A Generalized Approach for the Modeling of Goodwin-Type Cycles

Matteo Madotto; Marcellino Gaudenzi; Fabio Zanolin

doi:10.1515/ans-2015-5050

Article Open Access

A Generalized Approach for the Modeling of Goodwin-Type Cycles

Matteo Madotto , Marcellino Gaudenzi and Fabio Zanolin

Published/Copyright: August 28, 2016

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From the journal Advanced Nonlinear Studies Volume 16 Issue 4

Abstract

Goodwin’s celebrated growth cycle model has been widely studied since its introduction in 1967. In recent years several contributions have appeared with the aim of amending the original model so as to improve its economic coherence and enrich its structure. In this article we propose a new and generalized approach, within the theory of planar Hamiltonian systems, for the modeling of Goodwin-type cycles. This new approach, which includes and improves various attempts by the recent literature, is very general and fulfills the essential requirement that the orbits lie entirely in the economically feasible interval. We provide a necessary and sufficient condition for all solutions to be cycles lying entirely in the unit box. In addition, we study the period length of the cycles near the equilibrium and close to the boundary of the domain. Finally, we discuss an example of how small perturbations of the model may affect the qualitative behavior of the solutions.

Keywords: Goodwin Model; Nonlinear Systems; Growth Cycle; Period; Stability

MSC 2010: 91B64; 34C25

1 Introduction

Goodwin’s celebrated growth cycle model (also known as Goodwin’s class struggle model), which was first proposed in 1967 [13] and subsequently restated in 1972 [14], is a schematized, yet very elegant, dynamic formalization of Marx’s theory of distributive conflict and an interesting example of how nonlinear dynamical systems can be used to model important economic phenomena like growth cycles. Adopting a linear investment function and a linear real wage bargaining function, Goodwin obtains two nonlinear differential equations of the Lotka–Volterra type in the state variables “wage share in national income” and “proportion of labor force employed”.

Since its first formulation, the model has proved to be a useful framework for combining economic growth and endogenous fluctuations in a simple nonlinear model. Many authors have extended it in many different directions, especially during the 1970s and 1980s, trying to generalize the model by adopting less stringent hypotheses and introducing new economic phenomena into the analysis. Some relevant contributions include Atkinson [2], Desai [6], Medio [19], Desai and Shah [8], van der Ploeg [24], Di Matteo [9], Glombowski and Krüger [12], Sato [22], Mehrling [20], Asada [1] and Chiarella [4]. A more complete and recent survey of the literature can be found in [25].

In more recent years some authors have focused on developing models which, preserving Goodwin’s basic idea of a conflicting but at the same time symbiotic interaction between capitalists and workers, could overcome one of the major shortcomings of the original model. In particular, the state variables of the model (the wage share in national income and the employment proportion) cannot by definition exceed unity. By contrast, the Lotka–Volterra equations obtained by Goodwin generate cycles which lie in the entire first quadrant of the plane. Goodwin himself was aware of this issue (see [13]), but such a problem was rarely addressed by the early literature. However, it is essential to address it in order to obtain both a more realistic model and a coherent theoretical framework for the modeling of economic cycles in the two state variables considered by Goodwin. Some exceptions are Blatt [3], who suggests the use of a floor level for net investment, and Flaschel (see for instance [10]), who proposes the inclusion of additional elements such as money and fiscal policies by a state sector.

In the last decade relevant contributions to the solution of this problem have been provided by Desai, Henry, Mosley and Pemberton [7] and Harvie, Kelmanson and Knapp [16]. The novelty of these approaches consists in the idea of properly modifying the basic functional relationships considered by Goodwin without the need of introducing economic phenomena different from those considered in the original model, and without relying on ceiling or floor mechanisms whose economic interpretation is not always convincing.

Desai et al. [7] present a reformulation of the Goodwin model where the real wage bargaining function has a nonlinear form (as did Phillips originally for nominal wages) and Goodwin’s restrictive assumption that all profits are always reinvested is relaxed. Harvie et al. [16] propose a system of differential equations, inspired by mathematical models used in biology, in which each state variable has both a positive and a negative feedback effect on its own growth rate, allowing the modeling of several economic features.

In this article we propose a new and generalized approach, within the theory of planar Hamiltonian systems, for the modeling of the dynamic evolution of wage share and employment proportion where the solution trajectories, under certain conditions, are closed orbits which never stray outside the economically feasible interval. This new approach is very general, its economic interpretation is rich and it includes the recent attempts made by the literature as special cases.

In our generalized framework we obtain a necessary and sufficient condition for all solutions to be cycles lying entirely in the unit box. We show that the models by Desai et al. and Harvie et al. can be viewed as special cases of our generalized approach and that the assumptions made in these two papers are not sufficient to guarantee that the solution trajectories are closed orbits contained in the unit box as claimed. Then, we focus on the analysis of the period of the cycles in a neighborhood of the fixed point and near the frontier of the unit box. On the one hand we prove that the period of the cycles converges to the period of the corresponding linearized system as the starting point tends to the fixed point. On the other hand we prove analytically that the period tends to infinity as the distance of the starting point from the boundary goes to zero. The former result is known only for special cases like the basic Goodwin model, while the latter is, to the best of our knowledge, completely new.

Our results can also help to explain some of the empirical evidence on the Goodwin model. In particular, as shown by Harvie [15], there is evidence of a three-quarter cycle in the state variables wage share and employment proportion in many OECD countries. Our analysis shows that the period length in the missing quadrant grows to infinity as we approach the boundary, while this is not guaranteed in the other three quadrants.

Finally, we consider a slightly modified version of our generalized model so as to take inflation into account (see for example Desai [6], van der Ploeg [24], Flaschel [11]). We prove that in this case the equilibrium point ceases to be a center and becomes an asymptotically stable focus or node. This shows that our original system (like the basic Goodwin model) is structurally unstable. Our last result provides the condition to determine whether the equilibrium point is a focus or a node.

2 The Basic Goodwin Model and Some Recent Contributions

The basic Goodwin model [13, 14] is deliberately schematized. Its aim is to describe the conflicting but at the same time symbiotic relationship between capitalists and workers in a purely capitalist economy, adopting a framework which is as simple as possible.

The state variables of the model are u, the wage share in national income, and v, the proportion of labor force employed. Goodwin assumes a constant capital-output ratio σ, a constant exogenous labor productivity growth rate α and a constant labor force growth rate β. Moreover, he hypothesizes that the real wage bargaining function, which describes the growth rate of the real wage, is of the form -γ+ρ⁢v, where γ and ρ are positive parameters, and that capitalists reinvest all their profits while workers do not save. From these assumptions Goodwin obtains the following system:

(2.1) u ˙ u = - ( γ + α ) + ρ ⁢ v ,

(2.2) v ˙ v = 1 - u σ - ( α + β ) ,

where the dot indicates the time derivative.

Equations (2.1) and (2.2), which are the key result of Goodwin’s model, form a two-dimensional autonomous system in u and v. This dynamical system is a special case of the well-known Lotka–Volterra predator-prey model (see Lotka [18] and Volterra [27, 28]), which is used in mathematical biology to model the dynamic interaction between two populations. The system admits two equilibrium points: (0,0), which is a saddle, and (1-σ⁢(α+β),(γ+α)/ρ), where σ⁢(α+β)<1 is assumed. This second point is a center and all solution trajectories starting inside ℝ+×ℝ+ are closed orbits surrounding it.

