Single sourcing versus multiple sourcing.

With truthful menus, it is immediate that the equilibrium outcome must be efficient given that the schedule of payments offered to the retailer by either supplier reflects the respective supplier's marginal costs at all quantity levels. By symmetry and strict convexity of costs, efficiency requires that each supplier produces X/2. Another implication of the truthfulness requirement is that if the buyer rejects one of the two bids and thus ends up purchasing the entire quantity X from a single supplier, then the incremental price equals the respective supplier's incremental costs, namely C(X) - C(X/2). In equilibrium, it holds from optimality that each supplier chooses [t.sup.m] (Y/2) such that the buyer is just indifferent between acceptance and rejection. Taken together, this implies that [t.sup.m](X/2) is just equal to the incremental costs of procuring instead only from a single supplier, C(X) - C(X/2). Altogether, the buyer thus ends up paying [2t.sup.m](X/2), which is equal to 2[C(X) - C(X/2)].

If the buyer commits to single sourcing, then by symmetry, suppliers compete themselves down to zero profits. Consequently, the buyer has to compensate the winning supplier just for the respective costs of production, C(X). Single sourcing is then optimal if the respective total payment C(X) is strictly smaller than the respective payment under multiple sourcing, 2[C(X) - C(X/2)]. After rearranging expressions, this is the case if C(X) < 2C(X/2), which in turn holds from strict convexity of C.

[] Second benchmark: the case with two symmetric buyers. Our second benchmark case is that with two symmetric buyers such that [X.sub.B1] = [X.sub.B2] = X/2. We first extend the truthfulness requirement to this setting. (Again, we show below that our main results continue to hold if we do not impose this restriction.) For this, we first denote for a given equilibrium the resulting allocation, that is, the distribution of sales and purchases over suppliers and buyers, by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. That is, [[??].sup.m.sub.n] denotes the quantity that supplier m will sell to buyer n in the respective equilibrium. The respective transfers are denoted by [[??].sup.m.sub.n] := [t.sup.m.sub.n]([[??].sup.m.sub.n]). Note also that the supplier's total level of production is then [[??].sup.m] := [[summation].sub.n = B1, B2] [[??].sup.m.sub.n].

If a menu [t.sup.m.sub.n] truthfully reflects the incremental costs of supplier m, then given the supplier's (rationally anticipated) total production [[??].sup.m] we must have 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]) = C([[??].sup.m] + [[DELTA].sub.x]) - C([[??].sup.m]). (2)

In words, buyer n can purchase from m an incremental quantity [[DELTA].sub.x] above [[??].sup.m.sub.n] at m's incremental costs, where incremental costs are calculated on the basis of the supplier's rationally anticipated production volume [[??].sup.m] under the respective equilibrium. At this point, it may be useful to note that another way of expressing the truthfulness requirement (2), which then looks more similar to that in (1), is that

[t.sup.m.sub.n](x') - [t.sup.m.sub.n](x) = C(x' + [[??].sup.m.sub.n']) - C(x + [[??].sup.m.sub.n'])

must hold for any pair x' and x and for n' [not equal to] n.

With (2) it is again intuitive that without further restrictions on the allocation of sales and purchases, an equilibrium must again be efficient. Again, this requires that each supplier produces just one half of the total volume X/2. On the other hand, as the suppliers' products are homogeneous, it does not matter for efficiency how each buyer mixes and matches between the products from the two suppliers. Consequently, all allocations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are efficient as long as [[??].sup.S1] = [[??].sup.S2] = X/2. Intuitively, we can also support all of these (efficient) allocations as equilibrium outcomes. (3)

At one extreme of the characterized continuum of equilibrium allocations is the case where each buyer purchases exclusively from one supplier. (This clearly uses the symmetry of buyers and suppliers.) At the other extreme is the case where each buyer maximally spreads his purchases over both suppliers, thus purchasing X/4 from either supplier. Note once again, though, that in either case each supplier produces exactly X/2.

