Harutaka Takahashi

Department of Economics
Meiji Gakuin University
1-2-37 Shirokanedai,
Minato-ku,Tokyo
108-8636, Japan
Electric Mail:
haru@eco.meijigakuin.ac.jp
Telephone:
+81 3 5421-5641 ( Office )
+81 3 5421 5207 ( FAX )

The turnpike theory belongs to a research field of capital theory and is to investigate the property such that the efficient growth path would be in the vicinity of a balanced growth path (called a “turnpike”) for most of the planning periods. Suppose that you are planning to drive your car from Tokyo to Osaka to visit a friend of yours and want to minimize your driving time. Then the turnpike theory suggests you to take the following route: get on a nearest express highway as quickly as possible and keep driving on the expressway until you reach to a closest exit to your friend’s house in Osaka. In economic planning problems, the balanced growth path with a greatest growth rate is a kind of an express highway called the “turnpike.” But make sure that in the planning problem, an economic growth path need not get on the balanced growth path itself, but just stay in the vicinity of it for a most of the planning period. So Samuelson called this property a “turnpike theory”. Dorfman, Samuelson and Solow (in short DOSSO) putted this in their famous book as follows:

‘It is exactly like a turnpike paralleled by a network of minor roads. …., if origin and destination are far enough part, it will always to get on to the turnpike and cover distance at the best rate of travel, even if this means adding a little mileage at either end.’(Linear Programming and Economic Analysis (1958; p.331)).

2.A history of Turnpikes

The turnpike theory is originated in two famous papers. One is the paper by John von Neumann (1937) titled “A Model of General Economic Equilibrium”, and the other is the one by Frank Ramsey (1928) titled “A Mathematical Theory of Saving”. Later the turnpike property has been demonstrated in either of these models. McKenzie referred to them as the “Samuelson turnpike” and the “Ramsey turnpike” respectively. We will explain both turnpikes respectively. The reader who wants to quickly learn a brief history of the turnpike theory should consult an excellent survey article by McKenzie (1986).

2.1 The Samuelson Turnpike

According to McKenzie (1986), Samuelson is the first person who conjectured the existence of a turnpike in a von Neumann model. So he calls the turnpike in the von Neumann model as the “Samuelson turnpike.”

2.1.1 The von Neumann Model

The Samuelson turnpike is rooted in the growth model studied by von Neumann (1937). Let us start from introducing von Neumann’s model. First of all, define the following important notation:

• $${a_{ij}}$$ : the amount of the $$i$$th commodity needed as an input in a one-unit operation of the $$j$$th activity.
• $${b_{ij}}$$ : the amount of the $$i$$th commodity produced in a one-unit operation of the $$j$$th process.

Suppose there are $$n$$ commodities and $$m$$ processes in the economy. Then both processes will be expressed by the n-dimensional column vectors as $${a^j}\;and\;{b^j}\;(j = 1, \cdots ,m)$$ and they can also be expressed by the $$n \times m$$ matrices: $${\bf{A}} = \left[ {{a^1}, \cdots ,{a^m}} \right]\;and$$ ,$${\bf{B}} = \left[ {{b^1}, \cdots ,{b^m}} \right]$$. Let $${z_j}(t) \ge 0$$ be the level of operation of the $$j$$th process in period t. We will define the input-output vector $$({\bf{x}},{\bf{y}})$$ as follows:

$${\bf{x}}(t) = \sum\limits_{j = 1}^m {{a^j}} {z_j}(t)\;and\;{\bf{y}}(t) = \sum\limits_{j = 1}^m {{b^j}} {z_j}(t).$$

For our convenience, we often delete the time index t and the input-output vector will be denoted by $$({\bf{x}},{\bf{y}})$$. The von Neumann technology set $${\bf{D}}$$ is defined as follows:

$${\bf{D}} \equiv \left\{ {({\bf{x}},{\bf{y}}):{\bf{x}} \ge {\bf{Az}},{\bf{0}} \le {\bf{y}} \le {\bf{Bz}}\;for\;some\;{\bf{z}} \ge {\bf{0}}} \right\}.$$

The technology set $${\bf{D}}$$ which satisfied the following assumptions:

(A1) $${\bf{D}}$$ is a closed convex cone in $$\mathbb{R}_ + ^{2n}$$; the nonnegative quadrant of $${\mathbb{R}^{2n}}.$$

(A2) (Free disposal) $$({\bf{x}},{\bf{y}}) \in {\bf{D}},$$ $${{\bf{x}}^\prime } \ge {\bf{x}},\;and\;{\bf{0}} \le {{\bf{y}}^\prime } \le {\bf{y}}$$ implies $$({{\bf{x}}^\prime },{{\bf{y}}^\prime }) \in {\bf{D}}.$$

(A3) (The impossibility of the land of Cockaigne) $$({\bf{0}},{\bf{y}}) \in {\bf{D}}$$ implies $${\bf{y}} = {\bf{0}}.$$

(A4) (The productiveness) For any $$i,$$ there exists an $$({\bf{x}},{\bf{y}}) \in {\bf{D}}$$ such that $${y_i} > 0.$$

The economy transforms the stock of commodities at the beginning of period $${\bf{x}}(t)$$ to the stock of the commodities at the end of the period $${\bf{y}}(t)$$, such that $$({\bf{x}}(t),{\bf{y}}(t)) \in {\bf{D}}$$ and $${\bf{y}}(t) = {\bf{x}}(t + 1).$$  This type of economy is often called a “closed economy,” since all the goods are reproduced. The balanced growth paths are the paths of $$({\bf{x}}(t),{\bf{y}}(t)) \in {\bf{D}}\;and\;{\bf{y}}(t) = \alpha {\bf{x}}(t)\;for\;some\;\alpha \ge 0.$$ Note that a growth rate could be less than 1. The following theorem is first proved by von Neumann (1957) and later many economists tried to simplify his proof.

Theorem: Under the Assumptions (A1) to (A4), there exists a unique technology $$(\widehat {\bf{x}},\widehat {\bf{y}}) \in {\bf{D}}$$ such that $$\widehat {\bf{y}} = \hat \alpha \widehat {\bf{x}}$$ where $$\hat \alpha$$ is a maximal balanced growth rate.

This balanced growth path is often called the von Neumann balanced growth path or just the von Neumann ray.

2.1.2 The DOSSO Turnpike

Dorfman, Samuelson and Solow (1958), DOSSO in short, has set up a two capital good model with the following transformation function:$${K_2}(t + 1) = F\left[ {{K_1}(t + 1),{K_1}(t),{K_2}(t)} \right].$$ Making proper assumptions on the transformation function $$F( \cdot )$$ it is not difficult to define the technology set $${\bf{D}}$$ and show that (A1) to (A4) are also satisfied. As an additional assumption, they assume that $$F$$ is homogeneous of degree one. Then we may rewrite the transformation function as:

$$\frac{{{K_2}(t + 1)}}{{{K_1}(t + 1)}}\frac{{{K_1}(t + 1)}}{{{K_1}(t)}} = F\left( {\frac{{{K_1}(t + 1)}}{{{K_1}(t)}},1,\frac{{{K_2}(t)}}{{{K_1}(t)}}} \right)\;$$

$$k\lambda = F(\lambda ,1,k),\;where\;k = \frac{{{K_2}(t)}}{{{K_1}(t)}}\;and\;\lambda = \frac{{{K_1}(t + 1)}}{{{K_1}(t)}}.$$(1.1)

Considering the intertemporally efficient condition, we derive the following relation, which is referred to as the “Euler equation”;

$$\begin{array}{l}{F_2}\left[ {{K_1}(t + 1),{K_1}(t),{K_2}(t)} \right] + \\{F_3}\left[ {{K_1}(t + 1),{K_1}(t),{K_2}(t)} \right]{F_1}\left[ {{K_1}(t),{K_1}(t - 1),{K_2}(t - 1)} \right] = 0\;for\;all\;t.\end{array}$$(1.2)

Note that the von Neumann balanced growth path $$\widehat {\bf{x}} = \left( {{{\hat K}_1},{{\hat K}_2}} \right)$$ with the maximal growth rate satisfies Eq.(1.1) and Eq.(1.2) simultaneously. DOSSO has linearized these non-linear differential equations around the von Neumann solution $$\hat k = {\hat K_2}/{\hat K_1}$$ and the growth rate $$\hat \alpha$$ yields a two-dimensional linear difference equation system. Then by showing that the von Neumann solution point $$(\hat k,\hat \alpha )$$ will satisfy saddle-point stability, they tried to prove the turnpike theorem, which is often called the “catenary turnpike theorem.” However, through a Japanese translation process of DOSSO’s book, Yasui, Fukuoka, Watanabe and Koyama (1959) have found that a crucial term is missing from one of the difference equations and because of this missing term, we cannot decide the dynamical motion. In other words, DOSSO has failed demonstrating the Samuelson turnpike. It should be emphasized that their analysis has a serious defect. Since their proof depends on the local theory, it does not hold for paths starting from any distant point. McKenzie (1963) has generalize the original DOSSO model into the model with n goods and the homegenious first degree transformation function. The second property implies that there is a flat segment on the surface of the transformation function, called a "facet".

Radner (1961) has shown the Samuelson turnpike theorem successfully. He assumes (A1)’ instead of (A1).

