A Note on Lerner Index, Cross-Elasticity and Revenue Optimization Invariants

TL;DR

This paper explores invariance properties in retail pricing models, revealing that certain ratios remain constant over time and generalizing the relationship between elasticity and Lerner index, supported by practical examples.

Contribution

It introduces new invariance results in pricing models, extending the understanding of elasticity and Lerner index relationships within a calculus of variations framework.

Findings

Optimal de-seasoned revenue rate divided by elasticity is time invariant.

Generalization of the inverse relationship between price elasticity and Lerner index.

Illustrations through markdown optimization and continuous replenishment examples.

Abstract

We study common properties of retail pricing models within a general framework of calculus of variations. In particular, we observe that for any demand model, optimal de-seasoned revenue rate divided by price elasticity is time invariant. We also obtain a generalization of a well known inverse relationship between price elasticity of demand and Lerner index. These invariance results are illustrated by two contrasting examples of markdown optimization and optimal continuous replenishment