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Hi everybody,
I am building a DSGE for a small open economy (SOE) that borrows abroad, subject to a collateral constraint. If in the economy there is just one international good (so that debt is denominated in this good which is the same good produced by the SOE and the exchange rate is 1) the model works fine. Since I would like to give a role to the exchange rate, I have slightly changed the model by including a foreign good as an input in the production function; moreover, I assume that foreign debt is denominated in foreign goods. In this framework, I got 6 eigenvalues instead of 5, so the equilibrium is not stable and I do not understand why.

If I change the collateral constraint by assuming that the SOE can borrow against its future capital (instead of the current one) the equilibrium is stable, but the dynamic is really odd: output falls abruptly in the first period (around 5%), partially recovers in the second one and then comes back to the steady state very slowly.

I attach two mod files: i) FCSOE works fine, but I do not have the exchange rate; ii) In FCSOE_imp the SOE imports foreign inputs and the equilibrium is unstable.

Can anyone of you help me? I really would like to understand why BK conditions are not met (is it just a timing problem in my code or are there any other reasons?) .
Thank you very much,
Best,
Valerio

I would guess that there is an economic reason behind this. In the SOE with imports, you are having one unstable root too much. Therefore, the system is explosive. When you now shift the timing of capital to the future, you introduce an additional forward-looking variable that takes up this unstable root and you mechanically get a unique stable solution. But economically this is not a solution to your problem. You need to understand the dynamics that give rise to the explosiveness. For some reason, when you introduce the foreign imports, the economy will explode after shocks. What does prevent stability? Did you make something exogenous/predetermined that needs to be chosen endogenously?

Many thanks for your answer Prof. Pfeifer. Actually I am not shifting the timing of capital, but I am just changing the assumption of the collateral constraint. In the model with imports I specify two different kinds of collateral constraints:

1) S*B=gamma*K(-1)*(1-delta): where S is the Ex rate, B is the amount of debt chosen in the current period and (1-delta)*K(-1) is capital which is left from the previous period before investment takes place.
2) S(1)*B*Rz(1)=gamma*(K*(1-delta)): where now S(1)*B*Rz(1) is the expected amount which has to be repaid in t+1 and (K*(1-delta)) is capital which is left in t+1 before investment takes place.

Specification 1) in FCSOE_imp does not work: there are 6 eig for 5 fwd looking variables. In Specification 2), which is commented in FCSOE_imp, BK conditions are met: 4eig and 4fwd looking, since now the lagrangian multiplier phi of the collateral constraint becomes a static variable. However dynamics gets really odd.

I think both collateral constraints have an ecominic sense. I also tried to specify the constraint in term of current capital:

S*B=gamma*K

However, also in this case BK conditions are not met (5eig vs 4fwd).
I think there is an economic reason too, but I really do not see any mistakes in the model.
Thanks again Prof. Pfeifer, if you have any additional comments, they are really welcome.

Best,
Valerio

Your model currently is explosive. So ask yourself how agents in your model would bring back capital to steady state when after a shock hits.