Stochastic model with "endogenized" shock

Dear All,

I wrote a DSGE model, where an agent endogenizes one of the exogenous shock, z. That is, the variance of the shock \sigma_z^2, appears in-its first order conditions.

So my first question: is there any issue with this?

My second problem concerns the steady state in such a set-up. To make the model deterministic and solve the model, we set \sigma=0, i.e. there is no shocks. Does this imply that I also have to put \sigma_z=0 in the FOC ? How wrong would it be to leave \sigma_z>0, which means that the agent maximizes by taking into consideration a shock which will never realize since we are in steady state ?

Thank you very much for your help.

Kind regards

Could you please formally outline the problem here? The devil is in the details. Why/how does the \sigma_z appear in the FOC?

Thank you for your help. I will try to formulate the problem more precisely.

I have the following shock z_t \sim N(0,\sigma_{z}^2)

An agent in my model is penalized if z_t is below a threshold \bar{z}. Its objective contains the probability of paying a penalty times the penalty since the \bar{z} threshold is affected by the controls of this agent. Thus, the FOC looks like this :

\mathbb{E} \left\{[...] - \phi(\frac{\bar{z}_{t+1}}{\sigma_z})\cdot penalty \right\} = 0 ,
where \phi is the pdf of a standardized normal distribution.

I am not sure whether such a set-up makes sense. The idea is that when the agent maximizes he does not know the realization of the shock, but its distribution. At the steady state, I am not sure what should be the value of \sigma_z in the FOC. I know that the shock is nil, but since the agent maximizes ex-ante, the idea is that he takes into account the possibility of a shock (which never occurs in steady state). Or does the agent knows there is no variance since we are in steady state ? The problem is that if I put \sigma_z=0 at steady state, I will run into issues when dynare computes the Jacobian.

I hope this clarifies my question. Thank you very much for your help.

If that is your FOC, I would interpret sigma_z as a parameter of the model that you can use and that is not 0 in steady state. After all, it is the perturbation parameter that is set to 0, not the variance of shocks itself.

Thank you for your clarifications!

I think I got confused in the specification of the stochastic process for the exogenous variables. Everything is more clear if I specify this process as
\mathbf{x_2'} = C(\mathbf{x_2})+\sigma \eta_\epsilon \epsilon'
where \eta_\epsilon is matrix that determines the var-cov of the innovation and \sigma the perturbation parameter.

From this equation, I see more clearly that the agent’s FOCs contains the variance included in \eta_\epsilon, but not the perturbation parameter \sigma. They are two different objects.

Thank you again for your help.