3.1 Households

There is a representative household that consumes $c_t$ , supplies labor $H_t$ , and saves using one-period government bonds $B_t$ in zero net supply to maximize the expected lifetime utility in every period

(1) \begin{align} E_0\sum _{t=0}^{\infty }\beta ^t\xi _t\bigg [\frac {(c_t-hc_{t-1}^A)^{1-\sigma }}{1-\sigma }-\chi \frac {H_t^{1+\eta }}{1+\eta }\bigg ] \end{align}

subject to the budget constraint

(2) \begin{align} P_tc_t+B_t = P_tw_tH_t+R_{t-1}B_{t-1}+T_t+P_tdiv_t \end{align}

where $c_t^A$ is aggregate consumption,Footnote ⁶ $P_t$ is the price level of final goods, $w_t$ is the real wage, $R_t$ is the gross nominal interest rate, $T_t$ is lump-sum tax and $div_t$ is real profits from the firm sectors. The parameter $h$ captures the degree of consumption habits. $\xi _t$ is the preference shock and follows an AR(1) process in logs $ln(\xi _t)=\rho _{\xi }ln(\xi _{t-1})+\sigma ^{\xi }\epsilon _t^{\xi }$ , where $\epsilon _t^{\xi }\sim N(0,1)$ . $\sigma ^{\xi }$ is the standard deviation of the preference shocks.

3.2 Firms

Final goods firms buy the intermediate goods from intermediate goods firms and produce the final goods using CES technology:

(3) \begin{align} y_t = \bigg (\int _{0}^{1}y_t(j)^{\frac {\epsilon -1}{\epsilon }}dj\bigg )^{\frac {\epsilon }{\epsilon -1}} \end{align}

where $y_t(j)$ is the amount of the good from firm $j$ and $y_t$ is the amount of the final goods produced. Since the final goods firm is in the competitive market, from the profit maximization problem, the demand for the differentiated good $j$ is derived as

(4) \begin{align} y_t(j)=\bigg (\frac {P_t(j)}{P_t}\bigg )^{-\epsilon }y_t \end{align}

where $P_t(j)$ is the price of firm $j$ ’s goods and $P_t$ is the price of the final goods. The price for the final goods is expressed as

(5) \begin{align} P_t=\bigg [\int _{0}^{1}P_t(j)^{1-\epsilon }dj\bigg ]^{\frac {1}{1-\epsilon }} \end{align}

Given the demand for the intermediated good $j$ , the firm $j$ produces output with labor, $y_t(j) = H_t(j)$ . Because of its monopolistic power, the firm $j$ sets its own price $P_t(j)$ so as to maximize expected lifetime profits subject to the price adjustment costs in units of the final goods:

(6) \begin{align} \max _{P_{t}(j)} E_t\sum _{s=0}^{\infty }\beta ^{t+s}\frac {\lambda _{t+s}}{\lambda _t}\frac {\xi _{t+s}}{\xi _t}\bigg [&\frac {P_t(j)}{P_{t+s}}\bigg (\frac {P_t(j)}{P_{t+s}}\bigg )^{-\epsilon }y_{t+s}-mc_{t+s}\bigg (\frac {P_t(j)}{P_{t+s}}\bigg )^{-\epsilon }y_{t+s}\nonumber \\ &-\frac {\varphi }{2}\bigg (\frac {P_t(j)}{\Pi P_{t-1}(j)}-1\bigg )^{2}y_{t+s}\bigg ] \end{align}

3.3 Government

The monetary authority sets the nominal interest rate $R_t$ responding to the deviations of the average inflation rate and output from their own targets. The average inflation rate, $\hat {\Pi }_t$ , is defined as an exponential moving average inflation rate:Footnote ⁷

(7) \begin{align} \hat {\Pi _t} = \alpha \Pi _t+(1-\alpha )\hat {\Pi }_{t-1} \end{align}

where $\Pi _t$ is the one-period inflation rate. This specification reduces the endogenous state variables related to the average inflation rate to the unity regardless of the averaging window.Footnote ⁸ The parameter $\alpha$ is determined in a way that given the averaging window, for example, 4 years(16 quarters), the sum of weights accrued to recent 16 periods inflation rates is equal to 99.9%.Footnote ⁹ When $\alpha =1$ , the monetary policy is the standard inflating targeting that does not care about past inflation losses or gains. As $\alpha$ approaches 0, the aim of the central bank converges to stabilize the price level.Footnote ¹⁰ The monetary policy is constrained by a ZLB.