(A1)’ D is a closed cone in $$\mathbb{R}_ + ^{2n}.$$

Furthermore he introduced a utility function $$U$$ to evaluate the final state which satisfies the following assumptions:
(A.5) The function U(x) is nonnegative and continuous and there exists an $$\bar x > 0,\;\bar x \in {\mathbb{R}^n},\;such\;that\;U\left( {\bar x} \right) > 0.$$

(A6) The function U(x) is homogeneous of degree one. If the planning period is N and the final state is denoted by $${x_N}$$, then the optimal path is the one that maximize $$U({x_N}).$$ So a path is called “U-optimal.” The distance between two vectors $$x\;and\;y$$ will be measure by the following measure:

$$d(x,y) = \left\| {\frac{x}{{\left\| x \right\|}} - \frac{y}{{\left\| y \right\|}}} \right\|\;where\;\left\| \bullet \right\|$$ is the Euclidean norm.

The following lemma is very important to show the turnpike.

Radner’s Lemma: Suppose that (A1)’, (A2), (A3) and (A4) hold, and let $$\left( {\hat x,p,\hat \alpha } \right)$$ be a von Neumann path, a von Neumann price and a von Neumann growth rate, often called a “von Neumann triplet.” Then for any $$\varepsilon > 0$$, there exists a $$\delta ,\;0 < \delta < \widehat {\alpha ,}$$ such that $$(x,y) \in D\;and\;d(x,\hat x) \ge \varepsilon \;implies\;py \le (\hat \alpha - \delta )px.$$

Radner’s Lemma asserts that whenever the distance from the process $$(x,y) \in D$$) to a given von Neumann ray exceeds some number $$\varepsilon$$, then the value of the output falls short of $$\lambda$$times the value of the input for such a proces $$(x,y) \in D$$, by some proportion $$\delta$$, as long as $$px > 0.$$ In other words, there is a certain “value loss”  associated with such a process. Now it is straightforward to demonstrate the turnpike theorem. Suppose that a feasible program $$\left\{ {x(t)} \right\}_{t = 0}^N$$ deviates from a given von Neumann ray in many periods. Then the sum of the values lost will be excessive. By linking the value loss to the optimality criterion, we would be able to obtain a turnpike theorem. McKenzie call this method of proof as the “value loss approach” and has extended and applied to his turnpike theorems.

(A7) An initial commodity vector $${\overline x _0}$$ is given such that there exists real number $$k > 0\;such\;that\;\left( {{{\bar x}_0},k\bar x} \right) \in D.$$

(A8) There exists a real number $$a > 0\;such\;that\;U(x) \le a(px)$$ for all commodity vectors $$x.$$

(A9)  $$U\left( {\bar x} \right) > 0.$$

Radner’s Turnpike Theorem: Let $$(\hat x,p,\hat \alpha )$$ be a von Neumann triplet. Suppose that assumptions (a1) to (A9) hold. Let $$\left\{ {\bar x(t)} \right\}_{t = 0}^\infty$$ be an optimal feasible path. Then, for any $$\varepsilon > 0$$ there is number $$\overline {N{\kern 1pt} }$$ such that the number of periods in which $$d\left( {x(t),\hat x} \right) \ge \varepsilon$$ cannot exceed $$\overline {N.}$$

Note that one of the difficulties in Radner’s turnpike theorem is that it does not preclude the possibility that an optimal path may run out of the neighborhood of the von Neumann ray around the halfway point of the entire programming period. In other words, the optimal path may enter and leave the neighborhood several times. Scheinkman (1976) prove a similar turnpike theorem in a Ramsey model and called it the “visit lemma.” Following him, Radner’s turnpike may be referred to as a “visit turnpike theorem.”

3.Generalizations

0. Introduction

The theoretical perspective based on the external effects and endogenous growth of increasing returns to scale technologies stressed by Romer (1986) and Lucas (1988) has in the past decade had a considerable influence on macroeconomics as well as growth theory and development theory, and a massive thesis has been produced pertaining to these matters. Aghion and Howitt (1998) has compactly surveied these results. Though this sort of theoretical analysis is in vogue, most of the empirical research supporting these theories used aggregated macro data. Due to restrictions on data, little research has been conducted at the industrial level. Among these researchers, Hall (1988,1990) measured the economy of scales of American industries. These measurements indicated the existence of a considerably large scale economy and externality of production, providing support for the theories of Romer et al. With respect to these results, Bartelsman (1995) indicated an ommited variable bias of these measurements and concluded that the large-scale economy shown in Hall’s results was invalid. Also, the results of a series of investigations conducted by Basu and Fernald (1995,1997) showed that constant returns were the rule for the economy of scale in most American industries, with almost no externality of production among industries. The thing that can be understood from this empirical research is that when aggregated data is used, economy of scale can become conspicuous because of aggregation. This suggests that there are serious problems with Solow-type exogenous growth models and Romer-type endogenous growth models, which take only one sector into consideration as one type of capital good or representative industrial sector. From the perspective of analytical technique, it will be necessary to rely on a phase-diagram-based analysis to conduct a global analysis. This means that one must restrict the state variables that indicate the capital stock to two variables or less.

The purpose of this text is to, based on the results of the above empirical analysis, use constant returns to scale technology to construct an exoginous optimal growth model and to investigate the global nature of that optimal path (equilibrium path) from the view of multi -capital-goods economy. Also, these results will be applied to the global analysis of a multi-sector endogenous growth model. Ever since the 1970s, optimimal growth models that include multi capital goods have been reported in research results as consumption turnpike theories. The study of McKenzie (1984,1990,1998) can be cited as a good compilation of these results. However, these results are unrelated to structural models built from production functions and utility functions; instead, they pertain to reduced form models derived from the structural models. This text aims to apply the consumption turnpike research results of a reduced form model to multi-sector neoclassical optimal growth, and to explain as concisely as possible results and application of the study of Takahashi (1985, 1992, 1999, 2008), in which he conducted a global qualitative analysis of the optimal path. At first glance, this global qualitative analysis of the multi-sector model’s optimal path seems overly complicated and unmanageable. However, as explained below, provided the production technology of each sector is constant returns to scale technology, it is possible to restrict the global analysis to a local analysis. Also, by analyzing only the linear system, it is possible to analyze the local behavior of the optimal path. Putting this in greater detail, the plane from the non-substitution theorem described later, known as a “von Neumann-McKenzie facet,” which includes the optimal steady state path as an interior point, exists in the objective function surface of the reduced form model. By analyzing these planar dynamics, it is possible to simultaneously clarify the global dynamics of the optimal path as well as the local dynamics. An important point is that by analyzing only the linear system included in the non-linear system, it is possible to analyze the dynamics of the non-linear system itself.

In Section 1, we explain the neoclassical optimal growth model, which includes multi capital goods, and is derived from neoclassical production functions; the transformations to the reduced model are also explained. Section 2 pertains to the explanation of the methods for proving the consumption turnpike theorem demonstrated by Scheinkman (1976) and McKenzie (1983). Also, the case in which the essentials of the von Neumann-McKenzie facet, which plays an important role in the next part, became a two-sector model and is explained using figures. In Section 3, we postulate a two-sector neoclassical optimal growth model, and the optimal path behavior in the vicinity of the optimal steady state path (modified golden rule path) are classified using the characteristics of von Neumann-McKenzie facet. Also, we will use these results to prove, based on a weaker hypothesis, that the theorem that the optimal path local stability and the optimal path attained by Benhabib and Nishimura（1985）becomes a two-term periodic solution. In Section 4, the generalization of the global asymptotic stability conclusion achieved with two divisions into a case that includes two or more types of capital goods. In Addendum, the important fundamental principles used in the main text will be defined, and a number of theorems will be proved.

1. McKenzie-Scheinkman Turnpike Theory

Scheinkman (1976) established the asymptotic stability of an optimal path at a stationary optimal path in two-steps. The first step is to shaow that an optimal path must enter at least once, into any neighborhood of the stationary optimal path (the “visit lemma”). Then he demonstrates, as a second step, local stability of the optimal path, which is acomplished in two additional steps: Firstly, he demonstarates, by applying the stable manifold theorem of Hirsh and Pugh (1970) and taking into conasideration the result of Levhari and Liviatan (1972), that a locally stable manifold can be constructed around the stationary optimal path when the discount rate is sufficiently close to one. Secondly, given any capital stock near the stationary optimal path, it is possible to find the capital stock of the next period such that it belongs to the stable manifold.

McKenzie (1982,1984; Section 8) has established a stronger theorem which staes that an optimal path is eventually trapped in any neighborhood of a stationary optimal path (the “neighborhood turnpike theorem”), this being obtained without the differentiability assumption. Then as an analogous second step to Scheinkman’s, McKenzie (1984; Section 8) proves the local stability of the optimal path at the stationaly optimal path by using differentiability. Both local theorem crucially depend upon the locally strict concavity assumption of the “reduced form utility function” which is expressed by initial and terminal capital stocks.

McKenzie (1983) has extended the neighbourhood turnpike theorem and the global asymptotic stability stability to the case where the objective function is not strictly concave. In so doing, to obtain the global asymptotic stability, he makes the following two basic assumptions: (i) there are no cyclic paths on the Neumann-McKenzie facet, which we will define laer, when the discount rate is 1 and (ii) the second differential of the reduced form utility function is negative for any direction leading out of the facet at the stationary optimal path with the discount rate one.

The purpose of this chapter is a preparatory one for the following chapters. Most of the results achived here will be applied to the neoclassical multi-sector optimal growth model with no-joint production . For these applications, we need some more elaboration of the local stability theorem and the neighbourhood turnpike theorem.

Section 2.2 is devoted to obtaining a sufficient condition for local stability without the strict concavity assumption-this to be achieved by applying a modified version of Scheinkman’s original argument in Scheinkman (1976). We then illustrate this result by applying it to McKenzie’s example which includes a non-trivial Neumann-McKenzie facet.

In Section 2.3 we introduce the Neumann-McKenzie facet and discuss McKenzie’s neighbourhood turnpike theorem. Combining this result and local stability we obtain global asymptotic stability. All the above arguments will be performed in the model with a reduced form utility function.

