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❯♥❜✉♥❞❧✐♥❣ ▲❛❜♦r

❈❤r✐s ❊❞♠♦♥❞ ❯♥✐✈❡rs✐t② ♦❢ ▼❡❧❜♦✉r♥❡ ❙✐♠♦♥ ▼♦♥❣❡② ❯♥✐✈❡rs✐t② ♦❢ ❈❤✐❝❛❣♦ ❏✉♥❡ ✷✵✷✵

❘❆✿ ❆❧❡① ❲❡✐♥❜❡r❣

❚❤✐s ♣❛♣❡r

Pr♦✈✐❞❡ ❛ ♥❡✇ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❤♦✇ ❝❤❛♥❣❡s ✐♥ ✇✐t❤✐♥✲♦❝❝✉♣❛t✐♦♥ ✇❛❣❡ ✐♥❡q✉❛❧✐t② ❝❛♥ ❜❡ ❞✉❡ t♦ ❝❤❛♥❣❡s ✐♥ t❡❝❤♥♦❧♦❣②

✶

❚❤✐s ♣❛♣❡r

✶✳ ❉❛t❛ ✲ ❚✇♦ ♥❡✇ ❢❛❝ts

❆✳ ❲✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥ r❡s✐❞✉❛❧ ✇❛❣❡ ✐♥❡q✉❛❧✐t② ✲ ❈P❙ ↑ ❍✐❣❤ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s , ↓ ▲♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ❇✳ ❙✐♠✐❧❛r✐t② ♦❢ ♦❝❝✉♣❛t✐♦♥s ✐♥ t❡r♠s ♦❢ t❤❡✐r s❦✐❧❧ ✐♥♣✉ts ✲ ❖✯◆❊❚ ↑ ❍✐❣❤ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s , ↓ ▲♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s

✷✳ ❚❤❡♦r② ✲ ❯♥❞❡rst❛♥❞ ❆✳ ✈✐❛ ❛ ❝♦♠♣❛r❛t✐✈❡ st❛t✐❝ ✐♥❢♦r♠❡❞ ❜② ❇✳

✲ ❊①t❡♥❞ ♠♦❞❡❧ ♦❢ ❘♦s❡♥ ✭✶✾✽✸✮✱ ❍❡❝❦♠❛♥ ❙❝❤❡✐♥❦♠❛♥ ✭✶✾✽✼✮ ✲ ❊♥❞♦❣❡♥✐③❡ ❇✳ ❛s ❛♣♣r♦♣r✐❛t❡ t❡❝❤♥♦❧♦❣② ❝❤♦✐❝❡ ✭❈❛s❡❧❧✐ ❈♦❧❡♠❛♥✱ ✷✵✵✻✮

✸✳ ❊①t❡♥s✐♦♥ ✲ ❙❤♦✇ t❤❛t ❇✳ r❛t✐♦♥❛❧✐③❡s ♦t❤❡r ♥❡✇ ❢❛❝ts

✲ ❉❡❝❧✐♥✐♥❣ ❡①♣❡r✐❡♥❝❡ ♣r❡♠✐✉♠ ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ✲ ❉❡❝❧✐♥✐♥❣ ♦✈❡rt✐♠❡ ♣r❡♠✐✉♠ ✴ ♣❛rt✲t✐♠❡ ♣❡♥❛❧t② ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ✲ ■♥❝r❡❛s✐♥❣ ♦❝❝✉♣❛t✐♦♥ s✇✐t❝❤✐♥❣ ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s

✷

❋❛❝t ❆✳ ✲ ❲✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥ ✇❛❣❡ ✐♥❡q✉❛❧✐t②

❲♦r❦❡rs ✐♥ ❧♦✇ ✭❤✐❣❤✮ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ❛r❡ ♥♦✇ ♣❛✐❞ ♠♦r❡ ✭❧❡ss✮ s✐♠✐❧❛r❧② ❆♣♣r♦❛❝❤ ✲ ❙♣❧✐t ✸ ❞✐❣✐t ♦❝❝✉♣❛t✐♦♥s ✐♥t♦ ▲♦✇ s❦✐❧❧ ❛♥❞ ❍✐❣❤ s❦✐❧❧

✲ ❘❛♥❦ ❜② ❢r❛❝t✐♦♥ ✇✐t❤ ❝♦❧❧❡❣❡ ❡❞✉❝❛t✐♦♥✱ s♣❧✐t ❜② ❡♠♣❧♦②♠❡♥t ✲ ❘❡✲❝❧❛ss✐❢② ❡❛❝❤ ②❡❛r

✲ ❘❡s✐❞✉❛❧ ✇❛❣❡s

✲ ❘❡s✐❞✉❛❧s ❢r♦♠ r❡❣r❡ss✐♦♥ ♦❢ ❈P❙ ❛♥♥✉❛❧ ❡❛r♥✐♥❣s log yit ♦♥ ♦❜s❡r✈❛❜❧❡s

• Y eart, NAICS1it, Edit, Raceit, Sexit, FirmSizeit, Expit, Exp2

it, Hoursit

• ✲ ❉❡❝♦♠♣♦s✐t✐♦♥

Vt

• eijt
• ❆✳ ❚♦t❛❧ ✈❛r✐❛♥❝❡

=

• j

ωjtVt

• eijt
• j
• ❇✳ ❲✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥

+

• j

ωjt

• Et
• eijt
• j
• − Et [eijt]

2

• ❈✳ ❇❡t✇❡❡♥ ♦❝❝✉♣❛t✐♦♥

✸

❋❛❝t ❆✳ ✲ ❲✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥ ✇❛❣❡ ✐♥❡q✉❛❧✐t②

❱❛r✐❛♥❝❡ ♦❢ r❡s✐❞✉❛❧s✳ ❘❡❞ ❂ ❍✐❣❤ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s✱ ❇❧✉❡ ❂ ▲♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s

✶✳ ▲❡✈❡❧ ❲✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥ ✐♥❡q✉❛❧✐t② ✐s ✐♠♣♦rt❛♥t ✷✳ ❈❤❛♥❣❡ ▲♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥ ✇♦r❦❡rs ♣❛✐❞ ♠♦r❡ s✐♠✐❧❛r❧② ✸✳ ❉❡❝♦♠♣♦s✐t✐♦♥ ❉r✐✈❡♥ ❜② ❞❡❝❧✐♥❡ ✐♥ ✇✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥ ✐♥❡q✉❛❧✐t②

❘♦❜✉st ❛❝r♦ss ④❆❧❧✱▼❛❧❡✱❋❡♠❛❧❡⑥×④❋✐① ♦❝❝✉♣❛t✐♦♥s ✐♥ ✶✾✽✵✱✷✵✶✵⑥

❉❡t❛✐❧s ❘♦❜✉st ✲ ✶✾✽✵ ❝❧❛ss✐✜❝❛t✐♦♥ ❘♦❜✉st ✲ ✷✵✶✵ ❝❧❛ss✐✜❝❛t✐♦♥

✹

❋❛❝t ❇✳ ✲ ❚❡❝❤♥♦❧♦❣②

▲♦✇ ✭❤✐❣❤✮ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ❤❛✈❡ ❜❡❝♦♠❡ ♠♦r❡ s✐♠✐❧❛r ✭♠♦r❡ ❞✐✛❡r❡♥t✮ ✐♥ ❆♣♣r♦❛❝❤ ✶✳ J × K ♠❛tr✐① ♦❢ s❦✐❧❧ ♠❡❛s✉r❡s At ❢r♦♠ ❖✯◆❊❚✿ ✷✵✵✸✲✷✵✵✾✱ ✷✵✶✵✲✷✵✶✽ ✷✳ ❘❡❞✉❝❡ t♦ J × K∗ ♠❛tr✐① ♦❢ s❦✐❧❧s A∗

t ✭▲✐s❡ P♦st❡❧✲❱✐♥❛②✱ ✷✵✷✵✮

✸✳ ❉✐st❛♥❝❡ ❜❡t✇❡❡♥ ♦❝❝✉♣❛t✐♦♥s ✭●❛t❤♠❛♥♥ ❙❝❤ö♥❜❡r❣✱ ✷✵✶✵✮ ✹✳ ❈♦♠♣❛r❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡s❡ ❞✐st❛♥❝❡s ϕj,j′ ♦✈❡r t✐♠❡

❉❡t❛✐❧s ✲ ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

✺

❋❛❝t ❇✳ ✲ ❚❡❝❤♥♦❧♦❣②

✶✳ ▲♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ✲ ▼♦r❡ s✐♠✐❧❛r ✲ ↓ ϕ ✷✳ ❍✐❣❤ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ✲ ▼♦r❡ ❞✐✛❡r❡♥t ✲ ↑ ϕ

✻

▲♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s✿ ❚❤❡♥ ✈s✳ ♥♦✇

❉✐✛❡r❡♥t✐❛t❡❞ t❡❝❤♥♦❧♦❣✐❡s ❙✐♠✐❧❛r t❡❝❤♥♦❧♦❣✐❡s ❍♦✇ ❞♦❡s t❤❡ r❡❧❛t✐✈❡ s❦✐❧❧ ❜✐❛s ♦❢ t❡❝❤♥♦❧♦❣✐❡s ❛❝r♦ss ♦❝❝✉♣❛t✐♦♥s ❞❡t❡r♠✐♥❡ ✇❛❣❡ ✐♥❡q✉❛❧✐t② ✇✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥s❄

✼

▼♦❞❡❧

• ●❡♥❡r❛❧ ❡q✉✐❧✐❜r✐✉♠ ❡♥✈✐r♦♥♠❡♥t

✕ ■♥❞✐✈✐❞✉❛❧ s❦✐❧❧s l(i) =

• lA(i), lB(i)
• ✕ ❚✇♦ ♦❝❝✉♣❛t✐♦♥s j ∈ {1, 2}✱ ✇✐t❤ ❞✐✛❡r❡♥t s❦✐❧❧ ✐♥t❡♥s✐t✐❡s
• ❈♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠ ✇❛❣❡s

wj(i) = ωjAlA(i) + ωjBlB(i) → var

• log wj(i)
• j
• ❲✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥ ✐♥❡q✉❛❧✐t② ❞❡t❡r♠✐♥❡❞ ❜② t✇♦ ❢♦r❝❡s

✶✳ ❉✐str✐❜✉t✐♦♥ ♦❢ s❦✐❧❧s ❝♦♥❞✐t✐♦♥❛❧ ♦♥ s❡❧❡❝t✐♦♥ ✷✳ ●r❛❞✐❡♥t ♦❢ ♦❝❝✉♣❛t✐♦♥ s❦✐❧❧ ♣r✐❝❡s

