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

A.3 The decentralized economy

We start the analysis of the decentralized case by rewriting the representative agent’s
optimization problem. In every period t the states At , kt, ht, and ha,t are given and next
period’s states kt+1 and ht+1 have to be chosen. The agent also knows the government’s
balanced budget restriction (36). This means that the agent’s earnings generated by
human and physical capital income can not exceed the economy’s per capita production.
Then the maximization problem is given by:

∞

β^tF (k_t, h_t, k_t+1, h_t+1; A_t, h_a,t)
t=0

such that

0 < ht+1 < Bht,

0<kt+1 <Atkt^αht^1-αh^γa,t,
0 < ha,t+1 <Bha,t,
ln A_t+1 = ρlnA_t + ε_t+1,
ht = ha,t.

We have argued in Section 4 that the representative agent does not exploit the external
effect, because the market mechanism prevents agents from coordinating their actions.
However, the path of ha,t is predictable and the representative agent treats this path as
given.

Let (ht, kt) ∈ X = R²₊₊ and (At, ha,t) ∈ Z = R²₊₊ with the Borel sets X and Z.⁹ Let
us now turn to the policy correspondence Γ, which is given by:
for all t ≥ 0. The exogenous shocks are serially correlated with Et [ln At+1] = ρ lnAt.
Again we want to verify that Assumptions 1 and 2 hold; find (v, G) and construct the
plan π*(∙; x, z) all (x, z) ∈ S, and show that the two hypotheses (a) and (b) of Theorem
3 hold.

First, note that Assumption 1 holds: Γ(ht, kt ; At, ha,t) is non-empty and there are
lots of measurable selections, for example:

h(ht, kt; At, h_a,t) = ( 1 Bht, 2Atkah1^-ah'^γ_a,t) ∈ Γ(ht, kt; At, h_a,t).

In order to show that Assumption 2 holds, note first that the per-period return function
F π_t¹_-1(z^t-1),π_t²_-1(z^t-1),π_t¹(z^t),π_t²(z^t), z_t] is μ^t(z₀, ∙)-integrable and second that for any

(xt , zt ) and any π ∈ Π(xt , zt ) for all t ∈ N:

ln π_t¹_-1 (z^t-1 ) < t ln B + ln h0 ,

t-1

ln∏t²-i(z^t-1) < ^αⁱ {lnAt-i-i + (1 - α)ln∏1-2-i(z^t-2-i) + Ylnha,t-i-i} + α^t lnko
i=0

⁹ Note that this Borel set differs from that in the previous section.

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