2.1. The demand side

We define consumer utility as a function of consumption $C_t$ and hours worked, $N_t$ , and maximize the utility of a representative agent with a standard budget constraint.Footnote ¹ Log-linearizing the optimizing conditions yield the following relationships, with lower-case letters denoting the natural logarithm of the corresponding variables:

(1) \begin{equation} mrs_t = \frac{\sigma }{1-h}(c_t-h c_{t-1})+vn_t \end{equation}

(2) \begin{equation} c_t = \frac{1}{1+h}E_tc_{t+1} + \frac{h}{1+h}c_{t-1} - \frac{(1-h)}{\sigma (1+h)}(i_t-E_t\pi _{t+1}) \end{equation}

where $mrs_t$ is the marginal rate of substitution between consumption and hours worked, and $h \in (0,1)$ measures (external) habit persistence. In equation (1), the parameter, $v$ , denotes the inverse of real wage elasticity of labor supply. Equation (2) suggests that current consumption is positively linked to lagged and future consumption and inversely related to real rate of interest ( $i_t-E_t\pi _{t+1}$ ). Using the market clearing condition ( $x_t = c_t$ ) in (2), we derive the following equation of the forward-looking IS curve:

(3) \begin{equation} x_t = \frac{1}{1+h}E_tx_{t+1} + \frac{h}{1+h}x_{t-1} - \frac{(1-h)}{\sigma (1+h)}(i_t-E_t\pi _{t+1})+u^x_t \end{equation}

where $x_t$ is the logarithm of the output gap, $E_t\pi _{t+1} = E_t p_{t+1} - p_t$ is the expected inflation rate, $p_t$ is the price of the good, and $u^x_t$ is the demand shock. $\sigma$ measures the intertemporal substitution of households or the degree of relative risk aversion.

2.2. The supply side

The key equation of the model is the forward-looking Phillips curve, derived in Galí (Reference Galí2015):

(4) \begin{equation} \pi _t = \chi E_t\pi _{t+1} + \lambda \text{rmc}_t \end{equation}

where $\text{rmc}_t$ is the log of real marginal cost of the firm (which measures the cost of producing an extra unit of output), $\chi$ is the firm discount rate, and $\lambda = \frac{(1-\eta )(1-\chi \eta )}{\eta }$ , where $\eta$ is the proportion of firms which cannot change their prices (or the probability that a firm cannot change the price). The derivation of the Phillips curve is primarily based on Calvo (Reference Calvo1983) and Galí and Gertler (Reference Galí and Gertler1999); the key difference in our model is the inclusion of a cost channel of monetary policy (or supply-side effect of monetary policy). To do so, we assume that total real cost of the firm $\left(\frac{\text{TC}_t}{P_t}\right)$ captures the fact that firms may borrow money from financial intermediaries at the effective interest rate $(1+i_t)^{\theta }$ to finance their wage bill, with the parameter $\theta$ measuring the strength of the cost channel:Footnote ²

(5) \begin{equation} \frac{\text{TC}_t}{P_t} = \frac{W_t(1+i_t)^{\theta } N_t}{P_t} \end{equation}

From equation (5), we derive the following real marginal cost ( $\text{RMC}_t$ ) equation:

(6) \begin{equation} \text{RMC}_t = \frac{W_t(1+i_t)^{\theta }}{\text{MPN}_t P_t} \end{equation}

where $\text{MPN}_t$ is defined as the marginal productivity of labor. While the mechanism is intuitive, several studies have confirmed that the nominal interest rate factors into firms’ cost of production. For instance, Ravenna and Walsh (Reference Ravenna and Walsh2006), Tillmann (Reference Tillmann2008) and, more recently, Abo-Zaid (Reference Abo-Zaid2020) find evidence of an active cost channel in the US economy and its impact on estimates of the government spending multiplier. Ravenna and Walsh (Reference Ravenna and Walsh2006) estimate a range of values for the cost channel, $\theta$ , which were as low as 1.276 and as high as 11.831. Chowdhury et al. (Reference Chowdhury, Hoffmann and Schabert2006) detect a cost channel in Canada, France, Italy, the UK, and the USA. Dedola and Lippi (Reference Dedola and Lippi2005) find evidence of a cost channel in France, Germany, Italy, and the UK, while Barth III and Ramey (Reference Barth and Ramey2001) confirm these results. With $\text{mpn}_t = \ln \text{MPN}_t$ and $\text{rmc}_t = \ln \text{RMC}_t$ , we derive the following relationship:

(7) \begin{equation} \text{rmc}_t = w_t - p_t + \theta \ln\!(1+i_t)-\text{mpn}_t \end{equation}

Using the log-linear production function ( $x_t = a_t+n_t$ ), where $a_t$ measures log of technology, and $\ln\!(1+i_t) \approx i_t$ , we can write equation (7) as follows:

(8) \begin{equation} \text{rmc}_t = w_t - p_t + \theta i_t-\text{mpn}_t \end{equation}

Combining equations (4) and (8) gives us the following Phillips curve, where we assume $a_t = 0$ for convenience:

(9) \begin{equation} \pi _t = \chi E_t\pi _{t+1} + \lambda (w_t - p_t + \theta i_t) \end{equation}

Equation (9) derives the key mechanism in our model, which links the cost channel of monetary policy with current inflation, and plays an important role in determining the NFE as demonstrated by the analytical results in the next section.