• ωjA, ωjB
• ✽

❊♥✈✐r♦♥♠❡♥t

• ❲♦r❦❡rs i ∈ [0, 1] ❡♥❞♦✇❡❞ ✇✐t❤ t✇♦ s❦✐❧❧s k ∈ {A, B}

l(i) =

• lA(i), lB(i)
• ,
• lA(i), lB(i)
• ∼ H
• lA, lB
• ❋✐♥❛❧ ❣♦♦❞

U

• Y1, Y2
• ❚❛s❦ ✴ ❖❝❝✉♣❛t✐♦♥ j t❡❝❤♥♦❧♦❣②✿ α1 = (1 − α2) > 0.5

Yj = Fj

• LjA, LjB
• = Zj
• αjLσ

jA

+ (1 − αj) Lσ

jB

1

σ

, σ < 1 LjA =

• lA(i)φj(i) di , LjB =
• lB(i)φj(i) di , φj(i) ∈ {0, 1}

❇✉♥❞❧❡❞ ✲ ❲♦r❦❡r i ♠✉st ❛❧❧♦❝❛t❡

• lA(i), lB(i)
• t♦ t❤❡ s❛♠❡ t❛s❦

▼❛♥❞❡❧❜r♦t ✭✶✾✻✷✮✱ ❘♦s❡♥ ✭✶✾✽✸✮✱ ❍❡❝❦♠❛♥ ❙❝❤❡✐♥❦♠❛♥ ✭✶✾✽✼✮

✾

❊✣❝✐❡♥t ❛❧❧♦❝❛t✐♦♥

max

φ1(i)∈{0,1}

U

• F1(L1A, L1B), F2(L2A, L2B)
• s✉❜❥❡❝t t♦

▲❡t ωjk ❜❡ t❤❡ s❤❛❞♦✇ ♣r✐❝❡ ♦❢ Ljk L1A =

• φ1(i) lA(i) di

− → ω1A = U1F1A L2A = 1 − φ1(i)

• lA(i) di

− → ω2A = U2F2A L1B =

• φ1(i) lB(i) di

− → ω1B = U1F1B L2B = 1 − φ1(i)

• lB(i) di

− → ω2B = U2F2B

✶✵

❊✣❝✐❡♥t ❛❧❧♦❝❛t✐♦♥

max

φ1A(i)∈{0,1},φ1B(i)∈{0,1}

U

• F1(L1A, L1B), F2(L2A, L2B)
• s✉❜❥❡❝t t♦

▲❡t ωjk ❜❡ t❤❡ s❤❛❞♦✇ ♣r✐❝❡ ♦❢ Ljk L1A =

• φ1A(i) lA(i) di

− → ω1A = U1F1A L2A = 1 − φ1A(i)

• lA(i) di

− → ω2A = U2F2A L1B =

• φ1B(i) lB(i) di

− → ω1B = U1F1B L2B = 1 − φ1B(i)

• lB(i) di

− → ω2B = U2F2B

❛♥❞ ♣❡rs♦♥✲❜②✲♣❡rs♦♥ ❜✉♥❞❧✐♥❣ ❝♦♥str❛✐♥ts φ1A(i) = φ1B(i) ❢♦r ❛❧❧ i ∈ [0, 1]

✶✵

❋❡❛s✐❜❧❡ ❛❧❧♦❝❛t✐♦♥s

❘❡♣❧❛❝❡ ❝♦♥t✐♥✉✉♠ ♦❢ ✐♥❞✐✈✐❞✉❛❧ ❝♦♥str❛✐♥ts ✇✐t❤ ❛ s✐♥❣❧❡ ❝♦♥str❛✐♥t✿ ❇✉♥❞❧✐♥❣ ❝♦♥str❛✐♥t✿ L1B ∈

• B
• L1A
• , B
• L1A
• ✲ ●✐✈❡♥ s♦♠❡ L1A ✇❤❛t ✐s t❤❡ ♠✐♥✐♠✉♠ L1B ❜✉♥❞❧❡❞ ✇✐t❤ ✐t❄

✲ ❈♦♥str✉❝t L1A ✉s✐♥❣ ✇♦r❦❡rs ✇✐t❤ ❤✐❣❤❡st lA(i)

• lB(i) ✜rst

L1A = i∗ lA(i) di , B(L1A) = i∗ lB(i) di ✲ ❊①❛♠♣❧❡ ▲❡t lk(i) ∼ ❋ré❝❤❡t(θ) ❢♦r ❡❛❝❤ s❦✐❧❧ k B

• L1A
• =

 1 −

• 1 −

L1A LA

• θ

θ−1 θ−1 θ 

 LB

✶✶

❋❡❛s✐❜❧❡ ❛❧❧♦❝❛t✐♦♥s

❋❡❛s✐❜❧❡ ❛❧❧♦❝❛t✐♦♥s ♠✉st s❛t✐s❢② ❛❣❣r❡❣❛t❡ ❜✉♥❞❧✐♥❣ ❝♦♥str❛✐♥ts L1B ∈ [B(L1A), B(L1A)] ❉❡t❡r♠✐♥❡❞ ❜② ❞✐str✐❜✉t✐♦♥ ♦❢ s❦✐❧❧ ❡♥❞♦✇♠❡♥ts ♦♥❧②✳ ❊①❛♠♣❧❡✿ lk(i) ∼ ❋ré❝❤❡t(θ)✳

❋❡❛s✐❜❧❡ ❛❧❧♦❝❛t✐♦♥s

❋❡❛s✐❜❧❡ ❛❧❧♦❝❛t✐♦♥s ♠✉st s❛t✐s❢② ❛❣❣r❡❣❛t❡ ❜✉♥❞❧✐♥❣ ❝♦♥str❛✐♥ts L1B ∈ [B(L1A), B(L1A)]✳ ❉❡t❡r♠✐♥❡❞ ❜② ❞✐str✐❜✉t✐♦♥ ♦❢ s❦✐❧❧ ❡♥❞♦✇♠❡♥ts ♦♥❧②✳ ❊①❛♠♣❧❡✿ lk(i) ∼ ❋ré❝❤❡t(θ)✳

❋❡❛s✐❜❧❡ ❛❧❧♦❝❛t✐♦♥s

❋❡❛s✐❜❧❡ ❛❧❧♦❝❛t✐♦♥s ♠✉st s❛t✐s❢② ❛❣❣r❡❣❛t❡ ❜✉♥❞❧✐♥❣ ❝♦♥str❛✐♥ts L1B ∈ [B(L1A), B(L1A)]✳ ❉❡t❡r♠✐♥❡❞ ❜② ❞✐str✐❜✉t✐♦♥ ♦❢ s❦✐❧❧ ❡♥❞♦✇♠❡♥ts ♦♥❧②✳ ❊①❛♠♣❧❡✿ lk(i) ∼ ❋ré❝❤❡t(θ)✳

❋❡❛s✐❜❧❡ ❛❧❧♦❝❛t✐♦♥s

❋❡❛s✐❜❧❡ ❛❧❧♦❝❛t✐♦♥s ♠✉st s❛t✐s❢② ❛❣❣r❡❣❛t❡ ❜✉♥❞❧✐♥❣ ❝♦♥str❛✐♥ts L1B ∈ [B(L1A), B(L1A)]✳ ❉❡t❡r♠✐♥❡❞ ❜② ❞✐str✐❜✉t✐♦♥ ♦❢ s❦✐❧❧ ❡♥❞♦✇♠❡♥ts ♦♥❧②✳ ❊①❛♠♣❧❡✿ lk(i) ∼ ❋ré❝❤❡t(θ)✳

❋❡❛s✐❜❧❡ ❛❧❧♦❝❛t✐♦♥s

❋❡❛s✐❜❧❡ ❛❧❧♦❝❛t✐♦♥s ♠✉st s❛t✐s❢② ❛❣❣r❡❣❛t❡ ❜✉♥❞❧✐♥❣ ❝♦♥str❛✐♥ts L1B ∈ [B(L1A), B(L1A)]✳ ❉❡t❡r♠✐♥❡❞ ❜② ❞✐str✐❜✉t✐♦♥ ♦❢ s❦✐❧❧ ❡♥❞♦✇♠❡♥ts ♦♥❧②✳ ❊①❛♠♣❧❡✿ lk(i) ∼ ❋ré❝❤❡t(θ)✳

❊✣❝✐❡♥t ❛❧❧♦❝❛t✐♦♥

max

L1A,L1B

U

• F1
• L1A, L1B
• , F2
• LA − L1A, LB − L1B
• s✉❜❥❡❝t t♦

L1B ≥ B(L1A)

• ▼✉❧t✐♣❧✐❡r✿ β

L1B ≤ B(L1A)

• ▼✉❧t✐♣❧✐❡r✿ β

❋✐rst ♦r❞❡r ❝♦♥❞✐t✐♦♥s L1A : ω1A = ω2A + β B′(L1A) L1B : ω1B = ω2B − β ❘❡s✉❧ts ✲ ✶✳ ❙❛♠❡ ❛❧❧♦❝❛t✐♦♥ ❛s ❵❢✉❧❧✬ ♣r♦❜❧❡♠✱ ✷✳ ❉❡❝❡♥tr❛❧✐③❛t✐♦♥ ❊①❛♠♣❧❡ ✲ ❋r❡❝❤❡t ✰ ❈♦❜❜✲❉♦✉❣❧❛s → ❈❧♦s❡❞ ❢♦r♠ ❝♦♠♣✳ st❛ts✳ ❢♦r β

✶✸

❯♥❜✉♥❞❧❡❞ ❛❧❧♦❝❛t✐♦♥

❵❈♦♥tr❛❝t ❝✉r✈❡✬ ❡q✉❛t❡s ♠❛r❣✐♥❛❧ r❛t❡s ♦❢ t❡❝❤♥✐❝❛❧ s✉❜st✐t✉t✐♦♥✿ F1A/F1B ❂ F2A/F2B✳ ❯♥❜✉♥❞❧❡❞ ❛❧❧♦❝❛t✐♦♥ ❡q✉❛t❡s U1/U2 t♦ ♠❛r❣✐♥❛❧ r❛t❡ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥ F2k/F1k✳

❯♥❜✉♥❞❧❡❞ ❛❧❧♦❝❛t✐♦♥

❵❈♦♥tr❛❝t ❝✉r✈❡✬ ❡q✉❛t❡s ♠❛r❣✐♥❛❧ r❛t❡s ♦❢ t❡❝❤♥✐❝❛❧ s✉❜st✐t✉t✐♦♥✿ F1A/F1B ❂ F2A/F2B✳ ❯♥❜✉♥❞❧❡❞ ❛❧❧♦❝❛t✐♦♥ ❡q✉❛t❡s U1/U2 t♦ ♠❛r❣✐♥❛❧ r❛t❡ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥ F2k/F1k✳

