2.1. Private sector behavior

I use a prototypical forward-looking NK model with the real cost channel as in Beaudry et al. (Reference Beaudry, Hou and Portier2022) and Nie (Reference Nie2021).Footnote ⁸ The behavior of the aggregate demand (AD) side of the economy can be summarized in the standard log-linearized Euler equation:

(1) \begin{equation} c_t=\mathbb{E}_t c_{t+1}-\frac{1}{\sigma _c}\left [R_t+\log\!(\beta )-\mathbb{E}_t\pi _{t+1}-r_t^n\right ], \end{equation}

where $c_t$ is private consumption, $\sigma _c$ is the risk aversion coefficient, $R_t$ is the nominal interest rate in level, $\pi _t$ is inflation, $\mathbb{E}_t$ is the rational expectation operator, and $r_t^n$ is the demand shock (also the natural rate shock).

The linear resource constraint in this economy is

(2) \begin{equation} y_t=(1-s_g)c_t+g_t, \end{equation}

where $y_t$ is the output gap, $s_g$ is the fraction of government spending in total production, and $g_t$ is government spending.Footnote ⁹ In this case, one can obtain the path of $y_t$ by substituting the resource constraint into the Euler equation below:

(3) \begin{equation} y_t=\mathbb{E}_t y_{t+1}-\frac{1}{\sigma }\left [R_t+\log\! (\beta )-\mathbb{E}_t\pi _{t+1}-r_t^n\right ]+g_t-\mathbb{E}_tg_{t+1}, \end{equation}

where $\sigma =\frac{\sigma _c}{1-s_g}$ .

The aggregate supply (AS) side of the economy can be summarized with the following log-linearized NK Phillips Curve.Footnote ¹⁰

(4) \begin{equation} \pi _t=\beta \mathbb{E}_t \pi _{t+1}+\kappa\! \left[\gamma _y y_t+\gamma _g g_t+\gamma _r(R_t+\log\! (\beta )-\mathbb{E}_t \pi _{t+1})\right ], \end{equation}

where $\beta$ is the subjective discount factor, and $\kappa$ is the elasticity of inflation with regard to marginal cost. $\gamma _y$ , $\gamma _g$ , and $\gamma _r$ are the elasticity of marginal cost elasticity with regard to the output gap, government spending, and the expected real interest rate, respectively.Footnote ¹¹ It is of note that $\gamma _r$ in equation (4) can be seen as the strength of the cost channel which controls the impact of this channel. This model with the real cost channel can collapse to the conventional one without the cost channel if $\gamma _r=0$ .Footnote ¹² In addition, it can nest the model with the nominal cost channel [represented by equation (6)] if we assume that nominal interest rates are introduced in firms’ borrowing costs as in Ravenna and Walsh (Reference Ravenna and Walsh2006), and then, the expected disinflation term ( $-\mathbb{E}_t\pi _{t+1}$ ) of equation (4) disappears.

It is assumed that the central bank sets the nominal interest rate following the (truncated) Taylor (Reference Taylor1993)-type rule with the ZLB:

(5) \begin{equation} R_t=\max\! \left \{0,-\log\! (\beta )+\phi _\pi \pi _t\right \}. \end{equation}

2.1.1. Real versus nominal cost channel

The Phillips Curve with the nominal cost channel is

(6) \begin{equation} \pi _t=\beta \mathbb{E}_t \pi _{t+1}+\kappa\! \left [\gamma _y y_t+\gamma _g g_t+\gamma _r(R_t+\log\! (\beta ))\right ]. \end{equation}

Equation (6) is used in most previous papers working as nominal cost channel specifications such as Ravenna and Walsh (Reference Ravenna and Walsh2006) and Surico (Reference Surico2008). In these papers, the nominal interest rate is introduced in borrowing costs. Even though the specifications in equations (4) and (6) can be seen as the cost channel, the real cost channel specification in equation (4) features expected real interest rates in borrowing costs and highlights the additional role of expected disinflation denoted by the negative $\mathbb{E}_t\pi _{t+1}$ term.

Note that the two cost channels can have similar effects in normal times if the central bank follows a simple Taylor rule in equation (5). In this case, a cost-push shock endogenously emerges in the two cases as in Ravenna and Walsh (Reference Ravenna and Walsh2006), and the two channels both increase firms’ marginal costs and inflation.

An interesting insight at the ZLB with $R_t=0$ is that the nominal cost channel as in Ravenna and Walsh (Reference Ravenna and Walsh2006) and Surico (Reference Surico2008) cannot influence the slope of the Phillips Curve. However, note that, at the ZLB, the real cost channel as in Beaudry et al. (Reference Beaudry, Hou and Portier2022) can rotate the Phillips Curve with expectations of disinflation. As a result, the Phillips Curve with the real cost channel is flatter than in the standard NK model without this channel, and this may explain a declining slope of the empirical Phillips Curve.