One of the major shortcomings of the Goodwin model which has been addressed and analyzed in some recent contributions is the fact that the periodic solutions of the system can exceed unity. This is inconsistent with the fact that the state variables u and v represent fractions of unity. The most recent attempts to overcome this issue are provided by Desai et al. [7] and Harvie et al. [16].

Using a nonlinear real wage bargaining function of the form -γ+ρ⁢(1-v)-δ, where δ>0, and a logarithmic investment function, Desai et al. obtain a system of the form

(2.3) u ˙ u = - ( γ + α ) + ρ ⁢ ( 1 - v ) - δ ,

(2.4) v ˙ v = ( - λ ⁢ log ⁡ ( 1 - u ¯ ) - ( α + β ) ) + λ ⁢ log ⁡ ( u ¯ - u ) ,

where α, β, γ and ρ have the same meaning as above, λ is a positive parameter and u¯∈(0,1) is the maximum wage share the capitalists can tolerate which ensures them their reservation rate of profit. Desai et al. assume ρ<γ+α and α+β<λ⁢log⁡(u¯/(1-u¯)) to ensure that the center lies in (0,u¯)×(0,1).

Desai et al. show that system (2.3)–(2.4) exhibits closed orbits lying entirely within the (0,u¯)×(0,1) interval. In Section 4, however, we prove that their assumptions are not sufficient to guarantee this result. In particular, it is necessary to impose a further condition on the primitives of the functions.

Another interesting proposal is that of Harvie et al. [16]. They consider a system of the form

(2.5) u ˙ u = k 1 ⁢ u μ 1 ⁢ ( 1 - u ) η 1 ⁢ ( - ( α + γ ) + ρ ⁢ v ) ,

(2.6) v ˙ v = k 2 ⁢ v μ 2 ⁢ ( 1 - v ) η 2 ⁢ ( 1 σ - ( α + β ) - 1 σ ⁢ u ) ,

where ρ>α+γ, σ⁢(α+β)<1 and

k 1 > 0 , k 2 > 0 , μ 1 ≥ 0 , μ 2 ≥ 0 , η 1 > 0 , η 2 > 0 .

System (2.5)–(2.6) allows the modeling of several economic features not captured by the original formulation. Indeed, the two state variables, u and v, now have both a positive and a negative feedback effect on their own growth rates, while in the Goodwin model each state variable only affects the growth rate of the other.

Harvie et al. show that all the solution trajectories of system (2.5)–(2.6) are closed orbits lying entirely in the unit box. But, as we prove in Section 4, the restrictions they impose on the size of the parameters are again not sufficient to guarantee the validity of this conclusion and further restrictions are needed.

3 The Model

In this section we propose a generalized approach for the modeling of Goodwin-type cycles fulfilling the essential requirement of lying entirely in an economically feasible interval.

Consider the following system of differential equations in the state variables u, the wage share in national income, and v, the proportion of labor force employed:

(3.1) u ˙ = u ⁢ f ⁢ ( u ) ⁢ ψ ⁢ ( v ) ,

(3.2) v ˙ = - v ⁢ g ⁢ ( v ) ⁢ ϕ ⁢ ( u ) ,

where we make the following assumptions:

(3.3) f : ( 0 , u 1 ) → ( 0 , + ∞ ) , g : ( 0 , v 1 ) → ( 0 , + ∞ ) , ϕ : ( 0 , u 1 ) → ℝ , ψ : ( 0 , v 1 ) → ℝ ,

where u1,v1∈[0,1], f,g∈C and ϕ,ψ∈C1;

(3.4) ϕ ′ ⁢ ( u ) > 0 , ψ ′ ⁢ ( v ) > 0

for all u∈(0,u1) and v∈(0,v1);

(3.5) lim s → 0 + ⁡ ϕ ⁢ ( s ) = L 1 < 0 , lim s → 0 + ⁡ ψ ⁢ ( s ) = L 2 < 0 , lim s → u 1 - ⁡ ϕ ⁢ ( s ) > 0 , lim s → v 1 - ⁡ ψ ⁢ ( s ) > 0 ,

where L1,L2∈ℝ.

Assumptions (3.3)–(3.5) are very general. The four functions which define the effects of the variables on the growth rates are defined on open intervals within the economically feasible range of values and are sufficiently regular. The functions ϕ and ψ, which define the effect of u on the growth rate of v and the effect of v on the growth rate of u, respectively, are increasing. This simply generalizes Goodwin’s idea that the employment proportion (the prey in the Lotka–Volterra model) has a positive effect on the growth rate of the wage share (the predator) and that the latter variable has a negative impact on the growth rate of v. As u goes to zero, v increases at finite rates, while as u goes to u1, v decreases at finite or infinite rates. The opposite holds for u as v goes to zero and v1.

From assumptions (3.3)–(3.5) it follows that there exists a unique value of u in the (0,u1) interval, say u*, such that ϕ⁢(u*)=0, and a unique value of v in the (0,v1) interval, say v*, such that ψ⁢(v*)=0.

We now define

A ⁢ ( u ) = ∫ ϕ ⁢ ( u ) u ⁢ f ⁢ ( u ) ⁢ 𝑑 u and B ⁢ ( v ) = ∫ ψ ⁢ ( v ) v ⁢ g ⁢ ( v ) ⁢ 𝑑 v ,

making the additional assumption that

(3.6) A ⁢ ( 0 + ) = A ⁢ ( u 1 - ) = + ∞ , B ⁢ ( 0 + ) = B ⁢ ( v 1 - ) = + ∞ .

System (3.1)–(3.2), which can be equivalently written as

u ˙ u ⁢ f ⁢ ( u ) = ψ ⁢ ( v ) ,

v ˙ v ⁢ g ⁢ ( v ) = - ϕ ⁢ ( u ) ,

has the first integral (the Hamiltonian)

H ⁢ ( u , v ) = A ⁢ ( u ) + B ⁢ ( v ) ,

whose level lines are given by

A ⁢ ( u ) + B ⁢ ( v ) = c ,

where c is a real constant. All the trajectories arising from system (3.1)–(3.2) lie on the level lines of this first integral.

The A function has a minimum when ϕ⁢(u)=0, which occurs only at u=u*. Similarly, B has a unique minimum at v*, where ψ⁢(v*)=0. It follows that H⁢(u,v) has a minimum, say c*, at (u*,v*), which therefore is the fixed point of system (3.1)–(3.2).

Now consider c>c*. The above assumptions guarantee that the equation H⁢(u*,v)=c has exactly two solutions, v¯ and v¯, which satisfy

0 < v ¯ < v * < v ¯ < v 1 .

Analogously, the equation H⁢(u,v*)=c has exactly two solutions, u¯ and u¯, which satisfy

0 < u ¯ < u * < u ¯ < u 1 .

Every pair (u,v) belonging to the level line H⁢(u,v)=c satisfies the following conditions:

0 < u ¯ ≤ u ≤ u ¯ < u 1 , 0 < v ¯ ≤ v ≤ v ¯ < v 1 .

Indeed, if u>u¯ and H⁢(u,v)=c, then A⁢(u)+B⁢(v)=c and A⁢(u¯)+B⁢(v*)=c. But since u,u¯>u* and A⁢(u) is increasing in the interval (u*,u1), we would have B⁢(v)<B⁢(v*), contradicting the fact that B⁢(v*) is the minimum of B. It is thus easy to prove that the level line H⁢(u,v)=c is contained in the box [u¯,u¯]×[v¯,v¯].

From the above discussion it is also clear that each level line H⁢(u,v)=c>c* is the union of (four) graphs of smooth functions and thus has a finite length (indeed, its length is less than the perimeter of the box [0,u1]×[0,v1]).

