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 kt + ψh ln ht + ψha ln ha,t, (34)
where the φi’s, with i ∈ {k, h, ha,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) , 7A := (1-ρβ)(1-αβ) ∙
The optimal controls implied by V are the following:
ct = (1 — αβ) yt 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+τw(1-β) ha,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:
(1-α)(1-β) τw (1-β)
1-α+βY β+τw (1-β)
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-βj'β +
(1-α) ln[τw] _ (1-α+βγ) ln[β+τm (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) +