Capacity Constaint, Price Discrimination, and Oligopoly

In the presence of market power in oligopolistic environment, price discrimination is a natural phenomenon. Surprisingly this setting has not been analyzed in depth in the literature. In contrast with existing literature, e.g., Hazledine (2006) and Kutlu (2009), we consider quantity setting games where …rms compete in two stages. In the …rst stage …rms decide on the choice of capacity and in the second stage they decide on the structure of price discrimination where the level of price discrimination is exogenous. In contrast to Hazledine (2006) we …nd that in the Cournot framework the quantity-weighted average price depends on the level of price discrimination. We also …nd that in the Stackelberg framework both the leader and the follower price discriminate as opposed to Kutlu (2009) which concludes that the leader doesn’t price discriminate. Moreover, it is discovered that both the players (even the follower) prefer to be in the Stackelberg framework rather than the Cournot framework when price discrimination exists. Comparing welfare under various settings, it is found that competition is not always good for the total welfare if price discrimination exists. 1 Introduction The strategic interactions of the …rms in industries have been analyzed in many settings. The literature essentially has many strands originating from Cournot, Bertrand and Stackelberg. On the one hand, the outcome of the Cournot is more realistic, but on the other hand, the setup of price competition in Bertrand is more close to reality. The extremes of Cournot and Bertrand has been put together in the seminal paper by Kreps & Scheinkman [16] where capacity competition followed by price competition justi…es the Cournot outcome. In many industries the existence of leaders and followers is a natural phenomenon. This is the source of another strand originating from Stackelberg [23]. Yet another dimension of …rms’behavior when they have market power is that of price discrimination. Various kinds of price discrimination in monopoly and their e¤ect on social welfare have interested economists from as early as Robinson [20] who considers third degree price discrimination. This question has been reexamined by Schmalensee [22] and Varian [27] where they …nd that increase in output is necessary for price discrimination to be welfare increasing. Formby and Millner [9] consider the relationship between price discrimination and competition.1 More precisely, they compare the social welfare of price discrimination (in Varian’s framework) and Cournot competition; and they …nd out that when the demand curve2 is concave (convex, linear), price discrimination with n 1Formby and Millner [9] call it Stackelberg price discrimination. 2Whenever we mention “demand curve”we mean “inverse demand curve.” 1 prices produces greater (lesser, equal) output and welfare than a Cournot oligopoly with n competitors. Next natural setup to analyze is the coexistence of price discrimination and oligopolistic competition. This happens invariably in all industries with airline industry being a good example. There is a class of recent literature focussing on this aspect [15, 13, 12, 14, 17].3 Hazledine [13] analyzes the Cournot competition with Varian’s framework of price discrimination. He …nds out that the contrast from the single-price standard Cournot model is in the quantity produced in the market. He also …nds out that the average price in the market is independent of the degree of price discrimination and thus the standard models’prediction is not misleading in terms of the average price. Hazledine [12] considers his earlier model of Cournot competition in the limiting case of in…nitely many prices. He demonstrates that the total surplus is maximized. However, there is a di¤erence from the monopoly case as the consumer surplus is non-zero.4 Finally, Kutlu [17] incorporates price discrimination in the Stackelberg model and …nds a counterintutive result that leader does not price-discriminate. Our paper di¤ers from the earlier works in that we analyze this situation as a two stage game. In the …rst stage, the …rms compete on quantities that they put in the market and in the second stage they decide what fraction of the quantity they sell to di¤erent group of buyers. In other words, in the second stage of competition for price discrimination there is a capacity constraint. The mode of price discrimination based on the valuation of consumers is standard in the literature. Firms may have many instruments at their disposal for discrimination between buyers. As Varian [27] considers the valuation of consumers to be a function of age, …rms may have discounts for senior citizens and students. We consider the example of airline industry where the valuation of the buyers is a function of the time when they are buying the tickets. The business travellers whose plans are generally last moment have less elastic demand whereas the tourists whose plans are almost always ‡exible have relatively more elastic demand. Thus di¤erent bins (groups) of buyers can be grouped based on the day they want to buy a particular airline seat. For example, higher bins consists of the likes of business travellers. The airline example also a good motivation for our two stage setting. In the …rst stage, when the …rms enter the market, they buy certain number of planes; thus the total number of seats are decided for the second stage of the game. They cannot buy planes everyday but they can decide how to allocate the total number of tickets during a time frame. This critical assumption of the stages is missing in the literature that we just reviewed and we hope that this will explain the missing results. In contrast to most of the other works, e.g., [15, 13, 12, 17], we …nd results for a general demand function rather than linear demand. We consider two …rms in most of the paper (except in section 3.1) for simplicity. One of the main …ndings of our paper is that in the second stage both …rms are active in the higher bins. If there are K bins, the …rm with higher capacity is active in all the K bins. The smaller …rm is active in the top t bins. Moreover, in the bins 1; 2; :::; t 1 it matches the quantity sold by the bigger …rm. We characterize the behavior of the …rms up to …nding {̂ = t+ 1, i.e., the …rst bin where the smaller …rm is not active. Although the value of {̂ is not explicitly provided for a given demand function, we describe a recursive algorithm for …nding the unique {̂. In section 3, we consider linear demand for expositional simplicity. We show …rms’behavior in the benchmark Cournot case with n …rms and Stackelberg case with 2 …rms the leader and the follower. The total quantities sold by the Cournot oligopolist increase with the 3See Stole [24] and Armstrong [1] for two comprehensive surveys about price discrimination. 4Note that in the Varian’s price discrimination model when the number of prices goes to in…nity, it is equivalent to …rst degree price discrimination. Thus, the total surplus is maximized and goes to the monopolist.