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Essays on strategic divisionalization and decentralization Yuan, Lasheng 1999

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ESSAYS ON STRATEGIC DIVISIONALIZATION AND DECENTRALIZATION by Lasheng Yuan B.Sc, Bei j ing University, 1986 M.Sc., Bei j ing Universi ty, 1989 M.A., S imon Fraser University, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY i n T H E FACULTY OF GRADUATE STUDIES Department of Economics We accept th is thesis as conforming to the required s tandard THE UNIVERSITY OF BRITISH COLUMBIA September 1999 © Lasheng Yuan, 1999 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of the thesis for scholarly purposes may be granted by the head of the Department of Economics. It is understood that copying or publication of the thesis for financial gain shall not be allowed without my written permission. Department of Economics The University of British Columbia # 997-1873 East Mall Vancouver, B.C. , Canada V6T 1Y2 Date: 0 . 1 Abstract The objective of the three essays of this doctoral dissertation is to investigate the strategic choices of organizational forms by competing firms in various environments. The first essay, which is a joint work with Professor Guofu Tan, provides an alternative theory of divestitures that relies on product-line complementarities and product market competition. We consider a simple environment in which there axe two firms, each sup-plying a group of complementary products and the products across groups axe imperfect substitutes. We model the firms' choices of divesting and pricing as a two-stage game. The duopohsts simultaneously choose their divestiture strategies in the first stage of the game and the independent divisions compete by setting prices in the second. It is shown that, when competing with each other, firms with complementary product-lines have in-centives to split into multiple independent divisions supplying complementary products and services. Such divestitures increase prices and the parent firms' values but reduce aggregate social welfare. Moreover, the degree of divestiture, as we illustrate in the lin-ear demand case, depends on the severity of competition and the nature of product-lines. Then, intensified competition due to deregulation, trade liberalization and entry may trig-ger divestitures. We further show that if two firms axe able to coordinate their divestiture strategies, they can achieve the joint monopoly prices and profits in a non-cooperative price game. The second essay analyzes the strategic incentive of oligopolists to create autonomous rival divisions when products are differentiated. We consider a two stage game where firms choose the number of autonomous divisions in the first stage and all the divisions engage in Cournot competition in the second. It is shown that product differentiation ensures the existence of an interior subgame perfect Nash equiubrium, and the equilibrium number of divisions increases with the degree of substitution among products and the number of firms. Further, if divisions are allowed to further divide, they always will, which leads to total rent dissipation. Thus, parent firms have incentives to unilaterally restrict their divisions from further dividing. In the free entry equihbrium, it is found that the possibility of setting up autonomous divisions is a natural barrier to entry. Incumbents may persistently earn abnormally high profits. In the cases where product differentiation is difficult, the only pure strategy free entry equilibrium is the monopoly outcome even if ii the entry cost is relatively low. The th i rd essay develops a game theoretic model to analyze strategic leasing behaviors of landowners in a nonexclusively owned common o i l pool. The o i l field development is modeled as two more-or-less independent one-stage noncooperative game. The landown-ers choose leasing strategies i n the first stage, and independent lease operators choose extraction strategies in the second. It is found that, i n a nonexclusively owned o i l field, it is individual ly rational for a landowner to unilaterally subdivide his landholding and dele-gate production rights to multiple independent firms, even though more dispersed produc-tion control leads to heavier common pool losses. Moreover, the degree of landownership concentration determines the degree of production concentration. The more fragmented the land ownership, the lower is the degree of production concentration i n equil ibrium. The analysis offers an explanation for the puzzling landowners' leasing behaviors i n U . S . onshore o i l fields. in Content s 0.1 Abstract ii 0.2 Acknowledgment vi 1 Introduction 1 2 Complements, Substitutes and Strategic Divestiture 7 2.1 Introduction 7 2.2 The Model 12 2.3 Analysis 15 2.3.1 Two Benchmarks 15 2.3.2 Second-Stage Price Game 18 2.3.3 Strategic Incentives to Divest 21 2.3.4 Coordinated divestitures 25 2.4 Extensions 28 2.5 Concluding Remarks 30 3 Product Differentiation, Strategic Divisionalization and Persistence of Monopoly 33 3.1 Introduction 33 3.2 The Model 36 3.2.1 Demand and Technology . . 36 3.2.2 The Divisionalization Game 37 3.3 Analysis 38 3.3.1 The Second Stage Cournot Quantity Game 38 3.3.2 The First Stage Game . 40 3.4 Possibility of Further Divisionalization 44 * 3.5 Free entry equiHbrmm 46 3.6 Concluding Remarks 49 4 Divide and Conquer: Strategic Leasing in Common Pool Oil Fields 51 4.1 Introduction 51 4.2 O i l Extract ion Technology i n a Common Poo l 54 4.2.1 The Extract ion Rate 54 4.2.2 The Pressure Depletion Dynamics 55 4.2.3 The Ult imate Recovery 56 4.3 Analysis of Competit ive O i l Extract ion 57 4.4 A Game Theoretic Mode l of Strategic Leasing i n a Common Poo l 60 4.4.1 The Potential Ga in of a Mul t ip le Leasing Strategy: A n Example . . 61 4.4.2 Analysis of the Leasing Game 63 4.5 Conclusion 65 5 Conclusion 67 5.1 Appendix 2 .A 77 5.2 Appendix 2 .B: A n Integer Divisionalization Game 80 5.3 Appendix 3 84 5.4 Appendix 4.1 88 5.5 Appendix 4.2 91 v 0 . 2 Acknowledgment I would like to thank my thesis committee, Kenneth Hendricks, Margaret Slade and especially Guofu Tan for invaluable comments and guidance. I am indebted to my family, especially my lovely wife Jenny Chen and son Alexander for their support. VI Chap te r 1 I n t r oduc t i on The modern economy is dominated by large corporations with sophisticated structures. Instead of being centralized to a single entity, production and marketing decisions in a firm are often made by various levels of management, subsidiaries, divisions, franchises, etc.. The institutional arrangement and decision structure in firms are critical factors for economic performance. The objective of this dissertation is to investigate how firms strategically set up autonomous units when facing competition in several simplified envi-ronments. Coase is among the pioneers who first realized the importance of the study of the "institutional structure of production." Stimulated by the great debate of the advantages of a free market system over a central planning system in the early thirties, Coase started to rationalize the very existence of the firm in the free market economy. According to eco-nomic theory, a competitive market economic system coordinated by prices would deliver the Pareto-efficient outcome. Then, if the pricing system provides all the coordination necessary, why is there a need for firms whose function is to coordinate in a planning fashion? In his famous 1937 article "The Nature of the Firm," Coase acknowledged that there are transaction costs of using the price system. And, it is the avoidance of these costs that could explain the existence of the firm in which allocation of factors comes about as a result of administrative decisions. A firm could continue to exist if it performs its coordination function at a lower cost than would be occurred if it were achieved by means of market transactions and also at a lower cost than the same function could be performed by another firm. 1 Coase's theory convincingly justifies the emergence of the firm in the free market economy but stops short of explaining how and why a firm chooses a certain form of orga-nization. Williamson (1975) investigates the increased use of multidivisional organization form in big corporations documented by Chandler (1962). He argues that big organi-zations inevitably encounter monitoring difficulties and moral hazard problems. Multi-divisional form is an institutional innovation arising to reduce monitoring costs and to mitigate moral hazard problems by devolving operational decisions to smaller autonomous divisional units. Along a similar line of argument, Aron (1988) shows that stock mar-ket evaluation of separated units can reduce the costs of motivating managers. Meyer, Milgrom and Roberts (1992) argue that since influence costs, which arise from the possi-bility of manipulation of performance measures by poor performing units, reduce a firm's potential value, poor performing units should be divested. These theories based on asym-metric information or moral hazard problems do provide rationales for why firms set up multiple independently managed units. The explanation, however, is incomplete because the potential external effects of a firm's choice of organizational form on other firms in the market are neglected. Recently, many authors have realized that there is a strategic aspect to the choice of the organizational form by a firm. Vickers (1984), Fershtman and Judd (1987), Fersht-man, Judd and Kalai (1991), and Sklivas (1987) study the separation of ownership and control in management controlled firms in a duopohstic setting. The generic model is a two-period model. In the first period, each owner provides his manager with an incentive contract that is a linear combination of profit and sales revenue or output; in the second period, managers make production and sales decisions following their own interest in ei-ther a Cournot quantity game or a Bertrand price game. These articles show that the separation of ownership and management is the rational response of firms to oligopolistic competition. Moreover, the firms' goal-setting incentive contracts for managers are inter-dependent and are typically not the direct profit maximizing schemes. Under Cournot (Bertrand) competition, firms' profits are lower (higher) relative to the case where there is no separation of ownership and management. These strategic delegation models have been helpful in explaining the existence of management control firms and the related compensation schemes for managers. However, by construction, these models can not explain the widespread existence of multiple au-2 tonomously managed units (such as divisions, franchises, etc.) within a firm, for each owner is only allowed to hire a single manager. More recently, there has been a growing interest in strategic divisionalization, which refers to firms' choices of multidivisional forms under oligopolistic competition. Schwartz and Thompson (1986) and Veendorp (1991) demonstrate that an incumbent monopoly can forestall entry through setting up multiple identical autonomous divisions. They argue that divisionalization can be used as a commitment device for the monopoly to produce an entry-deterring level of output. Corchon (1991), Polasky (1992), Corchon and Gonzalez-Meastre (1993), and Baye, Crocher and Ju (1996) consider a case of a symmetric duopoly with homogeneous products. The basic model is a two-stage game, where each duopolist chooses the number of autonomous divisions in the first stage, and all divisions engage in a Cournot competition in the second stage. The main result is that the duopolists have incentives to unilaterally set up multiple autonomous divisions. Such a strategy commits a parent firm to a higher level of output, mimicking a Stackelberg-type outcome in pursuing Cournot competition (Baye, Crocker and Ju, 1996). In equilibrium, the profits of competing firms are lower and social welfare is higher than in the no-divisionalization case. This dissertation is an attempt to expand the investigation of how and why firms di-vide production among autonomous units into several different scenarios of competition. It consists of three essays. The first essay studies divisionalization of competing mul-tiproduct firms when product-lines consist of complementary products or services. The second essay deals with divisionalization of firms with differentiated products. Finally, the third essay investigates strategic leasing in common pool oil production, which can be viewed as a special form of divisionalization. Without doubt, divisionalization in this thesis is used in the broadest sense possible. It refers to firms setting up independently managed units in whatever forms, such as independent leases, franchises, spinoffs, outright sales, etc.. In the first essay, a joint work with my thesis supervisor Professor Guofu Tan, we develop a theory of divisionalization (divestitures) that relies on product-line complemen-tarities and product market competition. We consider a simple environment in which each of the competing firms supplies a group of complementary goods or services. Division-alization of a multi-product firm is modeled as a partition of its product space. Hence, 3 divisions of the same firm produce products that are complementary to each other. The competition is modeled as a two-stage game. Each firm chooses its number of divisions in the first stage, and all divisions engage in a Bertrand price game in the second stage. Under a general demand, it is shown that firms with complementary product-lines have incentives to set up multiple autonomous divisions unilaterally in competition. Prices and firms' profits are higher and social welfare is lower relative to the case where no division-alization is permitted. More interestingly, if firms are able to coordinate their divisional-ization decisions in the first stage, the subsequent noncooperative Bertrand competition in the second stage leads to monopoly prices and profits, as if all firms (or all divisions) are managed by a single agent. The number of divisions in a symmetric game is posi-tively related to both the degree of competition, which is measured by the degree of the substitutability among the groups of products, and the number of groups. Therefore, this analysis builds a link between firms' business restructuring decisions or choices of organi-zational form and the competitive environment. Intensified competition that results from deregulation, trade liberalization or entries may trigger divisionalization. This may pro-vide a partial explanation for the coincidence of the increased use of multidivisional forms (including divestitures) and the waves of deregulation and globalization in the developed economies since the 1980s. The key to these results is the two opposite effects that the pricing of a product inflicts on complements and substitutes. It is known that an increase in the price of a product has a negative effect on the demand of its complements and a positive effect on the demand of its substitutes. In absence of divisionalization, the prices of products of two competing firms are too low relative to the monopoly prices, which maximize the aggregate profits of these firms. After divisionalization, independent divisions of the same firm ignore the negative externality of pricing on each other. Thus, the resulting prices and aggregate profits rise towards the monopoly level. The possibility that firms can tacitly collude in pricing through divisionalization makes antitrust regulation a more complex issue. The second essay analyzes strategic divisionalization of oligopoly with product differ-entiation and the implication of free entry equilibrium. This analysis can be viewed as a generalization of the model of Corchon, Polasky and Baye et al., since it contains their model as a special case. Following convention, the competition is modeled as a two-stage game. Oligopolists choose the number of identical autonomous rival divisions in the first 4 stage, and all divisions engage in a Cournot competition in the second. Like Corchon (1991), Polasky (1992) and Baye et al (1996), I show that oligopolists have incentives to set up unilaterally autonomous rival divisions, despite the fact that profits are lower for each of them in equilibrium relative to the no-divisionalization case. The equilibrium number of divisions of each firm increases with the substitut ability of the differentiated products and with the number of competing firms. Divisionalization also has significant implication on entry. An entry can considerably intensify divisionalization and, consequently, the competition. The credible threat of divisionalization in Ihe case of an entry makes divisionalization a natural entry deterrent. As a result, an incumbent can enjoy an extremely high profit relative to the entry cost without worrying about an entry, when new products are difficult to differentiate from the existing ones. If an entry does occur, to soften competition, the entrant has much higher incentives to differentiate itself from the incumbent relative to the case where no divisionalization is allowed. The third essay investigates strategic leasing of production rights in U.S. onshore oil fields. Oil production is modeled as a two-stage game. First, landowners choose leasing strategies, and then, all lease holders choose extraction strategies. It is shown that landowners have incentives to subdivide unilaterally their landholdings and grant production rights to multiple independent operators, despite the fact that more dispersed production concentration leads to a more serious rent dissipation in a common pool oil field. Multiple leasing is a commitment mechanism for landowners to gain the Stackleberg-leader-advantage in the production stage. This analysis provides an explanation to a long-standing puzzling phenomenon, i.e., the persistence of the widespread inefficient production organization in U.S. onshore oil fields. Moreover, it has significant implications on general common property problems. In the existing literature, the property owner is treated the same as the property operator. This article, however, shows that they are usually very different and that the operation structure can be much more dispersed than the ownership structure, because owners have incentives to grant operation rights to multiple independent agents. Therefore, distinguishing the ownership structure from the operation structure and modeling the relationship between them are essential to common property problems with privately controlled access (not free access). 5 The analysis in this dissertation is also closely related to, but distinct from the merger literature such as Salant et al. (1983) and Gaudet and Salant (1991), which examine the impact of mergers on firms' profits and on social welfare. Under plausible conditions, they show that merger of a subset of firms may result in profit loss for the merging firms, even though merger leads to a more concentrated oligopoly. This dissertation deals with the opposite issue: the incentive facing firms to divisionalize or to set up rival independent units. We show that, under several different and plausible scenarios, each firm has unilateral incentives to create rival competing units. Unlike most of the merger literature, our analysis permits one to analyze the equilibrium consequences of these incentives in a noncooperative setting that allows all firms to divisionalize freely. The merger literature, in contrast, implicitly views merger as a cooperative game among merging parties, for the set of firms that merge is usually exogenously selected. 