z y z y > 0 < 0 < 0 < 0 > 0 > 0. (B (B The ompaatve stats esults fo y on and and fo z on and ae ambguous. Note that y and z ae ndependent of and θ { }. To ompute the ompaatve stats fo γ β γ and β we use γ y β y y γ z β z z. (B3 Theefoe γ ( y y y y y 0 β ( y y ( > > 0 m m m m m m γ ( z z z z z 0 β ( z z ( < < 0 m m m m m m (B4 (B5 γ ( y y y y y 0 β ( y y ( < < 0 (B6 m m m m m m γ ( z z z z z 0 β ( z z ( > > 0 (B7 m m m m m m γ ( y y y 0 β ( y ( y y < < 0 (B8 y y y γ ( z z z z z 0 β ( z z ( > > 0 (B9

12 γ ( y y y y y 0 β ( y y ( > > 0 (B0 γ ( z z z 0 β ( z ( z z < < 0 (B z z z γ ( y y y ( y ( y y y β y y (B γ ( z z z z z 0 β ( z z ( < < 0 (B3 γ ( y y y y y 0 β ( y y ( < < 0 (B4 γ ( z z z ( z ( z z z β z z (B5 γ ( y y y β ( y ( y y (B6 y y y γ ( z z z z z 0 β ( z z ( > > 0 (B7 γ ( y y y y y 0 β ( y y ( > > 0 (B8 γ ( z z z β ( z ( z z. z z z (B9 Although the ompaatve stats fo γ on and n (B and fo γ on and n (B5 ae ambguous one an sgn these devatves by tansfomng the mplt equatons n (B n tems of γ and γ. Rewtng the equatons n (B usng z γ yelds the followng: y γ and J ( γ γ m γ γ γ γ 0 J ( γ γ m γ γ γ γ 0. (B30

13 Fo any paamete φ we have γ J J J J J J φ γ γ φ γ γ φ γ J J J J J J φ γ γ φ γ γ φ Afte some smplfaton ths an be ewtten as. (B3 γ γ γ J φ γ γ γ φ (B3 γ γ γ J φ γ γ γ φ whee the detemnant > 0. as follows: We an now ompute the ompaatve stats fo γ on and and fo γ on and γ γ γ γ γ γ γ γ γ γ γ γ (B33 γγ γ γγ γ γ γ γ γ γ γ γ γ γ γ γ (B34 γγ γγ γ 3

14 γ γ γ γ γ γ γ γ γ γ γ γ (B35 γγ γ γγ γ γ γ γ γ γ γ γ γ γ γ γ (B36 γγ. γγ γ Howeve the ompaatve stats of β on and and of β on and ae stll ambguous. Moeove note that γ β γ and β ae ndependent of and θ { }. We an now wte γ β γ β γ β γ β γ > 0 > 0 < 0 < 0 < 0 < 0 > 0 > 0 < 0 m m m m m m m m β γ β γ β γ β γ γ < 0 > 0 > 0 > 0 > 0 < 0 < 0 > 0 < 0 (B37 β γ γ β γ γ β γ γ β < 0 < 0 > 0 < 0 < 0 > 0 > 0 > 0 < 0 > 0. Fo the ompaatve stats esults fo α { } on the model paametes we note that ( ( ( (B38 α θ β θ γ θ β θ γ θ β θ γ ( ( ( (B39 α θ β θ γ θ β θ γ θ β θ γ α ( y θ y ( y θ y( z θ z (B40 4

15 α ( z θ z ( y θ y( z θ z. (B4 The ompaatve stats esults fo α { } on and ae easy to obtan. We have α α α α > 0 > 0 > 0 > 0. (B4 The ompaatve stats fo α { } on the othe model paametes ae ambguous gven the esults fo β and γ { }. The ompaatve stats fo y and z have the opposte sgns fo most of the model paametes so the devatves of the seond tem n equatons (B40 and (B4 ( y y( z z and θ θ ( y y( z z θ θ espetvely ae not easy to sgn. Fo the ompaatve stats fo the optmal band and gene advetsng desons we an see that sne u a and ( β γ S u ( β γ S a ( θ β θ γ and ( θ γ θ β most of the esults ae staghtfowad gven the ompaatve stats fo γ β { }. Fo the ompaatve stats of we ewte the optmal band advetsng desons as u w..t. and and of u w..t. and u ( β γ S y S u ( β γ S z S.(B43 Substtutng fo y n tems of mplt equatons: u and fo z n tems of S SS S S SS S u we have the followng G ( u u m u uu u 0 G ( u u m u uu u 0. (B44 The mplt funton theoem yelds 5

16 whee the detemnant > 0. Theefoe u G G G G G G u u u u u G G G G G G u u u u (B45 u u u u S SS S SS u u u u S SS SS S (B46 uu u SS S uu SS u u u u S SS S SS u u u u S SS SS S (B47 uu SS uu u SS S u u u u S SS S SS u u u u S SS SS S (B48 u uu u S SS S 0 6

