ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 ࢢͷूੵɾࢄϞσϧͷରশੑഁյذɿ ܈తذཧʹΑΔΞϓϩʔν ా ਗ਼1 ɾՏ ୡਔ2 ɾদ ོ3 ɾ༄ຊ জਔ4 ɾീ רढ़ೋ5 1 ਖ਼ձһ ౦େֶେֶӃڭत ֶڀݚՊֶઐ߈ʢ˟980-8579 ઋࢢ੨༿۠ߥࣈר੨༿ 6-6-06ʣ E-mail: [email protected] 2 ਖ਼ձһ ౦େֶେֶӃ।ڭत ֶڀݚՊֶઐ߈ʢಉ্ʣ 3 ਖ਼ձһ ౦େֶେֶӃڭत ใՊֶڀݚՊʢಉ্ʣ 4 ֶੜձһ ౦େֶେֶӃ ֶڀݚՊֶઐ߈ʢಉ্ʣ 5 ֶੜձһ ౦େֶେֶӃ ֶڀݚՊֶઐ߈ʢಉ্ʣ ۙɼنͷͱࡁܦ༌ૹඅ͓ΑͼཁૉҠಈͷ૬ʹ༻࡞ޓΑΔࢢͷूੵɾࢄΛੜతʹղ໌͢Δੳ͕ ΜʹߦΘΕ͍ͯΔɽ͜ͷੳϞσϧͷදྫͰ͋Δ Krugman ͷ Core–Periphery ϞσϧෳߧۉղΛ࣋ͪɼ ͦͷࢢूੵɾࢄաఔ༌ૹඅมԽʹΑΔذΛ͏͜ͱ͕ΒΕ͍ͯΔɽ͔͠͠ͳ͕Βɼͦͷूੵɾࢄ աఔओʹɼࢢ͕ 2 ͷ߹͔͠໌Β͔ʹ͞Ε͍ͯͳ͍ɽຊจͰɼCore–Periphery ϞσϧΛଟࢢϞσ ϧɼ۩ମతʹԁप্ʹۉʹҐஔ͢Δࢢ 22 , 23 , 24 Ϟσϧͱ֦ு͠ɼରশੑΛ࣋ͭܥͷҰൠͰ͋Δ܈ తذཧΛ༻͍Δ͜ͱʹΑΓɼߧۉղͷذաఔͷϝΧχζϜΛղ໌͢Δɽͦͷ݁Ռɼ༌ૹඅͷมԽʹΑ Δपظഒʹذද͞ΕΔଟஈ֊ͷରশੑഁյذΛ͏ࢢͷूੵɾࢄͷۭؒతͷมԽΛࣔ͢ɽ Key Words : bifurcation analysis, group-theoretic bifurcation theory, population analysis, Core– Periphery model, city accumulation 1. ͡Ίʹ ಛɼෆશڝ૪ͱੜ࢈ʹ͓͚Δنͷࡁܦੑ͔Β ࢢ͕ूੵͷࡁܦΛ࣋ͪɼҰํͰ༌ૹඅ͕ਓޱࢄྗ ͱͯ͠ಇ͘͜ͱͰ͋ΔɽͦͷޙɼCP Ϟσϧʹରͯ͠ɼ େنͳަ௨ࢪઃඋࢢؒͷਓޱΛେ͖͘ มԽͤ͞Δɽྫ͑ɼେنަ௨ࢪઃ͕ྡͨ͠ࢢ ࿑ಇऀࢿຊͱ͍ͬͨੜ࢈ཁૉͷҠಈՄೳੑޮ༻ؔ ؒʹඋ͞ΕΔͱɼยํͷࢢ͕ͯ͠͏Ұํ͕ ੜ࢈ؔΛมߋͨ͠όϥΤςΟʹΉϞσϧ͕ଟ ਰୀ͢ΔʮετϩʔޮՌʯ͕ੜ͡Δ͜ͱ͕͋Δɽ·ͨɼ ͘։ൃ͞Ε͍ͯΔɽ͜ΕΒͷϞσϧ New Economic ϦχΞϞʔλʔΧʔͷߴަ௨ࢪઃඋਓޱͷҰ ूۃதΛՃͤ͞ΔՄೳੑ͕͋ΔɽͦͷͨΊɼਓूޱ Geography (NEG) ϞσϧͱݺΕɼࢢͷूੵɾࢄ ϝΧχζϜʹؔ͢Δ ͕ݟNEG ϞσϧΛ༻͍ͯଟ͘ ੵͷ͋Δࢢ͓ΑͼͦͷूੵͷఔΛ༧ଌ͢Δ͜ͱ͕ɼ ੵ͞Ε͍ͯΔʢྫ͑ɼจݙ4),5),6),7) ʣɽ ަ௨ࢪઃඋͷޮࡁܦՌΛଊ͑Δ͏͑ͰඞཁͱͳΔɽͦ ͔͠͠ɼࢢͷूੵɾࢄաఔϝΧχζϜΛ NEG Ϟ ͜ͰࢢूੵݱΛ༧ଌ͢ΔͨΊʹɼࢢूੵϝΧχ σϧʹΑΓཧੳ͍ͯ͠Δैདྷڀݚͷଟ͘ɼࢢ ζϜΛղ໌͢Δཧߏங͕ॏཁͱͳͬͯ͘Δɽ Λ 2 ࢢʹ͍ͯͬݶΔɽ2 ࢢϞσϧͷੳͰɼΑΓ ަ௨ࢪઃඋʹ͏ࢢूੵϝΧχζϜΛઆ໌͢Δ ҰൠతͳଟࢢϞσϧʹ͓͚Δ༌ૹඅมԽʹ͏ࢢ දతͯ͠ͱڀݚɼKrugman1),2) ͕͋͛ΒΕΔɽKrugman ूੵɾࢄͷਐߦաఔಛੑʢi.e. ༌ૹඅΛύϥϝʔλͱ 3) Dixit and Stiglitz ͷಠతڝ૪ϞσϧΛ 2 ࢢϞ ͢Δܦذ࿏ͷنଇੑͱϝΧχζϜʣղ໌Ͱ͖ͳ͍ɽ σϧʹԾఆ͠ɼࢢؒͷ༌ૹඅมԽʹ͏ۀͷूੵɾ 2 ࢢΑΓଟ͘ͷࢢΛѻ͏ͯ͠ͱڀݚɼFujita et ࢄͷਐߦաఔΛੳͨ͠ɽ͜ͷ Krugman ͷϞσϧ Core–Periphery ϞσϧʢҎԼɼCP Ϟσϧͱུʣͱݺ Εɼࢢूੵʹ͍ͭͯࣔࠦʹΉੳ݁ՌΛͨΒ al.4) ͷ 6 ষʹ͓͍ͯ 3 ࢢ͓Αͼ 12 ࢢͷੳ͕ͳ͞ Ε͍ͯΔɽͨͩ͠ɼ༌ૹඅ༩݅ͷࢢؒਓޱͷۉ ߧղࣔ͞Ε͍ͯΔͷͷɼ༌ૹඅมԽʹ͏ܦذ ͍ͯ͠Δɽͦͷද͕ɼ༌ૹඅΛύϥϝʔλͱ͢Δ ࿏ੳ͞Ε͍ͯͳ͍ɽ·ͨɼKrugman8),9) Ͱɼ࿈ ݱذͰ͋Δɽ۩ମతʹɼରশͳۭؒ݅Λ࣋ͭ 2 ଓۭؒΛରͱͨ͠ੳߦΘΕ͍ͯΔɽͨͩ͠ɼू ࢢʹ͓͍ͯ༌ૹඅ͕͋ݮΔͱɼࢢूੵྗ͕ͳ͘ڧ ੵɾࢄϝΧχζϜΛଊ͑Δཧੳͱͯ͠ɼਓ͕ޱ Γରশ͕ߧۉෆ҆ఆͱͳΔɽͦͷ݁ՌɼʹذΑͬͯ Ұ༷Ͱ͋Δ͍͓ͯʹߧۉઢॴہͨ͠ࣅۙܗత ͍ͣΕ͔ͷࢢʹਓू͕ޱத͢Δʢi.e.ɼยํͷࢢਓ ੳ͕ͳ͞Ε͍ͯΔͷΈͰɼਓ͕ޱҰ༷Ͱͳ͍έʔ ޱθϩʹͳΔʣ͜ͱ͕ࣔ͞Ε͍ͯΔɽ CP Ϟσϧͷ εΛؚΉେҬతੳʹͳ͍ͬͯͳ͍ɽ·ͨɼFujita et 553 ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 al.10) ʹ͓͍ͯɼෳࢢͷ֊ߏܗϝΧχζϜ ͕ਓޱΛύϥϝʔλͱͨ͠ذղੳΛ༻͍ͯࣔ͞Εͯ લऀͷࢉܭذཧͱɼඇઢํܗఔࣜͷॴہతذ ܦ࿏๏ʹͮ͘جࢉܭख๏Ͱ͋ΔɽஶऀΒط ͍Δɽ͔͠͠ɼܥ౷తͳܦذ࿏ߦΘΕ͓ͯΒ ʹ͜ͷํ๏Λ 4 ࢢͷ CP Ϟσϧʹద༻ͯ͠ܦذ࿏ ͣɼಛఆͷܦذ࿏͕͍ࣔͯͮ͞جʹࢉܭΕ͍ͯ ΛٻΊ͍ͯΔ15) ɽऀޙͷ܈తذཧɼߏྗֶ Δ͚ͩͰ͋Δɽ݁ہɼ͍ͣΕͷੳܦذ࿏Λܥ౷ ܥͰɼରশੑΛ࣋ͭܥͷذϝΧχζϜͷղੳʹ త͔ͭཏతʹٻΊ͍ͯͳ͍ɽͦͷͨΊɼैདྷͰڀݚ ར༻͞Ε͍ͯΔ12),14) ɽ۩ମతʹɼରশੑԼΛ͍ ଟࢢϞσϧʹ͓͚Δ༌ૹඅมԽʹ͏ࢢूੵɾ ͳ͕Βਐߦ͢ΔذʢରশੑഁյذʣʹΑΓɼύλʔ ࢄͷਐߦաఔͷҰൠతنଇ໌Β͔ʹͳ͍ͬͯͳ͍ɽ ϯྲྀͷൃੜߏࡐྉͷ࠲ذ۶͕Ҿ͖͜͞ىΕΔ ͜ͱ͕ղ໌͞Ε͍ͯΔɽ Ҏ্ͷഎܠͷͱɼຊจɼଟࢢϞσϧʹ͓͚Δ ༌ૹඅมԽʹ͏ࢢूੵɾࢄͷਐߦաఔͷੑ࣭ʢi.e. ຊจɼ͜ͷ܈తذཧΛԁप্ʹִؒʹ ࢢूੵɾࢄύλʔϯมભͷنଇੑͱϝΧχζϜʣΛ Ґஔ͢Δ n(= 22 , 23 , . . .) ࢢͷ CP Ϟσϧద༻͢Δɽ ໌Β͔ʹ͢Δ͜ͱΛతͱ͢ΔɽͦͷͨΊʹɼ CP Ϟ ͦͷ݁Ռɼରশੑഁյͱذपظഒ͕ذಉ༷ʹൃੜ σϧΛରশͳۭؒ݅Λ࣋ͭ n(= 22 , 23 , . . .) ࢢϞσ ͢Δ͜ͱΛࣔ͠ɼͦΕΒͷ͕ذଟஈ֊ʹੜ͡Δ͜ͱ ϧͱ֦ு͠ɼ༌ૹඅΛύϥϝʔλͱͨ͠ذղੳΛ ͰݱΕΔࢢूੵɾࢄͷਐߦաఔΛ໌Β͔ʹ͢Δɽ ߦ͏ɽຊͰڀݚੜతʹܾఆ͞ΕΔࢢूੵɾࢄύ ຊจͷߏҎԼͷͱ͓ΓͰ͋Δɽୈ 2 ষɼCP λʔϯɼਖ਼ͷਓޱΛ࣋ͭࢢͷҐஔɼͦͷࢢ͓ ϞσϧΛ؆୯ʹઆ໌͢Δɽୈ 3 ষɼඇઢܗ࿈ཱํఔ ΑͼنͰ͋Γɼͦͷࢢूੵɾࢄύλʔϯͷ༌ૹඅ ࣜͷύϥϝʔλมԽʹ͏ߧۉղͷมભΛతʹ มԽʹ͏มભ͕ੳରͰ͋Δɽ ͢ΔҰൠతํ๏Λड़Δɽୈ 4 ষɼҎ߱ͷٞͷ 11) ͳ͓ɼຊ΅΄ͱڀݚฒߦͯ͠ɼTabuchi and Thisse ͨΊʹߧۉղͷྨΛࣔ͢ͱͱʹɼղͷ҆ఆੑͷఆ ຊͱڀݚಉ༷ͷڀݚతͰɼԁप্ͷରশͳ n(= ٛΛߦ͏ɽୈ 5 ষɼ܈తذཧΛ༻͍ͯରশੑ 2 3 2 , 2 , . . .) ࢢΛؚΉ NEG Ϟσϧʹؔͯ͠༌ૹඅΛ ഁյذΛઆ໌͢Δɽୈ 6 ষɼୈ 5 ষͷҰൠཧΛ ύϥϝʔλͱͨ͠ذղੳΛߦ͍ͬͯΔɽ͔͠͠ͳ͕ Θ͔Γࣔ͘͢͢ྫͱͯ͠ 4 ࢢϞσϧΛରͱͨ͠ Βɼຊ ͱڀݚTabuchi and Thisse Λൺֱ͢Δͱɼ1) ੳΛࣔ͢ɽୈ 7 ষɼୈ 5 ষͷ܈తذཧͱܭ ذղੳ๏ɼ2) ܦذ࿏ͷཏੑɼ3) ੳରϞσϧͷ ࢉذཧΛΈ߹Θͤͯɼଟࢢͷ CP Ϟσϧͷ 3 ʹҧ͍͕͋Δɽ·ͣ 1) ذղੳ๏ʹ͍ͭͯɼຊ ڀݚޙड़͢ΔΑ͏ʹࢉܭذཧͱ܈తذཧ ΛΈ߹ΘͤͨղੳΛ༻͍ΔɽҰํɼTabuchi and ذղੳΛߦ͏ɽୈ 8 ষɼذղੳͰಘΒΕͨ CP Ϟ Thisse ղੳతʹܦذ࿏ΛٻΊ͍ͯΔɽຊڀݚͷ ղੳ๏ղੳతʹٻղ͢ΔΑΓ͋ͰྗڧΓɼKrug- ֶ͍ͯࡁܦతؚҙΛซͤͯٞ͢Δɽୈ 