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SLIDE 1

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❆ ❙tr✉❝t✉r❛❧ ▼♦❞❡❧ ♦❢ ■♥❢♦r♠❛❧✐t② ✇✐t❤ ❈♦♥str❛✐♥❡❞ ❊♥tr❡♣r❡♥❡✉rs❤✐♣

P✐❡rr❡ ◆❣✉✐♠❦❡✉

  • ❡♦r❣✐❛ ❙t❛t❡ ❯♥✐✈❡rs✐t② ✲ ❯❙❆

✭♥♥❣✉✐♠❦❡✉❅❣s✉✳❡❞✉✮ ❯◆❯✲❲■❉❊❘ ❈♦♥❢❡r❡♥❝❡ ♦♥ ✏P✉❜❧✐❝ ❊❝♦♥♦♠✐❝s ❢♦r ❉❡✈❡❧♦♣♠❡♥t✑

▼❛♣✉t♦✱ ❏✉❧② ✻✱ ✷✵✶✼

✶ ✴ ✹✵

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SLIDE 2

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦♥t❡①t

❊♥tr❡♣r❡♥❡✉rs❤✐♣ ✲❃ ❝r✉❝✐❛❧ ✐♥❣r❡❞✐❡♥t ✐♥ ♣r♦♠♦t✐♥❣ ❛♥❞ s✉st❛✐♥✐♥❣ ❡❝♦♥♦♠✐❝ ❣r♦✇t❤✿ ♣♦t❡♥t✐❛❧ ❢♦r ❝r❡❛t✐♥❣ ❥♦❜s✱ ❞❡❧✐✈❡r✐♥❣ ✐♥♥♦✈❛t✐♦♥ ❛♥❞ r❛✐s✐♥❣ ♣r♦❞✉❝t✐✈✐t②✳ ■ss✉❡ ✇✐t❤ ❞❡✈❡❧♦♣✐♥❣ ❝♦✉♥tr✐❡s ✲❃ ❡①✐st❡♥❝❡ ♦❢ ❛ ❧❛r❣❡ ✐♥❢♦r♠❛❧ s❡❝t♦r ✭✻✵✲✾✵✪ ♦❢ ✇♦r❦❢♦r❝❡✮ ❋♦r♠❛❧ ❡♥t❡r♣r✐s❡s ✲❃ ❤✐❣❤❡r ❣r♦✇t❤ ♣r♦s♣❡❝ts✱ ❤✐❣❤❡r ♣r♦❞✉❝t✐✈✐t② ❛♥❞ ✐♥❝♦♠❡ ♣♦t❡♥t✐❛❧✱ ❣❡♥❡r❛t❡ t❛① r❡✈❡♥✉❡s ■♥❢♦r♠❛❧ ❡♥t❡r♣r✐s❡s ✲❃ ❣❡♥❡r❛❧❧② ❧♦✇✲s❝❛❧❡✱ ❧❛r❣❡❧② ✉♥t❛①❡❞ ✭s❤♦rt❢❛❧❧s ✐♥ ❣♦✈❡r♥♠❡♥t r❡✈❡♥✉❡s✮✳ ✲❃ ■♠♣♦rt❛♥t ❛✈❡♥✉❡ ❢♦r ❥♦❜ ❝r❡❛t✐♦♥✱✐♥❝✉❜❛t♦r ❢♦r ❜✉s✐♥❡ss ♣♦t❡♥t✐❛❧❀ st❡♣♣✐♥❣ st♦♥❡ ❢♦r ❛❝❝❡ss✐❜✐❧✐t② t♦ t❤❡ ❢♦r♠❛❧ ❡❝♦♥♦♠② ✭■▲❖ ✷✵✵✷✱ ❈❛♥♦✲❯r❜✐♥❛ ✷✵✶✺✮✳

✷ ✴ ✹✵

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SLIDE 3

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❊①❛♠♣❧❡s ♦❢ ■♥❢♦r♠❛❧ ❆❝t✐✈✐t✐❡s

■♥❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉rs✿ s♠❛❧❧ ❛♥❞ ♠❡❞✐✉♠ s✐③❡ ❡♥t❡r♣r✐s❡s✳

✸ ✴ ✹✵

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SLIDE 4

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❊①❛♠♣❧❡s ♦❢ ■♥❢♦r♠❛❧ ❆❝t✐✈✐t✐❡s

❙✉❜s✐st❡♥❝❡ ❛❝t✐✈✐t✐❡s✿ ❙tr❡❡t ✈❡♥❞✐♥❣❀ ❝❛❧❧ ❜♦①❡s

✹ ✴ ✹✵

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SLIDE 5

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❊①❛♠♣❧❡s ♦❢ ■♥❢♦r♠❛❧ ❆❝t✐✈✐t✐❡s

