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Munich Personal RePEc Archive Capital Investment as Real Options: A Note on Dixit-Pindyck Model. Termos, Ali American University of Sharjah 29 January 2008 Online at https://mpra.ub.uni-muenchen.de/9352/ MPRA Paper No. 9352, posted 28 Jun 2008 04:03 UTC

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Page 1: Capital Investment as Real Options: A Note on Dixit ... · Munich Personal RePEc Archive Capital Investment as Real Options: A Note on Dixit-Pindyck Model. Termos, Ali American University

Munich Personal RePEc Archive

Capital Investment as Real Options: A

Note on Dixit-Pindyck Model.

Termos, Ali

American University of Sharjah

29 January 2008

Online at https://mpra.ub.uni-muenchen.de/9352/

MPRA Paper No. 9352, posted 28 Jun 2008 04:03 UTC

Page 2: Capital Investment as Real Options: A Note on Dixit ... · Munich Personal RePEc Archive Capital Investment as Real Options: A Note on Dixit-Pindyck Model. Termos, Ali American University

�✁✂✄ ☎✂✆✝✞✟✠✝✂✟ ✁✞ ✡✝✁☛ ☞✌✟✍✎✂✏ ✑✒✎✟✝ ✎✂ ✓✍✔✍✟✕✖✍✂✗✘✙✄ ✚✎✗✝☛

✛✜✢ ✣✤✥✦✧★✩✪✫✬✭✮✯ ✰✱ ✲✳✳✴

✵✶✷✸✹✺✹✻✼✶✽ ✼✻✾ ✿✻❀❁✺❂✸✷❃✷❄❅❆❇❈❉❊❇

❋●❍■ ❏❑❏▲▼ ❑❏❏◆❍▲■ ❖P◗❘❍◗❙▲◗❘ ❖◆❑❍❚■ ❚P❯▲◆ ❑ ◆❑ ❱❍❲❍❘ ❑◗❯ ❳❍◗❯❨❩❬❭❪❫❫❴❵❛ P◗ ❜❑◗❬ ❍◗❝▲■❘❚▲◗❘❞ ❡❑◗❬■ ❑▼▲ ❍◗❯❍❢▲▼▲◗❘ ❜▲❘❣▲▲◗ ❍◗❝▲■❘❍◗❙❘●▲❍▼ ❑■■▲❘■ P◗ ❘●▲❍▼ P❣◗ ❑◗❯ ▲❲❘▲◗❯❍◗❙ ◆P❑◗■ ❘P ❍◗❝▲■❘P▼■❞ ❋●▲ ❩▼❍❘❍❩❑◆❯▲❩❍■❍P◗ ❤❑❩▲❯ ❜❨ ❘●▲ ❜❑◗❬▲▼ ❍■ ❘●▲ ❘❍❚❍◗❙ P❤ ❘●▲ ❍◗❝▲■❘❚▲◗❘ ❯▲❩❍■❍P◗❑◗❯ ❍❘■ ✐◗❩▲▼❘❑❍◗❘❨❞ ❥●▲◗ ❜❑◗❬■ ❚❑❬▲ ❑◗ ❍▼▼▲❝▲▼■❍❜◆▲ ❍◗❝▲■❘❚▲◗❘ ❯▲❦❩❍■❍P◗ ❘●▲❨ ▲❲▲▼❩❍■▲ ❘●▲ P❏❘❍P◗ ❘P ❍◗❝▲■❘ ❑◗❯ ❙❍❝▲ ✐❏ ❘●▲ P❏❏P▼❘✐◗❍❘❨P❤ ❣❑❍❘❍◗❙ ❤P▼ ◗▲❣ ❍◗❤P▼❚❑❘❍P◗ ❘P ❑▼▼❍❝▲❞ ❋●❍■ ◆P■❘ P❏❘❍P◗ ❝❑◆✐▲ ❍■ ❍◗❦❩P▼❏P▼❑❘▲❯ ❍◗ ❘●▲ ❍◗❝▲■❘❚▲◗❘ ❩P■❘❞ ❋●▲▼▲❤P▼▲❛ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘❚✐■❘ ▲❲❩▲▲❯ ❘●▲ ❍◗❝▲■❘❚▲◗❘ ❩P■❘ ❜❨ ❘●▲ ❝❑◆✐▲ P❤ ❬▲▲❏❍◗❙ ❘●▲ ❍◗❝▲■❘❚▲◗❘P❏❘❍P◗ ❑◆❍❝▲❞ ♠■❍◗❙ ❑ ❘●❍▼❯❦❚P❚▲◗❘ ❚▲❑◗❦▼▲❝▲▼■❍P◗ ❏▼P❩▲■■ P❤ ❘●▲ ❍◗❦❝▲■❘❚▲◗❘♥■ ❝P◆❑❘❍◆❍❘❨❛ ❘●▲ ❚P❯▲◆ ■●P❣■ ❘●❑❘ ❑ ●❍❙●▲▼ ❚▲❑◗❦▼▲❝▲▼■❍P◗❏❑▼❑❚▲❘▲▼ ▼▲❯✐❩▲■ ❜P❘● ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ P❏❘❍P◗ ❘P ❍◗❝▲■❘ ❑◗❯ ❘●▲ ❩▼❍❘❦❍❩❑◆ ❝❑◆✐▲ ❑❘ ❣●❍❩● ❘●▲ ❏▼P❧▲❩❘ ❯▲▲❚■ ❤▲❑■❍❜◆▲❞

♦♣qqrqst✉s ✈✇①②③qq①✇④ ⑤③✈t✇s⑥③✉s ①② ⑦⑧①✉①⑥r⑧q④ ♣⑥③✇r⑧t✉ ⑨✉r⑩③✇qrs❶ ①② ❷❸t✇❹t❸④ ❺✉rs③❻♣✇t❼ ⑦⑥r✇ts③q❽ ❾⑦⑥tr❿➀ ts③✇⑥①q➁t⑨q❽③❻⑨➂ ➃❽➄❽➅➄➆ ➇➈➈➈➈ ❷❸t✇❹t❸④ ❺♣⑦❽➉♣⑧➊✉①➋❿③❻➌⑥③✉s➀ ➍ s❸t✉➊ ➃t⑨❿ ➎t⑧➊❿③✇ ②①✇ ❸rq ⑩t❿⑨t❼❿③ r✉qr➌❸sq ①✉ s❸③ ✉⑨⑥③✇r⑧t❿q①❿⑨sr①✉❽ ♣❿❿ ③✇✇①✇q t✇③ ⑥r✉③❽

Page 3: Capital Investment as Real Options: A Note on Dixit ... · Munich Personal RePEc Archive Capital Investment as Real Options: A Note on Dixit-Pindyck Model. Termos, Ali American University

➐ ➑➒➓➔→➣↔↕➓➙→➒ ➓→ ➓➛➜ ➝→➣➜➞❋●▲ ◗▲P❦❩◆❑■■❍❩❑◆ ❘●▲P▼❨ P❤ ❍◗❝▲■❘❚▲◗❘ ❏P■❘✐◆❑❘▲■ ❘●❑❘ ❑ ➟▼❚ ■●P✐◆❯❍◗❝▲■❘ ❍◗ ❑ ❏▼P❧▲❩❘ ❑■ ◆P◗❙ ❑■ ❘●▲ ❏▼▲■▲◗❘ ❝❑◆✐▲ P❤ ❘●▲ ▲❲❏▲❩❘▲❯ ■❘▼▲❑❚P❤ ❏▼P➟❘■ ❘●❑❘ ❘●❍■ ❏▼P❧▲❩❘ ❣❍◆◆ ❙▲◗▲▼❑❘▲ ■●P✐◆❯ ▲❲❩▲▲❯ ❭P▼ ❑❘ ◆▲❑■❘▲➠✐❑◆❵ ❘●▲ ❏▼▲■▲◗❘ ❝❑◆✐▲ P❤ ❘●▲ ▲❲❏▲◗❯❍❘✐▼▲■ ■❘▼▲❑❚ ▼▲➠✐❍▼▲❯ ❘P ❜✐❍◆❯❘●▲ ❏▼P❧▲❩❘❞ ❋●✐■❛ ❘●▲ ◗▲❘ ❏▼▲■▲◗❘ ❝❑◆✐▲ ❭➡❳➢❵ P❤ ❘●▲ ❍◗❝▲■❘❚▲◗❘❏▼P❧▲❩❘ ❍■ ❙▼▲❑❘▲▼ ❘●❑◗ ➤▲▼P❞ ❋●❍■ ❩◆❑■■❍❩❑◆ ❘●▲P▼❨ ❤❑❍◆■ ❘P ❩❑❏❘✐▼▲❘●▼▲▲ ❚❑❍◗ ❩●❑▼❑❩❘▲▼❍■❘❍❩■ ❍◗ ❚P■❘ ❍◗❝▲■❘❚▲◗❘ ❯▲❩❍■❍P◗■❞ ➥❍▼■❘ ❍■ ❘●▲❍■■✐▲ P❤ ➦➧➧➨➩➨➧➫➦➭➦➯➦➲➳❞ ➵ ❩◆❑■■❍❩❑◆ ➡❳➢ ❩❑◆❩✐◆❑❘❍P◗ ❣P✐◆❯ ❍❚❏◆❍❩❍❘◆❨❑■■✐❚▲ ❘●❑❘ ❍❤ ▲❩P◗P❚❍❩ ❩P◗❯❍❘❍P◗■ ❘✐▼◗ ❘P ❜▲ ■◆✐❙❙❍■●❛ ❘●▲ ➟▼❚ ■●✐❘■❯P❣◗ ❘●▲ ❏▼P❧▲❩❘ ❑◗❯ ▼▲❩P❝▲▼■ ❘●▲ ❚P◗▲❨ ❘●❑❘ ❍■ ❍◗❝▲■❘▲❯❞ ➸P❣▲❝▲▼❛❘●❍■ ❍■ ◗P❘ ❑ ▼▲❑◆❍■❘❍❩ P✐❘❩P❚▲❞ ➺▲❩P◗❯❛ ❍■ ❘●▲ ❍■■✐▲ P❤ ➻➼➽➨➧➲➾➦➼➲➳❞➵❙❑❍◗❛ ◗▲P❩◆❑■■❍❩❑◆ ➡❳➢ ❩❑◆❩✐◆❑❘❍P◗ ❯P▲■ ◗P❘ ❑■■▲■■ ❏▼P❜❑❜❍◆❍❘❍▲■ P❤❑◆❘▲▼◗❑❘❍❝▲ P✐❘❩P❚▲■ P❝▲▼ ❘●▲ ❤✐❘✐▼▲ P❤ ❘●▲ ❏▼P❧▲❩❘ ◆❍❤▲❞ ❋●❍▼❯ ❍■ ❘●▲➲➦➚➦➼➪ P❤ ❍◗❝▲■❘❚▲◗❘❞ ➥❍▼❚■❛ P▼ ❜❑◗❬■ ❍◗ ❘●❍■ ❩❑■▲❛ ❩❑◗ ❏P■❘❏P◗▲ ❘●▲❍▼❍◗❝▲■❘❚▲◗❘■ ❘P P❜❘❑❍◗ ❚P▼▲ ❍◗❤P▼❚❑❘❍P◗ ❑❜P✐❘ ❘●▲ ❤✐❘✐▼▲❛ ❑◆❜▲❍❘ ◗▲❝▲▼❩P❚❏◆▲❘▲ ❍◗❤P▼❚❑❘❍P◗❞ ❋●▲ ❑❜❍◆❍❘❨ ❘P ❯▲◆❑❨ ❑◗ ❍▼▼▲❝▲▼■❍❜◆▲ ❍◗❝▲■❘❚▲◗❘❏▼P❧▲❩❘ ❩❑◗ ❑❢▲❩❘ ❘●▲ ❯▲❩❍■❍P◗ ❘P ❍◗❝▲■❘❞ ❋●❍■ ❍■ ❘●▲ ❚❑❍◗ ❘●▲❚▲ P❤ ❘●▲▼▲❑◆ P❏❘❍P◗ ❑❏❏▼P❑❩●❛ ➟▼■❘ ❯▲❝▲◆P❏▲❯ ❜❨ ➶❩❱P◗❑◆❯ ❑◗❯ ➺❍▲❙▲◆ ❭❪❫➹➘❵❞➥❍▼❚■ P▼ ❜❑◗❬■ ❣❍❘● ❑◗ P❏❏P▼❘✐◗❍❘❨ ❘P ❍◗❝▲■❘ ❑▼▲ ●P◆❯❍◗❙ ❑◗ ➴P❏❘❍P◗➴❑◗❑◆P❙P✐■ ❘P ❑ ➟◗❑◗❩❍❑◆ ❩❑◆◆ P❏❘❍P◗❞ ❥●▲◗ ❑ ➟▼❚ ❚❑❬▲■ ❑◗ ❍▼▼▲❝▲▼■❍❜◆▲❍◗❝▲■❘❚▲◗❘❛ ❍❘ ▲❲▲▼❩❍■▲■ ❍❘■ P❏❘❍P◗ ❘P ❍◗❝▲■❘❞ ❱P❍◗❙ ■P❛ ❘●▲ ➟▼❚ ❙❍❝▲■✐❏ ❘●▲ ❏P■■❍❜❍◆❍❘❨ P❤ ❣❑❍❘❍◗❙ ❤P▼ ◗▲❣ ❍◗❤P▼❚❑❘❍P◗ ❘P ❑▼▼❍❝▲❞ ❋●❑❘ ❚❍❙●❘❑❢▲❩❘ ❘●▲ ❯▲❩❍■❍P◗ ❘P ❍◗❝▲■❘ ◗P❣❞ ❋●❍■ ◆P■❘ P❏❘❍P◗ ❝❑◆✐▲ ❍■ ❑◗ P❏❏P▼❘✐❦◗❍❘❨ ❩P■❘ ❘●❑❘ ❚✐■❘ ❜▲ ❍◗❩◆✐❯▲❯ ❑■ ❏❑▼❘ P❤ ❘●▲ ❩P■❘ P❤ ❘●▲ ❍◗❝▲■❘❚▲◗❘❞❋●▲▼▲❤P▼▲❛ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘ ❚✐■❘ ▲❲❩▲▲❯ ❘●▲ ❍◗❝▲■❘❚▲◗❘ ❩P■❘❛❜❨ ❑◗ ❑❚P✐◗❘ ▲➠✐❑◆■ ❘P ❘●▲ ❝❑◆✐▲ P❤ ❬▲▲❏❍◗❙ ❘●▲ ❍◗❝▲■❘❚▲◗❘ P❏❘❍P◗❑◆❍❝▲❞ ❡▲■❍❯▲■ ❍❘■ ❤▲❑■❍❜❍◆❍❘❨❛ ❘●▲ ▼▲❑◆ P❏❘❍P◗■ ❚P❯▲◆ ▼▲➷▲❩❘■ ❘●▲ ❝❑▼❍❑❦❘❍P◗■ ❍◗ ❝❑◆✐▲ ❤P▼ ❑◗❨ ❍◗❝▲■❘❚▲◗❘ ❏▼P❧▲❩❘ ❍◗ ❑ ❩P◗❘❍◗✐P✐■ ■▲❘❘❍◗❙ ❘●❑❘❍■ ▼▲➷▲❩❘❍❝▲ P❤ ▼▲❑◆ ▲❩P◗P❚❍❩ ❝❑▼❍❑❘❍P◗■❛ ❤P▼ ▲❲❑❚❏◆▲❛ ➷✐❩❘✐❑❘❍P◗■ ❍◗❍◗❘▲▼▲■❘ ▼❑❘▲■❞

