Note that we obtained exactly the same optimal control $\pi^{i,*}$ as in the problem without consumption, and we can conclude as in the proof of [@lacker2017mean Theorem 14] that $\pi^{i,*} = \pi_i$ for all $i = 1, ..., n$ if and only if $\pi^{i,*}$ is as in . In addition, we prove the identity just as in [@lacker2017mean Theorem 14]. Recall that $\rho_i$ defined in (\[rhoi\]) depends on the investment policies $\pi_k$ of the other agents but not on the consumption policies; in particular, in equilibrium we have $$\begin{split}
  \rho_i := \Big(1 - \frac{1}{\delta_i}\Big) &\Bigg\{ \frac{(1 - \theta_i/n)(\mu_i - \sigma_i\theta_i(1 - 1/\delta_i)\frac{1}{n}\sum_{k \ne i} \sigma_k \pi^{k,*})^2}{2 (\sigma_i^2 + \nu_i^2)(1 - (1 -\theta_i/n)(1 - 1/\delta_i))} \\
  &\quad+ \frac{1}{2}\theta_i^2\Big(1 - \frac{1}{\delta_i}\Big)\Bigg(\Big(\frac{1}{n} \sum_{k\ne i} \sigma_k \pi^{k,*} \Big)^2 + \frac{1}{n} \sum_{k \ne i} (\nu_k \pi^{k,*})^2\Bigg)  \\
  &\quad -   \theta_i \frac{1}{n} \sum_{k \ne i} \mu_k \pi^{k,*} + \frac{\theta_i}{2n} \sum_{k \ne i} ( \sigma^2_k + \nu_k^2)(\pi^{k,*})^2 \Bigg\}
  \end{split} \label{pf:rhoi-def}$$

Next, we address the consumption policies. In light of our arguments above, in order to have an equilibrium, we must simultaneously solve the following system of equations, for $i=1,\ldots,n$: $$\begin{aligned}
c_i(t) &=  \epsilon_i^{-\gamma_i} (\bar{c}_{-i}(t))^{- \gamma_i \theta_i(1- 1 / \delta_i)} (f_i(t))^{- \gamma_i} \label{systeq1} \\
0 &= f_i'(t) + 