Unsystematic risk and coalition formation in product markets

International Journal of Industrial Organization 20 (2002) 313–338 www.elsevier.com / locate / econbase

Unsystematic risk and coalition formation in product markets Murray Brown a , Shin-Hwan Chiang b , * a

Department of Economics, State University of New York at Buffalo, Amherst, NY 14260, USA b Department of Economics, York University, Toronto, Ontario, Canada M3 J 1 P3

Received 27 September 1999; received in revised form 15 June 2000; accepted 6 August 2000

Abstract We study the conjecture that increasing market volatility leads to larger coalitions in an oligopoly. Here, coalition formation decisions are made in a noncooperative game by risk averse firms. They use a sequential offer–counter-offer procedure initiated by Selten and Rubinstein. We find that the conjecture generally fails in a small oligopoly whose firms play a unanimity game, but it is validated in an oligopoly that allows open membership. However, it is valid in a small oligopoly if market volatility is sufficiently high, whatever the rule of membership.  2002 Elsevier Science B.V. All rights reserved. JEL classification: D8; L1; C72 Keywords: Uncertainty; Market structure; Firm strategy; Market performance; Noncooperative games

1. Introduction We wish to study the relationship between unsystematic risk in the product market and coalition formation of oligopolistic firms. Specifically, we conjecture that increasing market volatility leads to larger coalitions. Starting from the assumption that firms are risk averse, firms attempt to avoid risk — which we

* Corresponding author: Tel.: 11-416-736-5083; fax: 11-416-736-5987. E-mail address: [email protected] (S.-H. Chiang). 0167-7187 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 00 )00099-0

314

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

identify with market volatility — by sharing it, and they do that by forming coalitions. Coalitions here are taken to be a subset of firms that coordinate their courses of action. Examples include complete mergers and cartels whose selfenforcing actions are coordinated. Our model is based on the noncooperative game that was initiated by Selten (1981) and Rubinstein (1982): players make sequential proposals to form coalitions. Their decision to participate in a coalition rests on an extensive form game of offers and counter offers. In another branch of the literature, externalities are permitted in such an extensive form game. Bloch (1995) studied associations in a two-stage model that yielded externalities to the participating firms which compete with each other and with outsiders in the market in the second stage. The sequential coalition formation decision is made in the first stage. In another paper, Bloch (1996) considers a more general sequential coalition model, assuming that the division of a coalition’s payoff is fixed. Brown and Chiang study the role of externalities in the form of cost saving in mergers and cartel formation (1997); they also examine managerial incentives and other externalities in mergers and cartels (1998). Conditions for a unique subgame perfect equilibrium of a sequential proposal game with externalities are given by Brown and Chiang (1999). Yi (1997), and Ray and Vohra (1997) make valuable contributions to this literature; they categorize various rules under which noncooperative sequential games proceed, analyze efficiency, stability, and similar properties of a sequential proposal model that involve externalities.1 In the externality literature of which this paper is a part, the payoff for each firm in a coalition depends on the coalition structure as a whole. To illustrate the coalition structure, suppose that, ex ante, there are three identical firms in an oligopoly. Label them sequentially. Four potential coalition structures can potentially form. One includes Firms 2 and 3 in a coalition and Firm 1 as a singleton; a second has all three firms are singletons; a third coalition structure includes Firms 1 and 2 in a coalition, while Firm 3 is a singleton; and the last is the grand coalition itself. (Symmetry obviates the need to consider the coalition structure in which Firms 1 and 3 coalesce, while Firm 2 remains a singleton.) Externalities can be expressed as shifts in the payoffs to all three firms in the four coalition structures for given output rates of the firms — for example, the payoff to Firm 1 in the first coalition structure may differ from the payoff to Firm 1 in the second coalition structure. The reason that externalities appear in the sequential coalition formation game is because firms are assumed to be farsighted and consider the consequences of their moves. As their turn to propose or respond to a coalition proposal comes up, they evaluate the effects of the various coalition

1

In an offshoot, Baye et al. (1996) analyze divisionalization and franchising using a simultaneous model rather than a sequential one but also allowing for externalities.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

315

structures on any coalition decision they make. This essential property is developed more fully in the literature on sequential coalition formation.2 Some studies have touched on the conjecture. Dewey (1961) claims that mergers are simply an efficient way to transfer assets from failing to successful firms; and there appears to be some empirical evidence for that hypothesis. If all firms are risk averse with declining risk aversion successful firms would tend to be less risk averse than failing firms and hence their acquisition can be interpreted as a risk spreading device. Gort (1969) argues that economic disturbances generate discrepancies in valuation of the type needed to produce mergers. One consequence is that the variance in valuations increases. It is a standard implication of risk aversion that the pooling of resources and the spreading of risk in an attempt to realize a rate of return that approaches the expected rate. We take this to be a natural motive for coalition formation. Of course, it is a consequence of the conjecture that firms and industries in relatively more risky endeavors — like those whose demand is more subject to cyclical factors than others in less volatile environments — are more likely to attempt to share risk and diversify, an empirical implication that we have not seen systematically tested. It is sufficient, both to validate and refute our conjecture, that we consider a simple, symmetric, two-stage, quantity-setting oligopoly model. In the first stage, firms make their self-enforcing coalition decisions, while in the second stage, they choose outputs as their strategies. All firms are identical, ex ante. Of course, in such a symmetric model the intra-coalitional share of the coalition’s payoff requires no cooperative game to determine it since all firms in the coalition receive the same payoff. Hence, the self-enforcing property is preserved in all aspects of the model. Though we focus on the symmetric model, some examples are given in which firms’ markets differ because of different random disturbances. These examples support the symmetric results and offer more suggestions for further inquiry. Central to our results is the rule by which coalitions can form. In a symmetric

2

We note here that a sequential coalition formation game has dramatically different outcomes than a similar game in which firms make their coalition decisions simultaneously. In a classic paper, Stigler (1964) showed that in a standard oligopoly model of simultaneous coalition decisions (firms are unaware of rivals’ actions), a prisoner’s dilemma occurs and the equilibrium coalition structure is Cournot–Nash. All coalitions are comprised of singleton firms. In contrast to this, a sequential coalition formation decision procedure for the same oligopoly model produces an equilibrium coalition structure that includes a single large firm with a competitive fringe of small firms around it. If there are only three firms, ex ante, the difference between the simultaneous and sequential procedures is starker: for the latter, the equilibrium coalition structure is comprised of a single firm, the grand coalition, whereas for the simultaneous procedure the outcome is Cournot–Nash. Another way to state this is that for the sequential procedure, the prisoner’s dilemma is resolved in favor of full cooperation among the firms (Bloch, 1996; Brown and Chiang, 1998; and Ray and Vohra, 1997).

