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According to Belongia and Ireland (2020), nominal interest rate management has been the Federal Reserve's approach to stabilise the economy's output gap and inflation. The Taylor rule is a good predictor of the nominal interest rate management. Interest rate management has been endorsed by previous literature and has shown advantages over the management of money stock, however, economists have only argued against a constant monetary growth rule but have neglected the potential benefits of a flexible monetary growth rule. Belongia and Ireland (2020) is a reconsideration of money growth rules in an estimated New Keynesian model. This paper provides a replication of the New Keynesian model by Belongia and Ireland (2020) as well as extending the authors' work via calibration, as opposed to estimation. This paper examines the responses of macroeconomic variables under the Taylor rule, Interest Rate rule, Flexible Money Growth rate rule as well as the Constant Money Growth rate rule, in response to the different shocks. The shocks are a preference shock, a productivity shock, a money demand shock and a cost push shock. The macroeconomic variables focused on are output growth rates, inflation, nominal interest rates, money growth rates and output gap.

In section 2, the model is explained. The first order conditions and the log-linearisations of the model are then presented in sections 3 and 4, respectively. Section 5 provides an explanation of the data and calibration methods used. The paper presents and discusses the calibration results, in the form of impulse response functions in section 6. Lastly, this paper concludes that of the two money growth rules considered, the Constant Money Growth rate rule is outperformed by the Flexible Money Growth rate rule with regards to efficient responses to the shocks, in particular the money demand shock. Additionally, the Flexible Money Growth rate rule presents comparable results to the interest rate rules and may offer more efficient stabilisation with further assumptions.

The model economy from Belongia and Ireland (2020) includes the representative household, a representative finished goods-producing firm, "i" intermediate goods-producing firms (such that $i \in [0; 1]$, is a continuum) and a central bank. For each period, a unique intermediate good is produced by each intermediate goods-producing firm $i$, such that the intermediate goods adopt the same indexing notation, $i \in [0; 1]$. The symmetry of this model suggests that the focus can be narrowed to a representative intermediate goods-producing firm instead. This representative firm then produces good intermediate good $i$.

The household preferences are described by their expected utility function. These preferences, along with the incomplete indexation of sticky nominal goods prices that are determined by the monopolistically competitive intermediate goods-producing firms suggests a New Keynesian IS and Phillips Curve that are both forward, and backwards-looking. Monetary Policy is assumed to follow a version of the Taylor (1993) rule, in line with the Federal Reserve behaviour over the sample period from 1983 to 2019. To allow for alternative monetary policy rules, the money demand curve used in the paper is chosen such that it may be consistent with US data ranging over the same sample period. With the basic model set up in this way, the derivations of the model's first order conditions for each representative is replicated in the following sections, followed by the complete system of linearised equations.

Each period $t=0,1,2,...$ is entered by the representative household with $M_{t-1}$ units of money and $B_{t-1}$ bonds. The household receives a lump-sum monetary transfer $T_{t}$ from the central bank at the beginning of period $t$. In addition, the household's bonds mature, yielding $B_{t-1}$ additional units of money. The household uses some of this money to purchase $B_{t}$ new bonds at the price of $1=rt$ units of money per bond. Therefore, $r_t$ represents the gross nominal interest rate between $t$ and $t + 1$. The household provides $h_{t}{(i)}$ units of labour to each intermediate goods-producing firm $i\in{[0; 1]}$ during period $t$. The household is paid at the nominal wage rate $W_{t}$. Households therefore earn $W_{t}h_{t}$ in labour income, where

$$h_{t}=\int h_{t}{(i)} ;di$$

represents the total hours worked during the period. In period $t$, the household consumes $C_{t}$ units of the finished goods. Finished goods are purchased at the nominal price, $P_{t}$ from the representative finished goods-producing firm. The household receives nominal profits $D_{t}(i)$ from each intermediate goods-producing firm $i \in {[0; 1]}$, at the end of period $t$. The household then carries $M_{t}$ units of money into period $t + 1$, which is subject to the following budget constraint:

$$\frac{M_{t-1}+T_{t}+B_{t-1}+W_{t}h_{t}+D_{t}}{P_{t}} \ge C_{t} + \frac{M_{t} + \frac{B_{t}}{r_{t}}}{P_{t}}$$

for all $t=0,1,2,...,$ where

$$D_{t}= \int_{0}^{1}D_{t}{(i)};di$$

represents total profits received for the period. The household's preferences are described by the following expected utility function:

$$E_{0}\sum_{t=0}^{\infty}\beta^{t}a_{t}[ln(C_{t} - \gamma C_{t-1}) + v(\frac{M_{t}}{P_{t}Z_{t}}, u_t)-(\frac{\phi_m}{2})(\frac{M_t/P_t}{zM_{t-1}/P_{t-1}}-1)^{2}(\frac{M_{t}}{P_{t}Z_{t}})-h_{t}]$$

Both the discount factor and the habit formation parameter lie between zero and one, with $0<\beta<1$ and $0 \le \gamma \le1$. The preference shock $a_{t}$ follows the stationary autoregressive process below:

$$\ln(a_{t})=\rho_{a} \ln(a_{t-1}) + \varepsilon_{at} \tag{2}$$

for all $t=0,1,2,...,$ with $0 \le \rho_{a} \le 1$, where the serially uncorrelated innovation $\varepsilon_{at}$ is normally distributed with a mean of zero and standard deviation $\sigma_{a}$. Total utility is a combination of the utility obtained from consumption, real balances, and hours worked, which implies additive seperability, so as to imply a specification for the model's IS curve that does not include terms involving money and employment. As shown by King, Plosser, and Rebelo (1988), additive separability also implies that a logarithmic specification over consumption is necessary for the model to be consistent with balanced growth. Also for balanced growth, real balances $M_{t}=P_{t}$ enter utility through the function $v$ after being scaled by the aggregate productivity shock $Z_{t}$. This follows a random walk with drift:

$$\ln(Z_{t})= \ln{z} + \ln(Z_{t-1}) + \varepsilon_{zt} \tag{3}$$

where the serially uncorrelated error term or innovation is normally distributed with zero mean and standard deviation $\sigma_{z}$. The shock $u_{t}$ to money demand follows the stationary autoregressive process:

$$\ln(u_{t})= \rho_{u} \ln(u_{t-1}) + \varepsilon_{ut} \tag{4}$$ for all $t=0,1,2,...$ with $0 \le \rho_{u} <1$, where the serially uncorrelated innovation $\varepsilon_{ut}$ is normally distributed with mean zero and standard deviation $\sigma_{u}$. Finally, the parameter $\phi_{m} \ge 0$ governs the magnitude of the adjustment cost for real balances, adapted from Nelson (2002) and Andres, Lopez-Salido, and Nelson (2004, 2009) to take the quadratic functional form used here. Since these costs subtract from utility along with hours worked, they have the interpretation as a time cost, and are scaled by the average growth rate parameter $z$ from (3) so as to equal zero in the model's steady state. Thus, the household chooses $Ct, ht, Bt$, and $Mt$ for all $t = 0, 1, 2,...,$ to maximize expected utility subject to the budget constraint (1) for all $t = 0,1,2,...,$

The Household preferences are characterised by the following expected utility function:

$$ EU(\cdot_t) = E_{0} \sum_{t=0}^\infty \beta^ta_t U(\cdot_t) = E_{0} \sum_{t=0}^\infty \beta^ta_t[\ln(C_{t}-\gamma C_{t-1})+v(\frac{M_t}{P_tZ_t},u_t)-\frac{\phi_m}{2}(\frac{\frac{M_t}{P_t}}{\frac{zM_{t-1}}{P_{t-1}}}-1)^2(\frac{M_t}{P_tZ_t})-h_t]$$

Given the Household budget constraint:

$$ \left[ \frac{M_{t-1} + T_t +B_{t-1} + W_th_t+D_t}{P_t} \ge C_t + \frac{M_t+\frac{B_t}{r_t}}{P_t} \right]$$

Combining the expected utility and budget constraint above, we can write the LaGrangian for the Household:

$$ \mathcal{L}(\cdot_t) = a_tU( \cdot_{t}) + \beta E_t \mathcal{L}(\cdot_{t+1}) + \Lambda_t \left[ \frac{M_{t-1} + T_t +B_{t-1} + W_th_t+D_t}{P_t}-C_t - \left(\frac{M_t+\frac{B_t}{r_t}}{P_t} \right) \right] $$

where:

$$\begin{aligned} \mathcal{L}(\cdot_{t}) = \mathcal{L}(C_t, h_t, B_t, M_t) = &E_{0} \sum_{t=0}^\infty \beta^ta_t \left[\ln(C_{t}-\gamma C_{t-1})+v(\frac{M_t}{P_tZ_t},u_t)-\frac{\phi_m}{2}(\frac{\frac{M_t}{P_t}}{\frac{zM_{t-1}}{P_{t-1}}}-1)^2(\frac{M_t}{P_tZ_t})-h_t \right]\\ & + E_{0} \sum_{t=0}^\infty\left[ \Lambda_t \left( \frac{M_{t-1} + T_t +B_{t-1} + W_th_t+D_t}{P_t}-C_t - \left(\frac{M_t+\frac{B_t}{r_t}}{P_t} \right) \right) \right] \end{aligned}$$

$$\frac{\partial \mathcal{L}(C_t, h_t, B_t, M_t)}{\partial C_t}= a_t(\frac{1}{C_t-\gamma C_{t-1}})-\Lambda_t + E_t\beta a_{t+1}(-\gamma)(\frac{1}{C_{t+1}-\gamma C_t})$$

Equate to zero and solve:

$$0= a_t(\frac{1}{C_t-\gamma C_{t-1}})-\Lambda_t + E_t\beta a_{t+1}(-\gamma)(\frac{1}{C_{t+1}-\gamma C_t})$$

$$\Lambda_t = a_t(\frac{1}{C_t-\gamma C_{t-1}}) + E_t\beta a_{t+1}(-\gamma)(\frac{1}{C_{t+1}-\gamma C_t}) \tag{5}$$

$$\frac{\partial \mathcal{L}(C_t, h_t, B_t, M_t)}{\partial h_t}= (-1)( a_t) + (\frac{W_t}{P_t})(\Lambda_t)$$

Equate to zero and solve:

$$0 = (-1)( a_t) + (\frac{W_t}{P_t})(\Lambda_t)$$ $$( a_t) = (\frac{W_t}{P_t})(\Lambda_t) $$ $$ a_t = (\frac{W_t}{P_t})(\Lambda_t) \tag{6}$$

$$\frac{\partial L(C_t, h_t, B_t, M_t)}{\partial B_t}= (\frac{-1}{r_t P_t})(\Lambda_t) + E_t\beta \Lambda_{t+1}(\frac{1}{P_{t+1}})$$

Equate to zero and solve:

$$ 0=(\frac{-1}{r_t P_t})(\Lambda_t) + E_t\beta \Lambda_{t+1}(\frac{1}{P_{t+1}})$$ $$ (\frac{1}{r_t P_t})(\Lambda_t) = E_t\beta \Lambda_{t+1}(\frac{1}{P_{t+1}})$$ $$ \Lambda_t = (r_t P_t) E_t\beta\Lambda_{t+1}(\frac{1}{P_{t+1}})$$ $$ \Lambda_t = \beta(r_t) E_t(\frac{P_t\Lambda_{t+1}}{P_{t+1}})$$

Because: $$\pi_t=\frac{P_t}{P_{t-1}}$$

Therefore: $$\frac{P_t}{P_{t+1}}=\frac{1}{\pi_{t+1}}$$

As such, the equation can be rewritten as:

$$\Lambda_t = \beta(r_t)E_t(\frac{\Lambda_{t+1}}{\pi_{t+1}}) \tag{7}$$

$$\frac{\partial L(C_t, h_t, B_t, M_t)}{\partial M_t}=a_t[v_1(\frac{M_t}{P_tZ_t}, u_t) \frac{1}{P_t Z_t}-\phi_m(\frac{\frac{M_t}{P_t}}{\frac{z_tM_{t-1}}{P_{t-1}}}-1)(\frac{\frac{1}{P_{t}}}{\frac{ZM_{t-1}}{P_{t-1}}})(\frac{M_t}{P_tZ_t})-\frac{\phi_m}{2}(\frac{\frac{M_t}{P_t}}{\frac{z_tM_{t-1}}{P_{t-1}}}-1)^2(\frac{1}{P_tZ_t})]$$ $$ + \Lambda_t[-\frac{1}{P_t}] + E_t\beta a_{t+1} \left[-\phi_m(\frac{M_{t+1}/P_{t+1}}{ZM_t/P_t}-1)(-\frac{M_{t+1}/P_{t+1}}{zM_{t}^2/P_{t}})(\frac{M_{t+1}}{P_{t+1}Z_{t+1}}) \right]+E_t \beta \Lambda_{t+1}(\frac{1}{P_{t+1}})$$

where:

$$ v_1(\cdot) = \frac{\partial}{\partial M_t} v(\cdot) $$

We can now use equation (7) and the fact that $\frac{P_t}{P_{t-1}} = \pi_t$, to yield:

$$ \beta E_t \Lambda_{t+1} \frac{1}{P_{t+1}} = \frac{1}{P_tr_t} \Lambda_t $$

multiply by $\frac{1} {P_t Z_t}$ throughout and set equal to zero:

$$\begin{aligned} \therefore 0 = & a_tv_1(\frac{M_t}{P_tZ_t}, u_t) - a_t\frac{\phi_m}{2}(\frac{\frac{M_t}{P_t}}{\frac{zM_{t-1}}{P_{t-1}}}-1)^2 - a_t \phi_m(\frac{\frac{M_t}{P_t}}{\frac{zM_{t-1}}{P_{t-1}}}-1)(\frac{\frac{M_t}{P_{t}}}{\frac{zM_{t-1}}{P_{t-1}}})\\ & - \Lambda_t Z_t + \frac{Z_t}{r_t} \Lambda_t + E_t\beta a_{t+1} \left[\phi_m(\frac{M_{t+1}/P_{t+1}}{ZM_t/P_t}-1) \left(\frac{M_{t+1}/P_{t+1}}{zM_{t}/P_{t}} \right)^2(\frac{zZ_t}{Z_{t+1}}) \right] \end{aligned}$$

Rearranging yields the final result:

$$\begin{aligned} &a_{t} v_{1}\left(\frac{M_{t}}{P_{t} Z_{t}}, u_{t}\right)-a_{t}\left(\frac{\phi_{m}}{2}\right)\left(\frac{M_{t} / P_{t}}{z M_{t-1} / P_{t-1}}-1\right)^{2} \\ &-a_{t} \phi_{m}\left(\frac{M_{t} / P_{t}}{z M_{t-1} / P_{t-1}}-1\right)\left(\frac{M_{t} / P_{t}}{z M_{t-1} / P_{t-1}}\right) \\ &+\beta \phi_{m} E_{t}\left[a_{t+1}\left(\frac{M_{t+1} / P_{t+1}}{z M_{t} / P_{t}}-1\right)\left(\frac{M_{t+1} / P_{t+1}}{z M_{t} / P_{t}}\right)^{2}\left(\frac{z Z_{t}}{Z_{t+1}}\right)\right] \\ &=Z_{t} \Lambda_{t}\left(1-\frac{1}{r_{t}}\right) \end{aligned} \tag{8}$$

The representative finished goods-producing firm uses $Y_t(i)$ units of each intermediate good $i \in [0,1]$. These intermediate goods are bought at the nominal price $P_t(i)$ to produce $Y_t$ units of the final good according to the technology described by

$$[\int_0^1Y_t(i)^{\frac{(\theta_t-1)}{\theta_t}} di]^{\frac{\theta_t}{\theta_t-1}} \ge Y_t$$

Here, $\theta_t$ represents a random shock to the intermediate goods-producing firms' preferred markup of price over marginal cost. It acts like a cost push shock of the kind introduced into the New Keynesian model by Clarida, Gali & Gertler (1999). This markup shock follows the stationary autoregressive process

$$\ln(\theta_t)=(1-\rho_{\theta})\ln(\theta)+\rho_{\theta}\ln(\theta_{t-1})+\varepsilon_{\theta t} \tag{10}$$

for all $t=0,1,2,...,$, where the serially uncorrelated innovation $\varepsilon_{\theta t}$ follows a normal distribution with a mean of zero and a standard deviation $\sigma_{\theta}$. Therefore, in each period $t$, the finished good-producing firm chooses $Y_t(i)$ for all $i \in [0,1]$ to maximize its profits, which are given by

$$P_t[\int_0^1Y_t(i)^{\frac{\theta_t-1}{\theta_t}} di]^{\frac{\theta_t}{\theta_t-1}} - \int_0^1P_t(i)Y_t(i) ;di$$

