On this basis we arrive at the following sign structure for the Jacobian of the 3D dynamics
at the interior steady state of the above reduced model:
J=
We therefrom immediately get that the trace of this matrix is negative, the sum a2 of
principal minors of order two is positive and a determinant of the whole matrix that is
negative. The coefficients ai ,i =1, 2, 3 of the Routh Hurwitz polynomial of this matrix
are therefore all positive as demanded by the Routh Hurwitz stability conditions. The
remaining stability condition is
a1a2 - a3 = (-traceJ)a2 - detJ > 0.
With respect to this condition we first of all see that the determinant of J is given by:
J33(J11J22 - J12J21) + J31(J12J23 - J13J22).
With respect to this expression we see that the first term is dominated by (-traceJ)a2
and can thus be canceled from the calculation of a1a2 - a3. The same holds true for the
term -J31J13J22) in the determinant of J, while the remaining, non-neutralized term
J31J12 J23 in this determinant can be made arbitrary small if the dependence of the
interest rate policy rule on the unconventional influence of the real wage on this interest
rate setting is made sufficiently small. the may however exist a variety of other situations
where the above sign structure of the Jacobian of the considered 3D dynamics will lead to
asymptotic stability, in particular if the actual size of the estimated parameters is taken
into account in addition. The real wage effect that is now included into the dynamics
of the private sector therefore seems to create not much harm for the stability of the
steady state of the considered dynamics, in particular due to its negative influence on
the rate of change of economic activity.
Increasing price flexibility may however change this situation, since growth rate of eco-
nomic activity can thereby be made to depend positively on the level of economic activity,
leading to an unstable dynamic multiplier process in the trace of J under such circum-
stances. Furthermore, such increasing price flexibility will also give rise to a negative
dependence of the real wage on economic activity and thus lead to further sign changes
in the Jacobian J. A further destabilizing mechanism is introduced if we add again the
law of motion for the inflationary climate surrounding the current evolution of price
inflation.
Under this latter extension to a 4D dynamical system the Jacobian J is augmented in
its sign structure in the following way:
/---+ ʌ
J = + _ + +
J= +0- 0 .
+0±0
As the positive entries J14, J41 show there is now a destabilizing feedback chain, lead-
ing from increases in economic activity to increases in inflation and expected inflation
23