consumption possibility sets and nonconvex preferences. In such cases, the supply and demand functions may be discontinuous with respect to price vector, thus a general equilibrium may not exist. However, we may "convexify" the economy, find an equilibrium for it, then by the Shapley–Folkman–Starr theorem, it is an approximate equilibrium for the original economy. In detail, given any economy satisfying all the assumptions given, except convexity of and , we define the "convexified economy" to be the same economy, except that iff . where denotes the convex hull. With this, any general equilibrium for the convexified economy is also an approximate equilibrium for the original economy. That is, if is an equilibrium price vector for the convexified economy, thenwhere is the Euclidean distance, and is any upper bound on the inner radii of all (see page on Shapley–Folkman–Starr theorem for the definition of inner radii). The convexified economy may not satisfy the assumptions. For example, the set is closed, but its convex hull is not closed. Imposing the further assumption that the convexified economy also satisfies the assumptions, we find that the original economy always has an approximate equilibrium. Accounting for time, space, and uncertainty The commodities in the Arrow–Debreu model are entirely abstract. Thus, although it is typically represented as a static market, it can be used to model time, space, and uncertainty by splitting one commodity into several, each contingent on a certain time, place, and state of the world. For example, "apples" can be split into "apples in New York in September if oranges are available" and "apples in Chicago in June if oranges are not available". Given some base commodities, the Arrow–Debreu complete market is a market where there is a separate commodity for every future time, for every place of delivery, for every state of the world under consideration, for every base commodity. In financial economics the term "Arrow–Debreu" most commonly refers to an Arrow–Debreu security. A canonical Arrow–Debreu security is a security that pays one unit of numeraire if a particular state of the world is reached and zero otherwise (the price of such a security being a so-called "state price"). As such, any derivatives contract whose settlement value is a function on an underlying whose value is uncertain at contract date can be decomposed as linear combination of Arrow–Debreu securities. Since the work of Breeden and Lizenberger in 1978, a large number of