Theorem. Suppose that $lg$ is the unique opportunity constraint for $l$. An increase in $v_{lg}$ raises $\underline{w}_l$ and decreases $\bar\pi_{\sigma(l)}$. A decrease in $v_{lg}$ lowers $\underline{w}_l$ and increases $\bar\pi_{\sigma(l)}$.

Proof. Replace $v_{lg}$ by $v'_{lg}>v_{lg}$ such that $\sigma$ remains an optimal match. Then the binding opportunity constraint on $w_l$ is tighter, so $w_l$ increases. Replace $v_{lg}$ by $v_{lg}'\lt v_{lg}$ such that $\sigma$ remains an optimal match. Then there is no binding opportunity constraint for $l$, and the argument of the Lemma's proof shows that the new greatest lower bound $\underline w_l'$ on $w_l$ is less than $\underline w_l,$ and that $\bar\pi'_{\sigma(l)}>\bar\pi_{\sigma(l)}$.

Similarly, if $lg$ is the unique opportunity constraint for $f$. Then raising $v_{lg}$ lowers $\underline\pi_f$ and raises $\bar w_{\sigma^{-1}(f)}$.

So if we increase $v_{lf}$ from a very low value for $f\neq\sigma(l)$, nothing happens until $lf$ becomes the opportunity constraint for $l$. Then $\underline w_{l}$ rises until it becomes optimal to assign $l$ to $f$. Then $\underline w_l$ holds constant. Along this same trajectory, $\bar\pi_{\sigma(l)}$ holds constant, then falls, and then holds constant when it is no longer optimal to match $l$ with $\sigma(l)$. ∎