❙✉❜s✐st❡♥❝❡ ❛❝t✐✈✐t✐❡s✿ ❜❛❝❦②❛r❞ ♠❛♥✉❢❛❝t✉r✐♥❣ ✲ ✉♥♣❛✐❞ ❢❛♠✐❧② ✇♦r❦

✺ ✴ ✹✵

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SLIDE 6

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❘❡s❡❛r❝❤ ◗✉❡st✐♦♥

❍♦✇ ❞♦ ✐♥❞✐✈✐❞✉❛❧ ❛♥❞ ✐♥st✐t✉t✐♦♥❛❧ ❛ttr✐❜✉t❡s ❞r✐✈❡ ❡♥tr❡♣r❡♥❡✉r✐❛❧ ❝❤♦✐❝❡ ❛♥❞ t❤❡ ❢♦r♠❛t✐♦♥ ♦❢ ✐♥❢♦r♠❛❧ ✈❡rs✉s ❢♦r♠❛❧ ✜r♠s ✐♥ ❞❡✈❡❧♦♣✐♥❣ ❝♦✉♥tr✐❡s❄ ❲❤❛t ♣♦❧✐❝✐❡s ❝❛♥ ♣r♦♠♦t❡ ❡♥tr❡♣r❡♥❡✉rs❤✐♣ ❛♥❞ ✐♥❝r❡❛s❡ t❤❡ s❤❛r❡ ♦❢ ❢♦r♠❛❧ ✜r♠s r❡❧❛t✐✈❡ t♦ ✐♥❢♦r♠❛❧ ✜r♠s❄ ❲❤❛t ❛r❡ t❤❡♥ t❤❡ ❝♦♥s❡q✉❡♥❝❡s ✐♥ t❡r♠s ♦❢ ♦✉t♣✉t ❛♥❞ ♣r♦❞✉❝t✐✈✐t②❄

✻ ✴ ✹✵

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SLIDE 7

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❖✈❡r✈✐❡✇ ♦❢ ▼❡t❤♦❞♦❧♦❣②

■ ❞❡✈❡❧♦♣ ❛ t❤❡♦r❡t✐❝❛❧ ♠♦❞❡❧ ♦❢ ❢♦r♠❛❧ ❛♥❞ ✐♥❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉rs❤✐♣ ✉♥❞❡r ♣❛rt✐❛❧ ❡q✉✐❧✐❜r✐✉♠✱ ✇❤✐❝❤ ✐s t❤❡♥ ❡st✐♠❛t❡❞ ❜② ●▼▼ ✉s✐♥❣ ❞❛t❛ ❢r♦♠ ❈❛♠❡r♦♦♥✳ ❉❡❝✐s✐♦♥✲♠❛❦✐♥❣ ♣r♦❝❡ss ❞❡♣❡♥❞s ♦♥ ❜♦t❤ ✐♥❞✐✈✐❞✉❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ✭s❦✐❧❧s ❛♥❞ ✐♥✐t✐❛❧ ✇❡❛❧t❤ ❡♥❞♦✇♠❡♥t✮ ❛♥❞ ✐♥st✐t✉t✐♦♥❛❧ ❢❛❝t♦rs ✭❡♥tr② ❝♦sts✱ t❛①❛t✐♦♥✱ ❡♥❢♦r❝❡♠❡♥t ❛♥❞ ❞❡❣r❡❡ ♦❢ ✜♥❛♥❝✐❛❧ ❢r✐❝t✐♦♥s✮✳ ❚❤❡ ❡st✐♠❛t❡❞ ♠♦❞❡❧ ✐s t❤❡♥ ✉s❡❞ ✐♥ ❝♦✉♥t❡r❢❛❝t✉❛❧ ♣♦❧✐❝② s✐♠✉❧❛t✐♦♥s t♦ q✉❛♥t✐❢② t❤❡ ✐♠♣❛❝t ♦❢ s❡✈❡r❛❧ ♣♦❧✐❝✐❡s ✭r❡❣✐str❛t✐♦♥ ❛♥❞ t❛① r❡❢♦r♠s✱ ✐♥❝r❡❛s❡❞ ❡♥❢♦r❝❡♠❡♥t✱ ❡t❝✳✮✱ ♦♥ ✐♥❢♦r♠❛❧✐t② ❛♥❞ ❛❣❣r❡❣❛t❡ ✐♥❝♦♠❡✳

✼ ✴ ✹✵

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SLIDE 8

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

Pr❡✈✐❡✇ ♦❢ ❋✐♥❞✐♥❣s

❊✈✐❞❡♥❝❡ ♦❢ ❛ ♥♦♥✲♠♦♥♦t♦♥✐❝✱ ❯✲s❤❛♣❡❞✱ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ❡♥tr❡♣r❡♥❡✉rs✬ ❡❞✉❝❛t✐♦♥ ❛♥❞ t❤❡✐r ❞❡❝✐s✐♦♥ t♦ ❢♦r♠❛❧✐③❡✳ ❊✈✐❞❡♥❝❡ t❤❛t ✐♥✐t✐❛❧ ✇❡❛❧t❤ ❛♥❞ ❛✈❡r❛❣❡ ❡❞✉❝❛t✐♦♥ ❞r✐✈❡ ✐♥❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉rs❤✐♣ ✇❤✐❧❡ ❤✐❣❤❡r ❡❞✉❝❛t✐♦♥ ❛♥❞ ♣❛r❡♥t✬s ❡♥tr❡♣r❡♥❡✉r✐❛❧ st❛t✉s ❞❡t❡r♠✐♥❡ ❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉rs❤✐♣✳ ❈♦✉♥t❡r❢❛❝t✉❛❧ ❡✈✐❞❡♥❝❡ t❤❛t ✇❤✐❧❡ r❡❣✐str❛t✐♦♥ ❛♥❞ t❛① r❡❢♦r♠s ❣❡♥❡r❛t❡ s✉❜st❛♥t✐❛❧ ❡♥t❡r♣r✐s❡ ❝r❡❛t✐♦♥✱ ✐♥❝r❡❛s❡❞ ❢♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ ❛❣❣r❡❣❛t❡ ✐♥❝♦♠❡ ❣❛✐♥s✱ ❛ ♣✉r❡ ❡♥❢♦r❝❡♠❡♥t ♣♦❧✐❝② ❛❣❛✐♥st ✐♥❢♦r♠❛❧✐t② ❤❛s ❛♥ ♦✈❡r❛❧❧ ♣❡r✈❡rs❡ ❡✛❡❝t✳

