How to introduce government adjustment cost in the DSGE model?

Hello everyone!
I’m trying to construct a DSGE model wich incorporates both private saving and government bond, but in the most models they have the same risk-free nominal interest rates, so when solving the consumer utility maximization problem, they have the same first order condition, and this will cause multiple collinearity.
In order to slove this problem, I want to introduce the bond adjustment cost in the DSGE model, but I don’t know how to introduce this adjustment cost, and what the formula will the bond adjustment cost will be?
Would you please suggest me some paper about the goverment bond adjustment? I’d appreciate some help. Thank you!

People typically use quadratic adjustment costs for bond holdings of the form
\frac{\phi_B}{2}(\frac{B_t}{P_t}-\bar B)^2
where \bar B is the target debt level. This is quite common in the small open economy literature. See e.g. Model 3 in Schmitt-Grohé/Uribe (2003): “Closing small open economy models”, Journal of International Economics, 61, pp. 163-185. The code is at

This helps me a lot! Thank you very much!:blush: