A
competitive equilibrium with transfers for the economy $\mathcal E$ is an
allocation $(x^∗,y^∗)$, a price vector $p∗$ and an assignment of
wealths $(w_1^∗, . . . , w_I^∗)$ to consumers such that
1. For every firm $m$, $y^*_m$ maximizes profits among all feasible production plans in $Y_m$:
\begin{equation*}
p^*y^*_m\geq p^*y_m\quad\mbox{for all}\quad y_m\in Y_m.
\end{equation*}
2. For every consumer $n$, $x^*_n$ is preference-maximal among all affordable consumption plans. That is, $x^*_n\succeq_n x_n$ for all $x_n$ in the set
\begin{equation*}
\{x_n:x_n\in X_n\quad\mbox{and}\quad p^*x_n\leq w^*_n\}.
\end{equation*}
3. $(x^*,y^*)\in A$.
4. $\sum_nw^*_n=\sum_np^*\omega+\sum_mp^*y^*_m$.
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