Promote Cooperation. Job Market Paper

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1 Divide and Conquer? Decentralized Firm Structure May Promote Cooperation Job Market Paper Michal Goldberg December 12, 2013 Abstract I consider a model in which an entrepreneur s objective is to maximize cooperation in a multidivisional organization. In order to do so, the entrepreneur can choose the number of production units in a profit center. The profit centers may rely on their long-term relationships with one another for cooperation. I use a repeated games setting that endogenizes the cooperation decisions. I show that in certain cases, in contrast to the common wisdom, decentralization may promote cooperation. In particular, I show that the stronger the interdependencies among units are, the smaller the optimal number of units in each profit center. I further use this setting to examine efficiency gains of mergers and acquisitions. Efficiency gains of a merger are below the value of synergies between the acquiring firm and the target when the ties are not strong enough to sustain cooperation. However, the potential synergy between an acquiring firm and a target may be an underestimate of the efficiency gains of a merger when the ties are sufficiently strong to cause a firm to shift from a noncooperative to a cooperative equilibrium. I characterize these cases and draw policy implications. Carnegie Mellon University, I thank Heski Bar-Isaac, Serguey Braguinsky, Boyan Jovanovic, Alessandro Lizzeri, Yaron Yehezkel, and seminar participants at Carnegie Mellon University for valuable comments. All errors are mine. 1

2 1 Introduction The economics of organizations literature has long been concerned with the optimal structure of firms (e.g., [Cremer, 1980], [Williamson, 1967], and [Hart and Moore, 2005]). However, while in other fields dynamic considerations are a key in understanding economic forces, the models that analyze the optimal structure of firms rely mostly on static considerations. Dynamic incentives are particularly important when studying the relationship among different parts of a firm because one of the attributes that distinguishes a large firm from a collection of firms is the long-term relationships among its parts. This paper studies the optimal firm s structure, and in particular, the optimal level of decentralization, that would promote the highest level of cooperation among different parts of a firm in a dynamic setting. The common wisdom is that centralization enhances intra-firm cooperation while decentralization provides better incentives within a unit (e.g., [Roberts, 2007]). The optimal firm structures then balances cooperation against incentives. In contrast, this paper shows that decentralization may increase intra-firm cooperation. In the model, intra-firm cooperation generates profits to the firm because potential synergies can be realized. Some of these synergies can be realized efficiently by using a contract and transfer pricing (such as supply of a good that does not require special adaptation), or by coordinating a move to a more efficient equilibrium (such as coordination on a standard adaptation). However, some other synergies cannot be efficiently realized using transfer pricing or through within-period coordination, but can be realized by an implicit contract, relying on the long-term relationships among parties in a firm. The structure of the firm affects the incentives of the units to cooperate with each other. In this paper, I study the optimal firm structure that maximizes profits from cooperation. In particular, I study the optimal level of centralization that maximizes cooperation. In the most decentralized firm structure, each production unit forms an independent profit center, and all decisions are made at the production unit level. In more centralized firm structures, a few production units are grouped together to form a profit center. All units in a profit center 2

3 maximize the profits of the profit center, and all decisions are made at the profit center level. I allow for a range of profit center sizes, to account for various levels of centralization. To avoid the obvious case in which a perfect centralization achieves the highest degree of cooperation, i.e. that all decisions are made by a single authority, I assume that a firm has to be divided into at least two profit centers. 1 When dynamic incentives are considered, decentralization may promote cooperation, as a profit center that is small and can achieve little without cooperation may anticipate gaining more from cooperation than a larger profit center. However, centralization may reduce the cost associated with cooperation, as the larger the size of the profit center, the fewer units outside the profit center, and consequently, a larger profit center is required to accommodate fewer needs. The optimal level of centralization is determined by balancing this trade off. More specifically, I initially assume a model in which the number of production units is exogenous, but the number of units in each profit center is determined by the entrepreneur. Cooperation is essential because each period each of the production units requires some need that one or more of the other units may be able to accommodate. This could be a pricing or output strategy in a market shared by more than one unit, adoption of some technology standard, or information regarding some market that is of interest to multiple units. Accommodating that need is costly, but the benefit to the requester exceeds the cost. That is, accommodating needs is always beneficial for the firm, but in a static world if the units maximize only their own profit, and in the absence of additional incentives, the needs of other profit centers are never accommodated, as this is a prisoner s dilemma game. However, I assume that the units are in long-term relationships among themselves. The units may be able to utilize these relationships and sustain cooperation. Although in a static world the optimal firm structure is the most centralized possible in this situation, in a dynamic setting, the optimal firm structure varies with the probability that 1 The exact reason for why organizing the firm in one profit center is not possible is beyond the scope of this paper. There could be many reasons for that, for example, monitoring may not be adequate when a large firm is organized into a single profit center. 3

