policy parameter γω). Then: The interior steady state of the resulting 4D
dynamical system (where the state variable ω is now included)
|
Vc |
= -αv c ( Vc — Vc ) — αlω ( ω — ) — αr (( r — p) — ( ro — π)) |
(9) |
|
Vl |
= βvi ( Vc — Vc ) |
(10) |
|
r |
= —Yr ( r — ro ) + Yp ( p — ∏) + Yv c ( Vc — V/c ) |
(11) |
|
ω |
= κ [(1 — κp ) βw 1 ( V l — V l ) — (1 — κw ) βP 1 ( V c — V c ) |
(12) |
is locally asymptotically stable.
Sketch of proof: It suffices to show in the considered situation that the determinant
of the resulting Jacobian at the steady state is positive, since small variations of the
parameter αω must then move the zero eigenvalue of the case αω = 0 into the negative
domain, while leaving the real parts of the other eigenvalues - shown to be negative in
the preceding proposition - negative. The determinant of the Jacobian to be considered
here - already slightly simplified - is characterized by
0+--
+000
J = 0 + - 0
0+00
This can be simplified to
/ 0 0 0 - ∖
J +000
J = 0 0 - 0
0+00
without change in the sign of the corresponding determinant which proves the proposi-
tion. ■
We note that this proposition also holds where βp2 >βw2 κp holds true as long as the
thereby resulting real wage effect is weaker than the one originating from αω . Finally -
and in sum - we can also state that the full 5D dynamics must also exhibit a locally
stable steady state if βπm is made positive, but chosen sufficiently small, since we have
already shown that the full 5D dynamics exhibits a negative determinant of its Jacobian
at the steady state under the stated conditions.
A weak Mundell effect, (here still) the neglect of Blanchard-Katz error correction terms,
a negative dependence of aggregate demand on real wages, coupled with nominal wage
and also to some extent price level inertia (in order to allow for dynamic multiplier
stability ), a sluggish adjustment of employment towards actual capacity utilization
and a Taylor rule that stresses inflation targeting therefore are (for example) the basic
ingredients that allow for the proof of local asymptotic stability of the interior steady
state of the dynamics (1) - (5). We expect however that indeed more general situation
of convergent dynamics can be found, but have to leave this here for future research and
numerical simulations of the model. Instead we now attempt to estimate the signs and
sizes of the parameters of the model in order to gain insight into the question to what
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