3 The centralized solution of the model
In the centralized economy, the social planner internalizes the contribution of the econ-
omy’s average level of human capital to goods production. That is, the planner is able to
reach the efficient allocation of resources without the instrument of taxation. Therefore,
we assume τr =τw = 1 throughout this section.
The central planner internalizes the social returns of human capital when choosing his
optimal controls. This means that he exploits the symmetry condition stated in equation
(6) and writes his DOP as follows:
∞
U = sup E0 βt ln ct
{ct ,ut }t∞=0 t=0
with respect to the state dynamics
|
kt+1 = Aktαut1-αht1-α+γ -ct, |
∀t ∈ N0, |
(11) |
|
ht+1 = B (1 - ut) ht, |
∀t ∈ N0, |
(12) |
|
ln At+1 = ρ ln At + εt+1 , |
∀t ∈ N0, |
(13) |
|
kt ≥ 0 and ht ≥ 0, |
∀t ∈ N0. |
Since the social planner uses the symmetry from (6), he has simply dropped the index
a. Furthermore, the initial values k0 , h0 , and A0 > 0 are assumed to be given and the
social planner has to ensure that
ct > 0 and 0 ≤ ut ≤ 1
hold for all t ∈ N0 . He defines the value function as the solution to his optimization
problem from time t onwards:
V (kt, ht, At) ≡ sup Et
{cs ,us }s∞=t
∞
∑ βs-t ln Cs
s=t
s.t. (11), (12), and (13).
The Bellman equation associated with the planner’s DOP is given by:
(14)
V (kt, ht, At) = sup {ln ct + βEt [V (kt+1, ht+1, At+1)]} .
ct,ut
The first-order necessary conditions for the optimal consumption choice and the optimal
allocation of human capital between the two sectors are given by:
where Vt stands for V (kt, ht, At) and the asterisk denotes optimality. Equation (15)
describes the behavior along the optimal consumption path. When shifting a marginal
unit of today’s output from consumption to investment, today’s marginal change in
utility should equal the expected discounted marginal change of wealth with respect to
tomorrow’s capital stock. Equation (16) states that the weighted expected marginal
change of wealth with respect to physical capital equals the weighted expected marginal
ct
ut
ɪ = βEt Γ dVM ,
c* L ∂kt+ι J ,
„ *
ut
= ( Et[∂⅛] (1-α)At А 1
I Et[∂ht+1]в h hαα
(15)
(16)