Single sourcing versus multiple sourcing.

Suppose thus that without loss of generality, buyer n = B1 is always the (weakly) larger buyer as [X.sub.B1] [greater than or equal to] X/2. Even with asymmetric buyers it is still intuitive that without a commitment to single sourcing, all efficient outcomes can still be supported as equilibrium outcomes, that is, any allocation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [[??].sup.S1] = [[??].sup.S2] = X/2 is an equilibrium outcome. Moreover, under the truthfulness requirement (2) also the converse holds again, namely that any equilibrium must give rise to an efficient allocation. With these observations, we can thus fully characterize any equilibrium allocation by the quantity [[??].sup.m.sub.B2] [greater than or equal to] [X.sub.B2]/2 that buyer B2, who is also the smaller of the two buyers, purchases from his larger supplier. We can then simply trace out the continuum of all equilibria satisfying the truthfulness requirement (2) by varying the respective purchases [[??].sup.m.sub.B2] between [X.sub.B2]/2 and [X.sub.B2]. For a given choice of [[??].sup.m.sub.B2], all other purchases are then uniquely pinned down by the requirement that equilibrium allocations are efficient such that [[??].sup.m'.sub.B2] = [X.sub.B2] - [[??].sup.m.sub.B2] as well as [[??].sup.m.sub.B1] = X/2 - [[??].sup.m.sub.B2] and [[??].sup.m'.sub.B2]. Of course, as the purchases of buyer B2 become more concentrated, this is also the case for buyer B1 and vice versa.

We turn next to single sourcing. Suppose first that only buyer B2 would commit to single sourcing. Intuitively, as this is the smaller buyer, the equilibrium outcome will still be efficient. The supplier m from whom buyer B2 exclusively purchases will in addition sell the quantity X/2 - [X.sub.B2] to the other buyer, B1. This is clearly different if the larger buyer commits to single sourcing. In this case, one supplier will end up producing more than the efficient share of total supply. Moreover, as B1 accounts for a larger share of total purchases given that [X.sub.B1]/X increases, the difference [X.sub.B1] - [X.sub.B2] between the two suppliers' sales becomes larger. Importantly, as [X.sub.B2] becomes smaller, the supplier m who sells only to buyer B2 would be willing to supply additionally to B1 at a lower price, reflecting his lower average incremental costs that he would have to incur. This essentially reduces the rent that the exclusive supplier can extract from B1. In fact, we already know that in the extreme case where [X.sub.B1]/X = 1, the respective supplier makes zero profits.

These observations suggest more generally that the large buyer will prefer single sourcing only if his purchases account for a sufficiently large fraction of the total procurement market. It is now convenient to state this result first only for the case where without commitment to single sourcing, purchases would be allocated symmetrically over both suppliers.

Proposition 2. With two buyers of different size, the smaller buyer never prefers single sourcing. In contrast, if without commitment to single sourcing, purchases are evenly distributed over both suppliers as [[??].sup.S1.sub.n] = [[??].sup.S2.sub.n] = [X.sub.n]/2, then the large buyer prefers single sourcing if and only if his purchases account for a sufficiently large fraction of the total procurement market as [X.sub.B1]/X is sufficiently large.

Proof See the Appendix.

Proposition 2 only considers the ease where without single sourcing, buyers' purchases would be evenly distributed over both suppliers. To extend the result we now proceed as follows. As we increase the large buyer's share of the total procurement market, [X.sub.B1]/X, we want to keep unchanged the degree to which purchases are concentrated without single sourcing. We do this by holding constant for the smaller buyer B2 the fraction [beta] [greater than or equal to] 1/2 that he purchases from his larger supplier. (Note that for [beta] = 1/2 we are back to the case of Proposition 2.)

Proposition 3. Generalizing Proposition 2, even if without single sourcing a buyer's purchases are not evenly distributed over both suppliers, the large buyer still prefers single sourcing if and only if he controls a sufficiently large fraction of the total procurement market. More formally, this is the case if [X.sub.B1]/X [greater than or equal to] y, where the respective threshold 1/2 < [lambda] < 1 is strictly lower if without single sourcing, purchases are also more concentrated (given that [beta] is higher).

Proof See the Appendix.