4 some unit is able to accommodate a need of another unit, (the interdependence degree). The interdependence degree can be viewed as a measure of the interdependency of the units in a firm with each other. I first analyze the case in which the ability to accommodate needs is identically and independently distributed across production units. I show that when the interdependence degree among the units is higher, the optimal firm structure is less centralized. The reason is that an increase in the interdependence degree among production units increases the probability that a large profit center can accommodate its needs on its own. This reduces the continuation payoff of cooperation and hence reduces the incentive of a profit center to cooperate. Meanwhile, the average cost of cooperation is lower as the profit center grows larger for two reasons. First, each profit center has to take into account the cost of accommodating fewer needs in total. Second, units that are in the same profit center are able to coordinate among themselves offerings to accommodate certain needs, and thus are able to avoid considering the cost of accommodating the same need. These effects are prominent when the interdependence among production units is low, and hence the profit centers expect to gain little from cooperation, but are still required to be willing to accommodate all needs they can. In addition, I find that when the interdependence of units is identical, the optimal number of units in a profit center does not change with the size of the firm. An increase in the number of production units in a firm increases the cost of cooperation as well as the reward from cooperation as each need is more likely to be accommodated. However, in the symmetric case, these two effects offset each other. A by-product of this result is that the number of profit centers increases with the size of a firm. These results have a few implications. First, one should expect that when the units in a firm are highly interdependent of each other, they will be organized in a decentralized way. This is a counterintuitive result because in a static model, interdependent units should form a profit center together. Another implication is that the number of profit centers should be higher in larger firms. 4

5 Next, I consider alternative assumptions regarding the distribution of the degrees of interdependence. Because it is complicated to analyze the general case for this problem, I consider extreme assumptions in order to understand the economic forces present in each case. I find that in cases where the units are heterogeneous and one unit is very likely to be able to accommodate all needs, the most decentralized structure achieves the best cooperation. The reason is that the incentive of a profit center that includes such a unit is even lower than in the symmetrical case because a large part of the profit center s needs can be accommodated by the dominant unit. Furthermore, in order to analyze the effect of correlation in the interdependence across units, I analyze two extreme cases of perfect negative and perfect positive correlations. In the extreme negative correlation case I assume that at most one unit is able to accommodate any given need. The extreme positive correlation case is the opposite, and if one unit is able to accommodate some need, all other units are able to accommodate it as well. In both of these cases, the most centralized structure possible achieves the best cooperation. I conjecture from this that in order for decentralization to promote cooperation, there must be a chance that more than one unit would be able to accommodate at least some of the needs, but the units cannot be perfect substitutes for each other. A next natural step is to treat the interdependence among units as an endogenous choice. To do so, I characterize the maximum cost an entrepreneur may be willing to bear to increase the level of interdependence among the units in a firm. At any level of interdependence among the units, it is optimal to pay a positive cost to increase the level of interdependence among the units. However, the maximum cost that an entrepreneur will be willing to pay to increase the level of interdependence varies. In particular, this maximal cost increases at the level of interdependence around the critical level of interdependence, above which full cooperation can be sustained. One way to increase interdependence among units is to extend the boundaries of the firm by mergers and acquisitions. Synergies are often cited as a reason for mergers and acquisitions, and are sometimes used to justify the approval of mergers and acquisitions ([U.S. Department 5

6 of Justice and the Federal Trade Commission, 2010]). However, the effect of a potential synergy on the performance of the merged firm varies. For example, Moeller, Schlingemann, and Stulz [2004] find that the larger the firm size the lower the gains of the shareholders of the acquiring firm. This paper provides a new way to evaluate the increased economic efficiency in mergers and acquisitions. The value of the synergy between the target firm and the acquiring firm may not correctly represent the added economic efficiency of the merger. Although there may be the potential for a synergy, it may not be realized because the target and acquiring firm cannot sustain cooperation. In particular, I show that when cooperation cannot be sustained in the acquiring firm prior to the acquisition, and the target firm has the same potential synergies with the acquiring firm as the rest of the units in the firm, cooperation will not be sustainable in the merged firm, and the synergies will not be realized. In that case, the added economic efficiency of mergers can be below the value of the potential synergy. However, there are cases in which the added economic efficiency of mergers exceeds the potential synergy between the acquiring firm and the target. That can happen when the target has very strong potential synergies with the acquiring firm. In that case, the acquisition can lead the entire firm to shift from a noncooperative to a cooperative equilibrium. I show that when all units in the acquiring firm are symmetrical in their interdependence of each other, and the potential synergies with the target are stronger, shifting to a cooperative equilibrium is more likely when the size of the acquiring firm is smaller. This is consistent with Moeller et al. [2004]. The rest of this paper proceeds as follows. Section 2 discusses the related literature. Section 3 describes an example of interdependencies among units in a firm. Section 4 describes the formal model and discusses the assumptions made. Section 5 analyzes the equilibrium when the interdependence degrees among units are independent. Section 6 analyzes the equilibrium under a few alternative assumptions regarding the correlation of interdependencies. Section 7 relaxes the assumption that the interdependencies are exogenous, and discusses the value of 6

