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

Authors: Minoru Tabata; Nobuoki Eshima
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## 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.


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## Body

## 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 theNew 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, theDixit-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 N points, where N is a natural number and each point represents aregion. The population consists of farmers and workers. The DSK model is described by thewage 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 adouble 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 numberN increases. In fact, there are ample analytical results for N=2, whereas there are much fewer ones for N=3 (see, e.g., [11–14]). For N=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 numberN is large. This article deals with existence ofshort-run equilibrium of the DSK model with no restriction on N, 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 whenN=2,3 [15, 16]. However, it is difficult to apply the methods used in those studies to the DSK model when N 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 N≥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 N≥2, it has not been proved yet under general conditions.In this paper, under the most general assumptions (Condition1 and (7)–(10)), we prove that the DSK model has a short-run equilibrium for every N≥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 Section2, 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

ByD we denote a finite set consisting of N points, where N is an arbitrary integer such that(1)N≥2.Each point of D represents a region where manufacturing and agriculture are conducted. By L we denote the set of all real-valued functions v=v(r) of r∈D. We regard L as an N-dimensional Euclidean space. Hence, each v=v(r)∈L can be regarded as a point of this Euclidean space. However, in order to avoid the confusion of elements of L with points of D, we refer to v=v(r)∈L asfunction of r∈D. We define the following norm instead of the usual Euclidean norm in L:(2)v≔maxs∈D⁡vs.We define the following closed subset of L: (3)L0+≔v=vr∈L;vr≥0∀r∈D. We divide L0+ into two disjoint subsets as follows:(4)L0+=L+∪L0,where(5)L+≔v=vr∈L0+;vr>0∀r∈D,(6)L0≔v=vr∈L0+;vr=0for  somer∈D.The wage equation contains theelasticity of substitutionσ, themanufacturing expenditureμ, and thetransport cost functionc=c(r,s), which is a known function of (r,s)∈D×D. We assume that(7)0<μ<1,(8)σ>1,(9)cr,s≥0∀r,s∈D,(10)cr,s=cs,r∀r,s∈D.Moreover, the wage equation contains the distribution of workersλ=λ(r) and the distribution of farmers ϕ=ϕ(r), which are known functions of r∈D. We assume the following condition in addition to (1) and (7)–(10).Condition 1.
One has(11)λ=λr∈S,(12)ϕ=ϕr∈S,where(13)S≔v=vr∈L0+;∑s∈Dvs=1.Condition1, (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)]:(14)wr=fwr1/σ,where w=w(r) denotes an unknown function that describes the distribution of nominal wages and f=f(u) denotes the following discrete nonlinear operator:(15)fu=fur≔∑s∈DYusGu;sσ-1e-σ-1cr,s.Here Y=Y(u(s)) and G=G(u;s) are operators defined as follows:(16)Yus≔μλsus+1-μϕs,(17)Gu;sσ-1≔1∑t∈Dλt1/utσ-1e-σ-1cs,t,which denoteincome andprice index at point s∈D, 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 (1/u(t))σ-1, we define that if w=w(r)∈L+ satisfies the wage equation (14) for all r∈D, then w=w(r) is asolution. Each short-run equilibrium is defined as a solution to the wage equation. Define(18)Sλ≔v=vr∈L0+;∑s∈Dλsvs=1.Making use of (4), we divide S(λ) into two disjoint subsets as follows:(19)Sλ=S+λ∪S0λ,where(20)S+λ=Sλ∩L+,(21)S0λ=Sλ∩L0.The following theorem is the main result, which will be proved in Sections4 and 6 with the assist of Section 5.Theorem 1.
If Condition1, (1), and (7)–(10) hold, then the following statements (i)–(iii) hold:
(i) The wage equation (14) has a solution w=w(r)∈L+.
(ii) If the wage equation (14) has a solution w=w(r)∈L+, then(22)w=wr∈S+λ.
