Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 9341502, 7 pages
https://doi.org/10.1155/2017/9341502

Research Article

Existence and Uniqueness of Solutions to the Wage Equation of Dixit-Stiglitz-Krugman Model with No Restriction on Transport Costs

Minoru Tabata¹ and Nobuoki Eshima²

¹Department of Mathematical Sciences, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan
²Center for Educational Outreach and Admissions, Kyoto University, Kyoto 606-8501, Japan

Correspondence should be addressed to Minoru Tabata; mnrtabata@luck.ocn.ne.jp

Received 18 April 2017; Revised 6 June 2017; Accepted 7 June 2017; Published 11 July 2017

Academic Editor: Pavel Rehak

Copyright © 2017 Minoru Tabata and Nobuoki Eshima. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In spatial economics, the distribution of wages is described by a solution to the wage equation of Dixit-Stiglitz-Krugman model. The wage equation is a discrete equation that has a double nonlinear singular structure in the sense that the equation contains a discrete nonlinear operator whose kernel itself is expressed by another discrete nonlinear operator with a singularity. In this article, no restrictions are imposed on the maximum of transport costs of the model and on the number of regions where economic activities are conducted. Applying Brouwer fixed point theorem to this discrete double nonlinear singular operator, we prove sufficient conditions for the wage equation to have a solution and a unique one.

1. Introduction

A large number of discrete dynamic models have been constructed in order to describe various dynamic phenomena observed in nature and society. In particular, the applications of discrete dynamic models have been broadened to various disciplines of economics. The applications help the advance of economics. In turn, it accelerates the progress of theory of discrete dynamics itself to study new discrete dynamic models constructed in economics.

In light of the close and cooperative interaction between discrete dynamics and economics, we should extend applications of discrete dynamics further to new disciplines of economics. In particular, noting that many mathematical sciences (game theory, financial engineering, and so on) have been born from the Nobel Prize research in economic sciences, we should focus on applications of discrete dynamics in spatial economics.

Spatial economics is an interdisciplinary area between economics and geography. Its purpose is to study the location, distribution, and self-organization of economic activities. In about 1990, Krugman began important seminal research and established a useful analytical framework with emphasis laid on the formation of a large variety of economic agglomeration and the clustering of economic activities. A large number of economists have conducted various types of research within his new framework. Their research has since grown into one of the most major branches of spatial economics. Now it is known as New Economic Geography (NEG). In 2008, Krugman was awarded the Nobel Memorial Prize in Economic Sciences for his great contribution to spatial economics (see, e.g., [1–4]).

In NEG a large number of discrete dynamic models have been constructed, and many of them are quite new and very impressive. Hence, NEG is one of the most promising fields of applications of discrete dynamics. Among those many discrete models in NEG, Dixit-Stiglitz-Krugman model (DSK model) is one of the most fundamental models. Hence, this paper deals with DSK model. In this model, economic activities are conducted in a set of regions, the economy consists of agriculture and manufacturing, and the population consists of farmers and workers [5, pp. ]. There have been important related studies, and several analytically solvable models have been developed, in order to analyze economic agglomeration and bifurcation (see, e.g., [6–11]).

If we regard the distributions of workers and farmers as known functions, then we define short-run equilibrium of DSK model as a solution to the wage equation. The wage equation is a discrete nonlinear singular equation whose unknown function denotes the distribution of wages. It should be noted that the wage equation has a double nonlinear singular structure in the sense that the equation contains a discrete nonlinear operator whose kernel itself is expressed by another discrete nonlinear operator with a singularity.

This strong nonlinearity causes great difficulty when attempting to solve the wage equation. In particular, this difficulty increases as the number of regions increases. In fact, there are ample analytical results when , but there are much fewer ones when (see, e.g., [12–16]). For or , it is customary to deal mainly with a specific case where the competition is between uniform distribution and a complete agglomeration. For these reasons we should seek sufficient conditions for the existence and uniqueness of solutions to the wage equation with no restriction on the number of regions.

In [17], as a first step, we proved that if the maximum of transport costs of DSK model is sufficiently small, then the wage equation has a unique solution for an arbitrary integer However, this sufficient condition is very restrictive, since it cannot be applied to DSK model whose transport costs are high. Hence, in this article, we prove sufficient conditions for the existence and uniqueness of solutions to the wage equation with no restriction on the maximum of transport costs for an arbitrary integer The main result of this article is Theorem 3. Theorem 3(i) gives a sufficient condition for the wage equation to have a solution, Theorem 3(ii) gives estimates for each solution to the wage equation, and Theorem 3(iii) gives a sufficient condition for the wage equation to have a unique solution.