7 an increase in interdependencies among units in a firm. Section 8 analyzes one way in which interdependencies could be increased namely by mergers and acquisitions, characterizes the cases in which an acquisition could increase firm value by more than the synergies between the acquiring firm and the target, and draws policy implications. Section 9 concludes. Finally, all proofs are in the appendix. 2 Related Literature There is a large body of literature on organizational design (see [Harris and Raviv, 2002] for a survey). However, the existing literature primarily uses static considerations, whereas in this paper dynamic incentives are in the basis of the analyses. A major focus of the literature on organizational design is information transmission and aggregation within an organization (e.g., [Bolton and Dewatripont, 1994], [Harris and Raviv, 2002], [Maskin et al., 2000], [Baccara and Bar-Isaac, 2008], [Stein, 2002], [Qian et al., 2006], and [Alonso et al., 2008]. My model can be applied to information transmission within an organization, if the need that can be accommodated is information. However, the model is not specific for information and may also be useful in analyzing other attributes. A part of the literature on organizational design is concerned with the optimal functions of each unit that is, a unitary form ( U form ) versus a multidivisional form ( M form ). Another part of the literature (e.g., [Alonso et al., 2008], [Stein, 2002]) essentially takes the optimal function of each unit as given, and is concerned with the optimal level of decentralization. This paper adds to the latter. I take the function of each unit, and the degree of interdependence among them as given and analyze the optimal level of decentralization. Alonso et al. [2008] is similar to this paper in the sense that in some cases, better cooperation can be achieved through a less centralized firm structure. However, their result was driven by how organizations aggregate dispersed information and relied on the divisions interests being sufficiently aligned to each other. My model does not assume any correlation in the interests of the unit managers, but achieves cooperation through long term relationships among the units. 7

8 The literature that uses repeated games in order to analyze the optimal structure of a firm includes Baker, Gibbons, and Murphy [2002]. Baker et al. analyzed the optimal ownership structure of a firm when a downstream party uses a good produced by an upstream party. The ownership structure affects the outside options of the contracting parties, and as a result affects the level of cooperation among them. The difference between Baker et al. and my model is that my model analyzes horizontal relationships, while in Baker et al. the units are related vertically. My model relies on the fact that the units depend mutually on each other. 3 An Example of Interdependence Among Units This section will describe an example of interdependence among units in a firm that will apply to this model. Consider, for example, four of the subsidiaries of Amazon.com: Quidsi, Zappos, Woot, and A9.com. Quidsi is an online retailer for household items; Zappos is an online retailer that specializes in shoes; Woot is an online daily deals retailer; A9.com is an IT company that specializes in search technology, used by Amazon.com, some of its subsidiaries, and other online retailers, and in online advertisements. There are many potential sources for synergy among these subsidiaries. The three online retailers could coordinate prices, share shipping facilities and shipping contracts, advertise on each other s websites, and adopt user interface technology developed by one of the other companies. A9.com could adapt their software for the use of one or more of the retailers. It could also benefit from obtaining sales data from the retailers in order to improve its advertising. Part of these synergies can be easily realized using explicit agreements and transfer pricing. Other potential synergies are a result of a coordination problem, in which cooperation benefits all parties within the period, such as in taking advantage of economies of scale in order to reduce shipping costs. Finally, some synergies cannot be realized using the methods above, but could be realized by using the long-term relationships among the units. One of these synergies is user interface technology. For example, suppose that Quidsi is seeking to improve its search engine. An efficient way to do so may involve software developed by A9. One could argue that Quidsi should simply buy the software from A9. However, as 8

