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Sustainability 2013, 5(2), 440-455; doi:10.3390/su5020440
Abstract: A neoclassical growth model is examined with a special mound-shaped production function. Continuous time scales are assumed and a complete steady state and stability analysis is presented. Fixed delay is then assumed and it is shown how the asymptotic stability of the steady state is lost if the delay reaches a certain threshold, where Hopf bifurcation occurs. In the case of continuously distriubuted delays, we show that with small average delays stability is preserved, then lost at a threshold, then it is regained if the average delay becomes sufficiently large. The occurence of Hopf bifurcation is shown at both critical values.
The examination of economic growth models is one of the most frequently discussed issues in mathematical economics. Day [1,2] has investigated a neoclassical growth model, and a productivity and population growth model and showed the emergence of complex behavior even under simple economic structure. His models were based on discrete time scales and a mound-shaped production function that represented the negative effect of pollution resulting from increasing capital. It was demonstrated by numerical computations that these models could generate cyclic and even chaotic behavior. Following Day’s pioneering works, a lot of effort has been given to the understanding of complex economic dynamics. Day , Puu  and Bischi et al.  present the earlier contributions of this field. A large number of studies assumed discrete time scales. Li and Yorke  have introduced the “period-three condition” to detect chaos, which has many applications in first-order nonlinear difference equations. The papers collected by Rosser  offer many applications. Only a few studies are devoted to the case of continuous time scales, since there is no general criterion to detect chaos and the system must have at least three dimensions.
In this paper, we will examine an extension of the neoclassical growth model, which can be traced back to the early works of Solow  and Swan . The neoclassical growth model is constructed with the two (usually implicitly mentioned) assumptions; one is the full-employment of labor and capital and the other is instantaneous adjustment in the output market. Thus, it is suitable for describing the long-run behavior of the economy. Due to the well-behaved production function, the steady state of the model is usually asymptotically stable. However, it is often observed in reality that growth path exhibits persistent fluctuations. The neoclassical model could be the good point of departure to show how such persistent behavior can emerge when nonlinearities and a production lag are present. Matsumoto and Szidarovszky  attempt to fill the gap and have introduced a neoclassical model with a mound-shaped production function that was assumed to be a Cobb-Douglas type function of the form F(x)=Axa(1 - x)b with x being the capital per unit labor. Although they show emergence of erratic fluctuations in the capital accumulation process, the production function is restrictive in the sense that it is defined only in the unit interval. This paper modifies this drawback, considers another type of mound-shaped production function and will examine the stability of the steady state with and without time delays in the continuous-time framework. Two kinds of delays will be discussed, fixed and continuously distributed (continuously hereafter) delays. We keep the relatively simple model of Matsumoto and Szidarovszky  in order to be able to compare the results and to illustrate that complex dynamics can be generated under simple economic assumptions with both function types.
This paper develops as follows. First the mathematical model is formulated without time delays, and complete steady state and stability analysis is presented. Then, models with fixed delays and then with continuous delays are introduced and complete stability analysis is given. The last section concludes the paper.
2. The Mathematical Model
Matsumoto and Szidarovszky  have introduced a special growth model of the form
Assume first that γ < 1. The steady states are the solutions of f(x) = 0. Notice that f(0) = 0, so zero is a steady state. f(x) converges to - ∞ as x → ∞ Since
Assume next that γ = 1. Then the steady state equation has the form
Consider finally the case of γ > 1. The steady state equation has now the form
(i) if g( ) < 0, then there is no positive steady state and with arbitrary x(0)>0, x(t) decreases and converges to zero.
(ii) if g( ) < 0, then = is the unique positive steady state and f(x) < 0 for all 0 < x ≠ . If x(0) < , then x (t) decreases and converges to 0, and if x(0) > then x (t) decreases again and now converges to . If x(0) = , then x (t) = for all t > 0.
(iii) If g( ) > 0, then equation (7) has two positive solutions, 1 < and 2 > . Relation (6) implies that f(x) < 0 as x < 1 or x > 2, and f(x) > 0 if 1 < x < 2. Therefore if x(0) < 1, then x (t) decreases and converges to zero, if 1 < x(0) < 2, then x (t) increases and converges to 2, if x (0) > 2, then x (t) decreases and converges to 2. That is, 1 is locally unstable and 2 is locally asymptotically stable. If x(0) = 1 or x (0) = 2, then x (t) remains at that steady state level for all t > 0.
3. Model with Fixed Delay
The fixed delay T > 0 is assumed in the second term of the right hand side of equation (1), so we have the following equation:
Lemma 1 Assume that |h´( )| < α. Then is locally asymptotically stable.
