The Model

We adopt a simple model of two stage innovation in which the second product can be considered an application of the first innovation, that is a product that incorporates the first innovation. The model, which is specified in detail in the rest of this section can be summarized in the following diagram:

Innovations are described by their net value v (revenue minus costs). Because our interest lies in examining the trade-off between innovation at different stages we make no distinction between social and private value (i.e. there are no deadweight losses) and v may be taken to be either.

We assume the base (first) innovation takes two values: low ($$v_{L}^{0}$$) and high ($$v_{H}^{0}$$) with probability $$q, (1-q)$$ respectively. We assume that $$v_{L}^{0} \lt 0$$ so that without some additional source of revenue, for example from licensing (see below), the innovation will not be produced. Applications (2nd stage innovations) also take two values: low ($$v_{L}$$) and high ($$v_{H}$$) with probability p, (1-p) respectively. While the value of a second stage innovation is known to the innovator who produces it, the value is not known to the owner of the first stage innovation which it incorporates (this could occur because of imperfect information regarding revenue, costs or both).

Solving

A second-stage innovator of type X faces a payoff of v_{X} - r if she invests and 0 if she does not. Given this a second-stage innovator will only invest if r \leq v_{X}. Given this the first innovator faces a straightforward selection/pooling problem with a choice between a low royalty rate r_{L} = v_{L} (all applications produced) or a high royalty rate r_{H} = v_{H} (only high value applications produced). If there are N possible applications then these result in expected payoffs for the first stage innovator of respectively:

Rearranging we have a low royalty rate is chosen if:

Intuitively: the extra revenue from low value producers (LHS) when you have a low royalty must be greater than the revenue foregone (compared to high royalty regime) from high value innovators. Let us denote the low royalty situation RL and the high royalty situation RH.

Welfare

To determine welfare we need to know the 'trade-off' between first and second stage innovations that occurs when revenue is allocated from one to the other by licensing. As stated above without revenue from second stage innovations a proportion q of first stage innovations are not produced. We assume that when royalty income is forthcoming it is sufficient to ensure that all first stage innovations are produced (i.e. $$r > v_{L}^{0}$$). Thus in the absence of royalty income there are q first-stage innovations that are not produced, and these have average value v_{L}^{0}. The remaining innovations ($$1-q$$) are produced irrespective of whether royalty revenue is received and have average value v_{H}^{0}.

Let us now consider social welfare in the four possible situations given by (IP, RL), (IP, RH), (NIP, RL), (IP,RH). Note that due to our earlier assumption welfare is determined by calculating total net value. Let v_{0} = q v_{L}^{0} + (1-q) v_{H}^{0} be average first stage innovator costs (if all innovate) and v = p v_{L} + (1-p) v_{H} average second stage innovator costs (if all innovate).

Optimal Policy

It is immediately clear that in the RL situation a patenting regime is preferable. The reason for this is straightforward: in the RL situation all applications will be produced whether there are patents or not. In that case one wishes to maximize returns to the first innovator and patents do this by transferring rents via licensing.

The situation with RH is less clear. Intellectual property (monopoly) rights (IP) will be preferable to no intellectual property rights (NIP) if:

Now v_{L}^{0} (which is negative) must be less in absolute terms than the royalty received N (1-p) r_{H} (we are assuming that the royalty enables low value first stage innovators to produce). Thus define $$\beta$$ as the proportion of the total royalty payment which is used up in paying for first stage innovator net losses (which should be interpreted as corresponding to their higher costs) -- and $$1-\beta$$ is the proportion of the total royalty left as welfare -- then we may write v_{L}^{0} = -\beta (N (1-p) r_{H}), \beta \in (0,1] . Note that $$\beta = 1$$ corresponds to all the royalty being used up to fund first stage innovators higher costs while $$\beta \approx 0$$ means all of the royalty payment is being retained as extra profits (and welfare).

