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

In spatial economics, the distribution of wages is described by a solution to the wage equation ofDixit-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.


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][2][3][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.61-77].
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  = 2, but there are much fewer ones when  ≥ 3 (see, e.g., [12][13][14][15][16]).For  = 2 or 3, 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  ≥ 2. 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  ≥ 2. 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).

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  0+ by  0 , 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.45-49, 61-65], [15], and [17, (2.3), (2.4), (2.8)-(2.15)]).
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  ∈  0 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 (; ) −1 as a given function, we can regard the right-hand side of ( 14) as a nonlinear operator acting on  = ().We can regard  −(−1)(,) (; ) −1 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.

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) where then the wage equation has a solution.Moreover, it is proved in [17, Lemma 3.2, Theorem 3.3] that if C is so small such that then the wage equation ( 14) has a unique solution, where (, ) is a function of (, ) ∈ (0, 1) × (1, +∞) such that inf It follows from ( 23) and (25) that the previous research [17] cannot be applied to the case where C 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 C.Under Condition 1 we prove the following main theorem.
and  is so small such that then the following statements (i)-(iii) hold.
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 C strongly in [17].Hence, Theorem 3 can be applied to the case where C 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.
From (18) 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  ∈ (0, 1) and  − 1 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. 59].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 noblack-hole.

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 ()() 1/ .For this purpose we obtain estimates for (15).
Making use of Lemma 4, we obtain estimates for (37).
Proof of Theorem 3(ii).We make use of Lemma 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).

Uniqueness
The purpose of this section is to prove Theorem 3(iii).For this purpose we make use of the following lemma.
where  and  are constants such that then where where Subtracting (15) with  = V from ( 15) with  = , we see that where Add this inequality to (76).Applying the inequality thereby obtained to (74), we obtain (63).