Matching with contracts under status quo

We consider a matching with contracts problem where there is an initial matching a priori. Both sides of the market can veto the new matching whenever they would rather keep their initial contract. We propose a stability notion, called ``conditional stability." Our first mechanism---``Non-Vetoed Deferred Acceptance" ($NDA$)---is non-vetoed, conditionally stable, and strategy-proof. These three properties are incompatible with (two-sided) constrained efficiency; hence $NDA$ is not constrained efficient. However, no mechanism that is non-vetoed, conditionally stable, and strategy-proof improves on $NDA$. Lastly, we propose a non-vetoed, conditionally stable, and constrained efficient mechanism that dominates $NDA$.