O impacto da ampliação da cobertura previdenciária sobre a poupança na China

FUNDAÇÃO GETULIO VARGAS

ESCOLA DE PÓS -GRADUAÇÃO EM

ECONOMIA

Marcelo Lyra Machado de Carvalho

O impacto da ampliação da cobertura

previdenciária sobre a poupança da China

Marcelo Lyra Machado de Carvalho

O impacto da ampliação da cobertura

previdenciária sobre a poupança da China

Dissertação submetida a Escola de Pós-Graduação em Economia como requesito parcial para a obtenção do grau de Mestre em Econo-mia.

Orientador: Pedro Cavalcanti Gomes Ferreira

Ficha catalogháfica elabohada pela Biblioteca Mahio Henhiqre Simonsen/FGV

Cahvalho, Mahcelo Lyha Machado de

O impacto da ampliação da cobehtrha phevidenciáhia sobhe a porpança na China / Mahcelo Lyha Machado de Cahvalho. – 2012.

35 f.

Dissehtação (mesthado) - Frndação Getrlio Vahgas, Escola de Pós- Ghadração em Economia.

Ohientadoh: Pedho Cavalcanti Fehheiha. Inclri biblioghafia.

1. Segrho social - Modelos econômicos. 2. Porpança – Modelos econômicos. I. Fehheiha, Pedho Cavalcanti. II. Frndação Getrlio Vahgas. Escola de Pós-Ghadração em Economia. III. Títrlo.

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♣.❡✈✐❞❡♥❝✐2.✐❛ 3♦❜.❡ ❛ ♣♦✉♣❛♥*❛ ♥❛ ❈❤✐♥❛

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▼❛② ✶✱ ✷✵✶✷

❆❜"#$❛❝#

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❯♠❛ ♠❛✐♦* ❝♦❜❡*%✉*❛ ♣*❡✈✐❞❡♥❝✐7*✐❛ ♣✐♦*❛ ♦ ❜❡♠✲❡)%❛* )❡ ❛ ❝♦♥✲ %*✐❜✉✐8@♦τ ❢♦* ❡①K❣❡♥❛✱ ✉♠❛ ✈❡③ /✉❡ ♥@♦ ❤7 ✉♠ ♠❡❝❛♥✐)♠♦ ❞❡ ❛❥✉)%❡ ♥❛ ♦❝♦**B♥❝✐❛ ❞❡ ♠✉❞❛♥8❛) ♣❛*❛ ❧❡✈❛* ❛ ❡❝♦♥♦♠✐❛ ❞❡ ✈♦❧%❛ ❛♦ K%✐♠♦✳

✶ ■♥#$♦❞✉()♦

❉❡"❞❡ ✶✾✼✽✱ ❝♦♠ ❛ ❛❜❡./✉.❛ ❞❛ ❡❝♦♥♦♠✐❛ ♣♦. ❉❡♥❣ ❳✐❛♦♣✐♥❣✱ ❛ ❈❤✐♥❛ ✈❡♠ ♣❛""❛♥❞♦ ♣♦. ✈9.✐❛" /.❛♥"❢♦.♠❛;<❡" ♣❛.❛ "❡ ❛❞❡=✉❛. > ❡❝♦♥♦♠✐❛ ❞❡ ♠❡.❝❛❞♦ ♠✉♥❞✐❛❧✳ ❯♠ ❞♦" ♣❛""♦" ❛ "❡.❡♠ "❡❣✉✐❞♦" B ❛ ❢♦.♠❛;C♦ ❞❡ ✉♠ "✐"/❡♠❛ ❞❡ "❡❣✉.✐❞❛❞❡ "♦❝✐❛❧ ❛♠♣❧♦ ❡ ♠♦❞❡.♥♦✳

❚❛❧ "✐"/❡♠❛ /.❛❞✐❝✐♦♥❛❧♠❡♥/❡ ❝♦❜.✐❛ ♦" ❡♠♣.❡❣❛❞♦" ❞❛" ❡♠♣.❡"❛" ❡"/❛/❛✐"✱ =✉❡ ❞✐"♣✉♥❤❛♠ ❞❡ /♦❞♦" ♦" /✐♣♦" ❞❡ ❛""✐"/E♥❝✐❛ ♣❡❧❛ ✈✐❞❛ ✐♥/❡✐.❛✱ ❞❡"❞❡ ❛✉①G❧✐♦✲"❛I❞❡ ❛ ♣❡♥"<❡"✳ ❙❡♥❞♦ ❛""✐♠✱ ❛" ♣.K♣.✐❛" ❡♠♣.❡"❛" ❡"/❛/❛✐" ❡.❛♠ .❡"♣♦♥"9✈❡✐" ♣♦. "✉"/❡♥/❛. ♦" ♣❡♥"✐♦♥✐"/❛"✳

❙❡❣✉♥❞♦ ❖❦"❛♥❡♥ ❬✶✷❪✱ ❡♠ ✷✵✵✽✱ ♦ "✐"/❡♠❛ ♣.❡✈✐❞❡♥❝✐9.✐♦ ❝❤✐♥E" ❝♦❜.✐❛✱ /❡♦.✐❝❛♠❡♥/❡✱ ♦" ✸✵✷ ♠✐❧❤<❡" ❞❡ ❡♠♣.❡❣❛❞♦" ✉.❜❛♥♦"✱ ♠❛" ❛♣❡♥❛" ✺✺✪ ❞❡❧❡" ❞❡ ❢❛/♦ ♣❛❣❛✈❛♠ ❛" ❝♦♥/.✐❜✉✐;<❡"✳ ◆♦ ♠❡✐♦ .✉.❛❧ ♦ ♣.♦❜❧❡♠❛ ❡.❛ ♣✐♦.✿ ❞♦" ✹✼✸ ♠✐❧❤<❡" ❞❡ ❡♠♣.❡❣❛❞♦"✱ ❛♣❡♥❛" ✶✷✪ ❡"/❛✈❛♠ ❝♦❜❡./♦"✳ X♦./❛♥/♦✱ ✉♠ ❞♦" ❣.❛♥❞❡" ❞❡"❛✜♦" ❝❤✐♥❡"❡" B ❡①♣❛♥❞✐. ❛ ❝♦❜❡./✉.❛ ♣.❡✈✐❞❡♥❝✐9.✐❛✳

❚♦❞❛✈✐❛✱ ❝❤❡❣❛✲"❡ ❛ ✉♠ ♣♦♥/♦ ✐♥/❡.❡""❛♥/❡✿ =✉❛❧ "❡.✐❛ ♦ ✐♠♣❛❝/♦ ❞✐""♦ ♥❛ ♣♦✉♣❛♥;❛ ♣.✐✈❛❞❛ ❞♦ ♣❛G"✱ =✉❡ B ✉♠❛ ❞❛" ♠❛✐♦.❡" ❞♦ ♠✉♥❞♦❄ ➱ ❞❡ "❡ ❡"♣❡.❛. =✉❡ ❡❧❛ ❝❛✐❛✳ ◗✉❛♥❞♦ ✉♠❛ ♣❡""♦❛ ♥C♦ /❡♠ ❛❝❡""♦ ❛ ✉♠ "✐"/❡♠❛ ♣.❡✈✐❞❡♥❝✐9.✐♦✱ ❡❧❛ /❡♠ =✉❡ ♣♦✉♣❛. ♠❛✐" ❛♦ ❧♦♥❣♦ ❞❛ ✈✐❞❛ ♣❛.❛ ❣❛.❛♥/✐. ✉♠ ✢✉①♦ ❞❡ .❡♥❞❛ ❛♦ "❡ ❛♣♦"❡♥/❛.✳ ❆ ♣❛./✐. ❞♦ ♠♦♠❡♥/♦ ❡♠ =✉❡ ♦ ❣♦✈❡.♥♦ ♣❛""❛ ❛ ❣❛.❛♥/✐. /❛❧ ❛♣♦"❡♥/❛❞♦.✐❛ ♠❡❞✐❛♥/❡ ♦ ♣❛❣❛♠❡♥/♦ ❞❡ ✉♠❛ ❝♦♥/.✐❜✉✐;C♦✱ ❛ ♣❡""♦❛ ♣♦✉♣❛ ♠❡♥♦"✳ X♦./❛♥/♦✱ > ♠❡❞✐❞❛ =✉❡ ♠❛✐" ♣❡""♦❛" ♣❛""❛♠ ❛ ✐♥/❡❣.❛. ♦ "✐"/❡♠❛ ♣.❡✈✐❞❡♥❝✐9.✐♦✱ .❡❞✉③✲"❡ ❛ ♣♦✉♣❛♥;❛ ❛❣.❡❣❛❞❛✳

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X❛.❛ ✜♥" ❝♦♠♣❛.❛/✐✈♦"✱ ❑✉✐❥" ❬✶✵❪ ♠♦"/.❛ ❛ ♣.♦♣♦.;C♦ ❞❛ ♣♦✉♣❛♥;❛

❞♦♠#$%✐❝❛ $♦❜*❡ ♦ ,■❇ ♣❛*❛ ❛❧❣✉♥$ ♣❛4$❡$ ♥♦ ❛♥♦ ❞❡ ✷✵✵✶ ❛ ✷✵✵✸✳ ❊♥;✉❛♥%♦ ❛ ❈❤✐♥❛ ❛♣❛*❡❝✐❛ ✐$♦❧❛❞❛ ❝♦♠ ✹✷✱✺✪ ✭✷✵✵✸✮✱ ♦$ ❊$%❛❞♦$ ❯♥✐❞♦$ %✐♥❤❛♠ ✶✹✱✸✪ ✭✷✵✵✷✮✱ ❋*❛♥F❛ ✷✵✱✼✪ ✭✷✵✵✷✮✱ ❈♦*#✐❛ ✸✶✱✵✪ ✭✷✵✵✷✮✱ ❏❛♣I♦ ✷✺✱✺✪ ✭✷✵✵✷✮ ❡ ▼#①✐❝♦ ✷✵✱✽✪ ✭✷✵✵✶✮✳

❖ ♦❜❥❡%✐✈♦ ❞❡$%❡ %*❛❜❛❧❤♦ # ;✉❛♥%✐✜❝❛* ♦ ✐♠♣❛❝%♦ ❞❡ ✉♠ ❛✉♠❡♥%♦ ❞❛ ❝♦❜❡*%✉*❛ ♣*❡✈✐❞❡♥❝✐Q*✐❛ $♦❜*❡ ❛ ♣♦✉♣❛♥F❛ ♥❛ ❈❤✐♥❛✳ ❚❛❧ *❡$♣♦$%❛ # ❛❧✲ ❝❛♥F❛❞❛ ❛%*❛✈#$ ❞❡ ✉♠ ♠♦❞❡❧♦ $✐♠♣❧❡$ ❞❡ ❣❡*❛FT❡$ $✉♣❡*♣♦$%❛$ ❞❡ ❞♦✐$ ♣❡*4♦✲ ❞♦$ ♣❡❧♦ ;✉❛❧ ❡①✐$%❡♠ ❞♦✐$ %✐♣♦$ ❞❡ ❛❣❡♥%❡$✿ ✉♠ ;✉❡ ❡$%Q ❢♦*❛ ❞❡ ✉♠ $✐$%❡♠❛ ❞❡ ♣*❡✈✐❞W♥❝✐❛ ❞❡ *❡♣❛*%✐FI♦✱ ❞❡ ♠♦❞♦ ;✉❡✱ ;✉❛♥❞♦ ❥♦✈❡♠✱ ❡❧❡ %*❛❜❛❧❤❛✱ ❝♦♥✲ $♦♠❡ ❡ ♣♦✉♣❛ ✉♠❛ ♣❛*%❡ ❞❛ *❡♥❞❛✱ ❡ $❡ ❛♣♦$❡♥%❛ ;✉❛♥❞♦ ✐❞♦$♦✱ ❝♦♥$✉♠✐♥❞♦ ♦ ;✉❡ ♣♦✉♣♦✉ ❛♥%❡*✐♦*♠❡♥%❡❀ ♦✉%*♦ ;✉❡ ❡$%Q ❞❡♥%*♦ ❞♦ $✐$%❡♠❛✱ ❧♦❣♦✱ ♣❛❣❛ ✉♠ ✐♠♣♦$%♦✴❝♦♥%*✐❜✉✐FI♦ ;✉❛♥❞♦ ❥♦✈❡♠ ;✉❡ ❧❤❡ ❞Q ❞✐*❡✐%♦ ❛ *❡❝❡❜❡* ✉♠❛ ❛♣♦$❡♥%❛❞♦*✐❛ ;✉❛♥❞♦ ✐❞♦$♦✳

❊♠ *❡❣✐♠❡$ ❞❡ *❡♣❛*%✐FI♦✱ ❛ ❝♦♥%*✐❜✉✐FI♦ ♣❛❣❛ ♣❡❧♦$ ❥♦✈❡♥$ %*❛❜❛❧❤❛❞♦*❡$ # ❛✉%♦♠❛%✐❝❛♠❡♥%❡ ❝♦♥✈❡*%✐❞❛ ♣❛*❛ ♦ ♣❛❣❛♠❡♥%♦ ❞❡ ❛♣♦$❡♥%❛❞♦*✐❛$ ♣❛*❛ ♦$ ✐❞♦$♦$ ❛%✉❛✐$ ;✉❡ ♥I♦ ♠❛✐$ %*❛❜❛❧❤❛♠✳ ,♦* $❡*❡♠ ♠❛✐$ ❢Q❝❡✐$ ♣♦❧✐%✐❝❛♠❡♥%❡ ❞❡ $❡*❡♠ ❝*✐❛❞♦$✱ *❡❣✐♠❡$ ❞❡ *❡♣❛*%✐FI♦ $I♦ ♦$ ❛❞♦%❛❞♦$ ♥❛ ♠❛✐♦*✐❛ ❞♦$ ♣❛4$❡$✿ ♥❛ $✉❛ ✐♥%*♦❞✉FI♦ ❡♠ ✉♠❛ ❡❝♦♥♦♠✐❛✱ ❛ ♣*✐♠❡✐*❛ ❣❡*❛FI♦ ❞❡ ✐❞♦$♦$ ♥I♦ ♣*❡✲ ❝✐$❛ ♣❛❣❛* ❝♦♥%*✐❜✉✐FI♦ ❡ *❡❝❡❜❡ ♦ ❜❡♥❡❢4❝✐♦❀ ❛$ ❣❡*❛FT❡$ $❡❣✉✐♥%❡$ ♣❛❣❛*I♦ ❡ *❡❝❡❜❡*I♦ ♥♦*♠❛❧♠❡♥%❡✳

❊$$❛ ❝♦♥%*✐❜✉✐FI♦ # ❡①Z❣❡♥❛ ♥♦ ♠♦❞❡❧♦✱ ✐$%♦ #✱ ❞❛❞♦ ♦ ✈❛❧♦* ❡$❝♦❧❤✐❞♦ ♣❛*❛ ❡❧❛✱ $❡*Q $❡♠♣*❡ ❛;✉❡❧❡ ✈❛❧♦*✳ ■$$♦ # ❢❡✐%♦ ♣❛*❛ ❡✈✐%❛* ;✉❡ ♦❝♦**❛♠ ♠✉❞❛♥F❛$ ♥❛ ❝♦♥%*✐❜✉✐FI♦ ❞❡ ❛❝♦*❞♦ ❝♦♠ ❛❧%❡*❛FT❡$ ♥❛ ♣*♦♣♦*FI♦ ❞❡ ♣❡$$♦❛$ ✐♥❝❧✉4❞❛$ ♥♦ $✐$%❡♠❛✳

