CBDC, cash, and financial intermediary in HANK

Abstract

The introduction of interest-bearing Central Bank Digital Currency (CBDC) presents central banks with an additional instrument for implementing monetary policy. This article develops a Heterogeneous Agents New Keynesian model, incorporating financial frictions, to investigate the transmission mechanism and wealth distribution effects of digital currency interest rate from two distinct perspectives: (i) the interaction between the central bank and financial intermediaries and (ii) the substitution dynamics between cash and digital currency. By comparing models that include and exclude financial intermediaries, our research uncovers that in a society where CBDC is fully implemented, the reverse actions of financial intermediaries can hinder the efficacy of the central bank’s monetary policy and result in elevated costs associated with policy formulation. Additionally, the leverage effect of financial intermediaries exacerbates wealth inequality and contributes to the expansion of investment, thereby promoting an increase in economic output. Comparing societies where CBDC entirely replaces cash with those where CBDC coexists alongside cash, this paper demonstrates that the presence of cash mitigates significant economic fluctuations triggered by CBDC, while the complete elimination of cash amplifies wealth inequality. Consequently, it is crucial for the central bank to account for the behavior of financial intermediaries when adjusting digital currency interest rate and explore the development of an appropriate interest rate mechanism tailored to digital currency. Simultaneously, maintaining cash circulation for a specific period can act as an economic stabilizer.

Appendices

Appendix A. Derivations for models 1 and 2

In this section, we introduce the algorithm to solve the model of Mode 1 and Model 2 based on Kaplan et al. (2018).

Heterogeneous households’ optimization is given by,

$$\begin{aligned} \mathbb {E}_{0} \int _{0}^{\infty } e^{- \rho t} u\left( c_{t}, l_{t}\right) d t \end{aligned}$$

(A1)

Subjected to the budget constraints:

$$\begin{aligned} \begin{aligned}&\dot{b}_t=\left( 1-\tau _w\right) w_t z_t l_t+ r_{bt,pos} b_t\ (b_t\ge 0) + r_{bt,neg}b_t\ (b_t< 0)+ T_t -d_t-\chi (d_t, a_t) - c_t, \\&\dot{a}_t= r_{at} a_t + d_t , \qquad b_t \ge \underline{b} \quad a_t\ge 0 \end{aligned} \end{aligned}$$

(A2)

Here we assume that $r_{bt} = r_{bt,pos} (b_t\ge 0) + r_{bt,neg}\ (b_t< 0)$ denotes the interest rate matrix of households’ CBDC and loan. Then, the households’ optimization problem is given by:

$$\begin{aligned} \begin{aligned}&\quad \max _{c_{t}, l_t} \int _{t=0}^{\infty }u(c_{t},l_t)dt \qquad u(c_{t}, l_t) = \frac{(c_{t})^{1-\gamma _c}}{1-\gamma _c} - \frac{(l_t)^{1+\gamma _l}}{1+\gamma _l}\\ s.t.&\\&\dot{b_t} = (1-\tau _w)w_t l_t z_t + r_{bt} b_t + T_t - d_{t} - \chi (d_{t}, a_{t}) -c_{t},\\&\dot{a_{t}}=r_{at} a_{t} + d_{t}, \\&d z_{t}=-\mu _{j} z_{t} d t+\epsilon _{t} d W_{t}, \qquad \epsilon _{t} \sim \mathcal {N}\left( 0, \sigma _{z}^{2}\right) ,\\&\chi (d_{t}, a_{t})=\chi _{0}|d _{t}|+\chi _{1}\left| \frac{d_{t}}{\max \{ a_{t}, \underline{ a}\}}\right| ^{{2}} \max \{ a_{t}, \underline{ a}\} \end{aligned} \end{aligned}$$

(A3)

This optimization problem can be solved by the HJB equation:

$$\begin{aligned} \begin{aligned} V(b_t,a_{t},z_t)&= \max _{c_{t}, l_t} u(c_{t},l_t) + V_b \dot{b_t} + V_{a} \dot{a_{t}}+ \mu _z V_z + \frac{\sigma _z^2}{2} V_{zz} \\ s.t. \qquad \dot{b_t}&= (1-\tau _w)w_t l_t z_t + r_{bt} b_t + T_t - d_{t} - \chi (d_{t}, a_{t}) - c_t, \\ u(c_{t},l_t)&= \frac{\left( c_t\right) ^{1-\gamma _c}}{1-\gamma _c} - \frac{(l_t)^{1+ \gamma _l } }{1+\gamma _l}. \end{aligned} \end{aligned}$$