316

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

oligopoly firms can play an open membership game in which membership is open to all players who find it in their interest to adopt a joint output strategy. Alternatively, they can play by a unanimity rule in which a coalition forms if and only if all potential members agree to form it. It turns out the conjecture is sensitive to these rules. Our principal findings show the conjecture, increasing risk leads to larger coalitions, is not robust with regard to the rules of coalition formation. It fails, generally, in an oligopoly whose firms play by a unanimity rule but it is validated when they allow open membership. Interpretations and examples are given for these and other results that pertain to the inter-correlations of market volatility. Nevertheless, in spite of the different outcomes due to the two rules, we can say that if the volatility is sufficiently high, the conjecture holds irrespective of the rule by which coalitions form. Our model, which incorporates coalition formation with externalities and market volatility, is very simple. It is a complete information model in which unsystematic risk is represented by an additive random disturbance to demand. The model itself has an illustrious history, the demand specification first being used by Bowley (1924) and later by Spence (1976) and Dixit (1979). Products are taken to be imperfectly substitutable, with more differentiation raising the demand curves facing all firms. Thus, the implied market size increases as the degree of differentiation rises. This formulation is different from the other comparable formulation (see Shubik and Levitan, 1980). Modelling demand uncertainty has drawn considerable attention since Weitzman (1974). In their important contributions, Klemperer and Meyer (1986, 1989) adopt the supply function approach and show that additive demand uncertainty can give rise to supply function equilibria. This endogenizes the firms’ decisions to set quantity as in the Cournot model, or price as in the Bertrand model. When products are differentiated, it turns out that the supply function equilibria resemble a Cournot or a Bertrand equilibrium, depending on the nature of technology. More specifically, when marginal costs slope upward (downward), the unique Nash equilibrium involves firms choosing quantities (prices). Since we assume that marginal cost is constant in our model, the following Klemperer and Meyer (1986), pp. 1269–1270) results are especially pertinent: ‘ . . . with differentiated products, even in the limit as marginal cost becomes constant . . . , the equilibrium supply functions remain upward-sloping.’ In this framework, the quantity-setting-Cournot model is selected by firms. With differentiated products, firms possess some monopoly power in their own market and consequently, the supply function does not tend to be flat even when marginal costs are constant. Prices are above Bertrand levels. In short, the use of the quantity-setting, Cournot model and the assumption of constant marginal costs on which our analysis is based are not inconsistent. Needless to say, caution should be exercised when technology exhibits decreasing returns. In this case, the Bertrand model is more appropriate, so that whether the conjecture holds generally is an open question.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

317

Though there is a large literature that assumes uncertainty as we do, it does exact a price — namely, the assumption that firms are risk averse.3 This runs counter to a result from classical portfolio theory where it is known that the strategy of diversification — investing small fractions of wealth in each of a large number of noncorrelated securities — reduces risk. In the limit, any portfolio in which an investor’s wealth is invested equally in N securities with independently distributed returns achieves a riskless return as N increases beyond bound (Werner, 1997). A risk averse investor would prefer the limit riskless return to the return of any portfolio. In effect, all investor risk is diversified away in equilibrium if all investors are small, there are perfect capital markets in which all securities are not perfectly correlated, and agency problems in firms are non-existent. A large number of studies in various fields — for example, contract, portfolio, and insurance theory — implicitly or otherwise justify the assumption that firms are risk neutral by appealing to the classical result that investors, qua owners, achieve a risk neutral allocation in equilibrium. This risk neutrality property rests on the assumption that investors are small. If for some reason N does not increase beyond bound all risk need not be diversified away, under-diversification is the result, at least for some investors. Thus the assumption of risk neutrality on the part of all investors in equilibrium is not valid. In particular, a branch of the portfolio literature studies concentrated ownership, where it is found that the ownership structure of firms affects their payoffs by influencing the amount of monitoring of management (Admati et al., 1994; Zhang, 1998; Black, 1992).4 Firms then are subject not only to systematic risk but also to unsystematic risk. The results concerning coalition formation and unsystematic risk in this paper apply to such firms. Managerial ownership has also been adduced as a reason that those firms are risk averse. Because managers have a personal portfolio that is underdiversified after their commitments of human and financial capital, it is argued (Treynor and Black, 1976) that management’s risk aversion results in efforts to reduce risk.

3

Notable examples are Leland (1972), Batra and Ullah (1974), and Anam and Chiang (2000). The assumption of complete information is generally made with regard to demand disturbances. But there is a number of papers on private information of own costs. While we assume that firms are risk averse because of the presence of large investors, the papers on asymmetric information use both the risk neutrality and risk aversion assumptions (see Cramton and Palfrey, 1990, and Baron, 1989). The development of a model that involves market volatility and coalition formation based on asymmetric information remains to be done. 4 It is well documented that concentrated ownership is prevalent in the United States, Germany, and Japan (see Brancato, 1991; Brickley et al., 1988). These results generally use the convention that firms have at least one shareholder owning five percent of the shares. Of course, ownership in small firms tends to be generally more concentrated. It should also be noted that anecdotal and other evidence indicate that institutional and governmental restrictions on investors’ actions — for example, registered retirement savings in Canada are restricted to only 20% of foreign securities — prevent full diversification.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

318

Empirical evidence consistent with this has been found by Amihud and Lev (1981), May (1995) and Chen et al. (1998). Another segment of the literature leads to the conclusion that concentrated ownership should not be unexpected. The general equilibrium literature under uncertainty (Arrow and Hahn, 1971) — assuming complete information and no oligopoly problems — justifies the canonical assumption of risk neutrality on the part of firms as follows. In this uncertainty model, the commodities consumed and the outputs produced by firms are defined not only by their physical properties but also by an event conditional upon when it is available, the states of the world. In a two period world of uncertainty, the true state of the world is unknown in the first period but it is revealed in the second. Under standard convexity assumptions, a competitive equilibrium results that yields a set of prices such that each price is associated with a claim on producers. The producers promise to supply a unit of a commodity if a certain state occurs and nothing otherwise. If there is a price for each good in each state, the firm need only maximize its profits, acting as if it were risk neutral. All the risk bearing is done by consumers as their risk preferences allow. That means that consumers effectively write and sell insurance policies. The result is that a competitive equilibrium exists in a world of uncertainty and all firms are risk neutral, though consumers need not be. However, change the model to allow for stock ownership as follows: retain the complete information assumption, and allow for a mixture of insurance markets, markets of shares of ownership in firms, and markets for other assets. Then, even the general equilibrium model produces a potential mismatching of owners’ preferences and the production possibilities of firms (Dreze, 1974). In a two-firm, two-state, two-consumer example, Dreze allows each risk neutral firm to produce optimally, given the consumption preferences of its owner. With the firms’ production plans taken as given, and with shares in both firms selling at the same price, he shows that the optimal portfolios and the production plans are inefficient. One way to achieve efficiency is to have major changes in the composition of ownership. As Dreze explains: ‘The initiative now lies with the consumers. But this requires additional information on their part: each consumer must know the production set of the other firm, not only its production plan. Furthermore, major changes in ownership fractions are again required: in order to exert enough influence on the decision within the firm to move its production plan (in the appropriate direction), the consumer may have to acquire a majority interest.’ (Dreze, 1974, pp. 149–150). In short, the introduction of uncertainty into a general equilibrium model in which shares of firms are owned by consumers can result in a mismatching inefficiency; and hence, there is an incentive to concentrate ownership of firms in order to bring production plans more in line with consumer preferences.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

319

Summing up, concentrated ownership can arise from the incentive to monitor management in order to avoid agency problems or to align production plans with owners’ preferences. In either event, ownership can be expected to be concentrated, leading to under-diversification of owners’ financial portfolio and, when firms are managed by owners, under-diversification, also, of human capital endeavors. For this reason, firms themselves attempt to reduce market risk. The paper is organized as follows. The assumptions and notation of the Cournot-Nash model under uncertainty immediately follows. The second stage output decisions, along with certain results involving coalitional output and market volatility, are then presented. This is followed by the first stage analysis of sequential coalition formation and our principal findings. Concluding remarks complete the paper.

