The Impact of Dual-Channel Investments and Contract Mechanisms on Telecommunications Supply Chains

Abstract

This study examines how contract structures influence coordination and innovation incentives in dual-channel telecommunications supply chains. We consider a setting where a mobile network operator (MNO) supplies services both directly to consumers and indirectly through a mobile virtual network operator (MVNO), which competes in the retail market. Using a game-theoretic framework, we evaluate how different contracts—single wholesale pricing, revenue sharing, and quantity discounts—shape strategic decisions, particularly in the presence of investment spillovers between parties. A key coordination problem emerges from the externalized gains of innovation, where one party’s investment generates value for both participants. Our results show that single wholesale and revenue sharing contracts often lead to suboptimal investment and profit outcomes. In contrast, quantity discount contracts, especially when combined with appropriate transfer payments, improve coordination and enhance the total performance of the supply chain. We also find that innovation led by the MVNO, while generally less impactful, can still yield reciprocal benefits for the MNO, reinforcing the value of cooperative arrangements. These findings emphasize the importance of contract design in managing interdependence and improving efficiency in decentralized supply chains. This study offers theoretical and practical implications for telecommunications providers and policymakers aiming to promote innovation and mutually beneficial outcomes through well-aligned contractual mechanisms.

1. Introduction

From a supply chain perspective, in telecommunications, the relationship between a mobile network operator (MNO) and a mobile virtual network operator (MVNO) provides a unique market structure. Unlike the traditional supply chain structure that follows a wholesale–retail–consumer sequence, the relationship between these two partners represents a cooperative dynamic where the MNO becomes a supplier to the MVNO, and the bargaining power of the supplier (MNO) is dominant. Simultaneously, it constitutes a dual-channel supply chain structure where the MNO and the MVNO compete in the retail market. Governments in many countries are encouraging MVNOs to participate in the telecommunications market to promote competition and moderate market concentration. The market share of MVNOs varies by country, ranging from 0.9% to 47.5%. Notably, Germany (47.5%), Denmark (33.5%), and Canada (28.8%) have substantial MVNO market shares. As of 2019, the average market share of MVNOs across 36 OECD countries stood at 11.7% [1]. The number of MVNOs also varies by country, with the United States having over 200, while Germany, Japan, and the United Kingdom each have around 100 MVNOs operating. These MVNOs often offer more competitive pricing to customers compared to MNOs, making them appeal to price-sensitive consumers [2].

To stimulate competition in telecommunications markets characterized by oligopolistic structures with only three to four operators per country, governments around the world have consistently pursued strategies to foster the sustained growth of MVNOs. Key elements of various policy approaches include designating MNOs as mandatory wholesale providers and compelling them by law to enter into wholesale agreements with MVNOs. In some countries, such as Austria, Japan, Spain, and South Korea, the government intervenes by setting the pricing of wholesale services between MNOs and MVNOs, rather than leaving it to voluntary negotiations between operators. This approach promotes transparency and equitable access to wholesale services, ultimately aiming to invigorate market competition.

While not widely disclosed in individual agreements, the contractual arrangements between MNOs and MVNOs can be broadly categorized into three main types, as classified by [3]. The first type is called the “Retail-Minus” contract, where the wholesale price is established by applying a fixed discount rate to the MNO’s retail price, excluding avoidable costs. This approach is akin to the “single wholesale price” contract format, where the rates for voice and data services are specified and used to establish wholesale prices. The second widely employed approach is “Revenue Sharing”, wherein MNOs and MVNOs divide the revenue generated by an MVNO according to a predetermined percentage. This method ensures proportional sharing of revenues between the two parties. The third approach, adopted in certain countries such as Japan, involves “Quantity Discount” contracts, where bulk discounts are negotiated between operators. This strategy often hinges on the volume of services exchanged and can lead to cost savings for MVNOs based on the quantity of services they purchase.

Another concern facing MNOs within the framework of the dual-channel supply chain is the incentive for investment when such a market structure is in place [4,5]. The market structure of MNOs and MVNOs differs from the traditional addition of a direct channel (online) to the retail channel (brick-and-mortar). Instead, the crucial difference is that it involves the emergence of another retail channel alongside the MNOs’ existing direct channel. From the perspective of MNOs, investments in innovation typically aim to maximize their profits through enhancements in network quality and speed and launching next-generation communication services, such as 5G, 6G, satellite communication, and AI-based network technologies, and so on. However, in a dual-channel supply chain, the investments made by the supplier (MNO) can lead to spillover effects on the retailer’s (MVNO) demand and revenue. There are arguments that this dynamic, where MVNOs also reap the rewards of investments, can lead to concerns and dampen the enthusiasm of MNOs for investment in future innovation. In innovation investment, a spillover effect refers to the impact or influence that one party’s innovation efforts have on the other party within a supply chain or a collaborative setting [6,7]. It represents how an investment made by one participant affects the performance, decisions, or outcomes of the other participant in the supply chain, often in terms of increased demand, efficiency, or profitability. This effect can be either positive, where the innovation benefits both parties, or negative, where it may create challenges or conflicts. Therefore, spillover effects are crucial to understanding the coordination within supply chains involving innovation investments.

Based on the example above, this study was conducted under the premise of a formed dual-channel supply chain rather than a choice of supply chain channels. We mainly explore effective contract mechanisms among the contracting parties while considering the spillover effects of investment for innovation on supply chain coordination. First, we examine efficient wholesale contract methods that facilitate coordination between the two supply chain partners who maintain a cooperative relationship while simultaneously competing in the retail market, taking into account the characteristics of the telecommunications market. We also investigate the resulting changes in the profitability of the market participants and the supply chain. Furthermore, regarding the spillover effects of innovation, this study inherently addresses the spillover of MNO (supplier) investments given the tens of billions of dollars of substantial annual investments made by MNOs in reality. In addition, we analyze the impact on supply chain profitability when retailers engage in innovative activities such as launching new services (e.g., partnerships with Over-the-Top (OTT) providers) or investing in enhancing their own services.

Our model consists of three main aspects: First, we analyze a situation where two companies are either centralized or in a parent–subsidiary relationship, with perfect monitoring between contracting parties. This serves as the benchmark case for our analysis. Second, we incorporate three contract models commonly observed in the current market, including single wholesale price contracts, revenue sharing contracts, and quantity discount contracts, among others. By comparing these contract models with the benchmark case, we analyze the changes in the profit functions of the contracting parties, the expected profits of the entire supply chain, and overall efficiency. This allows us to examine more realistic and efficient contract mechanisms between the business partners in a dual-channel supply chain among the various contract options available. Third, we examine the spillover effect of self-initiated investments for innovation on both the MNO and the MVNO and investigate the coordination dynamics between these two partners when investment decisions are under consideration.

This study makes several contributions to the literature on dual-channel supply chain coordination in the telecommunications sector. First, it demonstrates that implementing quantity discount mechanisms alongside innovation investments can significantly enhance the overall supply chain performance. Such coordination not only improves the alignment between suppliers and retailers but also protects the supplier’s returns by mitigating the negative effects of investment spillovers. In addition, we show that innovation investments by the retailer—enabled through spillovers—can positively affect the entire supply chain, providing reciprocal benefits to the supplier.

Unlike prior studies that primarily focus on single-channel coordination or channel selection with multiple retailers, our research emphasizes the contractual dynamics between the supplier (MNO) and a single retailer (MVNO), particularly in the context of innovation-driven investments such as 5G, 6G, satellite communication, and AI-based network technologies. This focus allows us to highlight the unique coordination challenges arising when the supplier bears the innovation cost. Moreover, while much of the existing literature limits itself to price and profit optimization, our work explores a broader range of contract mechanisms and considers the investment incentives of both parties. By doing so, we offer a more comprehensive view of supply chain coordination that reflects the strategic interests of telecommunications stakeholders.