The monetary policy rule is implemented as follows

(8) \begin{align} R_t=\max (1,R_t^n) \end{align}

where $R_t^n$ denotes the notional rate that would be set without the ZLB constraint.

(9) \begin{align} R_t^n &= (R_{t-1}^n)^{\rho _R}\bigg [R\bigg (\frac {\hat {\Pi }_t}{\Pi }\bigg )^{\phi _{\pi }}\bigg (\frac {y_t}{y}\bigg )^{\phi _{y}}\bigg ]^{1-\rho _R}exp(\sigma ^{m}\epsilon _t^m) \end{align}

where $R$ is the nominal interest rate in the steady state. $\Pi$ is the monetary authority’s inflation target and $y$ is the level of output in the deterministic steady state. The inertial term is included and the parameter $\rho _R$ is associated with the persistence of the monetary policy. $\epsilon _t^m$ is a i.i.d monetary policy shock, where $\epsilon _t^{m} \sim N(0,1)$ . $\sigma ^{m}$ is the standard deviation of the shocks. The fiscal authority just runs balanced budget constraints in every period, which results in the resource constraint below:

(10) \begin{align} c_t&=y_t\bigg [1-\frac {\varphi }{2}\bigg (\frac {\Pi _t}{\Pi }-1\bigg )^2\bigg ] \end{align}

3.4 Model equilibrium

An aggregate production function of the model is $y_t=H_t$ . The equilibrium of the model is the sequence of $\{\lambda _t, c_t,\xi _t, R_t, \Pi _t, w_t, H_t, mc_t, y_t, R_t^n, \hat {\Pi }_t, W_t\}_{t=0}^{\infty }$ which satisfies the all first-order conditions derived from the agents’ maximization problems and the monetary policy related equations along with the aggregate production function given the states of the economy.

3.5 Model solution and calibration

Following Richter et al. (Reference Richter, Throckmorton and Walker2014), we solve the model globally with time iteration and linear interpolation. Expectations are evaluated with Rouwenhorst (Reference Rouwenhorst1995), which Kopecky and Suen (Reference Kopecky and Suen2010) show outperforms other methods for approximating autoregressive processes. A detailed description of the solution method is provided in Appendix C.

We set $\beta =0.994$ to match a real interest rate at the steady state of roughly 2.5 percent on an annual basis, $\eta =1$ to deliver a Frisch elasticity of 1, and $\chi =1.73$ , a normalization of the hours worked. The relative risk aversion parameter $\sigma$ is set to 1, and the Rotemberg parameter $\varphi$ is set to 667 so that the slope of the New Keynesian Phillips curve is 0.01. The elasticity of substitution among good varieties, $\epsilon =7.67$ , implies a steady-state markup of 15 percent. The habit parameter $h$ is set to 0.5. We set the inflation target to 2% annually.

Whereas the persistence of the monetary policy rule $\rho _R$ is set to 0.8, there are no clear guidelines on how to set $\phi _{\pi }$ and $\phi _y$ under AIT due to the limited experience of implementing AIT in the real world. Following the literature, for example, Coenen et al. (Reference Coenen, Montes-Galdon and Schmidt2021), $\phi _{\pi }$ is set equal to the length of the averaging window on an annual basis.Footnote ¹¹ This specification keeps the volatilities of the policy rates comparable across all averaging windows. For the baseline case, we set $\phi _y=0$ and explore the effects of different values later. For the exogenous shock processes, we follow Bianchi et al. (Reference Bianchi, Melosi and Rottner2021) and set $\rho _{\xi }=0.7$ , $100\sigma _{\xi }=2.16$ , and $100\sigma _{m}=0.42$ .

