Qualification-Mismatch and Long-Term Unemployment in a Growth-Matching Model

=■ Φι(k)

Proposition 7 The balanced accumulation function is a concave function with Φι(0)

= 0. Φι(∞) = _'x..Φ∣ (0) = ∞. Φ'ι(∞) = _«₅,Φi(k)

and Φ₁(k) < 0.

(≥ ⁰

I <⁰

for ak^{a 1} {

≥ a₅

< ^a5

and therefore

Φ'₁(k)

(≥⁰

I <⁰

for

ak^{a 1}

≥ α₅

< a₅

Iim Φ₁ (k) = Iim (ak“ ¹ — a₅) = ∞
fc→0               fc→0 ^{v              ,}

Iim Φ₁(k) = Iim (ak“^{_ 1} — a₅) = _a₅.
fc→∞            fc→∞ `

Furthermore,

Φι(k) = (« _ 1) aa₄k“^-2 < 0

is implied. ■

Proposition 8 Suppose n — 2λ > 0. then < 0.

Proof. Take the derivative of (21) w.r.t. Λ and use n — 2Λ > 0, then

5Φι(k)

. . , , . . ʌ -,
(1 _ α)(1 _ £)(1 _ ω) Λ_o n _ 2Λ ± ɪkɑ"¹

k“

ʌ

∂Λ

c 2 Г c ,      ,

c_υ0λ  n _ λ ± -ɪk" ¹

is implied. ■

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