This paper presents a surprising example that shows that the lattice theoretic properties in Mas-Colell’s (1986) seminal work are relevant to the existence of equilibrium even when the commodity space is finite dimensional.
The example is a two-period securities model with a three-dimensional portfolio space and two traders. The paper identifies a non-marketed call option that fails to have a minimum cost super-replicating portfolio. Using this option, we construct an economy that satisfies all of Mas-Colell’s assumptions, except that the three-dimensional commodity space is not a vector lattice. In this economy, there is no Walrasian equilibrium and the second theorem of welfare economics fails.
Our example has important finite- as well as infinite-dimensional implications. It is also an example of a “well behaved” economy in which optimal allocations that are not supported by linear Walrasian prices are decentralized by the non-linear prices studied in Aliprantis–Tourky–Yannelis (2001).