Mean Field Games for Stochastic Growth with Relative Utility

This paper considers continuous time stochastic growth-consumption optimization in a mean field game setting. The individual capital stock evolution is determined by a Cobb-Douglas production function, consumption and stochastic depreciation. The individual utility functional combines an own utility and a relative utility with respect to the population. The use of the relative utility reflects human psychology, leading to a natural pattern of mean field interaction. The fixed point equation of the mean field game is derived with the aid of some ordinary differential equations. Due to the relative utility interaction, our performance analysis depends on some ratio based approximation error estimate.