2. Cournot–Nash model under uncertainty In the first stage of our two-stage model, the owners of firms play an offer–counter-offer, extensive form game to decide on the size and number of coalitions to be formed. The second stage involves a Cournot–Nash quantity game among the coalitions and the outsider firms. At the end of the second stage the owners realize their profits. There are n > 1 firms in the industry of which m j firms belong to the jth coalition and k is the number of coalitions. The feasible values of m j belong to [1, n]. All n firms have the option to join a coalition or to remain independent. Let N 5 h1, 2, 3, . . . , nj and M 5 hM1 , M2 , . . . , Mk j, where each Mi is a nonempty set k of firms such that the union < l 51 Ml 5 N and Ml > Mj 5 5, for all l ± j. Note that M is assumed to be the coalition structure under certainty. Each firm’s demand is subject to an additive random disturbance. We write the inverse demand function facing the ith firm as

O q 1e , n

pi 5 B 2 qi 2 a

i

i

i ±j

where B is a demand (scale) parameter and e i is a random variable with E(e i ) 5 0, var(e i ) 5 sii [ [0, `), and Cov(e i , el ) 5 sil . Note that 0 , a < 1 represents the degree of substitutability between goods, a 5 1 representing perfect substitutability. a 5 0 means that all goods are independent in the utility function common to all consumers. Cost functions for firms are of the form: C(qi ) 5 cqi where c is their constant marginal cost. Consider coalition Mj consisting of m j members. The total profit of the coalition is

O P 5 O FSB 2 q 2 a O q 1 e Dq 2 cq G. n

Vj 5

i [M j

i

i

i [M j

l

l ±i

i

i

i

(1)

320

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

The function is assumed to be twice continuously differentiable in quantities and to be strictly concave in own outputs. Total Vj is assumed to be divided among the coalition members according to a j fixed proportion s i , so that o mi 51 s i 5 1. For the coalition to be viable, s i must be such that the net benefit from participating in the coalition for firm i is at least non-negative. Assume that firm i’s expected utility is characterized by its mean and variance of the form: 5 EU [s iVj ] 5 E[s iVj ] 2 (R / 2) Var[s iVj ] which can be written as EU [s iVj ] 5 s i

O FSB 2 q 2 a O q 2 c Dq G 2 s (R / 2) O O q q s , i

l

i [M j

i

2 i

i

i l

il

i [M j l [M j

l ±i l [N

(2) where R is the absolute risk aversion index. Since this model is subgame perfect and is assumed to have finite play, backward induction is appropriate.6 Thus, in Stage 2, each coalition maximizes its profit with respect to own output, given the optimal size and number of coalitions that are determined in Stage 1.

3. Stage 2: The determination of the optimal output In addition to introducing the assumptions and the symmetric model, we want to give two results on the relationship between the output rate and coalitions. One shows that firms belonging to a large coalition may end up producing more, contrary to the conventional claim under certainty; and the other indicates that the output of a coalition increases with the size of a coalition. Firms in coalition Mj set qi (where i [ Mj ) to maximize E[s iVj ]. The first-order conditions are

O

O

l [N

l [M j

O

≠EU [s iVj ] ]]] 5 s i B 2 2qi 2 a ql 2 c i 2 a ql 2 s 2i R ql sil ≠qi l ±i l ±i l [M j

F

5 0, i [ Mj and j 5 1, 2, . . . ,k.

G

(3)

5 Though the mean-variance approach is limited — primarily to the normality hypothesis — it is easily managed, easily interpreted, and it has a large number of applications. 6 Note that the subgame perfect equilibria may not be unique. Brown et al. (1991) develop an extremely general condition for uniqueness. It is called the Index Condition. Given that demands are linear and marginal costs are constant, the Index Condition is satisfied.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

321

Note that there are n equations altogether. Solving yields 7 q *i 5 q *i (B, c i , m i , sil , R, s i ). Under uncertainty, identical risk-averse firms may gain by spreading risks through coalition formation. Such benefits add to their marginal benefit function, which for fixed quantities motivates them to produce more. We first consider the symmetric case where s 2ii 5 s 2 and sil 5 r < s 2 . It follows that s i 5 1 /m j for i [ Mj , and that firms within the same coalition produce the same level of output. Let q i* 5 q*( j) (for i [ Mj ) be the optimal output produced by a firm belonging to the jth coalition. Since m j ± 0, Eq. (3) can be reduced to k equations with k unknowns, q*( j) ( j 5 1, 2, . . . k).

F

G

1 B 2 2 1 2a(m j 2 1) 1 ] Rr j q*( j) 2 a mj

O m q*(l) 2 c 5 0 k

l

j

l 51 l ±j

5 1, 2, . . . , k.

(4)

where r j 5 s 2 1 (m j 2 1)r. The first result assures us that the model behaves well. It shows that both the certainty and uncertainty models behave in a similar way if market volatility is sufficiently low. It goes on to illustrate that the introduction of market uncertainty makes a difference if it is sufficiently large. Lemma 1. ( i) If goods are highly substitutable or market volatility is low, the optimal output of each firm in the coalition is less than that of a firm outside the coalition, that is, if a /(1 2 r )Rs 2 . 1 /m 2 , then m j . m h ⇔q*( j) , q*(h). ( ii) If goods are less substitutable or market volatility is high, the optimal output of each firm in the coalition is greater than that of a firm outside the coalition — that is, if a /(1 2 r )Rs 2 , 1 /m 2 , then m j . m h ⇔q*( j) . q*(h). 7

In the absence of coalitions, the Cournot–Nash solution, denoted as qˆ i , is determined according to the following first-order conditions: B 2 2qˆ i 2 a

O qˆ 2 c 2 Rqˆ s 5 0, l

i

i

ii

for i, l 5 1, . . . , n.

l ±i l [N

Evaluating (3) at qˆ i and using (4), we have ≠EU [s iVj ] ]]] ≠qi

U

qˆ i

5 s i qˆ i Rsii (1 2 s i ) 2 s i

FO

l ±i l [M j

ql (a 1 Rs i sil )

G

. 0 if a 1 Rs i sil , 0.

The inequality holds when a or sil is sufficiently negative. This inequality ensures that q i* . qˆ i .

322

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

Proof. See Appendix A. h To interpret (i), notice that under certainty all coalitions produce the identical output when firms are symmetrical — a result that can be confirmed by allowing s 2 → 0, implying from the lemma that 8 a 1 lim ]]] . ]2 2 2 ( s 2 r )R m s →0 Therefore, in an environment of certainty the firm belonging to a larger coalition will produce less than the one belonging to a smaller coalition, so that the optimal output of each individual firm is negatively related to the size of the coalition to which it belongs. But when demand is uncertain and goods are highly substitutable with market volatility sufficiently low, the gain to risk averse firms from spreading and diversifying risks through coalition formation is insufficient to induce firms to move their production rates away from those in the certainty model — just as one would expect. Thus when market volatility is low, say s 2 is close to zero, we wind up with something akin to the certainty result: firms belonging to a larger coalition produce less than those belonging to a smaller coalition. To intuit (ii), if market volatility is high, the benefit to risk averse firms from spreading risk is large, so that there is an incentive to increase output and firms belonging to a larger coalition produce at a higher rate than those belonging to a smaller coalition; this differs materially from the certainty result. In the next result, we show that a coalition’s output increases with the size of a coalition and the output rates of member firms in two differently sized coalitions diverge as market volatility increases. This is derived from the fact that a coalition provides externalities through which the variation in profit is reduced. A coalition’s members, as a group, then collectively increase their outputs. Specifically, we have Lemma 2. Total coalitional outputs increase with its size and the difference in outputs of firms in two coalitions increases as markets become more volatile. Proof. See Appendix B. h We observe that these two lemmas refer to Stage 2, the stage in which output is optimally determined. Hence, they apply to all models we consider here. In particular, since any differences in rules affect only Stage 1 decisions, the two

8

Bloch (1996), who studies the certainty case, gets the same result.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

323

lemmas apply whether firms are free to join a coalition or whether their entry requires unanimous consent of the members.

4. Stage 1: determination of the coalition structure We wish to study what effect uncertainty may have on the equilibrium size of coalitions. Two rules are considered. First, we consider a simple coalition formation game in which firms enter the coalition with no restriction imposed on admission — this is known as an open membership model. Second, we examine a noncooperative coalition game in which firms make the decision to coalesce if and only if all members agree on the composition of the coalition; we call this the unanimity model. It turns out that market volatility has decidedly different effects on coalition formation under the two rules. In the open membership model, we find sufficient conditions for the validation of the conjecture, whereas in the unanimity model, the conjecture cannot be generally validated. It is sufficient to specialize the unanimity model to three firms since that is all that is needed to refute the conjecture. We focus on the firm’s expected utilities in Stage 1; these can be found by writing output that is produced by a firm belonging to coalition Mj 9 B2c q*( j) 5 ]]]]]]] . k 1 ]] H(m j ) 1 1 a l51 h(m l )

F

O

G

(5)

Hence the expected utility is EU [Vj /m j ] 5 [q*( j)] 2 x*( j)

(6)

where x*( j) 5 [1 1 a(m j 2 1) 1 1 / 2m j Rr j ].10

9 Write (4) as B 2 c 2 h(m j )m j q*( j) 2 aQ T 5 0, where Q T 5 o lk51 ml q*(l) and h(m j ) 5 H(m j ) /m j . Then, m j q*( j) 5 (B 2 c 2 aQ T ) /h(m j ). Summing this over j yields

O O

k 1 ]] (B 2 c) l 51 h(m l ) T Q 5 ]]]] . k 1 ]] 11a l 51 h(m l )

By substitution, we have (5). 10 Using the expected utility given in (2) and the first-order conditions given in (4), we obtain (6).

