Define $Y=\sum_mY_m$.
The private ownership economy $\mathcal{E}$ has an optimum if
- for all $n$, $X_n$ is closed and bounded from below and $\omega_n\in X_n$,
- each $Y_m$ is closed, convex, and contains 0,
- $Y\cap\R^L_+=\{0\}$ and $Y\cap -Y=\{0\}$,
- for every $x'_n\in X_n$, the set $\{x_n\in X_n:x_n\succeq x'_n\}$ is closed.
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