MANAGEMENT SCIENCE doi /mnsc ec pp. ec1 ec6

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1 MANAGEMENT SCIENCE doi /mnsc ec pp. ec1 ec6 e-companion ONLY AVAILABLE IN ELECTRONIC FORM informs 2007 INFORMS Electronic Companion The Horizontal Scope of the Firm: Organizational Tradeoffs vs. Buyer-Supplier Relationships by Olivier Chatain and Peter Zemsy, Management Science 2007, 534) Online Appendix Proof of Proposition 1. First note that if the core is nonempty, then for any N there exist values min max such that x is part of a core allocation if and only if x min max. Moreover, a general property of the core is that no player s allocation is greater than its added value, that is, max AV N. i) Let x j = v j S for all j B and x i = 0 for all i S. Note that this allocation satisfies condition 1) by Assumption 2 A2). Consider any b B and s S such that b s = G N. We have that j b x j + i s x i = j b v j S. Because j b v j S = j B v j S j B\b v j S i B v i S i B\b v i S\s = v N v N\G = AV G, we have that condition 2) holds. This establishes that the core is nonempty. It also establishes that i min = 0 for i S. In addition, the fact that x i = 0 for all i S is part of a core allocation implies that i min = 0 for all i S. Hence, for part ii) it only remains to show that i max = AV i. Consider the allocation that sets x i = AV i and x = 0 for S\i and x j = AV j AV ij. This allocation satisfies conditions 1) and 2). Hence, i max AV i and this must hold with equality because AV i is an upper bound on a player s core allocation. Proof of Proposition 2. Because x j = v j S for all j B and x i = 0 for all i S is shown to be a core allocation in Proposition 1, we have that j max = v j S. Now let x i = AV i for i S and x j = AV j i S AV ij for j B. We have that x j 0 if and only if AV j i S AV ij. Without loss of generality, let S = 1 n for some n 1. We have that indeed AV j = v j S = AV 1 j 1 + AV 2 j AV n j S i S AV ij by Assumption 3 A3); hence, x j 0. We have that the total payments are i S x i + j B x j = AV j = v N by A2) hence, condition 1) is satisfied. We now show that these allocations satisfy condition 2). We restrict attention to sets G such that B G because the payments to buyers are less than the value they add to a group, so excluding them maes it harder to find a set G such that G x v G. Suppose that G 1 = N \i for some i S. We have that G 1 x = v G 1. Now suppose that for some G n = N \s, where s S, it holds that G n x v G n and consider excluding an additional supplier. That is, let G n+1 = G n \ for some S\G n. G n+1 x v G n+1 follows from A3), so by induction we have that there is no incentive to exclude any subset of suppliers and condition 2) holds. Thus, we have that i max AV i for i S and that j min AV j i S AV ij, but each of these must hold with equality because x i AV i and condition 1). Proof of Lemma 1. Recall that A3) holds for some buyer type j AA BB AB if for all s s S, AV i j s AV i j s. i) Suppose j = AA. Note that min s\i D 2 min s \i D 2 follows from s s. These bounds lead to three cases. If D i 2 min s \i D 2, then AV i AA s = AV i AA s = 0. If min s\i D 2 D i 2 < min s \i D 2, then AV i AA s = 0 <AV i AA s. If D i 2 < min s\i D 2, then AV i AA s AV i AA s = 2T min s \i D 2 min s\i D 2 0. Thus, A3) holds for j = AA. An analogous argument holds for j = BB. ii) Suppose that there is a set of S suppliers, all of whom have positive added value. Moreover, suppose that there exists s s S such that AV i AB s > AV i AB s ; that is, A3) does not hold for type AB. Note that the focal buyer i must be in subset s and AV i AB s > 0. For a buyer of type AB either there is a single source and then at most one supplier has added value, or the tass are split, in which case at most two suppliers have added value. If there is a single source, then the single source must be supplier i, because AV i AB s > 0. Moreover, i must be a single source ec1