324

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

4.1. Coalition formation in the open membership model In this simple model only one coalition is allowed to form and membership is open to all firms who wish to join.11 Those who choose not to join remain singletons and become the outsiders. Clearly, firms will continue to enter the coalition as long as the marginal benefits of doing so remain positive. Therefore, the equilibrium must be such that expected utilities for insiders (i.e. coalition members) and outsiders are equalized. The coalition structure consists of an m-member coalition and n 2 m outsiders; that is, M 5 hM1 , M2 , . . . , Mk j 5 hh1, 2, . . . , mj, hm 1 1j, hm 1 2j, hm 1 3j, . . . , hnjj.12 An example of this model would be an open-membership cartel. We write the expected utility of a coalition member and a singleton as EUm and EU1 :

F

G

2

1 R(s 1 (m 2 1)r ) (B 2 c)2 1 1 a(m 2 1) 1 ] ]]]]] 2 m EUm 5 ]]]]]]]]]]]]] R(s 2 1 (m 2 1)r ) 2 2 1 a(m 2 2) 1 ]]]]] (1 1 aA)2 m

F

G

and

F

G

1 (B 2 c)2 1 1 ] Rs 2 2 EU1 5 ]]]]]]] (2 2 a 1 Rs 2 )2 (1 1 aA)2 respectively, where m n2m A 5 ]]]]]]]]]] 1 ]]]]2 .13 2 R(s 1 (m 2 1)r ) 2 2 a 1 Rs 2 1 a(m 2 2) 1 ]]]]] m Clearly, free mobility will eliminate any difference between the two expected

11 This assumption is restrictive but it is not unusual in the literature. For example, Salant et al. (1983) also allow one coalition to form in their coalition formation game. In fact, it is known in the literature that the equilibrium may not exist for the open membership model without imposing some conditions on the coalition formation game. For details, see Yi (1997). 12 Allowing one coalition to form means that any firm making a decision to coalesce knows that the remaining firms have only two choices, to join or not to join the coalition. Moreover, that firm also knows that the next firm will never join the coalition if it does not join; but if it decides to join, all prior firms making the decision will have decided to join the coalition. Thus, the firm’s choice depends on how many of the prior firms have decided to join the coalition. The behavior of the remaining players has been fully anticipated. It is for this reason that the open membership, sequential game resembles a simultaneous decision game. 13 Note that m 1 5 m and m i 5 1 (i 5 2, 3, . . . , k). Thus, h(m i ) 5 2 2 a 1 Rs 2 and r i 5 s 2 for i 5 2, 3, . . . , k. EUm and EU1 can be obtained by substitution.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

325

utilities so that the equilibrium must be such that EU m* 5 EU 1* . For convenience, define EUm f (m; s 2 , r, R) 5 ]] 2 1 EU1

F F

G GF

1 R(s 2 1 (m 2 1)r ) ] 1 1 a(m 2 1) 1 ]]]]] (2 2 a 1 Rs 2 )2 2 m 5]]]]]]]]]]]]]]] R(s 2 1 (m 2 1)r ) 2 1 2 1 a(m 2 2) 1 ]]]]] 1 1 ] Rs 2 m 2 2 1 5 0. (7)

G

When f (m; s 2 , r ) . 0 means that marginal benefits of joining the coalition are positive, inducing some singletons to participate in a coalition. Conversely, when f , 0, some coalition members will opt out to become a singleton. Thus, the coalition will expand or contract depending on whether f (m; s 2 , r ) is positive or negative. In equilibrium, f (m*; s 2 , r ) 5 0. Stability requires that ≠f (m*; s 2 , r ) / ≠m , 0 in the neighborhood of the equilibrium point. Our main result with regard to the open membership model is summarized in the following proposition, where s 20 solves a zero-gain-from-coalescing function, G(s 2 ) 5 R(s 2 2 r )(s 2 R 1 2 1 a) 2 2a 2 5 0. Proposition 3. Assume that firms are identical and free to join a coalition.

( i) Firms have no incentive to join a coalition when s 2 , s 20 ; ( ii) A coalition begins to form when s 2 . s 02 . The equilibrium coalition size, m*, increases with market volatility; ( iii) The equilibrium coalition structure is a grand coalition if markets become sufficiently volatile. Proof. See Appendix C. h This proposition confirms the conjecture that greater market volatility encourages coalition formation. The intuition is straightforward. In the open membership coalition, risks are shared among member firms and sharing produces an average outcome, which is always preferred by the risk-averse firms. Marginal benefits resulting from risk-sharing among the coalition members increase with market variability. When markets become highly volatile, incentives to join a coalition are enhanced and since there are no restrictions on joining, a larger coalition is therefore formed. Numerical examples highlight this result. Examples. (Open Membership Model) When n 5 10, R 5 2, s 2 5 r 5 0, a 5 1, then m* 5 0 (all singletons); when n 5 10, R 5 2, s 2 5 1.25, r 5 0.5, a 5 1, then

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

326

m* 5 3 (a 3-firm coalition plus 7 singletons); when n 5 10, R 5 2, s 2 5 4, r 5 0.9, a 5 1, then m* 5 10 (a grand coalition).14 As the market volatility increases from s 2 5 0, which is the certainty case, to s 2 5 1.25, to s 2 5 4, the coalition size increases. Thus the examples and the Proposition confirm the conjecture for the open membership model.

4.2. Coalition formation in the unanimity model Here, we examine the equilibrium coalition structure of a sequential decision model in which entry to the coalition requires unanimous consent of all members. Before proceeding, we first examine the condition under which mergers take place. Let us start with a coalition structure containing all singletons, i.e., m j 5 1 for all j 5 1, 2, . . . , k. Clearly, Mj will merge with Mt if EU [Vj >t /(m j 1 m t )] . EU [Vj / m j ]. As the two coalitions — or firms in this case since all are singletons — merge, the number of coalitions reduces from k to k 2 1. Firms belonging to a coalition after merger will earn EU [Vj >t /(m j 1 m t )] 5 [q*( j > t)] 2 x*( j > t)

(8)

where r j >t 5 s 2 1 (m j 1 m t 2 1)r,

F

G

Rr j >t x*( j > t) 5 1 1 a(m j 1 m t 2 1) 1 ]]] , 2(m j 1 m t ) B2c q*( j > t) 5 ]]]]]]]]]] , k 1 ]] 1 aZ H(m j 1 m t ) 1 1 a l 51 h(m l )

F O

G

and Z 5 2 1 /h(m j ) 2 1 /h(m t ) 1 1 /h(m j 1 m t ). We must show that coalitions can indeed form in the unanimity model. The following result gives a sufficient condition under which they do so. It says that there is some critical level of market volatility above which mergers take place; and as such lays a foundation for our Proposition 5 below. Lemma 4. Mergers among firms requiring unanimous consent to form the

14

Maple, version 5, is used to obtain these. A print-out of the computations for these examples and others below is available from the authors.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

327

coalition take place when markets are sufficiently volatile. Specifically, mergers occur if s 2 . s 20 , where s 20 is determined according to ]]] H(m j ) Z œx*( j > t) ]]]] ]]] . ]] H(m 1 m ) 5 1 1 a ] A x*( j) j t œ Proof. See Appendix D. h Given this lemma, we can now proceed to examine the equilibrium coalition structure and ask whether the conjecture can be sustained when unanimity is required among the participating firms. It is sufficient to analyze a three-firm oligopoly, for even then the conjecture fails for certain levels of market volatility. Our task is to determine the equilibrium coalition structure for a given s 2 , using the results to examine how coalition size relates to the given s 2 . For the 3-firm case, there are five coalition structures: M ( 1 ) 5 hh1jh2, 3jj,

M ( 2 ) 5 hh1jh2jh3jj,

M ( 4 ) 5 hh1, 3jh2jj,

M ( 5 ) 5 h1, 2, 3j.

