Comparative Statics

r ˆ

t

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g ⁻

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( r

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Cost-saving Motive

Commitment Motive

Declare LOR (L)

Figure 1.20: Strategy Space of a Patent with Full Protection

the current period’s returns from full patent protectionr_t, renewal will only be optimal if g_t^l ≥ˆg_t^l−(r_t). These are all (r_t, g_t^l) combinations that lie above the ˆg^l−_t (r_t)-curve in Figure 1.21. All patents that lie below this function will not be renewed, since the revenues will be too low to justify even the reduced renewal fees.

Corollary 1.1. The probability of expiration in a given year is higher for patents endorsed LOR compared to patents with full patent protection.

Proof. See Appendix 1.7.1.

The intuition for the corollary is the following. As long as LOR has not been declared, the patentee can still choose among all three options. Thus, he will only let his patent expire if the expected returns in case of full protection as well as LOR are too low, i.e., g_t^l < gˆ^l−_t (r_t) and r_t < rˆ_t. In turn, if LOR has already been declared in the past he will let his patent expire even if full protection was profitable, since g_t^l ≥ gˆ_t^l−(r_t) is the only condition relevant for renewal.

r g

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_t^l

( r

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Figure 1.21: Strategy Space of a Patent Endorsed LOR Selection effect

The selection effect tells us how the probability of observing a LOR declaration or an expiration decision for a patent changes with the per period returns r_t.

Proposition 1.1. If the difference between the option values of strategies (K) and (L), E^hV^g_K(t+ 1, g_t+1^k r_t, g^l_t+1g^k_t+1r_t)ⁱ−E^hV^f_L(t+ 1, g^k_t+1r_t, g^l_t+1g^k_t+1r_t)ⁱ, is not concave inr_t, then (i) gˆ_t^l−(rt) will be weakly decreasing in rt and

(ii) gˆ^l+_t (r_t) will be weakly increasing in r_t. Proof. See Appendix 1.7.1.

Again, assume that an agent has renewed his patent up to period t. There are two cases to be considered. If the patentee has already declared LOR in one of the previous periods he has to decide whether to keep this patent right (L) or to let it expire (X). He will only renew if this period’s returns from patent protection yt =g_t^lrt and the option value βE^hV^f_L(t+ 1, r_t+1, y_t+1)|r_tⁱare high enough to cover the renewal fees ^f₂^t. We know from Lemma 1.1 that the option value is non-decreasing in r_t. Thus, the higher r_t, the lower can be the LOR factorg_t^l for the agent to still renew the patent. This is why the function ˆ

g_t^l−(r_t) in Figure 1.21 is decreasing. If we define the probability for a patent endorsed

LOR to expire in yeart asP r(g_t^l<gˆ_t^l−(r_t)), then this probability must also be decreasing inr_t.

Consider now the case that the patent has been renewed with full protection in all previous periods. If the returns from full patent protection are low, r_t < rˆ_t, i.e., the patent is of low value, he will decide not to keep full patent protection and the choice will be between (L) and (X). This is equivalent to the first case and the cut-off value function ˆg_t^l−(r_t) will be decreasing in r_t in the respective region (Figure 1.20). If, however, the returns from full patent protection are high enough, rt ≥ rˆt, the agent will choose between (K) and (L). Assume that the difference in the option values in case of full patent protection and LOR is not concave in r_t.⁴¹ Now, the higher the per period returns r_t, the less important will be the reduction in renewal fees relative to the reduction in expected future returns due to the loss of exclusivity. Therefore, for valuable patents a high LOR factor g_t^l is needed for the patentee to choose LOR in this period. This is represented by an increasing function ˆg^l+_t (r_t) in Figure 1.20. To sum up, if the current returns from full patent protection are low, then the probability of observing a declaration in periodt, defined as P r(g_t^l ≥ ˆg_t^l−(r_t) |r_t <rˆ_t) will be increasing in r_t. However, if the patent is of relatively high value, the probability of observing a declaration,P r(g^l_t≥gˆ_t^l+(rt)|rt <rˆt) will be increasing in rt.

Horizon effect

In our model, by assumption, not only the renewal fees but also the probability distribu-tion of the growth rates vary witht. Consequently, the patent age should have an impact on both the decision to declare LOR and the decision to let the patent expire. This is reflected in the following two propositions.

Proposition 1.2. The cut-off value ˆr_t is non-decreasing in t.

Proof. See Appendix 1.7.1.