Page 4: Capital Investment as Real Options: A Note on Dixit ... · Munich Personal RePEc Archive Capital Investment as Real Options: A Note on Dixit-Pindyck Model. Termos, Ali American University

➮➱➮ ✛✃ ✢✜✜❐★❒✥❮❒✢❰✤ ✤Ï❮✦Ð✜✤Ñ ➺✐❏❏P■▲ ❘●❑❘ ❘●▲ ❏▼P❧▲❩❘ ❙▲◗▲▼❑❘▲■ ❑ ■❘▼▲❑❚ P❤ ❩❑■● ➷P❣ ▲➠✐❑◆■ ❘P❘●▲ ❏▼❍❩▲ P❤ ❘●▲ P✐❘❏✐❘❞ ➵■■✐❚▲ ❤P▼ ◗P❣ ❘●❑❘ ❏▼❍❩▲ ❑❘ ❘❍❚▲ ➤▲▼P ❍■ÒÓ Ô ÕÖÖ ❑◗❯ ❘●❑❘ ❘●▲ ➟▼❚ ❏▼P❯✐❩▲■ P◗▲ P✐❘❏✐❘ ❏▲▼ ❨▲❑▼❞ ❋●❍■ ❏▼❍❩▲❍■ ■✐❜❧▲❩❘ ❘P ❙P ✐❏ ❘P ×ÖÖ ❣❍❘● ØÖÙ ❩●❑◗❩▲ P▼ ❙P ❯P❣◗ ❘P ❪ÖÖ ❣❍❘●ØÖÙ ❩●❑◗❩▲❞ ➺P ❘●▲ ▲❲❏▲❩❘▲❯ ❝❑◆✐▲ P❤ ❘●▲ ❏▼❍❩▲ ❍■ ÕÖÖ❞ ❋▼❨❍◗❙ ❘P❑❝P❍❯ ▼❍■❬ P◗▲ ❩❑◗ ●▲❯❙▲ ❑❙❑❍◗■❘ ❘●❍■ ❏▼❍❩▲ ➷✐❩❘✐❑❘❍P◗❞ Ú▲❘ ❘●▲ ❍◗❍❘❍❑◆❍◗❝▲■❘❚▲◗❘ ❜▲ ❪Û ➘ÖÖ❛ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘ ❘P❯❑❨ ❍■

Ü❪➘ÖÖ Ý ÞßàáÓÕÖÖ❭❪â❪Ö❵à

Ü❪➘ÖÖ Ý ÕÕÖÖ Ô ➘ÖÖã❤ Ò ❯▼P❏■ ❘P ❪ÖÖ ❘●▲◗ ❘●▲ ❍◗❝▲■❘P▼ ◗▲▲❯■ ❘P ■▲◆◆ ■●P▼❘ ääÓÓäÓÓ Ô ❪❪✐◗❍❘■❞ ➺✐❏❏P■▲ Ò ❘✐▼◗■ ❯P❣◗ ❘P ❜▲ ❪ÖÖ ❘●▲◗ ❘●▲ ❍◗❝▲■❘P▼ ❩P◆◆▲❩❘■❪ÖÖå❪❪ Ô ❪❪ÖÖ ❤▼P❚ ❘●▲ ●▲❯❙❍◗❙ ❩P◗❘▼❑❩❘ ❑◗❯ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘❘✐▼◗■ ❘P ❜▲

ÞßàáÓ❪ÖÖ❭❪â❪Ö❵à Ô ❪❪ÖÖ

■P ❘●▲ ◗▲❘ P✐❘❩P❚▲ ❍■ ❪❪ÖÖ Ý ❪❪ÖÖ Ô ÕÕÖÖ❞➺✐❏❏P■▲ ◗P❣ ❘●❑❘ ❘●▲ ❏▼❍❩▲ ❣▲◗❘ ✐❏ ❘P ×ÖÖ❞ ❋●▲ ❍◗❝▲■❘P▼ ●❑■ ■P◆❯■●P▼❘ ❪❪ ✐◗❍❘■ ❤P▼ ❪❪ åÕÖÖ Ô ÕÕÖÖ ❑◗❯ ■●P✐◆❯ ❏❑❨ ❪❪ å×ÖÖ Ô ×Û×ÖÖ❘●✐■ ❍◗❩✐▼ ❑ ◗▲❘ ◆P■■ P❤ ÕÕÖÖ Ü ××ÖÖ Ô Ü❪❪ÖÖ ➸P❣▲❝▲▼❛ ❘●▲ ❝❑◆✐▲ P❤❘●▲ ❏▼P❧▲❩❘ ❍■ ◗P❣

ÞßàáÓ×ÖÖ❭❪â❪Ö❵à Ô ××ÖÖ

❑◗❯ ❘●▲ ◗▲❘ P✐❘❩P❚▲ ❍■ ××ÖÖ Ü ❪❪ÖÖ Ô ÕÕÖÖ ■❑❚▲ ❑■ ❍◗ ❘●▲ ❏▼▲❝❍P✐■❩❑■▲❞ ❋▼❑◗■◆❑❘❍◗❙ ❘●❍■ ▲❲❑❚❏◆▲ ❍◗❘P ❑ ▼▲❑◆ P❏❘❍P◗ ❩❑■▲ ◆▲❘ æÑ ❜▲ ❘●▲❝❑◆✐▲ P❤ ❘●▲ ❍◗❝▲■❘❚▲◗❘ P❏❏P▼❘✐◗❍❘❨ ◗▲❲❘ ❨▲❑▼❞ ã❤ Ò Ô ×ÖÖ ❘●▲◗

æÑ Ô Ü❪➘ÖÖ Ý ÞßàáÓ×ÖÖ❭❪â❪Ö❵à Ô ❪çÖÖ

ã❤ Ò Ô ❪ÖÖ ❘●▲◗ ❘●▲ P❏❘❍P◗ ❍■ ✐◗▲❲▲▼❩❍■▲❯ ■P æÑ Ô Ö❞ ➡P❣ ❣▲ ◗▲▲❯ ❘P➟◗❯ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ P❏❘❍P◗ ❘P❯❑❨ æÓ❞ ➥P▼ ❘●❑❘ ❣▲ ❩▼▲❑❘▲ ❑ ❏P▼❘❤P◆❍Pèé❸③ ③êt⑥✈❿③ ❸③✇③ rq ❼①✇✇①➋③❻ ②✇①⑥ ⑤rêrs t✉❻ ➃r✉❻❶⑧➊ ëìííîï

ð

Page 5: Capital Investment as Real Options: A Note on Dixit ... · Munich Personal RePEc Archive Capital Investment as Real Options: A Note on Dixit-Pindyck Model. Termos, Ali American University

❣❍❘● ❘❣P ❩P❚❏P◗▲◗❘■ñ ➟▼■❘❛ ❘●▲ ❍◗❝▲■❘❚▲◗❘ P❏❏P▼❘✐◗❍❘❨ ❍❘■▲◆❤❞ ➺▲❩P◗❯❛❑ ❩▲▼❘❑❍◗ ❑❚P✐◗❘ P❤ P✐❘❏✐❘❞ ❋●▲ ❏P▼❘❤P◆❍P ❍■ ▼❍■❬❦❤▼▲▲ ❭◗P ❑▼❜❍❘▼❑❙▲❵❞❋●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏P▼❘❤P◆❍P ❘P❯❑❨ ❍■

òÓ Ô æÓ ÜóÒÓã❤ ÒÓ Ô ÕÖÖ ❘●▲◗ òÓ Ô æÓ ÜÕÖÖó❞ ❋●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏P▼❘❤P◆❍P ◗▲❲❘ ❨▲❑▼❍■ ñ òÑ Ô æÑ ÜóÒÑ ❣●❍❩● ❯▲❏▲◗❯■ P◗ ÒÑ❞ ã❤ ÒÑ Ô ×ÖÖ ❘●▲◗

æÑ Ô Ü❪➘ÖÖ Ý ÞßàáÓ×ÖÖ❭❪â❪Ö❵à Ô ❪çÖÖ

❋●✐■ òÑô Ô ❪çÖÖ Ü ×ÖÖó❞ ã❤ ÒÑ Ô ❪ÖÖ❛ ❘●▲◗ æÑ Ô Ö❛ ❘●▲ P❏❘❍P◗ ❙P▲■✐◗▲❲▲▼❩❍■▲❯❞ ❋●▲▼▲❤P▼▲ òÑõ Ô Ö Ü ❪ÖÖó❞ ➡P❣ ❣▲ ◗▲▲❯ ❘P ❩●P■▲ ó ■P❘●❑❘ ❘●▲ ❏P▼❘❤P◆❍P òÑ ❍■ ▼❍■❬❦❤▼▲▲❞ ➺▲❘ òÑô Ô òÑõñ

❪çÖÖ Ü ×ÖÖó Ô Ü❪ÖÖó■P ó Ô ➹âØ❞ òÑ Ô ❪çÖÖ Ü ×ÖÖ❭➹âØ❵ Ô Ü➹ØÖ❣●▲❘●▲▼ Ò ▼❍■▲■ ❘P ×ÖÖ P▼ ❤❑◆◆■ ❘P ❪ÖÖ❞ ❋P ❩❑◆❩✐◆❑❘▲ ❘●▲ ❩❑❏❍❘❑◆ ❙❑❍◗P❤ ❘●❍■ ❏P▼❘❤P◆❍P❛ ➟▼■❘ ❣▲ ■●P✐◆❯ ➟❙✐▼▲ ❘●▲ ❏❑❨❚▲◗❘ ❘●❑❘ ❚✐■❘ ❜▲ ❏❑❍❯❘P ❘●▲ ●P◆❯▲▼ P❤ ❘●▲ ■●P▼❘ ❏P■❍❘❍P◗ ❭❘●▲ P❏❘❍P◗ ❏▼▲❚❍✐❚❵❞

ö÷øùú÷û ü÷ùó Ô òÑ Ü òÓ Üýøúùýó øþÿ�ù✁�➡P ❍◗❝▲■❘P▼ ❣P✐◆❯ ❜▲ ❣❍◆◆❍◗❙ ❘P ●P◆❯ ❑ ◆P◗❙ ❏P■❍❘❍P◗ ❍❤ ❩❑❏❍❘❑◆ ❙❑❍◗ ❍■➤▲▼P❞ ❋●▲ ●P◆❯▲▼ P❤ ❑ ◆P◗❙ ❏P■❍❘❍P◗ ❣❍◆◆ ▼▲➠✐❍▼▲ ❘●▲ ▼❍■❬ ❤▼▲▲ ▼❑❘▲ ❣●❍❩●❍■ ❑■■✐❚▲❯ ❪ÖÙ❞ ➺❍◗❩▲ ▲❲❏▲❩❘▲❯ ❏▼❍❩▲ ❍■ ÕÖÖ ❏▲▼ ✐◗❍❘❛ ❘●▲ ▼▲➠✐❍▼▲❯▼▲❘✐▼◗ ❍■ Öâ❪ÒÓ Ô Öâ❪ å ÕÖÖ Ô ÕÖ ❏▲▼ ✐◗❍❘❞ ❳▼▲❝❍P✐■◆❨❛ ❘●▲ ◗✐❚❜▲▼P❤ ✐◗❍❘■ ❍◗ ❘●▲ ❏P▼❘❤P◆❍P ❣❑■ ❤P✐◗❯ ❘P ❜▲ ➹âØ❛ ❘●▲▼▲❤P▼▲ ❘●▲ P❏❘❍P◗❏▼▲❚❍✐❚ ❍■ ÕÖ å ➹âØ Ô ❪çÖ❞ ❋●▲◗ ❘●▲ ❩❑❏❍❘❑◆ ❙❑❍◗ P❤ ●P◆❯❍◗❙ ❘●❍■❏P▼❘❤P◆❍P P❝▲▼ ❘●▲ ❨▲❑▼ ❍■ ❙❍❝▲◗ ❜❨ñ

òÑ Ü òÓ Ü ❪çÖòÑ Ü ❭æÓ ÜóÒÓ❵ Ü ❪çÖ

Ü➹ØÖ Ü ✂æÓ Ü ❭➹âØ åÕÖÖ❵✄ Ü ❪çÖ☎

Page 6: Capital Investment as Real Options: A Note on Dixit ... · Munich Personal RePEc Archive Capital Investment as Real Options: A Note on Dixit-Pindyck Model. Termos, Ali American University

Ô ➘➹Ö ÜæÓ➺❍◗❩▲ ❘●▲▼▲ ❍■ ◗P ❑▼❜❍❘▼❑❙▲❛ ❑◗❨ ❩❑❏❍❘❑◆ ❙❑❍◗ ❚✐■❘ ▲➠✐❑◆ ❘P ❪ÖÙ P❤ ❘●▲❍◗❍❘❍❑◆ ❏P▼❘❤P◆❍P❛ ❘●❑❘ ❍■

➘➹Ö ÜæÓ Ô Öâ❪òÓ➘➹Ö ÜæÓ Ô Öâ❪❭æÓ Ü ❪çÖÖ❵

æÓ Ô çç×❘●❑❘ ❍■ ❘●▲ P❏❏P▼❘✐◗❍❘❨ ❩P■❘ P❤ ❍◗❝▲■❘❍◗❙ ❘P❯❑❨❞ ❋●▲ ❤✐◆◆ ❩P■❘ P❤ ❍◗❝▲■❘❦❍◗❙ ❘P❯❑❨ ❍■ ❪➘ÖÖ Ý çç× Ô Õ×ç× ❣●❍❩● ❍■ ❙▼▲❑❘▲▼ ❘●❑◗ ÕÕÖÖ❞

