|
φ n |
( (1 + β + δ) O —1 O ʌ |
Г φ — φ ^ | |||
|
12 |
— |
-cμ - (1 + δ) cμ o |
a — a |
(A8) | |
|
μ |
O OO —μf '' |
μ — μ | |||
|
∖k ' |
C Cμ 1 —Cμ β у |
к к — к у |
where z = (φ, a, μ, к, ),and Jz denotes the Jacobian matrix of (A8) in the Hxed employment
case. To determine the stability properties of the equilibrium, we Hrst consider the trace
and determinant of the Jacobian matrix
tr (jɔ = ψι + ψ2 + ψ3 + ' 'ɪ = 2β > O,
det (jɔ = ψιψ2ψ3⅜ = (β + δ)δcμμf" > O,
where ψ = 1
— 4 are the eigenvalues of J. The condition tr
> O rules out the case
of four negative eigenvalues, while det
(Jf)
> O eliminates, respectively, the cases of one
negative and three positive eigenvalues, and three negative and one positive eigenvalue.
As in the more general framework with endogenous employment, we must directly eval-
uate the characteristic equation to determine whether Jz has two negative and positive
eigenvalues or four positive eigenvalues. In this case the characteristic equation, denoted
by det (jz-ψl) = O equals:
det (jr-ψl) =
ψ4 - tr (jZ) ψ3 + [^2 - (1 + δX1 + β + δ) — cμμf "]ψ2 + β[(1 + δX1 + β + δ) + cμμf "]ψ∙
+ det (Jr) = O
(A9)
Matching the coefficients of (2.16a) and (A9), we observe that
Ψ1Ψ2Ψ3 + Ψ1Ψ2⅜ + Ψ1ψ3⅜ + Ψ2ψ3ψ4 = β[(ι + δ)(ι + β + δ) + cμμf"] < O,
which implies that we can rule out the case in which all the eigenvalues are positive.
Thus, the Hxed-employment equilibrium of (A9) is a saddlepoint with two negative and
3O