Convergence of Global Solutions to the Cauchy Problem for the Replicator Equation in Spatial Economics

We study the initial-value problem for the replicator equation of the -region Core-Periphery model in spatial economics. The main result shows that if workers are sufficiently agglomerated in a region at the initial time, then the initial-value problem has a unique global solution that converges to the equilibrium solution expressed by full agglomeration in that region.


Introduction
The applications of functional equation theory (FET) have been broadened from natural sciences to various disciplines of social sciences.In turn, it accelerates the progress of FET to mathematically study various functional equations considered in social sciences (see, e.g., [1]).If we attempt to study such new applications of FET in social sciences, then we should concentrate on spatial economics.Spatial economics is an interdisciplinary area between economics and geography.Its purpose is to study the location, distribution, and selforganization of economic activities.In about 1990, Krugman began important new research in this interdisciplinary area, and his research has since grown into a major branch of economics known as the New Economic Geography (NEG).In 2008, the Nobel Prize in Economic Sciences was awarded to Krugman for his remarkable contributions.In the NEG, many interesting nonlinear functional equations have been considered.Hence the NEG is one of the most promising fields of application for FET.However, little mathematical research has been conducted on these nonlinear functional equations (see [2][3][4][5][6][7][8][9][10][11][12]).
One of the most important models in the NEG is the region Core-Periphery (NCP) model (see [2][3][4][5][6][7][8][9][10][11][12][13]).The economy in this model consists of agriculture and manufacturing.Agriculture and manufacturing are conducted in a finite set of points that represent economic regions.In this model an equilibrium is studied fully in [2,3,14,15].However, it is more important to consider dynamic models (see, e.g., [16][17][18]).Hence, Krugman constructs an evolutionary game by combining the NCP model with replicator dynamics in [4, p. 77].This evolutionary game is referred to as the NCP evolutionary game (NCPEG), and the replicator equation thus combined is referred to as the NCPEG replicator equation.
This paper deals with the Cauchy problem for the NCPEG replicator equation.This Cauchy problem is studied numerically when the time variable is sufficiently small [4, Chapter 6].However, there are no full mathematical studies on the behavior of global solutions as the time variable tends to infinity.We can derive a simple nonlinear equation from the NCP model in an urban setting, that is, under the condition that the economy has no agriculture [15].By making use of this simplification, we have studied the NCPEG replicator equation [19].However, the condition of an urban setting is too restrictive.Hence, we should study the NCPEG in an urban-rural setting, that is, under the general condition that the economy consists of both manufacturing and agriculture.The method in [19] cannot be applied to the NCPEG in an urban-rural setting, since it essentially depends on the simplification attained by accepting the condition of an urban setting.In what follows, we simply refer to the NCP model in an urban-rural setting, the NCPEG in an urban-rural setting, and the NCPEG replicator equation in an urban-rural setting as the NCP model, the NCPEG, and the NCPEG replicator equation, respectively.No confusion should arise.
The pure-strategy set of NCPEG is defined as a finite set of points (economic regions) at which manufacturing and agriculture are conducted.The distribution of workers is defined as a solution to the Cauchy problem for the NCPEG replicator equation.The coefficient of the NCPEG replicator equation denotes that the growth rate of the worker population choosing a strategy (i.e., an economic region) is equal to the difference between the average payoff and the payoff in the chosen region.Payoffs are defined as the distribution of real wages.Hence, workers move toward regions that offer higher real wages and away from regions that offer below-average real wages [4, p. 62].
If we regard the distribution of workers as a given function in the nominal wage equation of the NCP model, then the equation has a unique solution [14].By calculating the distribution of real wages from this unique solution, we can construct an operator that maps the distribution of workers, which is contained as an unknown function in the NCPEG replicator equation, to the distribution of real wages (payoffs).Therefore, we can regard the NCPEG replicator equation as a nonlinear ordinary differential equation whose coefficient is expressed by the nonlinear operator acting on an unknown function of the equation.Hence, the NCPEG replicator equation is quite a new kind of nonlinear ordinary differential equation, in contrast to the coefficients of the usual replicator equations, which are expressed by explicit functions of an unknown function (see, e.g., [20]).
In this study we prove that if the transport costs are sufficiently small, then the Cauchy problem for the NCPEG replicator equation has a unique global solution (Theorem 1).Furthermore, we prove that if the black-hole condition and the triangular inequality for transport costs hold along with the sufficient conditions imposed on Theorem 1 and if workers are sufficiently agglomerated in a region at the initial time, then the initial-value problem has a unique global solution that converges to full agglomeration in that region as the time variable tends to infinity (Theorem 3).

