Economia Politica. Rivista di teoria e analisi Economia Politica. Rivista di teoria e analisi
Non-Technical Abstract
This paper provides a critical discussion of the important paper by Farkas and Kotsis (1992, FK since now on) on delays and stable limit cycles within Goodwin's model. Some inconsistencies of the formulation by Farkas and Kotsis are pointed out. Then the model is reformulated in a way which is fully consistent with Goodwin's framework. A discussion is finally provided of the mechanism by which stable or unstable limit cycles are generated by the economic forces acting within the model.
As it is well known, despite all its indisputable (not only pedagogical) merits for the developments of the endogenous theory of fluctuations in applied sciences, the classical Lotka-Volterra-Goodwin (LVG) model is not a good "fluctuation" model, because of its conservative dynamics. Since long time ago a common target to biomathematicians and mathematical macro-economists has been that of finding avenues to remove the limitations of the original model, in order to find the conditions for the generation of persistent periodic behaviours.
As it is well known, three main avenues have been beaten in the literature, to remove this difficulty of the basic LVG framework: a) introduction of nonlinearities "able" to generate limit cycles; b) introduction of further equations; c)introduction of time-delays. In the latter area, a key contribution in the literature is the recent paper by FK (1992). This contribution, which actually reformulates in economic terms an ecological model developed by Farkas (1984), has been defined as: "the more remarkable extension of Goodwin's growth cycle model. ..'' (Flaschel and Groth, 1995).
The model proposed by FK adds to the original Goodwin formulation two main complications: a)a logistic-type dynamics (rather than exponential as in the original Goodwin's formulation) of the employment rate in absence of the predatory action of the wage share; b) a delay in the reaction of the wages to the changes of the state of the employment rate (i.e.: a delay in the Phillips mechanism assumed by the model). The delay is assumed to be of two possible types in the paper by FK, exponential or "humped". The structure of the model is therefore that of a ''prey-predator''-type model with distributed delays. The analysis of the resulting integro-differential system is performed by reducing the system into the corresponding ordinary differential equation systems, thanks to the use of the so called "linear chain trick".
The formal properties of the involved systems are studied in detail in Farkas (1984) and Farkas and Kotsis (1992). Their complexity makes it very difficult to obtain global results, and rather suggests to look for local bifurcations analysis. Farkas (1984) and Farkas and Kotsis (1992) have studied the ''fine'' structure of the process of generation of limit cycles around the positive equilibrium of the model via Hopf bifurcations. In particular the problem of stability of the emerging limit cycles is attacked via a detailed analysis of the behaviour of the system on the corresponding center manifold. The analysis of the center manifold determines the exact conditions conditions of stability/instability of the involved periodic orbits. (the so called conditions of supercriticity/subcriticity).
The undisputable relevance of the work by Farkas-Kotsis lies in the fact that it represents one of the very few succesfull tentatives to explicitly determine the stability properties of limit cycles for systems of dimension higher than the second, via the analysis of the center manifold.
It is shown in the present paper that the formulation by Farkas and Kotsis, although undisputably important from both the mathematical and conceptual point of view, is nonetheless inconsistent with the leading philosophy of the economic world (à la Goodwin) that Farkas and Kotsis intend to represent. In fact, the special formulation of the logistic growth of the employment rate chosen by FK leads, if we preserve the traditional Goodwin's assumption that the supply of labour and the average productivity of labour grow at fully exogenous rates (as explicitly stated by FK), to a series of contradictions with the general "philosophy" of the model. This inconsistency gives to the formulation of the employment equation used by FK a rather "ad hoc" character. The choice of the logistic assumption appears in fact motivated essentially by the need to recover the form of the model studied by Farkas (1984), thereby preserving the conditions for the generation of periodic orbits.
