Existence and computation of solutions to the initial value problem for the replicator equation of evolutionary game defined by the Dixit–Stiglitz–Krugman model in an urban setting: Concentration of workers motivated by disparity in real wages

Highlights

• An evolutionary game whose payoffs are defined by a spatial economic model.

• The replicator equation contains an operator mapping an unknown function to a payoff function.

• We obtain a numerical solution to the initial value problem for this equation.

• A global solution converges to an equilibrium attained when all workers are concentrated at a point.

Abstract

Consider an evolutionary game whose payoffs are defined as the distribution of real wages. The distribution of real wages is determined by the Dixit–Stiglitz–Krugman model in an urban setting, and workers (players) move toward points that offer higher real wages and away from points that offer below-average real wages. This game is described by the replicator equation whose unknown function denotes the distribution of workers. The growth rate of population contains an operator that maps an unknown function to the distribution of real wages. We prove that if the elasticity of substitution and the transport costs are sufficiently small, then the initial value problem for this equation has a unique global solution. We obtain a numerical solution by making use of an iteration scheme. We prove estimates for approximation error in this numerical solution. Moreover we prove that if workers are concentrated at a point at the initial time, then the global solution converges to a long-run equilibrium attained when all workers are concentrated at the point. The highest growth rate is attained at the point and the pure best reply is given at the point.

Introduction

The purpose of spatial economics is to study the spatial allocation of scarce resources and the location of economic activities. In this area Krugman began new important researches with emphasis laid on the formation of a large variety of economic agglomeration and on the clustering of economic activities. His remarkable studies succeeded admirably in giving a new useful analytical framework to spatial economics, and attracted a large number of social scientists. These studies have grown as one of the major branches of spatial economics, and now this new branch is referred to as the New Economic Geography (the NEG). For his great contribution to spatial economics, Krugman was awarded the Nobel Prize in Economic Sciences (Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel) in 2008 (see [4], [5], [6], [10] and [11]). A large number of quite new kinds of functional equations have been considered in the NEG (see, e.g., [4], [5]). However, many mathematical issues on them are still unresolved. We consider that the NEG is a new frontier of the theory of functional equations.

The origin of the NEG is the Krugman’s core-periphery model (the CP model). A large number of mathematical models have been constructed as extensions of the CP model. For example, in [4], [13], extensions to the case of an infinite number of locations along a circle have been analyzed. Among these models, one of the most important models is the Dixit–Stiglitz–Krugman model (the DSK model). The DSK model is regarded as an extension of the CP model to the case of a finite number of points (see [9] and [4, p. 61]). Mathematical aspects and numerical methods of the DSK model have been studied in [14], [17], [18], and [19]. In these studies the DSK model is analyzed under the restrictive condition that the distribution of workers is a known function independent of the time variable. In the real world workers migrate in order to seek higher real wages, but such an important phenomenon is not considered in these studies. Hence we need to consider movement of workers in the DSK model.

Evolutionary game theory plays a critical role in almost all scientific disciplines such as biology, sociology, physics, and economics (see, e.g., [3], [16], [23], [24], [25], [26], [27], [28], [29], [30]). In order to consider movement of workers, Krugman constructs an evolutionary game by combining the DSK model with the replicator dynamics in [4, p. 62, p. 77]. We refer to this evolutionary game as the DSK evolutionary game (the DSKEG). This evolutionary game is studied fully by simulations in [4], [10]. In [1], [8] the dynamic properties of the CP model have been analyzed. However, there have been few studies on mathematical foundations for the DSKEG.

Let us discuss the DSKEG. In the DSKEG the pure-strategy set is defined as a finite set of points contained in an euclidean space. If the dimension of the euclidean space exceeds 2, then the model is unrealistic from economic point of view. We accept such a case for mathematical generality. The payoffs are defined as the distribution of real wages. The replicator dynamics is described by the replicator equation, which is one of the most important differential equations in evolutionary game theory (see [2], [16, pp. 55–57], [7, p. 147] and [30, p. 73]). The replicator equation contains the distribution of workers as an unknown function and the distribution of real wages as a known function, and the growth rate of population is defined as the difference between the distribution of real wages and the population average real wage (see [4, (5.1), (5.2)]). Hence, workers (players) move toward points that offer higher real wages and away from points that offer below-average real wages (see [4, p. 62]).

The distribution of real wages is given by the DSK model. The DSK model consists of two economic sectors, a monopolistically competitive sector (manufacturing) and a perfectly competitive sector (agriculture) (see [4, p. 45, p. 61]). In this paper for simplicity we assume that the economy has no agriculture, i.e., we consider the DSK model in an urban setting in the same way as [19, Assumption 1.2]. This assumption is sufficiently reasonable. Because this assumption is accepted in [4, p. 331] also, and the DSK model with such an assumption actually corresponds to Krugman’s model of international trade with imperfect competition (see [12], [15]). It is proved in [19] that if the DSK model has no agriculture, then the distribution of real wages is defined as a solution of the real wage equation. We see that the replicator equation contains the distribution of real wages that is obtained by solving the real wage equation which contains the distribution of workers as a known function (see [19, (5.1)]).

By making use of [19, Theorem 3.1], we can define an operator that maps the distribution of workers to the distribution of real wages by solving the real wage equation uniquely. By substituting this operator in the growth rate of the replicator equation, we can transform the replicator equation into a nonlinear ordinary differential equation whose coefficient contains the operator that maps an unknown function to the distribution of real wages. We refer to this equation as the DSKEG replicator equation in order to distinguish it from the replicator equation of usual evolutionary game theory that contains an unknown function explicitly in its coefficient (see, e.g., [30, (3.3)] and [16, pp. 55–57]). The DSKEG replicator equation is quite a new kind of nonlinear ordinary differential equation. The DSKEG is described by the initial value problem for the DSKEG replicator equation.

In [20] we study an extension of the CP model to the case of a bounded continuous domain. In [21] and [22] we study a spatially continuous evolutionary game constructed by combining this spatially continuous model with the replicator equation. However, this spatially continuous evolutionary game is described not by a nonlinear ordinary differential equation but by a nonlinear integro-partial differential equation. Hence, we cannot apply the methods developed in [21] and [22] to the DSKEG replicator equation.

In this paper, we prove that if the elasticity of substitution among varieties of manufactured goods and the transport costs incurred in dispatching manufactured goods are sufficiently small, then the initial value problem for the DSKEG replicator equation has a unique global solution (Theorem 8). By making use of an iteration scheme, we obtain a numerical solution to the initial value problem. Moreover, taking approximation error into account, we obtain estimates for the difference between this numerical solution and the solution obtained in Theorem 8. Numerical simulations performed by this iteration scheme help spatial economists to understand economic aspects of the DSKEG. Moreover we prove that if workers are concentrated at a point at the initial time, then the global solution converges to a long-run equilibrium attained when all workers are concentrated at the point (Theorem 9). It follows from Theorem 9 that the highest growth rate is attained at the point and the pure best reply is given at the point (see [30, p. 73]).

This paper has eight sections including this Introduction. In Section 2 we introduce the DSKEG and the DSKEG replicator equation. In Section 3 we state Theorem 8, Theorem 9. In Section 4 we discuss the main theorems. In Section 5 we analyze the real wage equation. In Section 6 we construct an iteration scheme in order to prove that the initial value problem for the DSKEG replicator equation has a global solution. In Section 7 we obtain numerical solutions by making use of this iteration scheme. Moreover we obtain estimates for the difference between the numerical solutions thus obtained and the solution proved in Theorem 8. In the last section we prove the main theorems.