Strategic interactions in service supply chain with horizontal competition

Abstract

We analyze strategic interaction issues that arise in service supply chain, such as consulting, service outsourcing, and travel service. To capture strategic interactions in service supply chain, we study a case where there are two service vendors providing competing service products and selling them through a common Service Integrator. In particular, we derive and compare equilibrium solutions (e.g. service prices, wholesale prices, service volumes) for the service supply chain under three different scenarios. We then study the effect of key parameters in the model upon the equilibrium solution using sensitivity analysis, and discuss our results along with a numerical experiment. Finally, future research direction is pointed out.

Appendix

Proof of Theorem 1

Given service prices p ₁,p ₂ by the Service Integrator and earlier decision variables w ₁,w ₂,S ₁,S ₂ by the service vendors, the first-order condition can be shown as

From $\frac{\partial \phi_{\mathrm{SIP}}}{\partial p_{i}}=0$, we get

To check the optimality, we have the Hessian matrix:

Since $\frac{\partial^{2}\phi_{\mathrm{SIP}}}{\partial p_{i}^{2}}=-2\beta_{p}-2\gamma_{p}<0$, $\frac{\partial^{2}\phi_{\mathrm{SIP}}}{\partial p_{i}\partial p_{3-i}}=\frac{\partial^{2}\phi_{\mathrm{SIP}}}{\partial p_{3-i}\partial p_{i}}=2\gamma_{p}>0$ and

$$\left|\begin{array}{c@{\quad }c} -2\beta_{p}-2\gamma_{p} & 2\gamma_{p}\\ 2\gamma_{p} & -2\beta_{p}-2\gamma_{p}\end{array}\right| =4(\beta_{p}+\gamma_{p})^{2}-4\gamma_{p}^{2}>0, $$

where β _p >0,γ _p >0, we have a negative definite Hessian. Therefore, the p _i calculated above are the optimal response functions for the Service Integrator. Solving for $p_{i}^{*}$ by plugging p _i into p _3−i , we have

Substituting (6) into (1) and simplifying, we get (7). □

Proof of Theorem 2

The service vendors in this game simultaneously announce their prices w _i and service volume S _i , respectively. Owing to Service Integrator’s response function, they can calculate the optimal w _i and S _i . From the Service Integrator’s response function, we can obtain that

(16)

Substituting (16) and (6) into (1) and simplifying, we get

(17)

From (17) and (4),

Given earlier decision variables w ₁,w ₂,S ₁,S ₂ by the service vendors, $\varphi_{\mathrm{SV}_{i}}$ is the profit of service vendor i at this stage. To obtain the optimal wholesale price w _i , above all, we check in the first-order condition.

Moreover,

Therefore, we have a negative definite Hessian. $\varphi_{\mathrm{SV}_{i}}$ is strictly jointly concave in w _i and S _i .

From $\frac{\partial \varphi_{\mathrm{SV}_{i}}}{\partial w_{i}}=0$, $\frac{\partial \varphi_{\mathrm{SV}_{i}}}{\partial S_{i}}=0$, we get

(18)

(19)

Substituting (19) into (18), we obtain (8). □

Proof of Theorem 3

Using a similar method as in proof of Theorem 2, one can obtain very easily the validity of Theorem 3. Here the proof is omitted. □

Proof to Theorem 4

From (3) and simplifying, we can show the first-order condition as

where

To check for optimality, we check the Hessian matrix:

Since $\frac{\partial^{2}\phi_{\mathrm{SIP}}}{\partial p_{i}^{2}}=-2\beta_{p}-2\gamma_{p}<0$, $\frac{\partial^{2}\phi_{\mathrm{SIP}}}{\partial p_{i}\partial p_{3-i}}=\frac{\partial^{2}\phi_{\mathrm{SIP}}}{\partial p_{3-i}\partial p_{i}}=2\gamma_{p}>0$ and

$$\left|\begin{array}{c@{\quad }c} -2\beta_{p}-2\gamma_{p} & 2\gamma_{p}\\ 2\gamma_{p} & -2\beta_{p}-2\gamma_{p}\end{array} \right|=4(\beta_{p}+\gamma_{p})^{2}-4\gamma_{p}^{2}>0, $$

where β _p >0, γ _p >0, we have a negative definite Hessian.

Thus, using the above conditions, we can calculate the optimal response functions for the Service Integrator.

The notation adopted in this proof is as below.

 □

Proof of Theorem 5

Using a similar method as in proof of Theorem 4, one can obtain very easily the validity of Theorem 5. Here the proof is omitted. The notation adopted in this proof is as below.

 □