2 ec2 Chatain and Zemsy: Organizational Tradeoffs vs. Buyer-Supplier Relationships pp. ec1 ec6; suppl. to Management Sci. 534) , 2007 INFORMS when the set of suppliers is s as well. Hence, AV i AB s = 2 + R T D i D i 2 v AB s\i and AV i AB s = 2 + R T D i D i 2 v AB s \i. However, then AV i AB s > AV i AB s would imply v AB s \i >v AB s\i, which contradicts the superadditivity of v G. Hence, it must be that the tass for a type AB buyer are split between two suppliers, one of which is supplier i. Denote the other supplier by q. Ifs = i q, then one can show that A3) holds. If not, there must exist a supplier r such that min D i D q D r max D i D q, as otherwise it would be handling one of the tass. However, supplier r has no added value for type AB for any S containing i q. Moreover, having added value with type AA or BB would require either D r < min S D or D r > max S D, which is a contradiction. This completes the proof by contradiction that A3) holds for any set S of suppliers all of whom have AV i S > 0. Proof of Lemma 2. Suppose that there is only a single supplier i.e., m = 1) and n buyers. Then AV i = n 2 + R T D i 2 T 1 D i 2 and the optimal organizational design is D i = 1/2. Henceforth, suppose that m>1. One possibility is that, given the organizational designs of other suppliers, there is no organizational design that leads to positive added value for firm i, in which case any D i in 0 1 is optimal. Henceforth, suppose that there exists a D i such that firm i has positive added value. Let D be the organizational design of competitor. The organizational design of the competitor closest to zero is then min S\i D. There are five cases to consider based on the types of buyers for which firm i has added value. First is the case when firm i only has added value for buyers of type AA. This implies D i < min S\i D. The surplus for a type AA buyer is v AA S = 2 + R 2T D i 2 and local optimality implies that D i = 0. Analogously, the only locally optimal organizational design for a supplier that expects to have added value for buyer type BB is D i = 1. In the second case, supplier i expects to have added value for buyer types AA and AB but not BB. One possibility is that the supplier only does the A tass, in which case supplier i sees to maximize 1/2 p/2 2 + R 2T D i 2 + p 1 T D i 2 ; again, the locally optimal organizational design is D i = 0. The other possibility is that the supplier does the all the A tass but also does the B tas when serving buyer type AB. Then the optimal D i maximizes 1/2 p/2 2 + R 2T D i 2 + p 2+R T D i 2 T 1 D i 2. Local optimality requires D i = D H p/ p + 1, which is strictly between 0 and 1/2. A similar argument establishes that a supplier that expects to have added value to buyer types AB and BB and does both A and B tass for buyers of type AB must have an organizational design of D i = 1 D H = 1/ p + 1. In the third case, supplier i expects to have added value for all buyer types. Then it must be doing all tass and D i maximizes 2 + R T D i 2 T 1 D i 2 so that local optimality requires that D i = 1/2. In the fourth case, supplier i only has added value for buyer type AB. It must be that supplier i does both tass EC1 and then D i maximizes p 2 + R T D i 2 T 1 D i 2. Again we have D i = 1/2. The fifth and final possibility, that the firm has added value for buyer types AA and BB but not for buyer type AB, is not possible: Supplier i having added value for buyer type AA implies that D i < min s\i D, but having added value for buyer type BB implies that D i > max s\i D, which implies that there are no other suppliers, a contradiction. Proof of Proposition 3. We start by establishing that at most three firms enter in a pure strategy Nash equilibrium. Any firm that enters must have AV i > 0; otherwise i = F and the firm would increase its profits by not entering. For buyer type AA at most one firm satisfies the condition D i < min s\i D required for having positive added value. Similarly, there is at most one supplier that has added value for buyer type BB. For buyer type AB, either one supplier does both tass and is the only one to have added value or the tass are split between two suppliers, both of which may have added value. However, if the tass are split, the two suppliers are the ones that have added value for buyer types AA and BB. Hence we have that in any PSNE m 3. If m = 0, we have the NE equilibrium. Suppose that m = 1, then the profits of the single supplier are 1 = n 2+R TD1 2 T 1 D 1 2 F, which is maximized at D1 = 1/2 and we have the G equilibrium. Suppose that m = 2. We now from Lemma 2 that D i 0 D H 1/2 1 D H 1 for i = 1 2. It cannot be that D 1 = D 2, because then AV 1 = AV 2 = 0. It cannot be that D 1 = 1 and D 2 = 1 D H, because then firm 2 has positive added value and is doing all tass for buyer types AA and AB, so1 D H does not EC1 Suppose supplier i has added value and its only tas is A in state AB. This implies that D i < min j S\i D j because otherwise surplus would not fall when supplier i is removed. However, this implies that i has added value in state AA as well, which is a contradiction.

