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
More intriguing information
1. Keystone sector methodology:network analysis comparative study2. Integrating the Structural Auction Approach and Traditional Measures of Market Power
3. The name is absent
4. The fundamental determinants of financial integration in the European Union
5. The name is absent
6. AN ECONOMIC EVALUATION OF THE COLORADO RIVER BASIN SALINITY CONTROL PROGRAM
7. From music student to professional: the process of transition
8. The changing face of Chicago: demographic trends in the 1990s
9. The name is absent
10. Public-private sector pay differentials in a devolved Scotland