2.2. Quick tour: normal times and ZLB

In this section, I employ a two-state static Markov chain as in Eggertsson et al. (Reference Eggertsson and Woodford2003) to deal with the policy shocks vector $[r_t^n, g_t]$ . It is assumed that a specific policy shock (for example, the demand shock $r_t^n$ in this section) remains at the current short-run state (which I label $r^n_S$ with a subscript $'S'$ for the short run) with a persistence $p$ and then reverts to the long-run steady state, that is $r^n_L=0$ with a probability $1-p$ .Footnote ¹³ Since the NK model with the real cost channel in this paper is forward-looking, one can compute the expected output gap and inflation as follows:

(7) \begin{equation} \mathbb{E}_S y_{t+1}=p y_S, \quad \quad \mathbb{E}_S \pi _{t+1}=p\pi _S. \end{equation}

Assumption 1. I assume that the NK Phillips Curve with the real cost channel is always upward sloping in a $(\pi _S, y_S)$ graph such that

(8) \begin{equation} p\lt \frac{1-\kappa \gamma _r \phi _\pi }{\beta -\kappa \gamma _r}=\overline{p}^c. \end{equation}

As proposed by Laubach and Williams (Reference Laubach and Williams2003), Cochrane (Reference Cochrane2017), Han et al. (Reference Han, Tan and Wu2020), and Nie and Roulleau-Pasdeloup (Reference Nie and Roulleau-Pasdeloup2023), there is an implicit condition that the NK Phillips Curve is upward sloping in a $(\pi _S, y_S)$ graph as in Assumption 1. Compared with the NK model with the nominal cost channel as in Christiano et al. (Reference Christiano, Eichenbaum and Evans2005) and Ravenna and Walsh (Reference Ravenna and Walsh2006), this paper utilizes a more empirically relevant real cost channel proposed in Beaudry et al. (Reference Beaudry, Hou and Portier2022) and further extended in Nie (Reference Nie2021).

In this section, we discuss two cases which are the economy in normal times without ZLB binding and also in liquidity traps. There is a threshold of demand shock to trigger ZLB constraint binding. From the Taylor (Reference Taylor1993)-type rule, one can see that if the item $\{-\log\! (\beta )+\phi _\pi \pi _S\}$ is less than or equal to zero, the NK economy can be binding with the ZLB state. If not, the economy is in normal times and nominal interest rates can be free to adjust with the central bank’s monetary policy regulation. If the (negative) natural rate shock is too large, the economy can be caught up in liquidity traps. Thereby there is a boundary condition for the natural rate shock $r_S^n$ in the short run to trigger the economy into a state with ZLB binding.

Proposition 1. The boundary condition relationship among the three models is

(9) \begin{equation} \underline{r_S^{n,B}}\lt \underline{r_S^n}\lt \underline{r_S^{n,N}}, \end{equation}

where $\underline{r_S^{n,B}}$ is the boundary condition without the cost channel, $\underline{r_S^n}$ is with the real cost channel, and $\underline{r_S^{n,N}}$ is with the nominal cost channel.

Proof. See Online Appendix B.

In Proposition 1, the boundary condition to trigger the ZLB binding with the real cost channel is larger than in the conventional NK model without the cost channel since the real cost channel features the supply-side effects of interest rates. In other words, the economy can get stuck more easily in liquidity traps with the real cost channel than in the traditional model. On the other hand, the boundary condition with the nominal cost channel is larger than in the model with the real cost channel due to expectations of disinflation. Therefore, among the three models, the model with the nominal cost channel is the most easily entrapped in liquidity traps.Footnote ¹⁴

In the following sections, I will focus on government spending shocks and discuss the issues of the output gap and inflation multipliers.

2.3. Government spending: theoretical analysis

To obtain transparent results, I abstract from demand shocks and focus only on the effects of government spending shocks with the real cost channel. First, I compare the multiplier relationship among the three models. Second, I deliver the general property of the strength of the cost channel on the multiplier by using simple Markov chain closed-form solutions.

2.3.1. Government spending multipliers in normal times

I assume that a positive government spending shock $g_S\gt 0$ follows the Markov process. It starts in the short run, stays with the persistence probability $p$ , and returns to the steady-state $g_L=0$ in the long run with a probability $1-p$ . If the short-run economy is in normal times, I can rewrite the Euler equation (3) and the Phillips Curve with the real cost channel [equation (4)]:

(10) \begin{align} y_S &=-\frac{1}{\sigma (1-p)}(\phi _\pi -p)\pi _S+g_S\end{align}

(11) \begin{align} \pi _S &=\kappa \frac{\gamma _y }{1-\beta p-\kappa \gamma _r(\phi _\pi -p)}y_S+\kappa \frac{\gamma _g}{1-\beta p-\kappa \gamma _r(\phi _\pi -p)}g_S. \end{align}

Fiscal multipliers

I find that a positive government spending shock can move the Euler equation upward and shift down the Phillips Curve in a $(\pi _t,y_t)$ graph. The solutions with the real cost channel of the output gap multiplier $\mathcal{M}_{S,N}^{O}$ and inflation multiplier $\mathcal{M}_{S,N}^{I}$ can be computed as follows:Footnote ¹⁵

