Existence of a Short-Run Equilibrium of the Dixit-Stiglitz-Krugman Model

Each short-run equilibriumof theDixit-Stiglitz-Krugmanmodel is defined as a solution to thewage equationwhen the distributions ofworkers and farmers are given functions.We extend the discrete nonlinear operator contained in thewage equation as a set-valued operator. Applying the Kakutani fixed-point theorem to the set-valued operator, under themost general assumptions, we prove that the model has a short-run equilibrium.


Introduction
Spatial economics is an interdisciplinary field between economics and geography.In about 1990, Krugman commenced breakthrough research by placing particular emphasis on the clustering of economic activities and the formation of economic agglomeration in this interdisciplinary area.He successfully established a useful theoretical framework, which has attracted many social scientists from various disciplines.Since then, his research has grown into one of the major branches of spatial economics, which is now known as the New Economic Geography (NEG).In 2008, Krugman was awarded the Nobel Memorial Prize in Economic Sciences for his great contribution to spatial economics [1,2].
A large number of discrete dynamic models are constructed in the NEG.Among such models, the Dixit-Stiglitz-Krugman model (DSK model) is one of the most important models (see, e.g., [3,Chapter 5], [4, 9.2], and [5, pp. 16-28]).In this model, economic activities (agriculture and manufacturing) are conducted in a set consisting of  points, where  is a natural number and each point represents a region.The population consists of farmers and workers.The DSK model is described by the wage equation, which is a discrete nonlinear equation of which the unknown function denotes the distribution of nominal wages [3, (5.3)- (5.5)].
The DSK model has a strong nonlinearity.In fact, the wage equation has a double nonlinear singular structure in the sense that the equation contains a discrete nonlinear operator of which the kernel itself is expressed by another discrete nonlinear operator with a singularity.This nonlinearity causes great difficulty when attempting to solve the wage equation.
The insolvability of the DSK model has led to the introduction of several analytical methods in the NEG.For example, the Turing approach has been used to analyze the emergence of agglomeration in the DSK model.Moreover, several analytically solvable models have been developed in order to analyze economic agglomeration and bifurcation (see, e.g., [6,7], [3, pp. 85-88], [5,[8][9][10]).
We note that the insolvability of the DSK model is increasingly problematic as the number  increases.In fact, there are ample analytical results for  = 2, whereas there are much fewer ones for  = 3 (see, e.g., [11][12][13][14]).For  = 2 or 3, it is customary to deal mainly with a specific case where the competition between uniform distribution and a complete agglomeration exists.
Hence, we should study the DSK model when the number  is large.This article deals with existence of short-run equilibrium of the DSK model with no restriction on , where each short-run equilibrium is defined as a solution to the wage equation when the distributions of workers and farmers are known functions.
Mathematical studies on the existence of short-run equilibrium were conducted when  = 2, 3 [15,16].However, it is difficult to apply the methods used in those studies to the DSK model when  is large.If restrictive conditions are imposed on the maximum of transport costs and the manufacturing expenditure, then it is proved that the DSK model has a shortrun equilibrium for every  ≥ 2 [17, (2.10), Theorem 3.1] [18, Theorem 3.1] [19, Theorem 3].However, these conditions are too restrictive to apply the results to various economic phenomena.Although it is important to prove that the DSK model has a short-run equilibrium for every  ≥ 2, it has not been proved yet under general conditions.
In this paper, under the most general assumptions (Condition 1 and ( 7)-( 10)), we prove that the DSK model has a short-run equilibrium for every  ≥ 2. No condition is imposed on this paper in addition to these assumptions.The main result is Theorem 1. Theorem 1(i) gives the existence of short-run equilibrium.Theorem 1(ii)(iii) gives accurate estimates for each short-run equilibrium.
This paper consists of six sections in addition to this introduction and Appendix.In Section 2, we state Condition 1 and introduce the DSK model.In Section 3, we state and discuss Theorem 1.In Section 4, we prove Theorem 1(ii)(iii).In Section 5, we extend the discrete nonlinear operator contained in the wage equation as a set-valued operator (Definition 6).In Section 6, applying the Kakutani fixed-point theorem to the set-valued operator, we prove Theorem 1(i).Section 7 is the conclusion section.In the Appendix, we prove Lemma 7, which shows that the set-valued operator satisfies the conditions of the Kakutani fixed-point theorem.
In this article, we do not use the methods developed in [17][18][19], and we make use neither of advanced theory of discrete nonlinear equations, of the NEG, nor of fixedpoint theory.We make use of only the Kakutani fixed-point theorem, which is one of the most fundamental fixed-point theorems (see, e.g., [20][21][22]).Hence, this article can be easily understood even without reading [17][18][19] carefully and even without having an advanced knowledge of discrete nonlinear equations, the NEG, and fixed-point theory.

Condition and Equation
By  we denote a finite set consisting of  points, where  is an arbitrary integer such that  ≥ 2. ( Each point of  represents a region where manufacturing and agriculture are conducted.By  we denote the set of all realvalued functions V = V() of  ∈ .We regard  as an dimensional Euclidean space.Hence, each V = V() ∈  can be regarded as a point of this Euclidean space.However, in order to avoid the confusion of elements of  with points of , we refer to V = V() ∈  as function of  ∈ .We define the following norm instead of the usual Euclidean norm in : We define the following closed subset of : We divide  0+ into two disjoint subsets as follows: where The wage equation contains the elasticity of substitution , the manufacturing expenditure , and the transport cost function  = (, ), which is a known function of (, ) ∈  × .We assume that (, ) =  (, ) ∀,  ∈ .

Result and Discussion
Noting that (17) contains the singular term (1/()) −1 , we define that if  = () ∈  + satisfies the wage equation ( 14) for all  ∈ , then  = () is a solution.Each shortrun equilibrium is defined as a solution to the wage equation.Define Making use of (4), we divide () into two disjoint subsets as follows: where The following theorem is the main result, which will be proved in Sections 4 and 6 with the assist of Section 5.
Applying (11) to the definition (18), we see that (22) implies that if the economy is in a short-run equilibrium, then the average of nominal wages is identically equal to 1. Theorem 1 is an extension of previous research.
Let us discuss the lemma above.Noting that (15) contains (17), we see that ( 29) is a singular nonlinear operator expressed in terms of the double summation.Hence, the left-hand side of ( 35) is expressed in terms of the singular triple summation.However, the right-hand side of (35) has no singularity and is expressed in terms of the single summation of (16).Hence, the equality (35) can transform the singular triple summation into the single summation with no singularity.

Set-Valued Operator
The purpose of this section is to extend the wage operator (29) as a set-valued operator from () to 2 () .Considering Theorem 1(ii), we find it reasonable to seek a fixed point of the wage operator () in (18).Let us obtain estimates for () in (20).

Convergence of a Sequence of Solutions
The following lemma shows that the set-valued operator F() satisfies the conditions of the Kakutani fixed-point theorem.
(iv) F() is a convex subset of () for every  ∈ ().This lemma is proved in the Appendix.
Applying this result and (62) to (64), when  → +∞, and performing the same calculations as those when proving (34), we see easily that Hence we obtain Theorem 1(i) when (59) holds.