The first order conditions are derived from the following Langrangian function:

$$L = \int_0^1P_t(i)Y_t(i) ; di +\lambda_t[Y_t - (\int_0^1Y_t(i)^{\frac{\theta_t-1}{\theta_t}} ;di)^{\frac{\theta_t}{\theta_t-1}}]$$

$$\frac{\partial L}{\partial Y_t(i)} = P_t(i) - \lambda_t[{\frac{\theta_t}{\theta_t-1}} (\int_0^1 Y_t(i)^{\frac{\theta_t-1}{\theta_t}} di)^{\frac{\theta_t}{\theta_t-1} -1} \frac{\theta_t-1}{\theta_t} Y_t(i)^{\frac{\theta_t-1}{\theta_t} -1}] = 0$$ $$P_t(i) - \lambda_t[{\frac{\theta_t}{\theta_t-1}} (\int_0^1 Y_t(i)^{\frac{\theta_t-1}{\theta_t}} di)^{\frac{1}{\theta_t-1}} \frac{\theta_t-1}{\theta_t} Y_t(i)^{\frac{-1}{\theta_t}}] = 0$$

$$ \lambda_t[\int_0^1 Y_t(i)^{\frac{\theta_t-1}{\theta_t}} di]^{\frac{1}{\theta_t-1}} Y_t(i)^{\frac{-1}{\theta_t}} = P_t(i)$$

$$ \lambda_t[\int_0^1 Y_t(i)^{\frac{\theta_t-1}{\theta_t}} di]^{\frac{1}{\theta_t-1}} = P_t(i)Y_t(i)^{\frac{1}{\theta_t}}$$ $$ \lambda_t[\int_0^1 Y_t(i)^{\frac{\theta_t-1}{\theta_t}} di]^{\frac{1}{\theta_t-1}}P_t(i)^{-1} = Y_t(i)^{\frac{1}{\theta_t}}$$

$$ \lambda_t^{\theta_t} [\int_0^1 Y_t(i)^{\frac{\theta_t-1}{\theta_t}} di]^{\frac{\theta_t}{\theta_t-1}}P_t(i)^{-\theta_t} = Y_t(i)$$

$$ \lambda_t^{\theta_t} Y_t P_t(i)^{-\theta_t} = Y_t(i)$$ Where $Y_t = [\int_0^1 Y_t(i)^{\frac{\theta_t-1}{\theta_t}} di]^{\frac{\theta_t}{\theta_t-1}}$

$$ (\frac{\lambda_t}{P_t(i)})^{\theta_t} Y_t = Y_t(i)$$ Sub $Y_t(i) = (\frac{\lambda_t}{P_t(i)})^{\theta_t} Y_t$ into $Y_t$

$$[\int_0^1 ((\frac{\lambda_t}{P_t(i)})^{\theta_t} Y_t)^{\frac{\theta_t-1}{\theta_t}} di]^{\frac{\theta_t}{\theta_t-1}} = Y_t$$ $$[\int_0^1 ((\frac{P_t(i)}{\lambda_t})^{-\theta_t})^{\frac{\theta_t-1}{\theta_t}} di]^{\frac{\theta_t}{\theta_t-1}} = 1$$ $$(\frac{1}{\lambda_t})^{-\theta_t}[\int_0^1 (P_t(i))^{-\theta_t+1} di]^{\frac{\theta_t}{\theta_t-1}} = 1$$

$$[\int_0^1 (P_t(i))^{1-\theta_t} di]^{\frac{\theta_t}{\theta_t-1}} = {\lambda_t}^{-\theta_t}$$ $$[\int_0^1 (P_t(i))^{1-\theta_t} di]^{\frac{1}{1-\theta_t}} = {\lambda_t}$$ Therefore:

$$\lambda_t = P_t$$ Sub $Y_t = P_t$ back in:

$$(\frac{P_t}{P_t(i)})^{\theta_t} Y_t = Y_t(i)$$

Therefore:

$$Y_t(i) = (\frac{P_t(i)}{P_t})^{-\theta_t} Y_t$$

In each period $t=0,1,2,...,$ the representative intermediate goods-producing firm uses $h_t(i)$ units of labour from the representative household to produce $Y_t(i)$ units of intermediate good $i$ according to the technology described by:

$$Z_th_t(i) \ge Y_t(i) \tag{11}$$

Here, $Z_t$ represents the aggregate productivity shock introduced in (3).

In order to change its nominal price between periods, the firm undergoes a quadratic cost of adjustment, given by:

$$\frac{\phi_p}{2}[\frac{P_t(i)}{{\pi_{t-1}^a \pi^{1-a}P_{t-1}(i)}}-1]^2Y_t$$

In order to maximise its total real market value proportional to:

$$E_0\sum_{t=0}^\infty \beta^t\Lambda_t[\frac{D_t(i)}{P_t}]$$ the firm chooses $P_t(i)$ for all $t = 0,1,2,...$.

Here, the firm's real profits are measured by:

$$\frac{D_t(i)}{P_t}=[\frac{P_t(i)}{P_t}]^{1-\theta_t}Y_t-[\frac{P_t(i)}{P_t}]^{-\theta_t}(\frac{W_t}{P_t})(\frac{Y_t}{Z_t})-\frac{\phi}{2}[\frac{P_t(i)}{\pi_{t-1}^a \pi^{1-a}P_{t-1}(i)}]^2Y_t\tag{12}$$

The Value Function of this optimisation problem is: $$v=maxE_t\sum_{t=0}^\infty\beta^t\Lambda_t{\frac{D_t(i)}{P_t}}$$

From this, the Bellm an Equation can be constructed as:

$$v=\beta^t\Lambda_t{\frac{D_t(i)}{P_t}} + \beta^{t+1}E_t\Lambda_{t+1}{\frac{D_{t+1}(i)}{P_{t+1}}}$$ $$v=\beta^t\Lambda_t[(\frac{P_t(i)}{P_t})^{1-\theta_t}Y_t-(\frac{P_t(i)}{P_t})^{-\theta_t}(\frac{W_t}{P_t})(\frac{Y_t}{Z_t})-\frac{\phi}{2}(\frac{P_t(i)}{\pi_{t-1}^a \pi^{1-a}P_{t-1}(i)})^2Y_t] + \beta^{t+1}E_t\Lambda_{t+1}[(\frac{P_{t+1}(i)}{P_{t+1}})^{1-\theta_{t+1}}Y_{t+1}$$

$$-(\frac{P_{t+1}(i)}{P_{t+1}}]^{-\theta_{t+1}}(\frac{W_{t+1}}{P_{t+1}})(\frac{Y_{t+1}}{Z_{t+1}})-\frac{\phi_p}{2}[\frac{P_{t+1}(i)}{\pi_{t}^a \pi^{1-a}P_{t}(i)}]^2Y_{t+1}]$$

$$ \frac{\partial v}{\partial P_t(i)} = \beta^t\Lambda_t(1-\theta_t)(\frac{P_t(i)}{P_t})^{-\theta_t}(\frac{1}{P_t})Y_t-\beta^t\Lambda_t[(-\theta_t\frac{P_t(i)}{P_t})^{-\theta_t-1}(\frac{1}{P_t})(\frac{W_t}{P_t})(\frac{Y_t}{Z_t})]$$ $$-\beta^t\Lambda_t[\phi_p(\frac{P_t(i)}{\pi_{t-1}^a \pi^{1-a}P_{t-1}(i)}-1)(\frac{1}{\pi_{t-1}^a \pi^{1-a}P_{t-1}(i)})Y_t] -\beta^{t+1}E_t\Lambda_{t+1}\phi_p[(\frac{P_{t+1}(i)}{\pi_t^a\pi^{1-a}P_{t}(i)}-1)(\frac{-P_{t+1}(i)Y_{t+1}}{\pi_t^a\pi^{1-a}P_{t}(i)^2})] =0$$

Next, in order to cancel out most $\Lambda_t$, $Y_t$ and $P_t$'s, multiply through by $\frac{P_t}{\Lambda_t Y_t}$:

$$\beta^t(1-\theta_t)(\frac{P_t(i)}{P_t})^{-\theta_t} + \beta^t[\theta_t(\frac{P_t(i)}{P_t})^{-\theta_t-1}(\frac{W_t}{P_t})(\frac{1}{Z_t})] - \beta^t\phi_p[(\frac{P_t(i)}{\pi_{t-1}^a \pi^{1-a}P_{t-1}(i)}-1)(\frac{P_t}{\pi_{t-1}^a \pi^{1-a}P_{t-1}(i)})]$$ $$ +\beta^{t+1}\phi_pE_t(\frac{\Lambda_{t+1}}{\Lambda_{t}})(\frac{P_{t+1}(i)}{\pi_t^a\pi^{1-a}P_{t}(i)}-1)(\frac{P_{t+1}(i)}{\pi_t^a\pi^{1-a}P_{t}(i)})(\frac{Y_{t+1}}{Y_t})(\frac{P_t}{P_t(i)})=0$$

$$(1-\theta_t)(\frac{P_t(i)}{P_t})^{-\theta_t} + \theta_t(\frac{P_t(i)}{P_t})^{-\theta_t-1}(\frac{W_t}{P_t})(\frac{1}{Z_t}) - \phi_p[(\frac{P_t(i)}{\pi_{t-1}^a \pi^{1-a}P_{t-1}(i)}-1)(\frac{P_t}{\pi_{t-1}^a \pi^{1-a}P_{t-1}(i)})]$$ $$ + \beta\phi_pE_t(\frac{\Lambda_{t+1}}{\Lambda_{t}})(\frac{P_{t+1}(i)}{\pi_t^a\pi^{1-a}P_{t}(i)}-1)(\frac{P_{t+1}(i)}{\pi_t^a\pi^{1-a}P_{t}(i)})(\frac{Y_{t+1}}{Y_t})(\frac{P_t}{P_t(i)})=0 \tag{13}$$

and (11) with equality for all $t=0,1,2,...,$

A social planner for this economy who can overcome the frictions associated with monetary trade, sluggish price adjustment, and the monopolistically competitive structure of the intermediate goods-producing sector chooses $Q_t$ and $n_t(i)$ for all $i \in [0,1]$ to maximize the social welfare function

$$E_0\sum_{t=0}^{\infty}\beta^ta_t[\ln(Q_t-\gamma Q_{t-1})-\int_0^1n_t(i) di]$$ subject to

$$Z_t[\int_0^1n_t(i) di]^{\frac{\theta_t}{\theta_t-1}} \ge Q_t$$

As such, the Bellman Equation of this model is:

$$\begin{aligned} v(Q_{t-1}) = & a_t[\ln(Q_t-\gamma Q_{t-1})-\int_0^1n_t(i) di] \\ & + \beta E_tv(Q_t,Q_{t+1}) + \Xi [Z_t(\int_0^1n_t(i) di)^{\frac{\theta_t}{\theta_t-1}} - Q_t] \end{aligned}$$

$$\frac{\partial v(Q_{t-1})}{\partial Q_t} = \frac{a_t}{Q_t-\gamma Q_{t-1}} + \beta E_t\frac{\partial v(Q_t, Q_{t+1})}{\partial Q_t} - \Xi =0$$ We know that

$$\frac{\partial v(Q_{t-1})}{\partial Q_{t-1}} = \frac{a_t}{Q_t-\gamma Q_{t-1}} (-\gamma)$$

Therefore, when iterating forward, we find that: $$\frac{\partial v(Q_{t})}{\partial Q_{t}} = \frac{a_{t+1}}{Q_{t+1}-\gamma Q_{t}} (-\gamma)$$

Subbing $\frac{\partial v(Q_{t})}{\partial Q_{t}}$ back in gives:

$$\frac{\partial v(Q_{t-1})}{\partial Q_t} = \frac{a_t}{Q_t-\gamma Q_{t-1}} + \beta E_t\frac{a_{t+1}}{Q_{t+1}-\gamma Q_{t}} (-\gamma) - \Xi =0$$ Multiply through by $(-1)$:

$$ \frac{a_t}{Q_t-\gamma Q_{t-1}} - \beta E_t\frac{a_{t+1}}{Q_{t+1}-\gamma Q_{t}} (\gamma) = \Xi \tag{14.0}$$

$$\frac{\partial v(Q_{t-1})}{\partial n_t} = a_t (-1) + \Xi Z_t \frac{\theta_t}{\theta_t -1} (\int_0^1n_t(i)^\frac{\theta_t -1}{\theta_t} di)^\frac{\theta_t}{\theta_t - 1} (\frac{\theta_t -1}{\theta_t}) n_t(i)^{\frac{\theta_t -1}{\theta_t}-1}=0$$ $$ a_t + \Xi Z_t (\int_0^1n_t(i)^\frac{\theta_t -1}{\theta_t} di)^\frac{1}{\theta_t - 1} n_t(i)^{\frac{-1}{\theta_t}}=0$$ $$ \Xi Z_t (\int_0^1n_t(i)^\frac{\theta_t -1}{\theta_t} di)^\frac{1}{\theta_t - 1} n_t(i)^{\frac{-1}{\theta_t}}=a_t$$

The Feasibility Constraint of this problem is:

$$(\int_0^1n_t(i)^\frac{\theta_t -1}{\theta_t} di)^\frac{\theta_t}{\theta_t - 1} \ge \frac{Q_t}{Z_t}$$ Rearranging the Feasibility Constraint gives:

$$(\int_0^1n_t(i)^\frac{\theta_t -1}{\theta_t} di)^\frac{1}{\theta_t - 1} \ge (\frac{Q_t}{Z_t})^\frac{1}{\theta_t}$$

Therefore

$$\Xi Z_t (\frac{Q_t}{Z_t})^\frac{1}{\theta_t}n_t(i)^{\frac{-1}{\theta_t}}=a_t$$

Given that the equation above is consistent for each i, we have $n_t(i)=n_t$, and

$$\begin{aligned}&(\int_0^1n_t(i)^\frac{\theta_t -1}{\theta_t} di)^\frac{1}{\theta_t - 1} = (\frac{Q_t}{Z_t})^\frac{1}{\theta_t}\\ &\Rightarrow n_t = \frac{Q_t}{Z_t} \end{aligned}$$

Which yields:

$$\Xi = a_t/Z_t$$

and (14.0) can be written as:

$$\frac{1}{Z_{t}}=\frac{1}{Q_{t}-\gamma Q_{t-1}}-\beta \gamma E_{t}\left[\left(\frac{a_{t+1}}{a_{t}}\right)\left(\frac{1}{Q_{t+1}-\gamma Q_{t}}\right)\right] \tag{14}$$

From the definition of the output gap (say, $x_t$) we can use the efficient level of output to write:

$$x_t = Y_t/Q_t \tag{15}$$

\newpage

In the following section, we show the steps to reach the final log-linearised equations (19) to (31) from in Section 2.7 of "Belongia and Ireland (2020)", starting from the final system of equations shown in the [Appendix:].

We can rewrite this equation, for the sake of simplicity:

$$\begin{aligned} y_{t} &= c_t + \frac{\phi_{p}}{2} \cdot \left[ \frac{\pi_{t}}{{\pi_{t-1}}^{\alpha} \pi^{1-\alpha}} -1 \right]^{2} \cdot y\\ &=c_{t} + \frac{\phi_{p}}{2} \cdot \left(\frac{\pi_{t}}{{\pi_{t-1}}^{\alpha} \pi^{1-\alpha}} \right)^2 \cdot y_{t} - \phi_{p} \cdot \frac{\pi_{t}}{{\pi_{t-1}}^{\alpha} \pi^{1-\alpha}} \cdot y_{t} + \frac{\phi_{p}}{2} \cdot y_{t}\\ &= c_t + p_{1} - p_{2} + p_{3} \end{aligned}$$

Now, we apply the Taylor method to each $p_{i}$, to get:

$$\begin{aligned} p_{1}^{SS}&=\frac{\phi_{p} \cdot y}{2}\\left[\frac{dp_{1}}{dy_{t}} \right]^{SS} &= \frac{\phi_{p}}{2}\\left[\frac{dp_{1}}{d\pi_{t}} \right]^{SS} &= \frac{\phi_{p} \cdot y}{\pi}\\left[\frac{dp_{1}}{d\pi_{t-1}} \right]^{SS} &= - \frac{\phi_{p} \cdot y}{\pi} \end{aligned}$$

Therefore, we can rewrite $p_{1}$ using the Taylor method, as:

$$\begin{aligned} p_{1} = \frac{\phi_{p} \cdot y}{2} + \frac{\phi_{p}}{2} y \cdot \hat{y_{t}} + \frac{\phi_{p} \cdot y}{\pi} \pi \cdot \hat{\pi_{t}} - \frac{\phi_{p} \cdot y}{\pi}\pi \cdot \hat{\pi_{t-1}}\ = \frac{\phi_{p} \cdot y}{2} + \frac{\phi_{p} \cdot y \cdot \hat{y_{t}} }{2} + \phi_{p} \cdot y \cdot \hat{\pi}{t} - \alpha \cdot \phi{p} \cdot y \cdot \hat{\pi}_{t-1} \ \end{aligned}$$

and for $p_{2}$

$$\begin{aligned} \left[ p_{2} \right]^{SS} &= \phi_{p} y \\ \left[ \frac{dp_{2}}{dy_{t}} \right]^{SS} &= \phi_{p} \\ \left[ \frac{dp_{2}}{d\pi_{t}} \right]^{SS} &= \frac{\phi_{p} y}{\pi} \\ \left[ \frac{dp_{2}}{d\pi_{t-1}} \right]^{SS} &= - \alpha \frac{\phi_{p} y}{\pi} \end{aligned}$$

Then, writing the Taylor approximation for $p_{2}$

$$\begin{aligned} p_{2} &= \phi_{p} y + \phi_{p} y \hat{y}{t} + \frac{\phi{p} y}{\pi} \pi \hat{\pi}{t} - \alpha \frac{\phi{p} y}{\pi} \pi \hat{\pi}{t-1}\ &= \phi{p} y + \phi_{p} y \hat{y}{t} + \phi{p} y \hat{\pi}{t} - \alpha \phi{p} y \hat{\pi}_{t-1} \end{aligned}$$

Lastly, repeating the same process for the last term $p_{3}$

$$\begin{aligned} \left[ p_{3} \right]^{SS} &= \frac{\phi_{p} y} {2}\\left[ \frac{dp_{3}}{dy_{t}} \right]^{SS} &= \frac{\phi_{p}}{2} \end{aligned}$$