1.1 Reduced-form model and local stability

In this and the next section we will mainly treat a reduced-form utility function expressed by an initial stock and a terminal stock.
Let us consider the following standard problem:

$$\left\{ \begin{array}{l}\mathop {\max }\limits_{\{ {\bf{k}}(t)\} } \;\sum\limits_{t = 0}^\infty {{\rho ^t}V({\bf{k}}(t),{\bf{k}}(t + 1))\;\quad (0 < \rho < 1)} \\s.t.\\({\bf{k}}(t),{\bf{k}}(t + 1)) \in {\bf{D}}\;and\;{\bf{k}}(0) = \overline {\bf{k}} \end{array} \right.$$

where $${\rm{V}}:{\rm{D}} \subset {\rm{R2n}} + \to {\rm{R}}$$ and $${\bf{k}}(0)$$ is a given initial stock vector.

The optimality of a solution $$\left\{ {{{\bf{k}}^\rho }(t)} \right\}_{t = 0}^\infty$$ of the above problem is defined as follows:
Definition. Optimality is defined by the “catching up” criterion. A path $$\left\{ {{\bf{k}}(t)} \right\}_{t = 0}^\infty$$ catches up to a path $$\left\{ {{{\bf{k}}^\prime }(t)} \right\}_{t = 0}^\infty$$ starting at the same time if

$\mathop {\lim }\limits_{T \to \infty } \;\inf \;\left[ {\sum\limits_{t = 0}^T {{\rho ^t}V({{\bf{k}}^\prime }(t),{{\bf{k}}^\prime }(t + 1)) - \sum\limits_{t = 0}^T {{\rho ^t}V({\bf{k}}(t),{\bf{k}}(t + 1))} } } \right] \ge 0$

A path $$\left\{ {{{\bf{k}}^\rho }(t)} \right\}_{t = 0}^\infty$$ is optimal if it catches up to every alternative path from the same initial stocks.[1]

The following basic assumptions are in orger now:

Assumption 1.  $${\bf{D}}$$ is convex and $$V$$ is concave and closed.[2]

Assumption 2. If $$({\bf{x}},{\bf{y}}) \in {\bf{D}}$$ and $$\left| {\bf{x}} \right| < \xi < \infty$$ there is $$0 < \eta < \infty$$ such that $$\left| {\bf{y}} \right| < \eta ,$$ where $$\left| \bullet \right|$$ is the Euclidean norm.

Assumption 3.  There is $$\varsigma > 0$$ such that $$\left| {\bf{x}} \right| \ge \varsigma$$ implies for any $$({\bf{x}},{\bf{y}}) \in {\bf{D}}$$, $$\left| {\bf{y}} \right| < \lambda \left| {\bf{x}} \right|,\;where\;0 < \lambda < 1.$$

Remark. Assumptions 2 and 3 imply $$({\bf{x}},{\bf{y}}) \in {\bf{D}}$$ is bounded. This will be shown by induction. Without loss of generality, we can assume $$\left| {{\bf{k}}({\bf{0}})} \right| = \left| {\overline {\bf{k}} } \right| < \varsigma$$ where $$\varsigma$$ is the same number used in Assumption 3. By Assumption 2, there is $$0 < \eta < \infty$$ such that $$\left| {{\bf{k}}(1)} \right| < \eta$$. Suppose $$\left| {{\bf{k}}(t)} \right| < \max \;(\varsigma ,\eta )$$. If $$\left| {{\bf{k}}(t)} \right| < \varsigma$$, then $$\left| {{\bf{k}}(t + 1)} \right| < \eta$$ by Assumption 2 again. If $$\varsigma < \left| {{\bf{k}}(t)} \right| < \eta$$, then by Assumption 3, $$\left| {{\bf{k}}(t + 1)} \right| < \left| {{\bf{k}}(t)} \right|$$. Hence $$\left| {{\bf{k}}(t + 1)} \right| < \max \;(\varsigma ,\eta )$$.

Asusumption 4.  If $$({\bf{x}},{\bf{y}}) \in {\bf{D}}$$, then $$({\bf{z}},{\bf{w}}) \in {\bf{D}}$$ for all $${\bf{z}} \ge {\bf{x}}\;and\;{\bf{0}} \le {\bf{w}} \le {\bf{y}}.$$

Moreover $$V({\bf{z}},{\bf{w}}) \ge V({\bf{x}},{\bf{y}}).$$

Assumption  5. There is $$(\overline {\bf{x}} ,\overline {\bf{y}} ) \in {\bf{D}}$$ such that $$\rho \overline {\bf{y}} > \overline {\bf{x}} .$$

To introduce Lemma 1 and Assumption 6, we need to define the notion of a sufficient capital stock and a stationary optimality.

Definition. A capital stock $${\bf{x}}$$ is called sufficient if there is a finite sequence

${\bf{k}}(0), \cdots ,{\bf{k}}(T)\;where\;{\bf{k}}(0) = {\bf{x}},\;({\bf{k}}(t - 1),{\bf{k}}(t)) \in {\bf{D}}\;for\;t = 1, \cdots ,T\;and\;{\bf{k}}(T)\;is\;\exp ansible.$

${\bf{k}}(T)\;is\;\exp ansible\;if\;there\;is\;{\bf{k}}(T + 1)\;such\;that\;{\bf{k}}(T + 1) > {\bf{k}}(T)\;and\;({\bf{k}}(T),{\bf{k}}(T + 1)) \in {\bf{D}}.$

Definition. $${{\bf{k}}^\rho }$$ is a stationary optimal path (S.O.P.) if $$({{\bf{k}}^\rho },{{\bf{k}}^\rho })$$ is an optimal solution to

$$\left\{ \begin{array}{l}\mathop {\max }\limits_{\{ {\bf{k}}(t)\} } \;\sum\limits_{t = 0}^\infty {{\rho ^t}V({\bf{k}}(t),{\bf{k}}(t + 1))\;\quad (0 < \rho < 1)} \\s.t.\\({\bf{k}}(t),{\bf{k}}(t + 1)) \in {\bf{D}}\;and\;{\bf{k}}(0) = {{\bf{k}}^\rho }\end{array} \right.$$

The following existence theorem has been proved by McKenzie (1983,1984);

Theorem (McKenzie). Under Assumptions 1 through 5, there is a stationary optimal path $${{\bf{k}}^\rho }$$, supported by price vectors $${\bf{p}}(t) = {\rho ^t}{\bf{q}}\;(0 < \rho \le 1)$$ in the sense that

$V({{\bf{k}}^\rho },{{\bf{k}}^\rho }) + \rho {\bf{q}}{{\bf{k}}^\rho } - {\bf{q}}{{\bf{k}}^\rho } \ge V({\bf{z}},{\bf{w}}) + \rho {\bf{qw}} - {\bf{qz}}\;for\;all\;({\bf{z}},{\bf{w}}) \in {\bf{D}}.$
Proof. See Theorem 1 of McKenzie (1983) and Theorem 8 of McKenzie (1984).
We will leave the existence problem and concentrate on a local stability argument.
We need the following assumptions for this argument.

Assumption 6. There is a unique

[1] See, in detail, Section 3 of McKenzie (1984).
[2] This means that
$$({{\bf{x}}^s},{{\bf{y}}^s}) \to ({\bf{x}},{\bf{y}}) \in boundary\;of\;{\bf{D}}\;implies\;\mathop {\lim }\limits_{s \to \infty } \;\sup \;V({{\bf{x}}^s},{{\bf{y}}^s}) = V({\bf{x}},{\bf{y}})\;if\;({\bf{x}},{\bf{y}}) \in {\bf{D}},\;and\;V({{\bf{x}}^s},{{\bf{y}}^s}) \to - \infty \;otherwise.$$

1.2 Structural and reduced-form model

The structural model researched here is the model of Burmeister and Graham (1975) generalized with the model of Srinivasan (1965) as multi capital goods.

$$Maximize\sum\limits_{t = 0}^\infty {{\rho ^t}u(c(t))}$$　　$$(0 < \rho < 1)$$(1.1)

subject to $$k(0) = \bar k$$

$${y_i}(t) + {k_i}(t) - {\delta _i}{k_i}(t) - (1 + n){k_i}(t + 1) = 0\quad (i = 1,2,....,n)$$(1.2)

$${c_t}(t) = {f^0}({k_{10}}(t),{k_{20}}(t),...,{k_{n0}}(t),{\ell _0}(t))$$(1.3)

$${y_t}(t) = {f^i}({k_{1i}}(t),{k_{2i}}(t),...,{k_{ni}}(t),{\ell _i}(t))$$　　$$(i = 1,2,....,n)$$(1.4)

$$\sum\limits_{i = 0}^n {{\ell _i}(t) = 1}$$　　$$(t = 0,1,2,...)$$(1.5)

$$\sum\limits_{i = 0}^n {{k_{ij}}(t) = {k_i}(t)}$$　　$$(i = 1,2,....,n;t = 0,1,2, \cdots )$$(1.6)

The symbols have the following meanings:

$$n$$ ＝ population growth rate $$(0 \le n \le 1),$$

$$\gamma$$ ＝ subjective discount factor,

$$\rho$$ ＝$$(1 + n)/(1 + \gamma )\;\quad (n < \gamma )$$,

$$u$$ ＝ representative individual utility function,

$$c(t) \in {\mathbb{R}_ + }$$ ＝ per-capita consumption goods consumed in period $$t$$,

$${\bf{y}}(t) \in \mathbb{R}_ + ^n$$ ＝ capital good per-capita production vector in period $$t$$,

$${\bf{k}}(0) \in \mathbb{R}_ + ^n$$ = initial capital stock vector,

$${f^j}:\mathbb{R}_ + ^{n + 1} \to {\mathbb{R}^ + }$$ ＝ per-capita production function of the j sector,

$${k_{ij}}(t)$$ ＝ capital good i used in sector j in period $$t$$,

$${\ell _j}(t)$$ ＝ labor input used in sector j in period $$t$$,

$${\delta _i}$$ depreciation rate of capital good i $$(0 < {\delta _i} < 1)$$.