Following Blanchard and Galí (Reference Blanchard and Galí2007) and Smith (Reference Smith2016), we further assume that real wages change sluggishly over time. Hall (Reference Hall2005) points out the importance of real wage rigidity to account for a number of important labor market facts. Blanchard and Galí (Reference Blanchard and Galí2007) document that the inclusion of real wage rigidity in the NK Phillips curve gives rise to a trade-off between stabilizing inflation and stabilizing the gap between output and desired output as perceived by most central banks, leading to the following equation:

(10) \begin{equation} w_t - p_t = \beta (w_{t-1} - p_{t-1})+(1-\beta )mrs_t \end{equation}

where $\beta = 0$ means complete real wage flexibility and $\beta = 1$ models complete real wage rigidity. Using equation (1) and the market equilibrium condition ( $x_t = c_t$ ) in equation (10) and again setting $a_t =0$ , the real wage equation can be written as follows:

(11) \begin{equation} w_t - p_t = \beta (w_{t-1} - p_{t-1})+(1-\beta )\left (\frac{\sigma +v(1-h)}{1-h}x_t - \frac{\sigma h}{1-h}x_{t-1} \right ) \end{equation}

Finally, combining (9) and (11) we arrive at the following forward-looking Phillips curve that includes both real wage rigidity ( $\beta$ ) and a cost channel of monetary policy ( $\theta$ ):Footnote ³

(12) \begin{equation} \pi _t = \chi E_t\pi _{t+1} + \lambda \left [\beta (w_{t-1} - p_{t-1})+(1-\beta )\left (\frac{\sigma +v(1-h)}{1-h}x_t - \frac{\sigma h}{1-h}x_{t-1} \right )+\theta i_t\right ] \end{equation}

2.3. The monetary authority

We close our model by defining the monetary policy reaction function.Footnote ⁴ The central bank uses the short-term interest rate ( $i_t$ ) to react to deviations of inflation and aggregate output from their respective targets. The inclusion of lagged interest rates gives rise to interest rate smoothing that replicates actual interest rate policy in many advanced economies (Galí (Reference Galí2015)). The parameter $\xi$ is the smoothing coefficient of the short-term interest rate; $\phi ^\pi$ is the responsiveness of the monetary authority to inflation, and $\phi ^x$ is the coefficient on the output gap:

(13) \begin{equation} i_t = \xi i_{t-1} + (1-\xi )(\phi ^\pi \pi _t + \phi ^x x_t)-u^i_t \end{equation}

Different from the inflation target shock considered in Garín et al. (Reference Garín, Lester and Sims2018), we study the impact of an expansionary unanticipated (or exogenous) monetary policy shock on the dynamics of interest rate, inflation, and output. We assume that the monetary policy shock follows an AR(1) process, where $u^i_t = \rho u^i_{t-1} + \varepsilon _t$ , enters with a negative sign in the policy rule to denote expansionary monetary policy, and $\varepsilon _t \sim N(0, \sigma ^2_{\varepsilon })$ and $\rho \in [0,1)$ . The parameter $\rho$ measures the persistence in the monetary innovation shock. If $\rho = 0$ , then the shock has no persistence and is purely stochastic; the shock is permanent when $\rho = 1$ , but this in turn induces non-stationarity in $u^i_t$ . Since the trajectory of all the endogenous variables depends on $u^i_t$ , inflation and output gap also become non-stationary. To avoid the non-stationarity problem, we follow Garín et al. (Reference Garín, Lester and Sims2018) and solve the model to derive closed-form solution for stationary cases only (i.e., when $\rho \in [0,1)$ ).

3.1. Unanticipated monetary policy shocks

To focus on the main mechanisms, we rely on three assumptions: (i) no external habit persistence ( $h=0$ ), (ii) wages are perfectly flexible ( $\beta =0$ ), and (iii) central bank does not smooth interest rates ( $\xi =0$ ). These frictions are added back to the model in Section 4. Our model, then, simplifies to the following structural form:

(14) \begin{align} x_{t} = E_{t}x_{t+1}-\frac{1}{\sigma }(i_{t}-E_{t}\pi _{t+1}) \end{align}

(15) \begin{align} \pi _{t} = \chi E_{t}\pi _{t+1}+\gamma _{1}x_{t} +\gamma _{2}i_{t} \end{align}

(16) \begin{align} i_{t} = \phi ^{\pi } \pi _{t}+\phi ^{x}x_{t}-u_{t}^{i} \end{align}

(17) \begin{align} u_{t}^{i} = \rho u_{t-1}^{i} +\varepsilon _{t} \end{align}

where $\gamma _{1}=\lambda (v+\sigma )$ ; $\gamma _2=\lambda \theta$ ; $\lambda = \eta ^{-1}(1-\eta )(1-\eta \chi )$ ; $\varepsilon \sim (0,\sigma _{\varepsilon }^{2})$ ; $0\leq \rho \lt 1$ . To solve the model analytically, we assume the following trial solution:

(18) \begin{equation} x_{t} = a u_{t}^{i} ; \; \pi _{t}=cu_t^i \end{equation}

Substituting equations (16) into (14) and (15), using the trial solution as given in (18), and finally using the method of undetermined coefficients, we derive the following set of restrictions on parameters $a$ and $c$ :

(19) \begin{align} (\gamma _1+\gamma _2+\phi ^{x})a-(1-\gamma _2 \phi ^{x}-\rho \chi )c=\gamma _2 \end{align}

(20) \begin{align} (1+\sigma ^{-1}\phi ^{x}-\rho )a+\sigma ^{-1}(\phi ^{\pi }-\rho )c=\sigma ^{-1} \end{align}

Applying Cramer’s rule on equations (19) and (20) and using the definition of the real rate of interest (i.e., $r_{t}=i_{t}-E_{t}\pi _{t+1}$ ), we derive the following multipliers:

(21) \begin{align} \dfrac{dx_t}{du_{t}^{i}} = a = \dfrac{\sigma ^{-1}[1-\rho \chi -\lambda \theta \rho ]}{\Delta } \end{align}

(22) \begin{align} \dfrac{d\pi _t}{du_{t}^{i}} = c = \dfrac{\lambda [\sigma ^{-1}(v+\sigma )-\theta (1-\rho )]}{\Delta } \end{align}

(23) \begin{align} \dfrac{di_t}{du_{t}^{i}} = \dfrac{\lambda (v+\sigma )\rho \sigma ^{-1}-(1-\rho )(1-\rho \chi )}{\Delta } \end{align}

(24) \begin{align} \dfrac{dr_t}{du_{t}^{i}} = \dfrac{(1-\rho )[\theta \lambda \rho -(1-\rho \chi )]}{\Delta } \end{align}

$\Delta = (1+\sigma ^{-1}\phi ^{x}-\rho )(1-\rho \chi )+\sigma ^{-1}\lambda (v+\sigma )(\phi ^{\pi }-\rho )-\lambda \theta (\sigma ^{-1}\rho \phi ^{x}+\phi ^{\pi }(1-\rho ))$ . We now focus on discussing the key mechanisms that give rise to and avert the NFE.

3.1.1. No cost channel of monetary policy

To first align the results of the baseline model with that of Garín et al. (Reference Garín, Lester and Sims2018), we illustrate the NFE with no cost channel. A key difference is that we consider an exogenous interest rate shock as compared to their inflation target shock.Footnote ⁵ One advantage of deriving their key results is to motivate how relaxing some of the underlying assumptions in their model can easily affect the existence of the NFE. Setting the cost channel, $\theta = 0$ , in equations (21)–(24), we can show that the response of key macroeconomic variables to a monetary policy shock, $\dfrac{dx_{t}}{du_{t}^{i}}\gt 0, \dfrac{d\pi _{t}}{du_{t}^{i}}\gt 0, \dfrac{dr_{t}}{du_{t}^{i}}\gt 0$ . With the assumption that $0\lt \rho \lt 1$ , we can also show that in the short run the nominal interest rate will increase and give rise to NFE if and only if the following condition holds:Footnote ⁶

(25) \begin{equation} \dfrac{(v+\sigma )}{\sigma } \gt \dfrac{(1-\rho )(1-\rho \chi )}{\rho } \dfrac{\eta }{(1-\eta )(1-\eta \chi )} \end{equation}

We extract additional insights from condition (25) to study the existence of the NFE. For example, for any given value of $\eta$ (which is the probability that a firm cannot change its prices, or a measure of price rigidity), if the shock is near permanent ( $\rho \rightarrow 1$ ), then $\dfrac{(1-\rho )(1-\rho \chi )}{\rho } \rightarrow 0$ , which implies that condition (25) is satisfied, that is, $\dfrac{(v+\sigma )}{\sigma }\gt 0$ . This result is consistent with that of Garín et al. (Reference Garín, Lester and Sims2018) who argue that in the NK model, the NFE occurs if the shock to inflation target is highly persistent ( $\rho \gt 0.60$ ).Footnote ⁷ Following Galí (Reference Galí2015), if we assume $v=\sigma =1$ and $\eta =0.70$ , we can show that the NFE will occur for $\rho \geq 0.61$ .

This result is also consistent with Schmitt-Grohé and Uribe (Reference Schmitt-Grohé and Uribe2014), who show that in a flexible price model, interest rate and inflation will co-move (i.e., the NFE) if the increase in the nominal interest rate is sustained for a prolonged period. When prices are almost rigid ( $\eta \rightarrow 1$ ), condition (25) will not hold and will consequently contradict the neo-Fisher hypothesis as long as $\rho \neq 1$ . To the contrary, if prices are almost perfectly flexible ( $\eta \rightarrow 0$ ) and $\rho \neq 0$ , condition (25) will also hold, a result consistent with Schmitt-Grohé and Uribe (Reference Schmitt-Grohé and Uribe2014). Extending these results, with perfectly flexible prices, inflation and interest rate move in the same direction even with low persistence in the monetary innovation shock. If we set $v=\sigma =1$ , then irrespective to the size of $\rho$ and $\eta$ , condition (25) will be satisfied automatically as long as $\rho =\eta$ .