Thus, using $$\beta$$ we can rewrite the previous condition as:

We can plot this result graphically as follows (recall that we have low royalties, RL, if $$p \geq \alpha$$):

Figure 1: The three different lines show in ascending order the marginal value of $$q$$ as a function of $$p$$ for $$\beta$$ equal to 0, 0.5, and 0.99.

N p is the 'number' of second stage innovations that do not occur with intellectual property rights (due to high royalties) while q is the number of first stage innovations that do not occur without intellectual property rights. As first stage innovations enable second stage ones when we lose a first stage innovation we lose all dependent second stage ones as well. Due to this, to choose no intellectual property rights over intellectual property rights p must be substantially higher (though not too high) relative to q, since only then will the cost of intellectual property rights, in terms of lost second stage innovations, outweigh the gains in terms of more first stage, and dependent second-stage, innovations.

As \beta increases the area in which no intellectual property rights are preferable will increase with the line seperating the two regions moving upwards. In the limit as \beta tends to 1 -- which corresponds to most royalty income being used by a first stage innovator to pay costs -- the marginal q tends to 1 -- i.e. it is optimal to have intellectual property rights only if all first stage innovations are of a low value type.

Introduction

This model extends the previous by the introduction of sampling by second stage firms. In this situation second stage firms can only use first stage innovations that they have encountered by some form of costly sampling (for example purchasing a good that embodies the innovation). The more products they sample the more likely it is a second stage firm comes up with a good idea of its own -- which is modelled, in this case, by having value innovation rather than a low value one.

Obvious real-world examples of such situations would be software and music. In software a new 'app' will likely combine many ideas (and even code) from previous products. But ideas can only come from applications that one has encountered (note that for re-use of code that means access to the source so the software must be open-source). In music, particularly modern music, re-use either explicit or implicit is ubiquitous. For example, in dance and hip-hop, 'sampling', whereby a small section of a previous work is directly copied and then repeated or reworked in some manner, is the very basis of the genre. More generally all composers whether classical or modern use previous musical, ideas, motifs, and melodies as parts of new works See e.g. Malcolm Gladwell, The New Yorker, 2004-11-22, Something Borrowed: Should a charge of plagiarism ruin your life?, also http://www.low-life.fsnet.co.uk/copyright/part3.htm#copyrightinfringement for information about sampling in dance and hip-hop music.

The Model

The model is exactly the same as the basic one except for the addition of an initial sampling period by second stage firms which influences whether they are high or low value. Define the following variables:

1. k, the number of stage 1 products stage 2 firms choose to sample (via purchase, observation etc)
2. \tau the cost of each sample
3. As before there are two types of stage 2 firms, high and low value: v_{H}, v_{L}
4. p(k), probability of being a low value firm given that $$k$$ products are sampled. Naturally p^{'} \leq 0 (otherwise no benefit of observing). We also assume diminishing returns for sampling so that p^{''} \geq 0 and that if no sampling takes place all firms are of low value type (p(0) = 1). The functional form p(k) is assumed to be common knowledge.
5. r, the lump-sum royalty rate set by stage 1 firms
6. N, the number of potential stage 2 firms

Regarding the sequence of actions the situation is the same as in the original model except for the fact that we now have two options as to when the royalty can be set: either after or before sampling (but as in the original model still before the second stage innovators take their investment decision). We shall refer to these two cases as royalty-after-sampling and royalty-before-sampling. Here we will confine our attention to royalty-after sampling The author has also examined the royalty-before-sampling but the situation is considerably more complex and yields fewer insights for policy. In the interests of brevity it has therefore been omitted. The interested reader who wishes to have the details is invited to contact the author.. The full sequence of decisions in that case is shown in the following diagram:

Solving the Model

We will solve for a subgame perfect nash equilibrium by recursing backwards through the game. Looking at the payoffs at the bottom of the previous diagram we see that the investment decisions by second stage firms are exactly the same as for the original model (this is because sampling costs are sunk and common).

