Predicting transaction outcomes under customized pricing with discretion: A structural estimation approach

We consider a firm selling heterogeneous products with prices customized for each customer and the final price is set by negotiations between the seller and the customer. This type of pricing modality is referred to as customized pricing with discretion and is commonly used in insurance, consumer loans, mortgages, and many business‐to‐business markets. We assume that each sales agent has a reserve price and each customer has a willingness‐to‐pay, which are jointly drawn from a distribution and, if the transaction is successful, they agree on a price between these two values based on their relative bargaining power. Given the outcomes of a series of negotiations, our goal is to estimate the underlying joint distribution of reserve price and willingness‐to‐pay, and predict the outcomes of future transactions. We assume that the price that prevails as the outcome of the negotiation can be represented as a generalized Nash bargaining equilibrium. We develop a structural estimation method based on the expectation‐maximization algorithm to estimate the parameters of reserve price and willingness‐to‐pay distribution. Using a real‐world data set from indirect auto‐lending industry, we show that our proposed method, which accounts for heterogeneity in sales agent's reserve price and customer's willingness‐to‐pay, improves the predictive accuracy of final price (APR) and take‐up probability (i.e., probability of a customer accepting the loan) on real‐world test data by about 8.70% and 3.68%, respectively, compared to a (Tobit‐based) model, which accommodates unobserved heterogeneity in customer's willingness‐to‐pay only. Using the structural EM estimates, we conduct counterfactual analyses to understand the impact of different pricing policies, which vary in the amount of discretion provided to the sales agent during the negotiation process. For example, we find that the lender may increase profits by 13.63% compared to the status quo by optimally imposing a customized minimum price (annual percentage rate) below which a loan should not be offered to the customer.