The NKM with a Supply Shock: Matrix Representation

Macroeconomics (M8674), May 04, 2026

Vivaldo Mendes, ISCTE-IUL

vivaldo.mendes@iscte-iul.pt

1. Introduction

The New Keynesian Model with a supply shock

The linearized baseline version of the NKM with supply shocks can be written with 7 equations:

\[\begin{array}{lllll} \text {IS : } & \hat{y}_{t}=\mathbb{E}_{t} \hat{y}_{t+1}-\frac{1}{\sigma}\left(i_{t}-\mathbb{E}_{t} \pi_{t+1}-r^{n}\right)\\[2pt] \text {Taylor rule : } & \ i_{t} = \pi_t + r^n +\phi_{\pi} (\pi_{t} - \pi^{_{T}})+\phi_{y} \cdot \hat{y}_{t} \\[2pt] \text {AS : } & \pi_{t} =\kappa \cdot \hat{y}_{t}+\beta \cdot \mathbb{E}_{t} \pi_{t+1} + s_t\\[2pt] \text {Supply shock : } & s_{t} =\rho_s s_{t-1}+\varepsilon_{t}^{s} \quad , \quad \varepsilon_{t}^{s} \sim {\cal{N}}(0,1) \\[2pt] \text {Output allocation : } & \hat{y}_{t}= \hat{c}_{t} \\[2pt] \text {Labor supply : } & \hat{\ell}_t=\left(\frac{1-\sigma}{1+\gamma}\right) \hat{y}_t \\[2pt] \text {Technology : } & \hat{a}_t=\left[1-\frac{\alpha(1-\sigma)}{1+\gamma}\right] \hat{y}_t \\[2pt] \end{array}\]

Variables and parameters

Endogenous variables: \(\{\hat{y}, \pi, i, \hat{c}, s, \hat{\ell}, \hat{a} \}\) are, respectively, the output-gap, inflation, nominal interest rate, real consumption, supply shock, labor supply, and technology.
Exogenous variables: \(\{r^n, \pi^{_{T}}, \varepsilon\}\), which represent, respectively, the natural level of the real interest rate, the target inflation rate, and a random disturbance.
Parameters: \(\{\sigma \ , \ \phi_{\pi} \ , \ \phi_{y} \ , \ \kappa=\frac{\psi(1-\mu)(1-\mu \beta)}{\mu} \ , \ \psi \ , \ \beta \ , \ \mu \ , \ \rho_s \ , \ \alpha \ , \ \gamma \}\)
Forward-looking variables: \(\hat{y}_{t} \ , \ \pi_{t}\)
Backward-looking variables: \(\ s_{t}\)
Static variables: \(i_t \ , \ \hat{\ell}_t \ , \ \hat{a}_t \ , \ \hat{c}_t\)

4 equations vs 4 unknowns

The model can be fully simulated only with 4 equations and 4 unknowns:
\[\begin{array}{lllll} \text {Supply shock : } & s_{t+1} =\rho_s s_{t}+\varepsilon_{t+1}^{s} \\[2pt] \text {Taylor rule : } & \ i_{t+1} = \pi_{t+1} + r^n +\phi_{\pi} (\pi_{t+1} - \pi^{_{T}})+\phi_{y} \cdot \hat{y}_{t+1} \\[2pt] \text {AS : } & \pi_{t} =\kappa \cdot \hat{y}_{t}+\beta \cdot \mathbb{E}_{t} \pi_{t+1} + s_{t}\\[2pt] \text {IS : } & \hat{y}_{t}=\mathbb{E}_{t} \hat{y}_{t+1}-\frac{1}{\sigma}\left(i_{t}-\mathbb{E}_{t} \pi_{t+1}-r^{n}\right)\\[4pt] \end{array}\]
The first two are non-forward looking variables, and the last two are forward looking variables.
To separate the two blocks of equations, the order we write them matters:
- First write down the non-forward looking block
- Then write down the forward looking block

Matrix representation

The 4 equations can be written as:

\[ \begin{aligned} {\color{blue}1 s_{t+1}+ 0i_{t+1}+ 0 \mathbb{E}_t \pi_{t+1}+0 \mathbb{E}_t \hat{y}_{t+1}} & =\rho_s s_t+ 0 i_t + 0 \pi_t+0 \hat{y}_t \ \ \ \ \ + {\color{red} 1 \varepsilon_{t+1}^s} \\[3pt] {\color{blue}0 s_{t+1}+ 1i_{t+1}- (1+ \phi_\pi) \mathbb{E}_t\pi_{t+1}- \phi_y \mathbb{E}_t\hat{y}_{t+1}}&=0 s_t + 0 i_t + 0 \pi_t+0 \hat{y}_t \ \ \ \ \ \ \ + {\color{red} 0 \varepsilon_{t+1}^i} + {\color{teal}r^n} - {\color{teal}\phi_\pi \pi^{_T}}\\[3pt] {\color{blue}0 s_{t+1}+0 i_{t+1} +\beta \mathbb{E}_t \pi_{t+1}+0 \mathbb{E}_t \hat{y}_{t+1}}&=-1 s_t + 0 i_t +1 \pi_t-\kappa \hat{y}_t \ \ \ \ + {\color{red}0 \varepsilon_{t+1}^\pi} \\[3pt] {\color{blue}0 s_{t+1}+0 i_{t+1} +(1/\sigma) \mathbb{E}_t \pi_{t+1}+1 \mathbb{E}_t \hat{y}_{t+1}}&=0 s_t+ (1/\sigma) i_t + 0 \pi_t+1 \hat{y}_t + {\color{red} 0 \varepsilon_{t+1}^y} + {\color{teal}(1/\sigma) r^n} \\ \end{aligned} \]

Passing the equations into matrices gives:

\[{\color{blue} \underbrace{\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & -(1+\phi_\pi) & -\phi_y \\ 0 & 0 & \beta & 0 \\ 0 & 0 & \frac{1}{\sigma} & 1 \end{array}\right]}_\mathcal{A}\left[\begin{array}{c}s_{t+1} \\ i_{t+1} \\ \mathbb{E}_t \pi_{t+1} \\ \mathbb{E}_t \hat{y}_{t+1} \end{array}\right]} = \underbrace{\left[\begin{array}{cccc} \rho_s & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & -\kappa \\ 0 & \frac{1}{\sigma} & 0 & 1 \end{array}\right]}_\mathcal{B}\left[\begin{array}{c} s_t \\ i_t \\ \pi_t \\ \hat{y}_t \end{array}\right] + {\color{red} \underbrace{\left[\begin{array}{lll} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right]}_{\cal{C}} \left[\begin{array}{c} \varepsilon_{t+1}^{s} \\ \varepsilon_{t+1}^{i} \\ \varepsilon_{t+1}^{\pi} \\ \varepsilon_{t+1}^{y} \end{array}\right]} + \underbrace{\left[\begin{array}{c} \color{teal} 0 \\ \color{teal} r^n - \phi_\pi \pi^{_T} \\ \color{teal} 0 \\ \color{teal} (1/\sigma)r^n \end{array}\right]}_{\cal{D}}\]