R&D Strategy space 

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

... can finally analyze what happens in zones III, which can be fairly small when spillovers are weak, i.e. θ approaches 0 . What is interesting about these areas is that the firm with smallest R&D investment cannot profitably outinvest the efficient firm this time: any increase in the R&D expenses would first put the competitor’s marginal cost to zero, as can be seen in Figure 1. Thus, one can think that it may be the case that interior asymmetric equilibria may arise, as opposed to the (unlikely) assymetric equilibria with no investment from the part of one firm. It is a matter of algebra to show that it is indeed possible that this happens (see the appendix); the parameter space where this might happen, however, is very small, and it shrinks as c grows, as can be seen in Figure 4. Importantly, the interior equilibria in zones III may co-exist together with point A being an equilibrium of the R&D game. For example if a = 385 128 , c = 1, v = 251 128 , θ = 128 3 , there are three equilibria: A ( c , θ ) and one in each zone III. Overall, then, where do we stand with the equilibria analysis? Figure 5 gives us an answer. ...

Context 3

... find pure-strategy equilibria points we perform a unilateral deviation analysis, that is, we consider all points ( x i , x j ) in the strategy space and see whether (at least) one firm has an incentive to deviate from it. For this, it is first convenient to divide the strategy space in different regions, which are represented in Figure 1. 2 The figure shows four relevant zones. If investments fall on any point of the diagonal between the origin and point A = ( c / ( 1 + θ ) , c / ( 1 + θ )) , this implies that firms invest the same, and therefore end up with identical marginal costs. The continuation game then will be, endogenously, a simultaneous Cournot, with marginal costs ranging from c to 0. If investments fall in zones III or IV, then firms will be asymmetric after investment, the firm with higher R&D investment will be more efficient and therefore will endogenously emerge as a Stackelberg leader; both firms end up with positive marginal costs. In zones II, there will be also a more efficient firm, but that firm will have zero marginal costs. Finally, in zone I both firms reach the minimum marginal cost possible, i.e. zero. Note that in zone I, despite the fact that R&D investments may be different, firms end up competing in Cournot fashion since they are both equally efficient. Figure 1 signals one of the complexities of solving for equilibrium, but also why the ...

Context 4

... find pure-strategy equilibria points we perform a unilateral deviation analysis, that is, we consider all points ( x i , x j ) in the strategy space and see whether (at least) one firm has an incentive to deviate from it. For this, it is first convenient to divide the strategy space in different regions, which are represented in Figure 1. 2 The figure shows four relevant zones. If investments fall on any point of the diagonal between the origin and point A = ( c / ( 1 + θ ) , c / ( 1 + θ )) , this implies that firms invest the same, and therefore end up with identical marginal costs. The continuation game then will be, endogenously, a simultaneous Cournot, with marginal costs ranging from c to 0. If investments fall in zones III or IV, then firms will be asymmetric after investment, the firm with higher R&D investment will be more efficient and therefore will endogenously emerge as a Stackelberg leader; both firms end up with positive marginal costs. In zones II, there will be also a more efficient firm, but that firm will have zero marginal costs. Finally, in zone I both firms reach the minimum marginal cost possible, i.e. zero. Note that in zone I, despite the fact that R&D investments may be different, firms end up competing in Cournot fashion since they are both equally efficient. Figure 1 signals one of the complexities of solving for equilibrium, but also why the ...