Single sourcing versus multiple sourcing.

* This article takes as a starting point the seminal analysis on optimal procurement strategies by Anton and Yao (1989). As in their basic setting, we explore a parsimonious static model with symmetric information. Our departure from Anton and Yao (1989) is that we allow for more than one buyer and that, in the second part of the article, we also consider auctions that are organized by suppliers instead of buyers. Interpreting the difference in procurement formats more generally as a difference in contracting power, our analysis provides, despite the simplicity of the setting, a rich set of results on when single sourcing as opposed to multiple sourcing may be an optimal strategy.

One key insight is the following. Whether single sourcing is optimal or not depends on a buyer's relative size, more precisely, on the fraction of the total procurement market that the buyer accounts for in equilibrium. Only sufficiently large buyers can substantially change the total allocation of production among suppliers if they commit to single sourcing. In contrast, single sourcing by a small buyer will merely lead to a reshuffling of purchases and sales without affecting any supplier's overall production. A second and related observation is that single sourcing serves different purposes under the two considered procurement formats. If suppliers post bids, then for a sufficiently large buyer, single sourcing makes competing bids more attractive. Instead, for smaller buyers, single sourcing makes their alternative options less attractive and thus allows suppliers to extract a higher price. If buyers post bids, committing to single sourcing affects now in a systematic way the attractiveness of suppliers' alternative options, namely to supply more to another buyer.

Appendix

* Proof of Proposition 2. For buyer n we have from (3), which still applies, that in case [[??].sup.S1.sub.n] = [[??].sup.S2.sub.n] = [X.sub.n]/2, the total purchasing price equals 2[C(X/2 + [X.sub.n]/2) - C(X/2)]. With single sourcing, the buyer pays likewise the incremental costs C(X) - C([X.sub.n']), where n' [not equal to] n. Note that in equilibrium the other supplier will sell [X.sub.n'] to buyer n'. (10) Comparing total purchasing costs, single sourcing is then (weakly) optimal for buyer n if

2[C(X/2 + [X.sub.n]/2) - C(X/2)] - [C(X) - C([X.sub.n'])] [greater than or equal to] 0. (11)

Note first that condition (11) holds strictly at [X.sub.n] = X, whereas the converse holds strictly at [X.sub.n] = [X.sub.n'] = X/2. To prove the assertion for the large buyer, it remains to show that (11) is strictly increasing in [X.sub.n] = [X.sub.B1]. Differentiating the left-hand side of (11) w.r.t. [X.sub.B1] while using that [X.sub.n'] = [X.sub.B2] = X - [X.sub.B1], the derivative is strictly positive if and only if C' (X/2 + [X.sub.B1]/2) > C' ([X.sub.B2]. This holds as [X.sub.B2] < X/2 < [X.sub.B1].

For the small buyer, note that with [X.sub.n] = [X.sub.B2] and [X.sub.n'] = [X.sub.B1] = X - [X.sub.B2] the derivative of the left-hand side of (11) w.r.t. [X.sub.B2] is now negative if and only if C' (X/2 + [X.sub.B2]/2) < C' ([X.sub.B1]) and thus [X.sub.B2] < X/3. As we also know that (11) does not hold at [X.sub.B2] = X/2 and as the value of the left-hand side of(l l) is clearly zero at [X.sub.B2] = 0, (11) does not hold for the smaller buyer. Q.E.D.

Proof of Proposition 3. We thus suppose now more generally that [[??].sup.m.sub.B2] = [beta] [X.sub.B2] and [[??].sup.m'.sub.B2] = (1 - [beta])[X.sub.B2] hold without single sourcing for some 1/2 < [beta] < 1. Using that [[??].sup.m.sub.B1] = X/2 - [[??].sup.m.sub.B2] and that [[??].sup.m'.sub.B1] = X/2 - [[??].sup.m'.sub.B2] together with (3), the total price of the large buyer is equal to

[C(X/2 + (X/2 - [beta][X.sub.B2])) - C(X/2)] + [C(X/2 + (X/2 - (1 - [beta])[X.sub.B2])) - C(X/2)],

implying that the large buyer prefers single sourcing if

[C(X - [beta] [X.sub.B2]) + C(X - (1 - [beta])[X.sub.B2]) - 2C(X/2)] - [C(X) - C([X.sub.B2])] [greater than or equal to] 0. (12)

We establish first that also for each [beta] > 1/2 there is a unique 0 < [gamma] < 1 at which (12) is satisfied with equality. To see this, note again first that at [X.sub.B2] = 0 and thus [gamma] = 1 the left-hand side in (12) is strictly positive, whereas at [gamma] = 1/2 and thus [X.sub.B2] = X/2 it is strictly negative. Note next that the derivative of the left-hand side of (12) with respect to [X.sub.B2] equals

- [beta]C'(X - [beta][X.sub.B2]) - (1 - [beta])C'(X- (1 - [beta])[X.sub.B2]) + C'([X.sub.B2]). (13)

Using that X - [beta] [X.sub.B2] > [X.sub.B1] > [X.sub.B2] and X - (1 - [beta])[X.sub.B2] > [X.sub.B1] > [X.sub.B2], it holds that C' (X - [beta] [X.sub.B2]) > C' ([X.sub.B2]) and that C' (X - (1 - [beta])[X.sub.B2]) > C' ([X.sub.B2]), by which (13) is strictly negative.