The key takeaway from this analysis is that in a simple model, which lacks key structural attributes such as the cost channel of monetary policy, habit persistence in consumption, interest rate smoothing policy of the central bank, etc., it is the persistence of the monetary innovation shock and the degree of price flexibility along with forward-looking nature of the agents, which give rise to NFE. In what follows, we argued that these structural factors, apart from the persistence of the monetary shock, may also give rise to or avert NFE.

3.1.2. Cost channel of monetary policy and the NFE

We now focus on the role of cost channel as one of the structural factors that can give rise to the NFE.Footnote ⁸ In the analysis below, we compare the role of the cost channel relative to the persistence of the monetary policy shock, connecting this particular mechanism with the NFE literature.

Using equations (21) and (22),Footnote ⁹ we show that under condition (26), a decrease in the interest rate will lead to an increase in the rate of inflation (i.e., no NFE):Footnote ¹⁰

(26) \begin{equation} \theta (1-\rho ) \lt \dfrac{v+\sigma }{\sigma }\lt \dfrac{(1-\rho \chi )(1-\rho )}{\rho } \dfrac{\eta }{(1-\eta \chi )(1-\eta )} \end{equation}

The violation of condition (26) implies the existence of the NFE occurring if $\theta (1-\rho )\gt \dfrac{v+\sigma }{\sigma }$ or, in more detail, when $\dfrac{v+\sigma }{\sigma }\gt \dfrac{(1-\rho \chi )(1-\rho )}{\rho }\dfrac{\eta }{(1-\eta \chi )(1-\eta )}$ , which suggests that the NFE in the NK model is much more pronounced in the presence of a cost channel ( $\theta \gt 0$ ). A rise in the interest rate works through the supply side by increasing the marginal cost of firms that borrow to finance production, which leads to a contemporaneous increase in the current and the expected inflation. As the proportion of firms who borrow money to finance their production increases, the supply-side effects can potentially overwhelm the contractionary demand-side effects of the increase in the interest rate and give rise to inflation.

An interesting observation to note here is that even though the persistence of the shock gives rise to NFE in the baseline case, but in fact undermines the effect of cost channel of monetary policy – with higher shock persistence, the impact of cost channel of monetary policy on domestic inflation is muted. We note in condition (26) that in special case of permanent shock ( $\rho =1$ ), the cost channel is completely infused.

In principle, condition (26) makes the key point in that with a stronger cost channel, the NFE arises even for low persistence in monetary policy shocks. A key insight from this section is that NFE can also arise due to the presence of a cost channel, working through a mechanism that differs from earlier studies such as Uribe (Reference Uribe2018), García-Schmidt and Woodford (Reference García-Schmidt and Woodford2019), and Cochrane (Reference Cochrane2016), who show that a permanent change in the nominal interest rate causes the NFE in NK models. The high degree of persistence in the monetary policy shock is not required per se in the presence of a cost channel.

We also note the role of price rigidity in driving the NFE in the presence of a cost channel of monetary policy. From condition (26) if $\theta \gt \dfrac{v+\sigma }{\sigma (1-\rho )}$ , the NK model yields the NFE even when prices tend to almost completely rigidity (as $\eta \rightarrow 1$ ). This result extends that found in Schmitt-Grohé and Uribe (Reference Schmitt-Grohé and Uribe2014) and Garín et al. (Reference Garín, Lester and Sims2018) who argue that NFE arises in a dynamic optimizing model with flexible prices when the interest rate increases for a prolonged period. On the contrary, our model suggests that if the supply-side effects are even calibrated at modest values (i.e, as long as they satisfy the following condition $\theta \gt \frac{v+\sigma }{\sigma }$ ), the NFE will occur.Footnote ¹¹

3.1.3. The role of real wage rigidity

A key point in Garín et al. (Reference Garín, Lester and Sims2018) is that NFE in the NK model is likely to occur when: (i) there is more persistent inflation rate target shock and (ii) prices are more flexible. Their discussion, however, is restricted only to cases when real wages are completely flexible, which may not be a very realistic assumption. For instance, Blanchard and Galí (Reference Blanchard and Galí2007) argue that real imperfections in the NK model (such as real wage rigidity) gives rise to a trade-off between inflation and unemployment, which is invariably faced by many central banks. Several other studies, such as Hall (Reference Hall2005), document the importance of real wage rigidity in business cycle models of the USA.