(A4)

where $V_b,V_{a},V_z$ are first-order partial derivatives of the value function V, and $V_{zz}$ is the second-order partial derivative. The HJB equations’ first order conditions (FOCs) are given by:

$$\begin{aligned} \begin{aligned} 0&= c_t^{-\gamma _c} - V_b \\ 0&= - l_t^\frac{1}{\varphi _l} + V_b (1-\tau _w)w_t z_t \end{aligned} \end{aligned}$$

(A5)

Obviously, the labor supply $l_t$ is the function of total consumption $c_t$:

$$\begin{aligned} l_t = \left( c_t\right) ^{-\frac{\gamma _c}{\gamma _l} } \left[ (1-\tau _w)w_t z_t\right] ^{\frac{1}{\gamma _l}} \end{aligned}$$

(A6)

In equilibrium, all assets’ change rates are equal to zero, and the deposits are also equal to zero. So in the equilibrium, the consumption $c_0$ and labor supply decision $l_0$ are given by:

$$\begin{aligned} c_0 = (1-\tau _w)w_t l_0 z_t + r_b b_t + T_t \end{aligned}$$

(A7)

Substitution $c_0$ into the previous equation:

$$\begin{aligned} l_0 = \left[ (1-\tau _w)w_t l_0 z_t + r_b b_t + T_t \right] ^{-\frac{\gamma _c}{\gamma _l} } \left[ (1-\tau _w)w_t z_t\right] ^{\frac{1}{\gamma _l}} \end{aligned}$$

(A8)

This equation shows how the labor supply $l_0$ is determined in equilibrium.

According to Achdou et al. (2022), the income flow of cash $\dot{b}_t$ can be split into two items:

$$\begin{aligned} \begin{aligned}S^c &= (1-\tau _w)w_t l_t z_t + r_{bt} b_t + T_t - c_{t}\\ S^{d} &= - d_{t} - \chi (d_{t}, a_{t})\\ \dot{b_t} &= S^c + S^{d} \end{aligned} \end{aligned}$$

(A9)

Let ${x}^{F+} \triangleq max (x,0)$ and ${x}^{B-} \triangleq min (x,0)$, obviously $x = {x}^{F+} + {x}^{B-}$. We can also separate constraint conditions into different parts. Let $b_t,a_{t},z_t$ be in different grids, and then we could mark a household as (i, j, k, nz) when $b_t = b_i, a_{t} = a_{j}, z_t = z_{nz}$. Then we define the approximate forward derivative $V^{F}_b$ and backward derivative $V^B_b$ as follows:

$$\begin{aligned} \begin{aligned}&V^{n+1}_b \approx \frac{V^{n}_{i+1,j,k,nz} - V^{n}_{i,j,k,nz}}{\Delta b} \approx \frac{V^{n}_{i,j,k,nz} - V^{n}_{i-1,j,k,nz}}{\Delta b} \\&V^{F}_b = \frac{V^{n}_{i+1,j,k,nz} - V^{n}_{i,j,k,nz}}{\Delta b}, \qquad V^B_b = \frac{V^{n}_{i,j,k,nz} - V^{n}_{i-1,j,k,nz}}{\Delta b} \end{aligned} \end{aligned}$$

(A10)

since $V_b \dot{b_t} = V^B_b (S^{c,B-} + S^{d,B-}) + V^{F}_b (S^{c,F+} + S^{d,F+})$, we can check the sign of $\dot{b_t}$ and choose the approximate derivative for computation. This approximate method makes it possible for the program to solve HJB equations quickly.

