Jones & Simmons introduce the  "spatial demand curve” [See Fig 2.3].  This can be simply defined as the likely relationship between the amount purchased by a household and its distance to the point of purchase.  It recognizes that the amount of a product that a consumer demands is inversely related to its price. (Why? How do economists express this?)  Given that we don’t all live, work, and shop in the same building, the effective cost is the cost of purchase plus the cost of getting to/from the purchase. The farther away the purchase point, the more the effective cost.

The amount of interaction (migration, travel, journey to shop) between two places depends upon:

• the distance between the two places
• the attractiveness of each place, and
• the availability of “intervening opportunities.”

I_ij = k P_iP_j / d_ij^a,  a.k.a the “gravity model” (J&S p. 127)

 where

I_ij is the amount of interaction per time period between i and j (migration per year, shopping trips per week or year, plane flights per year…)

k is a scaling factor, which provides a reasonable volume and unit (thus, the unit attached to k must be (interaction * distance^a) / population²

P is the amount of activity at i and at j:  the population, the size of the store or shopping center…

d_ij is the distance between the two potentially interacting points (expressed as geographic distance, or time)

a is a measure of the “friction of distance”;  it also is a way to recognize that we’re talking about the 2-dimensional distance around i and around j.

You’d get values for k and a by having data on actual interaction between lots of points, and then be able to use the formula to project the amount of interaction between points for which you don’t have data.

FURTHER READINGS ON SPATIAL INTERACTION MODELS

Peter Douglas Hall (1975)
An Assessment of the Calibration of Spatial Interaction Models
M.S. Thesis, McMaster University.
(Provides quite a bit of detail on the process of calibration:  estimating the values of the parameters, given sufficient empirical data on flows, origin-destination sizes, and distances.)
http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1137&context=opendissertations

Stan Openshaw (1975)
Some Theoretical and Applied Aspects of Spatial Interaction Shopping Models
(Provides quite a bit of detail on the process of calibration:  estimating the values of the parameters, given sufficient empirical data on flows, origin-destination sizes, and distances.)
http://qmrg.org.uk/files/2008/11/4-spatial-interaction-shopping.pdf

Jean-Paul Rodrigue (n.d.)
Spatial Interaction (section from an online book on Transportation Geography)
(Short introduction to the concepts.)
http://people.hofstra.edu/geotrans/eng/ch5en/meth5en/ch5m1en.html

Waldo Tobler (n.d., c.1975?)
Spatial Interaction Patterns
(Fairly short introduction to the concepts, with simple graphics to illustrate some examples.)
http://www.geog.ucsb.edu/~tobler/publications/pdf_docs/movement/Spatial_Interact.pdf

C.C. Yrigoyen and J.V. Otero (n.d., c. 1995?)
Spatial Interaction Models Applied to the Design of Retail Trade Areas
(Detailed presentation of the different types of spatial interaction models:  places much of what we’re doing into context;  then provides both theoretical bases for these models and much detail on their use.)
http://www.uam.es/personal_pdi/economicas/coro/investigacion/viena98.PDF