Recall the government’s budget constraint again,
$$ p_tg_t+d_t=\frac{d_{t+1}}{1+i_{t+1}}+T_t+(m_t-m_{t-1}) $$
Divide by \(p_t\), and assume for simplicity that \(m_t=m_{t+1}=…=\bar{m}\)
$$ g_t+\frac{d_t}{p_t}=\frac{d_{t+1}}{p_t (1+i_{t+1})}+\tau_t $$
denote \(\tilde{d_t}=d_t/p_t\), and recall that \((1+i_{t+1})\frac{p_t}{p_{t+1}}=1+r_{t+1}\).
Thus,
$$ g_t+\tilde{d_t}=\frac{\tilde{d}_{t+1}}{1+r_{t+1}}+\tau_t $$
Iterate forward, and impose the “no-Ponzi condition”, \( \lim_{s\rightarrow \infty} \frac{\tilde{d_{t+s}}}{\prod_{j=1}^s 1+r_{t+j}}=0\) to get,
$$ \tilde{d_t}=\sum_{s=0}^{\infty} \beta^s (\tau_{t+s}-g_{t+s}) $$
, where the equilibrium condition \( \beta=\frac{1}{1+r}\) has been imposed.
Implication:
$$ \frac{d_t}{p_t}=\sum_{s=0}^{\infty} \beta^s (\tau_{t+s}-g_{t+s}) $$
The “unpleasant arithmetic” stated that if the government has leadership, it can coerce expansions in money.
In contrast, FTPL says that the above restrictions are not a constraint to the CB or government. Instead, it is an equilibrium relation.
As a consequence, the CB and the government may choose policies independent of the above constraint. In the end, the price level \(p_t\) must then adjust such that the equation holds.