The Second Welfare Theorem
Let $P(x_n)=\{x'_n\in X_n:x'_n\succ_nx_n\}$ and $R(x_n)=\{x'_n\in
X_n:x'_n\succeq_nx_n\}$. A quasi-equilibrium for the economy
$\mathcal{E}$ is an allocation $(x^*,y^*)$ and a price vector $p^*$ such that
- 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*}
- For every consumer $n$, $x^*_n$ is expenditure-minimal on the 'no worse than' set. That is,
$p^*x^*_n\leq p^*x_n$ for all $x_n$ in the set $R(x^*_n)$.
- $(x^*,y^*)\in A$.
A quasi-equilibrium is sometimes called a compensated equilibrium.
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