Surplus division and investment incentives in supply chains: A biform-game analysis

European Journal of Operational Research 234 (2014) 763–773

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Decision Support

Surplus division and investment incentives in supply chains: A biform-game analysis Eberhard Feess, Jörn-Henrik Thun ⇑ Frankfurt School of Finance and Management, Sonnemannstraße 9-11, D-60385 Frankfurt, Germany

a r t i c l e

i n f o

Article history: Received 6 May 2013 Accepted 21 September 2013 Available online 7 October 2013 Keywords: Supply chain management Shapley value Biform game Underinvestment problem Incentive system Subsidies

a b s t r a c t In this paper, we use a biform-game approach for analyzing the impact of surplus division in supply chains on investment incentives. In the first stage of the game, firms decide non-cooperatively on investments. In the second stage, the surplus is shared according to the Shapley value. We find that all firms have inefficiently low investment incentives which, however, depend on their position in the supply chain. Cross-subsidies for investment costs can mitigate, but not eliminate the underinvestment problem. Vertical integration between at least some firms.yields efficient investments, but may nevertheless reduce the aggregated payoff of the firms. We show how the size of our effects depends on the structure of the supply chain and the efficiency of the investment technology. Various extensions demonstrate that our results are qualitatively robust. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Effective supply chain management can yield various benefits such as lower inventories by avoiding the well-known Bullwhip effect (Forrester, 1958; Lee et al., 1997; Größler et al., 2008). Consequently, supply chains are often defined as agreements for maximizing the joint surplus of the participating firms (Chopra and Meindl, 2001;Mentzer et al., 2001). In this paper, we analyze how the surplus division in supply chains influences the incentives for investments which improve the supply chain’s efficiency. As we find that investment incentives are inefficiently low, we discuss two possibilities to reduce this inefficiency, subsidies for the investments of other firms and costly binding contracts on investment levels. Recently, the so-called biform-approach which combines elements from non-cooperative and cooperative game theory (Brandenburger and Stuart, 2007) bas been applied to several problems in supply chain management (see the literature review in Section 2).1 We follow this approach by distinguishing two stages in the process of establishing a supply chain project. In the first stage, firms can make investments which increase the efficiency of the supply chain. For instance, a manufacturer might improve its IT system which saves costs for coordinating the information flow ⇑ Corresponding author. Tel.: +49 69 154 008 795. E-mail addresses: [email protected] (E. Feess), [email protected] (J.-H. Thun). In their paper on biform-games, Brandenburger and Stuart (2007) use the core rather than the Shapley value as solution concept for the cooperative part of their game. We use the term ‘‘biform game’’ more generally for all games combining noncooperative and cooperative stages. 1

0377-2217/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2013.09.039

with a supplier and a wholesaler, or a retailer may install an online order-entry system for the end consumer which improves the value of the supply chain. As these investments often take place before an agreement on the supply chain level is reached, we assume that they are made non-cooperatively. In the second stage, the supply chain project is conducted. For this stage, we apply cooperative game theory and assume that the surplus is shared according to the most widely accepted cooperative solution concept for more than two players, the Shapley value (Shapley, 1953). As a motivating example, we use the simplest straight supply chain often observed in practice which consists of a supplier, a manufacturer, a wholesaler and a retailer. Because of their positions in the supply chain, we refer to the manufacturer and the wholesaler as center firms, and to the supplier and the retailer as brink firms. We find that the Shapley value assigns less of the surplus to brink firms, and we show how this reduces their investment incentives in the first stage of the game. We believe that biform games are appropriate for our research question on the impact of surplus division on investment incentives as firms may often reach (approximately) efficient solutions after supply chains have been formed (ex post-perspective). Thus, cooperative game theory can be applied for the second stage of the game. By contrast, coordinating on ex ante-investments which maximize the overall future value of the supply chain is more difficult due to transaction costs (Williamson, 1975).2 Therefore, non-cooperative game theory is adequate for the first stage. 2 Most prominently, biform games are used in incomplete contract theory (Hart and Moore, 1999; Tirole, 1999). Incomplete contract theory assumes that binding contracts for investments are infeasible, but that the parties agree on the efficient solution and share the surplus cooperatively after the investments have been made.

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Our main findings can be summarized as follows: First and as already mentioned, we find that the surplus will be split asymmetrically among the firms involved, with brink firms getting less of the surplus. Second, we show that not only brink firms but all firms will make inefficiently low investments where our efficiency benchmarks are the investment levels which maximize the joint surplus of the participating firms. This result can be attributed to non-internalized positive externalities on other firms and is qualitatively robust with respect to the specification of the investment technology. Given this underinvestment result, we analyze wehther firms can increase their investment incentives by subsidies or vertical integration. For subsidies, we add an additional non-cooperative stage to the game where each firm can subsidize the investments of other firms. For instance, a wholesaler who knows that he might benefit from a manufacturer’s improved IT system in a supply chain may cover part of the cost of the new system.3 Our third result is that subsidies mitigate, but do not fully eliminate the underinvestment problem. We then discuss vertical integration by assuming that any two directly linked firms can integrate at some costs. Integration improves investment incentives as it eliminates the externality problem between any two integrating firms. Strikingly, however, incentives for vertical integration are, compared to our efficiency benchmark which is the aggregated payoff of all firms, excessively high whenever the value of the supply chain consisting of all firms is relatively low compared to the value of smaller supply chains. We provide an intuition for our finding which shows that the result is robust with respect to different ways of modeling the supply chain. The assumptions of our basic model allow a streamlined analysis, but some of them are restrictive from a practical point of view. We hence discuss several extensions with respect to the investment technologies and the value of different coalitions, and we show that our results are qualitatively robust. The remainder of the paper is organized as follows: Section 2 relates to the literature. Section 3 presents the model. Section 4 applies the Shapley value as a solution concept for surplus division, and derives investment incentives. Section 5 discusses subsidies as potential remedies for the underinvestment problem. Section 6 proceeds to vertical integration. Section 7 discusses the robustness of our results with respect to various extensions. We conclude in Section 8.

2. Related literature In recent years, game theory has gained importance for analyzing effective supply chain management. Many papers apply either non-cooperative (Cachon and Netessine, 2004;Nagarajan and Sošic´, 2008) or cooperative solution concepts (Leng and Parlar, 2005). For cooperative solution concepts, the Shapley value dominates. Raghunathan (2003) uses the Shapley value for analyzing information sharing among a manufacturer and several retailers in a supply chain. As in our paper, he shows that the role in the supply chain influences the surplus division and the incentives of forming supply chains. Accordingly, Leng and Parlar (2009) analyze the division of cost savings from sharing demand information. They compare different solution concepts of cooperative game theory, including the Shapley value. Rosenthal (2008) uses the Shapley value to determine transfer prices for intermediate products in a vertically integrated supply chain. Kemahliog˘lu-Ziya and Bartholdi (2011) use the Shapley value to allocate the expected excess profit 3

Of course, such an agreement implies some kind of enforceable contract, but is still less challenging than contracting on investment levels.