✽ ✴ ✹✵

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SLIDE 9

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

✶ ▼♦❞❡❧ ❉❡s❝r✐♣t✐♦♥ ✷ ❙tr✉❝t✉r❛❧ ❊st✐♠❛t✐♦♥ ❛♥❞ ❚❡st✐♥❣ ✸ ❈♦✉♥t❡r❢❛❝t✉❛❧ ❙✐♠✉❧❛t✐♦♥s ✹ ❈♦♥❝❧✉s✐♦♥ ✾ ✴ ✹✵

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SLIDE 10

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

▼♦❞❡❧ ❉❡s❝r✐♣t✐♦♥

✶✵ ✴ ✹✵

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SLIDE 11

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❋r❛♠❡✇♦r❦

❚❤r❡❡ ♦❝❝✉♣❛t✐♦♥s✿ ❲❛❣❡✲✇♦r❦✱ ❋♦r♠❛❧ ❊♥tr❡♣r❡♥❡✉rs❤✐♣✱ ■♥❢♦r♠❛❧ ❊♥tr❡♣r❡♥❡✉rs❤✐♣✳ ❆❣❡♥ts ❞✐✛❡r ✇✐t❤ t❤❡✐r ✐♥✐t✐❛❧ ✇❡❛❧t❤ ❡♥❞♦✇♠❡♥t z✱ ❛♥❞ t❤❡✐r ❡♥tr❡♣r❡♥❡✉r✐❛❧ s❦✐❧❧s θ ❞✐str✐❜✉t❡❞ ✇✐t❤ ❈❉❋ G(θ) ✳ ❲❛❣❡✇♦r❦❡rs r❡❝❡✐✈❡ ❛ ✜①❡❞ ✐♥❝♦♠❡ w > ✵✳ ❊♥tr❡♣r❡♥❡✉rs ♣r♦❞✉❝❡ ❣♦♦❞s ❛❝❝♦r❞✐♥❣ t♦ ❛ ❈♦❜❜✲❉♦✉❣❧❛s ♣r♦❞✉❝t✐♦♥ t❡❝❤♥♦❧♦❣② ❣✐✈❡♥ ❜② y = θkαlβǫ ✭✶✮ ✇❤❡r❡ α, β ∈ (✵, ✶)✱ ❛♥❞ γ = α + β < ✶✳

✶✶ ✴ ✹✵

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SLIDE 12

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❋♦r♠❛❧ ❊♥tr❡♣r❡♥❡✉rs❤✐♣

❚❛①❡s ❛♥❞ r❡❣✐str❛t✐♦♥ ❝♦st✳ ❚❤❡ ❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉r✬s ♣r♦❜❧❡♠✿ πF(θ) = ♠❛①

k≥✵,l≥✵

  • (✶ − τ)
  • θkαlβ − wl − rk
  • − rc
  • ✇❤❡r❡✿

✲ τ ✐s t❤❡ t❛① r❛t❡✱ r ✐s t❤❡ ✐♥t❡r❡st r❛t❡ ✲ c ✐s t❤❡ r❡❣✐str❛t✐♦♥ ❝♦st ✭s✉♥❦ ❝♦st✮✳ ■♥❝❧✉❞❡s ❢❡❡s✱ ❜r✐❜❡s✱ ❛❞♠✐♥✐str❛t✐✈❡ ❞❡❧❛②s✱ ❡t❝✳ ❚❤❡ ❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉r✬s ❡①♣❡❝t❡❞ ♣❛②♦✛ ✐s πF(θ) = (✶ − τ)(✶ − γ)θ

✶ ✶−γ

α r

  • α

✶−γ β

w

  • β

✶−γ

− rc

✶✷ ✴ ✹✵

slide-13
SLIDE 13

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

■♥❢♦r♠❛❧ ❊♥tr❡♣r❡♥❡✉rs❤✐♣

▲♦✇ ❛❝❝❡ss t♦ ❝r❡❞✐t✳ Pr♦❜❛❜✐❧✐t② p ♦❢ ❣❡tt✐♥❣ ❝❛✉❣❤t✳ ■♥❞✐✈✐❞✉❛❧s ❝❛♥ ❜♦rr♦✇ ♦♥❧② ✉♣ t♦ λz✱

❡♥❞♦❣❡♥❡✐③✐♥❣

❚❤❡ ✐♥❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉r✬s ❡①♣❡❝t❡❞ ♣❛②♦✛ ✐s πI(z, θ) = ♠❛①

✵≤k≤λz,l≥✵(✶ − p)

  • θkαlβ − wl − rk
  • ✇❤❡r❡

✲ λ ∈ [✶, ∞) ✐s t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❜♦rr♦✇✐♥❣ ❝♦♥str❛✐♥t✳

