Estimation of price sensitivity from real world data is typically complicated by a dependence between price and demand, or endogeneity of price, since prices are regularly varied in anticipation of changes to demand. This problem is particularly severe in customised pricing environments, where sellers have some freedom to quote different prices for different orders, based on information about each customer and their order. When endogeneity is left untreated and price is modelled as an independent predictor, this leads to underestimation of price sensitivity and sub-optimal pricing strategies.
Endogeneity bias is corrected by the inclusion of instrumental variables in the model; these are variables which are correlated with price and independent of demand, and allow us to separate the direct and indirect effects of price on demand. Whilst instrumental variable estimation has been well documented for retail pricing problems, their use in customised pricing is relatively under-represented in marketing literature.
Here we present a probit model of purchasing behaviour for these environments, whereby the price offered to a customer and their corresponding price threshold are represented by a bivariate Gaussian random variable. Recorded or known sources of dependence between the two are introduced via covariate effects on the mean and unrecorded sources of dependence are captured by a residual correlation parameter. The parameters are identified by the inclusion of instrumental variables in the model. Using a two-stage estimation procedure, the model is fitted to telesales data for heating oil, and price sensitivity estimates are compared to those of a naive model, which does not correct for endogeneity bias. Bayesian estimation of the model is then performed via MCMC, and the resulting sample of parameters are used to examine the impacts of various price changes on profit.