17 The ompaatve stats of u u u u S SS S SS u u u u S SS SS S (B49 a and fashon. The esults ae summazed below. 0. u uu u S SS S a w..t. and espetvely an be obtaned n a smla u u u u u u u u u > 0 < 0 < 0 > 0 < 0 > 0 > 0 < 0 > 0 m m m m u u u u u u u a a < 0 > 0 < 0 > 0 > 0 < 0 < 0 > 0 < 0 m m a a a a a a a a a < 0 > 0 < 0 > 0 > 0 < 0 < 0 < 0 < 0 m m a a a a a a a a a > 0 > 0 < 0 > 0 > 0 > 0 < 0 < 0 > 0. θ θ θ θ (B50 Gven the ambguous ompaatve stats esults fo α most of the esults fo V ae also ambguous. Howeve t s possble to sgn the ompaatve stats of V fo the paametes n equaton (B4. The esults ae as follows: V V V V > 0 > 0 > 0 > 0. We now ty to obtan the ompaatve stats esults fo the steady-state maet shaes. The long-un equlbum maet shaes ae gven by x ( βγ ( β γ x β γ β γ β γ β γ (B5. (B5 ( ( ( ( Fo all the paametes exept and the ompaatve stats of x and x ae staghtfowad to obtan gven the ompaatve stats fo y and z. Fo and we ewte equaton (B5 as 7

18 y z x x. y z y z (B53 Defne l y l z. Theefoe l l x x. l l l l l Fo the ompaatve stats of x w..t. any paamete φ note that x l l theefoe l x φ l l l (. φ φ φ l l ( l l (B54 (B55 Rewtng the mplt equatons n tems of l and l yelds H ( l l m l ll l 0 H ( l l m l ll l 0. (B56 As befoe one an obtan the ompaatve stats of l and l w..t. and usng whee φ s any paamete. l H H H φ l l φ l H H H φ l l φ (B57 Pefomng the ompaatve stats of l and l fo any paamete φ yelds whee the detemnant > 0. l H l l l φ φ l H l l l φ φ (B58 8

19 Pefomng the ompaatve stats of l and l w..t. we have l ( l l l 3 l ( l l. l (B59 l ( l l 0 l l Theefoe > 0 and < 0. Moeove l l ( l 3 ( l l ( l l ll 3 ( l l (B60 ( l ( l ( l l > 0. x Fom (B55 and (B60 one an see that > 0 3. The ompaatve stats of x on and an also be obtaned usng the mplt equatons n (B56. These esults ae summazed below. x x x x > 0 < 0 < 0 > 0. (B6 Fo all the othe paametes the esults fo y also hold fo x. Sne x x the ompaatve stats esults fo x an be obtaned by flppng the esults fo x. C. Poofs of Compaatve Stats Results fo Symmet Fms. Fo onvenene we epodue the ompaatve stats esults n Table. We now 4 4 ( 3m 8 m (6 m ( 6 m α (C m ( 6 m β (C 9 9

20 3 m ( 6 m γ. (C3 9 Table : Compaatve Stats Results fo the Symmet Case Vaables m α? β γ u a V?? x Legend: nease; deease; unhanged;? ambguous. Tang the fst devatves of β and γ n equatons (C-3 wth espet to eah of the paametes yelds the followng ompaatve stats: β γ β γ β γ β γ > 0 > 0 < 0 < 0 < 0 > 0 > 0 < 0. (C4 m m The esults n (C4 follow n a staghtfowad manne. The ompaatve stats fo the optmal advetsng expendtues u and a { } ae easy to obtan wheneve the esults fo β and γ ae n the same deton. The ompaatve stats β and γ wth espet to and ae not of the same sgn. Howeve even n these ases the sgns of the devatves an be obtaned as n the asymmet ase usng the mplt equatons. The ompaatve stats esults of V and x { } on the model paametes an also be obtaned as fo asymmet fms. We obtan the followng ompaatve stats fo the optmal advetsng desons: 0

21 u u u u u u u u a > 0 < 0 < 0 < 0 > 0 > 0 < 0 < 0 > 0 m m m a a a a a a a a a > 0 < 0 < 0 < 0 < 0 < 0 < 0 > 0 > 0. m (C5 D. Devatons fo the Extenson Seton. a. Maet Potental The Hamlton-Jaob-Bellman (HJB equaton fo fm { } s gven by V ps( bp dp3 ( a u ( u S3 3u3 S θ( a a QSS S V max (D u a p V ( 3u3 S u S3 θ3( a a Q S S. S3 Fom ths the fst-ode ondtons fo u and a yeld espetvely V V V V u ( S a ( θ θ QS S. 3 3 S S3 S S3 As befoe one an solve fo the optmal pes and vefy that the solutons ae the same as those n equaton (A4. Substtutng equatons (A4 and (D nto (D yelds V V V V V m S Q S S S ( θ θ j ( ( 3 S S3 S S3 3 V V V V 3 V V V V S θ θ θ θ QSS 3 S S S S 3 S S S S ( ( ( ( (. The lnea value funton V α β S γ S3 satsfes (D3. The optmal band and gene advetsng desons n (D an now be ewtten as u ( β γ S a ( θ β θ γ QS S. (D4 3 3 Substtutng V α β S γ S3 nto (D3 and smplfyng we have S S ms Q S S S (D5 ( ( ( ( (. α β γ 3 ( θβ θ3 γ ( ( β γ βγ β γ S θβ θγ θβ θγ QSS 3 3 (D (D3