9 ষ݁Ͱ σϧͷܦذ࿏ͷ͏ͪ҆ఆͳͷͷΈΛཧ͠ɼ༌ૹ අԼʹ͏ࢢͷूੵɾࢄͷਐߦաఔͷಛʹͭ ͋Δɽ man ͷ CP ϞσϧʹݶΒͣɼ͋ΒΏΔ NEG Ϟσϧͷ ܦذ࿏ΛՄೳͳҰൠతํ๏Ͱ͋Δɽ࣍ʹ 2) ܦذ ࿏ͷཏੑʹ͍ͭͯɼຊڀݚ CP Ϟσϧ͕࣋ͭ͢ 2. Core–Periphery Ϟσϧ ԁप্ʹۉʹҐஔ͢Δʹ͍ޓରͳ n ݸͷࢢͷ ͯͷܦذ࿏Λܥ౷త͔ͭཏతʹٻΊ͍ͯΔɽ͔͠ ͠ɼTabuchi and Thisse Ұ෦ͷܦذ࿏ʢ۩ମతʹ ਓੵूޱͷࢧํఔࣜΛ Core–Periphery ϞσϧʢCP पظഒذΛࣔ͢ܦ࿏ʣʹੳΛ͍ͯͬݶΔɽ ࠷ʹޙ Ϟσϧʣʹ͍ͯͮج༠ಋ͢Δɽ 3) ੳରϞσϧʹ͍ͭͯɼຊ͍༻ͰڀݚΔ CP Ϟ σϧͷফඅऀʹɼॴಘ૿ՃʹΑΓࡒফඅ͕૿Ճ͢Δޮ (1) ҰൠߧۉͷΈ CP ϞσϧҎԼʹࣔ͢ҰൠߧۉͷΈΛ࣋ͭɽৄ Ռʢࡒফඅʹ͓͚ΔॴಘޮՌʣ͕͋ΔɽҰํɼTabuchi ࡉ Krugman1) ·ͨ Fujita et al.4) ͷ 5 ষʹৡΔɽ and Thisse ղੳతʹܦذ࿏ΛٻΊΔత͔Β४ઢ • ࡁܦɼಠతڝ૪͕ߦΘΕΔۀ෦ͱશڝ ؔ༻ޮܗΛԾఆ͓ͯ͠Γɼࡒফඅʹ͓͚ΔॴಘޮՌ ૪తͳۀ෦ͷ 2 ͭͷ෦͔ΒͳΔɽ ͕ͳ͍ɽͦͷͨΊ CP Ϟσϧɼॴಘ૿Ճ͕ࡒফඅΛ ૿Ճͤͯ͞ࢢʹۀاΛूੵͤ͞ΔҰൠߧۉతٴϝ • ࡁܦશମͰɼۀ࿑ಇऀ μɼۀ࿑ಇऀ 1−μ ΧχζϜΛ࣋ͭɼTabuchi and Thisse ΑΓ๛ͳܦ ଘࡏ͢Δɽ • ۀ࿑ಇऀࣗͷޮ༻Λ࠷େԽ͢ΔΑ͏ʹࣗ༝ ʹࢢؒΛҠಈ͢Δ͜ͱ͕Ͱ͖Δ͕ɼۀ࿑ಇऀ ࡁֶతߏΛ࣋ͭɽͦͷ݁Ռ CP ϞσϧɼΑΓෳࡶ ͳܦذ࿏͕ಘΒΕΔ͜ͱ͕༧͞ΕΔɽ ຊจͰ༻͍Δذղੳ๏ɼࢉܭذཧ12) ͱ܈ 13),14) తذཧ 1 ҠಈෆՄೳͰɼͯ͢ͷࢢʹۉʹͯ͠ 1 ΛదʹΈ߹Θͤͨํ๏Ͱ͋Δ ɽ ͓Γɼۚ 1 Ͱ͋Δɽ ా16) ͕͛ڍΒΕΔɽ ͱذෳͷߧۉղͷଘࡏʹؔ͢Δೖॻͱͯ͠ɼాɾࣨ 554 ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 • ۀͷ༌ૹʹ༌ૹඅ͕͔͔Γɼۀͷ༌ૹ ʹ༌ૹඅ͔͔Βͳ͍͜ͱͱ͢Δɽ • ඇઢࢧܗํఔࣜ • ૿ࢧํఔࣜͷ༠ಋ ຊͰڀݚɼn ݸͷࢢ͕ԁपʹԊִͬͯؒʹଘࡏ • Β͔ͳղۂઢͷ • ಛҟͷఆ • ܦذ࿏ͷ୳ࠪ ͢Δɼ͍ΘΏΔɼٕڝࡁܦΛߟ͑Δɽ͢ͳΘͪɼ ࢢ r ͱࢢ s ͷؒͷ༌ૹඅ Trs Λ Trs = eτ |r−s| , (r, s = 1, . . . , n) ͱ͍͏߲ʹ͍ͭͯ·ͱΊΔ͜ͱͱ͢Δɽ (1) ͱ͢Δɽ͜͜Ͱɼτ ༌ૹඅύϥϝʔλͰ͋Γɼ|r − s| (1) ࢢ r ͔Βࢢ s ʹࢸΔ࠷͋ͰڑΔɽ (2) ඇઢࢧܗํఔࣜ ࣜ (7) ʹࣔͨ͠૬ิੑ݅ɼࣜ݅ ࢧํఔࣜͷఆࣜԽ ফඅऀͷޮ༻࠷େԽߦಈɼੜ࢈ऀͷར५࠷େԽߦಈɼ (ω̄ − ωr )λr = 0, (r = 1, . . . , n) (8) ණմ༌ૹΛߟྀͨ͜͠ͷ CP Ϟσϧʹ͓͍ͯɼࢢ r ͷ ۀ࿑ಇऀͷۚ wr ҎԼʹࣔ͢࿈ཱํఔࣜʹΑͬͯ ͱɼෆࣜ݅ࣜ ܾఆ͞ΕΔɽ Yr = μλr wr + (1 − μ)/n n λs (ws Tsr )1−σ ]1/(1−σ) Gr = [ s=1 n wr = [ 1−σ σ−1 1/σ Ys Trs Gs ] ω̄ − ωr ≥ 0, (2) (9) (3) ͱʹॻ͖͢͜ͱ͕Ͱ͖Δɽ (4) ઢࢧܗํఔࣜ (8) ͱਓޱҰఆͷ݅ࣜ (5) ͔ΒͳΔ 2.(1) અͰಋೖͨ͠ҰൠߧۉͷΈɼn ࣍ݩͷඇ s=1 n + 1 ࣍ݩͷඇઢܗ࿈ཱํఔࣜ2 ʹؼணͰ͖Δɽ ͜͜ͰɼYr ࢢ r ͷॴಘɼGr ࢢ r ͷۀՁ֨ ⎧ ⎪ F(λ, ω̄, f ) ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ ⎪ (ω̄ − ω1 (λ1 , . . . , λn , f ))λ1 ⎪ ⎨ ⎜ ⎟ .. ⎟=0 =⎜ . ⎝ ⎠ ⎪ ⎪ ⎪ ⎪ (λ , . . . , λ , f ))λ (ω̄ − ω ⎪ n 1 n n ⎪ ⎪ ⎩ F (λ) = λ1 + · · · + λn = 1T λ = 1 ࢦɼTsr ۀΛࢢ s ͔Βࢢ r ·Ͱ༌ૹͨ͠ͱ ͖ͷ༌ૹඅɼσ ҙͷࠩผԽ͞Εͨ 2 ࡒؒͷସ ྗੑͰ͋Δɽλr ࢢ r ʹ͓͚Δۀਓޱશମʹର͢ Δׂ߹Ͱ͋Γɼ͢ͳΘͪ࣍ࣜΛຬ͢Δɽ λ1 + · · · + λn = 1 λr ≥ 0, (r = 1, . . . , n) (5) (10) ࢢ r ͷ࣮࣭ۚɼ࣍ࣜʹΑΓද͞ΕΔɽ ωr = wr G−μ r ͜͜Ͱ λ = (λ1 , . . . , λn )T , 1 = (1, . . . , 1)T Ͱ͋Δɽ· (6) ͨɼf ͋ΔύϥϝʔλͰ͋Γɼ͜ͷจͰ༌ૹඅύ ফඅऀɼࢢ r Ͱܾఆ͞ΕΔ࣮࣭ۚ ωr Λൺֱ͠ ϥϝʔλ τ Λ༻͍Δɽ͢ͳΘͪɼf = τ Ͱ͋Δɽ ͯɼ࣮࣭ۚͷߴ͍ࢢʹҠॅ͢Δɽ͕ͨͬͯ͠ɼۉ ߧʹ͓͍ͯ࣍ͷ૬ิੑཱ͕݅ࣜ͢Δɽ (λr > 0) ω̄ − ωr = 0, ω̄ − ωr ≥ 0, (λr = 0) ඇઢܗ࿈ཱํఔࣜ (10) ͷղͷதͰɼෆࣜ݅ࣜ (9) Λຬͨ͞ͳ͍ղཧతʹ͜ىΓಘͳ͍ڐ༰͞Εͳ͍ (7) ղͱͯ͠ഉআ͢Δɽ۩ମతʹɼਓ͕ޱෛʹͳΔղΛ ഉআ͍ͯ͠ΔɽࢉܭذཧͰࣜ݅ͷΈΛऔΓ ͜͜Ͱɼω̄ ࣭࣮ߧۉۚͰ͋Δɽ͢ͳΘͪɼࢢ r ʹ ѻ͏ͷʹର͠ɼCP ϞσϧͷఆࣜԽʹ͓͍ͯෆࣜ ਖ਼ͷਓ͕͋ޱΔ߹ʹɼωr ࣭࣮ߧۉۚ ω̄ ͱ͠ ݅ࣜ (9) ༻͍Δ͜ͱ͕ಛతͰ͋Δɽ ͘ɼࢢ r ͷਓ͕ޱθϩͰ͋Δ߹ʹɼωr ࣮ߧۉ ࣭ۚ ω̄ ҎԼͱͳΔɽ (2) ࣜ (2)ʙ(7) ͷֶࡁܦతҙຯʹ͍ͭͯ Fujita et al.4) ·ͨಉஶͷຊ༁ͷ 5 ষΛࢀরͷ͜ͱɽ 3. ૿ࢧํఔࣜ CP Ϟσϧͷղۂઢͷʹઌཱͪɼͦͷ૿දࣔΛ ٻΊ͓ͯ͘ɽ ඇઢࢧܗํఔࣜͱղۂઢͷ ඇઢࢧܗํఔࣜ (10) Λ૿දࣔ͢Δͱɼ CP Ϟσϧͷඇઢܗ࿈ཱํఔࣜͱͦͷ૿දࣔΛࣔ ∂F ∂F δ ω̄ + δf = 0(11) ∂ ω̄ ∂f δF (δλ) = δλ1 + · · · + δλn = 1T δλ = 0 (12) δF(δλ, δ ω̄, δf ) = Jδλ + ͠ɼߧۉղۂઢͷ๏ʹ͍ͭͯ·ͱΊΔɽڍذಈ ΛٻΊΔࢉܭذཧ12) ͷ CP Ϟσϧͷద༻ͷৄࡉ ʹ͍ͭͯɼஶऀͷจ15) ʹৡΔɽҎԼɼඇઢํܗ 2 ఔࣜͷղۂઢͷʹؔ࿈ͨ͠ 555 ඇઢํܗఔࣜͷॖʹ۩ͮ͘جମతͳࣜͷ༠ಋ15) ʹৡΔɽ ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 ͱͳΔɽ͜͜ͰɼJ ϠίϏߦྻͰ͋Γɼ ⎛ ω̄ − ω1 0 ··· 0 ⎜ .. .. ⎜ 0 . ω̄ − ω2 . ⎜ J =⎜ . . . ⎜ . .. .. 