This model extends the previous by the introduction of sampling by second stage firms. The idea of this is that 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 a low cost of implementation rather than a high cost.
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
Malcolm Gladwell, The New Yorker, 2004-11-22, SOMETHING BORROWED: Should a charge of plagiarism ruin your life?. [[TODO: put in evidence that has been collected on bands such as Oasis and Led Zeppelin]]. See also this
article about sampling in dance and rap music.
Royalty-After-Sampling
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 - c_{H} and r_{H} = v - c_{L}. 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):
p(k) \geq 1 - \frac{v-c_{H}}{v-c_{L}} \equiv \alpha
\iff k \leq p^{-1}(\alpha) \equiv k_{\alpha}
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):
\Pi(k) = p(k)(v - r_{L} - c_{H} -k\tau) + (1-p(k))(v - r_{L} - c_{L} -k\tau) = - k\tau + (1-p(k))(c_{H}-c_{L})
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)(c_{H}-c_{L}) \leq 0 since, by assumption, p^{''}(k) \geq 0.
giving:
p^{'}(k_{2}) = \frac{-\tau}{c_{H}-c_{L}}
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)(c_{H}-c_{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):
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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}
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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:
r_{L}(1-x) + (1-(p(k)))r_{H} x = r_{L} + x ((1-(p(k)))r_{H} - r_{L})
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:
Payoffs for Innovators
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RL
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RH
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Stage 2
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- k\tau + (1-p(k))(c_{H}-c_{L})
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-k\tau
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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:
x p(k) * -k\tau + (1-x)(1-p(k))(-k\tau + c_{H} - c_{L}) = -k\tau + (1-x)(1-p(k)(c_{H}-c_{L})
The solution, k, is given implicitly by:
p^{'}(k) = \frac{-\tau}{(1-x)(c_{H}-c_{L})}
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 \lim 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, assuming that without the revenue from second stage innovation only a proportion 1-q of first stage innovations are realized, and that average costs for those realized (not realized) are c_{0}^{L} (c_{0}^{H}) respectively. To reduce the algebra define \beta(k) = (v - k\tau - c_{L} - p(k)(c_{H}-c_{L}) (that is average welfare generated by a second stage innovator), then total welfare is as follows:
Total Welfare
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k_{2} \leq k_{\alpha}
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k_{2} > k_{\alpha}
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Payoff to First Stage Firms (M)
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v_{0} - c_{0} + v - c_{H}
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v_{0} - c_{0} + v - c_{H}
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Payoff to Second Stage Firms (M)
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- k_{2}\tau + (1-p(k_{2}))(c_{H}-c_{L})
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-k_{\alpha} \tau + (1-x_{alpha})(1-p(k_{\alpha}))(c_{H}-c_{L})
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Welfare (M)
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(v_{0} - c_{0}) + N \beta(k_{2})
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(v_{0}-c_{0}) + N \beta(k_{\alpha}) - Nx_{alpha} (1-p(k_{\alpha}))(c_{H}-c_{L})
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Welfare (NM)
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(1-q) ((v_{0}-c_{0}^{L}) + N \beta(k_{2}))
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(1-q) ((v_{0}-c_{0}^{L}) + N \beta(k_{2}))
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Net Difference (M - NM)
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q ((v_{0}-c_{0}^{H}) + N \beta(\k_{2}))
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S (see below)
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S = q( v_{0} - c_{0}^{H} + N \beta(k_{\alpha})) - N(1-q) (\beta(k_{2}) - \beta(k_{\alpha})) - Nx_{alpha} (1-p(k_{\alpha}))(c_{H}-c_{L})
Some rearranging yields:
S = q(N\beta_{2} + c_{0}^{H}) - N(\beta_{2} - \beta_{\alpha}) - Nx_{alpha} (1-p(k_{\alpha}))(c_{H}-c_{L})
Thus for for monopoly rights to be preferable to no monopoly rights when k_{2} > k_{\alpha} requires:
q \geq q^{m] \equiv \frac{N(\beta_{2} - \beta_{\alpha}) + Nx_{alpha} (1-p(k_{\alpha}))(c_{H}-c_{L})} {(N\beta_{2} + c_{0}^{H})}
So q^{m} is the probability of a high cost innovation which leaves one indifferent between having and not having monopoly rights.