9 pointed out by Holmstrom and Tirole [1991], if the subsidiaries rely solely on transfer pricing for this transaction, some inefficiencies could occur. For example, A9 might not invest enough in relationship-specific capital, to increase the value of its product to the outside market, which would increase the price it could charge Quidsi. Contracting on the exact features of the product in all contingencies could be costly, which eliminates a detailed contract as a solution to the problem. In terms of the model, the accommodation required by Quidsi is to have Quidsi invest in a search engine that suits specifically the needs of Quidsi. It is important for this model that the relationship between A9 and the rest of the subsidiaries is not purely vertical, in the sense that some of the needs of A9 could be accommodated by other subsidiaries. An example of this would be sales data the retailers have that A9 could use to advance its online advertising business. Sales data could be useful for A9 to, for example, predict which consumers would be interested in a particular product. Using a contract and transfer pricing might not be possible in this case because of legal issues. The accommodation required here by A9 is to receive sales data from one or more of the retailers. This is costly to the retailers, but the benefit to A9 might exceed this cost. It should be noted here, that in this example, unlike with search engine improvement, it is likely that some of the subsidiaries would be substitutes for each other because if, for example, A9 is interested in kids shoes sales data, it could get them from either of the subsidiaries considered above. 4 The Model An entrepreneur seeks an optimal way to organize a firm. The firm consists of n production units. These are the most basic units of production, and cannot be further divided. However, the entrepreneur can choose whether the decisions will be made at the production unit level (decentralization), or at a higher level centralization. In the decentralized structure, each production unit is an independent division, and makes decisions on its own. In the centralized structure, each k production units are grouped together to form a profit center, and decisions are made at the profit center level. 9

10 The horizon is infinite. Time is discrete and is discounted by a common discount factor of δ. To focus the attention on cooperation within the firm, I assume that the value of the firm is driven solely by cooperation among the production units. Each period each of the production units develops some need. Unit i is able to accommodate a need of division j in a given period with probability p ij, the interdependence degree between production units i and j. 2 While accommodating this need is costly, the benefit to the requesting production unit exceeds this cost. The ability to accommodate a need is a public knowledge. However, the units cannot transfer money among them, so that market for these needs (i.e. transfer pricing) is unavailable. Also, the headquarters cannot directly motivate unit managers to accommodate these needs (e.g., the compensation contract of a unit manager cannot depend on whether the unit he manages had accommodated a certain need). 3 More precisely, I assume that period t profit of production unit i is: n (a tj,i b a ti,j c) (1) j=1 where a ti,j = 1 if production unit i accommodates a need of production unit j at period t. While it is possible that more than one production unit is able to accommodate some need, the benefit materializes at most once. That is, n j=1 a t i,j 1. b and c are constants that satisfy b > c > 0, and stand for the benefit from accommodating a need and its cost respectively. Accordingly, the total profit of the firm at period t is: n i=1 j=1 n (a ti,j b a ti,j c). (2) The profit of a profit center is the sum of the profits of its production units. I assume each production unit in a profit center maximizes the profit of the profit center it belongs to. 2 Different assumptions regarding the relations between the distribution of p ij across production units will be made in the following sections. 3 The setting is adopted from the trading favors literature ([Mobius, 2001], [Hopenhayn and Hauser, 2004]). However, this literature is mostly concerned with the case the opportunity to provide a favor is not a public knowledge, and consequently it is impossible to sustain a simple Nash reversion equilibrium. In this paper the opportunity to provide a favor is a public knowledge, but the trading favors setting is used to analyze cooperation in different firm structures. 10

11 In that I assume away free rider problems that bring about decentralization, but rather, any decentralization in this model can emerge only if it achieves better cooperation. 4.1 Discussion of Assumptions An important assumption is that profit centers cannot use transfer pricing to trade among themselves. One might argue that an intra-firm market would mitigate the cooperation problem, and will insure that the allocation is efficient. However, informational frictions may make the transaction costs too high, which in turn will prevent accommodations of some of the needs. Holmstrom and Tirole [1991] analyze a model in which the existence of an intra-firm market results in an inefficient allocation. Incentive problems arise in their model because of unobserved managerial investment in cost reduction. Their analysis distinguishes between the case in which profit centers are allowed to trade in outside markets and the case in which profit centers are not allowed to trade in outside markets. If profit centers are allowed to trade in outside markets, the market may monitor the quality of the product, but managers might invest in quality that is comparable to the market instead of relationship-specific investments. In addition, managers might invest in activities that enhance their outside options that may not be efficient otherwise. If the profit centers are not allowed to trade in outside markets, a profit center might not have sufficient incentive to invest in quality and cost reducing technology. Although the assumption in this model is that the profit centers cannot use pricing mechanisms, the model can be applied only for the needs, or the part of them, that cannot be accommodated otherwise, due to the reasons described above or due to any other inefficiency. That is, if it is assumed that the profit centers can use transfer pricing, there will be needs that are not accommodated in equilibrium because of different inefficiencies. These needs may still be accommodated if the profit centers utilize the long term relationships among themselves, as discussed in this paper. It has been also assumed that a firm cannot consist of a single profit center; a firm has to be divided into at least two profit centers. An analogous assumption is that the compensation 11

