The more the better? Optimal degree of supply-chain cooperation between competitors

Abstract

We consider the outsourcing strategy problem of two competing original equipment manufacturers (OEMs) whose products are each made up of two components. The OEMs have different specializations, and therefore the component that each firm can produce in-house is different. Each firm must decide whether to outsource the other component to the competing OEM or to a third-party supplier. Prior research has demonstrated that competitors can be better off cooperating as supply-chain partners; therefore, one might expect that, as long as the OEMs are not at a severe cost disadvantage, they should maximize their cooperation as supply-chain partners, especially when competition between products is strong. Interestingly, this study finds that more cooperation between competitors may actually be harmful. Under certain conditions, while one of the OEMs should outsource to the competing firm, the other should outsource to a third-party supplier, even when the third-party supplier is more expensive and the competition is intense.

Appendices

Appendix A

A.1. Derivation of equilibrium solutions

Strategy TT: π _i is concave in p _i . Solving the first-order conditions (FOCs) gives p _i as a function of w _i . Next, Π _i is concave in w _i . Solving the FOCs, we have w _i =((1−θ)(2+θ)+2Δc−Δcθ²)/(4−θ−2θ²).

Strategy TC: π _i is concave in p _i . Solving the FOCs gives p _i as a function of w _i . Next, Π _B is concave in w _B and π _G is concave in w _G . Solving the FOCs, we have w _G =((2+θ)(16−8θ−4θ²+3θ³−θ⁴)−Δcθ³(2−θ²))/(64−24θ²−5θ⁴+θ⁶) and w _B =(2Δc(2−θ²)(8+θ²)+(2−θ)(1−θ)(2+θ)²(4−θ+θ²))/(64−24θ²−5θ⁴+θ⁶).

Strategy CC: π _i is concave in p _i . Solving the FOCs gives p _i as a function of w _i . Next, π _i is concave in w _i . Solving the FOCs, we have w _i =(4−2θ+θ²)/(8+θ²).

Appendix B

B.1. Proof of propositions

In order to guarantee that demands of both products are positive, we consider the region where 0<Δc<UB, where UB≡((1−θ)(2+θ)(4−θ+θ²))/(8−θ²−θ⁴). This is the necessary and sufficient condition for D _G ^TC>0. It can be shown that UB>0 and (∂UB)/(∂θ)<0.

Proof of Proposition 1

• Part a:

  

  It can be shown that the numerator and the denominator are positive since 0⩽θ<1. Therefore,

  

  Lastly,

  

  Part b:

  
  

  □

Proof of Proposition 2

• Part a:

  

  The denominator is positive, and it can easily be shown that the numerator is positive if and only if .

  Part b: In the proof below, we need to show that a polynomial, f(Δc, θ), is positive within the region 0<Δc<UB(θ) and 0⩽θ<1 (cf Figure 1 for a visual presentation of the domain). We use the methodology proposed by Pun and Ghamat (2014) to solve this problem. In particular, the minimum of a polynomial is either at the boundaries or at the solutions of the FOCs. The solutions to the FOCs are numeric values, so it can be shown that f(Δc, θ)>0 at these solutions. The boundaries f(0, θ), f(UB(θ),θ) and f(Δc, 0) are single-variable polynomials that can be shown to be positive. Therefore, f(Δc, θ)>0 is true if f(Δc, θ) is positive at the solutions of the FOCs and at the boundaries

  To illustrate, consider the proof of π _G ^CC>π _G ^CT. Recall from Appendix A that

  

  and

  

