The minimal-time growth problem and ‘very strong’ turnpike theorem

. 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 { 𝑦𝑦 ∗ ( 𝑡𝑡 )} 𝑡𝑡=0𝑡𝑡 1∗ that leads the economy in the shortest possible time 𝑡𝑡 1∗ from the (initial) state of 𝑦𝑦 0 to the set of target/postulated states 𝑌𝑌 1 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 minimal-time growth problem) reaches a turnpike in a certain period of time 𝑡𝑡̌ < 𝑡𝑡 1∗ − 1 , then it stays on it everywhere else, except for, at most, final period 𝑡𝑡 1∗ . The obtained result confirms the well-known 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.


Introduction
There are several turnpike theorems (production, capital, consumption turnpikes, etc.) in the literature proved in various multi-product/multi-sector input-output models of economic dynamics -see e.g. Babaei (2019), Babaei et al. (2020), Giorgi and Zuccotti (2016), Jensen (2012), Khan and Piazza (2011), Majumdar (2009), Makarov and Rubinov (1977), Nikaido (1968, chap. 4), Panek (2003, part 2, chap. 5-6;2014, 2015, Takayama (1985, chap. 6-7). An extensive bibliography on the turnpike theory is presented in McKenzie (2005), Mitra and Nishimura (2009), Panek (2011), and others. The role of the growth criterion is most frequently embraced by the production utility generated in the economy either in the last period of defined horizon = {0, 1, … , − 1} or in all periods of the horizon. A paper by Panek (2021), however, presents a different approach -it proves a 'weak' turnpike theorem in the stationary Gale economy with products and with a single production turnpike. In that paper, the time needed for an economy starting from a fixed initial state (production vector) of 0 = ( 1 0 , … , 0 ) > 0 to reach the desirable target set of states (production vectors) 1 1 = { ∈ + | ≧ 1 }, 1 > 0 assumes the role of the growth criterion. According to this theorem, almost all optimal growth processes 2 − regardless of the distance of the target set of states 1 from initial vector 0take place in an arbitrarily close (in the angular measure sense) neighbourhood of the production turnpike, where the economy develops at the maximum rate and achieves the highest technological and economic efficiency. It is a state of a specific dynamic equilibrium (peak equilibrium of growth) in the Gale economy. This paper refers to the aforementioned article and contains a proof of a 'very strong' turnpike theorem. It states that if optimal process { * ( )} =0 1 * (the solution to the minimal-time growth problem) reaches the turnpike in a certain period of ̌< 1 * − 1, it remains on it everywhere else, except for, at most, the last period of horizon {0, 1, … , 1 * }. The potential precipitation of the economy from the turnpike in period 1 * results from the necessity to reach the target set of states 1 .
The paper further consists of Section 2, where a model of the stationary Gale economy is presented and selected properties of the production turnpike and the von Neumann equilibrium state are defined and discussed, Section 3, which presents the minimal-time growth issue and the conditions for the existence of a feasible stationary and optimal growth process, Section 4, which provides the formulation and proof of a 'very strong' turnpike theorem (Theorem 3), and Section 5, which features a certain specific version of the 'very strong' turnpike theorem in the stationary Gale economy with a single turnpike and a minimum-time growth criterion (Theorem 3'). The paper concludes with the author's indication of the possible directions for further development of the current research.

