Prof I have solved Mendoza’s model analytically in terms of the parameters that I am trying now to calibrate the parameters. One of the things that confuses me in Mendoza’s paper on the calibration part where he mentions: “The calibration is normalized by setting Y^T=1 and P_N =1.”
where: P_N is relative prices of non-tradable to tradable consumption.
As you have previously explained that introducing a normalization constant requires a linear homogeneity function and then estimation of the normalizing constant (similar to normalizing L and letting dis-utility parameter absorb the normalizing constant). Similarly, the case where “The calibration is normalized by setting Y^T=1 and P_N =1.” then I should assume that the author introduced normalization constants throughout the calibration stage in order to set P_N=1 ?
When we normalize to 1, it means total available endowment of a certain factor is 1. In the case of P_N =1 as a normalization it means a reference/benchmark rather than a maximum endowment ?
Solving analytically in terms parameters is a lousy way solve the model analytically ? Do you think I should have solved simultaneously for both parameters and variables ?
Thank you prof. , it is Mendoza 2002, P 23 (attached):
It would indeed facilitates the computation of steady state values. However, my concern is when I would able to normalize to 1. My understanding is that in order to normalize a variable then:
the equation on in which I will introduce a normalization in must be a homogeneity function and:
There must be a parameter that is calculated within the equation in order to achieve that normalization (similar to dis-utility parameter in the case of labor).
Is my understanding right prof ? If either 1 or 2 is not met then my normalization then I am not on the right track.
In that model, there are many degrees of freedom. Y^T,K are exogenously given and can be normalized as desired. My hunch is that p_N can be fixed because you also need to pin down A.
My hunch is yes. But I haven’t tried to compute the steady state.
Not necessarily. Models with linear homogeneity allow for normalization due to having dimensional constants that allow you to easily work in other units. But you can have normalizing constants also in other model even without linear homogeneity.
First I want to confirm that your hunch was right and the model has been solved by normalizing price by setting A. Many thanks, indeed.
Second, attached the solved model. I got an error “Unable to perform assignment because the left and right sides have a different number of elements.” After the model has been solved. Any advice please ?
One more thing please, regarding BK condition, I got 6 Eigenvalues larger than 1 for 5 forward variables. I suspect that it is the definition of the Infinite sum in eq.16 which is an embedded part of the model to define Uc given an endogenous discount factor and therefore I can skip it and move on, do you agree on that ?
D_t is current period debt stock that is chosen by agents given D_{t-1}. It is an endogenous state similar to capital stock convention. Since it depends on current realizations of states including D_{t-1} then my understanding that D_t is a Control variable which agents choose via maximizing Utility given a set states.
Note: In the original paper of Mendoza (2002) and in Schmitt-Uribe (2002) alike. Current period’s debt is referred to as D_{t+1} and last period (beginning of current period) is D_t. But given that Dynare works with end of period notation then I lagged D_{t+1} and D_{t} to preserve timing convention, Am I right here ?
In terms of S (Infinite Sum) definition:
This equation defines an infinite sum of discounted period utilities, which enters the U_c definition. The inifinte sum arises due to the presence of an endogenous discount rate B_{t}(C_t,N_t)) in utility specification.
Note: this equation is used come up with Euler equation used in Mendoza paper and I have checked that it will lead to the same Euler condition in Mendoza(2002) paper.
seems to have the timing of e.g. SGU and not the one Dynare would use.
2. What I meant is that it is unusual that a variable at time t that only depends on stuff in the future, i.e., a recursive variable being purely forward-looking. But I haven’t looked at the details.
Thank you for the remarks. I corrected this typo of timing for D in the BC (attached)
. However, the same issue still persists.
Is the error message Unable to perform assignment because the left and right sides have a different number of elements. is due to BK not satisfied ?
Any advise on how to carry on ?
The Uc_{t} is the lifetime utility. In principle, and generally speaking, isn’t the typical way to define an infinite sum like
:
One more thing, I have updated Dynare to version 6 and the error message disappeared, however, the BK condition not met and model_diagnostics points to a collinearity problem between the below variables. Do you agree on how to define a recursive sum as I define it in the latter response (which is also similar to what is explained in the Dynare Manual) if not then how to define U_c then ?
Collinear variables:
lambda
Uc
S
Collinear equations
16 17 21
Prof @jpfeifer I think I figure it out. I have defined Beta as a recursive sum while it is a finite sum. This is a grave mistake from my side. I will re-define it and revert if needed.
The \beta_{t}= \sum_{\tau=0}^{t-1} exp^{-v_t} where v_t = log (1+C_t-N_t^\delta/\delta)^{-\psi} is the endogenous discount factor used in Mendoza (2002). (I missed the t subscript for beta, sorry for the typo)
In the Euler equation of Mendoza paper: U_c= exp^{-v_t}*R*U_c(+1), \beta_{t} does not appear, but when I define U_c, the latter hinges on \beta_{t} which is I define it in my code in Dynare (attached) as \beta_t= \beta_{t-1} exp(-v_{t-1}). This term is clearly non-stationary as it change at different t and this is the cause of BK not being satisfied (please correct me if I am wrong). Do you agree on why BK is not being satisfied ?
If the above is true, then one solution is to follow Schmitt-Urbine (2003) methodology (the paper you have replicated) by taking the endogenous discount factor \theta_{t+1}, where \theta_{t+1}=\theta_t*B_t(C_t,N_t) as a constraint then the endogenous discount factor disappears from FOC and this is what you meant by " Only \beta_{t+1}/\beta_t will appear" rather than \beta_t. Is that a solution for the non-stationary issue of discount factor ?
SGU approach has solved the indeterminacy problem. The BK condition is now satisfied. However, I am trying to shock the model to induce the occasionally binding constraint (where D > = -\phi GDP) to bind with \phi=2 given Debt is -4 and GDP is 2.2 in the steady state (non-binding case), Dynare is unable to converge although I have increased simul_check_ahead_periods to 1000. when \phi=9 i.e. occ binding constraint never binds, I was able to get simulation results
Note: When Debt and -\phi GDP are two is negative values, then for debt to be higher in magnitude than -\phi GDP I should I switch the inequality ?
