2.1 Brief explanation of cognitive discounting

This subsection briefly explains the bounded rationality in our model. Our model is based on the framework developed by Gabaix (Reference Gabaix2020).Footnote ⁷ Infinitely-lived agents consider state-contingent plans regarding their infinite optimization problems. The standard NK model assumes a rational expectation model in which the agents’ cognitive capacities are unlimited. In a BNK model, agents form expectations through the perceived law of motion captured by the cognitive discounting parameter. In this model, the agents are fully rational for the steady-state values; however, they are partially myopic concerning the deviation from the steady state. Furthermore, following Benchimol and Bounader (Reference Benchimol and Bounader2023), when focusing on the interaction effect of cognitive discounting and the cost channel, we allow for the discrepancy of cognitive discounting between the household ( $h$ ) and firm ( $f$ ) sectors.

In the case of rational expectations, agents face the de-meaned and linearized law of motion as follows:

(1) \begin{align} \hat {X}_{t+1}=\Lambda \hat {X}_t + \varepsilon _{t+1}, \end{align}

where $\hat {X}_t$ is the state variable, and $\varepsilon _t$ denotes the vector of economic disturbances. If we allow for the presence of limited cognitive capacities in private sectors, behavioral agents follow the misperceived law of motion

(2) \begin{align} \hat {X}_{t+1}=m_l\left(\Lambda \hat {X}_t + \varepsilon _{t+1}\right), \end{align}

for $l=h,f$ . The parameter $m_l$ denotes the degree of cognitive discounting. The rational expectations are related to the expectations under bounded rationality as follows:

(3) \begin{align} E^{BR}_t \hat {X}_{t+1} = m_l\Lambda \hat {X}_t = m_l E_t \hat {X}_{t+1}, \end{align}

for $l=h,f$ . $E^{BR}_t$ represents the subjective behavioral expectations operator, and $E_t$ denotes the rational expectations. Iterating this relationship forward leads to

(4) \begin{align} E^{BR}_t \hat {X}_{t+k} = m^k_l\Lambda \hat {X}_t = m^k_l E_t \hat {X}_{t+k}, \end{align}

for $l=h,f$ . Thus, as the event occurs in the more distant future, the behavioral agents confirm events with an attenuated cognitive discount factor for any horizon $k$ .

2.2 Households

Households solve the standard intertemporal utility maximization problem. A representative household’s intertemporal utility is as follows:

(5) \begin{align} E^{BR}_0\sum _{t=0}^{\infty } \beta ^{t}\bigg [\frac {C_t^{1-\gamma }}{1-\gamma }-\frac {N_t^{1+\eta }}{1+\eta } \bigg ], \end{align}

where $C_t$ and $N_t$ denote the aggregate consumption and the household’s labor hours, respectively. Parameters $\gamma$ and $\eta$ denote the constant relative risk aversion (CRRA) parameter and the inverse of the elasticity of the labor supply, respectively. The aggregate consumption, $C_t$ , is defined as follows:

(6) \begin{align} C_t = \bigg [\int _0^1 c_t(j)^{\frac {\xi -1}{\xi }} dj \bigg ]^{\frac {\xi }{\xi -1}}, \end{align}

where $c_t(j)$ denotes the demand for goods $j$ , and $\xi$ is the elasticity of substitution for individual goods.Footnote ⁸ We obtain the following demand for good $j$ from the cost-minimization problem of households:

(7) \begin{align} c_t (j)=\left (\frac {p_t (j)}{P_t}\right )^{-\xi }C_t, \end{align}

where $p_t(j)$ represents the prices for goods $j$ . The aggregate price index $P_t$ is given as follows:

(8) \begin{align} P_t = \bigg [\int _0^1 p_t(j)^{1-\xi } dj \bigg ]^{\frac {1}{1-\xi }}. \end{align}

The representative household faces the following budget constraints:

(9) \begin{align} P_t C_t + M_{t+1} + B_{t+1}+D_t=(1+i_t)B_t+(1+i^d_t)D_t + M_t + W_t N_t + \Pi ^f_t+\Pi ^b_t+T_t, \end{align}

where $B_t$ represents nominal bonds held for one period, $M_t$ denotes money stock, and $i_t$ is the nominal interest rate. Furthermore, $\Pi ^f_t$ and $\Pi ^b_t$ are dividends earned from intermediate goods firms and financial intermediaries, respectively. $W_t$ and $T_t$ denote the nominal wage and lump-sum transfers, respectively. $D_t$ denotes the deposit service from financial intermediaries, and $i^d_t$ is the nominal interest rate on deposits.Footnote ⁹ The representative households also face the following cash-in-advance constraint:

(10) \begin{align} P_tC_t \leq M_t-D_t+W_tN_t+T_t. \end{align}

We obtain the following dynamic IS (DIS) curve by solving the household problem and using the goods market clearing condition:

(11) \begin{align} x_t = M E_t x_{t+1} - \sigma \left(\hat {i}_t - E_t \pi _{t+1} - r^n_t\right), \end{align}

where $M=m_h$ and $\sigma =1/\bar {r}\gamma$ , and $\bar {r}$ denotes the real interest rate at the steady state. $E_t\pi _{t+1}$ is the expected inflation rate defined as $E_t\log (P_{t+1}/P_t)$ , and the output gap is defined as $x_t=\hat {Y}_t-\hat {Y}^n_t$ . Here, $\hat {Y}_t$ is the current output, and $\hat {Y}^n_t$ denotes the natural rate of output, given by the following:

\begin{align*} \hat {Y}^n_t=\frac {1+\eta }{\gamma +\eta }A_t, \end{align*}

where $A_t$ denotes the productivity shock disturbance.Footnote ¹⁰ Finally, $r_t^n$ is the natural level of the real interest rate, given by the following:

\begin{align*} r^n_t = \sigma ^{-1}\left(M E_t \hat {Y}^n_{t+1}-\hat {Y}^n_t\right). \end{align*}

The cognitive discounting parameter $m_h$ affects the DIS curve, which mitigates future endogenous variables affecting the current inflation and output gap. That is, forwardly iterating Equation (11) leads to

(12) \begin{align} x_t &= -\sigma E_t\sum _{k=0}^{\infty } M^k \left(\hat {i}_{t+k} - \pi _{t+k+1} - r^n_{t+k}\right). \end{align}

As Gabaix (Reference Gabaix2020) argued, introducing cognitive discounting helps to ease several puzzles in the standard NK model, such as the forward guidance puzzle. The presence of $M$ in a DIS indicates that it discounts the effect of the future real interest rate on the current output gap.

2.3 Financial intermediaries

Following Chowdhury et al. (Reference Chowdhury, Hoffmann and Schabert2006), we consider the role of financial intermediaries.Footnote ¹¹ Financial intermediaries provide deposit services to households. If households deposit $D_t$ amount in period $t$ , they receive $(1+i_t)D_t$ at the end of the period. Financial intermediaries receive deposits from households, which they lend to firms; the lending rate is given by $i^L_t$ . Furthermore, financial intermediaries incur a monitoring cost, $\Gamma (i_t)$ , when lending funds to goods producers. We assume this monitoring cost is differentiable and satisfies the following properties: $\Gamma ^{'}(i_t) \geq 0$ and $\Gamma ^{''}(i_t) \geq 0$ .

Financial intermediaries face the following profit maximization problem:

(13) \begin{align} \Pi ^b_t = \left(1+i^L_t\right)[1-\Gamma (i_t)]L_t - (1+i_t)D_t -q L_t, \end{align}

subject to $L_t=D_t$ , where $L_t$ denotes loans to a firm. Parameter $q$ represents the management cost, which is constant. The equilibrium for the lending market is $D_t=W_tN_t^d$ , where $N_t^d$ denotes the demand for labor.

After solving the financial intermediaries’ profit maximization problem, the log-linearized lending rate is given as follows:

(14) \begin{align} \hat {i}_t^L = \psi _i\hat {i}_t, \end{align}

where $\psi _i$ represents the degree of cost channel strength.Footnote ¹² As Chowdhury et al. (Reference Chowdhury, Hoffmann and Schabert2006) and Castelnuovo (Reference Castelnuovo2007) argued, the cost channel amplifies the loan rate when $\psi _i$ exceeds unity, thereby increasing the inflation rate through increased real marginal costs.

2.4 Firms

A continuum of firms faces monopolistic competition and Calvo (Reference Calvo1983)’s nominal price rigidity. Like households, these firms have problems with future cognition—they cannot accurately recognize how the macroeconomic state will be realized.