0 ⎝ . ಛҟʹ͓͍ͯɼϠίϏߦྻ J ͷݻ༗Λ ei ͱ͠ɼ ⎞ ݻ༗ϕΫτϧΛ ψ i (i = 1, . . . , n) ͱ͢Δͱɼඪ४ݻ༗ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Jψ i = ei ψ i , (20) ͱͳΔɽ͜ͷͱ͖ɼಛҟͷ݅ɼগͳ͘ͱ 1 ݸ 0 ··· 0 ω̄ − ωn ⎛ ⎞ ω1,1 λ1 ω1,2 λ1 · · · ω1,n λ1 ⎜ .. ⎟ .. ⎜ω λ . . ⎟ ⎜ 2,1 2 ω2,2 λ2 ⎟ −⎜ . .. ⎟ .. .. ⎜ . ⎟ . . . ⎠ ⎝ . ωn,1 λn ··· · · · ωn,n λn Ҏ্ͷݻ༗͕θϩͱͳΔ͜ͱɼ͢ͳΘͪɼ ei = 0, (i = 1, . . . , M ) (21) ͕Γཱͭ͜ͱͰ͋Δɽ͜͜Ͱ M ಛҟͷଟॏͰ ͋Δɽ = diag(ω̄ − ω1 , . . . , ω̄ − ωn ) − diag(λ1 , . . . , λn )Ω ⎛ ⎞ (∂ω1 /∂f )λ1 ⎜ ⎟ ∂F ∂F .. ⎟ = λ, = −⎜ . ⎝ ⎠ ∂ ω̄ ∂f (∂ωn /∂f )λn (i = 1, . . . , n) (5) ܦذ࿏ͷ୳ࠪ ࢬ͍͓ͯʹذ͔Ε͢Δܦذ࿏Λ͢Δ͜ (13) ͱΛߟ͑Δɽ ୯७ذͷ߹ɼθϩݻ༗ʹରԠ͢Δݻ༗ϕ (14) Ϋτϧ ψ 1 ͷํʹذղΛ୳ࠪ͢ΕΑ͍ɽ 2 ॏͰذɼθϩݻ༗ʹରԠ͢Δ 2 ݸͷݻ༗ϕ Ϋτϧ ψ 1 ɼψ 2 ͕ଘࡏ͢ΔͷͰɼͦͷઢ߹݁ܗͷํ ʹ୳ࠪ͢Δɽશͯͷઢ߹݁ܗͷํʹղ͕ଘࡏ͢ΔΘ Ͱ͋Δɽ͜͜Ͱɼωi,j = ∂ωi /∂λj (i, j = 1, . . . , n)ɼΩ = (ωi,j | i, j = 1, . . . , n) Ͱ͋Γɼdiag(· · · ) ހׅͷ ͔ΒͳΔର֯ߦྻΛද͢ɽ ͚Ͱͳ͍ɽޙड़ͷ܈తذཧղͷํΛ͜ͷ ͷ͑Λ༩͑Δͱ͍͏ҙຯͰ༗༻Ͱ͋Δɽ (3) Β͔ͳղۂઢͷ 4. ඇઢܗ࿈ཱํఔࣜ (10) Λຬͨ͢ղΛɼ૿ํఔࣜ (11) ղͷྨͱղͷ҆ఆੑ ͱ (12) Λ༻͍ͯ Newton–Raphson ๏ͳͲͷ෮ղ๏ʹ ͷΛऔΓग़͢͜ͱʹΑΓɼཧతʹڐ༰͞ΕΔղΛ CP Ϟσϧͷذղੳʹ͓͍ͯɼ৭ʑͳछྨͷղʹૺ ۰͢Δɽ͜ͷষͰɼཧత ͯ͠ͱૅجCP Ϟσϧͷ ٻΊΔɽ ղΛྨ͠ɼղͷ҆ఆੑʹ͍͓ͭͯٞͯ͘͠ɽղͷ ΑΓٻΊɼͦͷղͷதͰෆࣜ݅ࣜ (9) Λຬ͢Δ ྨɼਓ͕ޱθϩͷࢢͷ༗ແͱࢧํఔࣜͷಛҟ ϠίϏߦྻ J ͕ਖ਼ଇͰ͋Δ߹ʹɼࣜ (11) Λ δλ ੑʹணͯ͠ߦ͏ɽ ʹ͍ͭͯղ͍ͨࣜ ∂F ∂F δ ω̄ − J −1 δf ∂ ω̄ ∂f Λɼࣜ (12) ʹೖ͢Δͱɼ δλ = −J −1 Aδ ω̄ + Bδf = 0 (15) (1) ͋Δ͔ɼਓ͕ޱθϩͷࢢ͕͋Δ͔ʹΑΓɼԼهͷΑ (16) ͏ʹྨ͞ΕΔɽ ղɿ શͯͷࢢͷਓ͕ޱਖ਼Ͱ͋Δղ ͱͳΔɽ͜͜Ͱɼ ∂F ∂F , B = 1T J −1 (17) ∂ ω̄ ∂f Ͱ͋Δɽࣜ (16) Λ༻͍ͯɼඇઢํܗఔࣜ (11) ͔Β δ ω̄ A = 1T J −1 Λফ͢ڈΔ͜ͱʹΑΓಘͨɼ ∂F B ∂F − δf = 0 Jδλ + ∂f A ∂ ω̄ ਓ͕ޱθϩͷࢢͷ༗ແʹΑΔྨ ඇઢํܗఔࣜ (10) ͷղɼશͯͷࢢͷਓ͕ޱਖ਼Ͱ ղɿ ਓ͕ޱθϩͷࢢ͕͋Δղ a) (22) ղͷੑ࣭ શͯͷࢢʹਓ͕ޱଘࡏ͢Δղʹରͯ͠ɼ λr > 0, (18) (r = 1, . . . , n) (23) Ͱ͋ΔͷͰɼࣜ (10) ΑΓɼ ͱ͍͏ δλ, δf ʹؔ͢Δ૿ํఔࣜͷղͱͯ͠ղۂઢΛ ω̄ − ωr = 0, (r = 1, . . . , n) ٻΊΕΑ͍ɽ (24) ͕Γཱͪɼෆࣜ݅ࣜ (9) ࣗಈతʹຬ͞ΕΔɽ (4) ղʹରͯ͠ɼϠίϏߦྻ (13) ԼهͷͳʹܗΔɽ ಛҟͷఆ J = −diag(λ1 , . . . , λn )Ω ඇઢࢧܗํఔࣜ (10) ͷղɼϠίϏߦྻ J = J(λ, ω̄, f ) ͕ಛҟͰ͋Δʹ͓͍ͯɼಛҟੑ݅ࣜ det J(λ, ω̄, f ) = 0 (25) ಛʹɼશͯͷࢢ͕ಉҰͷਓޱΛ࣋ͭਓޱҰ༷ ղ (λ1 = · · · = λn = 1/n) ্ͰɼϠίϏߦྻɼ 1 (26) J =− Ω n (19) Λຬ͢Δɽ 556 ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 ͱ͍͏ରশߦྻͱͳΔɽ b) ղͷੑ࣭ m ࢢҎ߱ͷਓ͕ޱফࣦͨ͠ղ λr > 0, (r = 1, . . . , m) λr = 0, (r = m + 1, . . . , n) (a) ਓޱҰ༷ղ (27) (b) Ұूۃதղ (c) ඇࣗ໌ղ ਤ–1 දతͳࣗ໌ղ (a),(b) ͱඇࣗ໌ղ (c)ʢ6 ࢢʣ Λߟ͑ΔɽҰൠతʹɼ࠷ॳͷ m ࢢ͚ͩਓ͕ޱଘࡏ ͢Δͱ͍͏ঢ়Ͱگͳ͍͕ɼదٓมͷॱ൪ΛೖΕସ͑ Δͱ͢Δͱɼ͜ͷΑ͏ʹԾఆͯ͠ҰൠੑΛࣦΘͳ͍ɽ ͜ͷͱ͖ɼ૬ิੑ݅ࣜ (7) ΑΓɼ ω̄ − ωr = 0, (r = 1, . . . , m) ω̄ − ωr ≥ 0, ⎜ ⎜ ⎜ ⎜ ⎜ J =⎜ ⎜ ⎜ ⎜ ⎝ ω̄ − ωm+1 0 .. O 0 . ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ (29) ⎟ ⎟ ⎟ ⎠ ຊϞσϧͷಛҟҎԼͷΑ͏ʹྨͰ͖Δɽ ⎧ ⎪ ύϥϝʔλ f ͷۃେ·ͨۃখ (M = 1) ⎪ ⎪ ⎪ ⎪ ⎪ ਓޱͷফࣦ (M = 1) ⎪ ⎨ ⎧ ⎪ ୯७( ذM = 1) (36) ⎪ ⎨ ⎪ ⎪ ⎪ 2 ॏذ (M = 2) ⎪ ذ ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎩ ⎩ . ω̄ − ωn ͱ͍͏ͳʹܗΔɽ͜͜Ͱɼ Φ = −diag(λ1 , . . . , λm )Ω̃ ⎞ ⎛ ω1,1 · · · ω1,n ⎜ . .. ⎟ .. ⎟ . Ω̃ = − ⎜ . . ⎠ ⎝ . ωm,1 · · · ωm,n ඇઢࢧܗํఔࣜ (10) ͷղɼϠίϏߦྻ J = (28) ⎞ Φ ࢧํఔࣜͷಛҟੑʹΑΔྨ J(λ, ω̄, f ) ͕ਖ਼ଇͰ͋Δ͔ಛҟͰ͋Δ͔ʹΑΓɼԼه ͷΑ͏ʹྨ͞ΕΔɽ ௨ৗɿ J ͕ਖ਼ଇ (35) ಛҟɿ J ͕ಛҟ (r = m + 1, . . . , n) ͕ΓཱͭͷͰɼϠίϏߦྻ ⎛ (3) ຊจͰऔΓѻ͏ n ࢢϞσϧͷΑ͏ʹਖ਼ n ֯ܗঢ়ͷ ରশੑΛ࣋ͭͰܥɼҰൠతʹଟॏ M = 1, 2 Ͱ (30) ͋Δ͜ͱΛ 6 ষͰ۩ମతʹࣔ͢ɽ·ͨɼਓޱͷফࣦ ɼ4.(1) અͰಋೖͨ͠ղͱղͷଘࡏ͕ɼCP (31) Ϟσϧݻ༗ͷੑ࣭Ͱ͋Δɽ a) Ͱ͋Δɽ ύϥϝʔλ f ͷۃେ·ͨۃখ ਓޱ λr ʹؔ͢Δύϥϝʔλ f ͷۃେ·ͨۃ খͰɼϠίϏߦྻ J ͷݻ༗ ei ͷූ߸͕มԽ͢Δ (2) ࣗ໌ղͱඇࣗ໌ղ ͷͰɼ҆ࡏݱఆͳղඞͣෆ҆ఆͱͳΓɼҰํɼࡏݱ ඇઢํܗఔࣜ (8) ɼҼղ͞Ε͓ͨͯͬͳͱܗΓɼ ෆ҆ఆͳղ҆ఆͱͳΔՄೳੑ͕͋Δʢͦͷ··ෆ҆ ޙड़͢ΔΑ͏ʹɼύϥϝʔλ f ΛมԽͤͯͦ͞ͷύ ఆͳ߹͋Δʣɽ͜ͷΑ͏ʹɼύϥϝʔλ f ͷۃେ λʔϯ่͕Εͳ͍ਓޱҰఆͷղΛ࣋ͭɽ͜ͷਓޱҰఆ ·ͨۃখͰɼ҆ఆͳղ్͕ઈ͑ͯ͠·͏ɽύ ͷղͷଘࡏ͕͜ͷϞσϧʹ͓͍ͯಛతͰ͋Δɽ͜ͷ ϥϝʔλ f ͷۃେͷҰྫΛਤ–2(a) ʹࣔ͢ɽ छͷղͷଘࡏʹରԠ͠ɼղΛԼهͷ 2 छྨʹྨ͢Δɽ ࣗ໌ղɿ ਓޱҰఆͷղ (32) ඇࣗ໌ղɿ ਓ͕ޱมԽ͢ΔҰൠͷղ b) ذɼ͞ΒʹԼهͷΑ͏ʹྨ͞ΕΔɽ ୯७( ذM = 1) 2 ॏ( ذM = 2) ࣗ໌ղͱͯ͠ɼԼهͷ 2 छྨͷղ͕දతͰ͋Δɽ ͱͳ͍ͬͯΔɽܦذ࿏Ͱɼ͋Δڸөରশੑճ సରশੑͷҰ෦·ͨશ͕ࣦͯΘΕΔ͜ͱʹΑΓɼܥ (33) ͷରশੑ͕ओܦ࿏ͱൺͯԼ͢Δɽ͜ͷରশੑͷ • Ұूۃதղɿ1 ͭͷҬʹશਓ͏·ͯ͠͠ੵू͕ޱ ղ λ1 = 1, λ2 = · · · = λn = 0 (37) ਤ–2(b) ʹࣔ͢Α͏ʹɼذෳͷղ͕ަࠩ͢Δ • ਓޱҰ༷ղɿશͯͷࢢ͕ಉҰͷਓޱΛ࣋ͭ ղ λ1 = · · · = λn = 1/n ذ ԼͷΈ 5 ষͰऔΓ্͛Δɽ c) (34) ਓޱͷফࣦ ࢢͷਓ͕ޱθϩͱͳΔͱ͍͏ਓޱͷফࣦ (λr = 0) 6 ࢢͷ߹ʹର͠ɼਤ–1(a) ʹਓޱҰ༷ղΛɼ (b) ʹҰूۃதղΛɼ(c) ʹඇࣗ໌ղͷҰྫΛͦΕͧΕ ͰɼԼهͷ 2 छྨͷಛҟ͕ൃੜ͢Δɽ ࣗ໌ղͱඇࣗ໌ղͱͷަ ਓ͕ޱਖ਼͔ΒෛͱมԽ͢Δ ࣔ͢ɽ 557 (38) ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 ͍ղͰ͋Δɽ ਓޱͷಈଶํఔࣜͱͯ࣍ࣜ͠ΛԾఆ͢Δɽ λ̇i = (ωi (λ1 , . . . , λn ) − ω̂)λi , (i = 1, . . . , n) ͨͩ͠ɼ (a) ύϥϝʔλ f ͷ ۃେ ω̂ = (b) ذ n λi ω i (39) (40) i=1 ͜͜Ͱ͍͓ͯʹߧۉࣜ (7) ཱ͕͢ΔͨΊɼࣜ (7) Λࣜ (40) ʹೖ͢Δͱ ω̂ = n λi ω̄ = ω̄ (41) i=1 ͕Ͱߧۉཱ͢Δɽ ࣜ (39) ͷҙຯ߹͍ɼฏۉΑΓߴ͍࣮࣭ۚͷҬ ͷਓޱ৳ͼͦͷฏ͔ۉΒͷဃʹґଘ͢Δͱ͍͏ ಈతํఔࣜͱͳ͓ͬͯΓɼཱ͕࣍ࣜ͢Δɽ n λ̇i = 0 (c) ਓޱͷফࣦʢࣗ໌ղͱඇࣗ໌ղͱͷަʣ ਤ–2 ಛҟͷྨ ҆ఆߦྻ ⎛ ω1 − ω̄ ⎜ ⎜ 0 ⎜ B=⎜ . ⎜ . ⎝ . ࣗ໌ղͱඇࣗ໌ղͱͷަͷࣜਤΛਤ–2(c) ʹࣔ ͢ɽԣ্࣠ͷࣗ໌ղ (λr = 0) Ͱɼω̄ − ωr ͷූ߸͕ม Խ͠ɼ࣮ઢͰࣔࣜ͢ (9)ɼ͢ͳΘͪɼ(ω̄ − ωr > 0) Λຬ ͨ͢ڐ༰͞ΕΔղ͔ΒɼઢͰࣔࣜ͢ (9) Λຬͨ͞ͳ͍ 0 ω2 − ω̄ .. . 0 ··· ⎛ ∂(ω1 −ω̄) λ1 ∂λ1 ⎜ ⎜ ∂(ω2 −ω̄) λ ⎜ 2 + ⎜ ∂λ1. ⎜ . . ⎝ ∂(ωn −ω̄) λn ∂λ1 ڐ༰͞Εͳ͍ղ (ω̄ − ωr < 0) ͱҠߦ͢ΔɽҰํɼඇ ࣗ໌ղͰɼλr ͷූ߸͕มԽ͠ɼ࣮ઢͰࣔ͢ڐ༰͞Ε Δղ (λr > 0) ͔ΒઢͰࣔ͢ڐ༰͞Εͳ͍ղ (λr < 0) ͱҠߦ͢Δɽ͜ͷಛҟʹ͍ͭͯɼԼهͷ 2 छྨ ͷղऍ͕Γཱͭ • ඇઢํܗఔࣜ (8) Λຬ͢Δશͯͷղʹண͢Δ ⎞ ··· .. . .. . 0 0 ωn − ω̄ 0 .. . ∂(ω1 −ω̄) λ1 ∂λ2 ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ∂(ω1 −ω̄) ∂λn λ1 . ··· .. . .. . ··· ··· ∂(ωn −ω̄) λn ∂λn ∂(ω2 −ω̄) λ2 ∂λ2 .. .. . .. . ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = diag(ω1 − ω̄, . . . , ωn − ω̄) + diag(λ1 , . . . , λn )Ω ͱɼ͜ͷಛҟ͋ͰذΔɽ = −J • ڐ༰͞ΕΔղ͚ͩʹண͢Δͱɼ͜ͷಛҟڐ (43) Ͱ͋Δʢࣜ (13) ͱ (41) ࢀরʣɽ҆ఆߦྻ B ͷݻ༗͕ ༰͞ΕΔղ͕ંΕ͕ۂΔͱͳ͍ͬͯΔɽ શͯෛͷ࣮෦Λ࣋ͯɼղॴہతʹ҆ఆͰ͋ΔɽҰ ຊจͰɼऀޙͷղऍΛ࠾༻͢Δɽ (4) (42) i=1 ํɼҰͭͰਖ਼ͷ࣮෦Λ࣋ͯɼෆ҆ఆͰ͋Δɽߦྻ B = −J Ͱ͋ΔͷͰɼϠίϏߦྻ J ͷݻ༗͕શͯਖ਼ ҆ॴہఆੑ ͷ࣮෦Λ࣋ͯɼղॴہతʹ҆ఆͰ͋Δɽ ຊϞσϧ Fujita et.al. ʹ͕͍ͨ͠ɼ࿑ಇऀฏۉҎ ্ͷ࣮࣭ۚΛఏ͢ڙΔҬʹྲྀೖ͠ɼฏۉҎԼͷ࣮ 5. ࣭ۚΛఏ͢ڙΔҬ͔Βྲྀग़͢ΔಈֶաఔΛԾఆ ܈తذཧʹΑΔذͷهड़ ͢ΔɽΑΓ۩ମతʹɼਐԽήʔϜཧͰ௨ৗ༻͍Β ରশੑΛ࣋ͭܥͷذͷҰൠཧͰ͋Δ܈తذ ΕΔ Replicator dynamics Ͱ͋Δɽ͜ͷաఔͷ҆ఆੑ ཧʹΑΓɼຊϞσϧͷذͷΈΛ໌Β͔ʹ͢Δɽ 17),18) ݅ɼCP Ϟσϧͷߧۉͷॴہత҆ఆੑ݅ Λ ୈ 3 ষͰհͨ͠ࢉܭذཧ͕ݱذͷղۂઢΛ ಋೖ͠ɼͦͷ҆ఆੑ݅Λํߧۉఔࣜ (8) ͷϠίϏߦྻ ٻΊΔ͜ͱΛ௨ͯ͠ఆྔతͳଆ໘Λଊ͑Δಓ۩ͱͯ͠ Λ༻͍ͯఆͰ͖ΔɽCP Ϟσϧͷߧۉͷ҆ॴہఆੑ ༗༻Ͱ͋Δͷʹର͠ɼ͜ͷষͰհ͢Δ܈తذཧ ݅ 17),18) Λಋೖ͠ɼͦͷ҆ఆੑ݅Λํߧۉఔࣜ (8) ʹذ͏ରশੑԼͷΈΛఆੑతʹଊ͑Δ ͷϠίϏߦྻΛ༻͍ͯද͢͜ͱͱ͢Δɽ҆ఆͳղͰ ͷʹ༗༻Ͱ͋Δɽ ෆࣜ݅ࣜ (9) ͕ࣗಈతʹຬ͞Εɼ͞Βʹɼذ (1) ղੳͱ߹͢Δ͜ͱ͕ɼ͜ͷ҆ఆ݅ͷ༏Εͨಛ࣭Ͱ ํఔࣜͷରশੑ ͋Δɽ҆ఆղ͕͜ىʹ࣮ݱΓ͑ΔղͰ͋ΓɼҰํɼ҆ Ұ༷ͳ҆ఆঢ়ଶʹ͋Δ͕ܥɼύϥϝʔλ͕͋ΔΛ ఆͰͳ͍ղɼ͢ͳΘͪෆ҆ఆղʹ࣮ݱ͜ىΓ͑ͳ ͑ΔͱҰ༷ͳঢ়ଶ͕ෆ҆ఆԽ͠ɼ͋ΔύλʔϯΛ࣋ 558 ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 ͭผͳ҆ఆղ͕ग़͢ݱΔ͜ͱ͕ΒΕ͍ͯΔɽ͜Ε ͜ͷϞσϧͷରশੑɼਖ਼ n ֯ܗͷରশੑΛද͢ 2 ໘ ύλʔϯܗͱݺΕΔݱͰ͋ΓɼͦͷΈΛه ମʹ܈ΑΓهड़͞ΕΔɽ 2 ໘ମ ܈Dn ड़͢Δཧͱͯ͠܈తذཧ͕ൃలͨ͠ɽ Dn = {e, rn , . . . , rnn−1 , s, srn , . . . , srnn−1 } ඇઢํܗఔࣜ (10) ͷରশੑɼ͋Δؔ͢ʹ܈Δಉมੑ T (g) F(λ, ω̄, f ) = F(T (g)λ, ω̄, f ), ∀ g∈G (44) (47) ͱఆٛ͞ΕΔɽ͜͜Ͱ͜ͷࣜͷӈลͷ֤ཁૉԼهͷ زԿֶతରশੑΛද͢࠲ඪม͋ͰΔɽ ʹΑΓද͞ΕΔɽ͜͜ʹ T (g) ରশੑΛද͢࠲ඪม • e : Կૢ࡞͠ͳ͍ม߃( ม) • rn : ਖ਼ n ֯ܗͷத৺ͷ·ΘΓʹ࣌ ʹํܭ2π/n ճస͢Δม( ฒਐม) ߦྻͰ͋Γɼg ڸөมճసมͲͳΛද͢࠲ඪม ཁૉͰ͋ΓɼG ͜ͷཁૉ͔ΒͳΔ͋Δ͋Ͱ܈Δɽͨ ͩ͠ɼω̄ ͱ F (λ) g ͕Ҿ͖࠲͢͜ىඪมཁૉʹର͠ ม͢Δ͜ͱͱɼࣜ F શମΛ T (g) Ͱมͨ͠ͷͱ • rni : มΛ rn Λ i ճߦ͏ɼ2iπ/n ճస͢Δม • ্࣠ʹڸΛஔ͍ͯөͨ͠૾ͷมڸ( өม ): s ͕ಉҰͰ͋Δ͜ͱΛද͢ҰൠతͳزԿֶతରশ݅Ͱ • srni : 2iπ/n ճసʹޙɼڸөΛߦ͏ม ͯෆมͰ͋Δɽࣜ (44) ɼม λ Λ T (g) ʹΑΓ࠲ඪ ਤ–3 ʹɼn = 6 ͷ߹ͷ࠲ඪมཁૉΛࣔ͢ɽ ͋Δɽ ಉม݅ࣜ (44) ͕Γཱͭͱ͖ɼ૿ํఔࣜ (11) ͷ n ࢢϞσϧʹରԠ͢Δ ܈Dn ෆมͳղ͔Βɼରশ ܈ಉมੑΑΓɼT (g)λ = λ Λຬ͢Δ G ෆมͳ͋Δղ ੑ͕͍ղɼ͢ͳΘͪذղ͕ʹذΑΓൃੜ͢Δɽ͜ (λ, f ) ʹରͯ͠ɼϠίϏߦྻɼରশ݅ࣜ ͷذղɼ ܈Dn ͷ෦ରশੑΛද͢෦܈ɼ͢ͳΘ T (g)J = JT (g), ∀ g∈G ͪɼ࣍ m ͕ n ͷͰ͋Δ 2 ໘ମ܈ (45) i i k−1 Dkm = {rm , srm rn | i = 0, 1, . . . , m − 1} Λຬ͠ɼ∂F/∂ ω̄ ͱ ∂F/∂f ɼෆม݅ࣜ ∂F ∂F = , ∂ ω̄ ∂ ω̄ Λຬ͢Δɽ T (g) T (g) ∂F ∂F = , ∂f ∂f ∀ g∈G (k = 1, 2, . . . , n/m) (46) ८ճ܈ i Cm = {rm | i = 0, 1, . . . , m − 1} ಉม݅ࣜ (44) Λຬ͢ΔܥͷʹذΑΔύλʔϯ ࣠ͷݸΛɼk ͕࣠ͷํΛද͢ɽͪͳΈʹɼn = 4 ʹ • ܥͷରশੑذΛͰ·͢͜ىอ࣋͞ΕΔɽ • ܥͷରশੑղͷݸͰذมԽ͢Δɽ ର͢ΔذղͷରশੑΛද͢෦܈Λਤ–4 ʹࣔ͢ɽਤ தͷ˓ͷେ͖͞ʹΑΓɼਓޱͷେ͖͞Λࣔ͢ɽਤதɼ෦ • ܥͷ҆ఆੑಛҟͰมԽ͢Δɽ • ௨ৗͰɼܥͷରশੑɾ҆ఆੑղͷݸม ܈D12 ࣼΊํͷ 2 