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



7 Appendix

Lemma 1 Using (12) and (13), then Fκ(k) = (1 + c1 ) r.

Proof. Differentiate (12) w.r.t. time and substitute it in (13), then Fχ (k) =
(1 +
ci) r is implied. ■

Lemma 2 Using (2), (9), (10), cv = cvQe'xt and λ = ʌoeʌt, then Fe(k) = w +
≡yθj
r - λ + β (Ù - V) + у .

Proof. Differentiate (2) w.r.t. V and (9) w.r.t. time, use cv := cvQe'xt and
λ := ʌoeʌt, then

e~rtcυ (r - λ) + 1-β ι + μ1β (Ù - V)] θ- = = 0
ʌ

^ -μι = γ-λ3βe~rtcυ (r - λ) θβ + μ1β (Ù - V) .

Substitute (10) for -/11, then
ʌ

γ-Γβe~rtcυ (r - λ) θβ + μ1β (/ - V) = e-rt [Fe(k) - w] - μ1n

and substitute (9) for μ1, then
ʌ ʌ

c l λi [β (/ - V) + n = Fe(k) - W - c /" θ (r - λ)

1 — βL / J                 1 — β    /

ʌ

^ Fe(k) = w + cυ ʌ r - λ + β - V) + y θβ.

1 — β L                   /     -

Therefore, (16) is implied. ■

Proposition 3 Using (1), (6), (7a), (15), (16) and (20), then the efficient factor

allocation function Φι(k) := —^λ°,∙, ka = θβ follows.
c^λ[{τ+rUka 1+"7-λ]

Proof. Differentiate (7a) w.r.t. E, substitute this and (6) in (16), then

θβ =   (1 - qQ (1 - β)(1 - ш) λ  ka

.             . .      .

cυλ r + β - V) + у - λ

Differentiate (7a) w.r.t. K and substitute this, (15), λ = λoext and cυ = cvqcxt in
the above equation, then

θβ =       (1 - qQ (1 - β) (1 - ʃ) λo      k“

cυ0λ [ɪkɑ-1 + β(/ - V) + n - λ
28



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