As indicated by Benhabib and Nishimura (1979a), it is possible[3]to consolidate equations (1.3) through (1.6) here as social transformation function $$c(t) = T({\bf{y}}(t),{\bf{k}}(t))$$. Also, the social transformation function becomes a concave and continuously differentiable function under our assumptions. The important thing to keep in mind is the fact that the transformation function does not become strictly concave. As we will discuss later, the existence of the Von Neumann-McKenzie facet is closely related to this fact

Assumption 1.1 1) All the goods are produced non-jointly with the per-capita production functions $${f^i}()\;(i = 0,1, \cdots ,n)$$ which are defined on$${\mathbb{R}^{n + 1}}$$, homogeneous of degree one, strictly quasi-concave and continuously differentiable for a positive inputs. 2) Any goods $$j\;(j = 0,1, \cdots ,n)$$ cannot be produced unless $${k_{ij}} > 0\;$$ for some $$i = 1, \cdots ,n$$. 3) Labor must be used directly in each sector. If labor input of some sector is zero, its sector’s output is zero.

Also, we make the following assumption so that the productive structure directly appears in the utility function. Of course, it is sufficient to assume the linearity of the utility function in the vicinity of the optimal steady state path, but here we will offer a more direct assumption.

Assumption 1.2  $$u(c(t)) = c(t)$$

First of all, in order to guarantee the existence of an optimal steady state path (expressed as “OSS”  henceforce), we must make a following assumption.

Assumption 1.3[4]With respect to all positive factor price vectors, the non-negative investment matrix $$\overline {\bf{a}}$$ written below is an indecomposable matrix2, and all the elements of the row vector element $${{\bf{a}}_{\bf{0}}} = ({a_{00}},{a_{01}},...,{a_{0n}})$$ are positive.

$$\overline {\bf{a}} \equiv \left[ {\begin{array}{*{20}{c}}{{a_{11}}}&{...........}&{{a_{1n}}}\\ \vdots & \ddots & \vdots \\{{a_{n1}}}&{............}&{{a_{nn}}}\end{array}} \right]$$

where $${a_{ij}} = {k_{ij}}/{y_i}$$ and $${a_{0i}} = {l_i}/{y_i}\;(i = 0,1,....,n;j = 1,2,....,n)$$.

Assumption 1.4 When discount factor γ and depreciation matrix $$\Delta$$ are given, the selected technology matrix $${{\bf{a}}^\rho }$$ satisfies $${\left( {{\bf{I}} - ({\bf{\gamma }} + \Delta ){{\bf{a}}^\rho }} \right)^{ - 1}} \gg {\bf{0}}$$. Also, matrices $${{\bf{a}}^\rho }$$, $$\Delta$$, and $${\bf{\gamma }}$$ are defined as follows.

${{\bf{a}}^\rho } = \left[ {\begin{array}{*{20}{c}}0&0&{...}&0\\{a_{10}^\rho }&{a_{11}^\rho }&{...}&{a_{1n}^\rho }\\ \vdots & \vdots & \ddots & \vdots \\{a_{n0}^\rho }&{a_{n1}^\rho }&{...}&{a_{nn}^\rho }\end{array}} \right]$

$\Delta = \left[ {\begin{array}{*{20}{c}}0&0& \cdots &0\\0&{{\delta _1}}& \ddots & \vdots \\ \vdots & \ddots & \ddots &0\\0& \cdots &0&{{\delta _n}}\end{array}} \right]$,

and

$${\bf{\gamma }} = \left( {\begin{array}{*{20}{c}}1&0& \cdots &0\\0&\gamma &0& \vdots \\ \vdots &0& \ddots &0\\0& \cdots &0&\gamma \end{array}} \right)$$

This assumption is referred to as “Viability.” Viability is a condition that requires the possibility of producing the goods in quantities equal or greater than the input. Please refer to Burmeister and Dobell (1970) for details.

Assumption 1.5  Population growth rate $$g \ge 0$$ will satisfy $$g < 1/\lambda '$$, where $$\lambda '$$ is the maximum unique root (Frobenius root)[5]of  matrix $${\bar a^\rho }{(I - \Delta {\bar a^\rho })^{ - 1}}$$.

Assumption 1.5 implies that the population growth rate should be small enoug to enable the steady-state equilibrium path. The following more fundamental theorem is proven on the basis of these assumptions.

Lemma 1. (Non-substitution Theorem) Suppose that the production functions are differentiable and linear homogenous, and subjective discount factor  γ is given[6], the cost functions of each sector, based on Assumptions 1.1, should become linear homogenous. They are expressed as follows.

$$p_0^{}/w_0^{} = {C^0}(w_1^{}/w_0^{}, \cdots ,w_n^{}/w_0^{},1)$$

$$\frac{{w_i^{}/w_0^{}}}{{{\delta _i} + \gamma }} = {C^i}(w_1^{}/w_0^{}, \cdots ,w_n^{}/w_0^{},1)\quad (i = 1,2, \cdots ,n)$$

Also, the positive relative price vector $${{\bf{p}}^\rho }/{w_0}$$[7]and the positive relative factor price vector $${{\bf{w}}^\rho }/{w_0}$$ are unambiguously determined ( in the following discussion, each price vector has been normalized as wage rate $${w_0} = 1$$). Thus, it follows that input coeficients $$\partial {C^i}/\partial {w_j} = {a_{ij}}\;(i = 1,2, \cdots ,n;j = 1,2, \cdots ,n)$$ corresponding to these prices are also unambiguously determined.

Proof.  Refering for detailed proof to Theorem 1 of Burmeister and Dobell (1972,p.242) and Burmeister and Kuga (1970). Also, each price to be a positive value can be shown as follows: The price equation of the steady path based on the non-substitution theorem and Euler’s theorem pertaining to homogeneous functions is expressed below.

$${{\bf{p}}^\rho } = {\bf{a}}_0^\rho + {{\bf{p}}^\rho }({\bf{\gamma }} + \Delta ){{\bf{a}}^\rho }$$ or $${{\bf{p}}^\rho } = {\bf{a}}_{\bf{0}}^\rho {({\bf{I}} - ({\bf{\gamma }} + \Delta ){{\bf{a}}^\rho })^{ - 1}}$$

where, $${\bf{a}}_0^\rho = (a_{00}^\rho ,a_{01}^\rho , \cdots ,a_{0n}^\rho )$$ and $${{\bf{p}}^\rho } = (p_0^\rho /w_0^\rho ,p_1^\rho /w_0^\rho , \cdots ,p_n^\rho /w_0^\rho )$$. Since $${\bf{a}}_{\bf{0}}^\rho \gg 0$$ holds from Assumption 1.3 and also the inverse matrix from Assumption 1.4 will satisfy $${({\bf{I}} - ({\bf{\gamma }} + \Delta ){{\bf{a}}^\rho })^{ - 1}} \gg {\bf{0}}$$ price vectors become positive.

Lemma 2. When discount factor $$\gamma$$ is given, the optimal steady state vector $${{\bf{k}}^\rho }( \gg 0)$$ exists uniquely.

Proof.  This is proven in Addendum A5.

Thus, when $${\bf{x}}$$ represents initial capital stock, and $${\bf{z}}$$ represents end of period capital stock, let us define each of the following.

Definition 1.1

$$V({\bf{x}},{\bf{z}}) \equiv T[(1 + g){\bf{z}} - ({\bf{I}} - \bar \Delta ){\bf{x}},{\bf{x}}]$$

$${\bf{D}} \equiv \left\{ {({\bf{x}},{\bf{z}}) \in \mathbb{R}_ + ^n \times \mathbb{R}_ + ^n:T[(1 + g){\bf{z}} - ({\bf{I}} - \bar \Delta ){\bf{x}},{\bf{x}}] \ge 0} \right\}$$

where$$\overline \Delta = \left[ {\begin{array}{*{20}{c}}{{\delta _1}}&0& \cdots &0\\0&{{\delta _2}}& \ddots & \vdots \\ \vdots & \ddots & \ddots &0\\0& \cdots &0&{{\delta _n}}\end{array}} \right]$$

Problems (1.1)–(1.6) from the above discussion can be rewritten as the following reduced-form  problems.

$$\left\{ {\begin{array}{*{20}{l}}{Maximize\sum\limits_{t = 0}^\infty {{\rho ^t}V({\bf{k}}(t),{\bf{k}}(t + 1))} }\\{}\\{s.t.\quad ({\bf{k}}(t),{\bf{k}}(t + 1)) \in {\rm{int}}{\bf{D}}}\end{array}} \right.$$

Now, considering period $$t$$ and period $$t - 1$$, let us try to find out the necessary conditions for $${\bf{k}}(t)$$ that maximize the following 2 periods’ Lagrange function $$L$$ when $${\bf{k}}(t - 1)$$ and $${\bf{k}}(t + 1)$$ are given.

$$L({\bf{k}}(t)) = V({\bf{k}}(t - 1),{\bf{k}}(t)) + \rho V({\bf{k}}(t),{\bf{k}}(t + 1))$$

For now, let us not consider only internal solutions. It can be easily found the following equation as a necessary condition.

$${{\bf{V}}_2}({\bf{k}}(t - 1),{\bf{k}}(t)) + \rho {{\bf{V}}_1}({\bf{k}}(t),{\bf{k}}(t + 1)) = {\bf{0}}$$(1.7)

and $${{\bf{V}}_2}$$ where $${{\bf{V}}_1}$$ are vectors that are established by the first and second factor as partial derivatives respectively. With respect to the optimal path, Equation (1.7) must hold for all t. Also, Equation (1.7) is frequently referred to as an ” Euler Equation.” The stability of the optimal steady state path returns us to the research pertaining to the simultaneous difference equations expressed here as vectors.

