As commonly discussed in Microeconomics courses, Walrasian equilibrium is guaranteed to exist under some restrictive conditions on consumers preferences.
A restrictive model proposed by Hatfield et al. (2011) explains that a Walrasian Equilibrium must rely on consumers viewing goods as a specific and unique type of preferences, sometimes being considered as either substitutes or complements, not both of them. However, we can use the model proposed by Eduardo M. Azevedo, E. Glen Weyl and Alexander White to show that:
“A competitive equilibrium exists even if agents have arbitrary preferences over bundles of items .” (Azevedo, Weyl, and White, 2013)
The authors of the paper use a model with a continuum of consumers scenario with two different groups: Charlies and Sonias, and two indivisible goods, as well as money to value the possible bundles. The model also explains that there is enough of each commodity for half of all consumers. For me, the key point of this paper is that the model uses Quasilinear utility to describe the consumers preferences, which can be either complements or substitutes properties.
The paper refers to Farrell’s (1959) insight that the impact of non convexities in any individual consumer’s demand diminishes as the number of consumers increases. In context:
“When there is an infinite number of consumers, aggregate demand exhibits the convexity necessary to prove the existence of an equilibrium price vector.” (Farrell, 1959)
Proposed model properties:
The paper explains the model results with far more ease. In this Topic I show the main results and examples that could help understand the paper.
Charlies group: Charlies view both goods as perfect complements and assigns $1 of value for consuming BOTH goods and 0 for consuming neither of them.
Sonias group: Sonias, on the contrary, view both goods as perfect substitutes, assigning $0.75 for consuming only one of them and 0 for consuming both of them.
For Charlies to consume (demand) both goods, it must be the case that p1+p2<=1 (the sum of both goods prices must be lower or equal to one). If that condition is satisfied, however, at least one of the two prices must be strictly less than $0.75, implying that Sonia will demand only one good, as the model proposes it.
The efficient allocation in this economy is as follows. One-half of the Sonias receive good 1 and half receive good 2; however, only half of all Charlies receive any good and those who do, receive both of the two goods. This is efficient because Sonias value each good at $0.75 per unit, while Charlies effectively value each at $0.50 and thus all Sonias should be satisfied before any Charlies are.
The efficient allocation is supported by a Walrasian price vector: each good costs $0.50. At these prices, Sonias strictly prefer to consume some good, but are indifferent as to which one, while Charlies are indifferent between consuming the bundle and consuming nothing. Thus, Charlies are happy to mix between demanding both goods and demanding neither, while Sonias are collectively content to mix between demanding one good or the other.
In case you are interested in reading more about the results of the paper, please refer to the following site:
Walrasian Equilibrium in Large, Quasilinear Markets
If you want to share your opinion on the topic, please comment on this site down below so we can have an interesting discussion, regarding the matter.
Azevedo, Eduardo M. and Weyl, E. Glen and White, Alexander, Walrasian Equilibrium in Large, Quasilinear Markets (April 13, 2012). Theoretical Economics, Vol. 8, No. 2, pp. 281-290, May 2013, DOI: 10.3982/TE1060.
Farrell, Michael J. (1959), “The convexity assumption in the theory of competitive markets.” Journal of Political Economy, 67, 377–391. 
Hatfield, John W., Scott D. Kominers, Alexandru Nichifor, Michael Ostrovsky, and Alexan- der Westkamp (2011), “Stability and competitive equilibrium in trading networks.” Unpublished paper. [281, 283]