❈❛♣✐t♦❧♦ ✶

P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

✶✳✶ P❡r❝❤é st✉❞✐❛♠♦ ✐❧ P■▲

❈♦rr❡❧❛③✐♦♥❡ ❢r❛ P■▲ ❡✿ ❧✐❢❡ ❡①♣❡❝t❛♥❝②❀ ❡①tr❡♠❡ ♣♦✈❡rt②❀ ❡❞✉❝❛t✐♦♥❀ ❞❡♠♦❝r❛❝②✭♥♦t❛ ❝❤❡ ✐ r✐❝❝❤✐ ♥♦♥ ❞❡♠♦❝r❛t✐❝✐ s♦♥♦ q✉❡❧❧✐ ❝♦♥ ❞♦t❛③✐♦♥❡ ✐♠♣♦rt❛♥t❡ ❞✐ r✐s♦rs❡♥❛t✉r❛❧✐✮❀ s❝♦♣❡rt❡ s❝✐❡♥t✐✜❝❤❡ ❡ t❡❝♥♦❧♦❣✐❝❤❡✳✳✳

✶✳✷ ❈♦♥t❛❜✐❧✐tà

❱❡❞r❡♠♦ ❝♦s✬è ✐❧ P■▲✱ ❝♦♠❡ s✐ ❝❛❧❝♦❧❛ ❡ ❝♦♠❡ s❡ ♥❡ s❡❣✉❡ ❧✬❛♥❞❛♠❡♥t♦ ♥❡❧ t❡♠♣♦❛ttr❛✈❡rs♦ ❡s❡♠♣✐ ❝♦♥❝r❡t✐ ❝❤❡ ❧♦ ✐♥q✉❛❞r❡r❛♥♥♦ ❞❛ ✈❛r✐❡ ❛♥❣♦❧❛③✐♦♥✐✳ P■▲ st❛♣❡r Pr♦❞♦tt♦ ■♥t❡r♥♦ ▲♦r❞♦✱ ❞♦✈❡ ■♥t❡r♥♦ s✐ ❝❛♣✐s❝❡ ✭s✐ r✐❢❡r✐s❝❡ ❛ ✉♥❛ ❡❝♦♥♦♠✐❛♥❛③✐♦♥❛❧❡✮✱ ▲♦r❞♦ ♣♦✐ ❧♦ ✈❡❞r❡♠♦✱ ❡ Pr♦❞♦tt♦ è ✐❧ ✈❛❧♦r❡ ❞❡✐ ❜❡♥✐ ✜♥❛❧✐ ♣r♦❞♦tt✐✳■♥ ■t❛❧✐❛ ✐❧ P■▲ è ❝✐r❝❛ ✶✺✵✵ ▼✐❧✐❛r❞✐ ❞✐ ❊✉r♦✳ ◗✉❡st♦ ♦✈✈✐❛♠❡♥t❡ ♥♦♥ ✈✉♦❧ ❞✐r❡♥✐❡♥t❡ ✜♥❝❤é ♥♦♥ s✐ s♣❡❝✐✜❝❛ ✐♥ q✉❛♥t♦ t❡♠♣♦ s✐ ♣r♦❞✉❝❡❀ ❧❛ ❢r❛s❡ ❝♦rr❡tt❛ è ❝❤❡♥❡❣❧✐ ✉❧t✐♠✐ ❛♥♥✐ ✐❧ P■▲ è st❛t♦ ❞❡❧❧✬♦r❞✐♥❡ ❞❡✐ ✶✺✵✵ ▼✐❧✐❛r❞✐ ❧✬❛♥♥♦✳ ◗✉✐♥❞✐✱ ♣r✐♠❛❝♦s❛ ❞❛ t❡♥❡r❡ ♣r❡s❡♥t❡✿ ✐❧ P■▲ è ✉♥ ✢✉ss♦✳ ➮ ✐❧ r❡❞❞✐t♦ ❛♥♥✉❛❧❡ ❞❡❧❧❛ ♥❛③✐♦♥❡✳

❊s❡♠♣✐♦ ✶✳✶ ✭❇❡♥✐ ■♥t❡r♠❡❞✐ ❡ ❱❛❧♦r❡ ❆❣❣✐✉♥t♦✮✳ ■♥ q✉❡st❛ ❡❝♦♥♦♠✐❛ ❝✐ s♦✲♥♦ ✸ s❡tt♦r✐ ❝❤❡ ♣r♦❞✉❝♦♥♦ ✉t✐❧✐③③❛♥❞♦ ✐❧ s♦❧♦ ❧❛✈♦r♦ ✭❡ ❧❛ ❝❛♣❛❝✐tà ♣r♦❞✉tt✐✲✈❛✴✐♠♣r❡♥❞✐t♦r✐❛❧❡✮✿ ❣❧✐ ❛❣r✐❝♦❧t♦r✐ ❝❤❡ ♣r♦❞✉❝♦♥♦ ❣r❛♥♦✱ ✐ ♠✉❣♥❛✐ ❝❤❡ tr❛s❢♦r✲♠❛♥♦ ✐❧ ❣r❛♥♦ ♣r♦❞♦tt♦ ✐♥ ❢❛r✐♥❛✱ ❡ ✐ ♣❛♥❡tt✐❡r✐ ❝❤❡ tr❛s❢♦r♠❛♥♦ ❧❛ ❢❛r✐♥❛ ✐♥ ♣❛♥❡✳◆❡❧ P■▲ ♥♦♥ ♣♦ss✐❛♠♦ ❝♦♥t❛r❡ ❣r❛♥♦ ❡ ❢❛r✐♥❛ ♣❡r❝❤é s♦♥♦ ✐♥t❡r❛♠❡♥t❡ ✏❞✐str✉tt✐✑♥❡✐ ✈❛r✐ st❛❞✐ ❞❡❧ ♣r♦❝❡ss♦ ♣r♦❞✉tt✐✈♦✱ ❝✐♦è s♦♥♦ ✐♠♣✐❡❣❛t✐ ❝♦♠❡ ✏❜❡♥✐ ✐♥t❡r♠❡❞✐✑❀❞♦❜❜✐❛♠♦ ❝♦♥t❛r❡ s♦❧♦ ✐❧ ✈❛❧♦r❡ ❞❡❧ ♣❛♥❡✱ ❝❤❡ è q✉❡❧❧♦ ❝❤❡ ❡✛❡tt✐✈❛♠❡♥t❡ ❧❡ ♣❡rs♦♥❡✐♥ q✉❡st❛ ❡❝♦♥♦♠✐❛ ❝♦♥s✉♠❛♥♦✳

✶

✷ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

❙❡tt♦r❡Pr♦❞✉tt✐✈♦

❘✐❝❛✈✐✭✶✮

❈♦st♦ ❇❡♥✐■♥t❡r♠❡❞✐

✭✷✮

❱❛❧♦r❡❆❣❣✐✉♥t♦✭✶✮✲✭✷✮

●r❛♥♦ ✷✹ ✷✹❋❛r✐♥❛ ✹✵ ✷✹ ✶✻P❛♥❡ ✼✺ ✹✵ ✸✺

P■▲ ✼✺ ✼✺

◗✉✐♥❞✐ ✐❧ P■▲ è 75✱ ✉❣✉❛❧❡ ❛❧ ✈❛❧♦r❡ ❞❡✐ ❜❡♥✐ ✜♥❛❧✐✱ ❝✐♦è ♥♦♥ ✉s❛t✐ ❝♦♠❡ ❜❡♥✐✐♥t❡r♠❡❞✐✳ ➮ ✉❣✉❛❧❡ ❛✐ r✐❝❛✈✐ ❞❡✐ ✈❛r✐ s❡tt♦r✐ ♠❡♥♦ ✐❧ ✈❛❧♦r❡ ❞❡✐ ❜❡♥✐ ✉s❛t✐ ♥❡✐♣r♦❝❡ss✐ ♣r♦❞✉tt✐✈✐✿ 75 = (24+40+75)− (24+40)✳ ▲❛ ❝♦s❛ ✐♥t❡r❡ss❛♥t❡ è ✈❡❞❡r❡q✉❛♥t♦ ❝♦♥tr✐❜✉✐s❝❡ ♦❣♥✐ s❡tt♦r❡✳ P❡r ❡s❡♠♣✐♦ ✐ ♠✉❣♥❛✐ ❤❛♥♥♦ tr❛s❢♦r♠❛t♦ ❣r❛♥♦✱✈❛❧♦r❡ ✷✹✱ ✐♥ ❢❛r✐♥❛ ❞✐ ✈❛❧♦r❡ ✹✵❀ ❝✐♦è ♥❡❧ ❧♦r♦ ♣r♦❝❡ss♦ ♣r♦❞✉tt✐✈♦ ❤❛♥♥♦ ❛❣❣✐✉♥t♦

✈❛❧♦r❡ ✶✻✳ ❊ ❝♦sì s✐ ❝❤✐❛♠❛ ❧❛ ❞✐✛❡r❡♥③❛ ❢r❛ r✐❝❛✈✐ ❡ ❝♦st✐ ❞❡✐ ❜❡♥✐ ✐♥t❡r♠❡❞✐✿❱❛❧♦r❡ ❆❣❣✐✉♥t♦✳ ➮ ❡s❛tt❛♠❡♥t❡ q✉❡❧❧♦ ❞❡❧❧✬■❱❆✱ ■♠♣♦st❛ s✉❧ ❱❛❧♦r❡ ❆❣❣✐✉♥t♦ ✲♥❡ r✐♣❛r❧❡r❡♠♦ ♥❡❧❧✬❡s❡♠♣✐♦ ✶✳✺✳ P❡r ❣❧✐ ❛❣r✐❝♦❧t♦r✐ è ✉❣✉❛❧❡ ❛✐ r✐❝❛✈✐✱ ♣❡r❝❤é ♥♦♥❤❛♥♥♦ ✉s❛t♦ ❜❡♥✐ ✐♥t❡r♠❡❞✐ ✭s♦❧t❛♥t♦ ❧❛✈♦r♦✮✳ ❉❛❧❧✬✉❧t✐♠❛ ❝♦❧♦♥♥❛ ✈❡❞✐❛♠♦ ❝❤❡ ✐❧P■▲ è ❧❛ s♦♠♠❛ ❞❡✐ ✈❛❧♦r✐ ❛❣❣✐✉♥t✐ ❞❛✐ ✈❛r✐ s❡tt♦r✐✿ 75 = 24+(40−24)+(75−40)✳❉✉♥q✉❡

PIL =∑

jVAj

❞♦✈❡ VAj è ✐❧ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦ ❞❡❧ s❡tt♦r❡ j✱ ❝❤❡ ♠♦str❛ ❞❛ ❞♦✈❡ ✈✐❡♥❡ ✐❧ r❡❞❞✐t♦♣r♦❞♦tt♦✳

❆ ❝❤✐ ✈❛ ❝✐ò ❝❤❡ ✈✐❡♥❡ ♣r♦❞♦tt♦❄ ▲❛ r✐s♣♦st❛ ♥♦♥ ♣♦trà ❝❤❡ ❡ss❡r❡✿ ❛ q✉❡❧❧✐❝❤❡ ♣❛rt❡❝✐♣❛♥♦ ❛❧ ♣r♦❝❡ss♦ ♣r♦❞✉tt✐✈♦✳ ❱❡❞✐❛♠♦❧♦ ❝♦♥ ✉♥ ❡s❡♠♣✐♦✳

❊s❡♠♣✐♦ ✶✳✷ ✭❉✐str✐❜✉③✐♦♥❡ ❞❡❧ ❘❡❞❞✐t♦✮✳ ◗✉✐ ❝✐ s♦♥♦ ❞✉❡ s❡tt♦r✐✱ q✉❡❧❧♦ s✐❞❡✲r✉r❣✐❝♦ ❡ q✉❡❧❧♦ ❛✉t♦♠♦❜✐❧✐st✐❝♦✱ ❡ ❞✐ ♥✉♦✈♦ s✐ ❛ss✉♠❡ ✭♠♦❧t♦ ✐rr❡❛❧✐st✐❝❛♠❡♥t❡✮❝❤❡ ❧✬✉♥✐❝♦ ❢❛tt♦r❡ ♣r♦❞✉tt✐✈♦ s✐❛ ✐❧ ❧❛✈♦r♦✳ ◆❡❧ s❡❝♦♥❞♦ s❡tt♦r❡ s✐ ✉s❛ ❧✬❛❝❝✐❛✲✐♦ ♣r♦❞♦tt♦ ❞❛❧ ♣r✐♠♦✳ ◗✉✐ ♣r❡♥❞✐❛♠♦ ✐♥ ❝♦♥s✐❞❡r❛③✐♦♥❡ ❧❛ r❡♠✉♥❡r❛③✐♦♥❡ ❞❡❧❢❛tt♦r❡ ♣r♦❞✉tt✐✈♦ ✭❧❛✈♦r♦✮✱ ❡ ❞❡✜♥✐❛♠♦ ♣r♦✜tt♦ ❝♦♠❡ r❡s✐❞✉♦✱ r✐❝❛✈✐ ♠❡♥♦ ❝♦✲st✐ ✭❝♦♠❡ ✐♥ ♠✐❝r♦❡❝♦♥♦♠✐❛✮❀ ✐❧ ♣r♦✜tt♦ è ❞✐ ❢❛tt♦ ❧❛ r❡♠✉♥❡r❛③✐♦♥❡ ❞❡❧❧✬❛tt✐✈✐tà✐♠♣r❡♥❞✐t♦r✐❛❧❡✳

✶✳✷✳ ❈♦♥t❛❜✐❧✐tà ✸

❙❡tt♦r❡Pr♦❞✉tt✐✈♦

❘✐❝❛✈✐✭✶✮

❈♦st♦ ❇❡♥✐■♥t❡r♠❡❞✐

✭✷✮

❈♦st♦▲❛✈♦r♦✭✸✮

Pr♦✜tt✐✭✶✮✲✭✷✮✲✭✸✮

❱❛❧♦r❡❆❣❣✐✉♥t♦✭✶✮✲✭✷✮

❙✐❞❡r✉r❣✐❛ ✶✵✵ ✽✵ ✷✵ ✶✵✵❆✉t♦♠♦❜✐❧✐ ✷✶✵ ✶✵✵ ✼✵ ✹✵ ✶✶✵

P■▲ ✷✶✵ ✷✶✵

◆♦t❛ ❝❤❡ ♣❡r ♦❣♥✐ s❡tt♦r❡✱ ❡ q✉✐♥❞✐ ❛♥❝❤❡ ✐♥ ❛❣❣r❡❣❛t♦✱ ❛❜❜✐❛♠♦

Profitti = Ricavi− Costo Beni Intermedi − CostoLavoro

= V aloreAggiunto− CostoLavoro

❞♦✈❡ ✐❧ ❈♦st♦ ▲❛✈♦r♦ ♥♦♥ s♦♥♦ ❛❧tr♦ ❝❤❡ ✐ ❙❛❧❛r✐✿ Costo Lavoro = Salari ✳❉❛ q✉❡st♦ ✈❡❞✐❛♠♦ ❝❤❡

PIL ≡ Somma Valori Aggiunti = Salari + Profitti

❝✐♦è✱ ✐❧ P■▲ è ❞✐str✐❜✉✐t♦ ❛ ❧❛✈♦r❛t♦r✐ ❡ ✐♠♣r❡♥❞✐t♦r✐✿

❙❡tt♦r❡Pr♦❞✉tt✐✈♦

❱❛❧♦r❡❆❣❣✐✉♥t♦

✭✶✮

❈♦st♦▲❛✈♦r♦✭✷✮

Pr♦✜tt✐✭✶✮✲✭✷✮

❙✐❞❡r✉r❣✐❛ ✶✵✵ ✽✵ ✷✵❆✉t♦♠♦❜✐❧✐ ✶✶✵ ✼✵ ✹✵

❙♦♠♠❛ ✷✶✵ ✶✺✵ ✻✵

❉✉♥q✉❡ ❧❡ s♣❡s❡ ♣❡r ❜❡♥✐ ✜♥❛❧✐✱ r✐❝❛✈✐ ❞❡❧❧❡ ✐♠♣r❡s❡✱ s♦♥♦ ✉❣✉❛❧✐ ❛✐ r❡❞❞✐t✐ ❞✐str✐✲❜✉✐t✐ ❛❧❧❡ ❢❛♠✐❣❧✐❡✳ ❖✈✈✐❛♠❡♥t❡ q✉❡st✐ ✢✉ss✐ ✜♥❛♥③✐❛r✐ ❤❛♥♥♦ ❝♦♠❡ ❝♦♥tr♦♣❛rt❡✢✉ss✐ r❡❛❧✐✿ ✐♥ ❝❛♠❜✐♦ ❞❡✐ r❡❞❞✐t✐ ❧❡ ❢❛♠✐❣❧✐❡ ❢♦r♥✐s❝♦♥♦ ❛❧❧❡ ✐♠♣r❡s❡ ✐ ❢❛tt♦r✐ ♣r♦✲❞✉tt✐✈✐✱ ❡ ❧❡ ✐♠♣r❡s❡ ❢♦r♥✐s❝♦♥♦ ❛❧❧❡ ❢❛♠✐❣❧✐❡ ✐ ❜❡♥✐ ✜♥❛❧✐ ♣❛❣❛t✐ ❞❛❧❧❡ ❢❛♠✐❣❧✐❡✳ ➮ ✐❧❝♦s✐❞❞❡tt♦ ✏❞✐❛❣r❛♠♠❛ ❞✐ ✢✉ss♦✑ r❛♣♣r❡s❡♥t❛t♦ ♥❡❧❧❛ ✜❣✉r❛ q✉✐ s♦tt♦ ✭❝❤❡ s✐ tr♦✈❛s✉ ❲✐❦✐♣❡❞✐❛ ❛❧❧❛ ✈♦❝❡ ✏❈✐r❝✉❧❛r ❋❧♦✇ ♦❢ ■♥❝♦♠❡✑✮✿

✹ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

➮ ✐♠♣♦rt❛♥t❡ ❝❛♣✐r❡ ❝❤❡ ❢❛♠✐❣❧✐❡ ❡ ✐♠♣r❡s❡ s♦♥♦ ❢❛tt❡ ❞❛❧❧❡ st❡ss❡ ♣❡rs♦♥❡✱♣❡r❝❤é ❧❡ ✐♠♣r❡s❡ s♦♥♦ ❞✐ ♣r♦♣r✐❡tà ❞❡❧❧❡ ❢❛♠✐❣❧✐❡✳ ❙❡ ❘♦s❛r✐♦ è ✉♥ ❝❛❧③♦❧❛✐♦ ❝✐s❛rà ❧❛ ❢❛♠✐❣❧✐❛ ❘♦s❛r✐♦ ❡ ❧✬✐♠♣r❡s❛ ❘♦s❛r✐♦✳ ❘♦s❛r✐♦✲❢❛♠✐❣❧✐❛ ❢♦r♥✐s❝❡ ❧❛✈♦r♦ ❡♣❡r❝❡♣✐s❝❡ ✐❧ s❛❧❛r✐♦✱ ❛ ❘♦s❛r✐♦✲✐♠♣r❡s❛ ❛rr✐✈❛ ✐❧ ♣r♦✜tt♦✳

P✐ù ✐♥ ❣❡♥❡r❛❧❡ ❢r❛ ✐ ❢❛tt♦r✐ ♣r♦❞✉tt✐✈✐ ❝✐ s♦♥♦ ❛♥❝❤❡ ✐ ❜❡♥✐ ❝❛♣✐t❛❧✐✱ ❡ ✐❧ r❡❞❞✐t♦❞❛ ❝❛♣✐t❛❧❡ ✭❛❧tr♦ ❝♦st♦✮ è r❛♣♣r❡s❡♥t❛t♦ ❞❛❣❧✐ ✐♥t❡r❡ss✐✳ ❊ ❝✬è ❛♥❝❤❡ ❧♦ ❙t❛t♦ ❝❤❡❡♥tr❛ ♥❡❧❧❛ ♣r♦❞✉③✐♦♥❡ ✐♥❞✐r❡tt❛♠❡♥t❡ tr❛♠✐t❡ ✐ ❜❡♥✐ ♣✉❜❜❧✐❝✐ ❞✐ ❝✉✐ ❧❡ ✐♠♣r❡s❡✉s✉❢r✉✐s❝♦♥♦✳ ■❧ ❝♦st♦ ❞✐ q✉❡st♦ ✐♥t❡r✈❡♥t♦ s♦♥♦ ❧❡ ■♠♣♦st❡ ■♥❞✐r❡tt❡ ✲ q✉❡❧❧❡ s✉❧✈❛❧♦r❡ ❛❣❣✐✉♥t♦ ♣❡r ❧✬❛♣♣✉♥t♦✳ ◗✉✐♥❞✐ ❧❛ r❡❧❛③✐♦♥❡ ❞✐ ❞✐str✐❜✉③✐♦♥❡ ❞✐✈❡♥t❛✿

PIL = SommaV alori Aggiunti

= Salari+ Profitti+ Interessi+ Imposte Indirette.

■❧ P■▲ è ❝♦sì ❞✐str✐❜✉✐t♦ ❛✐ s♦❣❣❡tt✐ ❝❤❡ ♣❛rt❡❝✐♣❛♥♦ ❛❧ ♣r♦❝❡ss♦ ♣r♦❞✉tt✐✈♦✳ ◆♦t❛❝❤❡ st✐❛♠♦ s❝r✐✈❡♥❞♦

Profitti = Ricavi − Costo Beni Intermedi − Costo Lavoro

− CostoCapitale− Imposte Indirette

❡ q✉✐✱ ❛ ♣❛rt❡ ❧✬❛❣❣✐✉♥t❛ ❞❡❧❧❡ ✐♠♣♦st❡✱ ❝✬è ❞❛ ♥♦t❛r❡ ❧❛ ✈♦❝❡ ❞✐ ❝♦st♦ ❞❡✐ ❜❡♥✐✐♥t❡r♠❡❞✐✱ ❝❤❡ ♥♦♥ è ♠❡ss❛ ✐♥ ❡✈✐❞❡♥③❛ ✐♥ ♠✐❝r♦❡❝♦♥♦♠✐❛ ♠❛ q✉✐ è ❢♦♥❞❛♠❡♥t❛❧❡♣❡r❝❤é ♣❡r ❛rr✐✈❛r❡ ❛❧❧❛ ♠✐s✉r❛ ❞❡❧ P■▲ ✈❛♥♥♦ ❡✈✐t❛t❡ ❧❡ ❞✉♣❧✐❝❛③✐♦♥✐✳

❊s❡♠♣✐♦ ✶✳✸ ✭❚❡❝♥♦❧♦❣✐❛ ❡ Pr❡③③✐✮✳ ◆❡❣❧✐ ❡s❡♠♣✐ ♣r❡❝❡❞❡♥t✐ ❛❜❜✐❛♠♦ s♦♠♠❛t♦✈❛❧♦r✐✱ ♦✈✈❡r♦ s♦♠♠❡ ✐♥ ❡✉r♦ ❝❤❡ r❛♣♣r❡s❡♥t❛✈❛♥♦ ✐❧ ✈❛❧♦r❡ ❞❡❧ P■▲✱ ✐❧ ✈❛❧♦r❡❛❣❣✐✉♥t♦ ❞❡✐ s❡tt♦r✐✱ s❛❧❛r✐✱ ♣r♦✜tt✐ ❡t❝✳ ■♥ q✉❡st♦ ❡s❡♠♣✐♦ ❞❡❝♦♠♣♦rr❡♠♦ ✐❧ ✈❛❧♦r❡✐♥tr♦❞✉❝❡♥❞♦ t❡❝♥♦❧♦❣✐❛ ❡ ♣r❡③③✐✱ ❝♦sì ♣♦ss✐❛♠♦ ❢❛r❡ ❘✐❝❛✈♦ ❂ Pr❡③③♦ × ◗✉❛♥t✐tà✳❈✐ s♦♥♦ ✷ s❡tt♦r✐✱ ●r❛♥♦ ❡ ❇r✐♦❝❤❡ ✭P❛♥❡ ♥♦♥ ♣✉ò ❡ss❡r❡ ♣❡r❝❤é ❝♦♠✐♥❝✐❛ ♣❡r P❝❤❡ ✐♥❞✐❝❛ ♣r❡③③✐✮✳ ■♥❞✐❝❛♥❞♦ ❝♦♥ G,B,L q✉❛♥t✐tà ❞✐ ❣r❛♥♦✱ ❜r✐♦❝❤❡ ❡ ❧❛✈♦r♦✱ ✐❞✉❡ s❡tt♦r✐ ✐♠♣✐❡❣❛♥♦ r✐s♣❡tt✐✈❛♠❡♥t❡ ❧❡ t❡❝♥♦❧♦❣✐❡

G(L) = 4L, B(L,G) = min{L,G/5}

❝✐♦è✿ ♥❡❧ s❡tt♦r❡ ●r❛♥♦ ✶ ✉♥✐tà ❞✐ ❧❛✈♦r♦ ♥❡ ♣r♦❞✉❝❡ ✹ ❞✐ ❣r❛♥♦✱ ♥❡❧ s❡tt♦r❡ ❇r✐♦❝❤❡✶ ❜r✐♦❝❤❡ è ♣r♦❞♦tt❛ ❝♦♥ ✶ ✉♥✐tà ❞✐ ❧❛✈♦r♦ ❡ ✺ ❞✐ ❣r❛♥♦✳ ▲❛ ♣r♦❞✉③✐♦♥❡ t♦t❛❧❡ è❞❡s❝r✐tt❛ ♥❡❧❧❛ s❡❣✉❡♥t❡ t❛❜❡❧❧❛✿

❙❡tt♦r✐◗✳tà ✉s❛t❛ ❞✐

L◗✳tà ✉s❛t❛ ❞✐

G◗✳tà ♣r♦❞♦tt❛ ❞✐

G◗✳tà ♣r♦❞♦tt❛ ❞✐

B

●r❛♥♦ ✷✺✵✵ ✶✵✵✵✵❇r✐♦❝❤❡ ✶✷✵✵ ✻✵✵✵ ✶✷✵✵

■ ♣r❡③③✐ s✐❛♥♦ ❞❛t✐ ❞❛ (pL, pG, pB) = (2, 1, 8)✳ ❉✉♥q✉❡ ❛❜❜✐❛♠♦

✶✳✷✳ ❈♦♥t❛❜✐❧✐tà ✺

❙❡tt♦r❡Pr♦❞✉tt✐✈♦

❘✐❝❛✈✐❈♦st♦ ❇❡♥✐■♥t❡r♠❡❞✐

❈♦st♦▲❛✈♦r♦

Pr♦✜tt✐❱❛❧♦r❡

❆❣❣✐✉♥t♦

●r❛♥♦ 10000× 1 2500× 2 ✺✵✵✵ ✶✵✵✵✵❇r✐♦❝❤❡ 1200× 8 6000× 1 1200× 2 1200 ✸✻✵✵

❙♦♠♠❛ ✶✾✻✵✵ ✻✵✵✵ ✼✹✵✵ ✻✷✵✵ ✶✸✻✵✵

❉❛ q✉❡st❛ t❛❜❡❧❧❛ s✐ ✈❡❞❡ q✉❡❧❧♦ ❝❤❡ ❝✐ ✐♥t❡r❡ss❛✳ ■❧ P■▲ è ✶✸✻✵✵✱ ❝❤❡ ♣♦ss✐❛♠♦❝❛❧❝♦❧❛r❡ ❝♦♠❡ r✐❝❛✈✐ ♠❡♥♦ ❝♦st✐ ❜❡♥✐ ✐♥t❡r♠❡❞✐✿ 10000× 1+ 1200× 8− 6000× 1❀♦ ❝♦♠❡ s♦♠♠❛ ❞❡✐ ✈❛❧♦r✐ ♣r♦❞♦tt✐ ❛❧ ♥❡tt♦ ❞❡❣❧✐ ✉t✐❧✐③③✐✿ (10000 − 6000) × 1 +12000 × 8✱ ❞♦✈❡ ✹✵✵✵ s♦♥♦ ❧❡ ✉♥✐tà ❞✐ ❣r❛♥♦ ♥♦♥ ✉t✐❧✐③③❛t❡ ♥❡✐ ♣r♦❝❡ss✐❀ ♦ ❝♦♠❡s♦♠♠❛ ❞❡✐ ✈❛❧♦r✐ ❛❣❣✐✉♥t✐✱ 10000 + 3600✳ ■♥♦❧tr❡✱ ❧❡ q✉♦t❡ ❞✐ s❛❧❛r✐ ❡ ♣r♦✜tt✐ ♥❡❧r❡❞❞✐t♦ ♥❛③✐♦♥❛❧❡ s♦♥♦ r✐s♣❡tt✐✈❛♠❡♥t❡ 7400/13600 = 54% ❡ 6200/13600 = 46%✳

❊s❡♠♣✐♦ ✶✳✹ ✭❆♥❝♦r❛ ♣r❡③③✐ ❡ q✉❛♥t✐tà✮✳ ◗✉❡st♦ ❡s❡♠♣✐♦ è ❝♦♠❡ ✐❧ ♣r❡❝❡❞❡♥t❡❝♦♥ ❧❛ ❞✐✛❡r❡♥③❛ ❝❤❡ ✐♥✈❡❝❡ ❞❡❧❧❡ ❢✉♥③✐♦♥✐ ❞✐ ♣r♦❞✉③✐♦♥❡ ❞❡✐ ✈❛r✐ s❡tt♦r✐ s♦♥♦ ✐♥❞✐✲❝❛t❡✱ ♣❡r ♦❣♥✐ s❡tt♦r❡✱ ❞✐r❡tt❛♠❡♥t❡ ❧❡ q✉❛♥t✐tà ✉t✐❧✐③③❛t❡ ❞✐ ❧❛✈♦r♦ ❡ ❜❡♥✐ ✐♥t❡r♠❡❞✐❡ ✐❧ ♣r♦❞♦tt♦ ✜♥❛❧❡✳ ❈✐ s♦♥♦ 4 s❡tt♦r✐ ✭❝✐♦è ❜❡♥✐✮ ❝❤❡ ❝❤✐❛♠✐❛♠♦ Si, i = 1, 2, 3, 4✳▲✬❡❝♦♥♦♠✐❛ è s♣❡❝✐✜❝❛t❛ ❞❛❧❧❛ s❡❣✉❡♥t❡ t❛❜❡❧❧❛✳ ■♥ ♦❣♥✐ r✐❣❛ s♦♥♦ ✐♥❞✐❝❛t❡ ❧❛ q✉❛♥✲t✐tà ❞✐ ❧❛✈♦r♦ ❡ ❞❡❣❧✐ ❛❧tr✐ ❜❡♥✐ ✉t✐❧✐③③❛t❡ ♣❡r ❧❛ ♣r♦❞✉③✐♦♥❡ ❞❡❧ ❜❡♥❡ i = 1, 2, 3, 4✱❡ ❧❛ q✉❛♥t✐tà ♣r♦❞♦tt❛ ❞❡❧ ❜❡♥❡✳