3 Chatain and Zemsy: Organizational Tradeoffs vs. Buyer-Supplier Relationships pp. ec1 ec6; suppl. to Management Sci. 534) , 2007 INFORMS ec3 satisfy local optimality because it does not maximize 1/2 p/2 2 + R 2T D i 2 + p 2 + R T D i 2 T 1 D i 2. Similar arguments rule out D 1 D 2 = 0 D H, D 1 D 2 = 0 1/2, and D 1 D 2 = 1/2 1. It cannot be that D 1 D 2 = D H 1/2 because Supplier 1 only has added value for buyer type AA, in which case D 1 = 0 is optimal; a similar argument rules out D 1 D 2 = 1/2 1 D H. This leaves only the SS equilibrium, where D i 0 1 and D j = 1 D i and the SH equilibria, where D 1 D 2 is one of D H 1, 1 D H, 1 D H 0, and 0 1 D H. Suppose that m = 3. Without loss of generality, suppose that D 1 <D 2 <D 3 and that D 2 1/2. Then Supplier 1 only has added value for buyer type AA, sod1 = 0. This implies that Supplier 2 has added value for buyer type AB and that it does both tass, in which case D2 = 1/2. Finally, Supplier 3 then only has added value for buyer type BB and D3 = 1. Thus, for m = 3 the only possible equilibrium is GSS. Proof of Propositions 4 and 5. We consider each of the possible equilibria in turn, first deriving conditions for existence and then establishing the comparative statics results. Existence of equilibria with entry of at least one supplier requires that the profits of all suppliers be nonnegative, that an additional supplier cannot have positive expected profits with entry, and that the organizational designs of the suppliers are globally optimal. After having detailed each equilibrium, we can establish the conditions on R that prove Proposition 4. Existence and Comparative Statics Results G equilibrium: The profits of the sole supplier are 1 = n 2 1/2 T + R F, and these must be nonnegative, which gives an upper bound on F of F n 2 1/2 T + R. It remains to be checed whether no other suppliers can profitably enter. The optimal organizational design for an entrant is D = 0orD = 1. The added value of such a supplier for buyer type AA is nt /2; the added value for buyer type AB is max 0 n 2 T/4 2 + R T/2 ; and there is no added value for buyer type BB. Hence, the expected profit of the potential entrant is n 1/2 p/2 T /2 + p max 0 T/4 R F = n max 1/2 p/2 T /2 T/4 pr F, and this must be nonpositive. In summary, G is a PSNE iff n max 1/2 p/2 T /2 T/4 pr F n 2 T/2 + R ) EC1) We have that the lower bound on F is decreasing in R and p and increasing in T, while the upper bound is increasing in R, decreasing in T, and independent of p. Hence, G is increasing in R and p and decreasing T. SS equilibrium: The profits of the two specialized firms are nonnegative if 1 = 2 = n T pr F 0, which gives an upper bound on F. The optimal organizational design for an additional supplier is D = 1/2, which results in expected profits of np max R T/2 0 F, which gives a lower bound on F. Finally, global optimality requires that neither specialist can do better than a hybrid, which would result in a profit of nt 1/ p + 1 F, and this deviation is not profitable if p + 1 R T.In summary, SS is a PSNE iff np max 0 R T/2 F n T pr p + 1 R T EC2) EC3) Consider the effect of R on existence. From inequality EC3), the set of F values that support the equilibrium becomes empty when R increases beyond T/ p+ 1. For lower values of R, the lower bound on F is either increasing or independent of R, while the upper bound is decreasing in R. Hence, SS is decreasing in R. Similar arguments establish that SS is increasing in T and decreasing in p. SH equilibrium: Suppose that Supplier 1 pursues the hybrid strategy and firm 2 is the specialist. We have that 1 = nt 1/ p + 1 F and 2 = nt 1 p / p F. Because 1 p / p + 1 <1 for p>0, we have that 1 > 2 and the upper bound on F is given by the condition 2 = 0. An additional entrant can have added value for at most one buyer type. Targeting the buyer type AB leads to D = 1/2, as the locally optimal organizational design and profits of n p 1 2 p/2 p T F. Targeting the open specialist location leads to profits of n 1 p p 2 / p T F, which are greater