Then we can get the approximate solution for the household sector by solving the discrete HJB equation. By the definitions above, the discrete HJB equation is given by:

$$\begin{aligned} \begin{aligned} \frac{V^{n+1}_{i,j,nz} - V^{n}_{i,j,nz}}{\Delta } + \rho V^{n+1}_{i,j,nz} &=\max _{c_{t},l_t} U(c_{t},l_t) + V_b \dot{b_t} + V_{a} \dot{a_{t}} + V_z \mu _z + V_{zz} \frac{\sigma _z^2}{2} \\ &=U + V^B_b (S^{c,B-} + S^{d,B-}) \\&\quad+ V^{F}_b (S^{c,F+} + S^{d,F+}) \\&\quad+ V^B_{a} d^{-}_t + V^{F}_{a} (d^{+}_t + r_{at} a_{t})\\&\quad+ V_z \mu _z + V_{zz} \frac{\sigma _z^2}{2} \end{aligned} \end{aligned}$$

(A11)

Note that approximate derivatives, $V^{F}_b = \frac{V_{i+1,j,nz} - V_{i,j,nz}}{\Delta b}$ and $V^B_b = \frac{V_{i,j,nz} - V_{i-1,j,nz}}{\Delta b}$, can be factorized. Then we rewrite the discrete HJB equation as below:

$$\begin{aligned} \begin{aligned} \frac{V^{n+1}_{i,j,nz} - V^{n}_{i,j,nz}}{\Delta } + \rho V^{n+1}_{i,j,nz} &=U + \frac{V_{i-1,j,nz}}{\Delta b}(-S^{c,B-} - S^{d,B-}) \\&\quad+ \frac{V_{i+1,j,nz}}{\Delta b}(S^{c,F+} + S^{d,F+} )\\&\quad+ \frac{V_{i,j,nz}}{\Delta b}(S^{c,B-} -S^{c,F+} + S^{d,B-} - S^{d,F+})\\&\quad+ \frac{V_{i,j-1,nz}}{\Delta a}( - d^{-}_t) + \frac{V_{i,j+1,nz}}{\Delta a}(d^{+}_t+r_{at} a_{t}) \\&\quad+ \frac{V_{i,j,nz}}{\Delta a}(d^{-}_t - d^{+}_t - r_{at} a_{t}) \\&\quad+ \sum _{nz' \ne nz}\lambda _{nz}(V_{i,j,nz'} - V_{i,j,nz}) \end{aligned} \end{aligned}$$

(A12)

which means $V_{i-1,j,nz}$ and $V_{i+1,j,nz}$ items are on different sub-diagonals of matrix V, $V_{i,j,nz}$ items are on the diagonal of V. According to Achdou et al. (2022), there are the same conclusions on other derivatives, so the discrete HJB equation above can be summarized as:

$$\begin{aligned} \frac{1}{\Delta }\left( V^{n+1}_{i,j,nz}-V^{n}_{i,j,nz}\right) +\rho V^{n+1}_{i,j,nz}=U_{i,j,nz}^{n}+\left( \textbf{A}^{n}+\mathbf {\Lambda }\right) V^{n+1}_{i,j,nz} \end{aligned}$$

(A13)

This means that we can get the result of period $n+1$ if we have the value at the period n. By iterating this equation until the value function V is converged, we could solve the distribution $g(b_t,a_{t},z_t) \triangleq g_{i,j,nz}$ for the households through the Kolmogorov Forward equation:

$$\begin{aligned} \left( \textbf{A}^{n}+\mathbf {\Lambda }\right) ^T g = 0 \end{aligned}$$

(A14)

Introducing the Pollution Method in the Achdou et al. (2022), as the transaction matrix $A = \left( \textbf{A}^{n}+\mathbf {\Lambda }\right) ^T$ is a singular matrix, we randomly choose a position (k, k), let $A[k,k] = 1$, and set the k’s value on a zero vector as one: $\epsilon [k] = 0.01$. Then we have the distribution matrix $g \approx A^{-1} \epsilon$. According to Achdou et al. (2022) and Kaplan et al. (2018), the error accuracy of g is close to $10^{-8}$. Then we can compute aggregated values of endogenous variables by the distribution function and treat the rest parts of the model like the classical DSGE model.

