SOE NK-DSGE incomplete asset markets

Good evening everyone,

I hope you all are safe amid the current situation.

Currently I am working on an SOE NK DSGE model a la Gali 2005 (not-linearized) which is working perfectly fine. I am seeking to add foreign debt holdings to the household and augmented the budget constraint by foreign debt holdings, which the HH has to pay with a debt elastic interest rate a la Schmitt-Grohe; and added the FOC for foreign bonds. I impose the HH to have a SS foreign debt holding; and the household balances this in SS by saving in domestic bonds in equal amount.

Case 1: The timing in the budget constraint is paying (receiving) interest on debt (savings), while the interest rate and amount of bonds have been determined one period earlier, i.e. Domestic: (1+r(-1)) * d(-1)), Foreign: (1+r*(-1)) d*(-1)) * real FX rate. The amount of domestic and foreign debt/savings for the next period is chosen today. In the FOCs for domestic and foreign bonds, the time subscript is then +1 ahead. Now, when using a CPI/domestic taylor rule, the model works fine like that.

Case 2: Once I try to implement an exchange rate peg, i.e. exchange rate growth to 1 and delete the taylor rule, the rank condition fails to hold. Since a couple of days I scanned the forum here and thought about timing issues. Once I adjust the timing in the budget constraint for ONLY the domestic bonds +1 ahead, i.e. Domestic: (1+r)) * d(-1)) and in FOC domestic bond also +1 ahead, while keeping foreign interest rate timing the same, the rank condition is satisfied, and the model runs.

My question: Which timing is correct? Usually, I would say Case 1 is correct and inline with my equations, but the exchange rate peg doesn’t work - maybe I miss something in comprehending the timing here. I am struggling with this for several days and tried many different solutions, and am not quite sure if any of that is correct. I would be grateful for some advice and would appreciate any help/suggestions.

Best regards,

I would recommend not approaching this issue from a mechanical perspective. Bonds and their interest rate are usually predetermined. By there is a degree of freedom in whether the risk premium is determined by the current or future expected indebtedness.
If an exchange rate peg does not work within that setup, you should try to understand why that is the case. I guess your model is not stable anymore. There must be a economic reason for that.

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Thank you, Professor Pfeifer!