On the Dalgaard-Strulik Model with Logistic Population Growth Rate and Delayed-Carrying Capacity
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On the Dalgaard-Strulik Model with Logistic Population Growth Rate and Delayed-Carrying Capacity Carlo Bianca · Luca Guerrini
Received: 29 August 2012 / Accepted: 5 February 2013 / Published online: 14 February 2013 © Springer Science+Business Media Dordrecht 2013
Abstract Recently Dalgaard and Strulik have proposed (in Resour. Energy Econ. 33:782– 797, 2011) an energy model of capital accumulation based on the mathematical framework developed by Solow-Swan and coupled with Cobb-Douglas production function (Solow in Q. J. Economics 70:65–94, 1956; Swan in Econ. Rec. 32(63):334–361, 1956). The model is based on a constant rate of population growth assumption. The present paper, according to the analysis performed by Yukalov et al. (Physica D 238:1752–1767, 2009), improves the Dalgaard-Strulik model by introducing a logistic-type equation with delayed carrying capacity which alters the asymptotic stability of the relative steady state. Specifically, by choosing the time delay as a bifurcation parameter, it turns out that the steady state loses stability and a Hopf bifurcation occurs when time delay passes through critical values. The results are of great interest in the applied and theoretical economics. Keywords Dalgaard-Strulik model · Energy · Time delay · Hopf bifurcation · Logistic model · Nonconstant carrying capacity 1 Introduction The modern economic growth theory has its origin in the seminal papers by Solow [31] and Swan [32], who contemporaneously and independently have proposed a new theoretical framework for understanding world-wide growth of output and the persistence of geographical differences in per capita output. The model they provided is known as the Solow-Swan model, or in brief the Solow model after that the most famous of the two economists was awarded by the Nobel prize in economics for his contributions. C. Bianca () Dipartimento di Scienze Matematiche, Politecnico, 10129 Torino, Italy e-mail: [email protected] L. Guerrini Dipartimento di Matematica per le Scienze Economiche e Sociali, Università di Bologna, 40126 Bologna, Italy e-mail: [email protected]
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C. Bianca, L. Guerrini
A standard assumption in the Solow-Swan model is that population grows at a positive constant rate (Malthusian model [29]), giving rise to an exponential population growth curve. However, nowhere in nature it is possible to have unlimited exponential growth of any population over the long run. Claiming that the model proposed by Malthus was too simplistic, as it only included linear terms, led to replacing the linear population model of Malthus by a non-linear one. The first model of this type (nowadays called the logistic model) was proposed by Verhulst [33], who introduced an additional quadratic term with a negative coefficient in the Malthusian model, so that any population growth rate followed an elongated S-curve. The introduction of logistic-type population growth law, within the Solow-Swan model and its extension the Ramsey model [30], has recently attracted a great attention in the literature
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