generated by inventory pooling in supply chains among a supplier and his retailers. All of these papers, however, do not extend to incentive problems which requires stages where decisions are taken non-cooperatively. As mentioned in the introduction, our approach is most closely related to biform games which apply cooperative and non-cooperative solution concepts in different stages of the game. Anupindi et al. (2001) use such a biform-approach for a game with multiple retailers. First, each retailer decides non-cooperatively on his stocking decision. Then, the retailers observe demand and decide cooperatively on how much inventory to transship among locations in order to better match supply and demand. For this decision and for the profit allocation, they use the core (Gillies, 1959) as solution concept. Contrary to this, we apply the Shapley value, and we introduce assumptions ensuring that the Shapley value is in the core. By extending the approach of Anupindi et al. (2001),Granot and Sošic´ (2003) allow retailers to hold back the residual inventory. Their model consists of three stages, the inventory procurement which is done non-cooperatively, the decision about how much inventory to share with others, and the transshipment stage (cooperative stages). The non-cooperative stage corresponds to the investment decision in the first stage of our model, whereby the cooperative stage equals the division of surplus in the second step of our model. Taylor and Plambeck (2007a), Taylor and Plambeck, 2007b analyze games between two firms who might pool their capacity and investments to maximize the overall value of the supply chain. As in our model, firms first decide non-cooperatively on their investments, and then bargain cooperatively over the division of the market and the respective profits. Contrary to this paper, we focus explicitly on the supply chain structure taking the respective position of firms into account so that the impact of the position on investment incentives can be analyzed. Chatain and Zemsky (2007) use a biform game for considering the advantages of coordinating suppliers, the optimal level of buyer power, and the desirability of subsidizing suppliers. Leng and Zhu (2009) discuss subsidies as side-payments which are potentially required when allocating the surplus. They provide a comprehensive literature review on different types of side payments. However, these side payments are transfers in supply chains for reaching the grand coalition, and as such are by definition of cooperative game theory assumed to be always feasible. This is different to the subsidies in our paper which are non-cooperative payments taken for influencing the investment decisions before the supply chain formation. More generally, our paper is related to biform games analyzing the impact of the surplus division on business decisions such as investments, advertising or mergers. The common feature of the literature is that firms first decide on how to ‘‘shape’’ the competetive environment which then defines the playing field for cooperative bargaining. Biform-games allow to account for two motives of business strategies, increasing efficiency on the one hand, and improving the own bargaining position on the other hand (see Brandenburger and Stuart, 2007).4 While we assume that the surplus is shared according to the Shapley value, some of these papers including Inderst and Wey (2003) and DeFontenay and Gans (2005) define a non-cooperative bargaining structure which yields the Shapley division as a Nash Equilibrium of the non-cooperative bargaining game. As we do in Section 6, both papers compare private and social incentives for integration. While Inderst and Wey (2003) discuss incentives for 4 In our analysis of the incentives for vertical integration, these two motives can clearly be disentangled and determine whether, from an overall efficiency perspective, there are over- or underincentives for vertical integration.

E. Feess, J.-H. Thun / European Journal of Operational Research 234 (2014) 763–773

horizontal integration of firms in a bilateral oligopoly of two suppliers and two retailers, DeFontenay and Gans (2005) show that the impact of vertical integration on the surplus division depends on the degree of upstream competition. DeFontenay and Gans (2008) analyze how a downstream firm’s decision whether to outsource an activity to an independent firm or to an established upstream firm which already owns another upstream asset is affected by both efficiency effects and the impact on the bargaining position. All of these papers are related to our approach as they consider the impact of the surplus division (cooperative stage) on preceding decisions (non-cooperative stage). The specific questions discussed in our paper as outlined in the introduction seem are novel. 3. The model We consider a game where firms agree on conducting a supply chain project. Before doing so, each firm can make a supply chainspecific investment in order to increase the supply chain’s efficiency. Following the general literature, we will also refer to these investments as relationship-specific to express that they are worthless outside of the supply chain.5 Investments are made non-cooperatively, that is, each firm maximizes its own payoff when deciding upon its investment level. Given these investment levels, the supply chain with the highest overall surplus is formed. By assumption, this is a supply chain consisting of all four firms (see below). The surplus is divided according to the Shapley value. Our model is thus a biform game which includes a non-cooperative stage (the investment stage) and a cooperative stage (the formation of the supply chain and the surplus division). We first describe the supply chain. Then we turn to the investment technology. 3.1. Structure of the supply chain In our model, four firms, i = 1, 2, 3, 4, can create a supply chain similar to a straight chain analyzed by Forrester (1958) in his seminal work on the Bullwhip effect. In our model, one might think of a supplier (1), a manufacturer (2), a wholesaler (3) and a retailer (4). For producing a certain good, the manufacturer purchases parts from the supplier. After the final component assembly, the good is delivered to the wholesaler who distributes it to the retailer. This simple setting is sufficient to make our points, but our main insights are qualitatively robust with respect to all supply chains where the Shapley value depends on the position in the chain, and where at least part of the investments is relationship-specific. Due to the information and material flow, we assume that only adjacent firms can conduct supply chain projects. Thus, supply chain structures with missing links, such as {1, 3, 4} are excluded. For many applications, this assumption is reasonable. For instance, suppliers and wholesalers cannot conduct supply chain projects effectively without manufacturers. Similarly, a coalition of the supplier and the retailer cannot improve the material flow if the manufacturer and the wholesaler do not participate. By contrast, supply chain projects involving only suppliers and manufacturers are often observed for implementing lean management practices such as just-in-time or just-in-sequences, for instance. As supply chains without direct links are excluded, the set U of feasible supply chain structures is

U ¼ ðf1; 2g; f2; 3g; f3; 4g; f1; 2; 3g; f2; 3; 4g; f1; 2; 3; 4gÞ:

5 In reality, a certain percentage of the return on investment may also be realized outside of the supply chain. As long as this percentage is below 100%, our results are qualitatively robust.

765

We adopt cooperative game theory for the surplus division, and we will refer to the firms participating in feasible supply chain projects as coalitions. As we are interested in the impact of the set of feasible coalitions on the surplus division in the whole supply chain, we assume that the supply chain including all four firms (the grand coalition N) is efficient. Following cooperative game theory, this implies that the grand coalition will be formed. For the grand coalition, we will refer to firms 1 and 4 as brink firms, and to firms 2 and 3 as center firms. In our basic model, we assume that all equally large supply chains yield identical surplus, and we denote the surplus of a supply chain with two firms by a. All supply chains with three firms yield surplus b, and the grand coalition yields surplus c. We assume that the surplus is increasing in the number of the participating firms, c > b > a. Furthermore, we assume c > max (b, 2a) to ensure that the grand coalition is efficient. Without loss of generality, we normalize each firm’s profit if no supply chain is formed to zero. The Shapley value is only convincing as a solution concept if the resulting payoffs are in the core, i.e. if no coalition exists that can block the efficient coalition. In addition, we therefore introduce the following Assumption: Assumption 1. c > 20 17 b. In Appendix 6, we prove that this condition is sufficient, albeit not necessary to ensure that the Shapley value is in the core. We also show in the Appendix that Assumption 1 is compatible with all results derived in our paper.6 Assuming in our basic model that all equally large supply chains yield identical surplus allows us to make our points clearly laid out. In reality, however, equally large supply chains may yield different surplus. For instance, a supply chain consisting of the supplier and the manufacturer might well create different efficiency gains compared to a supply chain consisting of the manufacturer and the wholesaler. Consequently, we consider the case where equally large supply chains yield different surplus in an extension (see Section 7.2). Our insights are qualitatively robust with respect to this generalization. Forming supply chains will often require transaction costs. For the sake of simplicity, we assume that the values a, b and c as defined above already include these transaction costs, so that the grand coalition is indeed efficient. For instance, if the gross surplus of a chain consisting of three firms is given by b b, and if forming the chain requires transactions costs of tb, then b  b b  tb . 3.2. Investments We refer to the values for the supply chains introduced above (a, b and c) as base values. In addition to these base values, firms can enhance the value of supply chains by relation-specific investments. For instance, the manufacturer can invest in an IT-project which improves the service interface with the supplier and the wholesaler. While the investment is made by the manufacturer, other firms may benefit as well since the manufacturer’s improved IT may also reduce the supplier’s and the retailer’s personnel required for the process coordination. Similarly, the other firms in the supply chain benefit if the supplier implements a radio-frequency identification system (RFID system) to improve its inventory control, when the wholesaler implements a track&trace application, and when the retailer provides an online order-entry system for the end consumer in order to improve the overall value of the supply chain. In our basic model, we consider the simplest case where the investment functions of the four firms are symmetric and given by 6 All Appendices are provided in an online Appendix and also available on request from the authors.

