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This paper examines optimal pricing in a two-tier product and service supply chain consisting of a manufacturer and a retailer in the context of vertical competition in extended warranty in two cases: one considering the retailer’s fairness concerns and one without considering the retailer’s fairness concerns. A manufacturer-dominated product and service supply chain game-theoretic model on the Stackelberg model is developed to analyse how the level of vertical competition in extended warranty service and the intensity of a retailer’s fairness concerns influence the optimal pricing of products and extended warranties for the manufacturer and retailer. This study finds the following: (i) Two parties of the supply chain employ differential pricing strategies for extended warranties when the retailer has fairness concerns. (ii) Compared to the same pricing strategies for extended warranty service when the retailer has no fairness concerns, the increase of competition intensity of vertical extended warranty service will enlarge the price difference of extended warranty service. Meanwhile, it is the intensity of fairness concerns that determines the influences of retailer’s fairness concerns on the price difference of extended warranties. (iii) If no fairness concerns are raised, an increase in the level of vertical competition in extended warranty service would benefit both supply chain parties, rather than hurting their profit. If the retailer is fair-minded, its fairness utility increases when the intensity of the fairness concerns rises in a reasonable range and decreases when the intensity exceeds the reasonable range, but for the manufacturer, its profits will be damaged as long as the retailer raises fairness concerns.

As market competition intensifies, the profits generated by manufacturers and retailers through traditional product production and sales are gradually reduced, which forces them to actively seek new profit drivers. A case in point is their shift to the innovative business model of offering products and related services at the same time [

It is worth noting that competition in the extended warranty service market is based on products. However, the relationship between a manufacturer and a retailer within a supply chain is much more about upstream and downstream partnership regarding production and sales of specific products. This requires ensuring that both the manufacturers and retailers within a product and service supply chain obtain the optimal benefits. The key to solving this issue lies in making optimal pricing decisions for products and services. In optimal pricing decisions, the dominant manufacturer tends to play a powerful role in setting prices [

At present, the research on extended warranties has gone beyond extended warranties offered by a single enterprise [

The paper also covers fairness concerns in supply chains. Camerer and Thaler [

Drawing on the extant literatures, this paper considers retailers’ fairness concern preference and the intensity of vertical competition in extended warranty service between retailers and manufacturers when studying the optimal pricing decisions for products and services in a product and service supply chain composed of a manufacturer and retailer. We intend to answer the following questions.

What are the optimal pricing decisions for products and services in a product and service supply chain when the retailer does not have fairness concerns?

What are the optimal pricing decisions for products and services in a product and service supply chain when the retailer has fairness concerns compared with no fairness concerns considered?

How do the level of the retailer’s fairness concerns and the level of vertical competition in extended warranty service affect the optimal pricing of products and services in a product and service supply chain?

The rest of this study is organized as follows. In Section

We consider a two-tier product and service supply chain consisting of a manufacturer and a retailer. In the supply, consumers purchase products from the retailer to satisfy their needs, the retailer orders products from the manufacturer based on the consumers’ needs, and the manufacturer strives to manufacture products to meet the retailer’s order requirements. This creates a supply-demand flow of products. Beyond that, the supply chain also has a supply-demand flow of extended warranties on the products. To better meet consumer demand and improve user experience, both the manufacturer and retailer can provide homogeneous extended warranties. This means that there is vertical competition in an extended warranty service between the manufacturer and its downstream retailer in the supply chain.

In Figure

Schematic diagram of a product and service supply chain.

