Location size

The size of the location can be determined by various characteristics ( area , number of buildings, etc.). As a rule, however, a distinction is made according to the number of inhabitants and in Germany a differentiation is made between cities and municipalities .

Mathematical modeling

For the mathematical modeling of the size of the place, let us denote the population of the kth place as . The share of the population of the place in the total population of a country is given by: ${\displaystyle N_{k}}$ ${\displaystyle N}$

${\displaystyle {\begin{aligned}m_{k}&={\frac {N_{k}}{N}}\end{aligned}}}$

The size of a place depends on how much the proportion grows over time. The growth of a place is essentially determined by two processes. On the one hand, the size of the place changes due to the birth and death of residents. On the other hand, it can vary due to events such as the arrival and departure of residents. If one considers these events to be random , the larger a place, the higher the number of birth or death events per unit of time. It is also true that, as a first approximation, the number of immigration and emigration events of a location is proportional to its size. (The emergence of a new place is regarded as a rare event and is not explicitly taken into account here.) The growth in the population of the kth place can therefore be regarded as ${\displaystyle m_{k}}$

${\displaystyle {\begin{aligned}{\frac {dN_{k}}{dt}}&=\alpha _{k}N_{k}\end{aligned}}}$

where is the growth rate of the population of the kth place. The same processes that lead to the growth of a place also determine the growth of a country's total population. The total number of residents grows with it ${\displaystyle \alpha _{k}}$

${\displaystyle {\begin{aligned}{\frac {dN}{dt}}&=\alpha N\end{aligned}}}$

where is the mean growth rate of a country's population. It is determined by the proportional sum across all locations: ${\displaystyle \alpha }$

${\displaystyle {\begin{aligned}\alpha &=\Sigma \alpha _{k}m_{k}\end{aligned}}}$

If you now calculate the time derivative of the share of a place in the total number of inhabitants (first equation), the result is:

${\displaystyle {\begin{aligned}{\frac {dm_{k}}{dt}}&={\frac {1}{N}}{\frac {dN_{k}}{dt}}-{\frac {N_{k}}{N^{2}}}{\frac {dN}{dt}}\end{aligned}}}$

Substituting the above equations for and we get: ${\displaystyle {\tfrac {dN_{k}}{dt}}}$ ${\displaystyle {\tfrac {dN}{dt}}}$

${\displaystyle {\begin{aligned}{\frac {dm_{k}}{dt}}&=(\alpha _{k}-\alpha )m_{k}\end{aligned}}}$

This is what is known as a replicator equation . It determines the temporal development of the size of the place and states that places are in competition for inhabitants. The growth rate (so-called fitness) determines the growth success. A high level of fitness is generally linked to the fact that more people are born than die and more people move into one place than away from it. ${\displaystyle \alpha _{k}}$

The distribution of the size of the place

The rate of growth of a place depends on many factors that can change over time. For example, in order to achieve a high growth rate, a place can ensure that good conditions are created for raising children. Economic, cultural and social factors are essential for the influx (departure) of residents, the changes of which over time lead to fluctuations in the growth rate . However, the replicator equation states that disadvantages in the growth of one place can mean advantages for other places. Although the size of a place can vary over time, the distribution of the place size remains relatively stable. To determine this, the difference is formed from the replicator equations of the city with the highest mean growth rate and the share , and any place with the index : ${\displaystyle \alpha _{k}}$ ${\displaystyle f(m)}$ ${\displaystyle \alpha _{0}}$ ${\displaystyle m_{0}}$ ${\displaystyle k}$

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} \ln(m_{k})}{\mathrm {d} t}}-{\frac {\mathrm {d} \ln(m_{0})}{\mathrm {d} t}}&=(\alpha _{k}-\alpha _{0})=-\Delta \alpha _{k}\end{aligned}}}$

with . In order to take into account the temporal changes in the growth rates one can use a fluctuating size of the shape ${\displaystyle \Delta \alpha _{k}>0}$ ${\displaystyle \Delta \alpha _{k}}$

${\displaystyle \Delta \alpha _{k}=\Delta \alpha -\chi _{k}}$

write. In this equation, the mean difference between the growth rates of the locations in relation to the greatest growth rate over the period under consideration and a fluctuating variable that vanishes on average and is determined by random, mutually independent events. With this, the above equation can be transformed to: ${\displaystyle \Delta \alpha }$ ${\displaystyle \chi _{k}}$

${\displaystyle {\begin{aligned}{\frac {d}{dt}}\left({\frac {m}{m_{0}}}\right)&=(-\Delta \alpha +\chi ){\frac {m}{m_{0}}}\end{aligned}}}$

