3 CDAP functions

3.1. Describing the market: the reach function

In order to capture these ideas in a model we define a variable which we call reach. This variable is an index of the share which one product takes from another. Reach is larger the greater the relative market strength and the lower the relative price. But the effect of either market strength or price is less as there is less similarity between the products.

Denote by rij the reach of the ith with respect to the jth of a set of n products. Since we intend to use this concept of reach to determine the volume shares of the various products, all n of the products must be similar in the sense that their quantities can be measured in some common unit such as litres or grams or, in the case of non-financial services, person-hours.

Though we will develop a formal measure of market strength presently, we simply assert at this stage that there is some consistent measure of the market strength of each brand and denote by si the market strength of the ith brand. The price is pi. The vector of indices of attribute intensities of the ith brand is Qi. Just how we obtain these and what they mean will be described in general terms below and by example in the next section. In the usual notation, the distance between two attributes vectors is |Qj - Qi|.

Formally,

(1)

where

(2)

(3)

(4)

(5)

Verbally, the reach of one brand with respect to another is determined by their relative market strengths and relative prices. Naturally, reach increases with relative strength (inequality (2)) and diminishes with relative price (inequality (3)). The sensitivity of reach with respect to relative strengths and to relative prices diminishes as the brands are less similar (inequalities (4) and (5)).

Because we represent the intensity of each attribute for each brand as a real number in the unit interval, the coordinates representing the position of a brand in attributes space is always in the unit hypercube of dimensionality equal to the number of attributes. The maximum distance between any two points (corresponding to the diagonal of the hypercube) is the square root of its dimensionality -- in this case the square root of the number of attributes. It is therefore natural to normalize the distances between brands' positions on the square root of the number of attributes. In this way, the model is not sensitive to the size of the chosen attribute set.

At the same time, we recognize that if two products are both very different from a third, how different they are from one another is not usually relevant to the consumers' brand choices. We therefore used a squashing function giving us a distance measure which made increases in small distances more important than the same increases in large distances. The function used in the model reported here was:

(6)