Now consider a solution of system (3.1)–(3.2) satisfying the initial conditions

u ⁢ ( 0 ) = u 0 ∈ ( 0 , u 1 ) , v ⁢ ( 0 ) = v 0 ∈ ( 0 , v 1 ) and ( u 0 , v 0 ) ≠ ( u * , v * ) ,

so that H⁢(u0,v0)=c0>c*. The length of the regular curve (u⁢(t),v⁢(t)), t∈[t1,t2] is given by

∫ t 1 t 2 u ˙ 2 ⁢ ( t ) + v ˙ 2 ⁢ ( t ) ⁢ 𝑑 t = ∫ t 1 t 2 u 2 ⁢ ( t ) ⁢ f 2 ⁢ ( u ⁢ ( t ) ) ⁢ ψ 2 ⁢ ( v ⁢ ( t ) ) + v 2 ⁢ ( t ) ⁢ g 2 ⁢ ( v ⁢ ( t ) ) ⁢ ϕ 2 ⁢ ( u ⁢ ( t ) ) ⁢ 𝑑 t .

Since ϕ and ψ vanish only at u* and v*, respectively, and the level line H⁢(u,v)=c0 is a compact set, there exists m>0 such that the integrand function is greater than m for all t≥0. Hence, the length of the curve (u⁢(t),v⁢(t)) is greater than m⁢(t2-t1). On the other hand, as already remarked, the length of the level lines of the Hamiltonian H is finite (less than 2⁢(u1+v1)). Thus, there exist t1<t2 such that (u⁢(t1),v⁢(t1))=(u⁢(t2),v⁢(t2)). Therefore we conclude that for every choice of the starting point (u0,v0), the corresponding solution (u⁢(t),v⁢(t)) is periodic.

The above results can be summarized in the following theorem.

Theorem 3.1

Under assumptions (3.3)–(3.6) all the solutions of system (3.1)–(3.2) are periodic and describe closed orbits strictly contained in the rectangle (0,u1)×(0,v1).

To complete Theorem 3.1 with further details, we can add the following remark.

Remark 3.2

Every solution (u⁢(t),v⁢(t)) of (3.1)–(3.2) is contained in the box [u¯,u¯]×[v¯,v¯]. The component u⁢(t) is increasing in the first quadrant [u*,u¯]×[v*,v¯], and in the second quadrant [u¯,u*]×[v*,v¯], while it is decreasing in the last two quadrants. The component v⁢(t) is increasing in the second and third quadrants, while it is decreasing in the first and fourth quadrants. In the time interval [t,t+T), where T is the period of the solution, u⁢(t)-u* and v⁢(t)-v* have exactly two zeros (see also Figure 1).

Figure 1

Goodwin-type cycles in the feasible interval.

We stress that Theorem 3.1 does not hold if any of the conditions in assumption (3.6) is not fulfilled.

For example, suppose that A⁢(0+)=A¯<+∞ while the other three conditions, A⁢(u1-)=+∞, B⁢(0+)=+∞, B⁢(v1-)=+∞, are fulfilled. In this case, for all c>c* the points (u,v) belonging to the level line H⁢(u,v)=c satisfy the condition 0<v¯<v<v¯<v1. The same condition, however, does not always hold for the u variable. Indeed, for c*<c<A¯+B⁢(v*) the equation H⁢(u,v*)=c has exactly two solutions, while it has only one solution for c>A¯+B⁢(v*).

Thus, the conclusions of Theorem 3.1 are still valid for c*<c<A¯+B⁢(v*), while for c>A¯+B⁢(v*) there are no periodic solutions. By the previous argument on the length of the curve, we can also conclude that for c>A¯+B⁢(v*) every solution starting in the open box (0,u1)×(0,v1) and lying on the level line H⁢(u,v)=c will reach the boundary of the box.

In the following it will be interesting to consider the case where we have A⁢(0+)=+∞, A⁢(u1-)=A¯<+∞, B⁢(0+)=+∞ and B⁢(v1-)=B¯<+∞. Using the same arguments as above, we can conclude that for

c * < c < min ⁡ { A ¯ + B ⁢ ( v * ) , B ¯ + A ⁢ ( u * ) }

the conclusions of Theorem 3.1 still hold, while for c>min⁡{A¯+B⁢(v*),B¯+A⁢(u*)} system (3.1)–(3.2) has no periodic solutions for every choice of the initial conditions. In this latter case the solution trajectories will reach either the level u=u1 or the level v=v1.

We can now state the following theorem.

Theorem 3.3

Assume that hypotheses (3.3)–(3.5) are fulfilled and that at least one of the four limits in assumption (3.6) is finite. Then, there exists a closed curve γ contained in [0,u1]×[0,v1] such that

for all starting points lying in the interior of the curve γ , the solutions of system (3.1)–(3.2) are periodic and describe closed orbits contained in the interior of γ,
for all starting points lying in (0,u1)×(0,v1) but not on γ or in its interior, the solutions of system (3.1)–(3.2) are not periodic and tend to the boundary of (0,u1)×(0,v1).

Fulfilling assumption (3.6) is therefore essential to our objective. Its economic interpretation is provided at the end of the next section. The following proposition instead describes curve γ more in detail.

Proposition 3.4

Consider the same hypotheses as in Theorem 3.3 and let

c * * = min ⁡ { A ¯ + B ⁢ ( v * ) , B ¯ + A ⁢ ( u * ) , A ¯ + B ⁢ ( v * ) , B ¯ + A ⁢ ( u * ) } .

We then have c*<c**<+∞. Consider the closure of the level set H⁢(u,v)=c** in [0,u1]×[0,v1]. Such a set is the trace of a closed curve γ which is contained in (0,u1)×(0,v1) except for at most four points (see Figure 2).

Figure 2

The curve γ.

4 Applying the Model

System (3.1)–(3.2) represents a generalized version of the Goodwin model where the closed orbits are bounded within the (0,u1)×(0,v1) interval. It is usually assumed that u1=v1=1, but different values are possible (see for example Desai et al. [7]), provided that they are positive and less than unity.

As mentioned before, assumptions (3.3)–(3.5) are quite general and, under proper values of the parameters, they are satisfied even by the basic Goodwin model. By contrast, assumption (3.6), which requires f, g, ϕ and ψ to have functional forms allowing the divergence of A⁢(u) and B⁢(v) on the boundaries, is not fulfilled by the original Goodwin model in u=u1 and v=v1. Indeed, in this case f and g are identically equal to one and both the investment function, (1-u)/σ, and the real wage bargaining function, -γ+ρ⁢v, are linear and defined for all u,v>0, so that A⁢(u) and B⁢(v) diverge at u=v=0+, but they do not diverge elsewhere. As remarked in the Introduction, Goodwin himself was aware of the limitations of his model. In particular, he anticipated the possibility of adopting a nonlinear real wage bargaining function defined on [0,1) and diverging at v=1 (see [13, Figure 1]), but he chose to adopt a linear approximation “in the interest of lucidity and ease of analysis”.

The generalized model presented in Section 3 includes the extensions of the Goodwin model proposed by Desai et al. [7] and Harvie et al. [16] (see Section 2) as special cases.

As for the model by Desai et al. (see system (2.3)–(2.4)), we have

v 1 = 1 , u 1 = u ¯ , f ⁢ ( u ) = g ⁢ ( v ) = 1 , ψ ⁢ ( v ) = - ( γ + α ) + ρ ⁢ ( 1 - v ) - δ

and

ϕ ⁢ ( u ) = ( λ ⁢ log ⁡ ( 1 - u ¯ ) + ( α + β ) ) - λ ⁢ log ⁡ ( u ¯ - u ) .