4 ec4 Chatain and Zemsy: Organizational Tradeoffs vs. Buyer-Supplier Relationships pp. ec1 ec6; suppl. to Management Sci. 534) , 2007 INFORMS than the profits from D = 1/2. Finally, the hybrid firm cannot do better as a specialist, which requires that p + 1 R T. In summary, SH is a PSNE iff nt 1 p p2 p + 1 F nt 1 p 2 p T p + 1 R EC4) EC5) Because both the lower and upper bounds on F move in the same direction with changes in T and p, the equilibrium is neither increasing nor decreasing in these parameters. For R<T/ p+ 1 the equilibrium does not exist, and for larger values of R there is an interval of F values independent of R) for which it does. Hence, the equilibrium is increasing in R. GSS equilibrium: With suppliers at 0, 1/2, and 1 it is not possible for an additional entrant to have added value nor for an existing supplier to increase its profits by changing organizational design. Hence, the equilibrium exists if and only if the profits of all suppliers are nonnegative. The profits of the specialist suppliers are n 1/2 p/2 T /2 F and the profits of the generalist are np min T /2 R T/2 F, with the T/2 corresponding to the case where removing the generalist leads both tass to be given to the same supplier for buyer type AB and the R T/2 corresponding to the case where the tass are split for buyer type AB. In summary, the GSS equilibrium exists iff F n min 1/2 p/2 T /2 pmin T /2 R T/2 EC6) The upper bound on F is nondecreasing in R; hence, the equilibrium is increasing in R. NE equilibrium: The optimal deviation from the NE equilibrium is for one firm to enter as a generalist. This is not profitable iff F n 2 1/2 T + R. This lower bound is increasing in R, decreasing in T, and independent of p; hence, the NE equilibrium is decreasing in R, increasing in T, and independent of p. Equilibrium Sequence Results i) Suppose that R<R S. We consider first the case where R<T/2 and then the case where T/2 R<R S. Suppose that R T/2. The possible PSNE are NE, G, and SS. The boundary between the NE and the G equilibria is at F / n = 2 1/2 T + R, with NE above the boundary and G below. The upper bound of the SS equilibrium is given by F / n = T pr, which is less than 2 1/2 T + R. The lower bound of the SS equilibrium is zero, while the lower bound of the G equilibrium is at F / n = max 1/2 p/2 T /2 T/4 pr > 0. This establishes that for R<T/2, the sequence of equilibria that are met by lowering the value of F / n from an arbitrarily high value is NE, G, and SS. Now suppose that T/2 <R<R S. The only difference with the preceding case is that the lower bound of the SS equilibrium is F / n = R T/2 > 0, which also defines the upper bound of the GSS equilibrium. The sequence of equilibria that obtain by lowering the value of F / n is then NE, G, SS, and GSS. ii) Suppose that R>R S. We consider two cases starting with R S <R<T 3 p /4. The possible PNSE are NE, G, SH, and GSS. Because T<1 and p 0 1, the boundary between NE and G is strictly superior to the upper limit of SH, which is defined by F / n = T 1 p / p In addition, the upper bound of GSS, defined for these R values by F / n = p R T/2, is strictly above the lower bound of SH, which is defined by F / n = T 1 p p 2 / p Finally, for these values of R, the upper bound of GSS is strictly lower than the lower bound of G. All together this implies that the SH equilibrium is overlapping the G and GSS equilibria, while G and GSS are disconnected. Thus by lowering the value of F / n from an arbitrarily high value, one meets first the NE equilibrium, followed by the G equilibrium. There is an interval where SH is the unique equilibrium, and for the lowest values of F / n, GSS is unique. Now suppose that T 3 p /4 R. The same argument as above applies, except that the disconnect between G and GSS disappears, which means that the SH equilibrium is not a necessary part of the sequence anymore, as it is not a unique equilibrium for any value of F / n. Proof of Proposition 6. i) This is shown by superimposing Figure 1 on Figure 2 or by examining the equilibrium conditions in the proof of Proposition 5 for T = = 1/2 and p = 1/3.