To discuss the implications of real wage rigidity in determining the NFE, we derive the following expressions for complete real wage rigidity, that is, when $\beta =1$ in equation (12), with the cases of partial wage rigidity discussed in the simulations in Section 4:

(27) \begin{align} \dfrac{d\pi _{t}}{du_{t}^{i}} = \dfrac{-\lambda \theta (1-\rho )}{(1-\rho \chi -\lambda \theta \phi ^{\pi })(1-\rho +\sigma ^{-1}\phi ^{x})+\lambda \theta \phi ^{x}(\phi ^{\pi }-1)} \leq 0 \end{align}

(28) \begin{align} \dfrac{di_{t}}{du_{t}^{i}} = \dfrac{-(1-\rho )(1-\rho \chi )}{(1-\rho \chi -\lambda \theta \phi ^{\pi })(1-\rho +\sigma ^{-1}\phi ^{x})+\lambda \theta \phi ^{x}(\phi ^{\pi }-1)} \leq 0 \end{align}

From expressions (27) and (28), as long as there is some degree of cost channel, $\theta \gt 0$ and $0 \leq \rho \lt 1$ , we observe a NFE. Thus, even with full real wage rigidity, we note a NFE in the presence of a cost channel.

An additional point here is that in this particular case, a negative shock to interest rate causes a contemporaneous decrease in both interest rate and the inflation rate. This result is true even for no persistence in the monetary policy shock ( $\rho=0$ ). However, in the absence of no cost channel of monetary policy, we note that a transitory monetary shock causes a decrease in the nominal rate of interest but has no impact on inflation.Footnote ¹²

Finally, from equations (27) and (28), we note that irrespective to cost channel, in case of near permanent shock ( $\rho = 1$ ), neither interest rate nor inflation will change. The result is different from Cochrane (Reference Cochrane2016) who argues that NFE in NK model arises if the interest rate shock is permanent. In sum, this section suggests that the inclusion of real wage rigidity is critical as far as the existence of the NFE is concerned.

3.2. Dampened expectations and the NFE

Expectations of inflation, in particular, the forward-looking nature of the NK model is a key feature that drives neo-Fisherism (Garín et al. (Reference Garín, Lester and Sims2018)). Recent work on inflation dynamics in the US and other advanced economies reveals that contemporaneous real variables such as output gap and employment have little impact on future inflation. In fact, inflation expectations, as found in data, do not commensurate with the prediction of NK Phillips curve, which depicts an almost one-to-one correspondence between the current inflation and the short-run inflation expectations (see, e.g., Kocherlakota (Reference Kocherlakota2016); Jorgensen and Lansing (Reference Jorgensen and Lansing2019)).Footnote ¹³ To account for this literature, we amend the baseline IS and Phillips curve as follows:

(29) \begin{align} x_{t} = E_{t}x_{t+1}-\frac{1}{\sigma }(i_{t}-\hat{\psi }E_{t}\pi _{t+1}) \end{align}

(30) \begin{align} \pi _{t} = \chi \psi E_{t} \pi _{t+1} +\gamma _{1}x_{t}+\gamma _{2}i_{t} \end{align}

where $0\lt \hat{\psi }\lt 1$ , $0\lt \psi \lt 1$ are the dampened inflation expectation factors, following Gabaix (Reference Gabaix2016). With dampening factors, model expectations become less volatile and align better with real-world expectations, as noted in Gabaix (Reference Gabaix2016). For convenience, we assume that $\hat{\psi }=\phi \psi$ where $0\lt \psi \leq 1$ (note that when $\psi =1$ , households and firms have same dampened inflation expectations factor). Using (29), (30), and the Taylor rule (16), we derive the following multipliers:

(31) \begin{align} \dfrac{dx_{t}}{du_{t}} = \dfrac{\sigma ^{-1}(1-\rho \psi \chi -\rho \psi \lambda \theta \phi )}{\Delta _{1}} \end{align}

(32) \begin{align} \dfrac{d\pi _{t}}{du_{t}} = \dfrac{\lambda [\sigma ^{-1}(v+\sigma )-(1-\rho )\theta ]}{\Delta _{1}} \end{align}

(33) \begin{align} \dfrac{di_{t}}{du_{t}} = \dfrac{\lambda (v+\sigma )\rho \phi \psi \sigma ^{-1}-(1-\rho )(1-\rho \psi \chi )}{\Delta _{1}} \end{align}

$\Delta _{1}=(1+\sigma ^{-1}\phi ^{x}-\rho )(1-\gamma _{2}\phi ^{\pi }-\rho \psi \chi )+(\gamma _{1}+\gamma _{2}\phi ^{x})(\phi ^{\pi }-\rho \phi \chi )\sigma ^{-1}$

From (32), we note that if $\sigma ^{-1}(v+\sigma )\gt (1-\rho )\theta$ , inflation will increase and from (33), we note that under following condition interest rate will decrease which rules out the NFE:Footnote ¹⁴

(34) \begin{equation} \psi = 1 \hspace{0.25cm}if \hspace{0.15cm} 1 \lt \dfrac{1-\rho }{\rho [\sigma ^{-1}\lambda (v+\sigma )\phi +(1-\rho )\chi ]} \hspace{0.15cm} else \hspace{0.15cm}\psi \lt \dfrac{1-\rho }{\rho [\sigma ^{-1}\lambda (v+\sigma )\phi +(1-\rho )\chi ]} \end{equation}

From (34), we can show that the rate of discount ( $\psi$ ) and degree of persistence of the shock ( $\rho$ ) are inversely related to each other. This outcome is quite intuitive. The higher persistence of the shock affects agents’ expected inflation, which catches up with the changes in the nominal rate of interest and give rise to NFE. Augmenting the model with discounted expected inflation as per Gabaix (Reference Gabaix2016) breaks the close correspondence between the nominal rate of interest and expected inflation, and which averts the NFE. One special case is also seen in equation (34), that is in the event of near permanent increase in the nominal rate of interest ( $\rho \approx 1$ ), the NFE in NK model arises for a theoretical feasible degree of dampened inflation expectation factor ( $\psi$ ).

