Payoff as a function of own investment 

Contexts in source publication

Context 1

... this Section, we will search for equilibria when the market competition mode emerges endogenously as in van Damme and Hurkens (1999). We will consider all possible points in the strategy space and, through analysis of unilateral deviations, show that for most relevant cases a sub-game perfect Nash equilibrium (SPNE) is sustained only by investments that leave both firms with zero marginal costs (point A in Figure 1); in other words, there is full R&D investment. Let us start then by considering zone I. If investments are in this zone, any of the two firms can decrease its R&D expenses without affecting its marginal costs because they are already at their minimum attainable; therefore no point in zone I can sustain a SPNE. If we consider zones II, the leader, i.e. the firm with the highest expenditure in R&D, has an incentive to decrease its R&D expenses because, on one hand, it will not affect its own marginal costs –which are already zero– and, on the other hand, it will increase its opponent’s marginal cost through diminished spillovers. Thus zones II cannot sustain a SPNE either. We now focus on zones IV, but excluding the frontiers with zones III and point A for the time being. Consider first the diagonal, where x i = x j ≤ 1 + c θ , i.e. both firms have positive marginal costs. In this case, each firm has an incentive to increase its R&D investment marginally, thus becoming more efficient and therefore a Stackelberg leader. The intuition is given by Figure 2: on point P1 –where market competition will be Cournot– a marginal increase in R&D expenses generates an incremental change in payoffs. And this will be true as long as there is a space for a firm to outinvest the other, i.e. on any point along the diagonal with the exception of point A , where marginal costs reach zero. We can thus conclude the following: The next obvious question is whether asymmetric points in zones IV can sustain a SPNE, since in those cases the follower firm may find it too expensive to outinvest the leader; in other words, the prize may not be large enough. Straightforward use of equations (2) and (3), mapped out in the Appendix, shows that from any point of the type 0 < x F < x L < 1 + c θ it will always be more profitable for the firm that has less R&D investment to invest marginally above what the competitor did. In other words, this does not sustain a SPNE, as the follower has a profitable deviation to x L + ε . However, one can show (see the Appendix) that for a very small interval of values of v and a very small market size –as measured by the ratio a / c – there might be a SPNE with x F = 0 and 0 < x < c . The parameter space where this can happen is very small though, as shown ...

Context 2

... interesting: payoff functions are not continuous and therefore, small changes in a firm’s R&D investment may have large impacts, ceteris paribus . To show this graphically, in Figure 2 we have graphed the payoff function of one firm as a function of its own investment, keeping constant (at a positive level x ) the investment of the competitor. 3 A simple analysis of Figure 2 helps to understand most of what will be happening in the next Sections. When firm i invests little (less than j ), between 0 and point x 1 (excluded) in Figure 2, it ends up being a Stackelberg follower. If firm i invests exactly x 1 , then market competition will be (endogenously) simultaneous, and therefore firm i’s profit will increase non-smoothly because the mode of competiton changed: firm i is no longer a Stackelberg follower and therefore it no longer suffer from the second mover disadvantage. But surpassing what j is investing, even if marginally, implies an even large increase in its payoff because the increase in investment costs from x 1 is small, but the mode of competition has changed again and i is now a Stackelberg leader. This is what we call the hyper-strategic effect of R&D investments. Further investments above x 1 would only lower the firms marginal costs, but without changing the mode of competition. But if firm i invests above x 2 , then it will reach zero marginal costs and further investments will decrease only the rival’s marginal cost through spillovers, inducing a stronger competitor; this is reflected in a downward- sloping payoff. If investment go further, then there is an extra drop in the payoff function as the rival would also achieve zero marginal cost and competition cease to be Stackelberg to become a simultaneous Cournot. It is important to note, to close this section, that most of the previous literature on R&D focused only on what we call here zones IV and, most specifically, the diagonal excluding point A , that ...

Context 3

... interesting: payoff functions are not continuous and therefore, small changes in a firm’s R&D investment may have large impacts, ceteris paribus . To show this graphically, in Figure 2 we have graphed the payoff function of one firm as a function of its own investment, keeping constant (at a positive level x ) the investment of the competitor. 3 A simple analysis of Figure 2 helps to understand most of what will be happening in the next Sections. When firm i invests little (less than j ), between 0 and point x 1 (excluded) in Figure 2, it ends up being a Stackelberg follower. If firm i invests exactly x 1 , then market competition will be (endogenously) simultaneous, and therefore firm i’s profit will increase non-smoothly because the mode of competiton changed: firm i is no longer a Stackelberg follower and therefore it no longer suffer from the second mover disadvantage. But surpassing what j is investing, even if marginally, implies an even large increase in its payoff because the increase in investment costs from x 1 is small, but the mode of competition has changed again and i is now a Stackelberg leader. This is what we call the hyper-strategic effect of R&D investments. Further investments above x 1 would only lower the firms marginal costs, but without changing the mode of competition. But if firm i invests above x 2 , then it will reach zero marginal costs and further investments will decrease only the rival’s marginal cost through spillovers, inducing a stronger competitor; this is reflected in a downward- sloping payoff. If investment go further, then there is an extra drop in the payoff function as the rival would also achieve zero marginal cost and competition cease to be Stackelberg to become a simultaneous Cournot. It is important to note, to close this section, that most of the previous literature on R&D focused only on what we call here zones IV and, most specifically, the diagonal excluding point A , that ...

Context 4

... interesting: payoff functions are not continuous and therefore, small changes in a firm’s R&D investment may have large impacts, ceteris paribus . To show this graphically, in Figure 2 we have graphed the payoff function of one firm as a function of its own investment, keeping constant (at a positive level x ) the investment of the competitor. 3 A simple analysis of Figure 2 helps to understand most of what will be happening in the next Sections. When firm i invests little (less than j ), between 0 and point x 1 (excluded) in Figure 2, it ends up being a Stackelberg follower. If firm i invests exactly x 1 , then market competition will be (endogenously) simultaneous, and therefore firm i’s profit will increase non-smoothly because the mode of competiton changed: firm i is no longer a Stackelberg follower and therefore it no longer suffer from the second mover disadvantage. But surpassing what j is investing, even if marginally, implies an even large increase in its payoff because the increase in investment costs from x 1 is small, but the mode of competition has changed again and i is now a Stackelberg leader. This is what we call the hyper-strategic effect of R&D investments. Further investments above x 1 would only lower the firms marginal costs, but without changing the mode of competition. But if firm i invests above x 2 , then it will reach zero marginal costs and further investments will decrease only the rival’s marginal cost through spillovers, inducing a stronger competitor; this is reflected in a downward- sloping payoff. If investment go further, then there is an extra drop in the payoff function as the rival would also achieve zero marginal cost and competition cease to be Stackelberg to become a simultaneous Cournot. It is important to note, to close this section, that most of the previous literature on R&D focused only on what we call here zones IV and, most specifically, the diagonal excluding point A , that ...