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This paper studies the equilibrium consumption under external habit formationin a large population of agents. We first formulate problems under two types ofconventional habit formation preferences, namely linear and multiplicativeexternal habit formation, in a mean field game framework. In a log-normalmarket model with the asset specialization, we characterize one mean fieldequilibrium in analytical form in each problem, allowing us to understand somequantitative properties of the equilibrium strategy and conclude distinctfinancial implications caused by different consumption habits from a mean fieldperspective. In each problem with $n$ agents, we then construct an approximateNash equilibrium for the $n$-player game using the obtained mean fieldequilibrium when $n$ is sufficiently large. The explicit convergence order ineach problem can also be obtained.

A mean field game approach to equilibrium consumption under external habit formation