❆$$✉♠✐♥❞♦ ✉♠ $✐$%❡♠❛ ❞❡ *❡♣❛*%✐FI♦ ;✉❡ ❝✉❜*❛ ❛♣❡♥❛$ ✉♠❛ ♣❛*❝❡❧❛ ❞❛ ♣♦♣✉❧❛FI♦✱ ♣❡❧♦ ;✉❛❧ ♦$ %*❛❜❛❧❤❛❞♦*❡$ ❥♦✈❡♥$ ✜♥❛♥❝✐❛♠ ♦$ ❛♣♦$❡♥%❛❞♦$ ✐❞♦$♦$✱ # ❞❡ $❡ ❡$♣❡*❛* ;✉❡ ✉♠❛ ♠❛✐♦* ❛❜*❛♥❣W♥❝✐❛ ❞♦ $✐$%❡♠❛ *❡❞✉③❛ ❛

♣♦✉♣❛♥%❛✱ ✉♠❛ ✈❡③ +✉❡ ❡❧❛ ♥-♦ ♣.❡❝✐1❛.✐❛ 1❡. 2-♦ ❣.❛♥❞❡ ❣.❛%❛1 5 2.❛♥1❢❡.✲ 8♥❝✐❛ ♣.♦♠♦✈✐❞❛ ♣❡❧♦ ❊12❛❞♦ ✈✐❛ ♣❡♥1:❡1 ♥♦ ❢✉2✉.♦✳ <♦.=♠✱ +✉❛♥2♦ ❝❛✐.✐❛❄ ❊12✐♠❛✲1❡ ✉♠❛ +✉❡❞❛ ❞❡ ✸✻✱✻✪ ♥❛ ♣♦✉♣❛♥%❛✳

<❛.❛ ❝❤❡❣❛. ❛ ❡11❡ .❡1✉❧2❛❞♦✱ ♦ ♠♦❞❡❧♦ = ❝❛❧✐❜.❛❞♦ ❞❡ ❛❝♦.❞♦ ❝♦♠ ✈❛❧♦.❡1 ❡12✐♠❛❞♦1 ♣♦. ❛❧❣✉♥1 ❛.2✐❣♦1 ♣❛.❛ ❛❧❣✉♥1 ♣❛.D♠❡2.♦1 ♥♦1 E❧2✐♠♦1 ❛♥♦1✱ ❛ ♣❛.2✐. ❞❡ ✷✵✵✹ +✉❛♥❞♦ ♣♦11I✈❡❧✳ <.✐♠❡✐.❛♠❡♥2❡✱ = ✉1❛❞❛ ✉♠❛ ♣.♦♣♦.%-♦ <♦✉♣❛♥%❛✲<.♦❞✉2♦ ❞❡ ✷✸✪✱ +✉❡ ❝♦..❡1♣♦♥❞❡ ❛♦ ✈❛❧♦. ♦❜1❡.✈❛❞♦ ♣❛.❛ ❛♣❡♥❛1 ❛ ♣♦✉♣❛♥%❛ ❞❛1 ❢❛♠I❧✐❛1 ♥♦ ♣❛I1✳ ❯1❛✲1❡ ✉♠❛ ♣.♦♣♦.%-♦ ❞♦ ❝❛♣✐2❛❧ 1♦❜.❡ ♦ ♣.♦❞✉2♦ ❞❡ α = 0,48✳ ❆ ❝♦♥2.✐❜✉✐%-♦ ♣.❡✈✐❞❡♥❝✐L.✐❛ 1♦❜.❡ 1❛❧L.✐♦✱ τ✱ = ❞❡

✷✵✪✱ ❛ ❞❡♣.❡❝✐❛%-♦ = 2♦2❛❧ ❡ ♦ β = 0,940✱ ✈❛❧♦. +✉❡ = ❝❛❧✐❜.❛❞♦ ♣❛.❛ +✉❡ 1❡❥❛ ♦❜2✐❞❛ ❛ ♣.♦♣♦.%-♦ <♦✉♣❛♥%❛✲<.♦❞✉2♦ ❡1♣❡.❛❞❛✳ ❉❡♣♦✐1✱ = ❢❡✐2❛ ✉♠❛ ♥♦✈❛ ❝❛❧✐❜.❛%-♦ ❝♦♥1✐❞❡.❛♥❞♦ 2❛♠❜=♠ ❛ ♣♦✉♣❛♥%❛ ❞❛1 ✜.♠❛1✱ 2♦2❛❧✐③❛♥❞♦ ✉♠❛ ♣.♦♣♦.%-♦ <♦✉♣❛♥%❛✲<.♦❞✉2♦ ❞❡ ✸✺✪✳

❈♦♠♦ ❜❛1❡ ❞❡ ❝♦♠♣❛.❛%-♦✱ ❛ ❝♦❜❡.2✉.❛ ♣.❡✈✐❞❡♥❝✐L.✐❛ ❛2✉❛❧ ❝❤✐♥❡1❛ = ❡12✐♣✉❧❛❞❛ ❡♠ ✷✾✪✱ ❝♦♥❢♦.♠❡ ❖❦1❛♥❡♥ ❬✶✷❪✳

❊♠ 1❡❣✉✐❞❛✱ 1❡❣✉✐♥❞♦ ♠❡2♦❞♦❧♦❣✐❛ ❞❡ ❋♦.♥✐ ❬✼❪✱ = ❢❡✐2❛ ✉♠❛ ❡①2❡♥1-♦ ♥❛ +✉❛❧ 1❡ ❞❡✐①❛ ❞❡ ❝♦♥1✐❞❡.❛. ❛ ❝♦♥2.✐❜✉✐%-♦ ❝♦♠♦ ❡①[❣❡♥❛ ♣❛.❛ ♦ ❣♦✈❡.♥♦❀ ❤L ✉♠ ♣❧❛♥❡❥❛❞♦. ❝❡♥2.❛❧ +✉❡ 1[ 1❡ ♣.❡♦❝✉♣❛ ❝♦♠ ♦ ❜❡♠✲❡12❛. ❞❛ ❣❡.❛%-♦ ❥♦✈❡♠ ❡ ❞❡✈❡ ❡1❝♦❧❤❡. ✉♠❛ ❝♦♥2.✐❜✉✐%-♦ ♣.❡✈✐❞❡♥❝✐L.✐❛τ ❛ 1❡. ❝♦❜.❛❞❛ ♣❛.❛ ✐♥❞✉③✐.

❛ ❡❝♦♥♦♠✐❛ ❛♦ [2✐♠♦✳ ❖✉ 1❡❥❛✱ ❛ ♣❛.2✐. ❞❡ ❛❣♦.❛✱ ♦ ❣♦✈❡.♥♦ ✈❛✐ ❞❡✜♥✐. ❛ ❝♦♥2.✐❜✉✐%-♦ ❞❡ ❛❝♦.❞♦ ❝♦♠ ❛ 1✐2✉❛%-♦ ❡❝♦♥]♠✐❝❛ ❞❡ ♠♦❞♦ ❛ ✐♠♣❡❞✐. +✉❡ ❤❛❥❛ ♣❡.❞❛ ❞❡ ❜❡♠✲❡12❛.✳

❆❝❤❛✲1❡✱ ❡♥2-♦✱ ✉♠ ❡+✉✐❧I❜.✐♦ ❞❡♣❡♥❞❡♥2❡ ❛♣❡♥❛1 ❞❛ ✈❛.✐L✈❡❧ ❞❡ ❡12❛❞♦ ♥♦ ♣❡.I♦❞♦ t ✲ ♥♦ ❝❛1♦✱ ♦ ❡12♦+✉❡ ❞❡ ❝❛♣✐2❛❧ ✲ ♣♦✐1 ♦ ❣♦✈❡.♥♦ ♣.❡❝✐1❛ ♦❧❤❛.

1♦♠❡♥2❡ ♣❛.❛ ❡11❛ ✈❛.✐L✈❡❧ ♣❛.❛ 1❛❜❡. ❛ 1✐2✉❛%-♦ ❡❝♦♥]♠✐❝❛ ❞♦ ♣❛I1✳ ❊11❡ ❡+✉✐❧I❜.✐♦ = ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❡+✉✐❧I❜.✐♦ ❞❡ ▼❛.❦♦✈ ♣❡.❢❡✐2♦ ❡♠ 1✉❜❥♦❣♦1✳

<❛.❛ +✉❡ ♥-♦ ❤❛❥❛ ✐♥❝❡♥2✐✈♦1 ♣❛.❛ ❛ ❣❡.❛%-♦ ❥♦✈❡♠ ❛2✉❛❧ ❞❡1✈✐❛. ❞♦ ❝♦♠✲

♣!♦♠✐%%♦ ❞❡ ♣❛❣❛! ❛ ❝♦♥,!✐❜✉✐/0♦ ❛♦ %✐%,❡♠❛ ♣!❡✈✐❞❡♥❝✐2!✐♦✱ ❋♦!♥✐ ❬✼❪ ✐♠♣8❡ ✉♠❛ !❡❧❛/0♦ ♥❡❣❛,✐✈❛ ❡♥,!❡ ❛ ♥:✈❡❧ ❞❡ ♣♦✉♣❛♥/❛ ❞❡ ❝❛❞❛ ❣❡!❛/0♦ ❡ ❛ ;✉❛♥✲ ,✐❞❛❞❡ ❞❡ ❛♣♦%❡♥,❛❞♦!✐❛% ;✉❡ !❡❝❡❜❡ ;✉❛♥❞♦ ✐❞♦%❛✳ ❈❛%♦ ❛ ❣❡!❛/0♦ !❡❞✉③❛ %✉❛ ❝♦♥,!✐❜✉✐/0♦ ❛❜❛✐①♦ ❞♦ ❡;✉✐❧:❜!✐♦✱ ,❡!✐❛ ✉♠❛ ♣♦✉♣❛♥/❛ ♠❛✐♦! ✭✐✳❡✳✱ ♠❛✐♦! ❡%,♦;✉❡ ❞❡ ❝❛♣✐,❛❧✮✳ ❈♦♠♦ ♣✉♥✐/0♦✱ !❡❝❡❜❡!✐❛ ✉♠❛ ♠❡♥♦! ❛♣♦%❡♥,❛❞♦!✐❛✳ ❉❡%%❛ ❢♦!♠❛✱ %❛❜❡♥❞♦ ❞❛ ♣✉♥✐/0♦✱ ♥0♦ ❢❛!✐❛ ❛ !❡❞✉/0♦ ✐♥✐❝✐❛❧ ❞❛ ❝♦♥,!✐❜✉✐/0♦✱ ♠❛♥,❡♥❞♦ ♦ %✐%,❡♠❛ ❡;✉✐❧✐❜!❛❞♦✳

➱ ✐♥,❡!❡%%❛♥,❡ ♥♦,❛! ;✉❡ ❤2 ✉♠❛ ♥♦✈✐❞❛❞❡ ❡♠ !❡❧❛/0♦ ❛♦ !❡%✉❧,❛❞♦ ❡♥✲ ❝♦♥,!❛❞♦ ♣♦! ❋♦!♥✐ ❬✼❪✱ ♣♦✐% ❡%,❡ ♥0♦ ❝♦♥%✐❞❡!❛ ❛ ❡①✐%,G♥❝✐❛ ❞❡ ❞♦✐% ,✐♣♦% ❞❡ ❛❣❡♥,❡%✳ ❆♦ ❤❛✈❡! ✉♠❛ ♣!♦♣♦!/0♦ (1−γ)✱ γ ∈ (0,1)✱ ❞❡ ❛❣❡♥,❡% ❞❡♥✲ ,!♦ ❞♦ %✐%,❡♠❛✱ ♦ ♣❧❛♥❡❥❛❞♦! ❝❡♥,!❛❧ ♣❛%%❛ ❛ ❝♦♥%✐❞❡!2✲❧❛ ♥❛ %✉❛ !❡❣!❛ ❞❡ ❡%❝♦❧❤❛ ❞❛ ❝♦♥,!✐❜✉✐/0♦ J,✐♠❛✳ ❙❡♥❞♦ ❛%%✐♠✱ ♣❛!❛ ;✉❛❧;✉❡! ✈❛❧♦! ❞❡ γ✱ ♦

❡%,♦;✉❡ ❞❡ ❝❛♣✐,❛❧ ♥0♦ ♠✉❞❛ ✭❧♦❣♦✱ ♦% ❝♦♥%✉♠♦%✱ ♣♦✉♣❛♥/❛ ❡ ✉,✐❧✐❞❛❞❡ ,❛♠✲ ❜L♠ ♥0♦✮✳ ❉❡ ❛❝♦!❞♦ ❝♦♠ ♦ ✈❛❧♦! ❞❡ γ✱ ♦ ♣❧❛♥❡❥❛❞♦! ♠✉❞❛ ❛ ❝♦♥,!✐❜✉✐/0♦

❞❡ ♠♦❞♦ ❛ ♠❛♥,❡! ❛ ❡❝♦♥♦♠✐❛ ♥♦ J,✐♠♦✳ ▲♦❣♦✱ ❛% N♥✐❝❛% ✈❛!✐2✈❡✐% ;✉❡ ♠✉✲ ❞❛♠ ❝♦♠ ❛❧,❡!❛/8❡% ❡♠ γ %0♦ ♦ ✐♠♣♦%,♦ ❡✱ ❝♦♥%❡;✉❡♥,❡♠❡♥,❡✱ ♣❛!❛ ;✉❡ ♦

♦!/❛♠❡♥,♦ ♣!❡✈✐❞❡♥❝✐2!✐♦ ♣❡!♠❛♥❡/❛ ❡;✉✐❧✐❜!❛❞♦✱ ❛ ❛♣♦%❡♥,❛❞♦!✐❛ !❡❝❡❜✐❞❛ ;✉❛♥❞♦ ✐❞♦%♦✳

❊%,❡ ❛!,✐❣♦ ,❡♠ ❞✉❛% ❝♦♥,!✐❜✉✐/8❡% ♣❛!❛ ❛ ❧✐,❡!❛,✉!❛✿ ❞❡✐①❛ ❝❧❛!♦ ;✉❡ ✉♠❛ ♠❛✐♦! ❝♦❜❡!,✉!❛ ♣!❡✈✐❞❡♥❝✐2!✐❛ ♣✐♦!❛ ♦ ❜❡♠✲❡%,❛! ✭❡ ;✉❛♥,✐✜❝❛ ,❛❧ ♣✐♦!❛✮ %❡ ❛ ❝♦♥,!✐❜✉✐/0♦τ ❢♦! ❡①J❣❡♥❛✱ ✉♠❛ ✈❡③ ;✉❡ ♥0♦ ❤2 ✉♠ ♠❡❝❛♥✐%♠♦ ❞❡ ❛❥✉%,❡

♥❛ ♦❝♦!!G♥❝✐❛ ❞❡ ♠✉❞❛♥/❛% ♣❛!❛ ❧❡✈❛! ❛ ❡❝♦♥♦♠✐❛ ❞❡ ✈♦❧,❛ ❛♦ J,✐♠♦❀ ❛♠♣❧✐❛ ♦ !❡%✉❧,❛❞♦ ❡♥❝♦♥,!❛❞♦ ♣♦! ❋♦!♥✐ ❬✼❪ ❛♦ ❝♦♥%✐❞❡!❛! ❛ ❡①✐%,G♥❝✐❛ ❞❡ ❞♦✐% ,✐♣♦% ❞❡ ❛❣❡♥,❡% ❡ ❧❡✈❛! ❡♠ ❝♦♥,❛ ♥❛ !❡❣!❛ ❞❡ ❞❡❝✐%0♦ ❞♦ ♣❧❛♥❡❥❛❞♦! ❝❡♥,!❛❧ ❛ ♣!♦♣♦!/0♦ ❞❡ ❝❛❞❛ ❛❣❡♥,❡ ♥❛ ❡❝♦♥♦♠✐❛✳

