Euler-Lagrange equations of stochastic differential games: application to a game of a productive asset

Suppose that some economic agents (firms, countries) share access to the same stock of a productive asset, such as fish or forest, and consume the common resource under conditions of individualistic competition. Assuming that this interaction is intertemporal, interesting questions arise: how can the individual plan consumption in an optimal way, taking into account the strategic behavior of the rival agents?  Can the competition between agents lead to an overexploitation of the asset, ending even in extinction? How does the existence of unexpected events that escape the control of the agent affect the planned policy? These are important questions in the economic theory that underlies the exploitation of common resources, and they are questions that have an impact not only on economics, but also on a sustainable management of the environment and on the commercial relations of the countries involved.

In a recent paper, Ricardo Josa-Fombellida (Universidad de Valladolid) and Juan Pablo Rincón-Zapatero carried out a theoretical study of a model where agents exploit non cooperatively a common productive asset, which is subject to random disturbances. They set conditions for the existence of Nash equilibrium in a rather general context and are able to positively answer such questions as the structure of the optimal strategies, the way the asset is distributed among the agents, how the stock evolves, or how the inefficiency of noncooperation may arise. The paper also addresses questions related to the sensitivity of the equilibrium with respect to the number of players (for instance, a large number of players mitigates inefficiency) and with respect to the characteristics of the players, such as time preference or risk aversion (e.g., low risk aversion tends to ameliorate inefficiency). The effect of the size of the uncertainty on the consumption pattern is also studied (greater uncertainty makes agents deplete the common resource more intensively).


Josa Fombellida, Ricardo; Rincón-Zapatero, Juan Pablo