We assume that the market demand for a specific product from the manufacturer and the retailer is

This paper focuses on analysing how the level of vertical competition in extended warranty service and a retailer’s fairness concern preference affect the optimal pricing of products and services of the retailer and manufacturer. For ease of exposition, the manufacturer’s cost of production and the retailer’s cost of sales can be set to zero [

From the above description and assumptions, the manufacturer's profit function can be expressed as follows:

The retailer’s profit function given as follows:

As discussed in the work of Wu and Niederhoff [

In the partnership between the manufacturer and the retailer, the manufacturer decides the wholesale price

The second-order derivative of equation (

It can be seen that both

Based on equations (

Substituting

The second-order derivative of equation (

Since

Solving equation (

Likewise, substituting

The second-order derivative of equation (

Since

Without fairness concerns, both the manufacturer and the retailer can arrive at optimal pricing decisions. The optimal wholesale price and extended warranty service price determined by the manufacturer are

The optimal retail price and extended warranty service price determined by the retailer are

Proposition 1 shows that when the retailer has no fairness concerns, both the manufacturer and the retailer can make their optimal pricing decisions, and both parties adopt the same extended warranty service pricing strategy. This means that the vertical competition in extended warranty service between the two parties without fairness concerns does not result in differential pricing of extended warranties. Based on the analysis of Proposition 1, this paper explores how the potential market size of a product and the level of competition in extended warranty service on the product affect optimal pricing decisions of both the manufacturer and the retailer. The first-order partial derivatives of

If the retailer has no fairness concern, the manufacturer’s product wholesale price

Inference 1 indicates that an increase in product market size leads both the manufacturer and the retailer choose to increase the price of their products and extended warranties. As the competition in extended warranty service between the two parties intensifies, they tend to lower the product price as optimal decisions. This price reduction, however, is often accompanied by an increase in the extended warranty service price, meaning that stiffer service competition without fairness concerns contributes little to increasing consumer benefits.

If the retailer has no fairness concerns, the relationship between the profits of the manufacturer and the retailer and the level of competition in extended warranty service satisfies the following conditions:

Substituting

Since

On the premise of satisfying constraint condition

The partial derivative of equation (

On the premise of satisfying constraint condition

From Inference 1 and Proposition 2, it is clear that increasing vertical competition in an extended warranty service between a manufacturer and a retailer which has no fairness concerns leads the manufacturer to lower its product wholesale price and raise its extended warranty service price as the optimal pricing strategy. The same is true of the retailer. The reduction by the manufacturer in product wholesale prices can motivate the retailer to order more products from the manufacturer, and a lower retail price means a higher demand for a specific product, thereby increasing the sales revenue of the two parties. On the other hand, as vertical competition in the extended warranty service intensifies, the optimal decision of both the manufacturer and the retailer is to increase the service price with the aim of increasing the revenue from the service. From the above analysis, we can reach a counterintuitive conclusion: tougher competition in an extended warranty service increases the benefits of both the manufacturer and the retailer, instead of harming their interests. However, this inevitably damages the benefits of end users.

If a retailer is sensitive to fairness, it determines its retail price and extended warranty service price of a product with the goal of maximizing its utility

As discussed in [

Following the same logic of analysing optimal pricing decisions without fairness concerns, calculate the first-order and second-order derivatives of equation (

Substituting

Following the same logic of solving

Substituting

Following the same logic of solving

Substituting

Based on the above analysis and the analysis of Proposition 1, we propose the following proposition.

If the retailer is fair-minded, both the manufacturer and the retailer can arrive at optimal pricing decisions. The optimal wholesale price and extended warranty service price determined by the manufacturer are

The optimal product retail price and extended warranty service price determined by the retailer are

Proposition 3 shows that when the retailer has fairness concerns, both the manufacturer and the retailer can make optimal pricing decisions and they adopt different extended warranty service pricing strategies. This indicates that the fairness concerns of the retailer which experiences vertical competition in an extended warranty service with the manufacturer lead to differential pricing of the service.

Based on the analysis of Proposition 3, this paper investigates how the potential market size of a product, the level of competition in extended warranty service on the product, and the retailer’s fairness preference influence the optimal pricing decisions of both the manufacturer and the retailer. The first-order partial derivatives of

If the retailer has fairness concerns, the manufacturer’s optimal product wholesale price and extended warranty service price and the retailer’s optimal product retail price and extended warranty service price exhibit the following relationships with product market size

The partial derivatives of

where

Solving the partial derivative of retail price

where

where

Solving the partial derivative of the retailer’s optimal extended warranty service price