The intermediate firms’ optimization problem is:

$$\begin{aligned} \begin{aligned} \min \quad\Pi _{jt} &= \frac{p_{jt}}{P_{t}} y_{jt}- l_{jt} w_t - k_{jt} r_{kt}\\ st:y_{jt} &= Z_t k^\alpha _{jt} l_{jt}^{(1-\alpha )}\\ \end{aligned} \end{aligned}$$

(A15)

So the marginal cost of the intermediate firm, wage, and capital return rate are:

$$\begin{aligned} \begin{aligned} mc_t&= \frac{1}{Z_t}\left( \frac{r_{kt}}{\alpha }\right) ^{\alpha }\left( \frac{w_{t}}{1-\alpha }\right) ^{1-\alpha } \\ r_{kt}&= \alpha \left( \frac{k_{jt}}{l_{jt}}\right) ^{\alpha -1} \\ w_t&= (1-\alpha ) \left( \frac{k_{jt}}{l_{jt}}\right) ^{\alpha } \end{aligned} \end{aligned}$$

(A16)

Appendix B. Derivation for model 3

In Model 3, households hold 3 assets: CBDC $b_t$, deposit $a_t$ and cash $b_{ct}$, and The HJB equation for Model 3 is given as:

$$\begin{aligned} \begin{aligned} V(b_t,a_t,b_{ct},z_t)&= \max _{c_{t},l_t} u(c_{t},l_t) + V_b \dot{b_{t}} + V_{a} \dot{a_t} + V_c \dot{b_{ct}} + \mu _z V_z + \frac{\sigma _z^2}{2} V_{zz} \\ s.t.\quad \dot{b}_{t}&= \left( 1-\tau _w\right) w_t z_t l_t+ r_{bt} b_{t}\ (b_{t}\ge 0) + r_{bt,neg}b_{t}\ (b_{t}< 0)+ T_t -d_{c t}-d_{a t}\\&\quad -\chi _{c}\left( d_{c t}, b_{ct}\right) -\chi \left( d_{t}, a_{t}\right) - c_t, \\ \dot{a}_t&= r_{at} a_t + d_{t}, \quad a_t\ge 0,\\ \dot{b}_{ct}&= r_{ct} b_{ct} + d_{ct},\quad b_t \ge \underline{b}, \quad b_{ct}, a_t \ge 0,\\ r_{ct}&\equiv r_{bt} - \varXi _c \end{aligned} \end{aligned}$$

(B17)

According to Achdou et al. (2022), the income flow can be split into three items:

$$\begin{aligned} \begin{aligned}S^b &\triangleq (1-\tau _w)w_t l_t z_t + r_{bt} b_t + T_t - c_{t}\\S^{a} &\triangleq - d_{at} - \chi _a(d_{at}, a_t), \qquad S^c \triangleq - d_{ct} - \chi _{c}(d_{ct}, b_{ct}),\\\dot{b_{t}} &= S^b + S^{a} + S^c \end{aligned} \end{aligned}$$

(B18)

Let ${x}^{F+} \triangleq max (x,0)$ and ${x}^{B-} \triangleq min (x,0)$. Then we define $x = {x}^{F+} + {x}^{B-}$. We can also separate constraint conditions into different parts. Let $b_t,a_t,b_{ct},z_t$ be in different grids, and then we could mark a household as (i, j, k, nz) when $b_{t} = b_{i}, a_t = a_{j}, b_{ct} = b_{c,k}, z_t = z_{nz}$. Then we define the approximate forward derivative $V^F_b$ and backward derivative $V^B_b$ as follows:

$$\begin{aligned} \begin{aligned}V^{n+1}_b &\approx \frac{V^{n}_{i+1,j,k,nz} - V^{n}_{i,j,k,nz}}{\Delta b} \approx \frac{V^{n}_{i,j,k,nz} - V^{n}_{i-1,j,k,nz}}{\Delta b} \\V^F_b &= \frac{V^{n}_{i+1,j,k,nz} - V^{n}_{i,j,k,nz}}{\Delta b}, \qquad V^B_b = \frac{V^{n}_{i,j,k,nz} - V^{n}_{i-1,j,k,nz}}{\Delta b} \end{aligned} \end{aligned}$$

(B19)

since $V_b \dot{b_{t}} = V^B_b (S^{b,B-} + S^{a,B-} + S^{c,B-}) + V^F_b (S^{b,F+} + S^{a,F+} + S^{c,F+})$, we can check the sign of $\dot{b_t}$ and choose the approximate derivative for computation. This approximate method makes it possible for the program to solve HJB equations quickly.