Then, writing the Taylor approximation for $p_{3}$

$$p_{3} = \frac{\phi_{p} y} {2} + \frac{\phi_{p}}{2}y \hat{y}_{t}$$

Putting it all together with $y_{t} = c + p_{1} - p_{2} + p_{3}$ we get:

$$\begin{aligned} y_{t} = & \ c_t +\left[ \overbrace{\frac{\phi_{p} \cdot y}{2}}^{A} + \overbrace{\frac{\phi_{p} \cdot y \cdot \hat{y_{t}} }{2}}^{B} + \overbrace{\phi_{p} \cdot y \cdot \hat{\pi}{t}}^{C} - \overbrace{\alpha \cdot \phi{p} \cdot y \cdot \hat{\pi}{t-1}}^{D} \right] \ & \ - \left[ \overbrace{\phi{p} y}^{2A} + \overbrace{\phi_{p} y \hat{y}{t}}^{2B} + \overbrace{ \phi{p} y \hat{\pi}{t}}^{C} - \overbrace{\alpha \phi{p} y \hat{\pi}{t-1}}^{D} \right] + \left[ \overbrace{\frac{\phi{p} y} {2}}^{A} + \overbrace{\frac{\phi_{p}}{2}y \hat{y}{t}}^{B} \right] \= &\ c{t} + A + B + C -D - 2A - 2B - C + D + A + B\ \end{aligned}$$

Now, using $y = c + \frac{\phi_{p} \cdot y}{2} - \phi_{p} y + \frac{\phi_{p} y} {2} = c$, we subtract y on both sides, such that

$$\begin{aligned} A + B + C &- D - 2A - 2B - C + D + A + B = 0 \text{ and thus, } \\ y_{t} - y &= c_t - c \\ \Rightarrow y \cdot \hat{y}_t &= c \cdot \hat{c}_t \\ \Rightarrow \hat{y}_t &= \hat{c}_t \end{aligned}$$

Using equation (5*) and again, rewriting for simplicity, we separate the equation into two parts:

$$\lambda_{t}=\frac{a_{t} z_{t}}{z_{t} c_{t}-\gamma c_{t-1}}-\beta \gamma E{t}\left(\frac{a_{t+1}}{z_{t+1} c_{t+1}-\gamma c_{t}}\right) = a_1 + \beta \gamma E_t (a_2)$$

Now we can determine the Taylor approximations individually:

$$\begin{aligned} \left[ a_{1} \right]^{SS} &= \frac{a z}{z c -\gamma c} = \frac{a z}{(z -\gamma)c } \\ \left[\frac{da_{1}}{da_{t}} \right]^{SS} &= \frac{z}{(z -\gamma)c} \\ \left[\frac{da_{1}}{dz_{t}} \right]^{SS} &= \frac{a z c- \gamma a c-a z c}{(z c- \gamma c)^{2}} \\ &= -\frac{ \gamma a c}{(z c- \gamma c)^{2}} \\ \left[\frac{da_{1}}{dc_{t}} \right]^{SS} &= \frac{da_{1}}{dc_{t}} \left[ \frac{a_{t} z_{t}}{(z_t c_{t}-\gamma c_{t-1})} \right]^{SS}\\ &= -a_{t} z_t\left(z_{t} c_{t} - \gamma c_{t-1}\right)^{-2} \cdot z t\\ &= - \frac {a z z} {(z c - \gamma c)^{2}}\\ \left[\frac{da_{1}}{dc_{t-1}} \right]^{SS} &= \left[ \frac{d}{d c_{t-1}} \left[ -a_{t} z_{t} \left( z_{t} c_{t} - \gamma c_{t-1} \right)^{-1} \right] \right]^{SS}\\ &= \left[ -a_{t} z_{t}\left(z_{t} c_{t}-\gamma c_{t-1}\right)^{-2} \cdot(-\gamma) \right]^{SS}\\ &= \frac{a z \gamma}{(z c-\gamma c)^{2}} \end{aligned}$$

The similarity between $a_1$ and $a_2$ means that some of the steps will yield very similar results:

$$\begin{aligned} \left[ a_{2} \right]^{SS} &= \left[ a_{1} \right]^{SS} / z\\ &= \frac{a}{(z -\gamma)c }\\ \left[\frac{da_{2}}{da_{t+1}} \right]^{SS} &= \left[\frac{da_{1}}{da_{t}} \right]^{SS} \cdot 1 / z\\ &= \frac{1}{(z -\gamma)c}\\ \left[\frac{da_{2}}{dz_{t+1}} \right]^{SS} &= \left{\frac{d}{d z_{t+1}}\left[\frac{a_{t+1}}{z_{t+1} c_{t+1}-\gamma c_{t}}\right]\right}^{SS}\\ &=\left{-a_{t+1}\left[z_{t+1} c_{t+1}-\gamma c_t\right]^{-2} \cdot c_{t+1}\right}^{SS}\\ &= - \frac{a c}{z c-\gamma c}\\ \left[\frac{da_{2}}{dc_{t+1}} \right]^{SS} &= \left[\frac{da_{1}}{dc_{t}} \right]^{SS} \cdot 1 / z \\ &= - \frac {a z} {(z c - \gamma c)^{2}}\\left[\frac{da_{2}}{dc_{t}} \right]^{SS} &= \left[\frac{da_{1}}{dc_{t-1}} \right]^{SS} \cdot 1 / z\&= \frac{a \gamma}{(z c-\gamma c)^{2}} \end{aligned}$$

Thus, we have $$\lambda_t = a_1 - \beta \gamma E_t (a_2) \text{ which we can expand, and } \ \lambda = \frac{a z}{(z -\gamma)c } - \beta \gamma \frac{a}{(z-\gamma) c}$$

$$\begin{aligned} \lambda_t &= \overbrace{ \left[ \mathbf{\frac{a z}{(z -\gamma)c }}

• \frac{z}{(z -\gamma)c} \cdot a \hat{a}_t

• \frac{ \gamma a c}{(z c- \gamma c)^{2}} \cdot z \hat{z}_t
• \frac{a z z}{(z c-\gamma c)^{2}} \cdot c \hat{c}_t

• \frac{a z \gamma}{(z c-\gamma c)^{2}} \cdot c \hat{c}_{t-1} \right] }^{a_1}\ & - \beta \gamma E_t \overbrace{ \left[ \mathbf{\frac{a}{(z-\gamma) c}}
• \frac{1}{(z-\gamma) c} \cdot a \hat{a}_{t+1}

• \frac{a c}{z c-\gamma c} z \hat{z}_{t+1}
• \frac{a z}{(z c-\gamma c)^{2}} \cdot c \hat{c}_{t+1}

• \frac{a \gamma}{(z c-\gamma c)^{2}} \cdot c \hat{c}_t \right]}^{a_2} \end{aligned}$$

subtracting $\lambda$ from both sides of the equations to get $\lambda_t - \lambda$ on the left side, we get:

$$\begin{aligned} \lambda \hat{\lambda}_t &= \left[ \frac{z}{(z -\gamma)c} \cdot a \hat{a}_t

• \frac{ \gamma a c}{(z c- \gamma c)^{2}} \cdot z \hat{z}_t
• \frac{a z z}{(z c-\gamma c)^{2}} \cdot c \hat{c}_t

• \frac{a z \gamma}{(z c-\gamma c)^{2}} \cdot c \hat{c}{t-1} \right]\ & - \beta \gamma E_t \left[ \frac{1}{(z-\gamma) c} \cdot a \hat{a}{t+1}

• \frac{a c}{z c-\gamma c} z \hat{z}_{t+1}
• \frac{a z}{(z c-\gamma c)^{2}} \cdot c \hat{c}_{t+1}

• \frac{a \gamma}{(z c-\gamma c)^{2}} \cdot c \hat{c}_t \right]\ &= \frac{az}{(z -\gamma)c} \cdot \hat{a}_t

• \frac{ \gamma a c z}{(z c- \gamma c)^{2}} \cdot \hat{z}_t
• \frac{a c z z}{(z c-\gamma c)^{2}} \cdot \hat{c}_t

• \frac{\gamma a c z }{(z c-\gamma c)^{2}} \cdot \hat{c}{t-1} \ & - \frac{\beta \gamma a}{(z-\gamma) c} \cdot E_t \hat{a}{t+1}
• \frac{a c z}{z c-\gamma c} E_t \hat{z}_{t+1}
• \frac{a c z}{(z c-\gamma c)^{2}} \cdot E_t\hat{c}_{t+1}

• \frac{ \gamma a c }{(z c-\gamma c)^{2}} \cdot \hat{c}_t \end{aligned}$$

From the paper we know that $a=1$, $E_t \hat{a}_{t+1} = (\rho_a \hat{a}t)$, and $E_t \hat{z}{t+1}=0$, and thus:

$$\begin{aligned} \lambda \hat{\lambda}_t &= \frac{z}{(z -\gamma)c} \cdot \hat{a}_t

• \frac{ \gamma c z}{(z c- \gamma c)^{2}} \cdot \hat{z}_t
• \frac{c z z}{(z c-\gamma c)^{2}} \cdot \hat{c}_t

• \frac{\gamma c z }{(z c-\gamma c)^{2}} \cdot \hat{c}_{t-1} \ & - \frac{\beta \gamma a}{(z-\gamma) c} \cdot \rho_a \hat{a}_t
• 0
• \beta \gamma \frac{c z}{(z c-\gamma c)^{2}} \cdot E_t\hat{c}_{t+1}

• \beta \gamma \frac{\gamma c }{(z c-\gamma c)^{2}} \cdot \hat{c}_t \end{aligned}$$

The common denominator can be removed by multiplying both sides by $(zc-\gamma c)^2$ and then we divide by c

$$\begin{aligned} \lambda \hat{\lambda}_t (zc-\gamma c)^2 &= z (zc-\gamma c) \hat{a}_t

• \gamma c z \hat{z}_t
• c z z \hat{c}_t

• \gamma c z \hat{c}_{t-1} \ & - \beta \gamma a \rho_a (zc-\gamma c) \hat{a}_t
• c zE_t\hat{c}_{t+1}

• \gamma c \hat{c}_t\ \lambda \hat{\lambda}_t (z-\gamma )^2c &= z (z-\gamma ) \hat{a}_t
• \gamma z \hat{z}_t
• z z \hat{c}_t

• \gamma z \hat{c}_{t-1} \ & - \beta \gamma a \rho_a (z-\gamma ) \hat{a}_t
• \beta \gamma zE_t\hat{c}_{t+1}

• \gamma \hat{c}_t \end{aligned}$$

We know $\hat{c}_t = \hat{y}_t$ and $c = y$

$$\begin{aligned} \lambda \hat{\lambda}_t (z-\gamma)^{2} y &= (z-\gamma) (z-\beta \gamma \rho_a ) \hat{a}_t -\gamma z \hat{z}t -\left(z^{2} + \beta \gamma^{2}\right) \hat{y}{t}

• \gamma z \hat{y}_{t-1}
• \beta \gamma z E_{t} \hat{y}_{t+1} \end{aligned}$$

From the Appendix of "Belongia and Ireland (2020)", we can use the fact that

$$y \lambda=\frac{(z-\beta \gamma)}{(z-\gamma)}$$

$$\begin{aligned} &(z-\gamma)(z-\beta \gamma) \hat{\lambda}{t}\ & = (z-\gamma)\left(z-\beta \gamma \rho_{a}\right) \hat{a}{t}-\gamma z \hat{z}{t}-\left(z^{2}+\beta \gamma^{2}\right) \hat{y}{t}+z \gamma \hat{y}{t-1} + \beta \gamma z E_{t} \hat{y}_{t+1} \end{aligned}$$

yielding the final result: $$(z-\gamma)(z-\beta \gamma) \hat{\lambda}{t}=(z-\gamma)\left(z-\beta \gamma p{a}\right) \hat{a}{t}-\gamma z \hat{z}{t}+z \gamma \hat{y}{t-1}+\beta \gamma z E{t} \hat{y}{t+1}-\left(z^{2}+\beta \gamma^{2}\right) \hat{y}{t}$$

From Equation (7*) we have

$$\lambda_{t}=\beta r_{t} E_{t}\left(\frac{\lambda_{t+1}}{z_{t+1} \pi_{t+1}}\right) \tag{7*}$$

The steady state equation follows as

$$\lambda = \beta r \frac{\lambda} {z \pi} \\ \Rightarrow 1=\frac{\beta r} {z \pi}$$

Now, substituting each variable in the fashion similar to $\lambda_t = \lambda e^{\hat{\lambda}_t}$

$$\begin{array}{l} \lambda_{t}=\beta r_{t} E_{t}\left[\frac{\lambda_{t+1}}{z_{t+1} \pi_{t+1}}\right]\ \Rightarrow \lambda e^{\hat{\lambda}t}=\beta r e^{\hat{r}{t}} E_{t}\left[\frac{\lambda e^{\hat{\lambda}{t+1}}}{z e^{\hat{z}{t+1}} \pi e^{\hat{\pi}{t+1}}} \right]\ \Rightarrow \lambda\left(1+\hat{\lambda}{t}\right)=\frac{\beta r \lambda}{z \pi} \quad E{t}\left[\frac{\left(1+\hat{\lambda}{t+1}-\hat{z}{t+}-\hat{\pi}{t+1}+\hat{r}{t}\right)}{1}\right]\ \Rightarrow \left(1+\hat{\lambda}{t}\right)= E{t}\left[1+\hat{\lambda}{t+1}-\hat{z}{t+}-\hat{\pi}{t+1}+\hat{r}{t}\right]\ \text{using }{ E_t }[\hat{z}{t+1}]=0\ \Rightarrow 1+\hat{\lambda}{t} = 1+E_{t}\left[\hat{\lambda}{t+1}-\hat{\pi}{t+1}+\hat{r}{t}\right]\ \therefore \hat{\lambda}{t}=\hat{r}{t}+E{t}\left[\hat{\lambda}{t+1} - \hat{\pi}{t+1}\right] \end{array}$$

From equation (14*) it follows that

$$1=\frac{z_{t}}{z_{t} q_{t}-\gamma q_{t-1}}-\beta \gamma E_{t}\left[\left(\frac{a_{t+1}}{a_{t}}\right)\left(\frac{1}{z_{t+1} q_{t+1}-\gamma q_{t}}\right)\right]$$

The LHS of the equation, instead of approximating, the log of 1 is zero, so that is used instead For the RHS, we continue as before, by separating the function into simpler terms

$$\begin{aligned} z_1 = \frac{z_{t}}{z_{t} q_{t}-\gamma q_{t-1}} \\ z_2 = \left(\frac{a_{t+1}}{a_{t}}\right)\left(\frac{1}{z_{t+1} q_{t+1}-\gamma q_{t}}\right) \end{aligned}$$

$$\begin{aligned} z_{1}^{SS}&= \frac{z}{z q-\gamma q} \\ \left[\frac{dz_{1}}{dz_{t}} \right]^{SS} &= \left[\frac{\left(z_{t} q_{t}-\gamma q_{t-1}\right)-z_{t}\left(q_{t}\right)}{\left(z_{t} q_{t}-\gamma q_{t-1}\right)^{2}}\right]^{SS}\\ &=\frac{(z q-\gamma q)-z q}{(z q-\gamma q)}\\ &=-\frac{\gamma q}{(zq-\gamma q)^2}\\ \left[\frac{dz_{1}}{dq_{t}} \right]^{SS} &= \left[-z_{t}\left(z_{t} q_{t}-\gamma q_{t-1}\right)^{-2} z_{t} \right]^{SS} \\ &= -\frac{z^{2}}{\left(z q-\gamma q\right)^2}\\ \left[\frac{dz_{1}}{dq_{t-1}} \right]^{SS} &= \left[-z_{t}\left(z_{t} q_{t}-\gamma q_{t-1}\right)^{-2}(-\gamma)\right]^{SS} \\ &=\frac{z \gamma}{(z q-\gamma q)^2} \end{aligned}$$

Now we can expand the linearisation equation for $z_1$

$$z_1 = \frac{z}{z q-\gamma q}

• \frac{\gamma q}{(z q-\gamma q)^2} z \cdot \hat{z}_t
• \frac{z^{2}}{(z q-\gamma q)^2} q \cdot \hat{q}_t

• \frac{z \gamma}{(z q-\gamma q)^2} q \cdot \hat{q}_{t-1}$$

$$\begin{aligned} z_{2}^{SS} &= \left(\frac{a}{a}\right)\left(\frac{1}{z q-\gamma q}\right) \ &= \frac{1}{z q-\gamma q}\ \left[\frac{dz_{1}}{da_{t+1}} \right]^{SS} &= \left(\frac{1}{a}\right)\left(\frac{1}{z q-\gamma q}\right)\ \left[\frac{dz_{1}}{da_{t}} \right]^{SS} &=

• \left(\frac{a}{a^{2}}\right)\left(\frac{1}{z q-\gamma q}\right)\ &= -\left(\frac{1}{a}\right)\left(\frac{1}{z q-\gamma q}\right)\ \left[\frac{dz_{1}}{dz_{t+1}} \right]^{SS} &= \left[-\left(\frac{a_{t+1}}{a_{t}}\right)\left(z_{t+1} q_{t+1}-\gamma q_{t}\right)^{-2} \cdot\left(q_{t+1}\right) \right]^{SS} \ &=-\left(\frac{a}{a}\right) \frac{q}{(zq-\gamma q)^2}\ &= -\frac{q}{(zq-\gamma q)^2}\ \left[\frac{dz_{1}}{dq_{t+1}} \right]^{SS} &= -\left[\left(\frac{a_{t+1}}{a_{t}}\right)\left(z_{t+1} q_{t+1}-\gamma q_{t}\right)^{-2}\left(z_{t+1}\right)\right]^{SS} \ &=-\left(\frac{a}{a}\right) \frac{z}{(z q-\gamma q)^{2}} \ &=-\frac{z}{(z q-\gamma q)^{2}} \ \left[\frac{dz_{1}}{dq_{t}} \right]^{SS} &= \left[\left(\frac{a_{t+1}}{a_{t}}\right)\left(z_{t+1} q_{t+1}-\gamma q_{t}\right)^{-2}(-\gamma)\right]^{SS} \ &=\frac{\gamma}{(z q-\gamma q)^{2}} \end{aligned}$$