6 Chapter 2 Complements, Substitutes and Strategic Divestiture 2 . 1 Introduction Since the early eighties, corporate divestitures have been a major form of corporate re-structuring in the United States, accounting for 40 — 50% of merger and acquisition activities. Divestitures (or sell-offs) involve the transfer of a part of a firm's business to a new owner, as opposed to the sale of the entire firm. The value of such sell-offs reached over $177 billion in the United States in 19961, and similar markets exist in the other industrialized economies. Both the volume and the nature of these transactions raise questions for the economics of organization and for corporate strategy. What determines whether the complex bundle of business activities that comprises a corporation should be either a freestanding, independent enterprise or, instead, a unit of a large firm? What determines whether a particular business unit should be divested, and if so, how? In this paper, we examine a possible answer to these questions. More specifically, we focus on firms' strategic incentives of divestitures that rely on product-line complemen-tarities and product market competition. We consider a simple environment in which there are two firms. Each firm supplies a group of complements, and the two groups of complements are imperfect substitutes. With this framework, we try to approximate competition among conglomerates, which are the main sources of divestitures. For ex-^ee Mergerstat, 1997, p28-30. 7 ample, two competing akline alliances, each of which consists of several regional airline companies, have a conceptually similar structure. Another example is two competing shopping centers, each consisting of a number of stores. The existence of transportation costs makes stores within a mall, even when supplying substitutes in the normal sense, transactional complements ex ante (Ayres, 1985; Stahl, 1987; and Beggs, 1994). We ad-dress only the issue of strategic divestitures in such an environment, while the issue of mergers can be analyzed analogously in a setting where each complement is supplied by an independent firm. We model the firms' choices of divesting and pricing as a two-stage game. The duopohsts simultaneously choose their divestiture strategies in the first stage of the game, and the independent divisions (the parent firms and the divested divisions) compete in prices in the second. In order to highlight purely strategic motives in our analysis, we strip away other factors that might affect the firms' decisions to divest, such as moral hazard problems and increasing return to scales. In particular, we assume that the technology of each firm exhibits constant returns to scale and scope. In this framework we first show that, when facing competition, firms producing a set of complementary products or services may find divesting some of their complementary products a profitable strategy. By divesting, a firm credibly commits not to coordinate the pricing of complements and, therefore, softens competition to its rival firm. On one hand, this strategy makes the divesting firm more vulnerable to its rival firm's potential aggressive pricing. On the other hand, softened competition will increase the total profit in the market. If the gain in total profits outweighs the loss due to the vulnerability from divestiture, then divestiture is a profitable strategy for a firm. Once its rival has divested, a firm is more likely to follow the suit since the vulnerability resulted from divesting is reduced. In the symmetric equilibrium, both firms divest and neither suffers any loss in terms of relative strength in competition. Both firms then gain from the increase in total profits. The results rely critically on product-line complementarity and competition between firms. In the absence of divestitures, two competing firms set prices lower than the monopoly prices, which maximize the aggregate profits, for each firm ignores the positive externality of its pricing on the rival firm's profit. After a firm divests into independent divisions producing complementary products, those divisions ignore the negative exter-nalities of their pricing on each other's profits. When the degree of divestiture is small, 8 the own-group negative externality, which denotes the externality of pricing among divi-sions of the same firm, is smaller than the cross-group positive externality, which is the externality of pricing between firms. Therefore, a small degree of divestiture by one firm moves the prices closer to the monopoly prices, leading to higher profits. This reasoning works for both firms. Indeed, when we characterize the subgame perfect equilibrium of the two-stage game, we find that both firms have incentives to divest when competing. The degree of divestiture in equiKbrium is determined by both the degree of substitution (or the severity of competition) and the degree of complementarity of each firm's product-line. The more severe the competition among firms, the higher the incentives to divest. Thus, changing the competition environment such as deregulation, trade liberalization and entry may trigger divestitures. By divesting, competing firms create own-group pric-ing externality to mitigate the opposite cross-group externality, resulting in higher prices and profits. Our analysis has important implications for the social welfare consequences of di-vestitures. Conventional wisdom holds that mergers of firms supplying similar products reduce competition, increase the prices of the products, and decrease consumers' welfare and total surplus. Consequently, mergers have been the main concern of regulatory au-thorities. When divestitures are motivated by product-line complementarities, however, we find that divestitures increase prices, lower quantities, and decrease consumers' welfare and social surplus. Therefore, from a social welfare point of view, divestitures involving complementary goods or services should be discouraged as much as mergers involving substitutes.2 The fact that firms can achieve tacit collusion in pricing through divestiture when competing product-lines consist of complements makes regulation a more complex and difficult issue. We also consider the situation in which firms are able to coordinate their divestiture decisions in the first stage. It is shown that there exists a pair of divestiture strategies such that the joint monopoly prices and profits are achieved in a non-cooperative price competition game in the second stage. In such cases, by choosing the appropriate degree of divestiture, competing firms create an own-group negative externality which exactly 2 Our results in this context have a close parallel with those of Spence (1976) and Economides and Salop(1992), who argue that for complementary products, mergers reduce prices and increase consumers' welfare. 9 offsets the cross-group externality. Such a coordinated number of divisions is greater than the number of divisions in the non-cooperative game. This is due to a positive externality between the firms' choices of divisions. After divesting, a firm commits not to coordinate the pricing of its divisions. This fat-cat strategy softens the second stage competition, raises prices, and thus benefits the rival firm. Coordination between the two firms internalizes the positive externality and thus results in more divisions than the non-cooperative equilibrium permits. With non-coordinated divestiture, the prices move up but remain below the monopoly prices. With coordinated divestiture, the equilibrium prices are exactly the same as monopoly prices, as if all divisions of both firms are operated by a single agent. Through divestitures, firms may achieve perfect collusion in pricing in a non-cooperative price game, which is a striking result. A variety of other arguments have recently been put forward to explain divestitures. For instance, divesting is rationalized by some as an institutional innovation arising in response to the loss due to moral hazard problems in large corporations (Williamson, 1975; Aron, 1988; Hart and Moore, 1990; Holmstrom and Milgrom, 1991; Meyer, Milgrom and Roberts, 1992; etc.). These theories based on asymmetric information axe certainly useful in explaining the decentralization of the control of assets or business activities. They have little power, however, in explaining that many divestitures are of units that had been previously acquired rather than started from scratch by divesting firms (Porter, 1987; Ravenscraft and Scherer, 1987; and Kaplan and Weisbach, 1990). Both Porter (1987) and Ravenscraft and Scherer (1987) interpret those sales as recognition of failure, which would account for the presumed performance-divestiture linkage. The evidence assembled by Kaplan and Weisbach (1990), however, suggests that less than one-third of acquisitions that were later divested could be considered failures ex post. Our theory differs from those arguments by relating divestitures to competition environments and the nature of product-lines. We predict that increased competition tends to intensify divestiture activities. Jensen's free-cash-flow hypothesis (1986) suggests that managers who are imperfect agents of stockholders will have a tendency to invest even in unprofitable businesses. The frequent asset sales following hostile takeovers can be interpreted as undoing excessive and unprofitable conglomeration. Increased discipline on managers from the strengthened market for corporate control in the 1980s reduced such investments and might also have 10 led managers to divest their previous bad investments to avoid having their companies become subject to hostile takeover. Jensen's hypothesis is consistent with the finding that the accounting performance of the divesting firm improves after the divestiture and that the announcement effects are positive (John and Ofek, 1995). The puzzling side of this story is why the bad investments can be sold for more than they are worth to the current owner. One of the prominent explanations advanced is that divestitures are motivated by a desire to increase the focus of a firm's business and establish core competencies. The underlying hypothesis is that perhaps more focused firms are easier to manage and so create greater values or that different managers are skilled in managing different types of assets. Such an explanation is not very satisfactory, however, since it is hard to refute. In this paper, we provide an alternative explanation for a class of divestitures where product-line complementarities exist between the divesting units. Casual empiricism suggests that such complementarities often exist3 Divestitures increase the total value of a firm because of the changed competition structure and the firms' strategic behaviors. Our analysis is related to, but distinct from, the literature on mergers involving firms supplying complements. Salop (1990) and Economides and Salop (1992) show that, in a Cournot duopoly model with complements, mergers of two firms supplying complements reduce prices, because a merger allows coalition firms to absorb a positive externality.4 3For example, on September 20, 1995, AT&T announced that it would split into three indepen-dent firms, with the first offering long-distance telephone and credit card services, the second supplying telecommunication equipment, and the third dealing with computer businesses. These three businesses can be viewed as complementary. For details, see "AT&T's three-way split," The Economist, September 23rd, 1995. Many other telecom-equipment companies such as Sweden's Ericsson, Finland's Nokia, and the U.S.'s 3Com entered the computer market not long ago, but have now left the computer business. Another example is shopping malls. We often observe that in-town shopping malls consist of many independently managed shops which usually sell ordinary complements. Different shopping malls compete by selling goods which are substitutes across malls. Furthermore, even if different stores in a mall offer similar products, consumers are not sure which product they would prefer prior to visiting stores. Their decisions are often based on expected prices. Therefore, the existence of shopping costs makes ordinary substitutes within a mall transaction complements (see Ayres, 1985; Stahl, 1987; and Beggs, 1994). Pashigian and Gould (1995) provide empirical evidence that positive agglomerate externalities exist when stores are located together in a mall. Our analysis explains why shops in shopping malls are not owned by a single firm. 4The results are also similar to those in the vertical integration literature such as Greenhut and Ohta 11 Our concern is w i th the opposite issue: competing firms' incentives to divest and the consequences. Further, unlike most of the merger l i terature, which usually models mergers in a cooperative game setting, our analysis of firms' choices of organization forms and pricing is conducted in a non-cooperative game setting. This paper is also closely related to the recent strategic divisionalization l i terature, such as Corchon (1991), Corchon and Gonzales-Maestre (1993), Polasky (1992) and Baye, Crocker, and Ju (1996). These authors analyze the strategic incentives for firms to form independent divisions when competing in a homogeneous product market. They find that firms form mult iple divisions in order to take a larger share of the market. Moreover, divisionalization leads to lower prices, lower profits and higher social welfare. This paper analyzes a differnt competit ive environment and identifies a different incentive for firms to break up. In our framework, firms w i th a set of complementary products set up independent divisions to soften the competit ion f rom their rivals. As a result, divestiture of this k ind increases prices and profits, and reduces the social welfare. For simplicity, the main part of our analysis is conducted in a setting where there are two firms, each supplying a group of perfect complementary products or services. However, the in tu i t ion and qualitative results carry over to more complex settings in which there are more than two competing firms, and product-lines are imperfect complements. The rest of the paper is organized as follows. The next section introduces a two-stage duopoly model w i th differentiated products. Each firm supplies a group of perfect comple-ments, and across firms, the products are imperfect substitutes. Section 3 characterizes the equil ibr ium outcomes and provides the main results. Section 4 discusses possible extensions of the basic model. Section 5 concludes the paper. 2.2 The Model Suppose that consumers demand a number of differentiated products, which are divided into two groups. W i t h i n each group the products are perfect complements and, across groups, they are imperfect substitutes. Let Nk denote the set of products in group fc, k = 1,2. The demand functions are given by (1979). 12 qki = D(pk,pi), i G Nk, (2.1) for k, I = 1,2 and / ^ fc, where p*, and denote the price and quantity of product i in group k, respectively, and pk = J2"=i Pki is the aggregate price of products in group k, and njt is the number of products in group k. Notice that the demand system (2.1) is symmetric both within and between groups.5 We make the following assumptions regarding the function D(pi, p_)-6 Let P = {(pi, P2) € (Al) P is convex and bounded, D(pi ,p_) is twice continuously differentiable in P , D2(pi,P2) > 0, and £>i(pi,p_) + D2(pi,P2) < 0 for {pi.pi) in P , where Dk(pi,P2) denotes the first-order derivative of D(p1,p2) with respect to pk, k = 1,2. The assumption that D2 > 0 represents demand substitutability between the two groups of the products. D\ + D2 < 0 states that the effect of the aggregate price within the group on the demand (own-group effect) dominates the effect of the aggregate price from the other group (cross-group effect). To illustrate our analysis, we frequently use the linear form of demand functions where a > 0, $ > 7 > 0. The ratio, 7//3, measures the degree of substitution between the two groups of products or the extent to which the own-group effect dominates the cross-group effect.7 When j//3 = 0, the two groups' products are independent; when 7//? = 1, the two groups are perfect substitutes. There are two firms, 1 and 2. Firm 1 supplies all the goods in the first group and firm 2 offers all the complements in the second.8 To focus on the strategic incentives, we 5The symmetry of demands across groups is not crucial for our discussions below. 6These assumptions are standard in the literature on differentiated products. See Friedman (1977) and Deneckere and Davidson (1985). 7Beggs (1994) uses the same linear demand function. There are only two complements within each group in his model. 8An alternative specification of the model is to assume that the total number of firms is ni + "2 and each firm supplies one product. The issue of strategic incentives to merge can be addressed in this setup. Rl\D{p1,p2)>0, Z>(p2,pi)>0}. D(pi,P2) = a- Ppi + 7 P 2 , (2.2) 13 assume that the production technology of each firm exhibits, const ant returns to scale and scope.9 That is, the total cost function for firm k is Ck(qki,---,qknk) = ^Ckiqki, k = 1,2, »=i where Cfc,- is the constant marginal cost of producing product i in group fc, and Cki > 0. We consider the subgame perfect equilibria of the following two-stage game with per-fect information. In the first stage, the two firms simultaneously choose their restructuring strategies. In stage two, all independent firms compete by simultaneously setting prices. By restructuring, the parent firm keeps a subset of its product lines and sells the rest of its operations to independent entrepreneurs (not to its rival firm). It should be noted that, in our framework, the restructuring strategies of the firms can be any of divestiture, spin-off, breakup, or divisionalization, as long as the resulting firms independently choose their pricing strategies. In the following analysis, we simply refer to a restructuring choice as divisionalization, namely, a firm setting up autonomous rival divisions. We model divisionalization of a firm as a partition of its product space. Let dk denote a partition of Nk, and each cell in the partition be a division of firm k. Let nik be the number of divisions in dk- Given a pair of strategies (di, J2), the profit function of division j in group k is Kkj(pkj,Pk,Pi)dk,di) = (pkj - ckj)D(pk,pi), I ^  k, (2.3) where, for notational simplicity, pkj and Ckj represent the aggregate price and the aggregate marginal cost of the products in division j of group fc, respectively, and pk = 2~2j=iPkj is the aggregate price of products in group k. For k = 1,2, let 3=1 where / ^  k, and Ck = YlJ=i ckj is the aggregate marginal cost of the products in group k. A divisionalization (or divestiture) strategy can be viewed as a set of take-it-or-leave-it contracts signed between firm k (or the parent firm) and independent entrepreneurs. Each 9The assumption of constant returns to scope implies that there is no operating synergy between dif-ferent lines of businesses. Our model is motivated to address divestiture issues concerning conglomerates. 