The organization of this paper is as follows. Section 2 studies previous studies related to our research and emphasizes the unique aspects of this study. Section 3 describes the dual-channel supply chain contract models. In Section 4, we study the coordination within a dual-channel supply chain without investment for innovation. Section 5 explores coordination while considering the spillover effects of investment. Section 6 concludes our study with a discussion of its limitations and potential research directions.

2. The Related Literature

Earlier studies, such as the work conducted by [8], primarily focused on the contract mechanisms and coordination among supply chain partners. Subsequent research has seen active analyses of various contract types and market structures, including single-price contracts [9], flexible contracts [10], revenue sharing contracts [11], and quantity discount contracts [12]. Comprehensive reviews of this body of research can be found in references such as [13,14]. In the context of dual-channel supply chain coordination, a range of contract types has been explored to foster cooperation among suppliers and retailers in decentralized settings. For instance, ref. [15] shows the effect of the channel structure of the supply chain and channel coordination through the channel-adding Pareto zone concept. Ref. [16] examined the coordination structures within decentralized supply chains, finding that contracts involving wholesale and direct channel prices benefited retailers, but proposed complementary contracts such as two-part tariffs for mutual benefit. Ref. [17] introduced a two-way revenue sharing contract tailored to dual-channel supply chains, combining traditional revenue sharing with a reverse revenue sharing contract. Similarly, ref. [18] proposed contracts for managing manufacturer–retailer competition, investigating their impact on the pricing and recycling rates in closed-loop supply chains. They also introduced reverse revenue sharing by allowing manufacturers to share cost savings. Ref. [19] found that cost sharing contracts encouraged improvements in retailer services and discouraged price competition. Ref. [20] explored revenue and profit sharing contracts in non-cooperative and cooperative game structures, highlighting their effectiveness in different customer scenarios. For a comprehensive review of this body of research, we refer to [21].

Among various contract mechanisms, quantity discount contracts have been particularly effective in achieving supply chain coordination. Ref. [22] introduced linear quantity discount contracts designed to manage manufacturer–retailer competition. These contracts proved effective in supply chain coordination, albeit resulting in reduced retailer profits compared to those in decentralized scenarios. Ref. [23] employed hybrid mechanisms that combined quantity discounts and franchise fees to mitigate conflicts within dual-channel supply chains, yielding benefits across the entire supply chain when the expected profits aligned with those in decentralized settings. It is noteworthy that the above diverse contract types and coordination mechanisms may not entirely align with the realities of the MNO and MVNO contract relationships we examine in this study. For instance, the application of two-way revenue sharing or profit sharing concepts may not be feasible from the perspective of MNOs, which often hold negotiating leverage and significant revenue disparities. Additionally, the unique characteristics of service-based industries can make it challenging to implement concepts such as closed-loop supply chains and return policies. While the research [22] shares some similarities with our study, our research differs in that we focus on the contractual relationships between suppliers and retailers, particularly in cases involving MNO investments for innovation. Unlike scenarios involving channel selection and multiple retailers, our study considers the coordination aspects when innovation-driven investments are made by MNOs, highlighting this as a key different aspect.

Despite growing interest, the interface between investment for innovation and supply chain management remains an understudied domain. Ref. [24] investigated a scenario where a supplier invested in process innovation to improve product quality and increase consumer value. They examined three supply contracts and showed that the revenue sharing contract was capable of achieving the optimal innovation levels and channel coordination. Ref. [25] explored how cost sharing concepts within supply chains could increase innovation investments upstream. Their research demonstrated that such contracts effectively encouraged innovative investments by upstream partners. Ref. [26] examined the innovation and retail channel dynamics in a dual-channel supply chain. The findings suggested that retailer benefits could result from a supplier’s entry into the retail sector, as this motivated the supplier to make cost-reducing investments. This, in turn, led to lower wholesale prices and improved the profits for both parties. Ref. [27] explored that innovators might strategically outsource to competitor CMs, aiming for market leadership in cases of technical innovations and introducing innovation uncertainties for non-technical innovations. Ref. [28] explored the collaborative innovation within supply chains, specifically focusing on products co-developed by an upstream supplier and a downstream manufacturer. Their study showed that when the manufacturer possessed sufficient resources, they were inclined to invest in new product development, whereas the supplier did not share the same inclination. This work emphasized the significance of innovation driven by manufacturers and highlighted the necessity of factoring in product the profit margins when engaging in collaborative innovation efforts. Recent studies have investigated innovation spillovers in supply chain settings from various perspectives. Ref. [29] provides empirical evidence that knowledge spillovers from customers significantly enhance supplier innovation, particularly under geographic proximity. Ref. [30] further confirms that buyer innovation positively affects supplier innovation, especially in long-term buyer–supplier relationships, though technological proximity shows limited moderating effects. While these studies deepen our understanding of the effects of spillover on innovation outcomes, our research is distinct in its focus on the coordination within a dual-channel supply chain involving innovation investments. Specifically, we investigate how contractual mechanisms can address both channel conflict and investment spillovers, facilitating strategic alignment between mobile network operators (MNOs) and mobile virtual network operators (MVNOs). While there are similarities with [24], their research primarily centered on single-channel coordination utilizing the Hoteling model, setting it apart from our study.

Research on the contract mechanisms between MNOs and MVNOs has been conducted in both economics and the telecommunications industry. Ref. [31] analyzed the legitimacy of MVNO market entry and the changes in the profits of each party when these companies compete in the telecommunications market. Ref. [32] investigated situations where facility-based vertically integrated firms compete independently with rivals on the broadband access market, studying the impact of government regulations. They argued that under regulatory conditions where the government sets wholesale prices, there is a reduction in firms’ investment incentives, and competing firms providing wholesale services tend to overinvest when creating new value-added services. Similarly, ref. [33] studied how the entry of MVNOs and regulatory access policies affect MNOs’ investment behavior, utilizing data from 58 MNOs across 21 OECD countries. Their findings suggested that mandated access provision is associated with a reduced intensity of MNO investments, emphasizing the importance of addressing the investment incentives when granting access to MVNOs. In addition, refs. [34,35] used non-cooperative game theory to analyze the equilibrium wholesale prices between MNOs and MVNOs, considering the market conditions. A noteworthy distinction of our research is that it extends beyond the mere computation of wholesale prices and profit functions and instead explores a range of contractual mechanisms. Furthermore, our study is conducted with a particular focus on the interests and profitability of telecommunication supply chain stakeholders, as well as the perspective of investment incentives. These factors set our research apart from the aforementioned studies.

3. The Model

We explore a dual-channel supply chain structure where a supplier (MNO) distributes products both directly to customers through their own direct channel and indirectly through an MVNO. These channels are labeled as the “direct” and “indirect” channels correspondingly. The MNO sets a wholesale price, denoted as w, for selling products to the retailer and sets the direct retail price, labeled as $p_s,$ in the direct channel. The retailer decides on the retail price, denoted as $p_r,$ in the indirect channel.

Table 1. Notation for variables.

Variables	Description
w	The unit wholesale price offered by the MNO to the MVNO
$p_s$	The retail price of the direct channel
$p_r$	The retail price of the indirect channel
$D_s$	Demand from the direct channel
$D_r$	Demand for the indirect channel
a	Customer preference for the direct channel (${\displaystyle \frac{1}{2}}$ < a < 1)
b	Cross-price elasticity $(0<b<1)$
c	The operating cost of the telecommunication service
x	Investment level for innovation
${\pi }_s$	The supplier’s (MNO’s) profit
${\pi }_r$	The retailer’s (MVNO’s) profit

summarizes the notation of the variables.