Therefore, we have $z_2$

{Recall a=1}

$$\begin{aligned} z_2 =& \frac{1}{z q-\gamma q}

• \left(\frac{1}{a}\right)\left(\frac{1}{z q-\gamma q}\right) \cdot a \hat{a}{t+1} -\left(\frac{1}{a}\right)\left(\frac{1}{z q-\gamma q}\right) \cdot a \hat{a}{t}\ &-\left(\frac{1}{a}\right)\left(\frac{1}{z q-\gamma q}\right) \cdot z\hat{z}{t} -\frac{q}{(zq-\gamma q)^2} \cdot q \hat{q}{t+1} +\frac{\gamma}{(z q-\gamma q)^{2}} \cdot q \hat{q}_{t}\ =& \frac{1}{z q-\gamma q}
• \left(\frac{1}{z q-\gamma q}\right) \cdot \hat{a}{t+1} -\left(\frac{1}{z q-\gamma q}\right) \cdot \hat{a}{t}\ &-\left(\frac{1}{z q-\gamma q}\right) \cdot z\hat{z}{t+1} -\frac{q^2}{(zq-\gamma q)^2} \cdot \hat{q}{t+1} +\frac{\gamma}{(z q-\gamma q)^{2}} \cdot q \hat{q}_{t} \end{aligned}$$

\newpage

Now for $\beta \gamma E_t z_2$, using $E_t \hat{a}_{t+1}=\rho_a\hat{a}_t$

$$\begin{aligned} \beta \gamma E_t z_2 =& \beta \gamma \left[ \frac{1}{z q-\gamma q}

• \left(\frac{1}{z q-\gamma q}\right) \cdot E_t\hat{a}{t+1} -\left(\frac{1}{z q-\gamma q}\right) \cdot E_t\hat{a}{t} -\left(\frac{1}{z q-\gamma q}\right) \cdot zE_t\hat{z}{t+1} \right.\ & \left. -\frac{q^2}{(zq-\gamma q)^2} \cdot E_t\hat{q}{t+1} +\frac{\gamma}{(z q-\gamma q)^{2}} \cdot q E_t\hat{q}_{t} \right]\ =& \beta \gamma \left[ \frac{1}{z q-\gamma q}
• \left(\frac{1}{z q-\gamma q}\right) \cdot \rho_a\hat{a}t -\left(\frac{1}{z q-\gamma q}\right) \cdot \hat{a}{t} -0 -\frac{q^2}{(zq-\gamma q)^2} \cdot E_t\hat{q}{t+1} +\frac{\gamma}{(z q-\gamma q)^{2}} \cdot q \hat{q}{t} \right]\ =& \frac{\beta \gamma}{z q-\gamma q}
• \left(\frac{\beta \gamma}{z q-\gamma q}\right) \cdot \rho_a\hat{a}t -\left(\frac{\beta \gamma}{z q-\gamma q}\right) \cdot \hat{a}{t} -\frac{\beta \gamma q^2}{(z q-\gamma q)^2} \cdot E_t\hat{q}{t+1} +\frac{\beta \gamma^2}{(z q-\gamma q)^{2}} \cdot q \hat{q}{t} \end{aligned}$$

Now, we can write out $0 = z_1 - \beta \gamma E_t z_2$

Using

$$z_1-z_1^{SS} - \beta \gamma (z_2 - z_2^{SS}) \approx z_1 \hat{z_1} - z_2 \hat{z_2}\\ \Rightarrow \text{subtract from the RHS: } \frac{z}{z q-\gamma q}-\beta \gamma \frac{a}{a z q-\gamma a q}$$

$$\begin{aligned} 0 = &\frac{z}{z q-\gamma q}

• \frac{\gamma q}{(z q-\gamma q)^2} z \cdot \hat{z}_t
• \frac{z^{2}}{(z q-\gamma q)^2} q \cdot \hat{q}_t

• \frac{z \gamma}{(z q-\gamma q)^2} q \cdot \hat{q}_{t-1}\ &-\left[ \frac{\beta \gamma}{z q-\gamma q}
• \left(\frac{\beta \gamma}{z q-\gamma q}\right) \cdot \rho_a\hat{a}t -\left(\frac{\beta \gamma}{z q-\gamma q}\right) \cdot \hat{a}{t} -\frac{\beta \gamma q^2}{(z q-\gamma q)^2} \cdot E_t\hat{q}{t+1} +\frac{\beta \gamma^2}{(z q-\gamma q)^{2}} \cdot q \hat{q}{t} \right] \end{aligned}$$

Multiply both sides by $(z q-\gamma q)^{2} \frac{1}{q}$

$$\begin{aligned} 0 = & z \cdot (z -\gamma )

• \gamma z \cdot \hat{z}_t
• z^{2} \cdot \hat{q}_t

• z \gamma \cdot \hat{q}_{t-1}\ &-\left[ \beta \gamma(z -\gamma )
• \beta \gamma \cdot \rho_a\hat{a}_t (z -\gamma )

• \beta \gamma (z -\gamma ) \cdot \hat{a}_{t}
• \beta \gamma q \cdot E_t\hat{q}_{t+1}

• \beta \gamma^2 \cdot \hat{q}_{t} \right]\ 0 = & z^{2}-\gamma z

• \gamma z \cdot \hat{z}_t
• z^{2} \cdot \hat{q}_t

• z \gamma \cdot \hat{q}{t-1}\ &-\beta \gamma z+\beta \gamma^{2}-\beta \gamma \rho{a} z \hat{a}{t}+\beta \gamma^{2} \rho{a} \hat{a}{t}\ &+\beta \gamma z \hat{a}{t}-\beta \gamma^{2} \hat{a}{t} +\beta \gamma q E{t} \hat{q}{t+1}-\beta \gamma^{2} \hat{q}{t} \end{aligned}$$

Finally, we subtract:

$$\left[\frac{z}{z q-\gamma q}-\beta \gamma \frac{a}{a z q-\gamma a q} \right] \cdot (z q-\gamma q)^{2} \frac{1}{q}\\ = z(z -\gamma) -\beta \gamma (z- \gamma)\\ =z^2 -\gamma z-\beta \gamma z + \beta \gamma^2$$

to yield the final solution:

$$\begin{aligned} 0 = & \begingroup\color{red}z^{2} \endgroup - \begingroup \color{red} {\gamma z} \endgroup - \gamma z \cdot \hat{z}_t - z^{2} \cdot \hat{q}_t

• z \gamma \cdot \hat{q}{t-1}\ & - \begingroup \color{red}{\beta \gamma z} \endgroup + \begingroup \color{red}{\beta \gamma^{2}} \endgroup - \beta \gamma \rho{a} z \hat{a}{t} + \beta \gamma^{2} \rho{a} \hat{a}{t}\ & + \beta \gamma z \hat{a}{t} - \beta \gamma^{2} \hat{a}{t} +\beta \gamma q E{t} \hat{q}{t+1}-\beta \gamma^{2} \hat{q}{t} \ & - \begingroup \color{red}{z^2} \endgroup + \begingroup \color{red}{\gamma z} \endgroup + \begingroup \color{red}{\beta \gamma z} \endgroup - \begingroup \color{red}{\beta \gamma^2} \endgroup \ =& - \gamma z \cdot \hat{z}_t

• z^{2} \cdot \hat{q}_t

• z \gamma \cdot \hat{q}{t-1}\ &-\beta \gamma \rho{a} z \hat{a}{t}+\beta \gamma^{2} \rho{a} \hat{a}{t}\ &+\beta \gamma z \hat{a}{t}-\beta \gamma^{2} \hat{a}{t} +\beta \gamma q E{t} \hat{q}{t+1}-\beta \gamma^{2} \hat{q}{t} \ = & \hat{z}{t}(-\gamma z)+\hat{q}{t}\left(-z^{2}-\beta \gamma^{2}\right)+\hat{q}{t-1}(z \gamma) \ &+\hat{a}{t}\left(-\beta \gamma \rho_{a} z+\beta \gamma^{2} p_{a}+\beta \gamma z-\beta \gamma^{2}\right) \ &+\beta \gamma q E_{t} \hat{q}{t+1}\ = & \gamma z \hat{q}{t-1}-\hat{q}{t}\left(z^{2}+\beta \gamma^{2}\right)+\beta \gamma q E{t} \hat{q}{t+1}+\hat{a}{t} \beta \gamma\left(1-\rho_{a}\right)(z-\gamma)-\gamma z \hat{z}_{t} \end{aligned}$$

From the definition of the output gap $x_t$

$$\begin{aligned} x_{t}=y_{t} / q_{t} \\ \text{Steady State: } x = y/p \end{aligned}$$

Now,

$$\begin{aligned} x_{t}=y_{t} / q_t\ \Rightarrow x e^{\hat{x} t}=\frac{y e^{\hat{y} t}}{q e^{\hat{q} t}}\ \Rightarrow\left(1+\hat{x}{t}\right)=\left(1+\hat{y}{t}-\hat{q}{t}\right)\ \therefore \quad \hat{x}{t}=\hat{y}{t}-\hat{q}{t} \end{aligned}$$

Starting from equation (13*)

$$\begin{aligned} \theta_{t}-1=& \theta_{t}\left(\frac{a_{t}}{\lambda_{t}}\right)-\phi_{p}\left(\frac{\pi_{t}}{\pi_{t-1}^{\alpha} \pi^{1-\alpha}}-1\right)\left(\frac{\pi_{t}}{\pi_{t-1}^{\alpha} \pi^{1-\alpha}}\right) \\ &+\beta \phi_{p} E_{t}\left[\left(\frac{\lambda_{t+1} y_{t+1}}{\lambda_{t} y_{t}}\right)\left(\frac{\pi_{t+1}}{\pi_{t}^{\alpha} \pi^{1-\alpha}}-1\right)\left(\frac{\pi_{t+1}}{\pi_{t}^{\alpha} \pi^{1-\alpha}}\right)\right] \end{aligned}$$

In Steady State this becomes

$$\begin{aligned} \quad \ \theta-1=& \theta\left(\frac{a}{\lambda}\right)-\phi_{p}\left(\frac{\pi}{\pi^{\alpha} \pi^{1-\alpha}}-1\right)\left(\frac{\pi}{\pi^{\alpha} \pi^{1-\alpha}}\right) \\ &+\beta \phi_{p} E\left[\left(\frac{\lambda y}{\lambda y}\right)\left(\frac{\pi}{\pi^{\alpha} \pi^{1-\alpha}}-1\right)\left(\frac{\pi}{\pi^{\alpha} \pi^{1-\alpha}}\right)\right]\\ \Rightarrow \theta-1=& \theta\left(\frac{1}{\lambda}\right)-0 + 0 \\ \therefore \quad \lambda = \frac{\theta}{\theta-1} \end{aligned}$$

Using the above result we can obtain and simplify further. Then, log-linearise using Uhlig's Method:

First we separate the four terms, for simplicity:

$$\begin{aligned} \theta_{t}-1 & = \theta e^{\hat{\theta}{t}} \ & = \theta\left(1+\hat{\theta}{t}\right) \end{aligned}$$

$$\begin{aligned} \theta_{t}\left(\frac{a_{t}}{\lambda_{t}}\right) &= \theta e^{\hat{\theta}{t}}\left(\frac{a e^{\hat{a}{t}}}{\lambda \hat{e}^{\hat{\lambda}t}}\right)\ &= \frac{\theta a}{\lambda}\left(1+\hat{\theta}{t}+\hat{a}{t}-\hat{\lambda}{t}\right)\ & = \frac{\theta}{\lambda}\left(1+\hat{\theta}{t} \right)+ \frac{\theta}{\lambda} \left(\hat{a}{t}-\hat{\lambda}_{t}\right)\end{aligned}$$

$$\begin{aligned} -\phi_{p}\left(\frac{\pi_{t}}{\pi_{t-1}^{\alpha} \pi^{1-\alpha}}-1\right)\left(\frac{\pi_{t}}{\pi_{t-1}^{\alpha} \pi^{1-\alpha}}\right) &= -\phi_{p}\left(\frac{\pi e^{\pi_{t}}}{\pi^{\alpha} e^{\alpha \hat{\pi}{t-1}} \pi^{1-\alpha}}-1\right)\left(\frac{\pi e^{\hat{\pi}t}}{\pi^{\alpha} e^{\alpha \hat{\pi}{t-1}} \pi^{1-\alpha}}\right)\ &= -\phi_p\left[\exp \left[2 \hat{\pi}{t}-2 \alpha \hat{\pi}{t-1}\right]-\exp \left[\hat{\pi}{t}-\hat{\pi}{t-1}\right]\right]\ &=-\phi_p\left[\hat{\pi}{t}-\alpha \hat{\pi}_{t-1}\right] \end{aligned}$$

$$\begin{aligned} &\beta \phi_{p} E_{t}\left[\left(\frac{\lambda_{t+1} y_{t+1}}{\lambda_{t} y_{t}}\right)\left(\frac{\pi_{t+1}}{\pi_{t}^{\alpha} \pi^{1-\alpha}}-1\right)\left(\frac{\pi_{t+1}}{\pi_{t}^{\alpha} \pi^{1-\alpha}}\right)\right] \ =& \beta \phi_{p} E_{t}\left[\left(\frac{\lambda e^{\hat{\lambda}{t+1}} \cdot y e^{\hat{y}{t+1}}}{\lambda e^{\hat{\lambda}t} y e^{\hat{y}t}}\right)\right]\left(\frac{\pi e^{\pi{t+1}}}{\pi^{\alpha} e^{\alpha \hat{\pi}{t}} \pi^{1-\alpha}}-1\right)\left(\frac{\pi e^{\hat{\pi}{t+1}}}{\pi^{\alpha} e^{\alpha \hat{\pi}{t}} \pi^{1-\alpha}}\right)\ =& \beta \phi_{p} E_{t}\left[\exp \left(\hat{\lambda}{t+1}+\hat{y}{t+1}-\hat{\lambda}{t}-\hat{y}{t}+2 \cdot \hat{\pi}{t+1}-2 \alpha \hat{\pi}{t}\right)\right.\ &\left.-\exp \left(\hat{\lambda}{t+1}+\hat{y}{t+1}-\hat{\lambda}{t}-\hat{y}{t}+\hat{\pi}{t+1}-\alpha \hat{\pi}{t}\right)\right] \ =& \beta \phi_{p} E_{t}\left[ \hat{\pi}{t+1}-\alpha \hat{\pi}{t}\right] \end{aligned}$$

Putting all the components together then yields:

$$ \theta\left(1+\hat{\theta}{t}\right)=\frac{\theta}{\lambda}\left(1+\hat{\theta}{t} \right)+ \frac{\theta}{\lambda} \left(\hat{a}{t}-\hat{\lambda}{t}\right) - \phi_p\left[\hat{\pi}{t}-\alpha \hat{\pi}{t-1}\right] + \beta \phi_{p} E_{t}\left[ \hat{\pi}{t+1}-\alpha \hat{\pi}{t}\right] $$

Simplify using $$\lambda=\frac{\theta}{\theta-1}, \quad \hat{e}{t}=-\left(\frac{1}{\phi{p}}\right) \hat{\theta}{t} \text {, and } \psi=\frac{(\theta-1)}{\phi{p}}$$ i.e. $$ 1 - \frac{1}{\lambda} = 1/\theta, \ \frac{\theta}{\lambda} = \theta - 1 $$

$$\begin{aligned} \theta \left(1+\hat{\theta}{t}\right)\cdot \left[ 1 - \frac{1}{\lambda} \right] =& \frac{\theta}{\lambda} \left(\hat{a}{t}-\hat{\lambda}{t}\right) - \phi_p\left[\hat{\pi}{t}-\alpha \hat{\pi}{t-1}\right] + \beta \phi{p} E_{t}\left[ \hat{\pi}{t+1}-\alpha \hat{\pi}{t}\right]\ \frac{\left(1+\hat{\theta}{t}\right)}{\phi{p}} =& \frac{\theta - 1}{\phi_{p}} \left(\hat{a}{t}-\hat{\lambda}{t}\right) - \left[\hat{\pi}{t}-\alpha \hat{\pi}{t-1}\right] + \beta E_{t}\left[ \hat{\pi}{t+1}-\alpha \hat{\pi}{t}\right] \end{aligned}$$

Ignoring the constant parameter $-1/\phi_p$, and rearranging the terms yields the final result:

$$\begin{aligned} &-\hat{e}t = \psi \left(\hat{a}{t}-\hat{\lambda}{t}\right) -(1+\beta \alpha)\hat{\pi}{t} +\alpha \hat{\pi}{t-1} + \beta E{t}\left[ \hat{\pi}{t+1}\right]\ \Rightarrow &(1+\beta \alpha) \hat{\pi}{t}=\alpha \hat{\pi}{t-1}+\beta E{t} \hat{\pi}{t+1}-\psi \hat{\lambda}{t}+\psi \hat{a}{t}+\hat{e}{t} \end{aligned}$$

From the monetary policy interest rate rule

$$\ln \left(r_{t} / r\right)=\rho_{r} \ln \left(r_{t-1} / r\right)+\rho_{\pi} \ln \left(\pi_{t-1} / \pi\right)+\rho_{x} \ln \left(x_{t-1} / x\right)+\varepsilon_{r t}$$