  The denominator of π _G ^CT is positive because θ<1. Then π _G ^CC>π _G ^CT⇔Γ _{CCC T} (Δc, θ)>0, where Γ _CCCT (Δc, θ)≡−4Δc²θ²(2−θ²)²(2+θ²)(8+θ²)²−4Δc(1−θ)θ(2−θ²)(2+θ²)(8+θ²)²(8+4θ−3θ²−θ³)+(1−θ)(16384−12288θ²+1024θ³+2432θ⁴+128θ⁵+576θ⁶−208θ⁷−300θ⁸+14θ⁹−11θ¹⁰+13θ¹¹+11θ¹²+θ¹³). There is no root to the FOCs (∂Γ _CCCT (Δc, θ))/(∂θ)=0 and (∂Γ _CCCT (Δc, θ))/(∂Δc)=0. Therefore, the minimum value of Γ _CCCT (Δc, θ) must be at the boundary. However, the boundary is positive because (1) Γ _CCCT (Δc, 0)=16384>0, (2) Γ _CCCT (0, θ)=(1−θ)(16384−12288θ²+1024θ³+243²θ⁴+128θ⁵+576θ⁶−208θ⁷−300θ⁸+14θ⁹−11θ¹⁰+13θ¹¹+11θ¹²+θ¹³)>0 because there is no root to (∂Γ _CCCT (0, θ))/(∂θ)=0, Γ _CCCT (0, 0)⩾0 and Γ _CCCT (0, 1)⩾0, and (3) Γ _CCCT (UB(θ), θ)=((1−θ)(64−24θ²−5θ⁴+θ⁶)²X(θ))/((8−θ²−θ⁴)²), where X(θ)≡256−128θ−64θ²+32θ³−12θ⁴+78θ⁵+29θ⁶+17θ⁷+7θ⁸+θ⁹. X(θ) is positive because X(0)=256, X(1)=216, (∂X(θ))/(∂θ)<0⇔<0.74639θ and X(0.74639)=160.412. As a summary, since there is no root to (∂Γ _CCCT (Δc, θ))/(∂θ)=0 and (∂Γ _CCCT (Δc, θ))/(∂Δc)=0 and the boundary is positive, Γ _CCCT (Δc, θ)>0 is true, and hence π _G ^CC>π _G ^CT is true.

  We use similar logic to show the other three inequalities π _G ^TC>π _G ^CT, π _G ^TC>π _G ^TT, and π _G ^CC>π _G ^TT, so we omit the detail to avoid redundancy.

  Part c:

  

  The denominator is positive. There are two roots to the numerator: Γ₁(θ)≡(θ(1−θ)(32−32θ−12θ²+40θ³−8θ⁴−12θ⁵+3θ⁶+θ⁷))/((2−θ²)(64−32θ−48θ²+28θ³−9θ⁴−θ⁵+5θ⁶−θ⁷)) and ((1−θ)(8−3θ²)(32−4θ−12θ²−6θ⁴+θ⁵+θ⁶))/((2−θ²)(64−96θ+20θ³−θ⁴+11θ⁵−3θ⁶−θ⁷)). Note that these two roots are always real for 0⩽θ<1. Using the logic presented above, it can be shown that the second root is outside of the feasible region and hence the only root is Γ₁(θ), so the result follows.

  Part d: Define

  
  

  There are two roots to this quadratic equation, but only Γ₂(θ) is smaller than UB. Note that the term inside the square root of Γ₂(θ) is positive when θ<0.445, that is, 32−96θ+60θ²−12θ³−15θ⁴+30θ⁵−5θ⁶+6θ⁷>0⇔θ<0.445. Therefore, Γ₂(θ) is only defined for θ<0.445. However, we consider the region for 0<Δc<UB(θ). Then it can be shown that . When , it is trivial to show that π _G ^CC<π _G ^TC. Therefore, the result follows. □

Proof of Proposition 3

• Note that the thresholds Γ₁(θ) and Γ₂(θ) are only a function of θ. Therefore we can easily take the derivative of Γ₁(θ) and Γ₂(θ). Then we can show that the derivative is always positive when 0⩽θ<1. Similarly, Γ₂(θ)>Γ₁(θ) can be shown by taking the difference between these two functions and show that the difference has no root (as a function of θ) when 0⩽θ<1.

  In order to derive the outsourcing structure, we solve the following two-by-two matrix:

  

  Recall that π _S ^TT=π _G ^TT, π _S ^TC=π _G ^CT, π _S ^CT=π _G ^TC and π _S ^CC=π _G ^CC. The equilibrium is strategy TT when π _G ^TT>π _G ^CT, which is true when Δc<Γ₁(θ) (cf Proposition 2(c)). Similarly, the equilibrium is strategy CC when π _G ^CC>π _G ^TC, which is true when (cf Proposition 2(d)). Lastly, the equilibrium is strategy TC or CT when π _G ^TT<π _G ^CT and π _G ^CC<π _G ^TC, which is true when Δc>Γ₁(θ) and . □