Slow Learning

Figure 1: Learning equilibrium in non-linear model
(a) Outcomes
(b) Beliefs
Note: Sub-figure (a) shows outcomes for inflation, consumption, and the real interest rate under learning and in an REE when the natural rate of interest is low (\(r_{t}=r^{\ell}<r^{ss}\)). The natural rate of interest is low for the entire simulation. This is a very low probability event because the simulations cover 20 quarters. People believe that the natural rate of interest will return to its steady state level with probability \(1-p\). We use the simple NK model with only sticky prices and Rotemberg-style nominal rigidities. Sub-figure (b) shows the values of \(\mathbf{E}_{t}\log\left(\pi_{t+1}\right)\) and \(\mathbf{E}_{t}\log\left(C_{t+1}/C^{ss}\right)\) over the course of the simulation.
Source: Authors’ calculations.

Panel (a): Outcomes

Panel (a) of Figure 1 displays three figures that are arranged next to each other horizontally.

• The figure on the left is titled labeled “Inflation.” The vertical axis label is “log (π^𝓁) × 400,” and the axis spans about -7 to about 3. The horizontal axis represents time and spans 0 to 20. Two lines are displayed. The first line, labeled “Learning equilibrium” in the legend, starts at about -0.5 at time 1. It declines steadily to about -3 in time 4. Thereafter, it declines gradually to about -4 in period 20. The second line, labeled “REE” in the legend, is a horizontal line at about -6.5.

• The figure in the center is titled labeled “Consumption.” The vertical axis label is “log (C^𝓁/C^ss) × 100,” and the axis spans about -9 to about 0. The horizontal axis represents time and spans 0 to 20. Two lines are displayed. The first line, labeled “Learning equilibrium” in the legend, starts at about -6 at time 1. It holds about steady for a period, before increasing in periods 3 and 4 to about -5.5. Thereafter, the line increases very gradually over the remaining periods shown, reaching only slightly higher values. The second line, labeled “REE” in the legend, is a horizontal line at about -7.75.

• The figure on the right is titled labeled “Real Rate.” The vertical axis label is “Annualized Percent,” and the axis spans about -1 to about 7. The horizontal axis represents time and spans 0 to 20. Two lines are displayed. The first line, labeled “Learning equilibrium” in the legend, starts at about 1.5 at time 1. It falls in period 2 to about 0.5. Thereafter, the line increases gradually over the remaining periods shown, reaching about 2.5 in period 20. The second line, labeled “REE” in the legend, is a horizontal line at about 5.5.

Panel (b) of Figure 1 displays two figures that are arranged next to each other horizontally.

• The figure on the left is titled labeled “Aggregate Inflation Expectations.” The vertical axis label is “Elog (π^𝓁) × 400,” and the axis spans about -7 to about 2.5. The horizontal axis represents time and spans 0 to 20. Two lines are displayed. The first line, labeled “Learning” in the legend, starts at about 0 at time 1. It declines gradually throughout the 20 periods, ultimately reaching about -3 in period 20. The second line, labeled “REE” in the legend, is a horizontal line at about -6.5. There is a red dot shown on the figure at about -5 over period 20. There is an arrow pointing to the dot labeled “t = 10, 000 (red dot).”

• The figure on the right is titled labeled “Aggregate Consumption Beliefs.” The vertical axis label is “Elog (C^𝓁/C^ss) × 100,” and the axis spans about -9 to about 0. The horizontal axis represents time and spans 0 to 20. Two lines are displayed. The first line, labeled “Learning” in the legend, starts at about 0 at time 1. It falls quickly to about -4.5 in period 5. Thereafter, the line decreases very gradually over the remaining periods shown, reaching only slightly lower values (about -5.5 in period 20). The second line, labeled “REE” in the legend, is a horizontal line at about -7.75. There is a red dot shown on the figure at about -6 over period 20. There is an arrow pointing to the dot labeled “t = 10, 000 (red dot).”

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Figure 2: Anticipated utility versus internalized learning
Note: The figure shows outcomes for inflation, consumption, and the real interest rate under anticipated utility and under internalized learning when the natural rate of interest is low (\(r_{t}=r^{\ell}<r^{ss}\)). The natural rate of interest is low for the entire simulation. This is a very low probability event because the simulations cover 20 quarters. People believe that the natural rate of interest will return to its steady state level with probability \(1-p\). We use the simple NK model with only sticky prices and Rotemberg-style nominal rigidities. Notice that the dashed black lines, labeled “Anticipated utility,” in this figure are the same as the dashed black lines in Figure 1(a).
Source: Authors’ calculations.