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RðIi Þ ¼ 2aI2i 8i

ð1Þ

with Ii as firm i’s investment. The return on this investment, R(Ii) occurs if and only if a supply chain including firm iis formed. Note carefully that R(Ii) is the return on firm i’s investment for the whole supply chain, and not the part of the return reaped by the investing firm i itself. In our example from above, R(I2) is the sum of the cost savings for all firms in the supply chain due to the manufacturer’s investment.7 The Shapley analysis will show that the manufacturer itself can only appropriate part of this surplus, and this reflects the common idea of externalities in relationship-specific investments. Some additional remarks are in order with respect to our assumptions on the investment functions: First, we assume diminishing marginal returns. This is not only required for an interior solution for the efficient investment levels, but also straightforward: With increasing marginal returns, the optimal investment levels would be infinitely high, and they would be indeterminate for constant marginal returns. Second, assuming that all firms have symmetric technologies allows us to conveniently isolate the effects caused by the asymmetric positions of brink and center firms in the supply chain from the consequences of asymmetries in technologies. In reality, however, investment technologies may often be asymmetric. We therefore analyze the case with different technologies in Section 7.1, and we find that results are qualitatively robust. Third, the firms’ investment technologies are additively separable. This is sometimes, but not always realistic. For instance, if the manufacturer invests in an IT-concept and the retailer in a management in-house training, then the returns on investments are likely to be additively separable. When both invest in similar ITprocesses, however, then one would expect complementaries, so that the cross-partial derivatives of the return functions should be positive. We briefly discuss the case with complementary investments in the end of Section 7.1. While the results are likely to be robust, we do not fully investigate this setting as the model becomes rather convoluted for reasons well-known from the literature. Fourth, recall that we assume that R(Ii) is realized whenever firm i participates in a supply chain; irrespective of the supply chain’s size. One might reasonably argue that, in reality, the return on investment will often be increasing in the number of participating firms. For instance, the overall return on the manufacturer’s ITinvestment may be higher when not only the supplier, but also the wholesaler and the retailer join the supply chain. We do not cover this case in our paper, but from the discussion of the other settings in Section 7.1, it will become clear that our results are robust with respect to this extension. We denote the vector of all investments as I  (I1, I2, I3, I4), and the sum of all investment costs by I  R4i¼1 Ii . Accordingly, we define R  (R(I1), R(I2), R(I3), R(I4)) and R  R4i¼1 RðIi Þ. For notational convenience in the tables presented below, we use Ri instead of R(Ii) for short. The overall value V(I) of the grand coalition consists of the base value c as defined above and the value created through investments:

VðIÞ ¼ c þ R  I:

ð2Þ

The investment levels which maximize the overall value of the grand coalition are implicitly given by the first order conditions

@VðIÞ 1 ¼ aIi 2  1 ¼ 08i: @Ii

ð3Þ

We refer to these investment levels as efficient, and we denote all efficient values by the superscript e. We immediately get 7

Recall that firm 2 is the manufacturer.

Iei ¼ a2 8i;

Rei ¼ 2a2 8i;

Rei  Iei ¼ a2 8i;

V e ðIe Þ ¼ c þ 4a2 :

ð4Þ

By contrast to the efficient levels which maximize the surplus of the whole supply chain, we refer to the investment levels which maximize a firm’s own payoff as payoff-maximizing. 3.3. Structure of the game We can now summarize our biform game as follows: In stage 1, each firm decides non-cooperatively on its payoff-maximizing investment level. In stage 2, the efficient supply chain consisting of all four firms is formed, and the surplus is divided according to the Shapley value. Below, we extend the model by including a stage in which firms can decide on subsidies for other firms’ investments (Section 5) or where they can agree on (costly) binding contracts on investments (Section 6). 4. Surplus division and investment incentives 4.1. Stage 2. Surplus division When firms invest in stage 1, they anticipate how the surplus will be divided in the supply chain formed in stage 2. Hence, we need to solve for the Subgame Perfect Equilibrium (SPE) by backwards induction. We start with the surplus division for any investment vector I given. As discussed in the introduction, we adopt the most widely accepted concept of cooperative game theory for the surplus division in games with more than two participants, the Shapley value. As any cooperative concept of game theory, the Shapley value assumes that the efficient outcome is reached. In our model, this is by assumption the supply chain consisting of all four firms. Each player’s share depends on the surplus of the efficient coalition, compared to the surplus of all coalitions that can be formed without him.8 With N players, define the surplus of a supply chain including the player subset k 2 N as v(k). The value added by player i to a coalition k is v(k)  v(k  i). The Shapley value for player i, Si, is then calculated as the sum of these values over all permutations, divided by the overall number of possible permutations:

Si ¼ Rk2N

ðk  1Þ!ðN  kÞ! ½v ðkÞ  v ðk  iÞ: N!

ð5Þ

Applying the Shapley value to our supply chain yields the results summarized in the first column of Table 1. These payoffs are derived in Appendix 1. Note carefully that investment costs do not enter the Shapley values as they are sunk when the supply chain is formed. Thus, they play no role for the surplus division. The next two columns are the equilibrium values for the investments, returns and payoffs and will be derived later. First, we discuss the results in column 1. Of course, the Shapley values for the two brink firms 1 and 4, and for the two center firms 2 and 3, respectively, are identical. Considering only the base values (that is, neglecting the shares in the returns on investment for a moment), center firms get a greater part of the surplus, S2  S1 ¼ S3  S4 ¼ b  12 a > 0. The reason is that supply chains with three firms can only be formed with both center firms, while either firm 1 or firm 4 is dispensable. This strengthens the positions of firms 2 and 3 in the bargaining process and is reflected in the Shapley value. The asymmetry between brink and center firms will drive many of our results. 8 The Shapley value is based on an axiomatic framework (see Myerson, 1991 for a detailed explanation) consisting of the four axioms Pareto efficiency (which ensures in our model that the grand coalition is formed), symmetry (which implies that both center firms and both brink firms, respectively, get identical payoffs), the irrelevance of unessential players (which ensures that firms with missing links get nothing) and additivity (ensuring that payoffs from composed games equal the sum of payoffs from the respective independent subgames).

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E. Feess, J.-H. Thun / European Journal of Operational Research 234 (2014) 763–773 Table 1 Basic model.

