On a certain version of the multilane turnpike theorem in Gale’s non-stationary economy

In the vast majority of papers concerning asymptotic (main) properties of the optimal growth processes in Neumann-Gale-Leontief’s stationary economies, the geometric image of a turnpike is expressed by a single ray, called von Neumann’s ray. Even though neither the postulate of stationariness nor unambigiousness of a production turnpike are consistent with the observations of real economic processes, the list of papers devoted to the effect of a multilane turnpike in Neumann-Gale-Leontief’s non-stationary economies (with changing technology and multiple lanes) is much more modest. These works include mainly papers by Panek (2017, 2018), where the author replaces a single production turn-pike in Gale’s non-stationary economy with a multilane turnpike. This paper draws directly upon the author’s earlier work (Panek, 2019), where two turnpike theorems were presented. Both of them were based on the assumption signifi-cantly weakened by this paper – that in Gale’s non-stationary economy the optimal pro-duction structure in period ݐ remains optimal also in the future.

Non-Stationary Gale Economy with Limited Technology and Multilane Turnpike. ”Weak”, ”Strong” and ”Very Strong” Turnpike Theorem

In the author’s previous papers (2016a, 2016b) the generalized concept of turnpike in the stationary Gale’s economy has been proposed – a single turnpike (single von Neumann’s ray) has been replaced with a compact bundle of turnpikes and it has been called multilane turnpike. It has been proven that the ”weak” turnpike theorem holds in (a) stationary Gale’s economy with fixed (unchangeable in time) production technology and in (b) non-stationary Gale’s economy with technology convergent with time to a limit technology. In this article, in reference to the aforementioned papers, alongside with the ”weak” turnpike theorem, the proof of the ”strong” and ”very strong” turnpike theorem has been presented for the partially modified assumptions in a non-stationary economy with multilane turnpike and with technology convergent with time to a limit technology.

Gale’s Economy with Multiple Turnpikes. “Weak” Turnpike Effect

In the vast literature on turnpike theory it is generally assumed that the model path – called the turnpike – to which in a long time period all the optimal processes are convergent, is uniquely determined. Its geometric image in the Gale’s model (in its stationary version) is a ray in the space of all states of the economy. We call it von Neumann’s ray. In this paper we evade the assumption of the uniqueness of this turnpike (von Neumann’s ray) and study the behaviour of the stationary Gale’s economy with the compact turnpikes’ bundle. We call it multilane turnpike. We present proofs for several variants of the “weak” multilane turnpike theorem in the stationary Gales’ economy.

“Strong” Turnpike Effect in the Non-Stationary Gale Economy with Limit Technology

This article, in reference to Panek (2013a) presents proof of the “strong” turnpike theorem in the non-stationary Gale economy with changeable technology convergent to some limit technology. In the proof of the theorem assumption, that production processes efficiency in the economy is the lower the more the investment/input structure in such processes differs the optimum, play significant roles.The paper is part of trend of few works of mathematical economics containing proofs of the turnpike theorems in the non-stationary dynamic Neumann-Gale economic models.

A-Statistical Cluster Points in Finite Dimensional Spaces and Application to Turnpike Theorem

In the first part of the paper, following the works of Pehlivan et al. (2004), we study the set of allA-statistical cluster points of sequences inm-dimensional spaces and make certain investigations on the set of allA-statistical cluster points of sequences inm-dimensional spaces. In the second part of the paper, we apply this notion to study an asymptotic behaviour of optimal paths and optimal controls in the problem of optimal control in discrete time and prove a general version of turnpike theorem in line of the work of Mamedov and Pehlivan (2000). However, all results of this section are presented in terms of a more general notion ofℐ-cluster points.