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+τwβ(1-β) ht, (48)
a , _ nR- αah1-α+γ ( τw(1-β) ʌ ('40'1
kt+1 = αβAtkt ht ^β+τw(1-β) J • (49)
Hence, given any initial state s0 = (k0, h0, ha,0, A0), the plan π1* [∙, s0] generated by the
first policy rule can be calculated explicitly. Using this plan we also can calculate the
second plan π2*[∙, s0]; in logs, they are:
ln∏t1*ιMo] = tlnB +1ln [β+τ√(1-β)] + lnho,
t-1
ln∏2*1[∙,so] = ∑ αi(ln[αβ] + (1 - α) ln + ' + (1 — α) ln ho + Yln ha,o)
i=0
t-1 t-1
+ ^ ʌ αii (1 — α + γ) βlιι [β+τwβ1-β) ] + ln B^ + αt ln ko +^ ʌ αt 1 iAi •
i=o i=o
It remains only to be shown that conditions (a) and (b) of Theorem 3 hold. In order
to verify that the plans π1*[∙, s0] and π2*[∙, s0] satisfy condition (a), we have to show that
(37) holds for all (s0) when applying π1*[∙, s0] and π2*[∙, s0], where v is given by (47). In
our case we consider:
V [∏1-*1 (At-1,ha-1) ,∏2-*1 (At-1,ha-1) ,At,ha,t]
= Ψ + ψΒ ln B + (ψh + ψha ) βt ln B + t ln [β+∙7-wβ1-β)] + + ψh ln h0 + ψha ln ha,0
t-1
+ψk ɑt ln k0 +ψk ^ αi (ln[αβ] + (1 - α) ln [ β+w,(,1(1-β) ] +(1 - α) ln h0+γ ln ha,0
i=0
t-1
+ψk ∑ αii (1 - α + γ) (ln [β+τwβ(1-β)]
i=0
t-1
+ ln B) + φk αt 1 iAi + ψA ln At •
i=0
Using the fact that 0 < β < 1, 0 < αβ < 1, and E0 [ln At] = ρt lnA0, it is straightforward
to show that condition (37) indeed holds12 .
In order to verify condition (b) we need to show that for any initial state (s0) ∈ S,
any plan in Π(s0) is weakly dominated by a plan in Π(s0). Let (s0) ∈ S and π ∈ Π(s0)
be arbitrary. By definition π ∈ Π∏(s0) if and only if (37) holds. With v given by (47),
condition (37) reads as follows:
lim βtE0[√ + ψa ln At + ψha ln hat + .. i ln πt1-1[At-1 X-1 ] + ψk ln п2л [At-1 ,ht-1]] =0,
t→∞
where φ' := φ + φB ln B. It follows from the assumptions on the At’s that
lim βtE0 ∣^φA ln At"∣ = lim (ρβ)t r'A ln A0 = 0.
t→∞ t→∞
Furthermore, note that the path of ln ha,t is bounded by t ln B +ln ha,0. Hence condition
(37) holds if and only if:
lim βtE0[^h ln πt1-1 [At-1] + φk ln π21[At-1]l = 0. (50)
t→∞
12Note, that for β ∈ (0, 1) limt→∞ tβt = 0 holds.
21
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