❇✉♥❞❧❡❞ ❛❧❧♦❝❛t✐♦♥

❇✉♥❞❧✐♥❣ ❝♦♥str❛✐♥t ❜✐♥❞s✳ ❈❛♥♥♦t ❵❜r❡❛❦ ♦♣❡♥✬ ✇♦r❦❡rs t♦ ❣❡t ❛t ✉♥❞❡r❧②✐♥❣ s❦✐❧❧ ❝♦♥t❡♥t✳ ①①①①①①①①①①①①①① U1

• F1A + B′(L1A)F1B
• = U2
• F2A + B′(L1A)F2B

❇✉♥❞❧❡❞ ❛❧❧♦❝❛t✐♦♥

❇✉♥❞❧✐♥❣ ❝♦♥str❛✐♥t ❜✐♥❞s✳ ❈❛♥♥♦t ❵❜r❡❛❦ ♦♣❡♥✬ ✇♦r❦❡rs t♦ ❣❡t ❛t ✉♥❞❡r❧②✐♥❣ s❦✐❧❧ ❝♦♥t❡♥t✳ ①①①①①①①①①①①①①① U1

• F1A + B′(L1A)F1B
• = U2
• F2A + B′(L1A)F2B

■♥❝♦♠♣❧❡t❡ ♠❛r❦❡ts ❛❧❧♦❝❛t✐♦♥

❇✉♥❞❧✐♥❣ ❝♦♥str❛✐♥t ❜✐♥❞s✳ ❈❛♥♥♦t ❵❜r❡❛❦ ♦♣❡♥✬ ❛ss❡ts t♦ ❣❡t ❛t ✉♥❞❡r❧②✐♥❣ ❛rr♦✇ s❡❝✉r✐t✐❡s ①①①①①①①①①①①①①①①①①① U1A + C′(C1A)U1B = U2A + C′(C1A)U2B

❲✐t❤✐♥✲♦❝❝✉♣❛t✐♦♥ s❦✐❧❧ ♣r✐❝❡s ❛♥❞ ✐♥❡q✉❛❧✐t②

✶✳ ❲❛❣❡s

❖❝❝✉♣❛t✐♦♥ ✶✿ w1

• lA, lB
• = ω1A lA + ω1B lB

✷✳ ❙♦rt✐♥❣

✲ ❖❝❝✉♣❛t✐♦♥ ✶ ❝❤♦s❡♥ ❜② ✐♥❞✐✈✐❞✉❛❧s ✇✐t❤ ❤✐❣❤

• lA
• lB

✸✳ ■♥❡q✉❛❧✐t②

✲ ■♥❝r❡❛s❡s ❛s ♣r✐❝❡ ♦❢ ♣r✐♠❛r②✴s❡❝♦♥❞❛r② s❦✐❧❧ ✐♥❝r❡❛s❡s

• ω1A
• ω1B

✲ ❉❡❝r❡❛s❡s ❛s ♣r✐❝❡ ♦❢ ♣r✐♠❛r②✴s❡❝♦♥❞❛r② s❦✐❧❧ ❞❡❝r❡❛s❡s

• ω1A
• ω1B

■♥ t❤❡ ♣❛♣❡r

✲ ❈❧♦s❡❞ ❢♦r♠ ❡①❛♠♣❧❡ ✉♥❞❡r

• lA(i), lB(i)
• =
• eα(1−i), eαi

✲ ▲♦❣✲❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ t♦ ❝♦♠♣✉t❡ ❝♦♥❞✐t✐♦♥❛❧ ✈❛r✐❛♥❝❡ ✲ ❉❡❝♦♠♣♦s❡s var (log w(i)|j) ✐♥t♦ ✭✐✮ ❊♥❞♦✇♠❡♥ts✱ ✭✐✐✮ Pr✐❝❡s

❘❡s✉❧ts ✲ ❈❧♦s❡❞ ❢♦r♠ ❡①❛♠♣❧❡

✶✻

❚✇♦ ❧✐♠✐t✐♥❣ ❝❛s❡s

■❧❧✉str❛t❡ ✇✐t❤ t✇♦ ♥❡st❡❞ ❝❛s❡s✿ ❑❛t③✲▼✉r♣❤②

• θj→1

❛♥❞ ❘♦②

• αj→1

✶✳ ❵❈♦♠♣❧❡t❡✬ s❦✐❧❧ s✉♣♣❧② ⇒ ❆❧✇❛②s ✉♥❜✉♥❞❧❡❞ Yj =

• AjLLσ

L

+ AjHLσ

H

1

σ

, l ∈

• lL, 0
• ,
• 0, lH
• ▲❛✇ ♦❢ ♦♥❡ ♣r✐❝❡ ❢♦r ❡❛❝❤ s❦✐❧❧✿ ωA✱ ωB

var

• log w(i)
• j
• = var
• log w(i)
• ✷✳ ❊①tr❡♠❡ ❢❛❝t♦r ❜✐❛s

⇒ ❆❧✇❛②s ❜✉♥❞❧❡❞ Y1 = ZjL1A , L1A =

• lA(i)φ1(i) di

❖♥❡ ♣♦s✐t✐✈❡ ♣r✐❝❡ ❢♦r ❡❛❝❤ ❵s❦✐❧❧✬✿ ω1A✱ ω2B var

• log w(i)
• j
• = var
• log lA(i)
• i < i∗

✶✼

❚✇♦ ❧✐♠✐t✐♥❣ ❝❛s❡s

■❧❧✉str❛t❡ ✇✐t❤ t✇♦ ♥❡st❡❞ ❝❛s❡s✿ ❑❛t③✲▼✉r♣❤②

• θj→1

❛♥❞ ❘♦②

• αj→1

✶✳ ❵❈♦♠♣❧❡t❡✬ s❦✐❧❧ s✉♣♣❧② ⇒ ❆❧✇❛②s ✉♥❜✉♥❞❧❡❞ Yj =

• AjLLσ

L

+ AjHLσ

H

1

σ

, l ∈

• lL, 0
• ,
• 0, lH
• ▲❛✇ ♦❢ ♦♥❡ ♣r✐❝❡ ❢♦r ❡❛❝❤ s❦✐❧❧✿ ωA✱ ωB

var

• log w(i)
• j
• = var
• log w(i)
• ✷✳ ❊①tr❡♠❡ ❢❛❝t♦r ❜✐❛s

⇒ ❆❧✇❛②s ❜✉♥❞❧❡❞ Yj = ZjLjA , LjA =

• lA(i)φA(i) di

, lA(i) = F1

• x(i)
• ❖♥❡ ♣♦s✐t✐✈❡ ♣r✐❝❡ ❢♦r ❡❛❝❤ ❵s❦✐❧❧✬✿ ω1A✱ ω2B

var

• log w(i)
• j
• = var
• log lA(i)
• i < i∗

❉❡t❛✐❧s ✲ ❘❡❧❛t✐♦♥s❤✐♣ t♦ t❤❡ ❵●❡♥❡r❛❧✐③❡❞✬ ❘♦② ♠♦❞❡❧

✶✼

✶✳ ❑❛t③✲▼✉r♣❤②

❊♥t✐r❡ s❡t ❢❡❛s✐❜❧❡✳ ❊q✉✐❧✐❜r✐✉♠ ❛❧✇❛②s ✉♥❜✉♥❞❧❡❞✱ r❡❣❛r❞❧❡ss ♦❢ t❡❝❤♥♦❧♦❣②✳ ❲♦r❦❡rs ♥♦t s♦rt❡❞✳ ❆❧❧ ✇♦r❦❡rs ✐♥❞✐✛❡r❡♥t✳ ◆♦ r❡♥ts ❞✉❡ t♦ ❝♦♠♣❛r❛t✐✈❡ ❛❞✈❛♥t❛❣❡✳ wj(i) = ωjlj(i)

✷✳ ❘♦②

❊q✉✐❧✐❜r✐✉♠ ❛❧✇❛②s ❜✉♥❞❧❡❞✳ ❲♦r❦❡rs s♦rt❡❞ ❜② ❝♦♠♣❛r❛t✐✈❡ ❛❞✈❛♥t❛❣❡✳ ❙❦✐❧❧ ♣r✐❝❡s ω1A/ω2B ♣✐♥♥❡❞ ❞♦✇♥ ❜② r❡❧❛t✐✈❡ s❦✐❧❧s ♦❢ ♠❛r❣✐♥❛❧ ✇♦r❦❡r✱ x∗✳ wj(i) = ωjlj(i)

❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s

✶✳ ❙②♠♠❡tr✐❝ ❝❤❛♥❣❡ ✐♥ ❢❛❝t♦r ❜✐❛s ✲ α ✷✳ ❚❛s❦✲❜✐❛s❡❞ ❝❤❛♥❣❡ ✲ Z1 ✸✳ ❙❦✐❧❧✲❜✐❛s❡❞ ❝❤❛♥❣❡ ✲ ψA ✹✳ ❚❛s❦✲s❦✐❧❧✲❜✐❛s❡❞ ❝❤❛♥❣❡ ✲ ζ1A U

• Y1, Y2
• =
• ηY

φ−1 φ

1

+ (1 − η)Y

φ−1 φ

2

• φ

φ−1

φ > 1 Y1 = Z1

• ζ1A ψA αLσ

1A + (1 − α)Lσ 1B

1

σ

Y2 = Z1

• ζ1A ψA (1 − α)Lσ

2A + αLσ 2B

1

σ

✷✵

✶✳ ❙②♠♠❡tr✐❝ ❝❤❛♥❣❡ ✐♥ ❢❛❝t♦r ❜✐❛s

❱❛r② αj ∈ [0.50, 0.85]✳ ❯♥❜✉♥❞❧❡❞✿ ω1A = ω2A✱ ω1B = ω2B✳ ❇✉♥❞❧❡❞✿ ω1A = ω2A + B′(L1A)β✱ ω1B = ω2B + β✳ ❊❝♦♥♦♠② s❤✐❢ts ❢r♦♠ ✉♥❜✉♥❞❧❡❞ ❡q✉✐❧✐❜r✐✉♠ t♦ ❜✉♥❞❧❡❞ ❡q✉✐❧✐❜r✐✉♠ ❛s ↑ β ❖t❤❡r ♣❛r❛♠❡t❡rs✿ σ = 0.20✱ φ = 1✱ θ = 2✱ LA = LB = 1✱ Z1 = 1✳

✶✳ ❙②♠♠❡tr✐❝ ❝❤❛♥❣❡ ✐♥ ❢❛❝t♦r ❜✐❛s

❱❛r② αj ∈ [0.50, 0.85]✳ ❯♥❜✉♥❞❧❡❞✿ ω1A = ω2A✱ ω1B = ω2B✳ ❇✉♥❞❧❡❞✿ ω1A = ω2A + B′(L1A)β✱ ω1B = ω2B − β✳ ❊❝♦♥♦♠② s❤✐❢ts ❢r♦♠ ✉♥❜✉♥❞❧❡❞ ❡q✉✐❧✐❜r✐✉♠ t♦ ❜✉♥❞❧❡❞ ❡q✉✐❧✐❜r✐✉♠ ❛s ↑ β ❲❛❣❡✿ w(i) = ω1AlA(i) + ω1BlB(i)