ຊͷڸө࣠ʹؔͯ͠ରশͳɼݪ ΛڬΜͰରቂ͢Δࢢ͕ಉҰͷਓޱΛ࣋ͭͷର Խ͠ͳ͍ɽ শੑΛද͋͢Ͱ܈Δɽ෦ ܈D11 ࣼΊํͷ 1 ຊͷڸ ۩ମతͳذͷΈɼରͱ͢ΔͲ͕ܥͷΑ͏ ө࣠ʹؔͯ͠ରশͳਓޱͷରশੑΛද͋͢Ͱ܈Δɽ ͳରশੑΛ͔࣋ͭʹΑΓҟͳΔݸผͰ͋Γɼܥຖʹ ෦ ܈D31 ࣼٯΊํͷ 1 ຊͷڸө࣠ʹؔͯ͠ରশͳ ௐΒΕ͍ͯΔɽͦͷ۩ମྫɼຊจͰऔΓѻ͏ԁ ਓޱͷରশੑΛද͋͢Ͱ܈Δɽ෦ ܈D21 ॎํ प্ʹִؒʹҐஔ͢ΔࢢͷରশੑΛද͢ 2 ໘ମ܈ ͷ 1 ຊͷڸө࣠ʹؔͯ͠ࠨӈରশͳਓޱͷରশੑ ʹରͯ͠հ͢ΔɽͪͳΈʹɼଞͷରশੑΛ࣋ͭࢢ Λද͋͢Ͱ܈Δɽ෦ ܈D41 ԣํͷ 1 ຊͷڸө࣠ʹ ͷذղੳʹରͯ͠܈తذཧద༻ՄೳͰ͋ ্ؔͯ͠ԼରশͳਓޱͷରশੑΛද͋͢Ͱ܈Δɽ Δ͕ɼਖ਼ n ֯ܗঢ়ͷରশੑΛද͢ 2 ໘ମ܈Ҏ֎ͷʹ܈ ؔ͢Δղੳ͕ඞཁͱͳΔɽ (3) ຊͰڀݚѻ͏Ϟσϧɼਤ–3 ʹࣔ͢Α͏ͳ n ݸͷ ࢢ͕ԁपʹԊִͬͯؒʹଘࡏ͢Δঢ়گΛߟ͍͑ͯΔɽ ༷ʹɼܦذ࿏͔Β͞Βʹ͢ذΔܦ࿏ͷରশੑΛٻ Ί͍ͯ͘͜ͱʹΑΓɼ֊తذͷنଇΛٻΊΔ͜ͱ ͕Ͱ͖Δ14) ɽ UT UT UT ˳T ଟஈ֊ͷରশੑഁյذ Dn ෆมͳܦ࿏ͷ͔ذΒޙͨ͠ذͷରশੑ Լ͠ɼͦͷରশੑ Dn ͷ෦ʹ܈ΑΓද͞ΕΔ3 ɽಉ 2 ໘ମ܈ UT (49) ʹؔͯ͠ෆมͰ͋Δɽ ܈Dkm ʹ͓͍ͯɼm ͕ڸөมͷ ܗԼهͷੑ࣭Λຬͨ͢14) ɽ (2) (48) ࢢ n = 4 ʹର͢ΔɼҬ͕શͯಉ͡ਓޱΛ࣋ͭ UT ࣗ໌ղ (λ1 = · · · = λ4 = 1/4) ͔ΒͷذͷࣜਤΛ U ਤ–5 ʹࣔ͢ɽਤதҰ൪ࠨଆͷ D4 ෆมͳ 4 ࢢ͕Ձ ͳղ͔Βɼ୯७ ͱذ2 ॏ ͏͍ͱذ2 छྨͷ͕ذ 3 ਤ–3 ਖ਼ 6 ֯ܗঢ়ʹҐஔ͢ΔࢢͱͦͷزԿֶతม 559 ͜ͷରশੑഁյذɼࢢͷذͰɼಉҰͷਓޱΛ࣋ ͭࢢͷݮগΛ௨ͯ͡ɼਓޱͷूੵΛ༠ൃ͢Δɽ ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 UT UT U UT 㧰 㧰 㧰 㧰 㧰 㧰 ਤ–6 ࢢ n = 8 ͷ߹ͷपظഒذͷࣜਤ 㧰 㧰 㧰 㧰 ͱ͔ΒपظഒݺͱذΕ͍ͯΔɽਤ–6 ʹࢢ n = 8 = 23 ͷ߹ͷपظഒذͷࣜਤΛࣔ͢ɽ (a) ذղͷରশ܈ 㧰 㧰 㧰 6. 㧰 㧰 4 ࢢϞσϧͰͷذͷهड़ྫ ࣜ (45) ͷϠίϏߦྻͷରশ݅ΛຊϞσϧʹରͯ͠ (b) ࣗ໌ղͷରশ܈ ద༻͢Δ͜ͱʹΑΓɼຊϞσϧͷذͷΈΛௐ ਤ–4 n = 4 ͷ߹ͷذղͱࣗ໌ղͷରশ܈ ͯΈΔɽ͜͜Ͱɼ؆୯ͳྫͱͯ͠ࢢ n = 4 ͷ ⥄⸃ߣߩὐ ߹Λߟ͑Δ͜ͱͱ͢ΔɽͪͳΈʹɼ͜ͷٞҙͷ 㧰 㧰 න⚐ಽጘὐ n(≥ 3) ʹରͯ͠༰қʹ֦ுՄೳͰ͋Δɽ ⥄⸃ߣߩὐ න⚐ಽጘὐ 㧰 㧰 㧰 ㊀ಽጘὐ 㧰 㧰 (1) ૿ࢧํఔࣜͷରশੑͱରশ࠲ඪܥͷม 㧰 㧰 n = 4 ͷ߹ʹɼະมϕΫτϧΛ λ = (λ1 , λ2 , λ3 , λ4 )T ⥄⸃ߣߩὐ 㧰 න⚐ಽጘὐ ͱͱΔͱɼࣜ (44) ͷදྻߦݱ࣍ࣜͱͳΔɽ ⎞ ⎛ ⎛ 0 0 0 1 0 1 0 ⎟ ⎜ ⎜ ⎜0 0 1 0 ⎟ ⎜0 0 1 ⎟ ⎜ T (s) = ⎜ ⎜0 1 0 0⎟ , T (r4 ) = ⎜0 0 0 ⎠ ⎝ ⎝ 1 0 0 0 1 0 0 㧯 ਤ–5 4 ࢢ (n = 4) ͷଟஈ֊ͷରশੑഁյذͷࣜਤ ͜ىΓ͑Δ͜ͱΛ͜ͷਤ͍ࣔͯ͠Δʢ͜ͷ 2 छྨͷ ذͷৄࡉɼ࣍ষͰ༩͑Δʣɽ୯७Ͱذɼ෦ ܈D12 ࣼΊํͷ 2 ຊͷڸө࣠ʹؔͯ͠ରশͳɼݪ (51) ⎞ 0 ⎟ 0⎟ ⎟ 1⎟ ⎠ 0 (52) Ұ༷ͳࣗ໌ղ (λ1 , λ2 , λ3 , λ4 ) = (1/4, 1/4, 1/4, 1/4) ΛڬΜͰରቂ͢Δࢢ͕ಉҰͷਓޱΛ͕࣋ͭ j = 1, . . . , 4) ࣼΊ ্ʹ͓͍ͯɼϠίϏߦྻ J = (Jij | i, j = 1, . . . , 4) Λ ํͷ 1 ຊͷڸө࣠ʹؔͯ͠ରশͳ 4 छྨͷਓޱ ༠ಋ͠ɼରশ݅ (45) Λ༻͍ΔͱɼϠίϏߦྻͷ۩ମ ੜ͢Δɽ2 ॏͰذɼ෦܈ Dj1 ͕ܗ ʢਤ–4 (a)ʣͷɼ͍ͣΕ͔Ұ͕ͭൃ͢ݱΔɽͦͷޙɼ࣮ ⎛ a ઢͷҹͰࣔ͢Α͏ͳଟஈ֊ͷذΛʹͱ͜͢͜ىΑ ⎜ ⎜b J =⎜ ⎜c ⎝ Γɼ৭ʑͳूੵύλʔϯ͕ੜΈग़͞Ε͍ͯΔɽ͜ͷछ ͷਤΑΓɼਓ ͕ޱn ࢢʹ͍ͯ͠Δঢ়ଶ͔ΒҰۃ b ूத͢Δ·ͰͷذաఔͷΈΛΔ͜ͱ͕Ͱ͖Δɽ ࢢ n = 4 ͷࣗ໌ղɼਤ–4 (b) ʹࣔ͢ 5 ͭͰ͋ b c b ⎞ a b b a ⎟ c⎟ ⎟ b⎟ ⎠ c a b (53) ͱ͍͏ͳʹܗΔɽ͜͜Ͱɼa = −ω1,1 /4 = · · · Ͱ͋Δɽ Δʢࢢ 1(ӈ্) ͕ਓ࠷ޱେͱͨ͠ͱ͖ʣɽ͜ͷϞσϧ ·ͨɼରশ݅ࣜ (46) ΑΓɼ ɼ͋ΔҬͷਓ͕ޱθϩͱͳΔ߹ʹࣗ໌ղͱަΘ ∂F ∂F = (d, d, d, d)T , = (e, e, e, e)T ∂ ω̄ ∂f Δ߹͕͋Δͱ͍͏ಛΛ͍ͬͯΔɽਤதʹઢͷ ҹʹΑΓɼ͜ͷछͷަΛࣔ͢ɽ (54) ͱͳΔɽ͜͜Ͱɼ (4) पظഒذ d=− ଟஈ֊ͷରশੑഁյذͷදྫͱͯ͠ɼपظഒ ͛ڍ͕ذΒΕΔɽ͜ͷذɼn = 2k ͷ߹ʹൃੜ͠ɼ D2k −→ D2k−1 −→ D2k−2 −→ · · · 1 ∂ω 1 ∂ω , e=− 4 ∂ ω̄ 4 ∂f (55) Ͱ͋Δɽ ରশ࠲ඪܥͷ࠲ඪม (50) λ = HQ ͱ͍͏ͰܗରশੑΛࣦ࣍͢ΔͰܗਐߦ͢Δɽذ (56) Λߟ͑Δɽ͜͜ͰɼQ = (Q1 , . . . , Q4 )T ରশ࠲ඪܥ ʹΑΓɼಉҰͷύλʔϯ͕ݱΕΔप͕ظഒʑʹͳΔ͜ 560 ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 Ͱ͋Γɼ࠲ඪม ྻߦH ࣍ࣜͱͳΔɽ ⎞ ⎛ 1/2 1/2 1/2 1/2 ⎟ ⎜ ⎜−1/2 1/2 −1/2 1/2⎟ ⎟ ⎜ H=⎜ ⎟ 1/2 −1/2 −1/2 1/2 ⎠ ⎝ −1/2 −1/2 1/2 1/2 㧰 㧰 (a) ୯७ذ (57) 㧰 㧰 㧰 㧰 㧰 ͜ͷ࠲ඪม͕ྻߦରশੑΛ࣋ͭྻϕΫτϧͱͳͬͯ ͍Δ͜ͱ͔ΒɼQ Λରশ࠲ඪͿݺͱܥɽ (b) 2 ॏذ ૿ࢧํఔࣜ (11) Λ͜ͷ࠲ඪʹܥม͠ɼϠίϏ ਤ–7 4 ࢢۉͳঢ়͔گΒͷରশੑഁյذͷࣜਤ ߦྻ J ͷ۩ମ( ܗ53) Λ༻͍ΔͱɼԼهͷΑ͏ʹมͰܗ ͖Δɽ (2) T H δF(δλ, δf ) ∂F ∂F δ ω̄ + H T δf = H T JHδQ + H T ∂ ω̄ ∂f ⎛ ⎞ e1 0 0 0 ⎜ ⎟ ⎜ 0 e2 0 0 ⎟ ⎜ ⎟ δQ =⎜ ⎟ ⎝ 0 0 e3 0 ⎠ 0 0 0 e4 ⎛ ⎞ ⎛ ⎞ 0 0 ⎜ ⎟ ⎜ ⎟ ⎜0⎟ ⎜0⎟ ⎜ ⎟ ⎟ +⎜ ⎜ 0 ⎟ δ ω̄ + ⎜ 0 ⎟ δf = 0 ⎝ ⎠ ⎝ ⎠ 2e 2d ࣜ (61)∼(64) ΑΓɼҰ༷ղ λ1 = · · · λ4 = 1/4 ্ͷ ղ a) ௨ৗղ ୯७ذ (65) 2 ॏذ ௨ৗղ ௨ৗղҰ༷ղ λ1 = · · · λ4 = 1/4 ͔ΒذΛআ֎ ͨ͠ͷͰ͋Γɼδ ω̄ ͱ δf ͷؔɼࣜ (64) ʹΑΓ༩ (58) ͑ΒΕΔɽ b) ⎞ 1/2 ⎟ ⎜ ⎟ ⎜ ⎜δλ2 ⎟ ⎜−1/2⎟ ⎟ ⎜ ⎜ ⎟ δλ = ⎜ ⎟ ⎟=⎜ ⎝δλ3 ⎠ ⎝ 1/2 ⎠ −1/2 δλ4 δλ1 (60) ͱͳΔɽ͜ͷ݅ࣜ δQ4 = 0 Λ༻͍Δͱɼ૿ํఔࣜ (58) ɼԼهͷ 4 ͭͷࣜʹղ͢Δ (e2 = e3 )ɽ (62) e3 δQ3 = 0 (63) 2d δ ω̄ + 2e δf = 0 (64) (66) 2 ॏذ c) ਓޱҰఆͷ݅ࣜ (5) ɼࣜ (56) ͷରশ࠲ඪͰܥɼ e2 δQ2 = 0 ⎛ ७ذͷࣜਤΛࣔ͢. ͱͳΔɽ (61) ⎞ ͷํʹ 1 ຊͷذղ͕ଘࡏ͢Δɽਤ–7(a) ʹ͜ͷ୯ Ͱ͋Γɼ͜ͷݻ༗͕θϩͱͳΔ ͕ߧۉ2 ॏذ e1 δQ1 = 0 ୯७ذ ୯७Ͱذɼ ⎛ Ͱ͋Δɽݻ༗ e2 , e3 ඞͣಉҰͷΛऔΔ 2 ॏݻ༗ 2 ॏͰذɼ ⎞ ⎛ √ ⎞ ⎛ 1/2 1/ 2 ⎟ ⎟ ⎜ ⎜ ⎜ 0 ⎟ ⎜−1/2⎟ ⎟, ⎜ ⎟ ⎜ √ ⎟, ⎜ δλ = ⎜ ⎟ ⎝−1/ 2⎠ ⎝−1/2⎠ 1/2 0 ⎛ ⎞ 0 ⎜ √ ⎟ ⎜ 1/ 2 ⎟ ⎜ ⎟ ⎜ 0 ⎟, ⎝ √ ⎠ −1/ 2 ⎞ ⎛ −1/2 ⎟ ⎜ ⎜−1/2⎟ ⎟ ⎜ ⎜ 1/2 ⎟ ⎠ ⎝ 1/2 (67) ͷ 4 ͭͷํʹ 4 ຊͷذղ͕ଘࡏ͢Δɽਤ–7(b) ʹ ͜ͷ 2 ॏذͷࣜਤΛࣔ͢. d) ଟஈ֊ͷذ ਤ–7 ͷΑ͏ͳذͷΈΛذղʹରͯ͠ٻΊ ͯߦ͘͜ͱʹΑΓɼਤ–5 ʹࣔ͢Α͏ͳଟஈ֊ͷذͷ ࣜ (61)∼(64) ϒϩοΫର֯ݺͱܗΕΔඪ४Ͱܗ ΈΛٻΊΔ͜ͱ͕Ͱ͖Δɽ ͋ΓɼҟͳΔରশੑΛ࣋ͭϒϩοΫຖ4 ʹ͕ࣜղ͢Δ ΈΛ͓ࣔͯ͠ΓɼରশੑΛ࣋ͭܥͷذͷٞʹ 7. ͓͍ͯॏཁͰ͋Δ19) ɽ 4 ⎧ ⎪ 0, e2 = e3 = 0 ⎨ e1 = e1 = 0 ⎪ ⎩ e2 = e3 = 0 ͱྨͰ͖Δɽ ͜͜ͰɼϠίϏߦྻ J ͷର֯ɼ͢ͳΘͪɼݻ༗ ⎧ ⎪ ⎨ e1 = a + c − 2b : (୯७ࠜ) (59) e2 = e3 = a − c : (2 ॏࠜ) ⎪ ⎩ e4 = a + c + 2b : (୯७ࠜ) δλ1 + δλ2 + δλ3 + δλ4 = 2δQ4 = 0 ذղͷྨ ذղੳ݁Ռ ຊষͰɼ༌ૹඅͷมԽʹ͏ɼຊϞσϧͷߧۉͷ ͜ͷ߹ʹɼ͝ͱʹ͕ࣜղ͍ͯ͠Δ͕ɼҰൠతʹϒ ϩοΫຖʹղ͢Δੑ࣭͕͋Δɽ มԽΛࢉܭذཧʹैͬͯ͢ΔɽҎԼʹࣔ͢ղੳ 561 ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 ݁Ռɼύϥϝʔλͷҧ͍ʹΑΔӨڹΛݟΔਤ–10 Λ 㧰 আ͍ͯɼμ = 0.4ɼσ = 10.0 ͱݻఆͯ͠ղੳΛߦͬͨɽ ͜͜Ͱɼμ ͱ σ ҙʹܾΊΒΕΔύϥϝʔλͰ͋Γɼ 㧰 㧯 㧯 㧰 1 㧰 μ ۀͷࢧग़ׂ߹Λࣔ͢ͱಉ࣌ʹࡁܦશମͰͷ ۀਓޱͷΊΔׂ߹Λࣔ͠ɼσ ۀͷଟ༷ੑΛ ੱญᲧ₸ બ͢Δ߹͍Λࣔ͢ύϥϝʔλͰ͋Δɽ (1) 0.5 㧰 㧰 4 ࢢϞσϧ 㧰 4 ࢢϞσϧͷذղੳ݁Ռʹ͍ͭͯ·ͱΊΔɽ a) 㧰 㧰 ෳߧۉղͷൃݱ 0 0 4 ࢢϞσϧͷذղੳʹΑΓٻΊͨղۂઢΛਤ– 0.5 ャㅍ⾌ 8(a) ʹࣔ͢ɽਤͷॎ࣠͋Δࢢͷਓޱʢྫ͑ λ1 ʣ Λࣔ͠ɼԣ࣠ʹ༌ૹඅΛࣔ͢ɽ͜͜Ͱɼԣ࣠ͷ༌ૹඅ (a) ߧۉղۂઢ ༌ૹඅύϥϝʔλ τ ͷؔ 1 − 1/eτ π ͷΛ͓ࣔͯ͠ 㧰 Γɼ༌ૹඅ 0 Ͱࢢؒʹશ͘༌ૹඅ͕͔͔Βͳ͍ঢ় න⚐ಽጘ ଶΛɼ༌ૹඅ 1 Ͱ༌ૹඅ͕େ͖͘ࢢؒͷަྲྀ͕গ ㊀ಽጘ ͳ͍ঢ়ଶΛ͍ࣔͯ͠Δɽ·ͨɼ࣮ઢ҆ఆͳܦ࿏ɼഁઢ 㧰 ෆ҆ఆͳܦ࿏Λ͍ࣔͯ͠Δɽ֤ܦ࿏ʹ͓͚Δਓޱ 㧰 ΛࣜਤʹΑΓࣔ͢ɽࣜਤʹ͓͍ͯɼࠇ৭Ͱࣔ͢ ूੵύλʔϯ҆ఆղͰ͋Γɼփ৭Ͱࣔ͢ूੵύλʔ 㧰 ϯෆ҆ఆղͰ͋Δɽذࣗ໌ղͱͷަࠩʹ͏ର න⚐ಽጘ শੑͷԼͷࣜਤΛਤ–8(b) ʹࣔ͢ɽ 㧰 㧰 ਤ–8 ͔Β໌Β͔ͳΑ͏ʹɼذΛ܁Γฦͯ͜͠͠ى න⚐ಽጘ 㧰 㧯 න⚐ಽጘ 㧰 ରশੑഁյذ ਤ–9(a) ɼਤ–8 ͷܦ࿏ͷத͔Βදతͳ҆ఆղΛ (b) ʹذΑΔରশੑͷԼͷࣜਤ औΓग़͠ɼਤதӈ্ͷ 2 ॏͷ࢛֯ͷʹࣔ͢ର֯ઢ ਤ–8 ༌ૹඅύϥϝʔλ τ ͷมԽʹ͏ߧۉղͷมԽ (4 ࢢ) ্ͷ 2 ࢢͷਓ ޱλ1 ɼλ3 ͷ༌ૹඅมԽʹΑΔूੵͷ ༷ࢠΛࣔ͢ 3 ࣍ݩϓϩοτͰ͋Δɽܦ࿏্ʹࣔͨ͠ ◦ ۀͷଟ༷ੑΛࣔ͢ύϥϝʔλ σ ͷӨڹ c) ͋ͰذΓɼ• ࣗ໌ղͱͷަΛࣔ͢ɽذ ਤ–9(a) ʹࣔͨ͠ܦ࿏ͷΈͰ 7 ͭͷͰ͓͖ͯىΓɼ ͦΕͧΕͷͰذରশੑΛॱࣦ࣍͠ɼूੵύλʔ ຊઅͰɼۀͷଟ༷ੑΛબ͢Δ߹͍Λࣔ͢ύ ϥϝʔλ σ ͕ߧۉղʹ͢΅ٴͷӨͱ·͍ͯͭʹڹΊΔɽ ͦ͜Ͱɼਤ–9 ʹࣔ͢ղੳ݁Ռʢσ = 10.0ʣͱਤ–10 ʹ ϯʹมԽ͕͍ͯͬ͜ىΔ͜ͱ͕Θ͔Δɽ ༌ૹඅ͕ߴ͍ঢ়ଶʢԖ࣠ͷ༌ૹඅ͕ 1 ʹ͍ۙ߹ʣ Ͱɼ4 ͭͷࢢʹਓۉ͕ޱ͢Δ D4 ෆมͳ 4 ࣠ ରশύλʔϯ͕҆ఆͰ͋Γɼଞͷෆ҆ఆղͷଘࡏ֬ ೝ͞Εͳ͍ɽ͔͠͠ɼ༌ૹඅͷԼʹΑΓ୯७ ذI ࣔ͢ղੳ݁Ռʢσ = 5.0ʣͱΛൺֱ͢Δɽ σ = 5.0ɼ10.0 ͱʹ 4 ࢢʹۉʹ͢Δঢ়ଶͰ ͋Δ D4 ͔ΒରশੑΛॱࣦ࣍͠ɼ1 ࢢͷूੵΛى ͓ͯ͜͠Γɼશͯͷ͕ذਤ–5 ʹࣔͨ͠نଇʹैͬͯ ͍Δɽ͔͠͠ɼσ = 5.0ʢਤ–10 ʣͷ߹ɼD12 ʹ͓͚Δ ʹ౸ୡ͢ΔͱɼذΛ͜͠ىɼ4 ࣠ରশύλʔϯ͕ෆ҆ 2 ࢢߧۉͷࣗ໌ղ͕ଘࡏͤͣɼࣗ໌ղҠߦ͢ΔҎલ ఆͱͳΓɼ2 ࣠ରশͷର֯ 2 ࢢʹۉʹ͢Δ D12 ʹ D11 ͱ͍ͯ͠ذΔɽ͔͠͠ɼ͜ͷ߹ (68) ͷ ෆมͳύλʔϯ͕҆ఆͳղͱͳΔɽ͞Βʹ༌ૹඅ͕ ଟஈ֊ͷذͷنଇʹै͍ͬͯΔɽ Լ͢Δͱɼ୯७ ذIII ʹ͓͍ͯɼରশੑഁյ͕ذ ͜ىΓɼ҆ఆঢ়ଶ͕ 1 ͭͷࢢʹूੵ͢Δ D11 ෆมͳύ (2) λʔϯΛͯܦҰूۃதͱҠߦ͢Δɽ͜ͷঢ়گͷࣜਤ पظഒذ ਤ–9(b) ɼਤ–10(b) ͕ࣔ͢Ұ࿈ͷूੵաఔͰɼ Λਤ–9(b) ʹࣔ͢ɽ͜ͷҰ࿈ͷूੵύλʔϯͷมԽ D4 −→ D12 −→ D11 㧰 㧯 ͓Γɼଟ͘ͷߧۉղ͔ΒͳΔෳࡶͳڍಈΛ͍ࣔͯ͠Δɽ b) 1 4 ࢢۉ˰ 2 ࢢͷूੵ˰Ұूۃத (68) (69) ͱ͍͏ɼ͍ΘΏΔɼपظഒݺͱذΕΔɼෳࡶܥͷ ͱ͍͏ଟஈ֊ͷʹذΑΓҾ͖͜͞ىΕ͍ͯΔɽ యܕతͳڍذಈ͕ൃੜ͍ͯ͠Δɽ 562 ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 㧰 ャㅍ⾌ 㧰 1 dz1 㧯 1 dz3 㧰 㧰 㧰 + 0 +8 dz3 1 dz1 1 㧰 㧰 ੱญᲧ₸ ++ +++ 㧰 0.5 㧯 (a) 3 ࣍ݩਤʢҰ෦ൈਮʣ 㧰 න⚐ಽጘὐ + 㧰 ⥄⸃ߣߩ ὐ ++ 㧰 න⚐ಽጘὐ +++ 㧰 0 㧰 ⥄⸃ߣߩ ὐ +8 0 0.5 ャㅍ⾌ (b) ʹذΑΔूੵύλʔϯͷมԽ (a) શܦ࿏ ਤ–9 4 ࢢϞσϧͷߧۉղͷ 3 ࣍ݩϓϩοτʢσ = 10.0ʣ 㧰 1 㧰 ャㅍ⾌ dz1 㧰 1 1 㧰 dz3 㧰 K ੱญᲧ₸ 㧰 㧰 㧰 KKK KK 㧰 KX 㧰 0.5 㧰 0 1 dz1 1 㧰 㧰 㧰 dz3 㧰 㧰 0 0 0.5 ャㅍ⾌ (a) 3 ࣍ݩਤʢҰ෦ൈਮʣ 1 (b) पظഒܦذ࿏ 㧰 න⚐ಽጘὐ K 㧰 න⚐ಽጘὐ KK 㧰 ⥄⸃ߣߩ ὐ KKK 㧰 ⥄⸃ߣߩ ὐ KX 㧰 න⚐ಽጘ (b) ʹذΑΔूੵύλʔϯͷมԽ න⚐ಽጘ 㧰 ਤ–10 4 ࢢϞσϧͷߧۉղͷ 3 ࣍ݩϓϩοτʢσ = 5.0ʣ 㧰 㧰 ࢢΛ 8 ࢢʹ૿Ճͤͯ͞ղੳͨ݁͠ՌΛਤ– 11(a) ʹࣔ͢ɽࢢͷ૿ՃʹΑΓɼ͞Βʹෳࡶͳ ඈ༂తʹ૿Ճ͍ͯ͠Δɽ͔͠͠ɼਤ–11(b) ʹࣔ͢Α ͏ʹɼ 㧰 㧰 න⚐ಽጘ ㊀ಽጘ න⚐ಽጘ ㊀ಽጘ 㧰 㧯 න⚐ಽጘ න⚐ಽጘ 㧰 ߏذΛͭΑ͏ʹͳΓɼ͋Δ༌ૹඅʹ͓͚Δߧۉղ 㧰 න⚐ಽጘ න⚐ಽጘ 8 ࢢۉ˰ 4 ࢢͷूੵ ˰ 2 ࢢͷूੵ˰Ұूۃத 㧰 㧯 㧰 㧰 㧰 㧰 㧯 㧰 න⚐ಽጘ (70) 㧰 㧰 ͱ͍͏पظഒൃ͕ذੜ͍ͯ͠Δɽ·ͨɼਤ–12 ʹࣔ න⚐ಽጘ ͢ 16 ࢢϞσϧʹରͯ͠पظഒൃ͕ذੜ͍ͯ͠Δɽ ͜ͷΑ͏ʹɼTabuchi and Thisse11) ͕ຊͱڀݚҟͳΔ 