We consider the extension of a tractable NEG model with a quasi-linear log utility function to continuous space, and investigate the behavior of its solution mathematically. The model is a system of nonlinear integral and differential equations describing the market equilibrium and the time evolution of the spatial distribution of population density. A unique global solution is constructed, and a homogeneous stationary solution with an evenly distributed population is shown to be unstable. Furthermore, it is shown numerically that the destabilized homogeneous stationary solution eventually forms spiky spatial distributions. The number of the spikes decreases as the preference for variety increases or the transport cost decreases.

我们考虑将具有准线性对数效用函数的易处理 NEG 模型扩展到连续空间，并从数学角度研究其解的行为。该模型是一个非线性积分和微分方程组，描述市场均衡和人口密度空间分布的时间演化。构造了唯一的全局解，并且具有均匀分布的群体的齐次平稳解被证明是不稳定的。此外，数值显示不稳定的齐次平稳解最终形成尖峰空间分布。随着对品种偏好的增加或运输成本的降低，尖峰的数量减少。