We investigate how an alternative procurement process alters the results of the initial process analyzed in the rest of this article. In each period the buyer now announces first the quantities to be sourced from the supplier posting the higher and the lower price, D = Qtlow+ Qthigh. Adapting the tie-breaking rule of proposition 1, at equal prices Qtlow is sourced from supplier 1. Next, the active suppliers post their prices pti and depending on the relative prices are awarded the respective quantities. All other model specifications are identical to the initial process described in section 2.2. While the alternative process thus allows to guarantee a non-zero quantity to the supplier with the higher price, it also encompasses the option to implement a myopic sourcing process by setting Qtlow = D.
The outcome in the second period is then given by the following lemma.
Lemma 7. For the alternative procurement process, equilibria in period 2 are unchanged from the initial process.
Proof of lemma 7. The proof is included in the appendix.
In the case of no active supplier and a monopoly the equivalence to the initial process is straightforward. When both suppliers are active, the buyer sets Q2Dlow= D, which yields
Figure 2.7: Probability of sustained competition for the numerical solution of the alterna-tive procurement process. Thereby D = 1.0, V = 1.0.
purchasing behavior identical to the initial model. Any Q2Dhigh > 0 entails purchases at prices above c2 and is thus inferior. In addition, a qualification regarding transaction prices in the first period can be made.
Lemma 8. For the alternative procurement process, the expected transaction price in period 1, p1T is never lower than the marginal cost of supplier 2.
Proof of lemma 8. The proof is included in the appendix.
Supplier 2 will not price below his cost as these prices are weakly dominated. Then there is no need for supplier 1 to set a price below c2 as any p11 ≤ p12 ensures that he sells Q1low. Hence, at least for (c1, c2) ∈ CL2, equilibrium prices in period 1 will differ from the initial process where p1∗T < c2 according to proposition 2.
We implement the alternative procurement process numerically using the approach described in section 2.4.1. In the subgame-perfect equilibrium, for δ = 1.0 the buyer then always dual sources by splitting demand in the first period. Accordingly, except for the highest values of c2, both suppliers mix over prices in equilibrium and as depicted in figure 2.7, competition is sustained with a probability fD ≥ 0.5, which is often close to one. However, when c2 is high, supplier 2 requires more than half of the demand to be sustained even at the highest price p12 = V . It is then optimal for the buyer to either set Q1low = D, realizing BS = D(V − c2) in period 1 or to post Q1high ≥ D/2 such that both suppliers price at p1i = V , thereby realizing the same surplus in period 2. The latter approach is selected due to assumption 3 as it sustains competition.
For δ = 0.5 meanwhile, the value of sustained competition for the buyer is markedly reduced and thus cannot increase surplus above BS = D(V −c2). Since now current surplus is valued higher, this leads the buyer to implement myopic purchasing by setting Qlow= D.
Accordingly, both suppliers price at c2, with the buyer sourcing only from supplier 1 and competition is never sustained (cf. figure 2.7).
We therefore find that compared to the initial process, the character of competition is changed away from the predominance of exclusion towards more accommodating outcomes when the buyer values sustained competition sufficiently high.
In addition, in figure 2.8 the difference of the expected buyer surplus for the alterna-tive and initial procurement processes, BSA− BSI is depicted. When the buyer values sustained competition highly for discount factors δ = 1.0, the alternative process can
Figure 2.8: Difference of expected buyer surplus between alternative and initial procure-ment processes for the numerical solution. Thereby D = 1.0, V = 1.0.
increase the surplus of the buyer for small differences of marginal costs and especially for (c1, c2) ∈ CH∪ CL2. While then in the initial process exclusion dominates, for the alternative process there exist equilibria where competition is nearly always sustained and the additional surplus from period 2 overcompensates foregone surplus in period 1. For higher cost differences however, there are equilibria such that exclusionary prices are so low or competition is sustained at prices which are so low that the initial process can be superior to the alternative process.
Also, when δ = 0.5, we found that there is an equilibrium in the alternative process which implements myopic buying by setting Q1low = D. Accordingly then, in accordance with the finding in section 2.3.3, buyer surplus is never higher than with the initial process for costs in CH∪ CL2. For the equilibria arising from the calculation this often also holds for (c1, c2) ∈ CL1. However, there also exist equilibria where competition is sustained and surplus is slightly increased by myopic purchasing as implemented in the equilibrium of the alternative process (cf. figure 2.8). Thus, the overall impact of the alternative procurement process can be described by the next result.
Numerical Result 3. For the alternative procurement process in a discretized version of the model, there often exist equilibria where the buyer dual sources and competition is sustained with positive probability when the value of sustained competition for the buyer is sufficiently high, including for (c1, c2) ∈ CH∪ CL2. Buyer surplus can, but need not, increase compared to the initial process, especially for small cost differences.
These results are robust for ∆Q= 0.2 as well as ∆Q = 0.05. With ∆Q= 0.5 however, splitting demand in the alternative process guarantees a quantity of D/2 to the higher-priced supplier. Accordingly, for δ = 1.0 both suppliers price at V and competition is always sustained but surplus is then never higher than in the initial process.
Hence, even though the alternative procurement process always yields the same out-come as the initial process in a myopic situation, equilibrium outout-comes in a strategic context can be quite different. Thus, the choice of the procurement process can substan-tially influence surplus values for the buyer as well as the structure of the future supplier market. Furthermore, the optimality of the process might critically depend on the exact situation as demonstrated by the results for different marginal cost combinations.