Pakes, Ariel (1986). Patents as Options: Some Estimates of the Value of Holding European Patent Stocks. Econometrica 54, 755–784.
Let denote the expected discounted value of a patent of age . Then, for ,
where is the limit on patent lifetimes imposed by law, is the current payoff of renewing the patent, is the information set in year , and is the renewal cost.
Assumption A1: , where is a vector of parameters.
This assumption implies that the process is Markov and does not depend on any unobserved attributes of the patent.
Assumption A2: In each period, agents have expectations about future renewal fees that are consistent with the actual published current fee schedule. This schedule is nondecreasing in patent age.
Assumption A3.1: There exists an such that for .
Assumption A3.2: is continuous in for every except possibly those values of where is discontinuous.
Assumption A3.3: is nonincreasing in .
Assumption A3.4: is nondecreasing in .
The value function can now be written as
There is a unique sequence of stopping values with such that it is optimal for the agent to renew the patent in period a if and only if . The sequence is nondecreasing in .
We can now derive the proportion of agents that drop out as predicted by the model (using the subsequent stochastic specification) and match this with the proportion observed in the data. If is the unconditional distribution of returns for patents of age then for ,
Then, the proportion of patent holders in a given cohort who drop out at age (not before) is given by the difference in the fraction who drop out at or before age and the fraction who drop out at or before age . Let the proportion dropping out at age be denoted by . Then,
If indexes cohorts, and we correct for left and right censoring of the renewal times, then the fraction of patent holders from cohort dropping out at age can be written as . Let denote the number of patend holders from cohort who dropped out at age . The log likelihood for an observed sequence is
The Markov process governing transitions in the return to holding patents is
where is an exponential random variable with density
and for . , , and are parameters to be estimated. The initial distribution of returns is lognormal, that is
Given a vector of parameters and a cost schedule , the sequence of cutoff values can be found analytically. The likelihood function is approximated by the simulated likelihood constructed using a simple frequency simulator for . This simulation proceeds by taking draws from the initial distribution of returns and passing them through the stochastic process defined by the agent’s problem and the transition probabilities.
The log likelihood is maximized using a quasi-Newton algorithm and the information matrix is formed by calculating numeric derivatives (one percent perturbations).
- Deng, Yi (2005). A Dynamic Stochastic Analysis of International Patent Application and Renewal Processes. Unpublished manuscript, Southern Methodist University.