  This paper studies the equilibrium consumption under external habit formation
in a large population of agents. We first formulate problems under two types of
conventional habit formation preferences, namely linear and multiplicative
external habit formation, in a mean field game framework. In a log-normal
market model with the asset specialization, we characterize one mean field
equilibrium in analytical form in each problem, allowing us to understand some
quantitative properties of the equilibrium strategy and conclude some financial
implications caused by consumption habits from a mean-field perspective. In
each problem with n agents, we construct an approximate Nash equilibrium for
the n-player game using the obtained mean field equilibrium when n is
sufficiently large. The explicit convergence order in each problem can also be
obtained.
