Certainty equivalence

Dear All,

I am working on a model in which the standard deviation appears in the decision rule of one of the agent.
I have already asked a question related to such a set-up.


An agent in my model is penalized if z_t is below a threshold \bar{z}_t . Its objective contains the probability of paying a penalty times the penalty since the \bar{z}_t threshold is affected by the controls of this agent. Thus, the FOC looks like this :
\mathbf{E} \left\{ [...] - \phi(\frac{\bar{z}_{t+1}}{\sigma_z}) \cdot penalty \right\} = 0
where \phi is the pdf of a standardized normal distribution.

Solving this model by a first order approximation around the steady state implies certainty equivalence. I understand that the certainty equivalence principle implies that the decision rule is not affected by stochastic variability. So that certainty equivalence precludes the standard deviation of the shock to appear as an argument.

Does this imply that the set-up described above cannot be solved with a first order approximation?
In this setup, the agent’s FOC explicitly includes the shock dispersion parameter, even in a non-stochastic context. In other words, even in steady state, the agent incorporates the probability that the shock is below a threshold, even though this will never be the case in this nonstochastic case.

Thank you for your help.

I would recommend to have a look at the risk shocks literature around Christiano/Motto/Rostagno. They show how underlying shock variances can be modeled to have an influence at first order. If you can somehow proceed in a similar way, it could work.