As noted by Buchanan (
1964), the outcome should not be treated as a market failure simply if no single individual has sufficient incentive to finance the full cost of an essentially indivisible operation because people may or may not decide to act collectively. Similarly, the scholars of the Bloomington School have argued that like other commons, knowledge can be managed cooperatively under the right conditions (Hess and Ostrom
2007). Thus, we now supplement the earlier analysis by assuming that a third strategy, i.e., a joint investment, becomes a possibility.
Various different mechanisms for the private provision of public goods have been proposed (e.g., Bagnoli and Lipman
1992; Tabarrok
1998). In particular, the present study most closely follows Dixit and Olson (
2000), which is a further development of an earlier model by Palfrey and Rosenthal (
1984). In addition to being a discrete public good model, it has two other suitable properties (cf. Bagnoli and Lipman
1989; Saijo and Yamato
1999). First, the mechanism is very simple and realistic compared with many other studies. Second, the individuals have a choice about whether to participate in the mechanism, i.e., it is not imposed upon them, which is crucial when non-excludability is assumed.
In the first stage of Dixit and Olson (
2000), all individuals decide simultaneously whether to participate in the provision of the public good. In the second stage, when the number of participants is known, the public good is provided if and only if the private benefit for participants exceeds their equal share of the cost; otherwise, the game ends with no public good provision or cost to anyone. The only possible symmetric pure strategy equilibrium is Pareto optimal, but Dixit and Olson (
2000) show that the expected social welfare in the mixed strategy equilibrium becomes very small as the share of the required contributors increases, or the distance between private cost and benefit decreases, which is consistent with the normal assumptions regarding public good provision (Olson
1965). Hence, they argue that the mere lack of transaction costs is not sufficient such that voluntary cooperation would reach the Pareto optimal outcome. In the present study, we show that partial excludability can help to achieve the Pareto optimal outcome. In other words, R&D cooperation and IPR work in a complementary manner.
If we can make cooperation self-enforcing, we will reach the Pareto optimal outcome where everyone participates in the optimal number of joint R&D investments. First, this requires that no single player has an incentive to free-ride on the investments of others. Second, no player should have an incentive for trying to acquire the patent through a non-cooperative investment. Thus, the socially optimal level of excludability becomes an interval, which is our main result. We consider this situation under deterministic innovation, where the optimal number of investments is one, before we show that the result extends to stochastic innovation, where the optimal number of investments can be greater.
4.1 Cooperation and deterministic innovation
Player
i can now make the investment (
I) or abstain (
A) as well as participating in a joint investment (
J). Given deterministic innovation and the level of excludability
\(\alpha ,\) the payoff from the joint investment is
$$u_i(J)=\frac{\alpha NV}{nm}+(1-\alpha )V-\frac{C}{m},$$
(7)
where
\(n\le N\) is the (expected) number of all investments and
\(m\le N\) is the (expected) number of players participating in the joint investment. The payoffs obtained from making a non-cooperative investment (
1) or abstaining altogether (
2) remain the same as before.
In Dixit and Olson (
2000), no joint investment is made in the second stage if (
7) is negative, and thus the payoff is given by (
2). To make the question of voluntary cooperation interesting, we assume that the non-negativity of (
7) does not require that everyone participates in the joint investment. In this case, the analysis is also more straightforward than in Dixit and Olson (
2000) because we need not consider the probability of the joint investment occurring as we focus on the symmetric equilibrium where it is certain. Thus, in the case of deterministic innovation, the Pareto optimal outcome is that everyone participates in the joint investment with a probability of 1. We now show that this is an equilibrium outcome for particular levels of excludability.
4.2 Cooperation and stochastic innovation
Hitherto, we have assumed deterministic innovation, where there is no uncertainty in the outcome of the research project. This has facilitated the analysis, because solving for the mixed strategy equilibrium when there is further uncertainty in R&D output would prove very difficult. In reality, of course, knowledge investments typically involve uncertainty, so we investigate this matter further in this subsection.