(12) \begin{align} \mathcal{M}_{S,N}^{O}=\frac{\sigma (1-p)[1-\beta p-\kappa \gamma _r(\phi _\pi -p)]-\kappa \gamma _g(\phi _\pi -p)}{\sigma (1-p)[1-\beta p-\kappa \gamma _r(\phi _\pi -p)]+\kappa \gamma _y(\phi _\pi -p)} \end{align}

(13) \begin{align} \mathcal{M}_{S,N}^{I}=\frac{ \bigg [1+\frac{\gamma _g}{\gamma _y}\bigg ] \kappa \gamma _y\sigma (1-p)}{\sigma (1-p)[1-\beta p-\kappa \gamma _r(\phi _\pi -p)]+\kappa \gamma _y(\phi _\pi -p)}. \end{align}

The expected real interest rate can be in both the denominator and numerator of the output gap multiplier since the real cost channel here can influence the inflation rate and further impact the output gap through expectations. Regarding the inflation multiplier, the expected real interest rate is included in the denominator since the cost channel can impact this multiplier directly. In normal times, since the denominator of the multiplier equation is higher than the numerator, the multiplier is less than one.Footnote ¹⁶

In normal times, the real cost channel reduces the output gap multiplier, consistent with Abo-Zaid (Reference Abo-Zaid2022). Positive government spending shocks elevate short-run inflation;Footnote ¹⁷ the real cost channel further heightens borrowing costs and inflation.Footnote ¹⁸ With adjustable nominal interest rates, the central bank increases rates by more than inflation.Footnote ¹⁹ The ensuing higher real interest rates curb private consumption, decreasing the output gap multiplier compared to the scenario where this channel is absent.

As in Online Appendix C, compared to the nominal cost channel, the inflation multiplier with the real cost channel can be smaller. If we compare the NK Phillips Curve in equations (6) and (4), less influence is triggered by the real cost channel, which can echo the fact of lower inflation in the Euro area in normal times [Koester et al. (Reference Koester, Lis, Nickel, Osbat and Smets2021)]. In this scenario, the nominal cost channel can further reduce the output gap multiplier compared to the real cost channel.

I also explore the effects of the strength of the real cost channel $\gamma _r$ on spending multipliers. The power of the real cost channel can directly leverage the increment of inflation. On the other hand, we can see that higher inflation can crowd out more private consumption, hence leading to a much lower output gap multiplier given a stronger real cost channel. Therefore, the output gap multiplier decreases in $\gamma _r$ , whereas the inflation multiplier increases in $\gamma _r$ . I summarize the above theoretical results in Proposition 2.

Proposition 2. In normal times, the output gap multiplier $\mathcal{M}_{S,N}^{O}$ decreases in $\gamma _r$ , whereas the inflation multiplier $\mathcal{M}_{S,N}^{I}$ increases in $\gamma _r$ . The output gap multiplier relationship among the three models is

(14) \begin{equation} \mathcal{M}_{S,N}^{O,N}\lt \mathcal{M}_{S,N}^{O}\lt \mathcal{M}_{S,N}^{O,B}, \end{equation}

and the inflation multiplier relationship is

(15) \begin{equation} \mathcal{M}_{S,N}^{I,B}\lt \mathcal{M}_{S,N}^{I}\lt \mathcal{M}_{S,N}^{I,N}, \end{equation}

where in normal times: $\mathcal{M}_{S,N}^{O,B}$ is the output gap multiplier without the cost channel and $\mathcal{M}_{S,N}^{O,N}$ is with the nominal cost channel; $\mathcal{M}_{S,N}^{I,B}$ is the inflation multiplier without the cost channel, and $\mathcal{M}_{S,N}^{I,N}$ is with the nominal cost channel.

Proof. See Online Appendix C.

2.3.2. Government spending multipliers at ZLB

In this part, I focus on the case when the short-run economy is stuck at the ZLB.Footnote ²⁰ The Euler equation (3) and the Phillips Curve with the real cost channel [equation (4)] can be rewritten as:

(16) \begin{align} y_S& =-\frac{1}{\sigma (1-p)}[\log\! (\beta )-p\pi _S]+g_S \end{align}

(17) \begin{align} \pi _S& =\frac{\kappa \gamma _y}{1-\beta p+\kappa \gamma _rp}y_S+\kappa \frac{\gamma _g }{1-\beta p+\kappa \gamma _rp}g_S+\frac{\kappa \gamma _r\log\! (\beta )}{1-\beta p+\kappa \gamma _rp}. \end{align}

Fiscal multipliers

In liquidity traps, one can use the Euler equation and the Phillips Curve to obtain the solutions with the real cost channel of the output gap multiplier $\mathcal{M}_{S,Z}^{O}$ and the inflation multiplier $\mathcal{M}_{S,Z}^{I}$ :Footnote ²¹

