SOME great economists are Aristotelians, discerning the logic of markets from tangible examples around them. Others are Platonists, using their powers of reasoning to grasp ideal economic forms, of which actually existing markets are but flickering shadows. Kenneth Arrow, who died on February 21st aged 95, was both. His ideas gave economics some of its most compelling abstractions and most fruitful applications.

The abstractions won him the Nobel prize at the age of 51. (He remains the youngest winner and the most cited by others in their prize lectures.) He established the conditions under which prices might successfully co-ordinate production and exchange, eliminating shortages and surpluses. Adam Smith provided the best metaphor for this underappreciated feat: the “invisible hand”, guiding resources to their best uses. Ken Arrow and his co-author, Gérard Debreu, provided the best algebra.

To economists versed in mathematics, a well co-ordinated economy is like a system of simultaneous equations, which all hold true at the same time. The solution to these equations is a set of prices that equates demand and supply for scarce commodities in every market, including the market for labour and capital. Earlier economists had breezily assumed that such a solution existed, making their case with “cheerful prose and appeals to common sense”, as E. Roy Weintraub of Duke University has put it. Mr Arrow and Mr Debreu spelled out precisely when that good cheer was justified.

Mr Arrow showed similar rigour in exploring one alternative to market co-ordination: collective decision-making. A colleague studying America’s strategic contest with the Soviet Union had asked him whether it was safe to treat an entire country as an individual “player”, with coherent preferences. What was required, Mr Arrow knew, was a robust, reasonable rule to translate the preferences of Americans, say, into the preferences of America. But to his surprise, he discovered that such a rule was “impossible” to find. “Most systems are not going to work badly all of the time,” he said. “All I proved is that all can work badly at times.”

Together, these two achievements showed when markets could work, and why collective decision-making could fail. Given these intellectual preoccupations, you might assume Mr Arrow was a man of the right. But the opposite was the case.

Born in New York in 1921, he remembered the “gasping struggles” of relatives during the Depression. He was struck by the paradoxical coexistence of unmet needs and unused resources, a simultaneous equation that prices failed to solve. The son of Jewish immigrants from Romania (his last name and “olive complexion” led an acquaintance to assume he was native American), he attended City University of New York, “the Harvard of the Proletariat”. Unlike many of his peers, he rejected Marxism early (put off by the horrors of Stalin’s 1930s show trials—as well as the inadequacies of the labour theory of value), but socialism rather late.

Precisely because he knew the conditions required for markets to work, he understood the ways they could fall short. In economics, the future impinges on the present; what might happen has an effect on what does. So to co-ordinate the economy seamlessly, markets need an impossible reach: they must price all the goods on offer today, all that will be on offer in the future, and all that might be on offer, if contingencies arise. In the absence of full insurance and futures markets, the state could do more to share risks and co-ordinate investments, he suggested in 1978. In fact, the state retreated in the decades that followed and markets expanded, creating derivatives partly inspired by his work.

A different market failure became clear when he trained as an actuary: buyers of insurance often know more about their condition and behaviour than the seller. To cover its risks, an insurer might raise premiums, but that will only drive away the safest customers, leaving an “adverse selection” of the riskiest buyers. These insights helped him write one of the founding articles of health economics in 1963. They also help explain why the Obamacare mandate is so hard to replace in 2017.

Fortunately for economics, Mr Arrow abandoned a career as an actuary, because there was “no music in it”. He was, famously, a polymath, steeped in philosophy and literature, who once held his own at a dinner party with a scholar of Chinese art. He spent a decade at Harvard, which he chose over MIT because of its strength in the humanities, and the bulk of his career at Stanford University in California, where “we plan and build on ground that may open beneath us”. Like his brother-in-law, Paul Samuelson (whom he once compared to Humphrey Chimpden Earwicker, the protagonist of James Joyce’s “Finnegan’s Wake”), he popped up in different places and guises, offering insights into prediction markets, learning-by-doing, antimalarial drugs, discrimination between the races, equality between the generations, petrol-price controls, arms reduction, advertising, public investment, the “carrying capacity” of the Earth and the cost-effectiveness of airframes.

Systematically agnostic

Whatever his political sympathies, he never had the certitude required for activism. He once called himself an “agnostic” in his beliefs, if a “systematiser” in his talents. Keenly aware that not everything could be known, he wanted what could be grasped to be known as systematically as possible. He summed up his vision in the words of the mathematician Hermann Weyl: “If the transcendental is accessible to us only through the medium of images and symbols, let the symbols at least be as distinct and unambiguous as mathematics will permit.” Or to put it in his terms, we should plan and build as solidly as we can, even if the intellectual ground may occasionally open up beneath us.

Visit our Free exchange economics blog