Then we can get the approximate solution for the household sector by solving the discrete HJB equation. By the definitions above, the discrete HJB equation is given by:

$$\begin{aligned} \begin{aligned} \frac{V^{n+1}_{i,j,k,nz} - V^{n}_{i,j,k,nz}}{\Delta } + \rho V^{n+1}_{i,j,k,nz} &=\max _{c_t,l_t} U + V_b \dot{b_{t}} + V_{a} \dot{a_t} + V_c \dot{b_{ct}} + V_z \mu _z + V_{zz} \frac{\sigma _z^2}{2} \\ &=U + V^B_b (S^{b,B-} + S^{c,B-} + S^{a,B-}) \\&\quad+ V^F_b (S^{b,F+} + S^{c,F+} + S^{a,F+}) \\&\quad+ V^B_{a} d^{a,-}_t + V^{F}_{a} (d^{a,+}_t + r_{at} a_t) \\&\quad+ V^B_c d^{c,-}_t + V^F_c (d^{c,+}_t + r_{ct} b_{ct}) \\&\quad+ V_z \mu _z + V_{zz} \frac{\sigma _z^2}{2} \end{aligned} \end{aligned}$$

(B20)

Note that approximate derivatives, $V^F_b = \frac{V_{i+1,j,k,nz} - V_{i,j,k,nz}}{\Delta b}$ and $V^B_b = \frac{V_{i,j,k,nz} - V_{i-1,j,k,nz}}{\Delta b}$, can be factorized. Then we rewrite the discrete HJB equation as below:

$$\begin{aligned} \begin{aligned} \frac{V^{n+1}_{i,j,k,nz} - V^{n}_{i,j,k,nz}}{\Delta } + \rho V^{n+1}_{i,j,k,nz} =&\, U + \frac{V_{i-1,j,k,nz}}{\Delta b}(-S^{b,B-} - S^{c,B-} - S^{a,B-}) \\&+ \frac{V_{i+1,j,k,nz}}{\Delta b}(S^{b,F+} + S^{c,F+} + S^{a,F+})\\&+ \frac{V_{i,j,k,nz}}{\Delta b}(S^{b,B-} -S^{b,F+} + S^{c,B-} - S^{c,F+}\\&+ S^{a,B-} - S^{a,F+}) \\&+ \frac{V_{i,j,k-1,nz}}{\Delta a}( - d^{a,-}_t ) + \frac{V_{i,j,k+1,nz}}{\Delta a}(d^{a,+}_t + r_{at} a_t) \\&+ \frac{V_{i,j,k,nz}}{\Delta a}(d^{a,-}_t - d^{a,+}_t - r_{at} a_t) \\&+ \frac{V_{i,j-1,k,nz}}{\Delta b_{c}}( - d^{c-}_t) + \frac{V_{i,j+1,k,nz}}{\Delta b_{c}}(d^{c+}_t+r_{ct} b_{ct}) \\&+ \frac{V_{i,j,k,nz}}{\Delta b_{c}}(d^{c-}_t - d^{c+}_t - r_{ct} b_{ct}) \\&+ \sum _{nz' \ne nz}\lambda _{nz}(V_{i,j,k,nz'} - V_{i,j,k,nz}) \end{aligned} \end{aligned}$$

(B21)

which means $V_{i-1,j,k,nz}$ and $V_{i+1,j,k,nz}$ items are on different sub-diagonals of matrix V, $V_{i,j,k,nz}$ items are on the diagonal of V. According to Achdou et al. (2022), there are the same conclusions on other derivatives, so the discrete HJB equation above can be summarized as:

$$\begin{aligned} \frac{1}{\Delta }\left( V^{n+1}_{i,j,k,nz}-V^{n}_{i,j,k,nz}\right) +\rho V^{n+1}_{i,j,k,nz}=U_{i,j,k,nz}^{n}+\left( \textbf{A}^{n}+\mathbf {\Lambda }\right) V^{n+1}_{i,j,k,nz} \end{aligned}$$

(B22)

This means that we can get the result of period $n+1$ if we have the value at the period n. By iterating this equation until the value function V is converged, we could solve the distribution $g(b_t,b_{ct},a_t,z_t) \triangleq g_{i,j,k,nz}$ for the households through the Kolmogorov Forward equation:

$$\begin{aligned} \left( \textbf{A}^{n}+\mathbf {\Lambda }\right) ^T g = 0 \end{aligned}$$

(B23)

The following part of Model 3 is the same as Model 1 and Model 2.

