# Derivation of stationary equilibrium

I am currently considering a DSGE model that includes multiple variables with trends. It is relatively easy to transform the original model into a stationary equilibrium model when the DSGE model only has one variable with a trend. However, I am facing difficulties and lack a generalized approach when it comes to models with multiple variables containing trends. I have read several papers, but they only provide the form of auxiliary variables and the model equations after detrending, without offering any guidance on how to transform the original model equations into stationary equilibrium equations. I have no idea how to derive and obtain a stationary equilibrium and I donâ€™t know where to start. I would like to ask about your general approach to deriving stationary equilibrium for models with multiple trend variables. Are there any specific notes or papers that provide guidance on the derivation process?

Did you already have a look at https://www.sas.upenn.edu/~jesusfv/benchmark_DSGE.pdf ?

I have just read your note, and it is similar to the academic papers I have come across regarding the derivation of stationary equilibrium. However, it does not provide specific guidance on the methodology. I am particularly interested in understanding the general and specific approach to deriving stationary equilibrium. For example, which equation should we generally start with? Why do we construct detrended variables in a similar way as shown in the papers? How can we determine if the chosen detrended variables make the system of equations stationary? Additionally, I would like to know how to derive the specific form of the removed trend. While I find these questions manageable when there is only one trend, I tend to worry about the correctness of my transformation method when there are multiple trends.

This is tricky and has AFAIK not been written down in a pedagogical way. Usually, you start from the primitives, i.e. technology trend components in the production function and then work out which variables are affected by which trend. Usually, you know that there is composite growth rate coming from the combined trends that e.g. output inherits. The only challenge in the end is to sort out the trends of the unusual variables like Lagrange multipliers.