# Solving a Lagrange optimization problem with forward-looking constraints

Dear all,
I am currently working on a model with CSV framework and long-term debt. Unfortunately, I get to a point where I simply have no clue how to go about. I have a standard constraint maximization problem but with forward-looking constraints. Any ideas and suggestions would be highly appreciated.

Best,
Peter

What exactly is the question?

Hi Johannes,
So here’s the dilemma. In models with capital-adjustment costs the law of motion for capital is yet another constraint for the households (assuming we have a model where households are the owners of capital). Gali, Lopez-Salido, Valles (2007) do it like that for example. Then we need to differentiate w.r.t both K and I, rather than just K. Ok.

Now, imagine we had the following problem to solve:

max U(C_t)
s.t. Budget constraint
s.t. collateral constraint as in Forlati and Lambertini (2011)
s.t. St = Zt + (1 − δ)EtΛ t,t+1 θ(¯ωt+1) St+1

Now, the last constraint is a forward-looking one, where S_t just represents a discounted infinite sum that pops up each period. Omega_t+1 is a choice variable (the standard CSV), the other variables aren’t so important.

But similar to the case with capital adjustments costs, I don’t see a reason why not differentiating also w.r.t. not only omega_{t+1}, but also S_t+1 and S_t. Then we’ll just have another Lagrange multiplier and another dynamic equation.

Does this make sense to you?

I am asking since I know that if one wants to have a Bellman equation and solve such a problem, forward-looking constraints pose a problem, and there are attempts to circumvent it, e.g. Marcet and Marimon have a paper about that. But unless one wants to go the dynamic programming path, I don’t see a problem solving the Lagrangian presented above.

I haven’t given this much thought, but isn’t the issue with forward-looking constraints that there are commitment issues? It seems you want to ignore them, which should be fine.

Thanks a lot