Firm 1 Firm 2 Firm 3 Firm 4

Shapley values Si(R)

Investment Ii ; return Ri

Payoff Pi⁄

6cþ2a4b þ 12 R1 þ 16 R2 24 ð6c2aþ4bÞ þ 12 R1 þ 23 R2 24 ð6c2aþ4bÞ þ 16 R2 þ 23 R3 24

1 4 4 9 4 9 1 4

6cþ2a4b 2 þ 17 24 36 ð6c2aþ4bÞ 7 2 þ6 24 ð6c2aþ4bÞ þ 76 2 24 6cþ2a4b 2 þ 17 24 36

6cþ2a4b 24

þ þ

1 6 R3 1 2 R4

þ 16 R3 þ 12 R4

Next, it can be seen that each firm receives only part of the surplus created by its own supply chain-specific investment. As other firms also benefit from these investments, there is a problem of positive externalities reflected in the Shapley values. This externality problem is most pronounced for brink firms which get only 12 of their own investments, while center firms can reap 23 of their investments. The reason for this discrepancy is that brink firms can only be part of a supply chain if their direct neighbors agree. By contrast, center firms have two potential coalition partners, one at each side. Of course, adding up over all firms in column 1 yields c + R, the gross surplus of the supply chain. 4.2. Stage 1: Investments Given the surplus division in the first column of Table 1, we can now proceed to the investment decisions in the first stage of the game. Define Pi(I) = Si(I)  Ii as firm i’s payoff, which is the difference between the firm’s share Si and its investment costs Ii. As an example, consider firm 1:

P1 ðIÞ ¼ S1 ðIÞ  I1 ¼

6c þ 2a  4b 1 1 þ R1 þ R2  I1 : 24 2 6

ð6Þ

Note first that neither 6cþ2a4b (its share in the base value of the 24 chain) nor 16 R2 (its share from the surplus created by firm 2’s investment) can be influenced by firm 1. As this is the case for all firms, the firms’ investment decisions are independent of each other.9 Firm 1’s payoff-maximizing investment follows from maxi1 mizing P1(I) with respect to I1. Recalling that R1 ¼ 2aI21 , straightforward calculations yield I1 ¼ 14 a2 as firm 1’s investment and R1 ¼ a2 as the respective return. Due to the positive externality on other firms, the payoff-maximizing investment level is below the efficient investment which maximizes the overall value of the supply chain given by Ie1 ¼ a2 (see Eq. (4) above). We will refer to this result as the underinvestment problem. Analogously, we can determine all investment levels. Substituting the respective values in the firms’ payoff functions then yields the equilibrium values shown in Table 1. The second column in Table 1 shows that all investment levels are inefficiently low, where the underinvestment problem is most pronounced for brink firms, Ib < Ic .10 The reason is that center firms have a better bargaining position,11 and hence know that they can reap a larger part of the total return on their own investments (23 Rc compared to 12 Rb ) in stage 2. Still, the investments of center firms are also inefficiently low. We summarize our results in Proposition 1. Proposition 1. (i) In the Subgame Perfect Equilibrium (SPE), all firms invest inefficiently low. The underinvestment problem is less pronounced for center firms. The overall surplus of the supply chain is e 2 2 below the efficient surplus, V  ¼ Ri Pi ¼ c þ 59 18 a < V ¼ c þ 4a .

9 As mentioned above, things are formally more involved when we allow for complementary investments expressed by @I@R > 0. In this case, we get a system of i @I j reaction curves, and need to solve for the Nash Equilibrium in stage 1 as part of the Subgame Perfect Equilibrium for the whole game. See our discussion in Section 7.1. 10 Subscript b (c) denotes brink firms (center firms). 11 In the language of the Shapley logic, they can contribute to more coalitions.

2

a ; a2 ; a2 ; a2 ;

2

a 4 2 3a 4 2 3a a2

a a a a

Proof. Follows from the calculations of the Shapley values in Appendix 1 and the text above. h

5. Subsidies In the last section, we have demonstrated that the degree of the underinvestment problem depends on the position in the supply chain. As firms benefit from their neighbors’ investments, a natural strategy for mitigating the underinvestment problem can be seen in subsidies for the investments of directly linked firms. Specifically, assume that firm j can commit to pay a certain percentage of firm i’s investment. We denote this percentage as sji. When deciding on sji, firm j maximizes its own payoff, i.e. the decisions on subsidies are taken non-cooperatively.12 Including the subsidy stage leads to the following extended game: Stage 0: Firms decide simultaneously and non-cooperatively on their subsidies for the investment costs of other firms. Stage 1: Given the vector of subsidies, firms decide simultaneously and non-cooperatively on their investment levels. Stage 2: The grand coalition (supply chain of four firms) is formed, and the surplus is divided according to the Shapley value. Note that subsidizing investments will often be cheaper than signing binding contracts on investment levels. The difference is that binding contracts on investment levels do not only require that those levels are observable, but also that the contracts are enforceable in court.13 In complex situations, the enforceability problem will often be serious, and convincing judges that contracts have been breached may be expensive. For these reasons, we will assume that signing binding contracts is costly when we discuss the possibility of vertical integration (see Section 6). Again, we solve qua backwards induction. As the surplus division in stage 2 is still given by Table 1, we can directly proceed to stage 1. 5.1. Stage 1. Investments for given subsidies For any given vector of subsidies s, each firm’s investment level follows again from maximizing its payoff function. Recall from Table 1 that each firm benefits only from the investments of adjacent firms, so that there are no incentives to subsidize the investments of other firms. Hence, for instance, firm 1 will not subsidize firms 3 and 4, and firm 2 will not subsidize firm 4. With this in mind, consider firm 1 as an example. As only firm 2 is directly linked to firm 1, firm 1’s payoff function is 12 For the comparison to the literature, note that these subsidies are different from side-payments potentially required to reach the grand coalition. The latter subsidies are always assumed to be feasible by definition of cooperative game theory (see Leng and Zhu, 2009). In our paper, subsidies are used as incentive devices before the coalition is formed; see also Rubin and Carter (1990). 13 This distinction between observability on the one hand and enforceability on the other hand is the key assumption of incomplete contract theory (see Hart and Moore, 1999; Tirole, 1999) which plays a predominant role in the microeconomic theory of the firm.

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Table 2 The basic model with subsidies. Ii(s) Firm 1

Ri(s)

1 a2 4 ð1s21 Þ2

a2 ð1s21 Þ

Ii ðs Þ; Ri ðs Þ

Pi ðs Þ

9 16

a ; a

6cþ2a4b 24

2 þ 43 72 a

2

2

a ; a

6c2aþ4b 24

2 þ 59 48 a

a2 ; 43 a2

6c2aþ4b 24

2 þ 59 48 a

6cþ2a4b 24

2 þ 43 72 a

Firm 2

4 a 9 ð1s12 s32 Þ2

4 a 3 ð1s12 s32 Þ

4 9

Firm 3

4 a2 9 ð1s43 s23 Þ2

4 a2 3 ð1s43 s23 Þ

4 9

Firm 4

1 a2 4 ð1s34 Þ2

P1 ðI; sÞ ¼

2

2

9 16

a2 ð1s34 Þ

2

3 2

4 3

2

a2 ; 32 a2

6c þ 2a  4b 1 1 þ R1 þ R2  ð1  s21 ÞI1  s12 I2 : 24 2 6

ð7Þ

The first difference to the payoff function without subsidies (see Eq. (6)) is that firm 1 now pays only (1  s21)I1, since the percentage s21 of its own investment is effectively borne by firm 2. Second, the own subsidy potentially paid to firm 2 needs to be subtracted. Recall that firm 1 will neither subsidize firm 3 nor firm 4. As we consider the1 investment stage, all subsidies are given. Recalling that R1 ¼ 2aI21 and maximizing Eq. (7) with respect to I1 2 immediately yields Is1 ¼ 14 ð1sa Þ2 as firm 1’s investment level.14 21 Similarly, firm 2’s payoff function is

