What guarantees that endowed agents have non-zero prices in an Arrow-Debreu Economy

Question

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.

Answer 1

A pretty trivial assumption would do it:

If the consumers preferences are strongly increasing, i.e. $$ x \succ_i y, \mbox{ if $x \geq y$ but $x \neq y$ }, $$ and unbounded for all $i$, then no $x$ can be an optimal bundle if $p \notin \mathbb{R}^C_{++}$ (assuming there are $C$ commodities). So zero price cannot arise in an Arrow-Debreu equilibrium.