㧯 න⚐ಽጘ 㧰 㧰 㧰 න⚐ಽጘ 㧯 㧰 ϞσϧͰͦͷൃੜΛࣔͨ͠पظഒ͕͜ذͷϞσϧͰ 㧰 ൃੜ͓ͯ͠Γɼࢢͷूੵɾࢄݱͷ NEG Ϟσϧ (c) ʹذΑΔରশੑͷԼͷࣜਤ ʹ͓͍ͯपظഒ͕ذҰൠతͳੑ࣭Ͱ͋ΔՄೳੑΛࣔ ͍ࠦͯ͠Δ. ਤ–11 8 ࢢϞσϧͷղੳ݁Ռ 563 ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 㧰 1 㧭 㧭 㧮 㧮 㧯 ੱญᲧ₸ 㧰 㧯 㧰 0.5 㧰 㧰 㧰 㧰 㧰 㧰 0 0 0.5 ャㅍ⾌ 㧰 㧰 㧱 㧱 㧲 0 1 ャㅍ⾌ 㧲 㧳 㧳 1 ਤ–13 ઢࢢܗͷ҆ఆύλʔϯͷมભʢ5 ࢢʣ(μ = 0.4ɼ σ = 5.0) ਤ–12 पظഒܦذ࿏ʢ16 ࢢϞσϧʣ ͍ɼύλʔϯ G ͔Βύλʔϯ F ͋Δ͍ E ʹࢢ ύλʔϯ͕มભ͢Δɽ࣍ʹɼύλʔϯ F ͔Βύλʔ 8. ༌ૹඅԼʹ͏ࢢͷूੵɾࢄաఔɿ ࢢͷҐஔɼࢢ͓Αͼͦͷن ϯ EɼC ͷ͍ͣΕ͔ʹɼύλʔϯ E ͔Βύλʔϯ Dɼ CɼB ͷ͍ͣΕ͔ʹมભ͢Δɽ͞Βʹɼύλʔϯ D ͔ Βύλʔϯ CɼBɼA ͷ͍ͣΕ͔ʹมભ͢ΔɽͲͷύ 7 ষͰɼଟࢢ͕ԁप্ʹִؒʹଘࡏ͢ΔϞσϧ ʹ͓͍ͯɼࢢͷूੵɾࢄաఔʹରশੑഁյपظ λʔϯؒͷมભʹ͓͍ͯɼ͢Δࢢʹྡͷ ഒ͏͍ͱذಛ͕͋Δ͜ͱΛ໌Β͔ʹͨ͠ɽຊষͰ ʮετ ͕ͬͯɼઢ ʹ্ܗ5 ࢢ͕Ґஔ͢ΔϞσϧͰɼ ɼଟࢢϞσϧʹݱΕͨࢢͷूੵɾࢄաఔʢଘ ϩʔޮՌʯ͕ɼԁप্ʹࢢ͕Ґஔ͢ΔϞσϧͱಉ༷ ଓ͢ΔࢢͷҐஔɼࢢ͓Αͼنʣʹ͍ͭͯɼಛ ʹੜ͍ͯ͡Δɽ ࢢਰୀɼ͋Δ͍ݩʑਓޱθϩͰมԽ͕ͳ͍ɽͨ͠ ʹ 2 ࢢϞσϧͰ͋Δ CP ϞσϧͰݟΒΕͳ͔ͬͨ աఔʹֶ͍ͭͯࡁܦత͔؍Βࢦఠ͢ΔɽࢢͷҐஔ (2) ͓Αͼࢢ 2 ࢢϞσϧͰੳෆՄೳͳ߲Ͱ ෳଘࡏ͢Δࢢूੵաఔʢߧۉύλʔϯͷมભ ύεʣͱ࣮ߧۉ͍ͳ͠ݱύλʔϯ ͋Δɽͳ͓ɼ༌ૹඅ༻ͷߴ͍ঢ়͔گΒ͍ঢ়گͷม ༌ૹඅԼʹ͏ࢢूੵաఔΛݟΔͨΊʹɼ4 ࢢ Խʹ͏ूੵɾࢄաఔΛΈΔɽಛɼେ͖͚͘ Ϟσϧɼ8 ࢢϞσϧʹ͍ͭͯɼͦΕͧΕਤ–14 ɼਤ– ͯԼهͷ 2 ͋Δɽ 15 ʹ҆ఆͳߧۉύλʔϯͷΈΛࣔͨ͠ɽ༌ૹඅԼʹ ͏҆ఆͳߧۉύλʔϯͷมભ͕ࢢूੵաఔͰ͋Δɽ (1) ͢Δࢢʹྡ͢Δࢢͷਰୀ (2) ෳଘࡏ͢Δࢢूੵաఔʢߧۉύλʔϯͷมભ ύεʣͱ࣮ߧۉ͍ͳ͠ݱύλʔϯ ਤ–14 ʹࣔͨ͠ 4 ࢢϞσϧΛͯݟΈΔͱɼ༌ૹඅ ͷߴ͍ঢ়Ͱگɼύλʔϯ E ͷਓޱࢄঢ়ଶ͕࣮͠ݱɼ ֤ʹ͍ͭͯͦΕͧΕҎԼͰઆ໌͢Δɽ ͦͷޙɼύλʔϯ D ʹมભ͢Δɽ࣍ɼύλʔϯ CɼB ͷ͍ͣΕ͔ʹมભ͠ɼ࠷ޙύλʔϯ A ͷҰूۃதঢ় (1) ͢Δࢢʹྡ͢Δࢢͷਰୀ ଶͱͳΔ͜ͱ͕Θ͔Δɽ͕ͨͬͯ͠ɼࢄ͔Βूத ͷύλʔϯͷมભύε͕ෳଘࡏ͍ͯ͠Δɽ ਤ–12 ɼཱՄೳͳࢢ͕ 16 ͷ߹ͷܦذ࿏ ͷҰ෦Λ͍ࣔͯ͠Δɽ͜ͷਤʹࣔ͢Α͏ʹɼ༌ૹඅ ࣍ʹɼਤ–15 ʹࣔͨ͠ 8 ࢢϞσϧΛͯݟΈΔɽ༌ Լʹͬͯɼྡͨ͠ࢢͷยํ͕ͯ͠͏Ұํ ૹඅ͕ߴ͍ঢ়Ͱگɼύλʔϯ J ͷਓޱࢄঢ়ଶͰ͋ ͕ਰୀ͢Δύλʔϯ͕ൃੜ͢Δ͜ͱ͕Θ͔Δɽ͜Ε 5 Δɽͦͷঢ়ଶ͔Β༌ૹඅ͕Լ͢Δͱɼύλʔϯ J ͷ ষ (4) અ͓Αͼ 7 ষͰࢦఠͨ͠पظഒ͋ͰذΔɽࡁܦ ࠨͷ༌ૹඅͰ҆ఆͰ͋Δύλʔϯ IɼGɼE ͷ 3 ͭͷ ֶతʹΈΔͱɼଘଓ͢Δࢢͷͱͱʹྡࢢ ύλʔϯ͕࣍ͷύλʔϯʹͳΓಘΔ͜ͱ͕Θ͔Δɽͦ ͕ਰୀ͢ΔʮετϩʔޮՌʯ͕ੜ͍ͯ͡Δͱ͍͑Δɽ͜ ΕҎ߱༌ૹඅԼʹΑΔࢢύλʔϯͷมભύε Εɼਓޱͷࢄྗͱͯ͠ಇ͘༌ૹඅ͕ͨͨ͠ݮΊ ༷ʑ͋Δ͜ͱ͕Θ͔Δɽྫ͑ɼύλʔϯ G ͔Βɼύ ʹɼूੵͷ͕ࡁܦ૬ରతʹͳ͘ڧΓੜ͡ΔݱͰ͋Δɽ λʔϯ HɼFɼE ͕͋ΓಘΔɽ͕ͨͬͯ͠ɼ4 ࢢͱಉ ͜͏͍ͬͨݱͷൃੜʹɼԁप্ͷଟࢢϞσϧͱ ༷ʹɼ8 ࢢϞσϧʹࢄ͔Βूதͷύλʔϯͷม ͍ۭͬͨؒతߏ͕Ө͍ͯ͠ڹΔ͔Ͳ͏͔Λ֬ೝ͢Δ ભύεͱͯ͠ෳଘࡏ͓ͯ͠Γɼ͞ΒʹΑΓଟ͘ͷύ ͨΊʹɼઢ ʹ্ܗ5 ࢢ͕Ґஔ͢ΔϞσϧͷ҆ఆղΛ ε͕͋Δ͜ͱ͕Θ͔Δɽ ͜͜Ͱɼύλʔϯ D ʹணͯ͠ΈΔͱɼύλʔϯ J ٻΊͨɽͦͷੳ݁ՌΛਤ–13 ʹࣔ͢ɽަ௨අݮগʹ 564 ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 ͔Βɼύλʔϯ E ͷมભมભඇ࣮ݱతͰ͋Γɼ 㧭 㧭 ύλʔϯ B ͷมભ࣮ݱతͱ͍͑Δɽ ਤ–13 ʹࣔͨ͠ઢࢢܗͷ 5 ࢢϞσϧʹ͓͍ͯɼ 㧮 㧮 ༌ૹඅԼʹΑΔࢢͷूੵաఔʢ҆ఆߧۉύλʔϯ ͷมભʣ༷ʑͳύε͕͋Γ͏Δ͜ͱ͕Θ͔Δɽͨͩ 㧯 㧯 ͠ɼͦͷύλʔϯͷมભʹෆࣗવͳͷ͕ 8 ࢢԁ ܗϞσϧΑΓ໌֬ʹଘࡏ͢Δɽྫ͑ɼύλʔϯ E 㧰 㧰 ͔Βύλʔϯ DɼCɼB ͕͋ΓಘΔɽ͔͠͠ɼύλʔ ϯ D C Ͱଘࡏ͍ͯ͠Δࢢʢਖ਼ͷਓޱΛ࣋ͭࢢʣ 㧱 0 0.3 ャㅍ⾌ ɼύλʔϯ E Ͱଘࡏ͍ͯ͠ͳ͍ɽ͜ͷΑ͏ʹɼશ 㧱 1 0.4 ͘ࢢ͕ແ͍ͱ͜Ζʹɼ༌ૹඅԼͰಥવࢢ͕ൃੜ ͢Δͱ͍͏͜ͱ࣮ݱతͰͳ͍ɽҎ্ʹࣔͨ͜͠ͱ ਤ–14 ༌ૹඅʹΑΔ҆ఆύλʔϯͷมભʢ4 ࢢʣ ͔Βɼ҆ఆղύλʔϯؒͷมભʹ͓͍ͯɼͳΜΒ͔ͷ ๏ଇ͕࣮ݱతʹଘࡏ͢Δ͜ͱ͕ߟ͑ΒΕΔɽ͜ͷม 㧭 㧭 ભʹؔ͢Δ๏ଇͷϞσϧԽ͜Ε·Ͱͷ NEG ϞσϧͰ 㧲 ߦΘΕ͓ͯΒͣɼࠓޙͷ՝Ͱ͋Δɽɹɹ 㧮 㧯 㧮 㧳 㧯 㧴 9. 㧰 㧱 㧲 ຊڀݚɼذղΛཏతʹௐΔ͜ͱ͕Ͱ͖Δ܈ 㧳˳ 㧴 㧵 㧶 0 0.4 ャㅍ⾌ 0.6 ͓ΘΓʹ 㧰 㧵 㧱 㧶 తذཧΛଟࢢΛؚΉ NEG Ϟσϧͷذղੳʹ ద༻ͨ͠ɽͦͷ݁Ռɼࢢ͕ଟ͍߹ͷ༌ૹඅݮ ʹ͏ࢢͷूੵɾࢄͷਐߦաఔɼ͢ͳΘͪࢢू ੵɾࢄύλʔϯͷมભΛ໌Β͔ʹ͢Δ͜ͱ͕Ͱ͖ͨɽ 1 ͜ͷมભɼTabuchi and Thisse11) Ͱੳରͱ͞Ε ͨपظഒذΛࣔ͢ܦذ࿏Ҏ֎ʹɼଟ͘ͷෳࡶͳ ਤ–15 ༌ૹඅʹΑΔ҆ఆύλʔϯͷมભʢ8 ࢢʣ ܦذ࿏ΛؚΜͰ͍Δ͜ͱ͕Θ͔ͬͨɽ ͷਓޱࢄঢ়ଶ͔Β༌ૹඅԼʹͬͯύλʔϯ A ͷ ຊڀݚɼCP ϞσϧͷߧۉղͷಛੑΛ໌Β͔ʹ͢Δ ਓूޱதঢ়ଶʹมભ͢ΔͲͷύεʹ͓͍ͯɼύλʔϯ ͜ͱʹযΛߜͬͨɽࠓޙɼCP Ϟσϧͷ࠷ળղͷ D ࣮͍ͳ͠ݱɽ͢ͳΘͪɼ͋Δ༌ૹඅͷͰݩɼύ λʔϯ D ҆ఆߧۉղͱͯ͠ଘࡏ͢Δͷͷɼਓޱ ੳΛ͡Ίɼنൣੳʹ͍ͭͯߦ͏ඞཁ͕͋Δɽ· ࢄ͔Βूதʹมભ͢Δ࣮ݱͷ͍͓ͯʹࡁܦ࣮ݱՄೳ ߟ͑ͨɽٕڝࡁܦɼڥք݅ͷӨڹΛड͚ͳ͍Ұ ੑ͕ͳ͍ɽ͜ͷ͜ͱ͔Βɼ͋Δ༌ૹඅΛԾఆͨ͠੩ֶ ࣍ߏݩΛ࣋ͭɽͦͷͨΊɼੳ͕ൺֱత༰қͰ͋Δɽ Ϟσϧͷͯͬ͋ͰߧۉͰݩ࣮͢ݱΔͱߧۉݶΒͣɼ ͔͠͠ɼ࣮ݱͷࢢߏ 2 ࣍ݩతߏΛ͍࣋ͬͯΔͨ ͨɼຊڀݚɼࢢͷۭؒతߏͱٕͯ͠ڝࡁܦΛ ͦͷ͕ߧۉʮਓޱࢄ͔Βूதͱมભ͢ΔࡁܦʯͰ ΊɼࠓޙੳΛ 2 ࣍ݩతߏ·Ͱ֦ு͍ͯ͘͠ඞཁ ࣮͢ݱΔ͔͏Ͳ͔ߧۉΛผ్ݕ౼͢Δඞཁੑ͕͋Δ͜ ͕͋Δɽ͞Βʹ CP Ϟσϧ͚ͩͰͳ͘ɼଞͷ NEG Ϟσ ͱΛࢦఠͰ͖Δɽͳ͓ɼ͜ͷΑ͏ͳ࣮͍ͳ͠ݱύλʔ ϧͷଟࢢέʔεͷಛੑੳࠓޙͷ՝Ͱ͋ΔɽҎ ϯͷൃݟͷͨΊʹຊ܈ͨ͠༻࠾Ͱڀݚతذཧ ্ͷࠓޙͷڀݚ՝Λߦ͏߹ຊͨࣔ͠Ͱڀݚذ ʹΑΔཏతͳذղੳ͕༗༻Ͱ͋Δɽ ղੳ๏༗༻Ͱ͋Δɽ ࣍ʹɼਤ–15 Λ༻͍ͯɼ͍͔ͭ͘ͷύλʔϯมભͷ ࢀߟจݙ ࣮ݱੑΛݕ౼͢Δɽύλʔϯ E ͔Β༌ૹඅԼͨ͠ͱ ͖ɼ࣍ʹ࣮͏͠ݱΔύλʔϯύλʔϯ B ͱ C Ͱ͋Δɽ ͔͠͠ɼύλʔϯ B Ͱ 2 ͳͱۃΔࢢͷҰํύλʔ ϯ E Ͱͱͱਓ͕ޱθϩͰ͋ͬͨࢢͰ͋Δɽਓ ޱθϩͰ͋ͬͨॴʹɼ༌ૹඅԼʹΑΓಥવɼେ͖ ͳਓޱΛ࣋ͭࢢ͕ൃੜ͢Δ͜ͱ࣮ݱతʹߟ͑ʹ ͍͘ɽ͕ͨͬͯ͠ɼύλʔϯ E ͔Βมભ͢Δύλʔϯ ͱͯ͠ύλʔϯ C ͷํ͕࣮ݱతͰ͋Δɽύλʔϯ F 565 1) Krugman, P.: Increasing returns and economic geography, J.of Political Economy, Vol.99, pp.483–499, 1991. 