5 Chatain and Zemsy: Organizational Tradeoffs vs. Buyer-Supplier Relationships pp. ec1 ec6; suppl. to Management Sci. 534) , 2007 INFORMS ec5 ii) NE is an equilibrium if it is not profitable for even a single supplier to enter. The optimal organizational design for a single supplier that deviates and enters is D = 1/2, and the resulting profits are the same as in the G equilibrium. If these are negative, then the profits in all other equilibria must also be negative. Hence, if NE exists, it is the unique PSNE. According to EC3) a necessary condition for SS is p+1 R T, while by EC5) a necessary condition for SH is p + 1 R T and the two equilibria only coexist at the boundary p + 1 R = T. Similarly, according to EC1) a necessary condition for G is F 1/2 p/2 T /2, and according to EC6) a necessary condition for GSS is F 1/2 p/2 T /2; these two equilibria can only coexist at a boundary. Finally, a necessary condition for SS is that F p R T/2, and a necessary condition for GSS is that F p R T/2 ; these equilibria can only coexist at a boundary. We conclude that there are at most two equilibria for any given parameter values and the following equilibrium pairs can coexist: G with SS, G with SH, and SH with GSS. Proof of Proposition 7. Suppose that SS exists. The profits of the buyer under G are j = 1 2 1/2 T + R, and under SS profits are j = 2 1 T + R p 1/2p + 1/2 and the difference between SS and G is 1 T / T + p + 2p R. This is either positive or decreasing in R. Setting R = T/ 1 + p, the maximum consistent with existence of SS, the difference is equal to 2 + T 1/ p + 1 p + 2p 5/2 + 1/2. This is either positive or decreasing in T. Setting T at its maximal value T = 1 yields p + 3p + 1 /2 p + 1, which has the sign of p + 3p + 1. This is positive if and only if > p 1 / 3p + 1, which is always true for p 0 1 and 0 1. Suppose that SH exists. The profits of the buyer are j = 2 + R T / p p 2 + p + 2 and subtracting from this the profits under G yields T p + 1 T 2 T p 1 p T 2 1 p p + 1 ) + ) R T R T p 2 4 p The first term is always positive, and the second term is increasing in R. Setting R = T/ p+ 1, the minimum consistent with existence of SH, yields p )) )) ) = ) 1 2 p p T = 2 T 2 p p and this is positive for any p in 0 1, because T 1. Therefore, the buyer s profits are higher under SH than G. Proof of Proposition 8. Suppose that G and SS both exist. The total supply chain profitability under SS is 2+ 1 p R 2F. Subtracting the total supply chain profitability under G, 2 1/2 T +R F, yields 1/2 T pr F. Define the equalizing level of setup costs by F 1 = 1/2 T pr. We want to show that F 1 can be in the interior of the support of the two equilibria. Consider R = T/ p+ 1. Then condition EC1) for the G equilibrium becomes 1/2 p + 1/2 F 1 F 2 + F 1 and condition EC2) for the SS equilibrium becomes pf 1 F / 1/2 p/2 F 1. Hence, F 1 is in the interior for this value of R if >1/2 p/2 and >1/4 because we now that F 1 = 1 p /2 p + 1 T < 1/2. Suppose that G and SH coexist. The total supply chain profitability under SH is 2 + R 2F 1/2 T 2p/ p + 1. From this subtract the supply chain profitability under G, 2 1/2 T + R F, which gives