Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: What's A "Market"?, published by johnswentworth on August 9, 2023 on LessWrong.

Economists have a very mathematically clean class of models of "markets", and spill lots of ink arguing about how well this class of models applies to the markets of the real-world economy. I personally give relatively few shits about how well the mathematical notion of a market applies to real-world economic markets; I'm relatively more interested in applying the same models to systems in biology or ML/AI. They're very generalizable models.

Unfortunately, the mathematical notion of a "market" tends to be presented in math-heavy econ courses, and the parts I'd consider most central typically see surprisingly little coverage in more conceptual intro courses. So, this post aims to explain what I consider the central concepts of the mathematical notion of a market, without all the associated notation and jargon and proofs, in a way which lends itself to generalization beyond economics.

The Story About Apples And Bananas

We've got two people, Alice and Bob. Each of them can produce two goods, apples and bananas. Alice can use her land to produce five tons of apples, or one ton of bananas, or some proportional combination of the two. Bob can use his land to produce five tons of apples or twenty tons of bananas, or some proportional combination of the two. Both want a varied diet of apples and bananas.

. and you remember from econ 101 roughly how this goes, right? If the two just produce food for themselves separately, then each grows a mix of apples and bananas. But then Alice' opportunity cost for one apple is 1/5 = 0.2 tons of bananas, whereas Bob's opportunity cost for one apple is 20/5 = 4 tons bananas. So, the two could produce a pareto gain of apples and bananas by specializing: Alice can specialize more toward apple production, Bob can specialize more towards banana production. For instance, if Alice shifts production toward 1 more ton of apples, while Bob shifts production toward 1 less ton of apples, then together they produce - 0.21 + 41 = 3.8 tons more bananas with the same number of apples.

Now the key question for this post: when does this sort of specialization reach equilibrium? Under what conditions do Alice and Bob together decide that they've both specialized the correct amount, and don't need to shift their production around any more?

In this exact example, they'll only hit equilibrium once one of them is fully specialized - either Bob fully specialized in apples, or Alice fully specialized in bananas. Otherwise, they could always do better by specializing more. But in general, decreasing marginal returns might mean that both should be less-than-fully specialized - e.g. maybe both have some land better suited to apples and some better suited to bananas, so as they shift production their opportunity costs change.

So when will the two "reach equilibrium"? Well, when their opportunity costs are the same - i.e. when they have the same tradeoff between producing apples vs bananas.

. and that's a market.

More generally, we have:

A bunch of agents, and a bunch of goods.

Each agent has their own opportunity cost for each good, or marginal trade-off rate between goods.

At equilibrium, the trade-off rates are the same for all agents (otherwise they can achieve a pareto improvement by specializing more).

The "market" is the set of agents at equilibrium, and the "market prices" are the (shared) trade-off rates between goods.

Another Story: Stock Traders

We have a bunch of stock traders, each with a portfolio of stocks and cash, and a utility function over their portfolio which they seek to maximize. (We'll assume, for simplicity, that the traders are not updating their beliefs over the course of this story, so we can ignore the "expectation" part of their expected utility maximization.)...