✶✸ ✴ ✹✵

slide-14
SLIDE 14

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

■♥❢♦r♠❛❧ ❊♥tr❡♣r❡♥❡✉rs❤✐♣

■♥❢♦r♠❛❧ ❊♥tr❡♣r❡♥❡✉r✬s ♣❛②♦✛ ❛r❡ ❣✐✈❡♥ ❜②✿

πI(z, θ) =            (✶ − p)(✶ − γ)θ

✶ ✶−γ α

r

  • α

✶−γ

β w

  • β

✶−γ ,

θ ≤ θc(z) (✶ − p)

  • (✶ − β)θ

✶ ✶−β

  • β

w

  • β

✶−β (λz) α ✶−β − λrz

  • ♦✴✇✳

✭✷✮

→ ✉♥❝♦♥str❛✐♥❡❞ ❛♥❞ ❝♦♥str❛✐♥❡❞ ✐♥❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉rs✳

✶✹ ✴ ✹✵

slide-15
SLIDE 15

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

▼♦❞❡❧ ■♠♣❧✐❝❛t✐♦♥s ✲ ❖❝❝✉♣❛t✐♦♥❛❧ ❝❤♦✐❝❡

❚❤❡ ❡①♣❡❝t❡❞ ❡❛r♥✐♥❣ ♦❢ ❛♥ ❛❣❡♥t ✇✐t❤ ✭z✱ θ✮ ✐s ❣✐✈❡♥ ❜② π(z, θ) = ♠❛①

  • w, πI(z, θ), πF(z, θ)
  • ❚❤❡ ❜❡❤❛✈✐♦✉r ♦❢ ♣❛②♦✛ ❢✉♥❝t✐♦♥s ✐♥ ❣✐✈❡♥ ✐♥ ❋✐❣✉r❡ ✶✳

❋✐❣✉r❡✿ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ P❛②♦✛ ❋✉♥❝t✐♦♥s

5 10 15 20 20 20 40 60 80 100 120

w F()

  • c

w F , w I()

Informal Entrepreneurs Workers Formal Entrepreneurs ✶✺ ✴ ✹✵

slide-16
SLIDE 16

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

▼♦❞❡❧ ■♠♣❧✐❝❛t✐♦♥s

Pr♦♣♦s✐t✐♦♥ ❈♦♥s✐❞❡r ❛♥ ❛❣❡♥t ✇✐t❤ ❝❤❛r❛❝t❡r✐st✐❝s θ ❛♥❞ z✳ ❚❤❡r❡ ❡①✐st t❤r❡❡ ❝r✐t✐❝❛❧ ❛❜✐❧✐t② t❤r❡s❤♦❧❞s θW (z)✱ θc(z) ❛♥❞ θF(z)✱ ✇✐t❤ θW (z), θc(z) < θF(z)✱ s✉❝❤ t❤❛t

✶ ■❢ θ < θW (z) t❤❡ ❛❣❡♥t ❝❤♦♦s❡s t♦ ❜❡ ❛ ✇❛❣❡✇♦r❦❡r ✷ ■❢ θW (z) ≤ θ < θF(z) t❤❡ ❛❣❡♥t ✐s ❛♥ ✐♥❢♦r♠❛❧

❡♥tr❡♣r❡♥❡✉r

✸ ■❢ θ ≥ θF(z) t❤❡ ❛❣❡♥t ✐s ❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉r✳

❚❤❡ ❝r✐t✐❝❛❧ t❤r❡s❤♦❧❞s θW (z) ❛♥❞ θF(z) ❛r❡ s♦❧✉t✐♦♥s t♦ t❤❡ ❡q✉❛t✐♦♥s πI(z, θ) = w ❛♥❞ πI(z, θ) = πF(z, θ)✱ r❡s♣❡❝t✐✈❡❧②✳

✶✻ ✴ ✹✵

slide-17
SLIDE 17

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

▼♦❞❡❧ ■♠♣❧✐❝❛t✐♦♥s

❚❤❡ ♥❛t✉r❡ ♦❢ t❤❡ s❡❧❡❝t✐♦♥ ✐♥t♦ ♦❝❝✉♣❛t✐♦♥s ✐s ❞❡♣✐❝t❡❞ ❜❡❧♦✇✳

❋✐❣✉r❡✿ ◆❛t✉r❡ ♦❢ t❤❡ ❙❡❧❡❝t✐♦♥ ✐♥t♦ ❖❝❝✉♣❛t✐♦♥s

=F(z)

Unconstrained Informal Entrepreneurs

=W(z) =c(z) z

Formal Entrepreneurs

  • Workers

u

W

l

W

z ∗∗ z ∗

Constrained Informal Entrepreneurs

✶✼ ✴ ✹✵

slide-18
SLIDE 18

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

▼♦❞❡❧ ■♠♣❧✐❝❛t✐♦♥s

❚❤❡ ♥❛t✉r❡ ♦❢ t❤❡ tr❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ ♦❝❝✉♣❛t✐♦♥s ✐s ❞❡♣✐❝t❡❞ ❜❡❧♦✇✳

❋✐❣✉r❡✿ ❚r❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ ❖❝❝✉♣❛t✐♦♥s

W m VW,F F

  • VW,I

VI,F V

✶✽ ✴ ✹✵

slide-19
SLIDE 19

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❉❛t❛✱ ❊st✐♠❛t✐♦♥ ❛♥❞ ❚❡st✐♥❣

