A Simple Micro-Founded Model of Repeat Buying Based on Continuous Information Tracking

Abstract

We present a micro-founded model of consumers' repeat buying behaviors based on their information tracking decisions in continuous time. The model is simple enough to be analytically tractable while flexible enough to be empirically plausible. Three segments of consumers arise endogenously in the model: buyers, trackers and defectors, who differ in their product valuations. We find that consumers who are about to drop out tend to purchase less frequently, and who are currently alive are more likely to stay alive and continue to make purchases. This endogenous customer loyalty is captured by a one-parameter retention rate formula that features an initial steep drop and an asymptotic slow decay at the rate of inverse square root of time. Moreover, we prove sub-exponential distribution of purchase instances and provide closed-form formula for the firm's expected revenue. We show that consumers may benefit from a higher information tracking cost due to a lower price set by the firm. For estimation, we leverage a unique feature of our model by showing that it can be transformed into a \textit{heat diffusion equation}---an extensively studied partial differential equation (PDE). We solve the PDE numerically by using both finite-difference and finite-element methods and benchmark the model estimates against a standard approach based on random walk simulations. A comprehensive study shows that our model provides a similar level of fit and prediction accuracy compared with popular existing probabilistic models of repeat buying. For the study, consumers' information tracking cost is estimated to be about $4.5 per annum. Based on the estimates, our model allows a marketing manager to conduct a rich set of counterfactual analysis as well as to infer the likelihood of which segment a consumer belongs to based on her past purchase history.

Keywords: information tracking, repeat buying, customer relationship management, Brownian motion, optimal stopping, heat diffusion equation

JEL Classification: M31