14 contract specifies a reserve price at which the parent firm is willing to sell the operations of a subset of its products. We assume that the contracts are restrictive so that no division can further divide or subcontract prior to the price decisions in the second stage of the game.10 It is then reasonable to assume that firm k sets a reserve price equal to the profit that division j can make in the second stage game, and that each entrepreneur is indifferent about accepting the contract or rejecting it. Therefore, the total profit of firm k is given by (2.4). The two-stage game can be solved via backward induction. In the second stage of the game, for any pair of strategies (d\, d2), each division j in group k chooses its price pkj to maximize (2.3), given the price choices of other divisions within and across groups. In the first stage, firm k chooses a divisionalization strategy, dk, to maximize (2.4), taking into account the divisionalization choice of the other firm and the equilibrium prices of the second stage game. To simplify our presentation and emphasize the effects of the nature of demands on divisionalization, we set all the marginal costs equal to zero. This simplification does not affect the qualitative results in the paper. 2.3 Analysis In this section, we first introduce two benchmarks. One deals with competition between two firms without divestiture. The other is the joint profit maximization. We then characterize the equilibrium outcomes of the two-stage games, which can be solved by backward induction. For each set of divisions chosen by the two firms, the equilibrium of the second stage price game is determined, and comparative static properties of the equilibrium are discussed. Our main findings are then presented. 2 . 3 . 1 T w o B e n c h m a r k s In the first benchmark, two firms directly engage in Bertrand price competition without divestiture. That is, d\ and d2 are singletons and equal to N\ and A^, respectively. Given the simplification of the marginal costs, firm k has the following profit function: 1 0 If the divisions can further divide before they choose prices, the problem becomes very complicated. We are currently working on this issue. 15 Hk(pk,pi;Nk,Ni) = pkD(pk,pl), (2.5) for k,l — 1,2, I ^ k, and pk — Y£=iPki- Notice that the profit functions depend only on the aggregate prices of the complements, pk and p\. Each firm only needs to decide on its aggregate price, and individual prices are indeterminate. Furthermore, the profit function of firm k increases with the price of the other firm, since the products across groups are substitutes. In other words, there exists a positive externality between the two prices. In equilibrium, firm k chooses pk to maximize (2.5), given the aggregate price of the other group, pi. The first-order conditions are D(pk,pi)+pkD1(pk,pl) = 0, (2.6) for k,l = 1,2, and I ^  k. The eqvulibrium price (pi,P2) is determined by (2.6). Let ek(pk,pi) = -pfcDi(pj t ,p/) /D(pfc,^) be the elasticity of demands in group k with respect to its own price, pk. Equations (2.6) then imply that, in equilibrium, the two firms set their prices such that their demand elasticities are equal to 1. Given the symmetry of demands across groups, in equilibrium Pi = P2i which we denote by p°. Later, we will provide sufficient conditions for the existence and uniqueness of the equilibrium for this game. In the case of linear demand function (2.2), the best-reply functions from (2.6) are easily computed as rk{Pi) = 2 / 3 , < + k, which are linear and strictly increasing. Therefore, two prices are strategic complements, in the sense of Bulow, Geanakoplos, and Klemperer (1985). The equilibrium price is P = and the equihbrium profit for each firm is 2 )9-7 ' 16 In the second benchmark, the two firms do not divide but collude by setting prices to maximize their joint profits: £ 2 ) = PlD(Pl,P2) + P 2 - D ( p 2 , P l ) . (2.7) Clearly, in this opt imizat ion problem only the aggregate prices matter. Assume for now that the global max imum is unique. Given the symmetry of H . (p i , p2 ) , the opt imal ag-gregate prices are identical and denoted by PM, which is determined by the first-order condit ion Let rjki(pk 1P1) = PkD2(PkiPi)lD(pk, pi) be the cross elasticity of demands between the two groups of products. Then (2.8) can be rewritten as Therefore, at the monopoly solution, the own price elasticity is set to be 1 plus the cross elasticity. Since the cross elasticity is posit ive, the own price elasticity at the monopoly solution is greater than 1. Compared to the first benchmark, the joint profit maximizat ion internalizes the ex-ternality between the prices of the two groups. As a result, the monopoly price, PM, is greater than the non-cooperative equihbrium price, p°. This point can be clearly i l lus-trated for the linear demand function (2.2). In this case, H(pi,p2) is strict ly concave. The monopoly price and joint profits are computed as D(PM,PM) + PMD^PMJPM) + PMD2(PM,PM) = 0. (2.8) £k{PM,PM) = 1 + r}ki(pMiPM)-PM = 20?-7)' a The monopoly price is greater than the duopoly price, and monopoly profits are higher. 17 2.3.2 Second-Stage Price Game We next analyze the equihbrium outcome of the second-stage price game. For a given pair of divisionalization strategies d2), the profit function (2.3) of division j in group k can be written as Kkj{Pkj,Pk,Pi;dk,di) = PkjD(pk,pi), (2.9) for k,l = 1,2, and I ^ k. Division j chooses its price, pkj, to maximize (2.9), given the price choices of the other divisions both within and across groups. The first-order condition for an interior solution is D(pk,Pi) + PkjD^pkyPi) = 0, j = l, . . . ,m f c , (2.10) for k,l = 1,2, and / ^ k. Pure-strategy Nash equilibria are then determined by equations (2.10). In equihbrium, each division sets its own elasticity of demand, Ckj(Pkj,Pk,Pi)i equal to 1, where *kj(Pkj,Pk,Pi) = -PkjD^pkiPi)/'D(pk,pi). Notice that, by (2.10), the equihbrium prices of the divisions within the group are identical. Thus, the equihbrium conditions (2.10) are equivalent to mkD(pkiPi) + PkD1(pk,pi) = 0, (2.11) for fc, / = 1,2, and I ^ k. Equations (2.11) determine the best-reply function for group k and, hence, the equihbrium aggregate prices. There are interesting properties of the equihbrium. The first is that the own price elasticity of group k equals the number of divisions in that group, i.e., £k(Pk,Pi) = mk- This implies that each firm can increase its own price elasticity by setting up autonomous rival divisions. The second property is that only the number of divisions matters, not the number of products in each division. Thus, a pair of divisionalization strategies (d\, d2) can be summarized by a pair of division numbers (mi, 7 7 1 2 ) , where mk < rik for k = 1 and 2. 18 Further assumptions on the demand function are stated to guarantee the existence and uniqueness of equilibrium in the price-setting game (see Friedman 1977, p.72). (A2) D11{p1,p2) < 0 for (pi,p2) € P-(A3) D11(p1,P2) + D12(p1,p2)<0ioI{p1,p2)eP. (A4) D2(p1,p2) + piD12(p1,p2) > 0 f o r ( p i , ^ ) eP. Here Dki(pk,Pi) denotes the second-order derivative of D(pk,pi) w i th respect to pk and pi, k,l = 1,2. (A2) states that the demand function is concave wi th respect to the own-group price. This condition is sufficient for the concavity of irkj w i th respect to Pkj, which is the second-order condition for a maximum. (A3) states that the difference between own-group effect and cross-group effect falls as the own-group price goes up. (A2) guarantees the existence, and (A3) the uniqueness of equil ibrium. (A4) implies that the best-reply functions determined by (2.11) are strictly increasing. Thus, the prices between the two groups are strategic complements. This is a common assumption i n the literature on price competition wi th differentiated products. Some interesting comparative-static results can be obtained i n this case. Let p\ and p\ denote the equihbrium aggregate group prices, respectively, and qk = D(pk,pi), for k,l = 1,2, and I ^ k denote the equil ibrium quantities. Notice that the above assumptions axe satisfied for the linear demand function (2.2). In this case, the best reply functions determined by (2.11) are r k { P , ) - (l + mh)P ' which are linear and strictly increasing. Figure 2.1 illustrates the best-reply lines. A s mk increases, the best-reply line for group k shifts up, but the best-reply line for group / does not change. Therefore, equihbrium prices for both groups increase wi th mk. These prices can be easily computed as m f co.[(l + m ^ + m f 7 ] , _ , ^ = 1 ? 2 . (1 + mk)(l + mi)p2 - mkmiY For the general form of demand functions, we have the following characterization and 19 comparative statics. Lemma 2.1: Suppose that (A1)-(A3) hold. Then, for each pair of ( m i , m 2 ) , there exists a unique equilibrium in the price-setting game. Furthermore, (a) the equihbrium prices are identical within a group; ( b ) the aggregate price in group k, f>k, increases with m* for k = 1,2; ( c ) the aggregate price in group Z, pz, increases with m^ for A;, 7 = 1,2 and / ^ k, if (A4) holds; (d) the equilibrium quantity of each complement in group k, <jk, decreases with m^ for & = 1,2; and (e) if m i = m 2 - m, then pi = pi increases with m, and = q\ decreases with ra. The proof of Lemma 2.1 is given in Appendix 2.A. The intuition for the monotonicity of the aggregate group price with respect to the number of divisions in the group is the following. Since the products within the group are complementary, there is a negative externality among the prices of these products. In other words, an increase in the price of one product reduces the demand for the other products within the same group and, hence, decreases the profits of those products. If the firm is divided into independent divisions, divisions will not take this externality into account. In equilibrium, the aggregate prices of the complements within the group increase when more divisions compete against each other. From (2.11), an increase in rrik shifts up the best-reply curve of group k, but does not change the best-reply curve of group I. Given (A4), the best-reply curves for the two groups are upward-sloping. Therefore, both prices increase as m& goes up (see Figure 2.1). Lemma 2.1(d) states that the equihbrium quantity of each complement decreases with the number of divisions in that group. As the number of divisions in the other group goes up, however, the equilibrium quantity of the complement does not necessarily fall. It depends on the sizes of both m i and m 2 . What we know is that when the firms divide symmetrically, i.e., m i = ra2 = m, the equilibrium quantity of each complement decreases as m increases. 20 2 . 3 . 3 S t r a t e g i c I n c e n t i v e s t o D i v e s t We now examine whether a divestiture improves the firms' profits. Given the characteri-zation of the equihbrium in the second-stage price game, we can write the reduced-form profit function of firm k as ~n.k(mk,mt) = pkD(pk,pi), for k, I = 1,2 and I ^  k, which depends only on the numbers of divisions mk and m / . In the rest of this section, we treat m i and m2 as continuous variables. In Appendix 2.B, we illustrate that the qualitative results in this paper still hold when m,\ and m2 are restricted to be integers. The profit of firm 1 depends on m i through the two prices. The derivative of firm l's profit function with respect to m i can be computed as follows: t ? n i ( m i , m 2 ) , « \ , » rtf - M ^  i * rw-* - (o in\ ^ = [D^ ,P2) + PiD1(Pl,p2)]— + p1D2(Pl,p2)—. (2.12) Using the first-order conditions (2.11), we can write the own-group effect of the price on the profit as D(p\,p2)+p\D1(p\,fa) = frD^T^^-^/rnx, (2.13) which is negative for m i > 1. Therefore, by Lemma 2.1(b) and (c), an increase in m i has two effects on firm l's profit. The first is the own-group effect: it increases the aggregate price of products in group 1, which reduces firm l's profit due to the negative externality of prices within the group. The second is the cross-group effect: it increases the aggregate price of products in group 2, which increases firm l's profit due to the positive externality of prices across groups. When m i is close to 1, the own-group effect is close to zero, and the cross-group effect is positive. This means that a small degree of divestiture by firm 1 increases its total profit. If the degree of substitution between the two groups of products is high, then each firm has an incentive to divide unilaterally into multiple divisions. Next, we determine the non-cooperative equihbrium of the first-stage division game, where the firms independently choose their numbers of divisions. Each firm chooses its number of divisions to maximize its profit, taking the other firm's number of divisions 21 and the second stage pricing behavior as given. Using (2.12) and (2.13), we can write the first-order conditions for an interior solution as follows: Di(Pk,Pi) -sr- + D2(pk,Pl)-— = 0 (2.14) mk 'Jmk 'Jmk for k,l = 1,2, and / ^ k. The non-cooperative equihbrium is determined by (2.14). Since the reduced-form profit functions are symmetric in and ra2, which we denote by V(mi,m2), i.e., JJ.! ( 7 7 7 , 1 , 7 7 1 2 ) = V(m1,m2), and Il2(m2,mi) = V ( r a 2 , rai), there exists a symmetric equihbrium. We denote by m the symmetric equihbrium number of divisions. To determine the size of the equihbrium number of divisions, m, we impose a further restriction on the demand function: (A5) D12(pi,P2) + D22(pi,P2) < 0 for (pi,J>2) in P-Similar to (A3), (A5) states that the difference between the own-group effect and the cross-group effect does not increase as the cross-group price goes up. Both assumptions impose a limit on how the net effect varies with prices. (A3) and (A5) together imply that the degree of substitution at the symmetric price, —D2(p,p)/Di(p,p), is non-increasing in price p. In the case of the linear demand function, (A5) is obviously satisfied. The following lemma illustrates the important properties of the reduced-form profit function when the firms divest symmetrically. Lemma 2.2: Suppose that (A1)-(A3) and (A5) hold. Then the reduced-form profit func-tion V(m, m) is a single-peaked function of m and reaches the maximum at ra = ra*, where ra* = DI(PM,PM) , 2 l gv DI(PM, PM) + D2(pM, PM) ' The proof of Lemma 2.2 is presented in Appendix 2.A. Lemma 2.2 implies that a small degree of divestitures by both firms increases their profits, but too many divestitures can reduce their profits, provided that ra* is less than n i and n2. We now state our main result. 22 Proposition 2.1: Suppose that (A1)-(A5) hold. Then the symmetric equihbrium num-ber of divisions, m, satisfies the inequalities 1 < m < rn*. The proof of Proposition 2.1 is presented in Appendix 2.A. The uniqueness of the equihbrium in the division game requires further restrictions on the demand function. We do not provide the exact restrictions here. Instead, we use the linear demand function (2.2) to illustrate Lemma 2.2 and Proposition 2.1. In this case, the equihbrium payoff from the price game can be computed as x r / x mia2ft[(l + m2)/3 + m 2 7J 2 , 0 V{m1,m2) = jr— r r — r-= r ^ . (2.16) [(1 + mi)(l + m2)p2 — m1m2'y2\2 It can be easily verified that V{m\,m2) increases strictly with m2. Thus, there exists a positive externality between the firms' choices of divisions. The best-reply functions in the division game are mi = R(m2) and m2 = i2(mi), where R ( m ) = '^rf2 „ (2.i7) (1 + m)p2 — my2 which is strictly increasing and always greater than 1. Figure 2.2 illustrates the best-reply curves that cross at the point (m, m), where fi m = It can be verified that V(mi,m2) is a single-peaked function of mi, which guarantees (m, m) to be a Nash equihbrium of the division game. Furthermore, the equilibrium is unique. It follows from Lemma 2.1 that the subgame perfect equihbrium of the two-stage game is unique and symmetric. To illustrate Lemma 2.1 and Proposition 2.1, we can easily verify that V(m,m) is single-peaked and reaches the maximum at * fi m = fi-1 Clearly, m is greater than 1, but less than m*. For any fixed (3, both m and m* increase with 7, which measures the degree of substitution between the groups of products. As 7 23 goes up, the competition between the two groups intensifies. Competing firms respond by divesting into more and finer independent divisions. We now discuss the welfare implications of the equihbrium divestitures. Suppose that the demand system (2.1) is derived from an aggregate utility maximization subject to a budget constraint, where the aggregate utility function takes the following form: u = qo + U(q11,...,qini,q2i,...,q2n2), where the good qo is a numeraire, and U(-) is a monotone increasing function. It follows that the consumers' surplus is 2 nk CS = U(q11,...,qlni,q2i,...-,q2n2) - Y^P^i, fc=li=l and the social surplus is the consumer's surplus plus the firms' total profits, which is simply SC = U(qu,#ini, #21, ? 2 n 2 ) - Applying Lemmas 1 and 2 and Proposition 1, we obtain the following welfare implications of the equilibrium divestitures: Corollary 2.1: Suppose that (A1)-(A5) hold. Then the equilibrium divestitures increase the firms' profits and reduce the consumers' welfare and social surplus. The implications for the social welfare consequences of divestitures are noteworthy. Conventional wisdom tells us that mergers of firms supplying homogeneous products or imperfect substitutes (with quantity competition) reduce competition, increase the prices of the products, and decrease consumers' welfare and social surplus. In our model, di-vestitures are motivated by product-line complementarities. Corollary 2.1 and Lemma 2.1 imply that such divestitures increase prices, lower quantities, and decrease consumers' welfare and social surplus. Therefore, from a social welfare point of view, divestitures involving complementary goods or services should be discouraged as much as mergers in-volving substitutes. In practice, however, regulatory authorities are usually concern only about mergers, but usually not about divestitures. The possibility that competing firms can achieve tacit collusion in pricing through divestitures, rather than through mergers, makes antitrust a more complex and difficult issue. 24 2 . 3 . 4 C o o r d i n a t e d d i v e s t i t u r e s In this subsection, we consider the situation in which the firms are able to coordinate their divestiture decisions in the first stage. We show that there exists a pair of division numbers (mi, m2) such that the joint profit maximizing prices (PM,PM) can be supported as a non-cooperative equihbrium outcome of the price-setting game. This pair of division numbers turns out to be (m*,m*), defined in (2.15). Indeed, comparing the equihbrium conditions (2.11) for the price game with the neces-sary condition (2.8) for the joint profit maximization problem, we find that the joint profit maximizing prices satisfy the Nash equihbrium conditions, if mi = m2 = m*. Notice that m* = €I-(PM,PM), i-e., the number of divisions is equal to the own price elasticity of each group at monopoly prices (or 1 plus the cross elasticity). A positive degree of substitu-tion between the two groups of products implies that m* > 1. Since Lemma 2.1 provides sufficient conditions for the existence and uniqueness of the equihbrium, it follows that the joint profit maximizing prices can be supported as a unique equihbrium outcome of the price game if both firms break up into m* number of independent divisions. In other words, when the firms are able to coordinate their choices of divisions, they can replicate the monopoly profit. This provides an alternative way for the firms to collude. Proposition 2.2: Suppose that (A1)-(A3) hold and Min{ni,n2} > m*. Then, by set-ting mi = m2 = m*, the second-stage price competition yields the joint profit maximizing prices and profits. The logic behind Proposition 2.2 can be described as follows. Since the joint profit function (2.7) can be rewritten as the summation of the profit functions (2.9) over all independent divisions from both groups, the joint profit maximization problem can be equivalently solved by choosing prices (pii, ...,Pimi) and (p2i, •••,P2m2)- Notice that we can decompose the effect of an increase in price p\j into the following three terms: an/apu- = [ufo,^  + PIJD1(Pi,P2)} + £ pLJ>D1(p1,p2) + p2D2(p2,p1). (2.18) The first term in the square brackets represents the own-group effect of the price on the profit of division j in group 1, TT\J, which is the same as the left-hand side of (2.10). 