Conclusion
From our calculations of welfare it follows that we should choose to have monopoly 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 high cost goes down with sampling (p^{'} \leq 0) but at a diminishing rate (p^{''} > 0) and that:
p(k_{\alpha}) = \frac{c_{H}-c_{L}}{v-c_{L}}
p^{'}(k_{2}) = \frac{-\tau}{c_{H}-c_{L}}
Then:
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Reducing transaction costs, \tau, increase the level of optimal sampling k_{2} but leaves k_{\alpha} unaffected.
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Increasing the advantage of low cost over high cost innovations (c_{H}-c_{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 cost firms is greater then the loss to a first stage innovator of 'losing' high cost firms by charging the high royalty rate is smaller. Therefore the number of low cost firms 1-p(k) at which the switch to a high royalty rate is made can be smaller).
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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:
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Reducing sampling costs make it more likely that a freer (no monopoly 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) and higher average cost levels for second stage firms outweigh the benefits of more first stage (and dependent second-stage) innovations.
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Increasing the differential between low cost and high cost types (which could be interpreted as sampling becoming more important for product quality) makes it more likely that a freer (no monopoly rights) regime will be optimal. Conversely increasing the value of an innovation while leaving cost differentials constant (which in some sense corresponds to cost differentials being less important) makes it more likely that a monopoly rights regime will be optimal.
Royalty-Before-Sampling
We solve by recursing backwards through the game, first solving for the investment decision, then for the level of sampling as a function of the royalty rate, and then, finally, using this for the optimal royalty rate for stage 1 firms. Once we have solved we examine the welfare situation using a similar setup to the models above.
Using a general function to link sampling to firm cost (p(k)) greatly limits our analysis and the conclusions we can draw (for example we are not able to solve for the optimal royalty rate). Thus in the final section we consider a special case where the sampling function takes the form of the exponential distribution. This allows us to derive explicitly how choice variables such as the sampling level and the royalty depend on the 'fundamentals'.
Optimal Sampling
The investment decision by stage 2 firms is straightforward to solve: since sampling costs are sunk they will be ignored and a firm with costs c_{X}, X=L,H invests if, and only if, v - r \geq c_{X}. Define r_{H}, r_{L} as usual and consider the three possible situations:
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r_{L} \geq r \geq 0, in which case both firm types invest
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r_{H} \geq r > r_{L}, in which only the low cost firm types invest
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r > r_{H}, in which neither firm type invests (this case is trivial and will be ignored)
Case 1: Here we would expect sampling to be independent of the royalty: both firm types will invest and so the royalty rate has the same impact on both high cost and low cost producers. This means the royalty has no effect on the advantage of low cost over high cost firms. Since sampling is a function of this advantage rather than of the levels of profits themselves this in turn means that sampling is independent of royalty level. Formally we have that profits as a function of sampling are:
\Pi(k) = p(k)(v - r - c_{H} -k\tau) + (1-p(k))(v - r - c_{L} -k\tau) = v - r - k\tau - p(k)(c_{H}-c_{L})
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)(c_{H}-c_{L}) \leq 0 since, by assumption, p^{''}(k) \geq 0.
. This gives:
p^{'}(k) = \frac{-\tau}{c_{H}-c_{L}}
which is equating the marginal gain in terms of lower costs (p^{'}(k)(c_{H}-c_{L}) from a marginal increase in sampling to the costs (\tau). For future reference define the solution of this equation by k_{2} (the '2' indicates that this is situation where both firms participate).
As a function of the exogenous variables we have, as one would expect, that an increase in transaction costs decreases the level of sampling (remember p^{''} > 0) and that an increase in the benefit of being low cost vs. high cost increases the level of sampling.
