economic geographical model of historical dynamics of

Economic Geographical Model of

Historical Dynamics of Countries

Yuri Yegorov (with F.Wirl, D. Grass, M. Mirescu,

G. Feichtinger)

Viennese Vintage Workshop “Heterogeneous Dynamic Models of Economic and Population

Systems”

5-6 December 2019

Yegorov Yuri, Vintage Workshop, 5-6 December 2019

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Abstract

The paper models spatial evolution of countries and accounts foreconomics, geography and military force. The economy isagricultural, without technological progress, and is modeled byspatially distributed AK model, where production is proportionalto the land size. There are two types of costs: defense (with IRS)and transport (with DRS). A country is an open system, and itsborders can change over time depending on military pressurefrom both sides. A king maximizes discounted utility flow fromtwo terms: country size (his wealth) and per capita consumptionof the population of his country. The models exhibits multipleequilibria, with a possibility to have both small and large countriesfor a rich set of parameters. Historical applications regardingdynamics of empires are discussed.

2Yegorov Yuri, Vintage Workshop, 5-6

December 2019

This presentation is based onThree papers:

1. Yegorov Y. (2018) Modelling of Growth and Collapse of Empires”, presentation at the 14th Viennese conference on Optimal Control and Dynamic Games, 3-6 July 2018.

Empire model set up, but without dynamic optimization

2. Yegorov Y., D. Grass, M. Mirescu, G.Feichtinger and F.Wirl (2019) Growth and Collapse of Empires: A Dynamic Optimization Model. - Vienna Institute of Demography, 05/2019.

Working paper in ORCOS and VID, written in 2019, submitted

3. Yegorov Y., Wirl F. (2019) Dynamics of Countries: Economic-Geographical Model. – Working paper (in progress)

Paper extends [1], includes basic results on [2] and is oriented on economists and historians


December 2019

Introduction

• The work on empire models was started by Yegorov (2018). It was non-optimization model but a spatial dynamics, given by ODE with one stable (large) and one unstable (small) equilibrium. The work was presented in Vienna conference.

• The 2nd step with formulation and solution of dynamic optimization problem for a king. This work was done with my coauthors – G.Feichtinger, D.Grass, M.Mirescu, F.Wirl. This is working paper in VID and is submitted to journal. But this version is too mathematical.

• This paper is oriented for economists and historians. It just summarizes DO results but moves deeper in the roots of the model and its historical applications.


December 2019

Motivation

• Empire can be defined as an open system, characterized by collective economic, political and military activity of its members, and determined by its territory and borders.

• The objective of this paper is to explain such stylized historical evolutions and patterns by accounting for economics, geography and military force.

• We contribute to the existing literature by integrating different aspects from different fields like: (i) socio-physics in the derivation of the core equations, (ii) economicsdetermining output and budget constraints, (iii) political economics by solving for an emperor's optimal intertemporal strategy, (iv) interpreting our results in terms of historical patterns.


December 2019

Roots of the Model

There are several backgrounds for this model.

1. German school of regional science. Works of Christaller and Losch about tessellation of space (1930s)

2. Microeconomic concepts of IRS and DRS

3. Macroeconomic model of endogeneous growth (Barro, Sala-i-Martin). Set up of dynamic optimization model. But there are several differences here:

• a) there is spatial distribution of production,

• b) there is no capital accumulation, but a possibility of territorial expansion using military sector.

4. Cliometrics and cliodynamics as an approach to use economic models for history


December 2019

Scale Economies

Hal Varian presents such picture about average cost in his textbook. Let y be size R. Since minimal defense grows slower than output, its average cost is like on Fig. A. Since transport cost grows faster (with R) than output, its average cost is like on Fig. B. The sum of these average costs behave as on Fig. C. For any price p of output, AC<p in the interval R1<R<R2.


December 2019

Assumptions of the Model

• Empire model has several blocks including economic and geographical. It is modeled as a square in continuous space.

• Territory is given by R2. The border length is 4R.

• Production is dispersed in space. A county can be viewed as a combination of many identical equidistant villages. Population density is constant. The territory S produces the output by AK technology (linear in land): Y=AkS. But here capital is not accumulated, while territory can be added.

• The output is spent for consumption C, reinvestment I, defense W and transport cost (cubic in R): Y = C + I +W + T.

• There is defence sector (described in the next slide).


December 2019

Defence• Historical nations were groups of people who engaged in

insurance contract and provided a collective public good called "defence“.

• Military activity also included a possibility of expansion.

• The speed of expansion (V ) is determined by the difference in military pressures on the border. The territorial gain per unit of time depends on military budget and external pressure Po.

• The speed of expansion, V, is a linear gain in territory, orthogonal to a border, per unit of time. Pressure P on the border is defined as defence density per unit of length: P=D/L.

• The speed of expansion is proportional to the difference in pressures. Finally, territorial evolution is given by:


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Geography and Dynamics of Empire


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R

Area S=R*R.

Area gain after

expansion

dS=2RdR

Difference of

pressures at the

border defines the

speed of expansion

The square shape is used for mathematical simplicity. For a struggle of

2 empires (future work) it is better to use a rectangular shape.

