Parameter Updating Transition Permanent Shock

Perfect foresight simulations
1

Hi Dynare Community,

I am running a PF simulation where I have a permanent shock that affects e.g. productivity. The model is a closed economy New Keynesian model that features households (HH), production firms, financial intermediaries and a government that consists of a fiscal authority and a central bank.

My problem comes from the HH side, in particular how can we allow for parameters to update along a deterministic transition?
On the context, HH maximise the sum of expected discounted utility with habit formation in consumption, they save through deposits, corporate securities and gov bonds. Importantly, I assume the access to securities and bonds to be subject to quadratic adjustment costs. Let us focus on the government bonds:

The FOC reads as:
E_t\left(\beta\Lambda_{t+1,t}\left\{\left(1+r_{t+1}\right)\frac{q_t}{q_t+\kappa\left(Bh_t-\bar{Bh}\right)}\right\}\right)=1
where for simplicity, q_t is the price of bonds, r_t>0 its return, Bh_t households’ holdings, \bar{Bh} their reference holdings, \Lambda_t is the HH stochastic discount factor, and \kappa > 0 the adjustment costs.

As I am doing a transition following a permanent shock, the value of the \bar{Bh} must change, because the value of Bh_{t} will vary as e.g. productivity changes.

I have a steady_state_model block in which I call a function to numerically solve for certain variables (meaning that I cannot have an analytical expression of some equations), and I set \bar{Bh}= Bh_{ss} - \frac{(\beta(1+r_{ss})-1)q_{ss}}{\kappa}. The initval/endval block only set the exogenous variables. The model finds an initial and end steady state, both of which satisfy rank condition, and are stable. Model diagnostics look also okay.
In terms of dynamics, after calling the PF solver, I receive the message
"
Entering the homotopy method iterations…

Iter. | Share | Status | Max. residual | Duration (sec)
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
1 | 1.00000 | failed (in endval steady) | 0.0
"
across all ten iterations, all of which instantaneously, suggesting a deeper issue.
Running the model without shocks, things work properly.

Would anyone have an idea why the model keeps on failing and how to properly fix \bar{Bh}?
Many thanks in advance for your help!
Jan

2

Is Bh even well-defined in that context? What is its value supposed to be along the transition?

3

Hello Johannes,
Thank you for your answer.
\bar{Bh} is a reference level of holdings and its value should be positive and possibly time varying. Along the deterministic transition, it is determined by the Euler equation, the budget constraint, and market clearing.

I am thinking there must be a way to capture that limited asset participation for HH while there is a permanent shock hitting the economy, and making it change its initial steady state. In that spirit, Bh_t is well defined, and so must be its reference level \bar{Bh}.
I am noticing that the linear_approximation solver is the only one that manages to solve the model.

4

From what you describe, \bar{Bh} should be an exogenous variable that you predefine.

5

Interesting, thank you!
I understand then that this “exogenous” \bar{Bh} is then dependent on endogenous variables, given that it is pinned down by the Euler equation, the budget constraint, and market clearing. I will give it a thought and experiment.
Best,
Jan

6

No, you cannot endogenize it. Bh is a target from outside of the model.