(iii) If the wage equation (14) has a solution w=w(r)∈L+, then(23)w_≤wr≤w¯∀r∈D,where(24)w_≔exp⁡-σ-1C,w¯≔exp⁡σ-1C,(25)C≔maxr,s∈D⁡cr,s.Let us compare this theorem with the previous research [17–19]. Indeed it is proved in [17–19] that (14) has a solution w=w(r)∈L+. However, it is assumed in [17, (2.10), Theorem 3.1] that μ=1in addition to (1) and (8)–(12). It is assumed in [18, Theorem 3.1] that C is so small that the following inequality holds in addition to Condition 1, (1), and (7)–(10): (26)C<log⁡1/μσ-1. Moreover, it is assumed in [19, Theorem 3] that μ is so small that the following inequality holds in addition to Condition 1, (1), and (7)–(10): (27)0<μ<minr∈D⁡λr1+minr∈D⁡λr. 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 w_ and w¯ are independent of μ∈(0,1). 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 Theorem1(ii)(iii). Making use of (15), we define that if(28)u∈L+,then(29)Fu=Fur≔fururσ-1∀r∈D.We refer to F=F(u) as thewage operator. Noting that (17) and (29) contain (1/u(r))σ-1, we see that (29) can be defined for all u∈L+, but that no u∈L0 can be substituted in (29) (recall (5) and (6)).We see easily that ifw=w(r)∈L+ satisfies that(30)wr=Fwr∀r∈D,then w=w(r) is a solution to (14) and that if w=w(r)∈L+ is a solution to (14), then it satisfies (30). Hence, we seek a fixed point of F(u). Let us inspect (29) closely. In the next section, we make use of the following lemma.Lemma 2.
If Condition1, (1), (7)–(10), and (28) hold, then(31)Fu∈L+,(32)limU∈L+,U→u⁡FUr=Fur∀r∈D.Proof.
Applying (7)–(12) to (15)–(17), we see easily that(33)fu∈L+∀u∈L+,(34)limU∈L+,U→u⁡fUr=fur∀u∈L+.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 Condition1, (1), (7)–(10), and (28) hold, then(35)∑r∈DλrFur=Yu,where(36)Yu≔∑s∈DYus.Proof.
Substituting (15) and (29) in the left-hand side of (35), we obtain (37)∑r∈DλrFur=∑r∈D∑s∈Dλr1urσ-1YusGu;sσ-1e-σ-1cr,s. Exchange ∑r∈D and ∑s∈D 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 1/G(u;s)σ-1 and G(u;s)σ-1, 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 Theorem1(ii).
Letw∈L+ be a solution to (14). Recalling that (14) implies (30), we substitute (30) in (35) with u=w. Hence we see that (38)∑r∈Dλrwr=Yw. Applying (12), (13), and (16) to (36) with u=w, we obtain (39)Yw=μ∑r∈Dλrwr+1-μ. 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 Theorem1(iii) (recall (24)).Lemma 4.
If Condition1, (1), and (7)–(10) hold, and u=u(r)∈L+ satisfies(40)u∈Sλ,(41)U1≤ur≤U2∀r∈D,where Ui,i=1,2, are constants such that 0<U1≤U2, then(42)w_U1σ-1≤fur≤w¯U2σ-1∀r∈D.Proof.
It follows from (8), (9), and (25) that(43)exp⁡-σ-1C≤exp⁡-σ-1cr,s≤1∀r,s∈D.Applying (24), (41), and (43) to (17), we obtain (44)U1σ-1∑t∈Dλt≤Gu;sσ-1≤w¯U2σ-1∑t∈Dλt∀s∈D. Applying (11) and (13) to this inequality, we obtain(45)U1σ-1≤Gu;sσ-1≤w¯U2σ-1∀s∈D.
Applying (7), (11), (12), and (13) to (16), we see that(46)each  term  ofYusis  nonnegative  for  allus∈L0+.Hence, each term contained in the summation (36) is nonnegative. Applying this result, (24), (43), and (45) to (15), we obtain(47)w_U1σ-1Yu≤fur≤w¯U2σ-1Yu∀r∈D.Applying (7), (12), (13), (16), and (18) to (36), we easily deduce that(48)Yu=1∀u∈Sλ.Applying this result and (40) to (47), we obtain (42).Proof of Theorem1(iii).
By (22) we can substitute(49)U1≔minr∈D⁡wr,U2≔maxr∈D⁡wr,ur≔wr,in (40)–(42). Substituting (14) in the inequality thus obtained, we see that (50)w_U1σ-1≤wrσ≤w¯U2σ-1∀r∈D. Applying (49) to w(r) contained in this inequality, we deduce that (51)w_U1σ-1≤U1σ,U2σ≤w¯U2σ-1. Solving these inequalities with respect to Ui,i=1,2, 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 S(λ) to 2S(λ). Considering Theorem 1(ii), we find it reasonable to seek a fixed point of the wage operator F(u) in (18). Let us obtain estimates for F(u) in (20).Lemma 5.
If Condition1, (1), and (7)–(10) hold, then(52)Fu∈S+λ∀u∈S+λ.Proof.
Substitute (48) with u∈S+(λ) in the right-hand side of (35). Applying the definition (18) to the equality thus obtained, we see that(53)Fu∈Sλ∀u∈S+λ.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 (1/u(r))σ-1, we see that F(u) can be defined for all u∈S+(λ)⊂L+, but that no u∈S0(λ)⊂L0 can be substituted in F(u). Hence, in order to analyze such a singularity, we extend the wage operator F(u) as a set-valued operator from S(λ) to 2S(λ). Noting (19), we define this set-valued operator, which is denoted by F=F(u):S(λ)→2S(λ), as follows.Definition 6.
(i)(54)Fu≔Fufor  everyu∈S+λ,
(ii)(55)Fu≔v∈S0λ;D0v⊇D+ufor  everyu∈S0λ, where(56)D+v≔r∈D;vr>0,(57)D0v≔r∈D;vr=0.