P2 ðI; sÞ ¼

ð6c  2a þ 4bÞ 1 2 1 þ R1 þ R2 þ R3  ð1  s12 24 2 3 6  s32 ÞI2  s21 I1  s23 I3 ;

which yields

Is2

¼

4 a2 . 9 ð1s12 s32 Þ2

ð8Þ

The four firms’ payoff-maximizing

investment levels, depending on the subsidy vector, and the respective returns on investment are shown in the first two columns of Table 2. The latter two columns are for later reference.

share of its investment itself, so that it would increase its investment level to a large extent when receiving a subsidy. But then, subsidizing firm 3 has higher costs than benefits for firm 2.15 By similar calculations, we can easily derive the payoff-maximizing subsidies of all firms. We find that all subsidies except s21 and s34 are zero, so that center firms are not subsidized in the Subgame Perfect Equilibrium. Brink firms receive subsidies of 13 from their directly linked center firms. Substituting these subsidies s21 ¼ s34 ¼ 13 into columns 1 and 2 of Table 2 yields the values shown in column 3 of the table. Accounting for the subsidies paid and received finally leads to the payoffs in the fourth column. We summarize our results for the game with subsidies: Proposition 2. Suppose firms decide simultaneously and non-cooperatively about their subsidies on the investment levels of other firms. Then, only brink firms will be subsidized: s21 ¼ s34 ¼ 13. With subsidies, investment levels of brink firms are higher than those of center firms; 9 2 Is1 ¼ Is4 ¼ 16 a > Is2 ¼ Is3 ¼ 49 a2 . Subsidies mitigate, but do not solve the underinvestment problem, that is, Isi < Iei 8i. Proof. Follows from the text above.

h

The reason why subsidies mitigate, but do not solve the underinvestment problem is that neither the investing nor the subsidizing firm considers all positive benefits for the supply chain as a whole. Nevertheless, part of the externality is now internalized, and this increases the supply chain’s efficiency. Interestingly, investment levels of brink firms are now higher than those of center firms. Subsidies lead to higher payoffs for all firms: For brink firms, the increase in their payoffs follows from the subsidies they receive, while center firms can extract half of the return on the higher investments by brink firms.

5.2. Stage 0. Payoff-maximizing subsidies When choosing the subsidies, the investment levels displayed in Table 2 will be anticipated. Thus, when for instance firm 2 decides on its subsidies, it substitutes R1 and R3 in its payoff function by the values for Rs1 and Rs3 as given in the second column of Table 2, and moreover takes the subsidy-costs into account. It follows that, in stage 0, firm 2 maximizes

P2 ðI; sÞ ¼

  ð6c  2a þ 4bÞ 1 a2 2 þ R2 þ 24 2 ð1  s21 Þ 3   1 4a2  ð1  s12  s32 ÞI2 þ 6 3ð1  s43  s23 Þ ! ! a2 4a2  s21  s23 4ð1  s21 Þ2 9ð1  s43  s23 Þ2

ð9Þ

with respect to s21 and s23. The two first partial derivatives are

@ P2 ðI; sÞ 1 a2 ¼ ð3s21  1Þ and @s21 4 ðs21  1Þ3 @ P2 ðI; sÞ 2 2 1 þ 3s23  s43 ¼ a : @s23 9 ðs23 þ s43  1Þ3

ð10Þ ð11Þ

ðI;sÞ From @ P@s221 ¼ 0, we get an interior solution with s21 ¼ 13. Thus, firm 2 has an incentive to increase firm 1’s investment by subsidizing each unit with 13. The reason is that, according to the Shapley value, firm 2 can extract half of firm 1’s return on investment when the supply chain is formed. 2 P2 ðI;sÞ 1þs23 s43 By contrast, @@ðs ¼  43 a2 ð1s < 0 for all s23 which shows Þ2 s Þ4 23

23

43

that firm 2 will not subsidize firm 3. Thus, there is a corner solution with s23 ¼ 0. This follows from the fact that firm 3 gets the lion’s 14

Superscript s denotes the case with subsidies.

6. Vertical integration (binding contracts) 6.1. The extended game structure A second option for mitigating or eliminating the underinvestment problem can be seen in vertical integration. We first restrict attention to the case where only any two brink and center firms can integrate.16 We will denote the respective integrated firms by 12 and 34, respectively. Assuming instead that also firms which do not have direct links could integrate as well would not only unnecessarily convolute the analysis, but seems also unrealistic in most cases. For instance, an integration of a supplier and a wholesaler not including the manufacturer seems unlikely. The following analysis is essentially not restricted to vertical integration but includes all cases where two firms behave as one entity, both in the investment stage and in the surplus division according to the Shapley value. It does not matter for our model whether such a behavior is implemented by a merger or just by a legally binding and enforceable agreement for the supply chain itself. To isolate the strategic effects of integration, we assume that integration has no impact on the investment technologies itself. If, for instance, firms 1 and 2 integrate, then the net return their investment contributes to the supply chain is still given by R1  I1 + R2  I2. The only difference is that I1 and I2 are now chosen so as to maximize the joint payoff of firms 1 and 2. 15 Note that the fact that firms 2 and 3 do not subsidize each other has nothing to do with the symmetry of center firms. It follows simply from the fact that center firms can appropriate a large share of the return on their investments anyway. Hence, the reason why firm 2 does not subsidize firm 3 is the same why firm 4 does not subsidize firm 3. 16 In the end of this section, we show that our results are qualitatively robust when we allow for vertical integration of three firms.

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E. Feess, J.-H. Thun / European Journal of Operational Research 234 (2014) 763–773 Table 3 Partial integration.

Table 4 Full integration. Ii ðPÞ; Ri ðPÞ

Pi(P) Firm 12 Firm 3 Firm 4

2cþbþa þ R1 þ 6 2c2aþb þ 23 R3 6 2cþa2b þ 16 R3 6

2

2

þ 12 R4  I3

4a2 9

þ 12 R4  I4

a2 ;

4a2 3 2

R2 þ

1 6 R3

 I1  I2  F

a ; 2a

4

;

a

Pi ðPÞ

Pi(F)

2cþaþb 2 þ 20  6 9 2c2aþb 17 2 þ 6 18 2 2cþa2b þ 1736a 6

c 2 c 2

a a

F

We take into account that binding agreements are likely to be costly, for instance due to legal advice when fixing the contracts. Specifically, we assume fixed transaction costs of F > 0 for the integrating parties.17 The extensive form of the game is now as follows: Stage 0: Firms 1 and 2, and firms 3 and 4, respectively, decide simultaneously on integration. Firms integrate if and only if this increases their joint payoff, taking transaction costs F into account. Stage 1: Each firm decides non-cooperatively on its investment. Vertically integrated firms maximize their joint payoff from investment. Stage 2: A supply chain of all firms is formed, and the surplus is divided according to the Shapley value.

