Arrow's contributions are really too many to describe in detail (A Fine Theorem has made a start - the first of four promised posts is here), but I see four main areas:
- Social choice theory - Arrow's impossibility theorem, which we discuss in ECON110, details how there can be no 'perfect' majority-rule voting system that satisfies a set of axioms that most people would expect from a 'fair' voting system;
- General equilibrium - Arrow, along with Gerard Debreu, proved the (theoretical) existence of a market-clearing equilibrium set of prices across all markets, and that the resulting equilibrium would be efficient (this is one of the bases for economists' claims that competitive markets are efficient);
- Health economics - Arrow wrote the first paper that teased out the complexities of health care as a good (which we discuss in ECON110), including the fundamentals of asymmetric information that Spence and others built on; and
- Measurement of risk - Arrow outlined the mathematical concepts that economists use to measure risk.
Of course, he made many other contributions than those outlined above. Tim Harford has a great post here, and the New York Times obituary is here. And here is a recent (2016) interview with the great man. He will be missed.
[Update: The second post from A Fine Theorem is here]