Figure 2 displays three figures that are arranged next to each other horizontally.

• The figure on the left is titled labeled “Inflation.” The vertical axis label is “log (π^𝓁) × 400,” and the axis spans about -7 to about 3. The horizontal axis represents time and spans 0 to 20. Three lines are displayed. The first line, labeled “Anticipated utility” in the legend, starts at about -0.5 at time 1. It declines to about -3 in time 4. Thereafter, it declines gradually to about -4 in period 20. The second line, labeled “Internalized learning” in the legend, starts at about -2.25 at time 1. It declines steadily to about -4 in time 4. Thereafter, it declines gradually to about -5 in period 20. The third line, labeled “REE” in the legend, is a horizontal line at about -6.5. The line labeled “Anticipated utility” is always above the line labeled “Internalized learning,” which is always above the line labeled “REE.”

• The figure in the center is titled labeled “Consumption.” The vertical axis label is “log (C^𝓁/C^ss) × 100,” and the axis spans about -9 to about 0. The horizontal axis represents time and spans 0 to 20. Three lines are displayed. The first line, labeled “Anticipated utility” in the legend, starts at about -6 at time 1. It holds about steady for a period, before increasing in periods 3 and 4 to about -5.5. Thereafter, the line increases very gradually over the remaining periods shown, reaching only slightly higher values. The second line, labeled “Internalized learning” in the legend, starts at about -6.5 at time 1. It holds about steady for a period, before increasing gradually over the remaining periods shown, reaching about -6 in period 20. The third line, labeled “REE” in the legend, is a horizontal line at about -7.75. The line labeled “Anticipated utility” is always above the line labeled “Internalized learning,” which is always above the line labeled “REE.”

• The figure on the right is titled labeled “Real Rate.” The vertical axis label is “Annualized Percent,” and the axis spans about -1 to about 7. The horizontal axis represents time and spans 0 to 20. Three lines are displayed. The first line, labeled “Anticipated utility” in the legend, starts at about 1.5 at time 1. It falls in period 2 to about 0.5. Thereafter, the line increases gradually over the remaining periods shown, reaching about 2.5 in period 20. The second line, labeled “Internalized learning” in the legend, starts at about 1.25 at time 1. Thereafter, the line increases gradually over the remaining periods shown, reaching about 3.5 in period 20. The third line, labeled “REE” in the legend, is a horizontal line at about 5.5. After period 1, the line labeled “Anticipated utility” is always below the line labeled “Internalized learning,” which is always below the line labeled “REE.”

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Figure 3: Equilibria with and without an increase in G_t
(a) Outcomes during ZLB
(b) G multiplier during ZLB
Note: Sub-figure (a) shows outcomes for inflation, consumption, and the real interest rate under internalized learning when the natural rate of interest is low (\(r_{t}=r^{\ell}<r^{ss}\)) and \(G_{t}\) is either high or at it steady state value while \(r_{t}=r^{\ell}\). Sub-figure (b) shows the value of the multiplier defined in equation (23). The natural rate of interest is low for the entire simulation. This is a very low probability event because the simulations cover 20 quarters. People believe that the natural rate of interest will return to its steady state level with probability \(1-p\). We use the simple NK model with only sticky prices and Rotemberg-style nominal rigidities.
Source: Authors’ calculations.

Panel (a) Outcomes during ZLB

Panel (a) of Figure 3 displays three figures that are arranged next to each other horizontally.

• The figure on the left is titled labeled “Inflation.” The vertical axis label is “log (π^𝓁) × 400,” and the axis spans about -7 to about 1. The horizontal axis represents time and spans 0 to 20. Four lines are displayed. The first line, labeled “REE” in the legend, is a horizontal line at about -6.5. The second line, labeled “learning” in the legend, starts at about -2.25 at time 1. It declines steadily to about -4 in time 4. Thereafter, it declines gradually to about -5 in period 20. The third line, labeled “REE, high G” in the legend, is a horizontal line at about -3. The fourth line, labeled “Learning, high G” in the legend, starts at about -2 at time 1. It declines steadily to about -3.5 in time 4. Thereafter, it declines gradually to about -4 in period 20.

