Overlapping generations economy, environmental externalities, and taxation
DEPOCEN Working Paper Series No 2011/02 Overlapping generations economy, environmental externalities, and taxation Nguyen Thang Dao* * Room Center for Operations Research and Econometrics (CORE), Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium; and Vietnam Center for Economic and Policy Research (VEPR), Hanoi, Vietnam Email: nguyen.dao@uclouvain.be The author is extremely grateful to Julio Dávila for his useful suggestions and intellectual guidance The author thanks funding from a research grant from the Belgian FNRS as part of a M.I.S Mobilité Ulysse F.R.S - FNRS The scienti c responsibility belongs to the author The DEPOCEN WORKI NG PAPER SERI ES dissem inat es research findings and prom ot es scholar exchanges in all branches of econom ic st udies, wit h a special em phasis on Viet nam The v iews and int erpret at ions expressed in t he paper are t hose of t he aut hor( s) and not necessarily represent t he views and policies of t he DEPOCEN or it s Managem ent Boar d The DEPOCEN does not guarant ee t he accuracy of findings, int erpret at ions, and dat a associat ed wit h t he paper , and accept s no responsibilit y what soever for any consequences of t heir use The aut hor( s) r em ains t he copyright ow ner DEPOCEN WORKI NG PAPERS are available online at ht t p: / / www.depocenwp.org ❖✈❡r❧❛♣♣✐♥❣ ❣❡♥❡r❛t✐♦♥s ❡❝♦♥♦♠②✱ ❡♥✈✐r♦♥♠❡♥t❛❧ ❡①t❡r♥❛❧✐t✐❡s✱ ❛♥❞ t❛①❛t✐♦♥ ◆❣✉②❡♥ ❚❤❛♥❣ ❉❆❖∗ ❏✉❧② ✽✱ ✷✵✶✵ ❆❜str❛❝t ■ s❡t ✉♣ ✐♥ t❤✐s ♣❛♣❡r ❛♥ ♦✈❡r❧❛♣♣✐♥❣ ❣❡♥❡r❛t✐♦♥s ❡❝♦♥♦♠② ✇✐t❤ ❡♥✈✐✲ r♦♥♠❡♥t ❞❡❣r❛❞✐♥❣ ✐ts❡❧❢ ❛♥❞ ♣♦❧❧✉t✐♦♥ r❡s✉❧t✐♥❣ ❢r♦♠ ❜♦t❤ ❝♦♥s✉♠♣t✐♦♥ ❛♥❞ ♣r♦❞✉❝t✐♦♥ t♦ s❤♦✇ t❤❛t t❤❡r❡ ❛❧✇❛②s ❡①✐sts ❛♥ ✐♥t❡r✲t❡♠♣♦r❛❧ ❡q✉✐✲ ❧✐❜r✐✉♠ ❛♥❞ t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❝♦♠♣❡t✐t✐✈❡ st❡❛❞② st❛t❡✳ ❚❤✐s st❡❛❞② st❛t❡ ✐s ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ ❡q✉✐❧✐❜r✐✉♠ st❡❛❞② st❛t❡ ✐♥ t❤❡ s♦❝✐❛❧ ❜❡♥❡✈♦❧❡♥t ♣❧❛♥♥❡r✬s ♣♦✐♥t ♦❢ ✈✐❡✇✳ ❚❤❡ ♣❛♣❡r s❤♦✇s t❤❡ ♦♣t✐♠❛❧ ✏❣♦❧❞❡♥ r✉❧❡✑ ❛❧❧♦✲ ❝❛t✐♦♥ ✇❤✐❝❤ ♠❛①✐♠✐③❡s t❤❡ t♦t❛❧ ✉t✐❧✐t② ♦❢ ❛❧❧ ❣❡♥❡r❛t✐♦♥s✱ ❛♥❞ ✇❤❡♥❡✈❡r t❤❡ ❝❛♣✐t❛❧ r❛t✐♦ ✐♥ t❤❡ ❝♦♠♣❡t✐t✐✈❡ ❢r❛♠❡✇♦r❦ ✐s ❤✐❣❤❡r t❤❛♥ t❤❡ ❣♦❧❞❡♥ r✉❧❡ ❝❛♣✐t❛❧ r❛t✐♦✱ t❤❡ ❡❝♦♥♦♠② st❛♥❞s ♦♥ t❤❡ ❞②♥❛♠✐❝❛❧❧② ✐♥❡✣❝✐❡♥t ♣♦✐♥t✳ ❚❤❡ ✇✐❞t❤ ♦❢ t❤❡ ✐♥❡✣❝✐❡♥t r❛♥❣❡ ♦❢ ❝❛♣✐t❛❧ r❛t✐♦ ❞❡♣❡♥❞s ♣♦s✐t✐✈❡❧② ♦♥ t❤❡ ❡♥✈✐r♦♥♠❡♥t ♠❛✐♥t❛✐♥✐♥❣ t❡❝❤♥♦❧♦❣② ❛♥❞ ❞❡♣❡♥❞s ♥❡❣❛t✐✈❡❧② ♦♥ t❤❡ ❝❧❡❛♥♥❡ss ♦❢ ♣r♦❞✉❝t✐♦♥ t❡❝❤♥♦❧♦❣②✳ ❋♦r s✉❝❤ ❛♥② ❝♦♠♣❡t✐t✐✈❡ ❡❝♦♥♦♠②✱ ■ ✐♥tr♦❞✉❝❡ s♦♠❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❛①❡s ❛♥❞ tr❛♥s❢❡r ✇✐t❤ ♣✉r♣♦s❡ ♦❢ ❞❡✲ ❝❡♥tr❛❧✐③✐♥❣ t❤❡ ❜❡st st❡❛❞② st❛t❡ ❛tt❛✐♥❛❜❧❡ t❤r♦✉❣❤ t❤❡ ❣♦♦❞ ❛♥❞ ❢❛❝t♦rs ♠❛r❦❡ts✳ ❑❡②✇♦r❞s✿ ♦✈❡r❧❛♣♣✐♥❣ ❣❡♥❡r❛t✐♦♥s✱ ❡♥✈✐r♦♥♠❡♥t❛❧ ❡①t❡r♥❛❧✐t②✱ t❛①❡s ❛♥❞ tr❛♥s❢❡r s❝❤❡♠❡✳ ❏❊▲ ❈❧❛ss✐✜❝❛t✐♦♥✿ ❉✻✷✱ ❊✷✶✱ ❍✷✶✱ ❍✹✶ ✶ ■♥tr♦❞✉❝t✐♦♥ ❈♦♥s✐❞❡r❛t✐♦♥s ♦♥ ❡♥✈✐r♦♥♠❡♥t❛❧ ❡①t❡r♥❛❧✐t✐❡s ✐♥ t❤❡ ❖✈❡r❧❛♣♣✐♥❣ ●❡♥❡r❛t✐♦♥s ✭❖▲●✮ ❢r❛♠❡✇♦r❦ ❤❛✈❡ ❜❡❡♥ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t s✐♥❝❡ ∗ ❈❡♥t❡r ❢♦r ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ❛♥❞ ❊❝♦♥♦♠❡tr✐❝s ✭❈❖❘❊✮✱ ❯♥✐✈❡rs✐té ❝❛t❤♦❧✐q✉❡ ❞❡ ▲♦✉✲ ✈❛✐♥✱ ❇✲✶✸✹✽ ▲♦✉✈❛✐♥✲❧❛✲◆❡✉✈❡✱ ❇❡❧❣✐✉♠❀ ❛♥❞ ❱✐❡t♥❛♠ ❈❡♥t❡r ❢♦r ❊❝♦♥♦♠✐❝ ❛♥❞ P♦❧✐❝② ❘❡✲ s❡❛r❝❤ ✭❱❊P❘✮✱ ❍❛♥♦✐✱ ❱✐❡t♥❛♠✳ ❊♠❛✐❧✿ ♥❣✉②❡♥✳❞❛♦❅✉❝❧♦✉✈❛✐♥✳❜❡✳ ❚❤❡ ❛✉t❤♦r ✐s ❡①tr❡♠❡❧② ❣r❛t❡❢✉❧ t♦ ❏✉❧✐♦ ❉á✈✐❧❛ ❢♦r ❤✐s ✉s❡❢✉❧ s✉❣❣❡st✐♦♥s ❛♥❞ ✐♥t❡❧❧❡❝t✉❛❧ ❣✉✐❞❛♥❝❡✳ ❚❤❡ ❛✉t❤♦r t❤❛♥❦s ❢✉♥❞✐♥❣ ❢r♦♠ ❛ r❡s❡❛r❝❤ ❣r❛♥t ❢r♦♠ t❤❡ ❇❡❧❣✐❛♥ ❋◆❘❙ ❛s ♣❛rt ♦❢ ❛ ✏▼✳■✳❙✳ ✲ ▼♦❜✐❧✐té ❯❧②ss❡ ❋✳❘✳❙✳ ✲ ❋◆❘❙✑✳ ❚❤❡ s❝✐❡♥t✐✜❝ r❡s♣♦♥s✐❜✐❧✐t② ❜❡❧♦♥❣s t♦ t❤❡ ❛✉t❤♦r✳ ✶ ✶✾✾✵s✳ ▼♦st st✉❞✐❡s ❧♦♦❦ ❛t t❤❡ ❡✛❡❝ts ♦❢ ❡♥✈✐r♦♥♠❡♥t ❡①t❡r♥❛❧✐✲ t✐❡s ♦♥ ❞②♥❛♠✐❝ ✐♥❡✣❝✐❡♥❝②✱ ♣r♦❞✉❝t✐✈✐t②✱ ❤❡❛❧t❤ ❛♥❞ ❧♦♥❣❡✈✐t② ♦❢ ❛❣❡♥ts✱ ❛s ✇❡❧❧ ❛s t❤❡ ❞❡s✐r❛❜❧❡ ✐♥t❡r✈❡♥t✐♦♥s ♦❢ s♦❝✐❛❧ ❛✉t❤♦r✐t✐❡s✳ ▼♦st ♣❛♣❡rs t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❛t ♣♦❧❧✉t✐♦♥ ❝♦♠❡s ❢r♦♠ t❤❡ ♣r♦✲ ❞✉❝t✐♦♥ ♣r♦❝❡ss ❛♥❞ t❤❛t ❡♥✈✐r♦♥♠❡♥t ♠❛② r❡❝♦✈❡r ♦r ❞❡❣r❛❞❡ ✐ts❡❧❢ ❛t ❛ ❝♦♥st❛♥t r❛t❡ ✭▼❛r✐♥✐ ❛♥❞ ❙❝❛r❛♠♦③③✐♥♦ ✶✾✾✺❀ ❏♦✉✈❡t ❡t ❛❧ ✷✵✵✵❀ ❏♦✉✈❡t✱ P❡st✐❡❛✉ ❛♥❞ P♦♥t❤✐❡r❡ ✷✵✵✼❀ P❛✉tr❡❧ ✷✵✵✼❀ ●✉t✐érr❡③ ✷✵✵✽✮✳ ❍♦✇❡✈❡r✱ s♦♠❡ ♦t❤❡r r❡s❡❛r❝❤❡rs ❛ss✉♠❡ t❤❛t ♣♦❧❧✉t✐♦♥ ❝♦♠❡s ♦♥❧② ❢r♦♠ ❝♦♥s✉♠♣t✐♦♥ ✭❏♦❤♥ ❛♥❞ P❡❝❝❤❡♥✐♥♦ ✶✾✾✹❀ ❏♦❤♥ ❡t ❛❧✳ ✶✾✾✺❀ ❖♥♦ ✶✾✾✻✮✳ ❖♥❡ ✐♥t❡r❡st✐♥❣ r❡♠❛r❦ ❢r♦♠ t❤❡ ❧✐t❡r❛t✉r❡ ✐s t❤❛t ❞✉❡ t♦ ♦♣♣♦s✐t❡ ❛ss✉♠♣t✐♦♥s ✐♥ ❞✐✛❡r❡♥t ♠♦❞❡❧s✱ t❤❡ ✜♥❞✐♥❣s ♦❢ ❡✛❡❝ts ♦❢ ❡♥✈✐r♦♥♠❡♥t❛❧ ❡①t❡r♥❛❧✐t✐❡s ♦♥ ❝❛♣✐t❛❧ ❛❝❝✉♠✉❧❛t✐♦♥ ❛r❡ ❞✐✛❡r❡♥t ❛❝r♦ss ♣❛♣❡rs✳ ❏♦❤♥ ❡t ❛❧✳ ✭✶✾✾✺✮ s❤♦✇❡❞ t❤❛t ✇❤❡♥ ♦♥❧② ❝♦♥s✉♠♣✲ t✐♦♥ ❞❡❣r❛❞❡s ❡♥✈✐r♦♥♠❡♥t✱ t❤❡ ❡❝♦♥♦♠② ❛❝❝✉♠✉❧❛t❡s ❧❡ss ❝❛♣✐t❛❧ t❤❛♥ ✇❤❛t ✇♦✉❧❞ ❜❡ ♦♣t✐♠❛❧❀ ♠❡❛♥✇❤✐❧❡ ●✉t✐érr❡③ ✭✷✵✵✽✮ s❤♦✇❡❞ t❤❛t ✇❤❡♥ ♣r♦❞✉❝t✐♦♥ ❝❛✉s❡s ❛ ❤✐❣❤❡r ♣♦❧❧✉t✐♦♥✱ t❤❡ ❡❝♦♥♦♠② ❛❝✲ ❝✉♠✉❧❛t❡s ✐♥st❡❛❞ ♠♦r❡ ❝❛♣✐t❛❧ t❤❛♥ t❤❡ ♦♣t✐♠❛❧ ❧❡✈❡❧✳ ❚❤✐s ✐s s♦ ❜❡❝❛✉s❡ ✐♥ ❏♦❤♥ ❡t ❛❧✳✬s ♠♦❞❡❧ ❛❣❡♥ts ❤❛✈❡ t♦ ♣❛② t❛①❡s t♦ ♠❛✐♥t❛✐♥ ❡♥✈✐r♦♥♠❡♥t ✇❤❡♥ ②♦✉♥❣✱ t❤❡r❡❢♦r❡ ❛♥ ✐♥❝r❡❛s❡ ✐♥ ♣♦❧❧✉t✐♦♥ r❡❞✉❝❡s t❤❡✐r s❛✈✐♥❣ ❢♦r t❤❡ ❢✉t✉r❡❀ ❤♦✇❡✈❡r✱ ✐♥ ●✉t✐érr❡③✬s ♠♦❞❡❧✱ ❤✐❣❤❡r ❡♥✈✐r♦♥♠❡♥t❛❧ ♣♦❧❧✉t✐♦♥ ✐♥❝r❡❛s❡s ❤❡❛❧t❤ ❝♦sts✱ ✇❤✐❝❤ ❛r❡ ♣❛✐❞ ✐♥ t❤❡ ♦❧❞ ❛❣❡✱ ❧❡❛❞s t♦ ❛❣❡♥ts ❤❛✈❡ t♦ s❛✈❡ ♠♦r❡✳ ❙♦ t❤❡ ❞✐✛❡r❡♥❝❡ s❡❡♠s t♦ ❝♦♠❡ ❢r♦♠ ✇❤❡♥ t❤❡ t❛①❡s ❛r❡ ♣❛✐❞ ✭②♦✉♥❣ ♦r ♦❧❞❄✮ r❛t❤❡r t❤❛♥ ❢r♦♠ ✇❤❡t❤❡r ✐t ✐s ♣r♦❞✉❝t✐♦♥ ♦r ❝♦♥s✉♠♣t✐♦♥ t❤❛t ♣♦❧❧✉t❡s✳ ❆♥♦t❤❡r ❞✐✛❡r❡♥❝❡ t♦ ❜❡ ♥♦t❡❞ ✐♥ t❤❡s❡ t✇♦ ♣❛♣❡rs ✐s t❤❡✐r ❞✐✛❡r✲ ❡♥t ❛ss✉♠♣t✐♦♥s ❛❜♦✉t t❤❡ ❛❜✐❧✐t② ♦❢ ❡♥✈✐r♦♥♠❡♥t t♦ r❡❝♦✈❡r ❢r♦♠ ♣♦❧❧✉t✐♦♥✳ ❏♦❤♥ ❡t ❛❧✳ ✭✶✾✾✺✮ ❛ss✉♠❡❞ t❤❛t ❡♥✈✐r♦♥♠❡♥t ❞❡❣r❛❞❡s ✐ts❡❧❢ ♦✈❡rt✐♠❡ ♠❡❛♥✇❤✐❧❡ ●✉t✐érr❡③ ❛ss✉♠❡❞ ♦♥ t❤❡ ❝♦♥tr❛r② t❤❛t ✐t ✐♠♣r♦✈❡s ❜② ❛ s❡❧❢✲♣✉r✐✜❝❛t✐♦♥ ♣r♦❝❡ss✳ ■t ✇♦✉❧❞ ❜❡ ✐♥t❡r❡st✐♥❣ t♦ ❦♥♦✇ t❤❡♥ ✇❤✐❝❤ ❞✐✛❡r❡♥❝❡ ❢♦❧❧♦✇s ❢r♦♠ ✇❤✐❝❤ ❛ss✉♠♣t✐♦♥✳ ❚❤✐s ♣❛♣❡r tr✐❡s t♦ ❞✐s❡♥t❛♥❣❧❡ t❤❡ ❡✛❡❝ts ♦❢ ❜♦t❤ ♣r♦❞✉❝t✐♦♥ ❛♥❞ ❝♦♥s✉♠♣t✐♦♥ ♦♥ ❡♥✈✐r♦♥♠❡♥t s✐♠✉❧t❛♥❡♦✉s❧②✳ ❆s ✐♥ ❏♦❤♥ ❡t ❛❧✳ ✭✶✾✾✹✱ ✶✾✾✺✮✱ ✇❡ ❛❧s♦ ❛ss✉♠❡ t❤❛t t❤❡ ❡♥✈✐r♦♥♠❡♥t ❞❡❣r❛❞❡s ✐ts❡❧❢ ♦✈❡r t✐♠❡ ✇✐t❤ ❛ ❝♦♥st❛♥t r❛t❡ ❛♥❞ t❤❡ ②♦✉♥❣ ❛❣❡♥ts s♣❡♥❞ ❛♥ ❛♠♦✉♥t ❢r♦♠ t❤❡✐r ✐♥❝♦♠❡ ✐♥ ♦r❞❡r t♦ ♠❛✐♥t❛✐♥ ❡♥✈✐r♦♥♠❡♥t✳ ▼♦r❡♦✈❡r✱ ✐♥ ❏♦❤♥ ❡t ❛❧✳✬s ♣❛♣❡rs✱ t❤❡② ❛ss✉♠❡ t❤❛t ♦♥❧② t❤❡ ❝♦♥✲ s✉♠♣t✐♦♥ ♦❢ ♦❧❞ ❛❣❡♥ts ❞❡❣r❛❞❡s t❤❡ ❡♥✈✐r♦♥♠❡♥t ❛♥❞ ②♦✉♥❣ ❛❣❡♥ts ❞♦ ♥♦t ❝♦♥s✉♠❡✱ ✇❤✐❧❡ ✐♥ t❤❡ ♣❛♣❡r ♦❢ ❖♥♦ ✭✶✾✾✻✮✱ ❤❡ ❛ss✉♠❡s ❝♦♥✲ s✉♠♣t✐♦♥ ♦❢ ❜♦t❤ ②♦✉♥❣ ❛♥❞ ♦❧❞ ❛❣❡♥ts ❞❡❣r❛❞❡ t❤❡ ❡♥✈✐r♦♥♠❡♥t✳ ✷ ❍♦✇❡✈❡r✱ ✐♥ ❤✐s ♣❛♣❡r✱ t❤❡ ❝✉rr❡♥t ❝♦♥s✉♠♣t✐♦♥s ❞♦ ♥♦t ❞❡❣r❛❞❡ t❤❡ ❝✉rr❡♥t ❡♥✈✐r♦♥♠❡♥t ❜✉t t❤❡② ❞❡❣r❛❞❡ t❤❡ ❡♥✈✐r♦♥♠❡♥t ✐♥ t❤❡ ♥❡①t ♣❡r✐♦❞ ♦♥✇❛r❞✳ ❍❡r❡✱ ✇❡ ❛❧s♦ ❛ss✉♠❡ t❤❛t t❤❡ ❡♥✈✐r♦♥♠❡♥t ✐s ❞❡❣r❛❞❡❞ ❜② t❤❡ ❝♦♥s✉♠♣t✐♦♥s ♦❢ ❜♦t❤ ♦❧❞ ❛♥❞ ②♦✉♥❣ ❛❣❡♥ts ❜✉t ✇❡ ❛ss✉♠❡ ✐♥st❡❛❞ t❤❛t t❤❡ ❡♥✈✐r♦♥♠❡♥t ✐s ♥♦t ♦♥❧② ❞❡❣r❛❞❡❞ ❜② t❤❡ ♣❛st ❝♦♥s✉♠♣t✐♦♥s ❛♥❞ ♣r♦❞✉❝t✐♦♥ ❜✉t ❛❧s♦ ❜② t❤❡ ❝✉rr❡♥t ❝♦♥✲ s✉♠♣t✐♦♥s ❛♥❞ ❝✉rr❡♥t ♣r♦❞✉❝t✐♦♥✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ❞②♥❛♠✐❝❛❧❧② ✐♥❡✣❝✐❡♥t r❛♥❣❡ ♦❢ ❝❛♣✐t❛❧ r❛t✐♦s ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❡♥✲ ✈✐r♦♥♠❡♥t❛❧ ❡①t❡r♥❛❧✐t✐❡s✳ ❚❤❡♥✱ ✇❡ s❤❛❧❧ ✐♥tr♦❞✉❝❡ s♦♠❡ t❛①❡s ❛♥❞ tr❛♥s❢❡r ♣♦❧✐❝✐❡s t❤❛t ♠❛❦❡ t❤❡ ❝♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠ st❡❛❞② st❛t❡ t♦ ❜❡ ❡✣❝✐❡♥t✳ ❚❤❡ r❡st ♦❢ t❤✐s ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ❙❡❝t✐♦♥ ✷ ✐♥tr♦✲ ❞✉❝❡s t❤❡ ♠♦❞❡❧ ❛♥❞ ❞❡✜♥❡ t❤❡ ❝♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠ ❛♥❞ t❤❡ ❝♦♠♣❡t✐t✐✈❡ st❡❛❞② st❛t❡✳ ❙❡❝t✐♦♥ ✸ ♣r❡s❡♥ts t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ s♦❝✐❛❧ ♣❧❛♥♥❡r ❛♥❞ ❞❡✜♥❡ t❤❡ ❡✣❝✐❡♥t ❛❧❧♦❝❛t✐♦♥ ❛♥❞ ♦♣t✐♠❛❧ ❛❧❧♦✲ ❝❛t✐♦♥❀ ❛♥❞ ✇❡ s❤♦✇ t❤❡ ❞②♥❛♠✐❝❛❧❧② ✐♥❡✣❝✐❡♥t r❛♥❣❡ ♦❢ t❤❡ ❝❛♣✐t❛❧ r❛t✐♦ ✐♥ t❤❡ ❝♦♠♣❡t✐t✐✈❡ ❢r❛♠❡✇♦r❦ ✭♣r♦♣♦s✐t✐♦♥ ✶✮✳ ❲❡ ✇✐❧❧ ❝♦♠♣❛r❡ t❤❡ ❝♦♠♣❡t✐t✐✈❡ st❡❛❞② st❛t❡ ❛♥❞ t❤❡ s♦❝✐❛❧ ♣❧❛♥♥❡r✬s st❡❛❞② st❛t❡ ✐♥ s❡❝t✐♦♥ ✹✱ ❛♥❞ ❤❡♥❝❡✱ ✐♥tr♦❞✉❝❡ s♦♠❡ t❛①❡s ❛♥❞ tr❛♥s❢❡r s❝❤❡♠❡s t♦ ❞❡❝❡♥tr❛❧✐③❡ t❤❡ ❜❡st st❡❛❞② st❛t❡ t❤r♦✉❣❤ ❣♦♦❞s ❛♥❞ ❢❛❝t♦r ♠❛r❦❡ts✳ ❙❡❝t✐♦♥ ✺ ❝♦♥❝❧✉❞❡s t❤❡ ♣❛♣❡r✳ ✷ ❚❤❡ ♠♦❞❡❧ ❛♥❞ ❝♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐❛ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♦✈❡r❧❛♣♣✐♥❣ ❣❡♥❡r❛t✐♦♥s ❡❝♦♥♦♠② ✐♥ ❉✐❛♠♦♥❞ ✭✶✾✻✺✮ ✇✐t❤♦✉t ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤ ✭❣r♦✇t❤ r❛t❡ n = 0✮ ❛♥❞ ♥♦r✲ ♠❛❧✐s❡ t❤❡ s✐③❡ ♦❢ ❡❛❝❤ ❣❡♥❡r❛t✐♦♥ t♦ ✉♥✐t②✳ ❊❛❝❤ ❛❣❡♥t ✐♥ t❤❡ ❡❝♦♥♦♠② ❧✐✈❡s t✇♦ ♣❡r✐♦❞s✱ s❛② ②♦✉♥❣ ❛♥❞ ♦❧❞ r❡s♣❡❝t✐✈❡❧②✳ ❲❤❡♥ ②♦✉♥❣✱ ❛♥ ❛❣❡♥t ✐s ❡♥❞♦✇❡❞ ✇✐t❤ ♦♥❡ ✉♥✐t ♦❢ ❧❛❜♦r ✇❤✐❝❤ ❤❡ s✉♣✲ ♣❧✐❡s t♦ t❤❡ ♣r♦❞✉❝✐♥❣ ✜r♠s ✐♥❡❧❛st✐❝❛❧❧②✳ ❍❡ ❞✐✈✐❞❡s ❤✐s ✇❛❣❡✱ wt ✱ ❜❡t✇❡❡♥ ❝♦♥s✉♠♣t✐♦♥ ✇❤❡♥ ②♦✉♥❣ ctt ✱ ✐♥✈❡st♠❡♥t ✐♥ ♠❛✐♥t❛✐♥✐♥❣ ❡♥✲ ✈✐r♦♥♠❡♥t mt ✱ ❛♥❞ s❛✈✐♥❣s kt+1 ✇❤✐❝❤ ✇✐❧❧ ❜❡ ❝♦♥s✉♠❡❞ ✇❤❡♥ ♦❧❞✳ ❍❡ s✉♣♣❧✐❡s ❤✐s s❛✈✐♥❣s ✐♥❡❧❛st✐❝❛❧❧② t♦ ♣r♦❞✉❝✐♥❣ ✜r♠s ❛♥❞ ❡❛r♥s t❤❡ ❣r♦ss r❡t✉r♥ rt+1 kt+1 t♦ ❝♦♥s✉♠❡ ✇❤❡♥ ♦❧❞✱ ✇❤❡r❡ rt+1 ✐s t❤❡ r❡♥t❛❧ r❛t❡ ♦❢ ❝❛♣✐t❛❧ ✐♥ t❤❡ ♣❡r✐♦❞ t + 1✳ ❆❣❡♥ts ❜♦r♥ ❛t ❞❛t❡ t ❤❛✈❡ ♣r❡❢❡r❡♥❝❡s ❞❡✜♥❡❞ ♦✈❡r t❤❡✐r ❝♦♥s✉♠♣t✐♦♥s ✐♥ ②♦✉♥❣ ❛♥❞ ♦❧❞ ❛❣❡s (ctt , ctt+1 ) ∈ R2+ ❛♥❞ t❤❡ ✐♥❞❡① ♦❢ t❤❡ ❡♥✈✐r♦♥♠❡♥t❛❧ q✉❛❧✐t②✱ Et+1 ∈ R, ✇❤✐❝❤ t❤❡② ❡①♣❡r✐❡♥❝❡ ✇❤❡♥ ♦❧❞✳ ❚❤❡ ♣r❡❢❡r❡♥❝❡ ✐s r❡♣✲ r❡s❡♥t❡❞ ❜② t❤❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ U : R2+ × R → R✳ ❲❡ ❛ss✉♠❡ t❤❛t ✸ U (ctt , ctt+1 , Et+1 ) = u(ctt ) + v(ctt+1 ) + s❡♣❛r❛❜❧❡ t+1 ) ✐s ❛❞❞✐t✐✈❡❧② φ(E t t ❛♥❞ t❤❛t Ui (·) > 0✱ Uii (·) < 0✱ i ∈ ct , ct+1 , Et+1 ✳ ❊♥✈✐r♦♥♠❡♥t❛❧ q✉❛❧✐t② ❡✈♦❧✈❡s ❛❝❝♦r❞✐♥❣ t♦ t Et+1 = (1 − b)Et − αF (Kt+1 , Lt+1 ) − β(ct+1 t+1 + ct+1 ) + γmt ❢♦r s♦♠❡ α, β, γ > ❛♥❞ b ∈ (0, 1]✱ ✇❤❡r❡ F (·, ·) ✐s t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❡❝♦♥♦♠②✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ♣r♦❞✉❝t✐♦♥ s❡❝t♦r ✐♥ t❤❡ ❡❝♦♥✲ ♦♠② ✉s✐♥❣ t✇♦ ❢❛❝t♦rs ♦❢ ♣r♦❞✉❝t✐♦♥ ❛s ❝❛♣✐t❛❧ K ❛♥❞ ❧❛❜♦r L✳ ■♥ ❡❛❝❤ ♣❡r✐♦❞ t✱ ✜r♠s ♣r♦❞✉❝❡ ❛ q✉❛♥t✐t② ♦❢ ♦✉t♣✉t Yt t❤r♦✉❣❤ ❛ ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡ ❈♦❜❜✲❉♦✉❣❧❛s ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✱ Yt = F (Kt , Lt ) = AKtθ Lt1−θ ✱ ❛♥❞ ❝❛♣✐t❛❧ ❢✉❧❧② ❞❡♣r❡❝✐❛t❡s ❡❛❝❤ ♣❡r✐♦❞✳ ■♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ♣❡r❢❡❝t ❝♦♠♣❡t✐t✐♦♥✱ t❤❡ r❡♣r❡s❡♥t❛t✐✈❡ ♣r♦✜t ♠❛①✐♠✐③✐♥❣ ✜r♠ ❝❤♦♦s❡s Kt ❛♥❞ Lt t♦ ♠❛①✐♠✐③❡ ✐ts ♣r♦✜t πt = M ax F (Kt , Lt ) − rt Kt − wt Lt Kt ,Lt ≥0 s♦ t❤❛t ✐♥ ❡❛❝❤ ♣❡r✐♦❞ t✱ t❤❡ ✇❛❣❡ r❛t❡ ❛♥❞ t❤❡ r❡♥t❛❧ r❛t❡ ♦❢ ❝❛♣✐t❛❧ ❛r❡ ❞❡t❡r♠✐♥❡❞ r❡s♣❡❝t✐✈❡❧② ❜② t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t✐✈✐t② ♦❢ ❧❛❜♦r ❛♥❞ ❝❛♣✐t❛❧✳ ❙✐♥❝❡ ❛t ❡q✉✐❧✐❜r✐✉♠ t❤❡ ❝❛♣✐t❛❧ Kt ❛✈❛✐❧❛❜❧❡ ❛t ❛♥② ♣❡✲ r✐♦❞ t✱ ❣✐✈❡♥ ♣♦♣✉❧❛t✐♦♥ ✐s ♥♦r♠❛❧✐③❡❞ ❛t 1✱ ✐s t❤❡ ♣r❡✈✐♦✉s ❛❣❣r❡❣❛t❡ s❛✈✐♥❣s kt ❛♥❞ ❛❣❣r❡❣❛t❡ ❧❛❜♦r ✐s Lt ✱ t❤❡♥ t❤❡ ✇❛❣❡ r❛t❡ ❛♥❞ r❡♥t❛❧ r❛t❡ ♦❢ ❝❛♣✐t❛❧ t❤❛t t❤❡ ❛❣❡♥t ❧✐✈✐♥❣ ✐♥ ♣❡r✐♦❞ t ❛♥❞ t + ❢❛❝❡s ❛r❡ θ−1 rt+1 = FK (Kt+1 , Lt+1 ) = FK (kt+1 , 1) = θAkt+1 ✭✶✮ wt = FL (Kt , Lt ) = FL (kt , 1) = (1 − θ)Aktθ ✭✷✮ ❲✐t❤♦✉t ❤✉♠❛♥ ❛❝t✐✈✐t②✱ t❤❡ ❡♥✈✐r♦♥♠❡♥t❛❧ q✉❛❧✐t② ✇✐❧❧ ❝♦♥✈❡r❣❡ ❛✉t♦♥♦♠♦✉s❧② t♦ t❤❡ ❧❡✈❡❧ ♦❢ ③❡r♦ ❛♥❞ t❤❡ ❞❡♣r❡❝✐❛t✐♦♥ r❛t❡ b ♠❡❛✲ s✉r❡s t❤❡ s♣❡❡❞ ♦❢ r❡✈❡rs✐♦♥ t♦ t❤✐s ❧❡✈❡❧✳ ❚❤❡ t❡r♠s αF (Kt+1 , 1) t ❛♥❞ β(ct+1 t+1 + ct+1 ) ❛r❡ t❤❡ ❞❡❣r❛❞❛t✐♦♥s ♦❢ t❤❡ ❡♥✈✐r♦♥♠❡♥t q✉❛❧✐t② r❡s✉❧t✐♥❣ ❢r♦♠ ♣r♦❞✉❝t✐♦♥ ❛♥❞ ❝♦♥s✉♠♣t✐♦♥✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ t❡r♠ γmt ♠❡❛s✉r❡s ❡♥✈✐r♦♥♠❡♥t❛❧ ✐♠♣r♦✈❡♠❡♥t ❢r♦♠ t❤❡ ❛❝t✐♦♥ ♦❢ ②♦✉♥❣ ❛❣❡♥ts ❛t ♣❡r✐♦❞ t✳ ❖♥❡ ❝❛♥ t❤✉s ✐♥t❡r♣r❡t ❡♥✈✐r♦♥♠❡♥t❛❧ q✉❛❧✐t② ❛s t❤❡ ❝❧❡❛♥♥❡ss ♦❢ r✐✈❡rs ❛♥❞ ❛t♠♦s♣❤❡r❡✱ ❛♥❞ t❤❡ q✉❛❧✐t② ♦❢ s♦✐❧ ♦r ❣r♦✉♥❞✇❛t❡r✱ ❡t❝✳ ■t ✐s ❛❧s♦ t❤❡ q✉❛❧✐t② ♦❢ ♣❛r❦s✱ ❣❛r❞❡♥s ❛♥❞ ③♦♦s ✹ ✇❤✐❝❤ t❤❡♠s❡❧✈❡s ❞❡♣r❡❝✐❛t❡ ❛♥❞ ❛❧s♦ r❡q✉✐r❡ ♠❛✐♥t❡♥❛♥❝❡✳ ❖♥❡ ❡①✲ ❛♠♣❧❡ ❢♦r ❛♥ ♦✉t♣✉t ♦❢ ❣♦♦❞ t❤❛t ❜♦t❤ ♣r♦❞✉❝✐♥❣ ✐t ❛♥❞ ❝♦♥s✉♠✐♥❣ ✐t ❞❡❣r❛❞❡ t❤❡ ❡♥✈✐r♦♥♠❡♥t ✐s ✇♦♦❞✳ P❡♦♣❧❡ ♣r♦❞✉❝❡ ✇♦♦❞ ❜② ❝✉t✲ t✐♥❣ tr❡❡s ✐♥ t❤❡ ❢♦r❡sts ❞❡❣r❛❞✐♥❣ t❤❡ ❡♥✈✐r♦♥♠❡♥t✳ ■❢ ✇♦♦❞ ✐s ✉s❡❞ t♦ ♠❛✐♥t❛✐♥ t❤❡ ♣❛r❦s ♦r ③♦♦s t❤❡♥ t❤❡ ❡♥✈✐r♦♥♠❡♥t ✐s ✐♠♣r♦✈❡❞✳ ❇✉t ✐❢ ❝♦♥s✉♠❡❞ ❛s ❢✉❡❧ ✭❢♦r ❤❡❛t✐♥❣ ♦r ❞♦♠❡st✐❝ ✉s❡s✮✱ t❤❡♥ t❤❡ ❡♥✈✐r♦♥♠❡♥t ✇✐❧❧ ❜❡ ♣♦❧❧✉t❡❞✳ ❙✐♥❝❡ ✇❡ ♥♦r♠❛❧✐③❡ t❤❡ s✐③❡ ♦❢ ❡❛❝❤ ❣❡♥❡r❛t✐♦♥ t♦ ✉♥✐t② ❛♥❞ ❦♥♦✇ t❤❛t t❤❡ s❛✈✐♥❣s ♦❢ ❛❣❡♥t ❜♦r♥ ✐♥ t❤❡ ♣❡r✐♦❞ t ❛r❡ ✉s❡❞ ❛s ❝❛♣✐t❛❧ ❢♦r ♣r♦❞✉❝t✐♦♥ ✐♥ ♣❡r✐♦❞ t + 1✱ t❤❡♥ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❡♥✈✐r♦♥♠❡♥t❛❧ q✉❛❧✐t② ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥ t Et+1 = (1 − b)Et − αF (kt+1 , 1) − β(ct+1 t+1 + ct+1 ) + γmt ✇❤❡r❡ F (kt+1 , 1) ✐s t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ♣❡r ❝❛♣✐t❛❧ ✐♥ ♣❡r✐♦❞ t + 1✳ ❋♦r♠❛❧❧②✱ t❤❡ ❧✐❢❡✲t✐♠❡ ✉t✐❧✐t② ♠❛①✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ♦❢ t❤❡ r❡♣✲ r❡s❡♥t❛t✐✈❡ ❛❣❡♥t ✐s ❛s ❢♦❧❧♦✇s M ax ctt ,ctt+1 ,kt+1 ,mt ≥0 e u(ctt ) + v(ctt+1 ) + φ(Et+1 ) ✭✸✮ e Et , Et+1 s✉❜❥❡❝t t♦ wt = ctt + kt+1 + mt ✭✹✮ ctt+1 = rt+1 kt+1 ✭✺✮ Et = (1 − b)Et−1 − αF (kt , 1) − β(ctt + ct−1 t ) + γmt−1 ✭✻✮ e t Et+1 = (1 − b)Et − αF (kt+1 , 1) − β(ct+1,e t+1 + ct+1 ) + γmt ✭✼✮ Et−1 ✱ ctt−1 ✱ kt ✱ mt−1 ✱ wt ❛s ✇❡❧❧ ❛s t❤❡ ❡①♣❡❝t❡❞ t+1,e ❝♦♥s✉♠♣t✐♦♥ ♦❢ t❤❡ ②♦✉♥❣ ❛❣❡♥t✱ ct+1 ✱ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❛♣✐t❛❧ r❡✲ t✉r♥✱ rt+1 ✱ ✐♥ t❤❡ ♣❡r✐♦❞ t + 1✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤✐s ❛❣❡♥t ✐s ♦♥❡ ♦❢ t❤❡ ♠❛♥② ♠❡♠❜❡rs ♦❢ ❣❡♥❡r❛t✐♦♥ t ❛♥❞ t❤❡r❡❢♦r❡ ❤✐s s❛✈✐♥❣s ❛r❡ ❢♦r ❣✐✈❡♥ ✈❛❧✉❡s ♦❢ ✺ ✈❡r② s♠❛❧❧ ❝♦♠♣❛r❡❞ t♦ t❤❡ ❛❣❣r❡❣❛t❡ s❛✈✐♥❣s ♦❢ t❤❡ ❡❝♦♥♦♠② ❛s ❛ ✇❤♦❧❡✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ ❤❡ ✐❣♥♦r❡s t❤❡ ✐♠♣❛❝t ♦♥ t❤❡ ❛❣❣r❡❣❛t❡ ❝❛♣✐t❛❧ ♦❢ t❤❡ ❡❝♦♥♦♠② ❢r♦♠ ❤✐s ♦✇♥ s❛✈✐♥❣s✳ ❚❤✐s ❛ss✉♠♣t✐♦♥ ✐♠✲ ♣❧✐❡s t❤❛t ❤❡ ❞♦❡s ♥♦t ✐♥t❡r♥❛❧✐③❡ t❤❡ ✐♠♣❛❝t ♦❢ ❤✐s s❛✈✐♥❣s ❤❛✈❡ ♦♥ t❤❡ ❡♥✈✐r♦♥♠❡♥t ✈✐❛ ♣r♦❞✉❝t✐♦♥✳ ❚❤❡ ✜rst ♦r❞❡r ❝♦♥❞✐t✐♦♥s ✭❋❖❈s✮ ❢♦r t❤❡ ❛❣❡♥t✬s ♣r♦❜❧❡♠ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ e u′ (ctt ) − [β(1 − b) + γ] φ′ (Et+1 )=0 ✭✽✮ e rt+1 v ′ (ctt+1 ) − [βrt+1 + γ] φ′ (Et+1 )=0 ✭✾✮ ✇❤✐❝❤ r❡❧❛t❡ t❤❡ ♠❛r❣✐♥❛❧ ✉t✐❧✐t✐❡s ♦❢ ❝♦♥s✉♠♣t✐♦♥s ✇✐t❤ ♠❛r❣✐♥❛❧ ✉t✐❧✐t② ❚❤❡ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ♦❢ t❤❡ ❛❣❡♥t✱ t t ♦❢ ❡♥✈✐r♦♥♠❡♥t❛❧ eq✉❛❧✐t②✳ ct , ct+1 , kt+1 , mt , Et , Et+1 ✱ ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ Et−1 ✱ ct−1 t ✱ kt ✱ mt−1 ✱ t+1,e ct+1 ✱ ❛♥❞ rt+1 ✐♠♣❧✐❝✐t❧② ❞❡✜♥❡❞ ❜② t❤❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ctt + kt+1 + mt − wt = ✭✶✵✮ ctt+1 − rt+1 kt+1 = ✭✶✶✮ Et − (1 − b)Et−1 + αF (kt , 1) + β(ctt + ct−1 t ) − γmt−1 = ✭✶✷✮ e t Et+1 − (1 − b)Et + αF (kt+1 , 1) + β(ct+1,e t+1 + ct+1 ) − γmt = ✭✶✸✮ e u′ (ctt ) − [β(1 − b) + γ] φ′ (Et+1 )=0 ✭✶✹✮ γ e φ′ (Et+1 )=0 − β+ rt+1 ✭✶✺✮ v ′ (ctt+1 ) ❛s ❧♦♥❣ ❛s t❤❡ ❏❛❝♦❜✐❛♥ ♠❛tr✐① ♦❢ t❤❡ ❧❡❢t✲❤❛♥❞✲s✐❞❡ ♦❢ t❤❡ s②st❡♠ e ❛❜♦✈❡ ✇✐t❤ r❡s♣❡❝t t♦ ctt , ctt+1 , kt+1 , mt , Et , Et+1 ✐s r❡❣✉❧❛r ❛t t❤❡ s♦❧✉t✐♦♥✳ ❚❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ❏❛❝♦❜✐❛♥ ♠❛tr✐① ✇✐❧❧ ❜❡ ✈❡r✐✜❡❞ ❛t t❤❡ ❡q✉✐❧✐❜r✐✉♠ ✐♥ ❆♣♣❡♥❞✐① ❆✶✳ ■♥ ♦r❞❡r t♦ ❣✉❛r❛♥t❡❡ t❤❡ ❋❖❈s ❛r❡ ♥♦t ♦♥❧② ♥❡❝❡ss❛r② ❜✉t s✉✣✲ ❝✐❡♥t ❢♦r t❤❡ s♦❧✉t✐♦♥ t♦ ❜❡ ❛ ♠❛①✐♠✉♠✱ ✇❡ ❤❛✈❡ ❝❤❡❝❦ t❤❡ s❡❝♦♥❞ ♦r❞❡r ❝♦♥❞✐t✐♦♥s ✭❙❖❈s✮ ✇❤✐❝❤ ❛r❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❆♣♣❡♥❞✐① ❆✷✳ ✻ ✷✳✶ ❈♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠ ❚❤❡ ♣❡r❢❡❝t ❢♦r❡s✐❣❤t ❝♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠ ❛❧❧♦❝❛t✐♦♥s ❛r❡ ❝❤❛r✲ ❛❝t❡r✐③❡❞ ❜② t❤❡ ❛❣❡♥t ♠❛①✐♠✐③✐♥❣ ✉t✐❧✐t② ✉♥❞❡r t❤❡s❡ ❜✉❞❣❡t ❝♦♥✲ str❛✐♥ts ❤♦❧❞✐♥❣ ❝♦rr❡❝t ❡①♣❡❝t❛t✐♦♥s✱ t❤❡ ❞②♥❛♠✐❝s ♦❢ ❡♥✈✐r♦♥♠❡♥t✱ ❛♥❞ t❤❡ ❞❡t❡r♠✐♥❛♥ts ♦❢ t❤❡ ❢❛❝t♦rs✬ ♣r✐❝❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡② ❛r❡ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ctt + kt+1 + mt − FL (kt , 1) = ✭✶✻✮ ctt+1 − FK (kt+1 , 1)kt+1 = ✭✶✼✮ t Et+1 − (1 − b)Et + αF (kt+1 , 1) + β(ct+1 t+1 + ct+1 ) − γmt = u′ (ctt ) − [β(1 − b) + γ] φ′ (Et+1 ) = v ′ (ctt+1 ) − β+ ✭✶✽✮ ✭✶✾✮ γ φ′ (Et+1 ) = FK (kt+1 , 1) ✭✷✵✮ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ ♣❡r❢❡❝t ❢♦r❡s✐❣❤t ❝♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠ ❢♦❧❧♦✇s ✉♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥s ❣✉❛r❛♥t❡❡✐♥❣ t❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ❏❛❝♦❜✐❛♥ ♠❛tr✐① ✇✐t❤ r❡s♣❡❝t t♦ t ct+1 t+1 , ct+1 , kt+1 , mt , Et+1 ♦❢ t❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡ ♦❢ t❤❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ❛❜♦✈❡ ✭s❡❡ ❆♣♣❡♥❞✐① ❆✳✶✮✳ ❆t t❤❡ ❝♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠✱ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s ✭✶✮ ❛♥❞ ✭✷✮ ❤♦❧❞✐♥❣ ✐♥ ❡✈❡r② ♣❡r✐♦❞✱ t❤❡ ❛❣❡♥t✬s ❜✉❞❣❡t ❝♦♥str❛✐♥ts ✭✶✻✮ ❛♥❞ ✭✶✼✮ ❣✉❛r❛♥t❡❡ t❤❡ ❢❡❛s✐❜✐❧✐t② ♦❢ t❤❡ ❛❧❧♦❝❛t✐♦♥ ♦❢ r❡s♦✉r❝❡s✳ ❙✐♥❝❡ ✐♥ t✱ ❛❞❞✐♥❣ ✉♣ t❤❡ ❜✉❞❣❡t ❝♦♥str❛✐♥ts ♦❢ t❤❡ ②♦✉♥❣ ❛❣❡♥t✱ ctt + kt+1 + mt = rt kt ✱ = wt ✱ ❛♥❞ t❤❡ ❝♦♥t❡♠♣♦r❛♥❡♦✉s ♦❧❞ ❛❣❡♥t✱ ct−1 t ❛♥② ♣❡r✐♦❞ ✐t ❤♦❧❞s t❤❛t ct−1 + ctt + kt+1 + mt = FK (kt , 1)kt + FL (kt , 1) = F (kt , 1) t ❚❤❡ ❝♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠ ❛❧❧♦❝❛t✐♦♥ ❝❛♥ ❛❧s♦ ❜❡ ❝♦♠♣❧❡t❡❧② ❝❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ♣❡r ❝❛♣✐t❛ s❛✈✐♥❣s ❛♥❞ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ♣❡r ❝❛♣✐t❛ ✐♥✈❡st♠❡♥t ✐♥ ❡♥✈✐r♦♥♠❡♥t ✇❤✐❝❤ r❡s✉❧t ❢r♦♠ t❤❡ ❛❣❡♥t✬s ✉t✐❧✐t② ♠❛①✐♠✐③❛t✐♦♥ ❛♥❞ t❤❡ ❞❡t❡r♠✐♥❛t✐♦♥s ♦❢ ❢❛❝✲ t♦r ♣r✐❝❡s✳ ✼ ✷✳✷ ❈♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠ st❡❛❞② st❛t❡ ❆ ♣❡r❢❡❝t ❢♦r❡s✐❣❤t ❝♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠ st❡❛❞② st❛t❡ ♦❢ t❤✐s ♦✈❡r✲ ❧❛♣♣✐♥❣ ❣❡♥❡r❛t✐♦♥s ❡❝♦♥♦♠② ✐s ❛ ❝♦♥st❛♥t s❡q✉❡♥❝❡ {k, m} ❝❤❛r❛❝✲ t❡r✐③❡❞ ❜② u′ (FL (k, 1) − k − m) = ′ v (FK (k, 1)k) = γ β+ FK (k, 1) β(1 − b) + γ ′ v (FK (k, 1)k) γ β + FK (k,1) ′ φ ✭✷✶✮ (γ + β)m − (α + β)F (k, 1) + βk b ✭✷✷✮ t❤❡ ❝♦♥s✉♠♣t✐♦♥s c0 , c1 ❛♥❞ ❡♥✈✐r♦♥♠❡♥t❛❧ q✉❛❧✐t② ✐♥❞❡① E ❛t t❤❡ st❡❛❞② st❛t❡ ❜❡✐♥❣ ❞❡t❡r♠✐♥❡❞ ❜② E= c0 = FL (k, 1) − k − m ✭✷✸✮ c1 = FK (k, 1)k ✭✷✹✮ (γ + β)m − (α + β)F (k, 1) + βk b ✭✷✺✮ ✸✳ ❊✣❝✐❡♥t ❛❧❧♦❝❛t✐♦♥ ❛♥❞ ♦♣t✐♠❛❧ ❛❧❧♦❝❛t✐♦♥ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❡✣❝✐❡♥t ❛❧❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ❜❡♥❡✈✲ ♦❧❡♥t s♦❝✐❛❧ ♣❧❛♥♥❡r✬s ♣♦✐♥t ♦❢ ✈✐❡✇✳ ❚❤❡ s♦❝✐❛❧ ♣❧❛♥♥❡r ❛❧❧♦❝❛t❡s r❡s♦✉r❝❡s ✐♥ ♦r❞❡r t♦ ♠❛①✐♠✐③❡ t❤❡ ✇❡❧❢❛r❡ ♦❢ ❜♦t❤ ❝✉rr❡♥t ❣❡♥❡r✲ ❛t✐♦♥ ❛♥❞ ❛❧❧ ❢✉t✉r❡ ❣❡♥❡r❛t✐♦♥s✳ ❆♥② ❛❧❧♦❝❛t✐♦♥ s❡❧❡❝t❡❞ ❜② ❤❡r ✐s ♦♣t✐♠❛❧ ✐♥ t❤❡ P❛r❡t♦ s❡♥s❡ ✭s❡❡ ❇❧❛♥❞❝❤❛r❞ ❛♥❞ ❋✐s❤❡r ✶✾✽✾✱ ❝❤❛♣t❡r ✸✱ ♣♣ ✾✶ ✲ ✶✵✹✮✳ ❲❡ ✇✐❧❧ ✜♥❞ t❤❡ ❡✣❝✐❡♥t ❛❧❧♦❝❛t✐♦♥s ❛♥❞ t❤❡ ♦♣t✐♠❛❧ ❛❧❧♦❝❛t✐♦♥ ❜② s♦❧✈✐♥❣ t❤❡ ❞②♥❛♠✐❝ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❜❡❧♦✇✳ ❆ss✉♠❡ t❤❛t t❤❡ ❝✉rr❡♥t ♣❡r✐♦❞ ✐s t = 0✱ ❣✐✈❡♥ k0 , E0 , c−1 ✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ s♦❝✐❛❧ ♣❧❛♥♥❡r ✐s ❛s ❢♦❧❧♦✇s✱ M ax ∞ {ctt ,ctt+1 ,kt+1 ,mt ,Et+1 }t=0 ∞ X u(ctt ) + v(ctt+1 ) + φ(Et+1 ) (1 + R)t+1 t=0 s✉❜❥❡❝t t♦✱ ∀t = 0, 1, 2, ✱ ✽ ✭✷✻✮ F (kt , 1) = ctt + ct−1 + kt+1 + mt t t Et+1 = (1 − b)Et − αF (kt+1 , 1) − β(ct+1 t+1 + ct+1 ) + γmt ✭✷✼✮ ✭✷✽✮ ✇❤❡r❡ R ≥ ✐s t❤❡ s✉❜❥❡❝t✐✈❡ ❞✐s❝♦✉♥t r❛t❡ ♦❢ t❤❡ s♦❝✐❛❧ ♣❧❛♥♥❡r✳ ❚❤❡ ❞✐s❝♦✉♥t r❛t❡ R ✐s str✐❝t❧② ♣♦s✐t✐✈❡ ✇❤❡♥ s❤❡ ❝❛r❡s ♠♦r❡ ❛❜♦✉t t❤❡ ❝✉rr❡♥t ❣❡♥❡r❛t✐♦♥ t❤❛♥ ❛❜♦✉t t❤❡ ❢✉t✉r❡ ❣❡♥❡r❛t✐♦♥s✱ ✇❤✐❧❡ R ❡q✉❛❧s t♦ ③❡r♦ ✇❤❡♥ s❤❡ ❝❛r❡s ❛❜♦✉t ❛❧❧ ❣❡♥❡r❛t✐♦♥s ❡q✉❛❧❧②✳ ❚❤❡ ✜rst ❝♦♥str❛✐♥t ✭✷✼✮ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✐s t❤❡ r❡s♦✉r❝❡ ❝♦♥str❛✐♥t ♦❢ t❤❡ ❡❝♦♥✲ ♦♠② ✐♥ ♣❡r✐♦❞ t✱ r❡q✉✐r✐♥❣ t❤❛t t❤❡ t♦t❛❧ ♦✉t♣✉t ✐s ❛❧❧♦❝❛t❡❞ t♦ t❤❡ ❝♦♥s✉♠♣t✐♦♥s ♦❢ t❤❡ ②♦✉♥❣ ❛♥❞ t❤❡ ♦❧❞✱ t♦ s❛✈✐♥❣s ❢♦r t❤❡ ♥❡①t ♣❡✲ r✐♦❞✬s ❝❛♣✐t❛❧ st♦❝❦✱ ❛♥❞ t♦ ❡♥✈✐r♦♥♠❡♥t❛❧ ♠❛✐♥t❡♥❛♥❝❡✳ ❚❤❡ s❡❝♦♥❞ ❝♦♥str❛✐♥t ✭✷✽✮ ✐s t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ❡♥✈✐r♦♥♠❡♥t❛❧ q✉❛❧✐t②✳ ❙♦❧✈✲ ✐♥❣ t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ s♦❝✐❛❧ ♣❧❛♥♥❡r ✐s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❆♣♣❡♥❞✐① ❆✸✳ ❆t t❤❡ st❡❛❞② st❛t❡✱ t❤❡ ❋❖❈s ❢♦r t❤❡ s♦❝✐❛❧ ♣❧❛♥❡r✬s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s✉♠♠❛r✐③❡❞ ❛s ❢♦❧❧♦✇s u′ (¯ c0 ) = γ(1 + R) + β(1 + R)2 ′ ¯ φ (E) b+R ✭✷✾✮ γ + β(1 + R) ′ ¯ φ (E) b+R ✭✸✵✮ 1+R − (1 + R)α/γ ✭✸✶✮ v ′ (¯ c1 ) = ¯ 1) = FK (k, ❚❤❡ ❡q✉❛t✐♦♥s ♦❢ r❡s♦✉r❝❡ ❝♦♥str❛✐♥t ❛♥❞ t❤❡ ❡♥✈✐r♦♥♠❡♥t❛❧ q✉❛❧✲ ✐t② ✐♥❞❡① ❜❡❝♦♠❡ ¯ 1) = c¯0 + c¯1 + k¯ + m F (k, ¯ ✭✸✷✮ ¯ 1) + β k¯ (γ + β)m ¯ − (α + β)F (k, E¯ = b ✭✸✸✮ ❚❤❡ ❡✣❝✐❡♥t st❡❛❞② st❛t❡ ♦❢ t❤✐s ♦✈❡r❧❛♣♣✐♥❣ ❣❡♥❡r❛t✐♦♥s ❡❝♦♥♦♠② ¯ m, ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② ❛ ❝♦♥st❛♥t s❡q✉❡♥❝❡ c¯0 , c¯1 , k, ¯ E¯ t❤r♦✉❣❤ s♦❧✈✐♥❣ t❤❡ s②st❡♠ ♦❢ ✜✈❡ ❡q✉❛t✐♦♥s ❢r♦♠ ✭✷✾✮ t♦ ✭✸✸✮✳ ✾ ❋♦r ❛♥ ♦✈❡r❧❛♣♣✐♥❣ ❣❡♥❡r❛t✐♦♥s ❡❝♦♥♦♠② s❡t ✉♣ ❛❜♦✈❡✱ ✐♥ ❛♥② ♣❡r✐♦❞ ♦❢ t❤❡ tr❛♥s✐t✐♦♥ ♣r♦❝❡ss✱ t❤❡r❡ ❛❧✇❛②s ❡①✐sts ❝♦♥s✉♠♣t✐♦♥ t❛①❡s✱ ❝❛♣✐t❛❧ ✐♥❝♦♠❡ t❛①✱ ❧✉♠♣✲s✉♠ t❛① ❛♥❞ tr❛♥s❢❡r s❝❤❡♠❡ t♦ ❛tt❛✐♥ t❤❡ ❜❡st ❝❛♣✐t❛❧ ✭s❛✈✐♥❣✮ r❛t✐♦ k¯ ❛♥❞ ❜❡st ❝♦♥s✉♠♣✲ t✐♦♥ c¯1 t❤r♦✉❣❤ ❝♦♠♣❡t✐t✐✈❡ ♠❛r❦❡ts✳ Pr♦♣♦s✐t✐♦♥ ✹✿ Pr♦♦❢✿ ❙❡❡ ❆♣♣❡♥❞✐① ❆✹✳ ❙✐♠✐❧❛r t♦ ♣r❡✈✐♦✉s s❝❤❡♠❡✱ t❤✐s s❝❤❡♠❡ ✐s ♠❡r❡❧② ✐♠♣❧❡♠❡♥t❛❜❧❡✳ Pr♦♣♦s✐t✐♦♥ ✺ st❛t❡s t❤❛t ❢r♦♠ t❤❡ ♣❡r✐♦❞ t + ♦♥✇❛r❞ t❤❡ ❣♦✈❡r♥✲ ♠❡♥t✬s ❜✉❞❣❡t ✇✐❧❧ st✐❧❧ ❜❡ ❛❧✇❛②s ❦❡♣t ❜❛❧❛♥❝❡❞ ❛♥❞ t❤❡ ♣❡r✐♦❞ t + ✐s ❛ st❡♣♣✐♥❣✲st♦♥❡ ❢♦r ❡❝♦♥♦♠② t♦ ❛❝❤✐❡✈❡ t❤❡ ♣❡r♠❛♥❡♥t ❜❡st st❡❛❞② st❛t❡ ✐♥ t❤❡ ♣❡r✐♦❞ t + ♦♥✇❛r❞✳ ❆❢t❡r ✜♥✐s❤✐♥❣ ♣❡r✐♦❞ t ✭t❤❡ ✜rst st❛❣❡ ♦❢ t❛①❛t✐♦♥✮✱ t❤❡ ❡❝♦♥♦♠② ❝❛♥ ❛❝❤✐❡✈❡ t❤❡ ❜❡st st❡❛❞② st❛t❡ ❢r♦♠ ♣❡r✐♦❞ t + ♦♥✲ ✇❛r❞ ❜② ✐♠♣❧❡♠❡♥t✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠❜✐♥❛t✐♦♥ Pr♦♣♦s✐t✐♦♥ ✺✿ τ¯c = β + (1 − b)(γ − βb) + βR(1 + b + R) (b + R)γ τ¯k = − (b + R)(γ − (1 + R)α)(1 + τ¯c ) (1 + R)(γ + β − βb) ✭✼✷✮ ✭✼✸✮ ¯ 1) − (1 + τ¯c )¯ τ¯ = FL (k, c0 − k¯ − m ¯ ✭✼✹✮ ¯ 1)k¯ + τ¯ σ ¯ = τ¯c (¯ c0 + c¯1 ) + τ¯k FK (k, ✭✼✺✮ ❆t s✉❝❤ t❤❡ st❡❛❞② st❛t❡ t❤❡ ❣♦✈❡r♠❡♥t✬s ❜✉❞❣❡t ✐s ❦❡♣t ❜❛❧❛♥❝❡❞ ❡✈❡r② ♣❡r✐♦❞✳ Pr♦♦❢✿ ❙❡❡ ❆♣♣❡♥❞✐① ❆✺✳ ✹✳✸ ❚❛①❡s ♦♥ ❝♦♥s✉♠♣t✐♦♥ ❛♥❞ ♣r♦❞✉❝t✐♦♥ ❲❡ st✐❧❧ ❦❡❡♣ t❤❡ ♥♦♥✲❞✐s❝r✐♠✐♥❛t♦r② t❛① r❛t❡ τc ♦♥ ❝♦♥s✉♠♣t✐♦♥s ❛♥❞ t❤❡ s②st❡♠ ♦❢ ❧✉♠♣✲s✉♠ t❛① τt ❛♥❞ ❧✉♠♣✲s✉♠ tr❛♥s❢❡r σt+1 ✳ ❲❡ ♥♦✇ ✐♥tr♦❞✉❝❡ ❛ P✐❣♦✉✈✐❛♥ t❛① ♦♥ ♣r♦❞✉❝t✐♦♥✳ ■♥ ❛♥② ♣❡r✐♦❞✱ ❧❡t τp ❜❡ t❤❡ t❛① ♣❛✐❞ ❜② ✜r♠s ♣❡r ♦♥❡ ✉♥✐t ♦❢ ♦✉t♣✉t ♣r♦❞✉❝❡❞✳ ❲❡ ❛❧s♦ s❤♦✇ t❤❛t ✐♥ t❤✐s s❝❡♥❛r✐♦ t❤❡ s♦❝✐❛❧ ♣❧❛♥♥❡r ✐s ❛❜❧❡ t♦ ❞❡s✐❣♥ t❛①❡s ❛♥❞ tr❛♥s❢❡r ♣♦❧✐❝② ❦❡❡♣✐♥❣ t❤❡ ❣♦✈❡r♥♠❡♥t✬s ❜✉❞❣❡t t♦ ❜❡ ❜❛❧❛♥❝❡❞ ✶✾