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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 [

In this paper, we will examine an extension of the neoclassical growth model, which can be traced back to the early works of Solow [^{a}^{b }

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.

Matsumoto and Szidarovszky [^{α}^{b}^{γ}e^{-δx},

Assume first that ^{γ-1}^{-δx}^{γ}e^{-δx}^{γ} ^{1-γ} + ^{-δx}) = 0
^{1-γ} = ^{-δx}

Assume next that ^{-δx}) = 0

Consider finally the case of ^{γ}^{-1}^{-δx}) = 0
^{γ}^{-1}^{-δx} = 0
^{γ}^{-2}^{-δx} - ^{γ}^{-1}^{-δx}
^{γ}^{-2}^{-δx}(

(i) if

(ii) if

(iii) If _{1} < _{2} > _{1 }or _{2}, and _{1} < _{2}. Therefore if _{1}, then _{1} < _{2}, then _{2}, if _{2}, then _{2}. That is, _{1 }is locally unstable and _{2 }is locally asymptotically stable. If _{1 }or _{2}, then

The fixed delay ^{γ}e^{-δx}
_{δ}_{δ}^{λt}u

^{λT}^{T}^{(Reλ)}^{iT}^{(Imλ)}
^{T}^{(Reλ)}(cos[^{λT}

Notice that
^{γ}^{-1}^{-δx}(γ - ^{γ}e^{-δx} = ^{γ}^{-1}^{-δx} = ^{2}

In order to give a complete stability analysis, we have to find the possible stability switches. Substituting any stability switch, ^{(n)} = _{+}^{(n) }with
^{(n)} = _{-}^{(n) }with

By selecting ^{(0)} and cannot be regained later. That is, if the steady state is unstable without delay, then it remains unstable with any delay of positive length. If the steady state is asymptotically stable without delay, then it remains asymptotically stable until the delay reaches a certain threshold, and then becomes unstable and the stability cannot be regained later.

Taking, _{-}^{(0) }curve to
_{-}^{(0) }and it is unstable in the white region above the curve. Setting _{c}_{c}_{c}- 0.05 to 8.5 with an increment of 0.01 and the local maximum and minimum of the corresponding trajectory are plotted against each value of

Dynamics with

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 [

Linearizing equation (22), we have
_{δ}_{δ}^{λt}u

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.

Assume first that

Assume next that

Assume now that ^{2 }and (26) holds if and only if

If -8 <

If the steady state is unstable without delay, then it remains unstable with continuous delay with any

We will next show that at the critical values ^{2}, at

We perform numerical simulations to illustrate the results obtained above. In ^{13},

Dynamics with

The cases of ^{λT}

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).

The authors declare no conflict of interest.