April 06, 2005
Robert Waldmann Talks About First-Order Approximations
g(x) is a valid first order approximation of another function f(x) around x0 if the limit as a goes to zero of (g(x0+a)-f(x0+a))/a is zero. This does not mean that g(x) is a good approximation... "0" is a valid first order approximation for "x squared" around [x0=0]. This does not mean that 0 is a good approximation for one million squared. However, economists, who like to play with math but don't always take it seriously, use "is true to first order" to mean "is a useful approximation to the truth"....
I recall a professor presenting a quadratic loss function saying it was arbitrary. I raised my hand and said something.... The professor said "sure but there is no way of knowing how close is close enough, The loss function could look like this (drawing)" I was chastened at the time, but pleased when he said my comment was good.
Recently reading [Mankiw], I noticed a third meaning... "according to the [simplest] neoclassical model of the phenomenon."... Note that this third usage is much further from the formal mathematical usage than the second is. No one could possibly believe that the first neoclassical model of something must be a good approximation.... The professor who wouldn't let me elide the difference was named N. Gregory Mankiw.
Ah. But, Robert, attachment to methodological principles of rational thought invariably lasts only until they begin to cause serious pain to positions taken for ideological reasons...
Posted by DeLong at April 6, 2005 07:20 PM