by Lacker and Zariphopoulou [@lacker2017mean] to include consumption. Our model can be loosely described as follows, with full details given in Section \[finite population\]. Each agent chooses consumption and investment policies, with access to a riskless bond and a lognormal stock. The stocks in which the different agents specialize can be correlated, and we cover the extreme case of perfect correlation, i.e., a single stock in which all agents trade. Each agent has a CRRA utility depending on both absolute and relative wealth at the (common) time horizon $T$ as well as absolute and relative consumption, the latter in a time-integrated sense. Agents have different levels of risk aversion and different preferences toward absolute versus relative performance. The relative performance criteria couple these $n$ optimization problems, and we find the (unique, in a sense to be clarified later) Nash equilibrium in terms of the various model parameters. As is natural in light of the classical Merton problem [@merton1969lifetime], in equilibrium each agent invests a constant fraction of wealth in the stock, and the consumption strategy is time-dependent but independent of the agent’s wealth.

The equilibrium behavior fits with Samuelson’s result [@samuelson1975lifetime]: the investment strategy is independent of the consumption strategy. 