Thus once again the first stage firm need only consider two royalty levels: r_{L} = v_{L} and r_{H} = v_{H}. A first stage firm chooses a low royalty rate over a high one if (to be precise the firm is indifferent when there is equality):

Next define k_{2} as the optimal sampling level chosen by a second stage firm when the royalty level is low (the '2' indicates that this is situation where both firms invest). Formally, this is the level of sampling that maximizes the expected payoff to a firm (as a function of the sampling rate):

To find the sampling level that maximizes profits impose the first order condition \Pi^{'}(k) = 0 The second order condition, \Pi^{''} \leq 0, is easily checked: \Pi^{''} = -p^{''}(k)(v_{H}-v_{L}) \leq 0 since, by assumption, p^{''}(k) \geq 0. giving:

This is as one would expect: here both firms engage in production and the effect of sampling will be on the cost type (i.e. it will not effect the royalty paid or whether a firm invests). Hence the sampling level will be chosen so that the marginal gain in terms of lower costs (p^{'}(k)(v_{H}-v_{L})) equals the marginal sampling costs (\tau).

We are now in a position to state the results. However first we must distinguish between two possibilities regarding knowledge of the sampling level available to first stage innovators. In the first case the first stage innovator does observe the sampling level. In the second case the first stage innovator does not know the sampling level. In what follows we focus on the case where the sampling level is unobserved though the results are little changed when it is observed.

Proposition: The (perfect bayesian nash) equilibria of the game where second stage innovators play pure strategies are as follows (omitting investment strategy for simplicity):

1. k_{2} \leq k_{\alpha}: then there is a unique pure-strategy equilibrium given by: first stage innovator chooses high royalty if the sampling level is greater than k_{\alpha} and a low royalty rate otherwise. The second stage innovator chooses sampling level k_{2}
2. k_{2} > k_{\alpha}: there is a unique equilibrium with second-stage innovators playing pure strategies. This equilibrium consists of: second-stage innovators sample at level k_{\alpha}; first stage innovators choosing a high royalty level if the sampling level is above k_{\alpha}, a low one if it is below and randomizing between a high and low royalty rate if it is equal to k_{\alpha} (the exact probabilities involved are defined below).

Proof: We are restricting to the case where second stage innovators choose pure strategies (allowing mixed strategies yield similar results but complicate the algebra).

A first stage innovator through dominance is restricted to playing a mixed strategy consisting of r_{H}, r_{L}. Let us suppose that she plays these with probability x, 1-x. Revenue is then:

Maximizing revenue requires x = 0 if the term in brackets is less than zero, x = 1 if the term in brackets is greater than 0, and allows any value of x if the term in brackets is zero. By the definition of k_{\alpha} (see above) these conditions correspond to the sampling level being less than, greater than or equal to k_{\alpha}.

Turning to second stage innovators. Payoffs as a function of the royalty level are as follows:

Thus a high royalty level always yields a negative payoff unless the sampling level is 0. Since the royalty will be high if the sampling level is above k_{\alpha} it is immediately apparent that no pure strategy equilibrium can exist in which the sampling level is above k_{\alpha}.

Recall that k_{2} was defined as the sampling level that maximized the second stage innovators payoffs when the royalty is low. Thus if k_{2} \leq k_{\alpha}, then k_{2} is the dominant strategy for second stage firms and is the unique pure-strategy equilibrium.

If k_{2} > k_{\alpha} the situation is more complex. If individual sampling levels are observable then k = k_{\alpha} is again a dominant strategy. However if the sampling levels is not observable -- as we are assuming -- then first stage firms can not offer a royalty specific to each firm but must offer a general one and we must solve for a perfect bayesian nash equilibrium.

