The Welfare Theorems with Externalities

Individuals care about the consumption of others. They have social preferences $V:X\to\R$ on entire consumption allocations.

An individual has private preferences on her own consumption if her preferences are separable in her own consumption: If for some $x_{-n}$ \begin{equation*}\mkern-60mu V(x_n,x_{-n})\,\substack {\displaystyle >\\ \displaystyle\geq} \, V(z_n,x_{-n})\text{ then for all $x'_{-n}$, }V(x_n,x'_{-n})\,\substack {\displaystyle >\\ \displaystyle\geq}\, V(z_n,x'_{-n}). \end{equation*} Theorem. Individual $n$ has private preferences on her own consumption iff there are functions $v:\R\times X_{-n}\to\R$ and $u:X_n\to\R$ such that $V(x_n,x_{-n})=v\left(u(x_n),x_{-n}\right)$.

This is a general result about separable preferences.