This notebook is part of a computational appendix that accompanies the paper.
MATLAB, Python, Julia: What to Choose in Economics?
Coleman, Lyon, Maliar, and Maliar (2017)
In this notebook we summarize the key equations for the stylized New Keynesian model we solved in the paper.
For more information on the model itself see section 5 of
Maliar, L., & Maliar, S. (2015). Merging simulation and projection approaches to solve high-dimensional problems with an application to a new Keynesian model. Quantitative Economics, 15(7), 424. http://doi.org/10.1186/s13059-014-0424-0
The model features Calvo-type price frictions and a Taylor (1993) rule. The economy is populated by households, final-good firms, intermediate-good firms, a monetary authority and government.
In the equations below, the following variables appear:
We also see the following parameters:
In our computation we will approximate $S_t$, $F_t$ and $C_t$ using a complete monomial of degree N. Given these three variables, we express the equilibrium conditions of the model in the following way:
\begin{eqnarray} \pi_{t} &=& \left(\frac{1-(1-\theta)}{\theta}\left(\frac{S_t}{F_t}\right)^{1- \varepsilon}\right)^{\frac{1}{\varepsilon-1}} \\ \Delta _{t} &=&\left[ \left( 1-\theta \right) \left[ \frac{1-\theta \pi _{t}^{\varepsilon -1}}{1-\theta }\right] ^{\frac{\varepsilon }{\varepsilon -1% }}+\theta \frac{\pi _{t}^{\varepsilon }}{\Delta _{t-1}}\right] ^{-1} \\ Y_t &=& \frac{C_t}{1-\frac{\bar{G}}{\exp\left(\eta_{G,t}\right)}} \\ L_t &=& \frac{Y_t }{\exp\left(\eta_{a,t}\right) \Delta_t} \\ Y_{N,t} &=& \left[ \frac{\exp \left( \eta _{a,t}\right) ^{1+\vartheta }\left[1- \frac{\bar{G}}{\exp \left( \eta _{G,t}\right)} \right] ^{-\gamma}}{\exp \left( \eta_{L,t}\right) }\right] ^{\frac{1}{\vartheta +\gamma }}\\ R_{t} &=& \max \left\{1, \frac{\pi^*}{\beta} \left(R_{t-1} \frac{\beta}{\pi^*} \right)^{\mu} \left(\left(\frac{\pi_t}{\pi^*}\right)^{\phi_{\pi}} \left(\frac{Y_t}{Y_{N,t}} \right)^{\phi_y} \right)^{1-\mu}\exp\left(\eta_{R,t}\right) \right\}. \end{eqnarray}The Euler equations are given by
\begin{eqnarray} S_{t} &=&\frac{\exp \left( \eta _{u,t}+\eta _{L,t}\right) }{\left[ \exp \left( \eta _{a,t}\right) \right] ^{\vartheta +1}}\frac{\left( G_{t}^{-1}C_{t}\right) ^{1+\vartheta }}{\left( \Delta _{t}\right) ^{\vartheta }}+\beta \theta E_{t}\left \{ \pi _{t+1}^{\varepsilon }S_{t+1}\right \} \\ F_{t} &=&\exp \left( \eta _{u,t}\right) C_{t}^{1-\gamma }G_{t}^{-1}+\beta \theta E_{t}\left \{ \pi _{t+1}^{\varepsilon -1}F_{t+1}\right \} \\ C_{t}^{-\gamma } &=&\beta \frac{\exp \left( \eta _{B,t}\right) }{\exp \left( \eta _{u,t}\right) }R_{t}E_{t}\left[ \frac{C_{t+1}^{-\gamma }\exp \left( \eta _{u,t+1}\right) }{\pi _{t+1}}\right] \end{eqnarray}