Appendix C. Derivation for NKPC

The product function for the final firm is given by:

$$\begin{aligned} Y_{t}=\left( \int _{0}^{1} (y_{jt})^{\frac{\epsilon -1}{\epsilon }} d j\right) ^{\frac{\epsilon }{\epsilon -1}}, \quad j\in (0,1) \end{aligned}$$

(C24)

where $\epsilon$ denotes the substitution elasticity between intermediate firms.

The price of the final product and intermediate products are $P_t$ and $p_{jt}$, respectively, then the optimization problem of the final firm is given by:

$$\begin{aligned} \begin{aligned} \min _{y_{jt}} \quad&Y_t P_{t} - \int _{0}^{1} y_{jt} p_{jt} dj \\ s.t. \quad&Y_{t}=\left( \int _{0}^{1} (y_{jt})^{\frac{\epsilon -1}{\epsilon }} d j\right) ^{\frac{\epsilon }{\epsilon -1}}, \quad j\in (0,1) \end{aligned} \end{aligned}$$

(C25)

Since all intermediate firms are homogeneous, then the relation between intermediate products and final products is given by:

$$\begin{aligned} y_{jt}=\left( \frac{p_{jt}}{P_{t}}\right) ^{-\epsilon } Y_{t}, \quad P_{t}=\left( \int _{0}^{1} (p_{jt})^{1-\epsilon } d j\right) ^{\frac{1}{1-\epsilon }}, \quad j\in (0,1) \end{aligned}$$

(C26)

Since intermediate firms are compete in labor market and capital market, the price of labor and capital is market level $w_t$ and $r_{kt}$, then the cost minimization problem of intermediate firms is:

$$\begin{aligned} \begin{aligned} \min _{k_{jt},l_{jt}}: \quad cost_t &= k_{jt} r_{kt} + l_{jt} w_t \\ s.t.\quad y_{jt} &= Z_t (k_{jt})^{\alpha } (l_{jt})^{1-\alpha } \end{aligned} \end{aligned}$$

(C27)

where $Z_t$ denotes the total factor productivity, $\alpha$ is the capital share in the Cobb-Douglas productive function. The intermediate firms’ optimal marginal cost is given by:

$$\begin{aligned} mc_{t}=\frac{1}{Z_{t}}\left( \frac{r_{kt}}{\alpha }\right) ^{\alpha }\left( \frac{w_{t}}{1-\alpha }\right) ^{1-\alpha } \end{aligned}$$

(C28)

Under symmetric equilibrium, all intermediate firms produce the same amount of products which is equal to the final product:

$$\begin{aligned} \begin{aligned}&Y_t \equiv y_{jt}\\&P_t \equiv p_{jt}, \quad j\in (0,1). \end{aligned} \end{aligned}$$

(C29)

Intermediate firms will pay Rotemberg price adjustment costs:

$$\begin{aligned} \Theta _{t}\left( \frac{\dot{p}_{jt}}{p_{jt}}\right) =\frac{\theta }{2}\left( \frac{\dot{p}_{jt}}{p_{jt}}\right) ^{2} y_{jt} = \frac{\theta }{2} (\pi _{jt})^2 Y_t, \end{aligned}$$

(C30)

where $\pi _{jt} \triangleq \frac{\dot{P}_{t}}{P_{t}}$ denotes the inflation rate of intermediate prices.

