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(∞) =
07 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) ∕cvqΛ , a2 := a∕ (1 + eʃ) 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) =
aχ [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 “ < 0 + .aaιa,k*^ a +a3k °1^2(1 — a)-{1 — a2'+ ■ O }’ > 0 '--------------s/--------------) < 0 |
29
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