Wednesday, September 9, 2009

how history can influence theory: a little-known angle on keynes' general theory

There is both an uplifting and depressing side to the place Keynes' General Theory has in the history of economic thought. The depressing side is easy to tell: many economists afterward used a "watered down" version of Keynes. The critics of these watered-down versions argue that Keynes' ideas were reduced to overly simplified mathematical concepts or cast against a classical backdrop that did little to change the policy implications or logical conclusions of the Keynesian model past the short run.

The uplifting side is that Keynes' General Theory really was a methodological revolution. While many economists who talk about Keynes like to criticize the second chapter for many things (it's incoherent, not an accurate description of later parts of the book on expectations, etc.) I was really impressed by that chapter when I read it for the first time. I think part of that has to do with my background in math, because he uses a very interesting analogy in that chapter which I can deeply appreciate.

He begins by describing the classical system as based on two fundamental postulates. Calling into question the second of these postulates -- the one concerning whether wage is ever equal to its marginal disutility, or what economic forces may determine it to be so -- he says the goal of the General Theory is to break this second classical postulate, and that this is analogous to the breaking of Euclid's parallel postulate in geometry.

I find this interesting because of the history of the parallel postulate in mathematical thought. From the beginning, Euclid found it the hardest to put in words: indeed, while the parallel postulate is the 5th postulate in his axiomatic system of geometry, it is the longest stated by far and it contains vague references to infinity, something none of the other axioms do. So, the first postulate is "To draw a straight line from any point to any point", etc., until you get to 5:

That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are lessthan the two right angles

The history of the parallel postulate springs from the fact that it was so difficult to put in words and, indeed, so difficult to prove! Euclid himself, offering demonstrations of his other postulates, simply assumed the postulate to be true. Essentially, since Euclid introduced his geometry mathematicians have wondered whether the parallel postulate is independent of the other axioms of Euclid's system. What this means is: can we construct logically consistent geometries that may require the first four postulates, but do not require the parallel postulate? Early arabic mathematics, which mostly concentrated on algebra and trigonometry, was also obsessed with proving the parallel postulate but never did. Later attempts were made by Italian mathematicians, and many others as the 19th century saw the most activity in the field of geometry. (Some of this information comes from Boyer's A History of Mathematics)

The result, however, was not an eventual proof. In fact, in the 19th century several mathematicians developed an entirely new field of gemoetry, appropriately named non-Euclidean geometry, which was based on using the first four postulates and breaking the parallel postulate and seeing the resulting properties of that system. One of the major contributers to this field was Riemann. Riemann is in turn important because Riemannian geometry is the basis for Einstein's theories of space-time (incidentally, also developed in the 30s when Keynes wrote the General Theory!). Einstein developed his theory of general relativity based on the idea that the universe operates on a large scale not according to the principles of Euclidean geometry, but by these non-Euclidean systems which can deal with all the geometric oddities of space.

Drawing this all back to Keynes, what this means is that Keynes saw the breaking of the second classical postulate of economics as similar to the development of a theory of general relativity which revolutionized our understanding of physics. We can debate on whether he actually accomplished this, but certainly the General Theory should be appreciated on a methodological level for its contribution to economic thought. What the implications of truly breaking an axiom of an economic system mean for the state of economic theory is a very exciting thought.

And finally, should appreciate it not because of what "New Keynesians" such as Mankiw have done to Keynes' original theory: that is, reduced it to a few assumptions of sticky factor prices in the short run while maintaining the classical long run assumptions.


  1. A nit-picky point about space. It does not have "geometric oddities" The fact that space time is curved is not an oddity unless you "know" that straight parallel lines cannot converge, and you know that space is flat, or the 3d equivalent of flat, straight.
    Under modern understanding of space time Euclidean geometry does hold up....None the less....Go economics! Fuck Mankiw!

  2. I know this isn't the greatest source but according to wikipedia's entry on non-Euclidean geometry space is mostly Euclidean but near points of matter, curvature is apparent which can only be described holistically according to Riemannian geometry. This recalls the common "rubber sheet" analogy to space-time.