❖ ,!❛❜❛❧❤♦ ❡%,2 ♦!❣❛♥✐③❛❞♦ ❞❛ %❡❣✉✐♥,❡ ❢♦!♠❛✿ ❛ %❡/0♦ ✷ ❢❛③ ❛ ❞❡%❝!✐/0♦ ❞❛ ❡❝♦♥♦♠✐❛✱ ❞❡%❝!❡✈❡♥❞♦ ❡♠ %✉❛% %✉❜%❡/8❡% ,♦❞♦% ♦% ❛❣❡♥,❡% ❡♥✈♦❧✈✐❞♦%✱

♦ ❡"✉✐❧&❜(✐♦✱ ❛+ ❝❛❧✐❜(❛-.❡+ ❡ ♦+ (❡+✉❧/❛❞♦+❀ ❛ +❡-2♦ ✸ ♠♦+/(❛ ❛ ❡①/❡♥+2♦ ❞♦ ♠♦❞❡❧♦ ♣❛(❛ ♦ ❝❛+♦ ❡♠ "✉❡ +❡ ❞❡✐①❛ ❞❡ ❝♦♥+✐❞❡(❛( ❛ ❝♦♥/(✐❜✉✐-2♦ ❡①8❣❡♥❛ ❡ ♦ ❣♦✈❡(♥♦ ♣❛++❛ ❛ ❡+❝♦❧❤❡( ❛ /❛①❛-2♦ 8/✐♠❛❀ ❛ +❡-2♦ ✹ ❢❛③ ❛ ❝♦♥❝❧✉+2♦✳

✷ ❉❡#❝%✐'(♦ ❞❛ ❊❝♦♥♦♠✐❛

❈♦♥+✐❞❡(❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ❣❡(❛-.❡+ +✉♣❡(♣♦+/❛+ ❜A+✐❝♦ ❞❡ ❞♦✐+ ♣❡(&♦❞♦+✳ ❊①✐+/❡♠ ❞♦✐+ /✐♣♦+ ❞❡ ❛❣❡♥/❡+✿ ✉♠✱ ❋✱ "✉❡ ❡+/A ❢♦(❛ ❞❡ ✉♠ +✐+/❡♠❛ ❞❡ ♣(❡✲ ✈✐❞F♥❝✐❛ ❞❡ (❡♣❛(/✐-2♦✱ ❞❡ ♠♦❞♦ "✉❡✱ "✉❛♥❞♦ ❥♦✈❡♠✱ ❡❧❡ /(❛❜❛❧❤❛✱ ❝♦♥+♦♠❡ ❡ ♣♦✉♣❛ ✉♠❛ ♣❛(/❡ ❞❛ (❡♥❞❛✱ ❡ +❡ ❛♣♦+❡♥/❛ "✉❛♥❞♦ ✐❞♦+♦✱ ❝♦♥+✉♠✐♥❞♦ ♦ "✉❡ ♣♦✉♣♦✉ ❛♥/❡(✐♦(♠❡♥/❡❀ ♦✉/(♦✱ ❉✱ "✉❡ ❡+/A ❞❡♥/(♦ ❞♦ +✐+/❡♠❛✱ ❧♦❣♦✱ ♣❛❣❛ ✉♠ ✐♠♣♦+/♦✴❝♦♥/(✐❜✉✐-2♦ "✉❛♥❞♦ ❥♦✈❡♠ "✉❡ ❧❤❡ ❞A ❞✐(❡✐/♦ ❛ (❡❝❡❜❡( ✉♠❛ ❛♣♦+❡♥/❛❞♦(✐❛ "✉❛♥❞♦ ✐❞♦+♦✳

❈♦♥+✐❞❡(❡✱ ❛✐♥❞❛✱ "✉❡ ❤A ✉♠❛ ♣(♦♣♦(-2♦ γ ∈(0,1) ❞❡ ❛❣❡♥/❡+ "✉❡ ❡+/A ❢♦(❛ ❞♦ +✐+/❡♠❛ ❞❡ ♣(❡✈✐❞F♥❝✐❛ ❡ ✭✶ ✲γ✮ "✉❡ ❡+/A ❞❡♥/(♦ ❞♦ +✐+/❡♠❛✳ ❙❡♥❞♦

✉♠❛ ♣♦♣✉❧❛-2♦ ❤♦❥❡ ❞❡ /❛♠❛♥❤♦ ✶✱ ♣♦❞❡✲+❡ ✐♥/❡(♣(❡/❛( γ ❝♦♠♦ ❛ ♣(♦❜❛❜✐❧✐✲

❞❛❞❡ ❞❡ ✉♠ ❞❛❞♦ ❛❣❡♥/❡ ❡+/❛( ❢♦(❛ ❞♦ +✐+/❡♠❛✱ ❡ ✭✶ ✲ γ✮ ❛ ♣(♦❜❛❜✐❧✐❞❛❞❡ ❞❡

❡+/❛( ❞❡♥/(♦ ❞♦ +✐+/❡♠❛ ❞❡ (❡♣❛(/✐-2♦✳

✷✳✶ ❋✐%♠❛(

❆ ❢✉♥-2♦ ❞❡ ♣(♦❞✉-2♦ ❞❛ ❡❝♦♥♦♠✐❛ O ❞❛❞❛ ♣❡❧❛ ❢✉♥-2♦ ❈♦❜❜✲❉♦✉❣❧❛+✿

Yt=AKtαL1t−α ✭✶✮

♦♥❞❡α∈(0,1)O ❛ ♣(♦♣♦(-2♦ ❞♦ ❝❛♣✐/❛❧ ♥♦ ♣(♦❞✉/♦✱K✱L❡Y +2♦ ♦+ ✈❛❧♦(❡+

❛❣(❡❣❛❞♦+ ❞❡ ❝❛♣✐/❛❧✱ /(❛❜❛❧❤♦ ❡ ♣(♦❞✉/♦✱ (❡+♣❡❝/✐✈❛♠❡♥/❡✳ ❖ ♣❛(Q♠❡/(♦ ❞❛ ♣(♦❞✉/✐✈✐❞❛❞❡ /♦/❛❧ ❞♦+ ❢❛/♦(❡+A ❃ ✵ O ❝♦♥+/❛♥/❡✳

T♦❞❡✲+❡ (❡❡+❝(❡✈❡( ❛ ❡"✉❛-2♦ ✭✶✮ ❡♠ /❡(♠♦+ ♣❡" ❝❛♣✐&❛✱ ❞❡ ♠♦❞♦ ❛ /❡(✿

yt=f(kt) =Akαt ✭✷✮

♦♥❞❡ ❛( ✈❛*✐,✈❡✐( ♠✐♥.(❝✉❧❛( ❡(23♦ ❡♠ 2❡*♠♦( ♣❡" ❝❛♣✐&❛✳

❈♦♥(✐❞❡*❛♥❞♦ ❛ 2❛①❛ ❞❡ ❞❡♣*❡❝✐❛83♦ ✐❣✉❛❧ : ✉♥✐❞❛❞❡✱ ♦ ♣*♦❜❧❡♠❛ ❞❡ ♠❛①✐✲ ♠✐③❛83♦ ❞❛ ✜*♠❛ ❣❡*❛ ❝♦♥❞✐8@❡( ❞❡ ♣*✐♠❡✐*❛ ♦*❞❡♠ A✉❡ ❞❡2❡*♠✐♥❛♠ ♦ *❡2♦*♥♦ *❡❛❧ ❞♦ ❝❛♣✐2❛❧ ❡ ♦ (❛❧,*✐♦ *❡❛❧✿

rt=α

yt

kt

−1 =αAkα−1

t −1 ✭✸✮

wt= (1−α)Aktα=

1−α α

(1 +rt)kt ✭✹✮

✷✳✷ ❆❣❡♥&❡'

❖( ❛❣❡♥2❡( ❋ ❡ ❉ ❥♦✈❡♥( *❡(♦❧✈❡♠ ♦( (❡❣✉✐♥2❡( ♣*♦❜❧❡♠❛( ♥♦ ♣❡*I♦❞♦ t✿

max

cF

1t,cF2t

uF(cF_1t, cF_2t) ✭✺✮ (✉❥❡✐2♦ ❛

cF_1t=wt−StF ✭✻✮

cF_2t=S_tF(1 +rt+1) ✭✼✮

❡

max

cD

1t,cD2t

uD(cD_1t, cD_2t) ✭✽✮ (✉❥❡✐2♦ ❛

cD_1t=wt(1−τt)−StD ✭✾✮

cD_2t=S_tD(1 +rt+1) + ˆbt+1 ✭✶✵✮

♦♥❞❡ cj_it $❡♣$❡&❡♥'❛ ♦ ❝♦♥&✉♠♦ ❞♦ ❛❣❡♥'❡ ♥♦ '❡♠♣♦t'❛❧ .✉❡✿

i=







1, se jovem

2, se idoso

j=







F, se f ora do sistema previdenciario´

D, se dentro do sistema previdenci´ario

❆❧1♠ ❞✐&&♦✱St1 ♣♦✉♣❛♥4❛✱τt1 ♦ ✐♠♣♦&'♦✴❝♦♥'$✐❜✉✐47♦ ❛❞✲✈❛❧♦&❡♠ ❡ˆbt+1

1 ❛ ❛♣♦&❡♥'❛❞♦$✐❛✱ .✉❡ ♣♦❞❡ &❡$ ❞❡✜♥✐❞❛ ❝♦♠♦ ✉♠❛ '❛①❛ &♦❜$❡ ♦ ✈❛❧♦$ ❞♦ &❛❧;$✐♦✿

ˆ_{bt+1 =bt+1wt} ❙❡❥❛ ❛ ✉'✐❧✐❞❛❞❡uj(cj₁, cj₂) ❞❡✜♥✐❞❛ ❝♦♠♦✿

uj(cj₁, cj₂) = lncj₁+βlncj₂

▲♦❣♦✱ ♣♦❞❡♠✲&❡ $❡❡&❝$❡✈❡$ ♦& ♣$♦❜❧❡♠❛& ❞♦& ❛❣❡♥'❡& ❝♦♠♦✿

max

cF

1t,cF2t

uF(cF_1t, cF_2t) = lncF₁ +βlncF₂ ✭✶✶✮

&✉❥❡✐'♦ ❛ ✭✻✮ ❡ ✭✼✮

max

cD

1t,cD2t

uD(cD_1t, cD_2t) = lncD₁ +βlncD₂ ✭✶✷✮

&✉❥❡✐'♦ ❛ ✭✾✮ ❡ ✭✶✵✮✳

❖ ❝♦♥&✉♠✐❞♦$ ❝♦♥&✐❞❡$❛ τt❡ bt+1 ❝♦♠♦ ❞❛❞♦&✳ ❘❡&♦❧✈❡♥❞♦✱ ❝❤❡❣❛✲&❡ ❡♠✿

cF_1t= wt

1 +β ✭✶✸✮

cD_1t= wt 1 +β

(1−τt) +

bt+1

1 +rt+1

✭✶✹✮

cF_2t= β

1 +β(1 +rt+1)wt ✭✶✺✮ cD_2t= β(1 +rt+1)wt

1 +β

(1−τt) +

bt+1

1 +rt+1

✭✶✻✮

S_tF = β

1 +βwt ✭✶✼✮

S_tD =wt

(1−τt)−

1 1 +β

(1−τt) +

bt+1

1 +rt+1

✭✶✽✮

❚❡♥❞♦ ♦, ❝♦♥,✉♠♦, ❡ ♣♦✉♣❛♥2❛ ❞❡ ❝❛❞❛ ✐♥❞✐✈5❞✉♦✱ ♣♦❞❡♠✲,❡ ❝❛❧❝✉❧❛9 ♦, ,❡✉, ✈❛❧♦9❡, ❛❣9❡❣❛❞♦,✿

C1t=γcF1t+ (1−γ)cD1t ✭✶✾✮

❙✉❜,?✐?✉✐♥❞♦ ✭✶✸✮ ❡ ✭✶✹✮ ❡♠ ✭✶✾✮✿

C₁t=

wt

1 +β

1 + (1−γ)

bt+1

1 +rt+1

−τt

✭✷✵✮

❆♥❛❧♦❣❛♠❡♥?❡✱ ♣❛9❛ ❛, ❞❡♠❛✐, ✈❛9✐E✈❡✐,✿

C₂t=

β

1 +β(1 +rt+1)wt

1 + (1−γ)

bt+1

1 +rt+1

−τt

✭✷✶✮

St=

wt

1 +β

β[1−(1−γ)τt]−(1−γ)

bt+1

1 +rt+1

✭✷✷✮

F♦9 ✜♠✱ ❝♦♠♦ ❛ ♣♦✉♣❛♥2❛ ❛❣9❡❣❛❞❛ H ✐❣✉❛❧ ❛♦ ❡,?♦I✉❡ ❞❡ ❝❛♣✐?❛❧ ❛❣9❡✲ ❣❛❞♦✱ ❛ ❡I✉❛2J♦ ❞✐♥K♠✐❝❛ ❞♦ ❝❛♣✐?❛❧ H ❞❛❞❛ ♣♦9✿

γLtStF + (1−γ)LtStD =Kt+1 LtSt=Lt+1kt+1

(1 +n)kt+1=St ✭✷✸✮

✷✳✸ ●♦✈❡'♥♦

❈♦♥#✐❞❡'❡ (✉❡ ♦ ♦'*❛♠❡♥-♦ ♣'❡✈✐❞❡♥❝✐1'✐♦ ❞♦ ❣♦✈❡'♥♦ ❡#-❡❥❛ #❡♠♣'❡ ❡♠ ❡(✉✐❧5❜'✐♦✱ ❞❡ ♠♦❞♦ (✉❡ ♦ -♦-❛❧ ❛''❡❝❛❞❛❞♦ ✈✐❛ ✐♠♣♦#-♦ ❞♦# ❥♦✈❡♥# ❞❡ ❤♦❥❡ #❡❥❛ ✐❣✉❛❧ ❛♦ -♦-❛❧ ❛ #❡' ♣❛❣♦ ❡♠ ❛♣♦#❡♥-❛❞♦'✐❛# ❛♦# ✐❞♦#♦# ❞❡ ❤♦❥❡ ✭❥♦✈❡♥# ❞❡ ♦♥-❡♠✮✿

(1−γ)Lt+1τt+1wt+1= (1−γ)bt+1wtLt

❈♦♥#✐❞❡'❛♥❞♦ (✉❡ ❛ ♣♦♣✉❧❛*<♦ ❤♦❥❡ -❡♥❤❛ -❛♠❛♥❤♦ ✶ ❡ ❝'❡#*❛ > -❛①❛ n✱

♣♦❞❡✲#❡ '❡❡#❝'❡✈❡' ❛ ❡(✉❛*<♦ ❛❝✐♠❛ ❝♦♠♦✿

ˆ

bt+1=bt+1wt=τt+1wt+1(1 +n)