M ( 3 ) 5 hh1, 2jh3jj,

When firms are symmetrical — there are no cost differences and market correlations are identical — M (4) is considered to be the same as M ( 3) . In that case, the proposer sets m to maximize E[s iVj ], where s i 5 1 /m j . This yields a refutation of the conjecture because more market volatility does not always increase the size of the coalition. Proposition 5. Assume that there are three identical firms producing a homogeneous output and that the coalition formation decisions require unanimous consent. Then

( i) M (1) is an equilibrium outcome if s 2 [ (s a2 , s b2 ), where 0 , s a2 , s b2 ; ( ii) M ( 5 ) is an equilibrium outcome if (a) s 2 [ [0, s a2 ] or ( b) s 2 . s b2 ; where 2 (s a , s 2b ) are defined in the proof. Proof. See Appendix E. h It is encouraging to know that the Proposition confirms Bloch’s (1996) result that when there are three identical firms and there is no uncertainty (s 2 5 0), a grand coalition is an equilibrium coalition structure in the sequential game. But in the presence of uncertainty, if market volatility is in some middle range — between s 2b and s 2a — that no longer holds and subcoalitions can be an equilibrium outcome. More importantly, the proposition indicates that lowering the market variability increases the coalition size, so that the conjecture fails to hold.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

328

Table 1 Examples of sequential equilibrium coalitions Identical firms

M (1)

M (2)

M (3)

M (5)

1. a 5 0.9999, B 5 10 c 1 5 1, c 2 5 1, c 3 5 1 R 5 1, s 2 5 r 5 0

p1 p2 p3

9.0000 4.5006 4.5006

5.0630 5.0630 5.0630

4.5006 4.5006 9.0000

6.7504* 6.7504* 6.7504*

2. a 5 0.9999, B 5 10 c 1 5 1, c 2 5 1, c 3 5 1 R 5 1, s 2 5 0.51, r 5 0.1

p1 p2 p3

7.0683* 5.0424* 5.0424*

4.9982 4.9982 4.9982

5.0424 5.0424 7.0683

6.4255 6.4255 6.4255

3. a 5 0.9999, B 5 10 c 1 5 1, c 2 5 1, c 3 5 1 R 5 1, s 2 5 2, r 5 0.5

p1 p2 p3

4.5985 5.4911 5.4911

4.5003 4.5003 4.5003

5.4911 5.4911 4.5985

5.9562* 5.9562* 5.9562*

*

Equilibrium payoffs.

Table 1 illustrates this with some numerical examples. As the variance increases from 0 to 0.51 and further to 2, the equilibrium coalition structure changes from M ( 5 ) to M ( 1 ) and then back to M ( 5 ) . We thus find that the conjecture holds when coalitions require unanimity to join coalitions when market volatility goes from moderate to high levels but it is not valid in general. This makes sense since the unanimity rule restricts the formation of coalition more than the open membership rule. The beneficial externality, which is affected by risk sharing, has to be sufficiently strong to overcome the dampening effect of the unanimity rule. To intuit why a sufficiently low market volatility produces the grand coalition under the unanimity rule, notice that when uncertainty is low or even nil, Firms 2 and 3 will never form a coalition. If Firm 1 chooses to remain a singleton in Stage 1, this would give rise to a Cournot–Nash outcome. But all three firms can do better by forming a grand coalition, which consequently forms.15 When uncertainty becomes moderate, the externality effect begins to strengthen. If it reaches the point where there are enough benefits for Firms 2 and 3 to form a coalition, they will effectuate it. Knowing this, Firm 1 will consequently choose to remain a

15

As noted above, this result differs fundamentally from that obtained by Stigler (1964) who used a homogeneous product, certainty, simultaneous decision model. Recall that he found that no nontrivial coalitions (coalitions with two or more members) would form. But the application of a sequential coalition formation procedure to a three firm oligopoly — even one with some imperfect substitutability and low volatility changes the Stigler result. As we see in the text, the equilibrium coalition structure becomes the grand coalition.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

329

singleton in order to take advantage of the outsider effect — which in the present context is the Stigler effect. Such an outsider effect is only possible when the first mover firm is given an exclusive right to act according to its best interest under the unanimity rule. In this case, the equilibrium coalition structure consists of a two-firm coalition and an outsider.16 Any further increase in uncertainty enhances the beneficial externality, which ultimately leads to a grand coalition outcome.

4.3. Effects on coalition formation of variances and covariances So far, we have limited ourselves to a symmetric case where market variances and correlations are identical across markets. Here, we give some examples that suggest how market volatilities and correlations affect the equilibrium coalition structure. The unanimity rule is assumed to apply. It is necessary to reconsider the intra-coalitional distribution of profits. In the symmetric case when market volatility and correlations are the same, it is easily treated. When they differ, the problem of the share-out requires more attention. Since we wish to retain the self-enforcing quality of the model in both stages, we limit ourselves to the assumption that the share-out depends on the market heterogeneity facing the firms. In particular, low-variance firms are assumed to receive relatively more of the coalition’s total profits than high-variance firms.17 Thus, for a two-firm coalition, the ith firm’s share-out is s i 5 sjj /(sii 1 sjj ); for a three-firm coalition, a firm’s share-out ratio is inversely related to his market volatilities, i.e., s 1 5 s33 / o i351 sii , s 2 5 s22 / o i351 sii , and s 3 5 s11 / o 3i 51 sii if s33 , s22 , s11 or if s11 , s22 , s33 . Table 2 provides some numerical examples to highlight the effects of market correlations on firms’ coalition decisions. In Example 1, M ( 5 ) 5 hh1, 2, 3jj is the equilibrium coalition structure since it provides the best possible outcome for all three firms. In Example 2, we assume that markets 1 and 2 are negatively correlated, while markets 1 and 3 are positively correlated. Given this, the

16

Though the grand coalition will form in a certainty, three-firm symmetric oligopoly, we observe that subcoalitions become possible in the presence of uncertainty. Thus, the presence of demand uncertainty softens the monopoly outcome of the three-firm certainty model. We suggest that a similar result holds for larger oligopolies in a certainty model; that is, when demand uncertainty is allowed, the coalition structure comprises a set of subcoalitions with a competitive fringe of smaller firms. That has yet to be shown. 17 For a model in which firms differ with respect to costs, we have tried other share-out assumptions. They vary from equal intra-coalitional payoffs to payoffs proportional to a firm’s size or costs — all this to avoid resorting to cooperative game solutions of the share-out problem. Thus, self-enforcing output decisions in Stage 2 and the sequential coalition formation decisions in Stage1 are preserved. All our experiments give the same qualitative results as those reported in the text.

