# Demographic-Fiscal Model of the Growth and Collapse of Agrarian States

##### Model Description

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The Demographic-Structural theory is an empirically derived theory of the breakdown and collapse of agrarian states, originally posited by Jack Goldstone's (1991) work on Early Modern Engalnd, France, the Ottoman Empire, and Ming China. Pulling together opposing Marxist and Malthusian expanations of state collapse, the Demographic Structural theory suggests that population pressure among the general populace and among elites places fiscal pressures on the state. When the state becomes insolvent, elite factions inaugurate civil war, which eases population pressure among elites and commoners.

The following model is a simple formalization of the demographic-structural theory (readers are reffered to Turchin, 2003 for a more full and more complicated formalization).

There are two state variables: population (N) and state finances (S). The model specifies an agricultural population (N, modeled via the logistic equation) that produces a surplus, and pays a proportion of that surplus to the state. The carrying capacity (K) of this population (N) grows and declines as a function of state finances (S). The state also expends its surplus at a rate proportional to the population size. As the surplus of the agrarian population grows, so too do state finances and the carrying capacity. When the population approaches the maximum carrying capacity enabled by the state, the surplus approaches zero, causing state revenues to fall precipitously as state expenditures on the large population skyrocket. This causes state finances to collapse to zero, which causes the carrying capacity to fall to its original level -- causing a massive decline in the agrarian population.

You can adjust total time of simulation below.
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This model has two state variables. You can set their initial values below.
 $N_{t}$ $S_{t}$
This model has four parameters. You can set their values below.
 $r$ $c$ $s_{0}$ $\beta$
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