12 of profit center managers cannot be tied only to the performance of the entire firm; it must, at least partially, be tied to the performance of a smaller unit. It should be noted that without this assumption in place, full cooperation is always achieved, regardless of organizational structure. This assumption would be justified if in case all employees are compensated according to the performance of the entire firm, a free rider problem occurs. As a result, the profit center managers do not maximize the value of their firm. Although this is not specifically modeled, the underlying assumption is that there is a bound on the size of a profit center such that only if a profit center is larger than this bound a free rider problem occurs. One can think of a situation in which a manager is able to monitor the actions of a group of a particular size, but is unable to monitor the actions of a larger group. This assumption is most appropriate for large firms. If a firm is small enough such that tying the compensation of all employees to the performance of the firm ensures perfect cooperation, the model is not relevant. The model is more relevant, however, for larger firms in which free riding might exist. The fact that firms compensate their employees according to performance measures of units that are smaller than the entire firm supports this assumption. Firms use profit centers, cost centers, expense centers, and investment centers in order to evaluate the performance of its units. Many firms use these measures in order to compensate their employees. For example, Briggs & Stratton adopted in 1990 a system in which 40% of the bonuses of managers are based on a measure of divisional performance, 40% on a measure of corporate performance, and 20% on a measure of personal performance. The paper analyzes different assumptions regarding the interdependence across units. Since the general case is complicated to analyze, specific assumptions are made. However, in order for the framework to be valid, it is sufficient that the interdependence among production units is bilateral in the sense that each profit center may require a need that some other unit may be able to accommodate. This does not require that the same two units will be able to accommodate each other s needs, but that each division can be both the recipient and the sender of some 12

13 needs. The framework does not apply in case that the relationships among units are purely vertical, as in a case in which a unit is able to accommodate another unit needs but cannot benefit from cooperation with other units. 5 The Optimal Firm Structure in Case the Interdependence Degrees Are Independent This section analyzes the optimal structure of a firm in the case in which the interdependence level is identically and independently distributed across production units. The firms are identical ex ante in their relations to each other, and the ability of one division to accommodate needs of other units does not increase nor decrease the probability any other division is able to accommodate that need as well. While this assumption is extreme, its purpose is to illustrate the optimal structure of a firm when units are interchangeable with each other in some of their capacities. That could be the case if the firm is organized in an M-form and units produce goods that compete in the same market. For example, suppose that Opel subsidiary of GM would like to enter the North American market, and requires some accommodation from a car manufacturer. This could be, for example, withholding of production for some time in order to allow for the entry. In this case, Buick and Chevrolet could have similar prospects in accommodating the requests of Opel. I consider a range of feasible firm structures. The firm structures are characterized by the number of production units in each profit center, which reflects the level of centralization: Definition 1. Firm structure A is called more centralized than firm structure B if the number of production units in each profit center in firm structure A is larger than the number of production units in each profit center in firm structure B. In the most decentralized firm structure, each production unit forms a separate profit center, and accordingly maximizes only its own profits. In this case, no needs are accommodated unless cooperation among the profit centers is sustained. As a result, if cooperation among the profit 13

14 centers cannot be sustained, the value of the firm is zero. In more centralized firm structures, more than one production unit forms a profit center. All decisions are made at the profit center level. It is assume that the profit center manager is able to enforce costlessly any action that is taken within his profit center. In the centralized firm structure, even if there is no cooperation among profit centers, each profit center may accommodate the needs of its own production units. This is due to the fact that the profit center manager maximizes the sum of the profits of its production units. If a production unit is able to accommodate a need of another member of the profit center, accommodating it increases the profit of the profit center directly. However, the profit centers may be able sustain cooperation among themselves, and accommodate each other s needs. It is assumed that there is a priority in accommodating needs within the profit center, such that a profit center may ask another profit center to accommodate a need only if it cannot accommodate it on its own. I analyze a grim trigger strategy for this game. In the cooperative phase, profit centers request the other profit centers to accommodate needs that cannot be accommodated within the profit center. A profit center accommodates all requested needs. Play remains in the cooperative phase as long as none of the profit centers has deviated. If a profit center has deviated, play switches to a punishment phase forever. In the punishment phase, none of the profit centers offer to accommodate needs of other profit centers. 5.1 Profit Centers of Equal Size I first restrict the analysis to profit centers of equal size. The number of production units in the firm, together with other restrictions, then determine the feasible number of production units in a firm. First, by assumption, a profit center cannot consist of more than n 2 production units. The set of feasible number of production units in a profit center is further restricted to be a divisor of the number of production units in the firm. Finally, the analysis allows for additional restrictions on the number of production units in a firm. The following definitions will be useful in order to characterize the optimal organizational 14

