Theoretical Model of the Declaration of the License of Right and

Right and Patent Renewal

In this section we develop a model that incorporates both, the decision of an agent to renew a patent and the decision to declare the willingness to license. It builds on previous work done by Pakes (1986) and Lanjouw (1998), who were the first to build a dynamic stochastic model of the patent renewal decision.34

General Set-up

The patent system In our model an agent can acquire a patent with full protection,

which allows him to exclude others from the patented invention. Preserving protection is

not free of charge, so he must pay renewal fees ct =ft at the beginning of every period

t ∈ 1, ..., T. Let T < ∞ be the maximum number of years a patent can be renewed.

34_{Serrano (2011) has also used a patent renewal model and incorporated a third decision, namely the}

We assume that the fees are rising with the patent’s maturity as it is common for most patent systems. Furthermore, if the agent refuses to pay the fees, the patent expires irrevocably. Additionally, we introduce a second type of protection, the license of right (LOR). Contrary to the full protection regime it is not allowed to exclude others any more, but the right for reasonable remuneration through licensing still exists. Since LOR might be seen as inferior to full protection, the patent owner has to pay only half of the usual renewal feesct= 1₂ftfor all future periods. Once LOR is chosen the patentee cannot

return to full protection anymore.

The agents We assume that every patent belongs to exactly one profit maximizing

agent and generates a per period returnzt.35 He can choose between up to three different

strategies in the beginning of every period. Dependent on his previous choices he can either keep full protection (K) and payft, declare LOR (L) and pay 1₂ft, or let the patent

expire (X) and avoid any payments.

Evolution of returns The type of protection determines the returns zt of an agent in

every period. We assume that the yearly returns in case of full patent protection,zt =rt,

differ from returns from a patent endorsed LOR,zt=yt, and are 0 if the patent expires,

zt = 0. The exact per period returns are assumed to be known to the agent only at the

beginning of each period.

We allow the returns to evolve in the following way over time:36

• In the beginning an initial return is assigned to each patent, r1, which is drawn i.i.d.

from a continuous distribution FIR on a positive domain.

• If full protection is kept (K), the returns in the next period will be multiplied by a yearly growth rate gk_t, t ∈ 2, ..., T. This means that the yearly returns in the second period will be r2 =gk2r1 and rt =gtkrt−1 in the following ones. In each year

t the growth rates g_tk ∈h0, BKi are drawn from a distribution with the cumulative density function F_gk(uk | t) = P r

h

gk_≤uk_|ti. Furthermore, we assume that the

probability to draw a high growth rate and increase the yearly returns from patent protection decreases with a patent’s maturity in the sense of first-order stochastic dominance (Fgk(uk | t) ≤ F_gk(uk | t+ 1)).37 Agents are assumed to know the true probability distributionsF_gk(uk|t),t∈2, ..., T already in the first year.

35_{Alternatively, one could assume that all patents are independent of each other.}

36_{The following stochastic specification fulfills the Markov property. This means that the returns in}

the future periods will be independent of past periods’ returns.

37_{Usually, the application and usage of an invention is determined early in a patent’s life. The proba-}

• During the LOR regime (L) the returns differ by a multiplicative factor from the ones in the K-regime, yt=gltrt, t ∈1, ..., T. The LOR factor represents the part of

returns from full patent protection that can be realized by the patentee if he gives up his right to exclude others. In each period the values gl

t ∈ h

0, BLi _{are drawn anew}

from a distribution with the cumulative density function Fgl(ul) = P r

h

gl ≤uli. The probability distributionF_gl(ul) is assumed to be known to all agents.

Maximization problem In the beginning of each period the agent chooses the strategy

with the highest expected value. Unlike in deterministic models, where the value functions consist only of returns in the current period, here, the option value of future periods also has to be taken into account. Assume the patentee has kept full patent protection in all previous years 1, ..., t−1. If the agent decides to keep full patent protection (K) in yeart, his value functionVeK(t, r_t) will consist of the current returns r_t, less the renewal feesf_t,

plus the value of having the option to choose the optimal strategy in the next period. This

option value will be determined by rt and is defined as the discounted expected value of

the optimal strategy in yeart+1, gV_K(t+1, r_t+1, y_t+1).38 Withβ representing the discount

factor between the periods it follows:

e VK(t, rt) =rt−ft+βE h g VK(t+ 1, rt+1, yt+1)|rt i (1.1) with EhVg_K(t+ 1, r_t+1, y_t+1)|r_t i = ˆ ˆ g VK(t+ 1, ukrt, ulukrt)dFgk(uk |t)dF_gl(ul)

Similarly, if he decides to choose strategy (L) instead, the yearly returns will be multiplied by the the factor gl

t in each period, yt =gtlrt, and the renewal fees reduced by one half.