Firm $j$ ’s production function is given as follows:

(15) \begin{align} Y_t(j)=A_tN_t(j), \end{align}

where $N_t(j)$ denotes the labor supply employed in firm $j$ . $A_t$ denotes an aggregate productivity disturbance, following an autoregressive (AR)(1) process given by $\log A_t=\rho _{a}\log A_{t-1}+\epsilon _t^{a}$ with $0\leq \rho _a\lt 1$ , where $\epsilon ^a_t$ is an independent and identically distributed (i.i.d.) shock with constant variance $\sigma _{a}^2$ .

Regarding the derivation of real marginal costs, we follow Chowdhury et al. (Reference Chowdhury, Hoffmann and Schabert2006) and Ravenna and Walsh (Reference Ravenna and Walsh2006) and assume that the firm must borrow an amount $W_t N_t$ from intermediaries at the gross nominal interest rate $1+i^L_t$ . The real marginal cost is as follows:

(16) \begin{align} mc_t = \frac {\left(1+i^L_t\right) w_t}{A_t}, \end{align}

where $w_t$ is the real wage. Thus, unlike the textbook NK model, the real marginal cost depends on a change in the loan rate in the presence of the cost channel. We then obtain the log-linearized real marginal cost as the expression using the output gap:

(17) \begin{align} \widehat {mc}_t = (\gamma +\eta )x_t + \psi _i \hat {i}_t. \end{align}

Following Calvo (Reference Calvo1983), we assume that firms face price rigidity in each period. More precisely, a fraction $1-\theta$ of all firms adjusts their price, while the remaining fraction $\theta$ does not. When revising their prices, firms contemplate uncertainty concerning when they can adjust prices again. The firm’s optimization problem is given as follows:

(18) \begin{align} E^{BR}_t \sum _{k=0}^{\infty }\theta ^k Q_{t,t+k}Y_{t+k|t}(P_t^*-P_{t+k}mc_{t+k}), \end{align}

subject to the following demand constraints:

(19) \begin{align} Y_{{t+k}|t}=\left (\frac {P^*_{t}}{P_{t+k}}\right )^{-\xi }Y_{t+k}, \end{align}

where $Q_{t,t+k}$ is the stochastic discount factor. $Y_{{t+k}|t}$ is the output in period $t+k$ expected by the firm in period $t$ , and $P^*_{t}$ is the optimal price. We log-linearize the first-order condition of the firm’s optimal condition (18) around the steady state to obtain the following:

(20) \begin{align} \hat {P}^*_t-\hat {P}_{t-1} = (1-\beta \theta )E^{BR}_t\sum _{k=0}^{\infty }(\beta \theta )^k \left[\widehat {mc}_{t+k} + \hat {P}_{t+k} - \hat {P}_{t+k-1}\right]. \end{align}

After some manipulation and using Equation (17), we obtain the following NKPC under cognitive discounting:

(21) \begin{align} \pi _t = M_f \beta E_t \pi _{t+1} + \delta \left[(\gamma + \eta ) x_t + \psi _i \hat {i}_t \right], \end{align}

where $M_f = m_f[\theta + (1-\theta )(1-\theta \beta )/(1-m_f\theta \beta )]$ , and $\delta = (1-\theta )(1-\theta \beta )/\theta$ .Footnote ¹³ Unlike Gabaix (Reference Gabaix2020), the nominal interest rate is augmented in the NKPC owing to a cost channel.

Compared with the standard NK model with rational expectations, the NKPC in the BNK model is also discounted as follows:

(22) \begin{align} \pi _t = \delta E_t\sum _{k=0}^{\infty } (\beta M_f)^k [(\gamma +\eta ) x_{t+k}+\psi _i \hat {i}_{t+k}]. \end{align}

Thus, cognitive discounting attenuates the effect of future output gaps on current inflation.Footnote ¹⁴ Without a cost channel, the discounted value expression of the NKPC indicates that monetary tightening can generate a negative response of inflation by successfully creating a negative response of the discounted value of the expected output gap. Conversely, as Surico (Reference Surico2008) argued, the cost channel generates a wedge into this relationship; thus, it may prevent monetary contraction from generating a negative response of inflation.Footnote ¹⁵ Sections 3 and 4 demonstrate how the interaction effect of cognitive discounting and the cost channel plays a significant role in accounting for monetary policy transmission.