22 Equatng the oeffents of S S3 and the onstant n equaton (D5 esults n the followng smultaneous equatons to solve fo α β and γ { } : ( Q ( ( Q (D6 3 α θβ θ3γ θβ θγ θβ θγ 3 ( ( ( ( ( (D7 3 3 β m θβ θ3γ β γ β γ θβ θγ θβ θγ 3 3 ( ( ( (.(D8 3 γ θβ θ3γ β γ θβ θγ θβ θγ 3 Wtng out these equatons expltly fo { } we get ( Q ( ( Q (D9 α θβ θγ θβ θγ θβ θγ ( ( ( ( ( (D0 β m θβ θγ β γ β γ θβ θγ θβ θγ ( ( ( ( (D γ θβ θγ β γ θβ θγ θβ θγ ( Q ( ( Q (D α θβ θγ θβ θγ θβ θγ ( ( ( ( ( (D3 β m θβ θγ β γ β γ θβ θγ θβ θγ ( ( ( (. (D4 γ θβ θγ β γ θβ θγ θβ θγ To solve subtat (D fom (D0 and (D4 fom (D3. Let y β γ and z β γ Smplfyng esults n the followng smultaneous quadat equatons n y and z :. y yz y m 0 (D5 z yz z m 0. (D6 Substtutng fo z n tems of y n equaton (D6 afte solvng fo z n (D5 yelds the followng quat equaton n y : η η η η η (D7 4 3 y y 3y 4y 5 0

23 4 whee η 3 η 4 ( η 4 ( ( m m 4 η 8 m( and η 4 m. 5 3 The oots of equaton (D7 an be omputed usng Mathemata v4.0. Of the fou solutons fo y two ae magnay one s negatve and only one s always eal and postve. Ths s the unque Nash equlbum of the dffeental game. That soluton s gven by η ω y 4η 4 ω ω ω ( 7 ω ω5 ω6 ω ω5 ω6 5 6 (D8 η 3 whee η ω 4η 3 ω η3 3ηη 4 ηη 5 ω3 η3 9ηηη 3 4 7ηη 4 7ηη 5 7 ηηη 3 5 3η ω ( ω ω 4 ω ω 3 ω ω 4 5 ω6 3η ω ηη 3 η 8η 4 and ω7. 3 η η η η Knowng y we an ompute z ( m y y. Knowng y and z γ and γ y an be obtaned by solvng the followng smultaneous quadat equatons: ( y y ( y (( z (D9 γ θ γ θ γ θ γ (( z z ( y(( z. (D0 γ θ γ γ θ θ γ Afte γ and γ ae obtaned one an obtan β and β usng β y γ and β z γ espetvely. These solutons fo γ γ β and β an be used n equatons (D9 and (D to solve fo α and α. b. Othe Extensons The uent-value Hamltonan fo fm { } s H ps ( bp dp3 ( a u S λ( ω ( ω ω ( ω ( (D u( S3 S 3u3( 3 S 3 S3 S S a a S3 ( 3u3( 3 S ( 3 S3 u( S3 ( S a a S S µ ω ω ω ω (. 3

24 The optmal pes ae the same as those obtaned n equaton (A4. Fo the optmal advetsng desons the fst-ode ondtons fo fm ae H u ( ( u λ ( ω S ( ω S µ ( ω S ( ω S 0 (D 3 3 H λ S µ S 0. a S S S S 3 a (D3 Solvng the fst-ode ondtons fo the two fms yelds the optmal advetsng desons to be λ S µ S u ( S ( S ( a ( ω ω λ µ S S S S (D4 λ S µ S u ( S ( S ( a (. ω ω µ λ S S S S The adjont equatons fo the shadow pes λ ( t and µ ( t fo fm ae H H u H a λ λ S u S a S ( ω u ω u ( a a S λ p ( b p d p ( λ µ 3 ( S S ( S S ω (( ω S ( ω S ( λ µ ( S λ S µ S S ( ( ( µ 3 λ3 S S S S S S S ( H H u H a µ µ S u S a S ( µ λ 3 3 ωu ( ω u ( a a S µ ( λ µ ( S S 3 3 ( S S ( ω (( ω S ( ω S ( λ µ ( ( µ 3 λ S3 λ S µ S S ( ( ( µ 3 λ3. S S S S S S S ( (D5 (D6 One an now substtute the optmal pes and advetsng desons nto the state equatons and the adjont equatons. Sne these dffeental equatons do not pemt losed-fom solutons we use numeal methods to solve fo λ ( t and µ ( t { }. These solutons an then be used n (D4 to obtan the optmal band and gene advetsng desons fo the two fms. 4