Optimal growth processes in a non-stationary Gale economy with a multilane production turnpike

Abstract The topic of the paper is relevant in the field of optimal growth theory and therefore might be seen as an intellectual underpinning for research and practice in the field of transition economies and sustainable long-time development as well. It refers to the papers Panek (2015a, 2018) devoted to asymptotic properties of optimal growth properties in the non-stationary Gale type economy with single and multi-lane turn-pikes in which it was assumed that changing production technology converges in time with certain limits of technology. As far as the postulate of a non-stationary economy (here: technology change) is consistent with real processes, the hypothesis of the existence of some limiting technology may raise controversies and be difficult to verify. In the paper, referring to the above mentioned publications and Panek (2014), a Gale-type economy with changing technology, multi-lane turnpike and time-increasing production efficiency, with no assumption concerning the existence of a limit technology will be examined.


Introduction
In mathematical economics a vast majority of the theorems focus on turnpike properties of optimal growth processes in stationary economies with constant technology and a single-lane turnpike. 3 Attempts to prove the turnpike effect 4 Economics and Business Review, Vol. 5 (19), No. 2, 2019 in non-stationary economies with changing technology are less frequent. 4 A general analysis of the results recognizing a non-stationary Gale economy with a multi-lane turnpike and limit technology is included in articles by Panek (2017Panek ( , 2018. The explicit assumption of production technology converging to a hypothetical limit technology may raise some objections, not to mention the fact that it is difficult to verify this convergence empirically and thus it was relaxed in this paper. The rest of the paper is as follows. In sections 1 and 2 a non-stationary Gale-type model of the economy and define multilane production turnpike is presented. The main result of section 3 is a temporary von Neumann equilibrium theorem (Th. 1). In section 4 feasible and stationary growth processes in the economy are defined. Sections 5, 6 contain proofs of the so called "weak" (Th. 2) and "fast very strong" (Th. 3) turnpike theorems for the economy under investigation. The paper closes with some final remarks.

The model. Basic assumptions
The basic version of the presented model was introduced in Panek (2014). It is assumed that time t is discrete, t = 0, 1, … . In the considered economy there are n goods used up in production or produced in period t. By x(t) = (x 1 (t), …, x n (t)) a vector of goods used in period t (input vector) is denoted while y(t) = (y 1 (t), …, y n (t)) stands for a vector of goods produced in period t (output vector). 5 If production technology in the economy enables to produce an output vector y(t) from an input vector x(t), it is said that the pair (x(t), y(t)) is a feasible production process in period t. By Z(t) ⊂ R + 2n the set of all feasible production processes in t. So (x, y) ∈ Z(t) (or equivalently (x(t), y(t))∈ Z(t)) is denoted which means that in the economy in period t one can produce output vector y from input vector x. The set Z(t) is called the production space (technology set) of the economy in period t. It is assumed that the production spaces Z(t), t = 0, 1, …, satisfy the following conditions: (homogeneity and additivity of production processes).
Panek, Optimal growth processes in a non-stationary Gale economy (costless waste possibility).
(each production process (x, y) ∈ Z(t) feasible in period t is also feasible in the next period). Production spaces which satisfy the above conditions are closed convex cones in R + 2n , with vertices at 0. If (x, y) ∈ Z(t) and (x, y) ≠ 0, then, according to (G2), x ≠ 0.
The paper is only interested in non-trivial (non-zero) processes, that is, in processes (x, y) ∈ Z(t)\{0}.

Technological and economic production efficiency. Multilane production turnpike
Let us fix a period t and a feasible process (x, y) ∈ Z(t)\{0}. A non-negative number α(x, y) = max{α| αx  y} is called the technological efficiency rate of the process (x, y) in period t. If conditions (G1)-(G5) are satisfied, the function α(•) is positively homogenous of degree zero on Z(t)\{0}. Moreover, there exists ( ) which is called the optimal efficiency rate in the non-stationary Gale economy in period t and, by (G5): The process ( ) = ≥ that satisfies condition (1) is called an optimal production process in period t. Let us observe that any positive multiple of an optimal process ( ) = ≥ is also an optimal process: 6 Economics and Business Review, Vol. 5 (19), No. 2, 2019 To exclude the unrealistic case of zero optimal technological production efficiency in any period t it is assumed that: (in the initial period t = 0 the economy has access to technology which enables any good i = 1, 2, …, n to be produced). With this assumption α M, 0 > 0 due to (2), ensures that ∀ t ≥ 0 (α M, t > 0). By Z opt the set of all optimal production processes in t is denoted: The sets Z opt (t), t = 0, 1, …, are convex cones in w R + 2n not containing 0. 8 If x α x Z t y α y Z t .

