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

The state-like variable can be interpreted as a weighted ratio of the two capital stocks. In
the deterministic model, the state-like variable xt remains constant along the balanced
growth path. Here, we consider a stochastic model such that x_t may sometimes be above
or below its balanced growth path where the dynamics stem from the physical capital
stock since our solutions imply that h evolves deterministically both in the centralized as
well as in the decentralized case. In Section 4 we have shown that the government is able
to force agents to make socially optimal decisions, i.e. to internalize the external effects
stemming from the economy-wide average stock of human capital. Therefore this section
focuses on the decentralized case. The representative agent’s solution is fully described
by the policy rules (35) together with the laws of motion for kt , ht , ha,t , and At . Using
our results, the dynamics of total factor productivity, of the state-like variable, and of
the control-like variable are described by the following equations:

ln Atl1

ρ ln A_t + ε_t_l1

^xtl1

₌ ^aeu¹^—^α _xα a

1 — α +γ 1 — α +γ ^xt ʃɪt

B ¹^—^α (1-u) ¹^—^α

₌ ^u _xα a

1 — α +γ 1 — α +γ xt ^j^-t ■

B ¹^—^α (1-u) ¹^—^α

Taking logarithms and using small letters with a hat in order to indicate this transfor-
mation, we arrive at:

^at+1 = ρat + ^εt+1^,

^t+1

______αβu¹^-^α______

1 — α+γ 1—α+γ

B ¹^—^α (1-u) ¹^—^α

(1-αβ)u1^-^α

1 — α+γ 1—α+γ

B ¹^—^α (1-u) ¹^—^α

+ α^t + at,

+ α^t + à_t.

The law of motion of total factor productivity is a first-order autoregressive process with
stable root ρ:
ε ^εt
at = 1-ρL ■

The evolution of the logged state-like variable x is described by a stochastic first-order
difference equation with stable root α and stochastic disturbance ^. Hence the logged
state-like variable x follows an AR(2) process:

xt+1 = 1-α^ln

______αβu¹^-^α______

________^εt_________

(1-ρL)(1-αL)^,

1 — α+γ 1 — α+γ

B ¹^—^α (1-u) ¹^—^α

where the constant term on the right-hand side is the unconditional mean of the log
state-like variable x. Since the control-like variable q_t is non-ambiguously determined by
At and xt, we conclude that the whole system is saddle-path stable. Furthermore, the
control-like variable q_t follows an AR(2) process:

^t = ln [1 - αβ∖ + ln u + 1-α ln

_______αβ_______

1 — α+γ 1 — α+γ

B ¹^—^α (1-u) ¹^—^α

_______£t—1_______

(1-ρL)(1-αL) ■

We conclude that the detrended output s_t := y_t -h_t — 1^γαh_a,_t is also AR(2). Note that
B(1 — u_bgp) in the decentralized case is equal to Bβ+~(1-β), such that optimal taxation
induces a human capital growth rate of Bβ ₁¹ /..+_l ^l...⅜'_γ, whereas a laissez-faire policy implies
Bβ, such that the growth rates in the centralized case or in the decentralized case with
optimal taxation are indeed higher than in the decentralized economy with suboptimal or
no taxation. This concludes the discussion of the time-series implications of our solutions.
In the next section we formally prove the uniqueness of the value functions found before.

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