✶✳ ❙②♠♠❡tr✐❝ ❝❤❛♥❣❡ ✐♥ ❢❛❝t♦r ❜✐❛s

❱❛r② αj ∈ [0.50, 0.85]✳ ❯♥❜✉♥❞❧❡❞✿ ω1A = ω2A✱ ω1B = ω2B✳ ❇✉♥❞❧❡❞✿ ω1A = ω2A + B′(L1A)β✱ ω1B = ω2B − β✳ ❊❝♦♥♦♠② s❤✐❢ts ❢r♦♠ ✉♥❜✉♥❞❧❡❞ ❡q✉✐❧✐❜r✐✉♠ t♦ ❜✉♥❞❧❡❞ ❡q✉✐❧✐❜r✐✉♠ ❛s ↑ β ❲❛❣❡✿ w(i) = ω1AlA(i) + ω1BlB(i)

✶✳ ❙②♠♠❡tr✐❝ ❝❤❛♥❣❡ ✐♥ ❢❛❝t♦r ❜✐❛s

❱❛r② αj ∈ [0.50, 0.85]✳ ❯♥❜✉♥❞❧❡❞✿ ω1A = ω2A✱ ω1B = ω2B✳ ❇✉♥❞❧❡❞✿ ω1A = ω2A + B′(L1A)β✱ ω1B = ω2B − β✳ ❊❝♦♥♦♠② s❤✐❢ts ❢r♦♠ ✉♥❜✉♥❞❧❡❞ ❡q✉✐❧✐❜r✐✉♠ t♦ ❜✉♥❞❧❡❞ ❡q✉✐❧✐❜r✐✉♠ ❛s ↑ β ❲❛❣❡✿ w(i) = ω1AlA(i) + ω1BlB(i)

▲♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s✿ ❚❤❡♥ ✈s✳ ♥♦✇

• ❙❦✐❧❧ ❜✐❛s →

❇✉♥❞❧❡❞ ✴ ❙♦rt❡❞ ❡q✉✐❧✐❜r✐✉♠ →

• ■♥❡q✉❛❧✐t②
• ❙❦✐❧❧ ❜✐❛s → ❯♥❜✉♥❞❧❡❞ ✴ ❯♥s♦rt❡❞ ❡q✉✐❧✐❜r✐✉♠ →
• ■♥❡q✉❛❧✐t②

❯♥❞❡r ✇❤❛t ❝♦♥❞✐t✐♦♥s ❞♦ t❤❡s❡ ❝❤❛♥❣❡s ✐♥ ❢❛❝t♦r ✐♥t❡♥s✐t✐❡s ❡♠❡r❣❡ ❡♥❞♦❣❡♥♦✉s❧② ❢r♦♠ ❛♥ ❡①♣❛♥s✐♦♥ ✐♥ t❤❡ s❡t ♦❢ ❛✈❛✐❧❛❜❧❡ t❡❝❤♥♦❧♦❣✐❡s❄

✷✷

❊♥❞♦❣❡♥♦✉s t❡❝❤♥♦❧♦❣②

❯♥❞❡r ✇❤❛t ❝♦♥❞✐t✐♦♥s ❞♦ t❤❡s❡ ❝❤❛♥❣❡s ✐♥ ❢❛❝t♦r ✐♥t❡♥s✐t✐❡s ❡♠❡r❣❡ ❡♥❞♦❣❡♥♦✉s❧② ❢r♦♠ ❛♥ ❡①♣❛♥s✐♦♥ ✐♥ t❤❡ s❡t ♦❢ ❛✈❛✐❧❛❜❧❡ t❡❝❤♥♦❧♦❣✐❡s❄ ✶✳ Pr♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ Yj =

• αj
• ajALjA

σ + (1 − αj)

• ajBLjB

σ1/σ , σ < 1 ✷✳ ▼✐♥✐♠✐③❡ ♠❛r❣✐♥❛❧ ❝♦st s✉❜❥❡❝t t♦ ❛✈❛✐❧❛❜❧❡ t❡❝❤♥♦❧♦❣✐❡s min

ajA,ajB

 

• ωjA

α1/σ

j

ajA

• σ

σ−1

+

• ωjB

(1 − αj)1/σajB

• σ

σ−1

 

σ−1 σ

s✳t✳

• aρ

jA + aρ jB

1/ρ = Aj, ρ > 1

✷✸

❆✈❛✐❧❛❜❧❡ t❡❝❤♥♦❧♦❣✐❡s

❚❡❝❤♥♦❧♦❣② ❢r♦♥t✐❡r

• aρ

jA + aρ jB

1/ρ = Aj✳ ❆s ρ ց 1 ❝❛♥ r❡❛❝❤ ♠♦r❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ajA, ajB ❢♦r ❣✐✈❡♥ Aj✳

❈♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠

• ❙❦✐❧❧ ♣r✐❝❡s ❞❡t❡r♠✐♥❡ t❡❝❤♥♦❧♦❣② ❛❞♦♣t✐♦♥

ωjk = ⇒ a∗

jk

❈❛s❡❧❧✐✲❈♦❧❡♠❛♥ ✭✷✵✵✻✮

• ❆❞♦♣t❡❞ t❡❝❤♥♦❧♦❣② ❞❡t❡r♠✐♥❡s s♦rt✐♥❣ ❛♥❞ s❦✐❧❧ ♣r❡♠✐❛

a∗

jk

= ⇒ β ≥ 0 = ⇒ ωjk ❘♦s❡♥ ✭✶✾✽✸✮✱ ❍❡❝❦♠❛♥ ❙❝❤❡✐♥❦♠❛♥ ✭✶✾✽✼✮

✷✺

❊①❛♠♣❧❡

• ❙②♠♠❡tr✐❝ s❡❝t♦rs
• ■♥♥❛t❡ s❦✐❧❧ ❜✐❛s αj = 0.8
• ❙❤♦rt✲r✉♥ ρ = ∞ =

⇒ ajk = 1

• ▲♦♥❣✲r✉♥ ρ = 1✱ ❝❤♦♦s❡ t❡❝❤♥♦❧♦❣✐❡s
• Pr♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❈❊❙ ✇✐t❤ ❡✳♦✳s✳ σ
• ❘❡s✉❧t

σ > 0 s❦✐❧❧s ❛r❡ s✉❜st✐t✉t❡s → ❜✉♥❞❧✐♥❣ ∼ ❍✐❣❤ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s σ < 0 s❦✐❧❧s ❛r❡ ❝♦♠♣❧❡♠❡♥ts → ✉♥❜✉♥❞❧✐♥❣ ∼ ▲♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s

✷✻

❇✉♥❞❧✐♥❣ ❧❛❜♦r✿ σ > 0

❙❦✐❧❧s ❛r❡ s✉❜st✐t✉t❡s✱ σ > 0✳ ❙❡❝t♦r ✶ ❛❞♦♣t❡rs ❝❤♦♦s❡ t❡❝❤♥♦❧♦❣② ♠♦r❡ s❦✐❧❧ ✶ ❜✐❛s❡❞✳ ❊♥❞♦❣❡♥♦✉s❧② ♠♦r❡ ❵❘♦②✲❧✐❦❡✬✳ ❇✉♥❞❧✐♥❣ ❝♦♥str❛✐♥ts t✐❣❤t❡r✳ ❲❛❣❡ ❣❛✐♥s ♣♦❧❛r✐③❡❞✳

❇✉♥❞❧✐♥❣ ❧❛❜♦r✿ σ > 0

❙❦✐❧❧s ❛r❡ s✉❜st✐t✉t❡s✱ σ > 0✳ ❈❤♦♦s❡ t❡❝❤♥♦❧♦❣② ♠♦r❡ s❦✐❧❧ ❜✐❛s❡❞✳ ❊♥❞♦❣❡♥♦✉s❧② ♠♦r❡ ❵❘♦②✲❧✐❦❡✬✳ ❇✉♥❞❧✐♥❣ ❝♦♥str❛✐♥ts t✐❣❤t❡r✳ ❙♣❡❝✐❛❧✐st ✇❛❣❡s ✐♥❝r❡❛s❡✳ ■♥❝r❡❛s✐♥❣ ✐♥❡q✉❛❧✐t②✳

❯♥❜✉♥❞❧✐♥❣ ❧❛❜♦r✿ σ < 0

❙❦✐❧❧s ❛r❡ ❝♦♠♣❧❡♠❡♥ts✱ σ < 0✳ ❙❡❝t♦r ✶ ❛❞♦♣t❡rs ❝❤♦♦s❡ t❡❝❤♥♦❧♦❣② ❧❡ss s❦✐❧❧ ✶ ❜✐❛s❡❞✳ ❇✉♥❞❧✐♥❣ ❝♦♥str❛✐♥ts s❧❛❝❦✳ ❙♣❡❝✐❛❧✐st ✇❛❣❡s ❞❡❝r❡❛s❡✳ ❉❡❝r❡❛s✐♥❣ ✐♥❡q✉❛❧✐t②✳

❯♥❜✉♥❞❧✐♥❣ ❧❛❜♦r✿ σ < 0

❙❦✐❧❧s ❛r❡ ❝♦♠♣❧❡♠❡♥ts✱ σ < 0✳ ❈❤♦♦s❡ t❡❝❤♥♦❧♦❣② ❧❡ss s❦✐❧❧ ❜✐❛s❡❞✳ ❇✉♥❞❧✐♥❣ ❝♦♥str❛✐♥ts s❧❛❝❦✳ ❲❛❣❡ ❣❛✐♥s ❢♦r ❣❡♥❡r❛❧✐sts✳ ❲❛❣❡ ❧♦ss❡s ❢♦r s♣❡❝✐❛❧✐sts✳ ❉❡❝r❡❛s✐♥❣ ✐♥❡q✉❛❧✐t②✳

❚❤✐s ♣❛♣❡r

✶✳ ❉❛t❛ ✲ ❚✇♦ ♥❡✇ ❢❛❝ts

❆✳ ❲✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥ r❡s✐❞✉❛❧ ✇❛❣❡ ✐♥❡q✉❛❧✐t② ✲ ❈P❙ ↑ ❍✐❣❤ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s , ↓ ▲♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ❇✳ ❙✐♠✐❧❛r✐t② ♦❢ ♦❝❝✉♣❛t✐♦♥s ✐♥ t❡r♠s ♦❢ t❤❡✐r s❦✐❧❧ ✐♥♣✉ts ✲ ❖❊❙✱ ❖✯◆❊❚ ↑ ❍✐❣❤ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s , ↓ ▲♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s