Case 2: Here only the low cost innovator invests and thus expected profits are given by:
\Pi(k) = - p(k) * k\tau + (1-p(k))(v - r - c_{L} -k\tau)
= -k\tau + (1-p(k))(v - r - c_{L})
Maximizing
\Pi^{''} = p^{''}(v - r-c_{L}) \leq 0 (term in brackets is positive) so the second order condition is satisfied and this is a maximum.
profits gives:
p^{'}(k) = \frac{-\tau}{v - r -c_{L}}
Again this corresponds with intuition: at the optimum the marginal gain from greater sampling: p^{'}(k)(v - r-c_{L}) equals the marginal cost: \tau.
Define k_{1}(r) as the solution of this equation, i.e. the optimal sampling level as a function of the royalty when only one firm type invests -- those that are low cost. Again, as one would expect, the optimal sampling rate for second stage firms goes down with increasing transaction costs or higher royalties. Conversely the sampling level increases with the level of surplus v-c_{L}.
Note that it is possible that k_{1}(r) = 0 for r \lt r_{H} (i.e. the RHS is less than p^{'}(0) (the most negative value possible for p^{'}(k)). Define r_{0} as the minimal royalty rate at which k = 0 (note that r_{0} \leq r_{H}). Then k_{1}(r) = 0 for r \geq r_{0}.
We summarize these results in a lemma and diagram:
Lemma: k(r) is defined as follows:
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0 \leq r \leq r_{L}: k(r) = k_{2}
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r_{L} \leq r \leq r_{0}: k(r) = k_{1}(r)
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r > r_{0}: k(r) = 0
Furthemore k(r) is continuous, and is also differentiable except at r_{L} and r_{0}. k^{'}(r) \leq 0 with the inequality strict for r_{L} \lt r \lt r_{0}.
Optimal Royalty
Finally we come to the problem for a first stage innovator: \max_{r} \Pi(r) where:
\Pi(r) = rv, r \leq r_{L}
\Pi(r) = (1-p(k(r))rv, r > r_{L}
Restricting to the first case it is obvious that maximum must be at r_{L}. Note that \Pi(r) is not necessarily continuous at r_{L}, in fact it is very unlikely to be continuous -- it is continuous if, and only if, the probability of being a high cost firm is 0 at a sampling level of k_{2} (p(k(r_{L})) = p(k_{2}) = 0).
Define r_{1}^{e} as the value of r that maximizes profits in the second case (this will exist as continuous function on a compact set -- if we include the endpoints). Then an optimal r, denoted r^{e} exists defined by choosing whichever of r_{L}, r_{1}^{e} gives the greater profits.
Welfare and Monopoly Rights
Define c = c(k) = p(k) c_{H} + (1-p(k))c_{L}. Also write p for p(k). Then letting R_{H}, R_{L} denote situations with a high/low royalty rate respectively we have (very similar to original):
Total Welfare
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RL
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RH
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M
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v_{0} - c_{0} + N (v - c(k_{2}) - k_{2}\tau)
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v_{0} - c_{0} + N (1-p(k_{1})) (v - c_{L}) - N k_{1}\tau
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NM
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(1-q)(v_{0} - c_{0}^{L} + N (v - c(k_{2}) - k_{2}\tau))
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(1-q)(v_{0} - c_{0}^{L} + N (v - c(k_{2}) - \k_{2}\tau))
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Notes: From earlier results we know that for r \leq r_{L} k = k_{2}. In RH case k_{1} is shorthand for k_{1}(r^{e}).
