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



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:

lnt1*ιMo] = tlnB +1ln [β+τ(1-β)] + lnho,

t-1

ln2*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 + h + ψha ) βt ln B + t ln [β+∙7-wβ1-β)] + + ψh ln h0 + ψha ln ha,0
t-1

k ɑt ln k0 k ^ αi (ln[αβ] + (1 - α) ln [ β+w,(,1(1-β) ] +(1 - α) ln h0+γ ln ha,0
i=0

t-1

+ψkαii (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 holds
12 .

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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