❖✉ ❛✐♥❞❛✿

bt+1=

ˆ

bt+1 wt

= τt+1wt+1(1 +n)

wt

= τt+1(

1−α

α )(1 +rt+1)(1 +n)kt+1

wt ✭✷✹✮

❙✉❜#-✐-✉✐♥❞♦ ✭✷✸✮ ❡♠ ✭✷✹✮✿

bt+1=

τt+1 1−αα

(1 +rt+1)St

wt ✭✷✺✮

❙✉❜#-✐-✉✐♥❞♦ ♦ ✈❛❧♦' ❞❡St ❝❛❧❝✉❧❛❞♦ ❡♠ ✭✷✷✮✿

bt+1=τt+1

1−α α

(1 +rt+1)

1

1 +β β[1−(1−γ)τt]−(1−γ) bt+1 1 +rt+1

■#♦❧❛♥❞♦ bt+1✿ bt+1 =

β[1−(1−γ)τt]

1−γ (1 +rt+1)

τt+1(1−αα)(1−γ)(1+1β)

1 +τt+1(1−_αα)(1−γ)(₁₊1_β)

H♦' #✐♠♣❧✐❝✐❞❛❞❡✱ ❝♦♥#✐❞❡'❡✿

φ(τt+1) =

τt+1(1−_αα)(1−γ)(₁₊1_β)

1 +τt+1(1−_αα)(1−γ)(₁₊1_β) ✭✷✻✮

❉❡ ♠♦❞♦ (✉❡✿

bt+1

1 +rt+1

=β

1 1−γ −τt

φ(τt+1) ✭✷✼✮

❆ ❡"✉❛%&♦ ❛♥)❡*✐♦* ♠♦-)*❛ ❛ *❡❧❛%&♦ ❡♥)*❡ ❝♦♥)*✐❜✉✐%&♦ ❡ ❛♣♦-❡♥)❛❞♦*✐❛ "✉❡ ♠❛♥)3♠ ♦ ♦*%❛♠❡♥)♦ ♣*❡✈✐❞❡♥❝✐5*✐♦ ❡"✉✐❧✐❜*❛❞♦✳

❙❡♥❞♦ ❛--✐♠✱ ♣♦❞❡✲-❡ -✉❜-)✐)✉✐* ❛ ❡①♣*❡--&♦ ❛❝✐♠❛ ♥♦- ✈❛❧♦*❡- ❛❝❤❛❞♦-♣❛*❛ ❝♦♥-✉♠♦- ❡ ♣♦✉♣❛♥%❛ ♥❛- ❡"✉❛%<❡- ✭✷✵✮✱ ✭✷✶✮ ❡ ✭✷✷✮✱ ♣❛--❛♥❞♦ ❛ )B✲❧♦-❡♠ ❢✉♥%&♦ ❞❡kt✱τt ❡ τt+1✿

C1t=

wt

1 +β[1−(1−γ)τt][1 +βφ(τt+1)] ✭✷✽✮ C2t=

β

1 +β(1 +rt+1)wt[1−(1−γ)τt][1 +βφ(τt+1)] ✭✷✾✮ St=

β

1 +βwt[1−(1−γ)τt][1−φ(τt+1)] ✭✸✵✮

❙✉❜-)✐)✉✐♥❞♦ ✭✹✮ ❡ ✭✸✵✮ ❡♠ ✭✷✸✮✱ ❛❝❤❛✲-❡ ❛ ❡"✉❛%&♦ ❞❡ ❛❝✉♠✉❧❛%&♦ ❞♦ ❝❛♣✐)❛❧ ✭❧❡♠❜*❛♥❞♦ "✉❡ τ 3 ❡①I❣❡♥♦✮✿

(1 +n)kt+1 = β

1 +β(1−α)Ak

α

t[1−(1−γ)τt][1−φ(τt+1)] ✭✸✶✮

✷✳✹ ❊$✉✐❧(❜*✐♦

◆♦ ❡-)❛❞♦ ❡-)❛❝✐♦♥5*✐♦✱ )❡♠✲-❡ kt = kt+1 = k ❡ τt =τt+1 =τ✳ ❙✉❜-)✐✲

)✉✐♥❞♦ ❡♠ ✭✸✻✮ ❡ *❡-♦❧✈❡♥❞♦ ♣❛*❛k✿

k=

β

1 +β

1−α

1 +nA[1−(1−γ)τ][1−φ(τ)]

₁1 −α

✭✸✷✮

❈♦♠ ♦ *❡-✉❧)❛❞♦ ❛❝✐♠❛✱ ❝❤❡❣❛✲-❡ ♥♦- ✈❛❧♦*❡- ❞♦- ❝♦♥-✉♠♦- ❡ ♣♦✉♣❛♥%❛ ♥♦ ❡-)❛❞♦ ❡-)❛❝✐♦♥5*✐♦✱ ❛ ♣❛*)✐* ❞❛- ❡"✉❛%<❡- ✭✷✻✮✱✭✷✼✮ ❡ ✭✷✽✮✳

✷✳✺ ❈❛❧✐❜*❛/0♦

❆ ❝❛❧✐❜*❛%&♦ 3 ❜❛-❡❛❞❛ ❡♠ ✈❛❧♦*❡- ❡-)✐♠❛❞♦- ♣♦* ❛❧❣✉♥- ❛*)✐❣♦- ♣❛*❛ ❛❧❣✉♥- ♣❛*O♠❡)*♦- ♥♦- P❧)✐♠♦- ❛♥♦-✱ ❛ ♣❛*)✐* ❞❡ ✷✵✵✹ "✉❛♥❞♦ ♣♦--Q✈❡❧✳ ➱ ✉-❛❞❛ ✉♠❛ ♣*♦♣♦*%&♦ S♦✉♣❛♥%❛✲S*♦❞✉)♦ ❞❡ ✷✸✪✱ "✉❡ ❝♦**❡-♣♦♥❞❡ ❛♦ ✈❛❧♦* ♦❜-❡*✈❛❞♦ ♣❛*❛ ❛♣❡♥❛- ❛ ♣♦✉♣❛♥%❛ ❞❛- ❢❛♠Q❧✐❛- ♥♦ ♣❛Q-✱ ❞❛❞♦ "✉❡

❡❝❡♥$❡% ✐♥❞✐❝❛♠ ❛❧❣♦ ❡♠ $♦ ♥♦ ❞❡ ✷✵✪ ❡ ✷✺✪✳ ❆ ♣ ♦♣♦ 45♦ ❞♦ ❝❛♣✐$❛❧ 6

α= 0,48✱ ❡♠ ❧✐♥❤❛ ❝♦♠ ❇❛✐ ❡$ ❛❧ ❬✶❪ ✱ =✉❡ ❛♣♦♥$❛♠ =✉❡✱ ❞❡ ✶✾✼✽ ❛ ✷✵✵✸✱ $❛❧ ♣❛ C♠❡$ ♦ ♦%❝✐❧♦✉ ❡♥$ ❡ ✵✱✹✻ ❡ ✵✱✺✵✳

❆ $❛①❛ ❞❡ ✐♠♣♦%$♦ %♦❜ ❡ %❛❧H ✐♦✱ τ✱ 6 ❞❡ ✷✵✪✱ ♠✉✐$♦ ♣ I①✐♠❛ ❞❡ ✈❛❧♦ ❡%

✉%❛❞♦% ♥❛ ❧✐$❡ ❛$✉ ❛✱ ❝♦♠♦ ❡♠ ❏✐♥ ❬✻❪✱ =✉❡ ♠♦%$ ❛ ❛ ❞✐✈✐%5♦ ❞♦% $ L% ♣✐❧❛ ❡% ❞♦ %✐%$❡♠❛ ♣ ❡✈✐❞❡♥❝✐H ✐♦ ❝❤✐♥L% ❡ ❛ $❛①❛ ❡♠ ❝❛❞❛ ✉♠✱ ✜❝❛♥❞♦✱ ❡♠ ♠6❞✐❛✱ ❡♠ $♦ ♥♦ ❞♦ ♥N♠❡ ♦ ✉%❛❞♦✳ ❆ ❞❡♣ ❡❝✐❛45♦ 6 $♦$❛❧ ❡ ♦β = 0,940✱ ✈❛❧♦ =✉❡

6 ❝❛❧✐❜ ❛❞♦ ♣❛ ❛ =✉❡ %❡❥❛ ♦❜$✐❞❛ ❛ ♣ ♦♣♦ 45♦ P♦✉♣❛♥4❛✲P ♦❞✉$♦ ❞❡ ✷✸✪✳ ❈♦♠♦ ❜❛%❡ ❞❡ ❝♦♠♣❛ ❛45♦✱ ❛ ❝♦❜❡ $✉ ❛ ♣ ❡✈✐❞❡♥❝✐H ✐❛ ❛$✉❛❧ ❝❤✐♥❡%❛ 6 ❡%$✐♣✉❧❛❞❛ ❡♠ ✷✾✪ ✭✐✳❡✳✱γ = 0,71✮✱ ❝♦♥❢♦ ♠❡ ❖❦%❛♥❡♥ ❬✶✷❪✳

✷✳✻ ❘❡%✉❧(❛❞♦%

❈♦♠♦ ❡ ❛ ❞❡ %❡ ❡%♣❡ ❛ ✱ ❛♦ %❡ ❡①♣❛♥❞✐ ❛ ❝♦❜❡ $✉ ❛ ♣ ❡✈✐❞❡♥❝✐H ✐❛ ♣❛ ❛ $♦❞♦% ♦% $ ❛❜❛❧❤❛❞♦ ❡% ❝❤✐♥❡%❡%✱ ❤♦✉✈❡ ✉♠❛ =✉❡❞❛ ♥❛ ♣ ♦♣♦ 45♦ P♦✉♣❛♥4❛✲ P ♦❞✉$♦✱ ♥❛ ♠❛❣♥✐$✉❞❡ ❞❡ =✉❛%❡ ✺ ♣♦♥$♦% ♣❡ ❝❡♥$✉❛✐%✱ ❞❡ ✷✸✪ ♣❛ ❛ ✶✽✱✶✪✳ ❆ ❡❝♦♥♦♠✐❛ ✜❝♦✉ ♠❛✐% ♣♦❜ ❡✿ ❛ ♣ ♦❞✉$♦ ❝❛✐✉ ✶✾✱✼✪✳ ❆ ♣♦✉♣❛♥4❛ ❛❣ ❡❣❛❞❛ ❝❛✐✉ ✸✻✱✻✪✳ ❖ ♣ ♦❝❡%%♦ %❡ ❞H ❞❛ %❡❣✉✐♥$❡ ❢♦ ♠❛✿ =✉❛♥❞♦ ✉♠❛ ♣❡%%♦❛ ❡%$H ❝♦❜❡ $❛✱ ❡❧❛ ❡❞✉③ %✉❛ ♣♦✉♣❛♥4❛ ✭♦✉ %❡❥❛✱ ❛❝✉♠✉❧❛ ♠❡♥♦% ❝❛♣✐$❛❧✮✱ ✉♠❛ ✈❡③ =✉❡ ❡%$❛ ♥5♦ ♣ ❡❝✐%❛ ✐❛ %❡ $5♦ ❡❧❡✈❛❞❛✱ ❣ ❛4❛% Z $ ❛♥%❢❡ L♥❝✐❛ ♣ ♦♠♦✈✐❞❛ ♣❡❧♦ ❊%$❛❞♦ ✈✐❛ ❛♣♦%❡♥$❛❞♦ ✐❛ ♥♦ ❢✉$✉ ♦✳ ❙❡♥❞♦ ❛%%✐♠✱ =✉❛♥❞♦ ❤H ✉♠ ♥N♠❡ ♦ ♠❛✐♦ ❞❡ ♣❡%%♦❛% ❞❡♥$ ♦ ❞♦ %✐%$❡♠❛✱ ❤H ✉♠❛ =✉❡❞❛ ♥♦ ❝❛♣✐$❛❧ ❛❣ ❡❣❛❞♦✳

❆ ♣♦✉♣❛♥4❛ ❛❣ ❡❣❛❞❛ ❝❛✐✉ ♣♦✐% $♦❞♦% ♦% ❛❣❡♥$❡% ✜❝❛ ❛♠ ♠❛✐% ♣♦❜ ❡%✿ ❝♦♠ ❛ =✉❡❞❛ ❞♦ ❝❛♣✐$❛❧✱ %❛❧H ✐♦% ❝❛] ❛♠ ❡ ❥✉ ♦% %✉❜✐ ❛♠✳ ❉❡%%❛ ❢♦ ♠❛✱ $❛♥$♦ ❛ ♣♦✉♣❛♥4❛ ❞♦% ❛❣❡♥$❡% ❞❡♥$ ♦ ❞♦ %✐%$❡♠❛ =✉❛♥$♦ ❞❛=✉❡❧❡% ❢♦ ❛ ❝❛] ❛♠ ❝♦♥✲ ❢♦ ♠❡ ❛ ❝♦❜❡ $✉ ❛ ❛✉♠❡♥$♦✉✿ ❛ ♣ ✐♠❡✐ ❛✱ ♣❡❧♦ ❢❛$♦ ❞❡ ♦% ❣❛♥❤♦% ❞♦ ❛✉♠❡♥$♦ ❞♦% ❥✉ ♦% %♦❜ ❡ ❛ ❛♣♦%❡♥$❛❞♦ ✐❛ ♥5♦ $❡ ❡♠ ❝♦♠♣❡♥%❛❞♦ ❛ =✉❡❞❛ ❞♦ %❛❧H ✐♦❀ ❛ %❡❣✉♥❞❛✱ ❛♣❡♥❛% ♣❡❧♦ %❛❧H ✐♦ $❡ ❞✐♠✐♥✉]❞♦✳ P♦❞❡✲%❡ ♣❡ ❝❡❜❡ ✱ ♣♦ $❛♥$♦✱ =✉❡

❋✐❣✉$❛ ✶✿ ❘❛③*♦ ,♦✉♣❛♥/❛✲,$♦❞✉2♦

❋✐❣✉$❛ ✷✿ ,♦✉♣❛♥/❛

❋✐❣✉$❛ ✸✿ ($♦❞✉+♦

♦ ✐♠♣❛❝+♦ /♦❜$❡ ❛ ♣♦✉♣❛♥3❛ ❢♦✐ ♠❛✐♦$ ❡♥+$❡ ♦/ ❛❣❡♥+❡/ ❋ ❞♦ 5✉❡ ❡♥+$❡ ♦/ ❉✿ ❡♥5✉❛♥+♦ ❛ ❞♦/ ♣$✐♠❡✐$♦/ ❝❛✐✉ ✶✾✱✼✪✱ ❛ ❞♦/ /❡❣✉♥❞♦/ ❝❛✐✉ ✶✼✱✷✪✳

❆♠❜♦/ ♦/ ❛❣❡♥+❡/ $❡❞✉③✐$❛♠ ♦ ❝♦♥/✉♠♦ ♥♦ ♣$✐♠❡✐$♦ ♣❡$@♦❞♦✱ +❛♠❜A♠ ♣♦$ +❡$❡♠ ✜❝❛❞♦ ♠❛✐/ ♣♦❜$❡/✿ 5✉❡❞❛ ❞❡ ✶✾✱✼✪ ♣❛$❛ ♦/ ❛❣❡♥+❡/ ❋ ❡ ❞❡ ✷✶✱✺✪ ♣❛$❛ ♦/ ❛❣❡♥+❡/ ❉✳ ❏E ♥♦ /❡❣✉♥❞♦ ♣❡$@♦❞♦✱ ♦/ ❛❣❡♥+❡/ ❋ ❛✉♠❡♥+❛$❛♠ ♦ ❝♦♥/✉♠♦ ✭✶✳✽✪✮✱ ♣♦✐/ ❛ 5✉❡❞❛ ❞♦ /❛❧E$✐♦ ❢♦✐ ♠❛✐/ ❞♦ 5✉❡ ❝♦♠♣❡♥/❛❞❛ ♣❡❧❛ ❛❧+❛ ❞♦/ ❥✉$♦/✳ ❊♥+$❡+❛♥+♦✱ ✈❛❧❡ ❞❡/+❛❝❛$ 5✉❡ ❛❧+❡$❛3M❡/ ♥♦ ♣❛$N♠❡+$♦α ♣♦❞❡♠ ❣❡$❛$