✷✳ ❚❤❡♦r② ✲ ❯♥❞❡rst❛♥❞ ❆✳ ✈✐❛ ❛ ❝♦♠♣❛r❛t✐✈❡ st❛t✐❝ ✐♥❢♦r♠❡❞ ❜② ❇✳

✲ ❊①t❡♥❞ ♠♦❞❡❧ ♦❢ ❘♦s❡♥ ✭✶✾✽✸✮✱ ❍❡❝❦♠❛♥ ❙❝❤❡✐♥❦♠❛♥ ✭✶✾✽✼✮ ✲ ❊♥❞♦❣❡♥✐③❡ ❇✳ ❛s ❛♣♣r♦♣r✐❛t❡ t❡❝❤♥♦❧♦❣② ❝❤♦✐❝❡ ✭❈❛s❡❧❧✐ ❈♦❧❡♠❛♥✱ ✷✵✵✻✮ ✲ ❆❞❞ ♣❛rt✐❝✐♣❛t✐♦♥ ❞❡❝✐s✐♦♥ (l1, l2) = (ψ, ψx)✳ ❙❤♦✇ ❡✣❝✐❡♥❝② ♣r♦♣❡rt✐❡s✳

✸✳ ❊①t❡♥s✐♦♥ ✲ ❙❤♦✇ t❤❛t ❇✳ r❛t✐♦♥❛❧✐③❡s ♦t❤❡r ♥❡✇ ❢❛❝ts

✲ ■♥❝r❡❛s✐♥❣ ♦❝❝✉♣❛t✐♦♥ s✇✐t❝❤✐♥❣ ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ✲ ❉❡❝❧✐♥✐♥❣ ❡①♣❡r✐❡♥❝❡ ♣r❡♠✐✉♠ ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ✲ ❉❡❝❧✐♥✐♥❣ ♦✈❡rt✐♠❡ ♣r❡♠✐✉♠ ✴ ♣❛rt✲t✐♠❡ ♣❡♥❛❧t② ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s

✷✾

✶✳ ❖❝❝✉♣❛t✐♦♥ s✇✐t❝❤✐♥❣

0.20 0.30 0.40 0.50 0.60 1980 1985 1990 1995 2000 2005 2010 2015

Year

Low skill occupations High skill occupations

❋r❛❝t✐♦♥ ♦❢ ♠❛❧❡ ✇♦r❦❡rs ❡①♣❡r✐❡♥❝✐♥❣

• EMarch, . . . , Um, . . . , EMarch′
• t❤❛t s✇❛♣ ✶✲❞✐❣✐t ♦❝❝✉♣❛t✐♦♥s ❛❝r♦ss
• EMarch, EMarch′

✶✳ ❖❝❝✉♣❛t✐♦♥ s✇✐t❝❤✐♥❣

0.36 0.38 0.40 0.42 1995 2000 2005 2010 2015 2020

Year

Low skill occupations High skill occupations

❋r❛❝t✐♦♥ ♦❢ ♠❛❧❡ ✇♦r❦❡rs ❡①♣❡r✐❡♥❝✐♥❣

• EMonth, EMonth+1
• t❤❛t s✇❛♣ ✶✲❞✐❣✐t ♦❝❝✉♣❛t✐♦♥s ❛❝r♦ss
• EMonth, EMonth+1

✷✳ ❊①♣❡r✐❡♥❝❡ ♣r❡♠✐✉♠

1.00 1.50 2.00 2.50 3.00 3.50 1980 1985 1990 1995 2000 2005 2010 2015

Year

Low skill occupations High skill occupations

❖♥❡ ❡①tr❛ ②❡❛r ❡①♣❡r✐❡♥❝❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✷ t♦ ✸ ♣❡r❝❡♥t ❤✐❣❤❡r ✇❛❣❡

log Incit = α + βτ

Hours log Hoursit + βτ ExpExpit + βτ Exp2Exp2 it + βτ SizeSizeit . . .

+βτ

X [Y eart, Raceit, NAICS1it, Edit, Sexit]

✸✳ ❍♦✉rs ♣r❡♠✐✉♠

0.95 1.00 1.05 1.10 1980 1985 1990 1995 2000 2005 2010 2015

Year

Low skill occupations High skill occupations

(= 1)✿ ✇❛❣❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❤♦✉rs✱ (≥ 1)✿ ✇❛❣❡ ✐♥❝r❡❛s✐♥❣ ✐♥ ❤♦✉rs

log Incit = α + βτ

Hours log Hoursit + βτ ExpExpit + βτ Exp2Exp2 it + βτ SizeSizeit . . .

+βτ

X [Y eart, Raceit, NAICS1it, Edit, Sexit]

■♥t❡r♣r❡t✐♥❣ ♦t❤❡r ❢❛❝ts

✶✳ ■♥❝r❡❛s✐♥❣ ♦❝❝✉♣❛t✐♦♥ s✇✐t❝❤✐♥❣ ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s

✲ ❯♥❜✉♥❞❧❡❞ ❡q✉✐❧✐❜r✐✉♠ ❢❡❛t✉r❡s ✐♥❞❡t❡r♠✐♥❛t❡ ♦❝❝✉♣❛t✐♦♥❛❧ ❝❤♦✐❝❡

✷✳ ❉❡❝❧✐♥✐♥❣ ❡①♣❡r✐❡♥❝❡ ♣r❡♠✐✉♠ ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s

✲ ❆❞❞ ❧❡❛r♥✐♥❣ ❜② ❞♦✐♥❣ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ♦❝❝✉♣❛t✐♦♥ s❦✐❧❧ ❜✐❛s

❈❛✈♦✉♥✐❞✐s ▲❛♥❣ ✭❏P❊✱ ✷✵✷✵✮

✲ ❊①♣❡r✐❡♥❝❡ ♣r❡♠✐✉♠ ↔ ■♥❢r❛♠❛r❣✐♥❛❧ r❡♥ts ✲ ❯♥❜✉♥❞❧✐♥❣ ❧❛❜♦r r❡❞✉❝❡s ❣r❛❞✐❡♥t ♦❢ ♣r✐♠❛r② ✴ s❡❝♦♥❞❛r② s❦✐❧❧ ♣r✐❝❡s ✲ ❘❡❞✉❝❡s ♦❜s❡r✈❡❞ ❡①♣❡r✐❡♥❝❡ ♣r❡♠✐✉♠

✸✳ ❉❡❝❧✐♥✐♥❣ ♦✈❡rt✐♠❡ ♣r❡♠✐✉♠ ✴ ♣❛rt✲t✐♠❡ ♣❡♥❛❧t② ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s

✲ ❘❡q✉✐r❡s ♠♦r❡ ✇♦r❦ t♦ ❡①t❡♥❞ t❤❡ ♠♦❞❡❧ ✲ ❯♥❜✉♥❞❧✐♥❣ ❧❛❜♦r ↔ ❲♦r❦❡rs ❛r❡ ♠♦r❡ ❵s✉❜st✐t✉t❛❜❧❡✬

✸✹

❈♦♥❝❧✉s✐♦♥s

• ❉❡✈✐❛t✐♦♥s ❢r♦♠ ❧❛✇ ♦❢ ♦♥❡ ♣r✐❝❡ ❢♦r s❦✐❧❧s ✐❢ ❡✐t❤❡r

✭✐✮ t❡❝❤♥♦❧♦❣✐❡s s✉✣❝✐❡♥t❧② ❢❛❝t♦r ❜✐❛s❡❞✱ ♦r ✭✐✐✮ ✇❡❛❦ ♣❛tt❡r♥ ♦❢ ❝♦♠♣❛r❛t✐✈❡ ❛❞✈❛♥t❛❣❡ ✐♥ s❦✐❧❧s

• ❈❛♥ ❣❡♥❡r❛t❡ ♦♣♣♦s✐t❡ tr❡♥❞s ✐♥ ✇✐t❤✐♥✲♦❝❝✉♣❛t✐♦♥ ✇❛❣❡ ✐♥❡q✉❛❧✐t② ❢r♦♠

t❡❝❤♥♦❧♦❣② ❛❞♦♣t✐♦♥

• ■❢ s❦✐❧❧s s✉❜st✐t✉t❡s✱ t❡❝❤♥♦❧♦❣② ❛❞♦♣t✐♦♥ t✐❣❤t❡♥s ❜✉♥❞❧✐♥❣ ❝♦♥str❛✐♥ts

↑ r❡t✉r♥s t♦ ❝♦♠♣❛r❛t✐✈❡ ❛❞✈❛♥t❛❣❡✱ ↑ s♦rt✐♥❣ ↑ ✇✐t❤✐♥✲♦❝❝✉♣❛t✐♦♥ ✇❛❣❡ ✐♥❡q✉❛❧✐t② ❈♦♥s✐st❡♥t ✇✐t❤ ❡①♣❡r✐❡♥❝❡ ♦❢ ❤✐❣❤ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s

• ■❢ s❦✐❧❧s ❝♦♠♣❧❡♠❡♥ts✱ t❡❝❤♥♦❧♦❣② ❛❞♦♣t✐♦♥ ❝❛♥ ❝❛✉s❡ ✉♥❜✉♥❞❧✐♥❣

↓ r❡t✉r♥s t♦ ❝♦♠♣❛r❛t✐✈❡ ❛❞✈❛♥t❛❣❡✱ ↓ s♦rt✐♥❣ ↓ ✇✐t❤✐♥✲♦❝❝✉♣❛t✐♦♥ ✇❛❣❡ ✐♥❡q✉❛❧✐t② ❈♦♥s✐st❡♥t ✇✐t❤ ❡①♣❡r✐❡♥❝❡ ♦❢ ❧♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s

✸✺

❆♣♣❡♥❞✐①

▲✐♥❦ t♦ ❇❛✐s✱ ❍♦♠❜❡rt✱ ❲❡✐❧❧ ✭✷✵✷✵✮

✲ ❙❡t✉♣ ✲ ❚✇♦ ❛❣❡♥ts j ∈ {1, 2} ❝♦♥s✉♠❡ ✐♥ t✇♦ st❛t❡s k ∈ {A, B} ✲ Pr❡❢❡r❡♥❝❡s ✲ ❊①♣❡❝t❡❞ ✉t✐❧✐t② ♦❢ ❝♦♥s✉♠♣t✐♦♥ Fj

• CjA, CjB
• = πAαj

C1−γ

jA

1 − γ + πB (1 − αj) C1−γ

jB

1 − γ , α1 > 1 2 > α2 ✲ ❚r❡❡s ✲ P❤②s✐❝❛❧ ❛ss❡ts ✐♥❞❡①❡❞ i ∈ [0, 1] ❤❛✈❡ ♣❛②♦✛s d(i) =

• dA(i), dB(i)
• ,

dA(i)

