Optimal pricing decisions of centralized dual-channel supply chains in a duopoly: a study on the influence of competition structure

A price competition amongst two centralized dual-channel supply chains (DCSCs) has been investigated in this study. Three scenarios of competition are studied—a simultaneous move game and two sequential games (one in which the first supply chain is the leader and the second supply chain is the follower, and vice versa). The optimal prices and profits for these scenarios have been derived and compared. It is observed that prices and profits are higher under Stackelberg (Sequential move) competition as compared to Simultaneous move competition. Finally, the present study determines which type of competition structure would persist in the market by using a Wait or Declare two-strategy game. The game depicts the market entry decision of the competing supply chains. The results reveal that self interest in competing dual-channel supply chains leads to a sequential move competition and consequently to higher product prices. To reduce the price and thus induce a customer favourable equilibrium, regulatory authorities like the government can step in and provide incentives. We derive a lower bound on the incentive which would implement a low market price equilibrium in Dominant Strategies. Further, we note that under competition, dual-channel supply chains outperform traditional supply chains only upto a certain threshold of the retail channel market share and we derive an expression for the same threshold. This threshold is found to be independent of the nature of the product.

1. We do not consider the possibility implementing (W, W) as a PSNE, as this equilibrium delays the market entry decisions.

Authors and Affiliations

1. Department of Mechanical Engineering, National Institute of Technology Calicut, Calicut, India

   Ranjit B Nair, K Ratna Kumar & R Sridharan

2. Department of Industrial and Systems Engineering, Indian Institute of Technology Kharagpur, Kharagpur, India

   Anand Abraham

Corresponding author

Correspondence to K Ratna Kumar.

Appendix I. Proof of Lemma 1

The first order conditions of (7) with respect to \( p_{ij} \), \( i \in \{1,2 \} \), \( j \in \{r,d \} \) are given in (A1) through (A4) respectively. These constitute the necessary conditions.

Next, it is found that \( \frac{\partial ^{2}\pi _{i}}{\partial p_{ir}^2}=-2a<0 \). With respect to \( p_{ir} \) and \( p_{id} \), the determinant of the Hessian of (7) i.e. \( \left| \begin{matrix}-2a&{}\frac{2a}{3}\\ \frac{2a}{3}&{}-2a\\ \end{matrix}\right| =\frac{32a^2}{9}>0 \). As a result, Eq. (7) is strictly jointly concave in both \( p_{ir} \) and \( p_{id} \), satisfying the necessary and sufficient conditions for optimality. Thus, solving (A1) through (A4) simultaneously, we obtain eqs. (8), (9), (10) and (11). Hence proved.

Appendix II. Proof of Lemma 2

To begin, determine the first order conditions of \(\pi _{2}\) with respect to \( p_{2r} \) and \( p_{2d} \), as shown in eqs. (A5) and (A6), respectively. These are the necessary conditions that must be met.

Following that, we find \( \frac{\partial ^{2}\pi _{2}}{\partial p_{2r}^2}=-2a<0 \). The determinant of \( \pi _{2} \)’s Hessian with respect to \( p_{2r} \) and \( p_{2d} \), i.e. \( \left| \begin{matrix}-2a&{}\frac{2a}{3}\\ \frac{2a}{3}&{}-2a\\ \end{matrix}\right| =\frac{32a^2}{9}>0 \). The second supply chain’s profit function is thus strictly jointly concave with respect to \( p_{2r} \) and \( p_{2d} \), and the necessary and sufficient conditions are met. Solving eqs. (A5) and (A6) yields two temporary equations for \( p_{2r} \) and \( p_{2d} \) at the same time. The optimal equations are then obtained using the backward induction method. As a result, these temporary equations are substituted in the profit function of the first supply chain, yielding a new profit function for the first supply chain that we refer to as \( \pi _{1new} \). Now it is time to double-check the necessary and sufficient conditions for optimality. The first order conditions of \(\pi _{1}\) with respect to \( p_{1r} \) and \( p_{1d} \) are determined by the equations (A7) and (A8), respectively. These are the necessary conditions that must be met.

After that, we discover that \( \frac{\partial ^{2}\pi _{1new}}{\partial p_{1r}^2}=-\frac{5a}{3}<0 \). The Hessian determinant of \( \pi _{1new} \) with respect to \( p_{1r} \) and \( p_{1d} \) is \( \left| \begin{matrix}-\frac{5a}{3}&{}a\\ a&{}-\frac{5a}{3}\\ \end{matrix}\right| =\frac{16a^2}{9}>0 \). As a result, \( \pi _{1new} \) is strictly jointly concave with respect to \( p_{1r} \) and \( p_{1d} \), and the necessary and sufficient conditions for optimality are met. We get eqs. (14) and (15) by solving eqs. (A7) and (A8) simultaneously. We get eqs. (16) and (17) by substituting the optimal prices of the leader in the temporary pricing equations of the follower found earlier. By considering the second supply chain as the leader and the first supply chain as the follower, eqs. (18), (19), (20), and (21) can be obtained. As a result, the proof.

