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!