2.5 Calibration

This subsection briefly describes the calibrated values used in this study. Except for several parameter values, we set the deep parameters based on Gabaix (Reference Gabaix2020). The discount factor $\beta$ is set to 0.99. We set the degree of nominal price stickiness, the Calvo parameter ( $\theta$ ), to 0.75 as a benchmark case. We then set the CRRA coefficient ( $\gamma$ ) to 2.0 and set the value of the Frisch elasticity of labor supply ( $\eta$ ) as 1.0. The elasticity of substitution ( $\xi$ ) is set to 7.88. The shock persistence for the demand and supply shocks is 0.6 and 0.5, respectively; the standard deviations for the demand and supply shocks are set to 0.01.

Next, we explain the value of the cognitive discounting parameters $m_h$ and $m_f$ . As noted, $m_h=m_f=1$ corresponds to the standard NK with full information and rational expectations. Following Benchimol and Bounader (Reference Benchimol and Bounader2023) and Gabaix (Reference Gabaix2020), we set the degree of cognitive discounting, $m_h$ and $m_f$ , to 0.85 as a benchmark calibration. Ilabaca et al. (Reference Ilabaca, Meggiorini and Milani2020) considered a different value of cognitive discounting for households and firms, reporting the corresponding values of it equal to 0.7 and 0.4. In contrast, Kolasa et al., (Reference Kolasa, Ravgotra and Zabczyk2025) estimated the degree of cognitive discounting equal to 0.53 using a two-country BNK model.Footnote ¹⁶ We consider the range of $m_h$ and $m_f$ from 0 to 1 to explore how the degree of cognitive discounting impacts the properties of several monetary policy rules. We adopt the calibrated value of a small value of $m_f$ , which may not be empirically supported; however, such parameterization helps obtain the intuitive mechanism of our model.

Finally, we provide the calibrated value for the cost channel parameter; several studies have reported the degree of a cost channel parameter, $\psi _i$ . For instance, Ravenna and Walsh (Reference Ravenna and Walsh2006) found that the parameter $\psi _i$ is 1.276, while Chowdhury et al. (Reference Chowdhury, Hoffmann and Schabert2006) estimated a value of $\psi _i$ of 1.32 for the United States. Patel (Reference Patel2021) estimated the cost channel parameter in a two-country NK model using a Bayesian technique, finding that the estimated values of the cost channel parameter ranged from 1.8 to 2.3.Footnote ¹⁷ Given these approaches, we use an empirically plausible value of $\psi _i\in [0, 1.8]$ to account for the degree of the cost channel.Footnote ¹⁸

3.1 Determinacy properties

First, we explore the determinacy analysis in our model. Bullard and Mitra (Reference Bullard and Mitra2002) showed that the Taylor principle is necessary to attain the unique REE in the standard NK model. Llosa and Tuesta (Reference Llosa and Tuesta2008) found that introducing the cost channel complicates the determinacy condition in the NK model.Footnote ¹⁹ Gabaix (Reference Gabaix2020) demonstrated that an increased degree of cognitive discounting will likely attain the unique REE, contrary to the rational expectations model. However, it remains unclear how the interaction effect of cognitive discounting and the cost channel alters the determinacy properties. In what follows, we employ two simple monetary policy rules: the current- and forward-looking policy rules.

3.1.1 Current-looking Taylor rule

First, we consider the current-looking Taylor rule, implying that the central bank responds to the current inflation rate and the current output gap (Taylor Reference Taylor1993). More specifically, we adopt the following Taylor rule:

(23) \begin{align} i_t = \phi _\pi \pi _t + \phi _x x_t, \end{align}

where $\phi _\pi$ and $\phi _x$ represent the degree of the policy rate’s responsiveness to inflation and the output gap, respectively.

Using Eqs. (11), (21), and (23), we obtain the following dynamic system for $z_t=[\pi _t,x_t]^{'}$ and $r^n_t$ :

(24) \begin{equation} z_{t} = \Omega E_t z_{t+1} + \Xi r^n_t, \end{equation}

where the coefficient matrices are

(25) \begin{equation} \Omega = \omega \begin{bmatrix} (\sigma +\phi _x)M_f\beta +\delta [(\sigma +\eta )+\psi _i \phi _x] \beta & \sigma \delta [(\sigma +\eta )+\psi _i \phi _x]M \\ 1-M_f \beta \phi _{\pi }-\delta \psi _{i}\phi _{\pi } & \sigma (1-\delta \psi _i\phi _\pi )M \\ \end{bmatrix}, \end{equation}

where $\omega = (\sigma + \phi _x - \sigma \delta \psi _i\phi _{\pi } + \delta (\sigma +\eta )\phi _{\pi })^{-1}$ and $\Xi$ denotes the coefficient matrix for the exogenous variables. The determinacy conditions of the current-looking Taylor rule are summarized as follows:Footnote ²⁰

