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The minimal-time growth problem and 'very strong' turnpike theorem

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Panek E.

Przegląd Statystyczny

2021

vol. 68

issue 4

45-58

This paper refers to the author's previous work, in which the ‘weak’ turnpike theorem in the stationary Gale economy was proved. This theorem states that each optimal growth process {y*(t)}t*1t=0 that leads the economy in the shortest possible time t*1 from the (initial) state of y0 to the set of target/postulated states Y1 almost always runs in the neighbourhood of the production turnpike, where the economy remains in a specific dynamic equilibrium (peak growth equilibrium). This paper presents a proof of the ‘very strong’ turnpike theorem in the stationary Gale economy, which states that if the optimal process (the solution to the minimaltime growth problem) reaches a turnpike in a certain period of time tˇ < t*1 - 1, then it stays on it everywhere else, except for, at most, final period t*1. The obtained result confirms the wellknown Samuelson hypothesis about the specific turnpike stability of optimal growth paths in multiproduct/multisectoral von Neumann-Leontief-Gale-type models, also in the case where the growth criterion is not the (normally assumed) utility of production but the time needed by the economy to achieve the postulated target level or volume of production.

The minimal-time growth problem and turnpike effect in the stationary Gale economy

88%

Panek E.

Economics and Business Review

2021

vol. 7

issue 1

7-25

In mathematical economics there is a number of so called “turnpike theorems” proved mainly on the basis of multiproduct models of economic dynamics. According to these theorems, all optimal paths of economic growth over a long period of time converge to a certain path (turnpike) in which the economy achieves the highest growth rate while remaining in a specific dynamic (von Neumann) equilibrium. The article refers to this trend and presents some properties of optimal growth processes in the Gale-type model of the stationary economy when the quality criterion of growth processes is not the utility of production-which is normally postulated in the turnpike theory-but the time needed by the economy to achieve the desired final state, e.g. the level of production or production value. According to the author's knowledge, the idea of using time as a criterion for growth in turnpike theory (especially in Gale-type economy) is innovative. It has been proven that changing the growth criterion does not deprive the Gale economy of its asymptotic / turnpike properties.

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1 Economics and Business Review

1 Przegląd Statystyczny

2 Panek E.

2 2021

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The minimal-time growth problem and 'very strong' turnpike theorem

The minimal-time growth problem and turnpike effect in the stationary Gale economy