✆ ✝✞✟➙↕ ➔➜✞➞ →✠➓➙→➒✟ ➝→➣➜➞➵■■✐❚▲ ❘●❑❘ ❘●▲ ❜❑◗❬ ❍◗❝▲■❘■ ❯❍▼▲❩❘◆❨ ❍◗ ❑ ❏P▼❘❤P◆❍P P❤ ❏▼P❧▲❩❘■ ❍◗P▼❯▲▼ ❘P ❯❍❝▲▼■❍❤❨ ▼❍■❬❞ ❋●▲ ❜❑◗❬♥■ ❍◗❝▲■❘❚▲◗❘■ ❑▼▲ ❏❑▼❘◆❨ ❍▼▼▲❝▲▼■❍❜◆▲❭■✐◗❬ ❩P■❘❵❞ ❋●▲■▲ ❍◗❝▲■❘❚▲◗❘■ ❩❑◗ ❜▲ ❯▲◆❑❨▲❯❛ ■P ❘●▲ ❜❑◗❬ ●❑■ ❘●▲P❏❏P▼❘✐◗❍❘❨ ❘P ❣❑❍❘ ❤P▼ ◗▲❣ ❍◗❤P▼❚❑❘❍P◗ ❘P ❑▼▼❍❝▲ ❍◗ P▼❯▲▼ ❘P ▼▲❯✐❩▲▼❍■❬❞Ú▲❘ ✡ ❜▲ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❍◗❝▲■❘❚▲◗❘ ❏▼P❧▲❩❘ P❤ ❘●▲ ❜❑◗❬❞ ❥▲ ❩❑◗ ❘●❍◗❬P❤ ✡ ❑■ ❘●▲ ❯❍■❩P✐◗❘▲❯ ❜❑◗❬♥■ ❩❑■● ➷P❣ ❤▼P❚ ❘●❍■ ❏▼P❧▲❩❘❞ ➥P◆◆P❣❍◗❙➶❩❱P◗❑◆❯ ❑◗❯ ➺❍▲❙▲◆ ❭❪❫➹➘❵❛ ❑■■✐❚▲ ❘●❑❘ ❍◗ ❑ ❙❍❝▲◗ ❏▲▼❍P❯ P❤ ❘❍❚▲ ✡❤P◆◆P❣■ ❘●▲ ❤P◆◆P❣❍◗❙ ❙▲P❚▲❘▼❍❩ ❡▼P❣◗❍❑◗ ❚P❘❍P◗ñ

☛✡ Ô ☞✡ ☛ú Ý ✌✡ ☛✍ ❭❪❵❣●▲▼▲ ☞ ❍■ ❘●▲ ❙▼P❣❘● ▼❑❘▲ P❤ ✡ P▼ ❘●▲ ▲❲❏▲❩❘▲❯ ❏▲▼❩▲◗❘❑❙▲ ▼❑❘▲ P❤❩●❑◗❙▲ ❍◗ ✡ ❞ ✌ ❍■ ❘●▲ ■❘❑◗❯❑▼❯ ❯▲❝❍❑❘❍P◗ ❭✐◗❩▲▼❘❑❍◗❘❨❵❛ ❑◗❯ ☛✍ ❍■ ❘●▲❍◗❩▼▲❚▲◗❘ P❤ ❑ ❥❍▲◗▲▼ ❏▼P❩▲■■❞ ❋●▲ ❜❑◗❬♥■ ❍◗❝▲■❘❚▲◗❘ P❏❏P▼❘✐◗❍❘❨❍■ ▲➠✐❍❝❑◆▲◗❘ ❘P ❑ ❩❑◆◆ P❏❘❍P◗❞ ➵◗ ▲❲❑❚❏◆▲ ❣P✐◆❯ ❜▲ ❍❤ ❘●▲ ❜❑◗❬ ❍■❩P◗❘▲❚❏◆❑❘❍◗❙ ❑ ❯▲❩❍■❍P◗ ❘P ❍◗❝▲■❘ ❍◗ ▲❍❘●▲▼ ❏▼P❧▲❩❘ ➵ P▼ ❏▼P❧▲❩❘ ❡❞❋●▲▼▲❤P▼▲❛ ❘●▲ ❍◗❝▲■❘❚▲◗❘ ❯▲❩❍■❍P◗ ❍■ ❩P◗■❍❯▲▼▲❯ ❑■ ❑ ❏▼P❜◆▲❚ P❤ P❏❘❍P◗❝❑◆✐❑❘❍P◗❞ ❋●▲ ❜❑◗❬ ❣❍◆◆ ❣❑◗❘ ❘P ❚❑❲❍❚❍➤▲ ❘●▲ ❝❑◆✐▲ P❤ ❍❘■ ❍◗❝▲■❘❚▲◗❘P❏❏P▼❘✐◗❍❘❨❛ æ ❭✡ ❵❞Ú▲❘ ✎ ❯▲◗P❘▲ ❘●▲ ❩P■❘ P❤ ❘●▲ ❍◗❝▲■❘❚▲◗❘ ❏▼P❧▲❩❘❞ ❋●▲ ❏❑❨P❢ ❤▼P❚

Page 7: Capital Investment as Real Options: A Note on Dixit ... · Munich Personal RePEc Archive Capital Investment as Real Options: A Note on Dixit-Pindyck Model. Termos, Ali American University

❍◗❝▲■❘❍◗❙ ❑❘ ❘❍❚▲ ú ❍■ ✡à Ü ✎❞ ã◗ ❑ ❩◆❑■■❍❩❑◆ ■▲❘❘❍◗❙ ❘●▲ ❜❑◗❬♥■ ❚❑❲❍❦❚❍➤❑❘❍P◗ ❏▼P❜◆▲❚ ❣P✐◆❯ ❜▲ ▼▲❏▼▲■▲◗❘▲❯ ❜❨ñæ ❭✡ ❵ Ô�÷✑ ✒✂❭✡✓ Ü ✎❵ÿ✔✕✓ ✄ ❭Õ❵

✖ ❍■ ❘●▲ ❤✐❘✐▼▲ ❘❍❚▲ ❣●▲▼▲ ❘●▲ ❍◗❝▲■❘❚▲◗❘ ❣❍◆◆ ❜▲ ❚❑❯▲ ❑◗❯ þ ❍■ ❘●▲❯❍■❩P✐◗❘ ▼❑❘▲❞ ❡✐❘ ■❍◗❩▲ ❣▲ ❑■■✐❚▲ ❘●❑❘ ✡ ▲❝P◆❝▲■ ■❘P❩●❑■❘❍❩❑◆◆❨❛❣▲ ❣❍◆◆ ◗P❘ ❜▲ ❑❜◆▲ ❘P ❯▲❘▲▼❚❍◗▲ ❑ ❘❍❚▲ ✖ ❞ ã◗■❘▲❑❯❛ ❘●▲ ❍◗❝▲■❘❚▲◗❘❩P◗❯❍❘❍P◗ ❣❍◆◆ ❘❑❬▲ ❘●▲ ❤P▼❚ P❤ ❑ ❩▼❍❘❍❩❑◆ ❝❑◆✐▲ ✡ ✗ ■✐❩● ❘●❑❘ ❍❘ ❍■ P❏❘❍❚❑◆❘P ❍◗❝▲■❘ ❑■ ◆P◗❙ ❑■ ✡ ✘ ✡ ✗❞ ❥▲ ❣❑◗❘ ❘P ❯▲❘▲▼❚❍◗▲ ❘●▲ ❏P❍◗❘ ❑❘ ❣●❍❩●❍❘ ❍■ P❏❘❍❚❑◆ ❤P▼ ❘●▲ ❜❑◗❬ ❘P ❍◗❝▲■❘ ✎ ❍◗ ▼▲❘✐▼◗ ❤P▼ ❑◗ ❑■■▲❘ ❣P▼❘● ✡ ❞❋P ■P◆❝▲ ❤P▼ ✡ ✗ ❣▲ ❩❑◗ ✐■▲ ❖P◗❘❍◗❙▲◗❘ ❖◆❑❍❚■ ❑◗❑◆❨■❍■❞✙➱➮ ✚✧✃❒✢✃✛✤✃❒ ✚✜❮✢✦★ ★✧✜❐❒✢✧✃❋P ❚❑❬▲ ✐■▲ P❤ ❩P◗❘❍◗❙▲◗❘ ❩◆❑❍❚■ ❑◗❑◆❨■❍■ ❣▲ ■●P✐◆❯ ❑■■✐❚▲ ❘●❑❘ ❘●▲■❘P❩●❑■❘❍❩ ❝❑▼❍❑❘❍P◗■ ❍◗ ✡ ❚✐■❘ ❜▲ ■❏❑◗◗▲❯ ❜❨ ▲❲❍■❘❍◗❙ ❑■■▲❘■ ❣❍❘●❑ ❏▼❍❩▲ ❘●❑❘ ❍■ ❏▲▼❤▲❩❘◆❨ ❩P▼▼▲◆❑❘▲❯ ❣❍❘● ✡ ■P ❘●❑❘ ✐◗❩▲▼❘❑❍◗❘❨ P❝▲▼❤✐❘✐▼▲ ❝❑◆✐▲■ P❤ ✡ ❩❑◗ ❜▲ ▼▲❏◆❍❩❑❘▲❯ ❜❨ ▲❲❍■❘❍◗❙ ❑■■▲❘■❞ ➥P◆◆P❣❍◗❙❘●▲ ❩P◗❝▲◗❘❍P◗❑◆ ❏▼P❩▲❯✐▼▲ P❤ ❘●▲ ■❏❑◗◗❍◗❙ ❘▲❩●◗❍➠✐▲❛ ◆▲❘ Ò ❜▲ ❘●▲❏▼❍❩▲ P❤ ❘●▲ ❑■■▲❘ ❘●❑❘ ❍■ ❏▲▼❤▲❩❘◆❨ ❩P▼▼▲◆❑❘▲❯ ❣❍❘● ✡ ❞ Ú▲❘ ✜✢✣ ❜▲❘●▲ ❩P▼▼▲◆❑❘❍P◗ P❤ Ò ❣❍❘● ❘●▲ ❚❑▼❬▲❘ ❏P▼❘❤P◆❍P ✤ ❛ ❘●▲◗❛ ✜✢✣ Ô ✜✥✣ ❞➺❍◗❩▲ Ò ❍■ ❏▲▼❤▲❩❘◆❨ ❩P▼▼▲◆❑❘▲❯ ❣❍❘● ✡ ❛ Ò ❍■ ❑■■✐❚▲❯ ❘P ▲❝P◆❝▲ ❘●▲■❑❚▲ ❣❑❨ñ ☛Ò Ô ✦Ò☛ú Ý ✌Ò☛✍ ❭×❵✦ ❍■ ❘●▲ ▼❍■❬❦❑❯❧✐■❘▲❯ ▼❑❘▲ P❤ ▼▲❘✐▼◗ P◗ ❘●❍■ ❑■■▲❘❞ ❡❨ ❘●▲ ❖❑❏❍❘❑◆➵■■▲❘ ❳▼❍❩❍◗❙ ➶P❯▲◆ ❭❖➵❳➶❵❛ ✦ ❑◆■P ▼▲➷▲❩❘■ ❘●▲ ❑■■▲❘♥■ ■❨■❘▲❚❑❘❍❩▼❍■❬❞ ❋●▲ ✦ ❍■ ❙❍❝▲◗ ❜❨ñ

✦ Ô þ Ý ò✜✢✣✌ ❭❴❵❣●▲▼▲ ò Ô ✧✕★✔✕✩✪ ❍■ ❘●▲ ❑❙❙▼▲❙❑❘▲ ❚❑▼❬▲❘ ❏▼❍❩▲ P❤ ▼❍■❬❞ þ✣ ❍■ ❘●▲▲❲❏▲❩❘▲❯ ▼▲❘✐▼◗ P◗ ❘●▲ ❚❑▼❬▲❘❞ ❥▲ ❑■■✐❚▲ ❘●❑❘ ☞ ✫ ✦ ❤P▼ ❘●▲ ❜❑◗❬❘P ❍◗❝▲■❘ ❍◗ ❘●▲ ❏▼P❧▲❩❘❞Ú▲❘ ✬ Ô ✦ Ü ☞❛ ■P ❘●❑❘ ✬ ✭ Ö❞ ã❤ ❣▲ ❘●❍◗❬ P❤ ✦ Ô ☞ Ý ✬ ❑■ ❘●▲ ❘P❘❑◆

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▲❲❏▲❩❘▲❯ ▼▲❘✐▼◗ P◗ ❘●▲ ❏▼P❧▲❩❘❛ ❘●❑❘ ❍■ ❛ ❘●▲ ❯❍❝❍❯▲◗❯ ▼❑❘▲ ❏◆✐■ ❘●▲▲❲❏▲❩❘▲❯ ▼❑❘▲ P❤ ❩❑❏❍❘❑◆ ❙❑❍◗❛ ❘●▲◗ ✬ ❍■ ❑◗ P❏❏P▼❘✐◗❍❘❨ ❩P■❘ P❤ ❯▲◆❑❨❍◗❙❍◗❝▲■❘❍◗❙ ❍◗ ❘●▲ ❏▼P❧▲❩❘ ❑◗❯ ❬▲▲❏❍◗❙ ❘●▲ P❏❘❍P◗ ❘P ❍◗❝▲■❘ ❑◆❍❝▲❞ ã❤ ✬ Ô Ö❛❘●❑❘ ❍■ ❍❤ ✦ Ô ☞❛ ❘●▲◗ ❘●▲▼▲ ❣P✐◆❯ ❜▲ ◗P P❏❏P▼❘✐◗❍❘❨ ❩P■❘ ❘P ❬▲▲❏❍◗❙❘●▲ P❏❘❍P◗ ❑◆❍❝▲❛ ❑◗❯ ❘●▲ ❜❑◗❬ ❣P✐◆❯ ◗▲❝▲▼ ❍◗❝▲■❘ ❍◗ ❘●❍■ ❏▼P❧▲❩❘❞ ➡P❘▲❘●❑❘ ❍❤ ✬ ❍■ ❝▲▼❨ ◆❑▼❙▲❛ ❘●▲ P❏❏P▼❘✐◗❍❘❨ ❩P■❘ P❤ ❣❑❍❘❍◗❙ ❍■ ◆❑▼❙▲❛ ❘●✐■❘●▲ ❝❑◆✐▲ P❤ ❘●▲ P❏❘❍P◗ ❣❍◆◆ ❜▲ ❝▲▼❨ ■❚❑◆◆❞ ☞ ❘●▲◗ ❩❑◗ ❜▲ ▲❲❏▼▲■■▲❯ ❑■ñ