Proposition 1. Under a current-looking Taylor rule, the necessary and sufficient conditions for determinacy are

(26) \begin{align} & \phi _\pi + \varpi \phi _x \gt 1-\kappa, \end{align}

(27) \begin{align} & \sigma (1+M_f \beta + \delta \psi _i)\phi _x + \delta [\sigma (\gamma + \eta )-(1+M)\psi _i]\phi _\pi + (1+M)(1+M_f\beta ) + \sigma (\gamma +\eta )\delta \gt 0, \end{align}

where

\begin{align*} & \varpi = \frac {\sigma (1-M_f \beta - \delta \psi _i)} {\delta [\sigma (\gamma + \eta ) -(1-M)\psi _i]}; \quad \kappa = \frac {(1-M)(1-M_f \beta - \delta \psi _i)}{\delta [\sigma (\gamma + \eta ) -(1-M)\psi _i]} . \end{align*}

Proof. See Appendix A.

These conditions are generalized versions of Gabaix (Reference Gabaix2020) and Llosa and Tuesta (Reference Llosa and Tuesta2009). Unlike Llosa and Tuesta (Reference Llosa and Tuesta2008), the condition (26) states the generalized Taylor principle, which contains the additional term of the right-hand side ( $-\kappa$ ). In the long run, this condition indicates that each percentage point of permanently high inflation reflects a permanent reduction in the output gap of $\varpi$ . When $ \partial \varpi / \partial \psi _i \lt 0$ holds, a rise in the output gap is less responsive to an increase in the inflation rate with a greater degree of the cost channel.Footnote ²¹ This term makes the REE less determinate in the case of a weak $\phi _{\pi }$ and $\phi _x$ response. The condition (27) is also a generalized version of Llosa and Tuesta (Reference Llosa and Tuesta2009) in a model with cognitive discounting; the condition (27) is trivially satisfied in the case of no cost channel.Footnote ²²

We address the role of the output gap response $\phi _x$ in accounting for the determinacy condition in our model. If we consider the value of the cognitive discounting parameters (namely, $m_f$ and $m_h$ , which are slightly smaller than unity), then the value of the second term of the right-hand side is less than unity in Equation (26). In this case, for a given value of $\phi _{\pi }$ and $\phi _x$ , a significant value of the cost channel parameter may violate the condition (26). When the values of $m_f$ and $m_h$ are slightly smaller than unity, we can easily confirm $ \partial \varpi / \partial \psi _i \lt 0$ ; therefore, for small coefficients of $\phi _{\pi }$ and $\phi _x$ , the determinacy condition (26) does not guarantee the unique REE when the cost channel parameter takes a larger value. Llosa and Tuesta (Reference Llosa and Tuesta2009) showed that for a larger value of the cost channel parameter, the determinacy condition requires a high $\phi _{\pi }$ in the case of $m_f=m_h=1$ ; however, Gabaix (Reference Gabaix2020) noted that for small coefficients of $\phi _{\pi }$ and $\phi _x$ , bounded rationality in agents enlarges the determinacy regions. Our finding indicates that even if the central bank considers a weak inflation stabilization, it must place a heavy weight on the output gap stabilization to achieve the unique REE in our model.

We concentrate on the role of the discrepancy in the cognitive discounting parameters between firms and households to gain further insights into this issue. Two key parameters— $\kappa$ and $\varpi$ —can characterize the above results. Differentiating $\kappa$ regarding $m_h$ and $m_f$ , we obtain $\partial \kappa /\partial m_h \lt 0$ and $\partial \kappa /\partial m_f \lt 0$ . Thus, the household’s cognitive discounting parameter $m_h$ counteracts the positive impact of increased future inflation on the output gap in the DIS curve. The inflation rate rise from the output gap expansion is also weakened by the firm’s cognitive discounting parameter $m_f$ .

