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AbstractThe methodology of differential games is a combination of optimal control theory and game theory. It is the natural framework for economic analysis with strategic interaction and dynamical optimization. The theory gained traction by seminal papers in the early seventies, and it gradually found its way into economics. The purpose of this paper is to make theory and applications of differential games easily accessible by explaining the basics and by developing some characteristic applications. The core of the theory focuses on the open-loop and the multiple Markov-perfect Nash equilibria that use the maximum principle and dynamic programming as the techniques to solve the optimal control problems. The applications are the game of international pollution control and the game of managing a lake, which is an example of an ecological system with tipping points. Finally, it is interesting to note that the discovery of time-inconsistency in the open-loop Stackelberg equilibrium had a huge impact on macroeconomics, since policy making under rational expectations is a Stackelberg differential game.

AbstractThis paper analyzes three models in environmental economics with the property that tipping can occur in the ecological part of the model: the pollution control model where tipping suddenly shifts up the damage, the fishery model where tipping suddenly lowers the carrying capacity, and the Ramsey growth model where tipping suddenly decreases the total factor productivity. The question is whether the possibility of tipping gives rise to precaution in the sense of lower production, lower harvest, or higher accumulation of capital, respectively. The paper shows that precaution always results in the pollution control model but in the other two models, it depends on the elasticity of intertemporal substitution. In these models, the tipping point is uncertain, and the analysis employs a hazard rate that can be constant or state dependent. The paper shows that the Hamilton-Jacobi-Bellman equations are the appropriate framework to implement this hazard rate. The results are mostly known, but this paper puts the results together in a transparent way, using the HJB equations consistently. In this way, it pays tribute to this technique in optimal control that proves to be useful when tipping points can occur. Extensions to differential games are discussed at the end of each section.

Paul Ehrlich has written a wonderful article with the purpose to convince economists to switch their research interests to what he calls ecological economics. His argument is twofold. If our intellectual capacity is not geared towards these issues, humanity may be heading for a big disaster. Secondly, it gives tremendous opportunities for economists: the ones who switch will end up in the centre of the profession, because society will be in big need of their expertise.

AbstractThis paper considers self-enforcing international environmental agreements when countries are asymmetric with respect to emission-related benefits and environmental damage. Considering these asymmetries simultaneously yields large stable coalitions, also without the option of transfers between signatories. However, these large stable coalitions are only possible if they include countries that have relatively high marginal benefits and a relatively low marginal environmental damage. This type of countries hardly contributes to the common good and the gains of cooperation from including this type of countries in the stable coalition are small. This confirms a persistent result in this literature that large stable coalitions usually go hand in hand with low gains of cooperation. Without the option of transfers it is always better to have a small stable coalition with countries that matter than a large stable coalition with countries that do not matter. Only with transfers might a large stable coalition be able to perform better.

We develop a simple model of managing a system subject to pollution damage under risk of an abrupt and random jump in the damage coefficient. The model allows the full dynamic characterization of the optimal emission policies under uncertainty. The results, that imply prudent behavior due to uncertainty, are compared with the ambiguous outcomes reported in the literature for similar models. The differences are explained in terms of the properties of the damage function associated with each model. The framework is used to analyze the adaptation vs. mitigation dilemma and provides a simple criterion to determine whether adaptation activities should be undertaken promptly, delayed to some future date, or avoided altogether.