Notes on an Endogenous Growth Model with two Capital Stocks II: The Stochastic Case

Indeed these coefficients imply that the function v(h, k, A) is below the upper bound.
The policy functions associated with v are given by:

ht+1 = B β+τw^β(1-β) ht,                                       (48)

a ,   _ _nR^- α^ah^1-α+γ ( ^τw(^1-β) ʌ                           ('40'1

^kt⁺1 = ^αβAt^kt ht      ^β+τ_w(1-β) J    •                   (49)

Hence, given any initial state s₀ = (k₀, h₀, h_a,₀, A₀), the plan π¹* [∙, s₀] generated by the
first policy rule can be calculated explicitly. Using this plan we also can calculate the
second plan π²*[∙, s₀]; in logs, they are:

ln∏t¹*ιMo] = tlnB +1ln [_β+τ√(_1-β)] + lnho,

t-1

ln∏2*1[∙,so] = ∑ αⁱ(ln[αβ] + (1 - α) ln +     '    + (1 — α) ln ho + Yln ha,o)

i=0

t-1                                                                 t-1

+ ^ ʌ αⁱi (1 — α + γ) ^βlιι [β+_τw^β1_-β) ] + ln B^ + α^t ln ko +^ ʌ α^{t 1 i}Ai •
i=o                                                         i=o

It remains only to be shown that conditions (a) and (b) of Theorem 3 hold. In order
to verify that the plans π¹*[∙, s₀] and π²*[∙, s₀] satisfy condition (a), we have to show that
(37) holds for all (s₀) when applying π¹*[∙, s₀] and π²*[∙, s₀], where v is given by (47). In
our case we consider:

V [∏1-*1 (A^t-1,ha^-1) ,∏2-*1 (A^t-1,ha^-1) ,At,ha,t]

= Ψ ⁺ ψΒ ^ln B ⁺ (ψh ⁺ ψh_a ) ^{βt ln} B ^{+ t ln} [β+∙₇-_w^β1_-β)] + ⁺ ψh ^ln h0 + ψh_a ^ln ha,0
t-1

+ψk ɑt ln k0 +ψk ^ ^αi (^ln[^αβ] ⁺ (^{1 - α}) ^ln [ β+w,(,¹(1-β) ] ^{+(1 - α}) ^ln h0+γ ln ha,0
i=0

Using the fact that 0 < β < 1, 0 < αβ < 1, and E0 [ln At] = ρ^t lnA0, it is straightforward
to show that condition (37) indeed holds¹² .

In order to verify condition (b) we need to show that for any initial state (s0) ∈ S,
any plan in Π(s0) is weakly dominated by a plan in Π(s0). Let (s0) ∈ S and π ∈ Π(s0)
be arbitrary. By definition π ∈ Π∏(s₀) if and only if (37) holds. With v given by (47),
condition (37) reads as follows:

lim β^tE0[√ + ψa ln At + ψha ln hat + .. _i ln π_t¹-1[A^t-1 X^-1 ] + ψk ln п2_л [A^t-1 ,ht-¹]] =0,
t→∞

where φ' := φ + φB ln B. It follows from the assumptions on the A_t’s that

lim βtE₀ ∣^φA ln A_t"∣ = lim (ρβ)^t _r'A ln A₀ = 0.
t→∞              t→∞

Furthermore, note that the path of ln ha,t is bounded by t ln B +ln ha,0. Hence condition
(37) holds if and only if:

lim β^tE0[^h ln π_t¹_-1 [A^t-1] + φk ln π²₁[A^t-1]l = 0.                   (50)

t→∞

¹²Note, that for β ∈ (0, 1) lim_t→∞ tβ^t = 0 holds.

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