Before proceeding with the analysis, we briefly comment on the issue of multiplicity of equilibria. One way to narrow down the set of equilibrium allocations is to introduce some heterogeneity in suppliers' products or services. This would, for instance, seem appropriate if buyers are retailers who can stock one or two goods in a particular category. It is then straightforward to show that if goods are not perfect substitutes, then without commitment to single sourcing, each buyer purchases X/2 from either supplier. (5) This is the outcome that we chose for comparison with single sourcing in Proposition 2.

5. Discussion

* Relaxing the truthfulness requirement. Our results so far were obtained under the truthfulness requirement, which ensured that all equilibrium allocations are efficient and that for a given allocation, transfers are uniquely pinned down. We show now that this requirement, though it narrows down the set of equilibrium allocations, is not crucial to obtain our results. Precisely, we show that even if we can support a larger set of outcomes, some of them even inefficient, then commitment to single sourcing is still optimal for the single large buyer but not so for the two symmetric smaller buyers.

Take again first the case with a single large buyer. We now replace the truthfulness requirement by the weaker requirement that neither supplier is willing to sell more than the equilibrium quantity [[??].sup.m] at an incremental price that is strictly below incremental costs. Formally, for given [{[[??].sup.m]}.sub.m=Sl,S2] we require that for all [[DELTA].sub.x] [greater than or equal to] 0, (6)

[t.sup.m]([[??].sup.m] + [[DELTA].sub.x]) - [t.sup.m]([[??].sup.m]) [greater than or equal to] C([[??].sup.m] + [[DELTA].sub.x]) - C([[??].sup.m]). (5)

The main implication of (5) is that for a given allocation it fully pins down the respective equilibrium transfers. Namely, each supplier will again extract a transfer that is exactly equal to the respective incremental costs at the other supplier. In difference to the truthfulness requirement, however, under the weaker condition (5), we can now support any allocation where 0 [less than or equal to] [[??].sup.m] [less than or equal to] X and thus no longer only the efficient allocation with [[??].sup.m] = X/2. Intuitively, requirement (5) still allows supplier m to make it extremely unattractive for the buyer to purchase any quantity lower than [[??].sup.m] from him, say by specifying an exorbitantly high transfer [t.sup.m](X) for all x < [[??].sup.m]. (7)

For any allocation with 0 [less than or equal to] [[??].sup.m] [less than or equal to] X, the single buyer will then pay the total price of

[summation over(m=S1,S2)] [C(X) - C([[??].sup.m])], (6)

which just sums up the incremental costs at the respective alternative supplier. Differentiating (6) with respect to [[??].sup.m] and noting that [[??].sup.m'] = X - [[??].sup.m] for m' [not equal to] m shows that the total price is still minimized at the two corners where [[??].sup.S1] = X or [[??].sup.S2] = X.

With two buyers, we have in analogy to (5) the requirement that for all [DELTA] x [greater than or equal to] 0,

[t.sup.m.sub.n]([[??].sup.m.sub.n] + [[DELTA].sub.x]) - [t.sup.m.sub.n] ([[??].sup.m.sub.n]) [greater than or equal to] C([[??].sup.m] + [[DELTA].sub.x]) - C([[??].sup.m]). (7)

With (7) instead of (2), we can again support a wider range of equilibrium allocations. However, buyers' competition on the procurement market now limits the market share that any given supplier can obtain. (See Corollary 1 below.) We are first interested in whether our result from Proposition 1, namely that each of the two small buyers strictly prefers multiple sourcing, still holds. This is indeed the case.

Proposition 4. The results from Proposition 1 carry over if we replace the stronger truthfulness requirement by the weaker requirement that incremental quantity is not offered below incremental costs. That is:

(i) If there is a single large buyer, then single sourcing is still uniquely optimal.

(ii) If there are two buyers, then both buyers are worse off under single sourcing.

Proof. See the Appendix.

As a byproduct of the proof of Proposition 4, we have the following characterization of all equilibria, including those that are inefficient.

Corollary 1. Replacing the truthfulness requirement by (5) and (7), respectively, the following equilibrium allocations can now be supported:

(i) If there is a single large buyer, then the market share of either supplier can range from zero to 100%.

(ii) Instead, if there are two symmetric buyers, then either supplier can only have a share between 1/3 to 2/3 of the total procurement volume.

Proof. See the Appendix.

Corollary 1 is interesting in itself. Put differently, it says that if a given procurement volume is distributed over more than one buyer, as in assertion (ii), then this provides tighter bounds on the inefficiencies in production that can arise in equilibrium as some suppliers sell more than others.