7 Appendix
7.1 Second-order conditions at the steady-state in the de-
centralized economy
To check, at the steady state, the second-order conditions of the problem of the firm
that guarantee the presence of a maximum it is possible to observe that we have (at
the steady-state):
∂ 2π
∂n2
∂2π
∂ (Td)2
∂ 2π
∂p2
α (α — 1) a [(1 - p) hTdL∖ 1-α 2 - ξ (ξ 1) (J p)' " nξ-2
-α (1 - α) A [(1 - p) hTdL∖ α 1 (1 - p)2 h2L2nα
-α (1 - α) A [(1 - p) hTdL∖ α 1 h2 (Td)2 L2nα +
nξ ■
-7 η
d L
(η - 1)(1 - p)θ pη-2 - θ (1 - p)θ-1 pη-1 +
-θ η (1 - p)θ-1 pη-1 - (θ - 1)(1 - p)θ-2 pη
∂ 2π
∂T d∂n
∂ 2π
∂ρ∂n
∂2π
∂n∂T d
= α (1 - α) A [(1 - p) hTdL] α (1 - p) hLnα-1
-α (1 - α) A [(1 - p) hTdL] α hTdLnα-1 +
∂ 2π
∂n∂p
∂2π
∂T d∂p
∂ 2π
∂p∂T d
η (1 - p)θ pη-1 - θ (1 - pi"-' pη
: - (1 - α)2 A [(1 - p) hL]-α h (Td)-α Lnα
The hessian matrix is then:
|
■ ∂2π |
∂2π |
∂2π | |
|
∂n2 |
∂T d∂n |
∂ρ∂n | |
|
H = |
∂2π |
∂2π |
∂2π |
|
∂n∂T d |
∂(T d)2 |
∂ρ∂T d | |
|
∂2π |
∂2π |
∂2π | |
|
∂n∂ρ |
∂T d∂ρ |
∂ρ2 |
and in order to have a maximum for the problem of the firm the sequence of the
signs of the north-west principal minors of this matrix must be:
H1 < 0 H2 > 0 H3 < 0
With reference to this aspect we have:
H1 < 0 ⇒ ∣2∏ < 0 ⇒ α (α - 1) A [(1 - p) hTdL∖1-α nα-2- ξ (ξ - 1) (1 - p) p^ nξ-2 < 0
∂n2 d
28
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