Since strategy (K) will not be possible once LOR has been declared, the option value

is now defined as the discounted expected value of the optimal strategy in year t + 1,

f

VL(t+ 1, yt+1).39 The expected value of choosing strategy (L) can now be written as

e VL(t, rt, yt) =yt− 1 2ft+βE h f VL(t+ 1, rt+1, yt+1)|rt i (1.2) with EhVf_L(t+ 1, r_t+1, y_t+1)|r_t i = ˆ ˆ f VL(t+ 1, ukrt, ulukrt)dFgk(uk |t)dF_gl(ul) 38_{SubscriptK means that strategy (K) was chosen in all previous periods.}

However, if he decides to let the patent expire he will lose all possible returns from patent protection, but also the obligation to pay renewal fees:

e

VX(t) = 0 (1.3)

Now, we can define gV_K(t, r_t, y_t), the value function of the optimal strategy in period t, if

the patent has been renewed with full patent protection throughout the periods 1, ..., t−1,

g VK(t, rt, yt) =max n e VK(t, rt),VeL(t, r_t, y_t),VeX(t) o

and Vf_L(t+ 1, r_t+1, y_t+1), the value function of the optimal strategy in period t, if the

strategy (L) was chosen in one of the previous periods,

f VL(t, rt, yt) =max n e VL(t, rt, yt),VeX(t) o

Since the maximum number of years a patent can exist is finite, there is no option value in the last period, such that βEhfV_i(T + 1, r_T+1, y_T+1)|r_T

i

= 0 for i∈ {K, L}.

Some properties of the value functions are provided in the following lemma.

Lemma 1.1. The value functions VeK(t, r) and VeL(t, r, y), with t= 1, ..., T, are

(i) increasing and (ii) continuous in the current returns r and y,

(iii) and weakly decreasing in t.

Proof. See Appendix 1.7.1.

The agent’s decision in year t whether to keep the patent with full protection (K), to

declare LOR (L), or to let it expire (X) will be fully determined by the per period returns rt and the non-exclusivity factor gtl. The optimal strategy will depend on whetherrt and

gl

Definition 1.1.

{rˆt}T_t=1 : patent returns that make the agent indifferent between choosing strategy (K) and

choosing strategy (X). It depends on the age of the patent t and is defined as the solution to

e VK(t, rt) =VeX(t) = 0. n ˆ gl_t+(rt) oT

t=1: LOR factor that makes the agent indifferent between declaring LOR (L) and

keeping full patent protection (K). It depends on the age of the patent t and the level of per period returns rt. It is defined as the solution toVeK(t, r_t) =VeL(t, r_t, y_t) =VeL(t, r_t, gl_tr_t). n

ˆ

gl_t−(rt) oT

t=1: LOR factor that makes the agent indifferent between declaring LOR (L) and letting

the patent expire (X). It depends on the age of the patent t and the level of per period returns

rt. It is defined as the solution toVeL(t, r_t, g_tlr_t) =VeX(t) = 0.

Lemma 1.1 guarantees that these cut-off values exist and are unique.40 They are functions

in the (rt, gtl)-space and divide it in three, respectively two, decision regions (see Figures

1.20 and 1.21).

Consider a patent that has been renewed with full protection up to periodt. The patentee

has three options. He can renew the patent with full protection (K), declare LOR (L), or let it expire (X). Letting the patent expire will be the optimal strategy if and only if the per period returns and the LOR factor are too low, rt < rˆt and gtl < ˆg

l−

t (rt). In

this case, renewal in whatever regime would not justify the statutory renewal fees. This corresponds to the region in the lower left part of Figure 1.20. If in turn the current per period returns are high enough, such that renewal is optimal in any case, rt ≥ rˆt, the

agent will renew the patent with full protection (K), as long as the LOR factor is not too high, g_tl < ˆgtl+(rt). These patents are located in the lower right part of the figure.

The last strategy to consider is LOR. A declaration of the willingness to license can become optimal out of two motives, the cost-saving and the commitment motive. The cost-saving motive will be relevant if the per period returns are too low to maintain full patent protection,rt<rˆt, while the LOR factor is still large enough, gtl≥gˆ

l−

t (rt). In this

case, as long as exclusivity is not too valuable, the reduction in renewal fees can induce the agent to declare LOR and renew the patent. The commitment motive will be relevant if the per period returns are such that the patent will be renewed in any case,rt ≥ˆrt, but

the commitment to abandon exclusivity is of similar value as the right to exclude others, g_tl ≥ˆgtl+(rt).

Consider now a patent that has been renewed up to periodt and for which LOR has been

declared in one of the previous periods. In this case the patentee has only two options. He can either renew the patent with LOR (L) for the next period or let it expire (X). Given

40_{Since the value functions are weakly increasing and continuous in y}

t, they must also be weakly

increasing ingl

t

rˆ

r

l

g

1

)

(

ˆ

l _t t

r

g

−

ˆ

(

t

)

l t

r

g

+ 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 protectionrt, renewal will only be optimal if

gl t ≥ˆgl

−

t (rt). These are all (rt, gtl) combinations that lie above the ˆgl

−

t (rt)-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., gl

t < gˆ l−

t (rt) and rt < rˆt. In turn, if LOR has already been declared in the past he will

let his patent expire even if full protection was profitable, since gl t ≥ gˆ

l−

t (rt) is the only

condition relevant for renewal.

In document Rudyk, Ilja (2013): Three essays on the economics and design of patent systems. Dissertation, LMU München: Volkswirtschaftliche Fakultät (Page 51-56)