The Welfare Theorems

Now show that $(x^*,y^*,p^*)$ is quasi-equilibrium. Anything at least as good costs at least as much, and profit maximization. For $n\neq k$, and for any $x'_n\in R(x^*_n)$, let \begin{align*} g^i_n&=\sum_{j\neq k,n}x^*_n+x'_n+x^i_k-\sum_m y^*_m.\\ \omega&=\sum_{j\neq k,n}x^*_n+x^*_n+x^*_k-\sum_m y^*_m. \end{align*} Each $g^i\in G$, so $p^*g^i\geq p^*\omega$. Taking limits and subtracting, $p^*x'_n\geq p^*x^*_n$. Apply the same argument to $y^*_m$ to see that $-p^*y^*_m\geq -p^*y'_m$ for all $y_m\in Y_m$; $y^*_m$ is profit-maximizing. For consumer $k$, we can see directly by subtraction that for all $x'_k\succ_k x^*_k$, $p^*x'_k\geq p^*x^*_k$, and the conclusion for all $x'_k\succeq x^*_k$ follows from local non-satiation. ∎