# preference convexity and existence of equilbria

Consider a production economy with $$L$$ goods, a single consumer and a single producer whose production set are given by $$Y\subset R^L$$. Question is to find the existence condition of equilibria of this economy.

I think the existence conditions are:

1. continuity and monotonicity for consumer preference

2. convexity for firm's technology.

Generally, convexity of consumer's preference seems to be required also for existence of equilibria. However, I cannot find the counter example that there is no equilibrium of this economy if consumer's preference is not convex and all other consumptions are satisfied.

Convexity of consumer's preference is really a condition of existence of equilibria in this one consumer-one producer economy?

• Strict convexity is the condition of uniqueness of equilibrium. However, this is question for existence. Dec 7, 2020 at 11:48
• I also know the counterexample that there is no equilibrium if firm's technology is not convex. However, this question is about convexity of consumer's preference, not firm. Dec 7, 2020 at 12:37

Consider an economy with two commodities. Production is trivial, $$Y=\{0\}$$, there is a single consumer with endowment $$(1,1)$$ whose preferences are represented by the utility function given by $$u(x_1,x_2)=\max\{x_1,x_2\}+1/2\cdot x_1 +1/2\cdot x_2$$. These preferences are continuous, strictly monotone, but not convex. You can verify that there is no equilibrium in this economy by showing that for no price system, the demand of the consumer equals the supply of $$(1,1)$$.