How to do a calbiration?

I am working with the New Keynesian DSGE model under Rotemberg’s price setting. After solving the firm’s problem and log linearizing around a zero-inflation steady state, I end up with the following inflation equation. π_t= βE_(t){π_(t+1)}+(ε-1)/γ〖mc〗_t

Given the duration that a price remains unchanged is 5 months and the annual nominal interest rate is 4 percent on average. The average markup is 10 percent. The model period is one quarter. I want to calibrate, the discount factor β, the elasticity of substitution; and the Rotemberg parameter.

Any kind of guidance is greatly appreciated! Thank you.

In a quarterly model with a zero inflation steady state,

  1. a four percent interest rate implies \beta\approx 0.99
  2. a ten percent markup implies \varepsilon=11 as the gross steady markup is \frac{\epsilon}{\epsilon-1}
  3. five months average price duration is 1.6667 quarters. With \theta=0.4, the average price duration 1/(1-\theta)=1.6667
  4. You need to make the slope of the PC equal, i.e. \frac{\varepsilon-1}{\gamma}=\frac{(1-\theta)(1-\beta\theta)}{\theta} and therefore \gamma =\frac{(\epsilon-1)\theta}{(1-\theta)(1-\beta\theta)} which should be 11.0375



Many thanks, @Jpfeifer for your help. I have a couple of follow-up clarifications. One, from where did we get theta=0.4? Two, as to my understanding, beta is the inverse of the interest rate1/0.04. if so β=25. Adding more, If I want to solve the log linearized model in Dynare after introducing a shock on the discount factor, what steps do I need to follow. The discount factor follows AR(1) with a correlation coefficient of 0.9 on a quarterly basis. I have also parameter values for σ, φ.

I am sorry if my questions are elementary. Just I am a bignner:).

  1. As written above, the average price duration in the Calvo model is 1/(1-\theta). You can solve this for \theta if you know the duration (5/3 quarters in your case)
  2. No, \beta=\frac{1}{1+r}. So with 4% annually, you would have \beta=\left(\frac{1}{1+0.04}\right)^{1/4}\approx 0.99
  3. Regarding a preference shock, take a look at DSGE_mod/Gali_2015_chapter_3.mod at master · JohannesPfeifer/DSGE_mod · GitHub where z is the preference/discount factor shock.

Many thanks, Professor Pfeifer.
It’s clear now. About the preference shock, I went through the given mod file and get some ideas. As I do not have the 2nd edition of Gali’s book it is a bit hard to follow the code and get a full understanding though. Is it possible to get the book, please?