Hybrid marketing channel strategies of a manufacturer in a supply chain: game theoretical and numerical approaches

Abstract

This paper analyzes how the market power in a supply chain affects a manufacturer’s hybrid marketing channel strategies, considering market transaction costs and the ratio of market size of the online market and the offline market. Also, this paper investigates a price-matching strategy when a manufacturer adds an online channel. This paper shows a number of interesting results: generally, when a manufacturer acts as a Stackelberg follower, the manufacturer chooses a hybrid marketing channel strategy as online costs become very much smaller and the size of the online market is much larger. However, when a manufacturer acts as a Stackelberg leader, the manufacturer has more chances to use a hybrid marketing channel strategy even when the online costs are relatively higher. In addition, a manufacturer may use a price-matching strategy by lowering its prices with the aim to eventually drive the retailer out of the market if the manufacturer perceives that a retailer has more advantages in the offline market and the size of offline market is larger.

Appendices

Appendix 1: Proof for the manufacturer Stackelberg leader model

In the first period, a manufacturer anticipates a retailer’s reaction function. A retailer’s reaction function can be obtained from the retailer’s profit function which is given by

$$\varPi_{R}^{MS} = m\left( {p_{R}^{MS} - w^{MS} } \right)\left( {\frac{{V - p_{R}^{MS} }}{t}} \right) + \left( {1 - m} \right)\left( {p_{R}^{MS} - w} \right)\left( {\frac{{p_{D}^{MS} + a - p_{R}^{MS} }}{t}} \right).$$

The retailer’s reaction function can be derived from the first-order condition as follows:

$$\frac{{\partial \varPi_{R}^{MS} }}{{\partial p_{R}^{MS} }} = \frac{{a + Vm - am + w^{MS} + p_{D}^{MS} - 2p_{R}^{MS} - mp_{D}^{MS} }}{t} = 0.$$

From the above condition, the retailer’s reaction function is derived as:

$$p_{R}^{MS} = \frac{{a + Vm - am + w^{MS} + p_{D}^{MS} - mp_{D}^{MS} }}{2}.$$

Using the above reaction function, a manufacturer chooses a wholesale price and a direct online price from the first-order conditions of the manufacturer’s profit maximization problem. The manufacturer’s profit function is given by:

$$\varPi_{M}^{MS} = m\left( {w^{MS} - c} \right)\left( {\frac{{V - p_{R}^{MS} }}{t}} \right) + \left( {1 - m} \right)\left( {w^{MS} - c} \right)\left( {\frac{{p_{D}^{MS} + a - p_{R}^{MS} }}{t}} \right) + \left( {1 - m} \right)\left( {p_{D}^{MS} - c} \right)\left( {1 - \frac{{p_{D}^{MS} + a - p_{R}^{MS} }}{t}} \right).$$

The first-order conditions are:

$$\begin{aligned} \frac{{\partial \varPi_{M}^{MS} }}{{\partial w^{MS} }} & = - \frac{{am - Vm - a - cm + 2w^{MS} - 2p_{D}^{MS} + 2mp_{D}^{MS} }}{2t} = 0 \\ \frac{{\partial \varPi_{M}^{MS} }}{{\partial p_{D}^{MS} }} & = - \frac{{\left( {1 - m} \right)\left( {a - 2t - Vm + am - cm - 2w^{MS} + 2p_{D}^{MS} + 2mp_{D}^{MS} } \right)}}{2t} = 0. \\ \end{aligned}$$

From the above equations wholesale and online prices are obtained as follows:

$$w^{{MS^{*} }} = \frac{t}{2m} + \frac{V + c - t}{2},\quad p_{D}^{{MS}^{*}} = \frac{t}{2m} + \frac{V + c - a}{2}.$$

Then, in the second period, the retailer finds an optimal retail price by using the manufacturer’s wholesale and direct online prices as follows:

$$p_{R}^{{MS}^{*}} = \frac{t}{2m} + \frac{V + c - t}{2} + \frac{a}{4} + \frac{V - c - a}{4}m.$$

Appendix 2: Proof for the retailer Stackelberg leader model

A retailer, as a market leader, takes the manufacturer’s reaction function into account for its own retail price decisions. So, a retailer uses a manufacturer’s reaction function, which is derived from the manufacturer’s profit maximization problem in the second period. The manufacturer decides its wholesale and online prices anticipating the retailer’s margin, $k = p_{r}^{N} - w$. Then, the manufacturer’s profit function is given by:

$$\varPi_{M}^{RS} = m\left( {w^{RS} - c} \right)\left( {\frac{{V - p_{R}^{RS} }}{t}} \right) + \left( {1 - m} \right)\left( {w^{RS} - c} \right)\left( {\frac{{p_{D}^{RS} + a - p_{R}^{RS} }}{t}} \right) + \left( {1 - m} \right)\left( {p_{D}^{RS} - c} \right)\left( {1 - \frac{{p_{D}^{RS} + a - p_{R}^{RS} }}{t}} \right).$$

Where subscript, RS, denotes the case where a retailer acts as a Stackelberg leader.Here,$p_{R}^{RS} = k + w$, then this profit function is rewritten as:

$$\begin{aligned} \varPi_{M}^{RS} & = m\left( {w^{RS} - c} \right)\left( {\frac{{V - \left( {k + w^{RS} } \right)}}{t}} \right) + \left( {1 - m} \right)\left( {w^{RS} - c} \right)\left( {\frac{{p_{D}^{RS} + a - \left( {k + w^{RS} } \right)}}{t}} \right) \\ & \quad + \left( {1 - m} \right)\left( {p_{D}^{RS} - c} \right)\left( {1 - \frac{{p_{D}^{RS} + a - \left( {k + w^{RS} } \right)}}{t}} \right). \\ \end{aligned}$$

