That is, π ∈ Π(s0) if and only if condition (50) holds. In addition, we know from (38)
and (39) that for all st and any π ∈ Π(st) for all t ∈ N
lim βtE0∣^ln π1-1[At-1, ha-1]^∣ ≤ 0 and lim βtE0 ∣"ln π2-1[At~1, hta~1]l ≤ 0 (51)
t→∞ t→∞
must hold. Now suppose that π ∈ ΠΠ(s0), i.e. (44) fails. It follows from the inequality in
(46) that
∞
u (π, h0, k0, A0) ≤ E0
∑ βt (ln At + (1 - α + γ)ln ∏t1-i [At-1 ] + α ln ∏2,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 (48) and (49).
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