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

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

Eo [ln∏2-i(z^t-1)]   <  ^ρρ-α^t ln Ao + (^1-α+₁^γ-α^i-αt) ln ho + α^t ln ko

⁺ ( ι-α ⁺ (^α-α)2 ) ^{(1 - α+γ})^ln B                ⁽³⁹⁾

Since F(ht, kt, ht+1, kt+1, At) ≤ F (ht, kt, 0, 0, At) holds for the per-period return func-
tion, we know that:

F π_t¹_-1(A^t-1),π_t²_-1(A^t-1),π_t¹(A^t),π_t²(A^t),At

≤ ln At + (1-α+γ) ln π_t¹_-1(A^t-1) + α ln π_t²_-1 (A^t-1)                (40)

must also hold. Hence for any triple (h_o , k_o , A_o) and for any feasible plan π, the sequence
of expected one period returns satisfies:

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

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

n

Iim     l^∖ ft^t^F( √^γ^l ^{β β}(1^-α+^γ) ln B I       ln Ao       I (^1-α+^γ)^ln ho I α ln ko

n→∞ E^o λJ^β F (∙,t)J ≤ (1-β)²(1-αβ) ⁺ (1-ρβ)(1-αβ) ⁺ (1-αβ)(1-β) ⁺ T-αβ.

t=o

We know from Section 3 that:

v(h, k, A) = θ+θhlnh+θklnk+θAlnA+θBlnB             (41)

is a solution to the functional equation. The coefficients θi, with i ∈ {h, k, A, B}, are
defined as follows:

Д, — __^1-α+Y__ Д, — ^α    Д . — _____¹_____ _aτlΛ —  (^1-α+γ)^β

θ^h :   (1-αβ)(1-β)^,  θ^k :   1-αβ^{,  θ}A :   (1-ρβ)(1-αβ)^,  and ^θB :   (1-β)² (1-αβ) '

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-α++γ²) ht^,                                (42)

kt+1 = αβAtkt^αht^1-α+γ ( (¹₁-α+¹-_γ^β) )¹-^α .                   (43)

Hence, given any initial state (h_o, k_o, A_o), the plan π¹* [∙, h_o, k_o, A_o] generated by the first
policy rule can be calculated explicitly. Using this plan we can also calculate the second
plan π²*[∙, h_o, k_o,A_o]; in logs, they are:

ln∏t¹*1[∙,ho,ko,Ao] = tlnB + tln ^βι-α++γ) +lnho,

-α γ

t-1

ln ∏t²*1[∙,ho,ko,Ao] = ∖ αⁱ (ln[αβ] + (1 - α)ln ∖      +(1 - α + γ) ln h^

i=o

t-1                                                                      t-1

+ ∖ αⁱi (1 - α + γ)   (ln [^β₁(-α++^γ)] +ln B) + α^t ln ko + ∖ α^t-1-iAi.

i=o                                                             i=o

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