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 αt 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 ∑ αt-1-iAi + Θa ln At.
i=0 i=0
Using the fact that 0 < β < 1, 0 < αβ < 1, and E0 [lnAt] = ρt 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 βtEo ∣θ + θh ln πt1-i[At-1] + θfc ln π2-i[At-1] + Θa ln At + Θb ln B1 = 0.
t→∞
It follows from the assumptions on the At ’s that:
lim βtE0fθA ln At1 = lim (ρβ)t Θa ln A0 = 0.
t→∞ t→∞
Hence (37) holds if and only if:
lim βtE0 θhlnπt1-1[At-1]+θklnπt2-1[At-1] =0.
(44)
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:
lim
t→∞
βtE0 ln πt1-1 [At-1] ≤ 0
and lim βtE0 fln π21[At-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:
∞
u (π, h0, k0, A0) ≤ E0
∑βt (ln At + (1 - α + γ)lnπt1-i[At-1] + αln∏t-ι [At-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), limt→∞ tβt = 0 holds.
18
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