25 The second term is the aggregate intra-group effect of price on the profits of the other divisions in the same group, 7 ^ •/, for all j' different from j. It is negative since divisions within the same group supply complementary products. The third term is the aggregate inter-group effect of price on the profits of the divisions in the other group. Since products are imperfect substitutes across groups, the aggregate inter-group effect is positive. Now, if there exists a number of divisions m such that the negative intra-group effect exactly offsets the positive inter-group effect, then the necessary condition for joint profit maximization is equivalent to the first-order condition (2.10). In other words, given the number of divisions m, the necessary conditions for joint profit maximization are identical to the first-order conditions for a non-cooperative equilibrium in the price game. This can be accomplished by setting the second plus the third term on the left-hand side of (2.18) equal to zero. Imposing symmetry and substituting PM for p\ and pz, we obtain (mi - \){pMlm1)D1{pM-,PM) + PMD2(PM,PM) = 0. (2.19) The solution to (2.19) is mi = m*. A similar argument determines m2 = rn*. As a result, the monopoly price PM satisfies (2.8) and (2.11) and, hence, consists of the unique solution to the joint profit maximization problem and of the equihbrium of the price game. The driving force behind this finding is the commitment power of divestiture combined with the extended product space, which includes substitutes as well as complements. Prior to divestiture, the prices of products in each group are set coordinately. After divestiture, the firm credibly commits not to set prices of the group coordinately, therefore inducing less competition from the rival group. As a result, the prices and profits increase. This implies that price coordination by a group of firms supplying complements does not nec-essarily benefit the firms and harm consumers. In our model, the lack of coordination among the prices of complements indeed leads to higher profits and lower consumer sur-plus. In other words, firms have incentives to tie their own hands in order to induce a better (more profitable) response from their rivals11. The importance of incorporating both substitutabihty and complementarity within the same framework can be seen clearly in terms of externalities of each division's pricing U A similar commitment effect works when firms use a most-favored-customer policy to raise the prices (see Cooper, 1986). 26 decision. There is a positive externality between the prices across groups due to substi-tutability. Divestiture in the first-stage creates a negative externality of prices among independent divisions within a group. This negative externality can offset the positive externality. When the degree of divestiture is small, the positive externality dominates. When the degree of divestiture is large, the negative externality outweighs the positive one. At m = m*, all externalities are neutralized, and consequently, the monopoly out-come is achieved. We now understand the second inequality in Proposition 2.1. It is driven by the positive externality between the firms' choices of divisions. Given (A5), an increase in m i will increase firm 2's profit, when both firms choose the same number of divisions. In the presence of such a positive externality, a lack of coordination between the firms results in a smaller equihbrium number of divisions than the coordinated number of divisions. What determines the size of the optimally coordinated number of firms? Notice that the ratio, —D2(PMIPM)ID\(PM-,PM)I represents the degree of substitution between the two groups of the products. The following corollary provides a comparative-static result: Corollary 2.2: The optimally coordinated number of divisions, m* , increases with the degree of substitution between the two groups of products. In one extreme case, where there axe no substitute goods, m* is equal to 1, and each firm should monopohze the supply of the complements and never divide. In the presence of competing substitutes, the firms have incentives to divide. As the relative degree of substitution between the two groups of complements increases, the positive externality increases and, hence, each firm should split into more divisions to increase the negative externality and mitigate the positive one. At the other extreme, if the number of complements in each group is small or the degree of substitution is large, further divestiture may not be possible. In this case, monopoly profit cannot be replicated, and the firms prefer complete divestiture in which each product is supplied by an independent firm. 27 2.4 Extensions In the previous section, we have discussed the effect of product market competition (or product differentiation) on divisionalization and shown that a higher degree of substitution between the two groups of products leads to a greater number of divisions i n both groups. There are other factors that determine the coordinated and non-cooperative divionaliza-tion strategies and the scopes of the firms. They include marginal costs, asymmetric demands, imperfect complements, and several groups of complements. In this section, we briefly discuss only two of these factors, imperfect complements and the number of groups of complements. For simplicity, we use linear demand functions. a) Imperfect Complements In our basic model, we have considered only perfect complements wi th in the group. Our analysis can be extended to the case of imperfect complements. To illustrate, we consider an example wi th the following linear demand functions: qki = a - fi'pki - fi Y PW + 71* > for k, I = 1,2, I ^ k, i = 1 , 2 , n f c , where a > 0, fi' > fi > 7 > 0, and pk = £ " = 1 pki. The assumption fi' > fi implies that product i and any other product wi th in the group are imperfect complements and that the effect of the price pk{ on the demand for product i dominates the intra-group effect on the demand for any other product wi th in the group. A s before, fi > 7 means that intra-group effects outweigh inter-group effects. The demand system is symmetric both wi th in the group and between groups. Notice that the demand function for product i i n the first group can be wri t ten as qu = -(fi' ~ fi)Pii + « - fiPi+ 7P2, which consists of two parts, the first depending only on the individual price pu and the second being the same as the demand function (2.2). Clearly, fi' — fi does not affect the demand externality arising from an increase i n pu. Remember that the opt imally coordinated number of divisions is determined by balancing the negative and positive externalities of the prices. Therefore, it is independent of fi' — fi and can be computed as 28 ra* = 8/(8 — 7). However, ft' — 8 affects the Nash equihbrium number of divisions, ra. It can be easily calculated that ra is determined by the following first-order condition ™2(/?2 - 72) - P2 + 2(8' - 8)h(m - l)/n = 0, where n\ and n 2 are assumed to be equal, for simplicity, and denoted by n. Notice that ra decreases with 8' — 8. When the demand elasticity with respect to own price increases (higher /?'), the incentives for each firm to divide fall because the second-stage equihbrium prices decrease. As a result, the equihbrium number of divisions decreases as 8' increases. b) Multiple Groups of Complements Our analysis can also be extended to the case of many groups of complements, where the products are imperfect substitutes across groups. Let K be the number of groups, K > 2. The demand function for a complement in group k is linear and presented by qki = a- /3pfc + 7 ] T P k >, for k = 1 , K and i = 1 , n k , where a > 0, 8 > (K — 1)7 > 0, nk is the number of complements in group k, and pk = ]T)"=i Pki is the total price of the complements in group k, k = 1 , K . The demand for other products is symmetric with the group and across groups. We can compute the optimally coordinated number of divisions for each group and the non-cooperative equihbrium number of divisions as follows: m - 8 - ( K - l W . _ y/{K - 2)»72 + 4Q8 + 7)03 - (K - 1)7) - (K - 2)7 2(/3 + i)(8-(K-l)7) As before, the optimally coordinated number of divisions and the equihbrium number of divisions increase when the degree of substitution between the groups of the products increases. An interesting comparative-static result is that both ra* and ra increase with K. As the number of groups of the complements goes up, competition across groups 29 increases and, hence, the magnitude of the positive externalities among the prices across groups also increases. To mitigate the increased positive externalities, the firms need to create more negative externalities by dividing the firms into more independent divisions. Therefore, both the optimally coordinated number of divisions and the non-cooperative equihbrium number of divisions increase. 2.5 Concluding Remarks In this paper, we have argued that a firm producing a group of complements can generate higher profits through divestitures when there is a competing firm supplying an imperfect substitute group of complements. By delegating pricing decisions to independent divi-sions, each firm credibly commits not to set the prices of the complements within the group coordinately, which softens the competition between the two firms. Our analysis suggests that industry structure and size of firms are closely related to the nature of heterogeneous products and the strategic considerations of the firms. The welfare implications of the divisionalizations in our models are significant. It is shown that firms supplying complements have incentives to divest when facing competi-tion, and the resulting divestitures raise prices and reduce consumers' surplus. However, the profit gains are not large enough to compensate the consumers' losses. As a result, the total surplus is reduced. This suggests that, from a social welfare point of view, divesti-tures involving complementary goods or services can be as harmful' as mergers involving substitutes. Thus, antitrust authorities should be concerned about not only mergers, but also divestitures. Yet, in practice, mergers are usually the sole focus of antitrust policies. Another implication of our analysis is that divestitures motivated by product-line complementarities can be viewed as a response towards entry. Suppose that, initially, there is a monopoly that supplies a group of complementary goods or services. Clearly, in our context the monopolist does not have any incentives to divest its operations. When potential rivals enter the market, the monopolist has two possible responses. One is to compete directly against the entrants. If the entrants do not make enough profits to cover their entry costs, the entries can then be deterred. If the products offered by the entrants are differentiated enough from the incumbent's products, and if the entry costs are relatively small, the entries cannot be deterred. In this case, an optimal strategy for 30 the incumbent firm is to divest some of its operations to soften the competition. It should be noted, finally, that our analysis is based upon a number of assumptions. One crucial assumption is that divisions cannot further divide before they choose their prices. This is reasonable in situations, where the parent firms still have major ownership control of the divisions, but do not make management decisions. Franchise contracts can be viewed as an example of such a divisionalization. In other situations, it can be difficult for firms or divisions to make this type of commitment. Divisions may then have incentives to divide further. This raises the issue of what determines a stable industry structure in the presence of both substitutes and complements. Further research along this line is needed. Our modeling of the demand structure also implies that products across groups are not compatible. When they are completely compatible, the demand function has to be adapted. Firms divesting decisions will be affected. Therefore, our analysis does not apply in the case of cross-group product compatibility. 31 Chapter 3 Product Differentiation, Strategic Divisionalization and Persistence of Monopoly 3.1 Introduction It is generally accepted in economics that, in markets with similar products, competi-tion reduces firms' profits. Yet, large firms often set up independently managed rival divisions1 supplying similar products and competing in the same market2. A similar puz-zling phenomenon is that, to a certain degree, franchises of the same parent firm are often 1For example, many automobile manufacturers have independently managed operating divisions. In the case of General Motors, Sloan (1963) notes "According to General Motors plan of Organization ... the activities of any specific operation are under absolute control of the General Manager of that Division, subject only to very broad contact with the general officers of the Corporation" (p.106). Moody's Industrial Manual (1993) describes that each of G M operating divisions "is a self-contained adininistrative unit with a general manager responsible for all functional activities of his divisions. 2Baye et. al. (1995) indicate that "For example, General Motors produce the LeSabre (Buick Motor Division) and the Olds 88 (Oldsmobile Division) which, while differing in styling, are built on the same chassis and, comparably equipped, sell for virtually identical prices. Similarly, the Ford Motor Co. produces the Sable (Lincoln-Mercury Division) and the Taurus (Ford Division), which are effectively the same car with different name plates, as are the Chrysler Corporation's Plymouth Voyager and Dodge Caravan. Indeed, in just about every price range, all of the major domestic manufacturers have several divisions producing competing products." 33 allowed to compete for the same customers. There are a few different explanations for such practices. First, production efficiency requires mult ip le production plants under de-creasing returns to scale technology. Second, the need to satisfy heterogeneous consumers, in terms of either tastes or location, forces firms to set up mul t ip le divisions producing different brands or mult iple outlets operating at different locations. Th i rd , i t is argued by Wil l iamson (1975) that setting up autonomous divisions alleviates incentive problems due to moral hazard w i th in large organizations. Nevertheless, these explanations do not just i fy why a f i rm allows its divisions to compete rather than to cooperate w i t h each other in production (or sales) strategies. To search for a more satisfactory explanation, recent studies have focused on firms' strategic incentives in divisionalization. Schwartz and Thompson (1986) and Veendorp (1991) show that the incumbent firm can forestall entry by setting up mul t ip le r iva l divisions prior to entry. Corchon (1991) and Polasky (1992) analyze a two stage division-alization game w i t h a duopoly supplying homogeneous products. They show that each firm has an incentive to set up r ival divisions, but there is no finite Nash equi l ibr ium. Using a similar framework, Corchon and Gonzalez-Maestre (1993) and Baye, Crocker and Ju (1996) rectify the nonexistence problem by imposing either an exogenous bound of permissible number of divisions or a fixed division set-up cost. The insight f rom these recent studies is the fol lowing: in the homogeneous product market, setting up a new independent division reduces the aggregate profi t due to the increased competit ion, but increases a firm's share in the aggregate output and prof i t . The second effect always dominates the first i f products of different firms are perfect substitutes. Consequently, firms have a private incentive to divisionalize. Furthermore, there is no finite equil ibr ium because of the dominance of the second effect. In this paper, we provide a model of differentiated products, which contains the pre-vious framework considered by Corchon, Polasky and Baye et. al. as a special case. I t is first shown that the problem of nonexistence of an interior subgame perfect Nash Equi l ibr ium (SPNE) in this l i terature is due to the assumption of homogeneous prod-ucts. Product differentiation alone ensures the existence of an interior SPNE. As shown in previous studies, divisionalization has two effects on a firm's prof i t ; setting up au-tonomous divisions increases a firm's market share, on one hand, and creates competit ion for the firm's existing divisions, on the other. We refer to the first effect as the business 34 stealing effect, which by itself enhances the firm's profitability, and the second as the competition effect, which reduces the firm's profitability. Product differentiation weakens the business stealing effect because of the reduced substitutability among products. The competition effect, however, increases with product differentiation, since the competition from additional divisions is borne more by existing divisions of the same firm when the substitutability among rival products decreases. Thus, a firm's incentive to divisionalize is reduced if products are differentiated, resulting in an interior SPNE. We then show that, if divisions are allowed to divide further, they always will. The final outcome of the divisionalization game without restriction on further dividing is equivalent to the perfectly competitive equihbrium, where each firm earns zero profit. To prevent this disastrous outcome of total profit dissipation, parent firms have a unilateral incentive to restrict their divisions from further dividing. This finding provides a theoretical jus-tification for a rather important assumption in the divisionalization literature that only parent firms are allowed to set up independent divisions, and that divisions themselves are not. Veendorp (1991) shows that parent firms delegate output decisions and the like to their divisions, but not investment decisions regarding divisions' capacity. We show why parent firms reserve the authority regarding the divisionalization decision. Our find-ing is also consistent with the general business practice in franchising. Franchisees are usually not allowed to sell franchises themselves. An alternative organizational set-up is that the parent firm functions as a holding company, allowing its subsidiaries to manage independently and, at the same time, taking away the authority of setting up subdivisions. Finally, we study the effect of divisionalization on the free entry equihbrium. Because entry induces incumbent firms to set up more divisions, and the numbers of divisions of firms are strategic complements, entry can enormously intensify competition under divisionalization. An entrant faces competition not only from the existing divisions, but also from entry-induced additional divisions of incumbent firms. As a result, potential firms are more reluctant to enter a market if divisionalization is possible. In other words, divisionalization is a natural entry barrier, potentially generating high and persistent profits for the incumbent firms. In the cases where product differentiation is difficult, the only pure strategy free entry SPNE is the monopoly outcome. More interestingly, in such circumstances, it is the credible threat of divisionalization after entry occurs that ensures the monopoly outcome. The incumbent does not actually have to set up divisions 35 prior to entry. This finding sharply contrasts to Eaton and Lipsey (1979, 1981), Gilbert and Newberry (1982), and Schwartz and Thompson (1986), who suggest the incumbent firm has either to build up capacities or to set up divisions prior to entry to prevent it effectively from happening. The paper is closely related to, but distinct from the merger literature such as Salant et al. (1983) and Gaudet and Salant (1991), which examine the impact of mergers on firms' profits and on social welfare. Under plausible conditions, they show that merger of a subset of firms may result in profit loss for the merging firms, even though merger leads to a more concentrated oligopoly. This paper deals with the opposite issue: the incentive facing firms to divisionalize or to set up rival independent units. We show that, in plausible scenarios, each firm has unilateral incentives to create rival competing units. Unlike most of the merger literature, our analysis permits one to analyze the equilibrium consequences of these incentives in a noncooperative setting that allows all firms to divisionalize freely. The merger literature, in contrast, implicitly views merger as a cooperative game among merging parties, for the set of firms that merge is usually exogenously selected. The rest of the paper proceeds as follows. In the next section, we set up the basic model. Section 3 characterizes the equihbrium of the two stage divisionalization game. Section 4 considers the implication of the possibility of further divisionalization by divi-sions. Section 5 examines the free entry equihbrium. A final section concludes. 