☞ Ô ❪☛ú✒✂☛✡✡ ✄❑◗❯ ✬ ❩❑◗ ❜▲ ▲❲❏▼▲■■▲❯ ❑■ ❑ ❤✐◗❩❘❍P◗ P❤ ✡ ñ

✬❭✡ ❵ Ô ✦ Ü ❪☛ú✒✂☛✡✡ ✄➡P❣ æ ❭✡ ❵ ❩❑◗ ❜▲ ❯▲❘▲▼❚❍◗▲❯ ❜❨ ❩P◗■❘▼✐❩❘❍◗❙ ❑ ▼❍■❬❦❤▼▲▲ ❏P▼❘❤P◆❍P❛❯▲❘▲▼❚❍◗❍◗❙ ❍❘■ ▲❲❏▲❩❘▲❯ ▼❑❘▲ P❤ ▼▲❘✐▼◗❛ ❑◗❯ ▲➠✐❑❘❍◗❙ ❘●❑❘ ▼▲❘✐▼◗ ❘P❘●▲ ▼❍■❬❦❤▼▲▲ ▼❑❘▲ P❤ ❍◗❘▲▼▲■❘ þ❞ ❋P ❩P◗■❘▼✐❩❘ ■✐❩● ❑ ❏P▼❘❤P◆❍P❛ ❩P◗■❍❯▲▼●P◆❯❍◗❙ ❑◗ P❏❘❍P◗ ❘P ❍◗❝▲■❘ ❣●❍❩● ❍■ ❣P▼❘● æ ❭✡ ❵❞ ➵■■✐❚▲ ❑ ■●P▼❘❏P■❍❘❍P◗ P❤ ✯ Ô æ ✰ ❭✡ ❵ ✐◗❍❘■ P❤ ❘●▲ ❏▼P❧▲❩❘❞ ❋●▲ ❝❑◆✐▲ P❤ ❘●❍■ ❏P▼❘❤P◆❍P❍■ ❙❍❝▲◗ ❜❨ñ ✱ Ô æ ❭✡ ❵ Üæ ✰ ❭✡ ❵✡ ❭Ø❵☛✱ Ô ☛æ ❭✡ ❵ Ü ☛æ ✰ ❭✡ ❵✡ Üæ ✰ ❭✡ ❵☛❭✡ ❵❣●▲▼▲

æ ✰ ❭✡ ❵ Ô ☛æ☛✡❋●❍■ ❍■ ❑ ❯❨◗❑❚❍❩ ❏P▼❘❤P◆❍P❛ ■P ❣●▲◗ ❘●▲ ❝❑◆✐▲ ✡ ❩●❑◗❙▲■❛ æ ✰ ❭✡ ❵ ❚❑❨❩●❑◗❙▲ ❤▼P❚ P◗▲ ■●P▼❘ ❍◗❘▲▼❝❑◆ P❤ ❘❍❚▲ ❘P ❘●▲ P❘●▲▼❞ ❋●▲▼▲❤P▼▲❛ ❘●▲❩P❚❏P■❍❘❍P◗ P❤ ❘●▲ ❏P▼❘❤P◆❍P ❣❍◆◆ ❜▲ ❩●❑◗❙▲❯❞➵◆❘●P✐❙● ✯ Ô æ ✰ ❭✡ ❵ ❚❑❨ ❩●❑◗❙▲ ❤▼P❚ P◗▲ ■●P▼❘ ❏▲▼❍P❯ ❘P ❑◗P❘●▲▼❛❍❘ ❍■ ➟❲▲❯ P❝▲▼ ▲❑❩● ■●P▼❘ ❍◗❘▲▼❝❑◆ P❤ ◆▲◗❙❘● ☛ú❞ ❋●▲ ■●P▼❘ ❏P■❍❘❍P◗ ❍◗❘●❍■ ❏P▼❘❤P◆❍P ❣❍◆◆ ▼▲➠✐❍▼▲ ❑ ❏❑❨❚▲◗❘ P❤ ✬✡ æ ✰ ❭✡ ❵ ❯P◆◆❑▼■ ❘P ❘●▲ ●P◆❯▲▼P❤ ❘●▲ ◆P◗❙ ❏P■❍❘❍P◗ ❭❘●▲ ❜❑◗❬❵ ▲❝▲▼❨ ❘❍❚▲ ❏▲▼❍P❯❞ ➵◗ ❍◗❝▲■❘P▼ ●P◆❯❍◗❙❑ ◆P◗❙ ❏P■❍❘❍P◗ ❍◗ ❘●❍■ P❏❘❍P◗ ❣❍◆◆ ❯▲❚❑◗❯ ❘●▲ ▼❍■❬❦❑❯❧✐■❘▲❯ ▼▲❘✐▼◗

✦✡ Ô ☞✡ Ý ✬✡✲

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❣●▲▼▲ ☞✡ ▼▲❏▼▲■▲◗❘■ ❘●▲ ❙▼P❣❘● P❤ ❘●▲ ❜❑◗❬♥■ ❏▼P❧▲❩❘ ❭❘●▲ ❩❑❏❍❘❑◆❙❑❍◗❵ ❑◗❯ ✬✡ ▼▲❏▼▲■▲◗❘■ ❍◗❘▲▼▲■❘ ▲❑▼◗❍◗❙■ P▼ ❘●▲ ❯❍❝❍❯▲◗❯ ■❘▼▲❑❚ ❘P❘●▲ ❯▲❏P■❍❘P▼■❞ ❋●▲ ❘P❘❑◆ ▼▲❘✐▼◗ ❤▼P❚ ●P◆❯❍◗❙ ❘●▲ ❏P▼❘❤P◆❍P P❝▲▼ ❑■●P▼❘ ❘❍❚▲ ❍◗❘▲▼❝❑◆ ☛ú ❍■ ❙❍❝▲◗ ❜❨☛✱ Ü ✬✡ æ ✰ ❭✡ ❵☛ú☛✱ Ô ☛æ ❭✡ ❵ Üæ ✰ ❭✡ ❵☛✡ Ü ✬✡ æ ✰ ❭✡ ❵☛ú ❭➘❵ã◗ ▲➠✐❑❘❍P◗ ❭➘❵ ❘●▲ ❘▲▼❚ ☛æ ✰ ❭✡ ❵✡ ❣❑■ P❚❍❘❘▲❯ ❜▲❩❑✐■▲ ❣▲ ❑■■✐❚▲❘●❑❘ ✯ Ô æ ✰ ❭✡ ❵ ❍■ ●▲◆❯ ❩P◗■❘❑◗❘ P❝▲▼ ☛ú❞ ❋P P❜❘❑❍◗ ❑◗ ▲❲❏▼▲■■❍P◗ ❤P▼☛æ ❣▲ ✐■▲ ã❘P♥■ ◆▲❚❚❑ñ

☛æ Ô æ ✰ ❭✡ ❵☛✡ Ý ❪Õæ ✰✰ ❭✡ ❵❭☛✡ ❵ä ❭ç❵❣●▲▼▲

æ ✰✰ ❭✡ ❵ Ô ☛äæ☛✡ ä❭☛✡ ❵ä Ô ❭☞✡ ☛ú❵ä Ý Õ☞✌☛ú☛✍ Ý ❭✌✡ ☛✍❵ä❑■ ☛✍ä Ô ☛ú ÷ó☛ ☛úä ✳ Ö

❭☛✡ ❵ä Ô ✌ä✡ ä☛✍ä❭☛✡ ❵ä Ô ✌ä✡ ä☛ú ❭➹❵

➺✐❜■❘❍❘✐❘❍◗❙ ▲➠✐❑❘❍P◗ ❭➹❵ ❍◗❘P ❭➘❵ ❙❍❝▲■ ❘●▲ ❘P❘❑◆ ▼❍■❬❦❤▼▲▲ ▼▲❘✐▼◗ P◗❘●▲ ❏P▼❘❤P◆❍Pñ ❪Õæ ✰✰ ❭✡ ❵❭☛✡ ❵ä Ü ✬✡ æ ✰ ❭✡ ❵☛ú ❭❫❵❑❙❑❍◗ ■✐❜■❘❍❘✐❘❍◗❙ ▲➠✐❑❘❍P◗ ❭➹❵ ❍◗❘P ❭❫❵ ❨❍▲◆❯■

❪Õ✌ä✡ äæ ✰✰ ❭✡ ❵☛ú Ü ✬✡ æ ✰ ❭✡ ❵☛ú ❭❪Ö❵ã◗ ▲➠✐❍◆❍❜▼❍✐❚ ▲➠✐❑❘❍P◗ ❭❫❵ ❚✐■❘ ▲➠✐❑◆ ▲➠✐❑◆ ❘●▲ ▼❍■❬❦❤▼▲▲ ▼▲❘✐▼◗ ❍◗❘●▲ ❚❑▼❬▲❘❛ ❘●❑❘ ❍■

þ✱☛ú Ô þ ✂æ ❭✡ ❵ Üæ ✰ ❭✡ ❵✡ ✄☛ú❪Õ✌ä✡ äæ ✰✰ ❭✡ ❵☛ú Ü ✬✡ æ ✰ ❭✡ ❵☛ú Ô þ ✂æ ❭✡ ❵ Üæ ✰ ❭✡ ❵✡ ✄☛ú

Page 10: Capital Investment as Real Options: A Note on Dixit ... · Munich Personal RePEc Archive Capital Investment as Real Options: A Note on Dixit-Pindyck Model. Termos, Ali American University

❱❍❝❍❯❍◗❙ ❘●▼P✐❙● ❜❨ ☛ú ❑◗❯ ▼▲❑▼▼❑◗❙❍◗❙ ❨❍▲◆❯■ ❘●▲ ❯❍❢▲▼▲◗❘❍❑◆ ▲➠✐❑❘❍P◗❘●❑❘ æ ❭✡ ❵ ❚✐■❘ ■❑❘❍■❤❨

❪Õ✌ä✡ äæ ✰✰ ❭✡ ❵ Ý ❭þ Ü ✬❵✡ æ ✰ ❭✡ ❵ Ü þæ ❭✡ ❵ Ô Ö ❭❪❪❵æ ❭✡ ❵ ❚✐■❘ ❑◆■P ■❑❘❍■❤❨ ❘●▲ ❤P◆◆P❣❍◗❙ ❜P✐◗❯❑▼❨ ❩P◗❯❍❘❍P◗■ñ

æ ❭Ö❵ Ô Ö ❭❪Õ❵æ ❭✡ ✗❵ Ô ✡ ✗ Ü ✎ ❭❪×❵

æ ❭✡ ✗❵ Ô ❪ ❭❪❴❵➵❙❑❍◗ ✡ ✗ ▼▲❏▼▲■▲◗❘■ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘ ❑❘ ❣●❍❩● ❍❘ ❍■ P❏❘❍❚❑◆ ❘P❍◗❝▲■❘❞ä❖P◗❯❍❘❍P◗ ❭❪Õ❵ ■❘❑❘▲■ ❘●❑❘ ❘●▲ P❏❘❍P◗ ❘P ❍◗❝▲■❘ ❣❍◆◆ ❜▲ P❤ ◗P ❝❑◆✐▲❣●▲◗ ✡ Ô Ö❞ ✵➠✐❑❘❍P◗ ❭❪×❵ ❍■ ❘●▲ ➩➾➯➻➨✶➚➾➲➽✷➦➼➪ ❩P◗❯❍❘❍P◗❛ ❘●❑❘ ❍■✐❏P◗ ❍◗❝▲■❘❍◗❙❛ ❘●▲ ❜❑◗❬ ▼▲❩▲❍❝▲■ ❑ ◗▲❘ ❏❑❨P❢ P❤ ✡ ✗ Ü ✎❞ ✸▲❣▼❍❘❍◗❙❭❪×❵ ❑■ ✡ ✗ Ü æ ❭✡ ✗❵ Ô ✎ ❍❚❏◆❍▲■ ❘●❑❘ ❣●▲◗ ❘●▲ ❜❑◗❬ ❍◗❝▲■❘■ ❍◗ ❘●▲❏▼P❧▲❩❘❛ ❍❘ ❙▲❘■ ❘●▲ ❏▼P❧▲❩❘ ❝❑◆✐▲ ✡ ❛ ❜✐❘ ❙❍❝▲■ ✐❏ ❘●▲ P❏❏P▼❘✐◗❍❘❨ ❘P❍◗❝▲■❘ æ ❭✡ ❵❞ ❋●▲ ❩▼❍❘❍❩❑◆ ❝❑◆✐▲ ✡ ✗ ❍■ P❜❘❑❍◗▲❯ ❣●▲◗ ❘●❍■ ◗▲❘ ❙❑❍◗✡ ✗ Ü æ ❭✡ ✗❵ ▲➠✐❑◆■ ❘●▲ ❯❍▼▲❩❘ ❩P■❘ P❤ ❍◗❝▲■❘❚▲◗❘ ✎❞ ã◗ P❘●▲▼ ❣P▼❯■❛❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘ ✡ ✗ ❍■ ■▲❘ ❘P ▲➠✐❑◆ ❘●▲ ❯❍▼▲❩❘ ❩P■❘ ✎ ❏◆✐■ ❘●▲P❏❏P▼❘✐◗❍❘❨ ❩P■❘ æ ❭✡ ✗❵❞✵➠✐❑❘❍P◗ ❭❪❴❵ ❍■ ❘●▲ ➫➚✹✹➲✷✶✺➾➫➲➦➼➪ ❩P◗❯❍❘❍P◗❞ ❋●❑❘ ❍■❛ ❍❤ æ ❭✡ ❵ ❣▲▼▲◗P❘ ❩P◗❘❍◗✐P✐■ ❑◗❯ ■❚PP❘● ❑❘ ❘●▲ ❩▼❍❘❍❩❑◆ ▲❲▲▼❩❍■▲ ❏P❍◗❘ ✡ ✗ ❛ ❍❘ ❍■❜▲❘❘▲▼ ❤P▼ ❘●▲ ❜❑◗❬ ❘P ❣❑❍❘ ✻ú ❘P P❜■▲▼❝▲ ❘●▲ ◗▲❲❘ ■❘▲❏ P❤ ✡ ❞ ❋P■P◆❝▲ ❤P▼ æ ❭✡ ❵ ❣▲ ❚✐■❘ ■P◆❝▲ ▲➠✐❑❘❍P◗ ❭❪❪❵ ■✐❜❧▲❩❘ ❘P ❘●▲ ❜P✐◗❯❑▼❨❩P◗❯❍❘❍P◗■ ❭❪Õ❵❛ ❭❪×❵❛ ❑◗❯ ❭❪❴❵❞ ➶❩❱P◗❑◆❯ ❑◗❯ ➺❍▲❙▲◆ ❭❪❫➹➘❵ ■✐❙❙▲■❘▲❯❘●❑❘ ❘●▲ ■P◆✐❘❍P◗ ❘●❑❘ ■❑❘❍■➟▲■ ❩P◗❯❍❘❍P◗ ❭❪Õ❵ ❚✐■❘ ❘❑❬▲ ❘●▲ ❤P▼❚ñ