❙❡tt♦r❡Pr♦❞✉tt✐✈♦

▲❛✈♦r♦ ❇❡♥✐ ■♥t❡r♠❡❞✐ Pr♦❞♦tt♦

S1 S2 S3

S1 ✶✵✵✵ ✵ ✵ ✵ ✷✵✵✵S2 ✽✵✵ ✷✵✵✵ ✵ ✵ ✶✽✵✵S3 ✼✺✵ ✵ ✼✺✵ ✵ ✼✺✵S4 ✶✵✵ ✵ ✷✵✵ ✹✵✵ ✶✵✵

■ ♣r❡③③✐ s♦♥♦ pL = 1.5, p1 = 1, p2 = 2, p3 = 4, p4 = 30✳ ❋❛❝❝✐❛♠♦ ❣❧✐ st❡ss✐❝♦♥t✐ ❞✐ ♣r✐♠❛✱ ❝❤❡ r✐♣♦rt✐❛♠♦ ♥❡❧❧❛ s❡❣✉❡♥t❡ t❛❜❡❧❧❛✿

❙❡tt♦r❡ ❘✐❝❛✈✐ ❈♦st✐ ❇❡♥✐ ■♥t✳ ❱❆ ❙❛❧❛r✐ Pr♦✜tt✐

S1 ✷✵✵✵ ✵ ✷✵✵✵ ✶✺✵✵ ✺✵✵S2 ✸✻✵✵ ✷✵✵✵ ✶✻✵✵ ✶✷✵✵ ✹✵✵S3 ✸✵✵✵ ✶✺✵✵ ✶✺✵✵ ✶✶✷✺ ✸✼✺S4 ✸✵✵✵ ✷✵✵✵ ✶✵✵✵ ✶✺✵ ✽✺✵

❚♦t❛❧❡ ✻✶✵✵ ✸✾✼✺ ✷✶✷✺

✻ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

■❧ P■▲ è ❧❛ s♦♠♠❛ ❞❡✐ ❱❛❧♦r✐ ❆❣❣✐✉♥t✐ q✉✐♥❞✐ ✻✶✵✵✱ ❞✐ ❝✉✐ ✸✾✼✺✴✻✶✵✵≈✻✺✪ ✐♥❙❛❧❛r✐✱ ❡❞ ✐❧ r❡st❛♥t❡ ✸✺✪ ✐♥ Pr♦✜tt✐✳ ■❧ ❝♦♥tr✐❜✉t♦ ❞✐ ❝✐❛s❝✉♥ s❡tt♦r❡ ❛❧ P■▲ è✿S1 ≈ 33%✱ S2 ≈ 26%✱ S3 ≈ 24% ❡ S4 ≈ 16% ✭◆♦♥ s♦♠♠❛♥♦ ❛❧ ✶✵✵✪ ♣❡r❝❤é❛❜❜✐❛♠♦ ❛♣♣r♦ss✐♠❛t♦✮✳

❊s❡♠♣✐♦ ✶✳✺ ✭❈♦♠❡ ❢✉♥③✐♦♥❛ ❧✬■❱❆✮✳ ▲✬■❱❆ ✲ ✐♠♣♦st❛ s✉❧ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦ ✲ èq✉❡❧❧♦ ❝❤❡ ♣♦❝♦ ❢❛ ❛❜❜✐❛♠♦ ❝❤✐❛♠❛t♦ ✐♠♣♦st❡ ✐♥❞✐r❡tt❡✱ ❡ ❢❛ ♣❛rt❡ ❞❡❧ P■▲✳✶ ▲✬■❱❆❧❛ ♣❛❣❛ ✐❧ ❝♦♠♣r❛t♦r❡✳ ❙❡ ♣♦✐ ✉♥ ❝♦♠♣r❛t♦r❡ A r✐✈❡♥❞❡ ❛ ✉♥ ❛❧tr♦ B✱ q✉❡st✬✉❧t✐♠♦♣❛❣❛ ❛❞ A q✉❡❧❧♦ ❝❤❡ A ❤❛ ♣❛❣❛t♦ ♣✐ù ❧✬✐♠♣♦st❛ s✉❧ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦ ❞✐ A✳ ■❧ s✐❣♥♦rA tr❛tt✐❡♥❡ q✉❡❧❧♦ ❝❤❡ ❛✈❡✈❛ ♣❛❣❛t♦ ❡ ✈❡rs❛ ❧❛ ♣❛rt❡ r✐❣✉❛r❞❛♥t❡ ✐❧ ♣r♦♣r✐♦ ✈❛❧♦r❡❛❣❣✐✉♥t♦ ❛❧❧♦ ❙t❛t♦✳ ❙❡ ✐❧ ♣r♦❞♦tt♦ ❛rr✐✈❛ ❛❧ ❝♦♥s✉♠❛t♦r❡ ✜♥❛❧❡ ❧✉✐ ♣❛❣❛ ❧✬✐♠♣♦st❛s✉❧ ✈❛❧♦r❡ ❞❡❧ ❜❡♥❡✱ ❝✐♦è s✉ t✉tt♦ ✐❧ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦❀ ❡ tr❛♠✐t❡ ✐ ✈❡rs❛♠❡♥t✐ ❞❡✐✈❡♥❞✐t♦r✐ q✉❡❧❧♦ ❝❤❡ ❧✉✐ ♣❛❣❛ ❛rr✐✈❛ ❛❧❧♦ ❙t❛t♦✳ ❱❡❞✐❛♠♦❧♦ ❝♦♥ ✐ ♥✉♠❡r✐✱ ❝♦♥ ■❱❆❛❧ ✶✵✪ ✭❈❇■ ♣❡r ❈♦st✐ ❇❡♥✐ ■♥t❡r♠❡❞✐✮✳

❙❡tt♦r✐ ❘✐❝❛✈✐ ❈❇■ ❱❆ ❊♥tr❛t❡ ■❱❆ ❯s❝✐t❡ ■❱❆ ❱❡rs❛♠❡♥t✐

A ✶✵✵ ✶✵✵ ✶✵ ✶✵B ✶✸✵ ✶✵✵ ✸✵ ✶✸ ✶✵ ✸C ✷✵✵ ✶✸✵ ✼✵ ✷✵ ✶✸ ✼

❙♣❡s❛ ●❡tt✐t♦

❈♦♥s✉♠❛t♦r✐ ✷✵✵ ✷✵❙t❛t♦ ✷✵

◗✉✐ B ❝♦♠♣r❛ ❞❛ A ♣❡r 100e✱ ❡ ❛❧❧❛ ✜♥❡ C ✈❡♥❞❡ ❛✐ ❝♦♥s✉♠❛t♦r✐ ✜♥❛❧✐ ♣❡r 200e✳P❡r ❧✬■❱❆✱ B ♣❛❣❛ 10 ❛❞ A ❝❤❡ ❧✐ ✈❡rs❛ ❛❧❧✬❊r❛r✐♦✱ ✐♥❝❛ss❛ 13 ❞❛ C✱ 10 s❡ ❧✐ t✐❡♥❡✭q✉❡❧❧✐ ❝❤❡ ❛✈❡✈❛ ❛✈❡✈❛ ♣❛❣❛t♦ ❛❞ A✮ ❡ 3 ❧✐ ✈❡rs❛✳ ■♥ ❣❡r❣♦✱ B ✏s❝❛r✐❝❛ 10 ❞✐ ■❱❆s✉ C✑✱ ♥❡❧ s❡♥s♦ ❝❤❡ ✐ 10 ❝❤❡ ❛✈❡✈❛ ♣❛❣❛t♦ ❧✐ r✐♣r❡♥❞❡ ❞❛✐ 13 ❝❤❡ ✐♥❝❛ss❛ ❞❛ C✳◗✉❡st✐ ❛ s✉❛ ✈♦❧t❛ ♣❛❣❛ 13 ❛ B s✉✐ 130 ❝❤❡ ❤❛ ❝♦♠♣r❛t♦✱ ❡ ❞❛✐ 20 ❝❤❡ ✐♥❝❛ss❛❞❛✐ ❝♦♥s✉♠❛t♦r✐ r❡❝✉♣❡r❛ ✐ 13 ❝❤❡ ❛✈❡✈❛ ♣❛❣❛t♦ ❡ ✈❡rs❛ 7✳ ■ ❝♦♥s✉♠❛t♦r✐ ✜♥❛❧✐♣❛❣❛♥♦ ✷✵ ❡ ♥♦♥ ✏s❝❛r✐❝❛♥♦✑ ♥✉❧❧❛✳ ■♥ ❣❡♥❡r❛❧❡ ❝❤✐ r✐✈❡♥❞❡ ♥♦♥ ♣❛❣❛ ■❱❆✱ ❝✐♦è❊♥tr❛t❡ ■❱❆❂❯s❝✐t❡ ■❱❆✰❱❡rs❛♠❡♥t✐ ❛❧❧✬❊r❛r✐♦❀ ❧✬■❱❆ ❧❛ ♣❛❣❛ t✉tt❛ ❧✬✉❧t✐♠♦ ❝❤❡❝♦♠♣r❛ ❡ ♥♦♥ r✐✈❡♥❞❡✳ ▲❛ ❝♦s❛ ❢✉r❜❛ ❞✐ q✉❡st♦ ♠❡❝❝❛♥✐s♠♦ è ❝❤❡ s❡ ♣❡r ❡s❡♠♣✐♦C ❢❛❧❧✐s❝❡✱ ❝✐♦è ♥♦♥ ❝r❡❛ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦✱ ❧♦ ❙t❛t♦ ❤❛ ❣✐à ✐♥❝❛ss❛t♦ ❧❛ ♣❛rt❡ ❞✐✐♠♣♦st❛ s✉❧ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦ ❝r❡❛t♦ ❞❛ A ❡ B ✭13 s✉ 130✮ ♣❡r❝❤é C ❧✐ ❤❛ ♣❛❣❛t✐❛❧❧✬❛tt♦ ❞❡❧❧✬❛❝q✉✐st♦✱ ❡❞ A ❡ B ❧✐ ❤❛♥♥♦ ✈❡rs❛t✐✳✷

✶ ❆❧ ❝♦♥tr❛r✐♦ ❞✐ q✉❡❧❧❡ ❞✐r❡tt❡✱ s✉ r❡❞❞✐t✐ ❞❛ ❧❛✈♦r♦ ✭s❛❧❛r✐✮✱ r❡❞❞✐t✐ ❞❛ ❝❛♣✐t❛❧❡ ❡ ♣r♦✜tt✐✳✷ ◆♦t❛ ❝❤❡ q✉✐ ❧✬■❱❆ è ✐❧ 10% ❞❡❧ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦✳ ❙❛r❡❜❜❡ ❞✉♥q✉❡ ❧✬11.1% s✉❧❧✬✐♠♣♦♥✐❜✐❧❡

✭❝❤❡ è ❝♦♠❡ s✐ ❝❛❧❝♦❧❛ ♥♦r♠❛❧♠❡♥t❡✮✳ P❡r❝❤é ❛❜❜✐❛♠♦ ❞❡tt♦ ✏s✉ 100 ♣❛❣❛ 10 ❞✐ ■❱❆✑✱ q✉✐♥❞✐❧✬✐♠♣♦♥✐❜✐❧❡ ❡r❛ 90 ❡ 10 ❞✐ ■❱❆ s✉ 90 ❢❛ 11.1%✳ ❆❧tr❡tt❛♥t♦ ✈❛rrà ♣❡r ❣❧✐ ❡s❡♠♣✐ ❝❤❡ s❡❣✉♦♥♦ ✲❢❛r❡♠♦ s❡♠♣r❡ ■❱❆ ❝♦♠❡ ♣❡r❝❡♥t✉❛❧❡ ❞❡❧ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦✳

✶✳✷✳ ❈♦♥t❛❜✐❧✐tà ✼

❊s❡♠♣✐♦ ✶✳✻ ✭■❧ P■▲ è ▲♦r❞♦✿ ❣❧✐ ❆♠♠♦rt❛♠❡♥t✐✮✳ ◗✉✐ t❛❜❡❧❧❛ ♥♦♥ ❝❡ ♥✬è✱ ❧❛s✐t✉❛③✐♦♥❡ è s❡♠♣❧✐❝❡✿ ❝♦♥ ❧❛✈♦r♦ ❡ ♠❛❝❝❤✐♥❛r✐ ♣r❡✲❡s✐st❡♥t✐ s✐ s♦♥♦ ♣r♦❞♦tt✐ ❜❡♥✐❛❣r✐❝♦❧✐ ❡ ♠❛❝❝❤✐♥❛r✐ ♥✉♦✈✐ ♣❡r ✉♥ ✈❛❧♦r❡ ❝♦♠♣❧❡ss✐✈♦ ❞✐ ✷✵✵✵✳ ◆♦♥ s♦♥♦ st❛t✐✉s❛t✐ ❜❡♥✐ ♣r♦❞♦tt✐ ♥❡✐ ♣r♦❝❡ss✐ ♣r♦❞✉tt✐✈✐✳ ◗✉✐♥❞✐ ✐❧ P■▲ è ✷✵✵✵✳ ▼❛ ✐ ♠❛❝❝❤✐♥❛r✐✈❡❝❝❤✐ s✐ s♦♥♦ ✐♥ ♣❛rt❡ ❛♠♠♦rt❛t✐ ✲ r❛❞✐❝❡ ❡t✐♠♦❧♦❣✐❝❛ ♠♦rt❡ ✲✱ ❝♦♥ ✉♥❛ ♣❡r❞✐t❛st✐♠❛t❛ ❞❡❧ ❧♦r♦ ✈❛❧♦r❡ ❞✐ ✸✵✵✳ ◆♦♥ ♥❡ ❞♦❜❜✐❛♠♦ t❡♥❡r❡ ❝♦♥t♦❄ ◆❡❧ ♣r♦❞♦tt♦ ❧♦r❞♦

♥♦♥ s❡ ♥❡ t✐❡♥❡ ❝♦♥t♦✳ ▼❛ ✐❧ ♣r♦❞♦tt♦ ♥❡tt♦✱ ❝✐♦è ❛❧ ♥❡tt♦ ❞❡❣❧✐ ❛♠♠♦rt❛♠❡♥t✐✱ è2000− 300 = 1700✳ ❙✐ ❝❤✐❛♠❛ ♣❡r ❧✬❛♣♣✉♥t♦ P■◆✱ ❞♦✈❡ ◆ st❛ ♣❡r ♥❡tt♦✳

❉✐ s♦❧✐t♦ s✐ ✉s❛ ✐❧ P■▲ ♥❡✐ ❝♦♥t✐ ❡❝♦♥♦♠✐❝✐ ♥❛③✐♦♥❛❧✐✱ ♣❡r❝❤é ✐❧ ✈❛❧♦r❡ ❞❡❣❧✐ ❛♠✲♠♦rt❛♠❡♥t✐ è ✉♥ ✈❛❧♦r❡ st✐♠❛t♦✱ ❡ s♣❡ss♦ è ❛♥❝❤❡ ✏♠❛♥✐♣♦❧❛t♦✑ ♥❡✐ ❜✐❧❛♥❝✐✳ P♦✐❝❤é♥❡❧❧❡ ✈❛r✐❡ ❡❝♦♥♦♠✐❡ s✐ ♣✉ò s✉♣♣♦rr❡ ❝❤❡ ❣❧✐ ❛♠♠♦rt❛♠❡♥t✐ s✐❛♥♦ s✐♠✐❧✐✱ ❧✬❡rr♦r❡✐♥ ❝✉✐ s✐ ✐♥❝♦rr❡ è ❧♦ st❡ss♦ ♣❡r t✉tt✐ ❡ ✐ ❞❛t✐ ✐♥t❡r♥❛③✐♦♥❛❧✐ s♦♥♦ ♣❛r❛❣♦♥❛❜✐❧✐✳

■♥❝❧✉❞❡♥❞♦ ❣❧✐ ❛♠♠♦rt❛♠❡♥t✐✱ ❧❛ r❡❧❛③✐♦♥❡ ❞✐ ❞✐str✐❜✉③✐♦♥❡ ❞✐✈❡♥t❛✿

PIL = SommaV A

= Salari+ Profitti+ Interessi+ Imposte Indirette+Ammortamenti.

❖✈✈✐❛♠❡♥t❡ ❣❧✐ ❛♠♠♦rt❛♠❡♥t✐ ♥♦♥ s♦♥♦ ✉♥ r❡❞❞✐t♦✱ ♠❛ s♦♥♦ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ♥❡❧♣r✐♠♦ ♠❡♠❜r♦ q✉✐♥❞✐ ✈❛♥♥♦ ♠❡ss✐ ❛♥❝❤❡ ❛❧ s❡❝♦♥❞♦ ♣❡r ❜✐❧❛♥❝✐❛r❡✳

❊s❡♠♣✐♦ ✶✳✼ ✭❘✐s♦rs❡ ❡ ■♠♣✐❡❣❤✐✮✳ ▲❛ r❡❧❛③✐♦♥❡ q✉✐ s♦♣r❛ r✐s♣♦♥❞❡ ❛❧❧❛ ❞♦♠❛♥❞❛❛ ❝❤✐ ✈❛ q✉❡❧❧♦ ❝❤❡ ✈✐❡♥❡ ♣r♦❞♦tt♦✳ P❡♥s❛♥❞♦ ❛❧ P■▲ ❝♦♠❡ ✈❛❧♦r✐ ❞✐ ❜❡♥✐ ♣r♦❞♦tt✐✱❧✬❛❧tr♦ ❛s♣❡tt♦ ✐♥t❡r❡ss❛♥t❡ è✿ ❝♦s❛ ♥❡ ❢❛❝❝✐❛♠♦ ❞✐ q✉❡st❡ r✐s♦rs❡❄ ❈✐♦è✱ ❝♦♠❡ ❧❡✐♠♣✐❡❣❤✐❛♠♦❄ ❈❤❡ t✐♣♦ ❞✐ ❜❡♥✐ ♣r♦❞✉❝✐❛♠♦❄ ◗✉✐ ♥♦♥ ❝✐ ✐♥t❡r❡ss❛ ✉♥❛ ❝❧❛ss✐✜❝❛✲③✐♦♥❡ t✐♣♦ ✏❜❡♥✐ ❞✐ ♣r✐♠❛ ♥❡❝❡ss✐tà ❡ ❜❡♥✐ ❞✐ ❧✉ss♦✑❀ ❝✐ ♦❝❝✉♣✐❛♠♦ ❞❡❧ P■▲ ♣❡r❝❤és✐❛♠♦ ✐♥t❡r❡ss❛t✐ ❛❧ s✉♦ ❛♥❞❛♠❡♥t♦ ♥❡❧ t❡♠♣♦✱ ✐♥ ♣❛rt✐❝♦❧❛r❡ ❛❧❧❛ s✉❛ ❝r❡s❝✐t❛✳ ❊q✉❡st❛ è ❞❡t❡r♠✐♥❛t❛ ❞❛❧❧❛ ❝r❡s❝✐t❛ ❞❡✐ ❢❛tt♦r✐ ♣r♦❞✉tt✐✈✐✱ ❢♦♥❞❛♠❡♥t❛❧♠❡♥t❡ ❝❛✲♣✐t❛❧❡ ❡ ❧❛✈♦r♦✳ ▼❛ ❧❛ ❢♦r③❛ ❧❛✈♦r♦ ❝r❡s❝❡ ♣❡r ❝❛✉s❡ ❞❡t❡r♠✐♥❛t❡ ✐♥ ❜✉♦♥❛ ♣❛rt❡ ❛❧❞✐ ❢✉♦r✐ ❞❡❧ s✐st❡♠❛ ❡❝♦♥♦♠✐❝♦✱ q✉✐♥❞✐ ❜✐s♦❣♥❛ ❝♦♥❝❡♥tr❛rs✐ s✉❧❧❛ ❛❝❝✉♠✉❧❛③✐♦♥❡❞✐ ❝❛♣✐t❛❧❡✱ ❝❤❡ è ❞❛t❛ ❞❛❧ ✢✉ss♦ ❞✐ ✐♥✈❡st✐♠❡♥t✐ ❝❤❡ s✐ ❢❛♥♥♦ ♥❡❧❧✬❡❝♦♥♦♠✐❛✳ ◗✉✐♥✲❞✐ ❝✐ ✐♥t❡r❡ss❛ s❛♣❡r❡ q✉❛♥t✐ ✐♥✈❡st✐♠❡♥t✐ ❢❛❝❝✐❛♠♦✱ ❡ q✉❛♥t♦ ❞❡❧ ♣r♦❞♦tt♦ ✐♥✈❡❝❡✉t✐❧✐③③✐❛♠♦ ♣❡r s♦❞❞✐s❢❛r❡ ✐ ♥♦str✐ ❜✐s♦❣♥✐ ❞✐ ❝♦♥s✉♠♦ ♣r❡s❡♥t❡✳ ▲❛ r✐s♣♦st❛ ❛❧❧❛❞♦♠❛♥❞❛ ❞✐ s♦♣r❛ è ❞✉♥q✉❡ ❝❤❡ ✐ ❜❡♥✐ ♣r♦❞♦tt✐ ✈❡♥❣♦♥♦ ✐♠♣✐❡❣❛t✐ ❢♦♥❞❛♠❡♥t❛❧✲♠❡♥t❡ ✐♥ ❞✉❡ ♠♦❞✐✿ ❝♦♠❡ ❜❡♥✐ ❞✐ ❝♦♥s✉♠♦ ❡ ❝♦♠❡ ❜❡♥✐ ❞✐ ✐♥✈❡st✐♠❡♥t♦✳✸ ◆♦t❛ ❝❤❡❣❧✐ ✐♥✈❡st✐♠❡♥t✐ s♦♥♦ ✐♥✈❡st✐♠❡♥t✐ ✐♥ ❝❛♣❛❝✐tà ♣r♦❞✉tt✐✈❛✱ q✉✐♥❞✐ ✐♥ ✉❧t✐♠❛ ❛♥❛❧✐s✐✐♥ ❝♦♥s✉♠♦ ❢✉t✉r♦✳ ❊ss✐ ✐♥❝❧✉❞♦♥♦✱ ♦❧tr❡ ❛❧ ❝❛♣✐t❛❧❡ ✜ss♦✱ ❛♥❝❤❡ ❧❛ ✈❛r✐❛③✐♦♥❡ ❞❡❧❧❡s❝♦rt❡ ❞✐ ♠❛t❡r✐❡ ♣r✐♠❡ ❡ ♣r♦❞♦tt✐ ✜♥✐t✐✱ ❝❤❡ è q✉✐♥❞✐ ✈✐st❛ ❝♦♠❡ ✉♥ ✐♠♣✐❡❣♦ ❞❡❧❧❡✐♠♣r❡s❡✳ ◗✉✐♥❞✐ ❣❧✐ ✐♠♣✐❡❣❤✐ ❧✐ ♣♦ss✐❛♠♦ ♣❡♥s❛r❡ ❝♦♠❡ ❝♦♥s✉♠♦ ♣r❡s❡♥t❡ ❡ ❝♦♥✲s✉♠♦ ❢✉t✉r♦✳ ❉❡♥♦t❛♥❞♦ ❝♦♥ C ❡❞ I ✐❧ ✈❛❧♦r❡ ❞✐ ❝♦♥s✉♠✐ ❡ ✐♥✈❡st✐♠❡♥t✐✱ ❛❜❜✐❛♠♦

✸ ◆♦t❛ ❝❤❡ q✉❡st❛ ❝❧❛ss✐✜❝❛③✐♦♥❡ è ♣✉r❛♠❡♥t❡ ❝♦♥t❛❜✐❧❡✳ ▲✬✐❞❡♥t✐tà ❝❤❡ ♥❡ ❞❡r✐✈❛ è q✉✐♥❞✐✉♥✬✐❞❡♥t✐tà ❝♦♥t❛❜✐❧❡✱ ♦✈✈❡r♦ è s❡♠♣r❡ ✈❡r❛ ♣❡r❝❤é ❛❜❜✐❛♠♦ ❞❡✜♥✐t♦ ✐❧ P■▲ ❝♦♠❡ ❡ss❡r❡ ❝♦♠♣♦st♦❞❛ ❜❡♥✐ ❞✐ ❝♦♥s✉♠♦ ❡❞ ✐♥✈❡st✐♠❡♥t♦✳

✽ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

❞✉♥q✉❡PIL = C + I.

▲❡ ❝♦s❡ s♦♥♦ ✉♥ ♠✐♥✐♠♦ ♣✐ù ❝♦♠♣❧✐❝❛t❡ ♣❡r❝❤é ♠❛♥❝❛♥♦ ❙t❛t♦ ❡ r❛♣♣♦rt✐ ❝♦♥❧✬❡st❡r♦✳ ●❧✐ ❛❝q✉✐st✐ ❞✐ ❜❡♥✐ ❡ s❡r✈✐③✐ ❞❛ ♣❛rt❡ ❞❡❧❧❡ ❆♠♠✐♥✐str❛③✐♦♥✐ P✉❜❜❧✐❝❤❡s♦♥♦ ❞✐ s♦❧✐t♦ ❝♦♥t❛❜✐❧✐③③❛t✐ ❛ ♣❛rt❡✱ ❡ s✐ ✐♥❞✐❝❛♥♦ ❝♦♥ ❧❛ ❧❡tt❡r❛ G✳ ➮ ✐♠♣♦rt❛♥t❡r✐❝♦r❞❛r❡ ❝❤❡ s♦♥♦ ✈❛❧✉t❛t✐ ❛❧ ❝♦st♦✿ ♣❡r ❡s❡♠♣✐♦ ✐❧ ❧❛✈♦r♦ ❞✐ ✉♥ ❞♦❝❡♥t❡ ✉♥✐✈❡r✲s✐t❛r✐♦ ✈❛❧❡ ♥❡❧ P■▲ q✉❛♥t♦ ✐❧ s✉♦ st✐♣❡♥❞✐♦✱ s✐❛ ❝❤❡ s✐❛ ❜r❛✈♦ s✐❛ ❝❤❡ ♥♦♥ ❧♦ s✐❛✳■❧ ♠♦t✐✈♦ ❞✐ q✉❡st❛ ❝♦♥✈❡♥③✐♦♥❡ è ❝❤❡ ♣❡r ♠♦❧t❡ ❛tt✐✈✐tà ❞❡❧❧♦ ❙t❛t♦ è ❞✐✣❝✐❧❡q✉❛♥t✐✜❝❛r❡ ❧✬♦✉t♣✉t ✭s✐ ♣❡♥s✐ ❛❧❧❛ ❞✐❢❡s❛ ❞❡❧ t❡rr✐t♦r✐♦✮✳ ■♥♦❧tr❡✱ s♣❡ss♦ ♥♦♥ ❡s✐✲st❡ ✉♥ ♠❡r❝❛t♦✱ ❡ q✉✐♥❞✐ ♥♦♥ ❝✬è ✉♥ ♣r❡③③♦ ❞✐ ✈❡♥❞✐t❛ ❛ ❝✉✐ ✈❛❧✉t❛r❡ ❧❛ q✉❛♥t✐tà♣r♦❞♦tt❛✳ ■ r❛♣♣♦rt✐ ❝♦♥ ❧✬❡st❡r♦ r✐❣✉❛r❞❛♥♦ q✉✐ ■♠♣♦rt❛③✐♦♥✐ ❡❞ ❊s♣♦rt❛③✐♦♥✐✱❞♦✈❡ ❧❡ ✐♠♣♦rt❛③✐♦♥✐ s♦♥♦ ❞✐ ❢❛tt♦ r✐s♦rs❡ ❛ ♥♦str❛ ❞✐s♣♦s✐③✐♦♥❡ ♠❛ ♣r♦❞♦tt❡ ❞❛❛❧tr✐✱ ♠❡♥tr❡ ❧❡ ❊s♣♦rt❛③✐♦♥✐ s♦♥♦ ✉♥ ♣♦ss✐❜✐❧❡ ✐♠♣✐❡❣♦ ❞✐ r✐s♦rs❡ ♣r♦❞♦tt❡ ❞❛ ♥♦✐✳◗✉✐♥❞✐ ❧❛ r❡❧❛③✐♦♥❡ ❣❡♥❡r❛❧❡ ❢r❛ r✐s♦rs❡ ❡ ✐♠♣✐❡❣❤✐ è ❧❛ s❡❣✉❡♥t❡✿

PIL+ Importazioni = C +G+ I + Esportazioni .

