stable corridor and the death of an unstable limit cycle as the Hopf-bifurcation value is approached or the birth of a stable periodic motion after the bifurcation
point has been passed. The center – type stability of the Goodwin 1967 growth cycle is thus in particular made a locally implosive dynamics if inside workers
dominate the real wage bargain and an explosive one in the opposite situation at least locally. In the explosive case we have to add — as in Ito 1980 — the
condition V 5 1 to the model. We shall later on also assume — and motivate — that the growth rate of the labor supply will increase appropriately as the
overemployment within firms becomes higher and higher which will take pressure from the labor market and thus help to avoid labor supply bottlenecks. This will
lead us to bounded dynamics, but in the present situation still one where the share of profits 1 − vy may fall below zero — which is not meaningful. We therefore
add in a next step, as in Ito 1980, smooth factor substitution and the neoclassical theory of employment before we come to the discussion of global constraints from
the side of labor supply and their implications.
Before closing this section let us stress that the above extension of the Goodwin 1967 growth cycle model is now based on two interacting cycles: the usual
outsider cycle which can be isolated by assuming b
l
: , b
w
2
= 0 and which might
be called the ‘US case’ of the model and a new insider cycle which can be isolated by b
l
= 0, b
w
1
= 0, the ‘Japanese case’ of the model. The astonishing thing is that
the interaction of these two Goodwin cycles contributes to the stability of the general model, i.e. the presence of both variable inside employment with strong
wage claim effects and the assumed lagged adjustment of outside employment produces convergence to the steady state in a Goodwin growth cycle setup. A first
explanation of this may be seen in the fact that Eq. 6 inserted into Eq. 4 adds a derivative term to the postulated Phillips curve 4 which is known to be
stabilizing. Note that this derivative term is already contained in Phillips’ 1958 original article and gave rise there to the explanation of so-called Phillips
loops.
3. Smooth factor substitution
In the case of a neoclassical production function with the usual properties and the marginal productivity theory of the rate of employment
Y = FK, L and
v = F
L
K, L 7
we have to recalculate K . and on this basis the law of motion for the variable V
w
. Denoting by k the actual capital intensity KL we know from the Solow model
of neoclassical growth that 7 gives rise to y = fk, x = fkk, v = fk − f kk = gk
for the now endogenous output — input ratios of the preceding section. For the function g there furthermore holds
gk = f k − f k − f ¦kk= −f ¦kk\0,
i.e. the function g is strictly increasing due to decreasing marginal products of labor. We denote by k = kv the inverse of g and by ov = k´vvkv \ 0 the elasticity
of this function k. On the basis of these equations we then get:
L . =K.−k.=s
c
x1 − vy − ovv ˆ = s
c
f kv − ovv ˆ .
This expression is now to be used in Eq. 5 in the place of only s
c
x1 − vy in order to describe the dynamics of inside employment V
w
= LL
w
under neoclassical factor substitution and the neoclassical theory of employment. This, however, is the only
change in the dynamical system 4 – 7 if such substitution is included.
Proposition 2.
1
. The interior steady state of the dynamical system
4
–
6
with smooth factor substitution is of the same type as before, but now with an endogenous
determination of y
o
= f
k
o
, x
o
= f
k
o
k
o
and k
o
= kv
o
, where v
o
is gi6en by the solution of the equation ns
c
= f kv
o
.
2
. This steady state is locally asymptotically stable if b
w
2
\ b
w
1
V − d holds for some suitably chosen d \ 0. The size of d can be chosen the larger, the larger the terms
o v,b
l
,b
w
2
become.
Proof 2. See Appendix A. As the expression for the Routh – Hurwitz condition a
1
a
2
a
3
\ 0 in the proof shows, stability is indeed significantly increased by the
inclusion of smooth factor substitution and the neoclassical theory of employment.