6.2. Equilibrium analysis Following backwards induction, we need to start with the surplus division in stage 2. There are three possible constellations for which we need to derive the Shapley values: non-integration, integration by just two firms (referred to as partial integration), or integration by both pairs of firms (referred to as full integration). For the case without integration, we already know the overall payoffs of all firms from Table 1, so that we are left with the analysis of partial and full integration. For partial integration, assume without loss of generality that firms 1 and 2 integrate, while firms 3 and 4 do not. For the Shapley value analysis, partial integration means that the grand coalition now consists of only three independent entities, the integrated firm 12 consisting of the supplier (firm 1) and the manufacturer (firm 2), the wholesaler (firm 3) and the retailer (firm 4). The payoff functions for partial integration (see the Shapley value analysis in Appendix 2) are shown in the first column of Table 3.18 The next two columns show the equilibrium values which follow from maximizing the respective payoff functions. Analogously, Table 4 shows the payoff functions and the equilibrium values for full integration. The Shapley value analysis with full integration is simple as we are left with only two (integrated) firms; see the values in Table 4. When analyzing the incentives for one pair of firms to integrate, the integration decision of the other two firms needs to be treated as given. In other words, we need to derive a Nash Equilibrium in stage 0 as part of the Subgame Perfect Equilibrium of the whole game. We start with the incentives of firm 1 and 2 to integrate under the assumption that firms 3 and 4 do not integrate (partial integration). Inspecting the joint payoff function immediately shows that vertical integration solves the underinvestment problem for the integrating firms as they can fully reap the return on their investments (see the first row in the first column of Table 3). 17 These costs have nothing to do with transaction costs of forming the supply chain, which we assume to be already included in the values a, b and c (see the model section). 18 P in brackets expresses partial integration.

Firm 12 Firm 34

Ii ðFÞ; Ri ðFÞ

þ R1 þ R2  I1  I2  F þ R3 þ R4  I3  I4  F

2

2

a ; 2a a2; 2a2

Pi ðFÞ c 2 c 2

þ 2a2  F þ 2a2  F

Thus, the investments of integrating firms are always efficient; I1 ðP Þ ¼ I2 ðPÞ ¼ a2 . This holds both for the case with partial and with full integration. If firms 1 and 2 do not integrate (given that firm 3 and firm 4 do not integrate as well), then the payoffs are known from Table 1 above. Hence, integration of firm 1 and 2 is a best response to non-integration of firms 3 and 4 if and only if

2c þ b þ a 20 2 þ a  F P P1ð3; 4Þ þ P2ð3; 4Þ 6 9 1 59 2 a ¼ cþ 2 36

P12ð3; 4Þ ¼

ð12Þ

which can be written as

F6

7 2 aþbc a þ : 12 6

ð13Þ

Let us denote e F 12 ð34Þ as the maximum value for the integration costs such that integration of firm 1 and 2 is a best response to non-integration of firms 3 and 4:

7 2 aþbc e F 12 ð34Þ ¼ a þ : 12 6

ð14Þ

Eq. (14) shows that the integration incentives are increasing in the technology efficiency parameter a. This is due to the fact that the integrated firm can fully reap the return on its investment, which becomes more important when the technology’s efficiency increases. Furthermore, the integration incentives are increasing in a and b (the surplus of smaller supply chains), but decreasing in c (the grand coalition’s value). The intuition is as follows: if firms 1 and 2 integrate while firms 3 and 4 do not, there are three independent entities when calculating the Shapley value. Thus, the integrated firm gets only c/3 instead of c/4 twice from the grand coalition. But as the integrated firm gets the values a and b in relatively more permutations, it depends on the value of smaller chains (a and b) compared to the grand coalition’s value (c) whether partial integration improves or weakens the bargaining position. The next requirement for partial integration being a Nash Equilibrium is that firms non-integration by firms 3 and 4 is a best response to integration by firms 1 and 2. If firms 3 and 4 do not integrate, their payoffs can be taken from Table 3. If they integrate, we have full integration, so that payoffs are shown in Table 4. The condition for non-integration by firms 3 and 4 to be a best response to integration by firms 1 and 2 is thus

P3ð12Þ þ P4ð12Þ ¼

2c  2a þ b 17 2 2c þ a  2b þ a þ 6 18 6 2 17a c þ P P34ð12Þ ¼ þ 2a2  F 2 36

ð15Þ

which can be written as

F6

7 2 aþbc a þ : 12 6

ð16Þ

F 34 ð12Þ as the minimum value such that non-inteLet us denote e gration by firms 3 and 4 is a best response to integration by firms 1 and 2:

7 2 aþbc e F 34 ð12Þ ¼ a þ : 12 6

ð17Þ

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E. Feess, J.-H. Thun / European Journal of Operational Research 234 (2014) 763–773

It can immediately be seen that e F 12 ð34Þ ¼ e F 34 ð12Þ  e F . This means that the integration incentives are effectively independent of whether the other pair of firms integrates or not. It follows that partial integration is only a Nash Equilibrium in the degenerate case where F ¼ e F ð34Þ ¼ e F 34 ð12Þ. Then, full integration, partial 12

integration and non-integration are all Nash Equilibria for stage 0 of the game. Otherwise, either full integration or no integration at all occurs. We summarize our findings in the following Proposition: 7 2 F ¼ 12 Proposition 3. If F < e a þ aþbc 6 , the unique Subgame Perfect F ¼ 7 a2 þ aþbc, the Equilibrium (SPE) is full integration. If F > e 12

6

7 2 F ¼ 12 a þ aþbc unique SPE is no integration. If F ¼ e 6 , full integration, no integration and partial integration are all SPE.

e , each pair of Proof. For the payoffs, see the tables above. For F< F firms has higher payoffs from integration, independently of what the other pair of firms does. Thus, integration is the unique F , the opposite holds. Hence, no integration is the equilibrium. If F> e unique equilibrium. If F ¼ e F , then the firms’ payoffs are independent from integration, and all three possibilities are equilibria. h In the following, we neglect the uninteresting case where F ¼ e F, and where three equilibria with identical payoffs exist. Proposition 3 then shows that full integration is the equilibrium if and only if integration costs are sufficiently small. Otherwise, firms do not integrate. h 6.3. Private vs. efficient incentives to integrate We now turn to the most interesting question with respect to binding contracts by comparing the payoff-maximizing decisions of firms (subsequently referred to as the private incentive) with the decisions that maximize the value of the supply chain. From the fact that integration solves the underinvestment problem, one might tentatively conclude that the possibility of integration should weakly increase the joint payoff of the supply chain. We will show that this presumption is wrong. Without integration, the aggregated payoff of the four firms is given from adding up in Table 1:

V  ðNÞ ¼ c þ 2

  3 2 8 2 a þ a : 4 9

ð18Þ

With full integration, we get from Table 4: 

V ðFÞ ¼ c þ 4a2  2F:

ð19Þ

It follows that full integration yields higher overall payoff if and only if DV  V⁄(F)  V⁄(N) > 0. This gives a threshold of

13 2 e Fe ¼ a: 36

occur. Proof. Part (i). c 6 a + b is a sufficient condition for DF > 0, which means that firms integrate too often. Part (ii). If c > a + b, then 1 solving DF = 0 for a yields a ¼ 12 ð3ðc  a  bÞÞ2 . Then, the result follows. Part (iii). Follows from the definition of the two thresholds. h The Proposition contradicts the intuition that the possibility of vertical integration is always good news as it solves the underinvestment problem. By contrast, in the plausible case where the value of the grand coalition (c) is not too high compared to the value of smaller chains (a and b), the incentives to integrate are always excessively high. By excessively high incentives, we mean that, whenever e Fe < F 6 e F , firms integrate even though this reduces their net joint payoff in the supply chain. For intuition, note that, from the perspective of the entire supply chain, only the transaction costs for integration and the improvement in the incentives to invest matter. From a private point of view, the division in the basic surplus and in the returns on investment matters in addition. Whether the change in the division of the basic surplus increases the incentives to integrate depends on the relation of c to a and b. The division in the returns on investment, however, ceteris paribus always sets overincentives for integration as the integrating firms can reap higher part of the return on their investments compared to the case without integration. Thus, the incentives for vertical integration are distorted, and the possibility of costly vertical integration reduces the net value of the supply chain. In other words, costly binding contracts are then signed too often as they improve the bargaining position vis-a-vis other firms. Part (ii) of the Proposition shows that, if c is large compared to the sum of a and b, then this effect may be outweighed by the impacts of integration on the surplus division according to the Shapley value. In this case, any pair of two firms might not wish to integrate even if transaction costs are zero. Then, non-integration might be the only equilibrium even in cases where integration is efficient. In an Online Appendix, we have shown that the result on the impacts of the different parameters is qualitatively robust when we also consider the possibility of three integrating firms. Summing up, our analyzes of subsidies and vertical integration lead to the following results: Subsidies can only mitigate, but not fully eliminate the underinvestment problem. By contrast, vertical integration fully restores efficient investment incentives, but payoff-maximizing choices again do not coincide with efficient behavior. For plausible parameter constellations, integration incentives are excessively high.

