# Mathematical Complexity of Simple Economics

I ran across an old article by Donald Saari from the Notices of the AMS,
Mathematical Complexity of Simple Economics, which explains how some simple models of the economy can have arbitrarily complicated dynamics.

The basic model in economics of the economy as a whole is that of general equilibrium (GE). General equilibrium is a model of the economy where goods are traded for money which are traded for goods. It’s assumed that all of the goods are used up, and no one has money left over; prices are assumed to take on a value such that both of these things occur. It’s also assumed that in equilibrium the supply and demand of each good are exactly equal: everyone who wants to buy or sell at the current price is able to. The implicit dynamical idea is that if demand exceeds supply, then prices will go up, and if supply exceeds demand prices will go down. In this model, markets as said to clear.

It’s not easy to show that such market-clearing prices even exist. Under certain convexity assumptions, they can be shown to exist using the Brouwer fixed point theorem. The model, as stated, has no dynamics: prices achieve their equilibrium values, and that’s where the story ends. But as I mentioned above, there is an implicit dynamic story that whenever prices are too high, they adjust downwards, and when they are too low, they adjust upwards.

Saari’s article is about the difficulties that arise when taking that model of dynamics seriously. Assuming that prices adjust in proportion to how far supply and demand are apart can lead to arbitrarily complicated dynamics, even chaotic dynamics.