We have employed prior research as the foundation for formulating the linear demand functions for both channels [36,37,38,39]. Specifically, we assume that the market demand in both the direct and indirect channels responds to price changes and is influenced by the level of investment made by the supplier. In the market composed of an MNO and an MVNO, we also assume that the products are homogeneous, but the customer demand varies in response to differences in price.

$$\begin{array}{cc}{D}_s=aA-p_s+b{(p}_r-p_s)+\gamma x\hfill & \\ D_r=\left(1-a\right)A-p_r+b{(p}_s-p_r)+\gamma x\hfill & \end{array}$$

Let Ds represent the demand from the direct channel and Dr represent the demand for the indirect channel. Parameter A characterizes the baseline demand, and a (0 < a < 1) represents the level of customer preference for the MNO’s direct channel. In the telecommunications market, which is typically a direct channel leading market, we assume $a>{\displaystyle \frac{1}{2}}$. Correspondingly, 1 − a characterizes the degree of customer preference for the indirect channel. The precise value of the initial market potential A is not a critical factor in our analytical model. The primary findings of our paper remain robust, even if the scale of the baseline demand is adjusted. Thus, for analytical tractability and without loss of generality, we standardize the value of A to one, aligning with the methodology of [36,37,40,41,42].

We also assume that the price elasticity coefficients for Ds and Dr are both set to one, according to [43].

The cross-price elasticity is denoted as $b$($b_s=b_r=b)$, where $0<b<1$. A value of b = 0 would imply that the two markets operate independently, while b = 1 would indicate perfect substitutability. In the telecommunications context, services such as voice and data are similar, so we assume b is strictly between 0 and 1. Given the consumer loyalty and pricing differences between MNOs and MVNOs, it is reasonable to expect that b remains closer to 0 than to 1. Furthermore, we represent the increase in demand due to investment in technological innovation as $\gamma x$ [26,28,32]. While the increases in demand for the indirect and direct channels may differ, assuming the sale of products with the same attributes, we set ${\gamma }_s={\gamma }_r$ as the initial assumption and subsequently analyze cases where these parameters differ.

The MNO incurs a quadratic network investment cost associated with investments in innovative services, such as enhancing speed, improving data quality, or adding additional features. This cost function is represented as $C\left(x\right)=k{\displaystyle \frac{x^2}{2}}$, where it satisfies the conditions C(0) = 0, ${\displaystyle \frac{dC\left(x\right)}{dx}}>0$ and ${\displaystyle \frac{d{C}^2\left(x\right)}{d{x}^2}}>0$ ([44,45]).

With the above notation, the supplier’s profit is determined by

$${\pi }_s=\left(p_s-c\right)D_s+\left(w-c\right)D_r-k{\displaystyle \frac{x^2}{2}}$$

(1)

and the profit of the retailer is determined as

$${\pi }_r=\left(p_r-w\right)D_r$$

(2)

where w is the wholesale price charged by the MNO.

In the telecommunications market, the marginal operating cost (c) approaches zero, while the investment costs for new services and innovations can be substantially high. Therefore, we assume that the demand in the indirect channel is $(1-a)\gg c$.

If the dual-channel supply chain undergoes vertical integration, the profit of the centralized dual-channel is given by

$${\pi }_I=\left(p_s-c\right)D_s+\left(p_r-c\right)D_r-k {\displaystyle \frac{x^2}{2}}$$

(3)

Regarding investments in innovation, we assume that a significant amount of capital is invested over an extended period. Investment decisions must be made many years before final products are launched on the market. Therefore, the sequence of events in our model is as follows: First, the supplier selects the investment level x. Second, the supplier determines the retail price ($p_s$) of their own direct channel and the wholesale price ($w$). Third, the retailer, based on the wholesale price and the direct channel’s retail price, determines the indirect channel’s retail price ($p_r$). We employ a backward induction approach to obtaining equilibrium for both the supplier and the retailer. Figure 1 shows a conceptual diagram summarizing the relationships among the contract types, coordination outcomes, and investment returns.

4. The Equilibrium and Coordination Analysis Without Innovation Investment

Before discussing the investments made by the supplier (MNO), this section focuses on the equilibrium and coordination within the supplier-driven dual-channel supply chain. We evaluate the system-wide profit, which refers to the aggregated profit of all firms involved in the supply chain under coordinated decision-making. This concept is widely used in the literature on supply chain coordination as a normative benchmark [46,47]. First, we consider a centralized system where all decision-making is consolidated to maximize the entire channel’s profit. Under this centralized system, the vertically integrated organization manages both the retail price ($p_r$) and the direct channel’s price ($p_s$). Next, we explore a decentralized system within the Stackelberg game, with the supplier in a leading role. In this decentralized case, both the MNO and the MVNO make choices to maximize their expected profits.

4.1. A Centralized Dual-Channel Supply Chain

We first examine a benchmark model where both the direct channel and the indirect channel are centralized within the supply chain. The profit function (${\pi }_c$) for each channel, considering the demand for both channels, is as follows:

$${\pi }_c=\left(p_s-c\right)\left[a-p_s+b{(p}_r-p_s)\right]+\left(p_r-c\right)\left[1-a-p_r+b{(p}_s-p_r)\right]$$

(4)

The profit-maximizing prices ($p_s$) and ${(p}_r$) and the expected profit (${\pi }_c$) are as follows:

Proposition 1.

In a centralized dual-channel supply chain, the optimal price in each channel and maximized profit is given by

$$p_s={\displaystyle \frac{a+b+\left(1+2b\right)c}{2\left(1+2b\right)}}$$

$$p_r={\displaystyle \frac{1-a+b+\left(1+2b\right)c}{2(1+2b)}}$$

$${\pi }_c={\displaystyle \frac{1-2\left(1-a\right)a+b{\left(1-2c\right)}^2-2\left(1-c\right)c}{4\left(1+2b\right)}}.$$

From Proposition 1, we obtain the prices and profit for the benchmark case in the centralized dual-channel system. Next, we explore a decentralized system employing three different kinds of contracts. We consider a contract as coordinating the dual channel if the equilibrium outcomes of this contract are equivalent to those in the benchmark case. In addition, to ensure both channels have positive demand, we assume a > 1 − a >> c, as mentioned in Section 3. In the telecommunications market, significant infrastructure investments are made, while the marginal operating cost for services approaches zero.