4. Effective supply constraints?
Having extended the Goodwin growth cycle approach by inside – outside labor market effects married with Solovian economic growth we now come to a specifica-
tion of the bounds that can limit the evolution of such an economy. To do so, let us assume as an example that overtime work of the employed workforce L
w
is legally restricted to L
w
, i.e. overtime work supplied by a person can be and will be at most as large as the normal working time of this person, i.e. we impose the inequality
L 5 2L
w
or V
w
5 2. Furthermore: V 5 1 must hold true as in the Ito 1980 model,
due to the external labor market constraint. Finally, the rate of profit x1 − vy should be positive at all times.
Summarizing, our dynamical system with smooth factor substitution then reads, including the bounds v B y, V
w
5 2, V 5 1:
v ˆ = b
w
1
V − V + b
w
2
V
w
− 1
8 V
.
w
= s
c
f kv − ov[b
w
1
V − V + b
w
2
V
w
− 1] − b
l
V
w
− 1 − n
9 V
. =b
l
V
w
− 1
10 The above constraints represent all constraints that are needed from the perspective
of non-Walrasian disequilibrium growth theory. Note here that v,V
w
,V \ 0 is automatically ensured in finite time due to the growth rate formulation of the
model, but that this does not yet exclude that limit points of trajectories which start in the positive orthant may lie on the boundary of it. This however is another topic
that is not related to our questioning of the validity of the regime switching methodology of Ito 1980 and others, which is the topic of this paper.
Note here first that the viability condition v B y a positive profit share is always fulfilled, since
v = fk − f kk B fk
holds at all times in the Solow model. Next we consider the condition V
w
5 2. In order to guarantee that this condition
holds true at each moment in time we make the following simple additional assumption on the growth rate n of labor supply:
n = nV
w
, n ] 0, n2 = .
This assumption states that the growth rate of labor supply increases when overtime work increases, up to infinity, when the legal barrier V
w
= 2 is approached in a
continuous time framework, due to rapid changes in the participation rate of households, due to the influx of labor from rural surroundings and also due to
migration from other parts of the world. Note that all these changes may be due solely to workers’ decision making, but they may also be due to systematic efforts
of the firms. Firms surely will attempt to recruit new labor force in case of significant labor shortage by influencing in particular the labor supply decisions of those
households that so far were not part of the external labor market of the economy. In short, if labor tends to become the short side of the market, firms in market
economies find ways to push up the growth rate of the labor supply sufficiently to avoid absolute supply bottlenecks. There may be many microeconomic aspects of
the points just mentioned that deserve more detailed discussion. Our basic argument here however simply is that macroeconomic dynamics has to take account of these
possibilities and flexibilities in the behavior of households and firms even on a preliminary level, instead of discussing the hypothetical consequences of hard
constraints that have never existed in this form in developed market economies.
4
With respect to the third restriction V 5 1 it is possible, but not necessary, to assume that firms must reduce recruiting efforts on the ‘traditional’ labor market
when the upper bound on the external employment rate is approached,
5
increasing instead their activities for recruiting new workers as just discussed. In fact this
constraint can become binding in the hard way, without any supply consequences as long as firm can use the internal labor market for more employment and
production which they always can if the above assumption on the rate n holds.
4
Note that this means that firms have to consider V 5 1 and n = L .
s
as two separate characteristics of the labor market.
5
Assuming b
l
0 as V 1 says that it becomes more and more difficult for firms to recruit further
workers from the existing labor market when the external rate of employment approaches 1. They then choose instead to operate more and more directly on labor supply decisions L
.
s
.
Proposition 3.
1
. The dynamical system
8
–
10
with smooth factor substitution and supply constraints exhibits the domain 0, × 0,2 × 0,1] as in6ariant subset which
it thus cannot lea6e.
2
. The classical regime of non-Walrasian disequilibrium analysis is the only regime that is possible in this domain, i.e. v = F
l
K,L holds true at all times.
Proof 3. See Appendix A. Adding three simple extrinsic nonlinearities to the linear growth model of Section 2 is thus sufficient to imply that there is no need for supply
constraints as in non – Walrasian disequilibrium growth dynamics in order to keep such a dynamical system economically viable, in particular in the case where local
asymptotic stability see proposition 2 does not hold. The numerical properties of the dynamics considered in proposition 3 will be briefly discussed in the following
section.
5. Numerical features of the dynamics
