Distance and the ‘Reaction’ to Distance as a Function of Distance

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Abstract
“Distance and the ‘Reaction’ to Distance as a Function of Distance” is a phrase designed to emphasize the need for more analysis of the behavioral significance of gravity functions. This paper, in pursuing the behavioral significance theme, concentrates on research related to the “inertia models” suggested by Wolfe. Actually, alternative approaches to understanding the significance of gravity functions used in modeling travel behavior are introduced. However, the particular alternative pursued in depth is the methodology endorsed for understanding travel decisions in marginal terms. The most important concept presented is that if an inertia model of travel behavior of the type Wolfe describes is accepted, the decision to visit a given location must involve a reaction to distance in marginal rather than absolute terms. To quantify concepts, the paper focuses on the effect of going one more unit of distance beyond a given point. This quantification being carried out leads to the conclusion that analysis of reaction to distance in marginal terms should be based on the properties of an “impedance due to distance function” IDF (d), which may be derived from the distance part of a gravity function g(x) by: In the paper, a discussion of five gravity functions provides a basis for illustrating points and methods introduced. IDF(d)'s are derived for the functions considered. As well, conjectures on the response to distance suggested by the functions are checked by reference to the impedance of distance functions derived for the gravity functions introduced. Important conclusions relate to the fact that it is very easy to misinterpret the significance of gravity functions as to how they imply decisions are made if one only looks at the shape of such functions. Actually, it is shown that in cases where one may think each new mile to be traveled offers more resistance than the last, each new mile offers less resistance or constant resistance. One of the cases presented, in fact, involves impedance which decreases and then increases to approach infinity.1

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