Christiano, Motto, Rostagno FOC's

Good morning everybody,
I am trying to come up with the log linear version of CMR DSGE model with financial friction; in particular I am not understanding how to go trough the log linearization of functions of the cut-off value delimiting the mass of bankrupt entrepreneurs. Here you find attacched the combined FOCs (E.2 and E.3 in CMR 2010); how can I approch the log linearization of GAMMA function and G function;
Thancs everybody for the gentle support
cmr_focs.pdf (69.3 KB)

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Hello MarcoUrbani

the equations involve partial derivatives of standard cumulative probability density

eqs_cmr.pdf (82.7 KB)

good luck

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Thanks for the kind support Aldo,
just a couple clarifications:

  1. can’t get why you have multiplicative omegabar(steady state) and omegabar(hat) in the passage between equation 4 and equation 5; shouldn’t I mantain the difference?
  2. as regards the linearization of equation 6, I should proceed by linearizing the expression and then plug in the linearized F and G? here you find attacched a continued version of the previous pdf what I mean in point 2
    cmr_focs.pdf (74.4 KB)
    Thanks so much
  1. Differences \bar{\omega}_{t+1} - \bar{\omega} are not maintained because those would be linearizations and what is wanted are log linearizations \hat{\bar{\omega}}_{t+1} = (\bar{\omega}_{t+1} - \bar{\omega})/\bar{\omega} \approx \ln \bar{\omega}_{t+1} - \ln \bar{\omega}. There is an additional step
    step5.pdf (83.2 KB)

  2. Yes, that would be an appropriate approach

Thanks Aldo,
I have come up with the following log-linearization of the entire FOC. This implies log linearizing first partial derivatives of GAMMA and G. I provide an expression for the log linear G’. To log linearize GAMMA, though, I can’t get how to express log \hat{F}_{t}
cmr_focs.pdf (78.3 KB)

Dear Marco
I attach an expression for \hat{F}_{t}
step5_f.pdf (94.1 KB)

Thank you so much,
basically I have to calibrate CMR model so I have computed all the necessary derivatives: if I want to focus just on the financial accelerator channel (ie. I reduce balance sheet structure of the bank by just including loan to entrepreneurs on the asset side and deposits on the liabilities side), is it correct to claim that calibrating the model just reduces to choose the appropriate value for \bar{\omega} (ie.cut-off value), sigma shock which is the variance of the lognormal, return on capital (at the steady state) and return on deposits(at the steady states)?
cmr_focs.pdf (91.4 KB)