Recall A Cash-in-Advance Model, the government deficits cause inflation.
Here, I would apply the equation of exchange and government budget constraint to explain how inflation is generated by government deficits. Recalling the government budget constraint,
\overbrace{p_t g_t}^{Gov Spending} + \overbrace{i_t d_t}^{Interest Payment} = \underbrace{(d_{t+1}-d_t)}_{Increase in Debt Position}+\underbrace{T_t}_{Tax Revenue}+\underbrace{m_t-m_{t-1}}_{Print Money}
devide by \( p_t\) to get the equation in the real term,
$$ g_t+i_t \frac{d_t}{p_t}=\frac{d_{t+1}-d_t}{p_t}+\tau_t+\frac{m_t-m_{t-1}}{p_t} $$
, where \( \tau_t=\frac{T_t}{p_t}\).
By denoting real government debt as \( \hat{d}_t=\frac{d_t}{p_{t-1}}\), and replace \( (1+r_t)=(1+i_t)\frac{P_{t-1}}{P_t}=\frac{1+i_t}{1+\pi_t} \) and \( m_t = p_t y_t \), then we get all variables are in real terms,
$$ g_t – \tau_t +(1+r_t)\hat{d}_t =\hat{d}_{t+1}+\frac{p_t y_t-p_{t-1}y_{t-1}}{p_t}$$
At the steady state \( g_t=g_{t+1}=g, \tau_t=\tau_{t+1}=\tau \) and so on, and thus,
$$ \underbrace{g+r\hat{d}-\tau }_{Growth\ of \ interest\ deficits}= \underbrace{\frac{p_t-p_{t-1}}{p_t}}_{Seignorage} \times y$$
From the above equation, we can find that if inflation increases then it means the RHS increases. The LHS consists of two parts. Government Spendings \( g + r\hat{d}\) and government revenues \( \tau \). That means the government is getting deficits if the LHS rises. Meanwhile, the RHS increases and so inflation grows.
In sum, we find that government deficits, in the long run, would induce inflation. The zero-inflation condition is to make the LHS of the equation equal to zero (government spendings offset government revenue).