This paper addresses the spatial evolution of countries accounting for economics, geography and (military) force. Economic activity is spatially distributed following the AK model with the output being split into consumption, investment, transport costs and military (for defense and expansion). The emperor controls the military force subject to the constraints imposed by the economy but also the geography (transport costs, border length) and the necessity to satisfy the needs of the population. The border changes depending on how much pressure the emperor can muster to counter the pressure of neighboring countries. The resulting dynamic process determines a country’s size over time. The model leads to multiple steady states, large empires and small countries being separated by a threshold and collapse. The resulting patterns can be linked to historical observations.

中文翻译：

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帝国的成长与崩溃：动态优化模型

本文讨论了考虑经济、地理和（军事）力量的国家的空间演变。经济活动按照 AK 模型进行空间分布，产出分为消费、投资、运输成本和军事（用于国防和扩张）。皇帝控制军事力量，受到经济和地理（运输成本、边界长度）和满足人口需求的必要性的限制。边界的变化取决于皇帝可以承受多大的压力来对抗邻国的压力。由此产生的动态过程决定了一个国家随着时间的推移的规模。该模型导致多个稳定国家、大帝国和小国被一道门槛隔开并崩溃。