Nodal Changes in Brand Locations in Product Space

In this chapter we turn to an analysis of rational brand relocations in a product space, extending the analysis of Chapter 10 to facility location in conditions of noncooperation as well as rivalrous consonance. To make the analysis feasible — and, more importantly, realistic — important constraints are placed on the movement potentials of brands. For such purposes, a relatively small number of nodes are designated between initial locations as potential relocation sites for brands which bound the intervals in which the nodes lie. For example, if brand 5 is sited at position 7 and brand 3 at position 8, we might designate a node at position 7.5, and ask if brand 3 or brand 5 (but not both) would find it profitable, ceteris paribus, to relocate there. The bounded interval is the distance between brands 3 and 5, or 1 in the example of Chapter 10. The upper bound of the interval is 8 and the lower bound 7. No firms other than the bounding brands 5 and 3 are allowed to consider movement to the node — that is, brands are not allowed to ‘‘leapfrog’’ other brands. Original sites that are abandoned by relocating firms are also treated as nodes. The number of nodes placed in such subintervals is arbitrary, and current algorithms will permit rather large numbers to be accommodated, so that close approaches to continuous location within the intervals can be approached, but for purposes of obtaining insights and illustrating methodology small numbers will suffice.1