We are restricted to equilibria in which the second-stage innovator plays a pure strategy. It immediately follows that we must have that the sampling level chosen is k_{\alpha} (PF: suppose not and that the sampling level is k'. Since we are in a PBE beliefs must be consistent so the first stage innovator expects this sampling level. But for any sampling level other than k_{\alpha} the first stage innovator has a dominating pure strategy of high or low royalty level depending on whether k' is higher or lower than k_{\alpha}. If this is the case then k' must either be 0 or k_{2} respectively but each of these is inconistent with the associated royalty level).

For k_{\alpha} to be an equilibrium it must maximize expected profits. Thus we must show the existence of mixed strategy (x) for first stage innovators such that k_{\alpha} is optimal for second stage innovators. Profits are given by:

The solution, k, is given implicitly by:

Since p^{'} \lt 0 we have, denoting k(x) as the solution of this as a function of x, that k^{'}(x) \lt 0. Since k(0) = k_{2} > k_{\alpha} and that as x \rightarrow 1 the RHS of the above takes arbitrarily large negative values by the intermediate value theorem there must exist a unique x_{\alpha} \in (0,1) such that k(x_{\alpha}) = k_{\alpha}. QED.

Welfare

For the calculation of welfare we proceed as in the original model. A proportion $$q$$ of first stage innovations are low value ($$v_{L}^{0} \lt 0$$) and only occur when there is royalty income. Similar to above define $$v^{0} = q v_{L}^{0} + (1-q) v_{H}^{0} and v(k) = - k\tau + (1-p(k)) v_{H} + p(k) v_{L} (the value generated by a second stage innovator sampling at level $$k$$). Further define $$v(\alpha) = v(k_{\alpha}), v(2) = v(k_{2})$$. Then total welfare is as follows:

Some rearranging yields:

Thus for for intellectual property rights to be preferable to no intellectual property rights when k_{2} > k_{\alpha} requires:

(Where q^{m} has been defined as the probability of a low value first stage innovation which leaves one indifferent between having and not having intellectual property rights.) In words this can be written as:

Policy Implications

From our calculations of welfare it follows that we should choose to have intellectual property rights if k_{2} \leq k_{\alpha} or if k_{2} > k_{\alpha} and q \geq q^{m}.

Now q^{m} increases in k_{2}, decreases in k_{\alpha}. Thus the policy choice is directly related to the relative sizes of these different sampling levels.

What can we say about these two values as a function of the underlying parameters? Recall that the probability of a low value innovation goes down with sampling (p^{'} \leq 0) but at a diminishing rate (p^{''} > 0) and that:

Then:

• Reducing sampling costs, \tau, increases the level of optimal sampling k_{2} but leaves k_{\alpha} unaffected.
• Increasing the advantage of high value over low value innovations (v_{H}-v_{L}) increases the optimal level of sampling, k_{2}, but reduces k_{\alpha} the sampling level at which a high royalty is charged (intuitively: if the the differential between high and low value firms is greater then the loss to a first stage innovator of 'losing' low value firms by charging the high royalty rate is smaller. Therefore the number of high value firms 1-p(k) at which the switch to a high royalty rate is made can be smaller).
• Increasing the value of a second stage innovation, v, while leaving all other parameters constant increases k_{\alpha} but leaves the optimal level of sampling unchanged (the intuition is exactly the same as for the previous item).

These results suggest the following, corresponding, conclusions for policy:

1. Reducing sampling costs make it more likely that a freer (no intellectual property rights) regime will be optimal. This is because the 'optimal' level of sampling (k_{2}) will be sufficiently greater than the restricted level of sampling (k_{\alpha}) that the loss in terms of hold-up of second stage products ($$x_{\alpha} * p(k_{\alpha})$$) and lower average value of second stage firms outweigh the benefits of more first stage (and dependent second-stage) innovations.
2. Increasing the differential between high and low value types (which could be interpreted as sampling becoming more important for product quality) makes it more likely that a freer (no intellectual property rights) regime will be optimal. Conversely increasing the value of an innovation while leaving value differentials constant (which in some sense corresponds to value differentials being less important) makes it more likely that a intellectual property rights regime will be optimal.