The index is omitted to simplify the notation. It should be taken into account that the difference is usually very small, i.e. with . The characteristic growth of a place depends largely on its size. For small locations with a share , the first term in the above equation can be neglected because it is very small in terms of magnitude . The replicator equation is reduced to for small places ${\displaystyle \Delta \alpha }$ ${\displaystyle \Delta \alpha \sim \epsilon }$ ${\displaystyle \epsilon \ll 1}$ ${\displaystyle {\tfrac {m}{m_{0}}}<\epsilon }$ ${\displaystyle \epsilon ^{2}}$

${\displaystyle {\begin{aligned}{\frac {d}{dt}}\left({\frac {m}{m_{0}}}\right)&=\chi {\frac {m}{m_{0}}}\end{aligned}}}$

This is a so-called Langevin equation , which describes a multiplicative increase in the number of inhabitants (Gibrat's law). Assuming that it can be described by white noise, the size distribution of small places is given by a log-normal distribution due to the central limit theorem : ${\displaystyle \chi }$

${\displaystyle f(m)={\begin{cases}\displaystyle {\frac {1}{{\sqrt {2\pi }}\sigma {\frac {m}{m_{0}}}}}\,\exp {\Big (}-{\frac {(\ln({\frac {m}{m_{0}}})-\mu )^{2}}{2\sigma ^{2}}}{\Big )}&m>0\\0&m\leq 0\end{cases}}}$

with the free parameters and . For large places, however, the term must be taken into account. In order to determine the resulting change in the size distribution, new variables are introduced. Let it be: ${\displaystyle \mu }$ ${\displaystyle \sigma }$ ${\displaystyle \Delta \alpha }$

${\displaystyle {\begin{aligned}x=\ln \left({\frac {m}{m_{0}}}\right)\end{aligned}}}$

and

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} V}{\mathrm {d} x}}&=\Delta \alpha \end{aligned}}}$

By inserting you get:

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=-{\frac {\mathrm {d} V}{\mathrm {d} x}}+\chi \end{aligned}}}$

This form of a Langevin equation is known from the diffusion of Brownian particles . It describes a fluctuating quantity in a potential . The distribution function is described over a longer period of time by a Maxwell-Boltzmann distribution : ${\displaystyle x}$ ${\displaystyle V(x)}$ ${\displaystyle f(x)}$

${\displaystyle {\begin{aligned}f(x)\sim \exp \left(-{\frac {V(x)}{D}}\right)\end{aligned}}}$

It is the noise amplitude of the stochastic function and ${\displaystyle D}$ ${\displaystyle \chi }$

${\displaystyle {\begin{aligned}V(x)=\int \Delta \alpha \mathrm {d} x.\end{aligned}}}$

Substituting in the original variables one obtains:

${\displaystyle {\begin{aligned}f(x)\mathrm {d} x=f(m)dm\sim \exp \left(-{\frac {1}{D}}\int \Delta \alpha d\ln \left({\frac {\mathrm {m} }{m_{0}}}\right)\right)d\ln \left({\frac {\mathrm {m} }{m_{0}}}\right)\end{aligned}}}$

Finally, the integration provides a Pareto (power-law) distribution of the form

${\displaystyle {\begin{aligned}f(m)\sim {\frac {1}{{\frac {m}{m_{0}}}^{1+a}}}\end{aligned}}}$

with the Pareto exponent . The distribution of large places with is thus described by a power law after a long enough time . For large cities is known to be in a Zipf distribution with passes. The distribution of the size of the place is therefore a lognormal distribution for small places and a Pareto distribution for large places (cities), as it is also found in empirical studies. The theory is that large places have a growth advantage simply because of their size. This so-called economies of scale comes about because large towns can benefit more from a country's population growth than small towns. Small towns, on the other hand, run the risk of disappearing completely due to minimal fluctuations (due to low birth rates and poor economic conditions). ${\displaystyle a={\tfrac {\Delta \alpha }{D}}}$ ${\displaystyle m>\epsilon }$ ${\displaystyle a=1}$

Places by size

See also the list of large and medium-sized cities in Germany .

literature

• X. Gabaix, Y. Ioannides: The evolution of city size distributions in: Handbook of Regional and Urban Economics , V. Henderson and J. Thisse, Eds., Vol. 4, North-Holland, Amsterdam, The Netherlands, 2004.
• Empirical distribution of local sizes. J. Eeckhout, "Gibrat's law for (all) cities," The American Economic Review, vol. 94, no. 5, pp. 1429-1451, 2004. Online
• Joachim Kaldasch: Evolutionary Model of the City Size Distribution , ISRN Economics, Article ID 498125, 2014 Online.

Individual evidence

1. ↑ Gibrat's law (English WP)