◗✉❡st❛ r❡❧❛③✐♦♥❡ ❞à ❧✉♦❣♦ ❛ q✉❡❧❧♦ ❝❤❡ s✐ ❝❤✐❛♠❛ ♣r♦♣r✐♦ ✏❈♦♥t♦ ❞❡❧❧❡ ❘✐s♦rs❡❡ ❞❡❣❧✐ ■♠♣✐❡❣❤✐✑✱ ❢♦♥❞❛♠❡♥t❛❧❡ ✐♥ t✉tt❡ ❧❡ st❛t✐st✐❝❤❡ ❡❝♦♥♦♠✐❝❤❡✳ ◗✉❡❧❧❛ ❞✐ s♦tt♦r✐❣✉❛r❞❛ ❧✬■t❛❧✐❛ ♥❡❧ ✷✵✶✹ ❡❞ è ♣✉❜❜❧✐❝❛t❛ ❞❛❧❧✬■st❛t✱ ✐ ✈❛❧♦r✐ s♦♥♦ q✉❡❧❧✐ ✈❡r✐✳ ■❧P■▲ ❡r❛ ❝✐r❝❛ 1600 ♠✐❧✐❛r❞✐✳ ❋r❛ ❣❧✐ ✐♠♣✐❡❣❤✐✱ C s♦♥♦ ✐ ❝♦♥s✉♠✐ ❞❡❧❧❡ ❢❛♠✐❣❧✐❡ ❡G ✐❧ r❡st♦ ❞❡✐ ❝♦♥s✉♠✐✳ ◆♦t❛ ❝❤❡ ❣❧✐ ✐♥✈❡st✐♠❡♥t✐ s♦♥♦ ❝♦♠♣♦st✐ ❞❛ ✐♥✈❡st✐♠❡♥t✐✜ss✐ ❧♦r❞✐ ♣✐ù ✈❛r✐❛③✐♦♥✐ ❞❡❧❧❡ s❝♦rt❡✱ ♣✐ù ♦❣❣❡tt✐ ❞✐ ✈❛❧♦r❡✳ ❙♦r✈♦❧✐❛♠♦ ♣❡r ❛❞❡ss♦s✉❧❧❛ ❝♦♠♣♦s✐③✐♦♥❡ ❞❡❣❧✐ ✐♥✈❡st✐♠❡♥t✐ ✜ss✐ ✲ ❝✐ t♦r♥❡r❡♠♦ ♥❡❧❧✬❡s❡♠♣✐♦ ✶✳✶✻✳

❈♦♥t♦ ❊❝♦♥♦♠✐❝♦ ❞❡❧❧❡ ❘✐s♦rs❡ ❡ ❞❡❣❧✐ ■♠♣✐❡❣❤✐✱ ❆♥♥♦ ✷✵✶✹ ✭▼✐❧✐♦♥✐ ❞✐ ❊✉r♦✮

❘✐s♦rs❡ ✷ ✵✹✷ ✷✸✽✳✵ ■♠♣✐❡❣❤✐ ✷ ✵✹✷ ✷✸✽✳✶Pr♦❞♦tt♦ ■♥t❡r♥♦ ▲♦r❞♦ ✶ ✻✶✸ ✽✺✾✳✶ ❙♣❡s❛ ♣❡r ❈♦♥s✉♠✐ ✶ ✸✵✶ ✺✼✻✳✵

■♠♣♦rt❛③✐♦♥✐ ✹✷✽ ✸✼✽✳✾ ❋❛♠✐❣❧✐❡ ✾✽✻ ✷✺✼✳✵

❆♠♠✐♥✐str❛③✐♦♥✐ P✉❜❜❧✐❝❤❡ ✸✶✺ ✸✶✾✳✵

■♥✈❡st✐♠❡♥t✐ ✷✻✸ ✹✺✼✳✻

■♥✈❡st✐♠❡♥t✐ ✜ss✐ ❧♦r❞✐ ✷✻✽ ✶✶✼✳✷

■♥✈❡st✐♠❡♥t✐ ✜ss✐ ♥❡tt✐ ✲✷✻ ✽✸✼✳✾

❆♠♠♦rt❛♠❡♥t✐ ✷✾✹ ✾✺✺✳✶

❱❛r✐❛③✐♦♥❡ ❞❡❧❧❡ s❝♦rt❡ ✲✻ ✺✹✽✳✵

❖❣❣❡tt✐ ❞✐ ✈❛❧♦r❡ ✶ ✽✽✽✳✹

❊s♣♦rt❛③✐♦♥✐ ✹✼✼ ✷✵✹✳✺

■♥ ❝♦♥❝❧✉s✐♦♥❡✱ ✐♥❞✐❝❛♥❞♦ ❧❡ ❡s♣♦rt❛③✐♦♥✐ ♥❡tt❡ ❝♦♥

NX ≡ Esportazioni − Importazioni

✶✳✷✳ ❈♦♥t❛❜✐❧✐tà ✾

♣♦ss✐❛♠♦ s❝r✐✈❡r❡ ✐❧ P■▲ ✲ ❝♦♠✐♥❝✐❛♠♦ ❛ ✉s❛r❡ ❧❛ ❧❡tt❡r❛ Y ❝❤❡ s✐ ✉s❛ s❡♠♣r❡ ♣❡r✐❧ P■▲ ✲ ❝♦♠❡

Y = C + I +G+NX.

❘✐❣✉❛r❞✐❛♠♦ ❧✬❡s❡♠♣✐♦ ✶✳✸ ❞✐ s♦♣r❛✳ ❈♦♠❡ è r✐♣❛rt✐t♦ ✐♥ q✉❡❧ ❝❛s♦ ✐❧ P■▲ ❢r❛❝♦♥s✉♠✐ ❡ ✐♥✈❡st✐♠❡♥t✐❄ ▲❡ ❜r✐♦❝❤❡ s♦♥♦ ❝❤✐❛r❛♠❡♥t❡ ❝♦♥s✉♠♦✱ ♠❡♥tr❡ ✐❧ ❣r❛♥♦♥♦♥ ✉t✐❧✐③③❛t♦ ♥❡❧❧❛ ♣r♦❞✉③✐♦♥❡ ❞✐ ❜r✐♦❝❤❡ ✈❡rrà ♣✐❛♥t❛t♦ ♣❡r ♦tt❡♥❡r❡ ❛❧tr♦ ❣r❛♥♦✐♥ ❢✉t✉r♦✱ q✉✐♥❞✐ è ✐♥✈❡st✐♠❡♥t♦✳ ❉✉♥q✉❡ ✐❧ PIL = 13600 è ❞✐✈✐s♦ ✐♥ C = 9600✭✈❛❧♦r❡ ❞❡❧❧❡ ❜r✐♦❝❤❡✮ ❡❞ I = 4000 ✭✈❛❧♦r❡ ❞❡❧ ❣r❛♥♦ ♥♦♥ ✉t✐❧✐③③❛t♦ ❝♦♠❡ ❜❡♥❡✐♥t❡r♠❡❞✐♦ ♥❡❧❧❛ ♣r♦❞✉③✐♦♥❡ ❞❡❧ ❜❡♥❡ ❞✐ ❝♦♥s✉♠♦✮✳

❊s❡♠♣✐♦ ✶✳✽ ✭❘✐s♣❛r♠✐✱ ■♥✈❡st✐♠❡♥t✐ ❡ ❉❡✜❝✐t P✉❜❜❧✐❝♦✮✳ ❈♦♠❡ ❞❡✜♥✐r❡ ❘✐s♣❛r✲

♠✐♦❄ ▲❛ r✐s♣♦st❛ è s❡♠♣❧✐❝❡✿ q✉❡❧❧♦ ❝❤❡ ♥♦♥ s✐ ❝♦♥s✉♠❛✳ ❙✐ ❞❡✜♥✐s❝❡ q✉✐♥❞✐r✐s♣❛r♠✐♦ S ✭S ♣❡r ✏❙❛✈✐♥❣s✑✮ ❧❛ ❞✐✛❡r❡♥③❛ S := Y − C✳✹ ◆❡❧ ❝❛s♦ s❡♠♣❧✐❝❡Y = C + I✱ ❞❛ ❝✉✐ ❞✐r❡tt❛♠❡♥t❡ ❞✐s❝❡♥❞❡ I = S✱ ❧✬✉❣✉❛❣❧✐❛♥③❛ ❢r❛ r✐s♣❛r♠✐ ❡✐♥✈❡st✐♠❡♥t✐✳ ◗✉❡❧❧♦ ❝❤❡ ♥♦♥ è ❝♦♥s✉♠❛t♦ è ♣❡r ❞❡✜♥✐③✐♦♥❡ ✐♥✈❡st✐t♦✱ s❡ ♥♦♥ ✐♥

♠❛❝❝❤✐♥❛r✐ ✐♥ ✈❛r✐❛③✐♦♥❡ ❞❡❧❧❡ s❝♦rt❡✳ ◆❡❧ ❝❛s♦ ❣❡♥❡r❛❧❡ ❛❜❜✐❛♠♦

S := Y − C −G− NX ,

❞❛ ❝✉✐ ❞✐ ♥✉♦✈♦ ❧❛ r❡❧❛③✐♦♥❡ S = I✳ ❆❧tr❛ ❝♦s❛ ✐♥t❡r❡ss❛♥t❡ è ❝❤❡ ✐♥❞✐❝❛♥❞♦ ❝♦♥ T❧❡ ✐♠♣♦st❡ ❝♦♠♣❧❡ss✐✈❡ ✭❞✐r❡tt❡ ❡ ✐♥❞✐r❡tt❡✮✱ ✐❧ r❡❞❞✐t♦ ❡✛❡tt✐✈❛♠❡♥t❡ ❞✐s♣♦♥✐❜✐❧❡❛✐ ❝✐tt❛❞✐♥✐ ♣❡r ❝♦♥s✉♠♦ ♦ r✐s♣❛r♠✐♦ è Y −NX−T ✱ ❞❛ ❝✉✐ ❞❛❧❧❛ r❡❧❛③✐♦♥✐ ❞✐ s♦♣r❛s❡❣✉❡

S = (Y − NX − T − C) + (T −G) = Risparmi Privati + Risparmi Pubblici ,

❞♦✈❡ q✉❡st✐ ✉❧t✐♠✐ s♦♥♦ ❧❛ ❞✐✛❡r❡♥③❛ ❢r❛ ❧❡ ❡♥tr❛t❡ ♣✉❜❜❧✐❝❤❡ T ❡ ❧❡ s♣❡s❡ G✳

■❧ ❞❡✜❝✐t ♣✉❜❜❧✐❝♦ ✐♥ ✉♥ ❝❡rt♦ ❛♥♥♦ è ❧❛ ❞✐✛❡r❡♥③❛ ❢r❛ ✉s❝✐t❡ ❡❞ ❡♥tr❛t❡ ♣✉❜✲❜❧✐❝❤❡✿ G − T ✳ ❘✐❝♦r❞❛♥❞♦ ❝❤❡ ✐❧ t♦t❛❧❡ ❞❡✐ r✐s♣❛r♠✐ è ✉❣✉❛❧❡ ❛❣❧✐ ✐♥✈❡st✐♠❡♥t✐✿(Y − NX − T − C) + (T −G) = I ♦tt❡♥✐❛♠♦

Y − NX − T − C = I + (G− T )

❝❤❡ ❞✐❝❡ ❝❤❡ ✐❧ r✐s♣❛r♠✐♦ ♣r✐✈❛t♦ ✜♥❛♥③✐❛ ✐♥✈❡st✐♠❡♥t✐ ♣r♦❞✉tt✐✈✐ ❡ ❞❡✜❝✐t ♣✉❜❜❧✐❝♦✳

❊s❡♠♣✐♦ ✶✳✾ ✭❱❛r✐❛③✐♦♥✐ ❞❡❧ P■▲ ♥❡❧ t❡♠♣♦✿ P■▲ ♥♦♠✐♥❛❧❡ ❡ P■▲ r❡❛❧❡✮✳ ❈♦♠❡❣✐à ♦ss❡r✈❛t♦✱ ✐❧ P■▲ s✐ ♠✐s✉r❛ ♣❡r ✈❡❞❡r❡ ❝♦♠❡ ✈❛r✐❛ ♥❡❧ t❡♠♣♦✳ ❊ q✉✐ ❝♦♠✐♥❝✐❛♥♦✐ ♣r♦❜❧❡♠✐✱ ♣❡r❝❤é ♥❡❧ t❡♠♣♦ ✈❛r✐❛♥♦ ❧❡ q✉❛♥t✐tà ♠❛ ❛♥❝❤❡ ✐ ♣r❡③③✐✱ q✉✐♥❞✐ s✐ ❞❡✈❡❝❛♣✐r❡ q✉❛♥t♦ ✉♥❛ ❞❛t❛ ✈❛r✐❛③✐♦♥❡ ❞✐ ✈❛❧♦r❡✱ ♣❡r ❡s❡♠♣✐♦ ✐♥ ❛✉♠❡♥t♦✱ r✐✢❡tt❛ ✉♥♣✐ù ❛❧t♦ t❡♥♦r❡ ❞✐ ✈✐t❛✱ ♦♣♣✉r❡ ♠❛❣❣✐♦r✐ ✐♥✈❡st✐♠❡♥t✐ ❡ q✉✐♥❞✐ ✉♥ ♣✐ù ❛❧t♦ t❡♥♦r❡❞✐ ✈✐t❛ ✐♥ ❢✉t✉r♦✱ ❡ q✉❛♥t♦ s✐❛ ✐♥✈❡❝❡ ❧✬❡✛❡tt♦ ❞✐ ✉♥ ❛✉♠❡♥t♦ ❞❡❧ ❧✐✈❡❧❧♦ ❞❡✐ ♣r❡③③✐✳❱❡❞✐❛♠♦ s✉❜✐t♦ ✉♥ ❡s❡♠♣✐♦✱ ❛♣♣r♦❢♦♥❞✐r❡♠♦ ❧✬❛r❣♦♠❡♥t♦ ♥❡❧❧❛ ♣r♦ss✐♠❛ s❡③✐♦♥❡✳✹ ▲❛ ♥♦t❛③✐♦♥❡ S := Y −C ❞❡✜♥✐s❝❡ ✐❧ ♠❡♠❜r♦ ❞✐ s✐♥✐str❛ ✲ ♥✉♦✈♦ ✲ ✐♥ t❡r♠✐♥✐ ❞✐ ✈❛r✐❛❜✐❧✐ ✐♥tr♦✲❞♦tt❡ ♣r❡❝❡❞❡♥t❡♠❡♥t❡ ✭♠❡♠❜r♦ ❞✐ ❞❡str❛✮✳ ❯s❡r❡♠♦ ❛♥❝♦r❛ q✉❡st❛ ♥♦t❛③✐♦♥❡✳ ❆❧t❡r♥❛t✐✈❛♠❡♥t❡s✐ ✉s❛ ✐❧ s✐♠❜♦❧♦ ✏≡✑✳

✶✵ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

❆♥♥♦◗✳tàP✐③③❡

Pr❡③③♦P✐③③❡

◗✳tàP❛♥❡

Pr❡③③♦P❛♥❡

P■▲♥♦♠✐♥❛❧❡

❚❛ss♦❈r❡✲s❝✐t❛

P■▲Pr❡③✲③✐

✷✵✵✺

❚❛ss♦❈r❡✲s❝✐t❛

❉❡✢❛t♦r❡P■▲

✷✵✵✺ ✶✵✵ ✺ ✷✵✵ ✶ ✼✵✵ ✼✵✵ ✶✷✵✵✻ ✶✺✵ ✼ ✶✵✵ ✸ ✶✸✺✵ ✾✸✪ ✽✺✵ ✷✶✪ ✶✳✺✾✷✵✵✼ ✶✷✵ ✶✵ ✷✵✵ ✷ ✶✻✵✵ ✶✽✪ ✽✵✵ ✲✻✪ ✷

■ t❛ss✐ ❞✐ ❝r❡s❝✐t❛ s♦♥♦ ❞❡✜♥✐t✐ ❝♦♠❡ ✐♥❝r❡♠❡♥t✐ r❡❧❛t✐✈✐✳ P❡r q✉❛❧✉♥q✉❡ ✈❛r✐❛❜✐❧❡x✱ ✐❧ t❛ss♦ ❞✐ ❝r❡s❝✐t❛ ❢r❛ t ❡ t+ 1 è ❞❡✜♥✐t♦ ❝♦♠❡

xt+1 − xtxt

.

P❡r ❡s❡♠♣✐♦✱ ♥❡❧❧✬❡❝♦♥♦♠✐❛ ❞❡❧❧❛ t❛❜❡❧❧❛ ♣r❡❝❡❞❡♥t❡ ❧❛ ❝r❡s❝✐t❛ ❞❡❧ P■▲ ♥♦♠✐♥❛❧❡

✭❝✐♦è ❝❛❧❝♦❧❛t♦ ❛ ♣r❡③③✐ ❝♦rr❡♥t✐✮ ❢r❛ ✐❧ ✷✵✵✺ ❡ ✐❧ ✷✵✵✻ è (1350−700)/700 = 0.928 =92.8%✳ ■❧ ♣r♦❜❧❡♠❛ è ❝❤❡ ♥❡❧ ❢r❛tt❡♠♣♦ s♦♥♦ ❛♥❝❤❡ ❛✉♠❡♥t❛t✐ ✐ ♣r❡③③✐✳ P❡r❛♣♣r❡③③❛r❡ ❧❡ ✈❛r✐❛③✐♦♥✐ ♥❡❧❧❛ ♣r♦❞✉③✐♦♥❡ ✏r❡❛❧❡✑ ✐s♦❧❛♥❞♦ ❧✬❡✛❡tt♦ ❞❡❧❧❛ ✈❛r✐❛③✐♦♥❡❞❡✐ ♣r❡③③✐ s✐ ❝❛❧❝♦❧❛ ✐❧ P■▲ ♥❡✐ ✈❛r✐ ❛♥♥✐ ❛ ♣r❡③③✐ ❝♦st❛♥t✐✱ ❝❤❡ s✐ ❝❤✐❛♠❛ P■▲ r❡❛❧❡✳✺

❱❛❧✉t❛♥❞♦ ❧❛ ♣r♦❞✉③✐♦♥❡ ❛ ♣r❡③③✐ ❝♦st❛♥t✐ ❞❡❧❧✬❛♥♥♦ ✷✵✵✺ s✐ ✈❡❞❡ ❝❤❡ ✐ t❛ss✐ ❞✐❝r❡s❝✐t❛ ❝❛♠❜✐❛♥♦ ✐♥ ♠♦❞♦ r✐❧❡✈❛♥t❡✳

P❡r s❝♦♠♣♦rr❡ ❧❛ ✈❛r✐❛③✐♦♥❡ t♦t❛❧❡ ✐♥ ✈❛r✐❛③✐♦♥❡ ❞❡❧ P■▲ r❡❛❧❡ ❡ ✈❛r✐❛③✐♦♥❡ ❞❡✐♣r❡③③✐ s✐ ❞❡✜♥✐s❝❡ ✉♥ ✐♥❞✐❝❡ Pt ❞❡❧ ❧✐✈❡❧❧♦ ❞❡✐ ♣r❡③③✐ ✲ ❞❡tt♦ ❞❡✢❛t♦r❡ ❞❡❧ P■▲ ✲❞✐✈✐❞❡♥❞♦ P■▲ ♥♦♠✐♥❛❧❡ PILt ♣❡r ✐❧ P■▲ r❡❛❧❡ PILb

t ✱ ❞♦✈❡ ✐❧ s♦♣r❛s❝r✐tt♦ ✐♥❞✐❝❛ ❝❤❡q✉❡st✬✉❧t✐♠♦ è ❝❛❧❝♦❧❛t♦ ❛ ♣r❡③③✐ ❞✐ ✉♥ ❛♥♥♦ ❜❛s❡ b✿

Pt :=PILt

PILbt

.

■❧ r❛♣♣♦rt♦ r✐✢❡tt❡ ❧❛ ✈❛r✐❛③✐♦♥❡ ❞❡✐ ♣r❡③③✐ ♣❡r❝❤é ❧❡ q✉❛♥t✐tà s♦♥♦ ✉❣✉❛❧✐ s✐❛❛❧ ♥✉♠❡r❛t♦r❡ ❝❤❡ ❛❧ ❞❡♥♦♠✐♥❛t♦r❡✳ ■❧ ❞❡✢❛t♦r❡ ♥❡❧ ♥♦str♦ ❡s❡♠♣✐♦ è r✐♣♦rt❛t♦♥❡❧❧✬✉❧t✐♠❛ ❝♦❧♦♥♥❛ ❞❡❧❧❛ t❛❜❡❧❧❛✳ ▲❛ s❝♦♠♣♦s✐③✐♦♥❡ ❛ q✉❡st♦ ♣✉♥t♦ è s❡♠♣❧✐❝❡✱♣❡r❝❤é ❞❛ PILt = Pt ·PIL

bt ✲ ❝❤❡ r✐❝♦r❞❛ ❧❛ r❡❧❛③✐♦♥❡ ✏✈❛❧♦r❡❂♣r❡③③♦ ♣❡r q✉❛♥t✐tà✑

✲ s✐ ♦tt✐❡♥❡ s✉❜✐t♦✱ ♣❡r ❞✉❡ ♣❡r✐♦❞✐ ❞✐✈❡rs✐ s ❡ t✱PILt

PILs=

Pt

Ps

PILbt

PILbs

✐♥ ❝✉✐ ✐❧ ♣r✐♠♦ ❢❛tt♦r❡ r✐✢❡tt❡ ❧❛ ✈❛r✐❛③✐♦♥❡ ❞❡❧ ❧✐✈❡❧❧♦ ❞❡✐ ♣r❡③③✐✱ ✐❧ s❡❝♦♥❞♦ ❧❛✈❛r✐❛③✐♦♥❡ ❞❡❧ P■▲ r❡❛❧❡✳ ◆❡❧ ♥♦str♦ ❡s❡♠♣✐♦✱ ❢r❛ ✐❧ ✷✵✵✻ ❡ ✐❧ ✷✵✵✼ ❛❜❜✐❛♠♦

1.18 =1600

1350=

2

1.59·800

850= 1.26 · 0.94.

◗✉✐♥❞✐ ✐❧ P■▲ ♥♦♠✐♥❛❧❡ è ❝r❡s❝✐✉t♦ ❞❡❧ ✶✽✪ ❡ q✉❡st❛ ✈❛r✐❛③✐♦♥❡ è ❞♦✈✉t❛ ❛❞ ✉♥❛✉♠❡♥t♦ ❞❡❧ ✷✻✪ ❞❡❧ ❧✐✈❡❧❧♦ ❞❡✐ ♣r❡③③✐ ❡❞ ✉♥❛ ❞✐♠✐♥✉③✐♦♥❡ ❞❡❧ ✻✪ ❞❡❧ P■▲ r❡❛❧❡✳

✺ ❈❛❧❝♦❧❛r❡ ✐❧ P■▲ ❛ ♣r❡③③✐ ❝♦st❛♥t✐ s✐❣♥✐✜❝❛ ✈❛❧✉t❛r❡ ❧❛ ♣r♦❞✉③✐♦♥❡ ❛❧ t❡♠♣♦ t, t+1, t+2 ❡t❝✳❛✐ ♣r❡③③✐ ❞✐ ✉♥ ❛♥♥♦ ❛r❜✐tr❛r✐❛♠❡♥t❡ s❝❡❧t♦✱ ❝❤❡ s✐ ❞❡✜♥✐s❝❡ ❛♥♥♦ ❜❛s❡✳ ◆❡❧❧✬❡s❡♠♣✐♦ ❧✬❛♥♥♦ ❜❛s❡è ✐❧ ✷✵✵✺✳

✶✳✸✳ Pr❡③③✐✱ q✉❛♥t✐tà✱ t❡♠♣♦ ❡ t❛ss✐ ✶✶

❊s❡♠♣✐♦ ✶✳✾ ✭❈♦♥t✐♥✉❛✿ ❈❛✈❡❛t✮✳ ▲❛ ❡str❛③✐♦♥✐ ❞❡❧❧❡ ✈❛r✐❛❜✐❧✐ ✏r❡❛❧✐✑ r✐❝❤✐❡❞❡s❡♠♣r❡ s❝❡❧t❡ ✐♥ q✉❛❧❝❤❡ ♠✐s✉r❛ ❛r❜✐tr❛r✐❡✳ ◆❡❧ ❝❛s♦ ❞❡❧ P■▲ r❡❛❧❡ ❧❛ s❝❡❧t❛ ❛r❜✐✲tr❛r✐❛ r✐❣✉❛r❞❛ ❧✬❛♥♥♦ ❜❛s❡✱ ❝❤❡ ♣✉ò ❝♦♥❞✐③✐♦♥❛r❡ ❛♥❝❤❡ ♣❡s❛♥t❡♠❡♥t❡ ✐ r✐s✉❧t❛t✐✳❱❡❞✐❛♠♦❧♦ ✐♥ ✉♥ ❡s❡♠♣✐♦ ❡str❡♠♦ ❝❛♠❜✐❛♥❞♦ ✐ ♥✉♠❡r✐ ♥❡❧❧❛ t❛❜❡❧❧❛ ❞✐ s♦♣r❛✳

❆♥♥♦◗✳tàP✐③③❡

Pr❡③③♦P✐③③❡

◗✳tàP❛♥❡

Pr❡③③♦P❛♥❡

P■▲❚❛ss♦❈r❡✲s❝✐t❛

P■▲Pr❡③✲③✐

✷✵✵✺

❚❛ss♦❈r❡✲s❝✐t❛

P■▲Pr❡③✲③✐

✷✵✵✻

❚❛ss♦❈r❡✲s❝✐t❛

✷✵✵✺ ✷ ✶ ✶ ✷ ✹ ✹ ✺✷✵✵✻ ✶ ✷ ✷ ✶ ✹ ✵ ✺ ✷✺✪ ✹ ✲✷✵✪

❘✐❝❛♣✐t♦❧❛♥❞♦

❆❜❜✐❛♠♦ ❝❛❧❝♦❧❛t♦ ✐❧ P■▲ ❝♦♠❡ s♦♠♠❛ ❞❡✐ ✈❛❧♦r✐ ❞❡✐ ❜❡♥✐ ♣r♦❞♦tt✐ ❛❧ ♥❡tt♦ ❞❡❣❧✐✉t✐❧✐③③✐✱ ❡ ❝♦♠❡ s♦♠♠❛ ❞❡✐ ✈❛❧♦r✐ ❛❣❣✐✉♥t✐✳ ▲♦ ❛❜❜✐❛♠♦ s❝♦♠♣♦st♦ ❝♦♠❡ s♦♠♠❛❞❡✐ r❡❞❞✐t✐ ❞❡✐ ❢❛tt♦r✐ ♣r♦❞✉tt✐✈✐ ✭❝♦♥ ♣r♦✜tt♦ ❝♦♠❡ r❡s✐❞✉♦✮✳ ❊ ❛❜❜✐❛♠♦ ❝❧❛ss✐✜✲❝❛t♦ ❣❧✐ ✐♠♣✐❡❣❤✐ ❞❡❧❧❡ r✐s♦rs❡ ♣r♦❞♦tt❡ ♥❡❧❧❡ ❞✉❡ ❝❛t❡❣♦r✐❡ ❣❡♥❡r❛❧✐ ❞✐ ❝♦♥s✉♠♦ ❡✐♥✈❡st✐♠❡♥t♦ ✭❝❤❡ s❡r✈❡ ❛❧ ❝♦♥s✉♠♦ ❢✉t✉r♦✮✳ ◆❡❣❧✐ ✉❧t✐♠✐ ❡s❡♠♣✐ ❛❜❜✐❛♠♦ ✐♥tr♦✲❞♦tt♦ t❡♠♣♦✱ ♣r❡③③✐ ❡ q✉❛♥t✐tà✱ ❡ ❞✐ q✉❡st♦ ❝♦♥t✐♥✉✐❛♠♦ ❛ ♣❛r❧❛r❡ ♥❡❧❧❛ s❡③✐♦♥❡❝❤❡ s❡❣✉❡✳

✶✳✸ Pr❡③③✐✱ q✉❛♥t✐tà✱ t❡♠♣♦ ❡ t❛ss✐

❈♦♠✐♥❝✐❛♠♦ ❝♦❧ r✐s❝r✐✈❡r❡ ✐❧ P■▲ ✐♥ ❢♦r♠❛ ♣✐ù ❝♦♠♣❛tt❛✳ ❈♦♠❡ ❛❜❜✐❛♠♦ ✈✐st♦P■▲ è ✈❛❧♦r❡✱ ❝✐♦è ♣r❡③③✐ ♣❡r q✉❛♥t✐tà✱ ❞❡❧ ✈❡tt♦r❡ ❞❡✐ ❜❡♥✐ ✜♥❛❧✐✳ ❈✐ s✐❛♥♦ n ❜❡♥✐i = 1, 2, . . . n✱ ❡❞ m ✐♠♣r❡s❡ j = 1, 2, . . .m✳ ❙✐❛♥♦ Xi(j), X

inti (j) ❧❡ q✉❛♥t✐tà

❞❡❧ ❜❡♥❡ i r✐s♣❡tt✐✈❛♠❡♥t❡ ♣r♦❞♦tt♦ ❡ ✉s❛t♦ ❝♦♠❡ ❜❡♥❡ ✐♥t❡r♠❡❞✐♦ ❞❛❧❧✬✐♠♣r❡s❛ j✳❉✉♥q✉❡ ❧❡ q✉❛♥t✐tà t♦t❛❧✐ ❞❡❧ ❜❡♥❡ i r✐s♣❡tt✐✈❛♠❡♥t❡ ♣r♦❞♦tt❡ ❡ ✉s❛t❡ ♥❡✐ ♣r♦❝❡ss✐♣r♦❞✉tt✐✈✐ s♦♥♦

Xi =∑

jXi(j), X int

i =∑

jX int

i (j).

