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

A.2 The centralized economy

The social planner’s optimization problem can be rewritten such that in every period t
the states kt and ht are given and next period’s states kt+1 and ht+1 have to be chosen,
i.e. we want to replace the variables ct and ut. Equation (9) can be solved for ct and
the resulting expression can be substituted into the utility function. Similarly, we solve
equation (4) for utht and insert the result into the production function. In terms of the
state variables, the planner’s maximization problem is now given by:

∞

^βtF (kt^, ht^{, k}t+1^, ht+1^, At)

t=0

such that

At) = ln Atkt (ht - ht+¹ ) 1 ^α hγ - kt+1

0 < ht+1 < Bht,

0<kt+1 < Atkt^αht^1-α+γ,

ln A_t+1 = ρlnA_t + ε_t+1.

Hence, let (h_t, k_t)^T ∈ X = R²₊₊ and A_t ∈ Z = R₊₊ with the Borel sets X and Z. Let
β ∈ (0, 1) and let:

^,ht+1) Atkt (ht-^⅛⁺¹ )   h^γ-kt+1 ∈ R++; ^kt+1^,ht+1 > 0|

where α ∈ (0, 1) and γ ∈ [0, α). Let the exogenous shocks be serially correlated with
E[ln A_t+1] = ρln A_t. In order to apply Theorem 3, we want verify that Assumptions 1
and 2 hold; find (v, G) and construct the plan π*(∙; x, z), for all (x, z) ∈ S, and show that
the hypotheses (a) and (b) hold.

Clearly Assumption 1 holds: Γ(k, h, A) = 0 and there are lots of measurable selections,
for example,

h(ht,kt,At) = ( 1 Bht, 1 Atkt^αht^1-α+γ) ∈ Γ(ht,kt,At).

To establish that Assumption 2 holds, note first that the per—period return func—
tion F [π1_-1(A^t-1), π2_-1(A^t-1), π_t1(A^t), π2(A^t),A_t] is μ^t(A₀, ∙)-integrable and second that
for any (ht, kt, At) and any π ∈ Π(ht, kt, At) for all t ∈ N:

ln π_t¹_-1 (A^t-1 ) < t ln B + ln h0                                                     (38)

t-1                                  t-1

ln π_t²_-1(A^t-1) <       αⁱ ln A_t-1_-i + (1 - α + γ)     αⁱ ln π_t1_-2-i(A^t-2-i) + α^t ln k₀

i=0                           i=0

holds. Using the first inequality (38), we may further simplify the second and finally
arrive at the following condition:⁷

t-1

ln∏t²-1(z^t-1)  <  ∑ αⁱ lnAt-1-i + (ɪ + (τ⅛p∙} (1 -α + γ) ln B + α^t ln ko

i=0

+ (1^-α+^γ)(1^-αt) ln ho.

1-α            ^0.

⁷Note, that Γ^t ∩ sα^s = α .^1-α. ₂ - ^α₁ ^{+ t} holds.

^{,          s=0}           (1-α)²     1-α ^.

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