I have a specific question about considering a representative agent. According to Rubinstein 1974, in an inter-temporal economy with $N$ individuals each one with initial endowment $e_i$ and everything else homogeneous. The intertemporal constraints of the investor are: $c_{0i}=e_{0i}-\eta_i p$ $c_{1i}=e_{1i}+\eta_i X$ Where $e_{0i},e_{1i}$ are present and future endownments, $c_{0i}, c_{1i}$ represents present and future consumption, $p$ is the price of the investment technology, and $\eta_i$ is the number of units bought. Finally $X$ is the payoff of the investment technology.
Assuming investors differ only on their endowments, we can solve for the equilibrium using a representative agent with the average endowment $\bar{e}=\frac{\sum e_i}{N}$.
Suppose now that the present endowment of each agent depends on how many units he/she decides to buy. $c_{0i}=e^i(\eta_i)-\eta_i p$
The function $e^i(\eta_i)$ is not necessarily the same for all investors.
My question is: Can we solve for the equilibrium using a representative agent with endowment $\bar{e(\eta)}=\frac{\sum e^i (\eta)}{N}$ ?
Thank you
Reference: Rubinstein, M., 1974. An aggregation theorem for securities markets. Journal of Financial Economics 1, 225-244.