Problem
Let us think about why we need positive endowments in the proof of Arrow-Debrew. The problem is for you to construct an Edgeworth box example where this condition is violated, and no competitive equilibrium exists. The intuition is: fix one good to have zero endowment (across both individuals). Find utilities where each good will always be demanded, even in tiny quantities, at any finite price. Then market clearing will imply prices are infinite.