4.2. A Decentralized Dual-Channel Supply Chain

Suppose that the MNO and the MVNO decide on a single wholesale price contract. The MNO determines the profit-maximizing wholesale price in the indirect channel and the retail price in its own direct channel; afterwards, the MVNO chooses the selling price in their indirect channel. The MNO’s profit, ${\pi }_s^{wp}$, and the MVNO’s profit, ${\pi }_r^{wp}$, are given as follows:

$${\pi }_s^{wp}=\left(p_s-c\right)\left[a-p_s+b{(p}_r-p_s)\right]+\left(w-c\right)\left[1-a-p_r+b{(p}_s-p_r)\right]$$

(5)

$${\pi }_r^{wp}=\left(p_r-w\right)\left[1-a-p_r+b{(p}_s-p_r)\right].$$

(6)

To obtain the MNO and the MVNO’s decisions in equilibrium, we solve through a backward induction approach. If the wholesale price of the MNO exceeds the MVNO’s retail price, the retailer cannot make any profit and therefore ceases to participate in the retail market. This condition arises when 1 − a < c, indicating that the demand in the direct channel approaches one, while the demand in the indirect channel nears zero. Consequently, when 1 − a > c, a dual-channel wholesale price contract becomes feasible, and we can determine the following supply chain decisions and profits in equilibrium.

$${\pi }_s^{wp}={\displaystyle \frac{1}{4(1+b)}}\left({\displaystyle \frac{{(a+b-c-2bc)}^2}{1+2b}}+{\displaystyle \frac{{(1-a-c)}^2}{2}}\right)$$

$${\pi }_r^{wp}={\displaystyle \frac{{(1-a-c)}^2}{16(1+b)}}$$

Second, suppose that the MNO and the MVNO agree to a revenue sharing contract. We denote the initial wholesale price as $w_0$. In addition, let the supplier’s share in a revenue sharing contract be denoted as ‘s’ and the retailer’s share be ‘1 − s’. The MNO’s profit, ${\pi }_s^{rs}$, and the MVNO’s profit, ${\pi }_r^{rs}$, are given by

$${\pi }_s^{rs}=\left(p_s-c\right)\left[a-p_s+b{(p}_r-p_s)\right]+\left(s{p}_r+w_0-c\right)\left[1-a-p_r+b{(p}_s-p_r)\right]$$

(7)

$${\pi }_r^{rs}=\left[(1-s)p_r-w_0\right]\left[1-a-p_r+b{(p}_s-p_r)\right].$$

(8)

Applying the same procedure, we obtain the following decisions and profits in equilibrium.

$${\pi }_s^{rs}={\displaystyle \frac{1}{4(1+b)}}\left({\displaystyle \frac{{(a+b-c-2bc)}^2}{1+2b}}+{\displaystyle \frac{{(1-a-c)}^2}{2-s}}\right)$$

$${\pi }_r^{rs}={\displaystyle \frac{{\left(1-a-c\right)}^2(1-s)}{4(1+b){(2-s)}^2}}$$

When the sharing ratio is s = 0, we observe the same results as those for the single wholesale price contract. However, when the sharing ratio is s = 1, the retailer’s profit becomes zero, leading to non-participation in the market, which is close to the centralized dual-channel structure, as outlined in Proposition 1.

Third, suppose that the MNO and the MVNO enter into a quantity discount contract. The initial wholesale price is denoted as $w_I$, and it results in a discount of $\delta$ based on the quantity ordered by the MVNO. The wholesale price is $w_d=w_I-\delta D_r$, where $D_r$ represents the MVNO’s order quantity. The MNO’s profits, ${\pi }_s^{qd}$, and the MVNO’s profits, ${\pi }_r^{qd}$, are calculated as follows:

$$\begin{array}{cc}\hfill {\pi }_s^{qd}=\left(p_s-c\right)D_s& +\left(w_d-c\right)D_r\hfill \\ & =\left(p_s-c\right)\left[a-p_s+b{(p}_r-p_s)\right]\hfill \\ & +\left(w_I-\delta D_r-c\right)\left[1-a-p_r+b{(p}_s-p_r)\right]\hfill \end{array}$$

(9)

$${\pi }_r^{qd}=\left[p_r-\left(w_I-\delta D_r\right)\right]\left[1-a-p_r+b{(p}_s-p_r)\right].$$

(10)

By following the same procedure, we derive the following profits in equilibrium.

$${\pi }_s^{qd}={\displaystyle \frac{1}{4(1+b)}}\left({\displaystyle \frac{{(a+b-c-2bc)}^2}{1+2b}}+{\displaystyle \frac{{(1-a-c)}^2}{2-\delta -b\delta }}\right)$$

$${\pi }_r^{qd}={\displaystyle \frac{{(1-a-c)}^2(1-\delta -b\delta )}{4(1+b){(2-\delta -b\delta )}^2}}$$

When the discount parameter is $\delta =0$, we have the same outcomes as those for the single wholesale price contract. However, when the discount parameter is $\delta \ge {\displaystyle \frac{1}{1+b}}$, the retailer cannot generate a positive profit. Therefore, we consider $\delta <{\displaystyle \frac{1}{1+b}}$ as a condition for maintaining the dual-channel structure. For instance, if the cross-price elasticity is b = 0.7, the maximum discount value for $\delta$ should be less than 58%.

Proposition 2.

In the dual-channel supply chain, the prices in both the direct and indirect channels, the wholesale price, and the profits in equilibrium under the single wholesale contract, the revenue sharing contract, and the quantity discount contract are presented in

Table 2. The equilibrium prices and profits for the decentralized supply chain based on the contract types.

	Single Wholesale Price	Revenue Sharing	Quantity Discount
w	${\displaystyle \frac{1-a+b+\left(1+2b\right)c}{2\left(1+2b\right)}}$	${\displaystyle \frac{(1-s)\left(2\right(1+b\left)\right(1-a+b+c+2bc)-(2-a(2+3b)+b(4+b+c+2bc)\left)s\right)}{2(1+b)(1+2b)(2-s)}}$	${\displaystyle \frac{2a-2(1+b+c+2bc)+b(a+b)\delta +(2+b)(1+2b)c\delta }{2(1+2b)(-2+\delta +b\delta )}}$
$p_s$	${\displaystyle \frac{a+b+\left(1+2b\right)c}{2\left(1+2b\right)}}$	${\displaystyle \frac{a+b+\left(1+2b\right)c}{2\left(1+2b\right)}}$	${\displaystyle \frac{a+b+\left(1+2b\right)c}{2\left(1+2b\right)}}$
$p_r$	${\displaystyle \frac{3-a(3+4b)+c+2b(3+b+2(1+b\left)c\right)}{4(1+b)(1+2b)}}$	${\displaystyle \frac{\frac{2+b(4+b)-a(2+3b)}{1+2b}+bc-\frac{1-a-c}{2-s}}{2(1+b)}}$	${\displaystyle \frac{\begin{array}{c}3+c+2b\left(3+b+2\left(1+b\right)c\right)-2d-b\left(6+c+b\left(5+b+3c+2bc\right)\right)\delta \\ -a(3+4b-(1+b\left)\right(2+3b\left)\delta \right)\end{array}}{2(1+b)(1+2b)(2-\delta -b\delta )}}$
$D_s$	${\displaystyle \frac{2a+b+ab-2c-3bc}{4(1+b)}}$	${\displaystyle \frac{a(2+b+s)+b(1-3c-s+2cs)-c(2-s)}{2(1+b)(2-s)}}$	${\displaystyle \frac{2a+b+ab-2c-3bc-(1+b)(a-c-b(1-2c\left)\right)\delta }{2(1+b)(2-\delta -b\delta )}}$
$D_r$	${\displaystyle \frac{4-6a+5b-7ab-2c-3bc}{4(1+b)}}$	${\displaystyle \frac{2-3a+3b-4ab-c-2bc-\frac{b(1-a-c)}{2-s}}{2(1+b)}}$	${\displaystyle \frac{2-3a+3b-4ab-c-2bc-\frac{b(1-a-c)}{2-d-b\delta }}{2(1+b)}}$
${\pi }_s$	${\displaystyle \frac{1}{4(1+b)}}\left({\displaystyle \frac{{(a+b-c-2bc)}^2}{1+2b}}+{\displaystyle \frac{{(1-a-c)}^2}{2}}\right)$	${\displaystyle \frac{1}{4(1+b)}}\left({\displaystyle \frac{{(a+b-c-2bc)}^2}{1+2b}}+{\displaystyle \frac{{(1-a-c)}^2}{2-s}}\right)$	${\displaystyle \frac{1}{4(1+b)}}\left({\displaystyle \frac{{(a+b-c-2bc)}^2}{1+2b}}+{\displaystyle \frac{{(1-a-c)}^2}{2-\delta -b\delta }}\right)$
${\pi }_r$	${\displaystyle \frac{{(1-a-c)}^2}{16(1+b)}}$	${\displaystyle \frac{{\left(1-a-c\right)}^2(1-s)}{4(1+b){(2-s)}^2}}$	${\displaystyle \frac{{\left(1-a-c\right)}^2(1-\delta -b\delta )}{4(1+b){(2-\delta -b\delta )}^2}}$

.

