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:
∞
sup E0
{kt+1 ,ht+1}t∞=0
βtF (kt, ht, kt+1, ht+1, At)
t=0
such that
F (kt, ht,kt+1,ht+1,
At) = ln Atkt (ht - ht+1 ) 1 α hγ - kt+1
0 < ht+1 < Bht,
0<kt+1 < Atktαht1-α+γ,
ln At+1 = ρlnAt + εt+1.
Hence, let (ht, kt)T ∈ X = R2++ and At ∈ Z = R++ with the Borel sets X and Z. Let
β ∈ (0, 1) and let:
Γ(kt, ht, At) = (kt+1
,ht+1) Atkt (ht-^⅛+1 ) hγ-kt+1 ∈ R++; kt+1,ht+1 > 0|
and
F(kt, ht, kt+1, ht+1, At) = ln
At kα (ht— B+1 ) hγ—kt+1
where α ∈ (0, 1) and γ ∈ [0, α). Let the exogenous shocks be serially correlated with
E[ln At+1] = ρln At. 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 Atktαht1-α+γ) ∈ Γ(ht,kt,At).
To establish that Assumption 2 holds, note first that the per—period return func—
tion F [π1-1(At-1), π2-1(At-1), πt1(At), π2(At),At] is μt(A0, ∙)-integrable and second that
for any (ht, kt, At) and any π ∈ Π(ht, kt, At) for all t ∈ N:
ln πt1-1 (At-1 ) < t ln B + ln h0 (38)
t-1 t-1
ln πt2-1(At-1) < αi ln At-1-i + (1 - α + γ) αi ln πt1-2-i(At-2-i) + αt 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(zt-1) < ∑ αi lnAt-1-i + (ɪ + (τ⅛p∙} (1 -α + γ) ln B + αt ln ko
i=0
+ (1-α+γ)(1-αt) ln ho.
1-α 0.
7Note, that Γt ∩ sαs = α .1-α. 2 - α1 + t holds.
, s=0 (1-α)2 1-α .
16
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