Knife-Edge Conditions in the Modeling of Long-Run Growth Regularities

Abstract

Abstract.Balanced (exponential) growth cannot be generalized to a concept which would not require knife-edge conditions to be imposed on dynamic models. Already the assumption that a solution to a dynamical system (i.e. time path of an economy) satisfies a given functional regularity (e.g. quasi-arithmetic, logistic, etc.) imposes at least one knife-edge assumption on the considered model. Furthermore, it is always possible to find divergent and qualitative changes in dynamic behavior of the model -strong enough to invalidate its long-run predictions -if a certain parameter is infinitesimally manipulated. In this sense, dynamics of all growth models are fragile and "unstable".

When economic growth is less than exponential

“…15 As noted in the introduction, the case ϕ = 1, combined with constant L, and the case ϕ < 1 combined with exponential growth in L, are well understood. And the case ϕ > 1 leads to explosive growth.…”

“…where Y is aggregate manufacturing output (net of capital depreciation), A society's stock of "knowledge", K society's capital, u the fraction of the labor force employed in manufacturing, and c per-capita consumption; σ, α, γ and ϕ are con- 8 However, an interesting paper by Growiec (2008) takes steps in this direction. We may add that this paper, as well as the constructive comments by its author on the working paper version of the present article, has taught us that reducing the number of problematic knife-edge restrictions is not the same as "getting rid of" knife-edge assumptions concerning parameter values and/or functional forms.…”

“…Moreover, this case is also of interest because its dynamics turn out to reappear as a sub-system of the more elaborate example with embodied technical change in the next section. The Bernoulli equation (15) has the solution…”

“…14 The entire spectrum of regular growth patterns can also be obtained in an elementary version of the Jones (1995) model with no capital, but two types of (immobile) labor, i.e., unskilled labor in final goods production and skilled labor in R&D. 15 As an alternative to our "learning-by-doing" interpretation of (15), one might invoke a "population-breeds-ideas" hypothesis. In his study of the very-long run history of population Kremer (1993) combines such an interpretation of (15) with a Malthusian story of population dynamics.…”

On the cyclicality of R&D activities

“…Growiec (2010) discusses the necessity of such a knife-edge condition for any model generating exponential growth.…”