Applying the expectations operator with respect to the information set available in period
0 to all At-1-i gives:
Eo [ln∏2-i(zt-1)] < ρρ-αt ln Ao + (1-α+1γ-αi-αt) ln ho + αt 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 πt1-1(At-1),πt2-1(At-1),πt1(At),πt2(At),At
≤ ln At + (1-α+γ) ln πt1-1(At-1) + α ln πt2-1 (At-1) (40)
must also hold. Hence for any triple (ho , ko , Ao) and for any feasible plan π, the sequence
of expected one period returns satisfies:
Eo [F(∙,t)] < ρtlnAo + α (ρρ~α ) lnAo + ( 1—α + α1-α)α) (1 -α + γ)lnB
(1-α+γ-α-αt+1 ) ln ho + αt+1 ln ko,
where F(∙,t) := F(kt,ht,kt+1,ht+1,At). Then for any feasible plan, the expected total
returns are bounded from above:
n
Iim l^∖ ftt^F( √γ^l β β(1-α+γ) ln B I ln Ao I (1-α+γ)ln ho I α ln ko
n→∞ Eo λJβ 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
θh : (1-αβ)(1-β), θk : 1-αβ, θA : (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 = αβAtktαht1-α+γ ( (11-α+1-γβ) )1-α . (43)
Hence, given any initial state (ho, ko, Ao), the plan π1* [∙, ho, ko, Ao] generated by the first
policy rule can be calculated explicitly. Using this plan we can also calculate the second
plan π2*[∙, ho, ko,Ao]; 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) + αt ln ko + ∖ αt-1-iAi.
i=o i=o
17