Churn vs. Diversion: An Illustrative Model

An important question in merger analysis is how much of a firm's lost output after a unilateral price increase will shift to the merger partner. To estimate this diversion ratio, antitrust agencies sometimes use data on consumer switching ("churn"), potentially caused by various reasons. This paper uses a tractable model of oligopoly competition to investigate the relation between churn and diversion, depending on what caused the churn. If the cause is an exogenous decrease in a firm's product quality and all prices remain constant, or an increase in its marginal cost that induces a price increase only by that firm, then churn ratios will equal the corresponding diversion ratios; for the same quality or cost shocks, if churn is observed after all prices adjust to the new equilibrium, churn ratios will generally differ from diversion ratios, but nevertheless will still track the ranking of diversion ratios across the firm's competitors. If the exogenous shock is an increase in a rival's product quality, or a decrease in its cost that leads to a price decrease, the churn ratio to that rival will always overstate the diversion ratio. We also consider churn caused by shifts in consumer preferences, broadly interpreted to include changed circumstances or learning about product attributes. Plausibly, churn ratios can then suggest a wrong ranking of how intensely the firm competes with various rivals.