As before, in the low royalty case it is always preferable to have monopoly rights. In the high royalty case we have that a monopoly rights regime is preferable to no monopoly rights if:
q (v_{0} - c_{0}^{H}) + N q (v-c(k_{2}) -k_{2}\tau) >
N ( v - c(k_{2}) - k_{2}\tau) - N((1-p(k_{1})) (v-c_{L}) + k_{1}\tau)
For simplication let us assume that v_{0} - c_{0}^{H}, which is negative, is small relative to other terms (e.g. N is big) then:
q > 1 - \frac{ (1- p(k_{1})) (v-c_{L}) - k_{1}\tau }
{ v-c(k_{2}) -k_{2}\tau }
= 1 - \frac{v-c_{L} - p(k_{1})(v-c_{L}) -k_{1}\tau}
{v-c_{L} - p(k_{2})(c_{H} - c_{L}) - k_{2}\tau}
Example: Exponential Probability Function
Define probability function as p(k) = e^{-\lambda k}
The level of sampling with a low royalty rate, k_{2}, is given by the solution of:
\lambda e^{-lambda k_{2}} = \frac{\tau}{c_{H}-c_{L}}
\implies k_{2} = - \frac{1}{\lambda} * \log(\frac{\tau}{\lambda (c_{H}-c_{L})})
Similarly the level of sampling with a high royalty rate, k_{1}(r), is the solution of:
\lambda e^{-lambda k_{1}} = \frac{\tau}{v - r-c_{L}}
Thus the highest possible royalty rate is:
r_{0} = \frac{\lambda (v-c_{L}) - \tau}{\lambda}
For r_{L} \leq r \leq r_{0} we have:
p(k_{1}(r)) = e^{-lambda k_{1}(r)} = \frac{\tau}{\lambda (v - r - c_{L})}
Define \alpha = v - r - c_{L} then the first order condition for maximization of \Pi(r) = (1-p(k(r)) rv reduces to:
\lambda \alpha^{2} - \tau (v-c_{L}) = 0
\implies r_{1} = \frac{\lambda (v-c_{L}) - \sqrt{\lambda \tau (v-c_{L})}}{\lambda}
Let us assume for the present that r_{1} is the optimal royalty (we would need to to check that r \in [r_{L}, r_{0}] and that \Pi(r_{1}) > \Pi(r_{L}))
To have r_{1} \geq 0 requires \lambda (v-c_{L}) \geq \tau. This condition also ensures that r \leq r_{0}. r_{1} \geq r_{L} requires:
\lambda (c_{H}-c_{L}) \geq \sqrt{\lambda \tau (v-c_{L})}
\implies lambda \geq \frac{\tau (v-c_{L})}{c_{H}-c_{L}}
In our situation higher \lambda corresponds to a higher probability of being low cost type for any given level of sampling. Thus this restriction makes intuitive sense: low lambda means high probability of a high cost type so a stage 1 firm will want to set royalties so that high cost firms invest (and therefore generate revenue).
. Compared to the standard high royalty case (v-c_{L}), which occurs when there is no sampling or royalties are set after sampling, the royalty here is 'discounted' by the amount of the the second, square root, term. A first stage innovator reduces the royalty in this way because by doing so she encourages more sampling which increases the proportion of low cost firms she can 'tax'.
This 'discounting' of the royalty is:
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Increasing in the value generated by low cost firms (v-c_{L}) and the level of transaction costs
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Decreasing in the rate of payoff to sampling (\lambda)
Sampling: By substituting for the optimal royalty we may derive an explicit expression for the level of sampling in the high royalty case:
k_{1} = -\frac{1}{2\lambda} \log (\frac{\tau}{\lambda (v-c_{L})})
What does this tell us of interest?
First it shows that the level of sampling is non-monotonic with respect to \lambda, which we may think of as the rate of payoff to sampling: initially the level of sampling is increasing with \lambda but it then peaks and decreases.
The reaason for this is that at low levels of the rate of payoff, sampling is low and thus the marginal effect of increasing sampling is high (remember that the marginal benefit of sampling decreases as sampling increases: p^{''} \geq 0). Thus when the rate of payoff increases marginally the marginal benefit of increasing sampling will be positive. However when the rate of payoff is high, even with a small level of sampling the marginal benefits of sampling are low, thus if the rate of payoff increases a second stage innovator will realize net gains by reducing the level of sampling. (Note that this result applies equally to the level of sampling in the low royalty case).
A second feature of interest is the comparison between the high and low royalty sampling levels when v - c_{L} \approx c_{H} - c_{L} -- which occurs, for example, when the surplus generated by high cost firms, v - c_{H}, is relatively small). In this case the level of sampling under a 'high' royalty is approximately half that under 'low' royalty regime
This is also a situation in which the high royalty regime will occur since the income to a first stage innovator from high cost firms will be small. Note however it is not necessarily the case that the royalty level with be that high: .
This translates into the probability of a high cost innovation under the high royalty regime being the square of the probability under the low cost regime. How important this is depends on the level of sampling (if the level of sampling is already high under the high royalty regime then this difference will be small).
