Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 409286, 10 pages

http://dx.doi.org/10.1155/2015/409286

## Stability Analysis of R&D Cooperation in a Supply Chain

^{1}College of Business Administration, Hunan University, Changsha 410082, China^{2}College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China

Received 18 May 2015; Accepted 13 August 2015

Academic Editor: Leonid Shaikhet

Copyright © 2015 Luyun Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

R&D outsourcing becomes the often-adopted strategy for firms to innovate. However, R&D cooperation often ends up with failure because of its inherent quality of instability. One of the main reasons for cooperation failure is the opportunistic behavior. As the R&D contract between firms is inherently incomplete, opportunistic behavior always cannot be avoided in the collaborative process. R&D cooperation has been divided into horizontal and vertical types. This paper utilizes game theory to study opportunistic behavior in the vertical R&D cooperation and analyzes the equilibrium of the cooperation. Based on the equilibrium and numerical results, it is found that the vertical R&D cooperation is inherently unstable, and the downstream firm is more likely to break the agreement. The level of knowledge spillovers and the cost of R&D efforts have different effects on firms’ payoffs. When the level of knowledge spillover is low or the cost of R&D efforts is high, mechanisms such as punishment for opportunism may be more effective to guarantee the stability of cooperation.

#### 1. Introduction

In the knowledge economy era, the competition of technical market is increasingly fierce, and firms are forced to accelerate the process of technical innovation. However, it is more difficult for firms to accomplish knowledge creation and technological innovation in isolation [1]. R&D outsourcing becomes the often-adopted strategy for firms to innovate. R&D cooperation becomes a common phenomenon, which helps firms in sharing risk and cost, accessing knowledge and technological know-how network, and internalizing the externalities created by knowledge spillovers [2–4]. Despite these advantages, the inherent quality of instability of R&D cooperation often may not be avoided, and R&D cooperation often ends with failure [5–8]. A main reason for cooperation failure is the opportunistic behavior by one party or the other [9, 10]. Opportunism appears due to the cooperative and competitive relationship of the two collaborative firms. Opportunism is defined as selfish behavior which means seeking a firm’s self-interest with deceit at the expense of its partners [11–13]. As the R&D contract between firms is inherently incomplete, firms in the cooperation are often not accessible to the detailed information about what the partners are expected to do, and it is impossible for a third party to keep watch on R&D efforts [14]. Therefore, opportunistic behavior always cannot be avoided in the collaborative process.

Based on the types of collaborative partnership, R&D cooperation has been divided into horizontal and vertical R&D cooperation. Many of research works have been done about these two types of R&D cooperation [15–20]. Although firms may arrange their R&D inputs to realize the maximization of the total profit of the two firms, opportunism may still prevail in these two types of R&D cooperation. Such noncooperative behavior may prevent a firm from losing its competitive knowledge. However, it would lead to the instability of the cooperation. Kesteloot and Veugelers study the stability of horizontal R&D cooperation in a repeated game and emphasize the important role of spillovers [21]. Cabon-Dhersin and Ramani use a noncooperative game to discuss the effect of trust on horizontal R&D cooperation, and they find that when opportunism cannot be avoided, the nature of firms, the configurations of trust, and the level of spillovers decide whether the horizontal R&D cooperation is successful or not [14]. Cassiman and Veugelers find that, in vertical cooperation, the effectiveness of strategic protection is important to induce cooperation [22]. Lhuillery and Pfister find that vertical R&D cooperation also faces a higher risk of failures [23].

Most of the existing literatures using game theoretical approach have studied the stability of horizontal R&D cooperation. Our paper uses a game theoretical approach to analyze the stability of the vertical R&D cooperation. We focus on opportunistic behavior in the vertical R&D cooperation. The results of this paper indicate that the vertical R&D cooperation is unstable, and the downstream firm is more likely to break the agreement. When building a partnership, firms need to consider the social statue and reputation of its partner and mutual trust between the two firms. This paper also identifies the roles of knowledge spillover and the cost of R&D effort in the stability of vertical R&D cooperation. These two factors influence the firms’ payoffs in different situations. And they play different roles in the decision process. It is found that when the level of knowledge spillover is low or the cost of R&D efforts is high, mechanisms such as punishment for opportunism may be more effective.

The rest of this paper is given as follows. Section 2 introduces the model of our paper. Section 3 gives the equilibrium analysis of the game and analyzes the effects of spillover level and R&D effort cost on stability of the game. Section 4 presents numerical illustration and Section 5 gives the conclusion of this paper.

