• Applied mathematics and spatial wealth condensation: a phenomenological modelling in geographic economy.
  
  • Context:
    The geography of a large part of human activities is characterized by a gathering in areas such towns for instance, toward which it can be noticed the rural flight. When talking about human activities, it can be dealt with wealth, means (political or economical center), art or else ideas. Behind the denomination wealth, it is understood everything that may be measured as describing a wealth itself.
    Another way to convince of the existence of the phenomenon is to have a look on everything that is implemented to struggle against this condensation: decentralisation, delocalisation, externalisation, redistribution...
    In economy, this kind of observation is quiet difficult to be theoretically modelled. Nonetheless, this constitutes the stake of the economic geography. This theme has recently been highlighted through the Nobel Price of Economy 2008, P. Krugman, whose works examinate the mechanisms. He has been considered as promoting the the New Economic Geography, a theory based on the essential role of production on the development of a geographical area. This theory is completed by the concept of the Base Economic Concept, reintroduced by L. Daveziès: the local development also depends on its external resources (tourism, retirement, subventions...)
    Though the use of mathematical models may be criticized, we have tried to develop a phenomenological modelling aiming at reproducing the spatial condensation.
    
  
  
  • Modelling:
    The key idea relies on the principle of accumulation, as noticed by most of the economists (Smith, Marx...), with difference that we deal with a spatial distribution and not that between agents. The underlaying idea is that if someone possesses more wealth that necessary, then he can invert it. Using other hypothesis, the following equation may be obtained:
    
    It is called inverse diffusion ( or backward heat equation), a mathematical problem famous for its instability: numerical solution can not be simulated for large time (More about this...). This equation governs the time and space evolution of wealth.
  
  
  • Results:
    Below are presented two illustrations starting from a gaussian (on the left) and a periodic function (on the right) initial conditions (dotted lines).
    
    Both correspond to initial situations where wealth is unequally distributed, meaning that some territories are richer that others. When computing the temporal evolution of the variable, the solutions evolves in the sense of condensation (solid lines on the plots). the region initially rich are getting richer with the detriment of the poorest that loose more and more wealth. The model validates thus the main hypothesis: wealth flux goes in the sense of its condensation.
    It os then possible to qualitatively check the impact of a redistribution politic, which may be linked to externalisation for instance. To this end, a tax is periodically taken from the global wealth and redistributed towards the poorest regions. The result is proposed on the next plot.
    
    It is noticed that the richest regions do not accumulate as many wealth as before, and that the poor are not getting poorer. the modelling validates the concept that redistribution may be efficient to struggle against poverty.

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