As illustrated in Table 2, in the supplier-led dual channel, all three contract types have the same retail price in their own direct channel. However, the wholesale price is different among the three contract types. In the single wholesale price contract, the MNO sets w to maximize the profits in both channels, and the MVNO also sets its retail price, $p_r$, to maximize its profits. Similar to the single channel, double marginalization also occurs in dual-channel supply chains. In the case of the revenue sharing contract, the MNO initially sets a lower $w_0$ (in extreme cases, $w_0\cong c$, and with a low s, the indirect market becomes the MVNO’s monopoly market) and takes a share of the revenue generated by the MVNO’s sales. In the quantity discount contract, the MNO starts with a higher initial $w_I$ (in extreme cases, $w_I\cong p_r$, making the indirect market the MNO’s monopoly market) and offers linear discounts based on the MVNO’s order quantity. The main difference is that revenue sharing is based on the realized profit, which allows the MVNO to set its retail price arbitrarily, while the outcome depends on the sharing ratio. In contrast, the quantity discount relies on the expected profit, as opposed to the realized profit, which results in differences. In other words, from the MNO’s perspective in a single channel, a quantity discount contract represents a stable contract, while revenue sharing may yield varying realized profits depending on the uncertainties in demand.

Furthermore, dual-channel contracts exhibit distinctive characteristics compared to those of single-channel contracts. The marginal expected profit of the indirect channel in the revenue sharing contract, as the demand increases, is ${\displaystyle \frac{\partial {\pi }_s}{\partial D_r}}=s{p}_r+w_0-c$, and it becomes a function depending on the MVNO’s retail price. In contrast, for the quantity discount contract, the marginal expected profit of the indirect channel is ${\displaystyle \frac{\partial {\pi }_s}{\partial D_r}}=w-2\delta D_r-c$. This equation depends on the difference between the retailer and supplier prices and the cross-price elasticity parameter, $\beta$, and is considered throughout the contract. It implies that in addition to the indirect channel retail price, the entire supply chain, including the supplier’s direct channel, should be considered. In dual-channel quantity discount contracts, there is more room for coordination between the MNO and the MVNO. In markets led by the supplier, the discount parameter can be set to reduce the retailer’s profit to zero in extreme cases. In single-channel contracts, both contract types depend on the retailer’s price, making them equivalent in terms of their effects. In dual-channel contracts, the quantity discount contract, which affects the relationship between the direct and indirect channels, may be a more favorable environment for coordination. The majority of the literature reports that competition between the online direct channel and the conventional retail channel results in channel conflict. However, from a supply chain coordination perspective, the quantity discount contract, with its consideration of both the direct and indirect channels, offers a more favorable environment for coordination.

Moreover, for the quantity discount contract, it is possible to determine the value of δ that equals the performance in the centralized case, with the centralized dual channel (${{\pi }_c=\pi }_s^{qd}+{\pi }_r^{qd}$). At the point where $\delta ={\displaystyle \frac{1}{1+b}}$, the entire supply chain profit is maximized. In contrast, for revenue sharing, only when the sharing ratio is s=1 does it align with the centralized ${\pi }_c$, in which case the retailer exits the market, indicating lower overall efficiency compared to that for the quantity discount contract.

Proposition 3.

When $a>{\displaystyle \frac{1}{2}}$ and $1-a>c$, the quantity discount contract effectively coordinates the dual-channel supply chain. The contract parameters and profits are presented in Table 2. Specifically, the quantity discount contract with $\delta ={\displaystyle \frac{1}{1+b}}$ and $w={\displaystyle \frac{2a-2(1+b+c+2bc)+b(a+b)\delta +(2+b)(1+2b)c\delta }{2(1+2b)(-2+\delta +b\delta )}}$ perfectly coordinates the dual-channel supply chain.

When the profit of the supply chain is maximized at $\delta ={\displaystyle \frac{1}{1+b}}$, in this scenario, the MVNO’s profit approaches zero, and the MNO’s profit aligns with the centralized case. This indicates that the quantity discount contract has limitations for the retailer, as it brings in lower profits compared to those for a single wholesale contract. However, considering that the overall performance of the supply chain is higher than that for a simple single wholesale price contract, applying a transfer payment mechanism by the MNO can coordinate the entire supply chain. Therefore, the quantity discount contract, when combined with a proper transfer payment, can achieve perfect coordination, benefiting all firms with a Pareto improvement. In the case of revenue sharing, while it is not as straightforward as a quantity discount, by setting the sharing ratio to below 1 and implementing a transfer payment mechanism, it is possible to enhance the overall performance of the dual-channel supply chain compared to that with a single wholesale price contract.

Theorem 1.

When $a>{\displaystyle \frac{1}{2}}$ and $1-a>c$, both a quantity discount contract with $\delta ={\displaystyle \frac{1}{1+b}}-\epsilon$ and a revenue sharing contract with the sharing parameter s approaching 1 can effectively coordinate the dual-channel supply chain, and this creates a Pareto improvement zone. For both contract types, a transfer payment exceeding the MVNO’s profit with a single wholesale price contract is required to achieve this coordination.

4.3. A Numerical Example

For a direct channel where the demand is a = 0.7, the price elasticity is b = 0.5, and the operating cost is c = 0.1, under a single wholesale price contract, the MNO’s profit is 0.1230, the MVNO’s profit is 0.0035, and the total profit is 0.1265. Under a revenue sharing contract with a sharing ratio of s = 0.9, the MNO’s profit is 0.1288, the MVNO’s profit is 0.0011, and the total profit is 0.1299. For the quantity discount contract with a discount factor of δ = 0.625, the MNO’s profit is 0.1292, the MVNO’s profit is 0.0007, and the total profit is 0.1300. In both cases, the quantity discount and revenue sharing contracts outperform the single wholesale price contract in terms of their overall efficiency.

Figure 2a illustrates how the total profit of the supply chain for each contract relationship changes with increasing operating costs, while Figure 2b represents the effect of an increase in the preference of the customer for the direct channel (a) on the total supply chain profit for each contract type.

The efficiency of the single wholesale price contract is the lowest (black line), followed by that of the revenue sharing contract with $s<1$ (blue line). The quantity discount contract with $\delta ={\displaystyle \frac{1}{1+b}}-\epsilon$ (green line) demonstrates the highest performance in terms of profit.

5. The Impact of Coordination on Investment and Spillover

In this section, we examine how investment for innovation is influenced by the coordination effect and how innovation investment affects the retailer in terms of spillover effects. As described in Section 3, the increase in demand caused by the investment is represented as $\gamma x$. While the demand increases in the direct channel and the indirect channel may differ, for the sake of this analysis, we assume ${\gamma }_s={\gamma }_r$ because both channels sell a homogeneous product of the same nature. This assumption also aids our understanding of the pure effects of the investments made by both the MNO and the MVNO on each channel.