Under the assumptions made by Harvie et al. (see Section 2), conditions (3.3)–(3.5) are readily fulfilled. Since

A ⁢ ( u ) = ∫ ( λ ⁢ log ⁡ ( 1 - u ¯ ) + ( α + β ) ) - λ ⁢ log ⁡ ( u ¯ - u ) u ⁢ 𝑑 u

and

B ⁢ ( v ) = ∫ - ( γ + α ) + ρ ⁢ ( 1 - v ) - δ v ⁢ 𝑑 v ,

We have that A⁢(0+) and B⁢(0+) diverge as required by assumption (3.6), while for B⁢(1-) to diverge we have to impose the additional condition δ≥1. As for A⁢(u), the presence of a logarithmic investment function prevents this from diverging in u¯. A possible alternative would be the use of a power investment function with exponent less than or equal to -1, like the functional form Desai et al. adopted for the real wage bargaining function.

Therefore, the conditions imposed by Desai et al. are not sufficient to guarantee that all trajectories lie within the (0,u¯)×(0,1) interval. To put it more precisely, our discussion in Section 3 ensures that when c>min⁡{A¯+B⁢(v*),B¯+A⁢(u*)}, the system has no periodic solutions in the (0,u¯)×(0,1) interval: for these values of c every solution will reach either the level u=u¯ or the level v=1.

As for the model by Harvie et al. (see system (2.5)–(2.6)), we have

u 1 = 1 , v 1 = 1 , f ⁢ ( u ) = k 1 ⁢ u μ 1 ⁢ ( 1 - u ) η 1 , g ⁢ ( v ) = k 2 ⁢ v μ 2 ⁢ ( 1 - v ) η 2 ,

ψ ⁢ ( v ) = - ( α + γ ) + ρ ⁢ v and ϕ ⁢ ( u ) = - 1 σ + ( α + β ) + 1 σ ⁢ u .

Under the assumptions made by Harvie et al. on the size of the parameters involved in the original Goodwin model (see Section 2), conditions (3.3)–(3.5) are readily fulfilled. In this case we have

A ⁢ ( u ) = ∫ α + β - ( 1 - u ) / σ k 1 ⁢ u μ 1 + 1 ⁢ ( 1 - u ) η 1 ⁢ 𝑑 u and B ⁢ ( v ) = ∫ - ( α + γ ) + ρ ⁢ v k 2 ⁢ v μ 2 + 1 ⁢ ( 1 - v ) η 2 ⁢ 𝑑 v .

By standard properties of improper integrals, it follows that for A⁢(u) to diverge to +∞ as u goes to zero, we have to impose k1>0 and μ1≥0; while for it to diverge as u goes to one we also need η1≥1. Likewise, for B⁢(v) to fulfill assumption (3.6) we need k2>0, μ2≥0 and η2≥1. Harvie et al. however, impose η1>0 and η2>0, which are not sufficient conditions to guarantee that all trajectories lie inside the unit box. More precisely, if one of the two exponents η1 or η2 lies between 0 and 1, then either A⁢(1-) or B⁢(1-) is finite and our discussion in Section 3 ensures that for sufficiently large values of c the system has no periodic solutions for every choice of the initial conditions.

Thus, even though assumption (3.6) may seem a merely technical condition, it clarifies what kind of relations are needed to obtain economically coherent Goodwin-type cycles. In particular, as the above examples suggest, in order to prevent the closed orbits from straying outside the required interval, we need a strong impact of a variable on the growth rate of the other near the boundary and/or a strong negative feedback effect of a variable on its own growth rate. More precisely, we can identify some key ingredients necessary for assumption (3.6) to be fulfilled.

First, the real wage bargaining function should exhibit a sufficient degree of nonlinearity, with the growth rate of real wages rapidly increasing as the economy approaches full employment. Indeed, Phillips originally obtained a similar relationship for nominal wages. Analogously, the investment function should be highly nonlinear, with investments falling sharply as firms’ profit share goes to zero. This behavior may be interpreted as reflecting several possible features of the economy. The presence of credit-market imperfections, for example, may allow firms to invest proportionally to their profits when these are high, but force them to reduce their investments more than proportionally (and eventually disinvest) during periods of low profits.

An alternative set of ingredients, along the lines suggested by Harvie et al., is represented by the assumption that the parameters of the basic Goodwin model (the growth rates of technology and population, the coefficients of the wage bargaining function and the capital-output ratio) are not constant, but depend nonlinearly on the values of the state variables, giving rise to the aforementioned negative feedback effects. Indeed, these parameters change over time and, as Harvie, Kelmanson and Knapp argue, the wage share and the employment rate can exert a significant influence on their dynamics. Thus, assumption (3.6) can reflect various and empirically relevant features of the macroeconomy.

5 Analysis of the Period

In this section we turn to the analysis of the period of the cycles generated by system (3.1)–(3.2). First, we study the period length near the equilibrium point, obtaining a general result which is well known only in the special case of the standard Lotka–Volterra/Goodwin system and may clarify the use of the period of the linearized system in the empirical studies. Then, we analyze the period of the cycles close to the boundary of the (0,u1)×(0,v1) interval. In this case, the result we obtain appears less obvious and it may help to explain some of the empirical evidence on the Goodwin model.

5.1 Period of Small Cycles

In order to simplify the analysis, we consider a system with a center at (0,0). We can always reduce ourselves to this case by a change of variables (see Corollary 5.2). We thus consider the bidimensional autonomous system

z ˙ = F ⁢ ( z ) , where ⁢ z = ( u , v ) ∈ ℝ 2 , F ⁢ ( z ) = F ⁢ ( u , v ) = ( F 1 ⁢ ( u , v ) , F 2 ⁢ ( u , v ) ) .

We obtain the following proposition.

Proposition 5.1

Let F:D⊂R2→R2 be a function of class C1 in D, where D is an open set containing 0, and suppose that F⁢(0)=0. Assume that

F1⁢(u,v)⁢v>0 and F2⁢(u,v)⁢u<0, for all (u,v)∈D∖{0},
every solution which has initial conditions in D is a periodic solution,
the linearized system z˙=F′⁢(0)⁢z has a periodic solution of minimum period τ>0.

Then, denoting by τs the period of the solution of system z˙=F⁢(z) with starting point (s,0), s>0, we have

lim s → 0 ⁡ τ s = τ .

Proof.

Denote by zs⁢(t) the solution of the Cauchy problem

{ z ˙ = F ⁢ ( z ) , z ⁢ ( 0 ) = ( s , 0 ) ,

with (s,0)∈D, and let ws⁢(t)=zs⁢(t)∥zs∥∞. We have ρs=∥zs∥∞→0 as s→0. To see this, note that the solution zs intersects the v-axis at the points (0,s1) and (0,s3), s1<0<s3, and the u-axis at the point (s2,0), s2<0<s. We have s1→0, otherwise there exists a sequence {sk} converging to 0 such that sk1→s^<0. In this case, the solution of the problem with initial conditions u⁢(0)=0, v⁢(0)=s^2 intersects the orbit of one of the solutions zs, contradicting the uniqueness of the solutions. Similarly, s2→0 and s3→0. Since the conclusions of our discussion in Section 3 are still valid under our assumptions, the orbit of the solution is contained in the box [s2,s]×[s1,s3], so that ρs→0 as s→0.