✉♠ $❡/✉❧+❛❞♦ ❝♦♥+$E$✐♦✳ ❈♦♠♦✱ ❝♦♠ ♦ ❛✉♠❡♥+♦ ❞❛ ❝♦❜❡$+✉$❛ ♣$❡✈✐❞❡♥❝✐E$✐❛✱ ❛❣❡♥+❡/ ❋ /P♦ ❝❛❞❛ ✈❡③ ♠❡♥♦/ ♣$❡/❡♥+❡/ ♥❛ ❡❝♦♥♦♠✐❛✱ ♦ ❝♦♥/✉♠♦ ❛❣$❡❣❛❞♦ ♥♦ /❡❣✉♥❞♦ ♣❡$@♦❞♦ ❢♦✐ ❝❛❞❛ ✈❡③ ♠❡♥♦$✳✶

(❛$+❡✲/❡ ❛❣♦$❛ ♣❛$❛ ♦ ❝E❧❝✉❧♦ ❞❛ ✈❛$✐❛3P♦ ❞♦ ❜❡♠✲❡/+❛$ ❞♦/ ❛❣❡♥+❡/ ❞❛ ❡❝♦♥♦♠✐❛ ❡♠ ❢✉♥3P♦ ❞❛ ❛♠♣❧✐❛3P♦ ❞❛ ❝♦❜❡$+✉$❛ ♣$❡✈✐❞❡♥❝✐E$✐❛✳ (❛$❛ +❛❧✱

✶_{❱❛❧❡ ❞❡%&❛❝❛( )✉❡✱ ♣❛(❛ ♦ ❝❛%♦ ❞❡ ❛❣❡♥&❡% ❢♦(❛ ❞♦ %✐%&❡♠❛ ♣(❡✈✐❞❡♥❝✐4(✐♦✱ ❛ ❝♦♠♣❛(❛56♦}

7 ❢❡✐&❛ ❝♦♠ ♦ ❝❛%♦ ❧✐♠✐&❡ ♥♦ )✉❛❧ ❤4 ❛♣❡♥❛% ✉♠ ✐♥❞✐✈9❞✉♦ ❛✐♥❞❛ ❢♦(❛ ❞♦ %✐%&❡♠❛ ✲ ✐%&♦ 7✱

γ♠✉✐&♦ ♣(;①✐♠♦ ❞❡ ③❡(♦✳ ◆6♦ ❢❛(✐❛ %❡♥&✐❞♦ ❝♦♠♣❛(❛( ❝♦♠ ♦ ❝❛%♦ ❡♠ )✉❡γ= 0✱ ♣♦✐% ♥6♦

❤❛✈❡(✐❛ ♠❛✐% ❡%%❡ &✐♣♦ ❞❡ ❛❣❡♥&❡ ♥❛ ❡❝♦♥♦♠✐❛✳

❋✐❣✉$❛ ✹✿ ❈♦♥+✉♠♦ ✶➸ ♣❡$0♦❞♦

❋✐❣✉$❛ ✺✿ ❈♦♥+✉♠♦ ✷➸ ♣❡$0♦❞♦

❋✐❣✉$❛ ✻✿ ❯)✐❧✐❞❛❞❡

❝♦♥0✐❞❡$❛✲0❡ 2✉❡ ❡①✐0)❛ ✉♠ ❣♦✈❡$♥♦ 2✉❡ 0❡ ♣$❡♦❝✉♣❡ ❡♠ ♠❛①✐♠✐③❛$ ♦ ❜❡♠✲ ❡0)❛$ ❛❣$❡❣❛❞♦ ❞❛ ♣♦♣✉❧❛9:♦✳ ❙❡♥❞♦ ❛00✐♠✱ ❡❧❡ )$❛)❛ ❛ ♣♦♣✉❧❛9:♦ ❝♦♠♦ ✉♠ ❛❣❡♥)❡ >♥✐❝♦ 2✉❡ ❝♦♥0♦♠❡C1t♥♦ ♣$✐♠❡✐$♦ ♣❡$?♦❞♦ ❡C2t ♥♦ 0❡❣✉♥❞♦✳✷

@$✐♠❡✐$❛♠❡♥)❡✱ ❝❛❧❝✉❧❛✲0❡ ♦ ❜❡♠✲❡0)❛$ ♣❛$❛ ❛ 0✐)✉❛9:♦ ♦$✐❣✐♥❛❧ ❞❡ ✷✾✪ ❞♦0 )$❛❜❛❧❤❛❞♦$❡0 ✐♥❝❧✉?❞♦0 ♥♦ 0✐0)❡♠❛✱ ✉0❛♥❞♦✲0❡ ♦0 ♠❡0♠♦0 ✈❛❧♦$❡0 ♣❛$❛ ♦0 ♣❛$E♠❡)$♦0 2✉❡ ♦0 ✉0❛❞♦0 ❛❝✐♠❛✿

U∗ _{= ln [C}∗

1t] +βln [C2∗t]

❊♠ 0❡❣✉✐❞❛✱ ❝❛❧❝✉❧❛♠✲0❡C1t❡C2t 2✉❡ $❡0♦❧✈❡♠ ♦ ♠❡0♠♦ ♣$♦❜❧❡♠❛ ♣❛$❛

❝♦❜❡$)✉$❛ ❛❧)❡$♥❛)✐✈❛ ✭♥♦ ❝❛0♦ ❞❡ ❝♦❜❡$)✉$❛ )♦)❛❧✱ ✶✵✵✪✮✳ ❚❡♥❞♦ ❡00❡0 ✈❛❧✲ ♦$❡0✱ ❛❝❤❛✲0❡ ♦ ❝✉0)♦x ❞❡ ❜❡♠✲❡0)❛$ $❡❧❛❝✐♦♥❛❞♦ ❛♦ ❛✉♠❡♥)♦ ❞❛ ❝♦❜❡$)✉$❛✿

U∗ _{= ln [(1 +x)C}

1t] +βln [(1 +x)C2t]

♦✉ 0❡❥❛✱ x $❡♣$❡0❡♥)❛ ❡♠ 2✉❛♥)♦ ♦ ❝♦♥0✉♠♦ ❞❡✈❡$✐❛ ❛✉♠❡♥)❛$ ♣❛$❛ ❧❡✈❛$ ❛

❡❝♦♥♦♠✐❛ ❞❡ ✈♦❧)❛ ❛♦ ❜❡♠✲❡0)❛$ ♦$✐❣✐♥❛❧✳

✷_{■♥"❡$♣$❡"❛'❛♦ ♣❛$❡❝✐❞❛ ❝♦♠ ❛ -✉❡ ❞❡ ❢❛"♦ ♠✉✐"♦0 ❣♦✈❡$♥♦0 ❢❛③❡♠ ❛"✉❛❧♠❡♥"❡✳}

❋✐❣✉$❛ ✼✿ ❈✉)*♦ ❞❡ ❜❡♠✲❡)*❛$✿ ①

❋✐❣✉$❛ ✽✿ ❈✉)*♦ ❞❡ ❜❡♠✲❡)*❛$✿ xCt

y∗

❈❛"♦ ❛ ❝♦❜❡'(✉'❛ ♣'❡✈✐❞❡♥❝✐/'✐❛ ❢♦""❡ ❞♦❜'❛❞❛✱ ✐"(♦ 2✱ ❡❧❡✈❛❞❛ ♣❛'❛ ✺✽✪✱ ♦ ❝♦♥"✉♠♦ ❞❡✈❡'✐❛ ❛✉♠❡♥(❛' ❡♠ ✽✱✷✪ ♣❛'❛ 9✉❡ ♦ ❜❡♠✲❡"(❛' ❝♦♥(✐♥✉❛""❡ ♦ ♠❡"♠♦ 9✉❡ ♦ ❛♥(❡'✐♦'✳ <❛'❛ ♦ ❝❛"♦ ❞❡ ❝♦❜❡'(✉'❛ (♦(❛❧✱ ❞❡✈❡'✐❛ ❛✉♠❡♥(❛' ✷✷✱✾✪✱ ❛❧❣♦ ❜❛"(❛♥(❡ ❡①♣'❡""✐✈♦✱ ♦ 9✉❡ ♠♦"('❛ ♦ 9✉@♦ ♠❛❧2✜❝♦ ♣❛'❛ ♦ ❜❡♠✲ ❡"(❛' "❡'✐❛ ❛ ❛♠♣❧✐❛B@♦ ❞♦ "✐"(❡♠❛✳

❖✉('❛ ♠❛♥❡✐'❛ ❞❡ "❡ ✈❡'✐✜❝❛' ♦ ✐♠♣❛❝(♦ ❞❛ ♠✉❞❛♥B❛ 2 ❝❛❧❝✉❧❛♥❞♦ ❛ '❛③@♦

xCt

y∗ ✱ !✉❡ ♠♦&'(❛ ❛ ♣(♦♣♦(+,♦ ❞❛ ✈❛(✐❛+,♦ ♥❡❝❡&&2(✐❛ ❞♦ ❝♦♥&✉♠♦ &♦❜(❡ ♦

♣(♦❞✉'♦ ♥❛ &✐'✉❛+,♦ ♦(✐❣✐♥❛❧✳ 7❛(❛ ♦ ❝❛&♦ ❡♠ !✉❡ ❛ ❝♦❜❡('✉(❛ 8 ❞♦❜(❛❞❛✱ '❛❧ ♣(♦♣♦(+,♦ 8 ❞❡ ✷✱✵✺✪ ❡ ✹✱✸✵✪ ♣❛(❛ C1t ❡ C2t✱ (❡&♣❡❝'✐✈❛♠❡♥'❡✳ 7❛(❛

❝♦❜❡('✉(❛ '♦'❛❧✱ ♦& ✈❛❧♦(❡& &,♦ ❜❡♠ ♠❛✐& ❡①♣(❡&&✐✈♦&✿ ✺✱✸✼✪ ❡ ✶✷✱✾✽✪✱ ♦ !✉❡ ❞❡✐①❛ ❛✐♥❞❛ ♠❛✐& ❝❧❛(♦ ♦ ✐♠♣❛❝'♦ ♥❡❣❛'✐✈♦ ❞❛ ❛♠♣❧✐❛+,♦ ❞♦ &✐&'❡♠❛ &♦❜(❡ ♦ ❜❡♠✲❡&'❛( ❞❛ ♣♦♣✉❧❛+,♦ ❝❤✐♥❡&❛✳

7♦('❛♥'♦✱ ❝♦♠♦ ♦& ❛❣❡♥'❡& ✜❝❛(❛♠ ♠❛✐& ♣♦❜(❡&✱ '♦❞♦& '✐✈❡(❛♠ ✉♠❛ !✉❡❞❛ ♥❛ ✉'✐❧✐❞❛❞❡ ❝♦♠ ♦ ❛✉♠❡♥'♦ ❞❛ ❝♦❜❡('✉(❛✳ ❚❛❧ ❢❛'♦ ❥2 ❡(❛ ❞❡ &❡ ❡&♣❡(❛(✱ ❤❛❥❛ ✈✐&'❛ !✉❡ ❛ ❛♠♣❧✐❛+,♦ ❞❡ ✉♠❛ ✧♣♦✉♣❛♥+❛ ❝♦♠♣✉❧&L(✐❛✧ ❝❤❛♠❛❞❛ ❛♣♦&❡♥'❛✲ ❞♦(✐❛ !✉❡ ❧❡✈❛ ❛ ❡❝♦♥♦♠✐❛ ♣❛(❛ ❝❛❞❛ ✈❡③ ♠❛✐& ❧♦♥❣❡ ❞❡ ✉♠❛ (❡❛❧✐❞❛❞❡ &❡♠ ❞✐&'♦(+N❡& ♥,♦ ♣♦❞❡(✐❛ ♠❡❧❤♦(❛( ❛ &✐'✉❛+,♦ ♦(✐❣✐♥❛❧✳

✷✳✼ ◆♦✈❛ ❝❛❧✐❜+❛,-♦

❈♦♥&✐❞❡(♦✉✲&❡ ❛'8 ❛!✉✐ !✉❡ ❛♣❡♥❛& ❛& ❢❛♠P❧✐❛& ♣♦✉♣❛♠✱ &❡♥❞♦ ♦ ♠♦❞❡❧♦ ❝❛❧✐❜(❛❞♦ ♣❛(❛ ♦ (❡&♣❡❝'✐✈♦ ✈❛❧♦( ♥❛ ❡❝♦♥♦♠✐❛ ❝❤✐♥❡&❛✳ ❙✉♣♦♥❤❛ ❛❣♦(❛ !✉❡ ❛& ✜(♠❛& '❛♠❜8♠ ♣♦✉♣❡♠✸_{✳ ❊♥'(❡'❛♥'♦✱ ❛♦ ✐♥✈8& ❞❡ (❡❢❛③❡( ♦ ♠♦❞❡❧♦ ✐♥❝❧✉✐♥❞♦} ❡&'❛ ♥♦✈❛ ✈❛(✐2✈❡❧✱ &✉♣♦♥❤❛ !✉❡ ❡❧❛ ❡&'❡❥❛ ✐♥❝❧✉P❞❛ ♥❛ ♣♦✉♣❛♥+❛ ❞❛ ❢❛♠P❧✐❛&✳ 7♦('❛♥'♦✱ ❛ S♥✐❝❛ ♠✉❞❛♥+❛ 8 ♥❛ ❝❛❧✐❜(❛+,♦✿ ♥,♦ &❡(2 ♠❛✐& ✉&❛❞♦ ♦ ✈❛❧♦( ❞❡ ✷✸✪ ♣❛(❛ ❛ ♣(♦♣♦(+,♦ ❞❛ ♣♦✉♣❛♥+❛ ❞❛& ❢❛♠P❧✐❛& ♥♦ 7(♦❞✉'♦✱ ♠❛& ✸✺✪✱ ♥S♠❡(♦ ❡♥❝♦♥'(❛❞♦ ❡♠ ❑✉✐❥& ❬✶✵❪✳

✸_{❖♣"♦✉✲&❡ ♣♦( ❡①❝❧✉✐( ❛ ♣♦✉♣❛♥/❛ ❞♦ ❣♦✈❡(♥♦ ♣♦( ♣(♦❜❧❡♠❛& ♥❛ ❝❛❧✐❜(❛/5♦ 6✉❛♥❞♦ &❡}

❝♦♥&✐❞❡(❛ ✈❛❧♦(❡& ♠✉✐"♦ ❡❧❡✈❛❞♦& ✲ ❛❝✐♠❛ ❞❡ ✹✵✪ ✲ ♣❛(❛ ❛ (❛③5♦ ♣♦✉♣❛♥/❛✴♣(♦❞✉"♦✳

❆ ❝♦❜❡%&✉%❛ ♣%❡✈✐❞❡♥❝✐.%✐❛ ❛&✉❛❧ ❝❤✐♥❡1❛ ❢♦✐ ♥♦✈❛♠❡♥&❡ ❡1&✐♣✉❧❛❞❛ ❡♠ ✷✾✪ ✭✐✳❡✳✱ γ = 0,71✮✱ ❛ ♣%♦♣♦%;<♦ ❞♦ ❝❛♣✐&❛❧ ❡♠ α = 0,20 ❡ ❛ &❛①❛ ❞❡ ✐♠♣♦1&♦ 1♦❜%❡ 1❛❧.%✐♦✱ τ✱ ❡♠ ✶✺✪✹✳ ❆ ❞❡♣%❡❝✐❛;<♦ ❢♦✐ &♦&❛❧ ❡ ♦ β = 0,99✱

