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On Euler Equations

From Quantitative Easing by Carola Binder:

Noah Smith calls the Euler Equation "The Equation at the Core of Macro." He explains, "For the uninitiated, the Consumption Euler (pronounced "oiler") Equation is sort of like the Flux Capacitor that powers all modern 'DSGE' macro models." He adds:

"This equation underlies every DSGE model you'll ever see, and drives much of modern macro's idea of how the economy works. So why is Eichenbaum, one of the deans of modern macro, pooh-poohing it?

Simple: Because it doesn't fit the data. The thing is, we can measure people's consumption, and we can measure interest rates. If we make an assumption about people's preferences, we can just go see if the Euler Equation is right or not!"

Noah points to a 2006 paper that uses US data to calculate the interest rate implied by the Euler equation under various assumptions about preferences. The authors compare this Euler equation rate with a money market rate, and find that the two series differ significantly. Noah writes:

"If this paper is right - and one paper is not enough to be conclusive - then essentially all modern DSGE-type macro models currently in use are suspect. The consumption Euler Equation is an important part of nearly any such model, and if it's just wrong, it's hard to see how those models will work... We need a whole literature of papers analyzing whether the Euler Equation is usable or not."

I just want to make the point that Euler equations are not only part of "modern DSGE-type macro models," and that we do have a pretty big empirical literature on the Euler equation. Here are lecture notes from a microeconomics course deriving the Euler equation.

DSGE stands for dynamic stochastic general equilibrium. It is really the D of DSGE that brings in the Euler equations. Optimization problems--which are what a lot of economists and even non-economists solve-- whether deterministic or stochastic, whether partial or general equilibrium, havefirst-order conditions. Dynamic optimization problems have more than one time period, so you can have first-order conditions in consumption in different time periods. Combine these first-order conditions in consumption with a budget constraint and you get an Euler equation--whether or not you are writing down a DSGE model. The most obvious ways to get rid of the Euler equation are to get rid of optimization (a la Campbell and Mankiw 1991), add other constraints (liquidity constraints, incomplete markets), or pick weird preference relations that make utility non-time-separable (a la Campbell Cochrane 1999). 

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