[3] Obtained by solving $$\left\{ {Max\;c = {f^0}({k_{10}}, \cdots ,{k_{n0}},{\ell _0}),\quad s.t.\;{y_i} = {f^i}({k_{1i}}, \cdots ,{k_{ni}},{\ell _i})\;(i = 1, \cdots ,n),\sum\limits_{i = 0}^n {{\ell _i}(t) = 1} ,\sum\limits_{i = 0}^n {{k_{ij}}(t) = {k_i}(t)} \;(i = 1, \cdots ,n)} \right.$$.
[6] Hereinafter, in place of writing “subjective discount factor $$\gamma$$ was given,” we will write “$$\rho$$ is given.”
[7] From now on, a superscript $$\rho$$ will be used for optimum steady-state path variables.

2. Consumption turnpike theory

Here, we offer a simple explanation of the theories of Scheinkman (1976) and McKenzie (1983) in which he generalized Scheinkman’s study. The method they used to prove the asymptotic stability of the optimal steady state path of the optimal path ( this property is often referred to as “turnpike theorem” ) is known as a “value loss approach.” Now, let us assume that the value function V satisfies the strong concavity. Then, it is possible to draw figure 1.

The point in response to the optimal steady state path is expressed as point M in the figure. Due to the nature of the strong concavity function, the supporting hyperplane that passes through point N unambiguously exists, and a price (shadow price) vector is obtained as a vector that orthogonal to that plane. Now, let us postulate that the optimal path continues to deviate from the $$\varepsilon -$$neighaborhood of optimal steady state path. For example, let us say that this period and the next period’s capital stock existed in point P. The value losses at this time can be measured as QR, the distance between function V and the hyperplane. If the optimal path continues to deviate from the $$\varepsilon -$$neighborhood, the sum of the value losses will clearly increase, creating untimely contradictions in the definition of the optimality of the optimal path. Thus, it is proven that the optimal path must visit this neighborhood within the planning period. This value loss method has originally been used to prove the turnpike theorems by Ramsey (1928) and Atsumi (1965). However, if, as is the case in our problem, the discount factor is entered into the objective function, the value loss will also be discounted. Therefore, leading into the aforementioned contradictions without accumulating value losses becomes difficult.

Scheinkman (1976) applied the value loss method to a reduced-form model in which the objective function is strongly concave and the discount rate is positive, proving the turnpike theorem in two stages. First stage involves applying the value loss method to prove a lemma known as the “Visit Lemma.”

Visit Lemma: For a given $$\varepsilon > 0$$, threre exists a suitable $$\rho ' > 0$$ such that the optimal path must at least once enter into the $$\varepsilon -$$neighborhood of the optimal steady state path. This is also established with respect to $$\rho ' < \rho < 1$$.

In figure 2, the fact that the optimal path visits the neighborhood in a finite time is depicted as a dotted line.

In stage 2, the local stability is proven using the differentiability of function V, developing the Euler Equation in the vicinity of the optimal steady state path, and investigating that characteristic root. The local stability is proven by the fact that the optimal steady state path is shown to become saddle point stability. If linear approximation is performed on Euler Equation (1.7) in the vicinity of the optimal steady state path, and determinant $${\bf{V}}_{12}^\rho$$[8] is not zero, following equation (2.1) is established.

$${{\bf{z}}_{t + 1}} = - {({\bf{V}}_{12}^\rho )^{ - 1}}({\bf{V}}_{11}^\rho + \rho {\bf{V}}_{22}^\rho ){{\bf{z}}_t} - \rho {({\bf{V}}_{22}^\rho )^{ - 1}}{\bf{V}}_{21}^\rho {{\bf{z}}_{t - 1}}$$(2.1)

Here, all the matrices are evaluated as optimal stable state paths, and $${{\bf{z}}_t} = {{\bf{k}}_t} - {\bf{k}}_t^\rho$$.

First of all, the fact that in the vicinity of the optimal steady state path, the absolute value of the n characteristic roots of the characteristic equation of the difference equation (2.1) that linearized the Euler Equation of equation (1.7) is greater than 1 and that of the other n roots are smaller than 1, proves that the optimal steady state path will attained saddle point stability. The following lemma 2.1 serves an important role at this time.

Lemma 2.1 If $$\lambda$$ is the characteristic root of equation (2.1), $$1/\rho \lambda$$ will also be its characteristic root.

Proof: Refer to Levhari and Livitan (1972). Alternately, for a more complete theorem and proof, refer to Lemma 12 of Becker and Boyd (1997).

When the discount rate $$\rho$$ is 1, or when it is sufficiently close to 1, if the absolute values of $$n$$ characteristic roots is greater (or smaller) than 1, the absolute values of the other $$n$$ characteristic roots will become smaller (or greater) than 1. Thus, the saddle point stability in the vicinity of OSS is proven. However, in the saddle point stability proof, let us take heed of the fact that this is insufficient for proving local stability. For example, for the stable manifold displayed in the CD of figure 3, the path starting from point A will, no matter the case, be unable to intersect the stable manifold. Therefore, the local stability is not established.

This point was first proposed as a problem by McKenzie (1963), but has been neglected for a long time. Based on the condition, that determinant $${\bf{V}}_{{\bf{22}}}^\rho$$ from the second differential coefficient is zero, Scheinkman proved that no case will occur in which there exists a perpendicular stable manifold like the one displayed in the CD of figure 3. If discount rate $$\rho$$ is sufficiently close to 1, based on the visit lemma, the optimal path must at least visit the vicinity of the optimal steady state path at least once. At this time, the fact that the path in the stable manifold is the optimal path is proven by the transversality conditions and the strict concavity. Thus the optimal path must converge to the OSS. Here, the continuous variations of $$\rho$$ and the continuous variations of the stable manifold are guaranteed by the Hirch-Pugh stable manifold theorem (refer to Palis and de Melo (1982), p.75).

McKenzie (1983) has extended the results of Scheinkman (1976) to the case in which  function V has a plane on its surface that includes OSS. Here, let us examine figure 4. Presently, function V is not strictly concave.

It has become a concave function with a flat portion. Point M indicates the optimal steady state path. That, which projects the flat portion including OSS into plane $$({\bf{x}},{\bf{z}})$$, is known as the von Neumann-McKenzie facet (hereinafter, referred to as the “NMF’). This exact definition is given next.

Definition 2.1 The von Neumann-McKenzie Facet (NMF) is defined as follow:

$$F({{\bf{k}}^\rho },{{\bf{k}}^\rho }) = (({\bf{x}},{\bf{z}}) \in {\bf{D}}:[{\bf{V}}({\bf{x}},{\bf{z}}) + \rho {{\bf{p}}^\rho }{\bf{z}} - {{\bf{p}}^\rho }{\bf{x}}] = {\bf{V}}({{\bf{k}}^\rho },{{\bf{k}}^\rho }) + \rho {{\bf{p}}^\rho }{{\bf{k}}^\rho } - {{\bf{p}}^\rho }{{\bf{k}}^\rho })$$

Like Scheinkman (1976), McKenzie (1983) proved the turnpike theorem in two stages. First, based on the condition that when $$\rho = 1$$, no periodic solution exists in the NMF, he applied the value loss method and proved the following neighborhood turnpike theorem (also known as the Lyapunov stability).

Neighborhood Turnpike Theorem: The optimal path is trapped in the arbitrary $$\varepsilon -$$neighborhood of OSS, and cannot escape the vicinity. Also, if the discount rate $$\rho$$ approaches 1, $$\varepsilon$$ can approach 0 irrescpective of its value (this is expressed in general terms in the solid line path of figure 2).

These results and the proving of the local stability explained before allow the turnpike theorem to be proven by setting the value of $$\rho$$ sufficiently close to 1. The thing to keep in mind here is that differentiability assumptions are totally unnecessary to prove the visit lemma as well as the neighborhood turnpike theorem. That is, including the turnpike theorem, the differentiability is only necessary in proving the local stability. The detailed and comprehensive discussion so far can be found in McKenzie (1984). Also, in McKenzie (1990,1998), these discussions are concisely explained without proofs.

[8] V function’s second order differential is defined in the following matrix that has the second- order differential as a factor: $${\partial ^2}V(x,z)/\partial {x^2} = {V_{11}}$$, $${\partial ^2}V(x,z)/\partial x\partial z = {V_{12}}$$, $${\partial ^2}V(x,z)/\partial {z^2} = {V_{22}}$$.

3. The Neumann-McKenzie facet of two-sector optimal growth models

As described in Section 1, the results of the consumption turnpike theory based on a reduced-form model have hardly been applied to general neoclassical optimal growth theories except in the studies of Benhabib and Nisimura (1979a, 1979b), and Yano(1990). The most substantial reason for this is that when neoclassical optimal growth models (structural models) are converted to reduced-form models, based on the constant-returns-to-scale assumption pertaining to the production functions of each sector, objective function V does not become strictly concave, and the planar portion that includes the OSS exists in the surface of the function V. Because of this, as can be understood from the analysis by McKenzie (1983) of this sort of case, the proof of the turnpike theorem will become rather complex.

