Money-financed fiscal expansion

I am trying to build a model which would allow for two options:

  1. financing productive/unproductive government spending by a combination of government debt and lump-sum taxation
  2. monetary financing of productive/unproductive government spending (using seigniorage)

I have succeded in building a working model for the first case.

In particular, what I have is (money demand, debt evolution, taxes and seigniorage definition):

m_t^{-\mu} = \frac{1}{\kappa} c_t^{\sigma} \frac{R_t}{R_t-1} \\ b_t = \frac{R_t}{\pi_t} b_{t-1} + g_t - s_t - t_t \\ t_t = t + \psi_b (\frac{b_t}{y_t} - \frac{b}{y}) \\ s_t = m_t - \frac{m_{t-1}}{\pi_t}

and the usual Taylor rule.

When I stich off the Taylor rule and replace it by an equation which describes the degree of monetary financing:

s_t = (\frac{g_t - g}{g})^{\phi_{MF}}

I get indeterminacy due to the violation of BK conditions. What am I doing wrong? While deriving the model I loosely followed this paper.

Apart from that, I am not sure how to properly write this down in Dynare. I cannot log-linearize s_t because in steady state it is zero (unless trend inflation is positive). That is why I leave it without the exp() in the equations. However, in this case IRFs for government debt tend to become strange: following a 1% of GDP shock to government spending, debt falls by 2 (which means 200%). Should I somehow modify my debt accumulation equation if I use both exp() and linear variables?

Many thanks for any suggestions!
model.mod (7.7 KB)

Are you sure that particular type of monetary fiscal interaction works. We know from e.g. Leeper (1991) that one of the two policies needs to be active and the other one passive to get a unique determinate equilibrium. It seems you turned monetary policy passive with the seignorage rule.