First, the MNO determines its investment level. As in the previous section, the supplier then determines the wholesale price and the retail price of its own direct channel using the following objective function:

$${\pi }_{sI}=\left(p_s-c\right)D_s+\left(w-c\right)D_r-k{\displaystyle \frac{x^2}{2}}$$

(11)

In this scenario, the MVNO’s profit function remains unaffected. First, we analyze the centralized case as a benchmark model. Then, we proceed to analyze the coordination effect and the spillover effect when the supplier invests in innovation. In Section 5.3, we explore the case where the retailer is the innovator.

5.1. A Centralized Dual-Channel Supply Chain with Investment

First, we examine a centralized case as the benchmark model, where both channels are centralized. The profit function for each channel, considering the increase in demand resulting from investment in innovation, can be expressed as follows:

$${\pi }_{cI}=\left(p_s-c\right)\left[a-p_s+b{(p}_r-p_s)+\gamma x\right]+\left(p_r-c\right)\left[1-a-p_r+b{(p}_s-p_r)+\gamma x\right]-k{\displaystyle \frac{x^2}{2}}$$

(12)

The prices ($p_s^{\mathrm{*}}$ and $p_r^{\mathrm{*}}$) and the resulting profit (${\pi }_{cI}$) that maximize revenue can be determined as follows:

Proposition 4.

In a centralized case, the optimal investment level and price in each channel, in addition to the maximized profit, are

$$x_{cI}^{*}={\displaystyle \frac{(1-2c)r}{2(k-{\gamma }^2)}}$$

$$p_s^{\mathrm{*}}=c+{\displaystyle \frac{1}{4}}\left[{\displaystyle \frac{(1-2c)k}{k-{\gamma }^2}}-{\displaystyle \frac{1-2a}{1+2b}}\right]$$

$$p_r^{*}=c+{\displaystyle \frac{1}{4}}\left[{\displaystyle \frac{(1-2c)k}{k-{\gamma }^2}}+{\displaystyle \frac{1-2a}{1+2b}}\right]$$

$${\pi }_{cI}={\displaystyle \frac{1}{8}}\left[{\displaystyle \frac{{(1-2a)}^2}{1+2b}}+{\displaystyle \frac{{(1-2c)}^{2}k}{k-{\gamma }^2}}\right].$$

5.2. A Decentralized Dual-Channel Supply Chain with Supplier Investment

Based on the results from the previous section, we next evaluate the single wholesale price contract with the quantity discount contract that allows for the most effective coordination. The procedure begins with the supplier selecting the level of innovation investment, denoted as x. Afterwards, the sequence of events follows the same procedure as that outlined in the preceding section. The MNO’s profit, ${\pi }_{sI}^{wp}$, and the MVNO’s profit, ${\pi }_{rI}^{wp}$, in a single wholesale price contract are given by

$$\begin{array}{cc}\hfill {\pi }_{sI}^{wp}=\left(p_s-c\right)[a& - p_s+b{(p}_r-p_s)+\gamma x]+\left(w-c\right)\left[1-a-p_r+b{(p}_s-p_r)+\gamma x\right]\hfill \\ & -k{\displaystyle \frac{x^2}{2}}\hfill \end{array}$$

(13)

$${\pi }_{rI}^{wp}=\left(p_r-w\right)\left[1-a-p_r+b{(p}_s-p_r)+\gamma x\right].$$

(14)

To find the decisions of the MNO and the MVNO in equilibrium, we solve through backwards induction. Compared to the case without investment, the MNO’s optimal retail and wholesale prices in the presence of investment for innovation increase by ${\displaystyle \frac{rx}{2}}$ ($p_s={\displaystyle \frac{a+b+c+2bc}{2(1+2b)}}+{\displaystyle \frac{\gamma x}{2}}$, $w={\displaystyle \frac{1-a+b+\left(1+2b\right)c}{2\left(1+2b\right)}}+{\displaystyle \frac{\gamma x}{2}}$). Consequently, we can obtain the profits of both the MNO and the MVNO as follows:

$$\begin{gathered} {\pi }_{sI}^{wp}={\displaystyle \frac{1}{8\left(1+b\right)\left(1+2b\right)}}\left[1-2a+3a^2+2b+2a^{2}b+2b^2-2c-2ac-8bc-4abc-8b^{2}c+3c^2+10b{c}^2 \\ +8b^2{c}^2+2\left(1+2b\right)\left(1+a+b\left(2-4c\right)-3c\right)\gamma x-(1+2b)\left(4\right(1+b)k+(3+4b\left){\gamma }^2\right)x^2\right] \end{gathered}$$

$${\pi }_{rI}^{wp}={\displaystyle \frac{{(1-a-c+\gamma x)}^2}{16(1+b)}}.$$

The MVNO benefits from the MNO’s investment, and its profit increases compared to that in the scenario where no investment takes place (${\displaystyle \frac{{(1-a-c)}^2}{16(1+b)}})$. For instance, if an MNO invests in a communication infrastructure with a doubled download speed, it becomes reasonable for the MNO to raise the direct channel price and generate additional profit. Meanwhile, an MVNO pays a higher wholesale price (w) but can also increase its retail price and profit because of the spillover effect. The retailer can also take an additional profit margin resulting from the supplier’s investment. As illustrated in Figure 2, it is possible to determine the supplier’s optimal investment level.

$$x_{sI}^{wp\mathrm{*}}={\displaystyle \frac{(1+a+b(2-4c)-3c)\gamma }{4\left(1+b\right)k-(3+4b){\gamma }^2}}.$$

Comparing the profit-maximizing investment levels in both centralized and decentralized scenarios, as expected, the investment level is always lower for the single wholesale price contract ($x_{cI}^{\mathrm{*}}>x_{sI}^{\mathrm{*}}$).

Next, consider the relationship between investment for innovation and the quantity discount contract. The supplier profits, ${\pi }_{sI}^{qd}$, and the retailer profits, ${\pi }_{rI}^{wp}$, are calculated as follows:

$$\begin{array}{cc}{\pi }_{sI}^{qd}=\left(p_s-c\right)[a\hfill & - p_s+b{(p}_r-p_s)+\gamma x]\hfill \\ & +\left(w_I-\delta D_r-c\right)\left[1-a-p_r+b{(p}_s-p_r)\right]+\gamma x-k{\displaystyle \frac{x^2}{2}}\hfill \end{array}$$

(15)

$${\pi }_{rI}^{wp}=\left[p_r-\left(w_I-\delta D_r\right)\right]\left[1-a-p_r+b{(p}_s-p_r)\right].$$

(16)

After obtaining the MNO and the MVNO’s decisions in equilibrium, the profits are as follows:

$$\begin{gathered} {\pi }_{sI}^{qd}={\displaystyle \frac{1}{4\left(1+b\right)\left(1+2b\right)\left(2-\delta -b\delta \right)}}[1+2b+2b^2-2c-8bc-8b^{2}c+3c^2+10b{c}^2+8b^2{c}^2-b^2\delta -b^3\delta \\ \begin{array}{cc}& +2bc\delta +6b^{2}c\delta +4b^{3}c\delta -c^2\delta -5b{c}^2\delta -8b^2{c}^2\delta -4b^3{c}^2\delta +a^2\left(3+b\left(2-\delta \right)-\delta \right)\hfill \\ & +2\left(1+2b\right)\left(1+2b-3c-4bc+\left(1+b\right)\left(c-b\left(1-2c\right)\right)\delta \right)\gamma x\hfill \\ & -\left(1+2b\right)\left(2\left(1+b\right)\left(2-\delta -b\delta \right)k+\left(3-\delta +b\left(4-3\delta -2b\delta \right)\right){\gamma }^2\right)x^2-2a(1+b\delta +b^2\delta \hfill \\ & +(1+2b)c(1-\delta -b\delta )+(1+2b)(1-\delta -b\delta )\gamma x\left)\right]\end{array} \end{gathered}$$

$${\pi }_{rI}^{wp}={\displaystyle \frac{{\left(1-a-c+\gamma x\right)}^2(1-\delta -b\delta )}{4(1+b){(2-\delta -b\delta )}^2}}.$$

As Figure 2 illustrates, the MNO’s profit function is strictly concave, allowing us to determine the optimal investment level for the supplier, given by

$$x_{sI}^{qd\mathrm{*}}={\displaystyle \frac{(1+a+2b-3c-4bc-(1+b\left)\right(a-c+b(1-2c)\left)d\right)\gamma }{2\left(1+b\right)\left(2-d-bd\right)k-(3-d+b(4-(3+2b)d\left)\right){\gamma }^2}}.$$

This demonstrates that the MVNO’s profit increases as x increases.

