Annals of Operations Research

, Volume 275, Issue 2, pp 297–319 | Cite as

On a strategic model of pollution control

  • Giorgio Ferrari
  • Torben KochEmail author
Original Research


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.


Pollution Stochastic impulse nonzero-sum game Verification theorem Diffusions 

JEL Classification

C61 C73 Q52 

Mathematics Subject Classification

93E20 91B70 91A15 91B76 



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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Mathematical EconomicsBielefeld UniversityBielefeldGermany

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