Technological and economic production efficiency. Von Neumann equilibrium 3
In the economy we have < +∞ consumed and/or produced commodities. We consider a model with discrete time = 0, 1, … . By = ( 1 , … , ) we denote the input vector that is used in the economy in a specific unit of time, e.g. for a year (we also call it a production factors vector), and by = ( 1 , … , ) the output vector that is produced in a unit of time (also called a production vector). If the technology at the disposal of the economy allows the achievement of production from inputs , then the pair ( , ) is said to create (describe) a technologically feasible production process. 4 Non-empty set ⊂ + 2 of all technologically feasible production processes is called the Gale production space (or the technological set) if the following conditions are met: (G1) ∀( 1 , 1 ) ∈ ∀( 2 , 2 ) ∈ ∀ 1 , 2 ≥ 0 ( 1 ( 1 , 1 ) + 2 ( 2 , 2 ) ∈ ) (inputs/outputs proportionality condition and the additivity of production processes), (possibility of wasting the inputs/outputs), (G4) production space is a closed subset of + 2 .
The Gale production set is a closed cone in + 2 with a vertex at 0. If ( , ) ∈ and = 0, then, according to (G2), also = 0. We are only interested in processes ( , ) ∈ \{0}. The number is called the index of the technological efficiency of process ( , ) ∈ \{0}. It follows from the definition that function (•) is non-negative and positively homogeneous of degree 0 on \{0}.
□ Theorem 1. If conditions (G1)-(G4) are satisfied, then a solution to the problem exists: For proof, see Panek (2022, th.1), Takayama (1985, th. 6.A.1). ∎ The number is called the optimal indicator of the technological production efficiency. Process ( ̅ , �) ∈ \{0} is called the optimal production process. In the stationary Gale economy it is determined with the accuracy of a multiplication by a positive constant (with a structure accuracy); if ( ̅ , �) = , then ∀ > 0( ( ̅ , �) = ). We are interested in an economy where in optimal production process ( ̅ , �) all commodities are produced and the production of the commodities exceeds (on all coordinates) the inputs. This is ensured by the following condition: ). An economy that satisfies condition (a) is called regular, and an economy which meets condition (b) is called productive. If condition (G5) is met, then due to (G3): Everywhere else, when we talk about optimal process ( ̅ , �), we mean the production process that meets the above-mentioned condition. We say that vector represents the production structure in optimal process ( ̅ , �) 5 . Ray is called the production turnpike (the von Neumann ray) in the stationary Gale economy. By = ( 1 , … , ) ≥ 0 we denote the commodity price vector in the Gale economy. Let ( , ) ∈ \{0}. Then 〈 , 〉 = ∑ =1 is the inputs value and 〈 , 〉 = ∑ =1 the production value in process ( , ). The number is called the index of the economic efficiency of process ( , ). Let ( ̅ , �) ∈ \{0} be the optimal production process in the Gale economy. Then □ Theorem 2. If conditions (G1)-(G5) are satisfied, then such a price vector ̅ ≥ 0 exists that For proof, see e.g. Panek (2003;chap. 5, th. 5.4). ∎ Since in optimal process ( ̅ , �) the production vector is positive and the price vector is at least semi-positive, then From (1) We say that the triple { , ( ̅ , �), ̅ } represents the (optimal) von Neumann equilibrium state in the stationary Gale economy. Price vector ̅ is called the von Neumann price vector. In the equilibrium state, the technological production efficiency matches its economic efficiency (at the maximum possible level of that can be achieved by the economy).
In the von Neumann equilibrium state production process ( ̅ , �) and price vector ̅ are determined with a structure accuracy (multiplication by a positive constant).
To ensure the uniqueness of turnpike , we assume that the economy satisfies the following condition: a certain production process the inputs structure differs from the turnpike structure, then according to (G6), the economic efficiency of such a process is lower than optimal. 6 □ Lemma 1. If conditions (G1)-(G6) are satisfied, then For proof, see: Radner (1961), Takayama (1985; chap. 7), Panek (2003;chap. 5, lemma 5. 2). ∎

Dynamics. Feasible, stationary and optimal growth processes
We assume that time is discrete, = 0, 1, … . We denote the input vector (or the production factors vector) that is used in the economy in period by ( ) = � 1 ( ), … , ( )�, and the output vector (or the production vector) that is produced in period by ( ) = � 1 ( ), … , ( )�. From the assumption that � ( ), ( )� ∈ \{0} it follows that it is possible to produce production vector ( ) from input vector ( ) in period . The economy is closed in the sense that inputs ( + 1) (that are incurred in the next period) come from production ( ) (produced in the previous period), i.e.
if it is a solution to the following minimal-time growth problem: min 1 subject to (4), (5), (7), in which vector 0 and set 1 are fixed.
If conditions (G1)-(G6) are satisfied, and particularly 0 = � ∈ , then a growth process exists (satisfying conditions (4), (5)) of the following form: which is called the stationary growth process with the rate. 7 Since in such a process the following condition is satisfied: we therefore say that it is characterised by a constant (turnpike) production structure. Each stationary growth process lies on turnpike . The production of all the commodities in such a process increases at a maximum rate of > 1 achievable by the economy. This fact still plays an important role in the proof of the 'very strong' turnpike theorem in the next section.