3.3. Anticipated monetary policy shocks

Existing studies on NFE assume that monetary policy and inflation target shocks are completely unanticipated. In practice, the central bank’s (and government) policy changes are often well communicated, supporting the notion of at least partial anticipation in future policy changes. For instance, Blinder et al. (Reference Blinder, Ehrmann, Fratzscher, De Haan and Jansen2008) argues that improved monetary policy communication since 1990s in the USA, has enhanced the predictability of monetary policy decisions, which has helped the monetary authority to achieve their macroeconomic objectives. Milani and Treadwell (Reference Milani and Treadwell2012) disentangle a monetary policy shock into a surprise and a news component, suggesting that (in relative terms) the news shocks have a much larger and more persistent effect on the economy.

Motivated by the work on anticipated shocks, (Ali & Qureshi, Reference Ali and Qureshi2021) the second thrust of our contribution focuses on examining the NFE from the lens of an anticipated monetary policy shock. Without loss of generality, we assume that agents anticipate a change in the policy rate $k$ -period in advance. To solve the model analytically, we assume the following trial solution:

(35) \begin{align} x_{t} = \alpha ^{x}+a\varepsilon _{t-k} \end{align}

(36) \begin{align} \pi _{t} = \alpha ^{\pi }+c\varepsilon _{t-k} \end{align}

(37) \begin{align} \varepsilon _{t} = \rho \varepsilon _{t-1} +v_{t}; \; \; \; 0 \leq \rho \lt 1, \; \; \; v_{t} \sim\!\left(0,\sigma _{v}^{2}\right) \end{align}

The first term on the right-hand side of each of the trial solutions illustrates the effect of the anticipated change in the policy rate on the corresponding variable at the time of announcement of the policy. The second term captures the effect of the actual change in the policy rate at the time of implementation of the policy (i.e., at time $t+k$ ). We solve the model for the case when $k=1$ , that is, agents anticipate the shock one period in advance and that shock follows an AR(1) process as given by expression (37).Footnote ¹⁵ Using the method of undetermined coefficients, restrictions on the values of the parameters $a$ , $c$ , $\alpha ^{x}$ , and $\alpha ^{\pi }$ can be derived as follows, where agents anticipate a change in the policy rate of size $m$ :

(38) \begin{align} (\gamma _{1}+\gamma _{2}\phi ^{x})a-(1-\rho \chi -\gamma _{2}\phi ^{\pi })c = \gamma _{2} \end{align}

(39) \begin{align} (1+\sigma ^{-1}\phi ^{x}-\rho )a+\sigma ^{-1}(\phi ^{\pi }-\rho )c=\sigma ^{-1} \end{align}

(40) \begin{align} (\gamma _{1}+\gamma _{2}\phi ^{x})\alpha ^{x}-(1-\chi -\gamma _{2}\phi ^{\pi })\alpha ^{\pi } = -\chi c m \end{align}

(41) \begin{align} \phi ^{x}\alpha ^{x}+(\phi ^{\pi }-1)\alpha ^{\pi }=(\sigma a + c)m \end{align}

We solve this system of equations in two stages. First, after solving equations (38) and (39) simultaneously, we assign values to $a$ and $c$ that are the same as that derived in the case of unanticipated shock. Using these expressions in (40) and (41) yield the values of $\alpha ^{x}$ and $\alpha ^{\pi }$ . It is worth noting here that the on-impact effect of the announced policy on output, inflation, and the interest rate depend on the effects of the policy at the time of execution. Using (38)–(41), we derive the following multipliers:

(42) \begin{align} \dfrac{d\alpha ^{x}}{dm} = \dfrac{(1-\rho (\chi +\lambda \theta ))(1-\chi -\lambda \theta \phi ^{\pi })+\lambda (1-\phi ^{\pi }(\chi +\lambda \theta ))(\sigma ^{-1}(v+\sigma )-\theta (1-\rho ))}{\Delta \Theta } \end{align}

(43) \begin{align} \dfrac{d\alpha ^{\pi }}{dm} = \dfrac{\lambda (v+\sigma )[1-\rho \chi +\sigma ^{-1}\lambda (v+\sigma )+\sigma ^{-1}\phi ^{x}\chi ]+\lambda \theta [\sigma ^{-1}\lambda (v+\sigma )(\phi ^{x}-\sigma )+\phi ^{x}(1-\chi -\lambda \theta )]}{\Delta \Theta } \end{align}

