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

hold. Using the first inequality, we may further simplify the second and finally arrive at
the following condition:¹⁰

t-1

ln ^πt²-ι(z^t-1) < ∑ ^αi ln At-ι-i + (₁-⅛ + (α-α)2) (1^{- α} + γ)^ln B
i=0

+ ι-^γ-(1 — α^t) ln ha,o + (1 — α^t) ln ho + α^t ln ko.

Applying the expectations operator with respect to the information set available in period
0 to all At-1-i gives:

E0 [^{ln π}t-₁(^zt1)]   < ^ρ_ρ-- ^ln A0 + ɪ ⁽1 ^{- α}t)^{ln h}0 + ⁽1 ^{- α}t)^ln ho

+ ( 1—— + (^-_-α)2 ) ⁽1 ^{- α} + ^γ)^ln B + ^αt ln ko.

Since F(ht, kt, ht+1, kt+1,At; ha,t) ≤ F(ht, kt,0, 0,At; ha,t) holds for the per-period re-
turn function we know that:

F π_t¹_-1(z^t-1), π_t²_-1(z^t-1), π_t¹(z^t), π_t²(z^t), zt

≤ lnAt + γ ln ha,t + (1 — α) lnπ_t¹_-1(z^t-1) + α ln π_t²_-1(z^t-1)            (46)

must also hold. Hence for any pair (ho , ko , Ao) and for any feasible plan π, the sequence
of expected one period returns satisfies:

Eo [F(∙^,t)] < Pt^lnAo + ^α (^ρ_ρ - ) ^lnAo + (1-- + -_1-α)^- ) (1 ^{- α} + ^γ)^lnB
(1-^α+^γ-(α-^αt+1 ) ln ho + α^t+1 ln ko,

where F(∙,t) := F(k_t, ht, kt+1, ht+1, A_t). Then for any feasible plan, the expected total
returns are bounded from above:

n

[∑βtF (∙,t)]
t=o

—    α ln ko 1   (1-α) ln ho ∣     γ ln ha,o     .

≤   1-αβ + (1-αβ)(1-β) + (1-αβ)(1-β) +

This concludes our search for an upper bound of the value function, i.e. we have shown
that the limit in Assumption 2 exists although it may be minus infinity.

We know from Section 4 that:

v(h, k, A) = φ + φk ln k + ψh ln h + _; h ln ha + ψA ln A + ψB ln B       (47)

solves the functional equation. The coefficients φ_i, with i ∈ {k, h, h_a, A, B}, were defined
as follows¹¹:

-                         1--                       γ

^k := 1-αβ,                Ψh := (1-β)(1-αβ) ,        Ψha := (1-β)(1-αβ) ,

_ (1-α+γ)β               _ 1

^ψB := (1-β)²(1-αβ) ^{,       7}A := (1-ρβ)(1-αβ) ^.

¹⁰Note that Y'^t ∩ sα^s = α 1^-—^α,, — ^α₁ + ^t holds.
^s=0           (1-α)²     1-α         ^.

11The constant in is OTVen bv- in — ^{ln[1 —αβ}] ₊ ^{(1 —α}) ^ln[1—β]

The constant ψ Is ^glv^en by: ψ :—   1-β ⁺ (1-β)(1-αβ)

20

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