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A Surprising Implication of
Gravity-Type Models of the Spatial Distribution of Transactions

Ever since the publication of Reilly's Law of Retail Gravitation in 1931 gravity type models have been a common feature of economic and geographic modeling. Reilly found that a formula analogous to Newton's Law of Gravitation served to explain the purchases of residents of one area in the stores of another area; i.e., the purchases were proportional to the product of the population of the two areas and inversely proportional to the square of the distance between the two areas. Reilly's formulation was subsequently refined by using the income of the spending area and the shopping space of the selling area instead of the populations of the two areas. Sometimes the distance between the two areas was replaced by the cost of travel, including the value of travel time, as the explanatory variable.

What is shown here is that models of this general type have the implication that there is one and only one equilibrium distribution of sales among the subunits of the region. This is something that has been perceived and mentioned when gravity-type models have been used. The new and surprising implication is that there is also one and only one equilibrium distribution of the sources of the purchasing among the subunits of the regions. These results are not presented as theoretical predictions about the behavior of a spatially distributed economy. Instead they are presented as a theoretical weakness of gravity-type models.

Suppose there is a set of cities with their spatial distribution of residents and of businesses. Let Pi be the total purchases made by residents of city i among the set of cities and let Sj be the total sales made in the businesses of city j. Let Tij be the purchases made by the residents of city i in the businesses in city j.

The general model being considered here is


Tij = kaijPiSj
 

where k is a constant of proportionality. In a gravity model


aij = 1/dij2
 

where dij is the average distance between residents of city i and the businesses of city j. The way dii is defined as the average distance between the residents of city i and the businesses of city i means that it will not be zero.

The class of models being considered here is not just the gravity models but the much more general class defined by some set of coefficients aij The level of total purchases made by residents of city i is given by:


Pi = ΣjkaijPiSj
but Pi is a factor of all terms in this equation
and can be cancelled out leaving for all i
ΣjkaijSj = 1.
 

Let A represent the matrix of coefficients aij and S the column vector of the sales Sj. The condition found above can be represented in matrix form as


kAS = U
 

where U is a column vector of all 1's. Therefore the sales must be such that


S = (1/k)A-1U
 

The evaluation of the proportionality constant k will be taken up shortly.

Now consider the total sales in the businesses of city j, Sj


Sj = ΣikaijPiSj
but Sj is a factor of all terms in this equation
and can be cancelled out leaving for all j
ΣikaijPi = 1.
 

This condition in matrix form is


kATP = U
 

where AT is the transpose of the matrix A. This means that


P = (1/k)(AT)-1U
 

Suppose the total purchases made by residents of the all the cities in the system is known and is p. The total purchases p is equal to the total sales s. Then


p = UTP = (1/k)UT(AT)-1U
so
k = UTP = (1/p)UT(AT)-1U
 

Thus the constant k is the sum of all the elements of (AT)-1 (which is the same as the sum of all the elements of A-1) divided by the sum of all the purchases p.

The equilibrium distribution of sales


S = (1/k)A-1U
 

is equivalent to taking the row sums of A-1 dividing by the sum of all elements of A-1 and then multiplying by total sales in the region.

Likewise the equilibrium distribution of purchases P


P = (1/k)(AT)-1U
 

is equivalent to taking the column sums and dividing by the sum of all the elements of A-1 and multiplying by total purchases p.


The implication of there being equilibrium distributions of purchases and sales among the subunits of a region has nothing to do with the coefficients being the reciprocals of the distances squared. It is entirely due to the transactions being proportional to the product of factors that proportional to the purchases of source and the sales of the destination subunits.

As stated before it is not being asserted that these equilibrium distribution are being predicted. They are the implications of the model that practioners say they are using. This is a weakness of such models. Actually when people say they are using the gravity model


Tij = kaijPiSj/dij2
 

they are really using the model


Tij = kiaijPiSj/dij2
 

where the ki's cancel out in the computation of the shares of purchases in the various locations.


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