Then the profit maximization problem for intermediate firms is:

$$\begin{aligned} \begin{aligned} \max _{p_{jt}}:&\int _{0}^{\infty } e^{-\int _{0}^{t} r_{kt} d s}\left\{ \frac{p_{jt}}{P_{t}} y_{jt} - mc_t y_{jt} -\frac{\theta }{2} \left( \frac{\dot{p}_{jt}}{p_{jt}}\right) ^2 Y_t\right\} d t \\&= \int _{0}^{\infty } e^{-\int _{0}^{t} r_{kt} d s}\left\{ (\frac{p_{jt}}{P_{t}} - mc_t) \left( \frac{p_{jt}}{P_{t}}\right) ^{-\epsilon } Y_{t} -\frac{\theta }{2} \left( \frac{\dot{p}_{jt}}{p_{jt}}\right) ^2 Y_t\right\} d t \end{aligned} \end{aligned}$$

(C31)

we assume that the discount rate of intermediate firms is $r_{kt}$, this question could be rewritten as:

$$\begin{aligned} \begin{aligned} \max _{p_{jt}}:\Pi _{jt}(p_{jt}) &= (\frac{p_{jt}}{P_{t}} - mc_t) \left( \frac{p_{jt}}{P_{t}}\right) ^{-\epsilon } Y_{t} -\frac{\theta }{2} \left( \frac{\dot{p}_{jt}}{p_{jt}}\right) ^2 Y_t \\ s.t.\quad \dot{p_{jt}} &= p_{jt}\pi _{jt} \\ \dot{t} &= 1, \quad j\in (0,1). \end{aligned} \end{aligned}$$

(C32)

According to Kaplan et al. (2018) lemma 1, this problem could be solve by

$$\begin{aligned} {r_{k}} J(p, t)=\max _{\pi }\left( \frac{p}{P}-mc\right) \left( \frac{p}{P}\right) ^{-\epsilon } Y-\frac{\theta }{2} \pi ^{2} Y+J_{p}(p, t) p \pi + {J_{t}(p, t)} \end{aligned}$$

(C33)

The FOCs of $\pi$ and p are:

$$\begin{aligned} \begin{aligned} J_{p}(p, t) p&={ \theta \pi Y }\\ r_k J_{p}(p, t)&=-\left( \frac{p}{P}-mc\right) \epsilon \left( \frac{p}{P}\right) ^{-\epsilon -1} \frac{Y}{P}+\left( \frac{p}{P}\right) ^{-\epsilon } \frac{Y}{P}\\&\quad+\pi J_{p}(p, t)+J_{p p}(p, t) p \pi +J_{t p}(p, t) \end{aligned} \end{aligned}$$

(C34)

substituting $p=P$:

$$\begin{aligned} \begin{aligned} J_{p}(p, t)&=\frac{\theta \pi Y}{p} \\ \left( r_{k}-\pi \right) J_{p}(p, t)&=-(1-mc) \epsilon \frac{Y}{p}+\frac{Y}{p}+ J_{p p}(p, t) p \pi + J_{t p}(p, t) \end{aligned} \end{aligned}$$

(C35)

The partial derivatives of $J_{p}(p, t)$ respect to t gives:

$$\begin{aligned} J_{p p}(p, t) \dot{p}+J_{p t}(p, t)=\frac{\theta Y \dot{\pi }}{p}+\frac{\theta \dot{Y} \pi }{p}-\frac{\theta Y \pi }{p} \frac{\dot{p}}{p} \end{aligned}$$

(C36)

notice that $J_{p p}(p, t) \dot{p}+J_{p t}(p, t) \equiv J_{p p}(p, t) p \pi + J_{t p}(p, t)$, substituting this equation into previous equation:

$$\begin{aligned} \left( r_{k}-\pi \right) \frac{\theta \pi Y}{p} =-(1-mc) \epsilon \frac{Y}{p}+\frac{Y}{p}+ { \frac{\theta Y \dot{\pi }}{p}+\frac{\theta \dot{Y} \pi }{p}-\frac{\theta Y \pi ^2}{p} } \end{aligned}$$

(C37)

Which could also be written as:

$$\begin{aligned} \left( r_{kt}-\frac{\dot{Y}_t}{Y_t}\right) \pi _t=\frac{1}{\theta }(-(1-mc_t) \epsilon +1)+ \dot{\pi }_t. \end{aligned}$$

(C38)

This is the Kew Keynesian Phillips Curve shown in Kaplan et al. (2018).

Appendix D. Steady state values

Table 3 Steady state values