## 6. Convergence of a Sequence of Solutions

The following lemma shows that the set-valued operatorF(u) satisfies the conditions of the Kakutani fixed-point theorem.Lemma 7.
If Condition1, (1), and (7)–(10) hold, and(58)λ=λr∈S∩L+,then the following statements (i)–(vi) hold:
(i)S(λ) is a nonempty, compact, and convex subset of the Euclidean space L.
(ii)F(u) is a singleton subset of S+(λ)⊂S(λ) for every u∈S+(λ).
(iii)F(u) is a nonempty subset of S0(λ)⊂S(λ) for every u∈S0(λ).
(iv)F(u) is a convex subset of S(λ) for every u∈S(λ).
(v)F=F(u) has a closed graph.
(vi)F=F(u) has no fixed point in S0(λ).This lemma is proved in the Appendix.Proof of Theorem1(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 F=F(u). Hence we see that F=F(u) has a fixed point in S(λ). Making use of (19) and Lemma 7(vi), we see that F=F(u) has a fixed point w=w(r)∈S+(λ). Applying Definition 6(i) to this result, we see that the fixed point w=w(r) is a solution to (30). Hence, w=w(r) is a solution to (14). Hence, we obtain Theorem 1(i) when (58) holds.
By making use of Theorem1(i) with (58), we prove Theorem 1(i) when(59)λ=λr∈S∩L0.Recall (16). We denote f(u) by f(u;λ) in order to emphasize that (15) contains λ=λ(x). Defining (60)λn=λnr≔λr+1/n1+N/n,for  eachn∈N, and making use of (11) and (13), we see easily that(61)λn=λnr∈S∩L+,∀n∈N,(62)λn-λ⟶0asn⟶+∞.Substitute λ(r)=λn(r) 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 (63)w=fw;λn1/σ, for each n∈N. Hence, we can define a sequence of solutions {wn} such that(64)wn=fwn;λn1/σ∀n∈N.Applying (23) to this sequence of solutions, we see that (65)wn=wnr∈Lw_,w¯,∀n∈N, where (66)Lw_,w¯≔u=ur∈L;w_≤ur≤w¯∀r∈D. Noting that L(w_,w¯) is a compact subset of L, we see that {wn} contains a convergent subsequence. Denoting this subsequence by the same symbol {wn}, we see that (67)w∞≔limn→∞⁡wn∈Lw_,w¯⊂L+. Applying this result and (62) to (64), when n→+∞, and performing the same calculations as those when proving (34), we see easily that (68)w∞=fw∞;λ1/σ. Hence we obtain Theorem 1(i) when (59) holds.


## 7. Conclusion

Under the most general assumptions (Condition1, (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)).



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