4.1 Social welfare in stochastic steady state

Figure 1 shows social welfare (blue line, left axis) in Equation (1) and the output gap (red line, right axis) as a function of the averaging windows of AIT. Social welfare increases as the window lengthens up to 6 years but begins to decline beyond that point. As is well known, the presence of the ZLB constraint causes welfare losses under IT because the possibility of the ZLB in the future hampers well-anchored inflation expectations. In an economy with AIT, the private sector expects a lower-for-longer policy at the ZLB, resulting in higher inflation and a positive output gap in the future. Since heightened inflation expectations reduce the likelihood of the ZLB binding in the future, welfare losses arising from the ZLB are somewhat alleviated. The window length determines the strength of these effects.

Figure 1.

Social welfare/output gap in stochastic steady state.

Note: The blue line (left axis) shows the social welfare in AIT with various averaging windows in each SSS. The red line (right axis) shows corresponding output gaps relative to a flexible price economy.

The blue line shows that for up to 6 years, the expectation effects are favorable enough to reduce welfare losses, causing social welfare to increase. However, as the window length exceeds six years, the optimistic expectations become so strong that output and inflation tend to grow beyond the desired targets. The central bank needs to raise the policy rate to bring inflation back to the target. While this policy action drags inflation closer to the target, output falls below the target due to well-anchored inflation expectations, resulting in higher real interest rates. In other words, overly strong stimulative expectation effects from longer averaging windows hamper social welfare in the long run through reduced output. The red line illustrates this dynamic. As inflation is anchored around the target, what ultimately determines social welfare in the SSS is the output gap.

4.2 The importance of solution method

To evaluate the importance of the solution method in capturing the nonlinear relationship between social welfare and the AIT window, we compare the welfare outcomes obtained using the global method with those derived from a linear approximation. For the latter, given the occasional binding of the ZLB constraint, we employ the method developed by Guerrieri and Iacoviello (Reference Guerrieri and Iacoviello2015), which utilizes a first-order perturbation approach implemented in a piecewise linear framework.

One issue with this method is that the certainty equivalence property holds by the nature of first-order approximation, meaning that agents base their decisions on the expected values of future variables while ignoring higher-order moments such as variance or skewness. As a result, the model’s SSS coincides with its DSS, eliminating the influence of uncertainty on equilibrium dynamics. Consequently, the social welfare in the SSS across different AIT windows is identical.

To address this limitation, we perform a long-run simulation of the model using the linear solution method. Specifically, the model is simulated over 100,000 periods with a series of random shocks, and the first 10,000 periods are discarded as a burn-in to eliminate the influence of initial conditions. The average of the simulated social welfare over the remaining periods is then calculated to approximate the SSS. This approach enables the estimation of the SSS without relying on the flawed certainty equivalence assumption, thereby capturing the effects of future uncertainty on the model’s dynamics more accurately.

For comparison, we apply the same procedure to the model solved using the nonlinear solution method. While this approach does not yield exact welfare values from the fully nonlinear solution, it provides a close approximation of the SSS. Consequently, comparing the SSS derived from the linear solution with the SSS from the nonlinear solution based on these simulations is a reasonable and consistent approach.

Figure 2 displays the results of the simulations. The red line represents the social welfare from the nonlinear solution method, while the blue line shows the social welfare from the linear solution method. The black dashed line represents the social welfare in the actual SSS shown in Figure 1. The red line and the black dashed line are very close to each other, which implies that social welfare based on simulations is a reasonable proxy for the values in the SSS.

Figure 2.

Linear vs nonlinear solution method.

Note: While the blue line (Occbin) represents the results from the method of a first-order perturbation approach, the red line (Nonlinear) represents the results from the global method. The social welfare is calculated based on simulations. The black dashed line (Nonlinear, SSS) shows the SSS social welfare.

The blue line demonstrates that when we solve the model linearly, social welfare increases as the AIT window gets longer. However, if the model is solved nonlinearly, social welfare decreases once the window exceeds 6 years. This occurs because a first-order approximation fails to capture the mechanism through which agents respond to uncertainty in the model. Consequently, the linear solution does not account for the varying probabilities of binding constraints in the future, depending on the window length. This highlights the importance of solving the model nonlinearly when comparing AITs with different windows.