ð20Þ

The threshold expresses that integration yields a higher net value of the supply chain if and only if the fixed costs of integration are below 13 a2 . By contrast, recall from Proposition 3 that full inte36 gration is the unique equilibrium if fixed costs are below 7 a2 þ aþbc .The difference between the two thresholds is 12 6

2 aþbc Fe F e ¼ a2 þ DF  e : 9 6

Inspecting the difference yields the following result: Proposition 4. (i) Suppose c 6 a + b. Then, the private incentive for e>e F e . (ii) Suppose c > a + b. Then, the integration is too high since F private

chain. For e Fe < F 6 e F , vertical integration occurs and reduces the joint F , vertical integration does not payoff of the supply chain. For F > e

incentive 1

for 

integration

is 1

too 

high

(too

low)

if

a > 12 ð3ðc  a  bÞÞ2 a < 12 ð3ðc  a  bÞÞ2 . (iii). For F < maxð eF e ; eF Þ, vertical integration occurs and increases the joint payoff of the supply

7. Robustness of the results In this section, we generalize our model in two directions, and we show that our results are qualitatively robust in these respects. 7.1. Firm-specific investment technologies 7.1.1. Extended model and equilibrium So far, we have assumed that the investment technologies of all firms are equally efficient. This streamlined the exposition, but it may often be counterfactual. It is hence important to see whether our main results are qualitatively robust with respect to different investment technologies. As for this, we now assume that the re1 turn on investment of firm i is given by Ri ¼ 2ai I2i , where ai is a firm-specific efficiency parameter. Following the logic of the basic model, the efficient investments are now Iei ¼ a2i . Returns on

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E. Feess, J.-H. Thun / European Journal of Operational Research 234 (2014) 763–773

investments are then Rei ¼ 2a2i . If all four firms invest efficiently, the overall value in the grand coalition is V e ðIe Þ ¼ c þ Ra2i . We summarize the equilibrium values in Table 5. Comparing the payoff-maximizing investment levels to the efficient levels confirms that the underinvestment result still holds. All investments are inefficiently low and the degree of inefficiency is again higher for brink firms. Straightforwardly, we can no longer say that brink firms invest less which is only the case if ab < 43 ac .19 If ab is sufficiently larger than ac, then the lower investment incentive due to the position in the supply chain is outweighed by the more efficient technology. However, the economically important result that the investment of brink firms is more inefficient pre I vails as Ieb ¼ 14 < IIec ¼ 49. 2 b As in the basic model, we now extend to subsidies and vertical integration. 7.1.2. Subsidies We can derive all results from substituting ain the basic model by the respective values for ai. We get Proposition 5. In the subsidy game with different technologies, (i) only brink firms will be subsidized, s21 ¼ s34 ¼ 13. (ii) All investment levels are inefficiently low (iii) The investment level of a brink firm is higher than the investment level of a center firm if and only if ab > 89 ac . Proof. Follows from the analysis in the basic model when substituting aby the respective values ai for the four firms. h Proposition 5 shows that our results on subsidies are robust: subsidies mitigate, but do not solve the underinvestment problem. Furthermore, subsidies are still 1/3 of investment costs and hence independent of the technology’s efficiency parameters. Part (iii) follows straightforwardly from the basic model by taking the different technology parameters into account.

7.1.3. Vertical Integration A crucial insight from our basic model is that the incentives to integrate are independent from whether the other two firms integrate or not, i.e.

7 2 aþbc e F 12 ð34Þ ¼ e F 12 ð34Þ  e F¼ a þ : 12 6 7 It can easily be seen from our basic model that the part 12 a2 comes only from firm 2’s investment itself.20 It follows that the efficiency in the technologies of firms 3 and 4 has no impact on the integration incentives of firms 1 and 2. The reason is that the part of the return generated by the investments of firms 3 and 4 which is allocated to firms 1 and 2 by the Shapley value is independent of whether firms 1 and 2 integrate or not. Following the same logic as in our basic model, we get:

Proposition 6. (i). The threshold values for F which only just trigger integration by firms 1 and 2, and by firms 3 and 4, are e 12 ¼ 1 a2 þ 1 a2 þ aþbc and e F 34 ¼ 1 a2 þ 1 a2 þ aþbc, respectively. F 4

1

3

2

6

4

3

3

4

6

(ii) Suppose without loss of generality that a1, a2 P a3, a4. Then, partial integration is the unique equilibrium if and only if e F 12 . F 34 < F < e

19

Subscript b (c) denotes brink (center) firms. In fact, comparing the payoff functions yields P12ð34Þ  P1ð34Þ þ P2ð34Þ ¼       P12 34  P1 34 þ P2 34 ¼ 16 R2 þ aþbc 6 . 20

Table 5 Model with different technologies. Ii Firm 1 Firm 2 Firm 3 Firm 4

1 4 4 9 4 9 1 4

a a a a

Ri 2 1 2 2 2 2 2 4

2 1

a 4 2 3 a2 4 2 3 a3 a24

Pi 6cþ2a4b 24 6c2aþ4b 24 6c2aþ4b 24 6cþ2a4b 24

þ 14 a21 þ 29 a22 þ 12 a21 þ 49 a22 þ 29 a23 þ 29 a22 þ 49 a23 þ 12 a24 þ 29 a23 þ 14 a24

Proof. Part (i). Follows from substituting the values for ain the firms’ payoff functions in the basic model by ai. Part (ii). Partial integration requires that firms 1 and 2 integrate, while firms 3 and 4 do not. This is expressed in part (ii). h Proposition 6 shows that, by contrast to the basic model, partial integration can now be the unique equilibrium of the vertical integration game. The reason is that integration incentives are increasing in the efficiency of the investment technologies for two reasons: first, the more productive the technology is, the more important it is that investments are efficient. Second, a larger share of the return on investment can be reaped by the investing firms. Hence, both pairs of firms integrate if F 6 e F 34 , no integration occurs for F > e F 34 , and partial integration is an equilibrium for all integration costs in-between. This is an interesting additional insight compared to the symmetric model. Of course, our main results with respect to the comparison of private and social incentives carry over to the case with asymmetric technologies: c 6 a + b is still a sufficient condition for overincentives for integration, and for c > a + b, overincentives occur if and only if the technologies are sufficiently productive.21 7.1.4. Complementary investments Throughout, we have assumed that the investment technologies are additively separable. This simplifies the exposition considerably, but one might wish to know what happens when investments are complementary. Complementary means that the marginal return on the investment of firm i is increasing in firm j’s investment, @R > 0 (positive cross-partial derivatives). We have considered @I @I i

