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



Furthermore, differentiate (1) w.r.t. time, then È = —U, use θ := V∕U, then
• ∙ ∙ ∙ ∙ ∙ ∙

θ = V∕U - VU∕U2; use (1) and (20), then È = 0 = -U, θ = 0 and V = 0 are
implied and therefore
U = V = 0. Substitute this in the above equation, the efficient
factor allocation function
Φι(k) follows. ■

Proposition 4 Suppose и Λ > 0 and к(a2∕a3)1^1 a, Φ1(k) is an increasing
concave function with
Φχ(0) = 07Φχ(∞) = ∞7Φz1(0) = a1∕a2, Φ1(∞) =
0
7 Φ'1(k) 0,Φι(k) 0.

Proof. Equation (21) is equivalent to

Φι(k) =

Φι(k) =


a∣k"

a2ka~1 + α3
aχk

a2 + a3k1-a

with a1 : =


ʌ ʌ

Λθ (1 — a) (1 — w) (1 — β) cv , a2 := a∕ (1 + ) and a3 := и — Λ,

then

Φι(0)

Φχ(∞)


0,

.


ʌ /

Using и Λ > 0, the properties of Φ1(k) follow directly from

Φ'1(k) =


[a2 + a3k1 “] — (1 a) a3a3k1

[a2 + a3k1 “]2


then

Φ'1(k)

Φ'1(0)

Iim Φ1(∞)
k→oo>


a1a2 + aa1a3k1 “   0

[a2 + a3k1^ “]2        ,


a1

— < ∞,
a2


aa1


Iim ,      .     ,1

fc→∞ 2 [a2 + a3k1 “]


= 0.


Furthermore using k > (a2∕a3)1^1 “,


the properties of Φ1(k) follow directly from

Φ'ι'(k) =

= —2a1a2 [a2 + a3k1 “] 3 (1 — a) a3k “
s----------------------------------V----------------------------------'

< 0

+ .aaιa,k*^ a +a3k °1^2(1 — a)-{1 — a2'+   ■ O }’

> 0                           '--------------s/--------------)

< 0

29



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