• dB(i) ❞❡❝r❡❛s✐♥❣ ✐♥ i

✲ ❇✉❞❣❡t ❝♦♥str❛✐♥ts ✲ P❡r✐♦❞✲✵ ❛♥❞ P❡r✐♦❞ ✶✱ ❙t❛t❡✲k

• Q(i)φj(i) di + qAajA + qBajB

≤ φ0

j

• Q(i) di

Cjk =

• φj(i)dk(i) di + ajk

✲ ■♥❝❡♥t✐✈❡ ❝♦♠♣❛t✐❜✐❧✐t② ✲ ❖♥❧② s❤♦rt ❛rr♦✇ s❡❝✉r✐t✐❡s ✉♣ t♦ (1 − δ) ♦❢ tr❡❡ ♣❛②♦✛s Cjk ≥ δ

• φj(i)dk(i) di

, k ∈ {A, B}

❙❧❛❝❦ ✐❢ δ = 0✳ ◆♦ s❤♦rts ✐❢ δ = 1

✲ ❋❡❛s✐❜✐❧✐t② ✲ ❲❤❛t ■❈ (C1A, C2A) ❝❛♥ ❜❡ s✉♣♣♦rt❡❞ ❜② ❛ s❡t ♦❢ tr❡❡s❄ C1A = δ k∗ dA(i) di → k∗(C1A) → C1B (C1A) ≥ δ k∗(C1A) dB(i) di

▲✐♥❦ t♦ ❇❛✐s✱ ❍♦♠❜❡rt✱ ❲❡✐❧❧ ✭✷✵✷✵✮

✲ ❍❡r❡ ✇✴♦✉t ■❈✱ tr❡❡s r❡❞✉♥❞❛♥t✳ ❚r❛❞❡ ✐♥ ❆rr♦✇ s❡❝✉r✐t✐❡s✳ Q(i) =

k qkdk(i)✳

✲ ■❢ ■❈ ❜✐♥❞s✱ r❛t✐♦s ♦❢ ♠❛r❣✐♥❛❧ ✉t✐❧✐t✐❡s ♥♦t ❡q✉❛t❡❞✿ ω1A/ω1B > ω2A/ω2B ✲ ❚❤❡ ♣r✐❝❡ ♦❢ tr❡❡ i ❞❡♣❡♥❞s ♦♥ ✇❤✐❝❤ ❛❣❡♥t j ❤♦❧❞s ✐t Q1(i) = qAdA(i) + (qB − δµ1B) dB(i) , Q2(i) = (qA − δµ1A) dA(i) + qBdB(i) ✲ ■♥ ❡q✉✐❧✐❜r✐✉♠ ω1A > ω2A ❛♥❞ ω1B< ω2B✱ ✇❤✐❝❤ ✐♠♣❧✐❡s ω1A > ω1B ✲ ❘❡s✉❧t ✲ ❙❡❝✉r✐t✐❡s ✇✐t❤ ♠♦r❡ ❡①tr❡♠❡ ♣❛②✲♦✛s ✭s♣❡❝✐❛❧✐sts✮ ❛r❡ ♠♦r❡ ❡①♣❡♥s✐✈❡ ✲ ❘❡s✉❧t ✲ Pr✐❝❡ ♦❢ tr❡❡ ❡♥❝♦❞❡s ❝♦♥str❛✐♥t✱ ❧♦✇❡r t❤❛♥ r❡♣❧✐❝❛t✐♥❣ ❛rr♦✇ s❡❝✉r✐t✐❡s

❈♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠

Π1 = max

L1A,L1B

P1F1

• L1A, L1B
• − Cost1
• L1A, L1B
• Cost1
• L1A, L1B
• =

min

• φ1(i)
• φ1(i)w1(lA, lB) di

s✉❜❥❡❝t t♦ L1A =

• φ1(i) lA di

− → ω1A = P1F1A

• MC1A = MRPL1A
• L1B

=

• φ1(i) lB di

− → ω1B = P1F1B

• MC1B = MRPL1B
• ▲❛❜♦r ❞❡♠❛♥❞ ❢♦r ❡❛❝❤ t②♣❡
• φ1(i) =

     1 , ✐❢ ω1AlA(i) + ω1BlB(i) > w1

• lA, lB
• ,

✐❢ ω1AlA(i) + ω1BlB(i) < w1

• lA, lB
• ∈ (0, 1)

, ✐❢ ω1AlA(i) + ω1BlB(i) = w1

• lA, lB
• ❇❛❝❦ ✲ ❚✇♦ ❛❧❧♦❝❛t✐♦♥s

❈♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠

• Pr✐❝❡s ♣❡r ❡✣❝✐❡♥❝② ✉♥✐t ♦❢ s❦✐❧❧

wj

• lA, lB
• =

ωjAlA + ωjBlB ωjk = PjFjk = UjFjk

• ❲♦r❦❡r (lA, lB) ❝❤♦♦s❡s ♦❝❝✉♣❛t✐♦♥ j = 1 ♦♥❧② ✐❢

w1

• lA, lB
• >

w2

• lA, lB
• ❈✉t♦✛ ✇♦r❦❡r ✐♥❞✐✛❡r❡♥t

ω1A − ω2A ω2B − ω1B

• ❇❡♥❡✜t ♦❢ j = 1

= lB lA ∗

• ❘❡❧❛t✐✈❡ s❦✐❧❧ ✐♥ j = 2

= B′ L1A

• ❯♥❞❡r {ωjk = UjFjk}✱ t❤✐s ✐s t❤❡ s❛♠❡ ❝♦♥❞✐t✐♦♥ ❛s ✐♥ t❤❡ ♣❧❛♥♥❡r✬s

♣r♦❜❧❡♠

❇❛❝❦ ✲ ❚✇♦ ❛❧❧♦❝❛t✐♦♥s

❈♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠

• ❇✉♥❞❧❡❞ ❡q✉✐❧✐❜r✐✉♠✿ ❙♦rt✐♥❣ ♣r❡♠✐❛ ❛r❡ ✐♥❝r❡❛s✐♥❣ ✐♥ β

ω1A − ω2A = β B′(L1A) ω2B − ω1B = β ✕ ■♥❢r❛♠❛r❣✐♥❛❧ ✇♦r❦❡rs ❡❛r♥ r❡♥ts ❞✉❡ t♦ ❝♦♠♣❛r❛t✐✈❡ ❛❞✈❛♥t❛❣❡✱ ❞❡t❡r♠✐♥❡❞ ❜② s♦rt✐♥❣ ♣r❡♠✐❛✳ ✕ ❆❞❞✐t✐♦♥❛❧ s♦✉r❝❡ ♦❢ ✇✐t❤✐♥✲♦❝❝✉♣❛t✐♦♥ ✇❛❣❡ ✐♥❡q✉❛❧✐t②

• ❯♥❜✉♥❞❧❡❞ ❡q✉✐❧✐❜r✐✉♠✿ ❙♦rt✐♥❣ ♣r❡♠✐❛ ❛r❡ ③❡r♦✱ ✐♥❞❡t❡r♠✐♥❛t❡ s♦rt✐♥❣

ω1A − ω2A = 0 ω2B − ω1B = 0 ✕ ❆❧❧ ✇♦r❦❡rs ❛r❡ ♠❛r❣✐♥❛❧✳ ◆♦ r❡♥ts ❞✉❡ t♦ ❝♦♠♣❛r❛t✐✈❡ ❛❞✈❛♥t❛❣❡✳

❇❛❝❦ ✲ ❚✇♦ ❛❧❧♦❝❛t✐♦♥s

• ❡♥❡r❛❧✐③❡❞ ❘♦② ♠♦❞❡❧

✲ ■♥❞✐✈✐❞✉❛❧✲♦❝❝✉♣❛t✐♦♥ s♣❡❝✐✜❝ ♦✉t♣✉t yj(i) = exp

• αjAlA(i) + αjBlB(i)
• ,

Yj =

• φj(i)yj(i) di

✲ ❚❤❡ ♦♥❧② ♣r✐❝❡❞ ♦❜❥❡❝ts ❛r❡ y1(i)✱ y2(i) ✇✐t❤ ♣r✐❝❡s w1, w2 log wj(i) = log wj + αjAlA(i) + αjBlB(i) ✲ ■♥ ♦✉r ❝❛s❡ log wj(i) ≈ log wj + ωjA lA(i) + ωjB lB(i) ✶✳ ❚❡❝❤♥♦❧♦❣② ❛✛❡❝ts ✇❛❣❡s ❞✐r❡❝t❧② t❤r♦✉❣❤ t❤❡ t❡❝❤♥♦❧♦❣② ❝♦❡✣❝✐❡♥ts ✷✳ ❲✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥ ✐♥❡q✉❛❧✐t② ❡✛❡❝ts ❛r❡ s✐❧♦✲❡❞✿

✲ ❙✉♣♣♦s❡ t❤❛t t❡❝❤♥♦❧♦❣② ❝❤❛♥❣❡s ✐♥ ♦❝❝✉♣❛t✐♦♥ ✷ ✲ ❆❧❧ ❝❤❛♥❣❡s ✐♥ t❤❡ ❡❝♦♥♦♠② ❛r❡ ❡♥❝♦❞❡❞ ✐♥ t❤❡ ♦❝❝✉♣❛t✐♦♥ s❦✐❧❧ ♣r✐❝❡ wj✱ ✐✳❡✳ t❤❡ ♦❝❝✉♣❛t✐♦♥ ✜①❡❞ ❡✛❡❝t ✲ ◆♦ ❝❤❛♥❣❡ ✐♥ ✐♥❝✉♠❜❡♥t ✇✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥ ✐♥❡q✉❛❧✐t② ✐♥ ♦❝❝✉♣❛t✐♦♥ ✶

❇❛❝❦ ✲ ❲❤❡♥ ✐s t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❜✉♥❞❧❡❞❄

❲❛❣❡ ✐♥❡q✉❛❧✐t② ✲ ❈❧♦s❡❞ ❢♦r♠ ❡①❛♠♣❧❡

✲ ❙❦✐❧❧s ❢♦r ✐♥❞✐✈✐❞✉❛❧s i ∈ [0, 1]

• lA(i), lB(i)
• =
• γeα(1−i), γeαi

→ lB(i)/lA(i) = eα(2i−1) ✲ ❆♣♣r♦①✐♠❛t❡ ❧♦❣ ✇❛❣❡ ❛r♦✉♥❞ ♠❡❛♥ ❧♦❣ s❦✐❧❧s ❝♦♥❞✐t✐♦♥❛❧ ♦♥ s❡❧❡❝t✐♦♥ i∗ log w(i, j) = log

• ω1Aelog lA(i) + ω1Belog lB(i)

✲ ❲✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥ ✐♥❡q✉❛❧✐t② var

• log(w(i))
• j∗(i) = 1
• =

 

• ω1A

ω1B

• eα(1−i∗) − 1
• ω1A

ω1B

• eα(1−i∗) + 1

 