The first thing that should be done here is to investigate in detail the “von Neumann-McKenzie facet (NMF)” of a two-sector optimal growth model that serves as a bridge between structural models made up of the production functions and the reduced-form models. Based on the definition of NMF in Section 2, each point of the NMF is supported by the hyperplane of price vectors that are the same as those of the OSS. In such instances, the non-substitution theorem is once again established in each point of the plane. Thus, it follows that in each point of the NMF, the same technology matrix $${{\bf{a}}^\rho }$$ (this vector is defined in Assumption 2) as is chosen in the OSS is selected uniquely. Now, we shall define the factor intensity comparing the consumption goods sector with the capital goods sector.

Definition 3.1 (Capital Intensity Conditions)

When $$a_{11}^\rho /a_{01}^\rho < a_{10}^\rho /a_{00}^\rho$$ is established, the consumption goods sector is capital intensive in comparison with the capital goods sector. For $$a_{11}^\rho /a_{01}^\rho > a_{10}^\rho /a_{00}^\rho$$, the consemption goods sector is labor intensive in comparison with the capital goods sector.

Based on the assumption that the consumption goods sector is capital intensive, it is possible to draw a graph on the coordinates $$(y(t),c(t))$$, which is often used in trade theory.

Here, $$({y^\rho },{c^\rho })$$ is a production vector corresponding to the OSS and is written as a point of intersection for the labor-constraint line and the capital-constraint line. Also, the fact that the labor-constraint line intersects the capital-constraint line from the above is due to the capital intensity assumed above. Note that production specialization occurs at points A and B.

Now, suppose that $$\bar k(t){\mkern 1mu} ( < 1/a_{10}^\rho )$$, which is greater than OSS, $$\{ {k^\rho }\}$$ were given. Then, if we leave the capital-constraint line and the price vector as they are, then move upward along the labor-constraint line, a new point of intersection “E” is obtained. Also, the corresponding production vector $$(\bar y(t),\bar c(t))$$ is obtained at point E. Also, by substituting this value for accumulation equation (1.2), the next period’s capital stock $$\bar k(t + 1)$$ can be attained. The capital stock pairs $$(\bar k(t),\bar k(t + 1))$$ obtained in this manner can be plotted as point E on plane $$(x,z)$$. By further altering $$k(t)$$ and repeatedly conducting the similar procedure, line AB can be drawn on plane $$(x,z)$$ in the manner of figure 6. Now, it can be understood that labor-constraint line AB on the production plane $$(y,c)$$ directly corresponds to line AB on plane $$(x,z)$$. The portions of this line AB excluding the ends are the von Neumann-McKenzie facet.

Based on the above discussion, it is possible to redefine NMF in a neoclassical model as follows.

Definition 3.2 (Characterization of NMF)

$$F({k^\rho },{k^\rho }) \equiv \{ (k(t),k(t + 1)) \in {\bf{D}}:$$ There exist $$c(t) \ge 0$$ and $$y(t) \ge 0$$ such that they satisfy following conditions (1) through (5): (1)$$1 = w_0^\rho a_{00}^\rho + w_{}^\rho a_{10}^\rho$$, (2)$${p^\rho } = w_0^\rho a_{01}^\rho + w_{}^\rho a_{11}^\rho$$, (3)$$1 = a_{00}^\rho c(t) + a_{10}^\rho y(t)$$, (4)$$k(t) = a_{01}^\rho c(t) + a_{11}^\rho y(t)$$, (5)$$k(t + 1) = y(t) + (1 - \delta )k(t))\} ,$$

where, consumer commodity prices are normalized as 1; also, for simplification, the population growth rate has been postulated as zero.

Equations (1) and (2) are cost minimization conditions; equations (3) and (4) are equilibrium conditions for labor and capital goods. Also, (5) is a capital good accumulation equation.

Based on (3) and (4),

$$y(t) = {b^\rho }k(t) + b_{01}^\rho$$(3.1)

Here, $${b^\rho }$$ and $$b_{10}^\rho$$ are defined below as elements of the following matrix.

$${{\bf{B}}^\rho } = {({a^\rho })^{ - 1}} = \left[ {\begin{array}{*{20}{c}}{b_{00}^\rho }&{b_{01}^\rho }\\{b_{10}^\rho }&{b_{}^\rho }\end{array}} \right]$$

Also, based on accumulation equation (5) and equation (3.1), we obtain the following difference equation:

$$k(t + 1) = [{b^\rho } + (1 - \delta )]k(t) + b_{10}^\rho$$(3.2)

By defining $$\eta (t + 1) = k(t) - {k^\rho }$$, difference equation (3.2) is rewritten as

$$\eta (t + 1) = [{b^\rho } + (1 - \delta )]\eta (t)$$(3.3)

It is clear that the behaviors of the path on the NMF can be obtained by investigating this differential equation.

Now, by making a suitable selection of units, it is possible to normalize the element $${b^\rho }$$ of the matrix $${{\bf{B}}^\rho }$$ as follows.

$$\left| {{b^\rho }} \right| = \left| {a_{00}^\rho /(a_{00}^\rho a_{11}^\rho - a_{01}^\rho a_{10}^\rho )} \right| < 1$$

Due to the local stability from the discussion in the previous section, it is a necessary condition that at the very least, the determinant $$\left| {V_{12}^\rho } \right|$$ made up of the second differential coefficient, should not be zero. In the two-sector model, $$V_{12}^\rho$$ is calculated as follows (see in detail Benhabib and Nishimura (1985)).

$$V_{12}^\rho = - {({({b^\rho })^{ - 1}})^2}\left( {\frac{{\partial w}}{{\partial k}}} \right)\left[ {{b^\rho } + (1 - \delta )} \right]$$(3.4)

Based on this, equation (3.4) does not become zero other than in $${b^\rho } = - (1 - \delta )$$.

In the two-sector reduced-form model, the question that in what manner does the optimal path behave in the global sense can be clearly answered by examining the neighborhood turnpike theorem and the local behaviors of the optimal path.

First, we shall rerecord the neighborhood turnpike theorem explained in Section 2 as Lemma 3. 1.

Lemma 3.1 (Neighborhood Turnpike Theorem) Provided that no periodic solution exists in the NMF of $$\rho = 1\;(\gamma = g)$$. Then, for $$\varepsilon < 0$$ there exists a $${\rho ^'} > 0$$ such that with respect to for $${\rho ^'} < \rho < 1$$, the optimal path $$\{ {k^\rho }(t)\} _{t = 0}^\infty$$, ultimately will be trapped in the $$\varepsilon -$$neighborhood of the optimal steady state path $$\{ {k^\rho }\}$$. Also, by making $$\rho$$ sufficiently close to 1, it is possible to make $$\varepsilon$$ close to zero.

Proof: please refer to Theorem 3 of McKenzie (1983).

The fact that Lemma 3.1 is established in the case of a multi-sector model that has neoclassical production functions has been demonstrated by Takahashi (1985). An important point of this proof is the fact that uniform value losses occur in the vicinity of the NMF, which is proven based on the fact that the NMF is a lower semi-continuity correspondence of $$\rho$$.

Based on the above theorem, by making $$\rho$$ sufficiently close to 1, it can be understood that the optimal path comes trapped in the neighborhood of the optimal steady state path. Therefore, it is possible to make clear the global behavior of the optimal path by investigating the actions of the optimal path in this neighborhood. In the case of a two-sector model, the behaviors of the optimal path in the neighborhood of the OSS will be made clear by the following lemma.

Lemma 3.2 Provided that $$\left| {{b^*} + (1 - \delta )} \right| \ne 0$$ (however, $${b^*}$$ indicates the $${b^\rho }$$ under $$\rho = 1$$ whenever). There exists $$\widetilde \rho > 0$$ such that for $$\rho \in [\widetilde {\rho ,}1]$$, optimal path $$\left\{ {{k^\rho }(t)} \right\}_{t = 0}^\infty$$ asymptotically converges to either OSS or the cyclic solution of period 2.

Proof: This is proven in Addendum (6).

From this lemma, it becomes clear that the local nature of the optimal path becomes subject to only cyclic solution of period 2 term matter how complicated the situation. The matter of how the optimal path will change in the neighborhood of the optimal steady state path is classified on the basis of dynamics of the NMF path when $$\rho = 1\;(namely\;\gamma = g)$$. In Section 3.1, we will consider cases in which the NMF becomes unstable manifold, and in Section 3.2, we will consider cases in which the NMF becomes stable manifold.

3.1  Case 1：$$\left| {{b^*} + (1 - \delta )} \right| > 1$$
In equation (3.3), all the NMF paths diverge. In other words, the NMF becomes unstable manifold. This means that the absolute value of the characteristic root of the NMF direction becomes $$\left| \lambda \right| > 1$$. In this instance, based on Lemma 2.1, the characteristic equation (2.1) has $$- 1 < 1/\left| \lambda \right| < 1$$ as an inherent root. Therefore, linear space spread within the corresponding inherent vector is a stable manifold. Thus by selecting $$\rho$$ close enough to 1, the optimal path, based on Lemma 3.2, asymptotically converges to the OSS, and a local stability is established. By combining the local stability and the neighborhood turnpike theorem, the following turnpike theorem is proven.

Theorem 1 (Turnpike Theorem[9])

Based on Case 1, there exists $$\rho ' > 0$$ such that for any $$\rho \in [\rho ',1)$$, $$\mathop {\lim }\limits_{t \to \infty } {k^\rho }(t) = {k^\rho }$$ holds with any sufficient initial stock $$k_0^{}$$stock [10].

Next, with case 1 established, let us add the following hypothesis.

Assumption 3.1 $$1 < \left| {{b^*} + (1 - \delta )} \right| < 1/\widetilde \rho$$ where $$\widetilde \rho$$  is obtained in lemma 3.2.

