Abstract:     With the competition of the business environment becoming more and more intense, a large number of firms are trying to make R&D investments to strengthen their cost competitive advantage. However, the R&D investment a firm made may exist spillover effect and benefit its competitors. Extant research primarily focuses on R&D investments under symmetric spillover effect. In practice, the asymmetric spillover effect is commonly seen because the degree of intellectual property protection varies across industries. Neither the role of asymmetric spillover effect in competitive environment nor the conditions where firms have incentives to make R&D investment have been explored in the literature. In this paper, we investigate the question of whether or not two competing firms should make R&D investments to reduce the production costs under the asymmetric spillover effect.
   We consider four possible cases: the case where both firms make R&D investment, the case in which a single firm invests whereas the other behaves as the free-rider, and the case without R&D investment. We develop decision models in each of these cases by game theory and solve the equilibrium outcomes by backward inductions. Then, we make two-party comparisons of the equilibrium outcomes and comprehensively discuss the optimal R&D investment strategies. Furthermore, we conduct sensitivity analysis to demonstrate the impact of asymmetric spillover effect on the firms′ optimal expected profits, consumer surplus, and social welfare. Our main contribution is we explicitly identify circumstances in which competitive firms have incentives to make R&D investments. In addition, we explore how do factors such as asymmetric spillover effect, absorptive capacity, the probability of innovation success, and product substitutability affect the optimal decisions and profits of the game players. The main findings are as follows.
    First, the asymmetric spillover effect plays a pivotal role in affecting the optimal R&D investment strategies of competitive firms. In particular, if both firms face a weak spillover effect, all of them make R&D investments. If the spillover effect of a firm is weak but the rival′s is strong, they will solely behave as the investor and the free rider, respectively. If the spillover effect is at an intermediate level, the two firms jointly make R&D investments, showing behavior of synchronization. Also, they may jointly opt-out of R&D investment and leads to a behavior of synchronization. If the spillover effect is strong, neither firm makes the R&D investment. Therefore, in the presence of an asymmetric spillover effect, we identify circumstances where there are two investors, a unique investor, and no investor at all. We give detailed explanations for these optimal R&D investment strategies from the perspective of effective demand, marginal revenue, and invest-related cost.
    Second, the impacts of asymmetric spillover effect and product substitutability on the two firms′ optimal R&D investment strategies are as follows. Under a weak spillover effect, both firms make R&D investments, regardless of the degree of product substitutability. However, when the spillover effect becomes sufficiently strong, the optimal R&D investment strategies depend critically on the product substitutability. More precisely, if the degree of product substitutability is high, then the two firms are willing to make R&D investments. If it is low, neither firm makes the R&D investment. If the degree of product substitutability is at a medium level, they either jointly invest or joint exit. This is because a strong spillover effect means that a firm can get more benefits from the R&D investment of its competitor. On the one hand, either firm has an incentive to invest to reduce its production cost under an intermediate level of product substitutability; otherwise, it will suffer a loss in demand and profit. Hence, joint investing emerges. On the other hand, moderate product substitutability may go against the growth of effective demand and marginal revenue of the investor. Thus, the two firms may be reluctant to make R&D investments.
    Third, an analysis of asymmetric spillover effect and absorptive capacity shows that both firms make R&D investments if the spillover effect is sufficiently weak. If the spillover effect is strong, then the absorptive capacity significantly influences the optimal R&D investment strategies. Specifically, neither firm makes R&D investment if the absorptive capacity is strong, but they jointly invest if the absorptive capacity is weak. When the absorptive capacity is at a medium level, the two firms either jointly invest or joint exit. As a result, the role that the absorptive capacity plays in affecting the optimal R&D investment strategies is similar to the effect of product substitutability.
   Forth, the joint effect of asymmetric spillover effect and the probability of innovation success are as follows. There is no R&D investment if the probability of innovation success is sufficiently low, and joint investment occurs if it is high. However, it is optimal for the two firms to either invest or exist if the probability of innovation success is at an intermediate level. This is because in this case, the expected improved production cost finally helps the investors get more revenue. It is also could be that a medium level of probability of innovation success means a high invest-related cost to the investor. In the end, the investor′s optimal expected profit will be lower than when they do not make R&D investments. Consequently, a moderate level of probability of innovation success may lead to either a synchronous or an asynchronous investment behavior.
     Finally, we reveal that the number of investors profoundly influences consumer surplus and social welfare. More precisely, both consumer surplus and social welfare are at the highest levels when the two firms jointly make R&D investments; they are in the lowest levels when there is no R&D investment, and they will be in the second place when a single firm makes R&D investment. Our sensitive analysis further indicates that customers and society are better off as long as the spillover effect is stronger, or any of the following factors becomes bigger: the probability of innovation success, absorptive capacity, and product substitutability.

Key words:  R&D investment, asymmetric spillover, absorptive capacity, consumer surplus, social welfare