3.2 The Model 3.2.1 Demand and Technology We consider an environment where n oligopolistic firms each produces a differentiated product. To simplify the analysis, demands for different brands are assumed to be repre-sented by the following linear system: pk = a - bxk - dYxt, (3-1) for k — 1 , 2 , n , where d £ (0,6), and xk and pk are the consumption and price of the kth brand respectively. Notice that d can be viewed as a parameter indicating the degree of 36 brand differentiation (or the degree of substitution between brands). A s d approaches zero, brands become more differentiated, and i n the l imit (d = 0), demands are independent. As d approaches b, brands become closer substitutes and, i n the l i m i t , become perfect substitutes. In order to isolate the strategic motivation for firms to divisionalize, we assume that ohgopohstic firms have constant-return-to-scale technologies w i t h the same marginal cost. T h e n , without loss of generality, we set the marginal cost equal to zero. Further, divisions of the same parent firm inherit the same technology. Thus, divisions of the same firm supply perfect substitutes, and divisions of different firms supply imperfect substitutes. B y construction, the intra-firm substitutabihty is greater than the inter-firm substitut ability. This feature of our model w i l l be shown to be crit ical to ensuring the existence of an interior S P N E . Moreover, divisionalization is assumed to be costless (i.e., the division setup cost is zero). However, to enter the market, a firm has to incur a fixed cost, F, which can be interpreted as a R & D cost for developing a product. 3.2.2 The Divisionalization Game Following the convention i n the l i terature 3 , we refer to a firm setting up autonomous r iva l divisions as divisionalization hereafter. The divisionalization game we consider is a simultaneous-move two-stage game w i t h perfect information. In the first stage, each ohgopohstic firm simultaneously chooses its number of autonomous divisions. In the second stage, a l l divisions engage i n a Cournot competition by setting levels of output simultaneously. Every market participant knows the demand structure and technologies. Divisions can be viewed as independent profit centers whose managers are compensated solely according to divisions' profits. The profit of a parent firm is the sum of the profits of its divisions. We solve the game by backward induction: solving first for the Cournot equihbrium outputs i n stage two for a given set of numbers of divisions, then for the equihbrium number of divisions chosen by each firm i n stage one. Notice that, like other models i n the divisionalization literature, we imphci t ly assume that autonomous divisions w i l l not further divide into more independent subdivisions. The analysis of divisionalization depends critically on this assumption. W e w i l l provide 3 For examples, see Schwartz and Thompson (1986) and Baye, Crocker and Ju (1995). 37 a theoretical justification for this assumption in Section 3.4. 3.3 Analysis In this section, we first solve the two stage subgame perfect Nash equihbrium by back-ward induction. Then we discuss the comparative static results of the equihbrium and summarize our main findings. 3 .3 .1 T h e S e c o n d S t a g e C o u r n o t Q u a n t i t y G a m e In the second stage, all independent divisions choose their levels of output simultaneously for a given set of numbers of divisions. Let denote the number of divisions of firm k, and Xki be the output level of division i of firm where i = 1 , 2 , a n d A; = 1 , 2 , n . Subscripts k and i denote firm k and division i of firm fc, respectively. Notice that the total output of the kth. firm = xki- Then, the profit function of division i of firm k is TTfci — PkXki = (a — xk,; — Y xtj)xki- (3.2) t'=l t^k j=l Division i of firm k chooses xki to maximize its profit, taking the output decisions of the other divisions as given. The first order condition yields = a - b(2xki + J2xks) -  dYY,xti = °> (3-3) ' J X k i s& tjtk j=l for all k and i. Pure strategy Nash equihbrium is then determined by the above equation system. Solving for xki, we have CL n xki = T - xk - * Y) xt, (3.4) b t+k where z = f, which measures the degree of substitution between products. When z = 1, products of different firms are perfect substitutes. When 0 < z < 1, products are imperfect substitutes and the degree of substitution increases with z. Notice that x^i is the same for all i. Summing the above equation over i yields xu = * ~ Z S f (3-5) 38 for all k = 1 , 2 , n . Equation (3.5) involves only the total output of each firm and, thus, can be viewed as the best response function at the firm level. Solving Xk in terms of the numbers of divisions ,we have X K = K T T ^ A ) ' ( 3 ' 6 ) where 8t = 1+_L_Z> Vt, and A = £ " = 1 6t. L e m m a 3.1 : Given the linear demand system, for each configuration of numbers of divi-sions of the oligopolistic firms rai,m2,ran, there exists a unique Nash equihbrium in the Cournot quantity game. Moreover, (a) a division's output decreases with the number of divisions of each firm; (b) a parent firm's total output increases with its number of divisions and decreases with the number of divisions of every other firm; and (c) outputs of every division and firm decrease with the number of firms. (Proof: see Appendix 3.) The driving force behind Lemma 3.1 is that divisions' outputs are strategic substitutes in the Cournot game. Define the market share of firm k as the ratio of its output to the total output of the industry: Sk = where x = Ylk=ixk- An interesting corollary of Lemma 3.1(b) is in order: Corollary 3.1: An increase in the number of divisions of one firm increases the market share of that firm and decreases that of every other firm. Corollary 3.1 highlights a firm's private incentive to set up competing divisions. The dual effects of divisionalization on a firm's profit can also be easily shown from the fol-lowing derivation. Rearranging equation (3.3) and using the inverse demand function gets n Pk = a — bxk — dy^xt = bxki-39 Since Xki is the same for all i, the profit of firm k can be written as TTfc = rnkTTki = mkpkxki - fem^a:2..-. Then, d-Kk 9lTk 9lTk '3-Kki <3irki -j = o r o o = m f c o r TTfci, amj jmjt y^,- i/mfc raj where | ^ = 2&:rfc;|m^  < 0. The first term of the above equation represents the effect of creating a new division on the profit of the existing divisions of the firm. We shall refer to this effect as the competition effect of divisionalization, since the new division increases the level of competition and reduces profits of the existing divisions. The second term represents the direct contribution of the new division towards the profit of the firm. We shall refer to it as the business stealing effect of divisionalization, since the output of the new division comes partly at the cost of other firms. The second effect highlights the motivation behind divisionalization. 3 . 3 . 2 T h e F i r s t S t a g e G a m e Given the characterization of the equihbrium in the second stage, we now consider firms' divisionalization decisions in the first stage. The parent firm fc's total profit is the sum of profits of all its divisions. Then, *k = £ * " = * X i = b m i { l + zA)*: ( 3 - 7 ) Firm k chooses mk to maximize its profit taking as given the number of other firms' divisions. We have the following first order condition: dmk om%(l + zA)3 mk where k = l,2,.. . ,n and Afe = 2~2t^k^t- Equation (3.8) defines the equilibria of the first stage divisionalization game4. 4The second order derivative evaluating at the solution of the first order condition is r[l + zA*(l-z)]<0; dml mKl + zAf1 40 Lemma 3.2 : The numbers of divisions of the ohgopohstic firms axe strategic comple-ments. Proof: Rearranging equation (3.8), we have the best response function of firm k mk = i 1 z 2 • (3.9) It is easy to verify that > 0 and > 0 for t ± k. Then, 9mk _ dmk d Ak > Q <3mt 9Ak dmt Q.E .D . Lemma 3.2 implies that a firm responds to an increase in the number of divisions of another firm by increasing the number of its own divisions. In doing so, a firm attempts to mitigate its market share loss due to other firms' aggressiveness in divisionalization. Because the complementarity shown here implies a positive chain reaction among the numbers of divisions of firms, a change in a factor which affects a firm's number of divisions might have a profound effect on the number of total divisions and, in turn, on the level of competition. It will be shown later that this insight will have significant imphcations on the entry deterrence property of divisionalization. Imposing a symmetry condition to the equation (3.8)5, we obtain the solution for the equihbrium number of divisions m: m2[(l - z)((n - l)z + 1)] - m(n - 2)z - 1 = 0 or, m (ra - 2)z + y/(n - 2)2*2 + 4(1 - z)((n - l)z + 1) 2 ( l - z ) ( ( n - l ) * + l) ' (3.10) that is, the second order condition is satisfied. 5It is shown in the Appendix that all possible solutions to the stage one game are symmetric; therefore, imposing the symmetric condition here does not constitute any real restriction on the solution. 41 P r o p o s i t i o n 3 .1 : W i t h linear demand and differentiated products, the two stage divi-sionalization game has a unique subgame perfect Nash equil ibrium, i n which each firm chooses a finite number of independent divisions. Furthermore, the equihbrium number of divisions of each firm increases wi th the degree of substitution between products and the number of firms (the number of differentiated products). Proof: see Appendix 3. Proposit ion 3.1 states that product differentiation ensures the existence of an interior S P N E for the divisionalization game. The effect of product differentiation on the deter-mination of the equihbrium number of divisions is evident from the positive relationship between the equihbrium number of divisions and the degree of substitution between firms. A s we have shown earlier, divisionalization enables a firm to steal business from its rivals, on one hand, and creates competition for the firm's existing divisions, on the other. The business stealing effect decreases wi th the degree of differentiation among products, since lower substitutability among products naturally l imits the firm's abi l i ty to shift the de-mand from its rivals ' products to its own product. The competition effect, on the contrary, increases wi th the degree of product differentiation, because additional divisions have to compete mainly wi th the existing divisions of the same firm when product substitutability is low. Thus, firms' incentive to divisionalize is reduced when products are differentiated, resulting in an interior S P N E . The existence result of interior S P N E wi th product differentiation can be clearly illus-trated wi th the help of equation (3.9), the best response function of firm k. W i t h product differentiation, i.e., z < 1, 'Jrrik _ z2 1 ~drn~t = (Akz(l -z) + l ) 2 (mt(l - z) + l ) 2 K ' Then, i f firm t increases its number of divisions, mt, firm k has an incentive to increase its number of divisions but w i l l not match the increase i n mt. Moreover, as mt — • co, IT24- —> 0 and rrik — • /, ^".'V*-.w < T ^ - Recal l from Lemma 3.2 that mjt increases dmt K (l-2)(l-|-z(n-l)) — 1-z with mt. Thus, the best response function of firm k is bound above. No matter how many divisions other firms set up, firm k w i l l not set up more than divisions. That guarantees the interior equihbrium. 42 Without product differentiation (z = 1), however, equation (3.9) becomes n mk = ^2mt + 1, and r3rnk _ ^ That is, firm k has an incentive to match any increase in the number of divisions of any other firms. There is no upper bound for the best response function mk. In fact, each firm has an incentive to set up more divisions than the number of divisions of all the other firms combined. That drives the equihbrium number of divisions to infinity. In the case of duopoly, the two best response functions can be plotted in a diagram. When z — 1, they are two parallel lines (Figure 3.1 (a)). One always lies above, and the other lies under the 45 degree line. No interior equihbrium exists. When z < 1, however, the best response curves both bend towards the 45 degree line and are bound above at (see Figure 3.1 (b)), resulting in an interior equihbrium with the number of divisions for each firm less than 7^ —. It is worth considering two extreme cases. If there is zero substitution among prod-ucts, each firm faces totally independent markets. Divisionalization then creates the competition effect, but not the business stealing effect. Thus, the firm has no incentive to divisionalize. If products are perfect substitutes, however, the business stealing effect is at its maximum and the competition effect is at its minimum6. In fact, the former always dominates the latter. Each firm in equihbrium will set up infinite divisions. We summarize these two cases in the following corollaries: Corollary 3.2: If there is zero substitution among products, firms do not divisionalize. Corollary 3.3: If ohgopohsts produce perfect substitutes, the equihbrium number of divisions for each firm is infinity. Corollary 3.3 is one of the main results of Corchon (1991), Corchon and Gonzalez-Maestre (1993), Polasky (1992), and Baye, Crocker and Ju (1995), who all consider a model of homogeneous products. Our results show that the nonexistence problem in the 6The extra competition is shared equally by all existing divisions of all firms. 43 previous studies is mainly due to the assumption that firms supply homogeneous products. Product differentiation automatically guarantees an interior SPNE. Proposition 3.1 also characterizes a positive relationship between the equihbrium num-ber of divisions and the number of differentiated products. That is, firms get more ag-gressive in divisionalization when they face more competing firms. The intuition is that the larger the number of competing firms, the more sources a firm can steal business from and spread the additional competition to. Consequently, firms have a higher incentive to divisionalize and, as a result, the equihbrium number of divisions of each firm will be higher. This result also has a significant implication for entry deterrence. It implies that an entrant has to compete not only with the divisions existing prior to entry, but also with additional divisions induced by the entry. We will come back to this issue in section 3.5. 3.4 Possibility of Further Divisionalization In the two stage divisionalization game, we implicitly assume that the original parent firms can set up divisions, but divisions themselves cannot. What happens if this assumption is relaxed? Will divisions then further divide? Will parent firms' profits decrease or increase? Do parent firms have incentives to restrict divisions unilaterally from further dividing? To answer these questions, consider a case where divisions are allowed to divide fur-ther into autonomous subdivisions. Let a division's profit be the sum of profits of its subdivisions. We will first show by contradiction that if divisions are allowed to divide further, they always will. Assume that there exists an equihbrium structure such that no divisions choose to divide further even though they are free to do so. Let mk be the number of divisions of firm k in equihbrium, where k = 1,2, ...,n. By definition, it is not profitable for any division to divide further in equihbrium. Now, imagine a hypothetical further breakup of divisions. Let be the number of independent subdivisions in division i of firm k. The total number of divisions of firm k will be Nk = J23jZTk skj- According to equation (3.7), 44 the total profit of firm k is a2 TTfc = bNk[(l + ±--z)(l + zA)]2-Then, the total profit of the ski subdivisions of division i of firm k is ski a2ski Xki = TTTfc = Nk b[Nk(l + ^-z)(l + zA)]^ Division i of firm k chooses ski to maximize irki taking the number of subdivisions of other divisions as given. After simplifying the first order condition, we have & 1 + (1 - z)zAk Notice that Yjj^Tk skj + 1 is the number of subdivisions division i of firm k would set up if there were no other firms besides k, and , . fe2^* A is the number of divisions induced ' l+(l-z)zAk by the existence of subdivisions of firms other than k. For mk > 2, the symmetric solution to the above equation is infinity, which is not surprising, since divisions of the same firm supply perfect substitutes, and, therefore, the business stealing effect dominates the competition effect in further divisionalization. This result obviously contradicts our assumption earlier that there exists an equilibrium structure in which no divisions have the incentive to divide further. Thus, if divisions are allowed to divide further, they always will. Furthermore, the total profit of the firm which does not restrict its divisions from further dividing is a2 lim 7T{. = Hm T, ; z~, = 0, M^oo k M - o o bM[(l + £ - Z)(l + ZA)]2 ' and the price of kth firm's product is i - T A hm pk = lim = — — = 0. M ^ o o ^ A/^oo (1 + JL _ z)(l + Z A) That is, allowing divisions to divide further leads to zero markup and zero profit for the firm. The result holds regardless of whether other firms allow their divisions to divide 45 further. In order to avoid the total dissipation of profit, each firm has an incentive to restrict its divisions unilaterally from further dividing. When a parent firm cannot effectively enforce this restriction, it is better not to set up divisions. The results are summarized as follows: Proposition 3.2: If divisions of a firm are allowed to divide further, they always will, resulting in total profit dissipation for the parent firm. Thus, each firm has an incentive to restrict its divisions unilaterally from further dividing. One imphcit but critical assumption in the strategic divisionalization literature is that only the parent firms can set up divisions, and the divisions themselves cannot. Without this assumption, the corresponding analysis and results do not hold. Proposition 3.2 pro-vides a theoretical justification for this assumption. This result complements Veen dorp's (1991) finding that a firm in a multidivisional structure delegates to its divisions deci-sions regarding output and the like, but reserves for itself investment decisions regarding capacities. We show that a parent firm will reserve the divisionalization decision. This finding is also consistent with the general practice in franchising where franchisees are usually not allowed to set up franchises themselves. 