##### 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.

$K(S_{t})=1+c S_{t}/(s_{0}+S_{t})$ |

This is an autonomous auxiliary function that determines how much the carrying capacity (K) changes as a function of state resources (S) |

$dN_{t}/dt = r N_{t} (1 - N_{t}/K(S_{t}))$ |

Logistic growth of the agrarian population as a function of the carrying capacity |

$dS_{t}/dt = N_{t} (1-N_{t}/K(S_{t})) - \beta N_{t}$ |

Change in state state finances (surplus, where S >= 0), where the first term is inflow from taxation, and the second term is outflow from expenditures |

## Turchin (2003) Demographic-Fiscal model ## r is maximum intrinsic rate of population growth ## p is the maximum per capita taxation rate (at low population densities) ## b is the per capita expenditure rate ## ko (k_0) is the carrying capacity of the stateless society ## c is the difference between the maximum carrying capacity achievable in a state society ##and the carrying capacity of the stateless society (i.e. c = k_max - k_0) ## so (s_0) indicates how improvement in k depends on S ###---------Un-Rescaled Equations--------- ## K(S)=ko*(1+c*S)/(so+S) ## dN/dt=r*N*(1-N/K(S)) ## f(S)=if(S<0)then(S*0)else(1) ## dS/dt=f(S)*(p*N*(1-N/K(S))-b*N) ## ## note that the f(s) equation only keeps S from becoming negative ## ###---------Rescaling--------------------- ## N' = N/k_0 ## S' = S/p ## this reduces the model to 4 parameters: ## r and c are unchanged ## b' = b/pa ## so' = so/p ##The rescaled primes are dropped in the following parameters and equations par r=0.02,c=3,so=10,b=0.25 ## Initial conditions for the model init N=0.1,S=0.1 K(S)=1+c*S/(so+S) dN/dt=r*N*(1-N/K(S)) f(S)=if(S<0)then(S*0)else(1) dS/dt=f(S)*(N*(1-N/K(S))-b*N) @ dt=0.5, total=800, xplot=N,yplot=S,axes=2d @ xmin=-0.1,xmax=3.5,ymin=-5,ymax=70 @ xlo=-0.1,ylo=-5,xhi=3.5,yhi=70 done

Cesaretti R, Arizona State University.