Appendix III. Proof of Proposition 1

\( p_{1j}^{seqled*} - p_{1j}^{seqfol*}=\frac{3D\rho }{16a} \ge 0 \). Hence, \( p_{1j}^{seqled*}\ge p_{1j}^{seqfol*} \). \( p_{1j}^{seqfol*}-p_{1j}^{sim*}=\frac{D+D\left( 1-\rho \right) }{16a}>0 \). Hence, \( p_{1j}^{sim*}<p_{1j}^{seqfol*} \). So, \( p_{1j}^{sim*}<p_{1j}^{seqfol*}\le p_{1j}^{seqled*},\ j \in \{r,d\} \). Similarly, it can be proven that \( p_{2j}^{sim*}<p_{2j}^{seqfol*}\le p_{2j}^{seqled*},\ j \in \{r,d\} \). Thus, Eq. (26) is proved.

\(\pi _{1}^{seqfol*}-\pi _{1}^{seqled*}=\frac{D^2}{64a}[3+3\left( 1-\rho ^2\right) ]>0 \). Hence, \( \pi _{1}^{seqfol*}>\pi _{1}^{seqled*} \). \( \pi _{1}^{seqled*}-\pi _{1}^{sim*}=\frac{D^2}{96a}(\rho +1)^2>0 \). Hence, \( \pi _{1}^{seqled*}>\pi _{1}^{sim*} \). Thus, \( \pi _{1}^{sim*}<\pi _{1}^{seqled*}<\pi _{1}^{seqfol*} \). Similarly, it can be proven that \( \pi _{2}^{sim*}<\pi _{2}^{seqled*}<\pi _{2}^{seqfol*} \). Hence, Eq. (27) is proved.

Appendix IV. Proof of Proposition 2

We have to prove that \(\hat{\theta }_{1}^{sim}\) lies in [0,1] for any value of \(\rho \). For \(\hat{\theta }_{1}^{sim} \le 1 \), the denominator of \(\hat{\theta }_{1}^{sim}\) should be greater than or equal to its numerator. For \(\hat{\theta }_{1}^{sim} \ge 0 \), the numerator of \(\hat{\theta }_{1}^{sim}\) should be greater than or equal to 0. Thus, the following two conditions should be satisfied:

1. 1.

   \( 2(8+16\rho -19\rho ^2) \ge 9(A_{2}-3\rho ^2+A_{1}) \)

2. 2.

   \( 9(A_{2}-3\rho ^2+A_{1}) \ge 0 \)

1. Let

Here, \( f_{1}(\rho ) \) is the difference between the denominator and the numerator of \(\hat{\theta }_{1}^{sim}\).

The first order necessary condition for optimality is:

Solving for \(\rho \), we get

(There are three values of \(\rho _{11}^{sim *}\). This is the only feasible value. The other two values are infeasible, one being negative and the other being greater than 1).

Again,

Now,

This means that the above value of \(\rho _{11}^{sim *}\) is a maximum. Since there is only a single feasible value of \(\rho _{11}^{sim *}\) (\(\rho _{11}^{sim *}\) is unimodal) and that value being a maximum, the minimum value(s) should lie at the extreme(s). Hence, the minimum value should lie at either 0 or 1.

The above values indicate that \( f_{1}(\rho ) \) has a minimum value at 1. Since the value of \( f_{1}(1) \) is positive and that value is a minimum, all the other values of \(f_{1}(\rho )\) should be greater than the value at \(\rho =1\). Thus, the denominator is greater than the numerator for all values of \(\rho \). Thus,

Thus, the first condition is satisfied. Hence, \(\hat{\theta }_{1}^{sim} \le 1 \).

2. Let

Here, \( g_{1}(\rho ) \) is the numerator of \(\hat{\theta }_{1}^{sim}\).

The first order necessary condition for optimality is:

Solving for \(\rho \), we get

Again,

Now,

This means that the above value of \(\rho _{12}^{sim *}\) is a maximum. Since there is only a single feasible value of \(\rho _{12}^{sim *}\) (\(\rho _{12}^{sim *}\) is unimodal) and that value being a maximum, the minimum value(s) should lie at the extreme(s). Hence, the minimum value should lie at either 0 or 1.

The above values indicate that \( g_{1}(\rho ) \) has a minimum value at 1. Since the value of \( g_{1}(1) \) is positive and that value is a minimum, all the other values of \(g_{1}(\rho )\) should be greater than the value at \(\rho =1\). Thus, the numerator is greater than or equal to 0 for all values of \(\rho \).

Thus, the second condition is satisfied. Hence, \(\hat{\theta }_{1}^{sim} \ge 0 \).

Since both the conditions are satisfied, it is proved that \(\hat{\theta }_{1}^{sim}\) lies in [0,1] for any value of \(\rho \).

Similarly, it can be proven that \(\hat{\theta }_{2}^{sim}\) lies in [0,1] for any value of \(\rho \). Table 6 provides a detailed list of notations and their descriptions

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