Next, differentiating $\varpi$ regarding $m_h$ and $m_f$ , we obtain $\partial \varpi /\partial m_h \lt 0$ and $\partial \varpi /\partial m_f \lt 0$ . To interpret the role of $\varpi$ , we imagine a situation where a rise in the nominal interest rate accompanies an increase in the inflation rate. When the value of $m_h$ is small, Equation (12) implies that the demand channel dampens the positive effect of a rise in inflation on the output gap; the effect of monetary contraction on the output gap is also weakened. Conversely, Equation (22) shows that an increase in the nominal interest rate leads to further inflation through the cost channel in the NKPC. It also indicates that a smaller value of $m_f$ attenuates the effect of the output gap and the policy rate on inflation in this curve. The parameter $m_f$ affects discounting in the DIS curve via the attenuated effect of the future output gap and the nominal interest rate on current inflation in the NKPC. These effects counteract the impact of $\partial \varpi /\psi _i\lt 0$ on the condition (26). Accordingly, Equation (26) states that, even for a smaller value of $\phi _{\pi }$ , the Taylor rule with a larger value of $\phi _x$ may alleviate the effect of the cost channel on inflation to avoid self-fulfilling expectations in our model.

The intuition of Proposition 1 is as follows. This proposition shows that the REE is more likely to be achieved even in the cost channel model if firms experience a higher degree of cognitive discounting. Suppose that a sunspot shock increases inflation and the output gap. In response to this shock, the central bank raises nominal interest rates, causing inflation to decrease via a decline in the output gap through the DIS curve. In contrast, nominal interest rates push up inflation if the cost channel increases the real marginal cost. In the rational expectations model, the cost channel substantially increases the nominal interest rate because the indeterminacy problem arises if the real interest rates do not rise sufficiently. Conversely, cognitive discounting weakens the impact of future endogenous variables on current inflation and the output gap. In particular, if firms’ cognitive discounting is stronger than households’ cognitive discounting, the reduction in the inflation rate due to the demand channel is stronger than the cost channel effect. Therefore, it mitigates the impact of sunspot shocks on inflation and the output gap. Consequently, the central bank may deliver a unique REE without strengthening the coefficients on inflation.

If the central bank ignores inflation and output gap fluctuations (namely, $\phi _{\pi }=\phi _x=0$ ), we have a unique REE if and only if the following conditions hold:

(28) \begin{align} (1-M)(1-M_f \beta )\gt \delta \sigma (\gamma + \eta ). \end{align}

This equation is similar to that derived by Gabaix (Reference Gabaix2020) because the cost channel does not matter for determinacy when the nominal interest rate is fixed.

Finally, we numerically confirmed the interaction effect of cognitive discounting and the cost channel on the determinacy regions. Figure 1 depicts how the cost channel affects the determinacy region under several parameterizations of cognitive discounting. Regarding the rational expectations assumption, Llosa and Tuesta (Reference Llosa and Tuesta2009) showed that including the cost channel in the model requires an aggressive inflation reaction coefficient to attain the unique REE when the output gap response is allowed. Conversely, once our benchmark model slightly departs from the assumption of the rational expectations in the firm sector (i.e., $m_f=0.85$ ), a small value of $\phi _{\pi }$ makes the REE determinate for the larger value of $\phi _x$ that is above 2.0. If the firm sector’s expectations are highly discounted ( $m_f=0.5$ ), the determinacy region remains unaffected by the presence of the cost channel. In the case of $m_h=m_f=0.5$ , the REE is determinate even if $\phi _{\pi }$ and $\phi _y$ are less than unity. Moreover, this figure shows that the degree of cognitive discounting in the NKPC primarily affects the determinacy region. In other words, our numerical experiments reveal that the degree of cognitive discounting in the firm sector plays an important role in equilibrium determinacy.

Figure 1.

Equilibrium determinacy in a BNK with a contemporaneous Taylor rule.

Note: Det denotes the area of determinacy. The magenta and cyan regions indicate the area of determinacy in the model without and with the cost channel, respectively.

3.1.2 Forward-looking Taylor rule

Next, we consider the case where the central bank responds to the expected inflation rate and output gap. Unlike the current-looking policy rule, as noted by Bullard and Mitra (Reference Bullard and Mitra2002) and Llosa and Tuesta (Reference Llosa and Tuesta2009), adopting the forward-looking rule significantly affects determinacy properties. More specifically, we consider the following monetary policy rule:

(29) \begin{align} i_t = \phi _\pi E_t\pi _{t+1} + \phi _x E_t x_{t+1}. \end{align}

As in the current-looking rule, using (29), (11) and (21), we obtain the dynamic system for the endogenous vector $z_t=(\pi _t,x_t)'$ . The coefficient matrices of the reduced system ( $\Omega$ ) are defined as follows:

