Additional Musings on Free Markets

Further elaboration on my previous post about free market economics and its flaws was noted.  There were a few things of note that would make the last article more clear.  Clarification on what it means for a problem to be non-convex was noted to be necessary.  It was also found that additional comments from the main cited publication should have been included, which were of importance.  Additionally, there were just more ideas to share on the topic, and what these structural issues mean.

Non-convexity in the case of economics refers to the set of preferences for a given market participant.  It means that some prices for goods support two separate optimal states.  Quoting from the Wikipedia article on non-convexity with regards to economics

For example, we can imagine that, for zoos, a lion costs as much as an eagle, and further that a zoo's budget suffices for one eagle or one lion. We can suppose also that a zoo-keeper views either animal as equally valuable. In this case, the zoo would purchase either one lion or one eagle. Of course, a contemporary zoo-keeper does not want to purchase half of an eagle and half of a lion (or a griffin)! Thus, the zoo-keeper's preferences are non-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.

The fact that two different mutually exclusive states can yield equally valuable outcomes, is not a stretch of the imagination.  It is this very property that is very troublesome for economics and the free market, because it is also what makes computing optimal decisions no better than random in the aggregate for a lot of common cases (failing you do not have an insider edge in the market in question.)  There are additional things of note about the findings of Papdimitriou¹

Firstly, it is NP-complete to distinguish between economies that have no price equilibria or in marginal prices, and those that have both.  Computing allocations that are optimal are FΔ^P₂-complete. Furthermore it is FΣ₃^P-complete to compute the optimal plan preventing defections from the economy.  Though optimal solutions exist, the inefficiencies caused by complexity equilibrium is unbounded.

The consequences of these results are that the natural state of free markets is a state of inefficiency.  Since there are infinite families of allocations, this means that polynomial time agents (agents operating within normal human time time constraints) will nearly always be stuck at inefficient allocations.

While there is mathematical tools for converting non-convex problems to approximate convex ones to obtain quasi-optimal solutions,, it is not an easy task for lay persons.  For more information on non-convexity and its role in producing many of the market conditions we observe empirically as well as providing prescriptive actions on the response of public agencies acting in the greater interest.  See  Salanié² on this issue.  The book makes a very strong case against deregulation in its contemporary promoted form.  Additionally, it points out that the solutions are not production ready for imperfect competition

At this stage of research it is very evident that the transposition of the Arrow-Debreu general equilibrium model to an economy where competition is imperfect poses too many problems that have not yet been solved.  p 138  Salanié²

Frankly these troubles will probably never be solved, since they are based upon the bounds of the physical universe and what we know about competition.  These problems place limits on what the meaning of being successful  within the market framework really represents.  Considering the large amount of uncertainty and intractability of deciding what the correct allocations of resources, courses of action, or even if the solutions exist in a given context.

Further Reading & References

¹Papdimitriou, Cahristos H., and Christopher A. Wilkens. "Economies with Non-Convex Production and Complexity Equilibria." (n.d.): 1-19. http://www.eecs.berkeley.edu/~cwilkens/pubs/PapadimitriouWilkensComplexityEquilibria.pdf. Computer Science Division, University of California at Berkeley, CA, 94720 cwilkensg@cs.berkeley.edu, christos@cs.berkeley.edu. Web. 25 Sept. 2012.

²Salanié, Bernard (2000). "7 Nonconvexities". Microeconomics of market failures (English translation of the (1998) FrenchMicroéconomie: Les défaillances du marché (Economica, Paris) ed.). Cambridge, MA: MIT Press. pp. 107–125. ISBN 0-262-19443-0, 978-0-262-19443-3.