<p>A pretty trivial assumption would do it:</p>

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

CC BY-SA 4.0
2019-11-22T22:10:11.957Z
2019-11-23T04:50:15.983Z
2019-11-23T04:50:15.983Z
Michael
<p>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.</p>

<p>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.</p>

<p>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.</p>

CC BY-SA 4.0
2019-10-03T21:07:03.037Z
Is it enough to point out that Arrow-Debreu prices are simply probability-weighted, time-discounted ratios of marginal utility of consumption?
heh
CC BY-SA 4.0
2020-04-21T08:11:04.630Z
@heh No. Marginal utility of consumption can be zero, e.g., in case of Leontief preferences.
Giskard
<microeconomics><mathematical-economics><general-equilibrium><competitive-equilibrium>
What guarantees that endowed agents have non-zero prices in an Arrow-Debreu Economy
Dupin
# What guarantees that endowed agents have non-zero prices in an Arrow-Debreu Economy

(Asked by: username_0 on 2019-10-02)

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.

**Tags:** microeconomics, mathematical-economics, general-equilibrium, competitive-equilibrium

> **Comments:**
> **username_1**: Is it enough to point out that Arrow-Debreu prices are simply probability-weighted, time-discounted ratios of marginal utility of consumption?  
> **username_2**: @username_1 No. Marginal utility of consumption can be zero, e.g., in case of Leontief preferences.  

---

## 1 answers

### Answer by username_3 on 2019-11-22. Score: 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.


---

