Single sourcing versus multiple sourcing.

[] Flexible adjustment of purchases. So far we have stipulated that buyers will always purchase exactly the same quantity, namely X for the large buyer and X/2 for the two smaller buyers. In particular, this was the case both on equilibrium, that is, if all offers were accepted, and off equilibrium, that is, after only one supplier's offer was accepted. A fixed purchase volume may sometimes be realistic, for example, if this is just one input for a Leontieff-type production function and if the buyer has already purchased the right amount of all other inputs. More generally, however, a buyer may have some flexibility in adjusting his optimal purchase volume.

We show now that our key insights still hold if we allow for this additional flexibility. For this, we stipulate that if there are two buyers, then each derives the payoff (or revenues) r(x) from purchasing the total quantity x of the input. (Recall that suppliers' goods are homogeneous.) We assume that r(x) is continuously differentiable and strictly concave with r' (0) > C' (0). We again denote the unique efficient level of total output by X such that X/2 = arg [max.sub.x] [r(x) - C(x)]. If there is a single large buyer, then this buyer simply controls both of these two firms. Note that in this case, given symmetry and strict concavity, the large buyer's (gross) payoff from purchasing the total quantity x is equal to

R(x) := 2r (x/2).

Take now first the case with a single large buyer. In equilibrium, the buyer purchases X/2 from either supplier. Compared to the case with fixed purchasing quantities, what changes now are the purchases off-equilibrium, that is, if one bid from the two suppliers is rejected. Given the truthfulness requirement, when rejecting the bid of some supplier m, the buyer will now purchase from the other supplier the total quantity

X' = arg max [R(x) - C(x)], (8)

which satisfies X/2 < X' < X, at a total price equal to the sum of [[??].sup.m'.sub.n] and the respective incremental costs, C(X') - C(X/2). By optimality, equilibrium transfers are again chosen so as to make the buyer just indifferent between accepting or rejecting the respective offer. As there are two symmetric suppliers, this yields

[[??].sup.m.sub.n] = [R(X) - R(X')] + [C(X') - C(S/2)]. (9)

If the single large buyer resorts to single sourcing, then suppliers again compete themselves down to zero profits. The outcome will be constrained efficient such that the buyer purchases the quantity X' as defined in (8) from the winning supplier. Total purchasing costs are just equal to the respective supplier's costs of production C(X').

The procedure to characterize an equilibrium allocation if there are two symmetric buyers is analogous. Again, off-equilibrium, a buyer will optimally adjust the respective incremental purchases. Proposition 5 now confirms that our previous results from Proposition 1 extend to the currently considered case with flexible quantities. Though the algebra is somewhat more involved, the intuition is fully analogous.

Proposition 5. Proposition 1 extends to the case where buyers will optimally adjust their purchased quantities according to their own revenue function r(x). Precisely, also in this case, a single large buyer strictly prefers single sourcing, whereas two smaller buyers strictly prefer to spread their purchases equally over the two suppliers.

Proof See the Appendix.

6. Buyers competing at suppliers' auctions

* Analysis. As noted in the Introduction, we also want to compare the results from the case where suppliers compete to that where it is now buyers that make bids. Hence, we now stipulate that buyers submit menus [t.sup.m.sub.n] (x) to suppliers. Clearly, if there is only a single large buyer, then the analysis is trivial: the equilibrium outcome is efficient and the single buyer extracts all profits.

Turning thus to the case with two buyers, for brevity we restrict consideration to the case where buyers are again symmetric. (8) However, to ensure that suppliers who reject a buyer's bid have always profitable alternative options, namely to supply more to the other buyer, we still specify that each of the two buyers can realize the payoff r(x) when purchasing the quantity x. We invoke now again the truthfulness requirement. That is, this time the respective menu [t.sup.m.sub.n] (x) must now truthfully reflect a buyer's marginal revenues r' (x). To formalize this, we again have to choose a particular equilibrium allocation with respective values [[??].sup.m.sub.n]. Given the respective (rationally anticipated) quantity [[??].sub.n] := [[summation].sub.m=S1,S2] [[??].sup.m.sub.n] that buyer n will purchase, truthfulness for the menu [t.sup.m.sub.n] (x) thus requires for all [[DELTA].sub.x] that

[t.sup.m.sub.n]([[??].sup.m.sub.n] + [[DELTA].sub.x]) - [t.sup.m.sub.n] ([[??].sup.m.sub.n]) = r ([[??].sub.n] + [[DELTA].sub.x]) - r ([[??].sub.n]). (10)

It is immediate that given (10), the set of supported allocations is again equal to that of all efficient allocations. Moreover, it is also straightforward to show that for any given allocation, both buyers now pay strictly less than if suppliers make bids. In other words, the right to make offers is clearly valuable. What is at first somewhat surprising, however, is that the ranking of the different outcomes from the perspective of both buyers is now exactly the opposite to that in the previous case, where suppliers made bids.

Proposition 6. Suppose now that buyers bid in auctions organized by suppliers and that the truthfulness requirement still applies. Then the ranking of equilibrium allocations is reversed compared to that in Proposition 1: both buyers are strictly better off the more a buyer's purchases are concentrated on a particular supplier. On the other side, a single buyer who can post bids to suppliers will always strictly prefer multiple sourcing.

Proof. See the Appendix.

If buyers make bids, then there are now two reasons for why average purchase prices are lowest under single sourcing. The first reason is analogous to that underlying Proposition 1, though now it applies symmetrically to suppliers instead of buyers. That is, whereas concentrating purchases more on one supplier reduces the total value of a buyer's alternative options across the two suppliers if suppliers make bids, if buyers make bids then it now also reduces the total value of suppliers' alternative options, namely to sell more to another buyer. This is now profitable for the buyer if he has all "contracting power" as he makes the bid in the respective auction.

If buyers make bids, there is also a second reason for why average purchase prices are now lower the more a competing buyer purchases from one supplier. If we ignore for a moment a supplier's option to sell more to another buyer, then a buyer's bid would just have to cover a supplier's respective incremental costs. If a buyer purchases all from one supplier, then he has to compensate the supplier for the respective costs C(X/2). Instead, if he purchases X/4 from either supplier, then he must compensate each of them for the incremental costs C(X/2) - C(X/4), given that the respective supplier then also sells X/4 to the other buyer. With strictly convex costs, the respective incremental costs in the latter case, namely two times C(X/2) - C(X/4), are strictly higher than C(X/2). The latter effect has already been recognized in Chipty and Snyder (1999).

[] Comparison of the cases where suppliers or buyers run auctions. The following Corollary brings together our results on the optimality of single sourcing for the different procurement formats.

Corollary 2. Summarizing results, a buyer should choose single sourcing

(i) if he is either sufficiently large and invites bids from suppliers,

(ii) or if he is sufficiently small and submits bids to suppliers.

If we interpret the choice between the two (auction) formats as one between different distributions of contracting power, we can rephrase Corollary 2 as follows: a buyer should then be more likely to choose single sourcing if (i) contracting power resides more with suppliers and the buyer accounts for a sufficiently large fraction of the procurement market or if (ii) contracting power is more on the side of buyers but the buyer is relatively small compared to the overall size of the respective procurement market.

Considering public procurement, civil servants may often lack the appropriate (financial) incentives in negotiations. What is more, the fear of corruption or the requirement to increase accountability by making the procurement process more transparent may even dictate a particular format such as an open tender. (9) Moreover, in some markets such as those of health services or certain segments of the construction industry, public agencies may indeed be the major (local) buyers. To the extent that our key assumption applies, namely that of increasing marginal costs (at least over the relevant range), Corollary 2 would thus prescribe that officials should try to design large lots and rely on one or only few suppliers as much as possible. By increasing competition for the one big lot, the procurement agency basically compensates for its lack of bargaining power. In contrast, in markets where the public body is less dominant, it should secure lower purchase prices by relying on (strategic) second sourcing.

7. Conclusion