Log-linearization and bonds which are zero in steady state


I have what is a relatively simple problem about log-linearization.

I have a NK model, which features bonds which are zero in steady state and has a Phillips curve. I want to use Dynare, so I am trying to work out what are my options.

I figure log-linearizing the entire model is out (bonds - can’t take the log of zero). I gather I can’t use a combination of log-linearization and linearization. So how then to deal with the Phillips Curve? Can I simply interpret the Phillips Curve as:

pi_t = BetaE_t((pi_t+1 - pi_bar) / pi_bar) + kappa*((mc_t-mc_bar) / mc_bar)?

It doesn’t feel right. But otherwise, how can I reconcile the use of an equation in percentage deviations from steady state and other equations in levels which feature variables with steady states of zero?

How is this commonly accommodated?

Thanks in advance.

  1. There is no reason to not perform a log-linearization for all variables except for bonds and then only do a linearization for bonds. See Remark 19 (Variables with Negative Steady States or already in Percent) in Pfeifer(2013): “A Guide to Specifying Observation Equations for the Estimation of DSGE Models” sites.google.com/site/pfeiferecon/Pfeifer_2013_Observation_Equations.pdf.
  2. If bonds must always be 0, this market clearing is typically imposed so that bonds do not even appear in the model.

You could check out Gali’s textbook. There private bonds are also in 0 net supply and the model is log-linearized. An example mod-file is on my homepage.

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Hi. I was wondering whether this generates any bias or problem since you are not performing the same operation in some equations. For instance: you will have phi_t=1/2(exp(I_t)/exp(I_t(-1))-1)^2. Since in steady state phi_t must be zero you write the code without the “exp” but i_t is a variable you would like to log-linearize so it is accompanied by the “exp”. To me, it looks inconsistent. What is the advantage of this over redefining variables? let’s say, exp(phi_t)=1+1/2(exp(I_t)/exp(I_t(-1))-1)^2, and the where-ever phi_t is on the FOC you write (exp(phi_t)-1)?

It’s not inconsistent at all. At first order, it is simply a scaling with the Jacobian, i.e. the steady state of the variable. You are expressing everything in percentages if you put it in exp(). But some variables are already in percent or a percentage of it does not make sense as the steady state is 0. In those cases, you don’t do the exp()-substitution. Also note that doing a full exp()-substitution is not recommended. See