Quadratic expression for optimal discretionary policy


I am trying to solve for the optimal discretionary monetary policy using a microfounded welfare function. The objective function is

u(C_t) - v(L_t) = - \frac{\sigma}{2}C^{1-\sigma}(c_t - \frac{1}{\sigma})^2 - \frac{\psi}{2}L^{1+\psi}(\ell_t + \frac{1}{\psi})^2 + t.i.p.

I can use this for the optimal Ramsey policy, but when solving for the optimal discretionary policy, dynare tells me “discretionary_policy: the objective function must have zero first order derivatives.”

Is it possible to just use auxiliary variables to fix this? (i.e. use \tilde{c}_t = c_t - \frac{1}{\sigma} and \tilde{\ell}_t = \ell_t + \frac{1}{\psi} so that the objective function is - \frac{\sigma}{2}C^{1-\sigma}\tilde{c}_t^2 - \frac{\psi}{2}L^{1+\psi}\tilde{\ell}_t^2 )

Thank you.

No, that is not feasible. The reason is that linear terms in the objective mean that a full second order approximation would be required, which cannot be computed for discretionary policy, as far as I know.

Great, thank you!