15 design: Definition 2. K {k 1, k 2,..., k s } where k 1 < k 2 <... < k s is a sequence that specifies the set of all feasible number of production units in a profit center. Definition 3. Denote by K x = {k i k i 1 < x, k i x}. Similarly, K x = {k i k i x, k i+1 > x}. The following proposition characterizes the optimal organizational design. Proposition 1. Assume the discount factor δ is large enough such that it is possible to sustain full cooperation in some organizational design. 4 The interdependence degree determines the optimal number of production units forming a profit center. The optimal number of production { } units in a profit center is given by k ln (p) max 1 2,. For large enough K ln(1 p) K ln 1 2 ln(1 p) increases in p, 5 the optimal number of units in a profit center k (p) decreases in p, the interdependence degree. The optimal number of production units in each profit center is independent of the number of production units in a firm. The optimal number of profit centers increases with the number of production units in a firm. The proof appears in appendix A. The proposition states the optimal firm structure for cooperation. The criteria used to judge among the possible structures is the lowest discount factor for a fixed set of parameters that is required in order to sustain cooperation. The minimal discount factor that is required in order to sustain cooperation depends on the cost of accommodating a need, the reward, the interdependence level, and finally the number of units in a profit center. The optimal number of units that encompasses a profit center is determined by choosing the number of units in a profit 4 The discount factor should satisfy δ > arg max k K c (1 p) k 1 (1 (1 p) k )(b c)+c. If the discount factor is below this value, the optimal organizational design is the most centralized possible, that is, each profit center encompasses of k s production units. In this structure, the highest degree of partial cooperation can be sustained. 5 A sufficient condition that guarantees that if p 2 > p 1 k (p 2 ) > k ln 2 (p 1 ), is 1 ln 2 K 1 ln(1 p 1 ) K. ln(1 p 2 ) For a small difference between p 1, p 2 it could be that p 1 > p 2 and k (p 1 ) = and k (p 2 ) =, and accordingly k (p 1 ) > k (p 2 ). ln 1 2 ln(1 p) ln 1 2 ln(1 p) 15

16 center that minimizes the discount factor for a fixed cost of accommodating a need, reward of accommodating a need, and interdependence level. The choice of lowest discount factor that sustains cooperation pertains to a situation where the entrepreneur faces some uncertainty regarding the discount factor at the point where he chooses the firm structure. In this case he will prefer the structure that requires the lowest discount factor possible 6,7. If the discount factor is too low it is impossible to sustain full cooperation in any organizational design. In that case, the optimal firm structure is that which yields the highest level of cooperation in a static model. The highest level of cooperation in a static model, as shown by Chandler [1977] and others, can be obtained in the most centralized design possible, namely when there are two profit centers of equal size. In this case, the production units in each profit center cooperate among themselves, but there is no cooperation between the profit centers. The loss in welfare relative to the case of full cooperation is that there may be needs that cannot be accommodated within a profit center, but can be accommodated by one of the units in a profit center. Once the discount factor is large enough such that it is possible to sustain cooperation in some organizational design, the optimal organizational design depends on the interdependence degree p. The higher is the independence level, the fewer units form a profit center, and the organizational structure is more decentralized. The optimal number of production units in a profit center does not depend on the total number of production units in a firm. Empirical evidence about the relation between interdependence and organizational structure is rare because of both the challenge in measuring interdependence and the scarcity of data on organizational structure. One paper that examines this relationship is Aiken and Hage [1968].Aiken and Hage find in a sample of health and welfare organizations from the 60 s that interdependence, approximated by the number of joint programs, is associated with decentralization. 6 Halonen [2002] uses a similar criterion to judge among possible ownership structures. 7 Identical results are obtained if the criterion is minimal highest cost of accommodating a need that can support cooperation, or maximal lowest reward of an accommodated need. 16