(18) \begin{align} \mathcal{M}_{S,Z}^{O}=\frac{\sigma (1-p)[1-\beta p+\kappa \gamma _rp]+\kappa \gamma _gp}{\sigma (1-p)(1-\beta p+\kappa \gamma _rp)-\kappa \gamma _yp} \end{align}

(19) \begin{align} \mathcal{M}_{S,Z}^{I}=\frac{ \bigg [1+\frac{\gamma _g}{\gamma _y}\bigg ] \kappa \gamma _y\sigma (1-p)}{\sigma (1-p)(1-\beta p+\kappa \gamma _rp)-\kappa \gamma _yp}. \end{align}

From the solutions, the expected real interest rate can be in both the denominator and numerator of the output gap multiplier. The real cost channel here can influence the inflation rate and further impact the output gap through expectations. The real cost channel can impact the inflation multiplier directly through the NK Phillips Curve.

In liquidity traps, the denominator of the multiplier equation is lower than the numerator; thus, the output gap at the ZLB is larger than one. At the ZLB, with no cost channel, nominal interest rates remain unchanged, and an increase in government spending can generate inflation. Therefore, government spending leads to a drop in real interest rates, which, in turn, gives incentives to households to save less and consume more. This can be seen as the crowding effects as in Bouakez et al. (Reference Bouakez, Guillard and Roulleau-Pasdeloup2017).

However, following positive spending shocks, the real cost channel can decrease short-run inflation through expectations of disinflation in firms’ real borrowing costs. Intuitively, lower marginal costs arise due to expected disinflation during liquidity trap episodes. Further, expectations of disinflation can translate into realized lower inflation rates through rational expectations and sticky prices. In this case, the real cost channel can make inflation rates rise by less than in the conventional model following positive spending shocks, and there is a smaller drop in real interest rates. Higher real interest rates due to the real cost channel can depress not only people’s appetite for consumption but also decrease production activity, which results in a decline in the effects of government spending on output. Hence compared to the standard NK model without the real cost channel, both the output gap multiplier and the inflation multiplier with the real cost channel are lower.

On the other hand, as explained in Section 2.1.1, the nominal cost channel cannot modify the NK Phillips Curve slope at the ZLB. Spending multipliers with this nominal cost channel can be invariant with the standard NK model since this channel in liquidity traps cannot be included in the partial derivative of government spending to the output gap/inflation in the calculation of fiscal multipliers.

I next discuss the effects of the strength of the real cost channel $\gamma _r$ on spending multipliers. The stronger the power of the real cost channel, the lower inflation rates due to expectations of disinflation in liquidity traps. In this case, higher real interest rates due to lower inflation can depress people’s appetite for consumption, which results in more decline in the output gap multiplier. In a nutshell, both the inflation and the output gap multipliers with the real cost channel decrease in $\gamma _r$ . I summarize the main theoretical results in Proposition 3.

Proposition 3. At the ZLB, both the output gap multiplier $\mathcal{M}_{S,Z}^{O}$ and the inflation multiplier $\mathcal{M}_{S,Z}^{I}$ decrease in $\gamma _r$ . The output gap multiplier relationship among the three models is

(20) \begin{equation} \mathcal{M}_{S,Z}^{O}\lt \mathcal{M}_{S,Z}^{O,N}=\mathcal{M}_{S,Z}^{O,B}, \end{equation}

and the inflation multiplier relationship is

(21) \begin{equation} \mathcal{M}_{S,Z}^{I}\lt \mathcal{M}_{S,Z}^{I,N}=\mathcal{M}_{S,Z}^{I,B}, \end{equation}

where in liquidity traps: $\mathcal{M}_{S,Z}^{O,B}$ is the output gap multiplier without the cost channel and $\mathcal{M}_{S,Z}^{O,N}$ is with the nominal cost channel; $\mathcal{M}_{S,Z}^{I,B}$ is the inflation multiplier without the cost channel and $\mathcal{M}_{S,Z}^{I,N}$ is with the nominal cost channel.

Proof. See Online Appendix D.

Comparison with the literature

Our study contrasts with that of Abo-Zaid (Reference Abo-Zaid2022) who shows that higher borrowing costs with the nominal cost channel can increase inflation which makes the output gap multiplier larger than in the standard model at the ZLB. As in Abo-Zaid (Reference Abo-Zaid2022), the Phillips Curve with the nominal cost channel at the ZLB is steeper in a $(y_S,\pi _S)$ graph than in the standard NK model.Footnote ²² In this paper, I stress that inflation in the short run with the real cost channel can be lower due to expected disinflation in real borrowing costs.Footnote ²³ Thus, I can use the real cost channel to explain lower spending multipliers in liquidity traps as in Ramey and Zubairy (Reference Ramey and Zubairy2018).

