Closed-Form Solutions to Dynamic Pricing Problems with Cauchy-Type-Tailed Willingness-to-Pay

Abstract

We obtain closed-form solutions to a dynamic pricing problem in which a capacity provider determines the price of a product and customers' willingness-to-pay follows a specific family of distributions. This family of distributions not only includes the exponential distribution as a special case but also accommodates some distributions with Cauchy-type tails, which are significantly heavier than that of the exponential distribution. We use these closed-form solutions to study the impact of willingness-to-pay's tail heaviness on the optimal price and find that willingness-to-pay with a heavier tail results in higher sensitivity of the price with respect to the time remaining for selling the product when capacity is small and in lower sensitivity when capacity is large. Furthermore, willingness-to-pay with a heavier tail leads to poorer performance of the quasi-static pricing heuristic when capacity is small and to better performance when capacity is large.

Keywords: dynamic pricing problems, willingness-to-pay, heavy tail, closed-form solutions