# Pakes (1986)

## Citation

Pakes, Ariel (1986).
Patents as Options: Some Estimates of the Value of Holding European
Patent Stocks.
*Econometrica* 54, 755–784.

## Model

Let $V(a)$ denote the expected discounted value of a patent of age $a-1$. Then, for $a=1,\dots ,L$, $$V(a)=\mathrm{max}\{0,{r}_{a}+\beta E[V(a+1)|{\Omega}_{a}]-{c}_{a}\},$$

where $L$ is the limit on patent lifetimes imposed by law, ${r}_{a}$ is the current payoff of renewing the patent, ${\Omega}_{a}$ is the information set in year $a$, and ${c}_{a}$ is the renewal cost.

**Assumption A1:**
$\mathrm{Pr}(z>{r}_{a+1}|{\Omega}_{a})=G(z|r={r}_{a},a,{\omega}_{g})$,
where ${\omega}_{g}$ is a vector of parameters.

This assumption implies that the $\{{r}_{a}\}$ 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 $\epsilon $ such that
$E[{r}_{a}^{1+\epsilon}|{r}_{1}]<\mathrm{\infty}$ for $1<a\le L$.

**Assumption A3.2:**
$G(z|r,a)$ is continuous in $r$ for every $z$ except possibly
those values of $z$ where $G(z|r,a)$ is discontinuous.

**Assumption A3.3:**
$G(z|r,a)$ is nonincreasing in $r$.

**Assumption A3.4:**
$G(z|r,a)$ is nondecreasing in $a$.

The value function can now be written as $$V(a,r)=\mathrm{max}\{0,r+\beta \int V(a+1,z)G(\mathrm{dz}|r,a)-{c}_{a}\}.$$

## Empirical Implications

There is a unique sequence of stopping values $\{{\overline{r}}_{a}{\}}_{a=1}^{L}$ with ${\overline{r}}_{a}\in [0,{c}_{a}]$ such that it is optimal for the agent to renew the patent in period a if and only if ${r}_{a}\ge {\overline{r}}_{a}$. The sequence $\{{\overline{r}}_{a}{\}}_{a=1}^{L}$ is nondecreasing in $a$.

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 $F(r,a)$ is the unconditional distribution of returns for patents of age $a$ then for $a=2,\dots ,L$,

$$F(r,a)=1-\mathrm{Pr}({r}_{a}\ge r,{r}_{a-1}\ge {\overline{r}}_{a-1},\dots ,{r}_{1}\ge {\overline{r}}_{1}).$$

Then, the proportion of patent holders in a given cohort who drop out at age $a$ (not before) is given by the difference in the fraction who drop out at or before age $a$ and the fraction who drop out at or before age $a-1$. Let the proportion dropping out at age $a$ be denoted by $\pi (a)$. Then,

$$\pi (a)=F({\overline{r}}_{a},a)-F({\overline{r}}_{a-1},a-1).$$

If $j=1,\dots ,J$ indexes cohorts, and we correct for left and right censoring of the renewal times, then the fraction of patent holders from cohort $j$ dropping out at age $a$ can be written as $\tilde{\pi}(a,j)$. Let $n(a,j)$ denote the number of patend holders from cohort $j$ who dropped out at age $a$. The log likelihood for an observed sequence $\{n(a,j){\}}_{a,j}$ is

$$\ell (\omega )=\sum _{j=1}^{J}\sum _{a\in {A}_{j}}n(a,j)\mathrm{log}\tilde{\pi}(a,j|\omega )$$

## Stochastic Specification

The Markov process governing transitions in the return to holding patents is

$${r}_{a+1}=\{\begin{array}{ll}0& \mathrm{exp}(-\theta {r}_{a})\\ \mathrm{max}\{\delta {r}_{a},z\}& 1-\mathrm{exp}(-\theta {r}_{a})\end{array}$$

where $z$ is an exponential random variable with density

$${q}_{a}(z)={\sigma}_{a}^{-1}\mathrm{exp}[-\frac{\gamma +z}{{\sigma}_{a}}]$$

and ${\sigma}_{a}={\varphi}^{a-1}\sigma $ for $a=1,\dots ,L-1$. $\sigma $, $\gamma $, and $\varphi $ are parameters to be estimated. The initial distribution of returns is lognormal, that is

$$\mathrm{log}{r}_{1}\sim \eta (\mu ,{\sigma}_{R}).$$

## Estimation

Given a vector of parameters $\omega =(\mu ,{\sigma}_{R},\theta ,\gamma ,\sigma ,\delta ,\varphi )$ and a cost schedule $c$, the sequence of cutoff values can be found analytically. The likelihood function is approximated by the simulated likelihood constructed using a simple frequency simulator $\hat{\pi}(a)$ for $\pi (a)$. 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).

## Cites

## Cited By

- Deng, Yi (2005). A Dynamic Stochastic Analysis of International Patent Application and Renewal Processes. Unpublished manuscript, Southern Methodist University.