question about welfare calculation in a JME paper

[zhanshuo]

Dear all,
I am confused on the calculation method in the paper of “Capital controls and optimal Chinese monetary policy“， which is published on JME 2015.
Here I attach the DYNARE CODE and the paper downloaded from the author's website.
And the key part of the code is below:

planner_objective(C + log(C_ss) - (Phi_l*L_ss^(1+eta)/(1+eta))*exp((1+eta)*L));
ramsey_policy(planner_discount=1, nograph, noprint, irf=20, periods=1000, instruments=(R));
welf = -(1/(1-beta))*C_ss*Phi_l*(eta/2)*L_ss^(eta-1)*oo_.var(2,2);

Here comes by question:
Since in they write their model in log_liearizing way, why they define welfare in a recursive way which Prof. Pleifer said should only be used in no-linear model? And what is the relationship between a recursively defined welfare function in the paper and the welfare calculation equation in the MOD file, which is "welf = -(1/(1-beta))*C_ss*Phi_l*(eta/2)*L_ss^(eta-1)*oo_.var(2,2);". And the latter seemingly only cares about the variance of labor.

Thanks a lot.

[zhanshuo]

Call for help, thanks!

[jpfeifer]

That is hard to tell. I haven't read the paper, but they are explicit about the limitations of their approach in footnote 9. Using a recursive definition of utility is always fine, the problem is that the constraints are linearized, which is not generally correct.

With respect to welfare only depending on the variance of labor, it might be some algebraic transformation that gives rise to this expression. You might need to ask the authors.