A (pedagogical) note on the log-linearization of functions of several variables

Dynamic stochastic general equilibrium (DSGE) models are analytically intractable and numerical methods must be used to approximate a solution. A key input shared by many solution methods is log-linearization. While the basics of the procedure have been extensively documented, applying the methodology to complicated functions of model variables remains uncharted territory, often resulting in cumbersome and error-prone calculations. This paper offers a procedure – the log-linear product approach – that automates and simplifies this task, as I show with a full working example. The procedure relies on the basic fact that the product of second order terms is zero when dealing with a linear expansion.