'Very strong' turnpike effect
Let us introduce the following (angular) distance measure of production vector ( ) from turnpike = { ̅ | > 0}: The stationary growth process exists if and only if condition ( �, �) ∈ \{0} is satisfied. This condition is fulfilled in our model. In a paper by Panek (2021, Th. 4), we proved the 'weak' turnpike theorem which states that if conditions (G1)-(G6) apply, and (*) such a number < +∞ exists that regardless of the distance between target states set 1 and initial state 0 , each vector 1 > 0 determining the shape of this set (see (6)) satisfies condition max 1 min 1 ≤ , then -regardless of the distance between target states set 1 from initial state 0the production structure in each optimal growth process ( 0 , 1 , 1 * ), i.e. the solution to problem (8), almost always 8 differs slightly, in an arbitrary way, from the turnpike production structure on which the economy develops at its maximum rate, achieving the highest possible technological and economic efficiency. According to condition (*), 1 is any production vector (greater than initial vector 0 ) in which with ‖ 1 ‖ → +∞, the distance (range) between the values of its coordinates does not increase 'too rapidly' (i.e. no faster than linearly). 9 We will now trace the trajectory of optimal growth process { * ( )} =0 1 * , which in a certain time period of ̌< 1 * − 1 reaches turnpike , when condition (*) is replaced with the following condition: 10 (G7) vector 1 > 0 , on which the set of target states 1 depends, satisfies For the proof of Theorem 3, the following lemma will be necessary.
Proof. 11 If assumptions (G1)-(G7) are satisfied, then there exists ( 0 , 1 , 1 ) − a feasible growth process (10), in which �( 1 ) ≧ 1 , and 1 is the smallest natural number which satisfies the condition In this process, �() = * () and ∀ ∈ {,̌+ 1, … , 1 } ( �( ) ∈ ). Let us denote by 1 the smallest number (not necessarily natural) for which the following inequality holds: Such a number exists (since 1 > 0 and > 1) and The proof is partially based on the proof of Lemma 3 from the paper by Panek (2021). The sequence elements (10) starting from =̌ belong to stationary growth process (9) with the rate and initial production vector Thus, each target production vector 1 that satisfies condition (G7) corresponds to such a non-negative number 1 ≤ 1 that condition (12) is satisfied. Particularly, and this concludes the proof. ∎ In Theorem 3 we prove that if ( 0 , 1 , 1 * ) − optimal growth process in a certain time period ̌ meets condition ̌< 1 * , then everywhere else, except possibly in the last period 1 * , it remains on the turnpike.

Final remarks
The necessity to leave the turnpike by optimal process { * ( )} =0 1 * , i.e. the solution to minimal-time growth problem (8), in final period 1 * results simply from the postulate that the economy should reach the set of target states 1 . In a particular case, when target production vector 1 that determines the form of target state set (6) is located on the turnpike, 1 ∈ , condition (13) holds also for = 1 * .
The following version of Theorem 3 without the postulate of uniqueness of the solution also remains true: □ Theorem 3'. Suppose the following applies: • conditions (G1)-(G7) are satisfied; 12 We deal with a similar situation in the paper by Panek (2021;Th. 3). When 1 = 1 * , then < − 1, which cannot be excluded, because > 1. Therefore, one of the assumptions of this theorem is the condition of the solution unique.
• a certain ( 0 , 1 , 1 * ) − optimal growth process in period ̌< 1 * − 1 reaches the turnpike, then there also exists such a ( 0 , 1 , 1 The proof here is exactly the same as the proof of Theorem 3. ∎ An example trajectory of ( 0 , 1 , 1 * ), i.e. the optimal growth process in ⊂ + 2 satisfying the conditions of Theorem 3 is illustrated in the Figure. Source: author's work.

Conclusions
In many papers devoted to the asymptotic/turnpike properties of the optimal growth processes in von Neumann-Gale-Leontief economies, production utility is assumed to be the growth criterion. The novelty of the approach proposed in this article, like in the earlier paper by Panek (2021), consists in replacing the utility of production as the standard quality criterion of economic growth processes by a minimum-time growth criterion (minimising the time needed by the economy to reach the postulated/desired target state). It was proven that changing the growth criterion does not deprive the Gale economy of its asymptotic/turnpike properties. It would be interesting to study the turnpike properties of the solutions to the minimal-time growth problems of type (8) also in a non-stationary Gale economy with changing technology and multilane production turnpike, especially in the Gale economy with an investment mechanism (see Panek, 2022).
Probably the solution to minimal-time growth problem (8) is also characterised by a 'strong' turnpike effect (as in the case of many other optimal growth processes in the Gale economy with the maximisation of the production utility criterion).