❙❡ p = (p1, p2, . . . pn) è ✐❧ ✈❡tt♦r❡ ❞❡✐ ♣r❡③③✐ ❛❜❜✐❛♠♦ ❞✉♥q✉❡ ✭❧❛ ♣r✐♠❛ r❡❧❛③✐♦♥❡♣❡r ❞❡✜♥✐③✐♦♥❡✱ ❧❛ s❡❝♦♥❞❛ s♦st✐t✉❡♥❞♦ ❞❛ s♦♣r❛ ❡ ✐♥✈❡rt❡♥❞♦ ❧❡ s♦♠♠❛t♦r✐❡✱ ❧❛t❡r③❛ ❞✐ ♥✉♦✈♦ ♣❡r ❞❡✜♥✐③✐♦♥❡✮

PIL =∑

ipiXi −

∑

ipiX

inti

=∑

j

[

∑

ipiXi(j)−

∑

ipiX

inti (j)

]

=∑

jVAj

✶✷ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

❞♦✈❡ VAj è ✐❧ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦ ❞❡❧❧✬✐♠♣r❡s❛ j✳ ❆❞❡ss♦ s❡♠♣❧✐✜❝❤✐❛♠♦ ✉♥ ♣♦✬ ❧❛

♥♦t❛③✐♦♥❡✳ ❉❡♥♦t✐❛♠♦ ❝♦♥ X = (X1, . . . , Xn) ✐❧ ✈❡tt♦r❡ ❞❡✐ ❜❡♥✐ ✜♥❛❧✐ ♣r♦❞♦tt✐♥❡❧❧✬❡❝♦♥♦♠✐❛ ✭❣✐à ❛❧ ♥❡tt♦ ❞❡❧❧❡ q✉❛♥t✐tà ✉s❛t❡ ♥❡✐ ♣r♦❝❡ss✐✮✱ ❝♦sì PIL =

∑

i piXi✳❊ ✉s✐❛♠♦ ❧❛ ♥♦t❛③✐♦♥❡ ♠❛tr✐❝✐❛❧❡ ♣❡r q✉❡st♦ t✐♣♦ ❞✐ s♦♠♠❡✿ pX :=

∑

i piXi✳ ❆q✉❡st♦ ♣✉♥t♦ s❡ Xt è ✐❧ ✈❡tt♦r❡ ❞❡✐ ❜❡♥✐ ✜♥❛❧✐ ♣r♦❞♦tt✐ ❛ t ❡ pt è ✐❧ ❝♦rr✐s♣♦♥❞❡♥t❡✈❡tt♦r❡ ❞❡✐ ♣r❡③③✐ ♣♦ss✐❛♠♦ s❝r✐✈❡r❡ s❡♠♣❧✐❝❡♠❡♥t❡

PILt = ptXt.

✶✳✸✳✶ ■♥❞✐❝✐ ❞✐ ♣r❡③③♦ ❡ ✐♥✢❛③✐♦♥❡

■❧ ❞❡✢❛t♦r❡ ❞❡❧ P■▲ ❝❤❡ ❛❜❜✐❛♠♦ ✈✐st♦ ♥♦♥ è ❛❧tr♦ ❝❤❡

Pt =ptXt

pbXt≡ PPA

t .

◆✉♠❡r❛t♦r❡ ❡ ❞❡♥♦♠✐♥❛t♦r❡ s♦♥♦ ❞✐ ❢❛tt♦ ♠❡❞✐❡ ❞❡✐ ♣r❡③③✐ ♣♦♥❞❡r❛t❡ ❝♦♥ ♣❡s✐Xit/

∑

iXit✳ ▲♦ ✐♥❞✐❝❤✐❛♠♦ ❝♦♥ PPAt ♣❡r❝❤é ❧✬❤❛ ✐♥✈❡♥t❛t♦ ✐❧ s✐❣♥♦r P❛❛s❝❤❡✳✻

■❧ ♣r♦❜❧❡♠❛ ❞✐ q✉❡st♦ ✐♥❞✐❝❡ è ❝❤❡ q✉❛♥❞♦ ❝♦str✉✐❛♠♦ ❧❛ s❡q✉❡♥③❛ ❞❡❣❧✐ ✐♥❞✐❝✐❝♦sì ♦tt❡♥✉t✐ ♥❡❧ t❡♠♣♦ ✐ ♣❡s✐ ✈❛♥♥♦ ❝❛♠❜✐❛♥❞♦✱ ♣❡r❝❤é s♦♥♦ q✉❛♥t✐tà ❛❧ t❡♠♣♦ t✳❯♥ ✐♥❞✐❝❡ ♣✐ù ✏♣✉r♦✑ ❞❡❧❧❛ ✈❛r✐❛③✐♦♥❡ ❞❡✐ ♣r❡③③✐ ✭q✉❡❧❧♦ ❝❤❡ ✉s❛ ✐❧ ❋♦♥❞♦ ▼♦♥❡t❛r✐♦■♥t❡r♥❛③✐♦♥❛❧❡✮ s✐ ♦tt✐❡♥❡ ❧❛s❝✐❛♥❞♦ ✜ss✐ ✐ ♣❡s✐✱ ✉s❛♥❞♦ ♣❡r ❡s❡♠♣✐♦ ❧❡ q✉❛♥t✐tà♣r♦❞♦tt❡ ❛❧ t❡♠♣♦ ❜❛s❡ b✱ ❝✐♦è Xb✱ ✐♥ ♠♦❞♦ ❝❤❡ ❝❛❧❝♦❧❛♥❞♦ ❧✬✐♥❞✐❝❡ ♥❡❧ t❡♠♣♦✈❛r✐❛♥♦ s♦❧t❛♥t♦ ✐ ♣r❡③③✐✳ ❙✐ ♦tt✐❡♥❡ ❝♦sì ❧✬✐♥❞✐❝❡ ❞✐ ♣r❡③③♦ ❞✐ ▲❛s♣❡②r❡s ✭✐❧ ♥♦♠❡❞❡❧ s✉♦ ✐♥✈❡♥t♦r❡✮✱ ❝❤❡ s✐ ❝❤✐❛♠❛ ❛♥❝❤❡ ■♥❞✐❝❡ ❞❡✐ Pr❡③③✐ ❛❧ ❈♦♥s✉♠♦✿

PLAt =

ptXb

pbXb.

❈♦♠❡ ✈❡❞r❡♠♦ è q✉❡st♦ ❧✬✐♥❞✐❝❡ ❝❤❡ ❡♠❡r❣❡ ✐♥ ♠♦❞♦ ♥❛t✉r❛❧❡ q✉❛♥❞♦ ✈♦❣❧✐❛♠♦♣❛r❛❣♦♥❛r❡ ✐❧ ♣♦t❡r❡ ❞✬❛❝q✉✐st♦ ✐♥ ♣❡r✐♦❞✐ ❞✐✈❡rs✐ ✭✈❡❞✐ ❡s❡♠♣✐♦ ✶✳✶✶✮ ♦ q✉❛♥❞♦❞♦❜❜✐❛♠♦ ❞❡✜♥✐r❡ ✐❧ t❛ss♦ ❞✐ ✐♥t❡r❡ss❡ r❡❛❧❡✳

◆♦t❛ ✶✳ ✼◆é ❧✬✐♥❞✐❝❡ ❞✐ ▲❛s♣❡②r❡s ♥é q✉❡❧❧♦ ❞✐ P❛❛s❝❤❡ s♦♥♦ ❛❞ ✬✉t✐❧✐tà ❝♦st❛♥t❡✬✱❝✐♦è ♥♦♥ t❡♥❣♦♥♦ ❝♦♥t♦ ❞❡❧❧✬❡✛❡tt♦ s♦st✐t✉③✐♦♥❡✳ P❡r q✉❡❧❧♦ ❞✐ ▲❛s♣❡②r❡s ♣❡r ❡s❡♠✲♣✐♦ s✐ ♦ss❡r✈❛ ❝❤❡ ptXb è ✐❧ r❡❞❞✐t♦ ♥❡❝❡ss❛r✐♦ ❛ ❝♦♥s✉♠❛r❡ Xb ❛❧ t❡♠♣♦ t✱ ♠❛ ❝♦♥q✉❡❧ r❡❞❞✐t♦ ❛✐ ♣r❡③③✐ pt ✐ ❝♦♥s✉♠❛t♦r✐ s❝❡❣❧✐❡r❡❜❜❡r♦ ✉♥ ❛❧tr♦ ♣❛♥✐❡r❡ ❝♦❧❧♦❝❛♥❞♦s✐s✉ ✉♥❛ ❝✉r✈❛ ❞✐ ✐♥❞✐✛❡r❡♥③❛ ♣✐ù ❛❧t❛✳ P❡r ♠❛♥t❡♥❡r❧✐ s✉❧❧❛ ❝✉r✈❛ ❞✐ ✐♥❞✐✛❡r❡♥③❛ ❞✐

✻ ➮ ✉♥ ❡s❡♠♣✐♦ ❞✐ ♥✉♠❡r♦ ✐♥❞✐❝❡✳ ❯♥ ✐♥❞✐❝❡ è ✐❧ r❛♣♣♦rt♦ tr❛ ✐❧ ✈❛❧♦r❡ ❞✐ ✉♥❛ ❞❡t❡r♠✐♥❛t❛✈❛r✐❛❜✐❧❡ ❛❞ ✉♥ ❝❡rt♦ t❡♠♣♦ ❡❞ ✐❧ ✈❛❧♦r❡ ❞❡❧❧❛ st❡ss❛ ✈❛r✐❛❜✐❧❡ ❛❞ ✉♥ t❡♠♣♦ ❞✐ r✐❢❡r✐♠❡♥t♦✳ ◗✉✐♥❞✐✱s❡ ❧✬✐♥❞✐❝❡ ❤❛ ✈❛❧♦r❡ ♠❛❣❣✐♦r❡ ❞✐ ✶ ❧❛ ✈❛r✐❛❜✐❧❡ ❝❤❡ st✐❛♠♦ ♠✐s✉r❛♥❞♦ è ❝r❡s❝✐✉t❛ r✐s♣❡tt♦ ❛❧ ♣❡r✐♦❞♦❞✐ r✐❢❡r✐♠❡♥t♦✳

✼ ■♥ q✉❡st♦ ❡ ♥❡✐ ❝❛♣✐t♦❧✐ ❝❤❡ s❡❣✉♦♥♦ ❧❡ ✏◆♦t❡✑ s♦♥♦ ❞✐ ❢❛tt♦ ✏❛♣♣r♦❢♦♥❞✐♠❡♥t✐✑✳ ▲❡✏❖ss❡r✈❛③✐♦♥✐✑ ✐♥✈❡❝❡ s❛r❛♥♥♦ ♣❛rt✐ ❡ss❡♥③✐❛❧✐✳

✶✳✸✳ Pr❡③③✐✱ q✉❛♥t✐tà✱ t❡♠♣♦ ❡ t❛ss✐ ✶✸

Xb ❜❛st❡r❡❜❜❡ ✉♥ r❡❞❞✐t♦ ✐♥❢❡r✐♦r❡ ❛❧ ♥✉♠❡r❛t♦r❡ ❞❡❧❧✬✐♥❞✐❝❡ ❞✐ ▲❛s♣❡②r❡s ✭r✐❝♦r❞❛✐❧ ❞✐s❝♦rs♦ s✉❧❧❛ ❝♦♠♣❡♥s❛③✐♦♥❡ ♠♦♥❡t❛r✐❛ q✉❛♥❞♦ ❤❛✐ st✉❞✐❛t♦ ❣❧✐ ❡✛❡tt✐ r❡❞❞✐t♦ ❡s♦st✐t✉③✐♦♥❡✮✳ ❆❧❧♦ st❡ss♦ ♠♦❞♦ s✐ ♣✉ò ♦ss❡r✈❛r❡ ❝❤❡ ♥❡❧❧✬✐♥❞✐❝❡ ❞✐ P❛❛s❝❤❡ è ✐❧ ❞❡✲♥♦♠✐♥❛t♦r❡ ❛❞ ❡ss❡r❡ ♣✐ù ❛❧t♦ ❞✐ q✉❛♥t♦ ❜❛st❡r❡❜❜❡ ♣❡r ♠❛♥t❡♥❡r❡ ✐ ❝♦♥s✉♠❛t♦r✐s✉❧❧❛ ❝✉r✈❛ ❞✐ ✐♥❞✐✛❡r❡♥③❛ ❞✐ Xt✳ ❱❡❞✐ ❡s❡r❝✐③✐♦ ✶✳✾✳

◆❡❧❧❛ r❡❛❧tà ❞❡❧❧❛ ❝♦♥t❛❜✐❧✐tà ♥❛③✐♦♥❛❧❡✱ q✉❛♥❞♦ s✐ ❝♦♥s✐❞❡r❛♥♦ ✐♥❞✐❝✐ ❞✐ ♣r❡③③✐s✉ ♣❡r✐♦❞✐ ❧✉♥❣❤✐ ✐ ♣❡s✐ s✐ ❢❛♥♥♦ ✈❛r✐❛r❡ ❝♦♠✉♥q✉❡✱ ♦tt❡♥❡♥❞♦ ✐♥❞✐❝✐ ✏❝♦♥❝❛t❡♥❛t✐✑✳❉❡tt❛❣❧✐ ♥❡❧❧✬❡s❡♠♣✐♦ ✶✳✶✹✳ �

◆♦t❛ ✷✳ ■♥❞✐❝✐ ❛♥❛❧♦❣❤✐ s✐ ♣♦ss♦♥♦ ❝♦str✉✐r❡ ♣❡r ♠✐s✉r❛r❡ ✈❛r✐❛③✐♦♥✐ ❞✐ q✉❛♥t✐tà✱♣❡s❛♥❞♦❧❡ ❝♦♥ ♣r❡③③✐✳ P❡r ❡s❡♠♣✐♦ ❧✬✐♥❞✐❝❡ ❞✐ q✉❛♥t✐tà ❞✐ ▲❛s♣❡②r❡s QLA

t t✐❡♥❡ ✐♣r❡③③✐ ✜ss✐ ❛ pb✿

QLAt =

pbXt

pbXb.

❖ss❡r✈❛ ❝❤❡ ✐❧ ❞❡✢❛t♦r❡ ❞❡❧ P■▲ PPAt è ❧❛ ✈❛r✐❛③✐♦♥❡ ❞❡❧ P■▲ ♥♦♠✐♥❛❧❡ ✐♥ r❛♣♣♦rt♦

❛❧❧❛ ✈❛r✐❛③✐♦♥❡ ✭▲❛s♣❡②r❡s✮ ❞❡❧❧❡ q✉❛♥t✐tà✿

PPAt =

ptXt/pbXb

QLAt

❝✐♦è ✈❛r✐❛③✐♦♥❡ ❞❡❧ P■▲ ✏❞❡♣✉r❛t❛✑ ❞❛❧❧❛ ✈❛r✐❛③✐♦♥❡ ❞❡❧❧❡ q✉❛♥t✐tà✳ ■♥ q✉❡st♦ s❡♥s♦è ✉♥ ✐♥❞✐❝❡ ❞❡✐ ♣r❡③③✐ ✏❝❛❧✐❜r❛t♦✑ s✉❧ P■▲✳ �

◗✉❛❧✉♥q✉❡ s✐❛ ❧✬✐♥❞✐❝❡ Pt ✉t✐❧✐③③❛t♦ ♣❡r r❛♣♣r❡s❡♥t❛r❡ ✐❧ ❧✐✈❡❧❧♦ ❞❡✐ ♣r❡③③✐✱ ✐❧t❛ss♦ ❞✐ ✐♥✢❛③✐♦♥❡ πt ❞❛ t ❛ t+ 1 è ❞❡✜♥✐t♦ ❝♦♠❡ ✐❧ t❛ss♦ ❞✐ ❝r❡s❝✐t❛ ❞❡❧❧✬✐♥❞✐❝❡✿

πt =Pt+1 − Pt

Pt.

◆♦t❛ ❝❤❡ 1 + πt = Pt+1/Pt✳ ❖✈✈✐❛♠❡♥t❡ ❧❛ ❞❡✜♥✐③✐♦♥❡ ❞✐♣❡♥❞❡ ❞❛ q✉❛❧❡ ✐♥❞✐❝❡❞❡❧ ❧✐✈❡❧❧♦ ❞❡✐ ♣r❡③③✐ s✐ ✉s❛✳ P❡r ❡s❡♠♣✐♦ s❡ s✐ ✉s❛ ▲❛s♣❡②r❡s✱ ❝♦♥ X = Xb✱ r✐s✉❧t❛1 + πt = pt+1X/ptX ✭❣✐✉st♦❄✮✳

❆♥❛❧♦❣❛♠❡♥t❡ s✐ ❞❡✜♥✐s❝❡ ✐❧ t❛ss♦ ❞✐ ✐♥✢❛③✐♦♥❡ ❢r❛ ❞✉❡ ♣❡r✐♦❞✐ s, t✿ ❧✬✐♥✢❛③✐♦♥❡πt,t+s ❞❛ t ❛ t+ s✱ è ❛❧❧♦r❛ ❞❡✜♥✐t❛ ❞❛

1 + πt,t+s =Pt+s

Pt.

◆♦t❛ ❝❤❡ πt = πt,t+1✱ ❧✬✐♥✢❛③✐♦♥❡ ❢r❛ t ❡ t + 1✳ ▲❛ r❡❧❛③✐♦♥❡ ❢r❛ πt,t+s ❡ ✐ t❛ss✐ ❞✐✐♥✢❛③✐♦♥❡ ❢r❛ t ❡ t+ s è ❧❛ s❡❣✉❡♥t❡✱ ❝♦♠❡ è ❢❛❝✐❧❡ ✈❡r✐✜❝❛r❡✿

1 + πt,t+s = (1 + πt) · (1 + πt+1) · . . . · (1 + πt+s−1).

✶✹ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

✶✳✸✳✷ ❙❝❛♠❜✐ ❡ ❜❡♥✐ ❛ ❞❛t❡ ❞✐✈❡rs❡

P❡r ❛♥❞❛r❡ ❛✈❛♥t✐ ❝♦♥✈✐❡♥❡ ✉s❛r❡ ✉♥❛ ❛❜❜r❡✈✐❛③✐♦♥❡ ❝♦♠♦❞❛✳ ❙❡ a è ✉♥ ❜❡♥❡ ♦ ✉♥♣❛♥✐❡r❡ ❛❜❜r❡✈✐❡r❡♠♦ ✏x ✉♥✐tà ❞✐ a✑ ❝♦♥ x

∣

∣ a❀ ❡ ♣❡r ❞✐r❡ ❝❤❡ q✉❡st❡ s✐ s❝❛♠❜✐❛♥♦❝♦♥ y ✉♥✐tà ❞✐ b ✭❝✐♦è ❝♦♥ y

∣

∣ b✮ s❝r✐✈❡r❡♠♦

x∣

∣ a ≃ y∣

∣ b.

❆ss✉♠❡r❡♠♦ ❝❤❡ q✉❡st❛ r❡❧❛③✐♦♥❡ s✐❛ ❧✐♥❡❛r❡✱ ♥❡❧ s❡♥s♦ ❝❤❡ x∣

∣ a ≃ y∣

∣ b s❡ ❡ s♦❧♦s❡ kx

∣

∣ a ≃ ky∣

∣ b ♣❡r ♦❣♥✐ k > 0✳ ▲✬✐♥t❡r♣r❡t❛③✐♦♥❡ è ♦✈✈✐❛✳ ❉❛❧❧❛ ❧✐♥❡❛r✐tà s❡❣✉❡♣❡r ❡s❡♠♣✐♦ ❝❤❡ s❡ ✐❧ ❜❡♥❡ c ❤❛ ♣r❡③③♦ p ✐♥ ❊✉r♦ ✭✉♥❛ ✉♥✐tà ❞✐ c s✐ s❝❛♠❜✐❛ ❝♦♥p ✉♥✐tà ❞✐ ❊✉r♦✮✱ ❝✐♦è 1

∣

∣ c ≃ p∣

∣e✱ ❛❧❧♦r❛ 3∣

∣ c ≃ 3p∣

∣e✱ ❝✐♦è ✸ ✉♥✐tà ❞✐ c s✐s❝❛♠❜✐❡r❛♥♥♦ ❝♦♥ 3p ✉♥✐tà ❞✐ ❊✉r♦❀ ❡ ❝♦sì ♣❡r ♦❣♥✐ ♠✉❧t✐♣❧♦ ♣♦s✐t✐✈♦✳ ❖ss❡r✈❛❛♥❝❤❡ ❝❤❡ ❞❛❧❧♦ st❡ss♦ ❛r❣♦♠❡♥t♦ s❡❣✉❡ 1

∣

∣e ≃ 1/p∣

∣ c ✭s❡ ✉♥❛ ✉♥✐tà ❝♦st❛ ✷ ❊✉r♦✱❝♦♥ ✶ ❊✉r♦ ♥❡ ✈✐❡♥❡ ♠❡③③❛ ✉♥✐tà✮✳

➮ ✐♠♣♦rt❛♥t❡ r❡❛❧✐③③❛r❡ ❝❤❡ ✉♥ ❜❡♥❡ è ❞❡✜♥✐t♦✱ ♦❧tr❡ ❝❤❡ ❞❛❧❧❡ s✉❡ s♣❡❝✐✜❝❤❡✜s✐❝❤❡✱ ❞❛❧❧✬✐st❛♥t❡ ✐♥ ❝✉✐ è ❞✐s♣♦♥✐❜✐❧❡✿ ✉♥ ❊✉r♦ ♦❣❣✐ è ✉♥ ❜❡♥❡ ❞✐✈❡rs♦ ❞❛ ✉♥❊✉r♦ ❢r❛ ✶✵ ❛♥♥✐✳ ❙❡ ❝✐ ♣❡♥s✐ ✉♥ ❛tt✐♠♦ q✉❡st♦ è ❞❡❧ t✉tt♦ ♦✈✈✐♦✿ ❛❝❝❡tt❡r❡st✐ ❞✐❞❛r ✈✐❛ ✉♥ ❊✉r♦ ♦❣❣✐ ✐♥ ❝❛♠❜✐♦ ❞✐ ✉♥ ❊✉r♦ ❢r❛ ❞✐❡❝✐ ❛♥♥✐❄ ▲♦ st❡ss♦ ✈❛❧❡ ♣❡r ✉♥❛♠❡❧❛✳ P❛r❧❛♥❞♦ ❞✐ t❛ss✐ ❞✐ ✐♥t❡r❡ss❡ q✉❡st❛ ♣r❡❝✐s❛③✐♦♥❡ s❛rà ✐♥❞✐s♣❡♥s❛❜✐❧❡✳ P❡r❡s♣❧✐❝✐t❛r❧❛ ♠❡tt❡r❡♠♦ ❝♦♠❡ ❛♣✐❝❡ ✐❧ t❡♠♣♦ ✐♥ ❝✉✐ ✐❧ ❜❡♥❡ è ❞✐s♣♦♥✐❜✐❧❡✳ ◗✉✐♥❞✐ ♣❡r❡s❡♠♣✐♦ ♣❛r❧❡r❡♠♦ ❞✐ et,et+1 ❡ ❝♦sì ✈✐❛❀ ❡ ❧♦ st❡ss♦ ♣❡r ✉♥ ♣❛♥✐❡r❡ X✱ ♣❛r❧❡r❡♠♦❞✐ Xt, Xt+1 ❡❝❝❡t❡r❛✳

▼❡tt❡♥❞♦ ❧❡ ❞❛t❡✱ s❡ pt è ✐❧ ♣r❡③③♦ ❞✐ c ❛❧ t❡♠♣♦ t ✲ ❝✐♦è ✐❧ ♣r❡③③♦ ❞✐ ct ✲ s❛rà♣❡r ❡s❡♠♣✐♦

1∣

∣et+1 ≃

1

pt+1

∣

∣ ct+1, α∣

∣et+1 ≃

α

pt+1

∣

∣ ct+1. ✭✶✳✶✮

❊s❡♠♣✐♦ ✶✳✶✵✳ ❋✐ss❛ ✉♥ ♣❛♥✐❡r❡ X = Xb✳ ▲✬✐♥❞✐❝❡ ❞✐ ▲❛s♣❡②r❡s è ❛❧❧❛ ❜❛s❡❞❡❧❧❡ ❝♦♥tr❛tt❛③✐♦♥✐ s✉❧❧❡ ✐♥❞✐❝✐③③❛③✐♦♥✐ ❞❡❧❧❡ r❡tr✐❜✉③✐♦♥✐✿ ✏X ♠✐ ❝♦st❛✈❛ pbX✱♦r❛ ♠✐ ❝♦st❛ ptX✱ ♠✐ ❞❡✈✐ ❞❛r❡ ❧❛ ❞✐✛❡r❡♥③❛✑ ✲ ✏◆♦ t✐ ❞❡✈♦ ❞❛r❡ ❞✐ ♠❡♥♦ ♣❡r❝❤é❛ ♣r❡③③✐ pt t✉ ❝♦♥s✉♠✐ ✉♥ ♣❛♥✐❡r❡ ❞✐✈❡rs♦ ❞❛ X✑✳ Pr❡❝✐s❛♠❡♥t❡✿ PLA

t è ✉❣✉❛❧❡

❛❧ ♥✉♠❡r♦ ❞✐ ❊✉r♦ ♥❡❝❡ss❛r✐ ❛ ❝♦♠♣r❛r❡ ❛❧ t❡♠♣♦ t ❧❛ st❡ss❛ ❢r❛③✐♦♥❡ ❞✐ X ❝❤❡

s✐ ❝♦♠♣r❛✈❛ ❛❧ t❡♠♣♦ b ❝♦♥ ✶ ❊✉r♦✳ ❆❜❜✐❛♠♦ ✐♥ ❡✛❡tt✐ 1∣

∣Xb ≃ pbX∣

∣eb ❞❛

❝✉✐ ♣❡r ❧✐♥❡❛r✐tà (1/pbX)∣

∣Xb ≃ 1∣

∣eb❀ ❞✬❛❧tr❛ ♣❛rt❡ 1

∣

∣Xt ≃ ptX∣

∣et ❞❛ ❝✉✐

(1/pbX)∣

∣Xt ≃ (ptX/pbX)∣

∣et = PLA

t

∣

∣et✳

❊s❡♠♣✐♦ ✶✳✶✶ ✭❈♦♥✈❡rs✐♦♥❡ ❞✐ s♦♠♠❡ ✐♥ ♣❡r✐♦❞✐ ❞✐✈❡rs✐✱ ✐♠♣♦rt❛♥t❡✮✳ ◗✉❛♥t♦✈❛rr❡❜❜❡ ♦❣❣✐ ✐♥ t❡r♠✐♥✐ ❞✐ ♣♦t❡r❡ ❞✬❛❝q✉✐st♦ ✉♥ st✐♣❡♥❞✐♦ ❞✐ 10000 ❊✉r♦ ❧✬❛♥♥♦❞✐ ✈❡♥t✬❛♥♥✐ ❢❛❄ ■♥t✉✐t✐✈❛♠❡♥t❡✱ ❧❛ r✐s♣♦st❛ è ✏❞❡✈✐ t❡♥❡r ❝♦♥t♦ ❞❡❧❧✬✐♥✢❛③✐♦♥❡✑✳■♥ ❣❡♥❡r❛❧❡ ❧❛ ❞♦♠❛♥❞❛ è✿ s✐ ❤❛ ✉♥❛ ❝❡rt❛ q✉❛♥t✐tà k ❞✐ ❊✉r♦ ❛❧ t❡♠♣♦ t✱ ♣❡r

✶✳✸✳ Pr❡③③✐✱ q✉❛♥t✐tà✱ t❡♠♣♦ ❡ t❛ss✐ ✶✺

❡s❡♠♣✐♦ ✉♥♦ st✐♣❡♥❞✐♦❀ q✉❛♥t♦ ✈❛rr❡❜❜❡ ✐♥ t❡r♠✐♥✐ r❡❛❧✐ q✉❡st❛ s♦♠♠❛ ❛ t + s❄■♥ t❡r♠✐♥✐ ♣✐ù ♣r❡❝✐s✐ ❧❛ ❞♦♠❛♥❞❛ ♣✉ò ❡ss❡r❡ ❢♦r♠✉❧❛t❛ ❝♦♠❡ s❡❣✉❡✿ ✜ss❛ ✉♥♣❛♥✐❡r❡ X = Xb❀ ❝♦♥ k

∣

∣et s✐ ❝♦♠♣r❛✈❛ ✉♥❛ ❢r❛③✐♦♥❡ α ❞✐ X✱ ❝✐♦è✱ ♥❡❧❧❛ ♥♦t❛③✐♦♥❡

❛♣♣❡♥❛ ✐♥tr♦❞♦tt❛✱ k∣

∣et ≃ α

∣

∣Xt❀ q✉❛♥t♦ ❝✐ ✈♦rr❡❜❜❡ ♦r❛ ♣❡r r✐❝♦♠♣r❛r❧❛✱ ❝✐♦è✱♣❡r q✉❛❧❡ k′ ✈❛❧❡ k′

∣

∣et+s ≃ α

∣

∣Xt+s❄ ▲❛ r✐s♣♦st❛ è ♣r♦♥t❛ ♥❡❧❧❛ ❞❡✜♥✐③✐♦♥❡❞✐ PLA

t ✱ ♣❡r ❧✐♥❡❛r✐tà✳ ■♥❢❛tt✐ ❞❛❧❧✬❊s❡♠♣✐♦ ✶✳✶✵ s❡❣✉❡ ❝❤❡ PLAt

∣

∣et ❡ PLA

t+s

∣

∣et+s

❝♦♠♣r❛♥♦ ❧❛ st❡ss❛ ❢r❛③✐♦♥❡ α ❞✐X✱ q✉❡❧❧❛ ❝❤❡ s✐ ❝♦♠♣r❛✈❛ ❛❧ t❡♠♣♦ b ❝♦♥ ✉♥ ❊✉r♦✳P❡r ❧✐♥❡❛r✐tà s❛r❛♥♥♦ q✉✐♥❞✐ ❡q✉✐✈❛❧❡♥t✐ ❛♥❝❤❡ 1

∣

∣et ❡ PLA

t+s/PLAt

∣

∣et+s✱ ❡ q✉✐♥❞✐

❛♥❝❤❡ k∣

∣et ❡ kPLA

t+s/PLAt

∣

∣et+s✳ ❖✈✈✐❛♠❡♥t❡ è ♦♣♣♦rt✉♥♦ s❝❡❣❧✐❡r❡ Pt = PLA

t ✱♥❡❧ q✉❛❧ ❝❛s♦ ✐❧ r✐s✉❧t❛t♦ ❞✐❝❡ ❝❤❡ k

∣

∣et s♦♥♦ ❡q✉✐✈❛❧❡♥t✐ ❛ k(1 + πt,t+s)

∣

∣et+s✿ ✐❧

♠♦❧t✐♣❧✐❝❛t♦r❡ è ❧✬❡✛❡tt♦ ✐♥✢❛③✐♦♥❡ ❢r❛ t ❡ t+ s✳

✶✳✸✳✸ ❚❛ss✐ ❞✐ ✐♥t❡r❡ss❡

■❧ t❛ss♦ ❞✐ ✐♥t❡r❡ss❡ ✏♥♦♠✐♥❛❧❡✑ è ✐❧ ♣r❡③③♦ ❞❡❧❧❛ ♠♦♥❡t❛ ♦❣❣✐ ✐♥ t❡r♠✐♥✐ ❞✐ ♠♦♥❡t❛❞♦♠❛♥✐✿ s❡ ✉♥ ❊✉r♦ ♦❣❣✐ è s❝❛♠❜✐❛❜✐❧❡ ❝♦♥ 1.02 ❊✉r♦ ❞♦♠❛♥✐ ✐❧ t❛ss♦ ❞✐ ✐♥t❡r❡ss❡♥♦♠✐♥❛❧❡ è 0.02 = 2%✳ P❡r ✐❧ t❛ss♦ ✏r❡❛❧❡✑ ✈❛❧❡ ❧♦ st❡ss♦ ❞✐s❝♦rs♦ q✉❛♥❞♦ ♣❡rò s✐s❝❛♠❜✐❛♥♦ ❜❡♥✐ r❡❛❧✐✳

■❧ t❛ss♦ ❞✐ ✐♥t❡r❡ss❡ ♥♦♠✐♥❛❧❡✳

■❧ t❛ss♦ ❞✐ ✐♥t❡r❡ss❡ ♥♦♠✐♥❛❧❡ it ❞❛ t ❛ t+ 1 è ❞❡✜♥✐t♦ ❞❛❧❧❛ r❡❧❛③✐♦♥❡

1∣

∣et ≃ 1 + it

∣

∣et+1

❝❤❡ ❞✐❝❡ ❝❤❡ ✉♥ ❊✉r♦ ❛ t è s❝❛♠❜✐❛❜✐❧❡ ❝♦♥ 1 + it ❊✉r♦ ❛ t+ 1✳◆♦t❛✱ ❞❛❧❧❛ ❧✐♥❡❛r✐tà✱ ❛♥❝❤❡ ❧♦ ✏s❝♦♥t♦✑✿

1∣

∣et+1 ≃

1

1 + it

∣

∣et

❝❤❡ ❞✐❝❡ ❝❤❡ ✐♥ ❝❛♠❜✐♦ ✉♥ ❊✉r♦ ❞♦♠❛♥✐ ♣✉♦✐ ❛✈❡r❡ s♦❧♦ 1/(1 + it) ❊✉r♦ ♦❣❣✐✳

❊s❡♠♣✐♦ ✶✳✶✷ ✭■❧ Pr❡③③♦ ❞❡✐ ❇❖❚✮✳ ❯♥ ❇✉♦♥♦ ❖r❞✐♥❛r✐♦ ❞❡❧ ❚❡s♦r♦ ✭❇❖❚✮ è ✉♥♣❡③③♦ ❞✐ ❝❛rt❛ ❡♠❡ss♦ ❞❛❧❧♦ ❙t❛t♦ ✐t❛❧✐❛♥♦ ❛❧ t❡♠♣♦ t s✉ ❝✉✐ st❛ s❝r✐tt♦ ✏❆❧ t❡♠♣♦

t+ 1 t✐ ❞❛rò ✉♥ ❊✉r♦✑✳ ◗✉❛♥t♦ ✈❛❧❡❄ ❙❡ PBt è ✐❧ s✉♦ ♣r❡③③♦ ✭✐♥ ❊✉r♦ ❛ t✮✱ st✐❛♠♦

❞✐❝❡♥❞♦PBt

∣

∣et ≃ 1

∣

∣et+1,

❝❤❡ ♣❡r ❧✐♥❡❛r✐tà è ❡q✉✐✈❛❧❡♥t❡ ❛ 1∣

∣et ≃ 1/PB

t

∣

∣et+1✳ ❉❛❧❧❛ ❞❡✜♥✐③✐♦♥❡ ❞✐ it

s❡❣✉❡ ❛❧❧♦r❛✿

PBt =

1

1 + it.