#### 2. The Model

In the part, we present the model in two subsections. Our game model is described in the first subsection. R&D expenditures and payoffs in different situations are solved in the second subsection.

##### 2.1. Description of Game Model

Studies show that single source brings long-term benefits if used appropriately [24] and one or two suppliers are usually [25, 26] enough for a manufacturer. Following Ge et al. [20], we consider cooperative R&D in a simple supply chain with a final-good manufacturer (denoted as Firm ) and an input supplier (denoted as Firm ) in our model.

We assume that Firm decides its production quantity based on the market demand and then submits its order to Firm . Firm sells inputs to Firm in the market. The two firms in the supply chain establish a vertical strategic R&D collaboration link. The reduction of marginal cost in our study is an R&D production function following d’Aspremont and Jacquemin [5]. In the vertical R&D cooperation, firms coordinate their R&D inputs and then allocate R&D resources to reduce the production costs. The reduction of marginal production cost comes from a firm’s own research. Meanwhile, R&D efforts of its cooperator also help reducing the firm’s cost due to positive spillovers. Let be the parameter reflecting the spillover level between firms. The spillover levels of the two firms in the vertical R&D cooperation are assumed to be symmetric. We also assume that the fixed costs of firms are set to be zero. Let and denote the original marginal cost and let denote the R&D efforts of firm . Therefore, the marginal cost after R&D investment of firm is written as follows:

We assume that R&D investment is costly. Given a level of R&D efforts, the cost of efforts is given by , and is a technological parameter and satisfies , indicating diminishing returns to R&D.

In the production market, let us suppose that Firm produces the inputs and sells them to Firm at price , and Firm uses the inputs to produce final goods. Let us suppose that the inverse demand function in the market is given by , where represents the price of a final product and represents the total production quantity of final goods produced by Firm . At the same time, the order quantity of Firm is equal to production quantity of final goods. Given an R&D profile of the two firms, the quantity of final goods and the input price of Firm and the net profits of Firm and Firm , , can be written as follows:

As the game is dynamic, we can use the backward introduction method. We start with the quantity of final goods. Firm chooses its output level independently to realize the maximization of its own profit. For any configuration of R&D efforts invested by Firm and Firm , the optimal condition for the profit maximization of Firm is given by

Then the production quantity of the final product can be got by

Substituting (5) into (3), the net profit function of Firm can be rewritten as

By solving partial derivatives of (6) about for profit maximization, the optimal input price can be got as follows:

Then the production quantity of the final product can be obtained as follows:

Therefore, the net profits of Firm and Firm are gained as

The total profit of the supply chain profit is

##### 2.2. R&D Expenditures and Payoffs in Different Situations

After establishing the R&D cooperation, firms choose their levels of R&D efforts. Each firm has two choices. One is to invest on R&D efforts to maximize the total profit of the two firms, which is treated as reciprocal behavior. Another is to invest on R&D efforts to maximize a firm’s own profit, which is regarded as opportunistic behavior. After that, each firm invests in its R&D efforts, and this is not observable. With the above discussion, we can now solve for the R&D efforts and the corresponding net profits in four different situations of vertical R&D cooperation.

First, Firms and both choose reciprocal behavior, and we will define this situation as situation . In this situation, each firm decides its R&D efforts by the maximization of the total profit. Then the optimal condition for situation is obtained as follows:where .

By solving (11), we can, respectively, define the R&D efforts and net profits of Firm and Firm in situation as follows:

Second, Firms and both choose opportunistic behavior, defined as situation . In this situation, each firm decides its R&D efforts by the maximization of its own profit. Then the optimal condition for situation is expressed by

And the R&D efforts and net profits of Firm and Firm in situation can be separately defined as follows:

Third, we will define this situation as situation , where Firm chooses reciprocal behavior and Firm chooses opportunistic behavior. At this time, Firm decides its R&D efforts for the best interest of the cooperation, while Firm chooses to cheat. Then the optimal condition for situation is written by

And the R&D efforts and net profits of Firm and Firm in situation are severally defined as follows:

Similarly, when chooses opportunistic behavior and Firm chooses reciprocal behavior, we can define R&D efforts and net profits in situation as follows:

The details in four situations are summarized in Table 1, where , and increases with the spillover level between cooperative firms but decreases with the cost of R&D efforts. We will assume that to make sure that the R&D investment and production quantity exist.