(44) \begin{align} \dfrac{d\alpha ^{i}}{dm} = \dfrac{\phi ^{x}[\lambda \sigma ^{-1}(v+\sigma )+(1-\rho \chi )(1-\chi -\lambda \theta )]+\phi ^{\pi }\lambda (v+\sigma )[1-\rho \chi -\lambda \theta +\sigma ^{-1}\lambda (v+\sigma )]}{\Delta \Theta } \end{align}

$\Delta =(1+\sigma ^{-1}\phi ^{x}-\rho )(1-\rho \chi )+\sigma ^{-1}\lambda (v+\sigma )(\phi ^{\pi }-\rho )-\lambda \theta (\sigma ^{-1}\rho \phi ^{x}+\phi ^{\pi }(1-\rho )) \,\,\Theta =\lambda (v+\sigma )(\phi ^{\pi }-1)+\phi ^{x}(1-\chi -\lambda \theta)$

Under an anticipated shock, the model yields the NFE even without a cost channel of monetary policy and without any minimum degree of shock persistence (see equation (45) below). Note that this outcome is quite different from the case when the shock is unanticipated (or exogenous). In the presence of cost channel, however, the signs of the derived multipliers depend on the parameter values of the model.

(45) \begin{equation} \dfrac{d\alpha ^{\pi }}{dm}\gt 0, \; \; \; \dfrac{d\alpha ^{x}}{dm}\gt 0,\; \; \; \dfrac{d\alpha ^{i}}{dm}\gt 0 \end{equation}

Using standard values used in the literature, such as $v=\sigma =1$ , $\rho =\eta =0.70$ , $\chi =0.99$ and different values of parameter $\theta \in (0,1,1.5,2,2.5)$ , an anticipated expansionary monetary shock invariably causes an increase in both the interest rate and the inflation rate at the time of announcement of the policy. Therefore, when the monetary shock is anticipated, the NK model yields NFE irrespective of the structure of the model. Intuitively, rational economic agents expect inflation at the outset as central bank announces credibly that it will change the interest rate in some future period. The higher expected inflation results in an increase in actual inflation, prior to the implementation of the policy.

The direction of the impact on output depends on the values of cost channel ( $\theta$ ) and monetary policy shock persistence ( $\rho$ ). For $\rho =0$ , output increases even for a relatively higher value of $\theta$ (e.g., $\theta =2.5$ ). But for $\rho \geq 0.60$ with $\theta =2.5$ , output decreases. In general, we find that output decreases for relatively higher values of $\rho$ .Footnote ¹⁶

4.1. No cost channel of monetary policy

We illustrate the simulations from the general model, focusing first on the case with no cost channel. Figure 1 plots the impulse response functions of inflation and interest rate to a one standard deviation unanticipated shock to the policy rate.

Figure 1. Impulse responses to a one standard deviation unanticipated shock to the policy rate with no cost channel. (a) $\rho =0.70$ , $\xi =0.75$ , $\theta =0$ , $\beta =0$ , $h=0$ . (b) $\rho =0.70$ , $\xi =0.75$ , $\theta =0$ , $\beta =0.90$ , $h=0$ . (c) $\rho =0.70$ , $\xi =0.75$ , $\theta =0$ , $\beta =0$ , $h=0.90$ . (d) $\rho =0.70$ , $\xi =0.75$ , $\theta =0$ , $\beta =0.90$ , $h=0.90$ .

The upper two panels of Figure 1(a and b) depict the two cases where we observe a NFE, whereas the lower sub-figures (c and d) highlight those combinations of parameter values which eliminate the NFE from our results. It is evident that in the absence of habit persistence ( $h=0$ ) in consumption, no real wage rigidity ( $\beta =0$ ), and in the presence of interest rate smoothing ( $\xi =0.75$ ), the NFE occurs due to a relatively high persistence in the shock (see Figure 1a). We continue to observe NFE even with a high degree of real wage rigidity ( $\beta =0.90$ ) but with no habit persistence in consumption (see Figure 1b). The NFE, however, disappears when we assume a high degree of habit persistence in consumption ( $h=0.90$ ) irrespective of the level of real wage rigidity ( $\beta =0$ or $\beta =0.90$ ).Footnote ¹⁷

In the absence of cost channel of monetary policy, the monetary policy shock becomes a purely demand-side phenomenon and habit persistence in consumption, simply makes this shock less effective. The key takeaway from the simulations so far is that—in the absence of cost channel of monetary policy—habit persistence in consumption can be sufficient to remove the NFE in the NK model. From this result, an additional conclusion is that averting the NFE may not only be contingent upon a certain mass of backward-looking firms in the model as suggested by Garín et al. (Reference Garín, Lester and Sims2018) but can be eliminated in a more general model setting as well, such as one that includes habit formation in consumption.

4.2. Cost channel of monetary policy

We now simulate the importance of the cost channel in determining the interest rate and inflation linkages. To do so, we assume that all firms borrow money at the market rate of interest.

