Qualitative Analysis of the Goodwin Model of the Growth Cycle

Goodwin’s model is a set of ordinary differential equations and is a wellknown model of the growth cycle. However, its four constants require an extensive numerical study of its two differential equations to identify all possible unsteady state behaviors, i.e. phase portraits, which corresponds to infinitely many combinations of numerical values of the constants. Qualitative interpretation of Goodwin’s model solves these problems by replacing all numerical constants and all derivatives by trends (increasing, constant and decreasing). The model has two variables — the employment rate V , and the labour share U . A solution of the qualitative Goodwin’s model is a scenario. An example of a Goodwin’s scenario is — V is increasing more and more rapidly, U is decreasing and the decrease is slowing down. The complete set of all possible 41 Goodwin’s scenarios and 168 time transitions among them are given. This result qualitatively represents all possible unsteady state Goodwin’s behaviours. It is therefore possible to predict all possible future behaviours if a current behaviour is known/chosen. A prediction example is presented in details. No prior knowledge of qualitative model theory is required.

Análisis cualitativo del modelo de Goodwin de ciclos de crecimiento

Introduction
Goodwin (1967) states a simple but useful model of the struggle between capital and labour for shares in national income, based on the classic Volterra-Lotka predator-prey model for fish populations.Since then, the model has been extended in many directions and as such has proved to be a useful framework for combining growth and cycles in a simple non-linear model (Desai et al., 2006).
Some researchers intended to add additional variables to the original model in order to make the model more generalized and realistic, see e.g.Sordi and Vercelli, (2014) or Sportelli, (1995).Other researchers studied the stability and other properties of the original model (Yoshida and Asada, 2007;Cao and Jiang, 2011;Veneziani and Mohun, 2006).Those who tried to evaluate the model empirically (Weber, 2005;Harvie, 2000;Moura Jr. and Ribeiro, 2013) and investigate the cyclical behaviour, used the data available from different sources or just solved the model numerically.
However, a numerical solution of Goodwin's model requires knowledge of all the four numerical constants.A qualitative interpretation of such parameters is possible and is studied in this paper.The paper presents a qualitative approach to the Goodwin model.
Deep knowledge items reflect undisputed elements of the corresponding theory.The law of gravity is an example.This law has no exceptions.This is a typical feature of deepknowledge items.Soft sciences as e.g.macroeconomics, are just very rarely based on deep-knowledge items.Goodwin's model is based partially on shallow knowledge.
A shallow-knowledge item is usually a heuristic or a result of a statistical analysis of observations and has usually many exceptions, see e.g.Oliveira and Rezende, (2013) or Ahn and Kim (2009).
Volterra-Lotka model is a set of ordinary nonlinear differential equations (ONDEs).A qualitative solution of ONDEs is specified if all its n qualitative variables: are described by the corresponding qualitative triplets: where i X is the i -th variable and An oriented graph is commonly used to represent graphically the set of all the transitions.If it is possible to transfer the r-th solution (3) into the s -th solution, then an oriented arc represents the corresponding transition from the node r to the node s .
A qualitative addition is represented by the matrix shown in Table 2.
It is sometimes possible to find more than one qualitative value.It is impossible to predict a sign of the result: A qualitative derivative of a sum of qualitative variables is a sum of their qualitative derivatives.

DX i + DX j = DX s DDX i + DDX j = DDX s (6)
A qualitative multiplication is described by Table 3.A known relation for the first qualitative derivative gives ONDEs are interpreted as a set of qualitative differential equations and solved using qualitative additions and multiplications.
A multiplication by a qualitative constant c is irrelevant: where c is a numerical constant and X is variable.

The qualitative Goodwin model of the growth cycle
The Goodwin model can be represented as follows (Desai et al., 2006;Sordi and Vercelli, 2014): where numerical constants δ α , ,γ and σ are considered to be positive; v is the employment rate and u is the labour share.
A numerical solution of the set of differential equations shown in (10) requires knowledge of all the four constants.Therefore a qualitative interpretation of the model represented by (10) generates a meaningful solution based on trends only.
A qualitative interpretation of Eq. ( 10) is, see (9): Algorithms used to solve qualitative models are combinatorial tasks and are not studied in this paper; for details see e.g.Dohnal (1991).Table 4 gives 41 scenarios, see (3), of the qualitative Goodwin model described by (10).There are 168 transitions among them.As an example, a set of 20 transitions among the scenarios included in Table 4, is given in Table 5.The time sequence shown in ( 12) corresponds to the cyclical behaviour of the model in Weber (2005).The described behaviour is not the only possible one, keeping in mind that there are 168 possible transitions among 41 scenarios.
The following examples give an interpretation of qualitative Goodwin's results, see Tables 4 and 5. Different qualitative answers are presented.The first qualitative question to be answered is the following: Is there a steady state?; i.e. is there the following scenario (see n = 2 in (2)?: The answer is solved just by searching through

Conclusion
The well-known Goodwin's model is the first model which tried to combine cyclical behaviour and economic growth.Therefore a complex system with its negative features must be studied.It means that business cycle analysis must be done under the following conditions: • Severe shortage of information • High level of subjectivity of available knowledge • Inconsistencies of information items of interdisciplinary nature Throughout the years the model has been tested both theoretically and numerically.Although it was mentioned above that constants δ α , ,γ and σ of the system (10) are hard to identify numerically, so this paper attempted to evaluate the model qualitatively and apply the achieved results into practice.
We have qualitatively described the trajectory of the cyclical behaviour of the Goodwin's model, studied in Weber (2005), by a sequence of one-dimensional scenarios and found out that this sequence represents only a subset of our qualitative model.
Our qualitative Goodwin model gives all the possible developments based on two variables.No statistical data sets are needed and all the possible solutions are identified.
the first and second qualitative derivatives with respect to the independent variable t (which is usually time).DX is the qualitative interpretation of a numerical value for dt dx / .The reason why the third and higher derivatives are ignored is that these derivatives are known just rarely.A qualitative model has m qualitative solutions i.e. scenarios.The j -th qualitative state is the n -triplet: i DX and i DDX are list of possible one-dimensional transitions, see Table 1.Multidimensional transitions must satisfy the Table1for n one-dimensional scenarios.However, this table is not a dogma.If a user feels/ knows/believes that a certain transition is not possible then this transition can simply be removed from the table.

Table 1 :
Some transition rules

Table 4 :
List of all scenarios of Goodwin model

Table 5 :
Some transitions of Goodwin model ) simulates the behaviour of Goodwin's model by applying randomly selected values.As a result, a graph is achieved, which represents the cyclical behaviour of the model.One-dimensional transitions, given in Table1, are used to identify some possible transitions among the scenarios set, see Table4.

Table 4 .
According to this table, scenario No. 21 is the qualitative steady state as both variables V and U are positive and have the first and the second derivatives equal to zero.It means that both employment rate and labour share are increasing more and more rapidly.Scenario No. 21 is the steady state.The list of transitions (see Table 5) indicates that it is possible to transfer Scenario No. 21 into Scenario No. 1.It is impossible to transfer Scenario No.1 into Scenario No. 21.For instance, if it is necessary to reach the steady state of the system, there are many paths being available.For example: