Abstract
This paper proposes a strategic model of pollution control. A firm, representative of the productive sector of a country, aims at maximizing its profits by expanding its production. Assuming that the output of production is proportional to the level of pollutants’ emissions, the firm increases the level of pollution. The government of the country aims at minimizing the social costs due to the pollution, and introduces regulatory constraints on the emissions’ level, which then effectively cap the output of production. Supposing that the firm and the government face both proportional and fixed costs in order to adopt their policies, we model the previous problem as a stochastic impulse two-person nonzero-sum game. The state variable of the game is the level of the output of production which evolves as a general linearly controlled one-dimensional Itô-diffusion. Following an educated guess, we first construct a pair of candidate equilibrium policies and of corresponding equilibrium values, and we then provide a set of sufficient conditions under which they indeed realize an equilibrium. Our results are complemented by a numerical study when the (uncontrolled) output of production evolves as a geometric Brownian motion, and the firm’s operating profit and the government’s running cost functions are of power type. An analysis of the dependency of the equilibrium policies and values on the model parameters yields interesting new behaviors that we explain as a consequence of the strategic interaction between the firm and the government.
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Notes
Stochastic impulse control problems naturally arise in many areas of applications. Among these we refer to optimal control of exchange and interest rates (Cadenillas and Zapatero 1999; Mitchell et al. 2014; Perera et al. 2016, among others), portfolio optimization with fixed transaction costs (Korn 1999), optimal inventory control (Bensoussan et al. 2010; Harrison et al. 1983), rational harvesting of renewable resources (Alvarez 2004), and optimal dividend problems (Cadenillas et al. 2006).
The interested reader may refer to the book by Bensoussan and Lions (1984) for the theory of QVIs.
Restrictions on the output of production can be achieved by the government in different ways. The interested reader may refer to the classical book by Pigou (1983).
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Acknowledgements
We wish to thank Giorgia Callegaro, Herbert Dawid, Frank Riedel and Jan-Henrik Steg for useful comments. Financial support by the German Research Foundation (DFG) through the Collaborative Research Centre 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications” is gratefully acknowledged by the authors.
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Ferrari, G., Koch, T. On a strategic model of pollution control. Ann Oper Res 275, 297–319 (2019). https://doi.org/10.1007/s10479-018-2935-7
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DOI: https://doi.org/10.1007/s10479-018-2935-7