In this instance, the following important theorem can be proven.

Theorem 2 (Existence of a stable periodic solution)
Under Assumption 3.1, there exists $$\widehat \rho \in [\widetilde \rho ,\rho ')$$ such that the optimal path $$\left\{ {k_t^\rho } \right\}_{t = 0}^\infty$$ for $$\rho \in [\widetilde \rho ,\widehat \rho ]$$converges to the cyclical solution of period 2.

Proof: Based on Assumption 5, by selecting $$\widehat \rho \;( > \widetilde \rho )$$, it can be made into $$1 < \left| {{b^{\hat \rho }} + (1 - \delta )} \right| < 1/\hat \rho$$.

In this instance, it is understood that the characteristic equation has the roots $$\left| \lambda \right| > 1\;and\;1/\left| {\hat \rho \lambda } \right| > 1$$ simultaneously from Lemma 2.1. Thus, it becomes totally unstable in OSS ($$\{ {k^{\hat \rho }}\}$$) (refer to Figure 2). This means that it is impossible for the optimal path to asymptotically converge to the OSS to which the optimal path corresponds. Based on Lemma 3.2, the optimal path must converge on the cyclical solution of period 2.

Thus, the matter of whether a turnpike theorem is established or convergence occurs upon the periodic solution of term 2 is proven in instances when the NMF becomes unstable manifold. The results of theorem 2 are in theorem 5 of Benhabib and Nishimura (1985) and are proven on the basis of more exacting assumption that follows: all points in the domain D of function V are such that $${V_{12}} \ne 0$$. Because they did not introduce a NMF, they obtained sufficient conditions that were the result of the existence of a cyclic solution of period 2 based on the fixed-point theorem and use those in their proof. The point that is different here is the fact that by introducing a NMF, our proof is made directly based on a weaker hypothesis.

It is understood that in the consumption turnpike theorem, if the discount rate $$\rho$$ is made sufficiently close to 1, the optimal path becomes stable, and if the discount rate $$\rho$$ becomes close to zero, periodic solutions or chaos occur. For example, if $$\rho$$ is set sufficiently close to zero, the breadth of the interval defined in Assumption 3.1 becomes unlimited in size, the breadth of the cyclic solution of period 2 will increase, and the possibility of a periodic solution will become high. However, conversely, irrespective of the discount rate being close to 1, as proven in Theorem 2, by suitably taking up $${b^\rho }$$ and $$\rho$$, it becomes possible to bring about a periodic solution.

3.2  Case 2： $$\left| {{b^*} + (1 - \delta )} \right| < 1$$

In this instance, the path of the NMF from equation (3.3) will once again become asymptotically stable.

Therefore, NMF itself will become stable manifold. Also, by taking $$\rho$$ sufficiently close to 1, it is possible to make the NMF stable manifold in the neighborhood of $$\rho = 1$$. Thus, a local stability is established, and similar to that in the previous section, by combining it with the neighborhood turnpike, a turnpike theorem is established.

[9] With respect to the matter of whether the Turnpike theorem can actually be observed, the fact that this sort of phenomenon could be observed in the various OECD countries (the formerWest Germany in particular) was reported by Brems (1985).
[10] The sufficiency of the capital stock is defined as follows. If the capital stock $$x$$ will establish $$(x,y) \in {\bf{D}}$$ and $$y > x$$, then $$x$$ is referred to as “expendable.”  Also, with respect to the finite period path $$\{ {k_0},{k_2}, \cdots ,{k_T}\}$$, when $$x = {k_0},({k_t},{k_{t + 1}}) \in {\bf{D}}\;(t = 1, \cdots ,T)$$ is established, the capital stock is described as satisfying “sufficiency.”

4. Generalization into a multi-sector model

In this section, we will expand the two-sector model demonstrated in Section 3 into a multi-sector model that includes multi capital goods, and also prove similar results that were attained in Section 3. However, we will advance our arguments by postulating that population growth rate $$g$$ is not zero.

First, we will define the “Generalized Capital Intensity Conditions” researched by Inada (1971).

Definition 4.1 The inverse matrix $${{\bf{B}}^\rho }$$ of the technology matrix $${{\bf{a}}^\rho }$$ of the OSS is referred to as the “SSS-I matrix” when the diagonal elements are positive ($$b_{ij}^\rho > 0$$) and the non-diagonal elements are negative ($$b_{ij}^\rho < 0$$).

Definition 4.2 The inverse matrix $${{\bf{B}}^\rho }$$ of the technology matrix $${{\bf{a}}^\rho }$$ of the OSS is referred to as the “SSS-II matrix” when the diagonal elements are negative ($$b_{ij}^\rho < 0$$) and the non-diagonal elements are positive ($$b_{ij}^\rho > 0$$).

If $$n = 1$$, the factor intensity conditions coincides with these matrices conditions. Also, when $$n = 2$$, it is possible to lead to the intensity conditions from these matrices conditions. That is, it is possible use the SSI-I matrix to lead to intensity conditions in which the capital good sector is more labor-intensive than the consumption good sector, or use the SSI-II matrix to lead to intensity conditions in which the consumption good sector is more capital intensive than the capital goods sector. However, if $$n \ge 3$$, the relationships between both become unclear.
Here, the $${\bf{V}}_{12}^\rho$$ matrix obtained in the last section is expressed as the following equation, in the manner of the proof provided by Takahashi (1990).

$${\bf{V}}_{12}^\rho = - (1 + g){({{\bf{b}}^\rho })^{ - 1}}\left[ {{{\bf{b}}^\rho } + ({\bf{I}} - \Delta )} \right]\left[ {\partial {\bf{w}}/\partial {\bf{k}}} \right]{({{\bf{b}}^\rho })^{ - 1}}$$(4.1)

Based on equation (3.4) of capital good model 1 from part 3, it is easy to imagine the expression of a multi capital goods case as the above sort of equation. Thus, based on the concavity of each production function and generalized intensity conditions, determinant $${\bf{V}}_{{\bf{12}}}^\rho$$ does not become zero. Therefore, it is possible to attain simultaneous difference equation (2.1), and lemma 1.2 is established.
In the same manner, based on equation (3.3) of part 3, the difference equation that expresses the behaviors of the path of the facet can be generally rewritten as follows.

$$\eta (t + 1) = 1/(1 + g)\left[ {{{\bf{b}}^\rho } + ({\bf{I}} - \Delta )} \right]\eta (t)$$(4.2)

This differs from equation (3.3) only in the sense that it includes the population growth rate $$g$$, and that the depreciation rate $$\delta$$ is rewritten as a matrix $$\Delta$$.

The following lemma is applied in the proof of the local stability.

Lemma 4.1 Let us consider the following difference equation system that possesses $${\bf{x}}$$ as a stationary solution.

$${\bf{x}}(t + 1) = ({\bf{C}} + {\bf{I}}){\bf{x}}(t)$$

where $${\bf{x}}(t) \in {\mathbb{R}^n}$$, and matrix $${\bf{C}}$$ become an $$n \times n$$ matrix.

In this instance, if matrix $${\bf{C}}$$ has a negative dominant diagonal[11], $${\bf{C}} + {\bf{I}}$$ becomes $${\bf{x}}(t) = {\bf{0}}$$ based on the norm defined below and is contractive, the system attains global asymptotic stability, and the Lyapunov function is $$V(x) = \left\| x \right\|$$. Here norm $$\left\| \cdot \right\|$$ is defined as $$\left\| {\bf{x}} \right\| = \max {c_i}\left| {{x_i}} \right|$$, and $${c_i}$$ is a positive constant. Conversely, if matrix $${\bf{C}}$$ has a positive dominant diagonal, the system becomes totally unstable.

Proof: The first half is proven in Newman (1961, pp. 27-24). We can easily be led to the total instability results of the latter half from the fact that if a positive dominant diagonal is possessed, the inherent root of that matrix possesses a positive real part.

Because we are applying Lemma 4.1, let us pay attention to the fact that it is possible for the coefficient matrix to change into the form written below.

$$1/(1 + g)\left[ {{{\bf{b}}^\rho } + ({\bf{I}} - \Delta )} \right] = 1/(1 + g)\left[ {{{\bf{b}}^\rho } - (g{\bf{I}} - \Delta )} \right] + {\bf{I}}$$(4.3)

Theorem 4 The NMF becomes stable manifold when matrix $${{\bf{b}}^\rho }$$ is SSS-II, and if it is SSS-I, it becomes unstable manifold.

Proof: Proving that matrix $$\left[ {{{\bf{b}}^\rho } - (g{\bf{I}} - \Delta )} \right]$$ has a negative dominant diagonal is sufficient.

The following is established based on the resource restrictions of the optimal steady state path.

$${{\bf{y}}^\rho } = {{\bf{b}}^\rho }{{\bf{k}}^\rho } + {\bf{b}}_0^\rho$$(4.4)

where $${\bf{b}}_{\bf{0}}^\rho = (b_{00}^\rho ,b_{10}^\rho ,....,b_{n0}^\rho )$$.

Also, from an accumulation equation,

$${{\bf{y}}^\rho } = (\Delta + g{\bf{I}}){{\bf{k}}^\rho }$$(4.5)

is established.

