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

4.2 The representative agent’s optimal decisions

Again, it is easy to check that a generalized version of Robinson Crusoe’s value function
and of the representative agent’s value function V found in Bethmann (2002) satisfies
the Bellman equation (27) and the first order necessary conditions (28) and (29) simul-
taneously:

V (^kt^, ht; ha,t) = ψ + ψB ^ln B ⁺ ΨA ^ln At ⁺ ψk ^{ln k}t ⁺ ψh ^ln ht ⁺ ψh_a ^ln ha,t^,     (³⁴)

where the φ_i’s, with i ∈ {k, h, h_a,B,A}, are defined as follows³:

^kk := 1-αβ ,               ^kh := (1-β)(1-αβ),       Ψha := (1-β)Y1-αβ),

_ (1-α+γ)β               _ 1

^ψB := (1-β)²(1-αβ) ^{,       7}A := (1-ρβ)(1-αβ) ∙

The optimal controls implied by V are the following:

ct = (1 — αβ) y_t       and       ut = _β+wT(1(--β).                 (35)

If the government sets τw and τr equal to 1, these results correspond exactly to the
deterministic case examined in Bethmann (2002)∙ V implies a constant allocation of
human capital between the two production sectors, i∙e∙ the evolution of the average
stock of human capital ha does not enter the first-order necessary condition for ut in (35)∙
Hence, there is no linkage between the representative agent’s decision and the economy-
wide average decision∙ Therefore the solution strategy of determining the evolution of the
agent’s stock of capital and then exploiting the symmetry condition (6) is equivalent to
the strategy of finding a fixed point where the representative agent’s policy rules coincide
with the economy-wide average decisions∙ Hence, the equation:

h . . B — B______^β______h .

ha,t⁺¹ B β+τ_w(1-β) h^a,t.

determines the path of the economy-wide average level of human capital in the decen-
tralized economy∙ Together with the agent’s optimal controls, this result implies that the
Euler equations (31) and (32) and the transversality conditions (33) are met∙

4.3 The government’s optimal policy

The government wants to reach the social planner’s solution by taxing, respectively
subsidizing the agent’s factor compensations∙ Note that the absence ofτr in the first order
conditions (35) implies that the planner’s solution can be reached by simply requiring ut
to be socially optimal∙ On the other hand, assumption (10) requires that the state has
to ensure that its budget is balanced in each period∙ These two requirements lead us to
the following two conditions:

and (τr - 1) α = (1 - τw) (1 - α) .          (36)

This implies the following optimal values of τw and τr :

Tw = ₁-_α+_γ and

³The constant is given by: φ ≡ ^ln[1-β_j'^β +
(1-α) ln[τ_w] _ (1-α+βγ) ln[β+τ_m (1-β)]
(1-αβ)(1-β)          (1-αβ)(1-β)²      '

_ α-α²+γ

^Tr    (1-α+γ)α .

(1-α)ln[1-β] . αβ ln α . (1-αβ+γ)β ln β _l
(1-β)(1-αβ) ⁺ (1-β)(1-αβ) ⁺  (1-β)²(1-αβ)  ⁺

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