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?


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