Since ∥ws∥∞=1, the sequence {ws⁢(t)} is equibounded. In addition, we have

(5.1) w ˙ s ⁢ ( t ) = F ⁢ ( z s ⁢ ( t ) ) ρ s = F ⁢ ( w s ⁢ ( t ) ⁢ ρ s ) ρ s .

Since the derivative of F⁢(z) is bounded in a neighborhood of 0, it is also Lipschitzian in such a neighborhood. Hence, there exists L>0 such that

F ⁢ ( w s ⁢ ( t ) ⁢ ρ s ) ρ s = F ⁢ ( w s ⁢ ( t ) ⁢ ρ s ) - F ⁢ ( 0 ) ρ s ≤ L ⁢ ∥ w s ⁢ ( t ) ⁢ ρ s ∥ ∞ ρ s = L ⁢ ∥ w s ⁢ ( t ) ∥ ∞ = L .

Thus, for s small enough the set of functions {ws⁢(⋅)} has a bounded derivative, hence it is equicontinuous. We can therefore apply the Ascoli–Arzelà Theorem and conclude that there exists a sequence wsk such that wsk converges uniformly on the compact sets on [0,+∞) to a continuous function y⁢(t):[0,+∞)→ℝ. By (5.1) we get

w s k ⁢ ( t ) = w s k ⁢ ( 0 ) + ∫ 0 t F ⁢ ( w s k ⁢ ( ξ ) ⁢ ρ s k ) ρ s k ⁢ 𝑑 ξ .

Since F⁢(wsk⁢(t)⁢ρsk)/ρsk converges uniformly to F′⁢(0)⁢y⁢(t) on the compact sets on [0,+∞), we conclude that

y ⁢ ( t ) = y ⁢ ( 0 ) + ∫ 0 t F ′ ⁢ ( 0 ) ⁢ y ⁢ ( ξ ) ⁢ 𝑑 ξ ,

and thus y⁢(t) is a (nontrivial) solution of the linear system

y ˙ ⁢ ( t ) = F ′ ⁢ ( 0 ) ⁢ y ⁢ ( t ) .

Let ϵ>0. Since y⁢(t) has (minimum) period τ, by Kamke’s theorem, for s small enough, ws⁢(t) has two zeros in the time interval (0,τ+ϵ), and hence τs<τ+ϵ (see also Remark 3.2). Passing to subsequences if necessary, we can assume that τsk→τ*. By the periodicity of wsk⁢(t) we have wsk⁢(t+τsk)=wsk⁢(t), hence y⁢(t+τ*)=y⁢(t) and therefore τ*≥τ. Hence, τ≤τ*≤τ+ϵ and we can conclude that τ=τ*.

By the above result, for every sequence sk→0 there exists a subsequence for which the period converges to the period of the linearized system. Therefore τs→τ as s→0. ∎

If we now turn to system (3.1)–(3.2), using the above result and considering the change of variables u~=u-u*, v~=v-v*, we obtain the following corollary.

Corollary 5.2

Suppose that assumptions (3.3)–(3.5) are fulfilled and that f,g∈C1. Denote by τs the period of the solution of system (3.1)–(3.2) with starting point (u*+s,v*). Then lims→0⁡τs=τ, where τ is the period of the nontrivial solutions of the linear system

u ˙ = ψ ′ ⁢ ( v * ) ⁢ v ,

v ˙ = - ϕ ′ ⁢ ( u * ) ⁢ u .

This result is quite standard in the field of dynamical systems. In particular, Volterra himself [27, 28] had already proved this for the basic Lotka–Volterra/Goodwin system. Proposition 5.1, however, is more general and it may be of some theoretical interest even beyond the specific applications considered here. Moreover, it has some interesting implications from an empirical point of view. In particular, in almost every contribution to the empirical literature on the Goodwin model (see for instance Harvie [15]) the period length of the cycles is approximated by the period of the linearized system, which is usually easier to compute. The above result guarantees that this approximation is valid near the equilibrium point and that this holds even for much more general models.

Clearly, the above analytical results only concern small oscillations in a neighborhood of the positive equilibrium point. The analysis becomes more complicated when studying closed (periodic) trajectories that are far from the equilibrium point. This is our goal for the next subsection.

5.2 Period of Large Cycles

Consider again system (3.1)–(3.2), i.e.

u ˙ = u ⁢ f ⁢ ( u ) ⁢ ψ ⁢ ( v ) ,

v ˙ = - v ⁢ g ⁢ ( v ) ⁢ ϕ ⁢ ( u ) .

In Section 3 we proved that under conditions (3.3)–(3.6) this system has a periodic solution for every c>c*, where c*=H⁢(u*,v*). We denote by τc such a period, which is the same for every choice of the starting point lying on the level line H⁢(u,v)=c.

Now consider the additional assumptions:

(5.2) f , g ∈ C 1 , ⁢ lim v → 0 + ⁡ v ⁢ g ⁢ ( v ) = 0 + ,

(5.3) f , g ∈ C 1 , ⁢ lim u → 0 + ⁡ u ⁢ f ⁢ ( u ) = 0 + .

We then have the following theorem.

Theorem 5.3

Assume that conditions (3.3)–(3.6) are satisfied and that either (5.2) or (5.3) holds. Then

lim c → + ∞ ⁡ τ c = + ∞ .

Proof.

We assume that (5.2) is satisfied (the proof is analogous when (5.3) holds). Consider the Cauchy problem

(5.4) { u ˙ = u ⁢ f ⁢ ( u ) ⁢ L 2 , u ⁢ ( 0 ) = u 0 ,

where u0∈(0,u1). The solution u⁢(t) is unique (since f∈C1⁢(0,u1)), decreasing and positive. We have

u ˙ ⁢ ( t ) u ⁢ ( t ) ⁢ f ⁢ ( u ⁢ ( t ) ) = L 2

and, integrating,

∫ 0 t u ˙ ⁢ ( s ) u ⁢ ( s ) ⁢ f ⁢ ( u ⁢ ( s ) ) ⁢ 𝑑 s = L 2 ⁢ t .

By the change of variable ξ=u⁢(t) we get

∫ u 0 u ⁢ ( t ) d ⁢ ξ ξ ⁢ f ⁢ ( ξ ) = L 2 ⁢ t .

Note that

∫ 0 x d ⁢ ξ ξ ⁢ f ⁢ ( ξ ) = + ∞ for all ⁢ x ∈ ( 0 , u 1 ) .

Indeed, by assumption (3.6),

∫ 0 x ϕ ⁢ ( ξ ) ξ ⁢ f ⁢ ( ξ ) ⁢ 𝑑 ξ = - ∞

and by assumption (3.5), limξ→0+⁡ϕ⁢(ξ)=L1<0.

Let [0,t^) be the maximal interval of existence of u⁢(t). If t^<+∞ then

∫ u ⁢ ( t ^ ) u 0 d ⁢ ξ ξ ⁢ f ⁢ ( ξ ) < + ∞ ,

and therefore u⁢(t^)>0. Since t^∈ℝ and u⁢(t^)>0, by the existence and uniqueness of the solution we have that the solution of (5.4) is prolongable after t^, contradicting the definition of t^.

We have proved that t^=+∞ and u⁢(t)→0 as t→+∞.

Now let T>0 and consider the solution u⁢(t) of the Cauchy problem (5.4) defined over the time interval [0,T]. Since limv→0+⁡v⁢g⁢(v)=0+, we can define v⁢g⁢(v) at v=0. In this way (u⁢(t),0) becomes a solution of the system

u ˙ = u ⁢ f ⁢ ( u ) ⁢ L 2 ,

v ˙ = 0 ,

satisfying the initial conditions u⁢(0)=u0, v⁢(0)=0.

