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

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.

• Is it enough to point out that Arrow-Debreu prices are simply probability-weighted, time-discounted ratios of marginal utility of consumption? – heh Oct 3 '19 at 21:07
• @heh No. Marginal utility of consumption can be zero, e.g., in case of Leontief preferences. – Giskard Apr 21 at 8:11

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.