## Complex Models with Additional Variables

A more comprehensive model of detail attraction would make use of the information in a gravity model and consider other attributes of store attractiveness. For retail markets in which all the stores are roughly the same in size and amenities, it may be possible to ignore these other variables.355

However, when store attributes differ significantly (such as one offering more product choices, more services, staying open later, having more expensive goods, or other factors), the location factor may still be the most important factor affecting sales, but it will not be the only factor.

### A Generalized Model

In such cases, if data are available for sales from multiple stores, and data are also available for the factors that distinguish the retail outlets, a general predictive model can be used. Such a model can take the functional form:

where Si denotes sales of store i in the region; d is a vector of distance variables, indicating the distance between store i and other stores; m is a vector of amenity variables such as size, hours, quality or price of merchandise, and atmosphere of the area surrounding the retail outlet; and y is a vector of data on the disposable income available to consumers in the area. We include the income variable because the data may come from retail outlets from different areas. With cross-sectional data on stores of a certain type, this equation could be specified and estimated in any number of ways.

The Huff Attractiveness Model: Similarity to Gravity Model

One specification for retail demand was created by David Huff in the 1960s. As will be shown below, it is very similar to a gravity model in its treatment of distance, and explicitly incorporates an additional variable. The additional variable is an important amenity: the size of the retail outlet.

The Huff specification defines the attractiveness a to consumer i of store j as:

355 Franchised retailers, who are normally contracted to meet certain uniform standards, are a subset of retailers where this assumption is more tenable. However, even in these cases, you will see variations due to the performance of the store personnel, management practices, and other individual factors.

where m / is the size of store j, the exponent a is a parameter; d'Jis the distance between consumer i and store j, and the exponent b is a parameter indicating the sensitivity of the consumer to distance.356

The model as defined in the equation has an unobservable variable ("attractiveness") equal to a formula with two unknown parameters. It is a model of individual choice and, therefore, could not be used with aggregate data.356a Thus, some transformation is necessary. To make it tractable, we first substitute the probability of purchasing goods from a certain store (which can be estimated from actual sales) for the unknowable term a • »357

"attractiveness."

The equation still involves a large number of parameters. In practice, the parameters a and b are often taken on the basis of past experience to be 1 and 2, respectively.358 With these series of simplifying assumptions, the equation becomes:

In this practical Huff equation, the size variable m is the single amenity, the distance variable is given the exponent 2, and sales by consumers (or group of consumers) from one area at one store is the dependent variable.

The Similarity between Practical Huff and Gravity Equations

We have seen this equation before. It is the gravity model equation of Equation (4) and Equation (5), with the exponent on distance (the decay parameter) set to 2 and the scale parameter set to the size of the store.

Thus, the practical Huff equation is a special case of a gravity model. Indeed, using the procedure described above, in which we estimate the decay parameter directly from the data, the scale parameter is an observed indication of the amenity vector of the particular retailer.359 In cases where size is known to be an important variable, it can be included in the specification as suggested in the general model discussion above.

356 D.L.Huff, "A probabilistic analysis of shopping center trade areas", LandEcon, 39, 1963; "Defining and estimating a trade area", J. Mark.., 28,1964; cited in A.Ghosh, Retail Management (Fort Worth, TX: Dryden, 1994), Chap. 11.

356a With aggregate data on consumers in different areas, we may be able to estimate the necessary parameters.

57 Economists might recognize this as using the "axiom of revealed preference." However, Huff relied upon a psychology concept known as Luce's Choice Axiom for this simplifying assumption. For a discussion of the uses of Luce's insight and the difficulties in applying an individual choice model to aggregate behavior, see the introduction and first chapter in A.Sen and T.E. Smith, Gravity Models of Spatial Interaction Behavior (New York: Springer, 1995). 359 See A.Ghosh, Retail Management (Fort Worth, TX: Dryden, 1994), Chap.11.

359 Size would be only one of the likely amenities, although for different-size stores, an important one.

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