4.3 AIT vs. PLT

Next, we compare AIT with PLT in terms of welfare. Since PLT closely approximates the optimal monetary policy, we use it as a benchmark to evaluate how well AIT performs relative to PLT.Footnote ¹² One challenge encountered is that solving the model with $\alpha = 0$ in this fully nonlinear framework rendered the model unsolvable. To address this issue, we employ a specific price-level targeting rule:

(11) \begin{align} R_t^n &= (R_{t-1}^n)^{\rho _R} \bigg [R \bigg (P_t \bigg )^{\phi _{\pi }} \bigg (\frac {y_t}{y}\bigg )^{\phi _{y}} \bigg ]^{1-\rho _R} \exp (\sigma ^{m} \epsilon _t^m), \end{align}

where $P_t = \Pi _{t} P_{t-1}$ represents the price level in the current period and the price level target is set to 1.Footnote ¹³

Figure 3 presents the results. Welfare under PLT is consistently higher than under AIT across all window lengths. However, the performance of the 6-year window AIT is notably close to the corresponding PLT outcomes. To provide a clearer interpretation, I translate the welfare losses into consumption-equivalent units.Footnote ¹⁴ A 6-year window AIT results in a welfare loss equivalent to a permanent reduction in consumption by 0.02 percent in the PLT economy.

Figure 3.

Social welfare of AIT vs. PLT.

Note: The blue line represents social welfare under AIT with various averaging windows in each SSS. The red line represents social welfare under PLT with the same coefficients for inflation responses in the policy rule as in the corresponding AIT framework.

4.4 Macroeconomic biases

Next, Figure 4 shows how the macroeconomic biases due to the ZLB vary as a function of AIT window length. We examine the biases of inflation, output, and the real interest rate in the SSS, which are computed by taking the difference between their values in the SSS and their corresponding values in the DSS. We observe that medium-length averaging window AITs are effective in mitigating all macroeconomic distortions. In particular, the 8-year window AIT almost closes all three gaps.

Figure 4.

Macroeconomic biases.

Note: The inflation bias (left plot), the output bias (middle plot), and the real interest rate bias (right plot) are computed by taking the difference between those values at the SSS and their value at DSS. The inflation and the real interest rate biases are expressed in annualized percentage points and the output gap in percentage points.

One noteworthy observation is that inflation bias and output bias move in opposite directions under longer-window AITs. In these regimes, the downside risk due to ZLB binding is low. However, when the ZLB binds, the private sector anticipates long periods of future inflation and output deviations in the positive direction. This expectation effect is so strong that it outweighs the downside risks. The point at which these two opposing effects balance occurs when the central bank raises the policy rate to mitigate the expectation effects, resulting in high real interest rates and a low output gap.

4.5 Determinants of the optimal 6-year averaging window

An important question in the design of AIT is: What determines the welfare-maximizing averaging period? Addressing this question is essential for understanding how AIT can be effectively implemented in practice. Given the baseline calibration, our numerical results suggest that the optimal averaging period is closely linked to the likelihood of the ZLB being binding.

Figure 5 compares the baseline result (blue line) with an alternative scenario (red line) where the standard deviation of preference shocks ( $\sigma ^{\xi }=0.05$ ) is more than doubled. As the standard deviation increases, the economy encounters the ZLB more frequently. Specifically, under the baseline parameterization, the economy is constrained by the ZLB in 10.7% of the state space, whereas in the alternative case, this increases to 11.4%. The figure illustrates that when the probability of the ZLB binding rises, the optimal averaging window lengthens to 10 years.

Figure 5.

Social welfare with different level of $\sigma ^{\xi }$ .

Note: The blue line (left axis) shows the social welfare in AIT with various averaging windows in each SSS with $\sigma ^{\chi }$ = 0.0216. The red line (right axis) shows the social welfare in AIT with various averaging windows in each SSS with $\sigma ^{\chi }$ = 0.05. to a flexible price economy.