Proposition 5.

The equilibrium prices in both the direct and indirect channels, the wholesale price, the investment level, and the profits in the decentralized dual-channel under single wholesale price contracts and quantity discount contracts, along with a comparison with the centralized dual-channel scenario, are presented in

Table 3. The optimal investment levels, equilibrium prices, and profits depending on the contract type.

	Centralized Dual Channel	Single Wholesale Price Contract	Quantity Discount Contract
x	${\displaystyle \frac{(1-2c)r}{2(k-{\gamma }^2)}}$	${\displaystyle \frac{(1+a+b(2-4c)-3c)\gamma }{4\left(1+b\right)k-(3+4b){\gamma }^2}}$	${\displaystyle \frac{(1+a+2b-3c-4bc-(1+b\left)\right(a-c+b(1-2c)\left)d\right)\gamma }{2\left(1+b\right)\left(2-d-bd\right)k-(3-d+b(4-(3+2b)d\left)\right){\gamma }^2}}$
w		${\displaystyle \frac{1-a+b+\left(1+2b\right)c}{2\left(1+2b\right)}}+{\displaystyle \frac{\gamma x}{2}}$	${\displaystyle \frac{b\left(4-\delta -2b\delta \right)\gamma x+2\left(1+b+c+2bc+\gamma x\right)-\left(b^2+\left(2+b\right)\left(1+2b\right)c\right)\delta -a\left(2+b\delta \right)}{2(1+2b)(2-\delta -b\delta )}}$
$p_s$	$c+{\displaystyle \frac{1}{4}}\left[{\displaystyle \frac{(1-2c)k}{k-{\gamma }^2}}-{\displaystyle \frac{1-2a}{1+2b}}\right]$	${\displaystyle \frac{a+b+c+2bc}{2(1+2b)}}+{\displaystyle \frac{\gamma x}{2}}$	${\displaystyle \frac{a+b+c+2bc}{2(1+2b)}}+{\displaystyle \frac{\gamma x}{2}}$
$p_r$	$c+{\displaystyle \frac{1}{4}}\left[{\displaystyle \frac{(1-2c)k}{k-{\gamma }^2}}-{\displaystyle \frac{1-2a}{1+2b}}\right]$	${\displaystyle \frac{3-a(3+4b)+c+3\gamma x+2b(3+b+2c+2bc+2(2+b\left)\gamma x\right)}{4(1+b)(1+2b)}}$	${\displaystyle \frac{\begin{array}{c}3+c+2b\left(3+b+2\left(1+b\right)c\right)-2\delta -b\left(6+c+b\left(5+b+3c+2bc\right)\right)\delta \\ -a\left(3+4b-\left(1+b\right)\left(2+3b\right)\delta \right)+3\gamma x-(2+b)(\delta -b(4-(3+2b)\delta \left)\right)\gamma x)\end{array}}{2(1+b)(1+2b)(2-\delta -b\delta )}}$
$D_s$	${\displaystyle \frac{2ak-2ck+{\gamma }^2-2a{\gamma }^2}{4(k-{\gamma }^2)}}$	${\displaystyle \frac{a\left(2+b\right)+b-2c-3bc+(2+3b)\gamma x}{4(1+b)}}$	${\displaystyle \frac{\begin{array}{c}a\left(2+b\left(1-\delta \right)-\delta \right)-b\left(1-\delta \right)\left(1-3c+3\gamma x\right)\\ -\left(2-\delta \right)\left(c-\gamma x\right)-b^2\delta (1-2c+2\gamma x)\end{array}}{2(1+b)(2-\delta -b\delta )}}$
$D_r$	${\displaystyle \frac{2\left(1-a-c\right)k-(1-2a){\gamma }^2}{4(k-{\gamma }^2)}}$	$\frac{1}{4}(1-a-c+\gamma x)$	${\displaystyle \frac{1-a-c+\gamma x}{2(2-\delta -b\delta )}}$
${\pi }_s$	${\displaystyle \frac{1}{8}}\left[{\displaystyle \frac{{(1-2a)}^2}{1+2b}}+{\displaystyle \frac{{(1-2c)}^{2}k}{k-{\gamma }^2}}\right]$	$$\begin{gathered} \begin{array}{c}\frac{1}{8\left(1+b\right)\left(1+2b\right)}\left[1-2a+3a^2+2b+2a^{2}b+2b^2-2c-2ac- \\ 8bc-4abc-8b^{2}c+3c^2+10b{c}^2+8b^2{c}^2+2\left(1+ \\ 2b\right)\left(1+a+b\left(2-4c\right)-3c\right)\gamma x-(1+2b)\left(4\right(1+b)k+ \\ (3+4b\left){\gamma }^2\right)x^2\right]\end{array} \end{gathered}$$	$$\begin{gathered} \begin{array}{c}{\displaystyle \frac{1}{4\left(1+b\right)\left(1+2b\right)\left(2-\delta -b\delta \right)}}\left[1+2b+2b^2-2c-8bc-8b^{2}c+3c^2+10b{c}^2+ \\ 8b^2{c}^2-b^2\delta -b^3\delta +2bc\delta +6b^{2}c\delta +4b^{3}c\delta -c^2\delta -5b{c}^2\delta - \\ 8b^2{c}^2\delta -4b^3{c}^2\delta +a^2\left(3+b\left(2-\delta \right)-\delta \right)+2\left(1+2b\right)\left(1+2b-3c- \\ 4bc+\left(1+b\right)\left(c-b\left(1-2c\right)\right)\delta \right)\gamma x-\left(1+2b\right)\left(2\left(1+b\right)\left(2-\delta - \\ b\delta \right)k+\left(3-\delta +b\left(4-3\delta -2b\delta \right)\right){\gamma }^2\right)x^2-2a(1+b\delta +b^2\delta +(1+ \\ 2b\left)c\right(1-\delta -b\delta )+(1+2b\left)\right(1-\delta -b\delta \left)\gamma x\right)\right]\end{array} \end{gathered}$$
${\pi }_r$		${\displaystyle \frac{{(1-a-c+\gamma x)}^2}{16(1+b)}}$	${\displaystyle \frac{{\left(1-a-c+\gamma x\right)}^2(1-\delta -b\delta )}{4(1+b){(2-\delta -b\delta )}^2}}$

.

As seen in Figure 3a, when a supplier makes an investment, the profit from a quantity discount contract (black line) is higher than that from a single wholesale price contract (blue line). The supplier’s optimal investment level can also be determined, and the optimal investment level for a quantity discount contract is higher compared to that for the single wholesale price contract ($x_{sI}^{qd\mathrm{*}}>x_{sI}^{wp\mathrm{*}}$). Figure 3b illustrates the retailer’s profit and highlights the spillover effect of innovation on the retailer. An interesting finding is that under the quantity discount contract, the spillover effect appears to be less influenced by the discount parameter $\delta$.

