<p>Strict convexity of preferences is not needed in existence results for competitive equilibria. Leontief preferences are quite well-behaved. They are continuous, convex, and strongly monotonic.   If all endowments are strictly positive, the existence of a competitive equilibrium in an exchange economy (or a production economy satisfying standard conditions) exists by the first result of the original <a href="https://web.stanford.edu/class/msande311/arrow-debreu.pdf" rel="noreferrer">Arrow-Debreu paper</a>.</p>

<p>Arrow-Debreu actually do not just require convexity, they make, as pointed out by denesp in a comment, the convexity assumption (III.c) on utility functions that $u(x)&gt;u(x')$ and $0&lt;t&lt;1$ implies $u(tx+(1-t)x')&gt;u(x')$. Plain convexity suffices for existence, but Leontief preferences do also satisfy condition (III.c).:  Assume $\min\{\alpha_i x_i\}&gt;\min\{\alpha_i x_i'\}$. Then
$$\min\big\{\alpha_i (tx_i+(1-t)x_i')\big\}&gt;\min\big\{\alpha_i tx_i\big\}+\min\big\{\alpha_i(1-t) x_i'\big\}$$ $$=t\min\{\alpha_i x_i\}+(1-t)\min\{\alpha_i x_i'\}&gt;\min\{\alpha_i x_i'\}.$$ </p>

CC BY-SA 3.0
2017-11-14T08:23:54.613Z
Doesn't Arrow-Debreu require strict convexity on [page 269 / III.c](https://web.stanford.edu/class/msande311/arrow-debreu.pdf#page=6)?
Giskard
CC BY-SA 3.0
2017-11-14T08:58:03.443Z
@denesp That assumption is somewhere between strict convexity and convexity; some people call it strong convexity. Notably, it is satisfied for Leontief preferences (while strict convexity is not).
Michael Greinecker
CC BY-SA 3.0
2017-11-14T18:04:05.903Z
So with Leontief preferencs CE always exists? This makes me wonder about the papers I read two years ago. AFAIR they claim that deciding whether CE exists is a difficult computational problem. How can this be a difficult problem if the answer is always yes? I have to re-read these papers to find out.
Erel Segal-Halevi
CC BY-SA 3.0
2017-11-14T19:07:18.283Z
@ErelSegal-Halevi Links to some of said papers would be nice!
Giskard
CC BY-SA 3.0
2017-11-14T22:06:17.843Z
@ErelSegal-Halevi I looked around and found that Yinyu Ye did some work on the issue. But in his framework, every consumer owns one unit of one good, so the interiority assumption does not hold. The problem consists in showing that a quasi-equilibrium (expenditure minimization) is actually an equilibrium,which reduces to the problem of showing that every consumer has positive wealth in equilibrium. This problem is not unique to Leontief preferences.
Michael Greinecker
CC BY-SA 3.0
2017-11-15T08:52:55.680Z
@denesp here are some links: https://doi.org/10.1145%2F1109557.1109629 and https://doi.org/10.1007%2F978-3-540-73814-5_9 and https://doi.org/10.1007%2F978-3-540-27836-8_33
Erel Segal-Halevi
CC BY-SA 3.0
2017-11-14T07:37:34.860Z
2017-11-14T10:07:38.710Z
2017-11-14T10:07:38.710Z
Michael Greinecker
<p>Consider an economy in which all consumers have, possibly different, <a href="https://en.wikipedia.org/wiki/Leontief_utilities" rel="nofollow noreferrer">Leontief utilities</a>. Since preferences are not strictly convex, it is not guaranteed that a competitive equilibrium exists. I found some papers that discuss the computational problem of deciding whether a Leontief economy has a competitive equilibrium, but I am interested in general existence results:</p>

<p>A. What conditions on Leontief economies guarantee that a competitive equilibrium exists?</p>

<p>B. In particular, if the initial endowments are equal (each of $m$ agents receives a fraction $1/m$ of each good), is a competitive equilibrium guaranteed to exist?</p>

CC BY-SA 3.0
2015-12-17T18:09:29.447Z
@denesp why did you delete your answer? It almost convinced me...
Erel Segal-Halevi
CC BY-SA 3.0
2015-12-17T18:59:00.010Z
It was incorrect. For $p_x = 0$ equilibria do exist. I will undelete it for a short while so you can see it. I will see if I can salvage it later.
Giskard
CC BY-SA 3.0
2015-12-17T19:52:16.983Z
@denesp Ah, I see! It is an interesting non-example :)
Erel Segal-Halevi
CC BY-SA 3.0
2017-10-30T17:35:11.930Z
It would seem that no _true_ equilibrium exists. only an approximate one when $x_1$ and $x_2$ are continuous. @denesp how does equilibria exist when $p_x=0$?
EconJohn
CC BY-SA 3.0
2017-10-30T19:10:18.323Z
@EconJohn An example: Let
$$
U_A(x_1,x_2) = \min(x_1;x_2) \mbox{ and } U_B(x_1,x_2) = \min(x_1;x_2).
$$
Assume initial endowments of $(3,2)$ for each player. For any $p_2 \in \mathbb{R}_{++}$ the pricevector $(0,p_2)$ is an equilibrium price vector. This means that given such a price vector each consumer has such an optimal consumption bundle that demand for each good does not surpass supply of respective good. The amount demanded of $x_2$ is trivially $2$ for both players. For $x_1$ it can be any number that is at least $2$. So e.g. $(2,2),(4,2)$ would constitute an equilibrium.
Giskard
CC BY-SA 3.0
2017-07-04T08:07:12.710Z
You can try papers on the existence of Nash equilibrium in aggregative games or large anonymous games. A Walrasian economy is such a game (the price vector is the aggregate action) and a Walrasian equilibrium is a Nash equilibrium. Generally existence theorems require compact action sets and continuous utilities.
Sander Heinsalu
<consumer-theory><competitive-equilibrium>
Competitive equilibrium in Leontief economies
Erel Segal-Halevi
# Competitive equilibrium in Leontief economies