In Figure 2, we note that under modest values of the cost channel ( $\theta =2.2$ ), we observe the NFE even when the unanticipated monetary policy shock is purely stochastic or transitory. Recall the condition (26) under which there will be no NFE in the presence of cost channel of monetary policy: $\theta (1-\rho )\lt \frac{v+\sigma }{\sigma }\lt \frac{(1-\rho \chi )(1-\rho )}{\rho }\frac{\eta }{(1-\eta \chi )(1-\eta )}$ . For $\rho \rightarrow 0$ , this condition can be violated easily for $\theta \gt 2$ given the fact that we assume $v=\sigma =1$ . On the other hand, we can note that as $\rho$ increases, since the right-hand side of the condition is difficult to hold, that is, $\frac{v+\sigma }{\sigma }\lt \frac{(1-\rho \chi )(1-\rho )}{\rho }\frac{\eta }{(1-\eta \chi )(1-\eta )}$ , the model entails the NFE.

Figure 2. Impulse responses to a one standard deviation shock to the policy rate with cost channel. Notes: In this figure, we plot the responses of inflation and interest rates to a to a one standard deviation shock to the policy rate with cost channel using different values of the structural parameters. (a) $\rho =0$ , $\xi =0$ , $\theta =2.2$ , $\beta =0$ , $h=0$ , $v=1$ . (b) $\rho =0$ , $\xi =0.75$ , $\theta =2.2$ , $\beta =0$ , $h=0$ , $v=1$ . (c) $\rho =0$ , $\xi =0$ , $\theta =2.2$ , $\beta =0.90$ , $h=0.90$ , $v=2$ . (d) $\rho =0.70$ , $\xi =0$ , $\theta =2.2$ , $\beta =0$ , $h=0.90$ , $v=1$ .

This result holds without any habit persistence in consumption and real wage rigidity, the combination of which yield very different results in a scenario without a cost channel. In Figure 2b, we note that NFE disappears once we assume interest rate smoothing by the central bank, which seem to tame individual expectations. Interestingly enough, for $\rho \rightarrow 0$ , we note that the NFE persists if we assume habit persistence in consumption and real wage rigidity ( $h\gt 0$ , $\beta \gt 0$ ) but no interest rate smoothing ( $\xi =0$ ). For $h\gt 0$ , $\beta =0$ , the NFE emerges even for $v \geq 2$ (see Figure 2c). Conditional on the persistence of the shock, the role of parameter $h$ is much more profound in averting the NFE (Figure 2d).

The key outcome from the simulation exercises in a more general model is that in the presence of cost channel, it is difficult to avoid the NFE but which, as demonstrated throughout this note, it can be easily averted in a standard model that includes additional frictions. In particular, we show that two parameters, that is, the degree of interest rate smoothing ( $\xi$ ) and the intensity of habit persistence in consumption ( $h$ ) are important in averting the NFE. In short, when the monetary policy shock is purely stochastic ( $\rho =0$ ) and real wages are not rigid ( $\beta = 0$ ), a higher value of $\xi$ is required. When the monetary policy shock persistence is relatively high ( $\rho \gt 0.60$ ), a higher value of habit formation $h$ defuses the NFE. Intuitively, habit persistence in consumption weakens the sensitivity of aggregate demand to changes in the real rate of interest and also affects the dynamics of the inflation stemming from the Phillips curve, which eventually helps in diminishing the NFE.

4.3. Anticipated monetary policy shocks

The results for an expansionary monetary policy shock, which is anticipated by the agents three periods in advance, are reported in Figure 3. We simulate the baseline model which does not include either cost channel or habit persistence in consumption or real wage rigidity. In subsequent panels, we relax these assumptions and simulate the general model including these frictions.

Figure 3. Impulse response of the interest rate and inflation to an anticipated monetary expansionary shock. Notes: In this figure, we plot impulse response of the interest rate and inflation to an anticipated monetary expansionary shock. We set $k=3$ , which controls the periods in advance agents expect the shock to the policy rate. As before, we use different values of the structural parameters. (a) $\rho =0$ , $\xi =0$ , $\theta =0$ , $\beta =0$ , $h=0$ , $v=1$ . (b) $\rho =0$ , $\xi =0$ , $\theta =2.2$ , $\beta =0$ , $h=0$ , $v=1$ . (c) $\rho =0$ , $\xi =0.75$ , $\theta =2.2$ , $\beta =0$ , $h=0$ , $v=1$ . (d) $\rho =0$ , $\xi =0.75$ , $\theta =2.2$ , $\beta =0$ , $h=0.90$ , $v=1$ . (e) $\rho =0$ , $\xi =0$ , $\theta =2.2$ , $\beta =0.90$ , $h=0.90$ , $v=1$ .

Figure 3 reports the results with $\rho =0$ , which unravels the role of an anticipated shock in the determination of NFE. We note that on impact, both inflation and the interest rate increase. In the subsequent periods, apart from the Panel B, interest rate increases but inflation falls. Once the policy is implemented (at period 3), interest rate and inflation both fall at the same time for all cases.

These results hold regardless of the different degrees of real wage rigidity, cost channel, various intensities of habit persistence, and interest rate smoothing by the central bank.Footnote ¹⁸ Our main finding is that under an anticipated shock, the model always yields a NFE, both in the impact period and at the time of implementation of the policy, that is, period 3 in this particular case to illustrate the key model results, but can apply to more general values.