My understanding that Dynare, by default, always linearize FOC equations. If my model is in levels (Whether model is linear or Non-linear) then results are expressed as “absolute deviation”. If my variables are already in log-form (Whether model is linear or Non-linear) then results interpreted as “log deviation from SS”, Else I have to use the log linear option if variables are not in log scale to come up with log deviation variables. Do you agree ?
If I want to use the results of the above question to set SS values equal to 0 to elude the calculation of steady state values, then either I do the following:a. I log-linearize a Non-linear model by hand and impose SS values = 0 (log deviation from steady state equals 0) using linear option.b. Use the loglinear option and impose SS values = 0.c. However, when the model is non-linear in levels, then imposing SS values of the endogenous variables equals to zero (i.e. variables will be after linearization expressed as absolute deviations and are equal 0s), without using any option, then Dynare is unable to find the SS. Why ?
Only the manual linearization is feasible in that case. You need to clearly distinguish the steady states of the original level variables (which are generally non-zero) and the ones of the deviations from steady state (which are 0). Only the manual linearization may not require the steady state of the original level variables.
@jpfeifer I am referring to my reply here. The reply deleted below was by mistake Sir.
Thank you Prof. many thanks.
Regarding 2.: This means in order to use steady state equal to zero then I always have to write my model in dynare in linear form (linear or log linear) after I linearize the model by hand ?
Loglinearizing option is meant to express the variables in log but does not always lead to a log linear form of FOC unless the model is purely multiplicative or purely additive (and given variables in levels are not negative) Right ?
If the model’s equations have an equation of the following form Y = KL + C then manual log-linearization would leads to:
0 = (K_ss * L_ss)/Y_ss * (1+ k + l - y) + C_ss/Y_ss(1+ c - y)
Where _ss denotes Steady state value in level and lower case (k, l, y, c) denotes log deviation from steady state. Therefore, in this case, I have to declare all variables (Steady State in level variables and log deviations from steady state) and assign steady state values for all variables whether they are levels as K_ss or in log deviations from SS (which are zeros at SS) Is my understanding correct ?
The aforementioned case necessitates manual log linearization, however if FOC of the model are purely multiplicative such as Y=KL and/or purely additive such Y = C + I then I still need to do loglinearization manually if I am after imposing zeros for steady state values of the log deviations variables ?
For Y_{t} = K_{t}L_{t}+C_{t} in log-deviation you obtain y_{t} = \frac{KL}{Y}\left(k_{t}+l_{t}\right)+\frac{C}{Y}c_{t}
where lowercase means x_{t} = \ln(X_{t}/X) so that in steady state with X_{t} = X we have x_{t} = \ln (1) = 0.
Therefore, when the expression contains additive elements your log-linear approximation will contain steady state values.
For Y = KL we have y_{t} = \frac{KL}{Y}\left(k_{t}+l_{t}\right) = k_{t}+l_{t}
since KL = Y
But for your second expression we use the approach in point 1 above: y_{t} = \frac{C}{Y}c_{t}+\frac{I}{Y}i_{t} (note that Y = C+I so if you know two of these values in steady state, you know the third one).
If you forget how to compute the log-linear approximation just remember the approach in Obstfeld and Rogoff (1996): totally differentiate an equation and then use x_{t} = dX_{t}/X.
Thank you for your reply. Please note that I am using Uhlig method whereby X= X_ss (1+ x) where x = log(X/X_ss). After I simplified my expression above I have attained to the same expression you have mentioned.
Let me rewrite my question based on your reply please:
If FOC are multiplicative, then loglinearization will lead to variables in log deviations from SS (without any variables in steady state level). In this case, loglinearization by hand can be substituted by Dynare methods such as -loglinearize- option or writing variables in exp() and let Dynare do the job and, meanwhile, assign Steady State variables equal to zero for solving for Steady State. Do you agree ?
No, in the purely multiplicative case, you can work with steady state 0 variables if you enter the linearized equations. Whenever you enter nonlinear equations in Dynare, you need to provide the steady state values of the variables in question (which are generally not zero). That applies to both using an exp() substitution and the loglinear option.