◆♦t❛ ❜❡♥❡ ❝❤❡ ♣r❡③③♦ ❛❧t♦ ❞❡❧ ❇❖❚ ✈✉♦❧ ❞✐r❡ ✐♥t❡r❡ss❡ ❜❛ss♦ ✭❡ ✈✐❝❡✈❡rs❛✮✳

✶✻ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

■❧ t❛ss♦ r❡❛❧❡✳

P❡r ✐❧ t❛ss♦ r❡❛❧❡ rt✱ ❝♦♠✐♥❝✐❛♠♦ ❝♦❧ ❞❡✜♥✐r❧♦ ♣❡r ✉♥ s✐♥❣♦❧♦ ❜❡♥❡ c✳ ❈❤✐❛♠✐❛♠♦rct ✐❧ t❛ss♦ ❞✐ ✐♥t❡r❡ss❡ r❡❛❧❡ ✐♥ t❡r♠✐♥✐ ❞❡❧ ❜❡♥❡ c✳ ▲❛ ❞❡✜♥✐③✐♦♥❡ è ❛♥❛❧♦❣❛ ❛❧❧❛♣r❡❝❡❞❡♥t❡✿ ✶ ✉♥✐tà ❞✐ c ❛ t è s❝❛♠❜✐❛❜✐❧❡ ❝♦♥ 1 + rct ✉♥✐tà ❞✐ c ❛ t + 1✳ ❈✐♦è✿1∣

∣ ct ≃ 1 + rct∣

∣ ct+1✳ ◗✉❛❧ è ❧❛ r❡❧❛③✐♦♥❡ ❢r❛ rct ❡❞ it❄ ▲✬✐❞❡❛ è ❝❤❡ ✐❧ t❛ss♦ r❡❛❧❡è ✐❧ t❛ss♦ ♥♦♠✐♥❛❧❡ ✏❞❡♣✉r❛t♦✑ ❞❡❧❧✬✐♥✢❛③✐♦♥❡✳ P❡r ✈❡❞❡r❧♦ ♣r❡❝✐s❛♠❡♥t❡✿ ✶ ✉♥✐tà❞✐ c ❛ t ♣✉ò ❡ss❡r❡ s❝❛♠❜✐❛t❛ ❝♦♥ pt ❊✉r♦ ❛ t❀ ✐♥♦❧tr❡ pt ❊✉r♦ ❛ t ♣♦ss♦♥♦ ❡ss❡r❡s❝❛♠❜✐❛t✐ ❝♦♥ pt(1 + it) ❊✉r♦ ❛ t + 1✱ ❡❞ ✐♥✜♥❡ pt(1 + it) ❊✉r♦ ❛❧ t❡♠♣♦ t + 1♣♦ss♦♥♦ ❡ss❡r❡ s❝❛♠❜✐❛t✐ ❝♦♥ pt(1 + it)/pt+1 ✉♥✐tà ❞✐ c ❛ t + 1✳ ■♥ ❛❧tr✐ t❡r♠✐♥✐✭❣✉❛r❞❛ ❣❧✐ ❡s❡♠♣✐ ✐♥ ✭✶✳✶✮✮✿

1∣

∣ ct ≃ pt∣

∣et ≃ pt(1 + it)

∣

∣et+1 ≃

pt(1 + it)

pt+1

∣

∣ ct+1 =1 + itpt+1/pt

∣

∣ ct+1.

❉✉♥q✉❡ ✐❧ t❛ss♦ r❡❛❧❡ è 1 + rct = (1 + it)/(pt+1/pt)✱ ❝✐♦è ✐❧ t❛ss♦ ♥♦♠✐♥❛❧❡ 1 + it❞✐✈✐s♦ ❧✬✐♥✢❛③✐♦♥❡ s✉❧ ♣r❡③③♦ ❞✐ c✱ ❝❤❡ ♣♦ss✐❛♠♦ ❝❤✐❛♠❛r❡ πc

t ✿

1 + rct =1 + it1 + πc

t

.

■❧ t❛ss♦ r❡❛❧❡ rt ❞❡✜♥✐t♦ r✐s♣❡tt♦ ❛ ✉♥ ♣❛♥✐❡r❡ ✜ss♦ X ✭♣❡r ❡s❡♠♣✐♦ ✐❧ ✈❡tt♦r❡ ❞❡✐❜❡♥✐ ♣r♦❞♦tt✐ ♥❡❧❧✬❛♥♥♦ ❜❛s❡ b✮✱ è ❞❛t♦ ❞❛❧❧❛ r❡❧❛③✐♦♥❡

1∣

∣Xt ≃ 1 + rt∣

∣Xt+1.

❊ ❧❛ r❡❧❛③✐♦♥❡ ❢r❛ it ❡❞ rt è ♣r❛t✐❝❛♠❡♥t❡ ❣✐à ❢❛tt❛✿

1∣

∣Xt ≃ ptX∣

∣et ≃ (1 + it)ptX

∣

∣et+1 =

1 + itpt+1X/ptX

pt+1X∣

∣et+1

≃1 + it

pt+1X/ptX

∣

∣Xt+1.

◆♦t❛ ❝❤❡ q✉✐ p r❛♣♣r❡s❡♥t❛ ✐❧ ✈❡tt♦r❡ ❞✐ ♣r❡③③✐ ❞❡✐ ❜❡♥✐ ♣r❡s❡♥t✐ ♥❡❧ ♣❛♥✐❡r❡X. ❉❛❧❧✬✉❣✉❛❣❧✐❛♥③❛ pt+1X/ptX = PLA

t+1/PLAt = 1 + πt✱ ♦tt❡♥✐❛♠♦ ❧❛ r❡❧❛③✐♦♥❡

❣❡♥❡r❛❧❡

1 + rt =1 + it1 + πt

.

P❡r ❡s❡♠♣✐♦ s❡ i = 200% ❡ π = 100% s❛rà r = 50% ✭✈❡r✐✜❝❛ ❝♦♥ ❧❛ ❢♦r♠✉❧❛ ❡✐♠♠❛❣✐♥❛♥❞♦ ❧♦ s❝❛♠❜✐♦ ❞✐ c ✐♥ ♣r❛t✐❝❛✮✳

P❡r ♥✉♠❡r✐ ♣✐❝❝♦❧✐ ✈❛❧❡ ❧❛ s❡♠♣❧✐❝❡ ❛♣♣r♦ss✐♠❛③✐♦♥❡ rt = it−πt ✭❝❤❡ ♥♦♥ ❢✉♥③✐♦✲♥❛ ❝♦♥ ♥✉♠❡r✐ ❣r❛♥❞✐✱ ✈❡❞✐ ❧✬❡s❡♠♣✐♦ ❛♣♣❡♥❛ ❢❛tt♦ ❞♦✈❡ i−r = 100% 6= 50%✮✱ ❝❤❡s✐ ♦tt✐❡♥❡ ♣r❡♥❞❡♥❞♦ ✐ ❧♦❣❛r✐t♠✐ ❞✐ ❡♥tr❛♠❜✐ ✐ ♠❡♠❜r✐ ❡ ✉s❛♥❞♦ ❧✬❛♣♣r♦ss✐♠❛③✐♦♥❡ln(1 + x) ≃ x✳

✶✳✸✳ Pr❡③③✐✱ q✉❛♥t✐tà✱ t❡♠♣♦ ❡ t❛ss✐ ✶✼

❊s❡♠♣✐♦ ✶✳✶✸ ✭❱❛❧✉t❛③✐♦♥❡ ❞❡❣❧✐ ✐♥✈❡st✐♠❡♥t✐✮✳ ❯♥ ♣r♦❣❡tt♦ ❞✐ ✐♥✈❡st✐♠❡♥t♦❝♦♠♣♦rt❛ ❞❡✐ ❝♦st✐ ♣r❡s❡♥t✐ C ❛ t ❡ ✉♥ ✢✉ss♦ ❞✐ r✐❝❛✈✐ ❢✉t✉r✐ ❢r✉tt♦ ❞✐ ♣r♦❞✉③✐♦♥❡yt+1, yt+2, . . . yt+T ✱ ❡ ✐❧ ♣r♦❜❧❡♠❛ è ✈❛❧✉t❛r❡ q✉❡st♦ ✢✉ss♦ ♣❡r ✈❡❞❡r❡ s❡ ❝♦♣r❡ ✐❧❝♦st♦✳ Pr❡♥❞✐❛♠♦ ✐❧ ❝❛s♦ s❡♠♣❧✐❝❡ ❝❤❡ s✐ ♣r♦❞✉❝❛ s♦❧♦ ❛ t + 1 ✭❝✐♦è T = 1✮✳Pr❡♥❞❡♥❞♦ ❛ ♣r❡st✐t♦ C ♦❣❣✐ ❞❡✈♦ r✐❞❛r❡ (1 + it)C

∣

∣et+1✱ q✉❛♥❞♦ ❞✬❛❧tr❛ ♣❛rt❡

✐♥❝❛ss❡rò pt+1yt+1

∣

∣et+1❀ q✉✐♥❞✐ ❧✬✐♥✈❡st✐♠❡♥t♦ è ❞❛ r❡❛❧✐③③❛r❡ s❡ (1 + it)C ≤

pt+1yt+1✱ ❝✐♦è s❡ C ≤ pt+1yt+1/(1 + it)✳ P✐ù ❛❧t♦ è ✐❧ ❞❡♥♦♠✐♥❛t♦r❡ 1 + it = (1 +rt)(1+πt) ♣✐ù ❜❛ss♦ ✐❧ r❡♥❞✐♠❡♥t♦✱ q✉✐♥❞✐ s❡♠❜r❛♥♦ ❡ss❡r❝✐ ❞✉❡ ❡❧❡♠❡♥t✐ ✏❝♦♥tr♦✑✱rt ❡ πt✳ ▼❛ ❧✬✐♥✢❛③✐♦♥❡ è ❛♥❝❤❡ ✏❛ ❢❛✈♦r❡✑ ♣❡r❝❤é s✐ ✈❡♥❞❡ ❛ ♣r❡③③✐ pt+1✳ P❡r ✈❡❞❡r❧♦♠❡❣❧✐♦ ❛ss✉♠✐❛♠♦ ❝❤❡ y ❛✉♠❡♥t✐ ❞✐ ♣r❡③③♦ ♣✐ù ♦ ♠❡♥♦ ❝♦♠❡ ✐❧ ♣❛♥✐❡r❡ ✜ss♦ X ❝♦♥❝✉✐ s✐ ❝❛❧❝♦❧❛ ✐❧ ❧✐✈❡❧❧♦ ❞❡✐ ♣r❡③③✐✿ pt+1yt+1/ptyt+1 ≈ pt+1X/ptX = 1 + πt✳ ❆❧❧♦r❛❧❛ r❡❧❛③✐♦♥❡ ❞✐ ♣r✐♠❛ ❞✐✈❡♥t❛

C ≤ptyt+1(1 + πt)

(1 + rt)(1 + πt)= pt

yt+1

1 + rt,

❝❤❡ ❞✐❝❡ ❝❤❡ ✐♥ q✉❡st♦ ❝❛s♦ ❧✬✐♥✈❡st✐♠❡♥t♦ s✐ ♣✉ò ✈❛❧✉t❛r❡ ✉s❛♥❞♦ ♣r❡③③✐ ❝♦st❛♥t✐♣❡r ✐❧ ❜❡♥❡ ❡ s❝♦♥t❛♥❞♦ ❧❡ q✉❛♥t✐tà ❢✉t✉r❡ ❛❧ t❛ss♦ r❡❛❧❡✳ ❉❛t♦ ptyt+1✱ ♣✐ù ❜❛ss♦è rt ♣✐ù ❧✬✐♥✈❡st✐♠❡♥t♦ ❞✐✈❡♥t❛ ❝♦♥✈❡♥✐❡♥t❡✳ ◗✉❡st♦ è ✐♠♣♦rt❛♥t❡✳ ▲♦ r✐✈❡❞r❡♠♦♣✐ù ❛✈❛♥t✐ ♥❡✐ ♠♦❞❡❧❧✐ ❞✐ ♠❛❝r♦❡❝♦♥♦♠✐❛ ✐♥ ❝✉✐ s✐ ❛ss✉♠❡rà ❝❤❡ ❣❧✐ ✐♥✈❡st✐♠❡♥t✐❞✐♣❡♥❞♦♥♦ ❞❛❧ t❛ss♦ r❡❛❧❡✳

■♥t❡r❡ss❡ ❡ s❝♦♥t♦ ❢r❛ t ❡ t+ s ✳

❆❜❜✐❛♠♦ ❣✐à ❛✈✉t♦ ♠♦❞♦ ❞✐ ♥♦t❛r❡ ❝❤❡ 1/(1+ it) è ✐❧ ♣r❡③③♦ ❞✐ et+1 ✐♥ t❡r♠✐♥✐ ❞✐et✳ ❙✐ ❝❤✐❛♠❛ ❝♦♥✈❡♥③✐♦♥❛❧♠❡♥t❡ ✏s❝♦♥t♦✑✱ ❞❛ t+ 1 ❛ t✳ ❆♥❛❧♦❣❛♠❡♥t❡ ❧♦ s❝♦♥t♦

✏r❡❛❧❡✑ è ❞❛t♦ ❞❛ 1/(1 + rt)✳ ■❧ ❞✐s❝♦rs♦ ❞✐ ✐♥t❡r❡ss❡ ❡ s❝♦♥t♦ s✐ ❡st❡♥❞❡ ❢❛❝✐❧♠❡♥t❡❛ ♣✐ù ♣❡r✐♦❞✐✳ ❉❡✜♥✐❛♠♦ ✐♥t❡r❡ss❡ ♥♦♠✐♥❛❧❡ ❡ r❡❛❧❡ ❞❛ t ❛ t+ s ❝♦♥ ❧❡ r❡❧❛③✐♦♥✐

1∣

∣et ≃ 1 + it,t+s

∣

∣et+s , 1

∣

∣Xt ≃ 1 + rt,t+s

∣

∣Xt+s,

❞♦✈❡ ♣❡r ✐❧ ❝❛s♦ r❡❛❧❡ ❛❜❜✐❛♠♦ ❛♥❝♦r❛ ✜ss❛t♦ X = Xb ✭✈❡tt♦r❡ P■▲ ❛❧ t❡♠♣♦ b✮✳❉✉♥q✉❡ 1 + it,t+s ❬r✐s♣✳ 1 + rt,t+s❪ è ✐❧ ♣r❡③③♦ ❞✐ et ❬r✐s♣✳ Xt❪ ✐♥ t❡r♠✐♥✐ ❞✐ et+s

❬r✐s♣✳ Xt+s❪✳ ❆❜❜✐❛♠♦ ♣❡r ❡s❡♠♣✐♦

1∣

∣et ≃ 1 + it

∣

∣et+1 ≃ (1 + it) · (1 + it+1)

∣

∣et+2,

❝✐♦è 1+ it,t+2 = (1+ it) ·(1+ it+1)✳ ◆♦t❛ ❝❤❡ it = it,t+1 ❡ ❛♥❛❧♦❣❛♠❡♥t❡ rt = rt,t+1✳■♥ ♠♦❞♦ ❛♥❛❧♦❣♦ ✭s❡ ✈✉♦✐ ❡ss❡r❡ ♣r❡❝✐s♦✱ ♣❡r ✐♥❞✉③✐♦♥❡✮ s✐ tr♦✈❛♥♦ ❢❛❝✐❧♠❡♥t❡ ❧❡r❡❧❛③✐♦♥✐ s❡❣✉❡♥t✐✿

1 + it,t+s = (1 + it) · (1 + it+1) · . . . · (1 + it+s−1)

1 + rt,t+s = (1 + rt) · (1 + rt+1) · . . . · (1 + rt+s−1).

✶✽ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

❊ ❞❛ q✉❡st❡ s❡❣✉❡ ✐♠♠❡❞✐❛t❛♠❡♥t❡ ❝❤❡

1 + rt,t+s =1 + it,t+s

1 + πt,t+s.

P❡r ❧♦ s❝♦♥t♦ ❞❛ t+ s ❛ t s✐ ❞✐✈✐❞❡ ♣❡r 1 + it,t+s ♦ ♣❡r 1 + rt,t+s ❛ s❡❝♦♥❞❛ ❝❤❡ s✐✈♦❣❧✐❛ ✉s❛r❡ ✐♥t❡r❡ss❡ ♥♦♠✐♥❛❧❡ ♦ r❡❛❧❡✳ ▲❛ ❞✐✛❡r❡♥③❛ è ✐♠♣♦rt❛♥t❡✱ ❝♦♠❡ ✈❡❞r❡♠♦♥❡❧❧✬❡s❡♠♣✐♦ ❝❤❡ s❡❣✉❡✳

❖ss❡r✈❛③✐♦♥❡✳ ◆❡❧ ❝❛s♦ ❞✐ t❛ss✐ ❝♦st❛♥t✐ it+s = i, rt+s = r ♣❡r ♦❣♥✐ s ≥ 0 ❧❡ ❢♦r♠✉❧❡s✐ s❡♠♣❧✐✜❝❛♥♦✿

1 + it,t+s = (1 + i)s, 1 + rt,t+s = (1 + r)s.

◆♦t❛ ✸✳ ◆❡❧❧✬❡s❡♠♣✐♦ ✶✳✶✶ ❛❜❜✐❛♠♦ ❝♦♥❝❧✉s♦ ❝❤❡ 1∣

∣et è ✏❡q✉✐✈❛❧❡♥t❡✑ ❛❞ (1 +

πt,t+s)∣

∣et+s ♥❡❧ s❡♥s♦ ❝❤❡ ❡♥tr❛♠❜❡ ❧❡ q✉❛♥t✐tà ❝♦♠♣r❛♥♦ ❧❛ st❡ss❛ ❢r❛③✐♦♥❡ ❞✐

❞❛t♦ X✳ ➮ ❜❡♥❡ ♦ss❡r✈❛r❡ ❝❤❡ q✉❡st♦ ♥♦♥ ✐♠♣❧✐❝❛ ❝❤❡ ❧❡ ❞✉❡ q✉❛♥t✐tà ❞✐ ❊✉r♦ s✐❛♥♦s❝❛♠❜✐❛❜✐❧✐✳ ❙❡ ♣❡r ❡s❡♠♣✐♦ πt,t+s = 0✱ ❝♦♥ ✉♥ ❡✉r♦ s✐ ❝♦♠♣r❛♥♦ ❣❧✐ st❡ss✐ ❜❡♥✐ ❛ t❡ t+s✱ ♠❛ ✐♥ ❣❡♥❡r❛❧❡ it,t+s > 0✱ q✉✐♥❞✐ 1

∣

∣et è s❝❛♠❜✐❛❜✐❧❡ ❝♦♥ (1+it,t+s)

∣

∣et+s >

1∣

∣et+s✱ ♣❡r❝❤é ✈✐❡♥❡ r✐❝♦♠♣❡♥s❛t♦ ✐❧ s❛❝r✐✜❝✐♦ ❞✐ ❛s♣❡tt❛r❡ ❞❛ t ❛ t+s ♣❡r r✐❛✈❡r❡

❧✬❡✉r♦✳ �

❊s❡♠♣✐♦ ✶✳✶✹ ✭P■▲ r❡❛❧❡ ❛ ✈❛❧♦r✐ ❝♦♥❝❛t❡♥❛t✐✮✳ ■❧ P■▲ r❡❛❧❡ ♥❡❧❧❛ ♣r❛t✐❝❛ ❞❡✐❝♦♥t✐ ♣✉❜❜❧✐❝✐ s✐ ❝❛❧❝♦❧❛ ✐♥ ✉♥ ♠♦❞♦ ✉♥ ♣♦✬ ♣✐ù s♦✜st✐❝❛t♦ ❞✐ ❝♦♠❡ ❛❜❜✐❛♠♦ ✈✐st♦✱✉s❛♥❞♦ ✐♥❞✐❝✐ ✏❝♦♥❝❛t❡♥❛t✐✑✳ ▼❡tt✐❛♠♦ q✉✐ q✉❡st♦ ❡s❡♠♣✐♦ ❝♦sì s❡ t✐ ❝❛♣✐t❛ ❞✐❞♦✈❡r s❛♣❡r❡ ❝♦s❛ s♦♥♦ ❝❡ ❧✬❤❛✐ ❛ ♣♦rt❛t❛ ❞✐ ♠❛♥♦✳ ■❧ ♣✉♥t♦ ❞✐ ♣❛rt❡♥③❛ è ❝❤❡♣♦ss✐❛♠♦ s❡♠♣r❡ s❝r✐✈❡r❡

psXt =psXt

psXspsXs

❝✐♦è✿ ✐❧ P■▲ r❡❛❧❡ ✭❝♦♠❡ ❝❛❧❝♦❧❛t♦ ❞❛ ♥♦✐✮ ❛❧ t❡♠♣♦ t ❝♦♥ ❜❛s❡ t = s è ✐❧ ♣r♦❞♦tt♦❞❡❧ P■▲ ♥♦♠✐♥❛❧❡ ❛❧ t❡♠♣♦ s ♣❡r ❧✬✐♥❞✐❝❡ ▲❛s♣❡②r❡s ❞❡❧❧❛ ✈❛r✐❛③✐♦♥❡ ❞❡❧❧❡ q✉❛♥t✐tà✲ ❝❤❡ ❤❛ s❡♥s♦ s❡ ❝✐ ♣❡♥s✐ ✉♥ ❛tt✐♠♦✳ ◗✉❡st♦ ♠❡t♦❞♦✱ ❝♦♠❡ ❛❜❜✐❛♠♦ ✈✐st♦ ♥❡❧♣❛r❛❣r❛❢♦ ♣r❡❝❡❞❡♥t❡✱ è ♣❛rt✐❝♦❧❛r♠❡♥t❡ s❡♥s✐❜✐❧❡ ❛❧❧❛ s❝❡❧t❛ ❞❡❧❧✬❛♥♥♦ ❜❛s❡✱ ❝❤❡❞✐✈❡♥t❛ ✉♥ ♣r♦❜❧❡♠❛ r✐❧❡✈❛♥t❡ q✉❛♥❞♦ s✐ ✉t✐❧✐③③❛ ✉♥ ❛♥♥♦ ❜❛s❡ r❡❧❛t✐✈❛♠❡♥t❡ ❧♦♥✲t❛♥♦ ❞❛❧❧✬❛♥♥♦ ♣❡r ❝✉✐ st✐❛♠♦ ❝❛❧❝♦❧❛♥❞♦ ✐❧ P■▲✳ ▲✬✉s♦ ❞❡❣❧✐ ✐♥❞✐❝✐ ❝♦♥❝❛t❡♥❛t✐ s✐♣r♦♣♦♥❡ ❞✐ s✉♣❡r❛r❡ q✉❡st♦ ❧✐♠✐t❡ s♦st✐t✉❡♥❞♦ ❧✬✐♥❞✐❝❡ psXt/psXs ❝♦♥ ✉♥ ✐♥❞✐❝❡ ❞✐✈❛r✐❛③✐♦♥❡ ❞✐ q✉❛♥t✐tà ♠❡♥♦ ❛♥❝♦r❛t♦ ❛❧❧✬❛♥♥♦ ❜❛s❡✱ ❝❤❡ t✐❡♥❡ ❝♦♥t♦ ❞❡❧❧❡ ✈❛r✐❛③✐♦✲♥✐ ✐♥ t✉tt♦ ✐❧ ♣❡r✐♦❞♦ ❝♦♥s✐❞❡r❛t♦✳ ■❧ ♠❡t♦❞♦ ❞❡❣❧✐ ✐♥❞✐❝✐ ❝♦♥❝❛t❡♥❛t✐ ❝♦♥s✐st❡ ♥❡❧❝❛❧❝♦❧❛r❡ ✉♥❛ ✈❛r✐❛③✐♦♥❡ ♣❡r❝❡♥t✉❛❧❡ tr❛ ❛♥♥✐ ❛❞✐❛❝❡♥t✐✱ ♦tt❡♥❡♥❞♦ ✐♥ q✉❡st♦ ♠♦❞♦✉♥❛ s✉❝❝❡ss✐♦♥❡ ❞✐ t❛ss✐ ❞✐ ✈❛r✐❛③✐♦♥❡✱ ♣❡r ♣♦✐ ❝❛❧❝♦❧❛r❡ ✐❧ P■▲ r❡❛❧❡ r✐s♣❡tt♦ ❛ ✉♥❛♥♥♦ ❞✐ r✐❢❡r✐♠❡♥t♦ ❝♦♠❡ ♣r♦❞♦tt♦ tr❛ ✐❧ P■▲ ♥♦♠✐♥❛❧❡ ♥❡❧❧✬❛♥♥♦ ❞✐ r✐❢❡r✐♠❡♥t♦ ❡❞✐❧ t❛ss♦ ❞✐ ✈❛r✐❛③✐♦♥❡✳

✶✳✹✳ ❉✉❡ ❡s❡♠♣✐ ♣✐ù ❡❧❛❜♦r❛t✐ ✶✾

■❧❧✉str✐❛♠♦ s❝❤❡♠❛t✐❝❛♠❡♥t❡ ❧❛ ♣r♦❝❡❞✉r❛✽ ✐♠♠❛❣✐♥❛♥❞♦ ❞✐ ❞♦✈❡r ❝❛❧❝♦❧❛r❡ ✐❧P■▲ r❡❛❧❡ ♣❡r ❧✬✐♥t❡r✈❛❧❧♦ {1, .., T}.