Indeed the operator contained in the wage equation is very complicated. However, by applying only Brouwer fixed point theorem to this complicated operator, we can prove the main result. Hence, it is easy for readers without a full knowledge of discrete nonlinear equations to understand this article, since Brouwer fixed point theorem is one of the most elementary tools to prove existence of solutions of nonlinear equations (see, e.g., [18]). Moreover, in this paper we do not make use of the method developed in [17]. Hence, readers can understand this article without reading [17] carefully. This paper has four sections in addition to this introduction. In Section 2 we introduce the wage equation of DSK model. In Section 3 we state and discuss Theorem 3. In Section 4, we prove Theorem 3(i)(ii). In Section 5 we prove Theorem 3(iii).

2. Equation

By we denote a finite set consisting of points. Each point of represents a region where economic activities are conducted. By we denote the set of all real-valued functions of We can regard as an -dimensional Euclidean space and as a point of the -dimensional Euclidean space. In we define the following norm instead of the usual Euclidean norm for convenience:

We define the following subsets of :

and we denote the complement of in by , that is, we define

Moreover we define

where and are arbitrary constants such that

The wage equation contains the elasticity of substitution and the manufacturing expenditure as parameters, and the transport cost function as known function of Moreover, the wage equation contains the distribution of workers and the distribution of farmers as known functions of Throughout the paper we assume the following conditions, which are the most general conditions that are fully accepted in economics (see, e.g., [5, pp. , ], [15], and [17, , , –]).

Condition 1.

where

The wage equation is the following discrete nonlinear equation (see, e.g., [5, –], [8, pp. ], and [17, –]):

where denotes an unknown function that describes the distribution of wages, and and are operators that act on These operators are defined as follows:

In NEG (15) and (16) are referred to as price index and income, respectively.

Let us discuss the singularity of the wage equation (14). We see that all can be substituted in (14) and (15) but that (14) has a singularity such that no can be substituted in (15). Noting that describes the distribution of wages, which should be nonnegative-valued in economics, we reasonably seek a solution in .

Let us discuss the nonlinearity of (14). Regarding as a given function, we can regard the right-hand side of (14) as a nonlinear operator acting on We can regard as the kernel of this nonlinear operator. However, observing (15), we see that this kernel itself is a discrete nonlinear singular operator acting on Therefore, we see that (14) is a discrete double nonlinear singular equation.

We employ the following symbols in this paper.

Definition 2. (i)

where

(ii)

3. Result and Discussion

Let us review previous research on the wage equation (14). Restrictive assumptions are imposed on the previous research in addition to Condition 1. If the equality

holds instead of (7), that is, if DSK model has no agriculture, then the wage equation has a solution [19, Assumption  1.2, , Theorem  3.1]. This result is too restrictive, since it cannot be applied to DSK model with (7). It is proved in [17, Theorem  3.1] that if is so small such that

where

then the wage equation has a solution. Moreover, it is proved in [17, Lemma  3.2, Theorem  3.3] that if is so small such that

then the wage equation (14) has a unique solution, where is a function of such that

It follows from (23) and (25) that the previous research [17] cannot be applied to the case where is large. Hence, as mentioned in Section 1, we seek sufficient conditions for the existence and uniqueness of solutions to the wage equation with no restriction on Under Condition 1 we prove the following main theorem.

Theorem 3. If

and is so small such that

then the following statements (i)–(iii) hold.
(i) The wage equation (14) has a solution such that

(ii) If is a solution to (14), then the solution satisfies (30).
(iii) If and are so small that

then (14) has a unique solution in

We prove Theorems 3(i)(ii) and 3(iii) in Sections 4 and 5, respectively. Let us discuss this theorem in this section. It follows from (27) and (28) that workers and farmers live in all regions. Theorem 3 does not contain the transport cost function , in contrast to the fact that (23) and (25) restrict strongly in [17]. Hence, Theorem 3 can be applied to the case where is large. Moreover, Theorem 3(i)(ii) does not contain the elasticity of substitution , either. Hence, we can apply Theorem 3(i)(ii) also to the case where is large.

It follows from (6), (11)–(13), (19), and (20) that

Applying this inequality (7) and (27)–(29) to (18), we see easily that

Making use of this inequality (8) and (27), we can define (21).

From (18) we see that

Applying this result to (21), we see easily that

It follows from (27) and (28) that the right-hand side of this equality is positive and finite. Applying this result to (31), we see that if and are sufficiently small, then (31) holds.