330

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

Table 2 Effect of variances and covariances Firms in different markets

M (1)

M (2)

M (3)

M (4)

M (5)

1. a 5 0.9999, B 5 10, c 5 1 R 5 1, sii 5 2, sij 5 0.2

p1 p2 p3

4.5985 5.4911 5.4911

4.5003 4.5003 4.5003

5.4911 5.4911 4.5985

5.4911 4.5985 5.4911

5.9562* 5.9562* 5.9562*

2. a 5 0.9999, B 5 10, c 5 1, R 5 1, sii 5 2 s12 5 2 1.9, s13 5 1.9, s23 5 0.2

p1 p2 p3

4.5985 5.4991 5.4991

4.5003 4.5003 4.5003

7.3216* 7.3216* 3.3765*

4.5638 5.3187 4.5638

4.8032 4.8032 4.8032

3. a 5 0.9999, B 5 10, c 5 1 R 5 1, s11 5 1, s22 5 2, s33 5 3 sij 5 0.5

p1 p2 p3

7.0259* 4.7818* 3.5724*

6.9987 4.1475 2.9162

7.3550 4.3053 3.3915

6.2288 4.5312 2.9344

7.4357 5.5162 3.0376

4. a 5 0.9999, B 5 10, c 5 1 R 5 1, s11 5 3, s22 5 2, s33 5 1 sij 5 0.5

p1 p2 p3

3.3915 4.3053 7.3550

2.9162 4.1475 6.9987

3.5724* 4.7818* 7.0259*

2.9344 4.5312 6.2288

3.0376 5.5162 7.4357

*

Equilibrium payoffs.

equilibrium coalition structure turns out to be M (3) 5 hh1, 2jh3jj. This result is not surprising and can be explained as follows. Negative market correlations reduce the variability of firm’s profits, thus raising a firm’s expected utility. Though proposing a grand coalition remains an option (for which only the high-variance firm will benefit), Firm 1 as the first mover can further enhance the externalities by teaming up only with firms whose markets are negatively correlated to his own. These examples also refute the conjecture that coalition size rises with market volatility. But there are additional relationships when markets differ in terms of random shocks. Consider Example 3, which assumes that Firm 1 operates in the lowest risk market among all three. The optimal decision for Firm 1 is to remain a singleton, letting Firms 2 and 3 form a coalition. In so doing, Firm 1 is able to enjoy benefits accruable to an outsider because of the presence of the Stigler effect. However, there is a further benefit from remaining a singleton: it prevents a free-riding problem for a low-risk firm. Reverse the risk elements, as in Example 4, where the first mover is now the high-risk firm. Then, Firm 1 would rather form a two-firm coalition with Firm 2 so that risks can be diversified through coalition formation.18 Notice that two-firm coalitions are formed, whether Firm 1 is the high

18 A reason for Firm 1 not to team up with Firm 3 (whose risk is the lowest) is because risk differences between the two firms are so diverse that Firm 1’s relatively low share-out ratio puts it in an unfavorable position.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

331

risk or the low risk firm. In sum, not only relative market risks but also market correlations across markets can profoundly affect firms’ intentions to coalesce.19

5. Concluding remarks Our analysis applies to quantity-setting oligopolistic firms that are risk averse. Accepting this, our conjecture has considerable intuitive appeal. The desire to reduce risk by sharing it is a natural consequence of risk aversion, and coalition formation is a natural way to share risk. Hence, there is considerable force to the conjecture that more market volatility leads to larger coalitions. But the conjecture is not vacuous because it does not hold in all cases. A change in the rule of how a coalition forms, from open membership to a unanimity rule, can invalidate it. Our simple symmetric model suffices both to confirm the conjecture in the open membership rule case and to refute it when unanimity is required for a coalition to form. Though the conjecture does not hold, in general when the unanimity rule applies, neither does it always fail. In fact, the conjecture holds nicely for higher ranges of the market volatility measure. If the variance of demand increases from a level in a middle range to a higher level, coalition size increases, no matter what the rule. It only fails when the market volatility is sufficiently low and proceeds to a middle range. Considerable volatility is required to overcome coalition-destroying factors in unanimity based coalitions — the principal factor being the Stigler effect, which is the desire by firms to remain outside coalitions in order to free ride on the price rise effected by the increase in the monopoly power of a coalition. In sum, if market volatility starts at a low level and proceeds to a moderate level, do not expect a general increase in coalition size. But if it is sufficiently large, coalition size rises with market volatility, independently of the coalitionforming rule. Our analysis rests on a complete information model and the assumption that

19 What is done here is to examine how the coalition structure is affected by firms’ types — a type being related to the market in which a firm operates. In the symmetric case, the outcome is independent of a firm’s type but with heterogeneity, assigning different characteristics can change the equilibrium coalition structure. If types of firms are permuted, that in effect means that the order of moves is permuted. Hence, we are asking how the equilibrium coalition structure is independent of the order of moves. In preliminary work, about which we expect to report at a later date, we have found that there are fairly wide ranges of types for which permuting the order of moves yields the same coalition structure. In short, the equilibrium coalition structure may be independent of the order of moves for certain ranges of parameter values. Note that in Examples 3 and 4, that is not the case.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

332

uncertainty resides in the demand for the products. Whether the results hold for asymmetric information concerning demand uncertainty or cost uncertainty, this relaxation of the complete information assumption in its several directions remains to be pursued. Also, the paper raises immediate caveats concerning the assumptions that firms compete in quantities and that marginal costs are constant. Clearly, these have to be relaxed. It remains an open question whether our results can be extended to the decreasing marginal cost case, and especially to the subcase where only a Bertrand equilibrium can be obtained. We add that with regard to the latter, there is a need to examine coalition formation within Bertrand price-setting oligopolies. These are worthy subjects for future research.

Acknowledgements We wish to thank Michael Gort, Elmar Wolfstetter, the editor Stephen Martin, and an anonymous referee for valuable comments and suggestions.

Appendix A. Proof of Lemma 1 Note that (4) holds for any two coalitions ( j and t, say). Therefore, we have

F

G

F

G

1 1 2 1 a(m j 2 2) 1 ] Rr j q*( j) 5 2 1 a(m t 2 2) 1 ] Rr t q*(t). mj mt

Let H(m) 5 2 1 a(m 2 2) 1 1 /mRr where m 5 m j 5 m t . Thus, the relationship between q*( j) and q*(t) depends on whether H is increasing or decreasing in m. 2 2 . . ] Calculate ≠H(m) / ≠m 5 a 1 [(2s 2 1 r )R] /m 2 ] , 0, provided a /(s 2 r )R , 1 /m . 2 2 . . ] Evidently, H(m j ) ] , H(m h ) if a /(s 2 r )R , 1 /m . This along with (4) gives (i) and (ii) in the text.

Appendix B. Proof of Lemma 2 Rewrite (4) as H(m j ) H(m h ) ]] m j q*( j) 5 ]] m h q*(h). mj mh 2

2

Clearly, ≠(H(m) /m) / ≠m 5 2 [2(1 2 a) 1 2R /m(s 2 r ) 1 Rmr ] /m , 0 (m 5 m j or m h ). Higher m implies lower H(m) /m, which further implies higher mq*. Note that H(m j ) /m j and H(m t ) /m t have the same functional form, so that an increase in

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

333

m reflects a movement along the H(m) /m curve. A large coalition thus produces more in total relative to a smaller one. Calculate ≠ 2 (H(m) /m) / ≠m≠s 2 5 2 2R /m 3 , 0. Thus, H(m) /m becomes steeper as s 2 increases. This implies that the relative outputs of firms in two coalitions of different sizes becomes larger as s 2 increases.

