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Monday, April 04, 2005

The Third Meaning of "First Order"

Formally a function 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. For example 0 is a valid first order approximation for x squared around zero. 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". Similarly, economists use "second order" to mean economically insignificant.

This is not how the concepts are presented in first year graduate courses (I took only one undergraduate economics course so I don't know about them). Instead it is stressed that nth order approximations are valid only locally. I recall a professor presenting a quadratic loss function saying it was arbitrary. I raised my hand and said something like "Over in the physics department they have an argument which they admit is bogus. It is that, at a maximum, the derivative is zero so the second order Taylor series approximation to any smooth function is proportional to the square of the distance to the x which maximimizes (note I hadn't heard of the notation argmax). The professor said "sure but there is no way of knowing how close is close enough, The loss function could looke like this (drawing)" I was chastened at the time, but pleased when he said my comment was good.

Recently reading this, I noticed a third meaning of "first order" which is "according to the first neoclassical model of the phenomenon (that is the first model which featured rational optimization)." 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 rules of theoretical debate among economists often require great deference to simplifying assumptions which no one has ever believed to be approximately correct. Roughly, the rule seems to be "according first neoclassical model" must be accepted as meaning "true to first order" which much be interpreted as "a useful approximation." I really don't know why.

The professor who wouldn't let me elide the difference was named N. Gregory Mankiw.

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