= (c^i_tX^i_t)^{1-\theta_i}\left(\frac{c^i_tX_t^i}{\overline{cX}_t}\right)^{\theta_i}, \quad\quad X^i_T \overline{X}_T^{- \theta_i} = (X^i_T)^{1-\theta_i}\left(\frac{X_T^i}{\overline{X}_T}\right)^{\theta_i}.\end{aligned}$$ The ratios $c^i_tX_t^i/\overline{cX}_t$ and $X^i_T/\overline{X}_T$ measure the relative consumption rate and relative terminal wealth, respectively. In particular, the utility function in is applied to the log-convex combination between absolute and relative consumption rate and terminal wealth, with $\theta_i$ controlling the tradeoff between absolute and relative performance. For $\theta_i$ close to $1$, agent $i$ is more concerned with relative performance than absolute performance, and for $\theta_i=0$ agent $i$ is not at all competitive and ignores the rest of the population.

The goal is to find a Nash equilibrium, an investment strategy $(\vec{\pi}^*_t, \vec{c}^*_t)_{t \in [0,T]}$ such that $\pi_t^{i,*}$ and $c_t^{i,*}$ are respectively the optimal stock and consumption allocation exercised by agent $i$ in response to the strategy of all the other agents. With Merton’s problem and the recent findings of [@lacker2017mean] in mind, we might expect to find an equilibrium where the investment strategies $\vec{\pi}^*_t$ are constant and the consumption strategies $c_t^{i,*}$ are only time-dependent.

We say that a vector $(\pi^{i,*}, c^{i,*})_{i=1}^n$ of admissible strategies (i.e., $(\pi^{i,^*}, c^{i,*}) \in \A$ for each $i$) is an *equilibrium* if for each $i = 1, ..., n$ and each $(\pi, c) \in \A$ we have $$J_i((\pi^{i,*},c^{i,*})_{i=1}^n) 