1.4 Theoretical Foundations
1.4.2 Comparative Statics
In this section we analyze how the returns from patent protection (selection effect) and the age of a patent (horizon effect) determine the agent’s strategy choices.
r
lg
1
)
(
ˆ
tlr
tg
−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 rt.
Proposition 1.1. If the difference between the option values of strategies (K) and (L),
EhVgK(t+ 1, gtk+1rt, glt+1gkt+1rt) i
−EhVfL(t+ 1, gkt+1rt, glt+1gkt+1rt) i
, is not concave inrt, then
(i) gˆtl−(rt) will be weakly decreasing in rt and
(ii) gˆlt+(rt) will be weakly increasing in rt.
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 =gtlrt and the option value
βEhVfL(t+ 1, rt+1, yt+1)|rt i
are high enough to cover the renewal fees ft
2. We know from
Lemma 1.1 that the option value is non-decreasing in rt. Thus, the higher rt, the lower
can be the LOR factorgl
t for the agent to still renew the patent. This is why the function
ˆ
LOR to expire in yeart asP r(gl t<gˆ
l−
t (rt)), then this probability must also be decreasing
inrt.
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, rt < rˆt, 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 ˆgtl−(rt)
will be decreasing in rt 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 rt.41 Now, the higher the per period returns rt, 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 gl
t is needed for the patentee to choose LOR in this period. This is represented
by an increasing function ˆglt+(rt) 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(gtl ≥ ˆgtl−(rt) |rt <rˆt) will be increasing in rt. However, if the patent is of
relatively high value, the probability of observing a declaration,P r(glt≥gˆtl+(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 ˆrt is non-decreasing in t.
Proof. See Appendix 1.7.1.
The threshold value ˆrtis 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 (rt ≥ rˆt), 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(gl) < 1 and that the density function Fg0l(u
l) is decreasing fast
enough for higher values ofgl. The reduction in maintenance fees in each year is fixed and independent
ofrt, whereas the “expected loss” in returns from the declaration, [1−E(gl)]rt, increases withrt. These
assumptions are justified since we would observe far more declarations in the data ifE(gl)≥1 was the
1
ˆ
t+r
r
lg
)
(
ˆ
tlr
tg
+ t rˆ)
(
ˆ
tlr
tg
−)
(
ˆ
tl+−1r
t+1g
1
)
(
ˆ
tl++1r
t+1g
Figure 1.22: Selection and Horizon Effect
are increasing and the option value is decreasing witht, the per period returns rt needed
to belong to the second category will increase with the maturity of the patent. This is represented by the shift of ˆrt to the right for the older period in Figure 1.22.
Proposition 1.3. Given per period returns from full patent protection r
(i) gˆtl−(r) is non-decreasing in t and
(ii) gˆlt+(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 gl
t 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, ˆgtl−(r), must not exceed the ones of the subsequent period, ˆgtl−+1(r). This shifts
the threshold value function ˆgl−(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
to expire: P r(gl t<ˆg l− t (r))≤P r(glt+1 <ˆg l− t+1(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(gl t <gˆ
l−
t (r)∧r <rˆt). We know from Proposition 1.2 and Proposition 1.3 that ˆrt and
ˆ
gtl− 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, rt<ˆrt. However, if the LOR factor gtland 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 < rˆt) rises with a patent’s maturity, since ˆrt 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 (ˆglt−(r) is non-decreasing in t), making a declaration for this type of patents
less likely to occur. P r((gl t≥ˆg
l−
t (r)|r <rˆt)) will decrease with t.
The second group consists of patents with per period returns high enough for renewal with full protection, r ≥ˆrt, but for which declaring LOR is even more profitable. In this
case, maturity reduces the probability to fall into this category (again, ˆrt is increasing
with t),P r(r ≥ ˆrt). At the same time, a lower LOR factor is needed for a patent owner
to be willing to declare LOR regime (ˆgtl+(r) is non-increasing in t), which increases the
probability of this type of declaration, P r((glt ≥ gˆtl+(r)|r ≥ rˆt)). 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((gl t ≥gˆ l− t (r)∧r <rˆt)) +P r((glt≥gˆ l+ t (r)∧r≥rˆt)) is ambiguous.
42The losses in option value arise because once LOR (L) has been declared it is not possible to choose