Proof. Assume that Re λ ≥ 0. Then
In order to give a complete stability analysis, we have to find the possible stability switches. Substituting any stability switch, λ= iω with ω > 0 into equation (11) yields
By selecting T as the bifurcation parameter and implicitly differentiating the characteristic equation with respect to T, we have
Taking, α = 1, β = 25, γ = 1 and δ = 1, we give an illustrative numerical example in Figure 1. The critical value γ - δ is denoted by
4. Model with Continuously Distributed Delay
Assuming continuously distributed delays in the second term of equation (1) gives the following Volterra-type integro-differential equation:
A similar model is investigated by Fanti and Manfredi  where m = 2 is selected and the stability of the system with a cubic characteristic polynomial is examined based on the Routh-Hurwitz criterion. Stamova and Stamov  consider a generalized Solow model with endegenous labor growth and impulsive perturbations. Their stability analysis is based on the Lyapunov-Razumikhin sufficient stability conditions, which is a different approach than ours.
Linearizing equation (22), we have
Lemma 2 Assume that |h´( )| < α. Then is locally asymptotically stable.
Proof. Assume that Reλ≥ 0. Then
It is well known that the Routh-Hurwitz stability theorem provides necessary and sufficient conditions for all roots of a polynomial equation with real coefficients to have negative real parts. It is also known that it is difficult to locate the eigenvalues with analytic methods in general. However in some special cases, analytic results are still possible to obtain, as it will be next demonstrated.
Case I. T = 0
Assume first that T = 0, which reduces equation (22) with delays to equation (1) without delays. The asymptotic properties of this equation were already discussed earlier.
Case II. T > 0 and m = 0
Assume next that T > 0 and m = 0, when the kernel function becomes exponentially declining. Then characteristic equation (23) becomes quadratic,
Case III. T > 0 and m = 1
Assume now that T > 0 and m = 1, when the shape of the kernel function takes a bell-shaped form. Then we have a cubic characteristic polynomial:
If -8 < z < 0, then D < 0, so (26) holds and is asymptotically stable. The instability region is shown in Figure 2(A) where z is the horizontal axis and T is the vertical axis. If we start with a very small value of T with any given z < -8, then is asymptotically stable. If we gradually increase T then remains asymptotically stable until it reaches the critical value , when the steady state becomes unstable. It remains unstable until when stability is regained, and remains asymptotically stable for all T > .
If the steady state is unstable without delay, then it remains unstable with continuous delay with any T and m. If it is asymptotically stable without delay, then either it remains asymptotically stable with all T and m, or loses stability at a certain value of the average delay T and stability is regained with an even larger value of T and the steady state remains asymptotically stable afterwards. In such cases small and large average delays lead to asymptotically stable steady states.
We will next show that at the critical values and , Hopf bifurcation occurs giving the possibility of the birth of limit cycles. We select T as the bifurcation parameter. At the critical values (26) is satisfied with equality, so
We perform numerical simulations to illustrate the results obtained above. In Figure 2(A), the steady state is locally asymptotically stable in the dark-gray region with z > -1 due to Lemma 2. It is also locally asymptotically stable in the light gray region and unstable in the white region when z < - 1.The appearance and disappearance of a limit cycle can be observed in Figure 2(B) where we take α = 1, β = e13, γ = 1 and δ = 1 implying z = - 12,
The cases of m ≥ 2 result in fourth or larger degree polynomial equations. The stability of the steady states can be examined similarly, but the mathematical details become much more complicated. It can be mathematically confirmed that as m → ∞, equation (23) converges to the characteristic equation (11) of the model with fixed delay. In particular, if m → ∞, then expression
In this paper, a special neoclassical growth model was introduced and examined. A mound-shaped production function for capital growth was assumed in the dynamic equation. Zero is always a steady state, and depending on model parameters there is either no positive steady state, or one, or two positive steady states. A complete steady state analysis was followed by the derivation of stability conditions. By introducing fixed delay we demonstrated that stability can be lost at a certain value of the delay and the equilibrium remains unstable afterwards. In the case of continuously distributed delays it has been shown how stability can be lost at a certain value of the average delay and by further increasing the average delay it can be regained. At the critical values, Hopf bifurcation occurs giving the possibility of the birth of limit cycles. In our further study, more complex kernel functions will be considered and their effect on the asymptotic behavior of the steady state will be examined.
It is now well-known that any discrete-time dynamic model can generate complex dynamics involving chaos. However, in an aggregate model like the neoclassical growth model, it is natural to treat economic phenomena as continuous since the variables to be examined are the outcomes of a great number of decisions taken by different agents at different points of time. This is the main reason why this paper considers the neoclassical growth model in the continuous-time framework. Implication of the analysis is the following. Coexistence of nonlinearity and delay in production can be a source of persistent fluctuations in the capital-labor ration.
The authors are grateful to two anonymous referees for their constructive suggestions and helpful comments and highly appreciate financial supports from the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 24530202) and Chuo University (Grant for Special Research and Joint Research Grant 0981).
Conflict of Interest
The authors declare no conflict of interest.
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