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

It only remains to be shown that conditions (a) and (b) of Theorem 3 hold. In order to
verify that the plans π¹* [∙, h₀, k₀, A₀] and π²* [∙, h₀, k₀, A₀] satisfy (a), we have to show
that (37) holds for all (h₀,k₀,A₀) when applying π¹* [∙,h₀,k₀,A₀] and π²*[∙,h₀,k₀,A₀],
where v is given by (41). In our case we consider:

v [ni-! (At^-1) ,∏2-1 (At^-1) ,At]

= θ + Θb ln B + θh ft ln B +1 ln ^β₁(-^-++γ) +ln ho} + θk α^t ln ko
-α+βγ

t-1

+θk ∑ αⁱ (ln [αβ] + (1 - α) ln [ (¹1^-_-^+-ʃ) ] +(1 — α + γ)ln ho)

i=0

t-1                                                             t-1

+θk ∑ αⁱi (1 - α + γ) (ln [^βj(⅛+^γi] + ln b) + θk ∑ α^t-1-iAi + Θa ln At.

i=0                                                 i=0

Using the fact that 0 < β < 1, 0 < αβ < 1, and E0 [lnAt] = ρ^t lnA0, it is straightforward
to show that condition (37) indeed holds.⁸

In order to verify condition (b), we need to show that for any initial state (h0, k0, A0)
in S, any plan in Π(h0, k0, A0) is weakly dominated by a plan in Π(h0, k0, A0). Let
(h₀,k₀,A₀) ∈ S and π ∈ Π(h₀,k₀,A₀) be arbitrary. By definition π ∈ Π(h₀,k₀,A₀) if
and only if (37) holds. With v given by (41), the condition (37) reads as follows:

lim β^tEo ∣θ + θh ln πt¹-i[A^t-1] + θ_fc ln π2-i[At^-1] + Θa ln At + Θb ln B1 = 0.

t→∞

It follows from the assumptions on the At ’s that:

lim β^tE₀fθA ln A_t1 = lim (ρβ)^t Θa ln A₀ = 0.
t→∞              t→∞

Hence (37) holds if and only if:

lim β^tE0 θhlnπt¹-1[A^t-1]+θklnπt²-1[A^t-1] =0.

t→∞

That is, π ∈ Π(h₀, k₀, A₀) if and only if condition (44) holds. In addition, we know from
(38) and (39) that for all (ht, kt, At) and any π ∈ Π(ht, kt, At) for all t ∈ N:

and lim β^tE₀ fln π²₁[A^t-1]'∣ ≤ 0        (45)

t→∞

must hold. Now suppose that π ∈ Π(h₀, k₀, A₀), i.e. (44) fails to hold. It follows from
the inequality in (40) that:

∞

∑βt (ln At + (1 - α + γ)lnπt¹-i[A^t-1] + αln∏t-ι [A^t-1f)
t=0

Since (44) fails, the conditions in (45) imply that this series must diverge to minus infinity:
u (π, h₀,k₀,A₀) = -∞; in this case π¹* and π²* dominate π¹ and π². Thus condition
(b) is satisfied, and Theorem 3 applies. That is, v is indeed the value function and the
policy rules are given by (42) and (43). This ends the discussion of the centralized case.
In the next subsection we turn to the decentralized economy.

⁸Note, that for β ∈ (0, 1), lim_t→∞ tβt = 0 holds.

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