Single sourcing versus multiple sourcing.

[C(X +a - [[??].sup.m.sub.n]) - C(X/2 + a)] - [C(X - [[??].sup.m.sub.n]) - C(X/2)] < [C(X) - C(X- [[??].sup.m.sub.n])] - [C(X/2 - a +[[??].sup.m.sub.n]) - C(X/2 - a)]. (22)

Note next for the expressions in the first line of (22) that (X + a - [[??].sup.m.sub.n]) - (X/2 + a) = X/2 - [[??].sup.m.sub.n] and (X - [[??].sup.m.sub.n]) - (X/2) = X/2 - [[??].sup.m.sub.n]. With these observations, we can then transform the first line from (22) into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Likewise, after noting for the expressions in the second line of (22) that (X) - (X - [[??].sup.m.sub.n]) = [[??].sup.m.sub.n] and (X/2 - a + [[??].sup.m.sub.n]) - (X/2 - a) = [[??].sup.m.sub.n], this transforms to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, (22) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

We now distinguish between two cases. In the first case, buyer n purchases at leastX/4 from the large firm such that [[??].sup.m.sub.n] [greater than or equal to] X/2 - [[??].sup.m.sub.n]. Note next that X - [[??].sup.m.sub.n] [greater than or equal to] X/2 + a. To see this, substitute the highest possible value for [[??].sup.m.sub.n], which by the arguments from the proof of Claim 1 equals [[??].sup.m.sub.n] = X/2 - a, in which case this condition holds with equality. Hence, we can make (23) (weakly) stricter by replacing the boundary X - [[??].sup.m.sub.n] in the integral on the fight-hand side by the boundary X/2 + a. That the thereby "relaxed" condition (23) holds then still strictly follows as X2 > X/2 - a due to a > 0, as [[??].sup.m.sub.n] [greater than or equal to] X/2 - [[??].sup.m.sub.n] by assumption of the current case, and as C'' > 0. In words, we have thus shown that if a buyer purchases at least X/4 from the larger supplier, then he pays overall always less than under single sourcing.

We turn next to the case where [[??].sup.m.sub.n] < X/4. We proceed in analogy to the previous case, though this time we extend the second line in (21) in a different way, namely by writing instead

With this, (21) transforms to the requirement

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

and finally to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

Condition (24) now holds surely as [[??].sup.m.sub.n] < X/2 -[[??].sup.m.sub.n] holds due to [[??].sup.m.sub.n] < X/4 and as

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

holds from [[??].sup.m.sub.n] [greater than or equal to] 2a, which was shown in the proof of Claim 1. Q.E.D.

Proof of Corollary 1. The proof follows almost directly from Proposition 3. There, we have shown that if m is the larger supplier with [[??].sup.m] = X/2 + a, then it must hold that a [less than or equal to] X/6. Consequently, we have the upper boundary [X.sub.B2] [less than or equal to] X/2 + X/6 = 21(/3. Q.E.D.

Proof of Proposition 5. Take first the case with a single large buyer. From (9), the buyer's payoff without single sourcing equals

R(X) - 2[[R(X) - R(X')] + [C(X') - C(XJ2)]]

compared to R(X') - C(X') under single sourcing, where X' was defined in (8). Single sourcing is thus strictly more profitable whenever

R(X) - 2C(X/2) > R(X') - C(X')

holds as X maximizes R(x) - 2C(x/2) and as 2C(x/2) < C(x).

Consider next the case with two buyers. Observe first more formally that given strict convexity of C and strict concavity oft, the program to maximize total surplus is indeed strictly concave. It is also again straightforward that under the truthfulness requirement, an allocation is supported as an equilibrium if and only if it is efficient. Off- equilibrium, if buyer n rejects the bid of supplier m, his payoff is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As [[??].sup.m.sub.n] makes the buyer again indifferent, in equilibrium buyer n, when purchasing [[??].sup.m.sub.n] from the respective suppliers, will realize the payoff

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

We consider now a shift that makes a buyer's purchases more concentrated on a given supplier. For this it is convenient to denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

and to also define for the respective choice of [[DELTA].sub.x] the quantity [??](x) := x [[DELTA].sub.x]. As we increase [[??].sup.m.sub.n] marginally while thereby reducing [[??].sup.m'.sub.n] = X/2 - [[??].sup.m.sub.n], (25) increases strictly if

which for [[??].sup.m.sub.n] X/2 holds if [OMEGA] (x) is strictly concave. In the remainder of the proof, we show that this is indeed the case. Using from the envelope theorem that [OMEGA]'(x) = r'([??]), we have that [OMEGA]"(x) = r"([??]) d[??] / dx < 0 holds whenever d[??] / dx > 0. To see that this is finally the case, note that from implicitly differentiating the respective first-order condition d[OMEGA]/d[DELTA].sub.x] = 0 for (26), we have that

d[??] / dx = C"(X/2 + [[DELTA].sub.x]) / r" ([??]) - C"(X/2 + [[DELTA].sub.x]) > 0.