Directly applying Uhlig's method using $r_t=r\cdot e^{\hat{r}_t}$ for each variable of the equation yields the required result:

$$\begin{aligned} & \ln \left[\frac{r e^{\hat{r} t}}{r}\right]=\operatorname{\rho_r} \ln \left[\frac{r e^{\hat{r}{t-1}}}{r}\right]+\rho{\pi} \ln \left[\frac{\pi e^{\hat{\pi}{t-1}}}{n}\right)+\rho_x \ln \left(\frac{x e^{\hat{x}{t+1}}}{x}\right)+\varepsilon{rt} \ & \therefore \quad \hat{r}{t}=\rho{r} \hat{r}{t-1}+\rho{\pi} \hat{\pi}{t-1}+\rho x \hat{x}{t-1}+\varepsilon_{r t} \end{aligned}$$

Starting Form Equation (9*)

$$ \begin{aligned} &\frac{a_{t}}{\delta}\left[\ln \left(m^{*}\right)-\ln \left(m_{t}\right)+\ln \left(u_{t}\right)\right]-a_{t}\left(\frac{\phi_{m}}{2}\right)\left(\frac{z_{t} m_{t}}{z m_{t-1}}-1\right)^{2} \\ &-a_{t} \phi_{m}\left(\frac{z_{t} m_{t}}{z m_{t-1}}-1\right)\left(\frac{z_{t} m_{t}}{z m_{t-1}}\right) \\ &+\beta \phi_{m} E_{t}\left[a_{t+1}\left(\frac{z_{t+1} m_{t+1}}{z m_{t}}-1\right)\left(\frac{z_{t+1} m_{t+1}}{z m_{t}}\right)^{2}\left(\frac{z}{z_{t+1}}\right)\right] \\ &=\lambda_{t}\left(1-\frac{1}{r_{t}}\right) \\ \end{aligned}$$

\newpage

As indicated in the Appendix of "Belongia and Ireland (2020)", substituting for the steady state conditions in equation (9) yields

$$\begin{aligned} &\frac{a}{\delta}\left[\ln \left(m^{}\right)-\ln \left(m\right)+\ln \left(u\right)\right]-a\left(\frac{\phi_{m}}{2}\right)\left(\frac{z m}{z m}-1\right)^{2} \ &-a \phi_{m}\left(\frac{z m}{z m_{t-1}}-1\right)\left(\frac{z m}{z m_{t-1}}\right) \ & +\beta \phi_{m} E\left[a\left(\frac{z m}{z m}-1\right)\left(\frac{z m}{z m}\right)^{2}\left(\frac{z}{z}\right)\right] \ &=\lambda\left(1-\frac{1}{r}\right)\ \Rightarrow &\frac{1}{\delta}\left[\ln \left(m^{}\right)-\ln \left(m\right)\right]-0-0+0=\lambda/r\left(r-1\right)\ \therefore &\quad \ln (m) =\ln \left(m^{*}\right)-\delta_{r}(r-1) \ \text{where }\ &\delta_{r} =\left(\frac{\delta}{r}\right)\left(\frac{\theta}{\theta-1}\right) \end{aligned}$$

Now, to simplify the linearisation process, we consider the equation (9*) in parts:

$$\begin{aligned} \frac{a_{t}}{\delta}\left[\ln \left(m^{}\right)-\ln \left(m_{t}\right)+\ln \left(u_{t}\right)\right] = &\frac{a_{t}}{\delta}\left{\ln \left(m^{}\right)-\ln (m)-\hat{m}{t}+\hat{u}{t}\right} \ = &\frac{a_{t}}{\delta}\left[\delta_{r}(r-1)-\hat{m}{t}+\hat{u}{t}\right] \ = &\frac{a_{t}}{\delta}\delta_{r}(r-1) + \frac{a_{t}}{\delta} \left( \hat{u}{t}-\hat{m}{t} \right) \end{aligned}$$

$$\begin{aligned} \lambda_{t}\left(1-\frac{1}{r_{t}}\right) &=\lambda e^{\hat{\lambda}{t}}\left(1-\frac{e^{-\hat{r}{t}}}{r}\right)\ &=\lambda\left[e^{\hat{\lambda}{t}}-\frac{e^{\lambda{t}-\hat{r}{t}}}{r}\right]\ &=\lambda\left[1+\hat{\lambda}{t}-\frac{1+\hat{\lambda}{t}-\hat{r}{t}}{r}\right]\ &=\lambda+\hat{\lambda \lambda}{t} - \lambda/r + \lambda \cdot \hat{\lambda}{t}/r - \lambda \cdot \hat{r}_{t}/r \end{aligned}$$

Using $\lambda - \lambda/r = \frac{\delta_r(r-1)}{\delta}$ to add the previous two solutions together and changing the sign for carrying the term to the other side of the equality, we get:

$$\begin{aligned}&\lambda+\hat{\lambda \lambda}{t} - \lambda/r + \lambda \cdot \hat{\lambda}{t}/r - \lambda \cdot \hat{r}{t}/r -\frac{a{t}}{\delta}\delta_{r}(r-1)\ = &-\frac{e^{\hat{a}{t}}}{\delta}\delta{r}(r-1) + \lambda-\lambda/r+\hat{\lambda}_t \cdot \lambda/r \cdot (r-1) + \lambda \cdot \hat{r}_t/r\ =& -\frac{\delta_r(r-1) \hat{a}_t}{\delta} + \hat{\lambda}\frac{\delta_r (r-1)}{\delta} +\hat{r}_t \frac{\delta_r}{\delta} \end{aligned}$$

$$\begin{aligned}&a_{t}\left(\frac{\phi_{m}}{2}\right)\left(\frac{Z_t m_t }{z Z_{t-1}m_{t-1}}-1\right)^{2}\ =& \left(\frac{\phi_{m}}{2}\right) \left[ \frac{a_{t} z_{t}^{2} m_{t}^{2}}{z^{2} m_{t-1}^{2}} - 2 \frac{a_{t} z_{t} m_{t}}{z m_{t-1}} + a_{t} \right]\ =& \left(\frac{\phi_{m}}{2}\right) \left[ \exp(\hat{a}{t}+2 \hat{z}{t}+2 \hat{m}{t}-2 \hat{m}{t-1})-2 \exp(\hat{a}{t}+\hat{m}{t}+\hat{z}{t}-\hat{m}{t-1})+ \exp(\hat{a}{t}) \right] \ =& \left(\frac{\phi{m}}{2}\right) \left[ 1 + \hat{a}{t}+2 \hat{z}{t}+2 \hat{m}{t}-2 \hat{m}{t-1} - 2( 1 + \hat{a}{t}+\hat{m}{t}+\hat{z}{t}-\hat{m}{t-1}) + (1 + \hat{a}{t}) \right] \ =& \left(\frac{\phi{m}}{2}\right) \left[ -1 - \hat{a}_t + 1 + \hat{a}_t \right]\ =& 0 \end{aligned}$$

$$\begin{aligned} &-a_{t} \phi_{m}\left(\frac{Z_t m_t }{z Z_{t-1}m_{t-1}} - 1\right)\left(\frac{Z_t m_t }{z Z_{t-1}m_{t-1}}\right)\ =& e^{\hat{a}{t}} \phi{m}\left[\frac{z m e^{\hat{z}{t}+\hat{m}{t}}}{z m e^{\hat{m}{t-1}}}-1\right]\left[\frac{z m e^{\hat{z}{t}+\hat{m}{t}}}{z m e^{\hat{m}{t-1}}}\right]\ =& \phi_m \left[ \exp \left(2 \hat{z}{t}+2 \hat{m}{t}-2 \hat{m}{t-1}+\hat{a}{t}\right)-\exp \left(\hat{z}{t}+\hat{m}{t}-\hat{m}{t-1}+\hat{a}{t}\right) \right]\ =& \phi_m \left[\hat{z}{t}+\hat{m}{t}-\hat{m}_{t-1}\right] \end{aligned}$$

$$\begin{aligned}&\beta \phi_{m} E_{t}\left[e^{\hat{a}{t+1}}\left(\frac{z m e^{\hat{z}{t+1}+\hat{m}{t+1}}}{z m e^{\hat{m}{t}}}-1\right)\left(\frac{z m e^{\hat{z}{t+1}+\hat{m}{t+1}}}{z m e^{\hat{m}{t}}}\right)^{2} \left(\frac{z}{z e^{\hat{z}{t+1}}}\right)\right] \ =&\beta \phi_{m} E_{t}\left{\exp \left(\hat{z}{t+1}+\hat{m}{t+1}+\hat{a}{t+1}-\hat{m}{t}+2 \hat{z}{t+1}+2 \hat{m}{t+1}-2 \hat{m}{t}-\hat{z}{t+1}\right)\right. \ & \left.-\exp \left(\hat{a}{t+1}+2 \hat{z}{t+1}+2 \hat{m}{t+1}-2 \hat{m}{t}-\hat{z}{t+1}\right)\right} \ =&\beta \phi{m} E_{t}\left{\exp \left(2 \hat{z}{t+1}+3 \hat{m}{t+1}-3 \hat{m}{t}+\hat{a}{t+1}\right)-\exp \left(\hat{z}{t+1}+2 \hat{m}{t+1}-2 \hat{m}{t}+\hat{a}{t+1}\right)\right} \ =&\beta \phi_{m} E_{t}\left{\hat{z}{t+1}+\hat{m}{t+1}-\hat{m}{t}\right}\ =& \beta \phi{m} E_{t}\left{\hat{m}{t+1}\right} - \beta \phi_m \hat{m}{t} \end{aligned}$$

$$\begin{aligned} &\frac{a_{t}}{\delta} \left( \hat{u}{t}-\hat{m}{t} \right) - \phi_m \left[\hat{z}{t}+\hat{m}{t}-\hat{m}{t-1}\right] + \beta \phi{m} E_{t}\left{\hat{m}{t+1}\right} - \beta \phi_m \hat{m}{t} \ = & -\frac{\delta_r(r-1) \hat{a}_t}{\delta} + \hat{\lambda}\frac{\delta_r (r-1)}{\delta} +\hat{r}_t \frac{\delta_r}{\delta} \end{aligned}$$

Multiply by $\delta$ throughout:

$$\begin{aligned} & a_t \left( \hat{u}{t}-\hat{m}{t} \right) - \phi \left[\hat{z}{t} + \hat{m}{t}-\hat{m}{t-1}\right] + \beta \phi E{t}\left{\hat{m}{t+1}\right} - \beta \phi \hat{m}{t} \ =& -\delta_r(r-1) \hat{a}_t + \hat{\lambda}\delta_r (r-1) +\hat{r}_t \delta_r \\end{aligned}$$

Assuming $a_t = 1$ the final solution follows:

$$\begin{aligned}\therefore \quad & \delta_r (r-1) \hat{\lambda} - \delta_r(r-1) \hat{a}t -\hat{u}{t} +\phi \hat{z}{t} = \phi \hat{m}{t-1}-[1+(1+\beta) \phi] \hat{m}{t}+\beta \phi \hat{m}{t+1}-\delta_{r} \hat{r}_{t} \end{aligned}$$

From the growth rate of money, we have

$$\mu_{t}=z_{t}\left(m_{t} / m_{t-1}\right) \pi_{t}$$

Applying uhlig's Method:

$$\begin{aligned} &\mu_{t} =z_{t}\left[\frac{m_{t}}{m_{t-1}}\right] \pi_{t} \ &\mu e^{\hat{\mu}{t}}=z e^{\hat{z{t}}}\left[\frac{m e^{\hat{m}{t}}}{m e^{\hat{m}{t-1}}}\right] \pi e^{\hat{\pi}{t}} \ \Rightarrow & \left(1+\hat{\mu}{t}\right) =\left(1+\hat{z}{t}+\hat{m}{t}-\hat{m}{t-1}+\hat{\pi}{t}\right) \ \therefore \quad & \hat{\mu}{t} =\hat{z}{t}+\hat{m}{t}-\hat{m}{t-1}+\pi_{t} \end{aligned}$$

Starting from the output growth rate

$$g_{t}=\left(y_{t} / y_{t-1}\right) z_{t}$$

Now, using the Steady State solution

$$g = z$$

Yields the following solution, using Uhlig's method:

$$\begin{aligned} g_{t}&=\left(\frac{y_{t}}{y_{t-1}}\right) z_{t}\ \Rightarrow ge^{g_{t}}&=\left[\frac{y e^{\hat{y}{t}}}{y e^{\hat{y}{t-1}}}\right] z e^{\hat{z}{t}}\ \Rightarrow \left(1+\hat{g}{t}\right)&=\left(1+\hat{y}{t}-\hat{y}{t-1}+\hat{z}{t}\right)\ \therefore \hat{g}{t} &= \hat{y}{t}-\hat{y}{t-1}+\hat{z}_{t} \end{aligned}$$

Equation (2*) simply becomes

$$\begin{aligned} & \ln \left(a_{t}\right)=p_{a} \ln \left(a_{t-1}\right)+\varepsilon_{a t} \ & \ln \left(a e^{\hat{a}{t}}\right)=p{a} \ln \left(a e^{\hat{a}{t-1}}\right)+\varepsilon{a t} \ \Rightarrow & \ln (a)+\hat{a}{t}=\rho{a} \ln (a)+p_{a} \hat{a}{t-1}+\varepsilon{a t} \ &[a =1] \ \therefore \quad & \hat{a}{t} =\rho{a} \hat{a}{t-1}+\varepsilon{a t} \end{aligned}$$

The log-linearization of equation (3*) also follows simply

$$\begin{aligned} \ln(z_t)=\ln(z) + \varepsilon_{z t}\ \Rightarrow \ln\left(z e^{\hat{z}t}\right) =\ln (z)+\varepsilon{z t}\ \Rightarrow \ln (z)+\hat{z}{t}=\ln (z)+\varepsilon{z t}\ \therefore \hat{z}t = \varepsilon{z t} \end{aligned}$$

Similarly, equation (4*)...

$$\begin{aligned} & \ln \left(u_{t}\right)=p_{u} \ln \left(u_{t-1}\right)+\varepsilon_{u t} \ & \ln \left(u e^{\hat{u}{t}}\right)=p{u} \ln \left(u e^{\hat{u}{t-1}}\right)+\varepsilon{u t} \ \Rightarrow & \ln (u)+\hat{u}{t}=\rho{u} \ln (u)+p_{u} \hat{u}{t-1}+\varepsilon{u t} \ &[u =1] \ \therefore \quad &\hat{u}{t} =\rho{u} \hat{u}{t-1}+\varepsilon{u t} \end{aligned}$$

Lastly, equation (10*) yields

$$\begin{aligned} & \ln \left(\theta_{t}\right)=\left(1-\rho_{\theta}\right) \ln (\theta)+\rho_{\theta} \ln \left(\theta_{t-1}\right)+\varepsilon_{\theta t} \ \Rightarrow & \ln \left(\theta e^{\hat{\theta}{t}}\right)=\left(1-\rho{\theta}\right) \ln (\theta)+\rho_{\theta} \ln \left(\theta e^{\hat{\theta}{t-1}}\right)+\varepsilon_{\theta t} \ \Rightarrow & \ln (\theta)+\hat{\theta}{t}=\ln (\theta)-\rho{\theta} \ln (\theta)+\rho_{\theta} \ln (\theta)+\rho_{\theta} \hat{\theta}{t-1}+\varepsilon{\theta t} \ \therefore & \quad \hat{\theta}{t}=\rho{\theta} \hat{\theta}{t-1}+\varepsilon{\theta t} \end{aligned}$$

using the normalisation $$\hat{e}{t}=-\left(1 / \phi{p}\right) \hat{\theta}_{t}$$

and

$$\rho_{e}=\rho_{\theta}$$

We can rewrite:

$$\begin{aligned} &\hat{\theta}{t}=\rho{\theta} \hat{\theta}{t-1}+\varepsilon{\theta t}\ \Rightarrow - & \left(1 / \phi_{p}\right) \cdot \hat{\theta}{t} = \hat{e}t = -\left(1 / \phi{p}\right) \left[ \rho{\theta} \hat{\theta}{t-1}+\varepsilon{\theta t} \right]\ \Rightarrow \quad &\hat{e}t = \rho{e} \hat{e}{t-1}+\varepsilon{e t} \end{aligned}$$

\newpage

Impulse response functions show the movement of macroeconomic variables back to steady state in response to a shock. This paper makes use of these graphs to compare different approaches to monetary policy, namely the Taylor (TR) rule, the zero Interest Rate (IR) rule, the Flexible Money Growth (FMG) rule, and the Constant Money Growth (CMG) rule.

To construct these functions, parameter values are inserted into the log-linearised equations from the previous section. Some of the parameter values are found via our own calculations. Upon researching calibrated values of the Phillips Curve slope, we found Hazell, Herreño, Nakamura & Steinsson (2020)'s estimation of the Phillips Curve slope. For the rest, posterior parameter values from Belongia & Ireland (2020) are used. We use the mean posterior values as opposed to the prior values so that all parameters are estimated values, since calibration is based on estimates believed to be true. The prior values are those that match the steady state values. In Appendix, the different parameter values used in figure 9.1.

The steady state growth rate was calculated in accordance with Belongia & Ireland (2020)’s method. 1983-2019 quarterly GDP data from FRED is used, which is then converted into per capita terms by dividing by the civilian noninstitutional population. The trend of the quarterly GDP per capita data is then extracted using a Hodrick-Prescott filter with a smoothing parameter of 1600. After this, the average of the natural log change in quarterly GDP is calculated as 1.00037. In order to calculate the steady state interest rate, we use FRED’s quarterly interest rate data between 1983 and 2019. The average of this data is then divided by four to get 0.98783908. This value is very similar to that of Belongia & Ireland (2020) where annual data is divided by 400. The steady state interest rate is then used to calculate the value of the discount factor of 0.99023. In accordance to Gollier & Hammitt (2014), the discount factor is equal to $1/(1+r)$.

