Diagrammatic Approach to Capacity - Constrained Price Discrimination.

• Author: Reece, William S.

William S. Reece [*]

Russell S. Sobel [+]

This paper presents a diagrammatic solution to the firm's profit-maximizing price discrimination problem in the face of capacity constraints. Airlines, hotels, and other firms practice yield management, allocating fixed capacity to customer groups paying different prices. In these cases, the firm's short-run problem is not a decision about production levels, but it is one of allocating a fixed number of output units among customers. Our diagram shows that the conditions for profit-maximizing price discrimination are very different under these circumstances than in the standard model in which the firm is not constrained by capacity.

1. Introduction

   News media regularly report consumer frustration and confusion over pricing in travel and tourism. USA Today reports in an article on cruise discounts that "only a chump pays full fare" (Stoddart 1999). The New York Times laments complexity in airline pricing in an article entitled "So, How Much Did You Pay for Your Ticket?" (Wald 1998). The Wall Street Journal reports on techniques to get low airfares in "How Farebusters Play the Airlines" (Keates 1998). This complexity, confusion, and frustration are the inevitable result of a rational process designed to solve a very difficult and common business problem: optimal pricing when the firm has fixed capacity and faces classes of consumers with different demands. In the airline's cases, once the airline assigns a particular aircraft to a particular flight, the number of seats is fixed. It faces many different kinds of demand for those seats, including business travelers trying to fly on short notice, senior citizens considering a variety of travel alternatives, and college students planning months in advance to return home for a holiday. Under these circumstances, how will the airline price its seats to maximize profits?

   This paper examines the firm's profit-maximizing pricing problem in the face of capacity constraints using a simple diagram suitable for use in classes in intermediate microeconomics, industrial economics, and applied subjects such as the economics of transportation or tourism. The topic has important applications: Airlines, hotels, universities, theaters, and other firms practice price discrimination as part of systems to allocate fixed capacity to customer groups paying different prices. The airline industry was the first major industry to implement formal systems to solve this complex problem. [1] Airlines recognized that if they did not implement controls, a passenger willing to pay only a low price could, by reserving early, take a seat away from a passenger willing to pay much more for that seat. To prevent such revenue losses, the airlines implemented a control process known as yield management. These practices later spread to other areas such as the lodging industry. [2] To illustrate the relevance o f the capacity-constrained model, during 1997 the 10 largest U.S. air carriers denied boarding to more than one million ticketed passengers from flights filled to capacity. [3]

   Many previous authors have examined price discrimination in yield-management systems. Some of these authors, including many using mathematical programming approaches to carefully address problems associated with demand uncertainty over time, take the prices as predetermined exogenously. [4] Others give standard explanations of price discrimination but do not illustrate the fundamental changes to the problem created by the capacity constraint. [5] Kraft, Oum, and Tretheway (1986) in considering stochastic demand suggest that airlines may simulate various alternative discounts to find the profit-maximizing level taking the full fare as given. This paper shows a diagrammatic solution to a simplified version of the yield-management problem. Our diagram shows how both prices and quantities for each customer group can be determined endogenously, rather than taking prices as predetermined. [6]

   A simplified version of optimal capacity allocation with price discrimination can be described as a few standard steps. First, the seller must segment the market into groups of customers with different demands. Second, the seller creates restrictions that separate the categories of service offered to the customer groups. For example, requiring a Saturday night stay will in many cases separate business travelers from leisure travelers. In the third step, the seller establishes a price for each category based on anticipated demand. Finally, the seller allocates its fixed inventory among the categories. For example, if there are 130 coach seats on a particular flight, the airline might create 3 fare categories: deep discount, discount, and full fare, requiring, respectively, 14-day advance purchase and Saturday stay, 7-day advance reservation and Saturday stay, and no restrictions. The airline allocates some portion of the 130 coach seats to each of these categories. Over a period of months, as the flight time approaches the airline may reallocate seats to categories depending on sales. In principle, the prices may remain constant over time, while the availability of fare categories changes as seats in categories with lower fares become filled and the category becomes unavailable. This typically means that the discount and deep...