is a unique strong equilibrium $(\pi^{i,*}, c^{i,*})_{i=1}^n$, and it takes the following form: $$\begin{aligned}
\pi^{i,*} &= \frac{\mu}{\sigma^2}\left(\delta_i - \frac{\theta_i}{\theta_{\mathrm{crit}}}(\delta_i-1)\right) \label{optcontroleq-singlestock} \\
c^{i,*}_t &= \begin{cases}
            \left(\frac{1}{\newomega_i} + \left(\frac{1}{\newC_i} - \frac{1}{\newomega_i} \right)e^{- \newomega_i (T - t)} \right)^{-1} & \mathrm{if } \ \beta_i \ne 0, \\
            (T - t + \newC_i^{-1})^{-1} & \mathrm{if } \ \beta_i = 0. 
        \end{cases} \label{optconsumption-singlestock}\end{aligned}$$ For each $i$, the constant $\newC_i$ is given by , and $\newomega_i$ and $\theta_{\mathrm{crit}}$ are given by $$\begin{aligned}
\newomega_i &= \frac{\mu^2}{2\sigma^2}(1 - \delta_i)\left(1 - \frac{\theta_i}{\theta_{\mathrm{crit}}}\right)\left(\delta_i - \frac{\theta_i}{\theta_{\mathrm{crit}}}(\delta_i-1)\right), \\
\theta_{\mathrm{crit}} &= \frac{1 + \frac{1}{n}\sum_{k = 1}^{n} \theta_k(\delta_k -1)}{\frac{1}{n}\sum_{k = 1}^{n} \delta_k }.\end{aligned}$$

In Section \[se:mfg\] we simplify this further by sending $n\to\infty$. Then, in Section \[se:discussion\], we analyze in detail how the equilibrium behavior depends on the various parameters.

First, we fix an agent $i \in \{1,\ldots,n\}$ and suppose that all other players follow given strategies. That is, for $k \neq i$, let $\pi_k \in \R$ and $c_k : [0,T] \to \R_+$ denote fixed admissible strategies for the other agents, in which the investment policy $\pi_k$ is constant and the consumption policy $c_k$ is a deterministic continuous function. We will solve the optimization problem for agent $i$, determining the agent’s best response to the 