✈❛❧♦% @✉❡ ❢♦✐ ❝❛❧✐❜%❛❞♦ ♣❛%❛ @✉❡ ❢♦11❡ ♦❜&✐❞❛ ❛ ♣%♦♣♦%;<♦ A♦✉♣❛♥;❛✲A%♦❞✉&♦ ❞❡ ✸✺✪✳

▼❛✐1 ✉♠❛ ✈❡③✱ ❛♦ 1❡ ❡①♣❛♥❞✐% ❛ ❝♦❜❡%&✉%❛ ♣%❡✈✐❞❡♥❝✐.%✐❛ ♣❛%❛ &♦❞♦1 ♦1 &%❛❜❛❧❤❛❞♦%❡1 ❝❤✐♥❡1❡1✱ ❤♦✉✈❡ ✉♠❛ @✉❡❞❛ ♥❛ ♣%♦♣♦%;<♦ A♦✉♣❛♥;❛✲A%♦❞✉&♦✱ ♥❛ ♠❛❣♥✐&✉❞❡ ❞❡ @✉❛1❡ ✾ ♣♦♥&♦1 ♣❡%❝❡♥&✉❛✐1✱ ❞❡ ✸✺✪ ♣❛%❛ ✷✻✪✳ ❆ ❡❝♦♥♦✲ ♠✐❛ ✜❝♦✉ ♠❛✐1 ♣♦❜%❡✿ ❤♦✉✈❡ ✉♠❛ @✉❡❞❛ ♥♦ ♣%♦❞✉&♦ ❞❡ ✼✱✷✪ ❡ ❛ ♣♦✉♣❛♥;❛ ❛❣%❡❣❛❞❛ ❝❛✐✉ ✸✶✱✶✪✳ ▲❡♠❜%❛♥❞♦ @✉❡ ✐11♦ ♦❝♦%%❡ ♣♦%@✉❡✱ ✉♠❛ ✈❡③ ❝♦❜❡%&❛✱ ❛ ♣❡11♦❛ %❡❞✉③ 1✉❛ ♣♦✉♣❛♥;❛ ✭❛❝✉♠✉❧❛ ♠❡♥♦1 ❝❛♣✐&❛❧✮✳

❈♦♠ ✉♠ ❝❛♣✐&❛❧ ♠❡♥♦%✱ ♦1 1❛❧.%✐♦1 ❝❛❡♠ ❡ ♦1 ❥✉%♦1 1♦❜❡♠✳ ❙❡♥❞♦ ❛11✐♠✱ ♦ ❝♦♥1✉♠♦ ❞♦ ♣%✐♠❡✐%♦ ♣❡%O♦❞♦ ❝❛O✉ ♣❛%❛ ♦1 ❞♦✐1 ❛❣❡♥&❡1✱ ♣♦% &❡%❡♠ ✜❝❛❞♦ ♠❛✐1 ♣♦❜%❡1✿ @✉❡❞❛ ❞❡ ✼✱✷✪ ♣❛%❛ ♦1 ❛❣❡♥&❡1 ❋ ❡ ❞❡ ✶✷✱✽✪ ♣❛%❛ ♦1 ❛❣❡♥&❡1 ❉✱ 1❡♥❞♦ ✉♠❛ @✉❡❞❛ ❛❣%❡❣❛❞❛ ❞❡ ✻✱✶✪✳ ❏. ♥♦ 1❡❣✉♥❞♦ ♣❡%O♦❞♦✱ ❛♠❜♦1 ♦1 ❛❣❡♥&❡1 ❛✉♠❡♥&❛%❛♠ 1❡✉1 ❝♦♥1✉♠♦1 ✭✷✺✳✵✪ ❡ ✶✼✳✹✪✮✱ ♦❝♦%%❡♥❞♦ ✉♠ ❛✉♠❡♥&♦ ❞♦ ❝♦♥1✉♠♦ ❛❣%❡❣❛❞♦ ♥❡1&❡ ♣❡%O♦❞♦ ❞❡ ✷✻✱✺✪✺_✳

❆♣❡1❛% ❞❡ ✉♠❛ @✉❡❞❛ ♥❛ ♣♦✉♣❛♥;❛ ❛❣%❡❣❛❞❛✱ ♦ %❡1✉❧&❛❞♦ ❢♦✐ ❝✉%✐♦1♦✿ ❤♦✉✈❡ ✉♠ ❣❛♥❤♦ ❞❡ ❜❡♠✲❡1&❛%✳ ■11♦ 1❡ ❞❡✈❡ ❛♦ ❢♦%&❡ ❛✉♠❡♥&♦ ❞♦ ❝♦♥1✉♠♦ ♥♦ 1❡❣✉♥❞♦ ♣❡%O♦❞♦✱ @✉❡ ♠❛✐1 @✉❡ ❝♦♠♣❡♥1♦✉ ❛ @✉❡❞❛ ❞♦ ❝♦♥1✉♠♦ ♥♦ ♣%✐♠❡✐%♦✳ A♦%&❛♥&♦✱ ❡11❡ %❡1✉❧&❛❞♦ ❝♦♥✜%♠❛ ❛ ♦❜1❡%✈❛;<♦ ❢❡✐&❛ ❛♥&❡%✐♦%♠❡♥&❡ ❞❡

✹_{❊♠ ❢✉♥%&♦ ❞♦ ♥)♠❡+♦ ❧✐♠✐.❛❞♦ ❞❡ ♣❛+1♠❡.+♦2✱ ❛♣❡2❛+ ❞❡ ♥&♦ ❝♦♥❞✐③❡+ ❝♦♠ ♦ ♦❜2❡+✈❛❞♦}

♥❛ ❈❤✐♥❛✱ ♦ ✈❛❧♦+α= 0,20❢♦✐ ✉2❛❞♦ ♣❛+❛ :✉❡ 2❡ ♣✉❞❡22❡ .❡+ ✉♠❛ ❝❛❧✐❜+❛%&♦ ❢❛❝.;✈❡❧ ✲

❞♦ ❝♦♥.+=+✐♦✱ .❡+✲2❡✲✐❛ ✈❛❧♦+❡2 ❞❡β✱ ♦ ❢❛.♦+ ❞❡ ❞❡2❝♦♥.♦ ✐♥.❡+.❡♠♣♦+❛❧✱ ♠❛✐♦+❡2 ❞♦ :✉❡ ❛

✉♥✐❞❛❞❡✳ ?❡❧♦ ♠❡2♠♦ ♠♦.✐✈♦τ ❢♦✐ +❡❞✉③✐❞♦✱ ♠❛2✱ ♥❡2.❡ ❝❛2♦✱ ❛✐♥❞❛ ♣❡+.♦ ❞❛ +❡❛❧✐❞❛❞❡✳

✺_{❖ ❧❡✐.♦+ ❛.❡♥.♦ ♣♦❞❡ ❡2.+❛♥❤❛+ :✉❡ ♦2 ✈❛❧♦+❡2 ❞❛ ✈❛+✐❛%&♦ ❞❛2 ✈❛+✐=✈❡✐2 ❛❣+❡❣❛❞❛2}

❡2.❡❥❛♠ ❜❡♠ ❞✐❢❡+❡♥.❡2 ❞❛:✉❡❧❛2 ❞♦2 ❛❣❡♥.❡2 :✉❡ ❝♦♠♣C❡♠ ❛ ❡❝♦♥♦♠✐❛✳ ❊♥.+❡.❛♥.♦✱ ✈❛❧❡ ❧❡♠❜+❛+ :✉❡ ❡2.= 2❡♥❞♦ ❝♦♠♣❛+❛❞❛ ❛ 2✐.✉❛%&♦ ❡♠ :✉❡ ❤= ✉♠❛ ♣+♦♣♦+%&♦ ❣+❛♥❞❡ ❞❡ ❛❣❡♥.❡2 ❋ ♥❛ ❡❝♦♥♦♠✐❛ ❝♦♠ ♦✉.+❛ ❡♠ :✉❡ ❡❧❡2 ♥&♦ ❡①✐2.❡♠✳

❋✐❣✉$❛ ✾✿ ❈✉)*♦ ❞❡ ❜❡♠✲❡)*❛$✿ ①

❋✐❣✉$❛ ✶✵✿ ❈✉)*♦ ❞❡ ❜❡♠✲❡)*❛$✿ xCt

y∗

✉❡ ♦ ♠♦❞❡❧♦ ' (❡♥(*✈❡❧ ❛ ❛❧-❡.❛/0❡( ♥♦( ♣❛.2♠❡-.♦(α ❡ τ✱ ❡ ✉❡ ♠✉❞❛♥/❛(

♥❡❧❡( ♣♦❞❡♠ ❣❡.❛. .❡(✉❧-❛❞♦( ❜❡♠ ❞✐❢❡.❡♥-❡(✳

✸ ❊①#❡♥&'♦

❙✉♣♦♥❤❛ ❛❣♦.❛ ✉❡ (❡ ❞❡✐①❡ ❞❡ ❝♦♥(✐❞❡.❛. ❛ ❝♦♥-.✐❜✉✐/=♦ ♣.❡✈✐❞❡♥❝✐>.✐❛ ❝♦♠♦ ❡①?❣❡♥❛✳ ❈♦♥(✐❞❡.❡ ✉❡ ❤❛❥❛ ✉♠ ❣♦✈❡.♥♦ ✉❡ (❡ ♣.❡♦❝✉♣❡ ❛♣❡♥❛( ❝♦♠ ♦ ❜❡♠✲❡(-❛. ❞❛ ❣❡.❛/=♦ ❥♦✈❡♠ ❛ ❝❛❞❛ ♣❡.*♦❞♦✳ ❚❛❧ ❣♦✈❡.♥♦ ❞❡✈❡ ❡(❝♦❧❤❡. ✉❛❧ ' ♦ ✐♠♣♦(-♦ ❝♦❜.❛❞♦ ❛ ❝❛❞❛ ♣❡.*♦❞♦ ❞❡ ♠♦❞♦ ❛ ♠❛♥-❡. ♦ (✐(-❡♠❛ ♣.❡✈✲ ✐❞❡♥❝✐>.✐♦ ❡ ✉✐❧✐❜.❛❞♦✳ ❙❡❣✉✐♥❞♦ ♠❡-♦❞♦❧♦❣✐❛ ❞❡ ❋♦.♥✐ ❬✼❪✱ ❢♦✐ ❞❡(❡♥✈♦❧✈✐❞♦ ✉♠ ❡( ✉❡♠❛ ✉❡ .❡(✉❧-❛ ❡♠ ✉♠ ❡ ✉✐❧*❜.✐♦ ❞❡ ▼❛.❦♦✈ ♣❡.❢❡✐-♦ ❡♠ (✉❜❥♦❣♦(✱ ✐(-♦ '✱ ❞❡♣❡♥❞❡♥-❡ ❛♣❡♥❛( ❞❛ ✈❛.✐>✈❡❧ ❞❡ ❡(-❛❞♦ ♥♦ ♣❡.*♦❞♦ ❛-✉❛❧ ✲ ♥♦ ❝❛(♦✱ ♦ ❡(-♦ ✉❡ ❞❡ ❝❛♣✐-❛❧✳

J❛.❛ ✉❡ ♥=♦ ❤❛❥❛ ✐♥❝❡♥-✐✈♦( ♣❛.❛ ❛ ❣❡.❛/=♦ ❥♦✈❡♠ ❛-✉❛❧ ❞❡(✈✐❛. ❞♦ ❝♦♠✲ ♣.♦♠✐((♦ ❞❡ ♣❛❣❛. ❛ ❝♦♥-.✐❜✉✐/=♦ ❛♦ (✐(-❡♠❛ ♣.❡✈✐❞❡♥❝✐>.✐♦✱ ❋♦.♥✐ ❬✼❪ ✐♠♣0❡ ✉♠❛ .❡❧❛/=♦ ♥❡❣❛-✐✈❛ ❡♥-.❡ ❛ ♥*✈❡❧ ❞❡ ♣♦✉♣❛♥/❛ ❞❡ ❝❛❞❛ ❣❡.❛/=♦ ❡ ❛ ✉❛♥✲ -✐❞❛❞❡ ❞❡ ❛♣♦(❡♥-❛❞♦.✐❛( ✉❡ .❡❝❡❜❡ ✉❛♥❞♦ ✐❞♦(❛✳ ❈❛(♦ ❛ ❣❡.❛/=♦ .❡❞✉③❛ (✉❛ ❝♦♥-.✐❜✉✐/=♦ ❛❜❛✐①♦ ❞♦ ❡ ✉✐❧*❜.✐♦✱ -❡.✐❛ ✉♠❛ ♣♦✉♣❛♥/❛ ♠❛✐♦. ✭✐✳❡✳✱ ♠❛✐♦. ❡(-♦ ✉❡ ❞❡ ❝❛♣✐-❛❧✮✳ ❈♦♠♦ ♣✉♥✐/=♦✱ .❡❝❡❜❡.✐❛ ✉♠❛ ♠❡♥♦. ❛♣♦(❡♥-❛❞♦.✐❛✳ ❉❡((❛ ❢♦.♠❛✱ (❛❜❡♥❞♦ ❞❛ ♣✉♥✐/=♦✱ ♥=♦ ❢❛.✐❛ ❛ .❡❞✉/=♦ ✐♥✐❝✐❛❧ ❞❛ ❝♦♥-.✐❜✉✐/=♦✱ ♠❛♥-❡♥❞♦ ♦ (✐(-❡♠❛ ❡ ✉✐❧✐❜.❛❞♦✳

▲♦❣♦✱ ♦ ❡ ✉✐❧*❜.✐♦ ❞❡ ▼❛.❦♦✈ ♥❡❝❡((✐-❛.✐❛ ❛♣❡♥❛( ❞❛ ❝❤❡❝❛❣❡♠ ❞♦ ♥*✈❡❧ ❞❡ ❝❛♣✐-❛❧ ♣.❡(❡♥-❡ ❡ ❞❛ ♣✉♥✐/=♦ ❝.*✈❡❧ ❞♦( ❛❣❡♥-❡( ✉❡ ❞❡(✈✐❛((❡♠ ❞❡❧❡✳

✸✳✶ ◆♦✈♦ ♠♦❞❡❧♦

❖ ♠♦❞❡❧♦✱ ❡♥-=♦✱ (❡.> ♠✉✐-♦ ♣❛.❡❝✐❞♦ ❝♦♠ ♦ ❞❛ (❡/=♦ ❛♥-❡.✐♦.✳ ❖ ♣.♦❜✲ ❧❡♠❛ ❞♦( ❛❣❡♥-❡( ❡ ❞❛( ✜.♠❛( ♣❡.♠❛♥❡❝❡♠ ♦( ♠❡(♠♦(✳ ❆♦ .❡(♦❧✈❡.❡♠ (❡✉( ♣.♦❜❧❡♠❛(✱ ❛♠❜♦( ❝♦♥(✐❞❡.❛♠ ❛ -❛①❛/=♦ ❝♦♠♦ ❡①?❣❡♥❛✳ J♦. ✐((♦✱ ❝♦♥-✐♥✉❛♠

