#### Abstract

Each short-run equilibrium of the Dixit-Stiglitz-Krugman model is defined as a solution to the wage equation when the distributions of workers and farmers are given functions. We extend the discrete nonlinear operator contained in the wage equation as a set-valued operator. Applying the Kakutani fixed-point theorem to the set-valued operator, under the most general assumptions, we prove that the model has a short-run equilibrium.

#### 1. 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, 8–10]).

We note that the insolvability of the DSK model is increasingly problematic as the number increases. In fact, there are ample analytical results for , whereas there are much fewer ones for (see, e.g., [11–14]). For or , 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 [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 short-run equilibrium for every [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 , 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 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 [17–19], and we make use neither of advanced theory of discrete nonlinear equations, of the NEG, nor of fixed-point theory. We make use of only the Kakutani fixed-point theorem, which is one of the most fundamental fixed-point theorems (see, e.g., [20–22]). Hence, this article can be easily understood even without reading [17–19] carefully and even without having an advanced knowledge of discrete nonlinear equations, the NEG, and fixed-point theory.

#### 2. Condition and Equation

By we denote a finite set consisting of points, where is an arbitrary integer such thatEach point of represents a region where manufacturing and agriculture are conducted. By we denote the set of all real-valued functions of We regard as an -dimensional Euclidean space. Hence, each 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 as* function* of We define the following norm instead of the usual Euclidean norm in :We define the following closed subset of : We divide 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

Moreover, the wage equation contains the distribution of workers and the distribution of farmers , which are known functions of We assume the following condition in addition to (1) and (7)–(10).

*Condition 1. *One haswhere

Condition 1, (1), and (7)–(10) are the most general assumptions (see, e.g., [3, pp. 45–49, 61–65] and [4]). No condition is imposed on this paper in addition to these assumptions. The wage equation is the following discrete nonlinear equation [3, (5.5)]:where denotes an unknown function that describes the distribution of nominal wages and denotes the following discrete nonlinear operator:Here and are operators defined as follows:which denote* income* and* price index* at point , respectively [3, (5.3), (5.4)]. The nonlinearity of the wage equation is fully discussed in [19, Section 2].

#### 3. Result and Discussion

Noting that (17) contains the singular term , we define that if satisfies the wage equation (14) for all , then is a* solution*. Each short-run equilibrium is defined as a solution to the wage equation. DefineMaking 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.

Theorem 1. *If Condition 1, (1), and (7)–(10) hold, then the following statements (i)–(iii) hold:**(i) The wage equation (14) has a solution **(ii) If the wage equation (14) has a solution , then**(iii) If the wage equation (14) has a solution , thenwhere*

Let us compare this theorem with the previous research [17–19]. Indeed it is proved in [17–19] that (14) has a solution However, it is assumed in [17, (2.10), Theorem 3.1] that in addition to (1) and (8)–(12). It is assumed in [18, Theorem 3.1] that is so small that the following inequality holds in addition to Condition 1, (1), and (7)–(10): Moreover, it is assumed in [19, Theorem 3] that is so small that the following inequality holds in addition to Condition 1, (1), and (7)–(10): However, as mentioned in the previous section, no condition is imposed on Theorem 1 in addition to Condition 1, (1), and (7)–(10). We note that and are independent of Hence, (23) is more simple and more accurate than [18, (5.1), (5.7)–(5.9), (6.1)]. Moreover, (22) was not proved in [17–19]. 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.

In [17–19], we directly apply the Brouwer fixed-point theorem to the discrete nonlinear operator (15). For this purpose, in [17–19] we need the restrictive conditions in addition to Condition 1, (1), and (7)–(10). In Section 5, we extend the discrete nonlinear operator (29), which is defined in the next section, as a set-valued operator (see Definition 6). By applying the Kakutani fixed-point theorem to the set-valued operator, we prove Theorem 1(i) in Section 6. By this new method, we need no restrictive condition in addition to Condition 1, (1), and (7)–(10).

#### 4. Necessary Conditions

The purpose of this section is to prove Theorem 1(ii)(iii). Making use of (15), we define that ifthenWe refer to as the* wage operator*. Noting that (17) and (29) contain , we see that (29) can be defined for all , but that no can be substituted in (29) (recall (5) and (6)).

We see easily that if satisfies thatthen is a solution to (14) and that if is a solution to (14), then it satisfies (30). Hence, we seek a fixed point of Let us inspect (29) closely. In the next section, we make use of the following lemma.

