Explaining Fashion Cycles: Imitators Chasing Innovators in Product Space

This paper considers the problem of a fashion trend-setter confronting an imitator who can produce the same product at lower cost. A one- dimensional product space is considered, which is an abstraction of the key attribute of some consumer good. Three broad strategies can be optimal for the fashion-leader: (1) Never innovate; milk profits from the initially advantageous position but ulti- mately concede the market without a fight. (2) Innovate once but only once, which just temporarily defers conceding the market. (3) Cycle in- finitely around product space, never letting the imitator catch up and capture the market. Sometimes the cycles start immediately; sometimes the innovator should wait for a time before beginning the cycles. The optimal solution exhibits strong state-dependency, with so-called Skiba curves separating regions in state space where various of these strategies are optimal. There are even instances of intersecting Skiba curves. In most cases, analytical expressions can be stated that characterize these Skiba curves.