Tessellations

• The main tessellations: triangular (left), square (center), hexagon (right).

• While circles introduced by von Thunen (1826) are natural, they cannot model spatial division across countries, while tessellations can.

• Further a model of a country as a square will be used, due to lower complexity of formulae.


December 2019

Spatial filling of a country

• Horses here symbolize military force, while cows - consumption. A unit of land can give food either for 1 cow or for 1 horse. Horses are on the border and symbolize military protection. The ratio of cows/horses is 1/8 on the left picture (very small country), but grows to 4/16 for a larger country. This figure illustrates scale economies in defense.


December 2019

Transport costs

• Total transport costs are proportional to the average distance from the center, cR, and to the mass, proportional to territory, RxR. If the scale grows, R1<R2<R3, the total transport cost grow as a cube of this scale.


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R1

R2R3

Budget of small and large countries

Typically a small country has the largest fraction of budget on defense (left Fig.), while a large country – on transport (right Fig.)

Here consumption and investment are 20% in both cases, but consumption franction can differ.


December 2019

Preferences for a Country

• A country has a leader (king) who cares both about its land size and average consumption of people.

• It is quite a natural assumption. In old times practically all the value has been derived from land. Even castles were location specific and thus were an immobile investment. The trade and financial activities were small fractions of GDP before the 18th century.

• Country leader should have the following intertemporal preferences:


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Dynamic Optimization Problem

• This dynamic optimization problem of an emperor maximizes the discounted utility flow coming from territory and the logarithm of consumption per capita, subject to dynamic equation for territorial expansion.

• The feasible region [R1 < R < R2] for this DO problem is given by the equation:


December 2019

The Canonical System

• This DO problem generates the following canonical system.

• Its steady states are the solutions of the following 4th order polynomial.


December 2019

The Base Case and Skiba Points • We calibrate our model so that the unit of time is 1 year and

the unit of distance (R=1) corresponds to 1000 km. Base Case parameters are given by Table 1.

• We have a Skiba point for this case.

• The next slides show a set of different pictures that have been obtained by numerical calculations of optimal solutions using our analytical formulae above.

• First the Skiba point is shown along with attractor domains.


December 2019

Skiba Point


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Bifurcation Diagram for beta


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Bifurcation Diagram for Po


December 2019

Example of Empires. Roman


December 2019

Example of Empires. Russian


December 2019

Historical Maps. 1

Fig. H1. 3000 years ago (1000 BC) there were many countries, and there were no empires (R>1, or S>1 mln. sq. km). Source: History of the world: every year.https://www.youtube.com/watch?v=ymI5Uv5cGU4


December 2019

https://www.youtube.com/watch?v=ymI5Uv5cGU4

Historical Maps. 2

Fig. H2. 2500 years ago (500 BC) there was on large empire (Achaemenid, R>2) coexisting with many small countries, including Greece


December 2019

Historical Maps. 3

Fig. H3. 1700 years ago (300 AD) Europe was dominated by a large Roman empire, but there were also empires on the territories of modern Iran, India and China. Few small countries coexisted with them.


December 2019

Historical Maps. 4

Fig. H4. 1100 years ago (900 AD) Europe was very segmented (R<1), but there were empires in Asia (with 1<R<2). This is a typical historical pattern of coexistence of small and large countries.


December 2019

Policy Issues


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• We can observe 3 types of solutions: a) existence of only small country, b) existence of only large country (empire), c) coexistence of both small and large countries (most typical in history).

• Each type of solutions has its ecological niche (where it exists).

• The sizes of optimal small countries typically do not change much when parameters vary. The sizes of empires vary a lot with parameter change.

• It takes a lot of time (150-250 years) to build an empire. One emperor cannot do it, and similar preferences (with low discount) for a dynasty are important.

• Economic development goes in line with a decline in transport cost. As one can see from bifurcation diagram, for high costs, beta>0.33, only a small country exist. We see this before 500-800 BC, which is the moment of emergence of the first large empire (Achaemenid). Further in time (for lower costs) we have coexistence of empires and small countries.

• In some periods we observe higher military tension, and a small country becomes larger, in order to survive. Border solutions are unstable.

Conclusions


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• The objective of this paper is to offer an economic explanation of the formation (and collapse) of empires.

• The model uses a homogeneous space, a single production factor, and accounts for increasing and decreasing returns to geographical scale, economic constraints on military expenditures, and the preferences of an emperor for a large country and the consumption of his people.

• The crucial implication is that two viable long run outcomes exist: a small country and a large empire. Both outcomes are separated by a threshold. They have different ecological niches in the parameter space.

• Favorable for the build up of an empire are: low discounting, e.g., caring about one's dynasty, striving more for grandness than for caring about the people, and last but not least sufficient a sufficient size. Low productivity, high transport costs, a weak military, and strong pressure from neighboring countries limit the size of empires.

• Since we offer just a first try at what we think is an interesting and interdisciplinary topic, there are many areas for extensions.

Thank you for your attention!

You can write your comments to:

[email protected]

My papers can be found on Research Gate:

https://www.researchgate.net/profile/Yuri_Yegorov


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