Fundamental Equations
By  we denote the pure-strategy set of the NCPEG, where  is a finite set of points contained in an Euclidean space.Each point of  represents a pure strategy (region).No restriction is imposed on the total number of regions.For mathematical generality, we impose no restriction on the dimension of the Euclidean space.By () we denote the set of all real-valued functions of  ∈ .By  + () we denote the set of all positivevalued functions of  ∈ .We define the following norms in By Δ we denote the set of all nonnegative-valued functions  = () ∈ () such that |‖‖| = 1.By  1 0+ ([0, +∞)) we denote the set of all nonnegative-valued functions of (, ) ∈ [0, +∞) ×  that are continuous with respect to  ≥ 0 and continuously differentiable with respect to  > 0 for each  ∈ .
The NCP model is described by the nominal wage equation, which is the following nonlinear discrete equation (see [4, (5.5)]): where  = () is an unknown function that denotes the distribution of nominal wages.We denote the distribution of workers by  = () and the elasticity of substitution by .Income  = ((), ()) has the following form (see [4, (5.3)]): where we denote the share of manufacturing expenditure by  and the distribution of farmers by  = ().The price index   =   ((⋅), (⋅); ) is the following nonlinear operator that acts on  = () and  = () (see [4, (5.4)]): We reasonably accept the following condition in the NEG.

Condition 1.
> 1, (5) (, ) = 0 for each  ∈ , (, ) =  (, ) for each ,  ∈ , In [14, (3.8), Lemma 3.2] we define a positive-valued function  = (, ) such that By making use of this function, we assume that the transport costs are sufficiently small compared with the elasticity of substitution  and the share of manufacturing expenditure , as follows.
The NCPEG replicator equation has the following form (see [4, (5.2)]): where This equation is quite a new kind of nonlinear ordinary differential equation as mentioned in the Introduction.We consider the Cauchy problem for (19) with the initial condition, where  0 =  0 () is a given function.

Global Solutions
From [14, (5.11)], (5), and (6), we easily obtain the following inequality: where where = (, ) is defined by (18), and  is a constant dependent on (, , C) such that This theorem is proved in the last section, where the constant  is defined explicitly as a function of (, , C) ∈ (0, 1) × (1, +∞) × (0, +∞).We also use the constant  in Theorem 3. We must note that Conditions 1 and 2 are the same as those accepted in [14] to ensure the existence and uniqueness of a short-run equilibrium in the NCP model.Making use of ( 26) and ( 27), we obtain the following corollary.
From this corollary we see that if workers exist in a region at  = 0, then there are workers in the region for each  > 0.

Convergence of Global Solutions
We require the following condition to prove Theorem 3.

Condition 3.
(, ) ≤  (, ) +  (, ) for each , ,  ∈ , (33) Inequality (33) is the triangle inequality for transport costs.We accept this inequality, since transport costs generally and reasonably increase with distance.If (, ) = | − |, where  is a positive constant, then (8), ( 9), (10) where and  is the positive constant employed in Theorem 1, then the following inequalities hold for each  ≥ 0: where  = (, ) is defined by (18) and fl (1 − ) , (46) This theorem is proved in the last section.Each () ∈ Δ satisfies that It follows from ( 35) and (52) that Making use of ( 5), (6), and (10), we obtain Hence, recalling (32), we deduce that Applying (55) to (53), we can say that workers are sufficiently agglomerated in region  ∈  at  = 0 and that the total number of workers in other regions is sufficiently small at  = 0. Applying (35) and (54) to (46), we obtain Applying ( 56) to (45), we deduce that Λ() is a monotoneincreasing function of  ≥ 0 and that Substitute (49) in ( 2) and ( 4).Making use of Condition 1 and noting that () = 1, we can solve the equations thereby obtained as follows: () =  ∞ (, ).Substituting this solution in (15), we obtain Hence, ( ∞ (, ),  ∞ (, )) expresses the equilibrium of full agglomeration at  ∈ .Moreover, we see easily that (45) is a solution to the Cauchy problem for the logistic equation: Hence, it follows from (57), ( 41), (42), and (43) that the economy converges to a full agglomeration equilibrium such that  = (, ) increases with  ≥ 0 more rapidly than the logistic curve defined by (59).However, from Corollary 2, we see that the economy does not coincide with such an equilibrium completely within a finite time interval.From ( 44) and ( 56) we see that  gives the pure best reply and that the subpopulation associated with  has the highest growth rate.Note that no condition is imposed on  = () in addition to (7) in Theorems 1 and 3.
We make use of the following lemma in order to prove Theorems 1 and 3.
Proof of Theorem 1.In [21, (11) In what follows we can make use of Theorem 1 and Corollary 2. Let  = (, ) be a global solution proved in Theorem 1.