This finding underscores the critical role of the ZLB in determining the welfare-maximizing length of the AIT window. In scenarios where the ZLB represents a significant constraint, longer averaging windows are advantageous because they more effectively anchor inflation expectations and stabilize the economy. By doing so, they mitigate the adverse effects of constrained monetary policy at the ZLB. Conversely, in environments with a lower likelihood of the ZLB binding, shorter averaging windows are sufficient to achieve similar stabilization benefits without the risk of excessive inflation expectations or overly contractionary future monetary policy.

These results highlight the necessity for policymakers to account for the ZLB probability within their specific economic context when calibrating the averaging window for AIT. A one-size-fits-all approach may fail to address the nuanced trade-offs inherent in different economic environments. This discussion underscores the importance of tailoring AIT design to reflect the unique economic conditions faced by each central bank.

6.1 Multiplier outside the ZLB

In this subsection, we report the multiplier outside the ZLB. To examine how different window lengths affect the fiscal multiplier, we compare 6-year and 10-year window AITs to IT. Each economy with a different monetary policy rule begins at its own SSS, and government spending increases by 1% following the process described in (12). The results are plotted in Figure 8. The far-right panel in the second row shows the fiscal multipliers. We highlight two key points.

Figure 8.

GIRF to government spending shocks and fiscal multiplier outside the ZLB.

Note: Except for output, consumption and fiscal multiplier, the y-axis is the annualized percentage point difference between a path with and without government spending shocks. Output is the difference between two paths scaled by the SSS values. Fiscal multiplier is the cumulative multiplier computed by equation (13).

First, the multipliers under AIT are smaller than in the IT case. Increased demand due to higher government spending leads to a rise in inflation and output. Because of the well-known crowding-out effects, the multiplier remains below unity under IT. In the case of AIT, private agents anticipate a higher-for-longer policy stance from the central bank in response to the inflation gains from increased government spending. This expectation lowers inflation expectations and, consequently, inflation compared to IT. As a result, the nominal interest rate increases less on impact in response to the smaller rise in average inflation rates, but a higher-for-longer policy is implemented thereafter.

The real interest rate mimics the movement of the nominal interest rates. Even though the real rate under AIT is lower than IT on impact, it increases more after a few periods. This is why output under AIT increases less than in the IT case throughout the horizon. The smaller change in output, combined with a larger increase in real interest rates, results in a multiplier lower than in the IT case. In other words, the crowding-out effects become more pronounced under AIT. This is illustrated in the third panel of the second row, which displays the consumption responses.

Second, the multiplier declines as the window length increases. Since the expectation effects under longer-window AIT are stronger, the real interest rates rise more compared to shorter-window AIT. This leads to a greater decline in consumption, resulting in a smaller multiplier in the 10-year AIT case.

One notable point is that the fiscal multiplier outside the ZLB appears higher than reported in the literature. For example, Fernández-Villaverde et al. (Reference Fernández-Villaverde, Gordon, Guerrón-Quintana and Rubio-Ramirez2015) reported a fiscal multiplier of 0.54 on impact outside the ZLB when the model is solved nonlinearly. This discrepancy arises from the fact that the monetary policy rule in our model does not respond to the output gap. If we allowed the central bank to aim at stabilizing the output gap as well, the multiplier would be much lower.Footnote ¹⁷ Nevertheless, the points explained above remain robust to this adjustment.

6.2 Multiplier at the ZLB

Next, we investigate the government spending multiplier at the ZLB. Each economy begins at its own SSS in period 0. We inject a series of negative preference shocks in periods 1 and 2 into each economy, which sends the economy under IT to the ZLB, on average, for 5 consecutive quarters. Given the same series of shocks, the economies under AITs experience longer ZLB episodes due to their lower-for-longer policy feature. Government spending rises by approximately 1% in period 2, when the shock ends but the economy still remains at the ZLB. Figure 9 shows the results, and a couple of points are worth highlighting.

Figure 9.

GIRF to government spending shocks and fiscal multiplier at the ZLB.

Note: Except for output, consumption and fiscal multiplier, the y-axis is the annualized percentage point difference between a path with and without government spending shocks. Output is the difference between two paths scaled by the SSS values. Fiscal multiplier is the cumulative multiplier computed by equation (13).