The vector
y t s t y t is said to characterize the production structure in an optimal process ( )∈ ( ), ( ) ( ) opt x t y t Z t . 9 By S(t) the set of production structures in all optimal processes in period t is denoted: Assuming (G1)-(G6) sets S(t), t = 0, 1, …, are non-empty, compact and convex. 11 Let us now consider any period t and an optimal production structure s = s(t) ∈ S(t). The ray is called a single-lane production turnpike (von Neumann ray) in the non-stationary Gale economy starting from period t. The set 8 The proof is as in Panek (2016, Th. 1). 9 Here and on, if a ∈ R + n , then 10 Equivalently, Panek, Optimal growth processes in a non-stationary Gale economy is called a multi-lane production turnpike in the non-stationary Gale economy starting from period t. Observe that if y ∈  t and λ > 0, then λy ∈  t . If y 1 , y 2 ∈  t , so = ∈ s ∈ S(t), and y = y 1 + y 2 ∈  t . Hence each multi-lane turnpike  t is a convex cone in R + n not containing 0.  Lemma 1. If in the non-stationary Gale economy satisfying conditions (G1)-(G6), in some period t, the input structure x x or production structure y y in a process (x, y) ∈ Z(t)\{0} differ from the turnpike structure, then the technological efficiency of such process is less than the optimal efficiency rate: , so the technological efficiency of such a process is less than the optimal efficiency rate.

Von Neumann temporary equilibrium
Suppose p(t) = (p 1 (t), …, p n (t)) ≥ 0 is price vector of goods in economy in period t and (x(t), y(t)) ∈ Z(t)\{0}. The non-negative number 12 12 Here and on: if a, b ∈ R n , then Economics and Business Review, Vol. 5 (19), No. 2, 2019 ( ) is called the economic efficiency rate of the process (x(t), y(t)) (at prices p(t)). If a price vector ( ) 0 p t ≥ and a production process and then the triplet is said to be (characterize) temporary von Neumann equilibrium in the non-stationary Gale economy. The vector ( ) p t is called a temporary von Neumann equilibrium price vector in period t.
In the temporary von Neumann equilibrium (in period t) the economy attains not only the maximal technological efficiency rate ( ) but also the maximal economic efficiency rate ( ) ( ), ( ), ( ) β x y p t t t , which equals the technological efficiency rate α M, t 13 . The following condition (G7) together with (G1) (G6), ensures the existence of temporary von Neumann equilibrium in the non-stationary Gale economy in each period t: where ( ) p t satisfies condition (4). Equivalently: The condition states that in the Gale economy any process not attaining the maximal technological efficiency rate cannot attain the maximal economic efficiency rate.  Theorem 1. Under conditions (G1)-(G6) for each t = 0, 1, 2, …, there are prices ( ) p t ≥ 0 which satisfy (4). Moreover, if condition (G7) is fulfilled, then the triplet x t y t Z t , is a temporary von Neumann equilibrium (satisfies conditions (3)-(5)). The optimal x t y t Z t as well as temporary equilibrium prices ( ) p t (t = 0, 1, …) are defined up to the structure.
Proof is the same the proof of Theorem 1 in Panek (2018) 14 . 

Feasible and stationary growth processes
Let us fix a set of time periods T = {0, 1, …, t 1 }, t 1 < +∞. Named a horizon (of the economy). It is traditionally assumed that the economy is closed in the sense that the only source of inputs in the period t + 1 is the production (output) from the previous period t: which due to (G3) leads to condition: (y(t), y(t + 1)) ∈ Z(t + 1), t = 0, 1, …, t 1 -1.
Let y 0 represent the production vector in period t = 0: Every sequence of production vectors { } 1 0 ( ) t t y t = satisfying conditions (6), (7) is called a (feasible) (y 0 , t 1 ) -growth process (production trajectory) in the non-stationary Gale economy. The assumptions in this paper imply that such processes exist 0 The interest is in the economy in which each output vector ( ) y t of an optimal production process ( ) 14 After substituting ( , ) opt y , α M with α M, t and ( ) p t with ( ) p t .

t s t S t y t s t y t y t Z t y t
Such a sequence is characterized by invariant production structure, therefore it can be stated that it is a ( ( ) y t, t 1 ) -stationary growth process (with constant production structure) at variable rate α M, t , t = ( ) y t + 1, …, t 1 . Each positive multiple of ( ( ) y t, t 1 ) -stationary growth process (9) and the sum two such processes is also ( ( ) y t, t 1 ) -a stationary process. From (G8) S(t) ⊆ S(t + 1), i.e.  t ⊆  t + 1 , t = 0, 1, …, t 1 -1. 15 is also obtained. The set (bundle of turnpikes) y t of (9) satisfies the following condition: so throughout all periods (starting from ( ) y t = 0) it belongs to the multi-lane turnpike y t meets the following condition:

