## Model of Cumulative Innovation for Plant Breeding

The model of REs that we want to construct is related to the second strand in the literature previously discussed. In particular, we want to construct a simple model of innovation that captures some salient features of plant breeding. Plant breeding is a lengthy and risky endeavour that consists of 'developing new varieties through the creation of new genetic diversity by the reassembling of existing diversity' (International Seed Federation, 2003). Thus, the process is both sequential and cumulative, because new varieties would seek to maintain the desirable features of the ones they are based on while adding new attributes. As such, a critical input in this process is the starting germplasm, and that in turn is critically affected by whether or not one has access to other successful varieties, i.e. whether or not there is an RE. But in a dynamic context, the quality of the existing germplasm is itself the result of (previous) breeding decisions, and so it is directly affected by the features of the IPR regime in place. Industry views on the matter highlight the possibility that freer access to others' germplasm will create little incentive for pre-breeding germplasm enhancement, such as widening the germplasm diversity base by introducing exotic germplasm (Donnenwirth et al., 2004).

The stylized model considered in this chapter considers two firms that are competing to develop a new variety along a particular development trajectory. At time zero both firms have access to the same germplasm and, upon investing an amount c, achieve success with probability p. Thus the R&D process is costly and risky. Given one success, the firms then have the option to pursue the next improvement, again upon paying an initial cost c and with a probability p of a successful outcome for each firm. Whether or not both firms can attack the next innovation stage depends on the IPR regime (which is defined subsequently). However, following Bessen and Maskin (2002) we assume that each firm's outcome is independent of the other and that whenever both firms fail to achieve the next innovation no further innovation is possible. We are thus capturing the sequential nature of plant breeding, as well as the notion of what breeders sometimes call 'path dependency' (Donnenwirth et al., 2004), whereby successive improvements along a given path greatly benefit from the initial breakthrough.

Each successful innovation embeds all previous ones, thus reflecting the fact that breeding is a cumulative process whereby each new variety builds on the previous ones, and it is worth an additional A, per period, to society. What a success is worth to the innovator, however, depends on the IPR regime and on the possible constraining effects of competition among innovators. We make the simplifying assumption that only the best product is sold in this market, but what the owner can charge is the marginal value over what the competitor can offer (i.e. Bertrand competition). For example, if two firms have achieved n and m innovation steps, respectively, with m > n, the firm with m steps will be the one selling any product and will make an ex post per-period profit of (m - n)A. As for IPRs, we consider two regimes. For simplicity, the protection offered by both IPR regimes lasts forever (the more realistic alternative of a finite patent life adds nothing to the economic analysis but would make the exposition more cumbersome). The first regime, labelled 'full patent' (FP), does not allow an RE. The second regime, labelled 'research exemption' (RE), allows it (thus, the RE regime reflects the attributes of a PBR system).

Our ultimate goal is to compare incentives to innovate in an industry consisting of two firms and characterized by these two distinct IPR modes of protection. However, before proceeding to the direct comparison of these regimes, it is useful to analyse the incentive to innovate for a firm that has no competitors. This special case is useful in what follows, and also allows us to introduce the rest of the notation and the method of analysis. Thus, let VM denote the present expected value (at time zero) of the flow of profits to the (monopolist) firm. Assuming that the firm invests in every period in which it has an investment opportunity (i.e. after each successful innovation), VM0 satisfies the following recursive relation:

V0M =-c + p (^ + V1j where 8 e (0,1) denotes the discount factor (8 = 1/(1+ r), where r is the interest rate), such that we have:

The present value VM0 is positive if and only if:

Also, if this condition holds, the firm will choose to invest in every period.

### Patent protection mode

It is assumed that patents are of infinite length and with breadth defined by the innovation step (worth A). If two firms (e.g. firm A and firm B) invest c at time zero, four outcomes are possible: only firm A is successful, only firm B is successful, both are successful and neither is successful. If neither succeeds, the R&D contest ends. If both succeed, priority is assigned randomly with equal probability to either firm, such that we have a unique winner of the first stage of the R&D contest. With the full patent protection, we assume that the winner of the first stage is the only one that can attack the next research stages. As with Bessen and Maskin (2002), a critical assumption for this characterization is that licensing is not possible. Hence, the first firm to obtain a patent will become a monopolist starting from date one (from which the previous present-value discussion therefore applies). This implies that at time zero both firms will race to obtain this dominant position.

When both firms are involved in the first-stage R&D contest, the probability that either one is the sole winner is q = p(1 - p) + 0.5p2 < p. In such a situation, a firm that invests in the first period, and keeps investing if it is the winner of that stage as long as there is an investment opportunity, has a present value V0P that satisfies

Thus, we have:

"0

Under the assumption of risk neutrality, both firms will invest in period zero if

VnF > 0, that is if a> (i - s )(2 - Sp2) s t c p(2 - p) F

Note that tFP > tM, which means that competition to be the only firm in the industry in period one dissipates some of the incentive to innovate in period zero by lowering the probability of reaching stage one. Also, whenever A/c > tFP, so that both firms invest in the initial investment game, A/c > tM. Hence, the firm that wins the initial innovation contest (thereby becoming a monopolist) will keep investing in follow-up improvements (as assumed in the derivation of VFP). If tM < A/c < tFP, there are two pure-strategy Nash equilibria (a firm will invest provided the other does not) and, perhaps more interesting, there is also a (symmetric) mixed-strategy equilibrium in which each firm randomizes between investment and no investment (and earns a zero expected initial pay-off).