(30) \begin{equation} \Omega = \sigma ^{-1} \begin{bmatrix} \sigma M_{f}\beta -\delta (\sigma +\eta )(\phi _\pi -1) +\delta \phi _\pi \sigma \psi _i & \delta (\sigma +\eta )(\sigma M - \phi _x) + \sigma \delta \psi _i \phi _x \\ -\phi _\pi +1 & \sigma M -\phi _\pi \\ \end{bmatrix}. \end{equation}

The determinacy properties of the forward-looking Taylor rule are summarized as follows:

Proposition 2. Under interest rate rules with forward expectations, the necessary and sufficient conditions for determinacy are given as follows:

(31) \begin{align} & \phi _\pi +\varpi \phi _x \gt 1 - \kappa, \end{align}

(32) \begin{align} & \sigma (M_f\beta +\delta \psi _i)\phi _x-\delta \psi _iM\phi _\pi \lt 1+ MM_f\beta, \end{align}

(33) \begin{align} & -\sigma (M_f\beta +\delta \psi _i)\phi _x + \delta \psi _iM\phi _\pi \lt 1- MM_f\beta, \end{align}

(34) \begin{align} & \sigma (1+M_f \beta + \delta \psi _i)\phi _x + \delta [\sigma (\gamma + \eta )-(1+M)\psi _i]\phi _\pi \lt (1+M)(1+M_f\beta ) + \sigma (\gamma +\eta )\delta . \end{align}

Proof. See Appendix A.

The case for $m_h=m_f=1$ corresponds to the determinacy conditions obtained by Llosa and Tuesta (Reference Llosa and Tuesta2009), whose determinacy conditions are modified when cognitive discounting and cost channels interact in our model. First, the condition (31) represents the generalized Taylor principle, which holds even in the case of a current-looking rule. Second, the conditions (32) and (33) imply the upper bound on the coefficient for inflation and the output gap, respectively. Third, the inequality (34) requires that an upper bound restricts the total response to inflation and the output gap. This result reveals that even in the case of a severe cost channel, a small value of the cognitive discounting parameter can quickly satisfy conditions (32) and (33). A larger value of the cost channel parameter delivers the amplifying effect of the expectations channel; a small value of cognitive discounting captures its dampening effect. Our result indicates that the central bank can easily attain the unique REE when the latter effect dominates. In the extreme case of $\phi _{\pi }=\phi _x=0$ , the condition required to attain the unique REE corresponds to Equation (28).

Figure 2 illustrates the determinacy region under a forward-looking rule. In the case of $m_h=m_f=1$ , as noted earlier, Llosa and Tuesta (Reference Llosa and Tuesta2009) found that introducing a cost channel narrows the determinacy region compared with the standard NK model. Thus, unlike the model without the cost channel, a strong reaction to inflation is necessary to retain the unique REE with a small response to the output gap. Next, in the case of bounded rationality for slightly discounted expectations in the firm sector (namely, $m_f=0.85$ ), bounded rationality in agents easily expands the determinacy region in a model with and without the cost channel if the central bank keeps a smaller weight on the output gap stabilization. Figure 2 also shows that determinacy regions are slightly narrower in the NK model with the cost channel than without it. Furthermore, we observe that a high degree of cognitive discounting leads to a further expansion of the determinacy region despite introducing the cost channel. Accordingly, our result indicates that for a small value of $m_f$ , the determinacy region under a forward-looking rule remains unaffected by the cost channel.

Figure 2.

Equilibrium determinacy under a forward-looking Taylor rule.

Note: Det denotes the area of determinacy. The magenta and cyan regions indicate the area of determinacy in the model without and with the cost channel, respectively.

3.2 Model dynamics

This section examines the dynamic properties of our model, focusing on the impulse response of inflation to monetary tightening shocks to address the role of the cost channel in the BNK model. We analytically calculate the equilibrium response of the inflation rate to monetary policy shocks by adopting the current-looking rule with exogenous monetary policy shocks, which follows an AR(1) process. We consider the following monetary policy rule:

(35) \begin{align} i_t = \phi _\pi \pi _t + \phi _x x_t + \nu _t, \end{align}

where $\nu _t$ denotes an exogenous monetary policy shock specified by $\nu _t = \rho \nu _{t-1}+\varepsilon _t$ ; $\varepsilon _t$ is an i.i.d. shock $(0,\sigma ^2_{\varepsilon })$ .

