CONTENTS
What are the bases for interaction (the movement of goods, people, or information between places)?
How can we forecast the amount of interaction between two places?
How is this useful in predicting the relative "pull" of two competing market centers?
 

BASES  OF  INTERACTION
Question:  Why do goods or people move from one place to another?
Answer:  Distance between supply and demand, which can be overcome at a cost that makes the transportation worthwhile.

We can define spatial interaction quite simply as the flow of goods, people, or information among places, in response to localized supply and demand.

This has also been expressed, somewhat more formally, as complementarity:  a deficit in one place and a surplus in another.
Transferability is the possibility of transport at a cost that the market will bear (the cost of transport is less than the opportunity cost of not transporting).
Rational transport also assumes a lack of intervening opportunity, which is self-explanatory.
 

FORECASTING  INTERACTION
For forecasting the demand that will be placed on a transport or communications network (give examples), there is a very useful regularity in the determinants of interaction between two places:  communication flows, travel patterns, migration flows.

The simple relationship is usually called the gravity model (because of its analogy to gravitational force between two bodies in space), but the larger class of models are referred to as spatial interaction models.
 

GRAVITY MODEL
Any type of model that expresses interaction between two places as a function of the size of the two places and the distance between them.

These models recognize that complementarity generally increases with the size of the population in each place, that transferability decreases with distance, and that the likelihood of intervening opportunity increases with distance.

•  Why multiply the populations?  Because the potential pairs of interactions (the potential complementarity) multiply with the populations of the two places (if there are 2 people in each place, there are 4 potential inter-personal interactions).
•  Why divide by distance?  Clearly distance has some kind of inverse relationship (transferability) with interaction (we call this the distance decay in interaction).
•  Why have a constant?  For one thing, the units wouldn't work out without a constant (interaction per person per distance):  you’d have “squared people per square mile” — not a useful measure of interaction.  More substantively, there's no reason to expect the relationship of population and miles to work out exactly the same for every form of interaction (telephone calls, airline passengers, migrants), for every set of places, or for every time period.
•  Why square the distance (so that twice the distance reduces the interaction by 4)?  Because in most cases, the likelihood of intervening opportunity increases much more rapidly than simple distance (in part because the world is two-dimensional — intervening opportunities can occur in any direction, not just directly between the two places).  We don’t have to square the distance, we could use any exponential value a.  This yields a more general model of interaction:

The constant k can be interpreted as a scaling factor, specific to the relative volume of the given form of interaction (greater for telephoning, lower for migration).  The exponent a can be interpreted as the “friction of distance,” higher for forms of transport or interaction that become especially time-consuming or expensive with distance (greater for automobiles, less for airplanes, even less for telephoning).

Note the way that such a model is used:
1) Gather data on the actual interaction between lots of city-pairs, with each data point being a city pair, the population of each city, the distance between them, and the amount of interaction of a certain type (telephone calls, shopping trips, daily commuters, airline passengers, rail freight).
2) Use that data to find the values of the constant k and the exponent a, through statistical analysis.(see additional notes on this process).
3) Use the values of k and a to project the same kind of interaction between two particular places for which you don’t have data (e.g., in the future, when you have forecasts of population change, or for an airline route that doesn’t yet exist).
 

MARKET-AREA  BOUNDARIES

Reilly's law of retail gravitation (also see notes on trade-area analysis)
The likelihood that a city (or shopping center) will attract shoppers from a hinterland increases with the size of the city (or shopping center) and decreases with distance from the city (or shopping center).

The "law of retail gravitation" can be supplemented with a knowledge of the location and size of competing centers, to develop a boundary around each center beyond which it is more likely that a shopper will go to another center.

For each direction from a given center i in which there is a competing center [draw a simple diagram] the market-area boundary (or “breaking point”) is

The Stutz and deSouza textbook's Figure 4.23, page 183 (which will be presented in class), is an example of this principle applied to the relative market power of San Diego and Los Angeles.

For shopping centers, we often substitute the square footage of i for P_i and the square footage of j for P_j.

Note that a GIS can quickly compute these distances in all directions from a set of centers, creating the actual retail-market boundaries around each center.