The following condition is referred to as the assumption of no-black-hole, which is famous in NEG:

When this condition is violated, there is no agglomeration (core-periphery pattern). In order to avoid treating economies in which increasing returns are extremely strong, this inequality is assumed in [5, p. ]. We see easily that this condition is consistent with (29) and (31). Hence, we can apply Theorem 3 to DSK model that satisfies the assumption of no-black-hole.

4. Existence

The purpose of this section is to prove Theorem 3(i)(ii). Defining the operator,

we can rewrite (14) equivalently as follows:

Hence, in order to prove Theorem 3(i), we have to only seek a fixed point of the operator For this purpose we obtain estimates for (15).

Lemma 4. If (27) holds and satisfies that

where is a positive constant, then

Proof. Note that if (27) does not hold, then we cannot define the right-hand side of (40). Applying (9) to (15) with , we can divide as follows:

Dropping the second term from the right-hand side of this equality, we obtain the right-hand inequality of (40).
It follows immediately from (8)–(10) that

Making use of this inequality and (39), we replace with in the second term of the right-hand side of (41). Applying (11) and (13) to the inequality thereby obtained, we obtain the left-hand inequality of (40).

Making use of Lemma 4, we obtain estimates for (37).

Lemma 5. If (27) holds and satisfies

where and are constants that satisfy (5), then

where

Proof. Note that if (27) does not hold, then we cannot define (45). Applying (7), (11), and (12) to (16), we see that

Applying (7), (11)–(13), and (43) to

we obtain

Making use of (9), we divide (37) as follows in the same way as (41):

Applying (47) and the right-hand inequality of (40) to the right-hand side of this equality, we replace and with and , respectively. Moreover, applying (8), (19), (42), and (43) to the second term of the right-hand side of the inequality thereby obtained, we replace with Hence, we obtain

Applying (49) to the second term of the right-hand side of this inequality, we deduce that

Applying (47) and the left-hand inequality of (40) to the first term of the right-hand side of (50), and dropping the second term from the right-hand side, we obtain

Applying (7), (16), (17), and (43) to , we see easily that

Applying this inequality to (52) and (53), we obtain

Replace and with and in the left-hand side of (55). Replace with in the right-hand side of (55). Then we obtain (44).

Proof of Theorem 3(i). We prove Theorem 3(i) by making use of Lemma 5. Let

where and are constants that satisfy (5). Making use of (33), we can substitute in (45). Substituting (18) in (45) with and making use of (29), we see easily that

Substituting (57) in (44) with , we see that is an operator from to itself. Observing (37), we see easily that is continuous with respect to Moreover, we see easily that is a convex closed subset of Hence, applying Brouwer fixed point theorem to the operator (see, e.g., [18]), we see that the operator has a fixed point in Hence, we obtain Theorem 3(i).

Proof of Theorem 3(ii). We make use of Lemma 5. Let be a solution. Substituting (38) in (44), when

we obtain the following inequality:

Noting that and are independent of , we replace with and in this inequality. Hence we obtain the following inequalities:

Substitute (45) and (46) in these inequalities. Solving the inequalities thereby obtained with respect to and , making use of (29), and recalling (18), we obtain (30).

5. Uniqueness

The purpose of this section is to prove Theorem 3(iii). For this purpose we make use of the following lemma.

Lemma 6. If (27) holds and and satisfy

where and are constants such that

then

where

Proof. Let Applying the mean value theorem to the function of , we see easily that if , then

Applying (1) and (61) to this inequality with , we obtain

where

Subtracting (15) with from (15) with , we see that

where

It follows from (40) that

Applying this inequality to , we obtain

Applying (8), (11), (13), (42), and (66) with to , we obtain

Applying this inequality and (71) to (68), we see that

Substituting (37) in , we see that

where

Applying (11), (13), (42), and (70) to , we obtain

Applying (42), (47), (48), and (73) to , we see that

Applying (7) and (62) to the right-hand side of (49), we see that

Applying this inequality to (77), we obtain

Add this inequality to (76). Applying the inequality thereby obtained to (74), we obtain (63).

Proof of Theorem 3(iii). Assume that (14) has two solutions It follows from Theorem 3(ii) that

Subtracting (38) with from (38) with , we deduce that

Applying (8), (80), and (66) with to the left-hand side of this equality, we see that

Making use of (33) and (80), we can substitute in (62) and (63). Applying the inequality thereby obtained and (82) to (81), we see that

where

Applying (21) and (64) with to , we see easily that

Applying (31) to this inequality, we obtain Theorem 3(iii).

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by Grant-in-Aid for Scientific Research (15K05005, 26330045), Ministry of Education, Culture, Sports, Science and Technology of Japan.

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