Editing Arrow–Debreu model (section)

=== Accounting for nonconvexity ===
The assumption that production possibility sets are convex is a strong constraint, as it implies that there is no economy of scale. Similarly, we may consider nonconvex consumption possibility sets and nonconvex preferences. In such cases, the supply and demand functions <math>S^j(p), D^i(p)</math> 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 lemma#Shapley–Folkman–Starr theorem|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 <math>PPS^j, CPS^i</math> and <math>\succeq^i</math>, we define the "convexified economy" to be the same economy, except that

* <math>PPS'^j = \mathrm{Conv}(PPS^j)</math>
* <math>CPS'^i = \mathrm{Conv}(CPS^i)</math>
* <math>x \succeq'^i y</math> iff <math>\forall z \in CPS^i, y \in \mathrm{Conv}(U_+^i(z)) \implies x \in \mathrm{Conv}(U_+^i(z)) </math>.

where <math>\mathrm{Conv}</math> 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 <math>p^*</math> is an equilibrium price vector for the convexified economy, then<ref>(Starr 2011), Theorem 25.1</ref><math display="block">\begin{align}
d(D'(p^*) - S'(p^*), D(p^*) - S(p^*)) &\leq N\sqrt{L} \\
d(r, D(p^*) - S(p^*)) &\leq N\sqrt{L}
\end{align}</math>where <math>d(\cdot, \cdot)</math> is the Euclidean distance, and <math>L</math> is any upper bound on the inner radii of all <math>PPS^j, CPS^i</math> (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 <math>\{(x, 0): x \geq 0\}\cup \{(x,y): xy = 1, x > 0\}</math> 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.