✶✳ ❙✐ ❝❛❧❝♦❧❛ ✐♥ ♦❣♥✐ ❛♥♥♦ ❧✬✐♥❞✐❝❡ ❞✐ ▲❛s♣❡②r❡s ❞❡❧❧❡ q✉❛♥t✐tà ✉t✐❧✐③③❛♥❞♦ ✐♣r❡③③✐ ❞❡❧ ♣❡r✐♦❞♦ ✐♠♠❡❞✐❛t❛♠❡♥t❡ ♣r❡❝❡❞❡♥t❡ QLA

t,t−1 = pt−1Xt/pt−1Xt−1.

✷✳ ❙✐ ❝♦♥❝❛t❡♥❛♥♦ ❣❧✐ ✐♥❞✐❝✐✳ ✾ ■♥ ♦❣♥✐ ♣❡r✐♦❞♦ t✱ ❧✬✐♥❞✐❝❡ ❝♦♥❝❛t❡♥❛t♦ ❞❡❧❧❛✈❛r✐❛③✐♦♥❡ ❞❡❧❧❡ q✉❛♥t✐tà ❢r❛ 1 ❡ t è ❧❛ ♣r♦❞✉tt♦r✐❛ ❞❡❣❧✐ ✐♥❞✐❝✐ ❛❧ ♣✉♥t♦ ✶❞❛❧❧✬✐♥✐③✐♦ ❞❡❧❧❛ s❡r✐❡ ✜♥♦ ❛ t✿ ICt =

∏

i≤tQLAi ✳

✸✳ Pr❡s♦ s ❝♦♠❡ ❛♥♥♦ ❞✐ r✐❢❡r✐♠❡♥t♦✱ s✐ ♣r❡♥❞❡ ❝♦♠❡ ✐♥❞✐❝❡ ❞✐ ✈❛r✐❛③✐♦♥❡ ❞❡❧❧❡q✉❛♥t✐tà ❢r❛ s ❡ t ✐❧ r❛♣♣♦rt♦ ICt /ICs ✱ ❡ s✐ ❞❡✜♥✐s❝❡ P■▲ r❡❛❧❡ ❛ t✱ ❜❛s❡ s ✲❝❤✐❛♠✐❛♠♦❧♦ PILt,s ❛♥❛❧♦❣❛♠❡♥t❡ ❛ q✉❛♥t♦ ❢❛tt♦ ♣r✐♠❛ ❝♦♠❡ ♣r♦❞♦tt♦ ❢r❛P■▲ ♥♦♠✐♥❛❧❡ ❛❞ s✱ PILs✱ ♣❡r q✉❡st♦ ✐♥❞✐❝❡ ❞✐ ✈❛r✐❛③✐♦♥❡ ❞❡❧❧❡ q✉❛♥t✐tà✿

PILt,s =ICtICs

PILs.

✶✳✹ ❉✉❡ ❡s❡♠♣✐ ♣✐ù ❡❧❛❜♦r❛t✐

❊s❡♠♣✐♦ ✶✳✶✺ ✭❊❝♦♥♦♠✐❛ ❞❡✐ s❡r✈✐③✐✮✳ ■♥ q✉❡st♦ ❡s❡♠♣✐♦ ✐♥tr♦❞✉❝✐❛♠♦ ✐♥❝❡rt❡③③❛❡ r✐s❝❤✐♦ ♣❡r ✈❡❞❡r❡ ❝♦♠❡ ❧❡ ❛ss✐❝✉r❛③✐♦♥✐✱ ❝❤❡ s♦♥♦ ✉♥❛ ♣❛rt❡ ✐♠♣♦rt❛♥t❡ ❞❡❧s❡tt♦r❡ ❞❡✐ s❡r✈✐③✐✱ ❡♥tr❛♥♦ ♥❡❧ P■▲ ✲ ❝✐♦è ❝♦s❛ ♣r♦❞✉❝♦♥♦✳ P❡r ❢❛r❧♦ ❝♦♥s✐❞❡r❡r❡♠♦✉♥✬❡❝♦♥♦♠✐❛ ♣r✐♠❛ ❡ ❞♦♣♦ ❧✬❡♥tr❛t❛ ❞✐ ✉♥❛ ✐♠♣r❡s❛ ❞✐ ❛ss✐❝✉r❛③✐♦♥✐✳

▲❛ s✐t✉❛③✐♦♥❡ ❞✐ ♣❛rt❡♥③❛ è ❧❛ s❡❣✉❡♥t❡✳ ❈✐ s♦♥♦ q✉❛ttr♦ ❢❛♠✐❣❧✐❡✳ ▲❛ q✉❛rt❛❞✐❣✐✉♥❛✳ ▲❡ ♣r✐♠❡ tr❡ ♣r♦❞✉❝♦♥♦ ❝✐❛s❝✉♥❛ ✸ ♠❡❧❡✱ ♠❛ ✉♥❛ ❞✐ ❧♦r♦ ❛ ❝❛s♦ ♥❡ ♣❡r❞❡✷✳ ■❧ ❝❛❧❝♦❧♦ ❞❡❧ P■▲ è ❜❛♥❛❧❡✿ ❛❧❧❛ ✜♥❡ ❞❡❧❧✬❛♥♥♦ ❝✐ s♦♥♦ 9− 2 = 7 ♠❡❧❡✱ q✉✐♥❞✐ ✲✐♥ t❡r♠✐♥✐ ❞✐ ♠❡❧❡ ✲ ✐❧ P■▲ è ✼✳

◆❡❧❧❛ s❡❝♦♥❞❛ ❡❝♦♥♦♠✐❛ ❧❛ q✉❛rt❛ ❢❛♠✐❣❧✐❛ ✲ ❝❤❡ ❝❤✐❛♠❡r❡♠♦ ▲✐♥❛ ✲ ♣r♦♣♦♥❡❛❞ ♦❣♥✉♥❛ ❞❡❧❧❡ ❛❧tr❡ ✐❧ s❡❣✉❡♥t❡ ❝♦♥tr❛tt♦ ❞✐ ❛ss✐❝✉r❛③✐♦♥❡✿ ✏❚✉ ♠✐ ❞❛✐ ✉♥❛ ♠❡❧❛✱❡ s❡ ♥❡ ♣❡r❞✐ ✷ t❡ ❧❡ ❞ò ✐♦✑✳ ➮ ✉♥❛ ♣r♦♣♦st❛ r❛❣✐♦♥❡✈♦❧❡❄ ❙❡♥③❛ ❛ss✐❝✉r❛③✐♦♥❡ ❧❛❢❛♠✐❣❧✐❛ ♣r♦❞✉ttr✐❝❡ ❞✐ ♠❡❧❡ ❤❛ 3 ❝♦♥ ♣r♦❜❛❜✐❧✐tà 2/3✱ ❡ 1 ❝♦♥ ♣r♦❜❛❜✐❧✐tà 1/3❀ ✐❧✈❛❧♦r❡ ❛tt❡s♦ (2/3) ·3+(1/3) ·1 = 2+1/3 è ♣✐ù ❞✐ 2✱ ♠❛ ❝✬è ✐❧ r✐s❝❤✐♦ ❞✐ q✉❡❧❧✬✉♥♦❀❝♦♥ ❧✬❛ss✐❝✉r❛③✐♦♥❡ ✐♥✈❡❝❡ ❝✬è ❧❛ s✐❝✉r❡③③❛ ❞✐ ❛✈❡r❡ 2✳ ❙❡ s✐ è ❛❜❜❛st❛♥③❛ ❛✈✈❡rs✐❛❧ r✐s❝❤✐♦ s✐ ♣✉ò ❛❝❝❡tt❛r❡✳ ❆ss✉♠✐❛♠♦ ❝❤❡ q✉❡st♦ s✐❛ ✐❧ ❝❛s♦ ♣❡r ❧❡ tr❡ ❢❛♠✐❣❧✐❡♣r♦❞✉ttr✐❝✐ ❞✐ ♠❡❧❡✳ ❆❧❧♦r❛✿ ♦❣♥✉♥❛ ❞à ✉♥❛ ♠❡❧❛ ❛ ▲✐♥❛❀ ▲✐♥❛ ✐♥❝❛ss❛ ✸✱ ❡ ❞à ✷ ❛q✉❡❧❧❛ ❝❤❡ s♦✛r❡ ❧❛ ♣❡r❞✐t❛✳✶✵ ❆❧❧❛ ✜♥❡ ❞❡❧❧✬❛♥♥♦ ❧❡ tr❡ ❢❛♠✐❣❧✐❡ ❤❛♥♥♦ ✷ ❡ ▲✐♥❛ ❤❛✶✱ s✐❝❝❤é ❞✐ ♥✉♦✈♦ PIL = 7✳ ❚✉tt✐ st❛♥♥♦ ♠❡❣❧✐♦ ✭❧❡ ❢❛♠✐❣❧✐❡ ♣r♦❞✉ttr✐❝✐ ♣❡r❝❤é

✽ ❉❛❧ ✷✵✵✺ ❧✬■❙❚❆❚ ✉t✐❧✐③③❛ q✉❡st❛ ♣r♦❝❡❞✉r❛ ♣❡r ✐❧ ❝❛❧❝♦❧♦ ❞❡❧ P■▲ r❡❛❧❡✳✾ ■♥ ♣r❛t✐❝❛ s✐ ❢❛ ❧❛ st❡ss❛ ❝♦s❛ ❝❤❡ ❛❜❜✐❛♠♦ ❢❛tt♦ ♣❡r ❝❛❧❝♦❧❛r❡ ❧✬✐♥t❡r❡ss❡ ♦ ❧✬✐♥✢❛③✐♦♥❡ s✉ ♣✐ù

♣❡r✐♦❞✐✱ 1+rt,t+s = (1+r1)·(1+r2) . . . (1+rt+s−1) ♦ 1+πt,t+s = (1+πt)·(1+πt+1)·. . .·(1+πt+s−1)✳✶✵ ❙❡ ✈✉♦✐✱ ❧❛ ❢❛♠✐❣❧✐❛ ❝❤❡ s♦✛r❡ ♣❡r❞✐t❛ ♣♦tr❡❜❜❡ ❡ss❡r❡ ❛♥❝❤❡ ▲✐♥❛ ✲ ♥♦♥ ❝❛♠❜✐❛ ♥✐❡♥t❡✳

✷✵ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

❛❧tr✐♠❡♥t✐ ♥♦♥ ❛✈r❡❜❜❡r♦ ❛❝❝❡tt❛t♦ ✐❧ ❝♦♥tr❛tt♦✮ ♠❛ ✐❧ P■▲ ♥♦♥ è ❛✉♠❡♥t❛t♦✱ ès♦❧♦ r❡❞✐str✐❜✉✐t♦✿ ♠❡♥tr❡ ♣r✐♠❛ ✐ ✈❛❧♦r✐ ❛❣❣✐✉♥t✐ ❡r❛♥♦ (3, 3, 1, 0) ✲ ❝♦♥ ❧✬✶ ✐♥ ✉♥♦❞❡✐ ♣r✐♠✐ tr❡ ♣♦st✐ ✲ ❛❞❡ss♦ s♦♥♦ (2, 2, 2, 1)✳ ▲✐♥❛ s✐ è ❣✉❛❞❛❣♥❛t❛ ❧❛ s✉❛ ♣❛rt❡✳

❆ q✉❡st♦ ♣✉♥t♦ ✈✐❡♥❡ s♣♦♥t❛♥❡♦ ❝❤✐❡❞❡rs✐✿ ♠❛ ❛❧❧♦r❛ ✐❧ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦ ❞❡❧s❡tt♦r❡ ❛ss✐❝✉r❛t✐✈♦ è s❡♠♣r❡ ✏t♦❧t♦✑ ❛ q✉❛❧❝✉♥ ❛❧tr♦ s❡♥③❛ r❡❛❧❡ ✏❛❣❣✐✉♥t❛✑ ❛❧P■▲❄ ▲❛ r✐s♣♦st❛ è ◆♦✳ ❈❛♠❜✐❛♠♦ ✐ ♥✉♠❡r✐ ❞❡❧❧✬❡s❡♠♣✐♦ ❡ ✈❡❞✐❛♠♦ ❝♦♠❡ ✐♥♣r❛t✐❝❛ ✐❧ s❡tt♦r❡ ❛ss✐❝✉r❛t✐✈♦ ♣✉ò ❛❝❝r❡s❝❡r❡ ✐❧ P■▲✳ ■♥ q✉❡st❛ ❡❝♦♥♦♠✐❛ ♦❣♥✉♥❛❞❡❧❧❡ tr❡ ❢❛♠✐❣❧✐❡ ❞✐ ♣♦❝♦ ❢❛ ♣✉ò ♣r♦❞✉rr❡ 5 ♠❡❧❡✱ ♠❛ ✉♥❛ ❞✐ ❧♦r♦ ❛ ❝❛s♦ s✉❜✐s❝❡✉♥❛ ♣❡r❞✐t❛ ❞✐ 11 ❝❤❡ ❧❛ ♣♦rt❛ ❛ −6 ✭❝❤❡ ♣r❡♥❞❡r❡❜❜❡ ❞❛❧❧❡ s❝♦rt❡✱ ✐♥✈❡st✐♠❡♥t♦♥❡❣❛t✐✈♦✮✳ ◗✉✐ ✐❧ ✈❛❧♦r❡ ❛tt❡s♦ ❞❡❧❧❛ ♣r♦❞✉③✐♦♥❡ è (2/3)·5+(1/3)·(−6) = 1+1/3♠❛❝✬è q✉❡❧ −6 ❝❤❡ ♣❡s❛✱ ❡❞ ❡ss❡♥❞♦ ❧❡ ♥♦str❡ ❢❛♠✐❣❧✐❡ ❛✈✈❡rs❡ ❛❧ r✐s❝❤✐♦ ♣r❡❢❡r✐s❝♦♥♦♥♦♥ ♣r♦❞✉rr❡ ❛✛❛tt♦✱ ❝✐♦è ♣r❡❢❡r✐s❝♦♥♦ 0 ❝♦♥ ❝❡rt❡③③❛ ❛❧❧❛ ❧♦tt❡r✐❛ (5,−6) ❝♦♥♣r♦❜❛❜✐❧✐tà (2/3, 1/3)✳ ◗✉✐♥❞✐✱ PIL = 0✳ P♦tr❡❜❜❡ ❡ss❡r❡✱ s❡ ♣r♦❞✉❝❡ss❡r♦✱ 5 +5 − 6 = 4✱ ♠❛ è ③❡r♦ ♣❡r❝❤é ♥♦♥ ✈♦❣❧✐♦♥♦ r✐s❝❤✐❛r❡✳ ❊♥tr❛ ✐♥ ❝❛♠♣♦ ▲✐♥❛ ❝♦♥ ❧❛s✉❛ ♣r♦♣♦st❛✿ ✏❉❛♠♠✐ ✹ ♠❡❧❡✱ ❡ s❡ s♦✛r✐ ❧❛ ♣❡r❞✐t❛ t✐ ❞ò ✐♦ ❧❡ ✶✶✑✳ ❆❝❝❡tt❛♥❞♦❧❛ ❢❛♠✐❣❧✐❛ ❛✈rà ❛❞❡ss♦ ✶ ♠❡❧❛ ❞✐ s✐❝✉r♦✱ ❝❤❡ è ♣✐ù ❞✐ 0 ❞✐ s✐❝✉r♦ ❝❤❡ ❡r❛ ❣✐à♠❡❣❧✐♦ ❞❡❧❧❛ ♣r♦❞✉③✐♦♥❡ ❛❧❡❛t♦r✐❛ ❝♦♥s✐❞❡r❛t❛ ♣♦❝♦ ❢❛ ✲ q✉✐♥❞✐ ❛❝❝❡tt❛✱ ❡ ♣r♦❞✉❝❡tr❛♥q✉✐❧❧❛✳ ▲✐♥❛ ❞✬❛❧tr❛ ♣❛rt❡ ✐♥❝❛ss❛ 3 · 4 = 12 ❞✐ ♣r❡♠✐ ❡ ♣❛❣❛ 11 ✐♥ r✐♠❜♦rs✐q✉✐♥❞✐ ❤❛ ✉♥ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦ ❞✐ ✶✳ ❊ ❛♥❝❤❡ ♣❡r ❣❧✐ ❛❧tr✐ ✐❧ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦ ✭❧❛q✉❛♥t✐tà ❝♦♥t❛❜✐❧✐③③❛t❛ ❛❧❧❛ ✜♥❡ ❞❡❧❧✬❛♥♥♦✮ è ✶❀ q✉✐♥❞✐ ❛❞❡ss♦ PIL = 4 > 0✳ ■♥q✉❡st♦ ❡s❡♠♣✐♦ ♥♦♥ s♦❧♦ ❧❛ ❝♦♠♣❛❣♥✐❛ ❛ss✐❝✉r❛t✐✈❛ s✐ è ❣✉❛❞❛❣♥❛t❛ ❧❛ s✉❛ ♣❛rt❡✱♠❛ ❤❛ ❛♥❝❤❡ r❡s♦ ♣♦ss✐❜✐❧❡ ✉♥ ✐♥❝r❡♠❡♥t♦ r❡❛❧❡ ❞✐ P■▲✳

❊s❡♠♣✐♦ ✶✳✶✻ ✭❊❝♦♥♦♠✐❛ ❘❡❛❧❡✮✳ ❆♥❝❤❡ q✉❡st♦ ❡s❡♠♣✐♦ è ✉♥ ♣♦✬ ♣✐ù ❝♦♠♣❧❡ss♦❞❡✐ ♣r❡❝❡❞❡♥t✐✱ ♠❛ è ✐♠♣♦rt❛♥t❡ ♣❡r❝❤é ❢❛ ❞❛ ♣♦♥t❡ ♣❡r ✐❧ ♠♦❞❡❧❧♦ ❞✐ ❝r❡s❝✐t❛❝❤❡ st✉❞✐❡r❡♠♦ tr❛ ♣♦❝♦✳ ■♥ ♣r❛t✐❝❛ r✐❣✉❛r❞❛ q✉❡❧❧♦ s✉ ❝✉✐ ❛❜❜✐❛♠♦ s♦r✈♦❧❛t♦♥❡❧ ❝♦♥t♦ r✐s♦rs❡✴✐♠♣✐❡❣❤✐✿ ❣❧✐ ✐♥✈❡st✐♠❡♥t✐✱ ✐♥ ♣❛rt✐❝♦❧❛r❡ q✉❡❧❧✐ ✐♥ ❝❛♣✐t❛❧❡ ✜ss♦✳◆❡❧❧❛ t❛✈♦❧❛ ❛ ♣✳✽ s✐ ❧❡❣❣❡ ❝❤❡ ✐♥ ■t❛❧✐❛ ♥❡❧ ✷✵✶✹ ❣❧✐ ✐♥✈❡st✐♠❡♥t✐ ✜ss✐ ❧♦r❞✐ s♦♥♦st❛t✐ 268117.2 ✭≈ 268 ♠✐❧✐❛r❞✐ ❞✐ ❊✉r♦✮❀ ♠❛ ❣❧✐ ❛♠♠♦rt❛♠❡♥t✐ s♦♥♦ st❛t✐ ❛♥❝♦r❛s✉♣❡r✐♦r✐✱ ♣❡r ❧❛ ♣r❡❝✐s✐♦♥❡ 294955.1❀ ✐❧ r✐s✉❧t❛t♦ è st❛t♦ ❝❤❡ ❣❧✐ ✐♥✈❡st✐♠❡♥t✐ ♥❡tt✐s♦♥♦ st❛t✐ ♥❡❣❛t✐✈✐✿ 268117.2− 294955.1 = −26837.9✳ ❊ ❧❛ ❝♦s❛ ✐♠♣♦rt❛♥t❡ è ❝❤❡❣❧✐ ✐♥✈❡st✐♠❡♥t✐ ♥❡tt✐ s♦♥♦ ❧✬✐♥❝r❡♠❡♥t♦ ❞✐ ❝❛♣✐t❛❧❡✳ P❡r ❡s❡♠♣✐♦ s❡ ❞✉r❛♥t❡ ❧✬❛♥♥♦❝♦♠♣r✐ ✷✵ ❧❛♠♣❛❞✐♥❡ ❡ s❡ ♥❡ ❛♠♠♦rt❛♥♦ ✷✺ ✐❧ ❝❛♣✐t❛❧❡ ❞✐ ❧❛♠♣❛❞✐♥❡ s✐ è r✐❞♦tt♦❞✐ ✺ ✭❣❧✐ ✐♥✈❡st✐♠❡♥t✐ ♥❡tt✐ s♦♥♦ −5✮✳ ❊ ♦✈✈✐❛♠❡♥t❡ ❧✬❛♠♠♦rt❛♠❡♥t♦ ♣✉ò ♥♦♥r✐❣✉❛r❞❛r❡ ❧❛♠♣❛❞✐♥❡ ❝❤❡ s✐ ❢✉❧♠✐♥❛♥♦ ♠❛ ❜❡♥✐ ❝❛♣✐t❛❧✐ ❝❤❡ ❝♦♥t✐♥✉❛♥♦ ❛❞ ❡ss❡r❡✉t✐❧✐③③❛t✐ ♠❛ ♣❡r❞♦♥♦ ❞✐ ✈❛❧♦r❡ ♣❡r❝❤é ♥♦♥ ❢✉♥③✐♦♥❛♥♦ ♣✐ù t❛♥t♦ ❜❡♥❡ ♦ ❞✐✈❡♥t❛♥♦♦❜s♦❧❡t✐ ✭♣❡♥s❛ ❛❧❧❡ ❛ttr❡③③❛t✉r❡ ✐♥❢♦r♠❛t✐❝❤❡✮✳ ▲❛ r❡❧❛③✐♦♥❡ è s❡♠♣r❡

■♥❝r❡♠❡♥t♦ ❞✐ ❝❛♣✐t❛❧❡ ❂ ■♥✈❡st✐♠❡♥t✐ ✜ss✐ ❧♦r❞✐ ✲ ❆♠♠♦rt❛♠❡♥t✐✳

■♥ q✉❡st♦ ❡s❡♠♣✐♦ ✈❡❞r❡♠♦ ❝♦♠❡ q✉❡st❛ r❡❧❛③✐♦♥❡ ✈✐❡♥❡ ❢✉♦r✐ ✐♥ ❝♦♥❝r❡t♦ ✐♥✉♥✬❡❝♦♥♦♠✐❛ s❡♠♣❧✐❝❡✳ ◆❡✐ ♣r♦❝❡ss✐ ♣r♦❞✉tt✐✈✐ ❞✐ q✉❡st❛ ❡❝♦♥♦♠✐❛ s✐ ✐♠♣✐❡❣❛♥♦ ❧❛✲

✶✳✹✳ ❉✉❡ ❡s❡♠♣✐ ♣✐ù ❡❧❛❜♦r❛t✐ ✷✶

✈♦r♦ ❡ ♠❛❝❝❤✐♥❛r✐ ❞✐ ✈❛r✐❡ ❡tà✱ ❡ ❝♦♠❡ s❡♠♣r❡ s✐ ✉t✐❧✐③③❛♥♦ ❜❡♥✐ ♣r♦❞♦tt✐ ❝♦♠❡ ❜❡♥✐✐♥t❡r♠❡❞✐✳ ■ ❜❡♥✐ ♥❡❧❧✬❡❝♦♥♦♠✐❛ s♦♥♦✿ G ❣r❛♥♦✱ F ❢❡rr♦✱ L ❧❛✈♦r♦✱ B ❜r✐♦❝❤❡ ❡❞ Mi

♠❛❝❝❤✐♥❛r✐ ❣✐à ✉s❛t✐ ♣❡r i ❛♥♥✐ ✭M0 s♦♥♦ ♥✉♦✈✐✮✳ ❆❧❧❛ ✜♥❡ ❞❡❧❧✬❛♥♥♦ t−1 ❧♦ st♦❝❦ ❞✐❝❛♣✐t❛❧❡✱ ✐♥❝❧✉s❡ ❧❡ s❝♦rt❡ ❞✐ ❣r❛♥♦✱ è✿ 10M0, 18M1, 10M2, 50G ✭❝✐♦è ✶✵ ✉♥✐tà ❞✐M0

❡❝❝❡t❡r❛✮✳ ▲❛ t❛❜❡❧❧❛ s❡❣✉❡♥t❡ ❞❡s❝r✐✈❡ ✐ ♣r♦❝❡ss✐ ♣r♦❞✉tt✐✈✐ ❞❡❧❧✬❛♥♥♦ t ❡ r✐♣♦rt❛ ✐♣r❡③③✐✳ ■♥ r✐❣❛ ❝✐ s♦♥♦ ❧❡ q✉❛♥t✐tà ❞✐ ❢❛tt♦r✐ ♣r♦❞✉tt✐✈✐ ✉t✐❧✐③③❛t❡ ♥❡❧❧❛ ♣r♦❞✉③✐♦♥❡❞❡❧ s❡tt♦r❡ ❡ ❧❛ q✉❛♥t✐tà ♣r♦❞♦tt❛ ❞❡❧ ❜❡♥❡✳ P❡r ❡s❡♠♣✐♦ ♥❡❧ s❡tt♦r❡ ❋❡rr♦ s✐ ✉s❛✐❧ ✈❡tt♦r❡ ❞✐ ❢❛tt♦r✐ (G,F, L,B,M0,M1,M2M3) = (0, 10, 20, 0, 5, 8, 0, 0) ✭r✐❣❛✮ ❡ s✐♣r♦❞✉❝♦♥♦ ✶✵✵ ✉♥✐tà✳ ◆❡❧❧❡ ❝♦❧♦♥♥❡ s✐ ❧❡❣❣♦♥♦ ❧❡ q✉❛♥t✐tà ❞✐ ❜❡♥✐ ✉t✐❧✐③③❛t✐ ❝♦♠❡❢❛tt♦r✐ ♣r♦❞✉tt✐✈✐ ♥❡✐ ✈❛r✐ s❡tt♦r✐ ✭❝❤❡ s✐ ❞♦✈r❛♥♥♦ s♦ttr❛rr❡ ♣❡r ❝❛❧❝♦❧❛r❡ ✐❧ P■▲✦✮✱♣❡r ❡s❡♠♣✐♦ ✐❧ ❣r❛♥♦ è ✉s❛t♦ ♥❡✐ s❡tt♦r✐ ●r❛♥♦ ❡ ❇r✐♦❝❤❡ ✭❝♦❧♦♥♥❛✮✱ ❡ ❧❡ ❜r✐♦❝❤❡♥♦♥ s♦♥♦ ✉s❛t❡ ♣❡r ♣r♦❞✉rr❡ ❛❧tr✐ ❜❡♥✐✳ ◆♦t❛ ❝❤❡ s✐ ✉t✐❧✐③③❛♥♦ ✶✷ ♠❛❝❝❤✐♥❡ ♥✉♦✈❡✱❧❡ ✶✵ ✐♥ st♦❝❦ ♣✐ù ✷ ♣r♦❞♦tt❡ ♥❡❧ ♣❡r✐♦❞♦ ❝♦rr❡♥t❡✳

❋❛tt♦r✐ ♣r♦❞✉tt✐✈✐ ❡ ❇❡♥✐ ✐♥t❡r♠❡❞✐❙❡tt♦r✐

Pr♦❞✉tt✐✈✐▲ ● ❋ ❇ M0 M1 M2 M3

◗✉❛♥t✐tàPr♦❞♦tt❡

●r❛♥♦ ✶✵ ✺✵ ✺ ✼ ✷✵✵❋❡rr♦ ✷✵ ✶✵ ✺ ✽ ✶✵✵

❇r✐♦❝❤❡ ✷✵ ✼✺ ✺ ✷✵✵▼❛❝❝❤✐♥❡ M0 ✷✵ ✹✵ ✶✵ ✺ ✷✺

Pr❡③③✐ ✼ ✶ ✹ ✷ ✶✺ ✼ ✸ ✵

◆♦t❛ ❛♥❝❤❡ ❝❤❡ ✐ ❢❛tt♦r✐ ♣r♦❞✉tt✐✈✐ s♦♥♦ ❢❡rr♦ ❧❛✈♦r♦ ❡ ♠❛❝❝❤✐♥❡✱ ✐♥ ❛❧tr✐ t❡r♠✐♥✐♠❛t❡r✐❡ ♣r✐♠❡ ❧❛✈♦r♦ ❡ ❝❛♣✐t❛❧❡✳ ❘✐♣r❡♥❞❡r❡♠♦ q✉❡st❛ ♦ss❡r✈❛③✐♦♥❡ ♥❡❧ ♣r♦ss✐♠♦❝❛♣✐t♦❧♦ s❝r✐✈❡♥❞♦ ❧❛ ❢✉♥③✐♦♥❡ ❞✐ ♣r♦❞✉③✐♦♥❡ ❛❣❣r❡❣❛t❛ ❝♦♠❡ Y = F (K,L,M)✳

■♥t❛♥t♦ ❝❛❧❝♦❧✐❛♠♦ ✐❧ P■▲✱ ✉s❛♥❞♦ ❧❡ q✉❛♥t✐tà ♥❡tt❡ ❞❛❣❧✐ ✐♠♣✐❡❣❤✐ ✭♥♦t❛ ❝❤❡♥❡❧ ♣r♦❞♦tt♦ ❧♦r❞♦ ♥♦♥ s✐ s♦ttr❛❣❣♦♥♦ ❧❡ ♠❛❝❝❤✐♥❡ ♣r♦❞♦tt❡ ❡ ✉t✐❧✐③③❛t❡✮✿

PIL = 1× (200− 50− 75) + 4× (100− 5− 10− 40) + 2× 200 + 15× 25 = 1030.