17 In order to understand the reason for the differences in the optimal firm structure that are generated by the interdependence degree, consider the two extremes: when the interdependence degree is very high and when the interdependence degree is very low. When comparing the costs and rewards of conforming to the cooperation strategy for a profit center, it is convenient to compare the average cost and average reward per production unit. Although the profit center compares the total reward and total cost, comparing the average cost and reward per production unit is an equivalent and more intuitive way to compare the two. First consider the case the interdependence among production units is almost perfect, that is the probability a production unit is able to accommodate a need of some other production unit approaches 1. The expected cost of conforming to the cooperation strategy for a profit center that is able to accommodate the needs of all other production units equals to ck if the structure is such that each profit center consists of k production units. The reason is that each of the other profit centers are very likely to be able to accommodate that need as well. Since there are a total of n k k profit centers, and for each given need the probability that each of the profit centers is assigned to it is equal across profit centers, the probability a particular profit center is selected to accommodate a given need is k n k. When considering the average cost per production unit it is c independently of the size of a profit center. Notice here that although the probability a profit center is requested to accommodate needs of other profit centers approaches 0, in order to sustain cooperation a profit center must be willing to bear this cost once requested. The reward from cooperation, however, varies greatly with the organizational structure. In any organizational structure in which a profit center consists of more than one production units, the reward from cooperation approaches zero, since all needs are more likely accommodated within the profit center. It further approaches zero faster the more centralized is the organizational structure. In the most decentralized structure, i.e. that in which each profit center consists of a single production unit, the reward of each production unit approaches b, since the probability a requested need is accommodated approaches 1. Clearly, when the interdependence degree is very high, the higher the centralization, the harder is sustaining cooperation. 17

18 Next consider the case there is a very low level of interdependence among production units, that is the probability a production unit is able to accommodate a need of some other production unit approaches 0. The average reward per unit in a profit center from cooperation approaches 0 in all firm structures. The cost of conforming to the cooperation strategy for a production unit that is able to accommodate the maximal number of needs if there are k production units in each profit center approaches (n k)c. This is because if it offers to accommodate all needs it is likely to accommodate all of these, since the probability other production units are able to accommodate them as well is low. This cost is an average of (n k)c k per production unit. As the number of production units in a profit center k increases, the average cost per production unit decreases. That is, the average reward from cooperation does not change with the organizational structure, but the average cost decreases in the number of production units in a profit center. As a result, the more centralized is the organizational structure, the easier it is to sustain cooperation when the interdependence degree approaches 0. Another feature of the optimal firm structure is that the number of production units in a profit center is independent of the total number of production units in a firm n. The number of production units in a firm may restrict the number of feasible number of production units in a firm. However, holding the number of feasible number of production units in a firm constant, the number of production units in a profit center does not change with the number of units in a firm. An increase in the number of production units has two opposing effects on the incentive to cooperate. First, it increases the continuation value of cooperation. The higher the number of production units, the higher probability a unit s needs are accommodated in the cooperation phase, and accordingly the continuation value of cooperation increases. Second, an increase in the number of production units in a firm increases the expected number of needs the binding profit center is required to accommodate in equilibrium. These two numbers are proportional to the probability that at least one of the profit centers can accommodate a given need. That is, an increase in the number of units in a firm affects both the cost and the reward from cooperation similarly, and as a result cooperation incentives do not change with the number of production 18

19 units in a firm. A consequence of the independence of the number of production units in a profit center in the total number of production units in a firm n is that, holding everything else constant, the number of profit centers in a firm increases in the number of production units in a firm. The reason is that the number of profit centers under the assumption of profit centers of equal size is the number of production units divided by the size of a profit center. Since the size of a profit center does not change with the number of production units in a firm, the number of profit centers must increase in the number of production units in a firm. This proposition has a few implications. First, the model predicts that the higher the interdependence among production units in a firm the lower the decentralization in the sense of a lower number of production units in each firm. Second, the model predicts that the size of a profit center does not depend on the size of the firm, however, holding the interdependence among production units constant, the number of profit centers increases in the size of the firm. The following corollary is useful in case the interdependence is not observable. Corollary 1. Decentralization is associated with higher profits. Since a higher degree of interdependence increases profits, and interdependence is associated with decentralization according to the proposition, decentralization is associated with higher profits. 5.2 Profit Centers of Variable Size I now extend the analysis to profit centers of variable size. Profit centers are not restricted to be of the same size, but can rather take different sizes. Since this extension makes the problem cumbersome to solve for the general case, it will be solved for an example. In this example, there are four production units. The entrepreneur can choose among the following partitions: in the symmetrical centralized structure each two units form a profit center, in the symmetrical decentralized structure each unit forms a profit center, and in the asymmetrical structure there are two profit centers, one that encompasses of one unit, and 19

