in my research, I want to solve a medium-sized NK model with active fiscal policy (no feedback of fiscal policy to government debt) and passive monetary policy (Taylor coefficient is set to zero). When I solve the model with “stoch_simul”, everything works fine. For some simulations I need a perfect foresight solution. Yet, when I use a perfect foresight solver (either “simul” or “perfect_foresight_solver”) the model can either be not solved or it can be solved but government debt is non-stationary but exploding. As I reduced the complexity of my model to try and debug it, I found that this problem carries over to the plain vanilla NK model: With active fiscal policy and passive monetary policy, the model can be solved by perturbation but not under perfect foresight.
I attached the code of the version of the plain vanilla NK model which I used. As an example, I use a government expenditure shock. With stoch_simul the model can be solved under active fiscal policy (psitd=0 in the code) and passive monetary policy (theta_phi=0). Yet under perfect foresight, it either can’t be solved (for persistent government spending shocks, i.e., rho_G != 0) or it can be solved (for rho_G=0) but in this case, government debt is exploding. Does anyone know where the problem comes from and how it might be bypassed? Any help is highly appreciated:)
I always thought that in the case with active fiscal policy and passive monetary policy, inflation adjusts such that government debt does not explode (as it seems to work in the perturbation solution).
Yes, that is the case in perturbation solutions, because here you impose the transversality condition at infinity. In that case, the only solution not causing debt to explode is to inflate it away. But in perfect foresight, you are solving a two boundary problem. You are solving for a for an intermediate path for the variables satisfying an initial and terminal condition. The thing with perfect foresight simulations is that the debt equation is purely backward looking. It will simply explode and there is no way to impose that it is not allowed to explode.
Thanks a lot! Now, I see the problem. Is there a workaround for this problem that allows me to simulate a scenario in which the government is allowed to inflate away a too high initial debt level?
And also what I do not yet fully understand about the “exploding” solution is that it should violate the governments budget constraint in the finite period. Doesn’t the algorithm impose the equations in the finite period but only imposes them in the intermediate periods?
The government budget constraint is not violated. Debt is simply financed with new debt, i.e. the government runs a Ponzi scheme. Usually, you rule this out asymptotically, but here you have a finite horizon.
I don’t see an easy workaround as finite and infinite horizon problems are fundamentally different. You are running a finite horizon problem but want to impose an infinite horizon optimality condition.
Thanks a lot, this is very good to know, although its bad news.
Yet, I am still a little puzzled about the exploding solution. If I think in terms of a “backward” solution, the government budget constraint in the finite period T should be:
b(T-1)/piCPISS - bSS/RBSS +trSS = -GSS.
This would yield b(T-1) = bSS which should not be consistent with the exploding debt solution. Or in other words: I do not understand how an exploding debt solution meets the terminal condition (government debt being back at its steady state level after T periods) that I impose.
(Maybe this is equivalent to saying, I do not understand why imposing debt to be back at steady state in some finite periods does not rule out any solution in which the government runs a Ponzi scheme.)
You are misconstructing the timing structure of economic choices. You are considering 500 periods. The terminal condition relates to variables in period 501. In period 500, the government is free to chose a debt level as it sees fit. That debt level will of course have an implication for the feasible choices of variables in period 501, but these choices are not modeled here. The only choices of the terminal period that matter for the last endogenous period are the ones relating to forward-looking variables as they will be relevant for the FOCs at time 500.
The following variables appear with a lead and are therefore forward-looking variables:
ptilde
vl
vr
lambda
piCPI
You can see that debt does not appear with a lead. Hence, the debt choice in period 500 is not restricted by the debt level in period 501 (or only insofar as it influences the terminal period values of the forward-looking variables ).
Put differently, the solution does not care about whether the terminal condition is a steady state. Only the values for