2) Krugman, P.: History versus expectations, Quarterly Journal of Economics, Vol.106, pp.651–667, 1991. 3) Dixit, A.K. and Stiglitz, J.E.: Monopolistic competition and optimum product diversity, American Economic Review, Vol.67, No.3, pp.297–308, 1977. 4) Fujita, M., Krugman, P. and Venables, A.J.: The Spatial Economy: Cities, Regions, and International ᧁቇળ⺰ᢥ㓸㧰 Vol.63 No.4, 553-566, 2007. 12 5) 6) 7) 8) 9) 10) 11) 12) Trade, MIT Press, 1999. (ຊ༁ɿ౻ాɾΫϧʔάϚϯɾ ϕφϒϧζɿֶۭؒࡁܦɼ౦༸৽ࡁܦใࣾɼ2000ʣ Baldwin, R., Forslid, R., Martin, P., Ottaviano, G., and Robert-Nicoud, F.: Economic Geography and Public Policy, Princeton University Press, 2003. Fujita, M. and Thisse, J.F.: Economics of Agglomeration, Cambridge University Press, 2002. Henderson, J. V. and Thisse, J.F. eds.: Handbook of Regional and Urban Economics vol. 4, Cities and Geography, Elsevier, 2004. Krugman, P.: On the number and location of cities, European Economic Review, Vol.37, No.293298, 1993. Krugman, P.: The Self-Organizing Economy, Blackwell, 1996. ʢ·ͨ Fujita et al.4) ͷୈ 6 ষʣ Fujita, M., Krugman, P., and Mori, T: On the Evolution of Hierarchical Urban Systems, European Economic Review, Vol.43, pp.209-251, 1999.ʢ·ͨ Fujita et al. ͷ4) ୈ 11 ষʣ Tabuchi, T. and Thisse, J.F.: Self-organizing urban hierarchy, Preprint, 2006. ౻Ҫจɼେ࡚७ɼాਗ਼ɿߏͱࡐྉͷֶྗذ, ܭ ࢉֶγϦʔζ 3, ίϩφࣾ, 2005. 13) Golubitsky, M., Stewart, I. and Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, Vol. 2, Springer-Verlag, New York, 1988. 14) Ikeda, K. and Murota, K.: Imperfect Bifurcation in Structures and Materials, Springer-Verlag, New York, 2002. 15) ༄ຊজਔɼాਗ਼ɼদོɼՏୡਔɿࢉܭذཧ ʹΑΔࢢͷूੵࢄϞσϧͷܦذ࿏๏ͷఏҊɼ ܭըֶڀݚɾจू, No.24, pp.191–196, 2007. 16) ాਗ਼, ࣨాҰ༤ɿߏܥͷ࠲۶ͱذɼίϩφࣾɼ 2001ɽ 17) Tabuchi, T. and Zeng, D.-Z.: Stability of spatial equilibrium, J. Regional Sci., Vol.44, No.4, pp.641–660, 2004. 18) Zeng, D.-Z.: Equilibrium stability for a migration model, Regional Sci. & Urban Economics, Vol.32, pp.123–138, 2002. 19) Murota, K. and Ikeda, K.: Computational use of group theory in bifurcation analysis of symmetric structures, SIAM J. Sci. Statist. Comput., Vol.12, No.2, pp.273–297, 1991. (2007. 5. 28 ड) SYMMETRY-BREAKING BIFURCATION OF A CORE–PERIPHERY MODEL OF MANY CITIES: GROUP-THEORETIC APPROACH Kiyohiro IKEDA, Tatsuhito KONO, Takashi AKAMATSU, Akito YANAGIMOTO and Shunji YAMAKI The core-periphery model that expresses the city accumulation phenomenon in association with the change of transportation cost has multiple equilibriums for two or three cities. However, there is scarce knowledge on the pattern of spatial accumulation and decentralization of the population when the number of cities is increased further. In this research, the numerical analysis based on computational bifurcation theory and group-theoretic bifurcation theory is carried out on the Core–Periphery model for many cities with the same population that are located symmetrically along a circle. As a result, complex bifurcation behavior of the model has successfully been traced and the accumulation phenomenon of the city has been shown to be engendered via a phased loss of symmetry and spatial period-doubling bifurcation. 566
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