20 the other encompasses of three units. The following proposition characterizes the firm structure in that case: Proposition 2. In the case there are four production units, an asymmetrical assignment is never optimal. As a result, allowing for profit centers of variable sizes does not change the optimal firm structure. 6 Alternative Interdependence Relations Section 5 has analyzed the case the degrees of interdependence are identical and independent of each other. The purpose of this section is to analyze how changes in this assumption affect the optimal firm structure. One can think of two ways in which the relationships among units could be different from this assumption. One is that the interdependence degree is not symmetrical across units, that is, some units are more likely to accommodate needs of all or part of the other units. Another way in which the potential synergies could be different is that there is a correlation in the interdependence degree, positive or negative. If the ability of one division to accommodate some need of another division increases the chance other units will be able to do so as well, there is a positive correlation in the interdependence degree among units. If, however, the ability of one division to accommodate some need reduces the chance other units will be able to accommodate the same need, there is a negative correlation in the interdependence degree among units. Each of the assumptions made in this section is extreme, and shows how the optimal firm structure is affected by changes in these directions. Most of the interdependencies among units in a firm, however, would likely follow a mix of all assumptions made. In the example of Amazon.com s subsidiaries described above, for instance, it is likely that only A9 would be able to accommodate software needs of the retailers. As a result, A9 might be more likely than the rest of the subsidiaries considered to be able to accommodate needs of the retailers. When considering the sales data needs of A9, however, all retailers considered have similar abilities in being able to provide sales data to A9, which could depend on the market A9 would be interested 20

21 in at a certain period. Since some markets are common to all or a subset of the retailers, and some are not, it is not clear if the ability of one of the retailers to provide some data increases the chance other retailers would be able to provide similar data or not. Proposition 3 describes the optimal firm structure in the following cases. First, the dominant unit case assumes there is one unit that can accommodate all needs with certainty, while all other units can accommodate a need with a positive equal probability. This is an extreme case of asymmetry among the units. Second, the perfect negative correlation case assumes that at most one unit can accommodate a certain need. This assumption would be appropriate if each unit is very different from the other. It is assumed that the probability each unit is able to accommodate some need is symmetric across units 8. Finally, in the perfect positive correlation case, if one unit is able to accommodate some need all other units are able to accommodate that need as well. Proposition 3. Assume the discount factor δ is large enough such that it is possible to sustain cooperation in some organizational design. The optimal firm structure is: 1. The most decentralized structure possible in the dominant unit case. 2. The most centralized structure possible in the perfect negative correlation case. 3. The centralized firm structure in the perfect positive correlation case. The proof is in appendix B. In the dominant unit case, the more decentralized is the firm structure, the better the cooperation that can be achieved. The reason is that the dominant unit accommodates all needs of its profit center, other than its own need. The more units there are in a profit center, the lower the probability the profit center that includes the dominant unit requests that need, as it is more likely to be able to accommodate it on its own. At the same time, the cost of cooperation increases in the number of units in a profit center because there are fewer profit centers, and hence the probability it is selected to accommodate a need is higher. 8 Since at most one unit is able to accommodate a need, it has to be assumed that p 1 n 1. 21

22 In the perfect negative correlation case, the more centralized is the firm structure, the better the cooperation that can be achieved. I conjecture from this result that in order for the decentralized structure to achieve better cooperation, at least some of the units must be substitutes for each other in at least some of the cases. In the perfect positive correlation case, partial and full cooperation achieve the same level of cooperation. The reason is that once two units cooperate, adding more units to the alliance cannot increase the number of needs that are accommodated, as the additional units can only accommodate needs that are already accommodated. As a result, centralized firm structure is at least weakly better than the decentralized firm structure. It should be noted here, however, that in this case there is no advantage for more than two units to form a firm together, as the addition of another unit does not increase the total profits of the firm even if full cooperation can be achieved. 7 The Value of a Firm as a Function of the Interdependence Degree among its Production Units The purpose of this section is to analyze the effect of the interdependence among the units in a firm on its value. So far it has been shown that the level of interdependence among the units in a firm affects the optimal structure of a firm. It is interesting, then, to analyze how does the value of a firm responds to the level of interdependence among units. As expected, the value of a firm increases in the interdependence degree, since then there is a better opportunity for synergy among the units. However, the rate of increase in the value of the firm depends on the level of interdependence. Figure 1 illustrates the relations between the value of a firm and the level of interdependence among its units in the case the ability to accommodate needs is identically and independently distributed. When the interdependence degree is very low, full cooperation can not be sustained in any of the structures considered. The reason is that a unit that is able to accommodate a need 22

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