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Copy file name to clipboardExpand all lines: lectures/cagan_ree.md
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We'll use linear algebra first to explain and then do some experiments with a "fiscal theory of the price level".
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A fiscal theory of the price level was described by Thomas Sargent and Neil Wallace in chapter 5 of
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a 1981 article title "Unpleasant Monetarist Arithmetic". The theory has been extended, criticized, and applied by John Cochrane in {cite}`cochrane2023fiscal`.
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In another lecture, we'll describe some historical episodes in which the elemental forces at work in the fiscal theory help to explain some early twentieth century hyperinflations that occurred in the wake of World War I.
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According to this model, when the government persistently spends more than it collects in taxes and prints money to finance the shortfall (the "shortfall" is called the "government deficit"), it puts upward pressure on the price level and generates
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persistent inflation.
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While the log money supply portrayed in the bottom panel has a kink at $T_1$, the log price level does not -- it is "smooth" -- once again a consequence of the fact that the
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reduction in $\mu$ has been foreseen.
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To set the stage for our next experiment, we want to study the determinants of the price level a little more.
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### The log price level
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We can use equations {eq}`eq:caganmd` and {eq}`eq:ree`
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to discover that the log of the price level satisfies
In our next experiment, we'll study a "surprise" permanent change in the money growth that beforehand
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was completely unanticipated.
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At time $T_1$ when the "surprise" money growth rate change occurs, to satisfy
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equation {eq}`eq:pformula2`, the log of real balances jumps
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**upward* as $\pi_t$ jumps **downward**.
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But in order for $m_t - p_t$ to jump, which variable jumps, $m_{T_1}$ or $p_{T_1}$?
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### What jumps?
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What jumps at $T_1$?
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Is it $p_{T_1}$ or $m_{T_1}$?
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If we insist that the money supply $m_{T_1}$ is locked at its value $m_{T_1}^1$ inherited from the past, then formula {eq}`eq:pformula2` implies that the price level jumps downward at time $T_1$, to coincide with the downward jump in
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$\pi_{T_1}$
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An alternative assumption about the money supply level is that as part of the "inflation stabilization",
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the government resets $m_{T_1}$ according to
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$$
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m_{T_1}^2 - m_{T_1}^1 = \alpha (\pi^1 - \pi^2)
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$$ (eq:eqnmoneyjump)
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By letting money jump according to equation {eq}`eq:eqnmoneyjump` the monetary authority prevents the price level from **falling** at the moment that the unanticipated stabilization arrives.
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In various research papers about stabilizations of high inflations, the jump in the money supply described by equation {eq}`eq:eqnmoneyjump` has been called
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"the velocity dividend" that a government reaps from implementing a regime change that sustains a permanently lower inflation rate.
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#### Technical details about whether $p$ or $m$ jumps at $T_1$
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We have noted that with a constant expected forward sequence $\mu_s = \bar \mu$ for $s\geq t$, $\pi_{t} =\bar{\mu}$.
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A consequence is that at $T_1$, either $m$ or $p$ must "jump" at $T_1$.
At time $T_1$ when the "surprise" regime change occurs, to satisfy
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equation {eq}`eq:pformula2`, the log of real balances jumps
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**upward* as $\pi_t$ jumps **downward**.
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But in order for $m_t - p_t$ to jump, which variable jumps, $m_{T_1}$ or $p_{T_1}$?
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### What jumps?
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What jumps at $T_1$?
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Is it $p_{T_1}$ or $m_{T_1}$?
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If we insist that the money supply $m_{T_1}$ is locked at its value $m_{T_1}^1$ inherited from the past, then formula {eq}`eq:pformula2` implies that the price level jumps downward at time $T_1$, to coincide with the downward jump in
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$\pi_{T_1}$
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An alternative assumption about the money supply level is that as part of the "inflation stabilization",
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the government resets $m_{T_1}$ according to
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$$
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m_{T_1}^2 - m_{T_1}^1 = \alpha (\pi^1 - \pi^2)
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$$ (eq:eqnmoneyjump)
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By letting money jump according to equation {eq}`eq:eqnmoneyjump` the monetary authority prevents the price level
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from **falling** at the moment that the unanticipated stabilization arrives.
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In various research papers about stabilizations of high inflations, the jump in the money supply described by equation {eq}`eq:eqnmoneyjump` has been called
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"the velocity dividend" that a government reaps from implementing a regime change that sustains a permanently lower inflation rate.
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#### Technical details about whether $p$ or $m$ jumps at $T_1$
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We have noted that with a constant expected forward sequence $\mu_s = \bar \mu$ for $s\geq t$, $\pi_{t} =\bar{\mu}$.
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A consequence is that at $T_1$, either $m$ or $p$ must "jump" at $T_1$.
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