■ ✈❛❧♦r✐ ❛❣❣✐✉♥t✐ ❞❡✐ ✹ s❡tt♦r✐✿ ●r❛♥♦ 1×200−1×50−4×5✱ ❡❝❝❡t❡r❛✳ ◆♦t❛ ❜❡♥❡❝❤❡ ♥♦♥ s✐ s♦ttr❛❡ ✐❧ ✈❛❧♦r❡ ❞❡❧❧✬✉s✉r❛ ❞❡❧❧❡ ♠❛❝❝❤✐♥❡ ✉t✐❧✐③③❛t❡ ✲ st✐❛♠♦ ♣❛r❧❛♥❞♦❞✐ ♣r♦❞♦tt♦ ❧♦r❞♦✦ P❡r ❣❧✐ ❛♠♠♦rt❛♠❡♥t✐ D ✭D ♣❡r ✏❉❡♣r❡③③❛♠❡♥t♦✑✮ ❞♦❜❜✐❛♠♦❝❛❧❝♦❧❛r❡ ❧❛ ♣❡r❞✐t❛ ❞✐ ✈❛❧♦r❡ ❞❡✐ ✈❛r✐ t✐♣✐ ❞✐ ♠❛❝❝❤✐♥❛r✐✳ ■♥❞✐❝❛♥❞♦ ❝♦♥ pi ✐❧ ♣r❡③③♦❞❡✐ ♠❛❝❝❤✐♥❛r✐ ❞✐ ❡tà i ❛❜❜✐❛♠♦

D = 12× (p0−p1)+18× (p1−p2)+10× (p2−p3) = 12×8+18×4+10×3 = 198

❞♦✈❡ ✶✷ s♦♥♦ ❧❡ ♠❛❝❝❤✐♥❡ ♥✉♦✈❡ ✉t✐❧✐③③❛t❡ ♥❡❧❧❛ ♣r♦❞✉③✐♦♥❡ ❞❡❧❧✬❛♥♥♦ ✐♥ ❝♦rs♦✱ ✈❡❞✐❧❛ ❝♦❧♦♥♥❛ ❞✐ M0 ♥❡❧❧❛ t❛❜❡❧❧❛✱ ❞❛ ❝✉✐ PIN = PIL−D = 832✳ ◆♦t❛ ❝❤❡ ❧❛ ♣❡r❞✐t❛❞✐ ✈❛❧♦r❡ ❞✐ ✉♥ ♠❛❝❝❤✐♥❛r✐♦ è ❞❡✜♥✐t❛ q✉✐ ❝♦♠❡ ❧❛ ❞✐✛❡r❡♥③❛ tr❛ ✐❧ ♣r❡③③♦ ❞✐ ✉♥♠❛❝❝❤✐♥❛r✐♦ ✉s❛t♦ ♣❡r t − 1 ❛♥♥✐ ❡ ✐❧ ♣r❡③③♦ ❞✐ ✉♥ ♠❛❝❝❤✐♥❛r✐♦ ✉s❛t♦ ♣❡r t ❛♥♥✐✳

✷✷ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

P❡r ❝❛❧❝♦❧❛r❡ ❝♦♥s✉♠✐ ❡ ✐♥✈❡st✐♠❡♥t✐ r✐❝♦r❞✐❛♠♦ ❝❤❡ q✉❡st✐ ✉❧t✐♠✐ ❝♦♠♣r❡♥❞♦♥♦❧❡ ✈❛r✐❛③✐♦♥✐ ❞❡❧❧❡ s❝♦rt❡✱ ❡ s✉♣♣♦rr❡♠♦ ❝❤❡ ❧❡ s❝♦rt❡ ❞✐ ❣r❛♥♦ s♦♥♦ ❞✐♠✐♥✉✐t❡ ❞✐✶✵✳ ◗✉✐♥❞✐ ♦❧tr❡ ❛❧❧❡ ❜r✐♦❝❤❡ s✐ è ❝♦♥s✉♠❛t♦ ❣r❛♥♦ ♣❡r 75+ 10❀ ❝♦♠❡ s❛♣♣✐❛♠♦ ❧❡s❝♦rt❡ ❞✐ ❢❡rr♦ s♦♥♦ ❛✉♠❡♥t❛t❡ ❞✐ ✹✺✳

◗✉✐♥❞✐ C = 1×85+2×200 = 485 ❡❞ I = 15×25−1×10+4×45 = 545✳ ◆♦t❛✭❝♦♥tr♦❧❧❛✮ ❝❤❡ Y = C + I✳ ❙♦r✈♦❧✐❛♠♦ s✉❧❧❛ ❞✐str✐❜✉③✐♦♥❡ ❞❡❧ r❡❞❞✐t♦✱ ♣❡r❝❤é q✉✐è ❝♦♠♣❧✐❝❛t❛ ❞❛✐ r❡❞❞✐t✐ ❞❛ ❝❛♣✐t❛❧❡✳

❉❡♥♦t❛♥❞♦ ❝♦♥ Kt−1 ❡ Kt ✐❧ ✈❛❧♦r❡ ❞❡❧❧♦ st♦❝❦ ❞✐ ❝❛♣✐t❛❧❡ ❛❧❧✬✐♥✐③✐♦ ❡ ❛❧❧❛ ✜♥❡❞❡❧ ♣❡r✐♦❞♦✱ ❝❛❧❝♦❧✐❛♠♦ ❛❞❡ss♦ ❧❛ ✈❛r✐❛③✐♦♥❡ ∆K = Kt − Kt−1✳ ❘✐❝♦r❞❛♥❞♦ ❝❤❡❞✉❡ ♠❛❝❝❤✐♥❡ ♣r♦❞♦tt❡ ✭♥✉♦✈❡✮ s♦♥♦ st❛t❡ ✉s❛t❡ ❞✉r❛♥t❡ ❧✬❛♥♥♦ ❛❜❜✐❛♠♦

Kt−1 = 10p0+18p1+10p2+50×1 = 356, Kt = 23p0+12p1+18p2+40×1+45×4 = 703

q✉✐♥❞✐ ∆K = 703−356 = 347✳ ◆♦t❛ ❝❤❡ ❞✐ Kt ❢❛♥♥♦ ♣❛rt❡ 23 ♠❛❝❝❤✐♥❡ ♥✉♦✈❡ ✭❧❡25 ♣r♦❞♦tt❡ ♥❡❧❧✬❛♥♥♦ ♠❡♥♦ ❧❡ 2 ✉t✐❧✐③③❛t❡ ♥❡❧❧❛ ♣r♦❞✉③✐♦♥❡✮✳ ▲❛ ❝♦s❛ ✐♠♣♦rt❛♥t❡❞❛ ♥♦t❛r❡ è ❝❤❡ r✐s✉❧t❛ ✭❝♦♥tr♦❧❧❛✦✮

∆K = I −D

❝❤❡ ❞✐❝❡ ❝❤❡ ❧✬✐♥❝r❡♠❡♥t♦ ❞✐ ❝❛♣✐t❛❧❡ è ✉❣✉❛❧❡ ❛❣❧✐ ✐♥✈❡st✐♠❡♥t✐ ✭❧♦r❞✐✮ ♠❡♥♦ ❣❧✐❛♠♠♦rt❛♠❡♥t✐✿ ❝♦♠♣r✐ ✷✵ ❧❛♠♣❛❞✐♥❡✱ ✹ s✐ ❛♠♠♦rt❛♥♦✱ ❧♦ st♦❝❦ ❛✉♠❡♥t❛ ❞✐ ✶✻✳❘✐✈❡❞r❡♠♦ q✉❡st❛ ❡q✉❛③✐♦♥❡ ♥❡❧ ♣r♦ss✐♠♦ ❝❛♣✐t♦❧♦✳

✶✳✺ ■❧ ❝❛s♦ Y = G

P✉ò ❡s✐st❡r❡ ✉♥✬❡❝♦♥♦♠✐❛ ✐♥ ❝✉✐ Y = G❄ ❙✐✳ ❋♦r♠❛❧♠❡♥t❡ ❧✬✉❣✉❛❣❧✐❛♥③❛ ❞✐❝❡ ❝❤❡✐❧ ♣❛♥✐❡r❡ ❛❣❣r❡❣❛t♦ ❞✐ ❜❡♥✐ è ✐♥t❡r❛♠❡♥t❡ ♣r♦❞♦tt♦ ❞❛❧❧♦ st❛t♦✳ ▼❛ ❝❤❡ ✈✉♦❧ ❞✐r❡q✉❡st♦❄ ▲♦ st❛t♦ ♥♦♥ ✈❛ ❧❛ ♠❛tt✐♥❛ ✐♥ ❢❛❜❜r✐❝❛ ♦ ✐♥ ✉♥ ❝❛♥t✐❡r❡ ❛ ❝♦str✉✐r❡ ♣❛❧❛③③✐✱❝✐ ✈❛♥♥♦ ❧❡ ♣❡rs♦♥❡✳ Y = G ✈✉♦❧ ❞✐r❡ ❝❤❡ ❧♦ st❛t♦ ❞à ✐str✉③✐♦♥✐ ❞✐ ♣r♦❞✉③✐♦♥❡ ❞❛❡s❡❣✉✐r❡❀ ❡ st❛❜✐❧✐s❝❡ ♣❡r ❞❡❝r❡t♦ ❝♦♠❡ ❛❧❧♦❝❛r❡ ✐❧ ♣❛♥✐❡r❡ ❛❣❣r❡❣❛t♦ ♣r♦❞♦tt♦ ❢r❛ ✐♠❡♠❜r✐ ❞❡❧❧❛ ❝♦❧❧❡tt✐✈✐tà✳ ■♥ ❈♦r❡❛ ❞❡❧ ◆♦r❞ s✉❝❝❡❞❡ ♣✐ù ♦ ♠❡♥♦ ❝♦sì✳

➮ ✐♥t❡r❡ss❛♥t❡ ♦ss❡r✈❛r❡ ❝❤❡ Y ❡ G s♦♥♦ ♠✐s✉r❛t✐ ✐♥ ♠♦♥❡t❛ ❝♦rr❡♥t❡✳ ◗✉❛♥t♦✈❛❧❣♦♥♦ G ❡❞ Y ✐♥ ✉♥✬❡❝♦♥♦♠✐❛ s✐✛❛tt❛❄ ❉✐♣❡♥❞❡ ❞❛❧ ✈❡tt♦r❡ ❞❡✐ ♣r❡③③✐✱ s❡ ❧♦♠♦❧t✐♣❧✐❝❤✐ ♣❡r ✶✵✵ ❧♦ st❡ss♦ ♣❛♥✐❡r❡ ✜❣✉r❛ ✶✵✵ ✈♦❧t❡ ♣✐ù ❛❧t♦ ✐♥ t❡r♠✐♥✐ ♠♦♥❡t❛r✐✳❈♦s❛ ❞✉♥q✉❡ ❞❡t❡r♠✐♥❛ ✐❧ ❧✐✈❡❧❧♦ ❞❡✐ ♣r❡③③✐❄ ▲✬✐♥t✉✐③✐♦♥❡ ❞❛❧❧❛ ♠✐❝r♦❡❝♦♥♦♠✐❛❞✐❝❡ ❝❤❡ ✐❧ ✈❛❧♦r❡ s✐ ♠✐s✉r❛ ✐♥ t❡r♠✐♥✐ r❡❧❛t✐✈✐✱ ❝✐♦è ♣❡r ❛✈❡r❡ ✉♥ ✈❛❧♦r❡ ❝✐ ✈♦❣❧✐♦♥♦❞✉❡ ♣r❡③③✐ ✲ q✉✐ ❞✉❡ ❧✐✈❡❧❧✐ ❞✐ ♣r❡③③✐✳ ◗✉❛❧ è ❧✬❛❧tr♦❄ ■❧ ❧✐✈❡❧❧♦ ❞❡✐ ♣r❡③③✐ ♥❡❧ r❡st♦❞❡❧ ♠♦♥❞♦✳ P✉♦✐ ♠♦❧t✐♣❧✐❝❛r❡ ✐ t✉♦✐ ♣r❡③③✐ q✉❛♥t♦ ✈✉♦✐✱ ♠❛ s❡ ✈✉♦✐ ✉♥❛ ❋❡rr❛r✐♠✐ ❞❡✈✐ ❞❛r❡ ✐♥ ❝❛♠❜✐♦ ✉♥❛ q✉❛♥t✐tà ❞✐ ✈❛❧✉t❛ t❛❧❡ ❞❛ ❝♦♥s❡♥t✐r♠✐ ❞✐ ❝♦♠♣r❛r❝✐♥❡❧❧❛ t✉❛ ❡❝♦♥♦♠✐❛ ✉♥ ♣❛♥✐❡r❡ ❡q✉✐✈❛❧❡♥t❡ ❛ q✉❡❧❧♦ ❝❤❡ ♣♦ss♦ ❝♦♠♣r❛r❡ ❛ ❝❛s❛ s❡✈❡♥❞♦ ✉♥❛ ❋❡rr❛r✐✳ ◗✉❡st♦ ♠❡❝❝❛♥✐s♠♦ ❞❡t❡r♠✐♥❛ ✐❧ t❛ss♦ ❞✐ ❝❛♠❜✐♦✱ ❡ q✉✐♥❞✐ ✐❧✈❛❧♦r❡ ❞❡✐ ❜❡♥✐ ✐♥ q✉❡❧❧✬❡❝♦♥♦♠✐❛ ✐♥ t❡r♠✐♥✐ ❞❡✐ ❜❡♥✐ ♥❡❧ r❡st♦ ❞❡❧ ♠♦♥❞♦✳

✶✳✻✳ ❊s❡r❝✐③✐ ✷✸

◆♦✐ ♥♦♥ s✐❛♠♦ ❧❛ ❈♦r❡❛ ❞❡❧ ◆♦r❞✱ Y = C + I + G ❡ C ❡❞ I s♦♥♦ ❜❡♥ ❞✐✈❡rs✐❞❛ ③❡r♦ ✲ ♠❛ G è ❝✐r❝❛ ✉♥ q✉✐♥t♦ ❞❡❧ P■▲ ✭✈❡❞✐ t❛❜❡❧❧❛ r✐s♦rs❡✴✐♠♣✐❡❣❤✐ ♣✳✽✮✱ ♥♦♥è ♣r♦♣r✐♦ ✉♥♦ s❝❤❡r③♦✳ ◆♦♥ ❝✬è ♣r♦♣r✐♦ ♥✐❡♥t❡ ❝❤❡ ♣♦ss✐❛♠♦ ✐♠♣❛r❛r❡ ❞❛❧ ❝❛s♦❛♣♣❡♥❛ ❞✐s❝✉ss♦❄ ❙✐✱ ❝✬è✳ P❡r ❡s❡♠♣✐♦ ✐❧ s❡r✈✐③✐♦ ♣r♦❞♦tt♦ ❞❛ ❈❡s❛r❡ ♣r♦❢❡ss♦r❡✉♥✐✈❡rs✐t❛r✐♦ st❛ ♥❡❧ P■▲ ♣❡r q✉❛s✐ 4000e ❛❧ ♠❡s❡✳ ◆♦♥ s✐ ♣✉ò s♦st❡♥❡r❡ ❝❤❡ s✐❛❧♦ st❡ss♦ ❝❤❡ ♣❡r ✉♥❛ ♣❛rt✐t❛ ❞✐ ♠❡❧❡ ❝♦♥tr❛tt❛t❛ ❛❧ ♠❡r❝❛t♦ ❛❧❧✬✐♥❣r♦ss♦ ♣r✐♠❛❞❡❧❧✬❛❧❜❛✳ ■♥ q✉❛❧❝❤❡ ♠✐s✉r❛ è ❧♦ st❛t♦ ❝❤❡ ❞✐❝❤✐❛r❛ ✐❧ s✉♦ ✈❛❧♦r❡ ✭❝♦♠❡ ✐❧ ✈❛❧♦r❡❞❡✐ ♣♦❧✐t✐❝✐✱ s❡ ✈♦❣❧✐❛♠♦ ❢❛r❡ ✉♥ ❡s❡♠♣✐♦ ♣✐ù ❝♦❧♦r✐t♦✮✳ ❈❡s❛r❡ ♥♦♥ è ❝❡rt♦ ❝♦str❡tt♦❛❞ ❛♥❞❛r❡ ❛❧ ❧❛✈♦r♦✱ ♠❛ ❝❤✐ ❞❡✈❡ ♣r✐✈❛rs✐ ❞❡✐ ❜❡♥✐ ❝❤❡ ❧♦ st❛t♦ ✉t✐❧✐③③❛ ♣❡r ♣❛❣❛r❧♦❧♦ ❢❛ s♦❧♦ ♣❡r❝❤é ❣❧✐ è ❝♦♠❛♥❞❛t♦✱ ♣r♦♣r✐♦ ❝♦♠❡ ✐♥ ❈♦r❡❛✳ ❆tt❡♥③✐♦♥❡✱ ❛❜❜✐❛♠♦s❝❡❧t♦ ❝♦sì✱ ❛❜❜✐❛♠♦ s❝❡❧t♦ ❧♦ st❛t♦ ❡ ❧❡ t❛ss❡ ❡ ❧❡ r❡❣♦❧❡ ✲ ♠❛ ♣♦ss✐❜✐❧♠❡♥t❡ ❛♥❝❤❡✐♥ ❈♦r❡❛ ❧♦ ❤❛♥♥♦ ❢❛tt♦✦ ❊ ❞♦♣♣✐❛ ❛tt❡♥③✐♦♥❡✿ ♥♦♥ s♦♥♦ ✐♥ ❣✐♦❝♦ ❣✐✉❞✐③✐ ❞✐ ✈❛❧♦r❡✳❙✐ s❝❡❧❣♦♥♦ ❧❡ r❡❣♦❧❡ ❞❡❧ ❣✐♦❝♦✱ ♣♦ss✐❜✐❧♠❡♥t❡ ✐♥ ♠♦❞♦ ❞❡♠♦❝r❛t✐❝♦✱ ♠❛ ♣♦✐ ❛❧❧❡r❡❣♦❧❡ ❜✐s♦❣♥❛ st❛r❡✳

❈♦s❛ s✉❝❝❡❞❡ ❛❧❧♦r❛ q✉❛♥❞♦ ✐❧ ❣♦✈❡r♥♦ ❞❡❝✐❞❡ ❞✐ ❛✉♠❡♥t❛r❡ ❧❛ s♣❡s❛✱ ♣❡r ❡s❡♠✲♣✐♦ ❞❡❝✐❞❡ ❞✐ ❛✉♠❡♥t❛r❡ ✐❧ ❝❛r✐❝♦ ❞✐ ❧❛✈♦r♦ ❞✐ ❈❡s❛r❡❄ ❱❛ ❞❛ q✉❡❧❧♦ ❞❡❧❧❡ ♠❡❧❡❡ ❣❧✐ ❞✐❝❡ ✏❙❡♥t✐✱ ❞✐❡❝✐ ♣❡❞❛♥❡ ❧❡ ❞❡✈✐ ❞❛r❡ ❛ ♠❡ ❝❤❡ ♠✐ s❡r✈♦♥♦ ♣❡r ❈❡s❛r❡✳✑◗✉❡❧❧♦ ❣❧✐❡❧❡ ❞à s❡♥③❛ ❞✐s❝✉t❡r❡✱ ❛♥❝❤❡ ❧✉✐ ❤❛ ✈♦t❛t♦ ❧❡ r❡❣♦❧❡✱ ♠❛✳✳✳ ❜❡❤✱ ❈❡s❛r❡♣r❡❢❡r✐r❡❜❜❡ ♥♦♥ ✐♥❝♦♥tr❛r❧♦ q✉❡❧ ❣✐♦r♥♦✳✶✶ P❡r ✐♥❝✐s♦✱ ♠❛❣❛r✐ q✉❡❧❧♦ ❞❡❧❧❡ ♠❡❧❡♣♦✐ ❞❡❝✐❞❡ ❝❤❡ ✈✐st♦ ❝❤❡ ❣❧✐ s♦♥♦ ❛♥❞❛t❡ ✐♥ ❢✉♠♦ ❞✐❡❝✐ ♣❡❞❛♥❡ ❞♦♠❛♥✐ ❧❛✈♦r❛ ❞✐♣✐ù ♣❡r r❡❝✉♣❡r❛r❡✳ ❊ ✈❡❞r❡♠♦ ❝❤❡ q✉❡st✬✐❞❡❛ è ✐♥ ❡✛❡tt✐ ❛❧❧❛ ❜❛s❡ ❞✐ q✉❡❧❧♦ ❝❤❡✐♠♣❛r❡r❡♠♦ ❛❞ ❛♣♣r❡③③❛r❡ ❝♦♠❡ ✏♠♦❧t✐♣❧✐❝❛t♦r❡ ❞❡❧❧❛ s♣❡s❛ ♣✉❜❜❧✐❝❛✑✳ ▼❛ ♥♦♥❞✐♠❡♥t✐❝❤✐❛♠♦ ❝♦s❛ ❝✬è ❞✐❡tr♦✳

✶✳✻ ❊s❡r❝✐③✐

❊s❡r❝✐③✐♦ ✶✳✶✳ ▼♦❞✐✜❝❛ ❧✬❡s❡♠♣✐♦ ✶✳✶ ❛ss✉♠❡♥❞♦ ❝❤❡ ♣❡r ♣r♦❞✉rr❡ ✼✺ ❞✐ ♣❛♥❡s✐❛ ♥❡❝❡ss❛r✐♦ ✉t✐❧✐③③❛r❡ s♦❧♦ ✸✵ ❞❡❧❧❛ ❢❛r✐♥❛ ♣r♦❞♦tt❛✱ ✐ r✐♠❛♥❡♥t✐ ✶✵ ✈❡♥❣♦♥♦❝♦♥s✉♠❛t✐ ❝♦♠❡ ❜❡♥❡ ✜♥❛❧❡✳ ❚✉tt❡ ❧❡ ❛❧tr❡ ✐♣♦t❡s✐ ♥♦♥ ❝❛♠❜✐❛♥♦✳ ❘✐❝❛❧❝♦❧❛ ✐❧ P■▲❡ s♣✐❡❣❛ ❝♦s❛ è ❝❛♠❜✐❛t♦ ♥❡❧ ❝♦♥tr✐❜✉t♦ ❞❡❧ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦ ❞✐ ♦❣♥✐ s❡tt♦r❡✳

❊s❡r❝✐③✐♦ ✶✳✷✳ ■♥ q✉❡st❛ ❡❝♦♥♦♠✐❛ ❝✐ s♦♥♦ ❞✉❡ s❡tt♦r✐✱ ✐❧ s❡tt♦r❡ ❆✉t♦♠♦❜✐❧✐✉t✐❧✐③③❛ ❝♦♠❡ ✐♥♣✉t ✐❧ ♣r♦❞♦tt♦ ❞❡❧ s❡tt♦r❡ ❙✐❞❡r✉r❣✐❛ ❡ ✈❡♥❞❡ ✐❧ s✉♦ ♦✉t♣✉t ❛✐❝♦♥s✉♠❛t♦r✐ ✜♥❛❧✐✳ ❈♦♠♣❧❡t❛ ❧❛ s❡❣✉❡♥t❡ t❛❜❡❧❧❛ ❡ ❝❛❧❝♦❧❛ ❧❛ q✉♦t❛ ❞✐ P■▲ ❝❤❡ ✈❛✐♥ s❛❧❛r✐ ❡ ♣r♦✜tt✐✿

✶✶P❡r ❡ss❡r❡ ♣r❡❝✐s✐ ❧♦ st❛t♦ ❤❛ ✉♥✬❛❧tr❛ ♣♦ss✐❜✐❧✐tà✳ ❈❡rt❡ ✈♦❧t❡ ♣✉ò r❡♣✉t❛r❡ ♥♦♥ ♦♣♣♦rt✉♥♦♣r❡❧❡✈❛r❡ ❧❡ ♠❡❧❡ t♦✉t ❝♦✉rt ♣❡r❝❤é ♠❛❣❛r✐ ❣❧✐❡♥❡ ❤❛ ❣✐à ❝❤✐❡st❡ ❛❜❜❛st❛♥③❛✱ ❡ ❛❧❧♦r❛ s❡ ❧❡ ♣✉ò❢❛r❡ ♣r❡st❛r❡✳ ◆❡❧ ♣r✐♠♦ s❝❡♥❛r✐♦ ❛✈❡✈❛♠♦ ∆G = ∆T ✱ ♣✐ù t❛ss❡ s✉❜✐t♦❀ ✐♥ q✉❡st♦ ∆G = ∆B✱✐♥❝r❡♠❡♥t♦ ❞❡❧ ❞❡❜✐t♦✱ ❝✐♦è ♣✐ù t❛ss❡ ✐♥ ❢✉t✉r♦✳

✷✹ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

❙❡tt♦r❡Pr♦❞✉tt✐✈♦

❘✐❝❛✈✐❈♦st♦ ❇❡♥✐■♥t❡r♠❡❞✐

❈♦st♦▲❛✈♦r♦

Pr♦✜tt✐❱❛❧♦r❡

❆❣❣✐✉♥t♦

❙✐❞❡r✉r❣✐❛ ✶✺✵ ✵ ✶✵✵ ❄ ✶✺✵❆✉t♦♠♦❜✐❧✐ ✷✽✵ ✶✺✵ ❄ ✻✵ ❄

P■▲ ❄ ❄

❊s❡r❝✐③✐♦ ✶✳✸✳ ❆ss✉♠✐ ❝❤❡ ✉♥✬❡❝♦♥♦♠✐❛ ♣r♦❞✉❝❛ ●r❛♥♦✱ ❩✉❝❝❤❡r♦ ❡ ❇r✐♦❝❤❡❛ttr❛✈❡rs♦ ❧❡ ❢✉♥③✐♦♥✐ ❞✐ ♣r♦❞✉③✐♦♥❡

G(L) = 5L, Z(L) = 4L, B(L,Z,G) = min{L,Z/2, G/4},

❡ ❝❤❡ t✉tt♦ ✐❧ ❣r❛♥♦ ♣r♦❞♦tt♦ ❡❞ ✐ 5/8 ❞❡❧❧♦ ③✉❝❝❤❡r♦ ✈❡♥❣❛♥♦ ✉t✐❧✐③③❛t✐ ♥❡❧❧❛♣r♦❞✉③✐♦♥❡ ❞✐ ❜r✐♦❝❤❡✳ ❈♦♠♣❧❡t❛ ❧❛ s❡❣✉❡♥t❡ t❛❜❡❧❧❛✿

❙❡tt♦r✐◗✳tà

✉s❛t❛ ❞✐L

◗✳tà✉s❛t❛ ❞✐

G

◗✳tà✉s❛t❛ ❞✐

❩

◗✳tà♣r♦❞♦tt❛❞✐ G

◗✳tà♣r♦❞♦tt❛❞✐ ❩

◗✳tà♣r♦❞♦tt❛❞✐ B

●r❛♥♦ ✶✵✵✵ ❄❩✉❝❝❤❡r♦ ✶✵✵✵ ❄❇r✐♦❝❤❡ ✶✷✺✵ ❄ ❄ ❄

❊s❡r❝✐③✐♦ ✶✳✹✳ P❡r ❧✬❡❝♦♥♦♠✐❛ ❞❡❧❧✬❊s❡r❝✐③✐♦ ✶✳✸ ❛ss✉♠✐ ❝❤❡ ✐❧ ✈❡tt♦r❡ ❞✐ ♣r❡③③✐ s✐❛(pL, pZ , pG, pB) = (2, 3, 1, 12) ❡ ❝❛❧❝♦❧❛ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦✱ ❝♦st♦ ❞❡❧ ❧❛✈♦r♦✱ ♣r♦✜tt✐❡ P■▲✳ ❈❛❧❝♦❧❛ ❛♥❝❤❡ ❧❛ ♣❛rt✐t❛ ■❱❆ ✭❡♥tr❛t❡✱ ✉s❝✐t❡ ❡ ✈❡rs❛♠❡♥t✐✮ ❞❡✐ tr❡ s❡tt♦r✐✱❝♦♥ ❧✬■❱❆ ❛❧ 10% ❞❡❧ ✈❛❧♦r❡ ❛❣❣✐✉♥t♦✳

❊s❡r❝✐③✐♦ ✶✳✺✳ ■♥ q✉❡st❛ ❡❝♦♥♦♠✐❛ ❝✐ s♦♥♦ ❞✉❡ ✐♠♣r❡s❡ ❆ ❡ ❇ ✐ ❝✉✐ ❜✐❧❛♥❝✐ s♦♥♦❞❡s❝r✐tt✐ ♥❡❧❧❛ t❛✈♦❧❛ s❡❣✉❡♥t❡

■♠♣r❡s❛ ❆ ❇❈♦st✐ ❘✐❝❛✈✐ ❈♦st✐ ❘✐❝❛✈✐

❚❡ss✉t✐ ✺✵ ✸✵▲❡❣♥❛♠❡ ✶✵✵ ✼✵❚❛✈♦❧✐ ✷✵✵

P♦❧tr♦♥❡ ✶✺✵

❙❛❧❛r✐ ✽✵ ✶✽✵❆♠♠♦rt❛♠❡♥t✐ ✶✵ ✸✵

Pr♦✜tt✐ ✺✵ ✷✵

❈❛❧❝♦❧❛ ❧❛ ♣❡r❝❡♥t✉❛❧❡ ❞✐ ✐♠♣♦st❡ ✐♥❞✐r❡tt❡ ♥❡❧ P■▲✳

✶✳✻✳ ❊s❡r❝✐③✐ ✷✺

❊s❡r❝✐③✐♦ ✶✳✻✳ ❯♥❛ t❛ss❛ s✉❧❧❛ ✈❡♥❞✐t❛ ❛❧ ❞❡tt❛❣❧✐♦ ✭❘❡t❛✐❧ ❙❛❧❡s ❚❛①✱ ❘❙❚✮ s✐❛♣♣❧✐❝❛ s♦❧♦ ❛❧❧❡ tr❛♥s❛③✐♦♥✐ ❞✐ ❜❡♥✐ ✜♥❛❧✐✱ ❡ ♥♦♥ ❛ q✉❡❧❧❡ ❞✐ ❜❡♥✐ ✐♥t❡r♠❡❞✐✳ ✭✐✮❆ss✉♠❡♥❞♦ ❝❤❡ ✈♦❧✉♠✐ ❡ ♣r❡③③✐ ♥♦♥ ❝❛♠❜✐♥♦✱ q✉❛❧❡ ❛❧✐q✉♦t❛ ❞✐ ❘❙❚ ❣❛r❛♥t✐s❝❡❧♦ st❡ss♦ ❣❡tt✐t♦ ❞❡❧❧✬■❱❆ ❛❧ ✷✵✪❄ ❙♣✐❡❣❛✳ ✭✐✐✮ P❛r❛❣♦♥❛ ❧❡ ❞✉❡ t❛ss❡ ✐♥ t❡r♠✐♥✐ ❞✐✐♥❝❡♥t✐✈✐ ❛❧❧✬❡✈❛s✐♦♥❡✳