3.5 Free entry equilibrium In this section, we turn our attention to the free entry equihbrium. Formally, we consider the following simultaneous entry game with a large number of identical potential firms in the market: Stage 0: Firms make their entry decisions given the other firms' entry decisions. Stage 1: Firms which have decided to enter the market in the first stage choose their numbers of independent divisions. Stage 2: All the divisions engage in Cournot competition. Notice that the last two stages of the free entry game are the same as the two stage divisionalization game we have solved in Section 3.3. Thus, we only need to solve the entry game in stage 0. 46 Free entry, by definition, entails zero profit for each firm i n the market. That is, i n the free entry equihbrium, TTfc - F = 0, or, = F, (3.11) bm[l + ^ + z(n - l)]2 where m is the solution to equation (3.10), F is the fixed cost associated wi th entry, and 7Tfc is the gross profit of firm k. Equation (3.11) defines the number of firms i n the free entry equihbrium. P r o p o s i t i o n 3.3 For a given F, there exists a pure strategy S P N E for the free entry game. Moreover, the equihbrium number of firms (or differentiated products) de-creases wi th the degree of substitution between the products, as well as wi th the fixed entry cost, F. Similar to Salop (1979), we find that, i n equihbrium, the number of firms is negatively related to the magnitude of the fixed entry cost. However, our results imply a much stronger case for the persistence of high profits (except the case where the zero profit condition produces an exact integer solution for the number of firms). Since the equihb-r ium number of divisions of a firm increases wi th the number of firms, the competition an entrant faces comes not only from incumbents' existing divisions, but also from additional divisions induced by entry. The number of additional divisions induced by entry can be rather large, due to the complementarity among the numbers of divisions of different firms7. Thus, entry might induce very severe competition and profit dissipation. A s a result, w i th divisionalization, a potential firm is more reluctant to enter the market, and the incumbents may, i n turn, enjoy abnormally high profits (this is clearly il lustrated i n the following example). Like divisionalization itself, the entry deterrence effect of divisionalization increases wi th the degree of substitution among products. A s z approaches 1, even a duopoly w i l l generate so many divisions that the last stage Cournot game w i l l generate an outcome close 7See Lemma 2. 47 to the perfectly competitive equihbrium. Then, the only possible free entry equihbrium is the monopoly outcome, even if the entry cost is relatively low. Consequently, the incumbent can persistently earn the monopoly profit, which may be many times more than the entry cost, without worrying about entry. For example, in the case of a duopoly (n = 2), m = ^Aj-^j (from equation (3.10)) and the profit of a duopohst TT = ^1+^J^_z2y2 from (equation (11)). When z — .98, ra = 5, a duopolist's profit is 8.4% of that of a monopoly, which is Hence, if F, the fixed entry cost, is greater than or equal to 8.4% of the monopoly profit, the free entry equihbrium is a monopoly outcome, despite the fact that the monopoly may earn as much as 12 times of the entry cost. It is worth noting that, in this case, it is mainly the threat of divisionalization (not the fixed entry cost) that causes the natural monopoly outcome. Interestingly, unlike what the previous literature has suggested8, divisionalization does not need to occur to assure the monopoly result. The credible threat of divisionalization in case of entry is enough. The above result is summarized in the following proposition: Proposition 3.4: Under divisionalization, incumbents in free entry equihbrium may per-sistently earn abnormally high profits. In particular, for any given F, there exists a z* < 1 such that, for any z > z*, a natural monopoly is the unique pure strategy free entry SPNE, where z* satisfies (1+z^f!^)2 = F±. Remark: There is a paradoxical implication of Proposition 3.4. Given the number of firms in the industry, divisionalization intensifies the level of competition and alleviates the distortion in pricing. It seems that, from a social point of view, divisionalization should be encouraged. Under free entry, however, the threat of divisionalization by incumbents reduces potential entrants' incentive to enter a market. The competition level in free entry equilibrium with divisionalization might be much lower than when no divisionalization is allowed9. In summary, even though limiting the number of autonomous divisions of firms may weaken competition in the short run, it can strengthen competition in the long run. 8For example, Schwartz and Thompson (1986) show how the incumbent forestalls entry by setting up independent divisions just prior to the date of a potential entry. 9In the example where z = .98, if F is 8.4% of the monopoly profit, the free entry equihbrium with divisionalization is monopoly. If divisionalization is not allowed, it is easy to verify that there will be at least nine competing firms in equilibrium. 48 3.6 Concluding Remarks In this paper, we consider an environment in which competing oligopolistic firms with differentiated products can set up independent rival divisions. We analyze the strategic incentives for a firm to divisionalize, characterize the equihbrium of a divisionalization game and highlight the effect of product differentiation in ensuring an interior equihbrium. By allowing for product differentiation, we demonstrate that the existence of an interior equihbrium can be achieved without reliance on ad hoc assumptions, such as an exogenous bound of permissible number of divisions or a costly divisionalization. The existence of an interior equiUbrium depends heavily on the assumption that di-visions of the same firm produce closer substitutes than divisions of different firms do. However, the divisionalization result does not depend on this assumption. When all firms produce homogeneous products (the model of Corchon, Polasky and Baye et al.), firms still divisionalize. In fact, they set up an infinite number of divisions in equihbrium. We can infer that, if divisions of the same firm are able to be differentiated such that divisions of different firms produce closer substitutes than divisions of the same firm10, the business stealing effect will be strengthened, and the competition effect will be weakened. Thus, firms would have stronger incentives to divisionalize. Assuming there is no fixed set-up cost, each firm would set up as many such differentiated divisions as possible. However, differentiation of divisions of the same firm often requires differentiated products, and dif-ferentiated products often imply R & D costs. In such cases, according to Baye, Crocker and Ju (1996), fixed set-up costs may limit the number of divisions of a firm. To isolate the strategic aspect of divisionalization, we also assume a constant return to scale technology. Yet, the nature of technologies is an important factor when firms choose the number of divisions. Increasing return to scale technologies should reduce firms' in-centives to divisionalize (dividing production), and decreasing return to scale technologies should increase such incentives. For example, the increasing return of scale nature of ad-vertising in the cereal industry may prevent cereal firms from setting up differentiated divisions even though each cereal firm has a number of differentiated products. We also consider the consequences of allowing divisions to divide further. It is found 10For example, in the automobile industry, high-end market divisions compete more with other high-end market divisions than with the low-end divisions from the same firm. 49 that, if divisions are allowed to divide further, they always will. Then, the only possible outcome is the one in which the firm that allows its divisions to divide further has an infinite number of divisions and zero profit. Hence, each firm has an incentive to restrict its divisions unilaterally from further breakup. Our finding provides a theoretical justification for the assumption in the strategic divisionalization literature that only parent firms can set up divisions, and divisions themselves cannot. Finally, we discuss the free entry issue. We find that divisionalization has a natural entry deterrence property, for it can significantly magnify the severities of competition in the face of entry. As a result, incumbent firms may persistently earn abnormally high profits in free entry equihbrium, relative to the no divisionalization case. In fact, when firms have difficulty differentiating from each other because of either the concentration of consumers' tastes or technological reasons, the only pure strategy subgame perfect free entry equihbrium is the monopoly outcome, even if the entry cost is relatively low. By limiting the number of independent divisions or franchises, regulators can actually help to increase competition. In addition, in contrast to the previous literature which suggests that the incumbent actually has to set up divisions to deter entry, we show that the threat to divisionalize may be enough to ensure the monopoly outcome. Considering that the equihbrium we analyze is a free entry SPNE, this result is rather surprising. 50 Chapter 4 Divide and Conquer: Strategic Leasing in Common Poo l O i l Fields "The overdrillrng that has taken place ... represents a tremendous economic waste, not only in the expenditure of capital, but in the dissipation of natural resources." "The overdevelopment cannot be said to be entirely the fault of the operator, for the landowner is equally to blame." [Bulletin of the American Association of Petroleum Geologists, Vol. VIII, July-August, 1924] 4 .1 Introduction Since petroleum was first found in the U. S. in the middle of the 19th century, oil pro-duction has been plagued by serious common pool wastes1. They are usually attributed to excessive drilling, unnecessary surface storage, overextraction, and reduced ultimate oil recovery. Under the common law rule of capture, the property rights to oil are only assigned upon extraction. When multiple firms compete for migratory oil in a common pool reservoir, each has an incentive to drill competitively and drain oil from its neigh-bors. Common pool losses arise as capital costs axe driven up by the drilling of excessive numbers of wells (more than geologic and fluid conditions warrant) and the construction 1For example, in 1914, the U.S. Bureau of Mines estimated annual losses from competitive extraction at $50 million, approximately one-quarter of the total value of U.S. production; the Federal Oil Conservation Board (1926, p.30; 1929, p.10) estimated recovery rates of only 20-25 percent with competitive extraction, while 85-90 percent was possible with controlled withdrawal. 51 of surface storage. Rapid production also prematurely depletes the subsurface pressure and in turn reduces the total recovery. Given the extraordinary costs of competitive production in common pool fields, one would expect that landowners have incentives to limit the number of independent oper-ators in order to reduce the common pool loss associated with competitive extraction. However, a wide spread phenomenon in onshore oil production is that landowners in the same oil fields often divide their landholding into smaller pieces and grant production rights to multiple different operators2 (a phenomenom I shall refer to as multiple leasing hereafter). Landowners' multiple leasing practice can seriously aggravate the common pool problem. Given the competitive behavior of independent operators, why does a landowner grant more than one lease? This paper attempts a game theoretic explanation to this puzzling phenomenon. Table 4.1 (see Appendix 4.2) summarizes the landholding and leasing information of the 42 U.S. onshore oil fields, which I collected from Zingery and Southwest oil production maps in various issues of Oil Weekly from February 1938 to April 1940. Columns 2-4 are the number of landowners, the number of independent operators, and the number of leases in each field respectively3. The fifth column is the ratio of the number of operators to the number of landowners. The last column is the average number of leases per landowner. In terms of leases, Table I shows that, in all oil fields sampled, leases are more numerous than landowners. In 36 out of the 42 fields (or 86%), there are at least twice as many leases as landowners. Averaging across fields, each landowner has 3.8 leases. In terms of independent operators, similar results persist: 35 out of the 42 (or 83%) oil fields sampled have more operators than landowners, and the average operator-landowner ratio across fields is 2.6. The field level data shows that multiple leasing is a rather widespread practice in the early stage of U.S. onshore oil fields development. Formally, we view that the oil field development consists of two stages. In the first stage, the landowners simultaneously choose leasing strategies. In the second stage, in-dependent lease operators produce oil competitively by choosing extraction strategies simultaneously. Not surprisingly, it is found that competitive extraction by multiple in-2In the sample Libcap and Wiggins (1984) assembled, for example, Yates field has 16 independent operators but only 2 initial landowners and Hendrick field has 18 operators but only 3 landowners. 3In some cases, one operator owns more than one lease in one field so that the number of leases may differ from that of operators. That is why data on both of them are presented. 52 dependent operators leads to overdrilling, overextraction, and reduced recovery. More importantly, it is shown that, in a nonexclusively owned oil field, it is individually ra-tional for a landowner to subdivide his landholding unilaterally and delegate production rights to multiple independent lease operators. Consequently, the production lease own-ership is usually more dispersed than the landownership. Landowners are as responsible as the operators for the serious common pool wastes. Given that lower concentration of production leads to more serious rent dissipation in a common pool, it seems irrational for a landowner to grant leases to multiple operators. The key to understanding this puzzle is that operators of a multi-lease landowner as a whole are more aggressive in oil production than the landowner. This is because they ignore the externality of their extraction on each other, in addition to that on operators of other landowners. Essentially, the multiple leasing strategy enables a landowner to behave as a Stackleberg leader and credibly commit to a higher production level and, in turn, captures a bigger share of the aggregate output and of the fieldwide economic rent. Multiple leasing decreases the fieldwide rent, but increases the landowner's share. The effect of the increasing share can dominate that of the decreasing total rent. The tradeoff of these two effects determines how many leases a landowner will grant. Unfortunately for the landowners, if they all follow the same strategy, everyone will be worse off in equihbrium, for multiple leasing increases the extraction rate and, therefore, the common pool losses, but landowners' shares of output (or rent) remain unchanged in equihbrium. Nonetheless, in the non-cooperative gaine, given other landowners' strategies, a landowner has to pursue the multiple leasing strategy. Otherwise, the landowner will do even worse, for it will capture a smaller share of the shrunk total output and rent. This paper is closely related to, but distinct from, the common property literature. Much of the common property literature (Gorden, 1954; Hardin, 1968; and Dasgupta and Heal, 1979, for examples) focuses attention on the economic wastes under competitive production, but not on the determination of the organization of production itself. In this literature, resource owners are usually implicitly assumed to be the same as resource developers, and thus the property ownership structure is the same as the resource oper-ation structure. The traditional approach leaves the choices of inefficient organization of production unexplained. The main focus of our analysis is the strategic choices of the organization of production by property owners (landowners). More precisely, we show 53 that landowners have incentives to delegate production rights to multiple independent operators. As a result, the landownership structure is, in general, different from the pro-duction operation structure. Therefore, the traditional theory tends to under-estimate the tragedy of the commons, if resource ownership structure is used as an approximate of the operation structure. This analysis is also related to the literature on strategic divisionalization in Cournot competition setting. Many authors, Corchon (1991), Polasky (1992), and Baye, Crocker and Ju (1996), for example, analyze a two stage divisionalization game. In the game, parent firms choose the number of independently managed divisions in the first stage, and divisions compete in a Cournot fashion in the second stage. It is shown that each firm has an incentive to set up rival divisions. This shows that a parallel argument can be constructed to explain a long and puzzling phenomenon (multiple leasing) in oil field development and organization of production. The chapter proceeds as follows. Section 4.2 provides a brief review of oil production technology. Section 4.3 presents a model of oil field development. Section 4.4 provides an analysis of landowners' strategic leasing behaviors in a common pool. Section 4.5 concludes. 4.2 Oi l Extraction Technology in a Common Pool In this section, we discuss oil production in a common pool based on models of oil extrac-tion reviewed and developed by Ben-Zvi (1985). To simplify the analysis, we assume that an oil field consists of a perfectly-connected common pool without the stratification and separating faults. Thus, the oil can potentially flow to any corner of the field. Without a doubt, this is an oversimplification of the reality but it captures the essence of the common pool problem. 4.2.1 The Extraction Rate Oil reservoirs are usually compressed between a layer of natural gas and a layer of water. The underground pressure drives the oil to the surface when the surrounding formation is punctured by wells. The instantaneous extraction rate depends on the geological and fluid parameters of the reservoir formation and the number of wells in the field. Given the 54 geological and fluid characteristics of an oil reservoir, the extraction rate for the pool is proportional to the difference between the wellhead pressure and the underground pressure and is a concave function in the number of wells. That is, q(t) = ri*{N)(p(t)-p.{t)), (4.1) where q(t) is the instantaneous extraction rate; rj is a parameter which measures the effect of geological and fluid characteristics of the reservoir; N is the number of wells in the field and $ ( i V ) is a concave function in N; p(t) is the underground pressure; and pw(t) > 0 is the wellhead pressure. Theoretically, operators can control the extraction rate by choosing pw(t). Since we are mainly interested in competitive production, we assume that wells produce at full capacity, i.e., pw(t) = 0. To simplify the analysis, we also assume that $ ( i V ) = Nd, where d > 1 approximates the degree of concavity of the instantaneous production function with respect to the number of wells. The extraction function then becomes q(t) = riNL*p(t). (4.2) 4 . 