✶✾ ✴ ✹✵

slide-20
SLIDE 20

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❉❛t❛

❚❤❡ ✷✵✵✺ ◆❛t✐♦♥❛❧ ❙✉r✈❡② ♦❢ ❊♠♣❧♦②♠❡♥t ❛♥❞ ■♥❢♦r♠❛❧ ❙❡❝t♦r ✭❙❊■❙✮✳ ❆❞♠✐♥✐st❡r❡❞ ❜② t❤❡ ◆❛t✐♦♥❛❧ ■♥st✐t✉t❡ ♦❢ ❙t❛t✐st✐❝s ✭◆■❙✮ ✇✐t❤ t❤❡ ♣❛rt♥❡rs❤✐♣ ♦❢ ❚❤❡ ❲♦r❧❞ ❇❛♥❦ ●r♦✉♣✳ ❈♦✈❡r❡❞ ❤♦✉s❡❤♦❧❞s ✐♥ t❤❡ ✶✵ ❈❛♠❡r♦♦♥ r❡❣✐♦♥s ❞✐str✐❜✉t❡❞ ✐♥ ❜♦t❤ ✉r❜❛♥ ❛♥❞ r✉r❛❧ ❛r❡❛s✳ ❆ ❝r♦ss✲s❡❝t✐♦♥ ♦❢ ✻✶✶✷ ❛❝t✐✈❡ ❤♦✉s❡❤♦❧❞s ❤❡❛❞s ✿ ✶✳✶✪ ❋♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉rs✱ ✻✳✾✪ ■♥❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉rs✱ ✾✷✳✵✪ ❲♦r❦❡rs✳

✷✵ ✴ ✹✵

slide-21
SLIDE 21

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❉❛t❛ ✲ ❆ ◆❛t✐♦♥✇✐❞❡ ❙✉r✈❡②

✷✶ ✴ ✹✵

slide-22
SLIDE 22

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❉❡s❝r✐♣t✐✈❡ st❛t✐st✐❝s

❚❛❜❧❡✿ ❍♦✉s❡❤♦❧❞ ❈❤❛r❛❝t❡r✐st✐❝s ❜② ❖❝❝✉♣❛t✐♦♥s

Occupations Characteristics Formal Informal Wageworkers Entrepreneurs Entrepreneurs /Subsisters

  • Num. of obs.

65 424 5 623 % of sample 1.1% 6.9% 92.0% % of women 12.3% 37.3% 41.7%

  • Av. household size

6.0 6.1 5.9

  • Av. age of head

42.4 37.0 36.1 Years of schooling 0-6 years 11.1% 41.3% 48.4% 7-12 years 31.5% 48.6% 36.2% 13+ years 57.4% 10.1% 15.4% Parent Entrep. 41.5% 13.6% 3.5%

  • Av. monthly income*

353.3 75.3 77.2

  • Av. wealth*

21 792.9 4 569.7 3 007.4

*In thousands of local currency (CFA); 1, 000 CFA ∼ $2 US (in 2005)

✷✷ ✴ ✹✵

slide-23
SLIDE 23

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❉❡s❝r✐♣t✐✈❡ st❛t✐st✐❝s

❋✐❣✉r❡✿ ❉✐str✐❜✉t✐♦♥ ♦❢ ❊❞✉❝❛t✐♦♥ ❛♥❞ ❊❛r♥✐♥❣s ❜② ❖❝❝✉♣❛t✐♦♥

5 10 15 20 25 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Years of Education

Formal Entrepreneurs Informal Entrepreneurs Workers 200 400 600 800 0.005 0.01 0.015 0.02

Earnings

Formal Entrepreneurs Informal Entrepreneurs Workers ✷✸ ✴ ✹✵

slide-24
SLIDE 24

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❉❡s❝r✐♣t✐✈❡ st❛t✐st✐❝s

❋✐❣✉r❡✿ ❉✐str✐❜✉t✐♦♥ ♦❢ ▲♦❣ ■♥✐t✐❛❧ ❲❡❛❧t❤ ❜② ❖❝❝✉♣❛t✐♦♥

5 10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Log Initial Wealth

Formal Entrepreneusr Informal Entrepreneurs Workers ✷✹ ✴ ✹✵

slide-25
SLIDE 25

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❙tr✉❝t✉r❛❧ ❊st✐♠❛t✐♦♥ ✲ ❉✐str✐❜✉t✐♦♥ ♦❢ s❦✐❧❧s

❙❦✐❧❧s ❛r❡ ✉♥♦❜s❡r✈❛❜❧❡✳ ❆ss✉♠❡ ✭s❡❡ P❛✉❧s♦♥ ❡t ❛❧ ✷✵✵✻✮✿ ❧♥ θ = δ✵ + δ✶s + δ✷P + ǫ ✭✸✮ ✲s ✐s t❤❡ ❧♦❣ ♦❢ ②❡❛rs ♦❢ ❡❞✉❝❛t✐♦♥✱ ✲ P ✐s ❛ ❞✉♠♠② ❢♦r ♣❛r❡♥t ❡♥tr❡♣r❡♥❡✉r✐❛❧ st❛t✉s✳ ✲ ǫ ✐s ❛ss✉♠❡❞ ǫ|z,S,P ∼ IID(✵, σ✷)✳ ❆ss✉♠❡ ❧♦❣❧✐♥❡❛r s♣❡❝✐✜❝❛t✐♦♥ ❢♦r r❡❣✐str❛t✐♦♥ ❝♦sts✿ c(z) = c✵ ❡①♣(−c✶z) ❙tr✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs✿ ψ = [w, δ✵, δ✶, δ✷, α, β, σ, λ, c✶]

