The Fiscal Theory of the Price Level

Recall the government’s budget constraint again,

$$p_t{g}_t+d_t=\frac{d_{t+1}}{1+i_{t+1}}+T_t+(m_t-m_{t-1})$$

Divide by $p_t$, and assume for simplicity that $m_t=m_{t+1}=\dots =\overline{m}$

$$g_t+\frac{d_t}{p_t}=\frac{d_{t+1}}{p_t(1+i_{t+1})}+{\tau }_t$$

denote $\widetilde{d_t}=d_t/p_t$, and recall that $(1+i_{t+1})\frac{p_t}{p_{t+1}}=1+r_{t+1}$.

Thus,

$$g_t+\widetilde{d_t}=\frac{{\tilde{d}}_{t+1}}{1+r_{t+1}}+{\tau }_t$$

Iterate forward, and impose the “no-Ponzi condition”, $\underset{s\to \mathrm{\infty }}{lim}\frac{\widetilde{d_{t+s}}}{\prod _{j=1}^{s}1+r_{t+j}}=0$ to get,

$$\widetilde{d_t}=\sum _{s=0}^{\mathrm{\infty }}{\beta }^s({\tau }_{t+s}-g_{t+s})$$

, where the equilibrium condition $\beta =\frac{1}{1+r}$ has been imposed.

Implication:

$$\frac{d_t}{p_t}=\sum _{s=0}^{\mathrm{\infty }}{\beta }^s({\tau }_{t+s}-g_{t+s})$$

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.