Taking stock of the denominator of the multiplier

As discussed extensively in Bilbiie (Reference Bilbiie2008), Mertens and Ravn (Reference Mertens and Ravn2014), Borağan Aruoba et al. (Reference Aruoba, Cuba-Borda and Schorfheide2018), and Nie and Roulleau-Pasdeloup (Reference Nie and Roulleau-Pasdeloup2023), the denominator of the multiplier in the standard NK model at the ZLB can be negative due to a highly persistent $p$ .Footnote ²⁴ In this case, the output gap and inflation multipliers are negative in equations (18)–(19).

However, as described at length in Nie (Reference Nie2021), in the presence of the real cost channel, the denominator with the ZLB binding is less likely to be negative than in the standard model.Footnote ²⁵ I use the result from Nie (Reference Nie2021): The denominator can be always positive if assuming the elasticity of marginal costs w.r.t. the output gap $\gamma _y$ is sufficiently small and meets the following condition:

(22) \begin{align} \gamma _y\lt \Gamma (\gamma _r), \end{align}

where $\Gamma (\gamma _r)=\frac{(\beta -\kappa \gamma _r-1+\kappa \gamma _r\phi _\pi )\gamma _r\phi _\pi (\beta -\kappa \gamma _r)}{\sigma _r(1-\kappa \gamma _r\phi _\pi )}$ increases in $\gamma _r$ —See Online Appendix E.

As in condition (22), $\gamma _y$ is lower than the composite parameter $\Gamma (\gamma _r)$ , ensuring that the denominator is always positive. Assuming that $\gamma _y$ is small enough for a given $\gamma _r$ , this condition can always hold. In particular, the assumption with a low $\gamma _y$ is in line with the empirical finding as in Beaudry et al. (Reference Beaudry, Hou and Portier2022): $\gamma _y$ is very small while the key parameter $\gamma _r$ leveraging the strength of the real cost channel is much larger than $\gamma _y$ .

Bearing this in mind, in the following numerical exercise, I will consider only the case in which the denominator of the spending multiplier is positive.

2.4. Numerical results: benchmark

In this paper, I follow Budianto et al. (Reference Budianto, Nakata and Schmidt2020), Roulleau-Pasdeloup (Reference Roulleau-Pasdeloup2021), and Beaudry et al. (Reference Beaudry, Hou and Portier2022) to set the parameterization reported in Table 1.Footnote ²⁶

Table 1. Parameterization

I first provide numerical AS/AD figures with a contractionary natural rate shock in three NK Phillips Curves shown in Figure 1. The natural rate shock can move the Euler equation. If there is a temporary short-term natural rate shock $r^n_S$ (−2.2%), which exceeds the boundary condition, the economy can be in liquidity traps.Footnote ²⁷ In this case, the conventional monetary policy cannot work since the nominal interest rate is bounded at zero. Deflationary pressure can lead to higher real interest rates and further stimulate people to save but consume less. From Figure 1, it is clear that the real cost channel can change the Phillips Curve slope in liquidity traps and the Phillips Curve is locally flat as in Beaudry et al. (Reference Beaudry, Hou and Portier2022). The curve with the nominal cost channel at the ZLB shares the same slope as the conventional model. The numerical results also echo Proposition 1 that the case with the nominal cost channel is the most easily entrapped in liquidity traps among the three models.Footnote ²⁸ This Proposition can be further confirmed in the impulse response to a contractionary natural rate shock—See Online Appendix F.

Figure 1. AS/AD curves with natural rate shock in three models.

I compare spending multipliers of the models in normal times without the cost channel ( $\gamma _r=0$ ) vs. with the cost channel as in Ravenna and Walsh (Reference Ravenna and Walsh2006) vs. with the real cost channel in Figure 2.Footnote ²⁹ It can be seen that the output gap multiplier is lower than one as in Lewis (Reference Lewis2021), and it decreases in $\gamma _r$ . On the contrary, the inflation multiplier increases in $\gamma _r$ . It is observed that the output gap multiplier with the real cost channel is larger than that with the nominal cost channel while less than its counterpart without the cost channel. This can echo the theoretical analysis in Proposition 2. See Online Appendix G for numerical results w.r.t. the persistence $p$ in the three models. These numerical results robustly echo Proposition 2. I also find that the output gap multiplier decreases in $p$ , whereas the inflation multiplier is higher with the duration of increment of time.

Figure 2. Spending multipliers in normal times.

Figure 3 shows spending multipliers in liquidity traps. The simulation results can illustrate quantitatively our theoretical analysis in Proposition 3. On the one hand, one can see that the output gap multiplier is effective as in Christiano et al. (Reference Christiano, Eichenbaum and Evans2011) and Schmidt (Reference Schmidt2017). The real cost channel can attenuate the output gap and inflation multipliers simultaneously. The spending multipliers decrease in $\gamma _r$ . The nominal cost channel multipliers are invariant with the standard NK model without the cost channel.Footnote ³⁰ See Online Appendix G for numerical results w.r.t. $p$ in the three models, and these results further confirm our theoretical analysis. In addition, the output gap and inflation multipliers increase in $p$ .

Figure 3. Spending multipliers at ZLB.