First, the multipliers under IT seldom change compared to the outside ZLB case. One reason for this is the flat Phillips curve. The primary channel for government spending to stimulate the economy at the ZLB is through the rise in expected inflation. However, the flat Phillips curve attenuates this channel, limiting the decrease in the real interest rate.Footnote ¹⁸ In this sense, the flat Phillips curve weakens the substitution effects of increased government spending.Footnote ¹⁹

Another reason is the dominance of the wealth effect over the substitution effect, as evidenced by the decrease in consumption at impact, despite a decline in real interest rates. Since government spending continues to rise even after the economy exits the ZLB due to its persistence, the negative wealth effect dominates the attenuated substitution effect arising from lower real interest rates.

An additional factor is the Rotemberg price adjustment costs. As highlighted by Boneva et al. (Reference Boneva, Braun and Waki2016), these adjustment costs increase firms’ marginal costs in response to government spending shocks, thereby limiting the extent of output expansion.

Second, unlike the IT case, the multipliers under AITs increase for both window lengths. Numerically, the multipliers at impact are approximately 1.23 and 0.96 for 6-year and 10-year AIT, respectively, compared to values below 0.8 outside the ZLB for both. Under AIT, agents anticipate longer ZLB episodes, leading to a greater rise in inflation expectations. This partially restores the weakened substitution effect.

The restored substitution effect under 6-year AIT is strong enough to dominate the wealth effect, resulting in increased consumption and a fiscal multiplier above one at impact. Conversely, under a 10-year AIT, inflation expectations rise less than under a 6-year AIT, meaning the restored substitution effect cannot overturn the dominance of the wealth effect. As a result, the multiplier at impact does not exceed one, although it remains significantly higher than outside the ZLB.

To summarize, as the recession becomes more severe, fiscal policy under AIT proves more effective compared to IT. This result assumes a very flat Phillips curve and persistent government spending, which is broadly supported by empirical evidence

7.1 Social welfare in SSS

In Section 4, we show that social welfare increases non-monotonically with the AIT window length. In this subsection, we examine how robust this result is under different assumptions: no habit, the same value of $\phi _{\pi }(=2)$ , the response of the monetary policy rule to the output gap, and a steep Phillips curve.

Figure 10 shows the results. The first left panel in the first row corresponds to the case of no habit. The presence of habit does not affect the nonlinear property of social welfare, though it significantly changes the level of social welfare across all window lengths. One might speculate that the nonlinear property arises from the different responses to the inflation gap in the monetary policy rule. However, this turns out not to be the case.

Figure 10.

Social welfare in SSS in various environments.

Note: No habit means $h=0$ in the model. $\phi _{\pi }=2$ is the case when the response to the inflation gap in the monetary policy rule remains unchanged across AIT windows. $\phi _{y}=0.1$ is the case when the central bank responds to the output gap in addition to the inflation gap with baseline parameterization of $\phi _{\pi }$ . Steep NKPC corresponds to the case of smaller $\varphi$ .

The right panel of the first row shows the results when the response to the inflation gap in the monetary policy rule remains fixed at $\phi _{\pi }=2$ across all AIT windows. While the welfare gains from short to medium windows are minimal, the welfare losses from medium to long windows appear substantial. This occurs because, with a fixed low value of $\phi _{\pi }$ , the stability of inflation and the output gap deteriorates as the averaging window lengthens. This observation raises the possibility that, with a fixed but higher value of $\phi _{\pi }$ , the welfare under IT could exceed the welfare achieved under AIT for any window length. We confirm this conjecture, and the results are presented in Appendix E.Footnote ²⁰

Next, we investigate the case of $\phi _{y}=0.1$ , a form of extended AIT, and the steep Phillips curve. The second row of the figure corresponds to these scenarios. Neither of them alters the nonlinear properties. The lack of significant differences between the extended AIT and the baseline AIT could stem from the growing importance of addressing inflation gaps in both frameworks and the anchoring of expectations under credible policy.

