following proposition can be shown to hold true under mild - empirically plausible -
parameter restrictions. Note that we assume for the dynamics of the employment rate
the simple rule Vl = βvι (Vc — Vc) throughout this section.
Proposition 1:
Assume that the parameters γω ,γr are chosen sufficiently small and that the
parameters βw2 ,βp2,κp fulfill βp2 >βw2κp. Then: The 5D determinant of the
Jacobian of the dynamics at the interior steady state is always negative in
sign.
Sketch of proof: We have for the sign structure in this Jacobian under the given
|
assumptions the following initial situation (we here assume as limiting situation γr = | ||
|
γω =0): | ||
|
±+—±+ | ||
|
+0000 | ||
|
J= |
++00+ |
. |
|
—+0—0 | ||
|
++0+— | ||
We note that the ambiguous sigh in the entry J11 in the above matrix is due to the fact
that the real rate of interest is a decreasing function of the inflation rate which in turn
depends positively on current rates of capacity utilization.
Using second row and the last row in its dependence on the partial derivatives of p we
can reduces this Jacobian to
|
0 |
0 |
± |
+ | ||
|
+ |
0 |
0 |
0 |
0 | |
|
J= |
0 |
0 |
0 |
0 |
+ |
|
0 |
+ |
0 |
0 | ||
|
0 |
+ |
0 |
+ |
— |
without change in the sign of its determinant. In the same way we can now use the
third row to get another matrix without any change in the sign of the corresponding
determinants
|
0 |
0 |
± |
0 | ||
|
+ |
0 |
0 |
0 |
0 | |
|
J= |
0 |
0 |
0 |
0 |
+ |
|
0 |
+ |
0 |
0 | ||
|
0 |
+ |
0 |
+ |
0 |
The last two columns can under the considered circumstance be further reduced to
|
0 |
0 |
± |
0 | ||
|
+ |
0 |
0 |
0 |
0 | |
|
J= |
0 |
0 |
0 |
0 |
+ |
|
0 |
+ |
0 |
0 |
0 | |
|
0 |
0 |
0 |
+ |
0 |
12