3.1. Policy and shocks

I assume that government spending is longer-lived than demand shock. To be more specific, government spending in the short run $g_S$ can merge into medium-run government spending $g_M$ with a persistence $q$ and then collapse to the steady-state $g_L=0$ with a probability $1-q$ .Footnote ³¹ The natural rate shock $r^n_S$ operates in the short run with a probability $p$ and then returns to the long run $r^n_L=0$ with a probability $1-p$ . The graphical representation of our policy can be seen in Figure 4.

Figure 4. Long-run government spending: a graphical representation.

With this in mind, for current monetary policy, I use an adapted Taylor rule:Footnote ³²

(23) \begin{equation} R_t = \begin{cases} \max\! \left [0;-\log\! (\beta )+\phi _\pi \pi _S\right ]\,\, \, \, \, \, \, \, \, \, \, \,\,\,\text{In the short run} \\ -\log\! (\beta )+\phi _\pi ^q \pi _M\, \hphantom{max\left [0;+r_S^n\ \right ]}\, \text{In the medium run}\\ -\log\! (\beta )\ \hphantom{\max \left [0;+r_S^n+\phi _\pi \pi _S\, \right ]}\, \, \,\text{In the long run} \end{cases} \end{equation}

One can use the above equation (23) to trace the path of government spending. In the medium run, the output gap and inflation can be expressed as the product of medium-run multipliers and medium-run government spending as follows:

(24) \begin{equation} y_M=\mathcal{M}_{M}^{O}\cdot g_M \, \, \, \, \pi _M=\mathcal{M}_{M}^{I} \cdot g_M, \end{equation}

where $\mathcal{M}_{M}^{O}$ is the medium-run output gap multiplier and $\mathcal{M}_{M}^{I}$ is the medium-run inflation multiplier. See Online Appendix I for the medium-run spending multipliers.Footnote ³³ In this case, since the model is forward-looking, I can generate the expected output gap below, and one can show the expected inflation using the same manner.

(25) \begin{align} \notag \mathbb{E}_Sy_{t+1}&=py_S+(1-p)qy_M\\ \notag &=py_S+(1-p)q\mathcal{M}_{M}^{O}g_M\\ &=py_S+(1-p)q\mathcal{M}_{M}^{O}\zeta g_S. \end{align}

3.2. Long-run government spending: theoretical analysis

This section discusses the analytical results regarding the government spending multiplier with a three-state Markov structure.

3.2.1. Multipliers with long-run government spending in normal times

In normal times, one can use the Euler equation and the Phillips Curve in Online Appendix J to compute the output gap multiplier $\mathcal{M}_{S,N}^{O,\text{long}}$ and the inflation multiplier $\mathcal{M}_{S,N}^{I,\text{long}}$ :

(26) \begin{align} \mathcal{M}_{S,N}^{O,\text{long}}&=\frac{\Theta _{AD}\sigma (1-p)[1-\beta p-\kappa \gamma _r(\phi _\pi -p)]-\Theta _{AS}(\phi _\pi -p)}{\sigma (1-p)[1-\beta p-\kappa \gamma _r(\phi _\pi -p)]+\kappa \gamma _y(\phi _\pi -p)} \end{align}

(27) \begin{align} \mathcal{M}_{S,N}^{I,\text{long}}&=\frac{ \bigg (\kappa \gamma _y\Theta _{AD}+\Theta _{AS}\bigg ) \sigma (1-p)}{\sigma (1-p)[1-\beta p-\kappa \gamma _r(\phi _\pi -p)]+\kappa \gamma _y(\phi _\pi -p)}. \end{align}

Long-run government spending can increase short-run inflation through rational expectations and sticky prices. In normal times, the central bank increases nominal interest rates to combat higher prices caused by inflation. In that way, long-run spending policy can crowd out more private consumption due to higher short-run real interest rates. Additionally, the real cost channel can increase marginal costs and inflation. Thus, the inflation multiplier can increase depending on the strength of the real cost channel $\gamma _r$ , and this result is the same as the case in Section 2. Similarly, the real cost channel can lower the output gap. In summary, the output gap multiplier reduces in $\gamma _r$ whereas the inflation multiplier grows in $\gamma _r$ —For a formal proof, see Online Appendix K.

