The theory of quantity discounts and optimal pricing

Abstract

This paper explores a monopolist's optimal multi‐tier quantity‐discount prices and shows that only the last tier's marginal cost is relevant in determining the tier prices and each tier's price is equal to its preceding tier's marginal revenue. An increase in total output is associated with larger individual tiers’ own and cumulative outputs (the stretching effect), and the increases in their cumulative outputs are smaller if their tiers are closer to the first one (the ripple effect). The pricing structure is further characterized by the tier Lerner indices and the price elasticities defined on individual tiers’ own and their commutative outputs.

A Pedagogical Note on Multitier Pricing Scheme

“…As shown in Chang and Chen (2017), when Q is increased, there is a "stretching effect" on all tiers' outputs, and also the stretching exhibits a "ripple effect." Again, the condition in Equation A8b sheds more light.…”

“…Optimal Pricing Scheme in the n-Tier Case Under Linear Demand Chang and Chen (2017) characterized in detail the optimal quantity-discount pricing scheme with general demand and cost structures. For the convenience of the readers, we reproduce the relevant parts of the model in the n -tier case.…”

“…Earlier attempts in the economic literature to derive a monopolist's optimal tier-pricing structure and its implications for social welfare include Paine (1937); Buchanan (1952); Gabor (1955); Spence (1977); Willig (1978); Varian (1989); Yano (2001); Allen, Doherty, Weigelt, and Mansfield (2005); Wetzstein (2005); Baye (2010); Salvatore (2011); and Chang and Chen (2017). Whenever a seller has some market power, a multitier pricing scheme can generate a higher profit, output, and social welfare than a uniform pricing scheme.…”

“…Recently, Chang and Chen (2017) characterized in detail the optimal quantity-discount pricing scheme with general demand and cost structures. They showed that each tier's marginal cost (other than the last one) is irrelevant in determining the optimum, an increase in the equilibrium total output increases each tier's output as well as its cumulative output, and the amount of increase in each tier's cumulative output rises monotonically from the first tier.…”

Optimal Ordering Strategy for Goods at Multiple Retail Prices under Simultaneous Sales

“…To the best of our knowledge, it is very common for retailers to provide MRPS. Retailers who apply MRPS can always see an increase in the order quantity due to increasing demand and can obtain more consumer surplus [15]. Currently, the studies on MRPS of retailers focus on the different retail prices at different sales stages to handle excess inventory [16][17][18] instead of handling it simultaneously.…”