7 Appendix

Lemma 1 Using (12) and (13), then F_κ(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), c_v = c_vQe'^xt and λ = ʌoeʌ^t, then F_e(k) = w +
≡yθ^j r - λ + β (Ù - V) + у .

Proof. Differentiate (2) w.r.t. V and (9) w.r.t. time, use c_v := c_vQe'^xt and
λ := ʌoeʌ^t, then

e~^rtc_υ (r - λ) + ^1-β [μ_ι + μ₁β (Ù - V)] θ- = = 0
ʌ

^ -μι = γ-λ3βe~^rtc_υ (r - λ) θ^β + μ₁β (Ù - V) .

Substitute (10) for -/1₁, then
ʌ

γ-Γβe~^rtc_υ (r - λ) θ^β + μ₁β (/ - V) = e-^rt [F_e(k) - w] - μ₁n

and substitute (9) for μ₁, then
ʌ ʌ

c _l ^λ ∕ⁱ [β (/ - V) + n∖ = F_e(k) - W - c /" θ (r - λ)

1 — βL∖ / J                 1 — β ∖    /

ʌ

^ F_e(k) = w + c_υ ʌ r - ^λ + β (Ù - V) + y θ^β.

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°,∙, k^a = θ^β follows.
^c^^λ[{τ+rU^{ka 1}+"⁷-^λ]

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

θβ =   ^{(1 - q}Q ^{(1 - β})^{(1 - ш) λ}  k^a

.             . .      .

c_υλ r + β (и - V) + у - λ

Differentiate (7a) w.r.t. K and substitute this, (15), λ = λoe^xt and c_υ = c_vqc^xt in
the above equation, then

_θβ =       ^{(1 - q}Q ^{(1 - β}) ^{(1 -} ʃ) ^λo      k“

c_υ0λ [ɪkɑ-1 + β(/ - V) + n - λ
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