7.2 Fiscal multiplier

In Section 6, we show how the fiscal multiplier changes with the AIT window length both outside the ZLB and at the ZLB. In this subsection, we examine how robust these results are under different assumptions: a steep NKPC, the response of the monetary policy rule to the output gap, and less persistent government spending shocks. Figure 11 displays the results. The first column shows the multipliers outside the ZLB, while the second column corresponds to the ZLB case.Footnote ²¹

Figure 11.

Fiscal multipliers in various environments.

Note: Fiscal multiplier is the cumulative multiplier computed by equation (13). The first row shows the multipliers when the slope of the linearized Phillips curve is 0.03 instead of the baseline case of 0.01. The second row corresponds to the monetary policy rule responding to the output gap as well as the inflation gap. While the third row displays the case of less persistent government spending shocks, the last row shows the case of $\phi _{\pi }=2$ for AITs. The blue line, red line and black line represent IT, 6-year AIT, and 10-year AIT, respectively. The y-axis represents the size of the fiscal multiplier for all panels.

Steep Phillips curve The first row shows the multipliers when the slope of the linearized Phillips curve is 0.03, compared to the baseline value of 0.01. As mentioned earlier, the flat Phillips curve attenuates substitution effects. In contrast, with a steep NKPC, substitution effects are restored, causing the fiscal multiplier of IT at the ZLB to rise above two on impact, which is larger than the multipliers for 6-year or 10-year AIT.

Outside the ZLB, the multipliers are lower than in the baseline case. This is because inflation rises more sharply with the steep NKPC in response to the increase in government spending. The central bank reacts by raising nominal interest rates more than the increase in inflation, which drives real interest rates higher. As a result, the output increase is smaller compared to the baseline case for all three policy rules.

Response to output gap in the monetary policy rule The second row corresponds to the inclusion of a response to the output gap in the monetary policy rule. We assume $\phi _{y}=0.1$ , a value widely used in the literature.Footnote ²² Outside the ZLB, the multipliers are again lower than in the baseline case because the central bank raises nominal interest rates more aggressively, responding to both inflation and output gains. This reinforces crowding-out effects, causing the multipliers to decline.

At the ZLB, the inclusion of $\phi _{y}$ lengthens the ZLB episodes. As a result, the increase in government spending raises inflation expectations and lowers real interest rates further relative to the baseline case. This effect is substantial enough for the IT case to overturn the dominance of the wealth effect, causing the at-impact multiplier to exceed two. In contrast, under AIT, inflation expectations show little change compared to the baseline case, even with the inclusion of $\phi _{y}$ . This is reflected in the similar dynamics of multipliers in both cases. The reason is that the response to the output gap does not significantly lengthen the expected ZLB duration under AIT, where the duration is already extended by the adoption of AIT itself.

Less persistent government spending shocks The third row displays the multiplier with less persistent government spending processes, where we assume $\rho _{G}=0.5$ instead of 0.81. Less persistent processes imply a smaller wealth effect, regardless of whether the ZLB is binding. As shown in the left panel, the multipliers outside the ZLB are significantly higher and approach unity for the IT case due to the reduced wealth effect. While the multipliers under AIT also increase, they remain lower than those under IT.

At the ZLB, the multiplier also increases relative to the baseline case. Nevertheless, the multipliers are not significantly different from the baseline case, indicating that the attenuated substitution effect caused by the flat Phillips curve remains strong.

Smaller response to the inflation gap in the monetary policy rule The last row shows the case where $\phi _{\pi }=2$ for AITs in the monetary policy rule. The smaller response to the inflation gap arising from the increase in government spending results in a smaller rise in the policy rate and a greater increase in inflation expectations. This leads to a much lower real interest rate, mitigating the crowding-out effects. Consequently, unlike the baseline case, the fiscal multipliers under AIT at impact are higher than those under IT.

At the ZLB, the multipliers are not significantly different from those outside the ZLB, as shown in the baseline IT case. The smaller response to the inflation gap reduces the lower-for-longer policy effect, resulting in similar ZLB durations under IT and AIT. Therefore, the real interest rates fall less than in the baseline case, weakening the substitution effects. Given the flat Phillips curve and persistent government spending, the wealth effect dominates, and consumption under AIT decreases similarly to the IT case.