Based on (4.4) and (4.5),

$$\left[ {{{\bf{b}}^\rho } - (\Delta + g{\bf{I}})} \right]{{\bf{k}}^\rho } = - {\bf{b}}_0^\rho$$(4.6)

is obtained. It can be understood that if matrix $${{\bf{b}}^\rho }$$ is SSS-II, $$- {\bf{b}}_0^\rho < 0$$ is established, and the matrices of the problem has a negative dominant diagonal. Therefore, because all the conditions of Lemma 5 are satisfied, lemma can be used to prove the asymptotic stability of the path of the facet. Thus, the NMF becomes stable manifold. Also, it can be understood that if matrix $${{\bf{b}}^\rho }$$ is SSS-I,$$- {\bf{b}}_0^\rho > 0$$ is established, and the matrices of the problem has a positive dominant diagonals. Once again, the NMF becomes stable manifold based on the second results of Lemma 5.

It can be understood that based on Theorem 4 above, it is possible to generalize the Theorem 1 obtained using the two-sector model as a multi-sector case. Because Theorem 2 is dependent upon Lemma 3.1, which is established only with respect to the two-sector model, it cannot be generalized.

[11] Matrix A has a negative (positive) dominant diagonal when positive vector $${\bf{h}} \in {\mathbb{R}^n}$$ establishing sort of relationship below exists, and possesses negative (positive) diagonal elements. Refer to McKenzie (1959).

$$i = 1,2,....,n$$.

5. Conclusion

In this text, we have used the results of the latest empirical analyses to formulate a hypothesis that each industrial sector possesses constant return technology, which we have used as the foundation to create a multi-sector model based on industrial divisions. Then we conducted a global analysis of its optimal path. Exogenous optimal growth models to which population growth rates have been applied as a postulate have become central to our analyses. However, this discussion has, without making hardly any revisions, indicated that it is possible to apply it to a Lucas-type endogenous optimal growth model that does not include primary factors, like labor. In this way, it can be understood that there is no major difference between the analyses of a Lucas-type endogenous growth model including human capital and a neoclassical multi-sector endogenous growth model. Please refer to Takahashi (2008) for a more detailed discussion.

For readers’ convenience, the fundamental overall concepts and theorems used in the main text will be explained in this addendum; also, the important theorems of the main text will be proven.

A1. In this text, as long as there is no objection, the vectors will express column vectors. The row vectors are expressed with the transposition symbol “t.”

A2. The definitions of the symbols that express the major and minor relationships between the vectors and matrices are shown below.

(a) When $${\bf{x}}$$ expresses vectors and $${\bf{0}}$$ expresses zero vectors,

$${\bf{x}} \ge {\bf{0}}:{x_i} \ge 0$$ is established for all $$i$$.

$${\bf{x}} > {\bf{0}}:{x_i} \ge 0$$ is established for all , $$i$$ and for a certain $$i$$, $${x_i} > 0$$ is established.

$${\bf{x}} \gg {\bf{0}}:{x_i} > 0$$ is established for all $$i$$.

(b) When $${\bf{A}}$$ expresses arbitrary matrices, and $${\bf{0}}$$ expresses zero matrices,

$${\bf{A}} \ge {\bf{0}}:{a_{ij}} \ge 0$$ is established for all $$(i,j)$$.

$${\bf{A}} > {\bf{0}}:{a_{ij}} \ge 0$$ is established for all $$(i,j)$$, and $${a_{ij}} > 0$$ is established for certain $$(i,j)$$.

$${\bf{A}} \gg {\bf{0}}:{a_{ij}} > 0$$ is established for all $$(i,j)$$.

When a square matrix $${\bf{A}}$$ is converted into the following matrix by $${\bf{P}}$$

$${\bf{PA}}{{\bf{P}}^{\bf{T}}} = \left[ {\begin{array}{*{20}{c}}{{{\bf{B}}_{\bf{1}}}}&{{{\bf{B}}_{12}}}\\{\bf{0}}&{{{\bf{B}}_2}}\end{array}} \right]$$,

it is referred to as a “decomposable matrix,” and when such a conversion is impossible, it is referred to as “indecomposable.”  Here, $${{\bf{B}}_{\bf{1}}},\;{{\bf{B}}_{\bf{2}}}$$ expresses a square matrix, and $${\bf{0}}$$ expresses a zero matrix.

A4.Frobenius Theorem: if square matrix $${\bf{A}} > {\bf{0}}$$ is indecomposable, the following is established.

(i) The maximum characteristic root (Frobenius root) $${\lambda _F}$$ of $${\bf{A}} > {\bf{0}}$$ exists, and it is both a simple root and a real number.

(ii) $${(\lambda {\bf{I}} - {\bf{A}})^{ - {\bf{1}}}} \gg {\bf{0}}$$ is established for $$\lambda > {\lambda _F}$$.

A5.  Proof of Lemma 1.2 : it is clear that the following relationship is established.

$$({\bf{I}} - ({\bf{g}} + \Delta ){{\bf{a}}^\gamma }) = \left[ {{\bf{I}} - {\bf{g}}{{\bf{a}}^\gamma }{{({\bf{I}} - \Delta {{\bf{a}}^\gamma })}^{ - 1}}} \right]({\bf{I}} - \Delta {{\bf{a}}^\gamma })$$.

If the inverse matrix of the left hand side’s inverse matrix exists, the following is established.

$${[{\bf{I}} - ({\bf{g}} + \Delta ){{\bf{a}}^\rho }]^{ - 1}} = {({\bf{I}} - \Delta {{\bf{a}}^\rho })^{ - 1}}{\left[ {{\bf{I}} - {\bf{g}}{{\bf{a}}^\rho }{{({\bf{I}} - \Delta {{\bf{a}}^\rho })}^{ - 1}}} \right]^{ - 1}}$$.

Based on Assumption 1.3, the first inverse matrix of the left hand side exists and positive. Based on Assumption1.3, the Frobenius theorem (refer to Addendum A4) is established, and the second term of the left hand side also becomes positive. Therefore, we have established that $${[{\bf{I}} - ({\bf{g}} + \Delta ){{\bf{a}}^\rho }]^{ - 1}} > {\bf{0}}$$. Based on the steady state path and the market equilibrium conditions of the capital,

$${({c^\rho },{{\bf{y}}^\rho })^t} - ({\bf{g}} + \Delta ){{\bf{a}}^\rho }{({c^\rho },{{\bf{y}}^\rho })^t} = {({c^\rho },{\bf{0}})^t}$$

or

$${({c^\rho },{{\bf{y}}^\rho })^t} = {[{\bf{I}} - ({\bf{g}} + \Delta ){{\bf{a}}^\rho }]^{ - 1}}{({c^\rho },{\bf{0}})^t}$$,

are established. Therefore, if $${c^\gamma } > 0$$ is applied, then the $${{\bf{y}}^\rho }( > {\bf{0}})$$ vector is unambiguously determined.  Since, based on the equilibrium conditions of the labor market,

$${\bf{a}}_0^\rho {({c^\rho },{{\bf{y}}^\rho })^t} = {\bf{a}}_{\bf{0}}^\rho {[{\bf{I}} - ({\bf{g}} + \Delta ){{\bf{a}}^\rho }]^{ - 1}}{({c^\rho },{\bf{0}})^t} = 1$$

is established, $${c^\rho } > 0$$ is also unambiguously determined. Now, let us have $$k_i^\rho = \sum\nolimits_{j = 0}^n {k_{ij}^\rho = } 0$$ with respect to a certain $$i$$. This means that $$k_{ij}^\rho = 0$$ is established for all $$j$$. This also means that in the investment matrices $$a_{ij}^\rho = 0$$ is established for all $$j$$. Therefore, it runs into contradiction to the fact that $$A$$ is an indecomposable matrix. Therefore, $$k_i^\rho > 0$$ must be true for all $$i$$. Also, based on the accumulation equation, $$y_i^\rho = (g + {\delta _i})k_i^\rho$$ is established. Therefore, $$y_i^\gamma > 0$$ must also be established for all $$i$$. Therefore, $${{\bf{k}}^\rho } \gg {\bf{0}}$$, $${{\bf{y}}^\rho } \gg {\bf{0}}$$, and $${c^\rho } > 0$$ are obtained unambiguously.

A6. Proof of Lemma 3.2: $$V_{12}^*$$ is dependent upon intensity conditions, and takes positive and negative symbols. Now, let us suppose that $$V_{12}^* < 0$$. This same sort of proof will be possible for $$V_{12}^* > 0$$. Because of $$V_{12}^* < 0$$, there exists a $$\widetilde \rho > 0$$ such that $$V_{12}^\rho < 0$$ for $$\rho \in [\widetilde {\rho ,}1]$$. Now let us define the OSS neighborhood as follows.

$${N_\varepsilon }({k^\rho }) = \{ (x,z):\left\| {(x,z) - ({k^\rho },{k^\rho })} \right\| < \varepsilon \}$$

Here, it is possible to discover a neighborhood in which $$V_{12}^\rho < 0$$ is established for all corresponding $$\varepsilon (\rho )$$-neighborhood points with respect to each $$\rho \in [\widetilde {\rho ,}1]$$. That is, the following is established.

$$\varepsilon ' = \lim \sup \left\{ {\varepsilon (\rho ) > 0:{V_{12}}(x,z) < 0\;for\;all\;(x,z) \in N_{\varepsilon (\rho )}^{}({k^\rho })} \right\}$$.

Here, $$\lim \sup$$ is taken for all $$\{ {k^\rho }\}$$ for $$\rho \in [\widetilde {\rho ,}1]$$. Then, by applying the neighborhood turnpike theorem, there exists $$\bar \rho > 0$$ for $$\varepsilon ' > 0$$ such that any optimal path is trapped in neighborhood $${N_\varepsilon }({k^\rho })$$. Now by redefining that $$\widetilde \rho = Max\{ \widetilde \rho ,\overline \rho \}$$, it is possible to apply Theorem 3 of Benhabib and Nishimura (1985) to $${N_\varepsilon }({k^\rho })$$, and the proof is completed.

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