Then, a manufacturer’s reaction function can be derived from the following first-order conditions:

$$\begin{aligned} \frac{{\partial \varPi_{M}^{RS} }}{{\partial w^{RS} }} & = \frac{{\left( {a - k + Vm - am + cm - 2w + 2p_{D}^{RS} - 2mp_{D}^{RS} } \right)}}{t} = 0 \\ \frac{{\partial \varPi_{M}^{RS} }}{{\partial p_{D}^{RS} }} & = - \frac{{\left( {a - k - t - 2w + 2p_{D}^{RS} } \right)}}{t}\left( {1 - m} \right) = 0. \\ \end{aligned}$$

Thus, reaction functions are:

$$\begin{aligned} w^{RS} & = \frac{{\left( {t + Vm + cm - mt - mp_{R}^{RS} } \right)}}{m} \\ p_{D}^{RS} & = \frac{{\left( {t + Vm - am + cm} \right)}}{2m}. \\ \end{aligned}$$

Anticipating these reactions, the retailer maximizes its profits in the first period. The retailer’s profit function is given by:

$$\varPi_{R}^{RS} = m(p_{R}^{RS} - w^{RS} )\left( {\frac{{V - p_{R}^{RS} }}{t}} \right) + (1 - m)(p_{R}^{RS} - w^{RS} )\left( {\frac{{p_{D}^{RS} + a - p_{R}^{RS} }}{t}} \right).$$

The reaction function is derived from the first-order condition as follows:

$$\frac{{\partial \varPi_{R}^{RS} }}{{\partial p_{R}^{RS} }} = \frac{{\left( {2t + 2Vm + am + 2cm - 2mt - 4mp_{R}^{RS} + Vm^{2} - am^{2} - cm^{2} } \right)}}{mt} = 0.$$

Then, optimal prices are obtained as follows:

$$\begin{aligned} w^{RS*} & = \frac{{\left( {2t + 2Vm + 2cm - am - 2mt - Vm^{2} + am^{2} + cm^{2} } \right)}}{4m} \\ p_{D}^{RS*} & = \frac{{\left( {t + Vm - am + cm} \right)}}{2m} \\ p_{R}^{RS*} & = \frac{{\left( {2t + 2Vm + 2cm + am - 2mt + Vm^{2} - am^{2} - cm^{2} } \right)}}{4m}. \\ \end{aligned}$$

Appendix 3: Proof for the vertical Nash Model

A manufacturer conditions its wholesale and online prices on the retail margin $k = p_{r}^{N} - w.$ Then, the manufacturer’s profit function is given by:

$$\varPi_{M}^{N} = m\left( {w^{N} - c} \right)\left( {\frac{{V - p_{R}^{N} }}{t}} \right) + \left( {1 - m} \right)\left( {w^{N} - c} \right)\left( {\frac{{p_{D}^{N} + a - p_{R}^{N} }}{t}} \right) + \left( {1 - m} \right)\left( {p_{D}^{N} - c} \right)\left( {1 - \frac{{p_{D}^{N} + a - p_{R}^{N} }}{t}} \right),$$

where N denotes vertical Nash model.

The first-order conditions are:

$$\begin{aligned} \frac{{\partial \varPi_{M}^{N} }}{{\partial w^{N} }} & = a + Vm - am + cm - w^{N} + 2p_{D}^{N} - p_{R}^{N} - 2mp_{D}^{N} = 0 \\ \frac{{\partial \varPi_{M}^{N} }}{{\partial p_{D}^{N} }} & = a - t - w^{N} + 2p_{D}^{N} - p_{R}^{N} = 0. \\ \end{aligned}$$

From the above equations we obtain the following reaction functions:

$$w^{N} = \frac{1}{m}\left( {t + Vm + cm - mt - mp_{R}^{N} } \right),\quad p_{D}^{N} = \frac{1}{2}(p_{R}^{N} - a + t + w^{N} ).$$

The retailer’s profit function is given by:

$$\varPi_{R}^{N} = m\left( {p_{R}^{N} - w^{N} } \right)\left( {\frac{{V - p_{R}^{N} }}{t}} \right) + \left( {1 - m} \right)\left( {p_{R}^{N} - w^{N} } \right)\left( {\frac{{p_{D}^{N} + a - p_{R}^{N} }}{t}} \right).$$

The retailer chooses its retail price to maximise its profit function by anticipating the manufacturer’s wholesale price and direct online prices. The first-order condition is given by:

$$\frac{{\partial \varPi_{R}^{N} }}{{\partial p_{R}^{N} }} = a + Vm - am + w^{N} + p_{D}^{N} - 2p_{R}^{N} - mp_{D}^{N} = 0.$$

From the first-order condition, we obtain the following reaction function as follows:

$$p_{R}^{N} = \frac{{a + Vm - am + w^{N} + p_{D}^{N} - mp_{D}^{N} }}{2}.$$

Then, optimal prices are derived from the above reaction functions as follows:

$$\begin{aligned} w^{N*} & = \frac{{3Vm - Vm^{2} - am + 3cm + 3t - 3mt + am^{2} + cm^{2} }}{6m} \\ p_{D}^{N*} & = \frac{Vm + t - am + cm}{2m} \\ p_{R}^{N*} & = \frac{{3Vm + Vm^{2} + am + 3cm + 3t - 3mt - am^{2} - cm^{2} }}{6m}. \\ \end{aligned}$$