Optimal growth processes. "Weak" turnpike effect
Suppose u: R n + → R 1 + denotes the utility function defined on production vectors in the last period of horizon T and fulfills the following conditions: (G9) (i) u(•) is continuous, positively homogeneous of degree 1, concave and increasing, (2i) ( ) > . 16 Condition (i) is standard, (2i) states that irrespective of the length of horizon T the utility function may be approximated from above by a linear form with vector of coefficients a 1 ( ) p t proportional to a von Neumann price vector in the final period t 1 of horizon T.
Let us consider the following final state optimization problem (utility maximization of production obtained in the last period t 1 in horizon T): max u (y (t 1 )) subject to (6), (7) (with fixed y 0 ).
Its solution is called a (y 0 , t 1 , u) -optimal growth process (production trajectory) and denoted as { } 1 0 *( ) t t y t = . Under the above assumptions there exists a solution 0 1 0 y t ∀ ≥ ∀ < +∞ (Panek, 2003, ch. 5, Th. 5.7). 17 While proving turnpike theorems (Theorem 2 and 3) a significant role is played by the following lemma, which is a version Radner's Lemma (1961) adapted to the specific character of the model of a non-stationary Gale economy.
is the (angular) distance of a vector x from the multi-lane turnpike Proof. 18 If a process (x, y) ∈ Z(t) \ {0} fulfills the hypothesis then each process λ(x, y) with a λ > 0 fulfills it as well. Therefore while proving the lemma only consider feasible processes (x, y) from the set will be considered.
The distance (11) can be expressed equivalently as: Let us fix any t ∈ T and ε > 0. We shall demonstrate that the set V ε (t) is compact (bounded and closed in R 2n ).
(Closedness) Suppose (x i , y i ) ∈ V ε (t), i = 1, 2, …, and ( , ) ( , ) 18 The proof is based on the proof of Theorem 5 in Panek (2016). 13 E. Panek, Optimal growth processes in a non-stationary Gale economy The set S(t 1 ) is compact, so: . The set V ε (t) is thus closed in R 2n + . Condition (G5) entails in particular that V ε (t) ⊆ V ε (t + 1), t = 0, 1, …, t 1 -1. Then condition (G7) (due to Lemma 1) leads to ( ) and therefore: is continuous on a compact set V ε (t) (as a quotient of two linear functions with a non-zero function in denominator), so, according to the Weierstrass Theorem, a solution to the problem exists.
The inclusion V ε (t) ⊆ V ε (t + 1) results in The lemma does not exclude a highly unrealistic case when for some ε > 0 1 1 , lim 0 ε t t δ = , that is when the economic efficiency of a production process over time converges to the maximum rate, even though the input structure in the process permanently differs from the optimal (turnpike) structure by ε. To eliminate such a situation it is assumed that According to Lemma 2, , that is, ν ε ∈ (0, 1). In previous papers devoted to the turnpike effect in non-stationary Gale economies it was emphasized how important regular technological development for a stable economic growth is, which in Neumann-Gale-Leontief 's models of economic dynamics is expressed by means of technological production efficiency of economy and is reflected by the attained growth rate, see (Panek, 2015b). Now, along with the assumption of technology development (G5), which enables an increase in the technological efficiency rate, the following condition which eliminates rapid changes and fluctuations in technological production efficiency is also assumed and as a result, economic growth rate: To explain meaning of the condition let us denote  (2)), the sequence {Γ t } ∞ t = 1 is non-increasing and bounded: thus it has a limit c ≥ 0. (G11) requires c ≥ ρ > 0. It will be demonstrated that with this condition the (non-decreasing) sequence The last condition that is needed while proving the 'weak' multi-lane turnpike theorem in a Gale non-stationary economy simply says that there is at least one feasible growth process leading to the multi-lane turnpike: (G12) There is a (y 0 , ( ) The case lim t α M, t = +∞ is unrealistic. From the formal point of view, in the simplest onegood case economy with production spaces Z(t) ⊆ Z(t + 1) ⊂ R + 2 , t = 0, 1, …, it implies that the closure cl does not exceed k ε . The number k ε is independent of the horizon T length. Proof. As ( ) which, after considering (G9), leads to the following upper limit of production efficiency in the last period of horizon T, where a is a positive number. By (G12), a (y 0 , ( ) is a (y 0 , t 1 ) -feasible growth process. Moreover, From (17) By (G10), (G11) . Thus, since 0 < 1 -ν ε < 1, we obtain ln ln(1 ) It is enough to assume that k ε is the smallest positive integer greater than max{01 A}.  If it is assumed that in the (y 0 , t 1 , u) -optimal growth process { } 1 0 *( ) t t y t = the initial output vector y 0 is positive, then condition (G12) in Theorem 2 is redundant. 21 21 It is easy to demonstrate that in an economy with a positive initial production vector there a (y 0 , ( ) y t) -feasible growth process of the form (18) exists leading to a turnpike already in the first period (for ( ) y t = 1).

One special case
While proving the 'weak' multi-lane turnpike theorem the assumption of the existence of a feasible path from the initial y 0 to the multi-lane turnpike