Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 2193070, 8 pages
https://doi.org/10.1155/2018/2193070

Research Article

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

Minoru Tabata¹ and Nobuoki Eshima²

¹Department of Mathematical Sciences, Osaka Prefecture University, 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 6 March 2018; Accepted 10 April 2018; Published 25 June 2018

Academic Editor: Guang Zhang

Copyright © 2018 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.

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 that

Each 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 has

where

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. 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.

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 , then

where

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 if

then

We 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 that

then 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 that

Applying 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, then

where

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 satisfies

where , are constants such that , then

Proof. It follows from (8), (9), and (25) that

Applying (24), (41), and (43) to (17), we obtain

Applying (11) and (13) to this inequality, we obtain

Applying (7), (11), (12), and (13) to (16), we see that

Hence, each term contained in the summation (36) is nonnegative. Applying this result, (24), (43), and (45) to (15), we obtain

Applying (7), (12), (13), (16), and (18) to (36), we easily deduce that

Applying this result and (40) to (47), we obtain (42).

Proof of Theorem 1(iii). By (22) we can substitute

in (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 that

Combining 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, and

then 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) when

Recall (16). We denote by in order to emphasize that (15) contains Defining

and making use of (11) and (13), we see easily that

Substitute 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 that

Applying (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 that

Making use of Lemma 7(i), we see that

where

We have only to prove that

Recall (19). Let us prove (A.9) when

Recalling 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) when

Making use of (19), we can divide (A.3) into two disjoint subsequences as follows:

where

Applying Lemma 7(ii)(iii) and Definition 6 to (A.16) and (A.17), we can divide (A.4) into two disjoint subsequences as follows:

where

At 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 that

Let us obtain estimates for Applying (2), (36), (43), and (46) to (15) with , we see that

Applying this inequality to (29) with , we obtain

Let us obtain estimates for Recall (43) and (A.21). Replacing and with and , respectively, in (17) with , we see that

Replacing with

in the right-hand side of this inequality, we deduce that

Noting that is an infinite convergent subsequence of and applying (57), (A.7), and (A.21) to (A.25), we see easily that

It 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 that

Making use of (48), (A.3), and (A.16), we deduce that

Applying Lemma 8(i) to (A.14) in the same way as (A.21), we see that

It follows from this result, (20), (56), (A.7), and (A.16) that

Substitute (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 that

Note 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 , then

Taking the complements of both sides of (A.37) and recalling the definitions (56) and (57), we obtain

Applying (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).

References

P. Krugman, “The official homepage of the Nobel Prize in Economic Sciences,” 2008, http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/.

G. I. P. Ottaviano and D. Puga, “Agglomeration in the global economy: a survey of the ‘new economic geography’,” World Economy, vol. 21, no. 6, pp. 707–731, 1998.

M. Fujita, P. R. Krugman, and A. Venables, The Spatial Economy: Cities, Regions, and International Trade, MIT Press, 2001.

M. Fujita and J.-F. Thisse, Economics of Agglomeration: Cities, Industrial Location, and Globalization, Cambridge University Press, Cambridge, UK, 2013.

K. Ikeda and K. Murota, Bifurcation Theory for Hexagonal Agglomeration in Economic Geography, Springer, Tokyo, 2014.

T. Akamatsu, Y. Takayama, and K. Ikeda, “Spatial discounting, Fourier, and racetrack economy: a recipe for the analysis of spatial agglomeration models,” Journal of Economic Dynamics and Control, vol. 36, no. 11, pp. 1729–1759, 2012.

R. Forslid and G. I. P. Ottaviano, “An analytically solvable core-periphery model,” Journal of Economic Geography, vol. 3, no. 3, pp. 229–240, 2003.

K. Ikeda, K. Murota, T. Akamatsu, and Y. Takayama, “Agglomeration patterns in a long narrow economy of a new economic geography model: analogy to a racetrack economy,” International Journal of Economic Theory, vol. 13, no. 1, pp. 113–145, 2017.

P. Krugman, The Self-Organizing Economy Blackwell Publishers, Cambridge MA, 1996.

M. Pflüger, “A simple, analytically solvable, Chamberlinian agglomeration model,” Regional Science & Urban Economics, vol. 34, no. 5, pp. 565–573, 2004.

T. Ago, I. Isono, and T. Tabuchi, “Locational disadvantage of the hub,” Annals of Regional Science, vol. 40, no. 4, pp. 819–848, 2006.

S. B. S. D. Castro, J. Correia-da-Silva, and P. Mossay, “The core-periphery model with three regions and more,” Papers in Regional Science, vol. 91, no. 2, pp. 401–418, 2012.

P. Krugman and R. L. Elizondo, “Trade policy and the third world metropolis,” Journal of Development Economics, vol. 49, no. 1, pp. 137–150, 1996.

T. Mori and K. Nishikimi, “Economies of transport density and industrial agglomeration,” Regional Science & Urban Economics, vol. 32, no. 2, pp. 167–200, 2002.

P. Mossay, “The core-periphery model: a note on the existence and uniqueness of short-run equilibrium,” Journal of Urban Economics, vol. 59, no. 3, pp. 389–393, 2006.

N. G. Pavlidis, M. N. Vrahatis, and P. Mossay, “Existence and computation of short-run equilibria in economic geography,” Applied Mathematics and Computation, vol. 184, no. 1, pp. 93–103, 2007.

M. Tabata, N. Eshima, and Y. Sakai, “Existence, uniqueness, and computation of short-run and long-run equilibria of the Dixit-Stiglitz-Krugman model in an urban setting,” Applied Mathematics and Computation, vol. 234, pp. 339–355, 2014.

M. Tabata, N. Eshima, Y. Kiyonari, and I. Takagi, “The existence and uniqueness of short-run equilibrium of the Dixit–Stiglitz–Krugman model in an urban–rural setting,” IMA Journal of Applied Mathematics, vol. 80, no. 5, pp. 474–493, 2015.

M. Tabata and N. Eshima, “Existence and uniqueness of solutions to the wage equation of Dixit-Stiglitz-Krugman Model with no restriction on transport costs,” Discrete Dynamics in Nature and Society, vol. 2017, Article ID 9341502, pp. 1–7, 2017.

K. C. Border, Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, Cambridge, Uk, 1985.

I. Farmakis and M. Moskowitz, Fixed Point Theorems and Their Applications, World Scientific, Singapore, Singapore, 2013.

A. Granas and J. Dugundji, Fixed Point Theory, Springer Science & Business Media, 2013.