Economist Paul Krugman wrote
September 11, 2009, 4:48 am
Mathematics and economics
I’ve been getting some comments from people who think my magazine piece was an attack on the use of mathematics in economics. It wasn’t.
Contrary to the common assumption, mathematics does not tell you that 2 + 2 is always 4. That is true only under specific circumstances. Mathematicians are quite happy to consider entirely different circumstances, such as an arithmetic that goes in a circle with no 4 in it, where you count 0 1 2 0 1 2 0...so that 2 + 2 is 1. Or military time, which also goes around in a circle. For soldiers, 18:00 + 6:00 is 0-dark-hundred. Similarly in economics, we cannot prove Eternal Verities. We can prove that things will happen under certain circumstances, and woe betide any who forget (or worse, refuse) to verify and abide by the circumstances. That's how you blow up Space Shuttles and melt down economies.
Comments on this piece are closed on the site, so I am putting my comments here.
Krugman continued (and I added links)
Math in economics can be extremely useful. I should know! Most of my own work over the years has relied on sometimes finicky math — I spent quite a few years of my life doing tricks with constant-elasticity-of-substitution utility functions. And the mathematical grinding served an essential function — that of clarifying thought. In the economic geography stuff, for example, I started with some vague ideas; it wasn’t until I’d managed to write down full models that the ideas came clear. After the math I was able to express most of those ideas in plain English, but it really took the math to get there, and you still can’t quite get it all without the equations.
What I objected to in the mag article was the tendency to identify good math with good work. CAPM [Capital Asset Pricing Model] is a beautiful model; that doesn’t mean it’s right. The math of real business cycle models is much more elegant than that of New Keynesian models, let alone the kind of models that make room for crises like the one we’re in; that makes RBC models seductive, but it doesn’t make them any less silly.
And conversely, you can have great work in economics with little or no math. I can’t pull up papers now, but as I recall, Akerlof’s market for lemons [The Market for Lemons: Quality Uncertainty and the Market Mechanism] had virtually no explicit math in its main exposition; yet it was transformative in its insight.
So by all means let’s have math in economics — but as our servant, not our master.
Contrary to what is commonly taught in schools, math is not calculation of Right Answers or proofs of eternal verities. It is the derivation of consequences from premises (axioms and postulates). Anybody who forgets the premises, the preconditions for whatever success we might derive, is a fool or a charlatan. I will not speculate here on which of the possibilities is correct for Milton Friedman, Supply-Sider Arthur Laffer, or Ronald Reagan, except to note that George H. W. Bush was correct in characterizing their work as Voodoo Economics (until Reagan bought him off with the invitation to be his running mate, and Bush shut up permanently.) We humans are far too good at becoming both fools and charlatans, and at believing them.
I read Gerard Debreu's completely mathematical Theory of Value last year in pursuit of a moral and political principle which I had found earlier in Eugene Walras and others in the General Equilibrium tradition. General Equilibrium Theory posits certain conditions, given the conventional name Perfect Competition, and from them deduces the existence of a market equilibrium with certain optimal properties.
In particular, every product and service in the market clears. The market finds the price at which the amount offered for sale and the amount sought for purchase (supply and demand) are equal. In particular, in Perfect Competition there can be no unwanted unemployment. Furthermore, at the equilibrium, nobody can make a profit on any available trade, so that nobody can be made better off without making someone else worse off (Pareto Optimum). In other words, we can reduce Adam Smith's Invisible Hand and Milton Friedman's Magic of Markets to a specific set of requirements. From this we can draw two conclusions.
- If we want to make any of our models work at all well, we have to provide better approximations to Perfect Competition. For example, the Internet gives us unparalleled access to economic and financial information, a requirement of the theory. So we must make the Internet available to everybody on Earth, on the principle that it isn't a Free Market for you if you aren't in it. Contrary to Market Fundamentalism, the conditions for the proper functioning of a market cannot themselves be provided by the market. We must consider how they can be provided, given that governments are quite bad at doing that when corporate money dominates the political process.
- Although a market equilibrium is a Pareto optimum, which has some nice properties, the theory makes no choice among all possible Pareto optima. Any distribution of assets and income is possible, from equality to feudal aristocracy + serfdom. Any such outcome can be the result of some possible starting state of ownership of factors of production, including land, buildings, equipment, raw materials, and skills. If we have any reasonable preferences about the outcomes, such as preventing the formation of monopolies and the breaking down of the market, we have to think about how to restrict the possible states of the market. Governments are also not very good at antitrust, but something is required.
At this point, we can put the math aside and have a discussion about Political Economy, which is where we began before the pretense of making Economics into as pure a science as physics—What do we want, and how can we get it? Or if we can't get what we want most, then what else do we want instead? If that works out, we will be able to go back and forth between the math and the human elements, gaining greater insight at each step and coming closer to solutions desired by most of the population. Instead of the current process of generating more heat than light, of seeking to obfuscate, delay, and defeat rather than to inform and seek solutions, or of outright lying, of selling the souls of economists to Mammon at a bulk discount.