### Research exemption mode

Introducing RE in this model is equivalent to making any innovation (improvement of the existing product or variety) non-infringing. In such a situation a success by any one of the firms is a sufficient condition for both firms to be able to invest in the next period. A simplified analysis is presented by assuming that only two strategies are available to each firm: invest in every period (I) and never invest (N). In other words, each firm can either enter the market and try to innovate in each period or stay out of the market altogether. Let VqE (sA,sB) denote the pay-off (as of period zero) to firm j (j = A, B) when the two firms choose strategies (sA, sB) in every period at which there is an investment opportunity. Clearly, VRRA (IN) = VRRB (N, I) = VM, VRE (N, N) = 0 (j = A, B), VRB (I, N) = V0RA (N, I) = 0 and VRRA (I, I) = VRB (I, I) = VgE. A firm that chooses to stay out of the R&D contest gets a pay-off of zero, and a firm that enters the competition alone gets the monopolist's pay-off VM calculated earlier. Finally, when both firms engage in R&D at every date at which there is a research opportunity, they each have the same expected present value, which is labelled V§E.

Even with the simplified assumption that firms use the same strategy in every period, the characterization of V§E is not straightforward. This is because the return to a 'success' depends on where the rival stands on the ladder of quality improvements. For example, the winner of the first innovation stage (e.g. firm A) can charge A (because that is all that the innovation is worth). But under the RE regime both firms can then participate in the next innovation stage. If firm A wins the second stage as well, this firm can charge 2 A for the (twice improved) product. But if it is firm B that wins the second stage, this firm can charge only A because of our Bertrand competition assumption (given that firm A still owns the first innovation). Hence, what each firm can expect to earn in each period depends on two state variables (the highest number of innovation steps patented by the two firms), and as the time horizon progresses there is an infinite number of configurations of these state variables, whose probability distribution is implicitly defined by the initial stochastic assumptions (each firm has an independent probability of success equal to p).

Accounting for the number of all possible histories leading to a particular state configuration (m, n), where m and n denote the highest number of innovation steps achieved by firms A and B, respectively, it is possible to obtain the present value of the stream of expected profit of the two firms. The derivation of this result is somewhat lengthy and it is omitted. But it can be shown that the present value VRE can be written as:

where, again, 2q > [1 - (1 - p)2] > p is the probability that at least one firm is successful in a given stage.

From the initial (time zero) perspective, the R&D investment contest in which each firm chooses between I and N can be represented as a static game with the pay-off matrix given in Table 13.1. Several Nash equilibria are possible here. First, (I,I) is a Nash equilibrium if VRE > 0. Second, (I, N) and (N, I) are both Nash equilibria if VRE < 0 and VMM > 0. Finally, (N, N) is a Nash equilibrium if VMM < 0. Note that VRRE > 0 holds if and only if:

Thus, the equilibrium of the static game as previously defined will depend on the value of the benefit/cost ratio (A/c). If A/c is such that 0 < A/c < tM, no firm will invest. If tM < A/c < tRE, there are two pure-strategy Nash equilibria, (I, N) and (N, I) (and also a mixed-strategy equilibrium in which each firm randomizes between I and N, earning the expected initial pay-off of zero). Finally, if tRE < A/c, the unique Nash equilibrium is (I, I) with both firms receiving a pay-off equal to VRE.

Because q = 0.5p(2 - p) < p, it is verified that the threshold levels derived in the foregoing satisfy the following inequalities:

With these inequalities we can already conclude that the RE mode provides weaker ex ante incentives to invest than does the FP regime. For a given R&D cost c, there is a range of the benefit parameter A where the FP regime can support two firms in the (initial) R&D contest, each earning positive returns, whereas the RE mode cannot. Specifically, this outcome happens whenever tFP < A/c < tRE.

Table 13.1. Payoff matrix of the R&D game with a 'research exemption' (RE).

Firm B I N

Furthermore, from the ex ante pay-off formulae derived earlier we also conclude that dVFP = q > q jvqRE dA (1 - S) (1 - S) (1 - Sq) (1 - S) dA

Thus, not only does the FP model provide an R&D incentive for a range of A where the RE mode does not, but the ex ante returns to the firms increase faster with A under FP than under RE. In other words, vqF > VRE, where the inequality holds strictly whenever A/c > tFP. Hence, in our setting, a firm would never prefer weaker patent protection over stronger patent protection, unlike what may happen, for example, in the Bessen and Maskin (2002) framework. This result, illustrated in Fig. 13.1, shows the behaviour of the firms' ex ante expected profit for a range of the benefit/cost ratios.

We should note, before closing, that some limitations of our simplified analysis are readily apparent. Specifically, our identification of Nash equilibria does not address the question of whether such equilibria are subgame perfect. In other words, limiting our consideration to strategies that entail the same action at every period in which there is an investment opportunity is, admittedly, restrictive. Whereas this limitation of the analysis can be overcome, the more rigorous game-theoretic approach that is required is not pursued here but is left for future research.

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