that is always true (since it must be ξ>1). We then have:
μ ∂2∏ y2 ,,
- ∖∂Td∂n J >
∂2π ∂2π
H2 > 0 ⇒ —-∙ ∙ -----2
∂n2 ∂ (Td)2
that implies:
α (α - 1) A (1 - ρ)1-α (hL)1-α (Td)1 α nα-2 - ξ(ξ—1)(1—ρ^pnξ-2
-α (1 - α) A (1 - ρ)1-α (hL)1-α (Tdyα-1 nα] +
- α (1 - α) A (1 - ρ)1-α (hL)1-α (Td) -α nα-1i 2 > 0
that leads to:
α (1 - α) ξ (ξ - 1) A (1 - ρ)1-α+θ ρη (hL)1-α (Td¢ -α-1 nα+ξ-2 > 0
that is always true (since ξ>1). We finally have:
H3 < 0 ⇒ det H<0
that can be checked during the simulations (because it is not possible to obtain an
analytic solution for this inequality). In this case the second-order conditions of the
problem of the firm are satisfied and the value found is effectively a maximum.
7.2 Steady-state in the decentralized economy
The steady-state of the decentralized economy is obtained considering all the quan-
tities constant in the first-order conditions of the problem of the firm and of the
household and in the market equilibrium conditions, that is:
αA (hTdL} 1 α (1 - ρ)1-α nα = ∣nξ (1 - ρ)θ ρη (23)
(1 -α)A hTdL -α(1 - ρ)1-α nα =w
(1 - α) A (hTdL¢1 α (1 - ρ)1-α nα
= ndξ (1 - ρ)θ ρη P -
η (1 - ρ)
ρ
1β
- = -(1 + r)
cc
h _ βδ (1 - Ts)
(1 - δ) Ehδ (1 - Ts)-δ = 1 - δ
c = whTs + ra
h= Ehδ(1-Ts)1-δ
(24)
(25)
(26)
(27)
(28)
(29)
29