The threshold value ˆr_tis only relevant for patents that kept full protection in all previous periods. It divides these patents in two categories. The ones that would certainly have been dropped (if rt <ˆrt) and the ones that would certainly have been renewed with full protection (r_t ≥ rˆ_t), if the LOR system had not existed. Given that the renewal fees

41Simulations with different distribution functions (f.e. exponential, uniform, Rayleigh) have shown that it is sufficient to assume that E(g^l) < 1 and that the density function F_g⁰l(u^l) is decreasing fast enough for higher values ofg^l. The reduction in maintenance fees in each year is fixed and independent ofrt, whereas the “expected loss” in returns from the declaration, [1−E(g^l)]rt, increases withrt. These assumptions are justified since we would observe far more declarations in the data ifE(g^l)≥1 was the case andF_gl(u^l) was increasing for higher values.

ˆ_t+1

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Figure 1.22: Selection and Horizon Effect

are increasing and the option value is decreasing witht, the per period returns r_t needed to belong to the second category will increase with the maturity of the patent. This is represented by the shift of ˆr_t to the right for the older period in Figure 1.22.

Proposition 1.3. Given per period returns from full patent protection r (i) gˆ_t^l−(r) is non-decreasing in t and

(ii) gˆ^l+_t (r) is non-increasing in t.

Proof. See Appendix 1.7.1.

Consider patents that generate equal per period returns from full patent protection r, but at different ages. If a patent is already endorsed LOR, the factor g_t^l will determine whether the patent owner renews the patent (L) or let it expire (X). Compared to later periods, in earlier periods not only are the renewal fees lower, but the option values are higher, too. The minimum factor needed for the patentee to renew the patent in a given period, ˆg_t^l−(r), must not exceed the ones of the subsequent period, ˆg_t+1^l− (r). This shifts the threshold value function ˆg^l−(r) upwards for older patents (see Figure 1.22). The implication is that if you compare two patents of different age, both with equal per period returnsr and both endorsed LOR, the older one will have the same or higher probability

to expire: P r(g_t^l<ˆg_t^l−(r))≤P r(g^l_t+1 <ˆg_t+1^l− (r)).

For patents which have kept the right to exclude others, the patent’s maturity influ-ences not only the probability of expiration but also the probability of declaration.

The probability of expiration in year t for a patent with full protection is defined as P r(g_t^l <gˆ_t^l−(r)∧r <rˆ_t). We know from Proposition 1.2 and Proposition 1.3 that ˆr_t and ˆ

g_t^l− are increasing with a patent’s aget, thus unambiguously increasing the probability of expiration.

The effect on the probability of declaration is less clear. From above we know that there are two types of patents for which LOR will be declared.

The first group consists of patents for which the per period returns are too low for the agents to keep full protection, r_t<ˆr_t. However, if the LOR factor g_t^land the reduction in maintenance fees are high enough, they will choose to declare LOR instead. On the one hand, according to Proposition 1.2, the probability for a patent to fall into this category, P r(r < rˆ_t) rises with a patent’s maturity, since ˆr_t is increasing with age t. On the other hand, for older patents, a higher LOR factor will be necessary for the LOR regime to be profitable (ˆg^l−_t (r) is non-decreasing in t), making a declaration for this type of patents less likely to occur. P r((g^l_t≥ˆg_t^l−(r)|r <rˆ_t)) will decrease with t.

The second group consists of patents with per period returns high enough for renewal with full protection, r ≥ˆr_t, but for which declaring LOR is even more profitable. In this case, maturity reduces the probability to fall into this category (again, ˆr_t is increasing with t),P r(r ≥ ˆr_t). At the same time, a lower LOR factor is needed for a patent owner to be willing to declare LOR regime (ˆg_t^l+(r) is non-increasing in t), which increases the probability of this type of declaration, P r((g^l_t ≥ gˆ^l+_t (r)|r ≥ rˆ_t)). The reason is that the losses in option value the patentee will suffer if he chooses strategy (L) instead of strategy (K), will decrease with every period as the patent approaches yearT.⁴² Furthermore, the absolute reduction in maintenance fees soars, since the renewal fees are increasing in t. Overall, the effect of patent age on the probability of observing a LOR declaration P r((g_t^l ≥gˆ_t^l−(r)∧r <rˆ_t)) +P r((g^l_t≥gˆ_t^l+(r)∧r≥rˆ_t)) is ambiguous.

42The losses in option value arise because once LOR (L) has been declared it is not possible to choose strategy (K) anymore, but it is always possible to switch from full patent protection (K) to (L).

1.4.3 How to Profit from the Declaration of the Willingness to

Im Dokument Three essays on the economics and design of patent systems (Seite 56-61)