and (\[optcontrolNot\]’) as $$\label{optcontrolNotOne}
\begin{split}
\pi^{i,*} &=  \frac{\gamma_i(\mu_i - \sigma_i \widehat{\sigma \pi}_{-i}\theta_i (1 - 1 / \delta_i))}{\Sigma_i}, \\
c^{i,*}_t &= \epsilon_i^{-\gamma_i} (\bar{c}_{-i}(t))^{- \gamma_i \theta_i(1- 1 / \delta_i)} f_i^{- \gamma_i}(t).
\end{split}$$ [  ]{}\
**The case** $\bm{\delta_i=1}$: In the case $\delta_i = 1$ we must proceed differently, but we will ultimately derive optimal controls that are consistent with the formulas in . Note first that we may greatly simplify the form of , because the logarithmic utility function implies in particular that the other players no longer influence player $i$’s optimization. That is, player $i$ maximizes the simplified objective $$\label{valuefOne}
(1 - \theta_i /n) \E \left[ \int_0^T  \log (c^i_t X^i_t) dt + \epsilon_i  \log X^i_T \right].$$ Noting that $1-\theta_i/n > 0$, the value (\[valuefOne\]) is equal to $(1-\theta_i/n)\val2(X_0^i,0)$, where $\val2(x,t)$ solves the HJB equation $$\begin{split}
0 = \val2_t &+ \max_{\pi \in \R} \left[ \pi \mu_i x \val2_x + \pi^2 \frac{1}{2} \Sigma_i x^2 \val2_{xx}\right] + \max_{c \in \R_+} \left[- c x  \val2_x + \log(cx) \right],
\end{split} \label{HJB-One-simplified}$$ for $(x,t) \in \R_+ \times [0,T)$, with terminal condition $\val2(x,T) = \epsilon_i \log x$. The maximum is attained by $$\label{optconsone}
\begin{split}
    \pi^{i,*}(x,t) &= - \frac{\mu_i x \val2_x(x,t)}{\Sigma_i x^2 \val2_{xx}(x,t)}, \quad\qquad
    c^{i,*}(x,t) = \frac{1}{x \val2_x(x,t)}.
\end{split}$$ The 