#!/usr/bin/env python
# coding: utf-8

# # A stylized New Keynesian Model
# 
# 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:
# 
# - $\pi$: profit of intermediate good firms
# - $S$, $F$: intermediate variables introduced to help notation below
# - $C$: Consumption
# - $G$: Level of government spending
# - $Y$: Final goods production
# - $L$: Labor supply
# - $\Delta$: measure of price dispersion across firms (also referred to as efficiency distortion)
# - $Y_N$: Natural level of final goods production -- it is the level of $Y$ in the planner's solution, or the level of $Y$ in the model without distorting taxes.
# - $\bar{G}$: steady-state share of government spendint in output
# - $R$: gross nominal interest rate
# - $\eta_u$: exogenous preference shock to utility
# - $\eta_L$: exogenous preference shock to labor dis-utility
# - $\eta_B$: exogenous premium on bond returns
# - $\eta_a$: log of productivity of intermediate goods firms
# - $\eta_G$: government spending shock
# - $\eta_R$: monetary shock to interest rate
# 
# We also see the following parameters:
# 
# - $\beta$: discount factor
# - $\varepsilon$: Parameter in Dixit-Stiglitz aggregator over intermediate goods
# - $\theta$: Calvo-parameter -- a fraction (1-$\theta$) of firms set prices optimally each period.
# - $\gamma$, $\vartheta$: utility function parameters
# - $\pi^*$: target inflation
# - $\phi_y$, $\phi_{\pi}$, $\mu$: Taylor rule 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}