✈❛❧❡♥❞♦ ♦' (❡'✉❧*❛❞♦' ❛❝❤❛❞♦' ♥❛ '❡-.♦ ❛♥*❡(✐♦(✳

❆ 2♥✐❝❛ ♠✉❞❛♥-❛ 4 5✉❡✱ ❛❣♦(❛✱ ♦ ✐♠♣♦'*♦τ ♥.♦ '❡(9 ♠❛✐' ❡①;❣❡♥♦ ♣❛(❛ ♦

❣♦✈❡(♥♦✱ ♠❛' ❞❡♣❡♥❞❡(9 ❞♦ ❡'*♦5✉❡ ❞❡ ❝❛♣✐*❛❧✱ ❤❛❥❛ ✈✐'*❛ 5✉❡ ❛ '✉❛ ✐♥*❡♥-.♦ 4 ❛❥✉'*❛( ❛ ❝♦♥*(✐❜✉✐-.♦ ♣(❡✈✐❞❡♥❝✐9(✐❛ ♣❛(❛ ✐♥✢✉❡♥❝✐❛( ❛ ❞❡❝✐'.♦ ❞♦' ❛❣❡♥*❡' ❡ ♠❛♥*❡( ❛ ❡❝♦♥♦♠✐❛ ♥♦ ;*✐♠♦✳

❙❡♥❞♦ ❛''✐♠✱ ❞❡✈❡✲'❡ (❡❡'❝(❡✈❡( ❛ ❡5✉❛-.♦ ✭✷✼✮ ❝♦♠♦✿

bt+1

1 +rt+1

=β

1

1−γ −τ(kt)

φ(τ(kt+1)) ✭✸✸✮

♦♥❞❡

φ(τ(kt+1)) =

τ(kt+1)(1−_αα)(1−γ)(₁₊1_β)

1 +τ(kt+1)(1−αα)(1−γ)(1+1β)

✭✸✹✮

❖ ❣♦✈❡(♥♦ ✉'❛τt✭♣♦( ❝♦♠♦❞✐❞❛❞❡✱ ❝♦♥'✐❞❡(❡τt=τ(kt)✮ ❝♦♠♦ '❡✉ ✐♥'*(✉✲

♠❡♥*♦ ❞❡ ♣♦❧I*✐❝❛✿ ❛ ❝❛❞❛ ♣❡(I♦❞♦t✱ ❡❧❡ ❡'❝♦❧❤❡τtwt5✉❡ ♠❛①✐♠✐③❛ ❛ ✉*✐❧✐❞❛❞❡

✐♥*❡(*❡♠♣♦(❛❧ ❞♦ ❥♦✈❡♠✱ ❝♦♥❞✐❝✐♦♥❛❞♦ K ❡①♣❡❝*❛*✐✈❛ 5✉❛♥*♦ ❛τt+1wt+1✳

❈♦♠♦ '❡ *(❛*❛ ❞❡ ✉♠ ❡5✉✐❧I❜(✐♦ ❡♠ ❡'*(❛*4❣✐❛' ❞❡ ▼❛(❦♦✈✱ ❛ ❢✉♥-.♦ ♣♦❧I*✐❝❛ ✈❛✐ ❞❡♣❡♥❞❡( ❛♣❡♥❛' ❞❛ ✈❛(✐9✈❡❧ ❞❡ ❡'*❛❞♦ ♥♦ ♠❡'♠♦ ♣❡(I♦❞♦✱ ✐'*♦ 4✱τt=τt(kt)✳ ❆''✉♠✐♥❞♦ 5✉❡ ❛ ❢✉♥-.♦ '❡❥❛ ❝♦♥*I♥✉❛ ❡ ❞✐❢❡(❡♥❝✐9✈❡❧ ❡ '❡♥❞♦

♦ ♠♦❞❡❧♦ ❡'*❛❝✐♦♥9(✐♦✱ ♣♦❞❡✲'❡ ❢♦❝❛( ♥❛' ♣♦❧I*✐❝❛' 5✉❡ ♥.♦ ❞❡♣❡♥❞❡♠ ❞♦ ♣❡(I♦❞♦ ❡♠ 5✉❡ ♦ ❣♦✈❡(♥♦ ❡'*9 ♥♦ ♣♦❞❡(✱ ❞❡ ♠♦❞♦ 5✉❡τt=τ(kt)✳ ❆ ✐♥*✉✐-.♦

4 5✉❡✱ '❡ ❞♦✐' ❣♦✈❡(♥♦' '❡ ❞❡♣❛(❛♠ ❝♦♠ '✐*✉❛-P❡' ✐❞Q♥*✐❝❛'✱ ♠❡'♠♦ 5✉❡ ❡♠ ♠♦♠❡♥*♦' ❞♦ *❡♠♣♦ ❞✐❢❡(❡♥*❡'✱ *♦♠❛♠ ❞❡❝✐'P❡' ✐❣✉❛✐'✳

❖ ❣♦✈❡(♥♦ ❡'❝♦❧❤❡ '✉❛ ♣♦❧I*✐❝❛ ❡♠ t❞❡ ❛❝♦(❞♦ ❝♦♠ '✉❛ ❡①♣❡❝*❛*✐✈❛ ❝♦♠

(❡❧❛-.♦ K ♣♦❧I*✐❝❛ ❛❞♦*❛❞❛ ♣❡❧♦ ❣♦✈❡(♥♦ 5✉❡ ❡'*✐✈❡( ♥♦ ♣♦❞❡( ❡♠t+1✳ ❈❤❛♠❡

*❛❧ ❡①♣❡❝*❛*✐✈❛ ❞❡τ˜(kt+1)✳

❈♦♠♦ '❡(9 ♠♦'*(❛❞♦ K ❢(❡♥*❡✱ ♦ ❝♦♥'✉♠♦ ❛❣(❡❣❛❞♦ ❞♦' ❛❣❡♥*❡' ♥♦ ♣(✐♠❡✐(♦ ♣❡(I♦❞♦✱C1t✱ ♣♦❞❡ '❡( ❡①♣(❡''♦ ❡♠ ❢✉♥-.♦ ❞❡ kt✱τ(kt)❡ τ(kt+1)✳ ❙❡♥❞♦ ❛'✲

'✐♠✱ *❛♠❜4♠ ♦ ♣♦❞❡♠C₂t ❡ St✱ ❞❡ ♠♦❞♦ 5✉❡ ♦ ♣(♦❜❧❡♠❛ ♣♦❞❡ '❡( (❡❡'❝(✐*♦

❞❛ '❡❣✉✐♥*❡ ❢♦(♠❛✿

τ(kt) =T[˜τ(kt+1)] = arg max0<τ(t)<1U(C1t, C2t)

= arg max₀<τ(t)<1Ψ(kt, τ(kt),τ˜(kt+1))

❛❧ #✉❡

kt+1 = Φ(kt, τ(kt))

♦♥❞❡ Ψ(.) ) ❛ ❢✉♥+,♦ ✉ ✐❧✐❞❛❞❡ ✐♥❞✐.❡ ❛ ❡/❝.✐ ❛ ❡♠ ❢✉♥+,♦ ❞❡ kt✱ τ(kt) ❡

˜

τ(kt+1)✳ Φ(.) ) ❛ ❢♦.♠❛ ❡①♣❧6❝✐ ❛ ❞❛ ❡#✉❛+,♦ ❞❡ ❛❝✉♠✉❧❛+,♦ ❞❡ ❝❛♣✐ ❛❧✳

❉❡✈❡✲/❡✱ ❡♥ ,♦✱ ❛❝❤❛. ❛ ❢✉♥+,♦ ♣♦❧6 ✐❝❛τ(k)#✉❡ .❡/♦❧✈❡ ♦ ♣.♦❜❧❡♠❛ ❛❝✐♠❛✳

✸✳✷ ❚❛①❛&'♦ )*✐♠❛

❈♦♠♦ ❛ ❡/❝♦❧❤❛ ❞♦ ❣♦✈❡.♥♦ ❡♠ t ❞❡♣❡♥❞❡ ❞❛ ❡①♣❡❝ ❛ ✐✈❛ ❞❛ ♣♦❧6 ✐❝❛ ❞❡

❛①❛+,♦ ❞♦ ❣♦✈❡.♥♦ ❞❡ t+ 1 ✭❧❡♠❜.❛♥❞♦ #✉❡ ♦ ❣♦✈❡.♥♦ ❡♠ t /❡ ♣.❡♦❝✉♣❛

❛♣❡♥❛/ ❝♦♠ ♦ ❜❡♠✲❡/ ❛. ❞❛ ♣♦♣✉❧❛+,♦ ❥♦✈❡♠ ❡♠t✮ ❡ ❝♦♠♦ ♦ ❣♦✈❡.♥♦ /❛❜❡ ♦

❝♦♠♣♦. ❛♠❡♥ ♦ ❞♦/ ❛❣❡♥ ❡/ ❡ ❞❛/ ✜.♠❛/ ✭/✉❜/❡+B❡/ ✷✳✶ ❡ ✷✳✷✮✱ ❝❤❡❣❛✲/❡ ❛♦ /❡❣✉✐♥ ❡ ♣.♦❜❧❡♠❛ ❞❡ ♠❛①✐♠✐③❛+,♦ ❞♦ ♣❧❛♥❡❥❛❞♦. ❝❡♥ .❛❧✿

max

0<τ(t)<1U(C1t, C2t) = ln

wt

1 +β[1−(1−γ)τ(kt)][1 +βφ(τ(kt+1))]

+

+βln

_β

1 +β(1 +rt+1)wt[1−(1−γ)τ(kt)][1 +βφ(τ(kt+1))]

✭✸✺✮

❛❧ #✉❡

(1 +n)kt+1 = β

1 +βwt[1−(1−γ)τ(kt)][1−φ(τ(kt+1))] ✭✸✻✮

❈♦♠♦ .❡/✉❧ ❛❞♦✱ ❡♠✲/❡ #✉❡✿

!♦♣♦$✐&'♦ ✶✿

❊①✐#$❡ ✉♠ ❝♦♥$+♥✉♦ ❞❡ ❢✉♥./❡# ♣♦❧+$✐❝❛# ❡#$❛❝✐♦♥34✐❛#✱ ✐♥$❡4✐♦4❡# ❡ ❞✐❢❡4✲ ❡♥❝✐3✈❡✐# ❞❛ ❢♦4♠❛

τt+1 =

1 1−γ

α

1−α Ck

−1+1+βαβ

t+1 −1

✭✸✼✮

♦♥❞❡C ≥0 < ✉♠❛ ❝♦♥#$❛♥$❡ ❞❡ ✐♥$❡❣4❛.>♦ ❛4❜✐$434✐❛✳ A4♦✈❛✿ ❱✐❞❡ ❆♣E♥❞✐❝❡✳

❋✐❣✉4❛ ✶✶✿ ❈❛♣✐$❛❧ ❳ ❈♦♥$4✐❜✉✐.>♦

❆##✐♠✱ ♣❡❧❛ ❡J✉❛.>♦ ✭✸✼✮✱ ✜❝❛ ❝❧❛4❛ ❛ 4❡❧❛.>♦ ✐♥✈❡4#❛ ❡♥$4❡ τ ❡ k✻✳

➱ ✐♥$❡4❡##❛♥$❡ ♥♦$❛4 J✉❡ ❤3 ✉♠❛ ♥♦✈✐❞❛❞❡ ❡♠ 4❡❧❛.>♦ ❛♦ 4❡#✉❧$❛❞♦ ❡♥✲ ❝♦♥$4❛❞♦ ♣♦4 ❋♦4♥✐ ❬✼❪✳ ❆J✉✐✱ ❞✐❢❡4❡♥$❡♠❡♥$❡ ❞♦ 4❡❢❡4✐❞♦ ❛4$✐❣♦✱ ❤3 ❞♦✐# $✐♣♦# ❞❡ ❛❣❡♥$❡#✿ ✉♠ J✉❡ ❡#$3 ❞❡♥$4♦ ❞♦ #✐#$❡♠❛ ♣4❡✈✐❞❡♥❝✐34✐♦ ❡ ♦✉$4♦ J✉❡ ♥>♦ ❡#$3✳ ❋♦4♥✐ ❝♦♥#✐❞❡4❛ J✉❡ ❤3 ❛♣❡♥❛# ♦ ♣4✐♠❡✐4♦ $✐♣♦✳

❉❡##❛ ❢♦4♠❛✱ ❛♦ ❤❛✈❡4 ✉♠❛ ♣4♦♣♦4.>♦ (1−γ) ❞❡ ❛❣❡♥$❡# ♥♦ #✐#$❡♠❛✱ ♦ ♣❧❛♥❡❥❛❞♦4 ❝❡♥$4❛❧ ♣❛##❛ ❛ ❝♦♥#✐❞❡43✲❧❛ ♥❛ #✉❛ 4❡❣4❛ ❞❡ ❡#❝♦❧❤❛ ❞❡τ✳ ❈♦♥#❡✲

✻ _{❛"❛ ❝"✐❛%&♦ ❞♦ ❣"*✜❝♦ ❝♦♠♣❛"❛♥❞♦ ❝♦♥/"✐❜✉✐%&♦ ❡ ❝❛♣✐/❛❧✱ ❢♦"❛♠ ✉6❛❞♦6 ♦6 ♠❡6♠♦6}

✈❛❧♦"❡6 8✉❡ ♦6 ✉6❛❞♦6 ♠❛✐6 9 ❢"❡♥/❡✱ ❛ 6❛❜❡"✿ γ= 0_,71✱_β= 0_,9✱_α= 0_,25✱_C= 0_,42❡

n= 0_,0✳

✉❡♥$❡♠❡♥$❡✱ ♣❛)❛ ✉❛❧ ✉❡) ✈❛❧♦) ❞❡γ✱ ♦ ❡.$♦ ✉❡ ❞❡ ❝❛♣✐$❛❧ ♥1♦ ♠✉❞❛ ✭❧♦❣♦✱

♦. ❝♦♥.✉♠♦.✱ ♣♦✉♣❛♥4❛✱ ♦ ♣)♦❞✉$♦ ❡ ✉$✐❧✐❞❛❞❡ $❛♠❜6♠ ♥1♦✮✳ ❖ ♠♦$✐✈♦ 6 ❝❧❛)♦✿ ♦ ♣❧❛♥❡❥❛❞♦) ❝❡♥$)❛❧ ❧❡✈❛ ❡♠ ❝♦♥.✐❞❡)❛41♦ ❛ ❝♦❜❡)$✉)❛ ♣)❡✈✐❞❡♥❝✐<)✐❛ ♣❛)❛ ❛ .✉❛ ❞❡❝✐.1♦ ❞❡ $❛①❛41♦✱ ❤❛❥❛ ✈✐.$❛ ✉❡ $❛❧ ✈❛)✐<✈❡❧✱ ❝♦♥❢♦)♠❡ ✈✐.$♦ ♥❛ .❡41♦ ❛♥$❡)✐♦)✱ ✐♥$❡)❢❡)❡ ♥♦ ❝❛♣✐$❛❧ ❛❣)❡❣❛❞♦ ❞❛ ❡❝♦♥♦♠✐❛ ✲ ❧♦❣♦✱ ✐♥$❡)❢❡)❡ ♥❛. ❞❡❝✐.A❡. ❞♦. ❛❣❡♥$❡. ❞❡ ❝♦♥.✉♠♦ ❡ ♣♦✉♣❛♥4❛✱ ♣♦✐. ❛❢❡$❛ ❥✉)♦. ❡ .❛❧<)✐♦✳

