The theory of quantity discounts and optimal pricing

• Author: Winston W. Chang,Tai‐Liang Chen
• Published date: 01 June 2017
• Date: 01 June 2017
• DOI: http://doi.org/10.1111/ijet.12124

doi: 10.1111/ijet.12124

The theory of quantity discounts and optimal pricing

Winston W. Chang∗and Tai-Liang Chen†

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.

Key wor ds monopoly, quantity discount, multi-tier optimal pricing, social welfare, consumer

surplus

JEL classification D01, D21, D42, L12, L21.

Accepted 5 April 2016

1 Introduction

It is well known that a firm can employ imperfect price discrimination to extract part of the consumer

surplus. This type of discrimination has developed into a number of nonlinear pricing forms, –

including quantity discounts, two-part tariffs, block pricing, bundling, and tie-in-sales. (see, for

example, Allen et al. 2005; Baye 2010; Salvatore 2011). Quantity-discount pricing such as “buy one

and get the second at half price” is ubiquitous in the business world. It charges consumers declining

tier prices as the purchased quantity increases. Enke (1964) classifies quantity discounting as a form

of discrimination among units. By doing so, a firm can extract more, but not fully, of the consumer

surplus compared to uniform pricing.

Attempts havebeen made in the economic literature to derive a monopolist’s optimal tier-pricing

structure. Paine(1937) uses a g raphical method to determine the two-tieroptimal prices. His iterative

approach is rather involved and roundabout. However, his approach may not result in his series of

generated prices converging to the optimum. Gabor (1955) presents a new approach in his two-tier

structure to derive what he calls a pari passu marginal revenue curve. This, together with the marginal

cost curve, determines the total optimal output. He then equates the second tier’s price to the first

tier’s individual marginal revenue and obtains the first tier’s price. Buchanan (1953) considers the

income effect and emphasizes that a quantity discount is a device for the monopolist to increase

its profit by extracting surplus from consumers. More recently, Wilson (1993) examines a model in

∗Department of Economics, SUNY at Buffalo, Buffalo, NewYork, USA. Email: ecowwc@buffalo.edu

†WenlanSchool of Business, Zhongnan University of Economics and Law, Wuhan,China

Weare indebted to a referee for constructive comments and helpful suggestions. Weare also indebted to David Chiariello,

Zheng Han, Nicole Hunter, Eiichiro Kazumori, Takashi Mita, Benjamin Osenbach, and Yoshihiro Tomaru for helpful

comments. An earlier version of this paper was presented at the Asian Meeting of the Econometric Society, Taipei,

Taiwan,June, 2014.

International Journal of Economic Theory 13 (2017) 185–195 © IAET 185

International Journal of Economic Theory