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

(1b) πt(z^t) ∈ Γ[πt-1(z^t-1), zt].

Let Π(x0 , z0) denote the set of plans that are feasible from (x0 , z0). This set is
nonempty if the correspondence Γ is nonempty and a certain measurability constraint is
met.

Assumption 1. Γ is nonempty-valued and the graph of Γ is (X × X × Z)-measurable.
In addition, Γ has a measurable selection; that is, there exists a measurable function
h : (X, Z) → X such that h(x, z) ∈ Γ (x, z) for all (x, z) ∈ S.

Under this assumption, the set Π(x0, z0) is nonempty for all (x0, z0) ∈ S.⁶ A plan π
constructed by using the same measurable selection h from Γ in every period t is said to
be stationary or Markov, since the action it prescribes for each period t depends only on
the state [πt-1(z^t-1), zt] in that period. Nonstationary plans can be constructed by using
different measurable selections h_t in each period. Let a feasible plan and the transition
function Q on (Z, Z) be given. We want to calculate the total, discounted, expected
returns associated with this plan. Given the initial state (x0 , z0) ∈ S, we define the
following probability measures μ^t(z₀, ∙) : Z^t → [0,1]:

The domain of the per-period return function F is the set A, the graph of Γ. Then we
can define the set A as:

A = {C ∈X×X×Z : C ⊆ A}.

Under Assumption 1, A is a σ-algebra. Furthermore, if F is A-measurable, then for any
(x₀, z₀) ∈ S and any π ∈ Π(x₀, z₀),

F [πt-1(z^t-1), πt(z^t), zt] is Z ^t -measurable, ∀ t ∈ N.

This rationalizes our next assumption.

Assumption 2. F : A → R is A-measurable, and either (a) or (b) holds.

(a) F ≥ 0 or F ≤ 0

(b) For each (x₀, z₀) ∈ S and each plan π ∈ Π,

F[π_t-1(z^t-1), π_t(z^t), z_t] is μ^t-integrable, ∀ t ∈ N,

and the limit

exists (although it may be plus or minus infinity).

Assumption 2 ensures that, for each (x0, z0) ∈ S, we can define the functions un (∙, x0, z0) :
Π(x0,z0) →R,n ∈ N0, by:

⁶A proof of this result can be found in Lucas and Stokey (1989), page 243.

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