æ ❭✡ ❵ Ô ✼✡ ✽ ❭❪Ø❵❖P◗❯❍❘❍P◗■ ❭❪×❵ ❑◗❯ ❭❪❴❵ ❩❑◗ ❜▲ ✐■▲❯ ❘P ■P◆❝▲ ❤P▼ ✼ ❣●❍❩● ❍■ ❑ ❩P◗■❘❑◗❘❘P ❜▲ ❯▲❘▲▼❚❍◗▲❯❛ ❑◗❯ ❤P▼ ❘●▲ P❏❘❍❚❑◆ ❝❑◆✐▲ ✡ ✗ ❞ ✾✿❀①ê t✉❻ ❁①qq ëìí❂➈ï q❸①➋ s❸ts s❸③ qt⑥③ q①❿⑨sr①✉ rq ①❼str✉③❻ ⑨qr✉➌ ❻❶✉t⑥r⑧ ✈✇①➌✇t⑥❃⑥r✉➌ t✉t❿❶qrq r② ➋③ tqq⑨⑥③ s❸ts t❿❿ t➌③✉sq t✇③ ✇rq➊ ✉③⑨s✇t❿❽❄é❸③ ➌③✉③✇t❿ q①❿⑨sr①✉ s① ③❅⑨tsr①✉ ëììï tq q⑨➌➌③qs③❻ ❼❶ ❆⑧⑤①✉t❿❻ t✉❻ ❷r③➌❿③ st➊③q s❸③②①✇⑥ ❇ ë❈ ï ❉ ❊è❈ ❋●❍❊✿❈ ❋■ ❼⑨s s❸③ ❼①⑨✉❻t✇❶ ⑧①✉❻rsr①✉ ëì➇ï r⑥✈❿r③q s❸ts ë❏❑▲ï❈ ❇ ▼ë❈ ï ❉

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❖ ❍■ ❑ ❬◗P❣◗ ❩P◗■❘❑◗❘ ❣●❍❩● ❯▲❏▲◗❯■ P◗ ❘●▲ ❏❑▼❑❚▲❘▲▼■ ✌❛ þ❛ ❑◗❯ ✬P❤ ▲➠✐❑❘❍P◗ ❭❪❪❵ ❣●▲▼▲ ❖ ✭ ❪❞❋P P❜❘❑❍◗ ❝❑◆✐▲■ ❤P▼ ✼ ❑◗❯ ✡ ✗❛ ❣▲ ■✐❜■❘❍❘✐❘▲ ▲➠✐❑❘❍P◗ ❭❪Ø❵❍◗❘P ❭❪×❵❑◗❯ ❭❪❴❵ ■P ❘●❑❘

æ ❭✡ ✗❵ Ô ✼✡ ✗✽ Ô ✡ ✗ Ü ✎✼ Ô ✡ ✗ Ü ✎✡ ✗✽ ❭❪➘❵❡❨ ▲➠✐❑❘❍P◗ ❭❪❴❵❛ æ ✰ ❭✡ ✗❵ Ô ❖✼✡ ✗✽✔Ñ Ô ❪❞ ♠■❍◗❙ ▲➠✐❑❘❍P◗ ❭❪➘❵ ❘P■✐❜■❘❍❘✐❘▲ ❤P▼ ✼ ❣▲ P❜❘❑❍◗ñ

❖❭✡ ✗ Ü ✎✡ ✗ ❵ Ô ❪✡ ✗ Ô ❭ ❖❖ Ü ❪❵✎ ❭❪ç❵

➺✐❜■❘❍❘✐❘❍◗❙ ▲➠✐❑❘❍P◗ ❭❪ç❵❍◗❘P ❭❪➘❵ ❘P P❜❘❑❍◗ ❑ ❝❑◆✐▲ ❤P▼ ✼ ❑■P✼ Ô ❭❖ Ü ❪❵✽✔Ñ❖✽✎✽✔Ñ ❭❪➹❵

✵➠✐❑❘❍P◗■ ❭❪Ø❵❛❭❪ç❵❛ ❑◗❯ ❭❪➹❵ ❙❍❝▲ ❘●▲ ■P◆✐❘❍P◗ ❘P ❘●▲ ❩▼❍❘❍❩❑◆ ❝❑◆✐▲■P❤ ✡ ✗ ❑◗❯ æ ❭✡ ✗❵❞◗ ❘ ➝➜✞➒❙➔➜❚➜➔✟➙→➒ ➝→➣➜➞❯▲P❚▲❘▼❍❩ ❡▼P❣◗❍❑◗ ❚P❘❍P◗ ❚P❯▲◆■ ●❑❝▲ ❜▲▲◗ ❩▼❍❘❍❩❍➤▲❯ ❘●▲P▼▲❘❍❩❑◆◆❨P◗ ❘●▲ ❙▼P✐◗❯ ❘●❑❘ ❘●▲■▲ ❏▼P❩▲■■▲■ ❯P ◗P❘ ❯❍❝▲▼❙▲ P❝▲▼ ❘❍❚▲❞ ❋●❍■❍❚❏◆❍▲■ ❘●❑❘ ➟▼❚■ ❤❑❩❍◗❙ ❘●❍■ ■P▼❘ P❤ ❏▼❍❩▲ P▼ ❩❑■● ➷P❣ ❏▼P❩▲■■ ❩❑◗ ▲❑▼◗❍◗➟◗❍❘▲ ❏▼P➟❘ ❭➶▲❘❩❑◆❤ ❑◗❯ ➸❑■■▲❘❛ ❪❫❫Ø❵❞ ➶▲❑◗❦▼▲❝▲▼■❍P◗ ❏▼▲❩▲■■▲■ ❍◗❚❑◗❨ ❣❑❨■ ❏▼P❝❍❯▲ ❑ ❚P▼▲ ❏◆❑✐■❍❜◆▲ ❑❏❏▼P❲❍❚❑❘❍P◗ ❘P ❘●▲ ■❘P❩●❑■❘❍❩◗❑❘✐▼▲ P❤ ❑ ❏▼P❧▲❩❘ P▼ ❑◗ ❑■■▲❘ ❝❑◆✐▲❞ ❋●❍■ ■▲❩❘❍P◗ ❍◗❘▼P❯✐❩▲■ ❘●▲❚▲❑◗❦▼▲❝▲▼■❍P◗ ❏▼P❩▲■■ ❜❑■▲❯ P◗ ❘●▲ ❯▲▼❍❝❑❘❍P◗■ ❚❑❯▲ ❍◗ ❘●▲ ❏▼▲❝❍P✐■■▲❩❘❍P◗❞❱❲ s❸⑨q ❊✿ ❉ ❲④ s❸③✇③②①✇③ s❸③ q①❿⑨sr①✉ rq ⑧①✉❳✉③❻ s① s❸③ ②①✇⑥ ❇ ë❈ ï ❉ ❊❈ ❋● ①✇ ❹⑨qs❇ ë❈ ï ❉ ❊❈ ❋ ❽❨❷③③ t✈✈③✉❻rê ②①✇ q①❿⑨sr①✉❽❩é❸③ t✉t❿❶qrq ❸③✇③ ②①❿❿①➋q ⑤rêrs t✉❻ ➃r✉❻❶⑧➊ ëìííîï❽

➏❬

Page 12: Capital Investment as Real Options: A Note on Dixit ... · Munich Personal RePEc Archive Capital Investment as Real Options: A Note on Dixit-Pindyck Model. Termos, Ali American University

Ú▲❘ ✡ ◗P❣ ❤P◆◆P❣ ❘●▲ ❚▲❑◗❦▼▲❝▲▼■❍P◗ ❏▼P❩▲■■ ❙❍❝▲◗ ❜▲◆P❣❛ ❣●❍❩● ❍■❬◗P❣◗ ❑■ ❭➨✹➚➨➲➧➦➽ ❪➧➼➫➲➨➦➼✶❫✷➯➨➼➭➨➽❴ ✺➧✹➽➨➫➫

☛✡ Ô ❵❭❛✡ Ü ✡ ❵✡ ☛ú Ý ✌✡ ☛✍ ❭❪❫❵❣●▲▼▲ ❵ ❍■ ❘●▲ ▼❑❘▲ P❤ ❘●▲ ▼▲❝▲▼■❍P◗ ❘P ❘●▲ ❚▲❑◗❞ ➵❙❑❍◗❛ ✌ ❍■ ❘●▲ ■❘❑◗❦❯❑▼❯ ❯▲❝❍❑❘❍P◗ ❑◗❯ ☛✍ ❍■ ❘●▲ ❍◗❩▼▲❚▲◗❘ P❤ ❑ ❥❍▲◗▲▼ ❏▼P❩▲■■❞ ✵➠✐❑❘❍P◗❭❪❫❵ ❍❚❏◆❍▲■ ❘●❑❘ ❘●▲ ❩✐▼▼▲◗❘ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘ ❍■ ❬◗P❣◗❛ ❜✐❘ ❤✐❘✐▼▲❝❑◆✐▲■ ❑▼▲ ◆P❙◗P▼❚❑◆◆❨ ❯❍■❘▼❍❜✐❘▲❯ ❣❍❘● ❑ ❝❑▼❍❑◗❩▲ ❘●❑❘ ❙▼P❣■ ◆❍◗▲❑▼◆❨❣❍❘● ❘❍❚▲❞ ❋●▲ ❤✐❘✐▼▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘ ❍■ ❑◆❣❑❨■ ✐◗❩▲▼❘❑❍◗❞ ➥✐▼❘●▲▼❣▲ ❑■■✐❚▲ ❘●❑❘ ❘P❘❑◆ ▼▲❝▲◗✐▲ ❤▼P❚ ❘●▲ ❏▼P❧▲❩❘ ❩P❝▲▼■ ❍❘■ ❑❝▲▼❑❙▲ ❘P❘❑◆❩P■❘❛ ■P ❘●▲ ❜❑◗❬ ❣❍◆◆ ◗P❘ ▲❍❘●▲▼ ■●✐❘ ❯P❣◗ P▼ ❑❜❑◗❯P◗ ❘●▲ ❏▼P❧▲❩❘ ❍◗❘●▲ ■●P▼❘❦▼✐◗ ❑❘ ◆▲❑■❘❞ ➵◗P❘●▲▼ ❑✐❲❍◆❍❑▼❨ ❑■■✐❚❏❘❍P◗ ❍■ ◗▲▲❯▲❯ ❘P ❏▼▲❦■▲▼❝▲ ❘●▲ ❑■■✐❚❏❘❍P◗ P❤ ◆P❙◗P▼❚❑◆❍❘❨❛ ❍■ ❘●❑❘ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘❛✡ ❍■ ❑◆❣❑❨■ ❏P■❍❘❍❝▲❞❋●▲ ▲❲❏▲❩❘▲❯ ❏▲▼❩▲◗❘❑❙▲ ▼❑❘▲ P❤ ❩●❑◗❙▲ ❍◗ ✡ ❍■

❪☛ú✒✂☛✡✡ ✄ Ô ❪☛ú✒✂❵❭❛✡ Ü ✡ ❵✡ ☛ú Ý ✌✡ ☛✍✡ ✄Ô ❵❭❛✡ Ü ✡ ❵

❑■ ✒✂☛✍✄ Ô Ö❋●▲ ▲❲❏▲❩❘▲❯ ❑❜■P◆✐❘▲ ▼❑❘▲ P❤ ❩●❑◗❙▲ ❍■ ❙❍❝▲◗ ❜❨

❪☛ú✒✂☛✡ ✄ Ô ❪☛ú✒✂❭❵❭❛✡ Ü ✡ ❵✡ ☛ú Ý ✌✡ ☛✍✄❪☛ú✒✂☛✡ ✄ Ô ❵❭❛✡ Ü ✡ ❵✡

❪☛ú✒✂☛✡ ✄ Ô ❵ ❛✡ ✡ Ü ❵✡ ä Ô Ö❍❤ ✡ Ô Ö ýþ ✡ Ô ❛✡ã❘ ●❑■ ❑ ❚❑❲❍❚✐❚ ❑❘❜ ✡ Ô ❛✡Õ❝é❸ts rq ①❼str✉③❻ ❼❶ q③ssr✉➌ ❞❡❢ ❣❤ ❤ ✐❢❤ ■❥❞❤ ❉ ❲❦

➏➏

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❋●❍■ ❏❑▼❘❍❩✐◆❑▼ ❏▼P❩▲■■ ❯P▲■ ●❑❝▲ ❑◗ ❑◗❑◆❨❘❍❩❑◆ ■P◆✐❘❍P◗ ❘P ❘●▲ P❏❘❍❚❑◆❍◗❝▲■❘❚▲◗❘ ❏▼P❜◆▲❚❞ ❡❨ ■✐❜■❘❍❘✐❘❍◗❙ ❘●▲ ❝❑◆✐▲ P❤ ✬❭✡ ❵ Ô ✦Ü❵❭❛✡ Ü✡ ❵❍◗❘P ▲➠✐❑❘❍P◗ ❭❪❪❵ ❣▲ P❜❘❑❍◗ ❘●▲ ❤P◆◆P❣❍◗❙ ■▲❩P◗❯ P▼❯▲▼ ❯❍❢▲▼▲◗❘❍❑◆▲➠✐❑❘❍P◗ñ

❪Õ✌ä✡ äæ ✰✰ ❭✡ ❵ Ý ✂þ Ü✦ Ý ❵❭❛✡ Ü ✡ ❵✄✡ æ ✰ ❭✡ ❵ Ü þæ ❭✡ ❵ Ô Ö ❭ÕÖ❵➵❙❑❍◗ ❘●❍■ ▲➠✐❑❘❍P◗ ❚✐■❘ ■❑❘❍■❤❨ ❘●▲ ❜P✐◗❯❑▼❨ ❩P◗❯❍❘❍P◗■ ❭❪Õ❵❛ ❭❪×❵❛❑◗❯ ❭❪❴❵❞ ➡P❣ ❑■■✐❚▲ ❘●❑❘ ❘●▲ ■❏▲▲❯ P❤ ❘●▲ ❚▲❑◗❦▼▲❝▲▼■❍P◗ ❏▼P❩▲■■❍■ ❑ ◆❍◗▲❑▼ ❤✐◗❩❘❍P◗ P❤ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❍◗❝▲■❘❚▲◗❘ ❏▼P❧▲❩❘ ✡ ❞ ➵ ▼▲◆❑❦❘❍P◗■●❍❏ P❤ ❘●▲ ❤P▼❚ ❵ Ô ❧Ñ Ý ❧ä✡ ❭Õ❪❵❧ä ❍■ ❘●▲ ♠❑❩❩▲◆▲▼❑❘❍P◗➴ ❏❑▼❑❚▲❘▲▼ ❘●❑❘ ❚▲❑■✐▼▲■ ❘●▲ ■❏▲▲❯ P❤ ❑❯❦❧✐■❘❚▲◗❘ ❜❨ ❣●❍❩● ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘ ❩P◗❝▲▼❙▲■ ❘P ❘●▲ ◆P◗❙❦▼✐◗❚▲❑◗❞ ➺✐❜■❘❍❘✐❘❍◗❙ ▲➠✐❑❘❍P◗ ❭Õ❪❵❍◗❘P ❭ÕÖ❵ ❨❍▲◆❯■❪Õ✌ä✡ äæ ✰✰ ❭✡ ❵Ý ✂þ Ü✦Ý❭❧ÑÝ❧ä✡ ❵❭❛✡ Ü✡ ❵✄✡ æ ✰ ❭✡ ❵Üþæ ❭✡ ❵ Ô Ö ❭ÕÕ❵