Likewise, we can derive

The results from the analysis of Inference 1 and Inference 2 show that when the retailer has fairness concerns, market size and the level of vertical competition in extended warranty service influence the optimal pricing decisions of the manufacturer in a way similar to what they do for the retailer. This indicates that the retailer’s fairness concerns do not change the direction in which market size and vertical competition in extended warranty service influence optimal pricing decisions in the product and service supply chain. Further analysis reveals that in the face of increasing intensity of the retailer's fairness concerns, the manufacturer can offset the concerns and indirectly receive compensation by lowering its product wholesale price and the retailer can improve its profits by raising the retail price. In addition, both the manufacturer and the retailer can decrease the price of extended warranties to attract consumers to purchase the service, thereby increasing their revenue.

The findings from the analysis of Proposition 3 and Inference 2 suggest that the retailer's fairness concerns can lead to differential pricing of extended warranties, though they do not change the direction in which market size and vertical competition in extended warranty service affect the optimal pricing decisions of the retailer and manufacturer. It can be seen that the manufacturer and retailer may determine different extended warranty service price if the retailer is fair-minded. Then, how does the retailer's fairness concern affect this price difference? And is this price difference connected with the level of vertical competition in extended warranty service in the context of the retailer's fairness concerns? The following proposition will answer these questions.

When the retailer has fairness concerns, the difference in extended warranty service price between the retailer and manufacturer meets

The relationship between the difference

The relationship between the difference

Referring to the expressions of

In the discussion on the relationship between the difference in extended warranty service price and the level of competition in extended warranty service, the price difference is expressed as follows:

The partial derivative of

where

The partial derivative of

Proposition 4(i) shows that when the retailer has fairness concerns, the manufacturer tends to develop a differential pricing strategy in which it offers a specific extended warranty service at a price above that of the retailer, thereby shifting some of the consumer demand for the service to the retailer to compensate the retailer. Proposition 4(ii) implies that generally, the higher the level of vertical competition in an extended warranty service between the manufacturer and retailer, the greater the price difference in the extended warranty service provided by the two parties. This is mainly because increasing competition between the two parties means a greater negotiation power of the retailer, which forces the manufacturer to surrender part of its profits to the retailer by increasing its own extended warranty service price. From Proposition 4(iii), it can be seen that the intensity of the retailer's fairness concerns directly affects the differential pricing strategy for an extended warranty service subjected to vertical competition with the manufacturer. If the intensity of the retailer's fairness concerns is less than a certain threshold (

When the retailer has fairness concerns,

The relationship between the manufacturer’s profit function and the level of vertical competition in extended warranty service as well as the intensity of the retailer’s fairness concerns meets

The relationship between the retailer’s utility function and the level of vertical competition in extended warranty service as well as the intensity of the retailer’s fairness concerns is as follows: if the intensity of the fairness concerns meets

Substituting optimal product prices

The first-order partial derivative of equation (

where

Likewise, the first-order partial derivative of equation (

where

Substituting

To analyse the relationship between the retailer's utility function and the intensity of its fairness concerns, the first-order partial derivative of equation (

Following the logic of analysing the relationship between the retailer’s utility function and the intensity

Proposition 5(i) shows that when the retailer has fairness concerns, the manufacturer’s profit increases with the level of vertical competition in extended warranty service. Moreover, given Proposition 2, it can be seen that the retailer’s fairness concerns do not change the way that the competition level influences the manufacturer’s profit. On the other hand, as the retailer’s fairness concern intensifies, the profits obtained by the manufacturer decrease. Considering Inference 2, the main reason for this situation is that in the face of increasing intensity of the retailer’s fairness concerns, the manufacturer tends to lower product wholesale price so that the retailer receives indirect compensation. Another ideal solution for the manufacturer is to reduce extended warranty service price to attract consumers to purchase the service. In this case, a retailer that has a strong sense of fairness of concern may also choose to reduce extended warranty service price. However, the findings from the analysis of Proposition 4(i) show that retailers always offer extended warranties at a lower price than that of manufacturers. In summary, retailers’ fairness concerns negatively affect the profit of manufacturers. The greater the intensity of the fairness concerns, the lower the profit of manufacturers.