• The figure in the center is titled labeled “Consumption.” The vertical axis label is “log (C^𝓁/C^ss) × 100,” and the axis spans about -9 to about 0. The horizontal axis represents time and spans 0 to 20. Four lines are displayed. The first line, labeled “REE” in the legend, is a horizontal line at about -7.75. The second line, labeled “Learning” in the legend, starts at about -6.5 at time 1. It holds about steady for a period, before increasing gradually over the remaining periods shown, reaching about -6 in period 20. The third line, labeled “REE, high G” in the legend, is a horizontal line at about -3.9. The fourth line, labeled “Learning, high G” in the legend, starts at about -6.5 at time 1 (just above the line labeled “Learning”). It holds about steady for a period, before increasing gradually over the remaining periods shown, reaching about -5.9 in period 20.

• The figure on the right is titled labeled “Real Rate.” The vertical axis label is “Annualized Percent,” and the axis spans about -1 to about 7. The horizontal axis represents time and spans 0 to 20. Four lines are displayed. The first line, labeled “REE” in the legend, is a horizontal line at about 5.5. The second line, labeled “Learning” in the legend, starts at about 1.25 at time 1. Thereafter, the line increases gradually over the remaining periods shown, reaching about 3.5 in period 20. The third line, labeled “REE, high G” in the legend, is a horizontal line at about 2.5. The fourth line, labeled “Learning, high G” in the legend, starts at about 1 at time 1. Thereafter, the line increases gradually over the remaining periods shown, reaching about 3 in period 20.

Panel (b): G multiplier during ZLB

Panel (b) of Figure 3 displays one figure. The vertical axis spans about -1 to about 4. The horizontal axis represents time and spans about 0 to about 20. The figure title is “G Multiplier.” The figure displays two lines. The first line, labeled “REE” in the legend, is a horizontal line at about 4. The second line, labeled “Learning” in the legend, starts at about 1 in period 1. Thereafter, the line gradually rises, reaching about 1.25 in period 20.

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Figure 4: Forward guidance under learning and in the REE
Note: The figure shows inflation, consumption, and the real interest rate paths under internalized learning when the natural rate of interest is low (\(r_{t}=r^{\ell}<r^{ss}\)) with and without forward guidance about \(R_{t}\). Under forward guidance, the monetary authority credibly promising to set \(R_{t}=1\) for one period after \(r_{t}=r^{\ell}\). The natural rate of interest is low for the entire simulation. This is a very low probability event because the simulations cover 20 quarters. People believe that the natural rate of interest will return to its steady state level with probability \(1-p\). We use the simple NK model with only sticky prices and Rotemberg-style nominal rigidities.
Source: Authors’ calculations.

Figure 4 displays three figures that are arranged next to each other horizontally.

• The figure on the left is titled labeled “Inflation.” The vertical axis label is “log (π^𝓁) × 400,” and the axis spans about -7 to about 1. The horizontal axis represents time and spans 0 to 20. Four lines are displayed. The first line, labeled “REE” in the legend, is a horizontal line at about -6.5. The second line, labeled “learning” in the legend, starts at about -2.25 at time 1. It declines steadily to about -4 in time 4. Thereafter, it declines gradually to about -5 in period 20. The third line, labeled “REE, FG” in the legend, is a horizontal line at about -2.5. The fourth line, labeled “Learning, FG” in the legend, starts at about -2.1 at time 1. It declines steadily to about -3.5 in time 4. Thereafter, it declines gradually to about -4.5 in period 20.

• The figure in the center is titled labeled “Consumption.” The vertical axis label is “log (C^𝓁/C^ss) × 100,” and the axis spans about -9 to about 0. The horizontal axis represents time and spans 0 to 20. Four lines are displayed. The first line, labeled “REE” in the legend, is a horizontal line at about -7.75. The second line, labeled “Learning” in the legend, starts at about -6.5 at time 1. It holds about steady for a period, before increasing gradually over the remaining periods shown, reaching about -6 in period 20. The third line, labeled “REE, FG” in the legend, is a horizontal line at about -2.5s. The fourth line, labeled “Learning, FG” in the legend, starts at about -6.25 at time 1 (just above the line labeled “Learning”). It holds about steady for a period, before increasing gradually over the remaining periods shown, reaching about -5.5 in period 20.

