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Growth and Collapse of Empires: A Dynamic Optimization Model

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Abstract

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.

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Notes

  1. Territory is, however, considered in regional science, starting from the German school (work of [1]), followed by [2], and recently also in economic geography, see e.g., [3, 4].

  2. We owe the editor, Professor Franco Giannessi, the following quote from Volterra, “Empires die, the Theorems of Euclid maintain eternal youth,” when offered the presidency of the Academy of Italy countering Mussolini’s view about the unsure foundations of mathematics.

  3. “Der Krieg ist eine blosse Fortsetzung der Politik mit anderen Mitteln”— [9].

  4. Therefore, colonialism does not fit into our explanation.

  5. There exist mathematical models of military actions: for example, [18] laws about relative military strength and [19]. The law of [9] about the triple advantage for a successful attack is tactically correct, but here we want to model the long-term average territorial gain via the difference in military potentials on the border. Formally our law is similar to the physical law of oil moving in a pipeline; the speed of the viscous fluid is proportional to the difference in pressures.

  6. A cubic relationship also follows when computing the transport costs from all uniformly distributed points of the square to the center (the origin in \( \mathfrak {R}^{2}\) or respectively from the center to each point of the square) assuming the Euclidean distance,

    $$\begin{aligned} \underset{-R/2}{\overset{R/2}{\int }}\underset{-R/2}{\overset{R/2}{\int }} \sqrt{x^{2}+y^{2}}\mathrm{d}x\mathrm{d}y=\hbox {const}\;R^{3}. \end{aligned}$$

    The results do not depend on the order 3 of the transport costs but hold for all powers greater than 2.

  7. Logarithmic utility is the limiting case for unitary constant relative risk aversion and is frequently used as a simplifying assumption in many growth models.

  8. Therefore, the state constraint, \(R\ge 0\) corresponding to the nonnegativity of the distance, can be ignored.

  9. e.g., [29,30,31] are early, and indeed the first, representations of the shallow-lake model, e.g., [32, 33], are later examples.

  10. The same outcome is observed in the model of [11], where there is no concern about consumption. That dynamic model assumes spending of all surplus on defense and transport and gives convergence to \(R_{2}\) as the only stable solution of the dynamic equation (it is not a dynamic optimization problem!).

  11. [36] is a compilation of the major empires in history, of which the British one was the largest with \(S=33.7 \Longrightarrow R=5.80\), but it does not fit our assumption of pre-industrial economies. See also the related YouTube videos, see [37].

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Acknowledgements

The authors are grateful to Werner Richter for English corrections and to Andrij Halushka for historical comments. The paper has been presented by different authors at various conferences, seminars and workshops at the Vienna University of Technology, the Vienna Institute of Demography, the Vienna University of Economics, the University of Barcelona, the University College Dublin, and at the Moscow Financial University. We thank the participants for the helpful discussions. Last but not least, we thank two referees for their very helpful and in one case very detailed comments and also an Associate Editor for his/her suggestions.

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Authors and Affiliations

  1. University of Vienna, Vienna, Austria

    Yuri Yegorov, Magda Mirescu & Franz Wirl

  2. International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria

    Dieter Grass

  3. Vienna University of Technology, Vienna, Austria

    Gustav Feichtinger

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Correspondence to Yuri Yegorov.

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Communicated by Jean-Pierre Crouzeix.

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Yegorov, Y., Grass, D., Mirescu, M. et al. Growth and Collapse of Empires: A Dynamic Optimization Model. J Optim Theory Appl 186, 620–643 (2020). https://doi.org/10.1007/s10957-020-01719-5

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