This inconsistency motivated a first interesting target, i.e. the need to reformulate the model in a way which, while preserving the nice dynamical features of the model, be at the same time fully funded from the economic point of view and consistent with the general Goodwin's framework. The reformulation provided in the present paper has been based on the so called "discouraged worker hypothesis", quite popular in the literature on the labour market. The rate of change of the labour supply, which was constant in the original Goodwin's formulation, has therefore been written as:
The last formulation, as motivated in the paper, satisfies the targets: a) it preserves the formal structure of the model by FK; b) it is fully funded from the point of view of economic theory; c) it is fully consistent with the underlying modelling framework; d) it preserves the existence of a logistic growth of the prey population in absence of the predator in a totally natural way, as a direct consequence of the special economic forces acting in the model (rather than purely "ad hoc" as in FK). In other terms, our formulation embedding the discouraged worker assumption, is able to preserve the main feature of the underlying model by FK, the generation of persistent oscillations, being at the same time consistent with the the leading philosophy of Goodwin's model. These findings appear to be of a certain interest for the economic debate within the "Goodwinian" environment.
The importance of the results by FK lies also in a second fact: as they provide a fully explicit analytical treatment of the bifurcation process, they permit, in principle, to provide a rich interpretation of the economic mechanisms leading to the appearance of stable periodic orbits. This problem is, in practical terms, connected with the stability conditions of the periodic orbits emerging via the bifurcation process. In fact only if the orbits are stable they are physically observable, and the full investigation of the stability conditions is perhaps the main contribution in the paper by FK
This interpretation, which is not even sketched in Farkas and Kotsis, is provided in the second part of the paper and is based on the discussion of the bifurcation conditions followed in FK.
This discussion naturally leads to investigate the role played on the stability (and therefore on its losses leading to the appearance of oscillations) of the system by the two "new" parameters introduced by our formulation: the average amplitude of the time delay, the main effect of which on Goodwin's equilibrium is a destabilising one (locally destabilising, of course), and the "participation" parameter introduced via (1), which essentially plays a stabilizing effect.
With reference to this last point our discussion shows that the involved limit cycle are stable when, at the same time: a) the time delay is sufficiently large, thereby generating a sufficiently strong (localy) destabilising effect which permit to the trajectories to escape from the equilibrium, b) the strengths of the stabilizing pressures existing in the model, due to the dissipative term introduced via (1), are sufficiently small. The dynamical interpretation is therefore very clear and appealing: the stabilizing dissipative forces acting on the model become more and more effective as we take ourselves off from the equilibrium. This guarantees that trajectories can not escape "very far" from the equilibrium. But if they are sufficiently small they can not prevent, in presence of a strong local destabilising effect (due to the time-delay), a local escape of the trajectories. The outcome of such a local balancement between these forces is the convergence of the trajectories to a stable limit cycle.
Finally, from the more technical point of view, of some interest appears the explicit determination, permitted by the reformulation based on (1), of the threshold which separates the supercritical and subcritical behaviours of the involved limit cycles. This threshold appears, thank to our formulation, fully interpretable from the economic point of view. This last result appears therefore to have a potential empirical value also, in presence of adequate data to provide possible econometrical validations.
References
Farkas M. (1984), Stable Oscillations in a Predator-Prey Model with Time Lag, Journal of Mathematical Analysis and Applications, vol. 102, pp.175-188
Farkas M. - Kotsis M. (1992), Modelling Prey-Predator and Wage-Employment Dynamics, in Feichtinger G. (a cura di), Dynamic Economic Models and Optimal Control, New York, Elsevier, pp. 513-526
Flaschel P. - Groh G. (1995), The Classical Growth Cycle: Reformulation, Simulation and Some Facts, Economic Notes, vol. 24, no. 2, pp. 293-325
PIERO MANFREDI is researcher in economics at the Università di Pisa, Facoltà di
Economia, Via Ridolfi 10, 56100 Pisa
LUCIANO FANTI is researcher in economics at the Università di Pisa, Facoltà di
Economia, Via Ridolfi 10, 56100 Pisa
lfanti@ec.unipi.it
manfredi@ec.unipi.it
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