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We consider a temporary ZLB episode, as in Werning (2011). Between times 0 and

T > 0

⁠, the natural rate of interest is negative. In response, the central bank maintains its policy rate at 0. After time

T

⁠, the natural rate is positive again, and monetary policy returns to normal. This scenario is summarized in table 1A. We analyze the ZLB episode using the phase diagrams in figure 2.

Figure 2.

The timeline of a ZLB episode is presented in table 1A. The phase diagram of the NK model comes from figure 1C. The phase diagram of the WUNK model comes from figure 1D. The equilibrium trajectories are the unique trajectories reaching the natural steady state (where π=0 and y=yn) at time T. The figure shows that the economy slumps during the ZLB: inflation is negative, and output is below its natural level (A, B). In the NK model, the initial slump becomes unboundedly severe as the ZLB lasts longer (C). In the WUNK model, there is no such collapse: output and inflation are bounded below by the ZLB steady state (D).

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ZLB Episodes in the NK and WUNK Models

The timeline of a ZLB episode is presented in table 1A. The phase diagram of the NK model comes from figure 1C. The phase diagram of the WUNK model comes from figure 1D. The equilibrium trajectories are the unique trajectories reaching the natural steady state (where $π = 0$ and $y = y^{n}$ ⁠) at time $T$ ⁠. The figure shows that the economy slumps during the ZLB: inflation is negative, and output is below its natural level (A, B). In the NK model, the initial slump becomes unboundedly severe as the ZLB lasts longer (C). In the WUNK model, there is no such collapse: output and inflation are bounded below by the ZLB steady state (D).

Table 1.

ZLB Scenarios

	Timeline	Natural Rate of Interest	Monetary Policy	Government Spending
A. ZLB episode
ZLB	$t \in (0, T)$	$r^{n} < 0$	$i = 0$	–
Normal times	$t > T$	$r^{n} > 0$	$i = r^{n} + ϕ π$	–
B. ZLB episode with forward guidance
ZLB	$t \in (0, T)$	$r^{n} < 0$	$i = 0$	–
Forward	$t \in (T, T + Δ)$	$r^{n} > 0$	$i = 0$	–
guidance
Normal times	$t > T + Δ$	$r^{n} > 0$	$i = r^{n} + ϕ π$	–
C. ZLB episode with government spending
ZLB	$t \in (0, T)$	$r^{n} < 0$	$i = 0$	$g > 0$
Normal times	$t > T$	$r^{n} > 0$	$i = r^{n} + ϕ π$	$g = 0$

	Timeline	Natural Rate of Interest	Monetary Policy	Government Spending
A. ZLB episode
ZLB	$t \in (0, T)$	$r^{n} < 0$	$i = 0$	–
Normal times	$t > T$	$r^{n} > 0$	$i = r^{n} + ϕ π$	–
B. ZLB episode with forward guidance
ZLB	$t \in (0, T)$	$r^{n} < 0$	$i = 0$	–
Forward	$t \in (T, T + Δ)$	$r^{n} > 0$	$i = 0$	–
guidance
Normal times	$t > T + Δ$	$r^{n} > 0$	$i = r^{n} + ϕ π$	–
C. ZLB episode with government spending
ZLB	$t \in (0, T)$	$r^{n} < 0$	$i = 0$	$g > 0$
Normal times	$t > T$	$r^{n} > 0$	$i = r^{n} + ϕ π$	$g = 0$

This table describes the three scenarios analyzed in section IV: the ZLB episode, in section IVA; the ZLB episode with forward guidance, in section IVB; and the ZLB episode with government spending, in section IVC. The parameter $T > 0$ gives the duration of the ZLB episode; the parameter $Δ > 0$ gives the duration of forward guidance. We assume that monetary policy is active (⁠ $ϕ > 1$ ⁠) in normal times in the NK model; this assumption is required to ensure equilibrium determinacy (Taylor principle). In the WUNK model, monetary policy can be active or passive in normal times.

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