• The figure on the right is titled labeled “Real Rate.” The vertical axis label is “Annualized Percent,” and the axis spans about -1 to about 7. The horizontal axis represents time and spans 0 to 20. Four lines are displayed. The first line, labeled “REE” in the legend, is a horizontal line at about 5.5. The second line, labeled “Learning” in the legend, starts a little above 1 at time 1. Thereafter, the line increases gradually over the remaining periods shown, reaching about 3.5 in period 20. The third line, labeled “REE, FG” in the legend, is a horizontal line at about 2. The fourth line, labeled “Learning, FG” in the legend, starts at about 1 at time 1. Thereafter, the line increases gradually over the remaining periods shown, reaching about 3 in period 20.

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Figure 5: f(π^ℓ) Corresponding to the Target-Inflation Steady-State Equilibrium
Note: The zeros of the function \(f\), which is defined in the text, identify REE values for \(\pi^{\ell}\) assuming that the economy goes to the target-inflation steady state after \(r_{t}=r^{ss}\). The dashed line is the case when \(G^{\ell}\) is high. Source: Authors’ calculations.

The vertical axis is labeled f(π^𝓁), spans about -0.007 to about 0.003, and does not have units. The horizontal axis is labeled π^𝓁, and spans about 0.86 to about 1.02. The figure displays three lines. The first line is a horizontal line at zero. The second line, labeled “G^𝓁 = G^ss” in the legend, begins at about -0.005 on the left of the figure. The line rises to about 0.001 at about π^𝓁 = 0.94. Thereafter the line gradually declines to a little less than zero at about π^𝓁 = 0.985. At about π^𝓁 = 0.995, the line begins to decline more sharply, reaching about -0.007 at π^𝓁 = 1.02. The line crosses the zero line twice. At the left crossing (at about π^𝓁 = 0.91), the point is labeled “A.” At the right crossing (at about π^𝓁 = 0.985) the point is labeled “B.” The third line, labeled “G^𝓁 = 1.05 × G^ss” in the legend, is just slightly above the second line. The line crosses the zero line twice. At the left crossing, the point is labeled “A’.” At the right crossing, the point is labeled “B’.”

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Figure 6: Equilibria in Period of Switch from r_t = r^ℓ to r_t = r^ss Under One-Period Forward Guidance
Notes: Graph of the function, \(f^{I}\left(\pi^{I}\right),\) discussed after equation (96). The two crossings with the zero line correspond to REE in period \(I,\) the date when \(r\) switches from \(r_{t}=r^{\ell}\) to \(r_{t}=r^{ss}\). Monetary policy in period \(I\) corresponds to one-period forward guidance–that is, the interest rate is held at zero in period \(I\) and then reverts to \(R^{ss}\). The red star indicates the level of inflation in period \(I\) in the absence of forward guidance.

The vertical axis is labeled f^I(π^I), spans about -0.12 to about 0.03, and does not have units. The horizontal axis is labeled π^𝓁, and spans about 0.86 to about 1.25. The figure displays two lines. The first line is a horizontal line at zero. The second line, begins at about π^I = 0.97 at the top of the figure. It declines steadily until about π^I = 1.14, at which point it begins increasing. The line crosses the zero line twice. At the left crossing (at π^I just above 1), the point is labeled “A.” At the right crossing (at about π^I = 1.22) the point is labeled “B.” There is an inset box in the figure that enlarges the portion of the graph around the left crossing (point A). The inset box shows that 1 is less than π^I at the crossing with a red star shown at 1.

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Figure 7: REE Equilibria at the ZLB with and without Forward Guidance
Notes: The solid line reproduces the solid line in Figure 5 and corresponds to the case of no forward guidance. The dashed and dot-dashed lines correspond to the case of forward guidance. The dashed line corresponds to the case in which the economy goes to point \(\mathcal{B}\) in the period of the switch in \(r_{t}\) to \(r^{ss}\) (period \(I\)). It crosses the zero line more than once, but the other crossing involves very high inflation and is not an equilibrium because the present value of intermediate goods monopolists is negative. The dot-dashed line corresponds to the case in which the economy goes to point \(\mathcal{A}\) in period \(I\) (see Figure 6).