♥◗▲ ❑❯❝❑◗❘❑❙▲ P❤ ❑◆◆P❣❍◗❙ ❤P▼ ❑ ❘●❍▼❯ ❚P❚▲◗❘ ❏▼P❩▲■■ ❤P▼ ❯❍■❘▼❍❜✐❘❍P◗P❤ ✡ ❍■ ❘P ❙❑❍◗ ❍◗■❍❙●❘■ ❑❜P✐❘ ❘●▲ ■❬▲❣◗▲■■ P❤ ❘●❍■ ❯❍■❘▼❍❜✐❘❍P◗ ❑▼P✐◗❯❘●▲ ❚▲❑◗❞♦

♣➄✉③ ⑥③tq⑨✇③ ①② q➊③➋✉③qq ⑧t✉ ❼③ ❻③❳✉③❻ tq q❡❤ ✐❣❤ ❥rsr ❦➏➬

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t➱➮ ✛✃ ❮✃❮✜✉❒✢✈❮✜ ★✧✜❐❒✢✧✃❋●▲ ❑◗❑◆❨❘❍❩❑◆ ■P◆✐❘❍P◗ ❘P ▲➠✐❑❘❍P◗ ❭ÕÕ❵ ❍■ ❏▼▲■▲◗❘▲❯ ❍◗ ❯▲❘❑❍◆ ❍◗ ❘●▲❑❏❏▲◗❯❍❲❞ ➵■ ❯▲❚P◗■❘▼❑❘▲❯❛ ❘●▲ ■P◆✐❘❍P◗ ❍■ ▲❲❏▼▲■■▲❯ ❜❨ ❘●▲ ❤P◆◆P❣❍◗❙▲➠✐❑❘❍P◗ñ

✡ ✽✇❭✡ ❵✂❪Õ✌ä❖❭❖ Ü ❪❵ Ý ❭þ Ü✦ Ý ❵ ❛✡ ❵❖ Üþ✄Ý✡ ✽①Ñ✂❪Õ✌ä✡ ✇✰✰ ❭✡ ❵Ý ❭✌ä❖ÝþÜ✦Ý❵ ❛✡ Ü❵✡ ❵✇✰ ❭✡ ❵Ü❵❖✇❭✡ ❵✄ Ô Ö ❭Õ×❵✵➠✐❑❘❍P◗ ❭Õ×❵ ❚✐■❘ ●P◆❯ ❤P▼ ❑◗❨ ❝❑◆✐▲ P❤ ✡ ❛ ❘●✐■ ❘●▲ ❘❣P ❚❑❍◗ ❜▼❑❩❬❦▲❘▲❯ ❘▲▼❚■ ❍◗ ❘●▲ ▲➠✐❑❘❍P◗ ❚✐■❘ ▲➠✐❑◆ ➤▲▼P❞ ❋●▲ ➠✐❑◗❘❍❘❨ ❍◗ ❘●▲ ➟▼■❘

❜▼❑❩❬▲❘ ▼▲❏▼▲■▲◗❘■ ❑ ②➻➾③➧➾➲➦➽ ➨②➻➾➲➦✹➼ ❣●❍❩● ●❑■ ❘❣P ■P◆✐❘❍P◗■ ❤P▼ ❖❞♥◗▲ ❍■ ❏P■❍❘❍❝▲ ❑◗❯ ❘●▲ P❘●▲▼ ❍■ ◗▲❙❑❘❍❝▲❞ ♥◗◆❨ ❘●▲ ❏P■❍❘❍❝▲ ■P◆✐❘❍P◗■❑❘❍■➟▲■ ❘●▲ ❜P✐◗❯❑▼❨ ❩P◗❯❍❘❍P◗ æ ❭Ö❵ Ô Ö❞➺▲❘❘❍◗❙ ❘●▲ ➠✐❑❯▼❑❘❍❩ ▲➠✐❑❘❍P◗ ❘P ➤▲▼P ❑◗❯ ■P◆❝❍◗❙ ❤P▼ ❖ ❑■ ❤P◆◆P❣■ñ

❪Õ✌ä❖❭❖ Ü ❪❵ Ý ❭þ Ü✦ Ý ❵ ❛✡ ❵❖ Üþ Ô Ö❪Õ✌ä❖ä Ü ❭❪Õ✌ä Ü þ Ý ✦ Ü ❵ ❛✡ ❵❖ Üþ Ô Ö ❭Õ❴❵

❋●▲ ❏P■❍❘❍❝▲ ■P◆✐❘❍P◗ ❘P ▲➠✐❑❘❍P◗ ❭Õ❴❵ ❍■ ❙❍❝▲◗ ❜❨ ❖ Ô ✔õ①④õ⑤✔Pô⑥äô ❛❘●▲◗

❖ Ô ❪Õ Ý ❭✦ Üþ Ü ❵ ❛✡ ❵✌ä Ý ⑦✂❭þ Ü✦ Ý ❵ ❛✡ ❵✌ä Ü ❪Õ✄ä Ý Õþ✌ä ❭ÕØ❵❋●▲ ■▲❩P◗❯ ❜▼❑❩❬▲❘▲❯ ❘▲▼❚ P❤ ▲➠✐❑❘❍P◗ ❭Õ×❵ ❍■ ■▲❘ ❘P ▲➠✐❑◆ ➤▲▼Pñ

❪Õ✌ä✡ ✇✰✰ ❭✡ ❵ Ý ❭✌ä❖ Ýþ Ü✦ Ý ❵ ❛✡ Ü ❵✡ ❵✇✰ ❭✡ ❵ Ü ❵❖✇❭✡ ❵ Ô Ö ❭Õ➘❵❋P ❘▼❑◗■❤P▼❚ ▲➠✐❑❘❍P◗ ❭Õ➘❵ ❍◗❘P ❑ ■❘❑◗❯❑▼❯ ❤P▼❚ ▲➠✐❑❘❍P◗ ❣▲ ❏▼P❦

❩▲▲❯ ❑■ ❤P◆◆P❣■ñÚ▲❘ ✑ Ô ä⑧✥✪⑤ ❑◗❯ ✇❭✡ ❵ Ô ü❭✑❵ ❘●▲▼▲❤P▼▲❛

✇✰ ❭✡ ❵ Ô ü✰ ❭✑❵✑✰ Ô ü✰ ❭✑❵❭Õ❵✌ä ❵ ❭Õç❵✇✰✰ ❭✡ ❵ Ô ü✰✰ ❭✑❵✑✰ ❭Õ❵✌ä ❵ Ô ü✰✰ ❭✑❵❭Õ❵✌ä ❵ä ❭Õ➹❵

➏ð

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➺✐❜■❘❍❘✐❘❍◗❙ ▲➠✐❑❘❍P◗■ ❭Õç❵ ❑◗❯ ❭Õ➹❵ ❜❑❩❬ ❍◗❘P ❭Õ➘❵ ❨❍▲◆❯■ ⑨✑ü✰✰ ❭✑❵ Ý ❭⑩ Ü✑❵ü✰ ❭✑❵ Ü ❖ü❭✑❵ Ô Ö ❭Õ❫❵

❣●▲▼▲ ⑩ Ô Õ❖ Ý Õ❭þ Ü✦ Ý ❵ ❛✡ ❵✌ä✵➠✐❑❘❍P◗ ❭Õ❫❵ ❍■ ❬◗P❣◗ ❑■ ❶✐❚❚▲▼♥■ ▲➠✐❑❘❍P◗❞ ã❘■ ■P◆✐❘❍P◗ ❍■ ❙❍❝▲◗❜❨ ❘●▲ ❩P◗➷✐▲◗❘ ●❨❏▲▼❙▲P❚▲❘▼❍❩ ❤✐◗❩❘❍P◗ ❷✂✑Û❖Û⑩❭❖❵✄❛ ❣●❍❩● ●❑■ ❘●▲❤P◆◆P❣❍◗❙ ■▲▼❍▲■ñ❷✂✑Û❖Û⑩❭❖❵✄ Ô ❪Ý ❖⑩✑ Ý ❖❭❖ Ý ❪❵✑ä⑩❭⑩ Ý ❪❵Õ❸ Ý ❖❭❖ Ý ❪❵❭❖ Ý Õ❵✑✾⑩❭⑩ Ý ❪❵❭⑩ Ý Õ❵×❸ Ý ❹ ❹ ❹ ❭×Ö❵

❋●❍■ ❩P◗➟▼❚▲❯ ❘●❑❘ ❘●▲ ■P◆✐❘❍P◗ ❘P ▲➠✐❑❘❍P◗ ❭ÕÖ❵ ❍■ ❍◗❯▲▲❯ P❤ ❘●▲ ❤P▼❚❙❍❝▲◗ ❜❨

æ ❭✡ ❵ Ô ✼✡ ✽✇❭✡ ❵❑■ ❍◗❯❍❩❑❘▲❯ ❍◗ ❘●▲ ❑❏❏▲◗❯❍❲❞ ❋●▲◗ ❘●▲ ■P◆✐❘❍P◗ ❘❑❬▲■ ❘●▲ ❤P▼❚ñ

æ ❭✡ ❵ Ô ✼✡ ✽❷ ❭Õ❵✌ä ✡ ❺❖Û⑩❵ ❭×❪❵➵❙❑❍◗ ✼ ❍■ ❑ ❩P◗■❘❑◗❘ ❘P ❜▲ ❯▲❘▲▼❚❍◗▲❯❞ ✼ ❑◗❯ ✡ ✗ ❚✐■❘ ❜▲ ■P◆❝▲❯◗✐❚▲▼❍❩❑◆◆❨ ✐■❍◗❙ ❘●▲ ❜P✐◗❯❑▼❨ ❩P◗❯❍❘❍P◗■ ❭❪×❵ ❑◗❯ ❭❪❴❵❞❋P ❩●▲❩❬ ❣●▲❘●▲▼ ❘●▲▼▲ ▲❲❍■❘ ❑◗ ❑◗❑◆❨❘❍❩❑◆ ■P◆✐❘❍P◗ ❘P ▲➠✐❑❘❍P◗ ❭ÕÕ❵❣▲ ■✐❜■❘❍❘✐❘▲ ❵ Ô ❧Ñ Ý ❧ä✡ ❍◗❘P ❭Õ➘❵

❪Õ✌ä✡ ✇✰✰ ❭✡ ❵ Ý ✂✌ä❖ Ý þ Ü✦ Ý ❧Ñ ❛✡ Ü ❭❧Ñ Ü❧ä ❛✡ ❵✡ Ü❧ä✡ ä✄✇✰ ❭✡ ❵Ü✂❭❧Ñ Ü❧ä ❛✡ ❵❖ Ý ❧ä❖✡ ✄✇❭✡ ❵ Ô Ö

➥❑❩❘P▼❍◗❙ P✐❘ ❧Ñ ❑◗❯ ▼▲❑▼▼❑◗❙❍◗❙ ❘▲▼❚■ ❨❍▲◆❯■ ❘●▲ ❤P◆◆P❣❍◗❙ ■▲❩P◗❯P▼❯▲▼ ❯❍❢▲▼▲◗❘❍❑◆ ▲➠✐❑❘❍P◗❻✑ü✰✰ ❭✑❵ Ý ✂⑩ Ü ❭❪ Ü ❼Ñ❵✑ Ü ❼ä✑ä✄ü✰ ❭✑❵ Ü ❭❪ Ü ❼Ñ Ü ❼ä✑❵❖ü❭✑❵ Ô Ö ❭×Õ❵

❣●▲▼▲ ⑩ Ô Õ❖ Ý Õ✂❭þ Ü✦ Ý ❭❧Ñ Ý ❧ä✡ ❵ ❛✡ ✄✌ä❽❷③③ t✈✈③✉❻rê ②①✇ q①❿⑨sr①✉❾♣➌tr✉④ q①❿⑨sr①✉ rq ✈✇①⑩r❻③❻ r✉ s❸③ t✈✈③✉❻rê➏☎

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❼Ñ Ô ❧ä❧Ñ ❛✡❼ä Ô ✌ä❧äÕ❧äÑ❋●▲▼▲ ❍■ ◗P ❯❍▼▲❩❘ ❑◗❑◆❨❘❍❩❑◆ ■P◆✐❘❍P◗ ❘P ▲➠✐❑❘❍P◗ ❭×Õ❵ ❑■ ❍❘ ❣❑■ ❘●▲

❩❑■▲ ❤P▼ ▲➠✐❑❘❍P◗ ❭Õ❫❵ ❣●▲▼▲ ❘●▲ ■P◆✐❘❍P◗ ❣❑■ ❑❏❏▼P❲❍❚❑❘▲❯ ❜❨ ❘●▲❩P◗➷✐▲◗❘ ●❨❏▲▼❙▲P❚▲❘▼❍❩ ▲❲❏❑◗■❍P◗❞ ❋●▲▼▲❤P▼▲❛ ❑ ◗✐❚▲▼❍❩❑◆ ■P◆✐❘❍P◗❚✐■❘ ❜▲ ▲■❘❑❜◆❍■●▲❯ ▼▲❑◆❍➤❍◗❙ ❘●❑❘ ❘●▲ ◗✐❚▲▼❍❩❑◆ ■P◆✐❘❍P◗ P❤ ❭×Õ❵ ❍■▲➠✐❍❝❑◆▲◗❘ ❘P ❘●▲ ◗✐❚▲▼❍❩❑◆ ■P◆✐❘❍P◗ P❤ ❭ÕÕ❵❞ ❋●▲ ❤P◆◆P❣❍◗❙ ■✐❜■▲❩❘❍P◗❏▼P❝❍❯▲■ ❑◗ ▲◆❑❜P▼❑❘▲ ❯❍■❩✐■■❍P◗ P❤ ❘●▲ ◗✐❚▲▼❍❩❑◆ ■P◆✐❘❍P◗ ❘P ❘●▲ ❜❑◗❬❍◗❝▲■❘❚▲◗❘ ❏▼P❜◆▲❚ ❣❍❘● ❝❑▼❍P✐■ ■❩▲◗❑▼❍P■ ❑■ ●P❣ ❘●▲ ■❏▲▲❯ P❤ ❩P◗❦❝▲▼❙▲◗❩▲ P▼ ❘●▲ ❑❩❩▲◆▲▼❑❘❍P◗ P❤ ❘●▲ ❚▲❑◗❦▼▲❝▲▼❘❍◗❙ ❏▼P❩▲■■ ❑❢▲❩❘■ ❘●▲❝❑◆✐▲ P❤ ❘●▲ P❏❘❍P◗ ❤P▼ ❘●▲ ❜❑◗❬ ❘P ❍◗❝▲■❘❞t➱✙ ✛ ✃❐✦✤✥✢✈❮✜ ★✧✜❐❒✢✧✃ã◗ ■P◆❝❍◗❙ ▲➠✐❑❘❍P◗ ❭ÕÕ❵ ◗✐❚▲▼❍❩❑◆◆❨❛ ❘●▲ ❏❑▼❑❚▲❘▲▼■ P❤ ❘●▲ ❚P❯▲◆■ ❑▼▲❙❍❝▲◗ ❘●▲ ❤P◆◆P❣❍◗❙ ❝❑◆✐▲■ñ ✎ Ô ❪❛ þ Ô ÖâÖ❴❛ ✦ Ô ÖâÖ➹❛ ❑◗❯ ✌ Ô ÖâÕ❞➵❙❑❍◗❛ ✎ ❍■ ❘●▲ ❑❚P✐◗❘ P❤ ❍◗❍❘❍❑◆ ❍◗❝▲■❘❚▲◗❘❞ ã❘ ❍■ ◗P▼❚❑◆❍➤▲❯ ❘P P◗▲❞þ ❍■ ❘●▲ ▼❍■❬❦❤▼▲▲ ❍◗❘▲▼▲■❘ ▼❑❘▲❞ ✦ ❍■ ❘●▲ ▲❲❏▲❩❘▲❯ ▼❑❘▲ P❤ ▼▲❘✐▼◗ P◗ ❘●▲❍◗❝▲■❘❚▲◗❘❛ ❑◗❯ ✌ ❍■ ❘●▲ ❚▲❑■✐▼▲ P❤ ✐◗❩▲▼❘❑❍◗❘❨❞ P❤ ❋●▲■▲ ❝❑◆✐▲■ ❑▼▲❣❍❯▲◆❨ ❑❯P❏❘▲❯ ❍◗ ❘●▲ ▼▲❑◆ P❏❘❍P◗■ ◆❍❘▲▼❑❘✐▼▲❞❋●▼▲▲ ❩❑■▲■ ❑▼▲ P❤ ❑ ■❏▲❩❍❑◆ ❍◗❘▲▼▲■❘❺ ❛✡ ✭ ✎❛ ❛✡ ✫ ✎❛ ❑◗❯ ❛✡ Ô ✎❞ ❋●▲■P◆✐❘❍P◗■ ❑▼▲ ❯▲❏❍❩❘▲❯ ❍◗ ❘●▲ ➟❙✐▼▲■ ❭❪❵❦❭❿❿❵ ❏▼P❝❍❯▲❯ ❍◗ ❘●▲ ❑❏❏▲◗❯❍❲❞➵◗❑◆❨➤❍◗❙ ❘●▲■▲ ➟❙✐▼▲■❛ ❣▲ ❩❑◗ ▼▲❚❑▼❬ ❘●▲ ❤P◆◆P❣❍◗❙ ▼▲■✐◆❘■❞