$rho_\pi$, $rho_x$ and $rho_r$ are only used for the Taylor rule calibration. Under the FMG and CMG rules, rho_mm, rho_mpi and $rho_mx$ replace $rho_\pi$, $rho_x$ and $rho_r$, due to the switch from equation 24 to 33. When calibrating the CMG rule, these parameters are set to zero. Under the FMG rule, we set $rho_mm$, $rho_m\pi$ and $rho_mx$ equal to 1, 0 and -0.125 respectively. This is in accordance to Belongia & Ireland (2020).

To conduct robustness checks for $z$, $r$, $\beta$, and $\psi$, we replace them with 1.0046, 1.0121, 0.9987, and 0.1 respectively. The first three values are taken from Ireland (2011), and we have replaced the value for the Phillips Curve slope with the mean posterior value from Belongia & Ireland (2020). When comparing our impulse response functions to those using these values, we conclude that they are similar, implying that the values used are appropriate. The following section compares the impulse response functions under each rule after a small and positive productivity, preference, cost push, and money demand shocks are induced.

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This section presents the impulse response functions (IRFs) of macroeconomic variables under various rules, in response to the different shocks. In order to distinguish between the rules, different colours have been assigned to each rule in the IRFs. The trajectory of the Taylor (TR) rule is shown with a black line, that of the Interest Rate (IR) rule is green, the Constant Money Growth (CMG) rule is purple and the Flexible Money Growth (FMG) rule is blue.

The figure below shows the IRFs of the output growth, inflation, nominal interest rate, money growth rate and output gap after a positive productivity shock. The graphs depict the way in which each of these macroeconomic variables change under the TR rule, the IR rule, the FMG rule and the CMG rule.

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The IRFs of the output growth of the TR rule, IR rule, FMG rule, and CMG rule in response to a productivity shock are illustrated in the first of the graphs above. The effect of the TR rule and the IR rule on output growth is identical. The IRF of output growth under the FMG rule, TR rule and IR rule jumps above 0.002. Under these three rules, it then slowly moves down to zero at a decreasing rate. The IRF of output growth under the IR rule moves faster, and reaches zero after 12 periods, whereas under the FMG rule, it is more gradual and reaches zero at around 14 periods. Under both rules, the trajectory of output growth crosses the zero line, but not by much and moves upwards toward zero again. Under the CMG rule, however, the IRF starts below zero, and moves up to cross the zero line just past two periods. It then proceeds to increase until approximately six periods. Thereafter, it gradually moves back to the steady state. The trajectory of output growth of all three rules represented here takes more than 20 periods to settle around the steady state.

The IRF of inflation rate under both the CMG rule and FMG rule responds to a shock in productivity, however it follows different trajectories toward zero under the two rules. Under the FMG rule, the IRF jumps to just below the steady state, and crosses the zero line at around three periods. It rises minimally until a turning point at 10 periods, after which it returns to zero by 20 periods. The trajectory of the inflation rate under this rule does not stray far from the steady state. Alternatively, the IRF of inflation under the CMG rule experiences a bigger fluctuation in response to the productivity shock. It jumps down to approximately -0.0008 and declines further until two periods have passed. From there, it increases steadily at an increasing rate towards zero. It does not, however, reach zero within 20 periods, as the IRF under the FMG rule does.

Similar to the inflation rate, nominal interest rates under both the CMG rule and FMG rule respond to a shock in productivity. Both IRFs start above 0.0008. The IRF under the CMG rule declines steadily at an increasing rate towards zero, but under the FMG rule it first increases slightly before starting to decline after approximately two periods. The trajectory of the nominal interest rate then crosses the zero line just before 14 periods. At around 18 periods the IRF starts moving back to zero. Although the graph shows that the trajectory of the nominal interest rate moves close to zero under both scenarios, it takes more than 20 periods to reach the steady state.

The nominal money growth rate under the TR rule and IR rule in response to a productivity shock have identical IRFs. Both jump to above 0.001 and decline steadily to reach zero by 20 periods. Alternatively, the IRF under the FMG rule increases at an increasing rate, after reaching a point of inflection after about five periods, after which it increases at a decreasing rate. The trajectory of the money growth rate reaches a turning point between eight and nine periods, after which it declines to just above the steady state by 20 periods.

In response to a productivity shock, the IRF of output gap under the FMG rule first declines, and then returns to the steady state by eight periods. It then increases slightly, but almost reaches the steady state again after 20 periods have passed. The output gap under the CMG rule behaves differently in response to such a shock. The IRF also declines at first, but from a starting point of -0.004 (as opposed to -0.001). Just after four periods, it reaches a turning point from which it increases at a decreasing rate. The trajectory of the output gap does not reach the steady state within 20 periods.

Contractionary monetary policy refers to increases in interest rates so as to decrease the supply of money in the economy in order to lower inflation (Chen, 2020). After a small positive shock to productivity, the CMG rule displays signs of this policy, with nominal interest rates jumping up before slowly returning to the steady state. The FMG rule results in an increase in output growth, after which a rise in the money growth and a decrease in the nominal interest rate allows for the output growth to return to the steady state without significant changes to inflation. Under both the TR rule and the IR rule, money growth increases to accommodate the increase in output, so that the output gap remains stable. The CMG rule responds the least efficiently to the shock, with large fluctuations in inflation, as well as decreases in output growth, and a negative output gap.

The figure below shows the response of the chosen macroeconomic variables to a positive preference shock. As explained above, each panel shows the response of the given variable under each of the four rules. A preference shock refers to a shock to the utility aspect of the economy (Chugh, 2015:147). As such, it affects the macroeconomic variables from the demand side (Malik et al., 2019:12).

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The output growth of all four scenarios starts above zero in response to a preference shock, in order of IR rule, TR Rule, FMG rule and CMG rule from lowest to highest. The trajectory of the output growth then declines under all rules. However, under the TR rule, FMG rule and CMG rule, it becomes negative. Under the TR rule, the IRF stays very close to zero. The trajectory of the output growth of the FMG and CMG rules decrease further below zero but start to move back to the steady state between six and eight periods. The IRF under the FMG rule reaches zero around 12 periods, whereas under the CMG rule, it increases more gradually, nearing it at 20 periods. The IRF of output growth under the IR rule experienced the smallest jump up after the shock and returned to the steady state after four periods.

Inflation under the CMG rule jumps above 0.0002 in response to a preference shock. The IRF proceeds to increase further for two periods, after which it declines steadily past zero. It is still declining after 18 periods. The IRFs under both the FMG rule and the TR rule experience a small jump and start between zero and 0.0001. Under the TR rule, the IRF remains close to the steady state, and gradually proceeds to zero by 20 periods. Under the FMG rule, it sharply declines to cross the zero line and reaches a turning point between seven and eight periods. It increases to cross the zero line again at around 17 periods. It remains slightly above the steady state after 20 periods.

The nominal interest rate under the TR rule does not experience an immediate jump in response to a preference shock. Instead, the IRF slowly increases up to 0.0003 until it starts to decrease after six periods. It slowly moves down towards zero, but does not reach the steady state within 20 periods. The nominal interest rate under the FMG rule and CMG rule both jump down minimally in response to a preference shock, starting at -0.0001. The IRF of the nominal interest rate under the FMG rule, however, increases and then plateaus below 0.0001. That of the CMG rule remains level for approximately two periods, after which it starts to increase. After 14 periods, it reaches a turning point where it starts to return to zero, however, it does not reach the steady state within 20 periods.

The money growth rate under the FMG rule does not jump immediately after a preference shock. The IRF declines from zero at a decreasing rate. It reaches a turning point between seven and eight periods after the shock, after which it increases steadily. The IRF of the money growth rate crosses the zero line just before 16 periods and proceeds to rise. It is still moving away from the steady state at 20 periods. The money growth rate in response to the shock under the TR rule jumps to below -0.004 and remains there for approximately three periods. It then starts increasing and returns to the steady state between nine and 10 periods, after which it moves away from the steady state again in the positive direction. Unlike the money growth rate under the FMG rule, however, it starts moving back towards the steady state before 20 periods. It does not reach the steady state within 20 periods.

The output gap under the TR rule does not deviate from the steady state significantly in response to a preference shock. The IRF experiences a small positive jump, but returns to the steady state between 10 and 12 periods. The trajectory of the output gap under the FMG rule jumps up slightly more than under the TR rule, but instead of returning to zero immediately, it increases first. After two periods, it starts to decrease, and crosses the zero line after approximately seven periods. After this point, it deviates slightly negatively from the steady state, but slowly moves back. By 20 periods, it has returned close to the steady state.

The FMG rule displays signs of eventual contractionary policy. The nominal interest rate first decreases, after which it increases significantly. Along with this increase in the interest rate, the money growth rate declines. As a result, inflation minimally decreases, and the output growth and gap remain stable. The same cannot be said, however, for the CMG rule, where the money growth does not change to accommodate for the shock. Under this rule, inflation responds in a volatile manner. Additionally, the output growth and gap experience the biggest jumps under this rule. Under the TR rule, money growth decreases before it starts to increase. Interest rates do not respond immediately, but eventually increase. These changes result in inflation and the output gap being kept stable. The jump in output growth is also minimal. Macroeconomic factors under the IR rule do not respond strongly to a preference shock. There is only a very small positive change in output growth.

The figure below shows the response of the chosen macroeconomic variables to a positive money demand shock. As explained above, each panel shows the response of the given variable under each of the four rules. According to Lorenzoni (2006), output volatility may be attributed to demand shocks.

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As seen in the first impulse response function above, there is a negative effect on output growth under both the CMG and FMG rules after a positive money demand shock. The IRF of output growth starts at approximately -0.001 under the FMG rule and approximately -0.003 under the CMG rule. The IRF increases at a decreasing rate under both rules, with the trajectory of output growth under the FMG rule crossing the zero line first. Under both, the IRFs begin decreasing, with the FMG rule reaching the steady state at 14 periods, but then proceeds to slightly decrease below the steady state thereafter. Under the CMG rule, the IRF does not reach the steady state within twenty periods.

Inflation under both the CMG rule and the FMG rule is impacted negatively by the money demand shock. The negative effect under the CMG rule is larger than under the FMG rule. Under the CMG rule, the IRF starts at -0.0005 and approximately -0.00005 under the FMG rule. The IRF of inflation under the CMG rule initially decreases before increasing at two periods. It increases at a decreasing rate and reaches the steady state at 18 periods, before surpassing the steady state and continuing to increase. Under the FMG rule, the IRF increases until eight periods, then slightly decreases. The trajectory of inflation reaches, and remains at, the steady state after 18 periods.

The nominal interest rate under both the FMG rule and CMG rule experiences a positive shock and the IRFs start at 0.0008. Initially, the IRF under the CMG rule declines at a faster rate than under the FMG rule. However, at seven periods, the IRF under the FMG rule drops below the CMG rule. Under the FMG rule, the IRF of the nominal interest rate reaches the steady state at 13 periods but continues to decrease at a decreasing rate. Under the CMG rule, the IRF does not reach the steady state within 20 periods.

After the shock, the money growth rate is identically impacted by the TR rule and the IR rule. There is an initial positive effect on the money growth rate, however the IRF begins to decline, where it meets the steady state at 10 periods. Thereafter, the IRF of the money growth rate drops below the steady state and continues to decrease at a decreasing rate. Under the FMG rule, the IRF of the money growth rate starts at zero and initially increases. After nine periods, it begins to decrease and meets the steady state after 18 periods, thereafter, dropping below the steady state.

The output gap is negatively affected by the shock under both the CMG rule and the FMG rule. Under the FMG rule, the IRF of the output gap starts at -0.001 and initially decreases. After three periods, it increases to meet the steady state at eight periods. It increases at a decreasing rate and begins to stabilise after 13 periods. Under the CMG rule, the IRF of the output gap starts at 0.003 and decreases until the 4th period. It then increases at a decreasing rate and does not reach the steady state within 20 periods.

According to Belognia & Ireland (2020) and Poole (1970), money demand shocks result in higher volatility among macroeconomic variables under a rule that keeps money growth fixed. The inefficiency of the CMG rule's response to such a shock can be seen in the above IRFs as its effect on inflation is much larger than under the FMG rule. It is evident that an FMG rule adjusts the rate of money growth in response to movements in the output gap, which may assist in the pursuance of stablisation by monetary policy. As depicted in the IRFs above, the TR rule is most successful in accommodating the money demand shock as the effect of the shock on inflation, nominal interest rate and output is smallest.

The figure below shows the response of the chosen macroeconomic variables to a positive cost push shock. As explained above, each panel shows the response of the given variable under each of the four rules. A cost push shock results in higher inflation due to an increase in the price of production, causing a decrease in aggregate supply (Kenton, 2022).

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Under the four rules, output growth is not significantly affected by the cost push shock. However, the IRF of the output growth under the IR rule jumps up, starting at 0,01 and decreases to meet the steady state after approximately 12 periods.

As expected, inflation under the TR, CMG and FMG rules responds positively to a cost push shock. Under the FMG rule, the IRF of inflation starts at 0.015 and sharply decreases for the first three periods, thereafter, decreases more subtly until reaching the steady state after 12 periods. Under the CMG rule, the IRF starts at 0.00125 and decreases to surpass the zero line before four periods. It then increases but does not reach the steady state within 20 periods. Under the TR, the IRF starts at 0.0025 and decreases to meet the steady state after 4 periods.

The nominal interest rate under the CMG rule and the FMG rule is similarly affected by a cost push shock. Under both, there is a positive effect of the shock on nominal interest rate, where the IRF starts at 0.008. It declines more rapidly under the FMG rule initially, but after six periods, the IRF decreases at a faster rate under the CMG rule. Under the CMG rule, the IRF reaches the steady state sooner, at 13 periods, than the FMG, which does not reach the steady state within 20 periods. Under the TR, the IRF starts at zero and slightly increases before decreasing and merging with the steady state after eight periods.

There is a positive effect on the money growth rate under the IR rule and the TR rule. Under the IR rule, the IRF of the money demand growth rate starts at approximately 0.003 and decreases sharply to below the steady state after two periods. At the 4th period, the IRF increases to meet the steady state after 18 periods. Under the TR rule, the positive effect is smaller, as the IRF of the money growth rate starts at 0.0015. It then decreases to meet the steady state after three periods but increases again. After eight periods, the IRF decreases once again to meet the steady state after 18 periods. Under the FMG rule, the IRF starts at zero and steadily increases before decreasing after nine periods. The IRF of the money growth rate does not reach the steady state within 20 periods.

Under the TR rule, there is no significant impact of the shock on the output gap. Under the IR rule, the CMG rule and the FMG rule, there is a negative impact of the cost push shock on the output gap. The smallest effect on the output gap is under the FMG rule where the IRF of the output gap starts at just below zero and initially increases to cross the zero line. It then starts to decrease after 14 periods and moves towards the steady state, however, it does not reach it within 20 periods. Under the CMG rule, the IRF of the output gap starts at approximately -0.01 and initially decreases before increasing after four periods. It does not reach the steady state within 20 periods. Under the IR rule, the IRF of the output gap starts at -0.035 and remains there until two periods have passed. After that point, it increases at a decreasing rate and meets the steady state after 14 periods.

Although the responses of inflation follow similar trajectories under the CMG and FMG, the FMG rule is more successful than the CMG rule at minimising the volatility in the output gap in response to the cost push shock. Additionally, the FMG performs better at reducing the volatile behaviour of the output gap than the IR rule. Under the TR rule, the output gap trajectory is least volatile in response to the shock. The cost push shock also resulted in larger positive disturbances in the nominal interest rates under both of the money growth rules than the TR rule.

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Interest rate rules, such as the Taylor (TR) rule, have long been considered a successful policy decision tool of choice for the Federal Reserve Bank (Belongia & Ireland, 2020:2). Between 2009-2015, however, the Federal Reserve Bank’s response proved inadequate due to constraints from a nominal interest rate lower bound (Belongia & Ireland, 2020:2). Due to this, alternative rules, such as those that incorporate money growth, need to be investigated. This paper displays the success of interest rate rules in protecting the economy in response to money demand shocks and efficiently adjusting after real shocks, such as a productivity shock, preference shock and cost push shock. It further compares this success against alternative policy rules, namely the Flexible Money Growth rate (FMG) rule and the Constant Money Growth rate (CMG) rule.

The results discussed in the previous section illustrate that monetarily, the CMG rule does not respond appropriately to scenarios where adjusting output and inflation are necessary. The TR rule allows for the money growth rate to be changed. The FMG rule acts as an alternative to the TR rule but similarly lets money growth adjust slightly after changes to the output gap (Belongia & Ireland, 2020:34). As such, monetary policy can be used for stability in the short run without needing to alter long run money growth (Belongia & Ireland, 2020:34).

The first alternative to interest rate rules, the CMG rule, offers little to compare. The IRFs from this paper highlight the shortcomings of this rule towards achieving stability in output following a money demand shock. The IRFs of the second alternative, the FMG rule, provide comparable results to the interest rate rules. The response to a money demand shock by this rule achieves successful stabilisation of output alongside stabilisation of the nominal side of the economy. It achieves this despite there being a slight delay in the implementation of the policy. Belongia & Ireland (2020:29-35) mention this delay but argue that shocks similar to that following the Global Financial Crisis had a more predictable nature that would negate this gap. Therefore, applying this rule over the period following the Global Financial Crisis could have provided policy-makers with a tool comparable to the interest rate rule. Furthermore, it removes the need for unconventional policy rules when approaching the zero lower bound.

The historic success of the federal reserve bank based on an interest rate rule is not enough evidence for this policy rule to take preference over others. Lowering the interest rate to the zero lower bound and the decreasing marginal returns from doing so suggests an alternative (or additional) policy rule is required. The results from this paper suggest that the consideration of the FMG rule is necessary, and adds to efficiency where the interest rate rules fall short.

