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

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.