# Mathematical Location and Land Use Theory: An Introduction by Professor Dr. Tönu Puu (auth.) By Professor Dr. Tönu Puu (auth.)

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58) The factual layout of the assumed example is a situation where output price increases exponentially with distance from the origin of the location coordinate space. Input prices reflect a multiple best availability East and West of the origin for input 1, North and South for input 2. The various advantages among locations combine to sort out four local profit maxima. The very ambitious reader who would like to pass on to calculate the cost rate for transportation implied by the price distributions would find out that they are not entirely consistent.

Radial routes and circular price contours. 27). 30) are no longer needed. 48) which illustrates the logarithmic linearity of the production function. Fig. 12. Hyperbolic routes and price contours. 48) states that the firm should choose a location where the gradient of the logarithm of commodity prices equals the weighted sum of the 28 1 Location Theory gradients of the logarithms of the input prices, the weights being the exponents of the Cobb-Douglas function. 34), illustrates the elegance and economy in using a few concepts from vector analysis.

Flow lines and isovectures. In our case the transportation cost is zero at the common origin and attains a finite maximum value at the opposite pole. This latter point thus represents the most inaccessible point in space. The fact that a point at finite distance is more inaccessible than all locations at infinite distances is, of course, due to the fact that the transportation cost rate function f = 4 I ( 4 + r 2 ) has a maximum unitary value at the origin of coordinate space, and becomes almost zero sufficiently far away in the periphery.