Appl Math Optim (2011) 63: 11–44
Dynamic Bertrand Oligopoly
Andrew Ledvina · Ronnie Sircar
Published online: 17 July 2010
© Springer Science+Business Media, LLC 2010
Abstract We study continuous time Bertrand oligopolies in which a small number
of ﬁrms producing similar goods compete with one another by setting prices. We
ﬁrst analyze a static version of this game in order to better understand the strategies
played in the dynamic setting. Within the static game, we characterize the Nash equi-
librium when there are N players with heterogeneous costs. In the dynamic game
with uncertain market demand, ﬁrms of different sizes have different lifetime capaci-
ties which deplete over time according to the market demand for their good. We setup
the nonzero-sum stochastic differential game and its associated system of HJB partial
differential equations in the case of linear demand functions. We characterize certain
qualitative features of the game using an asymptotic approximation in the limit of
small competition. The equilibrium of the game is further studied using numerical
solutions. We ﬁnd that consumers beneﬁt the most when a market is structured with
many ﬁrms of the same relative size producing highly substitutable goods. However,
a large degree of substitutability does not always lead to large drops in price, for
example when two ﬁrms have a large difference in their size.
Keywords Stochastic differential games · Nash equilibrium · Bertrand oligopoly ·
Substitutable goods · Systems of HJB equations · Asymptotic expansions
Communicating Editor: Alain Bensoussan.
Work of A. Ledvina partially supported by NSF grant DMS-0739195.
Work of R. Sircar partially supported by NSF grant DMS-0807440.
A. Ledvina · R. Sircar (
ORFE Department, Princeton University, Sherrerd Hall, Princeton, NJ 08544, USA