2 . 2 T h e P r e s s u r e D e p l e t i o n D y n a m i c s Unlike many other exhaustible natural resources, the ultimate recovery of oil depends on the time path of output. With a high extraction rate, the ratio of natural gas and water to oil produced increases, leading to premature loss in subsurface pressure. Due to the loss of pressure, the natural gas dissolved in the oil leaves the solution, reducing the oil's mobility and leaving significant reserves permanently trapped. It is very costly to extract oil once the pressure is exhausted. Hence, characterizing the behavior of p(t) is an important issue in oil production. The rate of change in p(t) is generally believed to be a function of both the extraction rate, q(t), and the pressure, p(t). Following Ben-Zvi (1985), we adopt the following functional form for the rate of change in p(t): ^ = -*qb(t)p-°(t), (4.3) where is the rate of pressure depletion; a is a constant; and b ( a + 2 > 6 > l ) 55 measures the degree of the convexity of the rate of pressure depletion in terms of the extraction rate, where a > 0. Notice that this functional form imphes that there is no pressure depletion if no oil is extracted; otherwise, the pressure declines. Moreover, the marginal pressure loss increases with the extraction rate. 4 . 2 . 3 T h e U l t i m a t e R e c o v e r y The ultimate recovery is defined as f°° Q = / q(t)dt, Jo where Q is the ultimate recovery. Solving for dt in equation (4.3) yields dt = --q-b(t)pa(t)dp. a Substituting equations (4.2) and (4.5) into (4.4), we have Q = UN^r = UN-8, w W U = and s = ^ . Equation (4.6) shows that the ultimate recovery declines with the total number of producing wells in a common pool. When there is only one well (N = 1), the ultimate recovery reaches its maximum, where Q = U. Thus, U is the maximum recoverable oil reserve. The decline rate of the ultimate recovery is measured by s, which we shall refer to as the gross rent dissipation rate (GRDR). G R D R characterizes how fast the gross rent (ultimate recovery) decreases with the number of wells. Notice that, when s = 0 (or b = 1), the ultimate recovery is independent of the number of wells. The Federal Oil Conservation Board (1926, p.30; 1929, p.10) estimated that the ultimate recovery at competitive extraction is about 30% of that under controlled withdrawal. Since the number of wells under competitive extraction is usually many times greater than that under controlled withdrawal, it can be inferred that s is a number between zero and one, and perhaps relatively close to zero. We therefore assume hereafter that 0 < s < 1. 56 (4.4) (4.5) (4.6) 4.3 Analysis of Competitive Oi l Extraction In this section, we analyze the equihbrium of competit ive extraction of mult ip le indepen-dent operators in a common pool oi l field. In order to simplify the analysis, the following assumptions are made: A l : The dri l l ing cost function is assumed to be D(N) = NC, where C denotes the marginal cost of dri l l ing a wel l , and D is the total dri l l ing cost. A 2 : The crude o i l market is assumed to be perfectly competit ive. Wi thou t the loss of generality, we further assume the crude o i l price to be constant over t ime and normalize it to be one. 4 A 3 : The discount rate is assumed to be zero. Usual ly, dri l l ing cost is assumed to be proport ional to the dri l l ing depth. In assumption A l , I impl ic i t ly assume that al l wells in the same oi l field have the same depth. M y justif ication for it is that, before dri l l ing occurs, the relevant dri lhng cost for potent ial lessees is the expected value, which is more or less the same in the same field. Assumption A 2 is not unreasonable. F i rs t , in addit ion to numerous wor ld o i l produc-ers, there are in the U.S. alone thousands of o i l fields and many times more independent operating firms. Thus, oi l producers in a relatively small o i l pool face an approximately competit ive market, unless they mainly supply for a relatively independent local market. Second, in making leasing decisions, the relevant o i l price is the expected future price. It is not uncommon to assume the expected o i l price to be constant in the future. Assumption A 3 is obviously an oversimplif ication of reality. However, it helps to make a tedious derivation tractable. Under a zero discount rate, the present value of o i l product ion becomes the ult imate recovery. A nonzero discount rate does not affect the main results of the paper but only complicates the derivation. Consider a simple setting, where L independent operators extract o i l competit ively in a common pool o i l field. More specifically, we consider a game where L operators 4 The constant-price assumption seems contradictory to Hotelling pricing rule, but it is based on historical facts. One of the reason that hotelling rule fails, is that oil reserved estimated has been increasing over the last century. 57 choose their extraction strategy simultaneously. Since drilling is costly, each well drilled will produce to its full capacity. Since we assume that an oil field consists of a perfectly-connected homogeneous common pool without the stratification and separating faults, the oil can freely flow to any corner of the field, and the geological conditions are same for each well. Therefore, each well will produce at the same capacity. Thus, an operator's extraction strategies are simply choosing the number of wells. Let N( be the number of wells that operator / chooses to drill, where / — 1,2, The total number of wells in the common pool is N = YA=I N. Since each well produces at the same rate, operator Vs output share and instantaneous extraction rate are then jf- and j^-q(t), respectively, where q(t) is the total instantaneous extraction rate in the common pool. Operator Vs net return is the present value of its revenue flow, minus the drilling cost and the fixed lease fee, A, that is r°° Ni * = -^q(t)dt - N,C - A Jo JV = jjrQW - NtC - A. (4.7) Taking as given other operators' number of wells, operator / chooses JV} to maximize his net return. The first order condition yields On the left-hand-side of the above equation, each of the three terms represents a different effect of drilling an extra well on operator Vs profit: the first term measures the gain due to the subsequent increase in operator Z's output share; the second term measures the loss due to the decrease in ultimate recovery; and the third term measures the loss due to the increased drilling cost. The first order condition describes the state where these three effects are balanced. Rewriting equation (3.8), we have JV - JV } Q(N) + Nt dQ(N) -C N2 " v " / ' JV dN U((N - JV,)JV-( 2 + s) - sNtN-(2+s>>) c = o. (4.9) 58 Notice that the Ni that solves equation (4.9) is independent of the subscript /. That is, Ni is the same for / = 1 , 2 , L . Denote the equihbrium number of wells per operator by n. Then, the number of wells in the field is N = nL. Multiplying equation (4.9) by L and solving for N, we have N = t 1 " ^ ] ^ , (4.10) c for L > s + 1, where c = jj is the relative marginal drilling cost of a well. Then, 1 i ' _ » H . Ni = n = y[ (4-n) L c and 1 _ i±i Q(N) = UN~S = U[ (4.12) c Call the return before subtracting the fixed lease fee the gross return. Then, the field-wide gross return, denoted by IT, is H = (Q(N)-CN) Similarly, operator Vs gross return, denoted by II/, is Proposition 4.1: Under A1-A3, the following results hold: (a) both the ultimate recovery of oil and the aggregate gross return of the common pool decrease with the number of independent operators; (b) the value of a lease granted to an operator decreases with the number of indepen-dent leases in the common pool field. 59 The proof of Proposition 4.1 can be easily obtained by differentiating equation (4.12) to (4.14). Proposition 4.1 (a) states two separate claims regarding the ultimate recovery and the aggregate rent respectively. The ultimate recovery of oil declining with the number of operators can be said to be an oil extraction phenomenon, resulting from the underground pressure depletion dynamics of oil production. That a large number of independent op-erators causes low economic rent, however, is the usual outcome associated with common property problems. Here, the number of operators measures the degree of production frag-mentation. More operators represent a lower degree of production concentration, which results in higher rent dissipation. Proposition 4.1 (b) can be viewed as a corollary of Proposition 4.1 (a). More operators share a shrinking fieldwide rent. The value per lease of course declines with the number of leases in the field. 4.4 A Game Theoretic Model of Strategic Leasing in a Common Pool In this section, we analyze landowners' leasing behaviors. Consider a simple environment, where M landowners each owns a fraction of the land surface that covers a homogeneous common oil pool. In anticipation of the competitive behavior of lease operators, each landowner has to decide how to grant production rights to operators which specialize in oil production. If a landowner decides to produce oil himself, we can always view him as both the landowner and an operator, as if he signs a lease contract to himself. Our main focus is how the landowners take advantage of the common pool by strategically choosing the production organization (multiple leasing). More specifically, we model landowners' decisions as a game in which each of them chooses simultaneously the number of leases he or she will grant to independent operators. Following convention, we refer to landowners and operators as lessors and lessees respectively. In order to simplify the analysis further, the following assumption about lease contracts is made: A4: The lease contract takes the following form: 60 1. Having paid a fixed fee to a lessor, a lessee gains the full production right on a prespecified oil tract and retains the right to the oil produced. 2. Lessors have all the bargaining power; that is, the leasing market is perfectly com-petitive. Therefore, a lessor extracts all the economic rent of oil production through a fixed fee, and lessees make zero profit. 3. Leasing is costly. The marginal transaction cost of signing a lease is O. Assumption A4 attempts to capture the main characteristics of a typical oil lease contract. In order to make lessees aggressive in production, a lease contract usually requires a lessee to pay a large fee up front and grants the lessee up to 90 percent of the oil produced. The assumption of a fixed fee lease contract is an approximation of contracts of such a type. We can think of the contracting process of leasing as follows. A landowner first proposes contracts in a take-it-or-leave-it fashion to many potential independent operators with similar opportunity costs. The operators then decide whether they will accept the contracts. The competition among those operators will leave them with their opportunity costs and landowners with the entire rents. Finally, the leasing transaction cost is assumed to capture the cost and frictions associated with lease contracting. Before solving for the equihbrium of the leasing game, we show first, by the use of an example, the potential gains of the multiple leasing strategy to a landowner. 4.4.1 The Potential Gain of a Multiple Leasing Strategy: An Example Consider a common oil pool, where M — 5, U = $1,000,000, c = .001, and s = .5. Among the five landowners, we assume that four are strategically innocent, and one is strategically sophisticated. An innocent landowner either produces oil herself or grants only one production lease, whereas the sophisticated landowner may grant production leases to multiple independent operators. Denote by I the number of independent leases the sophisticated landowner grants; thus the total number of leases in the common pool is / + 4. Let ?TS and 7r' be the resulting profits of a typical sophisticated and innocent landowner, respectively. Using the equihbrium solution of the competitive extraction game, we obtain 61 7T Q(N) - nlC 1 + 4 Ucl l + 3-s 7 + 4 ( (7 + 4)c 1 + 3 + 3 - :), and = 77c 7 + 3 -s ) 1 / ( s + 1 ) l + s l + 4K(l + 4:)c) Z + 3 - 6 ' and n - 7 r s + 47r% (4.15) where IT is the total rent in the common pool. A simple calculation generates the following table: 1 I 2 3 4 5 10 6757 9172 9952 10047 9839 7947 TT*' 6757 4586 3317 2512 1938 1324 n 33785 28516 23220 20095 17591 13243 Table 4.2 Table 4.2 shows that, as the number of independent leases granted by the sophisticated landowner grows, the total rent in the common pool steadily decreases. The sophisticated landowner's profit, however, increases initially and then decreases with the number of independent leases she grants. When she has four leases, her profit reaches $10,047, which is $3290 more than the profit she makes if she grants only one lease strategy. Thus, 62 this example shows that, given that other landowners grant one lease each, the remaining landowner has incentive to grant unilaterally multiple leases. The question, then, is will landowners choose to grant multiple leases if each of them is free to do so? 4 . 4 . 2 A n a l y s i s o f t h e L e a s i n g G a m e Equipped with the characterization of the equihbrium in the competitive extraction game, we now turn to analyzing the equihbrium of the landowners' leasing game. Under the assumptions about leasing outlined earlier, lessors have all the bargaining power. Thus, lessors extract all the economic rent from leases through a fixed lease fee, leaving lessees to earn zero profits. Let M denote the number of lessors in a common pool, and Lm the number of leases the mth lessor chooses to grant, where m — 1,2, . . . , M . Denote the profit of lessor ra by 7 r m , which is her leasing revenue, netting the transaction cost of signing these leases. Hence, we have 7rm = ALm — flLm = If[UL-ML_^s + 1 ) ) M - ^ L m . (4.16) The example in section 4.4.1 shows that a landowner has an incentive to grant multiple independent leases, if others do not. We now turn to the symmetric Nash equihbrium of the leasing game. In the Nash leasing game, a typical lessor ra maximizes 7r m by choosing Lm, taking as given other lessors' number of leases. The first order condition yields (s + UUc* - ^ + ^ + A m ( A m - s - l ) , . 1 + J U X(2»+3)/(H-l)(£ _ 5 _ l)(2H-l)/(.+l) " ~ U ' ^ 0 for ra = 1,2, . . . , M . Notice that equation (4.16) is symmetric for all ras, which implies the equihbrium solution of Lm is the same for all ra. Denote the equihbrium number of leases of a typical lessor by I. Then, we obtain 63 ( M - 2)Ml - (s + 1)(M - 1) + 1 ^ - c - 4 r M T H i ^ [ M I - (s + 1)]^H = 0, (4.18) s -\- \ for M > 2, where to = ^ is the relative marginal transaction cost of leasing. Replacing / by ^ in the above equation yields (M - 2)L -(s + l)(M - 1) + 1 ^c-iTiML^[L - (s + 1)]^H = 0. (4.19) 5 + 1 Equation (4.17) determines the equihbrium number of leases per landowner. Equation (4.18) characterizes the relationship between the degree of production concentration and the degree of lan downer ship concentration. Proposition 4.2: If the marginal transaction cost of leasing is sufficiently low and M > 2, the subgame perfect equihbrium has the following properties: (a) each landowner grants multiple leases; (b) the number of independent leases increases with the number of landowners in the field; and, (c) the number of leases per landowner decreases with the number of landowners in the field. (See Appendix 4.1 for the proof) Proposition 4.2 (a) predicts landowners' multiple leasing behavior in a common pool with more than two landowners. It also implies that the degree of production concentra-tion is lower than the degree of landownership concentration. Proposition 4.2 (b) states that the degree of production concentration is determined by and positively related to the degree of landownership concentration. Hence, more fragmented landownership leads to more dispersed production control. Proposition 4.2 (c) indicates that there is a limit to landowners' strategic leasing. This is because the marginal transaction cost of leasing is a constant, but the total economic rent has an upper bound and declines as the number of leases increases. In order to understand the forces behind the landowners' leasing consideration, we rewrite the first order condition of the leasing game as follows: 64 Lm dQ(N) 9N 9Nr L dN <3Lm 9L, •m m C - 0 = 0. (4.20) The four terms on the LHS of the equation measure four different effects of granting an extra lease on the profit of lessor m: the first term measures the gain in lessor m's share of ultimate recovery, which we shall refer to as the share-enhancing effect; the second term measures the loss of profit due to reduced total recovery, which we shall refer to as the size-of-the-pie effect; the third term measures the profit loss due to increased drilling cost, which we shall refer to as the overinvestment effect; and the fourth term measures the loss due to the marginal transaction cost of leasing, which we shall refer to as the transaction effect. The last three effects are all negative. Obviously, the share-enhancing effect of multilateral leasing is what induces a lessor to create multiple leases. H a lessor's share in total output and rent were fixed under some rules, she would not grant multiple leases. Remark: Multiple leasing strategy benefits a lessor, not because lessees have superior technologies, but because lessees face different externalities. Lessees of a multi-lease lessor ignore the externality that extraction inflicts not only on other lessors, but also on other lessees of the lessor. This makes lessees more aggressive in extraction than the lessor. By granting production rights to multiple independent lessees, a lessor credibly commits to a higher level of extraction and, consequently, increases her share of the total output and economic rent. However, if all lessors pursue the same strategy, in equihbrium, everyone is worse off. Nevertheless, from an individual landowner's point of view, multiple leasing always dominates granting a single lease. The outcome of the leasing game is similar to that of the prisoner's dilemma. Without interventions from outsiders (the government, for example), individual rationality will defeat the common interest. This paper examines leasing behaviors of landowners in a common oil pool. We show that, despite the rent dissipation associated with nonconcentrated oil extraction, it is profitable for a landowner to grant production rights to multiple independent firms. The key to this puzzle is the power of commitment in a multi-stage noncooperative game. 4.5 Conclusion 65 Through, multiple leasing, a landowner credibly commits to a higher extraction rate and, consequently, captures a higher share of output and rent. When the gain in share domi-nates the loss due to the shrinking of fieldwide rent, it is rational for a landowner to grant an additional lease to an independent operator. This analysis provides an explanation for the puzzling leasing behaviors of landowners in U.S. onshore oil fields. Our results also indicate that insights from this phenomenon may have significant implications for the formation of effective regulation policies. Moreover, the structural characteristics that lead to landowners' multiple leasing strategies are also present in many other common property problems. A conceptually similar problem is that of fishery in international waters. In this case, each country would have incentives to license multiple fishing firms to increase its share in output. In prin-ciple, the framework of this paper can be applied to any common property problem with private access rights. 