# Competitive equilibrium in Leontief economies

Consider an economy in which all consumers have, possibly different, 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?

• @denesp why did you delete your answer? It almost convinced me... Dec 17 '15 at 18:09
• @denesp Ah, I see! It is an interesting non-example :) Dec 17 '15 at 19:52
• 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. Jul 4 '17 at 8:07
• 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
Oct 30 '17 at 17:35
• @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. Oct 30 '17 at 19:10

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.

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 $0<t<1$ implies $u(tx+(1-t)x')>u(x')$. 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 $$\min\big\{\alpha_i (tx_i+(1-t)x_i')\big\}>\min\big\{\alpha_i tx_i\big\}+\min\big\{\alpha_i(1-t) x_i'\big\}$$ $$=t\min\{\alpha_i x_i\}+(1-t)\min\{\alpha_i x_i'\}>\min\{\alpha_i x_i'\}.$$

• Doesn't Arrow-Debreu require strict convexity on page 269 / III.c? Nov 14 '17 at 8:23
• @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). Nov 14 '17 at 8:58
• 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. Nov 14 '17 at 18:04
• @ErelSegal-Halevi Links to some of said papers would be nice! Nov 14 '17 at 19:07
• @denesp here are some links: doi.org/10.1145%2F1109557.1109629 and doi.org/10.1007%2F978-3-540-73814-5_9 and doi.org/10.1007%2F978-3-540-27836-8_33 Nov 15 '17 at 8:52