Lemma 2. *If Condition 1, (1), (7)–(10), and (28) hold, then*

*Proof. *Applying (7)–(12) to (15)–(17), we see easily thatApplying these results and (28) to (29), we obtain (31) and (32) easily.

The following lemma is the key lemma of this paper.

Lemma 3. *If Condition 1, (1), (7)–(10), and (28) hold, thenwhere*

*Proof. *Substituting (15) and (29) in the left-hand side of (35), we obtain Exchange and in the right-hand side, and apply (10) to the right-hand side. Recalling the definition (17), we see that the right-hand side of the equality thus obtained contains both and , which cancel each other out. Hence, the right-hand side is equal to (36). We obtain (35).

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.

*Proof of Theorem 1(ii). *Let be a solution to (14). Recalling that (14) implies (30), we substitute (30) in (35) with Hence we see that Applying (12), (13), and (16) to (36) with , we obtain Combining these equalities, making use of (7), and recalling the definitions (18) and (20), we obtain (22).

By making use of the following lemma, we prove Theorem 1(iii) (recall (24)).

Lemma 4. *If Condition 1, (1), and (7)–(10) hold, and satisfieswhere , are constants such that , then*

*Proof. *It follows from (8), (9), and (25) thatApplying (24), (41), and (43) to (17), we obtain Applying (11) and (13) to this inequality, we obtainApplying (7), (11), (12), and (13) to (16), we see thatHence, each term contained in the summation (36) is nonnegative. Applying this result, (24), (43), and (45) to (15), we obtainApplying (7), (12), (13), (16), and (18) to (36), we easily deduce thatApplying this result and (40) to (47), we obtain (42).

*Proof of Theorem 1(iii). *By (22) we can substitutein (40)–(42). Substituting (14) in the inequality thus obtained, we see that Applying (49) to contained in this inequality, we deduce that Solving these inequalities with respect to , we obtain (23).

#### 5. Set-Valued Operator

The purpose of this section is to extend the wage operator (29) as a set-valued operator from to 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).

Lemma 5. *If Condition 1, (1), and (7)–(10) hold, then*

*Proof. *Substitute (48) with in the right-hand side of (35). Applying the definition (18) to the equality thus obtained, we see thatCombining this result and (31) and recalling the definition (20), we obtain (52).

Considering the definitions (5), (6), (20), and (21) and recalling that (17) and (29) contain , we see that can be defined for all , but that no can be substituted in Hence, in order to analyze such a singularity, we extend the wage operator as a set-valued operator from to Noting (19), we define this set-valued operator, which is denoted by , as follows.

*Definition 6. *(i) (ii) where

#### 6. Convergence of a Sequence of Solutions

The following lemma shows that the set-valued operator satisfies the conditions of the Kakutani fixed-point theorem.

Lemma 7. *If Condition 1, (1), and (7)–(10) hold, andthen the following statements (i)–(vi) hold:**(i) is a nonempty, compact, and convex subset of the Euclidean space **(ii) is a singleton subset of for every **(iii) is a nonempty subset of for every **(iv) is a convex subset of for every **(v) has a closed graph.**(vi) has no fixed point in *

This lemma is proved in the Appendix.

*Proof of Theorem 1(i). *Assume that (58) holds. Making use of Lemma 7(i)–(v), we can apply the Kakutani fixed-point theorem to the set-valued function Hence we see that has a fixed point in Making use of (19) and Lemma 7(vi), we see that has a fixed point Applying Definition 6(i) to this result, we see that the fixed point is a solution to (30). Hence, is a solution to (14). Hence, we obtain Theorem 1(i) when (58) holds.

By making use of Theorem 1(i) with (58), we prove Theorem 1(i) whenRecall (16). We denote by in order to emphasize that (15) contains Defining and making use of (11) and (13), we see easily thatSubstitute in the wage equation (14). Making use of (61), we can apply Theorem 1(i) with (58) to the wage equation thus obtained, that is, to for each Hence, we can define a sequence of solutions such thatApplying (23) to this sequence of solutions, we see that where Noting that is a compact subset of , we see that contains a convergent subsequence. Denoting this subsequence by the same symbol , we see that 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.

#### 7. Conclusion

Under the most general assumptions (Condition 1, (1), and (7)–(10)), we have proved that there exists a short-run equilibrium of the Dixit-Stiglitz-Krugman model (Theorem 1(i)). Moreover we have obtained the accurate estimates for each short-run equilibrium (Theorem 1(ii)(iii)).

#### Appendix

*Proof of Lemma 7(i)(ii). *Recalling that is an -dimensional Euclidean space and making use of (58), we see that (18) is a simplex contained in . Hence we obtain (i) easily. Note that if (59) holds, then (18) is not a simplex. Applying (52) to Definition 6(i), we obtain (ii).