➀ ❋●▲ ◆❑▼❙▲▼ ❍■ ❛✡ ❛ ❘●▲ ◆❑▼❙▲▼ ❍■ æ ❭✡ ❵ ❑◗❯ ❘●▲ ●❍❙●▲▼ ❍■ ✡ ✗❛ ❘●▲❩▼❍❘❍❩❑◆ ❝❑◆✐▲ ❑❘ ❣●❍❩● ❍❘ ❍■ P❏❘❍❚❑◆ ❤P▼ ❘●▲ ❜❑◗❬ ❘P ❍◗❝▲■❘ ❭➥❍❙✐▼▲■❭❪❵❦❭×❵❵❞ Ú❑▼❙▲▼ ❛✡ ❍❚❏◆❍▲■ ❑ ●❍❙●▲▼ ▲❲❏▲❩❘▲❯ ▼❑❘▲ P❤ ❙▼P❣❘● P❤ ✡■P ❘●❑❘ ❘●▲ ❝❑◆✐▲ P❤ ❑◗ P❏❘❍P◗ ❘P ❍◗❝▲■❘ ✡ ❣❍◆◆ ❜▲❩P❚▲ ●❍❙●▲▼❞➀ ➥P▼ ❛✡ ✫ ✎❛ ❑ ◆❑▼❙▲▼ ❝❑◆✐▲ P❤ ❧ä ❛ ❘●▲ ❚▲❑◗ ▼▲❝▲▼■❍P◗ ❏❑▼❑❚▲❘▲▼❛▼▲❯✐❩▲■ ❘●▲ ❍◗❝▲■❘❚▲◗❘ P❏❏P▼❘✐◗❍❘❨ æ ❭✡ ❵ ❭➥❍❙✐▼▲■ ❭❴❵❦❭➘❵❵❞ ❋●❍■❑◆■P ❍❚❏◆❍▲■ ❘●❑❘ ❑ ◆❑▼❙▲▼ ❝❑◆✐▲ P❤ ❧ä ▼▲■✐◆❘■ ❍◗ ❑ ◆P❣▲▼ ❝❑◆✐▲ P❤✡ ✗ ❞

➏✏

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➀ ➥P▼ ❑ ◆❑▼❙▲ ❛✡ ❛ ❘●❑❘ ❍■ ❤P▼ ❛✡ ✘ ✎ ❑■ ❧ä ❙▲❘■ ◆❑▼❙▲▼❛ æ ❭✡ ❵ ❜▲❩P❚▲■❩P◗❩❑❝▲ ❤P▼ ■❚❑◆◆ ❝❑◆✐▲■ P❤ ✡ ❛ ■P ❧ä ▼❍■▲■ ▼❑❏❍❯◆❨❛ ❘●▲▼▲❤P▼▲ æ ❭✡ ❵▼❍■▲■ ▼❑❏❍❯◆❨ ❑■ ❣▲◆◆ ❤P▼ ❑ ■❚❑◆◆ ❝❑◆✐▲ P❤ ✡ ❞ ➸P❣▲❝▲▼❛ ❘●▲ ❝❑◆✐▲P❤ æ ❭✡ ❵ ❯❍❚❍◗❍■●▲■ ❣❍❘● ◆❑▼❙▲ ❧ä ❑■ ✡ ❙▲❘■ ◆❑▼❙▲▼ ❭➥❍❙✐▼▲■ ❭ç❵❦❭❿❿❵❵❞

➵■■✐❚❍◗❙ ❘●❑❘ ■❚❑◆◆ ❜❑◗❬■ ❍◗❝▲■❘ ❍◗ ▼▲◆❑❘❍❝▲◆❨ ■❚❑◆◆ ❏▼P❧▲❩❘■ ❑◗❯ ◆❑▼❙▲❜❑◗❬■ ❍◗❝▲■❘ ❍◗ ◆❑▼❙▲ ❏▼P❧▲❩❘■❛ ❘●▲ ❍❚❏◆❍❩❑❘❍P◗ P❤ ❘●❍■ ▼▲❑◆ P❏❘❍P◗■ ■▲❘❦❘❍◗❙ ❘P ❜❑◗❬❍◗❙ ❍■ ❘●❑❘ ❘●▲ ■❍➤▲ P❤ ❘●▲ ❜❑◗❬ ❯P▲■ ◗P❘ ❚❑❘❘▲▼ ❣❍❘● ▼▲❦■❏▲❩❘ ❘P ❍❘■ ➟◗❑◗❩❍❑◆ ■P✐◗❯◗▲■■❛ ❣●❑❘ ❚❑❘❘▲▼■ ❍■ ❘●▲ ➠✐❑◆❍❘❨ P❤ ❜❑◗❬♥■❍◗❝▲■❘❚▲◗❘❛ ❣●▲❘●▲▼ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❍◗❝▲■❘❚▲◗❘ ▲❲❩▲▲❯■ ❍❘■ ❍◗❍❘❍❑◆ ❩P■❘❍■ ❘●▲ ❩▼❍❘❍❩❑◆ ➠✐▲■❘❍P◗❞

➁ ➂→➒↕➞↔✟➙→➒➥P◆◆P❣❍◗❙ ❱❍❲❍❘ ❑◗❯ ❳❍◗❯❨❩❬ ❭❪❫❫❴❵❛ ❑ ▼▲❑◆ P❏❘❍P◗■ ❑❏❏▼P❑❩● ❍■ ❑❯P❏❘▲❯❘P ❚P❯▲◆ ❜❑◗❬ ❍◗❝▲■❘❚▲◗❘■❞ ➵ ◗▲❣ ❯❍❚▲◗■❍P◗ ❍■ ❑❯❯▲❯ ❘P ❘●▲ ❚▲❑◗❦▼▲❝▲▼■❍P◗ ❏▼P❩▲■■ ❜❨ ❚P❯▲◆◆❍◗❙ ❘●▲ ❚▲❑◗❦▼▲❝▲▼■❍P◗ ❏❑▼❑❚▲❘▲▼ ❑■ ❑ ◆❍◗❦▲❑▼ ❤✐◗❩❘❍P◗ P❤ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘❞ ♠◗◆❍❬▲ ❱❍❲❍❘❦❳❍◗❯❨❩❬ ❚P❯▲◆❛❘●▲ ❏▼P❏P■▲❯ ❏▼P❩▲■■ ●❑■ ◗P ❯❍▼▲❩❘ ❑◗❑◆❨❘❍❩❑◆ ■P◆✐❘❍P◗ ■P ❘●❑❘ ❑ ◗✐❦❚▲▼❍❩❑◆ ■P◆✐❘❍P◗ ❣❑■ ❯▲▼❍❝▲❯❞ ➵■ ❘●▲ ❑❝▲▼❑❙▲ ❝❑◆✐▲ P❤ ❘●▲ ❍◗❝▲■❘❚▲◗❘❏▼P❧▲❩❘ ❙▲❘■ ◆❑▼❙▲ ■P ❯P▲■ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ P❏❘❍P◗ ❘P ❍◗❝▲■❘ ❍◗ ❘●▲❏▼P❧▲❩❘❞ ã❘ ❍■ ❑◆■P ■●P❣◗ ❘●❑❘ ❑■ ❘●▲ ❑❝▲▼❑❙▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘ ❙P▲■❜▲◆P❣ ❍❘■ ❍◗❍❘❍❑◆ ❩P■❘❛ ❑ ◆❑▼❙▲▼ ❝❑◆✐▲ P❤ ❘●▲ ❚▲❑◗❦▼▲❝▲▼■❍P◗ ❏❑▼❑❚▲❘▲▼▼▲❯✐❩▲■ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❍◗❝▲■❘❚▲◗❘ P❏❘❍P◗❞ ➸P❣▲❝▲▼❛ ❑■ ❘●▲ ❝❑◆✐▲ P❤❘●▲ ❏▼P❧▲❩❘ ▲❲❩▲▲❯■ ❍❘■ ❍◗❍❘❍❑◆ ❩P■❘❛ ❘●▲◗ ❑ ◆❑▼❙▲▼ ❚▲❑◗❦▼▲❝▲▼■❍P◗ ❏❑❦▼❑❚▲❘▲▼ ▼❍■▲■ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ P❏❘❍P◗ P◗◆❨ ❤P▼ ■❚❑◆◆ ❝❑◆✐▲■ P❤ ❘●▲ ❏▼P❧▲❩❘❑◗❯ ▼▲❯✐❩▲■ ❍❘ ❑■ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘ ❙▲❘■ ◆❑▼❙▲▼❞ ❋●▲ ❍❚❏◆❍❩❑❘❍P◗❤P▼ ❜❑◗❬♥■ ❍◗❝▲■❘❚▲◗❘■ ❑▼▲ ▲❲▲❚❏◆❍➟▲❯ ❜❨ ❘●▲ ❤❑❩❘ ❘●❑❘ ◆❑▼❙▲ ❜❑◗❬■✐■✐❑◆◆❨ ❍◗❝▲■❘ ❍◗ ▼▲◆❑❘❍❝▲◆❨ ◆❑▼❙▲ ❏▼P❧▲❩❘■ ❩P❚❏❑▼▲❯ ❘P ■❚❑◆◆ ❜❑◗❬■❞ã❤ ❣▲ ▼▲❙❑▼❯ ❘●▲ ▼❑❘▲ P❤ ■❏▲▲❯ P❤ ❘●▲ ❚▲❑◗❦▼▲❝▲▼■❍P◗ ❏❑▼❑❚▲❘▲▼ ❑■ ❑■❘❑❜❍◆❍➤❍◗❙ ❤❑❩❘P▼ ❤P▼ ❘●▲ ➟◗❑◗❩❍❑◆ ■P✐◗❯◗▲■■ P❤ ❘●▲ ❜❑◗❬❛ ❘●▲ ❚P❯▲◆■✐❙❙▲■❘■ ❘●❑❘ ❘●▲ ■❍➤▲ P❤ ❘●▲ ❏▼P❧▲❩❘ ■●P✐◆❯ ◗P❘ ▼▲❑◆◆❨ ❚❑❘❘▲▼❛ ❣●❑❘

➏✮

Page 18: Capital Investment as Real Options: A Note on Dixit ... · Munich Personal RePEc Archive Capital Investment as Real Options: A Note on Dixit-Pindyck Model. Termos, Ali American University

❚❑❘❘▲▼■ ❍■ ❣●▲❘●▲▼ ❘●▲ ❝❑◆✐▲ P❤ ❘●▲ ❏▼P❧▲❩❘ ❍■ ◆P❣▲▼ P▼ ●❍❙●▲▼ ❘●❑❘ ❘●▲❍◗❍❘❍❑◆ ❩P■❘ P❤ ❍◗❝▲■❘❚▲◗❘❞

➏✲

Page 19: Capital Investment as Real Options: A Note on Dixit ... · Munich Personal RePEc Archive Capital Investment as Real Options: A Note on Dixit-Pindyck Model. Termos, Ali American University

❘➃➃➄➅➆➑➇➈➉➊➋❇➌➉➍ ❇➉ ➎➏➋❉❇➌➉➍ ➐➑➒➓➔

✼ Ô ❭❖ Ü ❪❵✽✔Ñ❖✽✎✽✔Ñ✡ ✗ Ô ❭ ❖❖ Ü ❪❵✎✼ Ô ❭✽✔→✽✔Ñ ❵ Ü ✎

❭ ✽→✽✔Ñ❵✽ Ô ❭→✽ ❵ Ü ❪❭ ✽→✽✔Ñ❵✼ Ô ❭ ✎❖ Ü ❪❵❭

❖ Ü ❪❖✎ ❵✽ Ô ❭❖ Ü ❪❵✽✔Ñ✎Ñ✔✽❖✔✽✼ Ô ❭❖ Ü ❪❵✽✔Ñ❖✽✎✽✔Ñ➣↔➎ ❉➍❉➊↕❇➌❊❉➊ ❆➉➊➋❇➌➉➍ ❇➉ ➎➏➋❉❇➌➉➍ ➐➙➙➓➔

➥❍▼■❘❛ ❣▲ ❏▼P❩▲▲❯ ❜❨ ■P◆❝❍◗❙ ▲➠✐❑❘❍P◗ ❭ÕÖ❵ ■P ❣▲ ❯▲❤▲▼ ❘●▲ ■✐❜■❘❍❘✐❘❍P◗P❤ ❵ Ô ❧Ñ Ý ❧ä✡ ❘P ❘●▲ ▲◗❯❞❖P❲ ❑◗❯ ✸P■■ ❭❪❫ç➘❵ ■✐❙❙▲■❘▲❯ ❘●❑❘ ❘●▲ ■P◆✐❘❍P◗ ❘P ❑ ■▲❩P◗❯ P▼❯▲▼❯❍❢▲▼▲◗❘❍❑◆ ▲➠✐❑❘❍P◗ P❤ ❤P▼❚ ❭ÕÖ❵ ❣❍◆◆ ❘❑❬▲ ❘●▲ ❤P▼❚ P❤ ❘●▲ ❤P◆◆P❣❍◗❙❤✐◗❩❘❍P◗ñ