3.2.2. Multipliers with long-run government spending at ZLB

At the ZLB, I use the Euler equation and the Phillips Curve in Online Appendix J to produce the output gap multiplier $\mathcal{M}_{S,Z}^{O,\text{long}}$ and the inflation multiplier $\mathcal{M}_{S,Z}^{I,\text{long}}$ :

(28) \begin{align} \mathcal{M}_{S,Z}^{O,\text{long}}&=\frac{\Theta _{AD}\sigma (1-p)( 1-\beta p+\kappa \gamma _rp)+\Theta _{AS}p}{\sigma (1-p)(1-\beta p+\kappa \gamma _rp)-\kappa \gamma _yp} \end{align}

(29) \begin{align} \mathcal{M}_{S,Z}^{I,\text{long}}&=\frac{ \bigg (\kappa \gamma _y\Theta _{AD}+\Theta _{AS}\bigg ) \sigma (1-p)}{\sigma (1-p)(1-\beta p+\kappa \gamma _rp)-\kappa \gamma _yp}. \end{align}

Once the liquidity trap has subsided, the extension of government spending policy can cause inflation to increase. The nominal interest rate is zero, and higher inflation can help stimulate our NK economy since it can lower short-run real interest rates and then increase private consumption. At this time, the spending multiplier is larger, which is in line with Bachmann and Sims (Reference Bachmann and Sims2012) who empirically show that long-term spending effects are larger than in the short-term policy model. However, the real cost channel can reduce the effectiveness of long-run government spending. The cost channel can dampen inflation due to expectations of disinflation in the real borrowing cost. Thus, the output gap and inflation multipliers decrease as the real cost channel becomes stronger—See Online Appendix L for an analytical analysis.

3.3. Numerical results with long-run government spending

The numerical results of short-run and long-run spending policies that vary with the strength of the real cost channel $\gamma _r$ are presented in Figure 5.Footnote ³⁴ In normal times, it is observed that long-run government spending can decrease the output gap multiplier. Moreover, long-run government spending can result in a larger inflation multiplier compared to short-run policies. The output gap multiplier falls in $\gamma _r$ while the inflation multiplier increases in $\gamma _r$ . See Online Appendix M for numerical results w.r.t. the persistence $p$ . I find that the output gap multiplier can be negative with long-run spending.

Figure 5. Spending multipliers with long-run spending policy in normal times.

At the ZLB, the numerical results are reported in Figure 6. We find that long-run government spending can further increase the output gap and inflation multipliers. The two spending multipliers decrease in $\gamma _r$ . Noteworthy is that the output gap multiplier can be more effective for a prolonged spending policy in recessions. Similar results can be found in Online Appendix M for numerical results w.r.t. $p$ .

Figure 6. Spending multipliers with long-run spending policy at ZLB.

4.1. The NK model with Bounded Rationality

I use the model in Gabaix (Reference Gabaix2020) to show the behavior of the aggregate demand-side economy:

(30) \begin{equation} c_t=\bar m\mathbb{E}_t c_{t+1}-\frac{1}{\sigma _c}\left (i_t-\mathbb{E}_t\pi _{t+1}-r^n\right ), \end{equation}

where $\bar m \in [0,1]$ is the cognitive discounting parameter in Gabaix (Reference Gabaix2020).Footnote ³⁵

In this case, one can obtain the path of $y_t$ by substituting the resource constraint into the Euler equation:

(31) \begin{equation} y_t=\bar m(1-s_g)\mathbb{E}_t y_{t+1}-\frac{1}{\sigma }(R_t+\log\! (\beta )-\mathbb{E}_t\pi _{t+1}-r^n)+g_t-\bar m(1-s_g)\mathbb{E}_t g_{t+1}, \end{equation}

where $\sigma =\frac{\sigma _c}{1-s_g}$ .

As in Gabaix (Reference Gabaix2020), I can derive the NK Phillips Curve with the real cost channel:

(32) \begin{equation} \pi _t=\beta \bar m\bigg [\varphi +\frac{1-\beta \varphi }{1-\beta \varphi \bar{m}}(1-\varphi )\bigg ]\mathbb{E}_t \pi _{t+1}+\kappa \left [\gamma _y y_t+\gamma _g g_t+\gamma _r(R_t+\log\! (\beta )-\mathbb{E}_t \pi _{t+1})\right ], \end{equation}

where $\varphi \in (0,1)$ is the share of firms that cannot adjust their prices.

4.2. Government spending with bounded rationality: theoretical results

This section presents the analytical results regarding government spending multipliers with bounded rationality.

4.2.1. Multipliers with bounded rationality in normal times

In normal times, I use the Euler equation and the Phillips Curve with the consideration of bounded rationality—see Online Appendix N, to produce the solutions of the output gap multiplier $\mathcal{M}_{S,N}^{O,BR}$ and the inflation multiplier $\mathcal{M}_{S,N}^{I,BR}$ below:Footnote ³⁶

(33) \begin{align} \mathcal{M}_{S,N}^{O,BR}=\frac{\sigma [1-p\bar m(1-s_g)]\bigg \{1-\beta p\bar m\bigg [\varphi +\frac{1-\beta \varphi }{1-\beta \varphi \bar{m}}(1-\varphi )\bigg ]-\kappa \gamma _r(\phi _\pi -p)\bigg \}-\kappa \gamma _g(\phi _\pi -p)}{\sigma (1-p\bar m(1-s_g))\bigg \{1-\beta p\bar m\bigg [\varphi +\frac{1-\beta \varphi }{1-\beta \varphi \bar{m}}(1-\varphi )\bigg ]-\kappa \gamma _r(\phi _\pi -p)\bigg \}+\kappa \gamma _y(\phi _\pi -p)} \end{align}

