First-period problem in primal approach solution to optimal fiscal policy problem


I am trying to solve an optimal capital taxation problem similar to Chamley (1986). I’ve solved it using the primal approach, and I would like to obtain the (deterministic) transitional dynamics to a new steady-state in Dynare. The main issue is that the first order conditions for t=0 are different to the following peridos, i.e t=1,2,3… . An example of a similar problem being solved this way is “Optimal progressive taxation in a model with endogenous skill supply” by Angelopoulos et al. As shown in page 29, the response of the social planner is different in the first period.

Can you please suggest how I should introduce such cosntraints?

Doing this is not straightforward. In Dynare solves time-invariant problems where the equations do not change between periods. The way you can solve Ramsey problems is by having the initial state variables like Lagrange multipliers differ from the ones afterward. That should usually allow setting up such problems, but it may be tedious.

Thanks a lot for your answer. I guess it is not going to be easy to proceed in Dynare.

But I am not sure what you mean about treating the initial state variables like Lagrange multipliers. Do you have an example at hand of a case in which the Lagrange multipliers are set up to differ from the ones afterward?

I doubt that this model can be handled in Dynare. Ramsey policy set initial value of lagged Lagrange multipliers to zero because policy maker isn’t constrained by private agents expectations about policy under the previous policy regime, before the switch to Ramsey policy. But I don’t see how that device can be used here.

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What I meant is: it may be that the only difference between the FOCs in the first period and the others is having 0 multipliers in the first. Then you can use Dynare.

That’s right. What I am trying to do is the equivalent to solving the model with the dual apporach.

I’ve been searching a bit to try to understand my options as per your comments. I’ve tried solving Chamley’s problem with the command ramsey_model. I saw how you solve the Ramsey problem for Woodford (2003) using oo_.endo_simul(:,1)=0 to set the first period multiplier to zero, so I am hoping that this will do for me as well.

However, the code runs into troubles at finding the steady-state. As you can see if you run the attached .mod file, it struggles to find the steady state if I give initial values to the instrument under the initval block (it seems to pick a random starting value for taul). If I instead give the initial values for the instruments under the steady_state_model block, the code ends with what it seem to be the right steady-state and zero residuals, but still can’t find the static first order conditions for optimal policy. If I don’t provide the instruments, it does not work either.

Could you suggest what’s wrong?
dual_ramsey_pr_dynare.mod (2.6 KB)

Thanks a lot for your help!

  1. You need to provide a proper setup for perfect foresight simulations. That is: an initial and a terminal condition.
  2. You also need a proper conditional steady state.
  3. In a nonlinear model you cannot have

as that would set all variables to 0 in the first period, not just the multipliers.
4. Finally, your ordering of commands is incorrect.
Have a look at:
dual_ramsey_pr_dynare.mod (2.5 KB)
It does not find a terminal steady state but rather tries to get taxes to minus infinity. So maybe the main issue is no finite steady state existing.