æ ❭✡ ❵ Ô ✼✡ ✽✇❭✡ ❵ ❭➵❦××❵❣●▲▼▲ ✼ ❑◗❯ ❖ ❑▼▲ ❩P◗■❘❑◗❘ ❘P ❜▲ ❯▲❘▲▼❚❍◗▲❯ ❍◗ ❑ ❣❑❨ ■✐❩● ❘●❑❘✇❭✡ ❵ ■❑❘❍■❤❨ ❑ ❯❍❢▲▼▲◗❘❍❑◆ ▲➠✐❑❘❍P◗ ❣❍❘● ❬◗P❣◗ ■P◆✐❘❍P◗❞ ❥▲ ◗▲▲❯ ❘P■✐❜■❘❍❘✐❘▲ ▲➠✐❑❘❍P◗ ❭➵❦××❵ ❍◗❘P ❭ÕÖ❵❛ ❜✐❘ ➟▼■❘ ❣▲ ❯▲▼❍❝▲ ❘●▲ ➟▼■❘ ❑◗❯■▲❩P◗❯ ❯▲▼❍❝❑❘❍❝▲■ P❤ æ ❭✡ ❵ ❍◗ P▼❯▲▼ ❘P ❚❑❬▲ ❘●▲ ■✐❜■❘❍❘✐❘❍P◗ ❘▼❑❩❘❑❜◆▲❞

æ ✰ ❭✡ ❵ Ô ❖✼✡ ✽✔Ñ✇❭✡ ❵ Ý ✇✰ ❭✡ ❵✼✡ ✽æ ✰ ❭✡ ❵ Ô ✼✡ ✽ ✂❖✡ ✔Ñ✇❭✡ ❵ Ý ✇✰ ❭✡ ❵✄ ❭➵❦×❴❵

æ ✰✰ ❭✡ ❵ Ô ❭❖Ü❪❵❖✼✡ ✽✔ä✇❭✡ ❵Ý✇✰ ❭✡ ❵❖✼✡ ✽✔ÑÝ✇✰✰ ❭✡ ❵✼✡ ✽Ý❖✇✰ ❭✡ ❵✼✡ ✽✔Ñæ ✰✰ ❭✡ ❵ Ô ✼✡ ✽ ✂❖❭❖ Ü ❪❵✡ ✔ä✇❭✡ ❵ Ý ❖✡ ✔Ñ✇✰ ❭✡ ❵ Ý ✇✰✰ ❭✡ ❵ Ý ❖✡ ✔Ñ✇✰ ❭✡ ❵✄

æ ✰✰ ❭✡ ❵ Ô ✼✡ ✽ ✂❖❭❖ Ü ❪❵✡ ✔ä✇❭✡ ❵ Ý Õ❖✡ ✔Ñ✇✰ ❭✡ ❵ Ý ✇✰✰ ❭✡ ❵✄ ❭➵❦×Ø❵➡P❣ ✐■❍◗❙ ▲➠✐❑❘❍P◗■ ❭➵❦×❴❵ ❑◗❯ ❭➵❦×Ø❵ ❍◗ ❭ÕÖ❵ ❨❍▲◆❯■ñ

❪Õ✌ä✡ ä✼✡ ✽ ✂❖❭❖ Ü ❪❵✡ ✔ä✇❭✡ ❵ Ý Õ❖✡ ✔Ñ✇✰ ❭✡ ❵ Ý ✇✰✰ ❭✡ ❵✄Ý➏✴

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✂þ Ü✦ Ý ❵❭❛✡ Ü ✡ ❵✄✡ ✼✡ ✽ ✂❖✡ ✔Ñ✇❭✡ ❵ Ý ✇✰ ❭✡ ❵✄ Üþ✼✡ ✽✇❭✡ ❵ Ô Ö

❱▼P❏❏❍◗❙ ✼ ❤▼P❚ ❑◆◆ ❘▲▼❚■ ❑◗❯ ▼▲❑▼▼❑◗❙❍◗❙ ❘▲▼❚■ ▼▲■✐◆❘ ❍◗❪Õ✌ä✡ ✽ ✂❖❭❖ Ü ❪❵✇❭✡ ❵ Ý Õ❖✡ ✇✰ ❭✡ ❵ Ý ✇✰✰ ❭✡ ❵✡ ä✄Ý

✡ ✽ ✂þ Ü✦ Ý ❵❭❛✡ Ü ✡ ❵✄ ✂❖✇❭✡ ❵ Ý ✇✰ ❭✡ ❵✡ ✄ Üþ✡ ✽✇❭✡ ❵ Ô Ö✡ ✽✇❭✡ ❵✂❪Õ✌ä❖❭❖ Ü ❪❵ Ý ❭þ Ü✦ Ý ❵ ❛✡ ❵❖ Üþ✄Ý✡ ✽①Ñ➛❪Õ✌ä✡ ✇✰✰ ❭✡ ❵Ý✌ä❖✇✰ ❭✡ ❵Ý ✂þ Ü✦Ý❵❭❛✡ Ü✡ ❵✄✇✰ ❭✡ ❵Ü❵❖✇❭✡ ❵➜ Ô Ö

✡ ✽✇❭✡ ❵✂❪Õ✌ä❖❭❖ Ü ❪❵ Ý ❭þ Ü✦ Ý ❵ ❛✡ ❵❖ Üþ✄Ý✡ ✽①Ñ✂❪Õ✌ä✡ ✇✰✰ ❭✡ ❵Ý❭✌ä❖ÝþÜ✦Ý❵ ❛✡ Ü❵✡ ❵✇✰ ❭✡ ❵Ü❵❖✇❭✡ ❵✄ Ô Ö ❭➵❦×➘❵➣↔➎ ❆➉➊➋❇➌➉➍ ❇➉ ➎➏➋❉❇➌➉➍ ➐➙➝➓➔➺✐❜■❘❍❘✐❘❍◗❙ ▲➠✐❑❘❍P◗■ ❭Õç❵ ❑◗❯ ❭Õ➹❵ ❍◗❘P ❭Õ➘❵ ❨❍▲◆❯■❪Õ✌ä✡ ✂ü✰✰ ❭✑❵❭Õ❵✌ä ❵ä✄ Ý ❭✌ä❖ Ý þ Ü✦ Ý ❵ ❛✡ Ü ❵✡ ❵✂ü✰ ❭✑❵Õ❵✌ä ✄ Ü ❵❖ü❭✑❵ Ô Ö

Õ❵ä✡✌ä ü✰✰ ❭✑❵ Ý ❭✌ä❖ Ý þ Ü✦ Ý ❵ ❛✡ Ü ❵✡✌ä ❵Õ❵ü✰ ❭✑❵ Ü ❵❖ü❭✑❵ Ô ÖÕ❵ä✡✌ä ü✰✰ ❭✑❵ Ý ✂Õ❵❖ Ý Õ❵❭þ Ü✦ Ý ❵ ❛✡ ❵✌ä Ü Õ❵ä✡✌ä ✄ü✰ ❭✑❵ Ü ❵❖ü❭✑❵ Ô Ö❵➛Õ❵✡✌ä ü✰✰ ❭✑❵ Ý ✂Õ❖ Ý Õ❭þ Ü✦ Ý ❵ ❛✡ ❵✌ä Ü Õ❵✡✌ä ✄ü✰ ❭✑❵ Ü ❖ü❭✑❵➜ Ô ÖÕ❵✡✌ä ü✰✰ ❭✑❵ Ý ✂Õ❖ Ý Õ❭þ Ü✦ Ý ❵ ❛✡ ❵✌ä Ü Õ❵✡✌ä ✄ü✰ ❭✑❵ Ü ❖ü❭✑❵ Ô Ö➣↔➎ ❆➉➊➋❇➌➉➍ ❇➉ ➎➏➋❉❇➌➉➍ ➐➞➙➓➔

❪Õ✌ä✡ ✇✰✰ ❭✡ ❵ Ý ✂✌ä❖ Ý þ Ü✦ Ý ❧Ñ ❛✡ Ü ❭❧Ñ Ü❧ä ❛✡ ❵✡ Ü❧ä✡ ä✄✇✰ ❭✡ ❵Ü✂❭❧Ñ Ü❧ä ❛✡ ❵❖ Ý ❧ä❖✡ ✄✇❭✡ ❵ Ô Ö

❪Õ✌ä✡ ✇✰✰ ❭✡ ❵ Ý ✂✌ä❖ Ý þ Ü✦ Ý ❧Ñ ❛✡ Ü ❭❧Ñ Ü❧ä ❛✡ ❵✡ Ü❧ä✡ ä✄✇✰ ❭✡ ❵Ü✂❭❧Ñ Ü❧ä ❛✡ ❵ Ý ❧ä✡ ✄❖✇❭✡ ❵ Ô Ö

❪Õ✌ä✡ ü✰✰ ❭✑❵❭Õ❧Ñ✌ä ❵äÝ ✂✌ä❖ÝþÜ✦Ý❧Ñ ❛✡ Ü❭❧ÑÜ❧ä ❛✡ ❵✡ Ü❧ä✡ ä✄ü✰ ❭✑❵❭Õ❧Ñ✌ä ❵➏◆

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Ü✂❭❧Ñ Ü❧ä ❛✡ ❵ Ý ❧ä✡ ✄❖ü❭✑❵ Ô ÖÕ❧äÑ✡✌ä ü✰✰ ❭✑❵Ý✂Õ❧Ñ❖ÝÕ❭þ Ü✦ Ý ❧Ñ ❛✡ ❵❧Ñ✌ä ÜÕ❭❧Ñ Ü❧ä ❛✡ ❵❧Ñ✡✌ä ÜÕ❧Ñ❧ä✡ ä✌ä ✄ü✰ ❭✑❵

Ü✂❭❧Ñ Ü❧ä ❛✡ ❵ Ý ❧ä✡ ✄❖ü❭✑❵ Ô Ö➥❑❩❘P▼❍◗❙ ❧Ñ P✐❘ ▼▲■✐◆❘■ ❍◗ñÕ❧Ñ✡✌ä ü✰✰ ❭✑❵ Ý ✂Õ❖ Ý Õ❭þ Ü✦ Ý ❧Ñ ❛✡ ❵✌ä Ü Õ❧Ñ✡✌ä Ý Õ❧ä ❛✡ ✡✌ä Ü Õ❧ä✡ ä✌ä ✄ü✰ ❭✑❵

Ü✂❪ Ü ❧ä❧Ñ ❛✡ Ý ❧ä❧Ñ✡ ✄❖ü❭✑❵ Ô Ö✑ü✰✰ ❭✑❵Ý ❭⑩Ü✑Ý ❧ä❧Ñ ❛✡ ✑Ü ✌ä❧äÕ❧äÑ ✑ä❵ü✰ ❭✑❵Ü❭❪Ü ❧ä❧Ñ ❛✡ Ý ✌ä❧äÕ❧äÑ ✑❵❖ü❭✑❵ Ô Ö

✑ü✰✰ ❭✑❵ Ý ✂⑩ Ü ❭❪ Ü ❼Ñ❵✑ Ü ❼ä✑ä✄ü✰ ❭✑❵ Ü ❭❪ Ü ❼Ñ Ü ❼ä✑❵❖ü❭✑❵ Ô Ö➟❈❉➠↔❆➔

➬❬

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➡➢➤➥➦➧ ➏➨ ➩ ➫➭ ➯ ➲➤➲➢➳➵➸ ➭ ➺➻➦ ➼➭ ➽ ➾➚ ➪Ñ ➶ ❬➹➏➚ ➲➳➘ ➪ä ➶ ❬➹✏➴

➬➏

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➡➢➤➥➦➧ ➬➨ ➩ ➫➭ ➯ ➲➤➲➢➳➵➸ ➭ ➺➻➦ ➼➭ ➽ ➾➚ ➼➭ ➷ ➾➚ ➲➳➘ ➼➭ ➶ ➾➚ ➪Ñ ➶ ❬➹➏➚ ➲➳➘➪ä ➶ ❬➹✏➴

➬➬

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➡➢➤➥➦➧ ð➨ ➩ ➫➭ ➯ ➲➤➲➢➳➵➸ ➭ ➺➻➦ ➼➭ ➷ ➾➚ ➪Ñ ➶ ❬➹➏➚ ➲➳➘ ➪ä ➶ ❬➹✏➴

➬ð

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➡➢➤➥➦➧ ☎➨ ➩ ➫➭ ➯ ➲➤➲➢➳➵➸ ➭ ➺➻➦ ➼➭ ➽ ➾➚ ➪Ñ ➶ ❬➹❬✏ ➴

➬☎

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➡➢➤➥➦➧ ✏➨ ➩ ➫➭ ➯ ➲➤➲➢➳➵➸ ➭ ➺➻➦ ➼➭ ➽ ➾➚ ➪Ñ ➶ ❬➹➏➴

➬✏

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➡➢➤➥➦➧ ✮➨ ➩ ➫➭ ➯ ➲➤➲➢➳➵➸ ➭ ➺➻➦ ➼➭ ➽ ➾➚ ➪Ñ ➶ ❬➹✏➴

➬✮

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➡➢➤➥➦➧ ✲➨ ➩ ➫➭ ➯ ➲➤➲➢➳➵➸ ➭ ➺➻➦ ➼➭ ➶ ➾➚ ➪Ñ ➶ ❬➹❬✏➴

➬✲

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➡➢➤➥➦➧ ✴➨ ➩ ➫➭ ➯ ➲➤➲➢➳➵➸ ➭ ➺➻➦ ➼➭ ➶ ➾➚ ➪Ñ ➶ ❬➹➏➴

➬✴

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➡➢➤➥➦➧ ◆➨ ➩ ➫➭ ➯ ➲➤➲➢➳➵➸ ➭ ➺➻➦ ➼➭ ➶ ➾➚ ➪Ñ ➶ ❬➹✏➴

➬◆

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➡➢➤➥➦➧ ➏❬➨ ➩ ➫➭ ➯ ➲➤➲➢➳➵➸ ➭ ➺➻➦ ➼➭ ➷ ➾➚ ➪Ñ ➶ ❬➹❬✏➴

ð❬

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➡➢➤➥➦➧ ➏➏➨ ➩ ➫➭ ➯ ➲➤➲➢➳➵➸ ➭ ➺➻➦ ➼➭ ➷ ➾➚ ➪Ñ ➶ ❬➹➏➴

ð➏