• ❇✉♥❞❧✐♥❣

α2 i∗ 2 12

❘♦②

✶✳ ❘♦② ❆s ω1A/ω1B → ∞✱ ❜✉♥❞❧✐♥❣ t❡r♠s ❣♦❡s t♦ ③❡r♦ ✷✳ ❇✉♥❞❧✐♥❣ ❲✐t❤ ✜♥✐t❡ ω1A/ω1B✱ ✐♥❡q✉❛❧✐t② ✐♥❝r❡❛s✐♥❣ ✐♥ r❛t✐♦

❇❛❝❦ ✲ ❲❛❣❡ ✐♥❡q✉❛❧✐t②

✷✳ ❚❛s❦✲❇✐❛s❡❞ ❈❤❛♥❣❡

❊①♦❣❡♥♦✉s ↑ Z1✱ ✇✐t❤ φ > 1✿ ↑ Y1, ↓ Y2✳ ▼❛r❣✐♥❛❧ ✇♦r❦❡r ❤❛s ♠♦r❡ ❙❦✐❧❧ B✱ ♣✉s❤❡s ✉♣ ω1A/ω1B✳ ❖♣♣♦s✐t❡ ❢♦r t❛s❦ ✷✳ ❖t❤❡r ♣❛r❛♠❡t❡rs✿ α1A = α2B = 0.80✱ σ = 0.20✱ θ = 2✱ L1 = L2 = 1✱ Z2 = 1✳

❇❛❝❦ ✲ ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s

✷✳ ❚❛s❦✲❇✐❛s❡❞ ❈❤❛♥❣❡

❊①♦❣❡♥♦✉s ↑ Z1✱ ✇✐t❤ φ > 1✿ ↑ Y1, ↓ Y2✳ ▼❛r❣✐♥❛❧ ✇♦r❦❡r ❤❛s ♠♦r❡ ❙❦✐❧❧ B✱ ♣✉s❤❡s ✉♣ ω1A/ω1B✳ ❖♣♣♦s✐t❡ ❢♦r t❛s❦ ✷✳ ❖t❤❡r ♣❛r❛♠❡t❡rs✿ α1A = α2B = 0.80✱ σ = 0.20✱ θ = 2✱ L1 = L2 = 1✱ Z2 = 1✳

❇❛❝❦ ✲ ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s

✷✳ ❚❛s❦✲❇✐❛s❡❞ ❈❤❛♥❣❡

❊①♦❣❡♥♦✉s ↑ Z1✱ ✇✐t❤ φ > 1✿ ↑ Y1, ↓ Y2✳ ▼❛r❣✐♥❛❧ ✇♦r❦❡r ❤❛s ♠♦r❡ ❙❦✐❧❧ B✱ ♣✉s❤❡s ✉♣ ω1A/ω1B✳ ❖♣♣♦s✐t❡ ❢♦r t❛s❦ ✷✳ ❖t❤❡r ♣❛r❛♠❡t❡rs✿ α1A = α2B = 0.80✱ σ = 0.20✱ θ = 2✱ L1 = L2 = 1✱ Z2 = 1✳

❇❛❝❦ ✲ ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s

✸✳ ❙❦✐❧❧✲❇✐❛s❡❞ ❈❤❛♥❣❡

❊①♦❣❡♥♦✉s ↑ ψA✱ ✇✐t❤ φ > 1✱ σ > 0✿ ↑ Y1, ↓ Y2✳ ▼❛r❣✐♥❛❧ ✇♦r❦❡r ❤❛s ♠♦r❡ ❙❦✐❧❧ B✱ ♣✉s❤❡s ✉♣ ω1A/ω1B✳ ❖♣♣♦s✐t❡ ❢♦r t❛s❦ ✷✳ ❖t❤❡r ♣❛r❛♠❡t❡rs✿ α1A = α2B = 0.80✱ σ = 0.20✱ θ = 2✱ L1 = L2 = 1✱ Z1 = Z2 = 1✳

❇❛❝❦ ✲ ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s

✸✳ ❙❦✐❧❧✲❇✐❛s❡❞ ❈❤❛♥❣❡

❊①♦❣❡♥♦✉s ↑ ψA✱ ✇✐t❤ φ > 1✱ σ > 0✿ ↑ Y1, ↓ Y2✳ ▼❛r❣✐♥❛❧ ✇♦r❦❡r ❤❛s ♠♦r❡ ❙❦✐❧❧ B✱ ♣✉s❤❡s ✉♣ ω1A/ω1B✳ ❖♣♣♦s✐t❡ ❢♦r t❛s❦ ✷✳ ❖t❤❡r ♣❛r❛♠❡t❡rs✿ α1A = α2B = 0.80✱ σ = 0.20✱ θ = 2✱ L1 = L2 = 1✱ Z1 = Z2 = 1✳

❇❛❝❦ ✲ ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s

✸✳ ❙❦✐❧❧✲❇✐❛s❡❞ ❈❤❛♥❣❡

❊①♦❣❡♥♦✉s ↑ ψA✱ ✇✐t❤ φ > 1✱ σ > 0✿ ↑ Y1, ↓ Y2✳ ▼❛r❣✐♥❛❧ ✇♦r❦❡r ❤❛s ♠♦r❡ ❙❦✐❧❧ B✱ ♣✉s❤❡s ✉♣ ω1A/ω1B✳ ❖♣♣♦s✐t❡ ❢♦r t❛s❦ ✷✳ ❖t❤❡r ♣❛r❛♠❡t❡rs✿ α1A = α2B = 0.80✱ σ = 0.20✱ θ = 2✱ L1 = L2 = 1✱ Z1 = Z2 = 1✳

❇❛❝❦ ✲ ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s

❯♥❜✉♥❞❧✐♥❣ ▲❛❜♦r✿ ↓ ρ✱ σ < 0

❆s ρ ❢❛❧❧s✱ t❡❝❤♥♦❧♦❣✐❡s ❜❡❝♦♠❡ ❵♠♦r❡ s✉❜st✐t✉t❛❜❧❡✬✳ ■❢ σ < 0✱ ✜r♠s ✉♥❞♦ ❡①✐st✐♥❣ s❦✐❧❧ ❜✐❛s✱ ❜✉♥❞❧✐♥❣ ❝♦♥str❛✐♥ts ❧♦♦s❡♥✱ s❦✐❧❧ ♣r❡♠✐❛ ❢❛❧❧✱ ✇❛❣❡ ❣❛✐♥s ❢♦r ❣❡♥❡r❛❧✐sts✳ pA = ω1A − ω2A

shape of technology frontier, ;

1 2 3 4 5 6 7 8 9 10

log skill premia

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

< = !1:0 < = !0:5 < = !0:1

❊①t❡♥s✐♦♥s ■

• ❆❜s♦❧✉t❡ ✈s✳ ❝♦♠♣❛r❛t✐✈❡ ❛❞✈❛♥t❛❣❡
• l1, l2
• =
• ψ, ψx
• ,
• ψ, x
• ∼ H
• ψ, x
• + ✜①❡❞ ✉t✐❧✐t② ♦❢ ❜❡✐♥❣ ♦✉t ♦❢ t❤❡ ❧❛❜♦r ♠❛r❦❡t
• ❙❡❧❡❝t✐♦♥ ♦♥ x ♠❛r❣✐♥ ✭♦❝❝✉♣❛t✐♦♥✮ ❛♥❞ ♦♥ ψ ♠❛r❣✐♥ ✭♣❛rt✐❝✐♣❛t✐♦♥✮
• ❘❡s✉❧t✿ ❈♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠ ❛❧❧♦❝❛t✐♦♥ ✐s ❡✣❝✐❡♥t
• ❲❤❛t ❛r❡ t❤❡ ❡✛❡❝ts ♦❢ ❛❞❞✐♥❣ ❛ ♠❛ss ♦❢ ❧♦✇✲♣r♦❞✉❝t✐✈✐t②

✉♥s♣❡❝✐❛❧✐③❡❞ ✇♦r❦❡rs ✭↓ ψ✱ x ≈ 1✮❄

✭sr✮ ✇❛❣❡s ❛♥❞ ❛❧❧♦❝❛t✐♦♥s ❢♦r ✜①❡❞ t❡❝❤♥♦❧♦❣② ✭❧r✮ ✇❛❣❡s ❛♥❞ ❛❧❧♦❝❛t✐♦♥s ❢♦r ❡♥❞♦❣❡♥♦✉s t❡❝❤♥♦❧♦❣②

❊♠♣✐r✐❝s ✲ ❉❡t❛✐❧s

✲ ❆❧❧ ❞❛t❛ ❜❛s❡❞ ♦♥ ▼❛r❝❤ ❈P❙ ❵❧❛st ②❡❛r✬ q✉❡st✐♦♥s ✲ ❖❝❝✉♣❛t✐♦♥✱ ■♥❞✉str② ✲ ❉♦r♥✬s ✶✾✾✵ ❤❛r♠♦♥✐③❡❞ ❝r♦ss✲✇❛❧❦

✲ ❉r♦♣ ♠✐❧✐t❛r② ✲ ❖❝❝✉♣❛t✐♦♥ s❦✐❧❧ ❂ ❋r❛❝t✐♦♥ ♦❢ ✇♦r❦❡rs ✇✐t❤ ❤✐❣❤✲s❝❤♦♦❧ ♦r ❧❡ss ✲ ❖❝❝✉♣❛t✐♦♥s s♦rt❡❞ ♦♥ ♦❝❝✉♣❛t✐♦♥ s❦✐❧❧

✲ ❯s❡ ❍P❱ ✭❘❊❉✱ ✷✵✶✵✮

✲ ❊❛r♥✐♥❣s ❂ ❲❛❣❡ ✐♥❝♦♠❡ ✰ ✭✷✴✸✮× ❙❡❧❢ ❡♠♣❧♦②♠❡♥t ✐♥❝♦♠❡ ✲ ❆♥♥✉❛❧ ❤♦✉rs ❂ ❲❡❡❦s ✇♦r❦❡❞ ❧❛st ②❡❛r × ❯s✉❛❧ ❤♦✉rs ✇♦r❦❡❞ ♣❡r ✇❡❡❦ ✲ ❲❛❣❡ ❂ ❊❛r♥✐♥❣s ✴ ❆♥♥✉❛❧ ❤♦✉rs ✲ ❆❣❡ ✷✺✲✻✺✱ ❲❛❣❡ ❃ 0.5× ❋❡❞❡r❛❧ ♠✐♥✐♠✉♠ ✇❛❣❡✱ ❍♦✉rs ❃ ❖♥❡ ♠♦♥t❤ ♦❢ 8❤r ❞❛②s

