A Note on Lerner Index, Cross-Elasticity and Revenue Optimization   Invariants
  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 
Alexander Kushkuley, Su-Ming Wu
math.OC