✷✺ ✴ ✹✵

slide-26
SLIDE 26

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❙tr✉❝t✉r❛❧ ❊st✐♠❛t✐♦♥ ✲ ▼♦❞❡❧ ♣r❡❞✐❝t❡❞ ♠♦♠❡♥ts

❖❝❝✉♣❛t✐♦♥❛❧ st❛t✉s❡s ❛r❡ ❣✐✈❡♥ ❜② ✐♥❞✐❝❛t♦rs (W , I, F)✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ✐♥❝♦♠❡ ♦❢ ❋♦r♠❛❧ ❊♥tr❡♣r❡♥❡✉rs❤✐♣ ✐s

Pr[F = ✶|X] = Pr[θ ≥ θF(z)] = HF(ψ, X), E[y|F = ✶] = E[πF(z, θ)]

❚❤❡ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ✐♥❝♦♠❡ ♦❢ ♥♦♥✲❡♥tr❡♣r❡♥❡✉rs❤✐♣ ✐s

Pr[W = ✶|X] = Pr [❧♥ θ ≤ ❧♥ θW (z)] = HW (ψ, X), E[y|F = ✶] = w

❚❤❡ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ✐♥❝♦♠❡ ♦❢ ✐♥❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉rs❤✐♣ ✐s

Pr[I = ✶|X] = ✶−HW (ψ, X)−HF(ψ, X), E[y|W = ✶] = E[πI(z, θ)]

✷✻ ✴ ✹✵

slide-27
SLIDE 27

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❙tr✉❝t✉r❛❧ ❊st✐♠❛t✐♦♥ ✲ ▼♦❞❡❧ ♠♦♠❡♥ts ❛♥❞ s❛♠♣❧❡ ❛♥❛❧♦❣s

✷✼ ✴ ✹✵

slide-28
SLIDE 28

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❙tr✉❝t✉r❛❧ ❊st✐♠❛t✐♦♥✲■♥st✐t✉t✐♦♥❛❧ ♣❛r❛♠❡t❡rs

❚❛❜❧❡✿ ❈❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ ■♥st✐t✉t✐♦♥❛❧ ❊♥✈✐r♦♥♠❡♥t

Indicator Starting Indicator Paying a Business Taxes Number of procedures 12 Number of payments/year 44 Number of days 37 Number of days 90 Registration fees (% GNI/capita) 182.5 Total tax rate (% profit) 48.9

  • Min. capital (% GNI/capita)

232.0

GNI per capita = $640 ≈ CFA 320, 000

Source: Doing Business in 2005

❚❛① r❛t❡ ♦♥ ✜r♠ ♣r♦✜ts✿ τ = ✹✾% ❊♥tr② ❝♦st ✿ c✵ = ❘❡❣✳ ❢❡❡s + ◆✉♠❜❡r ♦❢ ❞❛②s × ▼❡❛♥ ❞❛✐❧② ❡❛r♥✐♥❣s = $✶✹✵✵ ❊♥❢♦r❝❡♠❡♥t✿ p = # t❛① ✐♥s♣❡❝t✐♦♥s # ✜r♠s ×❉❡❣r❡❡ ♦❢ ■♥t❡❣r✐t② ♦❢ t❛① ✐♥s♣❡❝t♦rs = ✵.✼✽%

✷✽ ✴ ✹✵

slide-29
SLIDE 29

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❙tr✉❝t✉r❛❧ ❊st✐♠❛t✐♦♥✲●▼▼ ❡st✐♠❛t✐♦♥ r❡s✉❧ts

❚❛❜❧❡✿ ❙tr✉❝t✉r❛❧ ●▼▼ ❊st✐♠❛t❡s ♦❢ t❤❡ ▼♦❞❡❧

✷✾ ✴ ✹✵

slide-30
SLIDE 30

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❘❡❞✉❝❡❞✲❋♦r♠ ❘❡s✉❧ts

❚❛❜❧❡✿ Pr♦❜✐t ❊st✐♠❛t❡s ♦❢ ❈❤♦✐❝❡ ❇❡t✇❡❡♥ ❖❝❝✉♣❛t✐♦♥s

✸✵ ✴ ✹✵

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SLIDE 31

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦✉♥t❡r❢❛❝t✉❛❧ P♦❧✐❝② ❆♥❛❧②s✐s

✸✶ ✴ ✹✵

slide-32
SLIDE 32

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥

◗✉❛♥t✐❢② t❤❡ ✐♠♣❛❝t ♦❢ ✈❛r✐♦✉s ♣♦❧✐❝✐❡s ♦♥ ❢♦r♠❛❧✐t②✱ ❡♥tr❡♣r❡♥❡✉rs❤✐♣ ❛♥❞ ❛❣❣r❡❣❛t❡ ❡❛r♥✐♥❣✳ ■♠♣❛❝t ♦❢ ❘❡❣✐str❛t✐♦♥ ❘❡❢♦r♠s✳ ✭❡✳❣✳ ❇r✉❤♥ ✷✵✶✶✱ ❑❛♣❧❛♥ ❡t ❛❧✳ ✷✵✶✶✮ ■♠♣❛❝t ♦❢ ❚❛① ❘❡❢♦r♠s✳ ✭❡✳❣✳ ▼♦♥t❡✐r♦ ❛♥❞ ❆ss✉♥çã♦ ✷✵✶✷✱ ❋❛❥♥③②❧❜❡r ❡t ❛❧✳ ✷✵✶✶✮✳ ■♠♣❛❝t ♦❢ ▲❛✇ ❊♥❢♦r❝❡♠❡♥t ✭❡✳❣✳ ❆❧♠❡❞❛ ✫ ❈❛r♥❡✐r♦ ✷✵✶✷✮