T /2 1 p / p + 1 F, and define the equalizing level of setup costs as F 2 = T /2 1 p / p + 1. We want to show that F 2 can be interior to the support of the G and SH equilibria. Consider R = T/ p+ 1. Condition EC1) supporting the generalist equilibrium becomes 1/2 p + 1/2 F 2 F 2 + F 2 and condition EC4) supporting the SH equilibrium becomes 2p2 p + 1 F 2 F 2F 2 p + 1 Hence, F 2 is in the interior for this value of R if > 1/2 p + 1 and >1/4 because we have F 2 = T /2 1 p / p + 1 <1/2.

6 ec6 Chatain and Zemsy: Organizational Tradeoffs vs. Buyer-Supplier Relationships pp. ec1 ec6; suppl. to Management Sci. 534) , 2007 INFORMS Proof of Proposition 9. Suppose EC1) holds with strict inequalities, which implies that G is a PSNE of the base model. If the first mover in the sequential entry game chooses D = 1/2, then no subsequent firm can enter profitably. Moreover, the first mover s expected profits are positive, and any other outcome yields a strictly lower profit. We conclude that G is the unique SPE. Now suppose that G is the unique SPE of the sequential entry game. Then the profits of a sole generalist are strictly positive and a firm that enters with D = 0 given a supplier at D = 1/2 must have negative profits. This establishes that EC1) holds strictly. Proof of Proposition 10. Except for GSS, buyer profits are given in the proof of Proposition 7. The buyer s profits under GSS are 2 + R pt /2 1/2 p/2 T p min T /2 R T/2. The derivatives of buyer profits w.r.t. R are 1 for SH, either 1 or 1 p for GSS, 1 for G, and 2p p + 1 for SS. The results follow from these expressions and the restrictions 0 p 1 and 1. Proof of Proposition 12. When R<T/2 there are only two possible equilibria: G and SS. Under SS, the buyer s profits are j = 2 1 T + R p 1/2 p + 1/2 and each specialist s profits are 1 = 2 = T pr F. We have that j is decreasing in the bargaining power of suppliers, while 1 and 2 increase in. The such that specialists profits are equal to zero is SS = F/ T pr. Notice that SS is increasing in R and decreasing in T. This implies that the buyer s optimal bargaining power, defined as 1 SS, is decreasing in R and increasing in T. The buyer s maximum profits under the constraint that SS exists are j SS = R 1 p 2F + 2. Under G, the buyer s profits are j = 1 2 1/2 T + R and the profits of the generalist supplier are 1 = 2 1/2 T + R F. The profits of the generalist are reduced to zero at G = F/ 2 1/2 T + R. The value of G is decreasing in R and increasing in T. This implies that the buyer s optimal bargaining power, defined as 1 G, is increasing in R and decreasing in T. We have j G = R 1/2 T F + 2. It is straightforward to show that SS > G for T<1. We have that j SS > j G if R 1 p 2F + 2 >R 1/2 T F + 2, which translates into the condition: F< F with F = 1/2 T Rp. Therefore, if F< F, the buyer chooses = SS and coordinates the suppliers on the SS equilibrium. If F> F, the buyer chooses = G and coordinates the suppliers on the generalist equilibrium. If F = F, it is indifferent between the two choices. Finally, because SS > G, there is an upward jump in the optimal buyer power 1 ) when the buyer switches at F = F from SS to G.