Having thus established existence of a threshold F for each [beta], we obtain next

d[gamma]/d[beta] = 1/x C'(X - (1 - [beta])][X.sub.B2]) - C'(X - [beta][X.sub.B2])/. C'([X.sub.B2]) - [[beta]C'(X - [beta][X.sub.B2]) + (1 - [beta])C'(X - (1 - [beta])[X.sub.B2])]

where we used that [beta] > 1/2. Q.E.D.

Proof of Proposition 4. Assertion (i) follows from the arguments in the main text. Turn thus to assertion (ii). We argue first that if one buyer single sources, then ignoring the identity of the two suppliers, the outcome is unique as each supplier sells to exactly one buyer. That is, we can rule out cases where for some (n, m) it holds that [x.sup.m.sub.n] = X/2 while also [x.sup.m.sub.n'] > 0. We argue to a contradiction, in which case we have the following requirements. First, from optimality for the supplier and from (7) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where we also use that by assumption m' only sells to n'. Second, to make supplying also to buyer n' profitable for supplier m, it must hold that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)

Taken together, (14) and (15) thus jointly require that

2C(X/2) [greater than or equal to] C(X/2 + [[??].sup.m.sub.n']) + C([[??].sup.m'.sub.n']). (16)

As this is just satisfied with equality in case [[??].sup.m.sub.n'] = 0 and thus [[??].sup.m'.sub.n'] = X/2, while the right-hand side of (16) is strictly increasing in [[??].sup.m.sub.n'] using also that [[??].sup.m'.sub.n'] = X/2 - [[??].sup.m.sub.n'], we have thus arrived at a contradiction.

To prove assertion (ii) we thus have to show that the now unique total price paid under single sourcing, which is

C(X) - C(X/2), (17)

is strictly above the total price paid under any other equilibrium allocation. Suppose without loss of generality that m is the larger supplier with [[??].sup.m] = X/2 + a such that [[??].sup.m'] = X/2 - a. From Proposition 1 we can restrict consideration to values a > 0. In a first step, we now derive the range of values a that can be supported in equilibrium. For future reference, it is helpful to prove a stricter assertion, providing a full characterization of all equilibria, and to state this as a separate result.

Claim 1. Any equilibrium allocation for the case without the truthfulness requirement is fully characterized by two parameters, namely (a, [[??].sup.m.sub.n]), where a [member of] [0, X/6]and[[??].sup.m.sub.n] [member of] [(X/2 + a)/2, X/2 - a]. All other supplies are then obtained by the requirements that [[??].sup.m] = X/2 + a and that [[??].sub.n], = [[??].sub.n'] = X/2.

Proof Consider some transfer [[??].sup.m.sub.n]. Also with the weaker requirement (7), the maximum (and consequently equilibrium) transfer that supplier m can demand is given by the respective incremental costs of the other supplier m', namely

C(X/2 - a + [[??].sup.m.sub.n]) - C(X/2 - a). (18)

This can only be supported as an equilibrium outcome if it also covers the own incremental costs of supplier m, which equal

C(X/2 + a) - C (X/2 + a - [[??].sup.m.sub.n]). (19)

Hence, for both n it must hold under an equilibrium allocation that (18) is not smaller than (19), which transforms further to the requirement that

C(X/2 - a + [[??].sup.m.sub.n]) + C(X/2 + a - [[??].sup.m.sub.n]) [greater than or equal to] C(X/2 + a) + C(X/2 - a). (20)

Condition (20) is only satisfied if we have for both n that [[??].sup.m.sub.n] > 2a. If n' is the buyer that purchases (weakly) less from m, that is, if [[??].sup.m.sub.n'] [less than or equal to] [[??].sup.m.sub.n], then setting [[??].sup.m.sub.n'] = 2a and thus [[??].sup.m.sub.n] = X/2 - a yields finally the restriction that a [less than or equal to] X/6.

Although we have thus shown that the conditions in Claim 1 are necessary to support an allocation as an equilibrium, that these conditions are also sufficient follows immediately, as under the weaker requirement (7) we can choose all [t.sup.m.sub.n] (x) for x < [[??].sup.m.sub.n] arbitrarily high. Q.E.D.

Using now the characterization of Claim 1 and that in equilibrium transfers are equal to the respective incremental costs at the other supplier, the total price that n pays is strictly lower than (17) under single sourcing if

[C(X/2 +a + [[??].sup.m'.sub.n]) - C(X/2 +a)] + [C(X/2-a + [[??].sup.m.sub.n]) - C(X/2-a)] < C(X) - C(X/2). (21)

Extending the expression in the second line of (21) by writing

[C(X) - C(X - [[??].sup.m.sub.n)] - [C(X - [[??].sup.m.sub.n]) - C(X/2)]

and substituting [[??].sup.m'.sub.n] = X/2 - [[??].sup.m.sub.n] into the first line, after rearranging expressions we have the requirement that