j

several variants for such a technology. As the simplest case, assume that the technology is of the CobbDouglas type, R ¼ P4i¼1 ðIi Þa where 0 < a < 1. All marginal returns are decreasing, and all cross-partial derivatives are positive. In the Cobb-Douglas case, all firms are equally indispensable as R = 0 if one of the firms does not participate. Each efficient investment level follows from maximizing P4i¼1 ðIi Þa  R4i¼1 Ii , and is thus given by 1  a1 Iei ¼  4 1 a  . By contrast, each payoff-maximizing investa Pj–i ðIj Þ   ment level follows from maximizing 14 P4i¼1 ðIi Þa  R4i¼1 Ii as each 1  a1 , firm gets 14 of the return on investment. Thus, Ii ¼  4 4 a  a Pj–i ðIj Þ so that all investment levels are inefficiently low. Our main results on underinvestments, subsidies and the incentives for vertical integration hold. 7.2. A more general view on coalitions In our basic model, we have assumed that all equally large coalitions generate the same surplus. For instance, the supply chains {1, 2} and {2, 3} yield identical surplus a. This helped to focus on the points we wish to make, but is not necessarily 21 Recall that the social benefit from integration coming from higher investment incentives is independent of whether the other two firms integrate or not. Hence, when comparing private and socially optimal incentives, the same logic applies to full and partial integration.

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Table 6 Shapley values with differently valuable coalitions. Shapley values Si Firm 1 Firm 2 Firm 3 Firm 4

6cþ4a12 2a23 þ2b123 6b234 þ 12 R1 þ 16 R2 24 ð6cþ4a12 þ2a23 8a34 þ2b123 þ2b234 Þ þ 12 R1 þ 24 ð6cþ4a34 þ2a23 8a12 þ2b234 þ2b123 Þ þ 16 R2 þ 24 6cþ4a34 2a23 þ2b234 6b123 24

2 3 R2 2 3 R3

þ 16 R3 þ 12 R4

þ 16 R3 þ 12 R4

realistic. In our motivating example, for instance, a supply chain of a supplier and a manufacturer might well yield a different surplus than a supply chain consisting of the same manufacturer and a wholesaler. In this section, we therefore assume that equally sized coalitions create different surplus. We denote the value of coalition {i, j} by aij, and the value of coalition {i, j, k} by bijk. We assume that larger coalitions yield larger surplus if the smaller coalition is a subset of the larger coalition, e.g. a12 < b123. However, we do not require that all three-player coalitions are superior to all two player-coalition, that is, we allow for a12 > b234 and a34 > b123. Still, we assume that the grand coalition is efficient. In Appendix 3, we derive the Shapley-values shown in Table 6. In our model with equally valuable coalitions, the Shapley values of brink firms were strictly below those of center firms. Table 6 shows that this still holds for adjacent firms (S1 < S2 and S3 < S4), but the brink firm to the left, for instance, may now have a higher Shapley value than the center firm to the right. This is straightforward as we now allow for the possibility that coalitions including firms 1 and 2 are more productive then coalitions including firms 3 and 4. Our main results, however, are robust: Most importantly, allowing for different values of equally large coalitions has no impact on investments which depend on the technology functions and the division of the return on investments, but not on the division of the base surplus of the supply chain. Since all investments are the same as in the basic model, the equilibrium payoffs differ from the basic model only with respect to the Shapley values for the base supply chain as given in Table 6, but not with respect to the division in the return on investments. Summarizing the additional insights from this extension, we restrict attention to the comparison of P1(I⁄) to P2(I⁄) and of P1(I⁄) to P3(I⁄); the other cases are analogous. Proposition 7. Suppose equally large coalitions yield different surplus. Then, P1(I⁄) < P2(I⁄). P1(I⁄) > P3(I⁄) if and only if 1 2 a12 > 25 18 a þ 3 ða23 þ a34 þ 2b234 Þ. 1 1 2 Proof. P1 ðI Þ  P2 ðI Þ ¼  25 as 36 a  6 a23 þ 3 ða34  b234 Þ < 0 a34  b234 < 0. From the entries in Table 6, it follows 1 2 P1 ðI Þ > P3 ðI Þ () a12 > 25 h 18 a þ 3 ða23 þ a34 þ 2b234 Þ. The Proposition shows the intuitive result that brink firm 1 can get a larger payoff than center firm 3 if the asymmetry in the value of equally large supply chains is sufficiently pronounced, that is if a12 is far above a23 and a34 (and if b234 is sufficiently small). This may well be the case in reality when a supply chain including the supplier and the manufacturer is more valuable than all supply chains not including the supplier, for instance. We do not need to redo our analysis for subsidies and vertical integration as the results follows straightforwardly from our former insights: For subsidies, nothing changes as the incentive to subsidize is independent from the Shapley values for the base game. It depends only on the investment technology. For the vertical integration-game (costly binding contracts), partial integration may now be an equilibrium. To see this, just recall that in the basic 7 model, each pair of firms integrates if F 6 12 a2 þ aþbc . Thus, if for 6

instance a12 is high while a34 is low, then there may be an asymmetric equilibrium in which firms 1 and 2 integrate while firms 3 and 4 do not. As a potential further extension, one might argue that, if for instance a12 > a23, then the return on firm 2’s investment should be higher in the coalition {1, 2} than in the coalition {2, 3}. This would lead to an additional asymmetry in the Shapley values. Applying the same procedure as usual, we have shown that our results are still robust: Subsidies again mitigate, but do not eliminate the underinvestment problem, and incentives for vertical integration are too high whenever the value of the grand coalition is sufficiently high compared to the values of the different sub-coalitions.

8. Conclusion We have analyzed investment incentives in a simple vertical supply chain consisting of four firms. In the first stage of the game, firms decide non-cooperatively on their investments. In the second stage, the surplus created by the supply chain is divided cooperatively. As solution concept for the surplus division, we use the Shapley value, and we introduce assumptions which ensure that the Shapley value is in the core. We have derived four main results: First, all investment incentives are ineffciently low since the surplus created by investments is shared among all firms (underinvestment problem). Second, the surplus division depends crucially on the position in the supply chain, and brink firms (the supplier and the retailer in our example) get less of the surplus than center firms (the manufacturer and the wholesaler) do. As a consequence, brink firms have the lowest investment incentives. Third, the underinvestment problem sets incentives for firms to subsidize the investments of their (potential) partners in the supply chain. We have found that subsidies can mitigate, but do not solve the underinvestment problem. Fourth, taking the possibility of vertical integration into account, we have shown that the incentives for vertical integration may be either too high or too low from the whole supply chains’ perspective. These results are robust with respect to various extensions to our basic model. For further research, we aim at testing our results experimentally. In our model, each firm is only concerned about its own payoff. Following general insights of the experimental literature, it is likely that results may depart in two directions: first, the position in the supply chain may be less important for the surplus division due to inequality aversion of the participants (Bolton and Ockenfels, 2000). Hence, brink firms might get more while center firms might get less of the surplus. Second, reciprocity considerations (Fehr and Schmidt, 1999) may yield higher shares for those who invest more, and may therefore mitigate the underinvestment problem even without subsidies. Acknowledgement We are grateful to three anonymous referees and the editor, Immanuel Bomze, whose comments considerably improved the paper. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.ejor.2013.09.039. References Anupindi, R., Bassok, Y., & Zemel, E. (2001). A general framework for the study of decentralized distribution systems. Manufacturing & Service Operations Management, 3, 349–368.

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