"S-shaped" Economic Dynamics. The Logistic and Gompertz curves generalized.
Over the years "S-shaped" evolutions have regularly been incorporated in economic models, and indeed in those of other sciences, by way of the logistic or Gompertz equations. However, both equations have noteworthy shortcomings when fitting some empirical features of economic growth: the logistic equation is characterized by strong symmetries, whilst the growth rate is decreasing in the case of both equations. In this paper, we have set out to overcome these limitations by defining a family of unimodal differential equations which includes the logistic and Gompertz equations and covers practically the whole spectrum of sigmoid curves. We have identified three sub-families of these differential equations, all offering good mathematical expressions. Using these, it is possible to obtain an acceptable fit for any S-shaped curve. The results are applied to various economic series, successfully replicating certain well-known economic phenomena. Mathematical analysis: Unimodal differential equations. Empirical analysis: Non-linear adjustment to the USA Capacity Index time series. We have defined a family of unimodal differential equations covering practically the whole spectrum of "S-shaped" curves. We have selected three sub-families mathematically manageable and which depend on five easily interpretable parameters. It is shown that any one of them may adequately replicate empirically relevant S-shaped phenomena (overcoming certain limitations of the logistic and Gompertz curves). In order to assess the power of these families to replicate real economic events we have calculate the fit with the USA Capacity Index for Total Industry (1967/02-2003/01) and the US Capacity Index series for Durables, Manufacturing, Computers and Primary processing. Then, we have drawn conclusions on capital accumulation and investment patterns in the period that would appear to be in line with recent historical facts in the US economy.
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