✸✷ ✴ ✹✵

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SLIDE 33

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❙✐♠✉❧❛t✐♦♥s ✲ ■♠♣❛❝t ♦❢ ❘❡❣✐str❛t✐♦♥ ❘❡❢♦r♠s

❆ss✉♠❡ b ❞❡❝r❡♠❡♥ts ✐♥ ❡♥tr② ❝♦sts t♦ t❤❡ ❢♦r♠❛❧ s❡❝t♦r✳ ❚❤❡♥ c′ = c − b, ✵ ≤ b < c,

❋✐❣✉r❡✿ ■♠♣❛❝t ♦❢ ❛ ❘❡❣✐str❛t✐♦♥ ❘❡❢♦r♠

b/c0 0.2 0.4 0.6 0.8 1 Proportion 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Formal Entrep Informal Entrep. Enterprise creation b/c0

0.2 0.4 0.6 0.8 1

Proportion

0.2 0.4 0.6 0.8

0.2 0.4 0.6 0.8 1 2 4 6 8

Aggregate Income gains Tax Revenue gains ✸✸ ✴ ✹✵

slide-34
SLIDE 34

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❙✐♠✉❧❛t✐♦♥s ✲ ■♠♣❛❝t ♦❢ ❚❛① ❘❡❢♦r♠s

❆ss✉♠❡ r❡❞✉❝t✐♦♥s ✐♥ t❛① r❛t❡s ♦❢ ♠❛❣♥✐t✉❞❡ d✱ s✉❝❤ t❤❛t τ ′ = τ − d, ✵ ≤ d < τ.

❋✐❣✉r❡✿ ■♠♣❛❝t ♦❢ ❛ ❚❛① ❘❡❞✉❝t✐♦♥ P♦❧✐❝②

d/ 0.2 0.4 0.6 0.8 Proportion 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Formal Entrep. Informal Entrep. Enterprise Creation

d/

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Proportion

0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.5

Aggregate Income gains Tax Revenue gains ✸✹ ✴ ✹✵

slide-35
SLIDE 35

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❙✐♠✉❧❛t✐♦♥s ✲ ■♠♣❛❝t ♦❢ ▲❛✇ ❊♥❢♦r❝❡♠❡♥t

❚❤❡ ❡✛❡❝t ♦❢ ✐♥❝r❡❛s❡❞ ❧❛✇ ❡♥❢♦r❝❡♠❡♥t ✐s q✉❛♥t✐✜❡❞ ❛s p′ = p + e, ✵ ≤ e < ✶ − p

❋✐❣✉r❡✿ ■♠♣❛❝t ♦❢ ▲❛✇ ❊♥❢♦r❝❡♠❡♥t

e 0.1 0.2 0.3 0.4 Proportion

  • 0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 Formal Entrep. Informal Entrep. Enterprise creation

e 0.1 0.2 0.3 0.4

  • 0.1
  • 0.05

0.05 0.1 0.15

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.02 0.04

Aggregate Income gains Tax Revenue gains

✸✺ ✴ ✹✵

slide-36
SLIDE 36

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❊st✐♠❛t❡❞ ❛ str✉❝t✉r❛❧ ♠♦❞❡❧ ♦❢ ♦❝❝✉♣❛t✐♦♥❛❧ ❝❤♦✐❝❡ ✇❤❡r❡ ❤❡t❡r♦❣❡♥♦✉s ❛❣❡♥ts ❝❤♦♦s❡ ❜❡t✇❡❡♥ ❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉rs❤✐♣ ❛♥❞ ✐♥❢♦r♠❛❧ ❡♥tr❡♣r❡♥❡✉rs❤✐♣ ❛♥❞ ♥♦♥✲❡♥tr❡♣r❡♥❡✉r✐❛❧ ✇♦r❦✳ ❚❤❡ ♠❛✐♥ r❡s✉❧ts ❛r❡✿ → ❊♥tr❡♣r❡♥❡✉rs ✇✐t❤ ❧♦✇ ♣r♦❞✉❝t✐✈✐t② ❝❤♦♦s❡ ✐♥❢♦r♠❛❧✐t② ✇❤❡r❡❛s t❤❡ ♠♦st ♣r♦❞✉❝t✐✈❡ ♦♥❡s ❝❤♦♦s❡ t❤❡ ❢♦r♠❛❧ s❡❝t♦r✳ → ❚❤❡ ❞❡❝✐s✐♦♥ t♦ ❢♦r♠❛❧✐③❡ ✐s ❤♦✇❡✈❡r ❯✲s❤❛♣❡❞ ✐♥ s❦✐❧❧s✳ → ❍✐❣❤ r❡❣✐str❛t✐♦♥ ❝♦sts ❛❝t ❛s ❛♥ ✐♠♣❧✐❝✐t ❡①❝❧✉s✐♦♥ ♠❡❝❤❛♥✐s♠ t♦ ❡♥t❡r♣r✐s❡s ✇✐t❤ ❧♦✇ ♣r♦❞✉❝t✐✈✐t②✳