❉❡ ❛❝♦)❞♦ ❝♦♠ ♦ ✈❛❧♦) ❞❡ γ✱ ♦ ♣❧❛♥❡❥❛❞♦) ❛❥✉.$❛ τ ❞❡ ♠♦❞♦ ❛ ♠❛♥$❡)

❛ ❡❝♦♥♦♠✐❛ ♥♦ ♠❡.♠♦ ♥C✈❡❧ D$✐♠♦ ❞❡ ❜❡♠✲❡.$❛)✳ ▲♦❣♦✱ ❛. F♥✐❝❛. ✈❛)✐<✈❡✐. ✉❡ ♠✉❞❛♠ ❝♦♠ ❛❧$❡)❛4A❡. ❡♠γ .1♦ ♦ ✐♠♣♦.$♦τ ❡ ❛ ❛♣♦.❡♥$❛❞♦)✐❛b✱ ❡ .1♦

♥❡❣❛$✐✈❛♠❡♥$❡ )❡❧❛❝✐♦♥❛❞❛. ❝♦♠ ❛ ❝♦❜❡)$✉)❛ ♣)❡✈✐❞❡♥❝✐<)✐❛✱ ❤❛❥❛ ✈✐.$❛ ✉❡ ✉♠❛ ♠❛✐♦) ❝♦❜❡)$✉)❛ ✐♠♣❧✐❝❛)✐❛ ♠❡♥♦) ❛❝✉♠✉❧❛41♦ ❞❡ ❝❛♣✐$❛❧ ♣❡❧♦ ♣F❜❧✐❝♦✳ ❈✐❡♥$❡ ❞✐..♦✱ ♦ ❣♦✈❡)♥♦ )❡❞✉③✐)✐❛ ❛ ❝♦♥$)✐❜✉✐41♦ τ✳ ❈♦♠♦ ✐..♦ ✐♠♣❧✐❝❛)✐❛

✉♠❛ ♠❡♥♦) ❛♣♦.❡♥$❛❞♦)✐❛ ♣❛)❛ ♦. ❥♦✈❡♥. ❞❡ ❤♦❥❡ ♥♦ ♣❡)C♦❞♦ .❡❣✉✐♥$❡✱ ♦❝♦))❡ ✉♠ ❛✉♠❡♥$♦ ❞❛ ❛❝✉♠✉❧❛41♦ ❞❡ ❝❛♣✐$❛❧ ♣♦) ♣❛)$❡ ❞❡❧❡.✱ ❝♦♥$)❛❜❛❧❛♥4❛♥❞♦ ❛ .✉❛ )❡❞✉41♦ ✐♥✐❝✐❛❧✳ ❙❡♥❞♦ ❛..✐♠✱ $❡♠✲.❡ ❝♦♠♦ )❡.✉❧$❛❞♦ ✉♠ ♥C✈❡❧ ❞❡ ❝❛♣✐$❛❧ ✐❣✉❛❧ ❛♦ ❛♥$❡)✐♦) J ♠✉❞❛♥4❛ ❡♠γ✱ ♠❛. ❝♦♠ ✉♠❛ $❛①❛ τ ♠❡♥♦)✳

❖✉ .❡❥❛✱ ❛♦ ❛❧$❡)❛) τ✱ ♦ ❣♦✈❡)♥♦ ❝♦♥.❡❣✉❡ ✐♥✢✉❡♥❝✐❛) ❛. ❞❡❝✐.A❡. ❞♦.

❛❣❡♥$❡. ♣❛)❛ ❛ ❞❡$❡)♠✐♥❛41♦ ❞♦ ❝❛♣✐$❛❧ D$✐♠♦✱ ✐♥✢✉❡♥❝✐❛♥❞♦✱ ♣♦)$❛♥$♦✱ ❛. ❞❡♠❛✐. ✈❛)✐<✈❡✐. ❞❛ ❡❝♦♥♦♠✐❛✱ ❞❡ ♠♦❞♦ ❛ ♠❛♥$L✲❧❛ ♥♦ ♥C✈❡❧ D$✐♠♦✳

❙✉❜.$✐$✉✐♥❞♦ ❛ ❢✉♥41♦ ♣♦❧C$✐❝❛τ ❡♥❝♦♥$)❛❞❛ ❛❝✐♠❛ ♥❛ ❡ ✉❛41♦ ❞✐♥M♠✐❝❛

❞♦ ❝❛♣✐$❛❧✱ ❛❝❤❛✲.❡ ❛ ❡.$)❛$6❣✐❛ ✉❡ ✐♥❞✉③ ♦ ❡ ✉✐❧C❜)✐♦ .✉.$❡♥$<✈❡❧ ❞❛ ❡❝♦♥♦✲ ♠✐❛✿

(1 +n)kt+1 =wt

β

1 +β 1−(1−γ)

1 1−γ

α

1−α

(Ck−υ t −1)

× ×   1

1 +₁₋1_γ α

1−α

(Ck−υ

t −1) 1−αα

(1−γ)₁₊1_β

❖✉ ♠❡❧❤♦'✿

(1 +β)kt+1

1 +

1 1 +β

(Ck−υ t+1−1)

=

= β

1 +n(1−α)Ak

α t 1− α

1−α

(Ck−υ t −1)

❘❡❛''✉♠❛♥❞♦✱ ❡♥❝♦♥/'❛✲1❡ ❛ 1❡❣✉✐♥/❡ '❡❧❛45♦✿

kt+1+Ck1t+1−υ= β

1 +n Ak

α

t −αCAkαt−υ

✭✸✽✮

♦♥❞❡υ= 1+₁₊βα_β ✳

❙❡❣✉♥❞♦ ❋♦'♥✐ ❬✼❪✱ ❛♣❡1❛' ❞❛1 ✈❛♥/❛❣❡♥1 ❞❡ 1❡ ✉1❛'❡♠ ❡1/'❛/B❣✐❛1 ▼❛'❦♦✈✱ E✉❛♥❞♦ 1❡ ✉1❛♠ ❢✉♥45♦ ✉/✐❧✐❞❛❞❡ ❧♦❣❛'G/♠✐❝❛ ❡ ❢✉♥45♦ ❞❡ ♣'♦❞✉45♦ ❈♦❜❜✲ ❉♦✉❣❧❛1✱ ♣♦❞❡♠✲1❡ ♣'♦❞✉③✐' ❞✐♥L♠✐❝❛1 ✐♥1/M✈❡✐1✳ N♦' ✐11♦✱ ♣❛'❛ ❡1/❛ 1❡45♦✱ ❝♦♠♦ ♥5♦ 1❡ E✉❡' ❝❤❡❣❛' ❡♠ ♥❡♥❤✉♠ '❡1✉❧/❛❞♦ ♣♦♥/✉❛❧ ♠❛1 ❛♣❡♥❛1 ♠♦1/'❛' ♦ ❝♦♠♣♦'/❛♠❡♥/♦ ❞❡τ ❡b✱ ❢♦'❛♠ ✉1❛❞♦1✱ ❛❧B♠ ❞❡γ = 0,71✱ ♦1 ♠❡1♠♦1 ✈❛❧♦'❡1

❞❡ ♣❛'L♠❡/'♦1 ✉1❛❞♦1 ♥♦ '❡❢❡'✐❞♦ ❛'/✐❣♦✱ ♣♦✐1 ❧❡✈❛♠ ❛ '❡1✉❧/❛❞♦1 ❡1/M✈❡✐1✱ ❛ 1❛❜❡'✿ β = 0,9✱ α = 0,25✱ C = 0,42 ❡ n = 0,0✳ ◆♦ ❝❛1♦ ❡♠ E✉❡ γ = 0✱

❝❛✐✲1❡ ❡①❛/❛♠❡♥/❡ ♥♦ ❝❛1♦ ❞❡ ❋♦'♥✐ ❬✼❪✳

❆ ❡E✉❛45♦ ❛♥/❡'✐♦' ❞❡1❝'❡✈❡ ❛ ❞✐♥L♠✐❝❛ ❞♦ ❝❛♣✐/❛❧ ♥❛ ❡❝♦♥♦♠✐❛ ❡ B ❢✉♥✲ ❞❛♠❡♥/❛❧ ♣❛'❛ ❛ ❞❡/❡'♠✐♥❛45♦ ❞♦ ❡E✉✐❧G❜'✐♦✳ ◆♦/❡ E✉❡✱ ❛♦ ❝♦♥/'M'✐♦ ❞♦ ♦❝♦''✐❞♦ ♥❛ 1❡45♦ ❛♥/❡'✐♦' ❝♦♠τ ❡①R❣❡♥♦✱ ❡❧❛ ♥5♦ ❞❡♣❡♥❞❡ ♥❡♠ ❞❡γ ❡ ♥❡♠

❞❡ τ✱ ❡①❛/❛♠❡♥/❡ ♣❡❧♦ ❢❛/♦ ❞❡ ♦ ❣♦✈❡'♥♦ 1❡ ❛❥✉1/❛' ✈✐❛ /❛①❛45♦ ♣❛'❛ ✐♠✲

♣❡❞✐' E✉❡ ♠✉❞❛♥4❛1 ♥❛ ❝♦❜❡'/✉'❛ ♣'❡✈✐❞❡♥❝✐M'✐❛ /✐'❡♠ ❛ ❡❝♦♥♦♠✐❛ ❞♦ R/✐♠♦✳ ❊11❡ ❛❥✉1/❡ ❡♠τ ✭❡✱ ❧♦❣♦✱ ❡♠b✱ ♣❡♥15♦ '❡❝❡❜✐❞❛ E✉❛♥❞♦ ✐❞♦1♦✮ 1❡ '❡❧❛❝✐♦♥❛

♥❡❣❛/✐✈❛♠❡♥/❡ ❝♦♠ ♦ ♥G✈❡❧ ❞❡ ❝❛♣✐/❛❧✳

N♦'/❛♥/♦✱ ❝♦♥1✐❞❡'❛♥❞♦ ♦1 '❡1✉❧/❛❞♦1 ❞❛ ♣'✐♠❡✐'❛ ❝❛❧✐❜'❛45♦ ❞❛ 1❡45♦ ✷ ❡♠ '❡❧❛45♦ ❛♦1 ♥❡1/❛ ❡♥❝♦♥/'❛❞♦1✱ ♣♦❞❡✲1❡ ❝♦♥❝❧✉✐' E✉❡ ✉♠❛ ♠❛✐♦' ❝♦❜❡'/✉'❛ ♣'❡✈✐❞❡♥❝✐M'✐❛ 1R ♣✐♦'❛ ❛ ❡❝♦♥♦♠✐❛ 1❡τ ❢♦' ❡①R❣❡♥♦✱ ✉♠❛ ✈❡③ E✉❡ ♥5♦ ❤M ✉♠

♠❡❝❛♥✐1♠♦ ❞❡ ❛❥✉1/❡ ♥❛ ♦❝♦''V♥❝✐❛ ❞❡ ♠✉❞❛♥4❛1 ♣❛'❛ ❧❡✈❛' ❛ ❡❝♦♥♦♠✐❛ ❞❡ ✈♦❧/❛ ❛♦ R/✐♠♦✳

❋✐❣✉$❛ ✶✷✿ ❉✐♥+♠✐❝❛ ❞♦ ❈❛♣✐2❛❧

❋✐❣✉$❛ ✶✸✿ ❈♦♥2$✐❜✉✐67♦ τ

❋✐❣✉$❛ ✶✹✿ ❆♣♦,❡♥/❛❞♦$✐❛b

✹ ❈♦♥❝❧✉'(♦

❊,/❡ /$❛❜❛❧❤♦ ♣$♦❝✉$♦✉ 6✉❛♥/✐✜❝❛$ ♦ ✐♠♣❛❝/♦ ❞❡ ✉♠ ❛✉♠❡♥/♦ ❞❛ ❝♦❜❡$✲ /✉$❛ ♣$❡✈✐❞❡♥❝✐;$✐❛ ,♦❜$❡ ❛ ♣♦✉♣❛♥<❛ ♥❛ ❈❤✐♥❛✱ ❞❛❞❛ ✉♠❛ ❝♦♥/$✐❜✉✐<?♦ ♣$❡✈✲ ✐❞❡♥❝✐;$✐❛ ❡①A❣❡♥❛✳ C❛$❛ /❛❧✱ ❢♦✐ ❝♦♥,/$✉❣❞♦ ✉♠ ♠♦❞❡❧♦ ,✐♠♣❧❡, ❞❡ ❣❡$❛<F❡, ,✉♣❡$♣♦,/❛, ❞❡ ❞♦✐, ♣❡$❣♦❞♦,✳

❆,,✉♠✐✉✲,❡ ✉♠ ,✐,/❡♠❛ ❞❡ $❡♣❛$/✐<?♦ 6✉❡ ❛❜$❛♥❣✐❛ ✉♠❛ ♣❛$/❡ ❞♦, /$❛✲ ❜❛❧❤❛❞♦$❡,✱ ❞❡ ♠♦❞♦ 6✉❡✱ ❛6✉❡❧❡, ✐♥,❡$✐❞♦, ♥❡❧❡ ♣❛❣❛✈❛♠ ✉♠❛ ❝♦♥/$✐❜✉✐<?♦ 6✉❛♥❞♦ ❥♦✈❡♥, 6✉❡ ❧❤❡, ❞❛✈❛ ❞✐$❡✐/♦ ❛ $❡❝❡❜❡$ ✉♠❛ ❛♣♦,❡♥/❛❞♦$✐❛ 6✉❛♥❞♦ ✐❞♦,♦,✳ C♦$ ❡,,❡ $❡❣✐♠❡✱ /$❛❜❛❧❤❛❞♦$❡, ❥♦✈❡♥, ✜♥❛♥❝✐❛✈❛♠ ♦, ❛♣♦,❡♥/❛❞♦, ✐❞♦,♦,✳ ▲♦❣♦✱ ❡$❛ ❞❡ ,❡ ❡,♣❡$❛$ 6✉❡ ✉♠❛ ♠❛✐♦$ ❛❜$❛♥❣I♥❝✐❛ ❞♦ ,✐,/❡♠❛ $❡✲ ❞✉③✐,,❡ ❛ ♣♦✉♣❛♥<❛✱ ✉♠❛ ✈❡③ 6✉❡ ❡❧❛ ♥?♦ ♣$❡❝✐,❛$✐❛ ,❡$ /?♦ ❣$❛♥❞❡ ❣$❛<❛, K /$❛♥,❢❡$I♥❝✐❛ ♣$♦♠♦✈✐❞❛ ♣❡❧♦ ❊,/❛❞♦ ✈✐❛ ♣❡♥,F❡, ♥♦ ❢✉/✉$♦✳

◆❛ ❝❛❧✐❜$❛<?♦ ❞♦ ♠♦❞❡❧♦✱ ❢♦✐ ✉,❛❞❛ ✉♠❛ ♣$♦♣♦$<?♦ C♦✉♣❛♥<❛✲C$♦❞✉/♦ ❞❡ ✷✸✹✱ 6✉❡ ❝♦$$❡,♣♦♥❞❡ ❛♦ ✈❛❧♦$ ♦❜,❡$✈❛❞♦ ♣❛$❛ ❛♣❡♥❛, ❛ ♣♦✉♣❛♥<❛ ❞❛, ❢❛♠❣❧✐❛, ♥♦ ♣❛❣,✳ ❊,/❡ ❡①❡$❝❣❝✐♦ ❛♣$❡,❡♥/♦✉ $❡,✉❧/❛❞♦, ❞❡♥/$♦ ❞♦ ❡,♣❡$❛❞♦✿