Let us evaluate the performance of the entire supply chain. We find that with the appropriate parameter settings for the quantity discount, as indicated in Theorem 1, we can achieve profits equivalent to those in a centralized dual channel setting. Due to tractability issues, it is challenging to express the optimal conditions analytically. However, a numerical investigation shows noteworthy insights. For instance, consider a scenario where the demand for the direct channel is a = 0.7, the cross-price elasticity is b = 0.7, the operating cost is c = 0.1, the innovation coefficient is k = 0.1, and the increase in demand due to investment is r = 0.1. In such a case, the total profit ties with that of the centralized dual channel, reaching 0.097 when applying the quantity discount contract with an optimal investment level of x = 0.444. Notably, the optimal investment level remains the same in both cases. In contrast, under the single wholesale price contract, the optimal investment level decreases to x = 0.405. Consequently, the profit for the entire supply chain diminishes to 0.095, indicating lower efficiency compared to that in the previous two scenarios. Figure 3 illustrates the performance of each supply chain. The appropriately coordinated quantity discount contract (blue line), with the parameters, demonstrates a performance close to that of the centralized dual channel (dashed line).

Theorem 2.

A quantity discount contract coordinates the dual channels with the supplier’s investment for innovation, leading to a Pareto improvement zone. Through the quantity discount contract, the supplier achieves an equivalent investment level to that in the centralized case, effectively addressing the spillover effect of the investment.

From the MNO’s perspective as the supplier, there may be concerns about the spillover effect of investment on the MVNO. However, it is shown that the profit of the entire supply chain can be increased through an appropriate quantity discount mechanism and investment, surpassing the level achievable by a simple single wholesale price contract as shown in Figure 4. This implies that a high level of coordination can compensate for the spillover effect of innovation, and it is observed that the MNO’s optimal investment level also increases.

The analysis shows that coordination mechanisms—particularly quantity discount contracts—enhance the supplier’s incentives to invest in innovation. In the absence of coordination, the supplier tends to underinvest due to its concerns about value appropriation by the retailer. However, coordination helps align the supplier’s investment incentives with the overall efficiency of the supply chain by ensuring a more equitable distribution of the innovation gains. This results in improved profitability for both the MNO and the MVNO, as the supplier captures a larger share of the returns while the retailer benefits from positive demand spillover.

5.3. A Decentralized Dual-Channel Supply Chain with Retailer Investment

Let us assume that the retailer makes an investment in launching new services (e.g., partnerships with OTT providers) or innovative customer services. In this case, it is supposed that this investment affects the indirect channel’s demand while not impacting the direct channel. Following the same analysis as that in the previous sections, it is observed that the retailer’s investment level results in lower $p_s$ and w values compared to those for the supplier’s investment. However, the retailer sets higher $p_s$ and w values when no investment is made. An essential point to note is that when the retailer invests in its indirect channel, it seems that only the demand in the indirect channel increases. Still, since there is an upstream supplier, the MNO, even in the indirect channel, there is also a spillover effect on the supplier. Therefore, from the supplier’s perspective, having an innovative retailer can provide similar benefits, leading to a desire to engage in transactions with such partners. Although the demand increase seems to be selfish, it affects the other party as well. It is also possible to determine the optimal investment level, denoted as $x_{rI}^{\mathrm{*}}$:

$$x_{rI}^{\mathrm{*}}={\displaystyle \frac{(1-a-c)\gamma }{8\left(1+b\right)k-{\gamma }^2}}.$$

The value of $x_{rI}^{\mathrm{*}}$ is lower than the supplier’s optimal level of investment. Nonetheless, the MNO’s profit also increases with the MVNO’s investment.

When the MVNO undertakes innovation, the resulting improvements in demand also extend upstream, generating positive spillover effects for the MNO. This mutual spillover dynamic highlights the potential for shared value creation within the dual-channel structure. Contractual coordination that accounts for these interdependencies can further encourage innovation efforts from both parties. Ultimately, such coordination fosters a more synergistic and strategically aligned relationship between the MNO and the MVNO.

We analyzed cases where the increase in demand for the MNO, ${\gamma }_s$, does not equal the increase in demand for the MNO (${\gamma }_s\ne {\gamma }_r$). For instance, when ${\gamma }_s>{\gamma }_r$, the spillover effect on the retailer was lower than that in symmetric cases (${\gamma }_s={\gamma }_r$). However, this difference in the parameter values does not invalidate the implications of Proposition 5; it merely shows a minor variation. In addition, it is noteworthy that while the assumption is unrealistic, we conducted an additional analysis in a scenario where the MNO’s investment had no effect on the demand in the indirect channel. In this case, the MNO’s investment still leads to an increase in wholesale prices and direct channel retail prices, which, in turn, affects the retailer’s indirect channel prices, causing them to rise. However, it is proven that the increase in demand in the direct channel does not have an effect on the MVNO’s profit in this scenario.

6. Conclusions

This study examined how contract design and investment spillover effects jointly influence the coordination and performance in dual-channel supply chains, with a focus on the telecommunications industry. Grounded in the supplier-dominated market structure often observed in mobile network operations, the model captures the strategic interaction between a mobile network operator (MNO) and a mobile virtual network operator (MVNO). The analysis centers on how investment in innovation by one party—particularly the supplier—can generate externalities that affect the other party’s performance, a phenomenon defined here as an investment spillover effect.

A key contribution of this work is the demonstration that quantity discount contracts can serve as an effective coordination mechanism in the presence of investment externalities. Compared to single wholesale price contracts, quantity discounts align the investment incentives better and enhance supplier profitability while maintaining competitive pricing on the retail market. In particular, this form of contract improves the joint profitability—interpreted as the system-level efficiency—by internalizing the externalities that arise from unilateral investment. These findings support the broader view that contractual mechanisms are not merely tools for setting prices but also function as strategic governance instruments in supply chains.

The results also reveal asymmetry in the spillover dynamics: supplier-led investments tend to generate stronger system-wide gains, while retailer-led investments yield more limited indirect benefits to the supplier. This asymmetry underscores the need for contract structures that address such imbalances, especially in sectors like telecommunications, where one party holds significant infrastructure and bargaining power.

Nevertheless, several simplifying assumptions in the model may limit the generalizability of the results. First, the assumption of product homogeneity abstracts from the practical differentiation strategies employed by MVNOs—such as customized pricing, bundling, or branding—that may influence the consumer demand across channels. Second, the model presumes symmetric and complete information between contracting parties, whereas in practice, strategic intentions and investment priorities may not be fully observable or shared. Third, the analysis is based on a bilateral MNO-MVNO relationship, while actual markets typically involve multiple players engaging in competitive and regulatory interactions. While these assumptions facilitate analytical tractability, relaxing them would allow for richer and more realistic insights. Future research could address these limitations by incorporating competitive market structures, asymmetric information, and dynamic investment behavior. In particular, analyzing the impact of regulatory interventions—such as mandated access, pricing controls, or innovation subsidies—could yield valuable implications for both policymakers and industry stakeholders. Lastly, although Theorems 1 and 2 are based on numerical observations due to analytical intractability, they are logically consistent within the model’s framework and serve as theoretically grounded hypotheses that can be tested in future empirical or simulation-based studies.

These findings also carry practical implications. For policymakers, contract design may serve as an indirect yet powerful policy lever to enhance investment incentives and promote market efficiency. For firms operating in capital-intensive and innovation-driven sectors, aligning the contract terms with spillover realities is critical for achieving sustainable collaboration and mutual profitability.