✸✻ ✴ ✹✵

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SLIDE 37

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❈♦✉♥t❡r❢❛❝t✉❛❧ s✐♠✉❧❛t✐♦♥s r❡s✉❧ts ✇✐t❤ ❈❛♠❡r♦♦♥ ❞❛t❛ s❤♦✇ t❤❛t✿ → ❘❡❞✉❝❡❞ ❡♥tr② ❝♦sts ❝❛♥ ✐♥❞✉❝❡ ♠♦r❡ ❢♦r♠❛❧ ✜r♠s ❛♥❞ ♠♦r❡ t❛① r❡✈❡♥✉❡s ♥❡t ♦❢ t❤❡ ❢♦r❡❣♦♥❡ ❝♦sts✳ →❚❤❡r❡ ❡①✐sts ❛♥ ♦♣t✐♠❛❧ t❛① r❛t❡✱ s❡t ❛t ❤❛❧❢ ♦❢ t❤❡ ❝✉rr❡♥t r❛t❡✱ t❤❛t ✇♦✉❧❞ ✐♥❞✉❝❡ t✇✐❝❡ ❛s ♠✉❝❤ ❢♦r♠❛❧ ❡♥t❡r♣r✐s❡s ❛♥❞ ♣r♦❞✉❝❡ t❤r❡❡✲❤❛❧✈❡s ♦❢ t❤❡ ❝✉rr❡♥t t❛① r❡✈❡♥✉❡s✳ → ■♥ ❝♦♥tr❛st✱ ❛ ❧❛✇ ❡♥❢♦r❝❡♠❡♥t ♣♦❧✐❝② ✇❤♦s❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✐♥❝r❡❛s❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞❡t❡❝t✐♦♥ ✇♦✉❧❞ ❤❛✈❡ ❛♥ ♦✈❡r❛❧❧ ♣❡r✈❡rs❡ ❡✛❡❝t ✐♥ t❡r♠s ♦❢ ✜r♠s ❛♥❞ ❥♦❜ ❧♦ss❡s✳

✸✼ ✴ ✹✵

slide-38
SLIDE 38

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❚❍❆◆❑ ❨❖❯✦✦

✸✽ ✴ ✹✵

slide-39
SLIDE 39

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❆ ❙tr✉❝t✉r❛❧ ▼♦❞❡❧ ♦❢ ■♥❢♦r♠❛❧✐t② ✇✐t❤ ❈♦♥str❛✐♥❡❞ ❊♥tr❡♣r❡♥❡✉rs❤✐♣

P✐❡rr❡ ◆❣✉✐♠❦❡✉

  • ❡♦r❣✐❛ ❙t❛t❡ ❯♥✐✈❡rs✐t② ✲ ❯❙❆

✭♥♥❣✉✐♠❦❡✉❅❣s✉✳❡❞✉✮ ❯◆❯✲❲■❉❊❘ ❈♦♥❢❡r❡♥❝❡ ♦♥ ✏P✉❜❧✐❝ ❊❝♦♥♦♠✐❝s ❢♦r ❉❡✈❡❧♦♣♠❡♥t✑

▼❛♣✉t♦✱ ❏✉❧② ✻✱ ✷✵✶✼

✸✾ ✴ ✹✵

slide-40
SLIDE 40

❖✈❡r✈✐❡✇ ▼♦❞❡❧ ❊st✐♠❛t✐♦♥ P♦❧✐❝② ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❊♥❞♦❣❡♥❡✐③✐♥❣ ❝r❡❞✐t ❝♦♥str❛✐♥ts

❊①♣❡❝t❡❞ ♣❛②♦✛ ❢♦r ❞❡❢❛✉❧t❡rs ✐s f (k, l) − φ(✶ + ν)z✱ ✇❤❡r❡ ✲ φ ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❜❡✐♥❣ ❝❛✉❣❤t✱ ✲ ν ✐s t❤❡ ❢r❛❝t✐♦♥ ♦❢ ✇❡❛❧t❤ ❢♦r❢❡✐t❡❞✳ ❚❤❡ ✐♥❝❡♥t✐✈❡ ❝♦♠♣❛t✐❜✐❧✐t② ❝♦♥str❛✐♥t ✐s t❤❡♥ f (k, l) − rk ≥ f (k, l) − φ(✶ + ν)z✳ ❙♦ ❧❡♥❞❡rs ♦♥❧② r❡♥t t♦ ❤♦✉s❡❤♦❧❞s ✇❤♦s❡ ✇❡❛❧t❤ s❛t✐s✜❡s✱ z ≥ r (✶ + ν)φk✳ ❊q✉✐✈❛❧❡♥t❧②✱ t❤✐s ♠❡❛♥s t❤❛t t❤❡ ❝❛♣✐t❛❧ ❛✈❛✐❧❛❜❧❡ t♦ ❜♦rr♦✇❡rs s❛t✐s✜❡s k ≤ φ(✶ + ν) r z✳

❇❛❝❦ ✹✵ ✴ ✹✵