We make use of the following lemma in order to prove Lemma 7(iii)–(vi).

Lemma 8. *If Condition 1, (1), (7)–(10), and (58) hold, then the following statements (i) and (ii) hold:**(i) If , then and are nonempty proper subsets of .**(ii) For each nonempty proper subset , there exists such that *

*Proof. *Making use of (58) and recalling the definitions (6), (18), (21), (56), and (57), we obtain (i) easily. By making use of (58), we can define the following function for each nonempty proper subset : where denotes the set of elements in but not in We see easily that and

*Proof of Lemma 7(iii)(iv). *Making use of Lemma 8(i), we can substitute in Lemma 8(ii) for each Hence, there exists such that Applying Definition 6(ii) to this result, we obtain (iii).

It follows immediately from the inclusion relation mentioned in Definition 6(ii) that is a convex subset of for every It follows from Lemma 7(ii) that is a convex subset of for every Combining these results, we obtain (iv).

*Proof of Lemma 7(v). *Consider arbitrary convergent sequences,such thatMaking use of Lemma 7(i), we see thatwhereWe have only to prove thatRecall (19). Let us prove (A.9) whenRecalling the definitions (5) and (20), we see easily that is the relative interior of the simplex Hence, we see that there exists a positive integer such that Applying this result and Definition 6(i) to (A.5), we see that Applying these results, (A.7), (A.8), and (A.10) to (32) with , we obtain Applying (A.10) and Definition 6(i) to this equality, we obtain (A.9).

Let us prove (A.9) whenMaking use of (19), we can divide (A.3) into two disjoint subsequences as follows: whereApplying Lemma 7(ii)(iii) and Definition 6 to (A.16) and (A.17), we can divide (A.4) into two disjoint subsequences as follows: whereAt least one of (A.16) and (A.17) is an infinite convergent sequence. Assume that (A.16) is an infinite convergent sequence. Applying Lemma 8(i) with to (A.14), we deduce thatLet us obtain estimates for Applying (2), (36), (43), and (46) to (15) with , we see thatApplying this inequality to (29) with , we obtainLet us obtain estimates for Recall (43) and (A.21). Replacing and with and , respectively, in (17) with , we see that Replacing within the right-hand side of this inequality, we deduce thatNoting that is an infinite convergent subsequence of and applying (57), (A.7), and (A.21) to (A.25), we see easily thatIt follows from (58) and (A.21) that the denominator of the right-hand side of (A.26) is a positive constant independent of Applying this result and (A.27) to (A.26), we see thatMaking use of (48), (A.3), and (A.16), we deduce thatApplying Lemma 8(i) to (A.14) in the same way as (A.21), we see thatIt follows from this result, (20), (56), (A.7), and (A.16) thatSubstitute (A.29) in (A.23), and let Noting that the denominator of the right-hand side of (A.23) is equal to and noting (A.30) and (A.31), we restrict the variable within Making use of (A.7), (A.8), (A.28), and (A.31), we see that that is, Recalling (21), (57), and (A.6) and applying (A.30) to (A.33), we see that Applying this result, (A.14), and (A.33) to Definition 6(ii), we obtain (A.9).

Assume that (A.17) is an infinite convergent sequence. Applying Definition 6(ii) to (A.17) and (A.20), we see thatNote that and are infinite subsequences of and , respectively. Recall the definition (56). It follows from (A.7) and (A.8) that there exists a positive integer such that if , thenTaking the complements of both sides of (A.37) and recalling the definitions (56) and (57), we obtainApplying (A.36) and (A.38) to (A.35), we obtain (A.33). In the same way as above we obtain (A.9). Therefore we obtain (v).

*Proof of Lemma 7(vi). *Assume that has a fixed point in , that is, that there exists such that Applying Definition 6(ii) to this assumption, we see that Applying Lemma 8(i) to , we see that both sides of this inclusion relation are nonempty proper subsets of However, it follows from the definitions (56) and (57) that both sides are disjoint from each other. This is a contradiction. Hence we obtain (vi).

#### Data Availability

Data and materials sharing is not applicable to this article as no datasets or materials were generated or analyzed during the current study.

#### Conflicts of Interest

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

#### Authors’ Contributions

All authors contributed equally to this article and read and approved the final manuscript.

#### Acknowledgments

Minoru Tabata and Nobuoki Eshima are supported in part by Grant-in-Aid for Scientific Research of Japan (15K05005, 26330045).