Since system (3.1)–(3.2) satisfies the assumptions of Kamke’s theorem (with (u,v)∈(0,u1)×[0,v1)), for every ϵ>0 there exists δ>0 such that for every v0∈(0,δ) the solution of system (3.1)–(3.2) with initial conditions u⁢(0)=u0, v⁢(0)=v0 is defined over [0,T] and, in addition,

| v ⁢ ( t ) | < ϵ for all ⁢ t ∈ [ 0 , T ] .

Taking δ,ϵ<v*, we can conclude that the period of the solution with starting point (u0,v0) is larger than T.

Let c⁢(δ)=H⁢(u0,δ2). If c>c⁢(δ) then there exists v0c such that H⁢(u0,v0c)=c and v0c<δ2. Therefore τc>T. This proves that limc→+∞⁡τc=+∞. ∎

The above result shows analytically that as we approach the frontier of the feasible interval and consider very large cycles, the period length of such cycles becomes arbitrarily large. More precisely, we have proved that (under assumption (5.2) or (5.3)) this occurs because the solution trajectories pass near the origin, which is always a saddle point.

5.3 Monotonicity of the Period: Simulation Results

Up to now we have studied analytically the period length of small cycles (i.e. cycles near the equilibrium point) and that of large ones (i.e. cycles passing near the frontier of the feasible interval). Now, we focus on the qualitative behavior of the period length as a function of the amplitude of the cycles (i.e. as a function of the value c taken by the Hamiltonian function H⁢(u,v)).

The monotonicity of the period has been proved in the special case of the basic Lotka–Volterra/Goodwin model (see, for instance, Rothe [21] and Waldvogel [29]). More generally, there is a great deal of research investigating the monotonicity of the period map of planar Hamiltonian systems related to the Lotka–Volterra model, and various interesting results have been obtained (see, for example, [5, 26] and the references therein). However, to the best of our knowledge, none of the results in the literature can be applied to a general model like system (3.1)–(3.2) satisfying assumptions (3.3)–(3.6).

In this section we propose a numerical example based on a special case of system (3.1)–(3.2) which provides evidence in favor of the monotonicity of the period map and, at the same time, highlights the difficulties in studying analytically this kind of models due to the presence of singularities at u=1 and v=1. In doing so, we restrict our attention to a model inspired by Desai et al. [7], but similar results could be obtained using a model à la Harvie et al. [16]. More specifically, Figures 3 to 7 show the results of a simulation based upon the model proposed in [7] where, however, we replaced their logarithmic investment function (which, as shown in Section 4, does not guarantee that all solution trajectories lie within the feasible interval) with a power function with exponent less than or equal to -1. More precisely, we considered the system

u ˙ u = f ⁢ ( u ) ⁢ ψ ⁢ ( v ) = - a 1 + b 1 ⁢ ( 1 - v ) - d 1 , v ˙ v = - g ⁢ ( v ) ⁢ ϕ ⁢ ( u ) = a 2 - b 2 ⁢ ( 1 - u ) - d 2 ,

with b1=b2=0.05, d1=1 and d2=1.2. We chose (u*,v*)=(0.6,0.9) as the fixed point. Consequently, a1=0.5 and a2≃0.15014. Figures 3 to 6 show the solution trajectories in each single quadrant and the corresponding time paths for specified time intervals, while Figure 7 shows the whole period length as a function of the starting point. From this last figure we see that the period length is very similar to that of the linearized system when we consider starting points close to the center. The period increases as we consider starting points far from the center and it eventually explodes as we approach the frontier of the unit box. Figure 7 supports the idea that the period is monotonically increasing (it increases as we take starting points closer to the frontier), but such a result has not been proved analytically and requires further research.

Our numerical simulations show that, in the case of our modified version of the model by Desai et al., the fact that the period is increasing is mainly due to its length in the third quadrant (low u and low v), which is greater than those in the other three quadrants and is increasing. The opposite holds in the first quadrant (high u and high v), where the speed of the solution trajectories increases as we approach the frontier. The results are mixed in the second and fourth quadrants. Note that while the whole period appears to be monotonic, this is not true at all in the case of the single quadrants. This shows the difficulty of proving a general theorem on the monotonicity of the period.

$Figure 3 First quadrant: solution trajectories in the (u,v)${(u,v)}$ phase plane for different initial values of v (left) and thecorresponding graphs of v as a function of t, where t∈[0,0.35]${t\in[0,0.35]}$ (right). Constant solution at v=0.9${v=0.9}$.$

Figure 3

First quadrant: solution trajectories in the (u,v) phase plane for different initial values of v (left) and thecorresponding graphs of v as a function of t, where t∈[0,0.35] (right). Constant solution at v=0.9.

$Figure 4 Fourth quadrant: solution trajectories in the (u,v)${(u,v)}$ phase plane for different initial values of u (left) and thecorresponding graphs of u as a function of t, where t∈[0,1.23]${t\in[0,1.23]}$ (right). Constant solution at u=0.6${u=0.6}$.$

Figure 4

Fourth quadrant: solution trajectories in the (u,v) phase plane for different initial values of u (left) and thecorresponding graphs of u as a function of t, where t∈[0,1.23] (right). Constant solution at u=0.6.

$Figure 5 Third quadrant: solution trajectories in the (u,v)${(u,v)}$ phase plane for different initial values of v (left) and thecorresponding graphs of v as a function of t, where t∈[0,2.5]${t\in[0,2.5]}$ (right). Constant solution at v=0.9${v=0.9}$.$

Figure 5

Third quadrant: solution trajectories in the (u,v) phase plane for different initial values of v (left) and thecorresponding graphs of v as a function of t, where t∈[0,2.5] (right). Constant solution at v=0.9.

$Figure 6 Second quadrant: solution trajectories in the (u,v)${(u,v)}$ phase plane for different initial values of u (left) and thecorresponding graphs of u as a function of t, where t∈[0,1.07]${t\in[0,1.07]}$ (right). Constant solution at u=0.6${u=0.6}$.$

Figure 6

Second quadrant: solution trajectories in the (u,v) phase plane for different initial values of u (left) and thecorresponding graphs of u as a function of t, where t∈[0,1.07] (right). Constant solution at u=0.6.

$Figure 7 Period length as a function of the starting point (0.6,v⁢(0))${(0.6,v(0))}$, where v⁢(0)∈[0.904,0.994]${v(0)\in[0.904,0.994]}$. The period of the linearized system is τ≃5.697${\tau\simeq 5.697}$.$

Figure 7

Period length as a function of the starting point (0.6,v⁢(0)), where v⁢(0)∈[0.904,0.994]. The period of the linearized system is τ≃5.697.

The above results may help to explain some of the empirical evidence on the Goodwin model, which suggests the presence of a three-quarter incomplete cycle in many OECD countries, starting from low values of u and high values of v and ending with high values of u and low values of v (see Harvie [15]). Indeed, the missing quarter of the cycle corresponds to the region where, as in the model considered in our simulation, the solution trajectories may become very slow. Thus, it is plausible that it will take some years before the observed cycles close, lending further support to the earlier findings of the literature, which suggest that Goodwin-type cycles can last several decades^[1]. Note that this is an unfavorable outcome from a socio-economic point of view, since it means that the employment proportion will rise very slowly after a period of high unemployment.