(Asked by: username_0 on 2015-12-17)

Consider an economy in which all consumers have, possibly different, [Leontief utilities](https://en.wikipedia.org/wiki/Leontief_utilities). Since preferences are not strictly convex, it is not guaranteed that a competitive equilibrium exists. I found some papers that discuss the computational problem of deciding whether a Leontief economy has a competitive equilibrium, but I am interested in general existence results:

A. What conditions on Leontief economies guarantee that a competitive equilibrium exists?

B. In particular, if the initial endowments are equal (each of $m$ agents receives a fraction $1/m$ of each good), is a competitive equilibrium guaranteed to exist?

**Tags:** consumer-theory, competitive-equilibrium

> **Comments:**
> **username_0**: @denesp why did you delete your answer? It almost convinced me...  
> **username_1**: It was incorrect. For $p_x = 0$ equilibria do exist. I will undelete it for a short while so you can see it. I will see if I can salvage it later.  
> **username_0**: @denesp Ah, I see! It is an interesting non-example :)  
> **username_3**: You can try papers on the existence of Nash equilibrium in aggregative games or large anonymous games. A Walrasian economy is such a game (the price vector is the aggregate action) and a Walrasian equilibrium is a Nash equilibrium. Generally existence theorems require compact action sets and continuous utilities.  
> **username_2**: It would seem that no _true_ equilibrium exists. only an approximate one when $x_1$ and $x_2$ are continuous. @denesp how does equilibria exist when $p_x=0$?  
> **username_1**: @username_2 An example: Let $$ U_A(x_1,x_2) = \min(x_1;x_2) \mbox{ and } U_B(x_1,x_2) = \min(x_1;x_2). $$ Assume initial endowments of $(3,2)$ for each player. For any $p_2 \in \mathbb{R}_{++}$ the pricevector $(0,p_2)$ is an equilibrium price vector. This means that given such a price vector each consumer has such an optimal consumption bundle that demand for each good does not surpass supply of respective good. The amount demanded of $x_2$ is trivially $2$ for both players. For $x_1$ it can be any number that is at least $2$. So e.g. $(2,2),(4,2)$ would constitute an equilibrium.  

---

## 1 answers

### Answer by username_4 on 2017-11-14. Score: 5 | Accepted answer

Strict convexity of preferences is not needed in existence results for competitive equilibria. Leontief preferences are quite well-behaved. They are continuous, convex, and strongly monotonic. If all endowments are strictly positive, the existence of a competitive equilibrium in an exchange economy (or a production economy satisfying standard conditions) exists by the first result of the original [Arrow-Debreu paper](https://web.stanford.edu/class/msande311/arrow-debreu.pdf).

Arrow-Debreu actually do not just require convexity, they make, as pointed out by denesp in a comment, the convexity assumption (III.c) on utility functions that $u(x)>u(x')$ and $0u(x')$. Plain convexity suffices for existence, but Leontief preferences do also satisfy condition (III.c).: Assume $\min\{\alpha\_i x\_i\}>\min\{\alpha\_i x\_i'\}$. Then
$$\min\big\{\alpha\_i (tx\_i+(1-t)x\_i')\big\}>\min\big\{\alpha\_i tx\_i\big\}+\min\big\{\alpha\_i(1-t) x\_i'\big\}$$ $$=t\min\{\alpha\_i x\_i\}+(1-t)\min\{\alpha\_i x\_i'\}>\min\{\alpha\_i x\_i'\}.$$

> **Comments:**
> **username_1**: Doesn't Arrow-Debreu require strict convexity on [page 269 / III.c](https://web.stanford.edu/class/msande311/arrow-debreu.pdf#page=6)?  
> **username_4**: @denesp That assumption is somewhere between strict convexity and convexity; some people call it strong convexity. Notably, it is satisfied for Leontief preferences (while strict convexity is not).  
> **username_0**: So with Leontief preferencs CE always exists? This makes me wonder about the papers I read two years ago. AFAIR they claim that deciding whether CE exists is a difficult computational problem. How can this be a difficult problem if the answer is always yes? I have to re-read these papers to find out.  
> **username_1**: @ErelSegal-Halevi Links to some of said papers would be nice!  
> **username_4**: @ErelSegal-Halevi I looked around and found that Yinyu Ye did some work on the issue. But in his framework, every consumer owns one unit of one good, so the interiority assumption does not hold. The problem consists in showing that a quasi-equilibrium (expenditure minimization) is actually an equilibrium,which reduces to the problem of showing that every consumer has positive wealth in equilibrium. This problem is not unique to Leontief preferences.  
> **username_0**: @denesp here are some links: https://doi.org/10.1145%2F1109557.1109629 and https://doi.org/10.1007%2F978-3-540-73814-5_9 and https://doi.org/10.1007%2F978-3-540-27836-8_33  

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