By Professor Dr. Tönu Puu (auth.)
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Extra info for Nonlinear Economic Dynamics
If the radius of the market area is denoted R, then the population (= labour force) in this market area equals: L= R. (2) Now suppose transportation costs are proportional to distance, measured in units of the output and denoted 'K per unit of distance. Then total transportation costs for supplying the market area are: (3) 34 The surplus of per capita output over transportation cost is: Q-T 2 -L-=(a f3 -'K)R-aR • (4) and obviously it is maximal when: (5) R K fa FIGURE I. Dependence of market radius on transportation cost.
In Figure 1 we show the various tessellation elements. The author proposed the square tessellation as the stable one in Puu (1979). This result was modified in Puu and Weidlich (1986) to admit hexagonal and triangular tessellations among the stable ones. FIGURE 1. The various tessellation elements. Actually there is just one hexagonal-triangular tessellation with hexagonal landscape around sinks and a triangular one around sources, or vice versa. This implies that the number of sources is then double or half the number of sinks.
Unlike the case of Hotelling, the contribution by Skellam made great success and was followed by a whole literature on "ecological diffusion". See Okubo (1980). To simplify notation, let us choose units of population, time, and space such that all three constants (growth rate, diffusivity, and saturation density) become unitary. Moreover we use a dot to denote the (partial) time derivative, and the Laplacian operator to denote the sum of the second direct spatial derivatives. The equation thus becomes: (2) Both Hotelling and Skellam dealt with the stationary solutions to this equation.
Nonlinear Economic Dynamics by Professor Dr. Tönu Puu (auth.)