In general, this outcome reflects two main features of the economy. First, the response of the growth rate of real wages to the employment proportion is increasing with respect to v. This may reflect, for instance, the presence of labor unions, which prevent wages from falling too much during periods of high unemployment. Second, the response of investments to firms’ profit share is decreasing with respect to 1-u. As mentioned above, this can reflect, for example, the presence of imperfections in the credit market, which prevent firms from dramatically increasing their capital stocks during good times but force them to reduce their investments significantly during bad times. It would be then interesting to test whether these ideas suggested by our model match actual data and to what extent they explain the period length of real-world cycles.

6 Introducing Inflation

Many authors over the years have extended the basic Goodwin model by introducing economic phenomena not considered in the original formulation. One of these phenomena is inflation. This has often been modelled introducing an additive term in the real wage bargaining function. This term is an increasing function of the wage share u, since firms are assumed to set their prices as a mark up over the unit labor cost of output (see for instance Desai [6], van der Ploeg [24], Flaschel [11]). The additive term is then multiplied by a constant, sometimes denoted by η, which in the case η>0 measures the degree of money illusion in the wage setting process. In this section we introduce inflation in system (3.1)–(3.2) by considering a model similar to those proposed in the literature, but in the context of our generalized approach. First, we study how the introduction of inflation affects the stability of the equilibrium point. Then, we provide a condition to determine whether the equilibrium point is a focus or a node.

6.1 Stability of the Equilibrium

Given the above premise, we now perturb the original Hamiltonian system (3.1)–(3.2) by considering the following model:

(6.1) u ˙ u = f ⁢ ( u ) ⁢ ( ψ ⁢ ( v ) - η ⁢ h ⁢ ( u ) ) ,

(6.2) v ˙ v = - g ⁢ ( v ) ⁢ ϕ ⁢ ( u ) ,

where f,g,ψ,ϕ satisfy assumptions (3.3)–(3.6), η is a real parameter, h:(0,u1)→ℝ is C1 and h′>0. As already remarked, there exists a unique value u* such that ϕ⁢(u*)=0. Assume that η⁢h⁢(u*)∈Im⁡(ψ) and let v* be the unique solution of ψ⁢(v*)=η⁢h⁢(u*), so that (u*,v*) is the unique fixed point of system (6.1)–(6.2).

Setting ψ~⁢(v)=ψ⁢(v)-ψ⁢(v*) and h~⁢(u)=h⁢(u)-h⁢(u*), equation (6.1) becomes

u ˙ u = f ⁢ ( u ) ⁢ ( ψ ⁢ ( v ) - η ⁢ h ⁢ ( u ) ) = f ⁢ ( u ) ⁢ ( ψ ⁢ ( v ) - ψ ⁢ ( v * ) + ψ ⁢ ( v * ) - η ⁢ h ⁢ ( u ) )

= f ⁢ ( u ) ⁢ ( ψ ⁢ ( v ) - ψ ⁢ ( v * ) + η ⁢ h ⁢ ( u * ) - η ⁢ h ⁢ ( u ) ) = f ⁢ ( u ) ⁢ ( ψ ~ ⁢ ( v ) - η ⁢ h ~ ⁢ ( u ) ) ,

and thus

u ˙ u = f ⁢ ( u ) ⁢ ψ ~ ⁢ ( v ) - η ⁢ f ⁢ ( u ) ⁢ h ~ ⁢ ( u ) .

Note that

ψ ~ ⁢ ( v * ) = 0 , h ~ ⁢ ( u * ) = 0 .

We set

A ⁢ ( u ) = ∫ u * u ϕ ⁢ ( ξ ) ξ ⁢ f ⁢ ( ξ ) ⁢ 𝑑 ξ , B ~ ⁢ ( v ) = ∫ v * v ψ ~ ⁢ ( ξ ) ξ ⁢ g ⁢ ( ξ ) ⁢ 𝑑 ξ .

System (6.1)–(6.2) can be written as

u ˙ ⁢ ϕ ⁢ ( u ) u ⁢ f ⁢ ( u ) = ϕ ⁢ ( u ) ⁢ ψ ~ ⁢ ( v ) - η ⁢ ϕ ⁢ ( u ) ⁢ h ~ ⁢ ( u ) ,

v ˙ ⁢ ψ ~ ⁢ ( v ) v ⁢ g ⁢ ( v ) = - ϕ ⁢ ( u ) ⁢ ψ ~ ⁢ ( v ) .

Therefore

d d ⁢ t ⁢ [ A ⁢ ( u ⁢ ( t ) ) + B ~ ⁢ ( v ⁢ ( t ) ) ] = - η ⁢ ϕ ⁢ ( u ) ⁢ h ~ ⁢ ( u ) .

In the following we will use LaSalle’s principle to prove the stability of the fixed point z*=(u*,v*) when η>0. For this purpose we need the concept of Liapunov function which, although classical, is recalled here for the reader’s convenience in the context of LaSalle’s principle.

Let Ω⊂ℝn be an open set and let F:Ω→ℝn be a continuous vector field. We assume the uniqueness of the solutions of the Cauchy problems associated to

(6.3) z ˙ = F ⁢ ( z )

and denote by z⁢(t,x) the solution of (6.3) with z⁢(0)=x∈Ω. We also assume that there is a unique equilibrium point z*∈Ω (so that F⁢(z*)=0). The function V:Ω→ℝ is a Liapunov function if it is positive definite, i.e. V⁢(x)>V⁢(z*)=0, for all x≠z*, and

V ˙ ⁢ ( x ) = d d ⁢ t ⁢ V ⁢ ( z ⁢ ( t , x ) ) | t = 0 ≤ 0 , for all ⁢ x ∈ Ω .

Here we assume that V⁢(x) is a C1 function, so that V˙⁢(x)=〈∇⁡V⁢(x),F⁢(x)〉 for all x∈Ω.

LaSalle’s principle can be stated as follows (see LaSalle [17]).

Theorem 6.1

Theorem 6.1 (LaSalle’s Principle)

Let Ω⊂Rn be an open set and let V:Ω→R be a positive definite Liapunov function on Ω. Suppose that for some c>0 the set Ωc={x∈Ω:V⁢(x)≤c} is a nonempty compact set. We define

S = { x ∈ Ω c : V ˙ ⁢ ( x ) = 0 } .

Then, for every starting point z0∈Ωc the solution z⁢(t,z0) tends, as t→+∞, to the largest invariant set inside S. In particular, if S contains no invariant sets other than x=z*, then z* is asymptotically stable.

In our case we define V⁢(u,v)=A⁢(u)+B~⁢(v). Since ϕ⁢(u) and h~⁢(u) are strictly increasing and they both vanish at u*, we conclude that for η>0, we have

d d ⁢ t ⁢ V ⁢ ( u ⁢ ( t ) , v ⁢ ( t ) ) ≤ 0

for every trajectory, so that V⁢(u,v) is a Liapunov function. The set S is given by {(u,v)∈ℝ2:u=u*}∩Ω and the only invariant set in S is {u*,v*). Therefore, recalling that (as proved in Section 3) for every c>0=V⁢(u*,v*) the set {(u,v):V⁢(u,v)≤c} is compact, we can apply LaSalle’s principle. We thus conclude that

if η>0, then (u*,v*) is a globally asymptotically stable equilibrium point,
if η<0, then (u*,v*) is a globally repulsive equilibrium point.

The above result shows that, like the basic Goodwin model, system (3.1)–(3.2) is structurally unstable, i.e. small perturbations of the model may affect the qualitative behavior of the solutions. In the literature on the Goodwin model the presence of inflation, combined with a certain degree of money illusion (η>0), has a stabilizing effect on the economy. Our result shows that this effect occurs even in the context of a much more general framework and, in addition, it holds globally.