Appendix C. Proof of Proposition 3 Take the derivative of f (m; s 2 , r ) with respect to m and evaluate it at m 5 1, yielding ≠f (1; s 2 , r ) / ≠m 5 [R(s 2 2 r )(s 2 R 1 2 1 a) 2 2a 2 ] / [(2 2 a 1 Rs 2 )(2 1 Rs 2 )) x 0 if G(s 2 ) 5 R(s 2 2 r )(s 2 R 1 2 1 a) 2 2a 2 x 0. Note that ≠G(s 2 ) / ≠s 2 5 2R 2 s 2 1 (2 1 a)R . 0. Therefore, there exists a s 02 such that G(s 20 ) 5 0. For s 2 , s 20 , G , 0; and hence no coalition will be formed. This proves (i). When s 2 . s 20 , G . 0. Then, a coalition consisting of m* firms will be formed, where m* satisfies the equilibrium condition: f (m*; s 2 , r, R) 5 0. The effect of s 2 on m* can be obtained by taking the total derivative of s 50 with respect to m and s 2 , yielding dm* / ds 2 5 2 [≠f (m*; s 2 , r, R) / ≠m] 21 [≠f (m*; s 2 , r, R) / ≠s 2 ], where ≠f (0; s 2 , r ) / ≠m 5 [R(s 2 2 r )(s 2 R 1 2 1 a) 2 2a 2 ] / [(2 2 a 1 R*s 2 )(2 1 Rs 2 )) x 0 and ≠f / ≠s 2 5 m(2 2 a 1 Rs 2 )R(m 2 1)h[s 4 r 1 r 2 (m 2 1)s 2 ]R 3 1 [(2 2 2a)s 4 1 (3am 2 1 (4 2 4a)m 1 3a)rs 2 1 ((2 1 a)m 2 a 2 2)r 2 ]R 2 1 [(2a 2 m 3 2 2a(23 1 2a)m 2 1 (6a 2 1 4 2 8a)m)s 2 1 (3a(2 1 a)m 2 2 4(2 1 a)(a 2 1)m) 1 (4 2 2a)(s 2 2 r )]R 1 2a 2 (2 1 a)m 3 2 2a(2 1 a)(23 1 2a)m 2 1 4(a 2 1)(a 2 2)mj / h[(s 2 1 (m 2 1)r )R 1 am 2 1 (2 2 2a)m] 3 (2 1 Rs 2 )2 j . 0.

334

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

Since ≠f / ≠m , 0 (for stability), we can conclude that dm* / ds 2 . 0. Higher s 2 implies higher m*, thus proving (ii) From (ii), we know that m* strictly increases with s 2 . It is evident that when s 2 becomes sufficiently large, m* will eventually hit its upper bound n. A grand coalition is formed, thus proving (iii).

Appendix D. Proof of Lemma 4 To ease calculations, take a monotonic transformation of the expected utility function, which does not affect the results. Compute ]]]]]]

]]]

œEU [Vj >t /(mj 1 mt )] 2œEU [Vj /mj ] ]]] ]] œx*( j > t) 1 œx*( j) 1 5 (B 2 c) ]]]] ]] 2 ]]] ] H(m j 1 m t ) A 1 aZ H(m j ) A

F

G

]]]

H(m j ) Z œx*( j > t) ]]]] _ 0 if ]]] ]] H(m 1 m ) _ 1 1 a ] A j t œx*( j) where A 5 1 1 a o kl 51 1 /h(ml ). ]]] ]] Let L(s 2 , r, a) 5 [œx*( j > t) /œx*( j)][H(m j ) /H(m j 1 m t )] and T(s 2 , r, a) 5 2 1 1 a(Z /A). Note that L is an increasing function of s , with L(`, r, a) 5 ]]]] ]]]] j 1 m t ) /m j and L(0, 0, a) 5œm j /(m j 1 m t ). That is, the L function starts with œ(m ]]]] ]]]] 2 gets larger. j 1 m t ) /m j as s œmj /(mj 1 mt ) , 1 and gradually approaches œ(m Also, note that T is an increasing function of s 2 for small or moderate s 2 and then becomes a decreasing function of s 2 afterward, with T(`, r, a) 5 1 and T(0, 0, a) 5 k / 1 1 k , 1. To highlight our assertion, we assume that L(0, 0, a) 5 ]]]] œmj /(mj 1 mt ) , T(0, 0, a) 5 k / 121 k. Thus, there2 is no incentive for coalitions j and t to merge under certainty (s 5 r 5 0). As s begins to deviate from zero, L increases with it, while T may increase and then decrease at some point. Therefore, there exists a s 20 such that L(s 20 , r, a) 5 T(s 20 , r, a). For s 2 , s 20 , then L , R, implying that ]]]]]]

]]]

œEU [Vj >t /(mj 1 mt )] 2œEU [Vj /mj ] , 0; and the two coalitions will not merge. But for s 2 . s 20 , L . T, implying that ]]]]]]

]]]

œEU [Vj >t /(mj 1 mt )] 2œEU [Vj /mj ] . 0; and hence the merger occurs. This holds even for any m j and m t , including m j 5 m t 5 1. The lemma is thus proven.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

335

Appendix E. Proof of Proposition 5 At the outset, let M 5 hh1j, h2j, h3jj. The game is initiated by Firm 1 which can choose to stay as a singleton, or form a coalition with Firm 2, or propose a grand coalition. If Firm 1 chooses to be a singleton, then Firm 2 becomes the next proposer that in turn can propose a coalition of size m (where m 5 1, 2). Choosing m 5 1 (m 5 2) gives rise to M (2) (M (1)). Since the game is sequential, we solve the problem by backward induction, beginning with the last proposer (i.e., Firm 2). If Firm 1 remains a singleton, Firm 2 makes its coalition decision by selecting optimal m ( 5 1, 2) to maximize expected utility; it does this by referring to Eq. (7) as given by EU [V2 (m) /m] 5

F

G

2

1 R(s 1 (m 2 1)r ) (B 2 c)2 1 1 a(m 2 1) 1 ] ]]]]] 2 m ]]]]]]]]]]]]]]]]]]]]]]. R(s 2 1 (m 2 1)r ) 2 2 1 a(m 2 2) 1 ]]]]] [1 1 a(1 /h(1) 1 1 /h(m) 1 1 /h(2 2 m))] 2 m

F

G

From Lemma 4, we know that there exists a critical value of s 2 above which a coalition between Firms 2 and 3 will be formed. To determine such a critical value, calculate EU [V2 (2) / 2] 2 EU [V2 (1) / 1] 2

6

2

2

5

4

2

2

5 (1 / 2)(B 2 c) [(s (s 2 r )(s 1 r )R 1 2s (7s 1 3r )(s 2 r )R

4

1 2s 2 (26r 2 1 31s 4 2 23rs 2 )R 3 1 (120s 4 2 8r 2 2 96rs 2 )R 2 1 (48s 2 2 64r )R 2 32)] / [(s 2 (s 2 1 r )R 2 1 (10s 2 1 2r )R 1 12)2 (Rs 2 1 4)2 ] Its sign depends on the terms in the first square bracket, which is increasing in s 2 . Therefore, there exists a s 2a . 0 such that EU [V2 (2) / 2] 2 EU [V2 (1) / 1] 5 0. For s 2 _ s a2 , EU [V2 (2) / 2] 2 EU [V2 (1) / 1] _ 0. Obviously, when s 2 . s a2 , Firms 2 and 3 form a coalition and this gives rise to M (1) . But, if s 2 , s a2 , Firm 2 remains independent, resulting in M ( 2 ) . Knowing what Firm 2 will do later in the game, we now go one step backward to consider Firm 1’s choices in Stage 1. Firm 1 sets m (m 5 1, 2, 3) to maximize its expected utility, given by

F

G

1 R(s 2 1 (m 2 1)r ) (B 2 c)2 1 1 a(m 2 1) 1 ] ]]]]] 2 m EU [V1 (m) /m] 5 ]]]]]]]]]]]]] 2 R(s 1 (m 2 1)r ) 2 2 1 a(m 2 2) 1 ]]]]] A 21 m

