Economies with non-convex production and complexity equilibria.

ABSTRACTThe convexity assumptions required for the Arrow-Debreu theorem are reasonable and realistic for preferences; however, they are highly problematic for production because they rule out economies of scale. We take a complexity-theoretic look at economies with non-convex production. It is known that in such markets equilibrium prices may not exist; we show that it is an intractable problem to achieve Pareto efficiency, the fundamental objective achieved by equilibrium prices. The same is true for core efficiency or any one of an array of concepts of stability, with the degree of intractability ranging from F Δ2P-completeness to PSPACE-hardness. We also identify a novel phenomenon that we call complexity equilibrium in which agents quiesce, not because there is no way for any one of group of them to improve their situation, but because discovering the changes necessary for (individual or group) improvement is intractable. In fact, we exhibit a somewhat natural distribution of economies that has an average-case hard complexity equilibrium.