Proposition 5(ii) shows that compared to the impact of vertical competition level on a retailer’s performance without fairness concerns, an increase in vertical competition level in the context of the retailer’s fairness concerns does not always enhance the utility of the retailer. When the intensity of the fairness concerns is lower than a certain threshold

To verify the propositions and inferences in Section

Relationships between wholesale prices

It can be seen from Figure

Figure

Relationships between extended warranty service prices

As can be seen from Figure

In the same way, let

Relationships between product wholesale price

As can be seen from Figure

Following the above method of assigning values to exogenous parameters, let

(a) Relationships between

As can be seen from Figures

Let

Relationships between

It can be seen from Figure

In the above parameter settings, the relationship between the utility

Relationship between the retailer’s utility

As can be seen from Figure

As the product service market grows rapidly, both manufacturers and retailers have begun to provide extended warranties. Consumers can choose to purchase an extended warranty service from either the manufacturer or the retailer. Since the manufacturers and retailers in a supply chain provide homogeneous extended warranties services, they face competition with each other. At the same time, besides considering its own earnings, a retailer pays attention to the distribution of profits in the supply chain. A lower-than-expected profit can lead the retailer to have fairness concern preferences. This paper proposes a manufacturer-led Stackelberg game model to investigate optimal pricing strategies of manufacturers and retailers for their products and extended warranty services when the retailer’s fairness concern preference. This paper further analyses how the level of vertical competition in extended warranty service and the intensity of a retailer’s fairness concerns influence the optimal pricing of products and extended warranties and profits for the manufacturer and retailer.

This paper presents the following findings. Firstly, when the retailer has no fairness concerns, the retailer and the manufacturer share the same optimal extended warranty service pricing strategies. And increasing competition in extended warranty service between the two parties leads to a higher consistent extended warranty service price and a lower retail price. In contrast, when the retailer has fairness concerns, the retailer and the manufacturer develop differential pricing strategies for an extended warranty service as their optimal choice. In this case, the manufacturer tends to set an extended warranty service price higher than that of the retailer to surrender part of its profits to the retailer and offset the retailer's concern about unfairness. Secondly, the difference in extended warranty service price between the two parties is affected by the level of competition in the extended warranty service and the intensity of the retailer’s fairness concerns. Increase in vertical competition in an extended warranty service will always widen the price gap in the extended warranty service between the manufacturer and retailer. The impact of the intensity of the retailer's fairness concerns on this price gap depends on the intensity itself. When the intensity of the retailer's fairness concerns is less than a certain threshold, an increase in the intensity of the fairness concerns will narrow the price gap in the extended warranty service between the manufacturer and retailer. When the intensity of the retailer’s fairness concerns is greater than the threshold, increasing intensity of the fairness concerns implies a greater difference in extended warranty service price between the two parties. Thirdly, the results from further analysis of the profit of both parties show that when the retailer has no fairness concerns, its competition with the manufacturer in extended warranty service will increase the income of both parties, rather than harming their interests. In comparison, a higher intensity of the retailer’s fairness concerns always harms the interests of the manufacturer. For the retailer, only when the intensity increases are within a certain threshold can it improve its utility.

This paper considers vertical competition in extended warranty service in a product and service supply chain consisting of only one retailer and one manufacturer. In reality, a manufacturer may also distribute the same product through multiple retailers, which may experience horizontal competition in an extended warranty service with each other. This issue can be further examined through vector equalization [

The data generated and/or analysed during the current study are available from the corresponding author on reasonable request.

The authors declare no conflicts of interest.

This research was funded by the National Key Research and Development Program (grant no. 2018YFB1701502), the National Natural Science Foundation of China (grant nos. 71572020 and 71501162), the Chongqing Basic Science and Frontier Technology Research Project (grant no. cstc2016jcyjA0528), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (no. KJQN201800715), the China Postdoctoral Science Foundation (grant no. 2015M580770), the Humanities and Social Science Research Program of Chongqing Municipal Education Commission (grant no. 19SKGH061), and the Chongqing Postdoctoral Science Foundation (grant no. Xm2015044).