66 Chapter 5 Conclusion Three essays comprise this thesis. They study how firms strategically set up autonomous units when facing competition. Our results illustrate the importance of the institutional arrangement and decisions structure in a firm, not only to its performance, but also to its rivals' profits and consumers' welfare. In the first essay, we show that a firm that produces a group of complements can generate higher profits through divestitures, when there is a competing firm supplying an imperfect substitute group of complements. By delegating pricing decisions to inde-pendent divisions, each firm credibly commits not to set the prices of the complements within the group collusively, which softens the competition. In other words, by dividing, the firms create negative externalities among the prices within the group that offset the positive externality between the prices across groups. The same insight also applies to the case when firms supplying imperfectly complementary goods or services compete by setting quantities. Our analysis suggests that industry structure and size of firms are determined by the nature of products and the strategic considerations of the firms. The welfare implications of the divisionalization in our model are significant. Divesti-tures by the firms supplying complements raise prices and reduce consumers' surplus. However, the profit gains are not large enough to compensate the consumers' losses. As a result, the total social surplus is reduced. Another implication of our analysis is that divestiture motivated by product-line com-plementarity can be viewed as a response towards entry. Suppose that, initially, there is a monopoly that supplies a group of complementary goods or services. Clearly, in our 67 context, the monopohst does not have any incentives to divest its operations. When po-tential entrants enter the market and supply differentiated products, the monopohst has two possible responses. One is to compete against the entrants. If the entrants do not make enough profits to cover their entry costs, the entries axe deterred. If the products offered by the entrants axe differentiated enough from the incumbent's products, and if the entry costs are relatively small, the entry cannot be deterred. In this case, an op-timal strategy for the incumbent firm is to soften competition by divesting some of its operations. Our analysis suggests that the optimal number of divisions for the incumbent increases as more entrants enter the market. It should be noted finally that our analysis is based upon a number of assumptions. One crucial assumption is that divisions cannot further divide before they choose their prices. This is reasonable in certain situations where the parent firms still have major ownership control of the divisions, but do not make management decisions. Franchise contracts can be viewed as an example of such a divisionalization. In other situations, it can be difficult for firms or divisions to make this type of commitment. Divisions may then have incentives to divide further. This raises the issue of what determines a stable industry structure in the presence of both substitutes and complements. Further research along this Une is needed. In the second essay, we consider an environment in which competing ohgopohstic firms with differentiated products can set up independent rival divisions. We analyze the strate-gic incentives for a firm to divisionalize, characterize the equihbrium of a divisionalization game, and highlight the effect of product differentiation in ensuring an interior equihb-rium. By allowing for product differentiation, we demonstrate that the existence of an interior equihbrium can be achieved without reliance on ad hoc assumptions such as an exogenous bound of permissible number of divisions or a costly divisionalization. We also consider the consequences of allowing divisions to further divide. It is found that, if divisions are allowed to further divide, they always will. Then, the only possible outcome is the one in which the firm that allows its divisions to further divide has infinite number divisions and zero profit. Hence, each firm has an incentive to unilaterally restrict its divisions from further breakup. Our finding provides a theoretical justification for the assumption in the strategic divisionalization literature that only parent firms can set up divisions, and divisions cannot. 68 Finally, we discuss the free entry issue. We find that divisionalization has a natural entry deterrence property, for it can significantly magnify the severities of competition in the face of entry. As a result, incumbent firms may persistently earn abnormally high profits in free entry equihbrium relative to the no divisionalization case. In fact, when firms have difficulty to differentiate from each other because of either the concentration of consumers' tastes or technological reasons, the only pure strategy subgame perfect free entry equihbrium is the monopoly outcome, even if the entry cost is relatively low. By hmiting the number of independent divisions or franchises, regulators can actually help to increase competition. In addition, in contrast to the previous literature, which suggests that the incumbent actually has to set up divisions to deter entry, we show that the threat to divisionalize is enough to ensure the monopoly outcome. The third essay examines leasing behaviors of landowners in a common oil pool. We show that, despite the rent dissipation associated with nonconcentrated oil extraction, it is profitable for a landowner to grant production rights to multiple independent firms. The key to this puzzle is the power of commitment in a multi-stage noncooperative game. Through multiple leasing, a landowner credibly commits to a higher extraction rate and, consequently, captures a higher share of output and rent. When the gain in share domi-nates the loss due to the shrinking of fieldwide rent, it is rational for a landowner to grant an additional lease to an independent operator. This analysis provides an explanation for the puzzling leasing behaviors of landowners in the U.S. onshore oil fields. Our results also indicate that insights from this phenomenon may have significant implications for the formation of effective regulation policies. Moreover, the structural characteristics that lead to landowners' multiple leasing strategies are also present in many other common property problems. A conceptually similar problem is that of fishery in international waters. In this case, each country would have incentives to license multiple fishing firms to increase its share in output. 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E . , 1979, "Transaction Costs Economics: The Governance of Con-tractual Relations," Journal of Law and Economics, 22, 233-61. 76 5.1 Appendix 2.A Proof of Lemma 2.1: The existence and uniqueness of the equihbrium follow from Fried-man (1977). The symmetry of the equihbrium prices within the group follows from the necessary conditions (2.9). In what follows, we show that the equihbrium aggregate prices, p\ and fa, increase with m,. Denoting xi = (pi,p 2) and x2 = (p2,p\) and applying standard comparative-static techniques to (11), we obtain A = [(1 + m!)I>i(xi) + fcl>u(zi)][(l + m2)D1(x2) + p2D11(x2)] -[m1D2(xx) + p\D12(x1)][m2D2(x2) + P2D2i(x2)]. Assumptions (A1)-(A3) imply that A > 0 and 'Jp\l'Jmx > 0. And dp2Jdmx > 0 follows Proof of Lemma 2.2: First notice that, by Lemma 1, when rax = ra2 = m the equihbrium dp\l9mx = -£>(*i)[(l + m2)D1(x2) + hDlx{x2)y dpi/dm-^ = D(a;i)[m 2 £> 2 (a: 2 ) + p2D 2 1 (x 2 )]/A, where from (A4). The proofs of statements (d) and (e) are analogous. Q.E.D. aggregate price for each group of the complements is symmetric, which we denote by p(m). Let x = (p(m),p(m)). Then, dV(m, m) dp(m) (D(x) + p{m)Dl(x) + p{m)D2(x)). dm dm From the first-order conditions (2.11), p(m) mD(x)lD\(x). It follows that (A5) imphes that —D2(p,p)/Di(p,p) is non-increasing with p, Lemma 1(e) imphes that p(m) increases strictly with ra, and (1 — ra)/ra strictly increases with ra. It follows that (1 — ra)/ra — D2(x)/Di(x) strictly decreases with ra and is equal to zero at ra = ra*. 77 Thus, dV(m, m)/dm is positive for ra < ra* and negative for ra > m*. The claim follows. Q . E . D . Proof of Proposition 2.1: Let ra be the number of divisions in the symmetric equihbrium. We first show that ra > 1. Using (2.12) and (2.13), we obtain, for any ra2, Lemma 2.1(c) imphes that dp2jdm\ > 0. The claim follows from the following inequality ^ni(ra 1,ra 2). drn\ -\mi=i > o. Next, we show that ra < rn*. Suppose to the contrary that ra > ra*. B y L e m m a 2.1, when rai = ra2 = ra the equihbrium aggregate price for each group of the complements is symmetric, which we denote by p(m). Let x = (p(m),p(rh)). It follows from (2.12) and (2.13) that, at rax = ra2 = ra, ^IIi(ra 1 ,ra 2 ) (m-1 &p\ dp\ \ ^ r a i \ m 'Jmi •jm1J where 'Jpkl'Jrni is the derivative of the equilibrium group price ph w i th respect to rai evaluated at rai - ra2 = ra, k = 1,2. B y the definition of ra*, p(m*) — pm and the following equation ra - 1 _ D2(p(m),p(m)) ra Di(p(m),p(m)Y holds at ra = ra*. The left-hand side of the above equation strictly increases wi th ra, (A5) imphes that —D2(p,p)/Di(p,p) is non-increasing i n p, and L e m m a 2.1(e) imphes that p(m) increases wi th ra. It follows that —D2(p(m),p(m))/Di(p(m),p(m)) is non-increasing i n ra. Thus, ra - 1 D2(p(m),p(rh)) ra ~~ D i ( p ( r a ) , £ ( r a ) ) ' Therefore, at rai = ra2 — ra, 78 ^ni(m1,m2) < p(m)D2(x) $Pi _j_ $ fa. p(m)D2D[D1 + m(Di + 7J2) + p(m)(Dxx + I>12)] [(1 + m)D1 + p{m)DlxY - [mD2 + £(m)D 1 2 p < 0 where the last inequality follows from (Al) and (A3) . This contradicts with the necessary condition for m to be the symmetric Nash equihbrium number of divisions. The claim follows. Q.E.D. 79 5.2 Appendix 2.B: A n Integer Divisionalization Game In this appendix we analyze an integer game in which the numbers of divisions, mi and ra2, chosen by the firms are restricted to be integers. The questions are whether it is still possible for the firms to achieve the monopoly profits by coordinating their divisionalization strategies and whether there exists a pure strategy Nash equihbrium in the non-cooperative division game. For simplicity, we consider only the linear demand function (2.2). The reduced-form payoff functions are represented in (2.16). Let m* and m be the optimally coordinated number of divisions and Nash equihbrium number of divisions in the continuous division game discussed in Section 2.3, respectively. We argue that for large ni and n 2 the firms may not in general be able to replicate the maximum joint profit by coordinating their division strategies, but can almost achieve the maximum joint profit. The optimally coordinated number of divisions in the integer game is close to m*. Moreover, there always exists a pure strategy Nash equihbrium in the integer game and the equihbrium number is close to h. Let [m] be the integer part of a real number m, i.e., the integer such that m — 1 < [m] < m. Consider first the non-cooperative division game. Denote [m] by J. Since the payoff for firm 1, V(mi,m 2), is a single-peaked function of mi, the best-reply of firm 1 to an integer m2 is either [_R(m2)] or [72(m2)] +1. Moreover, since R(m) is monotonically increasing and slopes of R(m) are always less than 1, the best-reply to I is either [i2(m)] or [.R(m)] + 1. Similarly, the best-reply to 7 + 1 is either [R(m + 1)] or [R(rh + 1)] + 1. It follows that (7,7) is an equihbrium if V(7,7)> V(7 + l,7), and (1+1,7-1-1) is an equihbrium if V(7+l ,7 + l )> V(7,7+l). If neither inequality holds, then both (7,7+1) and (7+1,7) are Nash equilibria. Therefore, there exists at least one pure strategy Nash equihbrium that is close to the equihbrium 80 Table 5.1: The Payoff Matrices in the Division Game Firm 2 mi \ m 2 1 2 3 4 1 9.38, 9.38 11.98, 9.07 13.5, 8 14.49, 7.01 2 9.07, 11.98 12, 12 13.78, 10.78 14.96, 9.56 3 8, 13.5 10.78, 13.78 12.5, 12.5 13.66, 11.16 4 7.01, 14.49 9.56, 14.96 11.16, 13.66 12.24, 12.24 in the continuous division game. The coordinated divisionalization works in a similar way. The joint profit is V(rai, m2)+ V(m 2,mi), denoted by V(mi,m 2), which is symmetric in mi and m 2 . It can be easily shown that, for any m 2 , V(mi,m 2) is single-peaked in mi. Suppose that the peak of V"(mi, m2) is reached at mi = R(m2) (i.e., the best reply function). Clearly, m* = R(m*). It can be verified that the slopes of R(m2) is between 0 and — l . 1 The single-peakedness of t^(mi,m2) imphes that for any m 2 the maximum is reached either at [i2(m2)] or at [i?(m2)] + 1. Since the slope of R(m2) is between 0 and —1, the symmetry then imphes that the maximum of V>(mi,m2) is reached at one of the following pairs: ([m*],[m*]), ([m*], [ro*] + 1), ([m*] + 1, [ro*]), or ([ro*] + 1, [ro*] + 1). In the following, we provide an example in which a = 10, /? = 6, 7 = 4, and n\ = n2 = 4. Notice that m* = 3 and m = 1.34. The payoff matrices in the integer division game are represented in Table 1. The joint profits are maximized at mi = m2 = 3. There are two pure strategy equilibria, (mi,m2) - (1,1) and (mi,m2) = (2,2). xlt should be noted that, when j//3 is greater than 8/9, the joint profit is strictly increasing in mi for small m2. In this case, R(m2) is equal to Ni for small m2. 81 Figure 2.1: The Best-Reply Lines in the Price Game (the Linear Demand Functions) 82 Figure 2.2: The Best-Reply Curves in the Division Game (the Linear Demand Functions) 83 5.3 Appendix 3 Proof of Lemma 3.1: The existence is t r ivia l ly shown from equation (3.6). (a) Divide equation (3.6) by rrik and rearrange it to obtain the output of a division in firm k: 1 6 ( m t + l ) + A f c ( ( l - z ) m f c + l ) ' Then , 'Jxki _ a o\n~k ~ ~ [b(mk + 1) + A f c ( ( l - z)mk + 1)]; ;(b + Ak(l-z))<0, and ((1 - z)mk + 1 ) ^ — < 0, 9mt [b(mk + l) + Ak((l-z)mk + l)]2XX ' ' dmt where t ^ k. (b) Differentiate equation (3.6) wi th respect to rrik and mt to get 9xk a(b+Ak) 9mk [6(m fc + l ) + A f c ( ( l - ^ ) m f c + l ) ] 2 > 0 , and 9xk = a 1 :?A f c 9mt [b{A-+ 1) + Ak((l - z) + ^))^ Z)^mk}9mt ' Proof of Proposition 3.1: Lemma 3.2 shows that numbers of divisions are strategic complements. Thus, the first stage solution must be symmetric. If not, without loss of generality, denote mi < ra2 < ... < m„ the solution, where at least one of the inequalities holds strictly. Then , nil < mn. B y the symmetry of the first order condition, any ordering of mt is a solution. In particular, consider the ordering m n , m 2 , m 3 , . . . , m n _ 1 , m i . In the new ordering, the number of divisions of firm 1 increases from mi to m „ and that of firm 2 to firm n — 1 stays the same. B y Lemma 3.2, the number of divisions of firm n should increase. However, it decreases from m „ to m\. It is a contradiction. Therefore, the solution for the first stage game is symmetric. The uniqueness is directly impl ied by equation (3.10). 84 To show the compaxative static results, define G(m, z, n) — m 2 ( l — z)((n — l)z + 1) m(n - 2)z - 1 = 0. Then, 9G — = 2 m ( l - z)((n - l)z + 1) - (n - 2)z = m ( l - z)((n - l)z + 1) + 1 > 0; _ = m * ( l - z ) ( m - _ ) ; and :?G 1 —- = m\(n — 2)((1 — z)(m — ) — mnz]. 'JZ 1 — z Recal l from section 3.2 that ra < r 1 - . Thus, 1°- < 0 and f£ < 0 for n > 2. Then , ,9™ dG(m,n,z) ,yn dG(m,n,z) ^ u' dm and 2 Z 3G(m,n,z) ^ U dm Q . E . D . Proof of Proposition 3.3: Rearrange equation (3.11) as the following: a2 F. [b-j= + (l + z ( n - l ) ) ^ Firs t , totally differentiate the above equation w.r.t. n and z to get x — ,z(n — 1) 1 3 . ~^s$mi * r y— / ^ ( r c - 1) 1 -5. / i \ \ ^ m i A Then, § j < 0. Similarly, totally differentiate equation (3.11) w.r.t. n and F to get [-1= + (1 + z(n - l ) ) y ^ ] A F + 2F[z^R + ( ^ = ^ + - 1 ) ) ^ ] A n = 0. 85 Then, g£ < 0. Q.E .D. Proof of Proposition 8.4-. A t n = 2, m = y/jzgr (from equation (3.10)) and the profit of a duopohst % = f ff^, From equation (3.11), we have _ jrJ> Then, the duopohst's profit is no greater than F i f z < z*. Q.E .D. 86 Figure 3.1 5.4 Appendix 4 . 1 S . O . C . o f t h e e x t r a c t i o n g a m e : 9Hi 9Nf S . O . C . o f t h e l e a s i n g g a m e : 5jV-( 2+*) - (2 + S)(N - (1 + s ) i V / ) i V - ( 3 + s ) < 0 dB 92*m -(* + l)Uc^(2Lm - 1) - g h L m *Lm B < U where B = U 2 s ^ ^ a + 1 \ L -s- i)(2»+i)/(.+i) > 0. Proof of Proposition 4-2: Define F(l,M,u,c,s) = (M-2)Ml-(s+l)(M-l)+l ^—c~^M2-^l^i[Ml-(s+l)]^ = 0 Differentiate F w . r. t. i ts arguments: F, = M(M - 2) - ( l ± j * + * + * m M s _ l ) ( M ( M - 2)1 - (. + - 1) + 1) < M(M-2)-j(M(M-2)1-(s + l)(M-1) + 1) 2M(M - 2)1 - (s + 1 ) ( M - 1) + 1 < < Z 2 M ( M - 2 ) Z - ( s + l ) ( M - l ) Z M(2(M - 2)1 -(s + 1)) Z < 0 for A f > 2. „ 1 o 2 « + 3 . » + 2 . • • 2a+l F w = - c " ^ i M «+i Z«+i [Ml -{s + 1)] »+i < 0 28+1 - -2a+3 .ii+2 . . - 2«+l Fc = use <+i M '+i Z«+i [MZ - (s + 1)] «+i > 0 88 FM = 2 ( M - 1)/ - ( . + 1) - ffi^L + | ± i M i _ >{s + x ) ) ( A f ( M - 2)i - ( . + l ) ( J f - 1) + 1 ; < 2 ( M - 1)/ - (s + 1) - ^(M(M - 2)1 - (a + 1)(M - 1) + 1) = - ^ ( 2 M 2 Z - M ( 6 Z + 3(s + l ) ) + 4 ( s + 2)) <r 2/ 6 / + 3(3 + 1) 2(^ + 2) 6/ + 3(* + 1 ) 2 M U 4/ ; / 1 4/ ; ; < 0 Then, dl F ^ = -^-<0,forM>2. 'JUJ Fi 9 1 F m <0,forM>2. •3M Fi Define D(L,M,u) = ( M - 2 ) i - ( s + l ) ( M - l ) + l ^ - c - 4 T M I H T [ I - ( 5 + 1 ) ] ^ = 0 Differentiating D(L,M-LJ) w.r. t . Z,M,c*>, yields 2 > L = M - 2 - ( ^ i + 2 ; + 1 1 ) ( ( M - 2 ) L - ( , + l ) ( M - l ) + l ) < M - 2 - | - ( ( A f - 2)X - (s + 1 ) ( M - 1) + 1) = _ l ( ( A f - 2)(2£ - + 1)) - a) < - y ( 2 L - 2 s - l ) < 0 for M > 2. D M L-(s + l)-^((M-2)L-(s + l)(M-l) + l) = ^^>0 1 « _ _ _ »+2 r _ . . . 2»+l „ c " ^ M i * [X - (s + 1)] »+i < 0 3 + 1 Then, (a)E = -gf < 0 , f o r M > 2 ; 89 ( b ) £ = - f < 0, for M > 2; (c) §s = -t£ > °» for M > 2; (d) ^ = - S f < 0 , f o r M > 2 . (c) and (d) validate (a) and (c) of Proposition 4.2 respectively. The equihbrium number of leases I is one, when u > u* = ('+1)((Jtf^f-('+1)(^^+1)c7tT . M «+i [ M - ( s + l ) ] »+i From (b) above, / > 1 i f u> < u*. Thus, we have Proposit ion 4.2 (b). Q . E . D . 90 5.5 Appendix 4.2 Table 4.1: Leases and Operators Distribution by Fields OH Field | * of landowner* * of Opmtoct UotLtws UnwlindownOT Op*r*torsAMdown Lh/errnore-Wright Loving ton Slaughter Keystone Mcknight North Cowden FHzslmnon Ordoviaan Pardue Aoe Lewis La Prelle Double Gum Foster North Cowden Oldham Denver Canyon Lime New Pool Wasson Bennett North tvy Happer.tex OflCfty Payton Riverside Walnut Bend JCMA Magnpda EHenberger HuBsflkSlke AOce Seymore NJones' Hatseel Cotton Lake Ivy Cotton Vatlay.tex Seminole HuQ-Snk Griffin Lisbon Sharon Flag Lake Average ~ 2 7 8 4.0 3.5 2 19 21 10.5 9.5 3 18 22 7.3 6.0 3 6 8 2.7 2.0 3 11 13 4.3 3.7 3 9 11 3.7 3.0 3 16 17 5.7 5.3 3 14 19 6.3 4.7 4 17 27 6.8 4.3 4 6 10 2.5 1.5 4 7 10 2.5 1.8 4 16 16 4.0 4.0 4 10 20 5.0 ZJ5 4 15 21 5.3 3.8 5 12 20 4.0 2.4 7 9 15 2.1 1.3 8 51 72 9.0 &4 8 23 28 3.6 ZS 8 22 29 3.6 ZJ& 8 14 23 2.9 1.8 9 40 69 7.7 AA 9 20 22 2.4 22 9 13 20 2 2 \A 9 7 18 1.8 0 8 9 4 10 1.1 OA 10 41 60 6.0 4.1 10 16 28 2.8 1.6 12 41 97 8.1 3.4 12 25 25 2.1 2.1 13 25 43 - 3.3 1.9 13 10 26 2.0 0.8 13 13 26- 2.0 1.0 15 15 26 1.7 1.0 18 22 40 2,5 1.4 18 22 45 2.5 1.2 20 11 28 1.4 0.6 22 15 44 2.0 0.7 27 101 154 5.7 3.7 28 26 45 1.6 0.9 35 42 84 2.4 1.2 40 59 94 2.4 1.5 44 32 66 1-6 0.7 11.5 21.5 35.2 3.8 2.6 91 

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