6.2 Focus or Node

Now we turn to the issue of determining whether the equilibrium point (u*,v*) is a focus or a node. We consider the same assumptions as above.

Consider the polar coordinates

{ u ⁢ ( t ) - u * = ρ ⁢ ( t ) ⁢ cos ⁡ θ ⁢ ( t ) , v ⁢ ( t ) - v * = ρ ⁢ ( t ) ⁢ sin ⁡ θ ⁢ ( t ) ,

so that

(6.4) u ˙ = ρ ˙ ⁢ cos ⁡ θ - ρ ⁢ θ ˙ ⁢ sin ⁡ θ ,

(6.5) v ˙ = ρ ˙ ⁢ sin ⁡ θ + ρ ⁢ θ ˙ ⁢ cos ⁡ θ .

Let us multiply (6.4) by ρ⁢sin⁡θ and (6.5) by -ρ⁢cos⁡θ. Summing the two terms, we get

u ˙ ⁢ ρ ⁢ sin ⁡ θ - v ˙ ⁢ ρ ⁢ cos ⁡ θ = - ρ 2 ⁢ θ ˙ ,

hence

- θ ˙ = u ˙ ⁢ ( v - v * ) - v ˙ ⁢ ( u - u * ) ρ 2 ,

from which

- θ ˙ = u ⁢ f ⁢ ( u ) ⁢ ( ψ ⁢ ( v ) - η ⁢ h ⁢ ( u ) ) ⁢ ( v - v * ) + v ⁢ g ⁢ ( v ) ⁢ ϕ ⁢ ( u ) ⁢ ( u - u * ) ( u - u * ) 2 + ( v - v * ) 2 .

Evaluating the differential of the second term at point (u*,v*), we get

- θ ˙ = 𝒩 ⁢ ( u - u * , v - v * ) ( u - u * ) 2 + ( v - v * ) 2 + o ⁢ ( u - u * , v - v * ) ( u - u * ) 2 + ( v - v * ) 2 ,

where

𝒩 ⁢ ( u - u * , v - v * ) = v * ⁢ g ⁢ ( v * ) ⁢ ϕ ′ ⁢ ( u * ) ⁢ ( u - u * ) 2 - η ⁢ u * ⁢ f ⁢ ( u * ) ⁢ h ′ ⁢ ( u * ) ⁢ ( u - u * ) ⁢ ( v - v * ) + u * ⁢ f ⁢ ( u * ) ⁢ ψ ′ ⁢ ( v * ) ⁢ ( v - v * ) 2 .

The numerator 𝒩 is a quadratic form whose associated matrix is

(6.6) ( v * ⁢ g ⁢ ( v * ) ⁢ ϕ ′ ⁢ ( u * ) - η 2 ⁢ u * ⁢ f ⁢ ( u * ) ⁢ h ′ ⁢ ( u * ) - η 2 ⁢ u * ⁢ f ⁢ ( u * ) ⁢ h ′ ⁢ ( u * ) u * ⁢ f ⁢ ( u * ) ⁢ ψ ′ ⁢ ( v * ) ) .

If such a form is positive definite then there exists k>0 such that for ρ small enough we have

(6.7) - θ ˙ ≥ k ⁢ ρ 2 ρ 2 = k .

Since the diagonal elements in (6.6) are positive, it is sufficient to consider the determinant, which is given by

u * ⁢ v * ⁢ f ⁢ ( u * ) ⁢ g ⁢ ( v * ) ⁢ ϕ ′ ⁢ ( u * ) ⁢ ψ ′ ⁢ ( v * ) - η 2 4 ⁢ ( u * ) 2 ⁢ f ⁢ ( u * ) 2 ⁢ h ′ ⁢ ( u * ) 2 .

Hence the quadratic form is positive definite when

v * ⁢ g ⁢ ( v * ) ⁢ ϕ ′ ⁢ ( u * ) ⁢ ψ ′ ⁢ ( v * ) > η 2 4 ⁢ u * ⁢ f ⁢ ( u * ) ⁢ h ′ ⁢ ( u * ) 2 ,

i.e. when

(6.8) η 2 < 4 ⁢ v * ⁢ g ⁢ ( v * ) ⁢ ϕ ′ ⁢ ( u * ) ⁢ ψ ′ ⁢ ( v * ) u * ⁢ f ⁢ ( u * ) ⁢ h ′ ⁢ ( u * ) 2 .

In view of (6.7), we conclude that when condition (6.8) is satisfied the equilibrium point (u*,v*) is a focus. By contrast, if the reverse strict inequality holds in (6.8), the quadratic form is indefinite and the equilibrium point is a node.

This result suggests that when the wage setting process is characterized by a low degree of money illusion (i.e. η is positive and small) the economy will fluctuate while approaching the equilibrium point. By contrast, if money illusion increases beyond a certain level and the wage share starts exerting a strong negative influence on its own growth rate, the economy will eventually converge monotonically to the equilibrium.

7 Conclusions

In this article we proposed a new and generalized approach for the modeling of Goodwin-type cycles fulfilling the essential requirement of lying entirely in the economically feasible domain. We provided a necessary and sufficient condition for this requirement to be satisfied. We proved that the period of the closed orbits generated by our model converges to the period of the corresponding linearized system as the starting point approaches the equilibrium point, whereas it becomes arbitrarily large as the distance of the starting point from the boundary approaches zero. Finally, we showed that our system, like the basic Goodwin model, is structurally unstable.

Our framework generalizes and enhances the economic coherence of the original growth cycle model. Indeed, Goodwin’s intuition that the interaction between wage share and employment rate gives rise to cyclical dynamics is preserved, but now it applies to a much wider range of cases within a unifying and formally well-grounded approach.

In our view, two main findings of our model are particularly worth noting. First, assumption (3.6), which guarantees that the cycles never stray outside the economically feasible range of values, requires the existence of strong nonlinearities in the wage bargaining function and in the investment function, and/or a nonlinear dependence of Goodwin’s original parameters on the state variables. As suggested above, these may indeed prove to be empirically relevant features that enrich the economic interpretation of the original model. Second, we proved that the period of the cycles becomes arbitrarily large in the third quadrant of the phase plane (low wage share and low employment proportion), while this is not guaranteed in the other quadrants. As we showed in our simulations, this result suggests that the responses of real wages to the employment rate and of investments to firms’ profit share are not constant as in the original Goodwin model, but vary significantly over the phase plane. This may provide an interesting and empirically testable explanation of the existence of three-quarter incomplete cycles in many OECD countries.

More generally, our framework may be employed as a starting point for further theoretical and empirical research. On the theoretical side, various contributions have recently tried to amend the original Goodwin model, but they have considered only specific cases. By contrast, our framework is very general and it can be easily employed to generate new, and perhaps more interesting and richer, models of fluctuations in the unit box, capturing several possible economic phenomena. On the empirical side, the estimation of the original Goodwin model is quantitatively unable to explain the time series of the state variables and predict the amplitude of the cycles and their length (see Harvie [15]). It would be then interesting to verify whether the econometric estimation of a specific model derived from our generalized approach (for example one with a power wage bargaining function and a power investment function) delivers significantly better results in terms of the size of the cycles and their length.

Communicated by Shair Ahmad

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Received: 2016-05-05

Revised: 2016-05-18

Accepted: 2016-05-19

Published Online: 2016-08-28

Published in Print: 2016-11-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2015-5050

Keywords for this article

Goodwin Model; Nonlinear Systems; Growth Cycle; Period; Stability

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