# General Equilibrium for Mathematicians

I've been reading up a lot on the recent financial crisis, and central to the story is the existence of general equilibrium models in economics, say, as proven by Arrow and Debreu (and MacKenzie?). Regardless of the real-world validity of these models (they're not looking so hot these days), I'm interested in them as a purely mathematical exercise. So, that said, does anyone know of a good (preferably available online) exposition of the existence of a general equilibrium aimed at mathematicians?

It would be hard to beat Debreu's Theory of Value

http://cowles.econ.yale.edu/P/cm/m17/index.htm

a masterpiece of economic and economical exposition.

One mathematically interesting development in general equilibrium theory is the application of tools from differential topology such as Sard's Lemma. Look at Balasko's The Equilibrium Manifold (Amazon) or Mas-Colell's Theory of General Economic Equilibrium. You can get a flavour of this approach by looking at this online paper by Geanakoplos and Polemarchakis.

I think it is worth pointing out that, from a purely mathematical point of view, economic general equilibrium theory is an exercise in fixed point theory.

The same may be said of the theory of non-cooperative games. John Nash invented the solution concept now known as the Nash equilibrium in his thesis. VonNeumann dismissed Nash's result as "just a fixed point theorem" but Nash eventually received the nobel prize in economics for this work.

The mathematical setup for economic general equilibrium theory focuses on constructing what is called the "excess demand correspondence". This is derived from assumptions about how consumers and producers formulate their plans to take best advantage of the prices they observe in the market place. The excess demand correspondence associates to any vector of market prices (one price for each commodity) the convex set of vectors of aggregate excess demands (one excess demand- possibly negative -for each commodity) that will arise when consumers and producers respond to that specified price vector.

The main idea of the proof is then to find another convex set of price vectors, each of which can be interpreted as a supporting hyperplane to the given convex set of excess demands, and each of which maximizes the market value of the excess demand.

This construction then can be shown to yield an upper semi-continuous, convex set valued function from the convex set of allowable vectors of market prices to the space of is convex subsets. One then applies an appropriate fixed point theory to deduce the existence of a price vector which, because of the structure of the excess demand correspondence, has the property that the value of excess demand in each market is zero. This is the market equilibrium price vector.

• ...''just a fixed point theorem.'' I don't find this criticism very trenchant. One of the fundamental problems of mathematics is to solve equations (of various sorts). A result that proves that a system of equations has at least one solution is a good result. How good depends on how non-obvious the existence of the solution is. Maybe von Neumann's criticism was somewhat more detailed? How non-obvious was Nash's result at the time? – Stephen Aug 11 '11 at 18:38

Kim Border has lots of excellent lecture notes related to mathematical economics, including some on general equilibrium theory. The lecture notes on general equilibrium theory by Nicholas Yannelis are very extensive and introduce a lot of the related mathematics.

The original paper by Arrow and Debreu, probably the most famous paper on the existence of an general equilibrium, can be found here. It is based on a result by Debreu to be found here. Ted Bergstrom has a nice survey on how to weaken several assumptions used in many existence proofs. In particular, how to substantially weaken the rationality requirements.

Mathematicians might be interested that existence of equilibrium has also proven for some infinite dimensional models that are important for finance, growth theory and various other fields. This literature makes much use of functional analysis, especially Riesz spaces. A good survey by Mas-Colell and Zame can be found here.

A short paper I've been wrestling with, providing a perspective on how do intertemporal equilibria relate to the information that economic agents need to make decisions?

Fortnow, Killian & Pennock (2006), Betting Boolean-style: A Framework for Trading in Securities based on Logical Formulas, provides a characterisation of what a market-maker must do, information-theoretically, in order to provide a coherent pricing if they offer what the authors call Arrow-Debreu secuirties, which are bets about future prices. The authors admit in their conclusions that they don't know whether equilibria in their model matches the Arrow-Debreu notion of equilibria.

The reswitching problem that was a central focus in the Cambridge Capital Controversy (CCC) led the anti-neoclassicals to claim that paradoxes in the valuation of assets with varying interest rates meant that there was no unitary notion of value of goods used in production, and hence "capital" was an incoherent notion in mainstream economics. The mathematics of basic reswitching examples is pleasingly simple, if the interpretation is not: Vienneau (2006) A Model For Exploring Manifestations of Capital-Theoretic ‘Paradoxes’ in Temporary Equilibria introduces a two-goods, three-processes reswitching example, and shows how the model has no single long-term equilibrium due to capital reversing. The author briefly mentions the defences of general equilibrium models in the CCC of Debreu and Samuelson, and doesn't reach any conclusions, except to argue that he has provided a useful model for looking at the effect of reswitching and capital reversing on the dynamics of equilibria.

To answer your request, you could start from a master: Smale S., Dynamics in General Equilibrium Theory'', American Economic Review, Vol.66, No.2, 1976.

Debreu is certainly a superb exposition, but I think a better choice is McKenzie, for two reasons:

1) Debreu's model assumes a fixed number of firms, which means that it's implicitly a model only of short-run equilibrium. McKenzie's model (which was contemporaneous with Debreu's) allows the number of firms to vary (at the cost of assuming constant returns to scale) and is therefore a better introduction to the kind of models you seem to be interested in.

2) Debreu's book, well written as it is, dates from the 1950s. Because McKenzie's book is much more recent, it's able to offer some historical perspective that I think will be helpful to you, as well as incorporating a lot of material (e.g. uncertainty) that wasn't well worked out until more recently.

Like Debreu, McKenzie was a good and careful expositor.

I recommend the paper " An elementary core equivalence theorem" by Robert M. Anderson. It is a very short paper that contains a fairly self contained description of the basic models and concepts. You can then try to guess how to prove the existence of equilibrium based on some fixed point theorems or look at papers that Anderson quote. There are various more advanced models including models with incomplete informations ("rational expectation is a useful buzz word). There are also plenty of examples of market failure.

(I do not see why the recent financial crisis has anything to do with these models being less hot...)

Acemoglu's new textbook Introduction to Modern Economic Growth includes a very thorough and rigorous exposition of the foundations of macroeconomic theory including general equilibrium and so on.

• Just a remark, but the general equilibrium theory is considered to be a part of microeconomics, not macro, even if in a sense it is about the economy as a whole. – Joël Apr 27 '11 at 1:58
• Well, not exactly. It's true that general equilibrium theory is first encountered by students on microeconomic courses. But most modern academic and policy-oriented macro is done within some kind of general equilibrium framework. – Tom Smith May 4 '11 at 5:37