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



(1b) πt(zt) Γ[πt-1(zt-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(zt-1), zt] in that period. Nonstationary plans can be constructed by using
different measurable selections
ht 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
μt(z0, ∙) : Zt → [0,1]:

μt(zo, Z)


Z1 ..  Zt-1   Zt


Q(zt-1, dzt)Q(zt-2, dzt-1)..Q(z0, dz1),


t N.


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(zt-1), πt(zt), 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[πt-1(zt-1), πt(zt), zt] is μt-integrable, t N,

and the limit

F[x0, π0, z0] +


lim

n→∞


n

t=1 Z


βtF[t-i(zt-1), ∏t(zt), zt]μt(xo, dzt)


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:

u0 (π, x0, z0)

un (π, x0, z0)


F[x0, π0, z0],


F[x0, π0,


n

z0]+

t 1 Zt


βtF [t-i(zt-1),∏t (zt),zt] μt(xo,dzt).


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

14



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