F

G

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

336

where (1) A 1 5 A (11 ) 5 1 1 a(1 /h(m) 1 1 /h(3 2 m)) if s 2 . s 2a , or (2) A 1 5 A (12 ) 5 1 1 a(1 /h(m) 1 (3 2 m) /h(1)) if s 2 , s 2a . Case (1) (Case (2)) refers to the case where a coalition (no coalition) will be formed between Firms 2 and 3 in Stage 2. These two cases are examined in what follows. Consider (1) first. In this case, A 1 5 A (1) 1 . Notice that the EU [V1 (m) /m] function is either upward sloping or U-shaped in m. If it is upward sloping, then Firm 1 will certainly propose a grand coalition, resulting in M ( 5 ) . But, if it is U-shaped, then m 5 2 will never be chosen. Therefore, the choice between m 5 1 (resulting in M ( 1 )) and m 5 3 (resulting in M ( 5 )) depends on whether EU [V1 (1) / 1] 2 EU [V1 (3) / 3] 5 2 (B 2 c)2 [s 2 (s 2 2 r )(s 2 1 r )2 R 4 1 2(s 2 2 r )( r 1 8s 2 )(s 2 1 r )R 3 1 4(20s 2 1 7r )(s 2 2 r )R 2 1 (56s 2 2 104r )R 2 72] / h[R(s 2 1 2r ) 1 18][s 2 (s 2 1 r )R 2 1 (10s 2 1 2r )R 1 12] 2 j is positive or negative. Clearly, its sign depends upon the terms in the first square bracket, which increases with s 2 . Thus, there exists a s b2 . 0 such that EU [V1 (1) / 1] 2 EU [V1 (3) / 3] 5 0. For s 2 + s 2b , EU [V1 (1) / 1] 2 EU [V1 (3) / 3] _ 0. Hence, for Firm 1 to stay independent, it requires that s 2 , s b2 . Note that s a2 , s b2 . This along with the above discussion allows us to conclude: (a) If s [ (s a2 , s b2 ), Firm 1 wishes to remain a singleton, and Firms 2 and 3 would form a coalition later in the game; M ( 1 ) is therefore the equilibrium coalition structure, proving (i); and (b) If s 2 . s 2b , EU [V1 (1) / 1] 2 EU [V1 (3) / 3] , 0, therefore, Firm 1 would propose a (5) grand coalition M , proving (ii) (b). Next, consider Case (2) (which occurs when s 2 , s a2 ). In this case, A 1 5 A 1(2 ) . Again, the EU [V1 (m) /m] function in this case can be either upward sloping or U-shaped. If it is upward sloping, M ( 5 ) results. But, if it is a U-shaped function, m 5 2 can never be the choice for Firm 1. Calculate [(s 4 2 rs 2 )R 2 1 (2s 2 2 2r )R 1 6](B 2 c)2 EU [V1 (3) / 3] 2 EU [V1 (1) / 1] 5 ]]]]]]]]]]]] [(4 1 Rs 2 )2 (R(s 2 1 2r ) 1 18)] . 0. Thus, Firm 1 proposes a grand coalition, proving (ii) (a).

References Admati, A., Pfleiderer, P., Zechner, J., 1994. Large shareholder activism, risk sharing, and financial market equilibrium. Journal of Political Economy 102, 1097–1130. Amihud, Y., Lev, B., 1981. Risk reduction as a managerial motive for conglomerate mergers. Bell Journal of Economics 12, 505–617.

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

337

Anam, M., Chiang, S.H., 2000. Export market correlation and strategic trade policy. The Canadian Journal of Economics 33, 41–52. Arrow, K., Hahn, F.H., 1971. General Competitive Analysis. Holden-Day, San Francisco. Baron, D.P., 1989. Design of regulatory mechanisms and institutions,. In: Schmalensee, R., Willig, R.D. (Eds.), Handbook of Industrial Organization. North-Holland, New York, Chapter 24. Batra, R., Ullah, A., 1974. Competitive firm and the theory of input demand under price uncertainty. Journal of Political Economy, 537-548. Baye, M.R., Crocker, K.J., Ju, J., 1996. Divisionalization, franchising, and divestiture incentives in oligopoly. American Economic Review 86, 223–236. Black, B.S., 1992. Agents watching agents: the promise of institutional investor voice. UCLA Law Review 39, 895–899. Bloch, F., 1995. Endogenous structures of association in oligopoly. Rand Journal of Economics 26, 537–556. Bloch, F., 1996. Sequential formation of coalitions in games with externalities and fixed payoff division. Games and Economic Behavior 14, 90–123. Bowley, A.L., 1924. The Mathematical Groundwork of Economics. Oxford University Press, Oxford. Brancato, C.K., 1991. The pivotal role of institutional investors in capital markets,. In: Sametz, A., Bicksler, J.L., Homewood, I. (Eds.), Institutional Investing: Challenges and Responsibilities of the 21st Century. Brickley, J.A., Lease, R.C., Smith, Jr. C.W., 1988. Ownership structure and voting on antitakeover amendments. Journal of Financial Economics 20, 267–291. Brown, M., Chiang, S.H., Yamamoto, K., 1991. Uniqueness of equilibrium for smooth multistage concave games. Games and Economic Behavior 3, 393–402. Brown, M., Chiang, S.H., 1997. Endogenous industry structure and noncooperative coalition analysis. Working Paper, State University of New York at Buffalo. Brown, M., Chiang, S.H., 1998. Managerial incentives and noncooperative coalitions. Working Paper, State University of New York at Buffalo. Brown, M., Chiang, S.H., 1999. The noncooperative kernel of a sequential coalition game. Working Paper, State University of New York at Buffalo. Chen, C.R., Steiner, T., Whyte, A.M., 1998. Risk-taking behavior and managerial ownership in depository institutions. Journal of Financial Research 19, 1–16. Cramton, P.C., Palfrey, T.R., 1990. Cartel enforcement with uncertainty about costs. International Economic Review 31, 17–47. Dewey, D., 1961. Mergers and cartels: some reservations about policy. American Economic Review 51, 255–262. Dixit, A., 1979. A model of duopoly suggesting a theory of entry barriers. Bell Journal of Economics 10 (1 Spring), 20–32. Dreze, J., 1974. Investment under private ownership: optimality, equilibrium and stability. In: Dreze, J. (Ed.), Allocation under Uncertainty: Equilibrium and Optimality. Macmillian, London. Gort, M., 1969. An economic disturbance theory of mergers. The Quarterly Journal of Economics 83, 624–642. Klemperer, P., Meyer, M., 1986. Price competition vs. quantity competition: the role of uncertainty. Rand Journal of Economics 17, 618–638. Klemperer, P., Meyer, M., 1989. Supply function equilibria in oligopoly under uncertainty. Econometrica 57, 1243–1277. Leland, H.E., 1972. Theory of the firm facing uncertain demand. American Economic Review 62, 278–291. May, D., 1995. Do managerial motives influence firm risk reduction strategies. Journal of Finance 50, 1291–1308. Ray, D., Vohra, R., 1997. Equilibrium binding agreements. Journal of Economic Theory 73, 30–78. Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50, 97–110. Salant, S.W., Switzer, S., Reynolds, R.J., 1983. Losses from horizontal merger: the effects of an

338

M. Brown, S.-H. Chiang / Int. J. Ind. Organ. 20 (2002) 313 – 338

exogenous change in industry structure on Cournot–Nash equilibrium. The Quarterly Journal of Economics XCVIII, 185–199. Selten, R., 1981. A non-cooperative model of characteristic function bargaining, in: Bohm and Nachtkamp, (Eds.), Essays in Game Theory and Mathematical Economics in Honor of O. Morgenstern, Mannheim, Bibliographisches Institut. Spence, A.M., 1976. Product differentiation and welfare. American Economic Review, pp. 407–414. Shubik, M., Levitan, R., 1980. Market Structure and Behavior. Harvard University Press, Cambridge, Massachusetts. Stigler, G.J., 1964. A Theory of Oligopoly. Journal of Political Economy 72, 44–61. Treynor, J., Black, F., 1976. Corporate Investment Decisions in Modern Developments in Financial Management. Praeger. Weitzman, M., 1974. Prices vs. quantities. Review of Economic Studies 41, 477–491. Werner, J., 1997. Diversification and equilibrium in securities markets. Journal of Economic Theory 75, 89–103. Yi, S., 1997. Stable coalition structures with externalities. Games and Economic Behavior 20, 201–237. Zhang, G., 1998. Ownership concentration, risk aversion and the effect of financial structure on investment decisions. European Economic Review 42, 1751–1778.