We employ the undetermined coefficient method to derive the equilibrium response to monetary policy shocks. We follow the method that McCallum (Reference McCallum1983) offered as the minimum state variable (MSV) solution to obtain the analytical solution in the model. We obtain the following reduced form for the inflation rate:

(36) \begin{align} \pi _t &= -\bigg \{\frac {\delta [(\gamma +\eta )\sigma -\psi _i(1-\rho M)]}{(1-\rho M+\sigma \phi _x)[(1-\beta \rho M_f)-\delta \psi _i\phi _{\pi }]+\sigma \delta [(\gamma +\eta )+\psi _i\phi _x](\phi _{\pi }-\rho )}\bigg \}\nu _t, \nonumber \\[4pt] &=-\Psi \nu _t. \end{align}

Unlike Castelnuovo (Reference Castelnuovo2007), our solution allows for a persistent monetary policy shock. Unlike the standard NK model with rational expectations, this MSV solution is affected by the degree of the cognitive discounting parameter through the DIS curve and the NKPC. The denominator never becomes a positive value under a plausible calibration.Footnote ²³

As noted, bounded rationality in the agents affects the shape of the DIS curve and NKPC. As Equation (12) shows, a high degree of cognitive discounting counteracts the future real interest rate’s effect on the current output gap. We show that $\partial \Psi /\partial m_h\gt 0$ ; thus, a smaller value of $m_f$ in the DIS curve mitigates the demand channel of monetary policy. Furthermore, from Equation (22), the effect of future interest rates on current inflation is attenuated by cognitive discounting. We can easily show that $\partial \Psi /\partial m_f\gt 0$ . These results indicate that cognitive discounting in both sectors helps to alleviate the dominance of the demand channel over the cost channel.Footnote ²⁴

In what follows, we explore how cognitive discounting affects the dynamic response of inflation rates to monetary tightening in our model. Figure 3 shows the impulse response of inflation rates to a contractionary monetary policy shock. Figure 3(i) plots the case without the cost channel, which helps to understand how cognitive discounting impacts the dynamic response of inflation. This figure displays that introducing bounded rationality in agents significantly induces the attenuated response of inflation to monetary contraction. Cognitive discounting counteracts the central bank’s ability to manipulate the expectations of the private sector; thus, equilibrium inflation is less responsive to monetary contraction. This figure also illustrates that the degree of cognitive discounting in the firm sector is essential in explaining inflation dynamics.

Figure 3.

Impulse response of inflation to a monetary tightening shock.

Next, we explore the interaction between cognitive discounting and the cost channel. Figure 3(ii) illustrates the inflation response to monetary tightening under a benchmark cost channel model. Unlike models without the cost channel, introducing a cost channel further attenuates the response of inflation to monetary contraction. This figure also displays the difference between rational expectations and bounded rationality because an increased degree of cognitive discounting dampens the demand channel of monetary policy relative to the cost channel. This figure also reveals that even in the case of a cost channel, the degree of cognitive discounting in the NKPC plays a significant role in explaining inflation dynamics in our model.

We further document the impact response of the inflation rate to monetary tightening to explore the interaction effect between cognitive discounting and cost channels. We concentrate on the degree of cognitive discounting in the firm sector by accounting for the results in Figures 1–3. Given the value of $m_h$ , Figure 4 plots how the degree of cognitive discounting in the firm sector influences the initial response of inflation to a monetary tightening shock, given a cost channel parameter. Even with a high value of $\psi _i$ , monetary contraction reduces the inflation rate. A sufficiently small value of the cognitive discounting parameter counteracts the impact of monetary tightening on inflation.Footnote ²⁵

Figure 4.

Impact response of inflation to a monetary tightening shock: the role of cognitive discounting.

Summing up, cognitive discounting makes the expectations channel of monetary policy less effective in our model. This implies that the central bank has difficulty fully manipulating the private sector’s expectations through demand and supply channels of monetary policy. In addition, we demonstrate that even if we allow for the role of bounded rationality in the household and firm sectors, the model captures the negative impact response of inflation to an increase in the nominal interest rate via increased real marginal costs. Notably, our numerical experiments also reveal the importance of the firm’s cognitive discounting parameter relative to the degree of the household’s cognitive discounting.

Figure 5.

Interaction effect of cognitive discounting and cost channel on inflation under optimal monetary policy: impulse response function to a demand shock.

Note: The solid line denotes the impulse response under discretion in a model without the cost channel. The dashed line denotes the impulse response under discretion in a model with the cost channel. The dotted line denotes the impulse response under commitment in a model without the cost channel. The line with (x) denotes the impulse response under commitment in a model with the cost channel.