The vertical axis is labeled f^I₋₁(π^𝓁), spans about -0.007 to about 0.005, and does not have units. The horizontal axis is labeled π^𝓁, and spans about 0.8 to about 1.02. The figure displays four lines. The first line is a horizontal line at zero. The second line, labeled “Going to A in period I” in the legend, begins at about -0.007 at about π^𝓁 = 0.85. The line rises and crosses zero at about π^𝓁 = 0.91. This crossing is labeled “A.” The line peaks at about 0.001 at about π^𝓁 = 0.94. The line then declines and crosses zero again at about π^𝓁 = 0.99. This crossing is labeled “B.” At about π^𝓁 = 0.995, the line starts to decline sharply and reaches about -0.007 at the right side of the figure. The third line, labeled “Going to B in period I” in the legend, begins at about -0.007 at about π^𝓁 = 0.81. The line rises sharply, crossing zero at about π^𝓁 = 0.825. This crossing is labeled “C.” The line rises out of the figure axis by π^𝓁 = 0.85 and does not appear again. The fourth line, labeled “No forward guidance” in the legend, runs just below the second line. Like the second line, the fourth line crosses the zero line twice.

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Figure 8: f^C(π^ℓ) Corresponding to the Target-Inflation Steady-State Equilibrium
Note: The function, \(f_{C},\) is defined in the text. The dashed line is the case when \(G_{t}\) is high. Source: Authors’ calculations.

The vertical axis is labeled f^C(π^𝓁), spans about -0.1 to about 0.175, and does not have units. The horizontal axis is labeled π^𝓁, and spans about 0.81 to about 1.02. The figure displays three lines. The first line is a horizontal line at zero. The second line, labeled “G^𝓁 = G^ss” in the legend, begins at about -0.07 on the left of the figure. The line rises to about 0.14 at about π^𝓁 = 0.91. Thereafter the line gradually declines to a little less than zero at about π^𝓁 = 0.985. At about π^𝓁 = 0.995, the line begins to decline more sharply, reaching about -0.08 at π^𝓁 = 1. The line crosses the zero line twice. The left crossing is at about π^𝓁 = 0.83. The right crossing is at about π^𝓁 = 0.985. The third line, labeled “G^𝓁 = 1.05 × G^ss” in the legend, begins just slightly below the second line. It rises above the second line at about π^𝓁 = 0.87 and runs slightly above the second line thereafter. The third line crosses the zero line twice. Both crossings are slightly to the right of the crossings of the second line.

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Figure 9: Size of Government Purchases Multiplier
(a) Euler Equation Learning
(b) Anticipated Utility
Note: The figure shows the size of the government purchases multiplier under three experiments in our simple NK model with only sticky prices. Experiment 1 is the REE. Experiment 2 changes intial expectations when \(G_{t}\) changes, as in . Experiment 3 leave initial expectations unchanged when \(G_{t}\) changes. Panel (a) is constructed under Euler equation learning, as in . Panel (b) is constructed under anticipated utility as outline in Appendix E. The figure displays the value of the government purchases multiplier in learning equilibria and the REE under the assumption that \(S_{t}=1\) for the entire duration of the period shown. This is a low probability event that would be surprising.
Source: Authors’ calculations.

Panel (a): Euler Equation Learning

Panel (a) of Figure 9 displays one figure. The vertical axis is labeled “Government purchases multiplier” and spans about 0 to about 1.3. The horizontal axis represents time and spans about 0 to about 20. The horizontal axis is labeled “Time during which S=1 (ZLB is binding).” The first line, labeled “Experiment 1: REE” in the legend, is a horizontal line at about 0.25. The second line, labeled “Experiment 2: when G changes also change m₀” in the legend, is very nearly a horizontal line that is just above the line labeled “Experiment 1: REE.” The third line, labeled “Experiment 3: when G changes keep the same m₀” in the legend starts at about 1 in period 1. Thereafter, the line gradually rises, reaching about 1.15 in period 20.

Panel (b): Anticipated Utility

Panel (b) of Figure 9 displays one figure. The vertical axis is labeled “Government purchases multiplier” and spans about -0.25 to about 2.25. The horizontal axis represents time and spans about 0 to about 20. The horizontal axis is labeled “Time during which S=1 (ZLB is binding).” The first line, labeled “Experiment 1: REE” in the legend, is a nearly horizontal line at about 0.4. The second line, labeled “Experiment 2: when G changes also change m₀” in the legend, starts at about 0.4 in period 1 and then is very slightly upward sloping, reaching about 0.5 in period 20. The third line, labeled “Experiment 3: when G changes keep the same m₀” in the legend starts just below 1 in period 1. Thereafter, the line gradually steeply reaching about 2.1 in period 7. The line is not shown for subsequent periods.

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