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S. Wrzaczek, E. Shevkoplyas, S. Kostyunin:
"Differential Game of Pollution Control with overlapping generations";
Talk: The Fifth International Conference Game Theory and Management, St. Petersburg, Russland (invited); 2011-06-27 - 2011-06-29; in: "Game Theory And Management", L. Petrosyan, N. Zenkevich (ed.); St. Petersburg University, V (2012), 310 - 320.

We consider a stable (not necessarily stationary) age-structured population
modeled by the McKendrick partial differential equation. In constrast to the resource
extraction model of Jorgensen and Yeung (1999), where new generations appear at
discrete time steps, we assume that at each instant in time a new generation enters the
game. Further the mortality as well as the fertility rate of the model are constant over
time and exogenous (implying a stable population). The maximal length of life of one
cohort equals omega (can be assured by an assumption, see e.g. Anita (2000)).
The cohorts maximize their lifetime utility by chosing the optimal emission rate
(i.e. age- and time-dependent control) over their life time. The emissions are aggregated
over time and cohorts (i.e. time-dependent state). The objective functions then consists
of three components. (i) Utility from the emissions (e.g. production), (ii) disutility from
emissions (e.g. pollution), (iii) altruism. (i) and (ii) are standard in economics. (iii) goes
back to Barro and Becker (1989) and includes the idea that also the utility of the
progenies has to be included in the objective functional. The last motive is very
important since otherwise all players behave without any care about the time after the
own life.
The resulting model looks quite similar to Shevkoplyas and Kostyunin (2010)
(see also Breton et al. (2005)), but works considerably different and has a different
interpretation. The duration of the game is no longer random, the differential game
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evolves over time, but includes overlapping generations and an altruistic motive is
included.
We calculate the open-loop Nash equilibrium for the differential game and
provide economic interpretations for the derived expressions. On the other hand we deal
with the cooperative solution for the differential game resulting in an age-specific
optimal control model (solved by the corresponding maximum principle presented in
Brokate (1985) or Feichtinger et al. (2003)). By comparing the outcome of both
solutions we are able to figure out the relevant differences and provide important
economic interpretations. Finally numerical simulations show the solution for both
outcomes over time and over the life-time of different cohorts. We are able to illustrate
the difference in the long-term behavior of the model compared to the model without an

Bidding Nash equilibrium,Cooperative solution, Overlapping generations, Agestructuredgame