In many countries holders of patents must pay an annual renewal fee in order to keep their patents in force. This paper uses data on the proportion of patents renewed, and the renewal fees faced by, post World War II cohorts of patents in France, the United Kingdom, and Germany, in conjunction with a model of patent holders' renewal decisions, to estimate the returns earned from holding patents in these countries. Since patents are often applied for at a nearly stage in the innovation process, the model allows agents to be uncertain about the sequence of returns that will be earned if the patent is kept inforce. Formally, then, the paper presents and solves a discrete choice optimal stochastic control model, derives the implications of the model on aggregate behaviour, and then estimates the parameters of the model from aggregate data. The estimates enable a detailed description of the evolution of the distribution of returns earned from holding patents over their life spans,and calculations of both; the annual returns earned from holding the patents still in force (or the patent stocks) in the alternative countries, and the distribution of the discounted value of returns earned from holding the patents in a cohort.

This paper introduces a stochastic algorithm for computing symmetric Markov perfect equilibria. The algorithm computes equilibrium policy and value functions, and generates a transition kernel for the (stochastic) evolution of the state of the system. It has two features that together imply that it need not be subject to the curse of dimensionality. First, the integral that determines continuation values is never calculated; rather it is approximated by a simple average of returns from past outcomes of the algorithm, an approximation whose computational burden is not tied to the dimension of the state space. Second, iterations of the algorithm update value and policy functions at a single (rather than at all possible) points in the state space. Random draws from a distribution set by the updated policies determine the location of the next iteration's updates. This selection only repeatedly hits the recurrent class of points, a subset whose cardinality is not directly tied to that of the state space. Numerical results for industrial organization problems show that our algorithm can increase speed and decrease memory requirements by several orders of magnitude.

A general central limit theorem is proved for estimators defined by minimization of the length of a vector-valued, random criterion function. No smoothness assumptions are imposed on the criterion function in order that the results might apply to a broad class of simulation estimators. Complete analyses of two simulation estimators, one introduced by A. Pakes (1986) and the other by D. McFadden (1989), illustrate the application of the general theorems. These examples illustrate how simulation can be used to circumvent two computational problems that arise frequently in applied econometrics: evaluating intractable aggregation formulae and evaluating discrete response probabilities.

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