Q.E.D.

Proof of Proposition 6. Suppose supplier m rejects the bid of buyer n. In this case, the supplier's payoff is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As the supplier's payoff on equilibrium is just [[??].sup.m.sub.n] + [[??].sup.m.sub.n'] - C(X/2), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consequently, a given buyer n's payoff, which is r(X/2) - [[??].sup.m.sub.n] - [[??].sup.m'.sub.n] , becomes now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Define now in analogy to the proof of Proposition 5 the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and denote for the respective choice of [[DELTA].sub.x] the quantity [??](x) := [[??].sup.m.sub.n'], + [DELTA].sub.x], which is the quantity produced optimally by supplier m after n rejected his offer. As we increase [[??].sup.m.sub.n] marginally while thereby reducing [[??].sup.m'.sub.n] = X/2 - [[??].sup.m.sub.n], the payoff of buyer n in (27) now decreases if

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

which for [[??].sup.m.sub.n] [greater than or equal to] X/2 is again the case if [??](x) is strictly concave. The argument for why [??] is strictly concave is now the same as that for [??] in the proof of Proposition 5.

Finally, note that the argument also holds if buyers always purchase X/2. In this case, we have [DELTA].sub.x] = 0 in all the preceding expressions such that [??](x) := r(X/2) - C(x), which is again strictly concave in x by strict convexity of C. Q.E.D.

References

ANTON, J. AND YAO, D. "Second Sourcing and the Experience Curve: Price Competition in Defence Procurement." RAND Journal of Economics, Vol. 18 (1987), pp. 57-76.

--. "Split Awards, Procurement and Innovation." RAND Journal of Economics, Vol. 20 (1989), pp. 538-552.

BERNHEIM, B. D. AND WHINSTON, M. D. "Menu Auctions, Resource Allocation, and Economic Influence." Quarterly Journal of Economics, Vol. 101 (1986), pp. 1-31.

BIGLAISER, G. AND VETTAS, N. "Dynamic Price Competition with Capacity Constraints and Strategic Buyers" Working Paper, University of North Carolina, 2005.

CHIPTY, T. AND SNYDER, C. M. "The Role of Outlet Size in Bilateral Bargaining: A Study of the Cable Television Industry." Review of Economics and Statistics, Vol. 81 (1999), pp. 326-340.

ELMAGHRABY, W. J. "Supply Contract Competition and Sourcing Policies." Manufacturing & Service Operations Management, Vol. 2 (2000) pp. 350-337.

LAFFONT, J.-J. AND TIROLE, J. "Repeated Auctions of Incentive Contracts, Investment and Bidding with an Application to Takeovers." RAND Journal of Economics, Vol. 19 (1988), pp. 516-537. RIORDAN, M. AND SAPPINGTON, D. "Second Sourcing." RAND Journal of Economics, Vol. 20 (1989), pp. 41-57.

TUNCA, T. I. AND WU, Q. "Multiple Sourcing and Procurement Process Selection with Bidding Events." Working Paper, Stanford Graduate School of Business, 2005.

WILSON, R. "Auctions of Shares." Quarterly Journal of Economics, Vol. 93 (1979), pp. 675-689.

(1) The theoretical literature on share or menu auctions, to which Anton and Yao (1989) as well as our article contributes, originates from work by Wilson (1979) and Bernheim and Whinston (1986).

(2) For instance, Anton and Yao (1987), Laffont and Tirole (1988), and Riordan and Sappington (1989) show how second sourcing reduces informational rents in a dynamic model. Recently, Biglaiser and Vettas (2005) have taken a different route by analyzing equilibrium purchasing strategies over time if suppliers have fixed capacity.

(3) To see this briefly, note first that the program to minimize total production costs [[summation].sub.m=S1,S2] C([x.sup.m.sub.1] + [x.sup.m.sub.2], where [[summation].sub.n=S1,S2] [x.sup.m.sub.n] = X/2 for both n = 1, 2, is strictly concave. Note next that by (2), it must hold for any (x, x') 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']). Consequently, holding all other [[??].sup.m'.sub.n'] fixed, a buyer's choice of [[??].sup.m.sub.n] must also satisfy the first-order condition of the preceding program.

(4) Importantly, we will show that what matters is the relative size, that is, relative to the size of the total procurement market, rather than a buyer's absolute size.

(5) This requires also to extend the truthfulness requirement (2) accordingly.