✲ ❘❡❣r❡ss✐♦♥ ❝♦♥tr♦❧s ❢♦r r❡s✐❞✉❛❧✐③❡❞ ✇❛❣❡✿

✲ ❲♦r❦❡r ❡❞✉❝❛t✐♦♥ ✭✸ ❧❡✈❡❧s✮✱ ■♥❞✉str② ✭✶ ❞✐❣✐t✮✱ ❊①♣❡r✐❡♥❝❡✱ ❊①♣❡r✐❡♥❝❡2 ❘❛❝❡✱ ▲♦❣ ❤♦✉rs✱ ✲ ❊①♣❡r✐❡♥❝❡ ❂ ✭❛❣❡ ✲ ♠❛①✭②❡❛rs ✐♥ s❝❤♦♦❧✱✶✷✮✮ ✲ ✻

❇❛❝❦ ✲ ▼♦t✐✈❛t✐♥❣ ❡♠♣✐r✐❝s

❊♠♣✐r✐❝s ✲ ❘❡❣r❡ss✐♦♥s

✶✳ ❲♦r❦❡rs ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ❣❡tt✐♥❣ ♣❛✐❞ ♠♦r❡ ❵s✐♠✐❧❛r❧②✬✳

• ❘❡❞✉❝❡❞ ❢♦r♠ ❡♠♣✐r✐❝❛❧ ❡✈✐❞❡♥❝❡ ❢r♦♠ t❤❡ ❈P❙

log Earningsi,t = γt + δOcc

period + β′ periodXi,t + εi,t

Xi,t =

• Y eart, NAICS1it, Edit, Raceit, Sexit, F irmSizeit, Expit, Exp2

it, Hoursit

• ▲♦✇ s❦✐❧❧✿ ❉❡❝❧✐♥❡ ✐♥ ↓

βperiod ❢♦r ✭✐✮ ❡①♣❡r✐❡♥❝❡✱ ✭✐✐✮ ❤♦✉rs✱ ✭✐✐✐✮ ❧❛r❣❡ ✜r♠

• ❍✐❣❤ s❦✐❧❧✿ ◆♦ ❝❤❛♥❣❡

✷✳ ❆♥❡❝❞♦t❛❧ ❡✈✐❞❡♥❝❡ ❢r♦♠ ❯❙ ❧❛❜♦r ♠❛r❦❡t

• ●♦❧❞✐♥ ❑❛t③ ✭✷✵✶✷✮ ✈s✳ ❉❛✈✐❞ ❲❡✐❧ ✭✷✵✶✹✮
• ❍❛r❞ t♦ ❡①♣❧❛✐♥ ❞❡❝❧✐♥✐♥❣ ❧❡✈❡❧ ♦❢ ❵❛tt❛❝❤♠❡♥t✬ ♦❢ ✇♦r❦✐♥❣ ❛❣❡ ♠❡♥

❇❛❝❦ ✲ ▼♦t✐✈❛t✐♥❣ ❡♠♣✐r✐❝s

❉❛t❛ ✲ ❲❛❣❡ ✐♥❡q✉❛❧✐t②

Vt

• log

yijt

• ❆✳ ❚♦t❛❧ ✈❛r✐❛♥❝❡

=

• j

ωjtVjt

• log

yijt

• ❇✳ ❲✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥

+

• j

ωjt

• Ejt [log

yijt] − Et [log yijt] 2

• ❈✳ ❇❡t✇❡❡♥ ♦❝❝✉♣❛t✐♦♥

✲ ❘❡❞ ❂ ❍✐❣❤ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s✱ ❇❧✉❡ ❂ ▲♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ✲ ✸ ❞✐❣✐t ♦❝❝✉♣❛t✐♦♥s ✲ ❈❧❛ss✐✜❡❞ ✐♥ ✷✵✶✵

Xi,t =

• Y eart, NAICS1it, Edit, Raceit, Sexit, F irmSizeit, Expit, Exp2

it, Hoursit

• ❇❛❝❦ ✲ ❘♦❧❧✐♥❣ ❝❧❛ss✐✜❝❛t✐♦♥

❉❛t❛ ✲ ❲❛❣❡ ✐♥❡q✉❛❧✐t②

Vt

• log

yijt

• ❆✳ ❚♦t❛❧ ✈❛r✐❛♥❝❡

=

• j

ωjtVjt

• log

yijt

• ❇✳ ❲✐t❤✐♥ ♦❝❝✉♣❛t✐♦♥

+

• j

ωjt

• Ejt [log

yijt] − Et [log yijt] 2

• ❈✳ ❇❡t✇❡❡♥ ♦❝❝✉♣❛t✐♦♥

✲ ❘❡❞ ❂ ❍✐❣❤ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s✱ ❇❧✉❡ ❂ ▲♦✇ s❦✐❧❧ ♦❝❝✉♣❛t✐♦♥s ✲ ✸ ❞✐❣✐t ♦❝❝✉♣❛t✐♦♥s ✲ ❈❧❛ss✐✜❡❞ ✐♥ ✶✾✽✵

Xi,t =

• Y eart, NAICS1it, Edit, Raceit, Sexit, F irmSizeit, Expit, Exp2

it, Hoursit

• ❇❛❝❦ ✲ ❘♦❧❧✐♥❣ ❝❧❛ss✐✜❝❛t✐♦♥

❋❛❝t ❇✳ ✲ ❚❡❝❤♥♦❧♦❣②

✲ ■♥♣✉t ✐s ❛ J × K ♥♦r♠❛❧✐③❡❞ ♠❛tr✐① ♦❢ s❦✐❧❧ ♠❡❛s✉r❡s A ❢r♦♠ ❖✯◆❊❚ ✶✳ ❆♣♣❧② ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts ✇✐t❤ K∗ ≪ K A[J×K] = A[J×K∗] P[K∗×K] + U[J×K] ✷✳ ❚♦ ♥❛♠❡ s❦✐❧❧s✱ r♦t❛t❡ ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts s✳t✳ s❛t✐s❢② K∗ ♦rt❤♦❣♦♥❛❧✐t② ❝♦♥❞✐t✐♦♥s A[J×K] =

• A[J×K∗]Ψ

Ψ−1 P[K∗×K]

• + U[J×K] → A∗ =

AΨ

= ⇒ ❋✐♥❛❧ s❦✐❧❧ ✶✱ ♣❧❛❝❡s ❛ ✇❡✐❣❤t ♦❢ ✶ ♦♥ k = 1✱ ❛♥❞ ③❡r♦ ♦♥ k ∈ {2, . . . , K∗}

✸✳ ❯s❡ ❛s K∗ ❵❛♥❝❤♦r✐♥❣✬ s❦✐❧❧s t❤♦s❡ ✉s❡❞ ❜② ❆❝❡♠♦❣❧✉ ❆✉t♦r ✭✷✵✶✶✮

✲ ◆♦♥✲r♦✉t✐♥❡ ❝♦❣♥✐t✐✈❡✿ ❆♥❛❧②t✐❝❛❧ ✲ ✏❆♥❛❧②③✐♥❣ ❞❛t❛ ✴ ✐♥❢♦r♠❛t✐♦♥✑ ✲ ◆♦♥✲r♦✉t✐♥❡ ❝♦❣♥✐t✐✈❡✿ ■♥t❡r♣❡rs♦♥❛❧ ✲ ✏▼❛✐♥t❛✐♥✐♥❣ r❡❧❛t✐♦♥s❤✐♣s✑ ✲ ❘♦✉t✐♥❡ ❝♦❣♥✐t✐✈❡ ✲ ✏■♠♣♦rt❛♥❝❡ ♦❢ r❡♣❡❛t✐♥❣ t❤❡ s❛♠❡ t❛s❦s✑ ✲ ❘♦✉t✐♥❡ ♠❛♥✉❛❧ ✲ ✏❈♦♥tr♦❧❧✐♥❣ ♠❛❝❤✐♥❡s ❛♥❞ ♣r♦❝❡ss❡s✑

❇❛❝❦ ✲ ❋❛❝t ❇✳ ❚❡❝❤♥♦❧♦❣②

✷

❉❡❝r❡❛s✐♥❣ s✐③❡ ♣r❡♠✐✉♠ ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝

0.05 0.10 0.15 0.20 0.25 1980 1985 1990 1995 2000 2005 2010 2015

Year

Low skill occupations High skill occupations

✶✵✵✵✰ ❡♠♣❧♦②❡❡ ✜r♠s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ✶✵ t♦ ✶✺ ♣❡r❝❡♥t ♣r❡♠✐✉♠

log Incit = α + βτ

Hours log Hoursit + βτ ExpExpit + βτ Exp2Exp2 it + βτ SizeSizeit . . .

+βτ

X [Y eart, Raceit, NAICS1it, Edit, Sexit]

❇❛❝❦ ✲ ▼♦t✐✈❛t✐♥❣ ❡♠♣✐r✐❝s

■♥❝r❡❛s✐♥❣ s✇✐t❝❤✐♥❣ ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝

0.50 0.55 0.60 0.65 0.70 1980 1985 1990 1995 2000 2005 2010 2015

Year

Low skill occupations High skill occupations

❋r❛❝t✐♦♥ ♦❢ ♠❛❧❡ ✇♦r❦❡rs ❡①♣❡r✐❡♥❝✐♥❣

• EMarch, . . . , Um, . . . , EMarch′
• t❤❛t s✇❛♣ ✸✲❞✐❣✐t ♦❝❝✉♣❛t✐♦♥s ❛❝r♦ss
• EMarch, EMarch′
• ❇❛❝❦ ✲ ▼♦t✐✈❛t✐♥❣ ❡♠♣✐r✐❝s

■♥❝r❡❛s✐♥❣ s✇✐t❝❤✐♥❣ ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝

0.36 0.38 0.40 0.42 1995 2000 2005 2010 2015 2020

Year

Low skill occupations High skill occupations

❋r❛❝t✐♦♥ ♦❢ ♠❛❧❡ ✇♦r❦❡rs ❡①♣❡r✐❡♥❝✐♥❣

• EMonth, EMonth+1
• t❤❛t s✇❛♣ ✶✲❞✐❣✐t ♦❝❝✉♣❛t✐♦♥s ❛❝r♦ss
• EMonth, EMonth+1
• ❇❛❝❦ ✲ ▼♦t✐✈❛t✐♥❣ ❡♠♣✐r✐❝s

■♥❝r❡❛s✐♥❣ s✇✐t❝❤✐♥❣ ✐♥ ❧♦✇ s❦✐❧❧ ♦❝❝

0.52 0.54 0.56 0.58 1995 2000 2005 2010 2015 2020

Year

Low skill occupations High skill occupations

❋r❛❝t✐♦♥ ♦❢ ♠❛❧❡ ✇♦r❦❡rs ❡①♣❡r✐❡♥❝✐♥❣

• EMonth, EMonth+1
• t❤❛t s✇❛♣ ✸✲❞✐❣✐t ♦❝❝✉♣❛t✐♦♥s ❛❝r♦ss
• EMonth, EMonth+1
• ❇❛❝❦ ✲ ▼♦t✐✈❛t✐♥❣ ❡♠♣✐r✐❝s