❊s❡r❝✐③✐♦ ✶✳✼✳ ✭PPP✮ ■❧ t❛ss♦ ❞✐ ❝❛♠❜✐♦ r❡❛❧❡ tr❛ ❞✉❡ ♣❛❡s✐ I ❡ J ✱ EJ/I ✱ è ❞❡✜♥✐t♦❝♦♠❡ ❧❡ ✉♥✐tà ❞✐ ✉♥ ♣❛♥✐❡r❡ X ❞✐s♣♦♥✐❜✐❧❡ ♥❡❧ ♣❛❡s❡ J ♥❡❝❡ss❛r✐❡ ❛❞ ❛❝q✉✐st❛r❡✉♥❛ ✉♥✐tà ❞❡❧ ♣❛♥✐❡r❡ ♥❡❧ ♣❛❡s❡ I. ❈♦♥ ❧❛ ♥♦t❛③✐♦♥❡ ♥❡❧ t❡st♦✱ 1

∣

∣XI ≃ EJ/I∣

∣XJ ✳■❧ t❛ss♦ ❞✐ ❝❛♠❜✐♦ ♥♦♠✐♥❛❧❡ EJ/I tr❛ ❧❡ ♠♦♥❡t❡ M I ❡❞ MJ ✉t✐❧✐③③❛t❡ ♥❡✐ ♣❛❡s✐ è❞❡✜♥✐t♦ ❞❛ 1

∣

∣M I ≃ EJ/I∣

∣MJ ✳ ■❧ ✈❡tt♦r❡ ❞❡✐ ♣r❡③③✐ ❞❡✐ ❜❡♥✐ ♥❡❧ ♣❛♥✐❡r❡ s♦♥♦r✐s♣❡tt✐✈❛♠❡♥t❡ pI ❡ pJ ✳ ✭✐✮ ■♥t❡r♣r❡t❛ ❧❛ r❡❧❛③✐♦♥❡ ❝❤❡ ❞❡✜♥✐s❝❡ ✐❧ t❛ss♦ ❞✐ ❝❛♠❜✐♦♥♦♠✐♥❛❧❡ ✭✐✐✮ ❚r♦✈❛ ❧❛ r❡❧❛③✐♦♥❡ ❞✐ s❝❛♠❜✐❛❜✐❧✐tà tr❛ ✶ ✉♥✐tà ❞✐ ♣❛♥✐❡r❡ X ♥❡❧ ♣❛❡s❡I ❡❞ M I . ✭✐✐✐✮ ❯t✐❧✐③③❛♥❞♦ ❧❛ ❧✐♥❡❛r✐tà✱ tr♦✈❛ ❧❛ r❡❧❛③✐♦♥❡ ❞✐ s❝❛♠❜✐❛❜✐❧✐tà tr❛ ✶✉♥✐tà ❞✐ ♣❛♥✐❡r❡ X ♥❡❧ ♣❛❡s❡ I ❡ ❧❛ ♠♦♥❡t❛ ❞❡❧ ♣❛❡s❡ J ✳ ✭✐✈✮ ❯t✐❧✐③③❛♥❞♦ ✐ r✐s✉❧t❛t✐♣r❡❝❡❞❡♥t✐✱ ❞✐♠♦str❛ ❝❤❡ EJ/I = EJ/I/PPP J/I ✱ ❞♦✈❡ PPP J/I ≡ pJX/pIX. ✭PPPè ❧✬❛❝r♦♥✐♠♦ ❞✐ P✉r❝❤❛s✐♥❣ P♦✇❡r P❛r✐t②✳✮

❊s❡r❝✐③✐♦ ✶✳✽✳ ❙✐ ❤❛♥♥♦ ❛ ❞✐s♣♦s✐③✐♦♥❡ ✐ s❡❣✉❡♥t✐ ❞❛t✐

❆♥♥♦◗✉❛♥t✐tà❈❛r♥❡

Pr❡③③♦❈❛r♥❡

◗✉❛♥t✐tà▲❛tt❡

Pr❡③③♦▲❛tt❡

✶✾✾✵ ✶✵✷✵ ✶✵ ✷✸✵✺ ✶✷✵✵✵ ✶✷✶✺ ✶✷ ① ②

❙✐ s❛ ✐♥♦❧tr❡ ❝❤❡ ♥❡❧ ✷✵✵✵ P■▲ ♥♦♠✐♥❛❧❡ ❡ r❡❛❧❡ ❛ ♣r❡③③✐ ✶✾✾✵ s♦♥♦ r✐s♣❡tt✐✈❛♠❡♥t❡✶✼✾✺✺ ❡ ✶✹✻✺✵✳ ❙✐ ❝❛❧❝♦❧✐♥♦ ♣r❡③③♦ ❡ q✉❛♥t✐tà ❞✐ ▲❛tt❡ ♥❡❧ ✷✵✵✵✳

❊s❡r❝✐③✐♦ ✶✳✾ ✭❊✛❡tt♦ s♦st✐t✉③✐♦♥❡✮✳ ❈♦♥s✐❞❡r❛ q✉❡st✐ ❞❛t✐ s✉❧❧❛ ♣r♦❞✉③✐♦♥❡ ❛♥✲♥✉❛❧❡ ❞✐ ✉♥✬❡❝♦♥♦♠✐❛✿

❆♥♥♦◗✉❛♥t✐tà❋r❛❣♦❧❡

Pr❡③③♦❋r❛❣♦❧❡

◗✉❛♥t✐tà▼❡❧❡

Pr❡③③♦▼❡❧❡

◗✉❛♥t✐tà❈✐❧✐❡❣❡

Pr❡③③♦❈✐❧✐❡❣❡

✶✾✾✾ ✽✵ ✶✷ ✶✷✵✵ ✵✳✽ ✻✵ ✺✷✵✵✵ ✶✵✵ ✶✵ ✶✵✵✵ ✶ ✺✵ ✼✷✵✵✶ ✶✶✵ ✽ ✽✵✵ ✶✳✷ ✹✵ ✶✵

✭✐✮ ❈❛❧❝♦❧❛ ❉❡✢❛t♦r❡ P■▲ ❡ ■♥❞✐❝❡ ❞❡✐ Pr❡③③✐ ❛❧ ❈♦♥s✉♠♦✱ ❝♦♥ ❛♥♥♦ ❜❛s❡ ✷✵✵✵✳✭✐✐✮ ❆ ❝♦s❛ è ❞♦✈✉t❛ ❧❛ ❞✐✛❡r❡♥③❛ tr❛ ✐ ❞✉❡ ✐♥❞✐❝✐❄ ✭❙✉❣❣✳ ✉♥♦ ♥♦♥ t✐❡♥❡ ❝♦♥t♦❞❡❧❧✬❡✛❡tt♦ s♦st✐t✉③✐♦♥❡✳✮

❊s❡r❝✐③✐♦ ✶✳✶✵✳ ❈♦♥s✐❞❡r❛ ❧❛ s❡❣✉❡♥t❡ ❡❝♦♥♦♠✐❛✱ ✐♥ ❝✉✐ s✐ ♣r♦❞✉❝♦♥♦ ✐ ❜❡♥✐A,B,C ❝♦♥ q✉❛♥t✐tà ❡ ♣r❡③③✐ q, p ❞❛t✐ ♥❡❧❧❛ s❡❣✉❡♥t❡ t❛❜❡❧❧❛✿

✷✻ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

❆♥♥♦ qA pA qB pB qC pC

✷✵✵✵ ✶✵✵✵ ✶✵✵ ✶✵✵ ✶✵✷✵ ✸✵ ✶✵✺✵✵✷✵✵✶ ✾✾✵ ✶✵✺ ✶✵✻ ✾✼✵ ✸✶ ✶✵✽✵✵

❈❛❧❝♦❧❛ ✐❧ t❛ss♦ ❞✐ ❝r❡s❝✐t❛ ❞❡❧ P■▲ r❡❛❧❡ ✭♣r❡③③✐ ❞❡❧ ✷✵✵✵✮ ❢r❛ ✐ ❞✉❡ ❛♥♥✐ ❡ ❧✬✐♥✢❛③✐♦♥❡♠✐s✉r❛t❛ ❝♦♥ ❧✬✐♥❞✐❝❡ ❞❡✐ ♣r❡③③✐ ❛❧ ❝♦♥s✉♠♦ ❡ ❝♦♥ ✐❧ ❞❡✢❛t♦r❡ ❞❡❧ P■▲✱ ❝♦♥ ❛♥♥♦❜❛s❡ ✷✵✵✵✳

❊s❡r❝✐③✐♦ ✶✳✶✶✳ ■♥ ✉♥ ♣❛❡s❡ s✐ ♣r♦❞✉❝❡ ♦r♦ ❣r❡③③♦ ❡ ❣✐♦✐❡❧❧✐ ❞✬♦r♦✳ ◆❡❧❧✬❛♥♥♦2000 ✐❧ s❡tt♦r❡ ♠✐♥❡r❛r✐♦ ♣r♦❞✉❝❡ 100kg ❞✐ ♦r♦ ✭✉s❛♥❞♦ s♦❧♦ ❧❛✈♦r♦✮✱ ✐❧ s❡tt♦r❡❣✐♦✐❡❧❧✐❡r♦ ✭❝❤❡ ✉s❛ 50kg ❞✐ ♦r♦ ❣r❡③③♦ ❡ ❧❛✈♦r♦ ♣❡r ❢❛❜❜r✐❝❛r❡ ❣✐♦✐❡❧❧✐✮ ❢❛tt✉r❛ 300✉♥✐tà✳ ■❧ ♣r❡③③♦ ❞❡❧❧✬♦r♦ è 1 ♣❡r ♦❣♥✐ kg✱ q✉❡❧❧♦ ❞✐ ✉♥ ❣✐♦✐❡❧❧♦ è 3✳ ◆❡❧❧✬❛♥♥♦ 2005❧❛ ♣r♦❞✉③✐♦♥❡ r❡st❛ ✐♥✈❛r✐❛t❛ ♠❛ ✐ ♣r❡③③✐ ❞✐✈❡♥t❛♥♦ 3 ❡ 3✱ r✐s♣❡tt✐✈❛♠❡♥t❡✳ ✭✐✮◗✉❛❧ è ✐❧ P■▲ ♥♦♠✐♥❛❧❡ ❞❡❧ ♣❛❡s❡ ♥❡❧ 2000❄ ✭✐✐✮ ◗✉❛❧ è ✐❧ P■▲ r❡❛❧❡ ♥❡❧ 2005 ❛♣r❡③③✐ ❞❡❧ 2000❄ ✭✐✐✐✮ ◗✉❛❧ è ✐❧ ❞❡✢❛t♦r❡ ❞❡❧ P■▲❄

❊s❡r❝✐③✐♦ ✶✳✶✷✳ ❉✐♠♦str❛ ❝❤❡ ❧✬✐♥❞✐❝❡ ❞❡✐ ♣r❡③③✐ ❛❧ ❝♦♥s✉♠♦ ♣✉ò ❡ss❡r❡ s❝r✐tt♦❝♦♠❡

PLAt =

∑

imiRi,t/

∑

imi,

❞♦✈❡ mi ≡ pi,bXi,b ❡❞ Ri,t ≡ pi,t/pi,b. ■♥t❡r♣r❡t❛✳

❊s❡r❝✐③✐♦ ✶✳✶✸✳ ❉✐♠♦str❛ ❝❤❡ ✐❧ t❛ss♦ ❞✐ ❝r❡s❝✐t❛ ❞❡❧ P■▲ r❡❛❧❡ ❛ ✈❛❧♦r✐ ❝♦♥❝❛t❡♥❛t✐❞❡❧❧✬❡s❡♠♣✐♦ ✶✳✶✹ ♥♦♥ ❞✐♣❡♥❞❡ ❞❛❧❧✬❛♥♥♦ ❞✐ r✐❢❡r✐♠❡♥t♦✳

❊s❡r❝✐③✐♦ ✶✳✶✹ ✭▼✐❝r♦❡❝♦♥♦♠✐❛✿ ✐❧ ✈✐♥❝♦❧♦ ❞✐ ❜✐❧❛♥❝✐♦ ✐♥t❡rt❡♠♣♦r❛❧❡✳✮✳ ◗✉❡st♦❡s❡r❝✐③✐♦ r✐♣r❡♥❞❡ ✐❧ ♣r♦❜❧❡♠❛ ❞✐ ❛❧❧♦❝❛③✐♦♥❡ ✐♥t❡rt❡♠♣♦r❛❧❡ ❞❡❧ ❝♦♥s✉♠♦ ❝❤❡ ❢❛♣❛rt❡ ❞❡❧ ❝♦rs♦ ❞✐ ♠✐❝r♦❡❝♦♥♦♠✐❛✱ ❛❣❣✐✉♥❣❡♥❞♦ ❧✬✐♥✢❛③✐♦♥❡✳ ■❧ ❝♦♥s✉♠❛t♦r❡ ❛❧t❡♠♣♦ t ❞❡✈❡ s❝❡❣❧✐❡r❡ ✉♥ ♣❛♥✐❡r❡ (ct, ct+1) ❞♦✈❡ ct ❡ ct+1 ❞❡♥♦t❛♥♦ ❝♦♥s✉♠♦ ♥❡❧♣❡r✐♦❞♦ ❝♦rr❡♥t❡ ❡ ❢✉t✉r♦✳ ■❧ r❡❞❞✐t♦ è ❝♦st✐t✉✐t♦ ❞❛ mt ✉♥✐tà ❞✐ ♠♦♥❡t❛ ❛ t ❡❞mt+1 ❛ t + 1✳ ■❧ ♣r❡③③♦ ❞✐ ct è pt✱ ❡ ✐♥t❡r❡ss❡ ♥♦♠✐♥❛❧❡ ❡ ✐♥✢❛③✐♦♥❡ s♦♥♦ it ❡ πt✳◗✉❛❧ è ✐❧ ✈✐♥❝♦❧♦ ❞✐ ❜✐❧❛♥❝✐♦ ❞❡❧ ♥♦str♦ ❝♦♥s✉♠❛t♦r❡❄

❊s❡r❝✐③✐♦ ✶✳✶✺✳ ◆❡❧❧❛ s❡❣✉❡♥t❡ ❡❝♦♥♦♠✐❛ s✐ ♣r♦❞✉❝♦♥♦ ❛✉t♦ ❡ ♠♦t♦✱ ❡ ♠♦t♦r✐✭♣❡r ❛✉t♦ ❡ ♠♦t♦✮✱ ❝♦♠❡ ❞❡s❝r✐tt♦ ♥❡❧❧❛ s❡❣✉❡♥t❡ t❛❜❡❧❧❛ ♣❡r ❣❧✐ ❛♥♥✐ ✷✵✵✵ ❡ ✷✵✵✶✳◆❡✐ s❡tt♦r✐ ❆✉t♦ ❡ ▼♦t♦ s✐ ✉t✐❧✐③③❛ ✉♥ ♠♦t♦r❡ ♣❡r ✉♥✐tà ♣r♦❞♦tt❛✳ ❈❛❧❝♦❧❛ ✐❧ t❛ss♦❞✐ ❝r❡s❝✐t❛ ❞❡❧ P■▲ r❡❛❧❡ ✭❜❛s❡ ✷✵✵✵✮ ♥❡❧ ♣❡r✐♦❞♦✳

▼♦t♦r✐ ❆✉t♦ ▼♦t♦

p q p q p q

✷✵✵✵ ✺ ✶✺✵ ✷✵ ✶✵✵ ✶✵ ✺✵✷✵✵✶ ✶✵ ✷✸✵ ✷✵ ✶✵✶ ✶✷ ✻✵

✶✳✻✳ ❊s❡r❝✐③✐ ✷✼

❊s❡r❝✐③✐♦ ✶✳✶✻✳ ❯♥ ✐♥❞✐✈✐❞✉♦ ♣❡r❝❡♣✐✈❛ ♥❡❧ ✷✵✵✵ ✉♥♦ st✐♣❡♥❞✐♦ ❞✐ 15000e❛❧❧✬❛♥♥♦✱✈♦❣❧✐❛♠♦ s❛♣❡r❡ ❛ ❝❤❡ st✐♣❡♥❞✐♦ ❡q✉✐✈❛❧❡ ♥❡❧ ✷✵✶✺ ✐♥ t❡r♠✐♥✐ ❞✐ ♣♦t❡r❡ ❞✬❛❝q✉✐st♦✳■❧ t❛ss♦ ❞✐ ✐♥✢❛③✐♦♥❡ ❞❛❧❧✬❛♥♥♦ t ❛ t+ 1✱ ❝♦♥ 2000 ≤ t ≤ 2014✱ è st❛t♦ ❞❡❧ 2% ❞❛❧✷✵✵✵ ❛❧ ✷✵✵✾ ❡ ❞❡❧❧♦ 0.5% ❞❛❧ ✷✵✶✵ ❛❧ ✷✵✶✺✳ ❘✐❝♦r❞❛ ❝❤❡ πt,t+1 ≡ Pt+1/Pt − 1✱ ❡❛ss✉♠✐ ❝❤❡ Pt = PLA

t ✱ ❧✬✐♥❞✐❝❡ ❞✐ ▲❛s♣❡②r❡s✳

❊s❡r❝✐③✐♦ ✶✳✶✼✳ ❈♦♥s✐❞❡r❛ ❞✉❡ ❇✉♦♥✐ ❖r❞✐♥❛r✐ ❞❡❧ ❚❡s♦r♦ ❛❞ ✉♥ ❛♥♥♦ ❡♠❡ss✐ ✐♥❞❛t❡ ❞✐✈❡rs❡✳ ■❧ ♣r✐♠♦ è st❛t♦ ❡♠❡ss♦ ✐❧ ♣r✐♠♦ ❣❡♥♥❛✐♦ ✷✵✶✵ ❡ ✈❡♥❞✉t♦ ❛❧ ♣r❡③③♦97 ✭♣❛❣❛ ✶✵✵ ✐❧ ♣r✐♠♦ ❣❡♥♥❛✐♦ ✷✵✶✶✮✱ ✐❧ s❡❝♦♥❞♦ ✐❧ ♣r✐♠♦ ❣❡♥♥❛✐♦ ✷✵✶✷ ❡ ✈❡♥❞✉t♦❛❧ ♣r❡③③♦ 96✳ ■❧ t❛ss♦ ❞✐ ✐♥✢❛③✐♦♥❡ ❛♥♥✉❛❧❡ ♥❡❧ ✷✵✶✵ è st❛t♦ ❞❡❧❧♦ 0.5%✱ ♠❡♥tr❡ ♥❡❧2012 è st❛t♦ ❞❡❧ 2%✳ ◗✉❛❧❡ ❞❡✐ ❞✉❡ t✐t♦❧✐ ❤❛ ❛✈✉t♦ ✉♥ r❡♥❞✐♠❡♥t♦ r❡❛❧❡ ♠❛❣❣✐♦r❡❄

❊s❡r❝✐③✐♦ ✶✳✶✽✳ ❊s❡r❝✐③✐♦ s❡♠♣❧✐❝❡ s✉ ✉♥❛ ❝♦s❛ ❝❤❡ ♥♦♥ ❛❜❜✐❛♠♦ ❢❛tt♦✿ r✐❞✉③✐♦♥❡❞❡❧ r❛♣♣♦rt♦ ❉❡❜✐t♦✴P■▲✳ ■♥❞✐❝❛♥❞♦ ❝♦♥ B ✐❧ ❞❡❜✐t♦ ♣✉❜❜❧✐❝♦ ❛❜❜✐❛♠♦ ♣❡r ❞❡✜♥✐✲③✐♦♥❡ ❝❤❡ ❧✬✐♥❝r❡♠❡♥t♦ ❞❡❧ ❞❡❜✐t♦ è ❞❛t♦ ❞❛ ❞❡✜❝✐t ♣r✐♠❛r✐♦ ♣✐ù ♦♥❡r✐ ❞❛ ✐♥t❡r❡ss✐♣❛ss✐✈✐✿ ∆B = G−T + iB✳ ❉❡♥♦t❛ ❝♦♥ PY ✐❧ P■▲ ♥♦♠✐♥❛❧❡ ✭❧✐✈❡❧❧♦ ❞❡✐ ♣r❡③③✐ ♣❡rr❡❞❞✐t♦ r❡❛❧❡✮ ❡ ♣♦♥✐ b = B/PY ❡ d = (G − T )/PY ✲ r✐s♣❡tt✐✈❛♠❡♥t❡ r❛♣♣♦rt♦❞❡❜✐t♦✴P■▲ ❡ r❛♣♣♦rt♦ ❞❡✜❝✐t ♣r✐♠❛r✐♦ s✉ P■▲✳ ❆❜❜✐❛♠♦ ❛❧❧♦r❛

∆B

PY= d+ ib.

✭✐✮ P❛rt❡♥❞♦ ❞❛ B = bPY ♣r❡♥❞✐ ✐❧ ❞✐✛❡r❡♥③✐❛❧❡ ❡ ♦tt✐❡♥✐ ✉♥❛ ❡s♣r❡ss✐♦♥❡❛♣♣r♦ss✐♠❛t❛ ♣❡r ∆B ✭❙✉❣❣✳ ❞❡✈✐ s♦❧♦ r✐❝♦r❞❛r❡ ❝❤❡ ∆f ≈ fx∆x+ fy∆y + fz∆z❡ s❝r✐✈❡r❡ ✉❣✉❛❧❡ ✐♥✈❡❝❡ ❞✐ ≈ tr❛s❝✉r❛♥❞♦ ❧✬❡rr♦r❡✮✳ ❉✐✈✐❞✐ ♣❡r PY ❡ ✉❣✉❛❣❧✐❛ ❧❡❞✉❡ ❡s♣r❡ss✐♦♥✐ ❝❤❡ ❛❜❜✐❛♠♦ ♣❡r ∆B/PY ✳ P♦♥❡♥❞♦ π = ∆P/P, γ(Y ) = ∆Y/Y❡❞ r = i− π ♠♦str❛ ❝❤❡ ♦tt✐❡♥✐

∆b = d+ (r − γ(Y ))b.

✭✐✐✮ P❡r ❧❛ ●r❡❝✐❛ ✷✵✶✹ ✐ ❞❛t✐ ❡r❛♥♦ ♣✐ù ♦ ♠❡♥♦✿ b = 175%, γ(Y ) = 0.8%, r =0.5%✳ ❈❛❧❝♦❧❛ d t❛❧❡ ❝❤❡ b ♣❛ss✐ ❞❛❧ 175% ❛❧ 173%✳ ✭❘✳ −1.475%✱ ✉♥ ❛✈❛♥③♦♣r✐♠❛r✐♦ ❞✐ ❝✐r❝❛ 1.5% ❞❡❧ P■▲✮❊s❡r❝✐③✐♦ ✶✳✶✾ ✭P❛rt✐t❛ ■❱❆✮✳ ▲❛ ❝♦♥t❛❜✐❧✐tà ❞❡❧❧✬❡❝♦♥♦♠✐❛ s✐❛ s♣❡❝✐✜❝❛t❛ ❞❛❧❧❛s❡❣✉❡♥t❡ t❛❜❡❧❧❛✱ ❝❤❡ ❞❡s❝r✐✈❡ q✉❛♥t✐tà✿

❙❡tt♦r❡Pr♦❞✉tt✐✈♦

▲❛✈♦r♦ ❇❡♥✐ ■♥t❡r♠❡❞✐ Pr♦❞♦tt♦

S1 S2 S3

S1 ✶✵✵✵ ✵ ✵ ✵ ✷✵✵✵S2 ✽✵✵ ✷✵✵✵ ✵ ✵ ✶✽✵✵S3 ✼✺✵ ✵ ✼✺✵ ✵ ✼✺✵S4 ✶✵✵ ✵ ✷✵✵ ✹✵✵ ✶✵✵

✷✽ ❈❛♣✐t♦❧♦ ✶✳ P■▲✱ ♣r❡③③✐ ❡ t❛ss✐

Pr❡③③✐✿ pL = 1.5, p1 = 1, p2 = 2, p3 = 4, p4 = 30✳ ❈♦♠♣❧❡t❛ ❧❛ s❡❣✉❡♥t❡t❛❜❡❧❧❛

❙❡tt♦r❡ ❘✐❝❛✈✐❈♦st✐ ❇❡♥✐■♥t❡r♠❡❞✐

❱❆ ❙❛❧❛r✐ Pr♦✜tt✐■❱❆ ❛❧ 10%

❊♥tr❛t❡ ❯s❝✐t❡ ❱❡rs❛♠❡♥t✐S1

S2

S3

S4

❚♦t❛❧❡

❊s❡r❝✐③✐♦ ✶✳✷✵ ✭❚❛♥t♦ ♣❡r ❡ss❡r❡ s✐❝✉r✐✮✳ ❙❡ 1Kg ❞✐ ♠❡❧❡ ❝♦st❛ ✐♥ ■t❛❧✐❛ 1.5e✱1e è ❝♦♥✈❡rt✐❜✐❧❡ ✐♥ 1.2$ ❡ ✐❧ ♣r❡③③♦ ❞✐ 1Kg ❞✐ ♠❡❧❡ ✐♥ ❯❙❆ è ❞✐ 1.6$✱ ✉♥ ❝❤✐❧♦ ❞✐♠❡❧❡ ✐t❛❧✐❛♥❡ è s❝❛♠❜✐❛❜✐❧❡ ❝♦♥ q✉❛♥t✐ ❝❤✐❧✐ ❞✐ ♠❡❧❡ ✐♥ ❯❙❆❄

❊s❡r❝✐③✐♦ ✶✳✷✶ ✭P■▲ ❡ ♠♦❞❡❧❧♦ ❞✐ ❝r❡s❝✐t❛✮✳ ■ ❜❡♥✐ ♥❡❧❧✬❡❝♦♥♦♠✐❛ q✉✐ ❝♦♥s✐❞❡r❛t❛s♦♥♦✿ g ❣r❛♥♦✱ f ❢❡rr♦✱ L ❧❛✈♦r♦✱ p ♣❛♥❡✱ mi ♠❛❝❝❤✐♥❛r✐ ✈❡❝❝❤✐ ❞✐ i ❛♥♥✐✱ i ≥ 0✭m0 s♦♥♦ ♥✉♦✈✐✮✳ ❆❧❧❛ ✜♥❡ ❞❡❧❧✬❛♥♥♦ t − 1 ❧♦ st♦❝❦ ❞✐ ❝❛♣✐t❛❧❡✱ ✐♥❝❧✉s❡ ❧❡ s❝♦rt❡❞✐ ❣r❛♥♦✱ è✿ 10m0, 18m1, 10m2, 50g ✭❝✐♦è ✶✵ ✉♥✐tà ❞✐ m0 ❡❝❝❡t❡r❛✮❀ s✐❛ Kt−1 ✐❧ s✉♦✈❛❧♦r❡✳ ▲❛ t❛❜❡❧❧❛ s❡❣✉❡♥t❡ ❞❡s❝r✐✈❡ ✐ ♣r♦❝❡ss✐ ♣r♦❞✉tt✐✈✐ ❞❡❧❧✬❛♥♥♦ t ❡ r✐♣♦rt❛ ✐♣r❡③③✐✳ ❈♦♠❡ st♦❝❦ ❞✐ ❝❛♣✐t❛❧❡ ❛❧❧❛ ✜♥❡ ❞❡❧❧✬❛♥♥♦✱ ❞✐ ✈❛❧♦r❡ Kt✱ ♦❧tr❡ ✐ ♠❛❝❝❤✐♥❛r✐♥✉♦✈✐ ❡ q✉❡❧❧✐ ♣r❡✲❡s✐st❡♥t✐ ✐♥✈❡❝❝❤✐❛t✐ ❞✐ ✉♥ ❛♥♥♦ tr♦✈✐❛♠♦ s❝♦rt❡ ❞✐ ❣r❛♥♦ ♣❡r60g✳

❇❡♥✐ ✐♠♣✐❡❣❛t✐ ♥❡✐ ♣r♦❝❡ss✐ ♣r♦❞✉tt✐✈✐

g f L p m0 m1 m2 m3 ❇❡♥✐ Pr♦❞♦tt✐

✺✵ ✺ ✶✵ ✺ 200g✶✵ ✷✵ ✺ ✽ 100f

✼✺ ✷✵ ✺ 200p✹✵ ✷✵ ✶✵ ✺ 25m0

Pr❡③③✐ ✶ ✹ ✼ ✷ ✶✺ ✼ ✸ ✵

❈❛❧❝♦❧❛✿ ✐❧ P■▲ Yt ❀ ❧❛ q✉♦t❛ ❞✐ r❡❞❞✐t✐ ❞❛ ❧❛✈♦r♦❀ ✐❧ ✈❛❧♦r❡ ❞✐ ❝♦♥s✉♠✐ Ct ❡✐♥✈❡st✐♠❡♥t✐ It ❀ ✐❧ ✈❛❧♦r❡ ❞❡❣❧✐ st♦❝❦ ❞✐ ♠❛❝❝❤✐♥❛r✐✱ Kt−1 ❡ Kt ❀ ✐❧ ✈❛❧♦r❡ ❞❡❣❧✐❛♠♠♦rt❛♠❡♥t✐ Dt✱ ✈❡r✐✜❝❛♥❞♦ ❝❤❡ Kt −Kt−1 = It −Dt❀ ❡ ✐❧ P■◆✳