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Belongia, M.T. & Ireland, P.N. 2020. A Reconsideration of Money Growth Rules. Journal of Economic Dynamics and Control, 135(104312):1-27.

Chen, J. 2020. Contractionary Policy [Online]. Available: https://www.investopedia.com/terms/c/contractionary-policy.asp [June, 2022].

Chugh, S.K. 2015. Modern Macroeconomics. Massachusetts: MIT Press.

Gollier, C & Hammitt, J. 2014. The Long-Run Discount Rate Controversy. Annual Review of Resource Economics, (6):273-295

Kenton, W. 2022. Cost_Push Inflation [Online]. Available: https://www.investopedia.com/terms/c/costpushinflation.asp [June, 2022].

Lorenzoni, G. 2006. A Theory of Demand Shocks. Working Paper No. 12477. Cambridge, Massachusetts: National Bureau of Economic Research.

Malik, K. Z., Ali, S.Z., Imtiaz, A. & Aftab, A. 2019. Preference shocks in an RBC model with intangible capital. Cogent Economics & Finance, 7(1):1-14.

Hazell, J., Herreño, J., Nakamura, E. & Steinsson, J. 2020. The Slope of the Phillips Curve: Evidence from U.S. States. Working Paper No. 28005. Cambridge, Massachusetts: National Bureau of Economic Research.

Ireland, P.N. 2011. A New Keynesian Perspective on the Great Recession. Journal of Money, Credit and Banking, 43(1):31-54.

Poole, W. 1970. Optimal Choice of Monetary Policy Instruments in a Simple Stochastic Macro Model. Quarterly Journal of Economics, 84: 197-216.

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To rewrite the equations into a usable system of equations that can be log-linearised, we make use of the following series of steady state, stationary or equilibrium equations and variables:

$$Y_{t}(i)=Y_{t}, h_{t}(i)=h_{t}, D_{t}(i)=D_{t}, \text { and } P_{t}(i)=P_{t} \text { for all } i \in[0,1] \text { and } t=0,1,2, \ldots$$

$$M_{t}=M_{t-1}+T_{t} \text { and } B_{t}=B_{t-1}=0$$

$$y_{t}=Y_{t} / Z_{t}, c_{t}=C_{t} / Z_{t}, m_{t}=\left(M_{t} / P_{t}\right) / Z_{t}, q_{t}=Q_{t} / Z_{t}, \lambda_{t}=Z_{t} \Lambda_{t}, \text { and } z_{t}=Z_{t} / Z_{t-1}$$

In steady state we can write the following, per their definitions:

$$\begin{aligned} & y_{t}=y,\ c_{t}=c,\ \pi_{t}=\pi,\ r_{t}=r, \ m_{t}=m,\ q_{t}=q,\ x_{t}=x,\ \\ & \mu_{t}=\mu,\ g_{t}=g,\ \lambda_{t}=\lambda,\ a_{t}=a=1,\ z_{t}=z,\ u_{t}=u=1, \\ & \text {and } \theta_{t}=\theta \end{aligned}$$

Firstly, we combine equation (11) and (12) from Section 3.3 [The Representative Intermediate Goods-Producing Firm] and impose the above conditions to obtain equation (1) from the appendix of "Belongia and Ireland (2020)":

Recall the results from equation (11), (12), and the Budget Constraint:

$$Z_th_t(i) \ge Y_t(i) \tag{11}$$ $$\frac{D_t(i)}{P_t}=[\frac{P_t(i)}{P_t}]^{1-\theta_t}Y_t-[\frac{P_t(i)}{P_t}]^{-\theta_t}(\frac{W_t}{P_t})(\frac{Y_t}{Z_t})-\frac{\phi}{2}[\frac{P_t(i)}{\pi_{t-1}^a \pi^{1-a}P_{t-1}(i)}]^2Y_t\tag{12}$$ $$\frac{M_{t-1}+T_{t}+B_{t-1}+W_{t} h_{t}+D_{t}}{P_{t}} \geq C_{t}+\frac{M_{t}+B_{t} / r_{t}}{P_{t}}$$

We can now apply the equilibrium conditions to obtain the results:

$$Z_th_t = Y_t $$ $$\begin{aligned} \frac{D_t}{P_t} & =[\frac{P_t}{P_t}]^{1-\theta_t}Y_t - [\frac{P_t}{P_t}]^{-\theta_t}(\frac{W_t}{P_t})(\frac{Y_t}{Z_t})-\frac{\phi}{2}[\frac{P_t}{\pi_{t-1}^a \pi^{1-a}P_{t-1}}]^2Y_t \\ & = Y_t - (\frac{W_t}{P_t})(\frac{Y_t}{Z_t}) - \frac{\phi}{2}[\frac{P_t}{\pi_{t-1}^a \pi^{1-a}P_{t-1}}]^2Y_t \end{aligned}$$ $$\begin{aligned} C_{t} &= \frac{M_{t-1}+T_{t}+\frac{W_tY_t}{Z_t}+D_{t} - M_{t-1} - T_t}{P_{t}}\\ & = \frac{W_tY_t}{Z_t P_t} + \frac{D_{t}}{P_t} \end{aligned}$$

$$\begin{aligned} -Y_t &= -\left[ \frac{D_t}{P_t} + (\frac{W_t}{P_t})(\frac{Y_t}{Z_t}) \right] - \frac{\phi}{2}[\frac{P_t}{\pi_{t-1}^a \pi^{1-a}P_{t-1}}]^2Y_t \\ \therefore Y_t & = C_t - \frac{\phi}{2}[\frac{P_t}{\pi_{t-1}^a \pi^{1-a}P_{t-1}}]^2Y_t \end{aligned}$$

Dividing both sides by $Z_t$ we can substitute for the necessary stationarity variables and rewrite as:

$$y_{t}=c_{t}+\frac{\phi_{p}}{2}\left(\frac{\pi_{t}}{\pi_{t-1}^{\alpha} \pi^{1-\alpha}}-1\right)^{2} y_{t} \tag{1*}$$

Directly from Equations (2)-(4) in Section 3.1 [The Representative Household]:

$$\ln \left(a_{t}\right)=\rho_{a} \ln \left(a_{t-1}\right)+\varepsilon_{a t} \tag{2*}$$

$$\ln \left(Z_{t}\right)=\ln (z) + \ln(Z_{t-1}) +\varepsilon_{z t}$$

Subtract both sides by $\ln(Z_{t-1})$

$$\ln \left(z_{t}\right)=\ln (z)+\varepsilon_{z t} \tag{3*}$$

$$\ln \left(u_{t}\right)=\rho_{u} \ln \left(u_{t-1}\right)+\varepsilon_{u t} \tag{4*} $$

From equation (5) and (7):

$$\Lambda_t = \frac{a_t}{C_t-\gamma C_{t-1}} - \beta \gamma E_t \left[ \frac{a_{t+1}}{C_{t+1} - \gamma C_t} \right] \tag{5}$$

$$\Lambda_t = \beta(r_t)E_t(\frac{\Lambda_{t+1}}{\pi_{t+1}}) \tag{7}$$

we can rewrite the above equation using the stationary variables:

$$\begin{aligned} \lambda_{t} / Z_t =\frac{a_{t}}{Z_{t} c_{t}-\gamma Z_{t-1} c_{t-1}}-\beta \gamma E_{t} \left(\frac{a_{t+1}}{Z_{t+1} c_{t+1}-\gamma Z_{t-1} c_{t}} \right)\\ \text{ Times each term with: } \left(Z_{t-1} / Z_{t-1}\right) \text{ and } (Z_{t} / Z_{t}) \text{ respectively} \\ \lambda_{t} =\frac{a_{t} z_{t}}{z_{t} c_{t}-\gamma c_{t-1}}-\beta \gamma E_{t}\left(\frac{a_{t+1}}{z_{t+1} c_{t+1}-\gamma c_{t}}\right) \end{aligned}$$

$$\Rightarrow \lambda_{t}=\frac{a_{t}z_{t}}{z_{t} c_{t}-\gamma c_{t-1}}-\beta \gamma E_{t}\left(\frac{a_{t+1}}{z_{t+1} c_{t+1}-\gamma c_{t}}\right) \tag{5*}$$

Similarly:

$$\lambda_{t}=\beta r_{t} E_{t}\left(\frac{\lambda_{t+1}}{z_{t+1} \pi_{t+1}}\right) \tag{7*}$$

Using the fact that:

$$v_1(\frac{M_t}{P_tZ_t},u_t) = \frac{1}{\delta}[\ln(m*) - \ln(\frac{M_t}{P_tZ_t}) + \ln(u_t)]$$

We can rewrite (8) to yield:

$$\begin{array}{l} \frac{a_{t}}{\delta}\left[\ln \left(m^{*}\right)-\ln \left(\frac{M_{t}}{P_{t} Z_{t}}\right)+\ln \left(u_{t}\right)\right]-a_{t}\left(\frac{\phi_{m}}{2}\right)\left(\frac{M_{t} / P_{t}}{z M_{t-1} / P_{t-1}}-1\right)^{2} \\ -a_{t} \phi_{m}\left(\frac{M_{t} / P_{t}}{z M_{t-1} / P_{t-1}}-1\right)\left(\frac{M_{t} / P_{t}}{z M_{t-1} / P_{t-1}}\right) \\ +\beta \phi_{m} E_{t}\left[a_{t+1}\left(\frac{M_{t+1} / P_{t+1}}{z M_{t} / P_{t}}-1\right)\left(\frac{M_{t+1} / P_{t+1}}{z M_{t} / P_{t}}\right)^{2}\left(\frac{z Z_{t}}{Z_{t+1}}\right)\right] \\ =Z_{t} \Lambda_{t}\left(1-\frac{1}{r_{t}}\right) \end{array} \tag{9}$$

And thus obtain:

$$\begin{aligned} & \frac{a_{t}}{\delta}\left[\ln \left(m^{}\right)-\ln \left(m_{t}\right)+\ln \left(u_{t}\right)\right]-a_{t}\left(\frac{\phi_{m}}{2}\right)\left(\frac{z_{t} m_{t}}{z m_{t-1}}-1\right)^{2} \ & -a_{t} \phi_{m}\left(\frac{z_{t} m_{t}}{z m_{t-1}}-1\right)\left(\frac{z_{t} m_{t}}{z m_{t-1}}\right) \ & +\beta \phi_{m} E_{t}\left[a_{t+1}\left(\frac{z_{t+1} m_{t+1}}{z m_{t}}-1\right)\left(\frac{z_{t+1} m_{t+1}}{z m_{t}}\right)^{2}\left(\frac{z}{z_{t+1}}\right)\right] \ & =\lambda_{t}\left(1-\frac{1}{r_{t}}\right), \ \end{aligned} \tag{9}$$

Equations (10) is used directly:

$$\ln(\theta_t)=(1-\rho_{\theta})\ln(\theta)+\rho_{\theta}\ln(\theta_{t-1})+\varepsilon_{\theta t} \tag{10*}$$

From equation (13), we rewrite:

$$\begin{array}{l} (1-\theta_t)(\frac{P_t(i)}{P_t})^{-\theta_t} + \theta_t(\frac{P_t(i)}{P_t})^{-\theta_t-1}(\frac{W_t}{P_t})(\frac{1}{Z_t}) - \phi_p[(\frac{P_t(i)}{\pi_{t-1}^a \pi^{1-a}P_{t-1}(i)}-1)(\frac{P_t}{\pi_{t-1}^a \pi^{1-a}P_{t-1}(i)})] \\ +\beta\phi_pE_t(\frac{\Lambda_{t+1}}{\Lambda_{t}})(\frac{P_{t+1}(i)}{\pi_t^a\pi^{1-a}P_{t}(i)}-1)(\frac{P_{t+1}(i)}{\pi_t^a\pi^{1-a}P_{t}(i)})(\frac{Y_{t+1}}{Y_t})(\frac{P_t}{P_t(i)})=0 \end{array}$$

$$\begin{aligned} (1-\theta_t) + \theta_t (\frac{W_t}{P_t})(\frac{1}{Z_t}) - \phi_p[(\frac{P_t}{\pi_{t-1}^a \pi^{1-a}P_{t-1}}-1)(\frac{P_t}{\pi_{t-1}^a \pi^{1-a}P_{t-1}})] \\ +\beta\phi_pE_t(\frac{\Lambda_{t+1}}{\Lambda_{t}})(\frac{P_{t+1}}{\pi_t^a\pi^{1-a}P_{t}}-1)(\frac{P_{t+1}}{\pi_t^a\pi^{1-a}P_{t}})(\frac{Y_{t+1}}{Y_t})=0 \end{aligned}$$

$$\begin{aligned} \Rightarrow \theta_t - 1 = & \theta_t \frac{a_t}{\Lambda_t Z_t} - \phi_p[(\frac{\pi_t}{\pi_{t-1}^a \pi^{1-a}}-1)(\frac{\pi_t}{\pi_{t-1}^a \pi^{1-a}})] \\ & +\beta\phi_pE_t(\frac{\Lambda_{t+1}Y_{t+1}}{\Lambda_{t}Y_t})(\frac{\pi_{t+1}}{\pi_t^a\pi^{1-a}}-1)(\frac{\pi_{t+1}}{\pi_t^a\pi^{1-a}}) \end{aligned}$$

$$\begin{aligned} \therefore \theta_{t}-1=& \theta_{t}\left(\frac{a_{t}}{\lambda_{t}}\right)-\phi_{p}\left(\frac{\pi_{t}}{\pi_{t-1}^{\alpha} \pi^{1-\alpha}}-1\right)\left(\frac{\pi_{t}}{\pi_{t-1}^{\alpha} \pi^{1-\alpha}}\right) \\ &+\beta \phi_{p} E_{t}\left[\left(\frac{\lambda_{t+1} y_{t+1}}{\lambda_{t} y_{t}}\right)\left(\frac{\pi_{t+1}}{\pi_{t}^{\alpha} \pi^{1-\alpha}}-1\right)\left(\frac{\pi_{t+1}}{\pi_{t}^{\alpha} \pi^{1-\alpha}}\right)\right] \end{aligned} \tag{13*}$$

From equation (14) and (15)

$$\frac{1}{Z_{t}}=\frac{1}{Q_{t}-\gamma Q_{t-1}}-\beta \gamma E_{t}\left[\left(\frac{a_{t+1}}{a_{t}}\right)\left(\frac{1}{Q_{t+1}-\gamma Q_{t}}\right)\right] \tag{14}$$

$$x_t = Y_t/Q_t \tag{15}$$

Which can be rewritten as:

$$1=\frac{z_{t}}{z_{t} q_{t}-\gamma q_{t-1}}-\beta \gamma E_{t}\left[\left(\frac{a_{t+1}}{a_{t}}\right)\left(\frac{1}{z_{t+1} q_{t+1}-\gamma q_{t}}\right)\right] \tag{14*}$$

and

$$x_t = y_t/q_t \tag{15*}$$

Lastly, the paper introduces: The monetary policy rule, The growth rate of money, and the growth rate of output with

$$\ln \left(r_{t} / r\right)=\rho_{r} \ln \left(r_{t-1} / r\right)+\rho_{\pi} \ln \left(\pi_{t-1} / \pi\right)+\rho_{x} \ln \left(x_{t-1} / x\right)+\varepsilon_{r t}, \tag{16*}$$

$$\mu_{t}=\left(\frac{M_{t} / P_{t}}{M_{t-1} / P_{t-1}}\right) \pi_{t}, \tag{17*}$$

and

$$g_{t}=Y_{t} / Y_{t-1} \tag{18*}$$

We apply the Taylor throughout the following section as follows:

For a function $f(x_t)$ input $x_t$ , we apply the Taylor method such that: $$f(x_t) = f(X^{SS}) + \frac{df(x)}{x_t} (x_t(i) - x^{SS})$$

Using the approximation: $$\begin{aligned} &x_t - x^{SS} \approx x^{SS} \hat{x}{t} \ &\text{where } \hat{x}{t} = \ln(x_t)-\ln(x^{SS}) \ &\text{For short, we write: } x^{SS} \equiv x \ &\text{(i.e. no time subscript)} \end{aligned}$$

Applying Uhlig's method requires the following:

\newpage

For the function $f(x_t)$ i.e., we use the fact that:

1. $$\hat{x}_t = \ln(x_t) - \ln(x) \ \text{ where x } \equiv x^{SS}$$

2. Then, $f(^{SS})$ is used to simplify $f(x_t)$ where $x_t$ is replaced accodring to $x_t = x \cdot e^{\hat{x}_t}$

3. Additionally, the approximation result $e^{x_t+y_t} \approx 1+x_t+y_t$ is used for further simplification.

4. For the last step, all constants can be dropped.

\newpage

{height="70%" width="90%"}

\newpage

//Hodrick Prescott on GDP

[Trend,Cyclical] = hpfilter(GDPC11,Smoothing=1600,DataVariables=3)

//Macros Project Calibration Taylor Rule

// Preamble var c yhat lambdahat ahat zhat rhat pihat qhat xhat ehat
uhat mhat mu ghat;

varexo epsilon_r epsilon_a epsilon_z epsilon_u epsilon_e;

parameters z_ss beta gamma rho_a alpha psi rho_r rho_pi rho_x delta_r
phi r_ss rho_u rho_e;

// parameters we need to add for scenario 3: rho_mm rho_mpi rho_mx;

// Parameter Declaration z_ss = 1.00037; // Detrended quarterly changes
in log of GDP r_ss = 0.98784; // From quarterly data divided by 4 beta =
0.99023; // 1/(r_ss+1) from Gollier & Hammitt (2014) gamma = 0.6741; // habit
formation parameter alpha = 0.2337; // price index psi = 0.0062; //
Phillips curve slope from Hazell et al. (2020) delta_r = 14.8716; //
money demand semi-elasticity phi = 14.8854; // money demand adjustment
cost rho_r = 0.8373; // interest rate smoothing rho_pi = 0.2296; //
policy response to inflation rho_x = 0.2388; // policy response to
output gap rho_a = 0.9263; // preference shock persistence rho_u =
0.9733; // money demand shock persistence rho_e = 0.3330; // cost push
shock persistence