We next turn to equilibrium transfers. By optimality, the required payment [[??].sup.m.sub.n] makes buyer n again just indifferent between procuring [[??].sup.m.sub.n] from supplier m or, instead, increasing his purchases from the alternative supplier m' by [[??].sup.m.sub.n]. Note that from the truthfulness requirement (2), the incremental price that n would have to pay to the other supplier m' equals the supplier's respective incremental costs C(X/2 + [[??].sup.m.sub.n]) - C(X/2), where we use that [[??].sup.m'], = X/2 holds by efficiency. Hence, we have that [[??].sup.m.sub.n] = C(X/2 + [[??].sup.m.sub.n]) - C(X/2).

Summing up over the payments made to the two suppliers, buyer n will thus end up paying the total price of

[summation over m=S1,S2] [C(X/2 + [[??].sup.m.sub.n]) - C(X/2)]. (3)

We analyze next how (3) changes as we move between different (efficient) allocations. Precisely, we are interested in how (3) changes as a buyer's purchases become more or less concentrated on one supplier. For this purpose, we specify now without loss of generality that a given buyer n purchases (weakly) more from some supplier m such that [[??].sup.m.sub.n] [greater than or equal to] [[??].sup.m'.sub.n]. That is, we can refer to supplier m as the larger supplier to buyer n. Differentiating then (3) with respect to [[??].sup.m.sub.n], while using that the buyer's total purchases remain constant at [[summation].sub.m=S1,S2] [[??].sup.m.sub.n] = X/2, we thus have that shifting more of the buyer's purchases to his larger supplier m increases total purchasing costs if

C'(X/2 + [[??].sup.m.sub.n]) > C' (X/2 + [[??].sup.m'.sub.n]). (4)

From strict convexity of C(.), we have that condition (4) holds indeed whenever [[??].sup.m.sub.n] > [[??].sup.m'.sub.n] holds strictly.

We can rephrase this result as follows. With two symmetric buyers, a buyer's total purchasing costs, as given by (3), are minimized if the buyer spreads his purchases evenly over the two suppliers. At the other extreme, purchasing costs are highest if the buyer's purchases are concentrated on only a single supplier.

[] Comparison. With two symmetric buyers, the outcome under single sourcing represents thus the worst possible outcome for either buyer. The crucial difference to the case with a single supplier is the following. With two buyers, if one buyer purchases more (or even all) from one supplier, then the other supplier simply sells more to the second buyer. In contrast, if there is only a single buyer and if this buyer commits to single sourcing, then one of the two suppliers will end up producing nothing. In the latter case, single sourcing therefore leads to "all-out" competition between suppliers, whereas this is not the case if there are two symmetric buyers and suppliers can consequently perfectly reallocate sales.

To put this differently, note first that what sustains a buyer's payoff in the auction above zero is the possibility to purchase the respective quantity instead from the other supplier. Under the truthtelling requirement, the buyer would then have to compensate the other supplier for the respective incremental costs of production. (See, however, Section 5 on a generalization of our results to all Nash equilibria of the auction.) By spreading purchases equally over the two suppliers, a buyer purchasing X/2 in total and thus X/4 at each supplier minimizes the "average dependency" on either supplier, where dependency is measured by the costs it takes to replace the respective volume of purchases. More formally, note that a buyer obtains a more attractive bid from a given supplier if the incremental purchasing costs that he would incur when procuring instead exclusively from the other supplier are low. If a large fraction of a buyer's purchases is concentrated on one supplier, while both suppliers still produce exactly X/2 in equilibrium, then this increases the incremental costs per unit when switching away from this supplier. Though on the other side it is then also less costly to replace the smaller volume that the buyer purchases from the other supplier, given strict convexity of suppliers' costs the average costs of substitution are still strictly higher the more unevenly the buyer's purchases are distributed over suppliers.

We summarize our results for the two benchmark cases as follows.

Proposition 1. A single buyer who conducts an auction strictly prefers to commit to single sourcing. In contrast, if there are two symmetric buyers, then either buyer strictly prefers to spread his purchases evenly over both suppliers. More generally, in the latter case, a buyer's costs of purchasing increase the more his purchases are concentrated on a single supplier, making single sourcing the worst outcome.

4. Analysis with asymmetric buyers

* In the preceding section, we compared the cases where a buyer either accounted for the whole of the respective procurement market or for only one half of it. The comparison of the two cases suggests, more generally, that single sourcing is more likely to be optimal, at least in the current case where buyers run auctions, if a buyer accounts for a larger fraction of the total procurement market. (4)