Suppose now that an R&D project is successful with probability \(p \in (0,1)\) and that it fails with probability \(q=1-p.\) In many cases these probabilities also depend on the actions of the players. Hence, we assume that a player or coalition of players i chooses the probability of success \(p_i\) and pays the R&D cost \(c(p_i),\) where \(\lim_{p_i \rightarrow 0}c(p_i)>0,\)
\(c'>0,\) and \(c^{\prime \prime}>0,\) which imply fixed costs and decreasing returns. (An equivalent approach would be to presume that the probability of success, \(p(c_i),\) increases, but at a diminishing rate, with R&D expenditures \(c_i.\))
Before proceeding to our main result concerning stochastic innovation, we consider the challenges presented by variable R&D costs and non-cooperative investment decisions. In this case a simple formation of joint investment projects may not ease the task of IPR policy. Suppose that there are
n non-cooperative joint R&D projects, each of which has
\(m\le N/n\) members. Then, coalition
i chooses
\(p_i\) to maximise
$$v_i=\alpha NV p_i F_i+(1-\alpha )mV(1-q_i\, \Pi q_j)-c(p_i),$$
(8)
where
\(q_i=1-p_i\) and
\(i\ne j.\) In Eq. (
8),
\(F_i\) is
i’s probability of winning the patent given its success in the R&D project, and hence it is a function of other probabilities of success
\(p_j.\) More specifically,
\(F_i=\sum_{j=0}^{n-1} \frac{P(s=j)}{j+1},\) where
P(
s) is the probability that
s other coalitions are successful. We assume that the coalitions’ members share the costs and benefits equally, and hence the expected net gain of each coalition member,
\(u_i(J),\) is one-
mth of (
8). For the given
n, the payoff from abstaining is
$$u_i(A)=(1-\alpha )V(1-\Pi q_j).$$
If player
i opts for private, non-cooperative investment instead, its payoff is given by
$$u_i(I)=\alpha NV p_i F_i+(1-\alpha )V(1-q_i\, \Pi q_j)-c(p_i).$$
Suppose for now that there are only joint investments with
m members each and no private investments. Equation (
8) gives the following first order condition:
$$\frac{\partial v_i}{\partial p_i}=\alpha NV F_i+(1-\alpha )mV\, \Pi q_j-c^{\prime}(p_i)=0.$$
(9)
Since the projects are identical, I assume that the equilibrium is symmetric. Since
\(p_i=p\) for all
\(i\in n,\)
$$F_i=\sum_{j=0}^{n-1} \left( {\begin{array}{c}n-1\\ j\end{array}}\right) \frac{p^j q^{n-1-j}}{1+j}=\frac{1}{pn}(1-q^n)\;\text{and}\; \Pi \, q_j=q^{n-1}.$$
We rewrite (
9) as
$$\frac{\partial v}{\partial p}=\alpha \frac{NV}{pn}(1-q^n)+(1-\alpha )mVq^{n-1}-c^{\prime}(p)=0,$$
(10)
which determines the equilibrium probability of success
p given the number of investments
n.
If all
N players are part of a coalition, i.e.,
\(m=N/n,\) then (
10) becomes
$$\alpha \frac{NV}{pn}(1-q^n)+(1-\alpha )\frac{NV}{n}q^{n-1}-c^{\prime}(p)=0.$$
(11)
Whether
n is composed of joint or private investments, the expected net social welfare is given by
As such, the optimal probability of success
\(p^{\ast}\) and number of investments
\(n^{\ast}\) are given by the first order conditions
$$\frac{\partial U}{\partial p}=nNVq^{n-1}-n\,c^{\prime}(p)=0 \leftrightarrow NVq^{n-1}-c^{\prime}(p)=0$$
(12)
and
$$\frac{\partial U}{\partial n}=-\ln q\, NVq^n-c(p)=0.$$
(13)
Consider now the socially optimal R&D investment given
n projects. Substituting (
12) into (
11) gives us the following proposition:
This result indicates that in the presence of uncertainty and variable R&D costs, it may not be a panacea that players simply form coalitions, because if these coalitions act non-cooperatively then the equilibrium R&D decisions are sensitive to the level of excludability. Hence, achieving the social optimum would require that the policy maker is able to set the correct level of excludability. Furthermore, it is not clear that the same level of excludability would sustain the socially optimal number of joint R&D projects in the equilibrium as required by the first-best optimum.
The case where the investment efforts are not sensitive to the level excludability or the number of projects arises when there is free entry of private projects. In this case all individual players choose the probability of success that minimizes the average cost. However, as Tandon (
1983) shows, this outcome is suboptimal. It can be shown that free-entry of identical joint projects gives the same outcome. This provides an additional reason for including all players into the joint projects in order to prevent excessive entry.
Taking into account the further challenges that stochastic innovation poses, the rest of the article concentrates on two cases under which R&D cooperation nevertheless achieves the social optimum given an interval of excludability. The first case is very straightforward, since in the case of fixed R&D costs alone the above-mentioned problem naturally disappears.