// In accordance to Belongia & Ireland (2020), the FMG rule requires the
following parameters to be set: // rho_mm = 1; // rho_mpi = 0; // rho_mx
= -0.125;

// Model Declaration

model(linear);

c = yhat; // A first-order Taylor approximation to Eq (1) implies this

(z_ss-beta*gamma)*(z_ss-gamma)*lambdahat = gamma*z_ss*yhat(-1) -
(z_ss\^2 + beta*(gamma\^2))*yhat + beta*gamma*z_ss*yhat(+1) +
(z_ss-beta*gamma*rho_a)*(z_ss-gamma)*ahat - gamma*z_ss*zhat; // Eq (19)

lambdahat = rhat + lambdahat(+1) - pihat(+1); // Eq (20)

0 = gamma*z_ss*qhat(-1) - (z_ss\^2+beta*(gamma\^2))*qhat +
beta*gamma*z_ss*qhat(+1) + beta*gamma*(z_ss-gamma)*(1-rho_a)*ahat -
gamma*z_ss\*zhat; // Eq (21)

xhat = yhat - qhat; // Output gap Eq (22)

// Eq (19)-(22) define the model's New Keynesian IS relationship

(1+beta*alpha)*pihat = alpha*pihat(-1) + beta*pihat(+1) -
psi*lambdahat + psi*ahat + ehat; // New Keynesian Phillips curve Eq (23)

// Scenario 1: Taylor rule rhat = rho_r*rhat(-1) + rho_pi*pihat(-1) +
rho_x\*xhat(-1) + epsilon_r; // Eq (24) Taylor rule for monetary policy

// Scenario 2: replace Eq (24) with Eq (32) // r = -ln(r_ss); // Eq (32)
zero interest rate condition

// Scenario 3: Money Growth Rules (replace Eq (24) with Eq (33)) // mu =
rho_mm*mu(-1) + rho_mpi*pi(-1) + rho_mx\*xhat(-1); // Eq (33) money
growth rate rule

delta_r*(r_ss-1)*lambdahat - delta_r*(r_ss-1)*ahat - uhat + phi*zhat =
phi*mhat(-1) - (1 + (1 + beta)*phi)*mhat + beta*phi*mhat(+1) -
delta_r\*rhat; // Eq (25) money demand relationship

mu = zhat + mhat - mhat(-1) + pihat ; // Eq (26) determines the growth
rate of the nominal money stock

ghat = yhat - yhat(-1) + zhat; // Eq (27) determines the growth rate of
aggregate output

ahat = rho_a\*ahat(-1) + epsilon_a; // Eq (28) preference shock

zhat = epsilon_z; // Eq (29) productivity shock

uhat = rho_u\*uhat(-1) + epsilon_u; // Eq (30) money demand shock

ehat = rho_e\*ehat(-1) + epsilon_e; // Eq (31) cost push shock

End;

steady (solve_algo=0); check;

// Define shocks: shocks; var epsilon_r; stderr 0.0016; // monetary
policy shock; value according to Belongia & Ireland (2020): Table 1 var
epsilon_a; stderr 0.0066; // preference shock; according to Belongia &
Ireland (2020): Table 1 var epsilon_z; stderr 0.0066; // productivity
shock; according to Belongia & Ireland (2020): Table 1 var epsilon_u;
stderr 0.0066; // money demand shock; according to Belongia & Ireland
(2020): Table 1 var epsilon_e; stderr 0.0016; // cost push shock;
according to Belongia & Ireland (2020): Table 1

End;

stoch_simul(order=1,irf=40);

//Macros Project Calibration Interest Rate Rule

// Preamble

var c yhat lambdahat ahat zhat rhat qhat xhat ehat uhat mhat mu ghat
pihat;

varexo epsilon_a epsilon_z epsilon_u epsilon_e; //epsilon_r; parameters
z_ss beta gamma rho_a alpha psi rho_x delta_r phi r_ss rho_u rho_e;
//rho_r rho_pi;

// parameters we need to add for scenario 3: rho_mm rho_mpi rho_mx;

// Parameter Declaration z_ss = 1.00037; // Detrended quarterly changes
in log of GDP r_ss = 0.98784; // From quarterly data divided by 4 beta =
0.99023; // 1/(r_ss+1) from Gollier (2014)

gamma = 0.6741; // habit formation parameter alpha = 0.2337; // price
index psi = 0.0062; // Phillips curve slope from Hazell et al. (2020)
delta_r = 14.8716; // money demand semi-elasticity phi = 14.8854; //
money demand adjustment cost //rho_r = 0.8373; // interest rate
smoothing //rho_pi = 0.2296; // policy response to inflation rho_x =
0.2388; // policy response to output gap rho_a = 0.9263; // preference
shock persistence rho_u = 0.9733; // money demand shock persistence
rho_e = 0.3330; // cost push shock persistence

// In accordance to Belongia & Ireland (2020), the FMG rule requires the
following parameters to be set: // rho_mm = 1; // rho_mpi = 0; // rho_mx
= -0.125;

// Model Declaration

model(linear);

c = yhat; // A first-order Taylor approximation to Eq (1) implies this

(z_ss-beta*gamma)*(z_ss-gamma)*lambdahat = gamma*z_ss*yhat(-1) -
(z_ss\^2 + beta*(gamma\^2))*yhat + beta*gamma*z_ss*yhat(+1) +
(z_ss-beta*gamma*rho_a)*(z_ss-gamma)*ahat - gamma*z_ss*zhat; // Eq (19)

lambdahat = rhat + lambdahat - pihat; // Eq (20)

0 = gamma*z_ss*qhat(-1) - (z_ss\^2+beta*(gamma\^2))*qhat +
beta*gamma*z_ss*qhat(+1) + beta*gamma*(z_ss-gamma)*(1-rho_a)*ahat -
gamma*z_ss\*zhat; // Eq (22)

xhat = yhat - qhat; // Output gap Eq (22)

// Eq (19)-(22) define the model's New Keynesian IS relationship

(1+beta*alpha)*pihat = alpha*pihat(-1) + beta*pihat(+1) -
psi*lambdahat + psi*ahat + ehat; // New Keynesian Phillips curve Eq (23)

// Scenario 1: Taylor rule //r = rho_r*r(-1) + rho_pi*pihat(-1) +
rho_x\*x(-1) + epsilon_r; // Eq (24) Taylor rule for monetary policy

// Scenario 2: replace Eq (24) with Eq (32) rhat = -ln(r_ss); // Eq (32)
zero interest rate condition

// Scenario 3: Money Growth Rules (replace Eq (24) with Eq (33)) // mu =
rho_mm*mu(-1) + rho_mpi*pi(-1) + rho_mx\*x(-1); // Eq (33) money growth
rate rule

delta_r*(r_ss-1)*lambdahat - delta_r*(r_ss-1)*ahat - uhat + phi*zhat =
phi*mhat(-1) - (1 + (1 + beta)*phi)*mhat + beta*phi*mhat(+1) -
delta_r\*rhat; // Eq (25) money demand relationship

mu = zhat + mhat - mhat(-1) + pihat ; // Eq (26) determines the growth
rate of the nominal money stock

ghat = yhat - yhat(-1) + zhat; // Eq (27) determines the growth rate of
aggregate output

ahat = rho_a\*ahat(-1) + epsilon_a; // Eq (28) preference shock

zhat = epsilon_z; // Eq (29) productivity shock

uhat = rho_u\*uhat(-1) + epsilon_u; // Eq (30) money demand shock

ehat = rho_e\*ehat(-1) + epsilon_e; // Eq (31) cost push shock

end;

steady; check;

// Define shocks:

shocks; //var epsilon_r; stderr 0.0016; // monetary policy shock; value
according to Belongia & Ireland (2020): Table 1 var epsilon_a; stderr
0.0066; // preference shock; according to Belongia & Ireland (2020):
Table 1 var epsilon_z; stderr 0.0066; // productivity shock; according
to Belongia & Ireland (2020): Table 1 var epsilon_u; stderr 0.0066; //
money demand shock; according to Belongia & Ireland (2020): Table 1 var
epsilon_e; stderr 0.0016; // cost push shock; according to Belongia &
Ireland (2020): Table 1

end;

stoch_simul(order=1,irf=40);

//Macros Project Calibration Constant Money Growth Rule

//Preamble:

var c yhat lambdahat ahat zhat rhat qhat xhat ehat uhat mhat mu ghat
pihat;

varexo epsilon_a epsilon_z epsilon_u epsilon_e; //epsilon_r

parameters z_ss beta gamma rho_a alpha psi rho_x delta_r phi r_ss rho_u
rho_e rho_r rho_pi rho_mm rho_mpi rho_mx; //rho_pi; //not in scenario 3

// Parameter Declaration z_ss = 1.00037; // Detrended quarterly changes
in log of GDP r_ss = 0.98784; // From quarterly data divided by 4 beta =
0.99023; // 1/(r_ss+1) from Gollier (2014)

gamma = 0.6741; // habit formation parameter alpha = 0.2337; // price
index psi = 0.0062; // Phillips curve slope from Hazell et al. (2020)
delta_r = 14.8716; // money demand semi-elasticity phi = 14.8854; //
money demand adjustment cost rho_r = 0.8373; // interest rate smoothing
//rho_pi = 0.2296; // policy response to inflation rho_x = 0.2388; //
policy response to output gap rho_a = 0.9263; // preference shock
persistence rho_u = 0.9733; // money demand shock persistence rho_e =
0.3330; // cost push shock persistence

//Model:

model(linear);

// A first-order Taylor approximation to Eq (1) implies that: c = yhat;

// Eq (19)-(22) define the model's New Keynesian IS relationship:

(z_ss-beta*gamma)*(z_ss-gamma)*lambdahat = gamma*z_ss*yhat(-1) -
(z_ss\^2 + beta*(gamma\^2))*yhat + beta*gamma*z_ss*yhat(+1) +
(z_ss-beta*gamma*rho_a)*(z_ss-gamma)*ahat - gamma*z_ss*zhat; // Equation
(19)

lambdahat = rhat + lambdahat(+1) - pihat(+1); // Equation (20)

0 = gamma*z_ss*qhat(-1) - (z_ss\^2+beta*(gamma\^2))*qhat +
beta*gamma*z_ss*qhat(+1) + beta*gamma*(z_ss-gamma)*(1-rho_a)*ahat -
gamma*z_ss\*zhat; // Equation (22)

xhat = yhat - qhat; // Equation (22) - Output Gap

//New Keynesian Phillips Curve: (1+beta*alpha)*pihat = alpha*pihat(-1) +
beta*pihat(+1) - psi*lambdahat + psi*ahat + ehat; // Equation (23)

// Scenario 1: Taylor rule //rhat = rho_r*rhat(-1) + rho_pi*pihat(-1) +
rho_x\*xhat(-1) + epsilon_r; // Equation (24) Taylor rule for monetary
policy

// Scenario 2: replace Equation (24) with Equation (32) //rhat =
-ln(r_ss); // Equation (32) (with zero interest rate condition)

// Scenario 3: Money Growth Rules (replace Equation (24) with Equation
(33)) mu = 0; // Eq (33) money growth rate rule;

delta_r*(r_ss-1)*lambdahat - delta_r*(r_ss-1)*ahat - uhat + phi*zhat =
phi*mhat(-1) - (1 + (1 + beta)*phi)*mhat + beta*phi*mhat(+1) -
delta_r*rhat; // Equation (25) money demand relationship //implies that:
mhat = delta_r*(r_ss-1)*yhat - delta_r*rhat + uhat

mu = zhat + mhat - mhat(-1) + pihat ; // Equation (26) determines the
growth rate of the nominal money stock

ghat = yhat - yhat(-1) + zhat; // Equation (27) determines the growth
rate of aggregate output

//Dynamics of preference, productivity, money demand and cost push
shocks: ahat = rho_a\*ahat(-1) + epsilon_a; // Equation (28) preference
shock

zhat = epsilon_z; // Equation (29) productivity shock

uhat = rho_u\*uhat(-1) + epsilon_u; // Equation (30) money demand shock

ehat = rho_e\*ehat(-1) + epsilon_e; // Equation (31) cost push shock

end;

steady; check;

// Define shocks:

shocks; //var epsilon_r; stderr 0.0016; // monetary policy shock; value
according to Belongia & Ireland (2020): Table 1 var epsilon_a; stderr
0.01; // preference shock; according to Belongia & Ireland (2020): Table
1 var epsilon_z; stderr 0.01; // productivity shock; according to
Belongia & Ireland (2020): Table 1 var epsilon_u; stderr 0.01; // money
demand shock; according to Belongia & Ireland (2020): Table 1 var
epsilon_e; stderr 0.01; // cost push shock; according to Belongia &
Ireland (2020): Table 1

end; stoch_simul(order=1,irf=40);

//Macros Project Calibration Flexible Money Growth Rule

//Preamble: var chat yhat lambdahat ahat zhat rhat qhat xhat ehat uhat
mhat mu ghat pihat;

varexo epsilon_a epsilon_z epsilon_u epsilon_e; //epsilon_r

parameters z_ss beta gamma rho_a alpha psi delta_r phi r_ss rho_u rho_e
rho_mm rho_mpi rho_mx; //rho_r rho_pi rho_x; //not in scenario 3

// Parameter Declaration z_ss = 1.00037; // Detrended quarterly changes
in log of GDP r_ss = 0.98784; // From quarterly data divided by 4 beta =
0.99023; // 1/(r_ss+1) from Gollier (2014)

gamma = 0.6741; // habit formation parameter alpha = 0.2337; // price
index psi = 0.0062; // Phillips curve slope from Hazell et al. (2020)
delta_r = 14.8716; // money demand semi-elasticity phi = 14.8854; //
money demand adjustment cost //rho_r = 0.8373; // interest rate
smoothing (not in scenario 3) //rho_pi = 0.2296; // policy response to
inflation (not in scenario 3) //rho_x = 0.2388; // policy response to
output gap (not in scenario 3) rho_a = 0.9263; // preference shock
persistence rho_u = 0.9733; // money demand shock persistence rho_e =
0.3330; // cost push shock persistence rho_mm = 1; rho_mpi = 0; rho_mx =
-0.125;

// Model:

model(linear);

// A first-order Taylor approximation to Eq (1) implies that: chat =
yhat;

// Eq (19)-(22) define the model's New Keynesian IS relationship:

(z_ss-beta*gamma)*(z_ss-gamma)*lambdahat = gamma*z_ss*yhat(-1) -
(z_ss\^2 + beta*(gamma\^2))*yhat + beta*gamma*z_ss*yhat(+1) +
(z_ss-beta*gamma*rho_a)*(z_ss-gamma)*ahat - gamma*z_ss*zhat; // Equation
(19)

lambdahat = rhat + lambdahat(+1) - pihat(+1); // Equation (20)

0 = gamma*z_ss*qhat(-1) - (z_ss\^2+beta*(gamma\^2))*qhat +
beta*gamma*z_ss*qhat(+1) + beta*gamma*(z_ss-gamma)*(1-rho_a)*ahat -
gamma*z_ss\*zhat; // Equation (22)

xhat = yhat - qhat; // Equation (22) - Output Gap

//New Keynesian Phillips Curve: (1+beta*alpha)*pihat = alpha*pihat(-1) +
beta*pihat(+1) - psi*lambdahat + psi*ahat + ehat; // Equation (23)

// Scenario 1: Taylor rule //rhat = rho_r*rhat(-1) + rho_pi*pihat(-1) +
rho_x\*xhat(-1) + epsilon_r; // Equation (24) Taylor rule for monetary
policy

// Scenario 2: replace Equation (24) with Equation (32) //rhat =
-ln(r_ss); // Equation (32) (with zero interest rate condition)

// Scenario 3: Money Growth Rules (replace Equation (24) with Equation
(33)) mu = rho_mm*mu(-1) + rho_mpi*pihat(-1) + rho_mx\*xhat(-1); //
Equation (33) money growth rate rule

delta_r*(r_ss-1)*lambdahat - delta_r*(r_ss-1)*ahat - uhat + phi*zhat =
phi*mhat(-1) - (1 + (1 + beta)*phi)*mhat + beta*phi*mhat(+1) -
delta_r*rhat; // Equation (25) money demand relationship //implies that:
mhat = delta_r*(r_ss-1)*yhat - delta_r*rhat + uhat

mu = zhat + mhat - mhat(-1) + pihat ; // Equation (26) determines the
growth rate of the nominal money stock

ghat = yhat - yhat(-1) + zhat; // Equation (27) determines the growth
rate of aggregate output

//Dynamics of preference, productivity, money demand and cost push
shocks: ahat = rho_a\*ahat(-1) + epsilon_a; // Equation (28) preference
shock

zhat = epsilon_z; // Equation (29) productivity shock

uhat = rho_u\*uhat(-1) + epsilon_u; // Equation (30) money demand shock

ehat = rho_e\*ehat(-1) + epsilon_e; // Equation (31) cost push shock

end;

steady; check;

// Define shocks:

shocks; var epsilon_a; stderr 0.01; // preference shock; according to
Belongia & Ireland (2020): Table 1 var epsilon_z; stderr 0.01; //
productivity shock; according to Belongia & Ireland (2020): Table 1 var
epsilon_u; stderr 0.01; // money demand shock; according to Belongia &
Ireland (2020): Table 1 var epsilon_e; stderr 0.01; // cost push shock;
according to Belongia & Ireland (2020): Table 1

end;

stoch_simul(order=1,irf=20);