In my research I am trying to find minimal conditions to guarantee a quasi-equilibrium must always be a typical Arrow-Debreu equilibria in a rather specific production setting.
This may be rather trivial/maybe I'm misunderstanding something, but I am struggling to show that agents endowed with goods will never have zero-prices on their goods. This seems to intuitively be true to me because it makes no sense for an agent to give away a good for free if they might make profit on it/can just keep it for themselves otherwise, but I can't see why market clearing and optimal production would imply that consumers never have zero prices at a quasi-equilibria.
Long story short, could somebody explain to me what stops agents from having zero prices on goods in an Arrow-Debreu production economy? What types of assumptions are necessary? Again, sorry if this is a silly question. I'm a mathematician by trade and have only recently started working with economic models.