Analyzing Household Brand Switching: A Stochastic Model

In a recent study, Kahn, Morrison and Wright showed that for exponentially distributed interpurchase times (of individuals), the household choice process approaches a zero-order process as the number of individuals in the household increases, even though the purchasing behavior of each individual is first order. We show that their result holds for any arbitrary interpurchase-time distribution that has a density over some interval. Thus, additional support is provided for their conclusion that the empirically observed zero-order choice behavior at the household level may not convey much information about individuals' choice behavior. We also derive a general formula that determines exactly the degree of dependence between two successive household purchases for any given family size and (well-behaved) interpurchase-time distribution.