(34) \begin{align} \mathcal{M}_{S,N}^{I,BR}=\frac{ \bigg [1+\frac{\gamma _g}{\gamma _y}\bigg ] \kappa \gamma _y\sigma [1-p\bar m(1-s_g)]}{\sigma [1-p\bar m(1-s_g)]\bigg \{1-\beta p\bar m\bigg [\varphi +\frac{1-\beta \varphi }{1-\beta \varphi \bar{m}}(1-\varphi )\bigg ]-\kappa \gamma _r(\phi _\pi -p)\bigg \}+\kappa \gamma _y(\phi _\pi -p)}. \end{align}

In normal times, cognitive discounting can reduce the expectation effects in our baseline model which can lower inflation following a positive spending shock. This means that there is a lower rise in real interest rates, hence a higher output gap multiplier compared to the baseline model. In this case, the inflation multiplier with bounded rationality can be lower, whereas the output gap multiplier can be higherFootnote ³⁷ —see Online Appendix O for a formal proof. Since the introduced bounded rationality is independent of the strength of the real cost channel $\gamma _r$ , the effects of this channel on spending multipliers are the same as the baseline model results. Specifically, the output gap multiplier decreases in $\gamma _r$ while the inflation multiplier increases.

I find that the output gap multiplier can be large (near one) in normal times if we suppose agents have relatively intense bounded rationality which means the expectation effects are extremely weak. Intuitively, agents tend to consume today but not to save since future consumption has less or no effect on today’s decisions. In this case, the crowding-out effects of government spending should be much attenuated, and the output gap multiplier can be near one. This result echoes the previous empirical evidence that the output gap multiplier can be large in normal times as in Auerbach and Gorodnichenko (Reference Auerbach and Gorodnichenko2012) and Acconcia et al. (Reference Acconcia, Corsetti and Simonelli2014).

4.2.2. Multipliers with bounded rationality at ZLB

At the ZLB, one can use the Euler equation and the Phillips Curve with bounded rationality—see Online Appendix N, to produce the output gap multiplier $\mathcal{M}_{S,Z}^{O,BR}$ and the inflation multiplier $\mathcal{M}_{S,Z}^{I,BR}$ :

(35) \begin{align} \mathcal{M}_{S,Z}^{O,BR}=\frac{\sigma [1-p\bar m(1-s_g)]\bigg \{1-\beta p\bar m\bigg [\varphi +\frac{1-\beta \varphi }{1-\beta \varphi \bar{m}}(1-\varphi )\bigg ]+\kappa \gamma _rp\bigg \}+\kappa \gamma _gp}{\sigma [1-p\bar m(1-s_g)]\bigg \{1-\beta p\bar m\bigg [\varphi +\frac{1-\beta \varphi }{1-\beta \varphi \bar{m}}(1-\varphi )\bigg ]+\kappa \gamma _rp\bigg \}-\kappa \gamma _y p} \end{align}

(36) \begin{align} \mathcal{M}_{S,Z}^{I,BR}=\frac{ \bigg [1+\frac{\gamma _g}{\gamma _y}\bigg ] \kappa \gamma _y\sigma [1-p\bar m(1-s_g)]}{\sigma [1-p\bar m(1-s_g)]\bigg \{1-\beta p\bar m\bigg [\varphi +\frac{1-\beta \varphi }{1-\beta \varphi \bar{m}}(1-\varphi )\bigg ]+\kappa \gamma _r p\bigg \}-\kappa \gamma _y p}. \end{align}

At the ZLB, inflation caused by government spending can be lower due to cognitive discounting. Thus, it can increase real interest rates by less than in the standard model, which, in turn, can lower the output gap multiplier.Footnote ³⁸ The effects on the strength of the real cost channel are the same as for the baseline model in liquidity traps. Specifically, the output gap and inflation multipliers decrease in $\gamma _r$ —see Online Appendix P for a proof.

4.3. Numerical results with bounded rationality

Figure 7 shows how spending multipliers with bounded rationality vary with the value of $\gamma _r$ for various values of $\bar m$ in normal times. The main takeaway is that the output gap multiplier can be larger with a lower $\bar m$ while the inflation multiplier can be lower under bounded rationality. In addition, the output gap multiplier decreases in $\gamma _r$ , whereas the inflation multiplier increases in $\gamma _r$ . See Online Appendix Q for numerical results w.r.t. $p$ . I find that the output gap multiplier can decrease in $p$ with rational expectations ( $\bar m=1$ ) but increase in $p$ with a small $\bar m$ .

Figure 7. Spending multipliers with bounded rationality in normal times.

The numerical simulation in Figure 8 shows that the inflation and output gap multipliers are lower with bounded rationality in liquidity traps. In addition, the role of the real cost channel with bounded rationality is the same as that of the baseline model. As in Online Appendix Q, the output gap multiplier increases numerically in $p$ .

Figure 8. Spending multipliers with bounded rationality at ZLB.