model, such as [@karatzas1997explicit; @sethi1992infinite], did so in an equilibrium context; each agent’s behavior depends on the others only through the price, which is determined in equilibrium. Agents are price-takers in our model, and we do not attempt to incorporate price equilibrium, as this would severely strain tractability.

In another direction, our work provides a new explicitly solvable mean field game model. Mean field games, introduced in [@huang2006large; @lasry-lions], are rarely explicitly solvable outside of linear-quadratic examples. See [@bayraktar2018large; @carmona2018dyson; @gueant-lasry-lions; @lacker2017mean; @sun2016systemic] for some notable exceptions and the book [@carmona-delarue-book] for further background on the active area of mean field games. From a mean field game perspective, our model is rather complex: It involves common noise, degenerate volatility coefficients, singular objective functions, and a mean field interaction through both the states and controls (i.e., an *extended mean field game* [@carmona-delarue-book Chapter I.4.6]). Nevertheless, the precise structure of the problem lends itself to an explicit solution. Our argument follows along the lines of [@lacker2017mean], treating the mean field term (geometric mean of wealth) as a state variable, which leads to a fixed point problem involving a single Hamilton-Jacobi-Bellman (HJB) equation as opposed to the $n$-dimensional HJB system often used for 