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 William J. Reilly's Law of Retail Gravitation

## Reilly's Law of Retail Gravitation

In the 1920's William J. Reilly of the University of Texas in Austin had access to data on the location of purchases by the residents of the various counties in Texas. He knew that generally the farther away two counties were from each other the fewer transactions would take place between them. On the other hand if a county had a big city with a lot of bigger retail stores it would act as a magnet attracting the shopping by residents of surrounding and even distant counties. Also a county that had a larger number of residents would have more transactions with a given county than one with a smaller number of residents.

Reilly concluded that a reasonable formula for explaining the purchases by residents of county A in the businesses of county B is:

#### kPopulationAxPopulationB/(DistAB) 2.

where DistA,B is the average distance from the residents of county A to the businesses of county B and k is just a constant of proportionality. This definition of distance in the formula means that DistA,A will not be zero.

The formula is analogous to Isaac Newton's Law of Gravitation that says that two bodies attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. Reilly published his results in 1931 in a book entitled Reilly's Law of Retail Gravitation.

Various Refinements of Reilly's Law have been made. Instead of population Reilly's Law is usually modified to use some measure of the attractiveness of the businesses of an location that represents the magnitude of its retail offering. Such a measure would be the total floor area of the stores but a simpler measure would be simply their total sales. Instead of the population of A the total spending by residents of an area is used. Instead of distance the cost of travel, including the value of tranvel time, is sometimes used.

Let the purchases by residents of A in B be denoted as TA,B. Then with the refinements noted above the Law of Retail Gravitation becomes

#### TA,B = k SpendingAxSalesB/(CostA,B)2.

```
Example:     Three cities  A, B, and C

Costs of Travel             Sales

A         B         C

A       0.5      2.0       3.0      100

B       2.0      0.3       5.0      200

C       3.0      5.0       0.6      400

The amounts of spending by residents of A in the stores of

A, B, and C are all proportional to k*SpendingA so if

what is sought is the proportions of spending in the three locations

then this factor cancels out.  The porportions are then given by:

SalesX/(CostAX)2       Share of Spending

Stores

A        100/(0.5)2 = 400.00         400.00/494.44 = 0.809

B        200/(2.0)2=   50.00         50.00/494.44  = 0.101

C        400/(3.0)2 =  44.44         44.44/494.44  = 0.090

Sum = 494.44                  Sum  = 1.000

Likewise the proportion of spending by residents of B in the

stores of A, B, and C are given by:

SalesX/(CostBX)2      Share of Spending

Stores

A        100/(2.0)2 =     25.00       25.00/2263.22 = 0.0110

B        200/(0.3)2 =   2222.22     2222.22/2263.22 = 0.9819

C        400/(5.0)2 =     16.00       16.00/2263.22 = 0.0071

Sum = 2263.22                 Sum = 1.000

And the proportions of spending by residents of C in the

stores of A, B, and C are given by:

SalesX/(CostCX)2 Share of Spending

Stores

A        100/(3.0)2 =    11.11    11.11/1130.22 = 0.0098

B        200/(5.0)2 =     8.00     8.00/1130.22 = 0.0071

C        400/(0.6)2 =  1111.11  1111.11/1130.22 = 0.9831

Sum =  1130.22            Sum    = 1.000

```

In the above computation the value of the constant k was not needed. Also the spending by the residents of A, B and C was not used. However the spending would be relevant in determining the sales in the three locations which then determine the proportions. Reilly's Law could be used to determine an equilibrium distribution of sales based upon a given distribution of income and spending among the various locations.
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