# Recursive preference, random walk shock, stationarity

Hi,

I have difficulty in detrending variables in a DSGE with recursive preference. There’re random walk technology and AR(1) discount factor shock.

Preference:

V_t = b_t u(c_t, l_t)+ beta ( E_t (V_{t+1})^(1-alpha))^(1/(1-alpha))

where V_t is the maximized utility, b_t is the preference shock, c_t is consumption, l_t is labor, beta is a constant, alpha is another constant.

Y_t= A_t (k_t)^theta (l_t)^(1-theta) is the production function.

ln(A_t)=ln(A_{t-1}) + e_t, random walk technology.

To get BGP, I need to detrend the variables,

If u(c_t,l_t) is non-separable, e.g. u(c_t,l_t)= c_t*(1-l_t), I can detrend by V_t_new = V_t/A_t, C_t_new=C_t/A_t.

If u(u_c,l_t) is additive, e.g. u(c_t,l_t)=log(c_t)-l_t^psi/psi, I cannot detrend the system by e.g. V_t_new = V_t- log(A_t), C_t_new=C_t/A_t.

There’re many functional forms of recursive utility. I also tried
V_t = b_t u(c_t, l_t)+ beta exp( E_t log(V_{t+1}) )
and
V_t = b_t u(c_t, l_t)+ beta log ( E_t exp(V_{t+1}). The latter can be detrended by e.g. V_t_new = V_t- log(A_t), C_t_new=C_t/A_t if there’s no b_t.

Any suggestions for the additive period utility? Thanks very much!

What matters is the period utility function/felicity function. That one needs to be compatible with balanced growth. As is known from King/Plosser/Rebelo (1988), there are only two basic versions. The additive one you have is the second one. You should be able to detrend it. Please outline more clearly where your problem is. When subtracting log A_t, you should get something like

V_t-ln A_t = b_t (log(c_t/A_t)-l_t^psi/psi)+ beta log ( E_t exp(V_{t+1})-(b_t-1)*log(A_t)

where the last term is an exogenous “constant” for utility maximization that can be omitted.

My question is:

I have recursive preference with preference shock: V_t = b_t (log(c_t)-l_t^psi/psi) + beta log ( E_t exp(V_{t+1}). Other parts of the model is the standard NK model with price rigidity.

If I detrend the variables as V_new_t= V_t-log A_t, c_new_t= c_t- log A_t,
I’ll end up with V_new_t = b_t (log(c_new_t)-l_t^psi/psi) + beta log ( E_t exp(V_new_{t+1}) + (b_t -1) logA_t,
since log A_t = log A_{t-1} +e_a_t is random walk, I’ll end up with a definition of V_new_t related to a random walk variable (log A_t). However, when I detrend other equations of my model, they turns out to be a function of either e_a_t or unrelated to e_a_t and log A_t,

e.g.
Y_t =A_t l_t, detrended by Y_new_t=Y_t/A_t, I get Y_new_t= l_t.

1. I’m not sure whether I can detrend the variables in that way, and end up with an equation related to logA_t instead of e_a_t.
2. I also don’t understand “the last term is an exogenous “constant” for utility maximization that can be omitted”. I cannot omit the last term when I try to look at the IRFs for the technology shock, where I assume technology shock and preference shock are correlated thus (b_t -1) logA_t not equal to 0.

Thanks!

[quote=“jpfeifer”]What matters is the period utility function/felicity function. That one needs to be compatible with balanced growth. As is known from King/Plosser/Rebelo (1988), there are only two basic versions. The additive one you have is the second one. You should be able to detrend it. Please outline more clearly where your problem is. When subtracting log A_t, you should get something like

V_t-ln A_t = b_t (log(c_t/A_t)-l_t^psi/psi)+ beta log ( E_t exp(V_{t+1})-(b_t-1)*log(A_t)

where the last term is an exogenous “constant” for utility maximization that can be omitted.[/quote]

1. I am not sure I understand your point. What i am saying is the following:

are exogenous processes that shift the objective function, but which the agents in your model take as given. As such, when you compute the first order conditions, this term will always drop out. For this reason, you can simply use
V_new_t = b_t (log(c_new_t)-l_t^psi/psi) + beta log ( E_t exp(V_new_{t+1})
The argmax of V_t and the V_new_t will exactly be the same (that’s why I said it can be omitted).

1. When you do this, there is no A_t showing up anymore. Depending on what you are interested in, you have to add the unit root in technology back to your IRFs later on. See e.g. github.com/JohannesPfeifer/DSGE_mod/blob/master/Aguiar_Gopinath_2007/Aguiar_Gopinath_2007.mod

[quote=“jpfeifer”]1. I am not sure I understand your point. What i am saying is the following:

are exogenous processes that shift the objective function, but which the agents in your model take as given. As such, when you compute the first order conditions, this term will always drop out. For this reason, you can simply use
V_new_t = b_t (log(c_new_t)-l_t^psi/psi) + beta log ( E_t exp(V_new_{t+1})
The argmax of V_t and the V_new_t will exactly be the same (that’s why I said it can be omitted).

1. When you do this, there is no A_t showing up anymore. Depending on what you are interested in, you have to add the unit root in technology back to your IRFs later on. See e.g. github.com/JohannesPfeifer/DSGE_mod/blob/master/Aguiar_Gopinath_2007/Aguiar_Gopinath_2007.mod[/quote]

Thanks Professor Pfeifer!

I understand that V_new_t = b_t (log(c_new_t)-l_t^psi/psi) + beta log ( E_t exp(V_new_{t+1}) + (b_t -1) logA_t, when I take first order derivatives, the last term (b_t -1) logA_t doesn’t affect the conditions.

In the model, ln(A_t)=ln(A_{t-1}) + e_t is a random walk technology process. My concern is that some other equations, e.g. the definition of stochastic discount factor, have V_new_{t+1} (it will have expectation of (V_new_{t+1} twisted by parameters) and twisted expectation of V_new_{t+1}). I’ll take 3rd derivative, and these V_new_{t+1}s won’t be cancelled out. If I drop this term (b_t -1) logA_t, then IRFs of stochastic discount factor to e_t will be different from not dropping it. These IRFs are also of my interests.

If I keep this (b_t -1) logA_t, will there be any problems running dynare since it’s a random walk?

Thank you very much!

Either my argument is correct that it drops out in all FOCs or you are correct and it does not drop out, but does affect the FOCs. It cannot be both. That is mathematically impossible. What may be true is that you may be interested in some non-detrended variables. However, as the detrended variables can be easily computed, you can then recover the non-detrended variables by adding the trend back. See github.com/JohannesPfeifer/DSGE_mod/blob/master/Aguiar_Gopinath_2007/Aguiar_Gopinath_2007.mod for an example of this.