# First order approximation of consolidated budget constraint describing the evolution of government debt

I am trying to derive a first order approximation of the government's budget constraint around a zero inflation steady state.

This is from Jordi Galís 2020 paper "The effects of a money-financed fiscal stimulus"

Government's consolidated budget constraint is given by $$P_t G_t+B_{t-1}(1+i_{t-1})=P_t T_t+B_t+\Delta M_t$$ where $$G_t$$ are real government purchases. $$\mathcal{B}\equiv B_t/P_t$$ is defined as real debt outstanding and $$\mathcal{R}\equiv(1+i_t)(P_t/P_{t+1})$$ is the (ex-post) gross real interest rate.

The consolidated budget constraint is rewritten as $$G_t+\mathcal{B}_{t-1}\mathcal{R}_{t-1}=T_t+\mathcal{B}_{t}+\Delta M_t/P_t$$ where $$\Delta M_t/P_t$$ is the seigniorage in period t.

It is assumed that there exists a steady state with constant government purchases $$G$$, taxes $$T$$ and government debt $$\mathcal{B}$$ and zero seigniorage $$\Delta M = 0$$.

Then $$T=G+\rho\mathcal{B}$$ must hold in steady state where $$\mathcal{R}=1+\rho$$ and $$\rho$$ is the discount rate for the households.

Seigniorage close to the steady state, as a fraction of steady state output, can be approximated as $$(\Delta M_t/P_t)/(1/Y)=(\Delta M_t/M_{t-1})(P_{t-1}/P_t)L_{t-1}/Y \simeq \varkappa \Delta m_t$$ where $$L_t\equiv M_t/P_t$$ are real balances, $$m_t\equiv \log M_t$$ and $$\varkappa\equiv L/Y$$ is the inverse income velocity of money.

$$\hat{b}_t\equiv (\mathcal{B}_t-\mathcal{B})/Y$$,$$\hat{g}_t\equiv (G_t-G)/Y$$,$$\hat{t}_t\equiv (T_t-T)/Y$$ are deviations of government debt, government purchases and government taxes.

A first order approximation of the rewritten budget constraint around the zero inflation steady state of debt is supposed to yield: $$\hat{b}=(1+\rho)\hat{b}_{t-1}+b(1+\rho)(\hat{i}_{t-1}-\pi_t)+\hat{g}_t-\hat{t}_t-\varkappa \Delta m_t$$ but I arrive at $$\hat{b}=(1+\rho)\hat{b}_{t-1}+\hat{g}_t-\hat{t}_t-\varkappa \Delta m_t$$

Where does the term $$b(\hat{i}_{t-1}-\pi_t)$$ come from?